Function index.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This file contains all the function calls in all the notebooks. It is currently updated manually, so it might be a little out of date."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# numpy\n",
    "\n",
    "* `numpy.abs`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `numpy.angle`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.arange`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Blocksim](Simulation/Blocksim.ipynb), [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.array`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Timing study](Simulation/Timing%20study.ipynb), [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.asmatrix`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `numpy.atleast_2d`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `numpy.bmat`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `numpy.clip`: [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `numpy.concatenate`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.convolve`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb)\n",
    "* `numpy.cos`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb)\n",
    "* `numpy.cumsum`: [Noise models](2_Control/6_Discrete_control_and_analysis/Noise%20models.ipynb)\n",
    "* `numpy.empty`: [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `numpy.exp`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.fft.rfft`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.fft.rfftfreq`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.hstack`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `numpy.interp`: [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb), [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [Timing study](Simulation/Timing%20study.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.linalg.det`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `numpy.linalg.eig`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `numpy.linalg.inv`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb)\n",
    "* `numpy.linalg.lstsq`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `numpy.linalg.norm`: [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `numpy.linalg.solve`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `numpy.linalg.svd`: [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `numpy.linspace`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb), [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb), [Timing study](Simulation/Timing%20study.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `numpy.log`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `numpy.logspace`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.matrix`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb)\n",
    "* `numpy.max`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.maximum`: [FOPDT fit](tclab/FOPDT%20fit.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb)\n",
    "* `numpy.meshgrid`: [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `numpy.minimum`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `numpy.ones`: [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `numpy.ones_like`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.polyfit`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `numpy.polyval`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb)\n",
    "* `numpy.rad2deg`: [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `numpy.random.normal`: [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.random.rand`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `numpy.random.randn`: [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb), [Noise models](2_Control/6_Discrete_control_and_analysis/Noise%20models.ipynb)\n",
    "* `numpy.random.random`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `numpy.random.uniform`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.roots`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb)\n",
    "* `numpy.sin`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `numpy.split`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `numpy.sqrt`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb), [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `numpy.tile`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `numpy.trapz`: [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb)\n",
    "* `numpy.zeros`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `numpy.zeros_like`: [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb)\n",
    "\n",
    "# scipy\n",
    "\n",
    "* `scipy.integrate.odeint`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `scipy.integrate.solve_ivp`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb), [Mixing system](1_Dynamics/2_Time_domain_simulation/Mixing%20system.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb), [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `scipy.integrate.trapz`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb)\n",
    "* `scipy.io.wavfile.read`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `scipy.linalg.toeplitz`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `scipy.optimize.curve_fit`: [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `scipy.optimize.fsolve`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `scipy.optimize.least_squares`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `scipy.optimize.minimize`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `scipy.signal.dlti`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `scipy.signal.lsim`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `scipy.signal.lti`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `scipy.signal.lti_conversion.ss2tf`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "* `scipy.signal.lti_conversion.tf2ss`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "* `scipy.signal.step`: [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb)\n",
    "\n",
    "# sympy\n",
    "\n",
    "* `sympy.And`: [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `sympy.Eq`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.Function`: [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `sympy.Heaviside`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `sympy.Matrix`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb), [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Multivariable closed loop transfer functions](2_Control/7_Multivariable_control/Multivariable%20closed%20loop%20transfer%20functions.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.N`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `sympy.Poly`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb)\n",
    "* `sympy.Rational`: [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.Sum`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `sympy.Symbol`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Numbers vs symbols](under_construction/Numbers%20vs%20symbols.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `sympy.cos`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb)\n",
    "* `sympy.denom`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb)\n",
    "* `sympy.diff`: [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb)\n",
    "* `sympy.dsolve`: [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `sympy.exp`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `sympy.expand`: [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `sympy.im`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.init_printing`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Linearisation](1_Dynamics/3_Linear_systems/Linearisation.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Block diagram simplification](1_Dynamics/5_Complex_system_dynamics/Block%20diagram%20simplification.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Multivariable closed loop transfer functions](2_Control/7_Multivariable_control/Multivariable%20closed%20loop%20transfer%20functions.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Numbers vs symbols](under_construction/Numbers%20vs%20symbols.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `sympy.integrate`: [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb)\n",
    "* `sympy.inverse_laplace_transform`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.lambdify`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.laplace_transform`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.limit`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb)\n",
    "* `sympy.log`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `sympy.nsimplify`: [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb)\n",
    "* `sympy.nsolve`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `sympy.numer`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `sympy.plot`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `sympy.plot_implicit`: [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `sympy.plotting.plot3d`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb)\n",
    "* `sympy.poly`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `sympy.re`: [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.roots`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `sympy.series`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Linearisation](1_Dynamics/3_Linear_systems/Linearisation.ipynb)\n",
    "* `sympy.simplify`: [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb), [Multivariable closed loop transfer functions](2_Control/7_Multivariable_control/Multivariable%20closed%20loop%20transfer%20functions.ipynb)\n",
    "* `sympy.sin`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `sympy.solve`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Block diagram simplification](1_Dynamics/5_Complex_system_dynamics/Block%20diagram%20simplification.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb), [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `sympy.sqrt`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb)\n",
    "* `sympy.symbols`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [The blending system](1_Dynamics/1_Modelling/The%20blending%20system.ipynb), [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Linearisation](1_Dynamics/3_Linear_systems/Linearisation.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Block diagram simplification](1_Dynamics/5_Complex_system_dynamics/Block%20diagram%20simplification.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb), [Multivariable closed loop transfer functions](2_Control/7_Multivariable_control/Multivariable%20closed%20loop%20transfer%20functions.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [IMC control design](under_construction/IMC%20control%20design.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `sympy.tan`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `sympy.var`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "\n",
    "# mpmath\n",
    "\n",
    "* `mpmath.cplot`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `mpmath.fourier`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `mpmath.fourierval`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `mpmath.plot`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "\n",
    "# pandas\n",
    "\n",
    "* `pandas.DataFrame`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb)\n",
    "* `pandas.DataFrame.from_records`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `pandas.read_csv`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Dealing with data](under_construction/Dealing%20with%20data.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `pandas.read_excel`: [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "\n",
    "# plt\n",
    "\n",
    "* `plt.axhline`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.axis`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.axvline`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.figure`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.grid`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `plt.legend`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Mixing system](1_Dynamics/2_Time_domain_simulation/Mixing%20system.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Blocksim](Simulation/Blocksim.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `plt.loglog`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `plt.pcolor`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `plt.plot`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Mixing system](1_Dynamics/2_Time_domain_simulation/Mixing%20system.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb), [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Noise models](2_Control/6_Discrete_control_and_analysis/Noise%20models.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [Blocksim](Simulation/Blocksim.ipynb), [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb), [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb), [Timing study](Simulation/Timing%20study.ipynb), [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Overshoot from two first orders](under_construction/Overshoot%20from%20two%20first%20orders.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `plt.savefig`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `plt.sca`: [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `plt.scatter`: [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb), [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.semilogx`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `plt.show`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.stem`: [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `plt.subplot`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.subplot2grid`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `plt.subplots`: [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `plt.text`: [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.tight_layout`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb)\n",
    "* `plt.title`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Root locus diagrams](2_Control/2_Laplace_domain_analysis_of_control_systems/Root%20locus%20diagrams.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `plt.twinx`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `plt.xlabel`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `plt.xlim`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `plt.ylabel`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `plt.ylim`: [Mixing system](1_Dynamics/2_Time_domain_simulation/Mixing%20system.ipynb), [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "\n",
    "# control\n",
    "\n",
    "* `control.bode`: [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb)\n",
    "* `control.bode_plot`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `control.feedback`: [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb)\n",
    "* `control.forced_response`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `control.impulse_response`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `control.nyquist_plot`: [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb)\n",
    "* `control.pzmap`: [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Root locus diagrams](2_Control/2_Laplace_domain_analysis_of_control_systems/Root%20locus%20diagrams.ipynb)\n",
    "* `control.rlocus`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `control.root_locus`: [Root locus diagrams](2_Control/2_Laplace_domain_analysis_of_control_systems/Root%20locus%20diagrams.ipynb)\n",
    "* `control.ss`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb)\n",
    "* `control.step_response`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `control.tf`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Transfer function matrices](1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb), [Root locus diagrams](2_Control/2_Laplace_domain_analysis_of_control_systems/Root%20locus%20diagrams.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Overshoot from two first orders](under_construction/Overshoot%20from%20two%20first%20orders.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "\n",
    "# tclab\n",
    "\n",
    "* `tclab.Experiment`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `tclab.Historian`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `tclab.TCLabModel`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `tclab.historian.TagDB`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "\n",
    "# tbcontrol\n",
    "\n",
    "* `tbcontrol.conversion.discrete_coeffs_pos_to_neg`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `tbcontrol.expectversion`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `tbcontrol.symbolic.evaluate_at_times`: [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb)\n",
    "* `tbcontrol.symbolic.linearise`: [Linearisation](1_Dynamics/3_Linear_systems/Linearisation.ipynb)\n",
    "* `tbcontrol.symbolic.pade`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `tbcontrol.symbolic.sampledvalues`: [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb)\n",
    "* `tbcontrol.symbolic.ss2tf`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "\n",
    "# blocksim\n",
    "\n",
    "* `blocksim.AlgebraicEquation`: [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `blocksim.Deadtime`: [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb)\n",
    "* `blocksim.Diagram`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `blocksim.DiscreteTF`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `blocksim.LTI`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `blocksim.PI`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `blocksim.PID`: [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `blocksim.simple_control_diagram`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `blocksim.step`: [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "\n",
    "# Other\n",
    "* `A.eigenvals`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `A.inv`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `A.solve`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `A.subs`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `Audio`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `ControlGame`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `DataFrame`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb)\n",
    "* `DiscreteController`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb)\n",
    "* `F.apart`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb)\n",
    "* `Fin`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb), [Read input from file](1_Dynamics/2_Time_domain_simulation/Read%20input%20from%20file.ipynb)\n",
    "* `G`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `G.det`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `G.inv`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `G.multiply_elementwise`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `G.pole`: [Overshoot from two first orders](under_construction/Overshoot%20from%20two%20first%20orders.ipynb)\n",
    "* `G.sample`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `G.step`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `G.to_ss`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "* `G.zero`: [Overshoot from two first orders](under_construction/Overshoot%20from%20two%20first%20orders.ipynb)\n",
    "* `G2.subs`: [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `G2.to_ss`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `G2ss.to_tf`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "* `G_CL.pole`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `G_CL.subs`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb)\n",
    "* `G_C_rational.simplify`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `G_p.adjugate`: [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb)\n",
    "* `G_p.applyfunc`: [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb)\n",
    "* `G_p.inv`: [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb)\n",
    "* `Gc.A.dot`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `Gc.B.dot`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `Gc.C.dot`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `Gc.D.dot`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `Gdc.minreal`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `Gdiscrete.step`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `Geval`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `Gnumeric`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb)\n",
    "* `Gp`: [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb)\n",
    "* `Gss.A.dot`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `Gss.B.dot`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `Gss.C.dot`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `Gss.D.dot`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `Gtf.pole`: [State space](1_Dynamics/6_Multivariable_system_representation/State%20space.ipynb)\n",
    "* `Gz.cancel`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `Gz.expand`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `Historian`: [FOPDT fit](tclab/FOPDT%20fit.ipynb)\n",
    "* `ISE.simplify`: [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb)\n",
    "* `K.multiply_elementwise`: [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb)\n",
    "* `L`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb)\n",
    "* `LHS.inv`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `MyClass`: [Classes](Simulation/Classes.ipynb)\n",
    "* `MyClass2`: [Classes](Simulation/Classes.ipynb)\n",
    "* `NotebookUI`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `PIController`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `PIDpos`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `PIDvel`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `Person`: [Classes](Simulation/Classes.ipynb)\n",
    "* `R`: [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb)\n",
    "* `S`: [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb)\n",
    "* `S.plot`: [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `T.subs`: [Decoupling](2_Control/7_Multivariable_control/Decoupling.ipynb)\n",
    "* `TankSystem`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `Tends.append`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `TestClass`: [Special functions in classes](Simulation/Special%20functions%20in%20classes.ipynb)\n",
    "* `X.append`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `Y.append`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `a.__add__`: [Special functions in classes](Simulation/Special%20functions%20in%20classes.ipynb)\n",
    "* `annotated_bode`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax.axvline`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `ax.plot`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `axT.plot`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `axT.set_ylabel`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `ax_complex.axhline`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_complex.axis`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_complex.axvline`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_complex.grid`: [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb)\n",
    "* `ax_complex.plot`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_complex.scatter`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_complex.set_title`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_e.plot`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb)\n",
    "* `ax_e.set_ylabel`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb)\n",
    "* `ax_gain.axhline`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_gain.axvline`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_gain.loglog`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_gain.set_ylabel`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_gain.set_ylim`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_impulse.plot`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_impulse.set_title`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_impulse.set_ylim`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_mag.loglog`: [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb)\n",
    "* `ax_phase.axhline`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_phase.axvline`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_phase.semilogx`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_phase.set_xlabel`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_phase.set_ylabel`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `ax_phase.set_ylim`: [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb)\n",
    "* `ax_step.axhline`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_step.plot`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_step.set_title`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_step.set_ylim`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_step.text`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `ax_time.axhline`: [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb)\n",
    "* `ax_time.plot`: [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb)\n",
    "* `ax_u.plot`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `ax_u.set_ylabel`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `ax_u.set_ylim`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `ax_y.axhline`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `ax_y.plot`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `ax_y.set_ylabel`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `ax_y.set_ylim`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `axcA.legend`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `axcA.plot`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `axcA.set_ylabel`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `axcv.plot`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `axgain.axvline`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axgain.grid`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axgain.loglog`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `axgain.set_ylabel`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axgain.text`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `axmv.plot`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `axphase.axvline`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axphase.grid`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axphase.semilogx`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `axphase.set_xlabel`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axphase.set_ylabel`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axphase.set_yticks`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `axphase.text`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `bigsymbolicA.solve`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `bode`: [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb), [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `bodegain`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `carl.display_name`: [Classes](Simulation/Classes.ipynb)\n",
    "* `ce`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `ce.expand`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `chareq.as_numer_denom`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `chareq.factor`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `closedloop`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `control_simulation`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `control_simulation_discrete`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb)\n",
    "* `controller`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `controller.change_input`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `controller.change_state`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `controller.run`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `controller.update`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb)\n",
    "* `cos`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb)\n",
    "* `cplot`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `cross_axis`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `curve`: [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb)\n",
    "* `d.items`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `dMdt`: [Mixing system](1_Dynamics/2_Time_domain_simulation/Mixing%20system.ipynb)\n",
    "* `data.plot`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb)\n",
    "* `data.sort_index`: [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb)\n",
    "* `db.get_sessions`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `decimal.Decimal`: [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb)\n",
    "* `denom.subs`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `df.plot`: [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `dhdt`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `diagram.simulate`: [RQ Online controller tuning](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20Online%20controller%20tuning.ipynb), [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `diagrams.items`: [Dead time compensation](2_Control/5_Advanced_control_methods/Dead%20time%20compensation.ipynb)\n",
    "* `dir`: [Special functions in classes](Simulation/Special%20functions%20in%20classes.ipynb)\n",
    "* `dsdt.append`: [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `enumerate`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb), [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `eq.coeffs`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `eq.diff`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `error.__name__.upper`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb)\n",
    "* `errorint.set_ylabel`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `errors.set_ylabel`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `evalf`: [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `exp`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb)\n",
    "* `experiment.clock`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `experiment.lab.Q1`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `expz`: [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb)\n",
    "* `extend`: [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `f`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb), [Classes](Simulation/Classes.ipynb), [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb), [Numbers vs symbols](under_construction/Numbers%20vs%20symbols.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `f.subs`: [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb)\n",
    "* `f.write`: [Dealing with data](under_construction/Dealing%20with%20data.ipynb)\n",
    "* `f2numeric`: [Visualising complex functions](1_Dynamics/3_Linear_systems/Visualising%20complex%20functions.ipynb)\n",
    "* `feedback`: [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb)\n",
    "* `filtersignal`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `fin`: [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `firstorder`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb)\n",
    "* `fit_and_plot`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `fiterror`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `flat.append`: [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `float`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `fopdtitae.parameters`: [ITAE parameters for FOPDT system](2_Control/3_PID_controller_design_tuning_and_troubleshooting/ITAE%20parameters%20for%20FOPDT%20system.ipynb), [RQ tuning relations](2_Control/3_PID_controller_design_tuning_and_troubleshooting/RQ%20tuning%20relations.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `frequencyplot`: [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb)\n",
    "* `func`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `fz.subs`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `g.subs`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb)\n",
    "* `ga.set_ylim`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `gainax.loglog`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `gainax.scatter`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `game.plot`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `game.ui`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `geval`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `h`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `h.get_sessions`: [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `h.load_session`: [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `help`: [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb)\n",
    "* `historian.to_csv`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `historian.update`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `hs.append`: [Hybrid system simulation](Simulation/Hybrid%20system%20simulation.ipynb)\n",
    "* `int`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb), [Classes](Simulation/Classes.ipynb), [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `interact`: [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Random response generator](1_Dynamics/5_Complex_system_dynamics/Random%20response%20generator.ipynb), [Dynamic model parameter estimation](1_Dynamics/7_System_identification/Dynamic%20model%20parameter%20estimation.ipynb), [Asymptotic Bode diagrams](1_Dynamics/8_Frequency_domain/Asymptotic%20Bode%20diagrams.ipynb), [Aliasing](1_Dynamics/9_Sampled_systems/Aliasing.ipynb), [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb), [Root locus diagrams](2_Control/2_Laplace_domain_analysis_of_control_systems/Root%20locus%20diagrams.ipynb), [Stability analysis](2_Control/2_Laplace_domain_analysis_of_control_systems/Stability%20analysis.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb), [ITAE parameters for FOPDT system](2_Control/3_PID_controller_design_tuning_and_troubleshooting/ITAE%20parameters%20for%20FOPDT%20system.ipynb), [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb), [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `interface.action_stop`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `interface.historian.get_sessions`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `interface.historian.load_session`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `intsys`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `invL`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [First order systems](1_Dynamics/4_First_and_second_order_system_dynamics/First%20order%20systems.ipynb), [Second order systems](1_Dynamics/4_First_and_second_order_system_dynamics/Second%20order%20systems.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb)\n",
    "* `itae`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb)\n",
    "* `keras.models.Sequential`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `l.append`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb)\n",
    "* `lab.Q1`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb), [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb)\n",
    "* `lab.Q2`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `lab.update`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `len`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `lim.subs`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `limit`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `lineq.subs`: [Linearisation](1_Dynamics/3_Linear_systems/Linearisation.ipynb)\n",
    "* `list`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `map`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `max`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `model.add`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `model.compile`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `model.fit`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `moving`: [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb)\n",
    "* `multiple_equations`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `my_solve_ivp`: [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `net.fit`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `network_output`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `noisy_and_original`: [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb)\n",
    "* `np.abs`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.arange`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.exp`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.fft.rfft`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.fft.rfftfreq`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.logspace`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `np.unwrap`: [Using the FFT to approximate the frequency response](under_construction/Using%20the%20FFT%20to%20approximate%20the%20frequency%20response.ipynb)\n",
    "* `num.collect`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `numer.all_coeffs`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `nyquist`: [Frequency response plots](1_Dynamics/8_Frequency_domain/Frequency%20response%20plots.ipynb)\n",
    "* `nyquistplot`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `objective`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `open`: [Dealing with data](under_construction/Dealing%20with%20data.ipynb)\n",
    "* `outputs.append`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `outputs.legend`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `outputs.plot`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `outputs.set_ylabel`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `p.append`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `p.extend`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `p.show`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `p1.extend`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `p1.show`: [Fourier series](1_Dynamics/8_Frequency_domain/Fourier%20series.ipynb)\n",
    "* `pa.set_ylim`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `pack`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `pade`: [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb)\n",
    "* `parameters`: [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb)\n",
    "* `patsy.dmatrices`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `phaseax.scatter`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `phaseax.semilogx`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `plot_approx`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `plot_continuous`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `plot_discrete`: [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb)\n",
    "* `plot_outputs`: [Simple discrete controller simulation](2_Control/6_Discrete_control_and_analysis/Simple%20discrete%20controller%20simulation.ipynb)\n",
    "* `plot_step_response`: [PID controller step responses](2_Control/1_Conventional_feedback_control/PID%20controller%20step%20responses.ipynb)\n",
    "* `plotcomplex`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `plotdiscrete`: [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `plotfit`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `plotfreqs`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `plotoutput`: [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `plotpz`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `plotresponse`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `plotsituation`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `plotstep`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `plotvector`: [Eigenvalues and Eigenvectors](2_Control/7_Multivariable_control/Eigenvalues%20and%20Eigenvectors.ipynb)\n",
    "* `plus_two`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `polynomial.diff`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `polynomial.expand`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `polynomial.integrate`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `polyvalfromroots`: [Frequency domain stability](2_Control/4_Frequency_domain_analysis_of_control_systems/Frequency%20domain%20stability.ipynb)\n",
    "* `positive`: [Control valve design](2_Control/8_Control_Practice/Control%20valve%20design.ipynb)\n",
    "* `predict`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `prediction`: [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb)\n",
    "* `print`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb), [Draining cup](1_Dynamics/1_Modelling/Draining%20cup.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb), [ITAE parameters for FOPDT system](2_Control/3_PID_controller_design_tuning_and_troubleshooting/ITAE%20parameters%20for%20FOPDT%20system.ipynb), [Optimal control - minimal integral measures](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Optimal%20control%20-%20minimal%20integral%20measures.ipynb), [Classes](Simulation/Classes.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb), [Matrix exponentials](under_construction/Matrix%20exponentials.ipynb), [Numbers vs symbols](under_construction/Numbers%20vs%20symbols.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `qpoly.all_coeffs`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `qsubs`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `queue.append`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `queue.pop`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `queue.reverse`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `range`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb), [Identifying discrete-time models](1_Dynamics/7_System_identification/Identifying%20discrete-time%20models.ipynb), [The z domain and continuous systems](1_Dynamics/9_Sampled_systems/The%20z%20domain%20and%20continuous%20systems.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Dahlin controller](2_Control/6_Discrete_control_and_analysis/Dahlin%20controller.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Multivariable Pairing](2_Control/7_Multivariable_control/Multivariable%20Pairing.ipynb), [Timing study](Simulation/Timing%20study.ipynb), [Difference Functions](under_construction/Difference%20Functions.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `regplot`: [Regression](1_Dynamics/7_System_identification/Regression.ipynb)\n",
    "* `residuals`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb)\n",
    "* `response`: [Sinusoidal response](1_Dynamics/4_First_and_second_order_system_dynamics/Sinusoidal%20response.ipynb)\n",
    "* `response.apart`: [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `result.append`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Noise models](2_Control/6_Discrete_control_and_analysis/Noise%20models.ipynb), [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `results.append`: [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `routh`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [SymPy Routh Array](2_Control/2_Laplace_domain_analysis_of_control_systems/SymPy%20Routh%20Array.ipynb), [Example 10.11](2_Control/4_Frequency_domain_analysis_of_control_systems/Example%2010.11.ipynb), [Multivariable stability analysis](2_Control/7_Multivariable_control/Multivariable%20stability%20analysis.ipynb)\n",
    "* `runexperiment`: [FOPDT fit](tclab/FOPDT%20fit.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [TCLab step test](tclab/TCLab%20step%20test.ipynb)\n",
    "* `s.say_my_name`: [Classes](Simulation/Classes.ipynb)\n",
    "* `sampledvalues`: [Cheatsheet](0_Getting_Started/Cheatsheet.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `savefig`: [RQ Freq basics](2_Control/4_Frequency_domain_analysis_of_control_systems/RQ%20Freq%20basics.ipynb)\n",
    "* `say_hello`: [Numbers vs symbols](under_construction/Numbers%20vs%20symbols.ipynb)\n",
    "* `score`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `self.G.to_ss`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.Gss.A.dot`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.Gss.B.dot`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.Gss.C.dot`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.Gss.D.dot`: [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.change_input`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.change_state`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.f`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `self.reset`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `self.update`: [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n",
    "* `setpoint`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `shownotes`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `sigmoid`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `sim`: [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb)\n",
    "* `simulate`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb), [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `simulation_results.items`: [Blocksim](Simulation/Blocksim.ipynb)\n",
    "* `sin`: [Laplace transforms](1_Dynamics/3_Linear_systems/Laplace%20transforms.ipynb)\n",
    "* `sinefit`: [Frequency domain](tclab/Frequency%20domain.ipynb)\n",
    "* `sklearn.neural_network.MLPRegressor`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `skogestad_half`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `smooth_1st`: [Difference Functions](under_construction/Difference%20Functions.ipynb)\n",
    "* `sol.sol`: [Solve IVP](2_Control/1_Conventional_feedback_control/Solve%20IVP.ipynb)\n",
    "* `song.sum`: [Sound and frequency](1_Dynamics/8_Frequency_domain/Sound%20and%20frequency.ipynb)\n",
    "* `sopdt`: [Filtering](1_Dynamics/9_Sampled_systems/Filtering.ipynb)\n",
    "* `ss`: [Nonlinear CSTR](1_Dynamics/2_Time_domain_simulation/Nonlinear%20CSTR.ipynb)\n",
    "* `statehistory.append`: [Timing study](Simulation/Timing%20study.ipynb)\n",
    "* `steptimer.start`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `steptimer.stop`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `sum`: [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Standard process inputs](1_Dynamics/4_First_and_second_order_system_dynamics/Standard%20process%20inputs.ipynb), [Regression](1_Dynamics/7_System_identification/Regression.ipynb), [Control game](2_Control/1_Conventional_feedback_control/Control%20game.ipynb), [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb), [Simple MPC](2_Control/7_Multivariable_control/Simple%20MPC.ipynb), [Frequency domain](tclab/Frequency%20domain.ipynb), [Triangular response](under_construction/Triangular%20response.ipynb)\n",
    "* `system.change_input`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `system.change_state`: [Object-Oriented simulation - Discrete](Simulation/Object-Oriented%20simulation%20-%20Discrete.ipynb), [Object-Oriented simulation](Simulation/Object-Oriented%20simulation.ipynb)\n",
    "* `tablefunction`: [ITAE parameters for FOPDT system](2_Control/3_PID_controller_design_tuning_and_troubleshooting/ITAE%20parameters%20for%20FOPDT%20system.ipynb)\n",
    "* `taylor`: [Approximation](1_Dynamics/5_Complex_system_dynamics/Approximation.ipynb)\n",
    "* `temp.removeO`: [Notebook introduction](0_Getting_Started/Notebook%20introduction.ipynb)\n",
    "* `time.sleep`: [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `time.time`: [Real time simulation](under_construction/Real%20time%20simulation.ipynb)\n",
    "* `tornado.ioloop.PeriodicCallback`: [TCLab PID](tclab/TCLab%20PID.ipynb)\n",
    "* `tqdm`: [Continuous PID on TCLab](tclab/Continuous%20PID%20on%20TCLab.ipynb)\n",
    "* `type`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Numeric representation](1_Dynamics/2_Time_domain_simulation/Numeric%20representation.ipynb), [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Classes](Simulation/Classes.ipynb), [Symbolic tools](under_construction/Symbolic%20tools.ipynb)\n",
    "* `u`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb), [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `u.set_ylabel`: [Closed loop controlled responses](2_Control/1_Conventional_feedback_control/Closed%20loop%20controlled%20responses.ipynb)\n",
    "* `u.subs`: [Convolution](1_Dynamics/3_Linear_systems/Convolution.ipynb)\n",
    "* `u_continuous.append`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `u_discrete.append`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `uaxis.plot`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `uaxis.set_ylabel`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `unitstep.doit`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `unpack`: [Neural networks](1_Dynamics/7_System_identification/Neural%20networks.ipynb)\n",
    "* `unsolvable_numeric`: [Equation solving tools](1_Dynamics/2_Time_domain_simulation/Equation%20solving%20tools.ipynb)\n",
    "* `xs.append`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `y_continuous.append`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `y_discrete.append`: [Discrete control](2_Control/6_Discrete_control_and_analysis/Discrete%20control.ipynb)\n",
    "* `y_func`: [Effect of Proportional Control](2_Control/1_Conventional_feedback_control/Effect%20of%20Proportional%20Control.ipynb)\n",
    "* `yaxis.legend`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `yaxis.plot`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `yaxis.set_ylabel`: [Discrete PI](2_Control/6_Discrete_control_and_analysis/Discrete%20PI.ipynb)\n",
    "* `ypred.append`: [Fitting example](under_construction/Fitting%20example.ipynb)\n",
    "* `ys.append`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `ys_statespace.append`: [Simulation of arbitrary transfer functions](1_Dynamics/5_Complex_system_dynamics/Simulation%20of%20arbitrary%20transfer%20functions.ipynb)\n",
    "* `yz.subs`: [The z transform](1_Dynamics/9_Sampled_systems/The%20z%20transform.ipynb)\n",
    "* `zeroeq.as_numer_denom`: [Direct synthesis PID design](2_Control/3_PID_controller_design_tuning_and_troubleshooting/Direct%20synthesis%20PID%20design.ipynb)\n",
    "* `zip`: [Extra Python](0_Getting_Started/Extra%20Python.ipynb), [Fed batch bioreactor](1_Dynamics/2_Time_domain_simulation/Fed%20batch%20bioreactor.ipynb), [ITAE parameters for FOPDT system](2_Control/3_PID_controller_design_tuning_and_troubleshooting/ITAE%20parameters%20for%20FOPDT%20system.ipynb), [Position vs velocity](tclab/Position%20vs%20velocity.ipynb)\n"
   ]
  },
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