Simple Model Predictive Control (MPC) for non-linear spring-mass system.
The system is illustrated below, with M masses (m_1=m_2...m_M=m) connected by (non-linear) springs, and with velocity-dependent friction under each mass. The system is controlled by appling forces u1 and u2 to the first and last masses. Noise is applied to the control signals. Objective is to hold masses as close to their "0"s as possible, while applying as little control force as possible.
For linear springs, the force is k*r, where r is displacement. For non-linear springs, the force is h(r) = k*r - 0.01 r^3. Friction force is 2*c*v, where v is velocity.
| k _______ k _______ k _______ k |
|__/\/\__| |__/\/\__| |_ ... _/\/\__| |__/\/\__|
| | m_1 | | m_2 | | m_M | |
| u1 -->|_______| |_______| |_______|--> u2 | x
============(c)==============(c)====== ... ========(c)================== -->
System dynamics is as following:
m*x_i'' + 2*c*x_i' + h(x_i - x_{i-1}) + h(x_i − x_{i+1}) = 0, for i=1..M,
where x_i is the position of mass i, x_0 = x_{M+1} = 0, x_i' and x_i'' are first and second derivatives of x_i.
Figure below shows:
- uncontrolled simulation,
- closed-loop simulation,
- thin lines indicate positions predicted by MPC.
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Make controller aware of spring non-linearity: Linearize non-linear terms around the solution from the previous MPC iteration.
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Consider more realistic noise: Add noise to all elements. Something that would perhaps mimic gusts of wind, or temperature effects on spring and friction coefficients. Also, try giving "noisy" system parameters (
k,m,c,x,v) to the MPC solver. -
Add latency of MPC calculations There should be a delay from the time MPC gets (x,v) and the time it outputs a solution.
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Can we learn system parameters from observations? Use optimal estimation, system identification, or machine learning to find system parameters (
k,m,c) from observing uncontrolled/controlled system. Can also be done as reinforcement learning. Could also balance exploration-vs-exploitation, as opposed to only doing exploration.
- YALMIP
- tight_subplot
- export_fig
- linprog (Optimization toolbox) or any YALMIP-supported, LP-capable solver (some QP/SOCP solvers handle LP just fine).
MIT.