Simple MPC Example

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Simple MPC Example

Consider the following linear MPC problem with lower and upper bounds on state and inputs, and a terminal cost term:

$$ \begin{align} \text{minimize} \ & x_N^T P x_N + \sum_{i=0}^{N-1} x_i^T Q x_i + u_i^T R u_i \\ \text{subject to} \ & x_0 = \color{#d22d33}{x} \\ & x_{i+1} = Ax_i + Bu_i \\ & \underline{x} \leq x_i \leq \bar{x} \\ & \underline{u} \leq u_i \leq \bar{u} \end{align} $$

This problem is parametric in the initial state $\color{#d22d33}{x}$, and the first input $u_0$ is typically applied to the system after a solution has been obtained. The following code generates a function that takes $-Ax$ as a calling argument and returns $u_0$, which can then be applied to the system:

Matlab
Python

%% FORCES multistage form % assume variable ordering zi = [ui, xi+1] for i=1...N-1 stages = MultistageProblem(N); % get stages struct of length N for i = 1:N % dimension stages(i).dims.n = nx+nu; % number of stage variables stages(i).dims.r = nx; % number of equality constraints stages(i).dims.l = nx+nu; % number of lower bounds stages(i).dims.u = nx+nu; % number of upper bounds % cost if( i == N ) stages(i).cost.H = blkdiag(R,P); % terminal cost (Hessian) else stages(i).cost.H = blkdiag(R,Q); end stages(i).cost.f = zeros(nx+nu,1); % linear cost terms % lower bounds stages(i).ineq.b.lbidx = 1:(nu+nx); % lower bound acts on these indices stages(i).ineq.b.lb = [umin; xmin]; % lower bound for this stage variable % upper bounds stages(i).ineq.b.ubidx = 1:(nu+nx); % upper bound acts on these indices stages(i).ineq.b.ub = [umax; xmax]; % upper bound for this stage variable % equality constraints if( i < N ) stages(i).eq.C = [zeros(nx,nu), A]; end if( i>1 ) stages(i).eq.c = zeros(nx,1); end stages(i).eq.D = [B, -eye(nx)]; end % RHS of first eq. constr. is a parameter: stages(1).eq.c = -A*x0 params(1) = newParam('minusA_times_x0',1,'eq.c');

Download MATLAB code with example simulation

# FORCES multistage form # assume variable ordering zi = [ui, xi+1] for i=1...N-1 stages = MultistageProblem(N) # get stages struct of length N for i in range(N): # dimension stages.dims[ i ]['n'] = nx+nu # number of stage variables stages.dims[ i ]['r'] = nx # number of equality constraints stages.dims[ i ]['l'] = nx+nu # number of lower bounds stages.dims[ i ]['u'] = nx+nu # number of upper bounds # cost if ( i == N-1 ): stages.cost[ i ]['H'] = np.vstack((np.hstack((R,np.zeros((nu,nx)))),np.hstack((np.zeros((nx,nu)),P)))) else: stages.cost[ i ]['H'] = np.vstack((np.hstack((R,np.zeros((nu,nx)))),np.hstack((np.zeros((nx,nu)),Q)))) stages.cost[ i ]['f'] = np.zeros((nx+nu,1)) # linear cost terms # lower bounds stages.ineq[ i ]['b']['lbidx'] = range(1,nu+nx+1) # lower bound acts on these indices stages.ineq[ i ]['b']['lb'] = np.concatenate((umin,xmin),0) # lower bound for this stage variable # upper bounds stages.ineq[ i ]['b']['ubidx'] = range(1,nu+nx+1) # upper bound acts on these indices stages.ineq[ i ]['b']['ub'] = np.concatenate((umax,xmax),0) # upper bound for this stage variable # equality constraints if ( i < N-1 ): stages.eq[i]['C'] = np.hstack((np.zeros((nx,nu)),A)) if ( i>0 ): stages.eq[i]['c'] = np.zeros((nx,1)) stages.eq[i]['D'] = np.hstack((B,-np.eye(nx))) # RHS of first eq. constr. is a parameter: stages(1).eq.c = -A*x0 stages.newParam('minusA_times_x0', [1], 'eq.c') # define output of the solver stages.newOutput('u0', 1, range(1,nu+1))

Download Python code with example simulation

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Simple MPC Example