/
nmpc.m
582 lines (549 loc) · 24.7 KB
/
nmpc.m
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function [t, x, u, comp_time] = nmpc(runningcosts, terminalcosts, ...
constraints, terminalconstraints, ...
linearconstraints, system, cov_propagation, ...
mpciterations, N, T, tmeasure, xmeasure, u0, ...
sig, beta, params, ...
varargin)
% nmpc.m routine by L. Gruene and J. Pannek (details: http://numerik.mathematik.uni-bayreuth.de/~lgruene/nmpc-book/matlab_nmpc.html)
% MPC optimization starts at line 313
% constraint tightening starts at line 430
% nmpc(runningcosts, terminalcosts, constraints, ...
% terminalconstraints, linearconstraints, system, ...
% mpciterations, N, T, tmeasure, xmeasure, u0, ...
% tol_opt, opt_option, ...
% type, atol_ode_real, rtol_ode_real, atol_ode_sim, rtol_ode_sim, ...
% iprint, printHeader, printClosedloopData, plotTrajectories)
% Computes the closed loop solution for the NMPC problem defined by
% the functions
% runningcosts: evaluates the running costs for state and control
% at one sampling instant.
% The function returns the running costs for one
% sampling instant.
% Usage: [cost] = runningcosts(t, x, u)
% with time t, state x and control u
% terminalcosts: evaluates the terminal costs for state at the end
% of the open loop horizon.
% The function returns value of the terminal costs.
% Usage: cost = terminalcosts(t, x)
% with time t and state x
% constraints: computes the value of the restrictions for a
% sampling instance provided the data t, x and u
% given by the optimization method.
% The function returns the value of the
% restrictions for a sampling instance separated
% for inequality restrictions c and equality
% restrictions ceq.
% Usage: [c,ceq] = constraints(t, x, u)
% with time t, state x and control u
% terminalconstraints: computes the value of the terminal restrictions
% provided the data t, x and u given by the
% optimization method.
% The function returns the value of the
% terminal restriction for inequality restrictions
% c and equality restrictions ceq.
% Usage: [c,ceq] = terminalconstraints(t, x)
% with time t and state x
% linearconstraints: sets the linear constraints of the discretized
% optimal control problem. This is particularly
% useful to set control and state bounds.
% The function returns the required matrices for
% the linear inequality and equality constraints A
% and Aeq, the corresponding right hand sides b and
% beq as well as the lower and upper bound of the
% control.
% Usage: [A, b, Aeq, beq, lb, ub] = linearconstraints(t, x, u)
% with time t, state x and control u
% system: evaluates the difference equation describing the
% process given time t, state vector x and control
% u.
% The function returns the state vector x at the
% next time instant.
% Usage: [y] = system(t, x, u, T)
% with time t, state x, control u and sampling interval T
% for a given number of NMPC iteration steps (mpciterations). For
% the open loop problem, the horizon is defined by the number of
% time instances N and the sampling time T. Note that the dynamic
% can also be the solution of a differential equation. Moreover, the
% initial time tmeasure, the state measurement xmeasure and a guess of
% the optimal control u0 are required.
%
% Arguments:
% mpciterations: Number of MPC iterations to be performed
% N: Length of optimization horizon
% T: Sampling interval
% tmeasure: Time measurement of initial value
% xmeasure: State measurement of initial value
% u0: Initial guess of open loop control
%
% Optional arguments:
% iprint: = 0 Print closed loop data(default)
% = 1 Print closed loop data and errors of the
% optimization method
% = 2 Print closed loop data and errors and warnings of
% the optimization method
% >= 5 Print closed loop data and errors and warnings of
% the optimization method as well as graphical
% output of closed loop state trajectories
% >=10 Print closed loop data and errors and warnings of
% the optimization method with error and warning
% description
% printHeader: Clarifying header for selective output of closed
% loop data, cf. printClosedloopData
% printClosedloopData: Selective output of closed loop data
% plotTrajectories: Graphical output of the trajectories, requires
% iprint >= 4
% tol_opt: Tolerance of the optimization method
% opt_option: = 0: Active-set method used for optimization (default)
% = 1: Interior-point method used for optimization
% = 2: Trust-region reflective method used for
% optimization
% type: Type of dynamic, either difference equation or
% differential equation can be used
% atol_ode_real: Absolute tolerance of the ODE solver for the
% simulated process
% rtol_ode_real: Relative tolerance of the ODE solver for the
% simulated process
% atol_ode_sim: Absolute tolerance of the ODE solver for the
% simulated NMPC prediction
% rtol_ode_sim: Relative tolerance of the ODE solver for the
% simulated NMPC prediction
%
% Internal Functions:
% measureInitialValue: measures the new initial values for t0
% and x0 by adopting values computed by
% method applyControl.
% The function returns new initial state
% vector x0 at sampling instant t0.
% applyControl: applies the first control element of u to
% the simulated process for one sampling
% interval T.
% The function returns closed loop state
% vector xapplied at sampling instant
% tapplied.
% shiftHorizon: applies the shift method to the open loop
% control in order to ease the restart.
% The function returns a new initial guess
% u0 of the control.
% solveOptimalControlProblem: solves the optimal control problem of the
% horizon N with sampling length T for the
% given initial values t0 and x0 and the
% initial guess u0 using the specified
% algorithm.
% The function returns the computed optimal
% control u, the corresponding value of the
% cost function V as well as possible exit
% flags and additional output of the
% optimization method.
% costfunction: evaluates the cost function of the
% optimal control problem over the horizon
% N with sampling time T for the current
% data of the optimization method t0, x0
% and u.
% The function return the computed cost
% function value.
% nonlinearconstraints: computes the value of the restrictions
% for all sampling instances provided the
% data t0, x0 and u given by the
% optimization method.
% The function returns the value of the
% restrictions for all sampling instances
% separated for inequality restrictions c
% and equality restrictions ceq.
% computeOpenloopSolution: computes the open loop solution over the
% horizon N with sampling time T for the
% initial values t0 and x0 as well as the
% control u.
% The function returns the complete open
% loop solution over the requested horizon.
% dynamic: evaluates the dynamic of the system for
% given initial values t0 and x0 over the
% interval [t0, tf] using the control u.
% The function returns the state vector x
% at time instant tf as well as an output
% of all intermediate evaluated time
% instances.
% printSolution: prints out information on the current MPC
% step, in particular state and control
% information as well as required computing
% times and exitflags/outputs of the used
% optimization method. The flow of
% information can be controlled by the
% variable iprint and the functions
% printHeader, printClosedloopData and
% plotTrajectories.
%
% Version of May 30, 2011, in which a bug appearing in the case of
% multiple constraints has been fixed
%
% (C) Lars Gruene, Juergen Pannek 2011
if (nargin>=17)
tol_opt = varargin{1};
else
tol_opt = 1e-6;
end;
if (nargin>=18)
opt_option = varargin{2};
else
opt_option = 0;
end;
if (nargin>=19)
if ( strcmp(varargin{3}, 'difference equation') || ...
strcmp(varargin{3}, 'differential equation') )
type = varargin{3};
else
fprintf([' Wrong input for type of dynamic: use either ', ...
'"difference equation" or "differential equation".']);
end
else
type = 'difference equation';
end;
if (nargin>=20)
atol_ode_real = varargin{4};
else
atol_ode_real = 1e-8;
end;
if (nargin>=21)
rtol_ode_real = varargin{5};
else
rtol_ode_real = 1e-8;
end;
if (nargin>=22)
atol_ode_sim = varargin{6};
else
atol_ode_sim = atol_ode_real;
end;
if (nargin>=23)
rtol_ode_sim = varargin{7};
else
rtol_ode_sim = rtol_ode_real;
end;
if (nargin>=24)
iprint = varargin{8};
else
iprint = 0;
end;
if (nargin>=25)
printHeader = varargin{9};
else
printHeader = @printHeaderDummy;
end;
if (nargin>=26)
printClosedloopData = varargin{10};
else
printClosedloopData = @printClosedloopDataDummy;
end;
if (nargin>=27)
plotTrajectories = varargin{11};
else
plotTrajectories = @plotTrajectoriesDummy;
end;
% Determine MATLAB Version and
% specify and configure optimization method
vs = version('-release');
vyear = str2num(vs(1:4));
if (vyear <= 2007)
fprintf('MATLAB version R2007 or earlier detected\n');
fprintf('Starting MPC routine\n');
if ( opt_option == 0 )
options = optimset('Display','off',...
'TolFun', tol_opt,...
'MaxIter', 2000,...
'LargeScale', 'off',...
'RelLineSrchBnd', [],...
'RelLineSrchBndDuration', 1);
elseif ( opt_option == 1 )
error('nmpc:WrongArgument', '%s\n%s', ...
'Interior point method not supported in MATLAB R2007', ...
'Please use opt_option = 0 or opt_option = 2');
elseif ( opt_option == 2 )
options = optimset('Display','off',...
'TolFun', tol_opt,...
'MaxIter', 2000,...
'LargeScale', 'on',...
'Hessian', 'off',...
'MaxPCGIter', max(1,floor(size(u0,1)*size(u0,2)/2)),...
'PrecondBandWidth', 0,...
'TolPCG', 1e-1);
end
else
% fprintf('MATLAB version R2008 or newer detected\n');
fprintf('Starting MPC routine\n');
if ( opt_option == 0 )
options = optimset('Display','off',...
'TolFun', tol_opt,...
'MaxIter', 10000,...
'Algorithm', 'active-set',...
'FinDiffType', 'forward',...
'RelLineSrchBnd', [],...
'RelLineSrchBndDuration', 1,...
'TolConSQP', 1e-6);
elseif ( opt_option == 1 )
options = optimset('Display','off',...
'TolFun', tol_opt,...
'MaxIter', 2000,...
'Algorithm', 'interior-point',...
'AlwaysHonorConstraints', 'bounds',...
'FinDiffType', 'forward',...
'HessFcn', [],...
'Hessian', 'bfgs',...
'HessMult', [],...
'InitBarrierParam', 0.1,...
'InitTrustRegionRadius', sqrt(size(u0,1)*size(u0,2)),...
'MaxProjCGIter', 2*size(u0,1)*size(u0,2),...
'ObjectiveLimit', -1e20,...
'ScaleProblem', 'obj-and-constr',...
'SubproblemAlgorithm', 'cg',...
'TolProjCG', 1e-2,...
'TolProjCGAbs', 1e-10);
% 'UseParallel','always',...
elseif ( opt_option == 2 )
options = optimset('Display','off',...
'TolFun', tol_opt,...
'MaxIter', 2000,...
'Algorithm', 'trust-region-reflective',...
'Hessian', 'off',...
'MaxPCGIter', max(1,floor(size(u0,1)*size(u0,2)/2)),...
'PrecondBandWidth', 0,...
'TolPCG', 1e-1);
end
end
%% MPC optimization
warning off all
t = [];
x = [];
u = [];
comp_time = []; % computation time
% Start of the NMPC iteration
mpciter = 0;
while(mpciter < mpciterations)
% Step (1) of the NMPC algorithm:
% Obtain new initial value
[t0, x0] = measureInitialValue ( tmeasure, xmeasure );
% Step (2) of the NMPC algorithm:
% Solve the optimal control problem
t_Start = tic;
[u_new, V_current, exitflag, output] = solveOptimalControlProblem ...
(runningcosts, terminalcosts, constraints, ...
terminalconstraints, linearconstraints, system, cov_propagation, ...
N, t0, x0, u0, T, ...
sig, beta, params, ...
atol_ode_sim, rtol_ode_sim, tol_opt, options, type);
t_Elapsed = toc( t_Start );
% Print solution
if ( iprint >= 1 )
printSolution(system, printHeader, printClosedloopData, ...
plotTrajectories, mpciter, T, t0, x0, u_new, params, ...
atol_ode_sim, rtol_ode_sim, type, iprint, ...
exitflag, output, t_Elapsed);
end
% Store closed loop data
t = [ t; tmeasure ];
x = [ x; xmeasure ];
u = [ u; u_new(:,1) ];
comp_time = [comp_time; t_Elapsed];
% Prepare restart
u0 = shiftHorizon(u_new);
% Step (3) of the NMPC algorithm:
% Apply control to process
[tmeasure, xmeasure] = applyControl(system, T, t0, x0, u_new, ...
atol_ode_real, rtol_ode_real, type, sig, params);
mpciter = mpciter+1;
end
end
function [t0, x0] = measureInitialValue ( tmeasure, xmeasure )
t0 = tmeasure;
x0 = xmeasure;
end
function [tapplied, xapplied] = applyControl(system, T, t0, x0, u, ...
atol_ode_real, rtol_ode_real, type, sig, params)
apply_flag = 1; % for prediction, no noise is applied (=0); for actual system noise is applied (=1)
xapplied = dynamic(system, T, t0, x0, u(:,1), ...
atol_ode_real, rtol_ode_real, type, apply_flag, sig, params);
tapplied = t0+T;
end
function u0 = shiftHorizon(u)
u0 = [u(:,2:size(u,2)) u(:,size(u,2))];
end
function [u, V, exitflag, output] = solveOptimalControlProblem ...
(runningcosts, terminalcosts, constraints, terminalconstraints, ...
linearconstraints, system, cov_propagation, N, t0, x0, u0, T, sig, beta, params, ...
atol_ode_sim, rtol_ode_sim, tol_opt, options, type)
x = zeros(N+1, length(x0));
x = computeOpenloopSolution(system, N, T, t0, x0, u0, ...
atol_ode_sim, rtol_ode_sim, type, sig, params);
% Set control and linear bounds
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
for k=1:N
[Anew, bnew, Aeqnew, beqnew, lbnew, ubnew] = ...
linearconstraints(t0+k*T,x(k,:),u0(:,k));
A = blkdiag(A,Anew);
b = [b, bnew];
Aeq = blkdiag(Aeq,Aeqnew);
beq = [beq, beqnew];
lb = [lb, lbnew];
ub = [ub, ubnew];
end
% Solve optimization problem
[u, V, exitflag, output] = fmincon(@(u) costfunction(runningcosts, ...
terminalcosts, system, N, T, t0, x0, ...
u, atol_ode_sim, rtol_ode_sim, type, sig, params), u0, A, b, Aeq, beq, lb, ...
ub, @(u) nonlinearconstraints(constraints, terminalconstraints, ...
system, cov_propagation, N, T, t0, x0, u, sig, beta, params, ...
atol_ode_sim, rtol_ode_sim, type), options);
end
function cost = costfunction(runningcosts, terminalcosts, system, ...
N, T, t0, x0, u, ...
atol_ode_sim, rtol_ode_sim, type, sig, params)
cost = 0;
x = zeros(N+1, length(x0));
x = computeOpenloopSolution(system, N, T, t0, x0, u, ...
atol_ode_sim, rtol_ode_sim, type, sig, params);
for k=1:N
cost = cost+runningcosts(t0+k*T, x(k,:), u(:,k));
end
cost = cost+terminalcosts(t0+(N+1)*T, x(N+1,:));
end
function [c,ceq] = nonlinearconstraints(constraints, ...
terminalconstraints, system, cov_propagation, ...
N, T, t0, x0, u, sig, beta, params, atol_ode_sim, rtol_ode_sim, type)
x = zeros(N+1, length(x0));
x = computeOpenloopSolution(system, N, T, t0, x0, u, ...
atol_ode_sim, rtol_ode_sim, type, sig, params);
c = [];
ceq = [];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% constraint tightening computation (& constraint generation)%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% compute covariance matrix propagation
sigma_e = cov_propagation(N, sig);
g = [1;0]; % constraint: g*x < h -> x1 < x1_limit
K = [0.2858 -0.4910]; % stabilizing state feedback law
for k=1:N % k=1 refers to the initial state, so (technically) no state constraint necessary, but k=1 necessary for input constraint
gamma = sqrt(2*g'*sigma_e(:,:,k)*g)*erfinv(2*beta-1); % constraint tightening
[cnew, ceqnew] = constraints(t0+k*T,x(k,:),u(:,k), gamma, K, params); % generate constraints
c = [c cnew];
ceq = [ceq ceqnew];
end
[cnew, ceqnew] = terminalconstraints(t0+(N+1)*T,x(N+1,:));
c = [c cnew];
ceq = [ceq ceqnew];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% end of constraint tightening %%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end
function x = computeOpenloopSolution(system, N, T, t0, x0, u, ...
atol_ode_sim, rtol_ode_sim, type, sig, params)
x(1,:) = x0;
for k=1:N
x(k+1,:) = dynamic(system, T, t0, x(k,:), u(:,k), ...
atol_ode_sim, rtol_ode_sim, type, 0, sig, params);
end
end
function [x, t_intermediate, x_intermediate] = dynamic(system, T, t0, ...
x0, u, atol_ode, rtol_ode, type, apply_flag, sig, params)
if ( strcmp(type, 'difference equation') )
x = system(t0, x0, u, T, apply_flag, sig, params);
x_intermediate = [x0; x];
t_intermediate = [t0, t0+T];
elseif ( strcmp(type, 'differential equation') )
options = odeset('AbsTol', atol_ode, 'RelTol', rtol_ode);
[t_intermediate,x_intermediate] = ode45(system, ...
[t0, t0+T], x0, options, u, apply_flag, sig, params);
x = x_intermediate(size(x_intermediate,1),:);
end
end
function printSolution(system, printHeader, printClosedloopData, ...
plotTrajectories, mpciter, T, t0, x0, u, params, ...
atol_ode, rtol_ode, type, iprint, exitflag, output, t_Elapsed)
if (mpciter == 0)
printHeader();
end
printClosedloopData(mpciter, u, x0, t_Elapsed);
switch exitflag
case -2
if ( iprint >= 1 && iprint < 10 )
fprintf(' Error F\n');
elseif ( iprint >= 10 )
fprintf(' Error: No feasible point was found\n')
end
case -1
if ( iprint >= 1 && iprint < 10 )
fprintf(' Error OT\n');
elseif ( iprint >= 10 )
fprintf([' Error: The output function terminated the',...
' algorithm\n'])
end
case 0
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' Warning IT\n');
elseif ( iprint >= 10 )
fprintf([' Warning: Number of iterations exceeded',...
' options.MaxIter or number of function',...
' evaluations exceeded options.FunEvals\n'])
end
case 1
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' \n');
elseif ( iprint >= 10 )
fprintf([' First-order optimality measure was less',...
' than options.TolFun, and maximum constraint',...
' violation was less than options.TolCon\n'])
end
case 2
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' Warning TX\n');
elseif ( iprint >= 10 )
fprintf(' Warning: Change in x was less than options.TolX\n')
end
case 3
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' Warning TJ\n');
elseif ( iprint >= 10 )
fprintf([' Warning: Change in the objective function',...
' value was less than options.TolFun\n'])
end
case 4
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' Warning S\n');
elseif ( iprint >= 10 )
fprintf([' Warning: Magnitude of the search direction',...
' was less than 2*options.TolX and constraint',...
' violation was less than options.TolCon\n'])
end
case 5
if ( iprint == 1 )
fprintf('\n');
elseif ( iprint >= 2 && iprint < 10 )
fprintf(' Warning D\n');
elseif ( iprint >= 10 )
fprintf([' Warning: Magnitude of directional derivative',...
' in search direction was less than',...
' 2*options.TolFun and maximum constraint',...
' violation was less than options.TolCon\n'])
end
end
if ( iprint >= 5 )
plotTrajectories(@dynamic, system, T, t0, x0, u, atol_ode, rtol_ode, type, params)
end
end
function printHeaderDummy(varargin)
end
function printClosedloopDataDummy(varargin)
end
function plotTrajectoriesDummy(varargin)
end