8.11. Controlling a DC motor using a FORCES PRO SQP solver

In this example our aim is to control a DC-motor using a FORCES PRO SQP solver. The model for the DC motor which we consider is borrowed from [BerUnb], to which we refer for further details. The dynamics of our model is described by the following set of ordinary differential equations:

\[\begin{split}\dot{x}_1(t) &= -\frac{R_a}{L_a}x_1(t) - \frac{k_m}{L_a}u(t)x_2(t) + \frac{u_a}{L_a} \\ \dot{x}_2(t) &= -\frac{B}{J}x_2(t) + \frac{k_m}{J}u(t)x_1(t) - \frac{\tau_l}{J}.\end{split}\]

The states \(x_1\) and \(x_2\) model the armature current and motor angular speed of out DC motor respectively and the control \(u\) models the input field current. The following values are chosen for our model constants

\begin{alignat*}{3} R_a &= 12.548 \Omega && \quad \text{(armature resistance)} \\ L_a &= 0.307 \text{H} && \quad \text{(armature inductance)} \\ k_m &= 0.23576 \text{Nm}/\text{A}^2 && \quad \text{(motor constant)} \\ u_a &= 60 \text{V} && \quad \text{(armature voltage)} \\ B &= 0.00783 \text{Nmsec} && \quad \text{(total viscuous damping)} \\ \tau_L &= 1.47 \text{Nmsec} && \quad \text{(Load torque)} \\ J &= 0.00385 \text{Nmsec}^2 && \quad \text{(total moment of inertia)} \end{alignat*}

The control objective is to track a piecewise constant angular speed. To test the robustness of out resulting controller we switch reference half way through our simulation. In the first half of the simulation we will track a constant angular speed \(x_2^{ref1} = 2\) and then switch to tracking a constant angular speed \(x_2^{ref2} = -2\). We collect the 2-dimensional state vector \(x = (x_1, x_2)^T\) and the 1-dimensional control \(u\) in the vector

\[\begin{split}z = \begin{pmatrix} u \\ x_1 \\ x_2 \end{pmatrix}\end{split}\]

You can download the following Matlab code of this example to try it out for yourself by clicking here.

8.11.1. Defining the MPC problem

8.11.1.1. The tracking objective function

Since we want to track a reference value it is natural to consider a least squared cost \(f(z,p) = \frac{|\!| r(z,p) |\!|^2}{2}\) for

\[r(z,p) = z_3 - p\]

Recall that \(z_3=x_2\) models the motor angular speed which is made to track a piecewise constant reference. The parameter \(p\) will be equal to \(x_2^{ref1}\) during the first half of the simulation and equal to \(x_2^{ref2}\) during the second.

The following code snippet reads in the least squared objective

model.LSobjective = @(z,p) sqrt(100) * (z(3) - p);
model.LSobjectiveN = @(z,p) sqrt(100) * (z(3) - p);

8.11.1.2. The dynamics

The following code snippet reads in the dynamics (8.1) of our model:

%% model parameters
% Armature inductance (H)
La = 0.307;
% Armature resistance (Ohms)
Ra = 12.548;
% Motor constant (Nm/A^2)
km = 0.23576;
% Total moment of inertia (Nm.sec^2)
J = 0.00385;
% Total viscous damping (Nm.sec)
B = 0.00783;
% Load torque (Nm)
tauL = 1.47;
% Armature voltage (V)
ua = 60;

model.E = [zeros(2,1), eye(2)];
model.continuous_dynamics = @(x,u) [(-1/La)*(Ra*x(1) + x(2)*u(1) - ua);...
                                           (-1/J)*(B*x(2) - km*x(1)*u(1) + tauL)];

8.11.1.3. Input and state constraints

The following constraints are to be met by out control and state vectors:

\[\begin{split}1 \text{A} \leq \ &u \leq 1.6 \text{A}\\ -5 \text{A} \leq \ &x_1 \leq 5 \text{A}\\ -10 \tfrac{\text{rad}}{\text{sec}} \leq \ &x_2 \leq 10 \tfrac{\text{rad}}{\text{sec}}\end{split}\]

This can be read into the FORCES PRO model as follows

model.lb = [1, -5, -10];
model.ub = [1.6, 5, 10];

8.11.1.4. Generating the FORCES PRO SQP solver

To generate a suitable SQP solver for out MPC problem one need a model struct as well as a codeoptions struct. Our model struct has been populated above and we now specify the codeoptions we want and generating the solver by calling FORCES_NLP. The following code-snippet shows how this can be done:

%% set codeoptions
codeoptions = getOptions('FORCESPROSolver');
codeoptions.solvemethod = 'SQP_NLP'; % generate SQP solver
codeoptions.nlp.integrator.type = 'ERK4';
codeoptions.nlp.integrator.Ts = 0.01;
codeoptions.nlp.integrator.nodes = 1;
codeoptions.nlp.hessian_approximation = 'gauss-newton';
codeoptions.server = 'https://forces.embotech.com';

%% generate FORCES PRO solver
FORCES_NLP(model, codeoptions);

8.11.1.5. Calling the solver

Once the solver has been generated it needs a struct containing an initial guess, initial condition of the ODE, the run-time parameters and the reinitialize field as explained in Sequential quadratic programming algorithm. The following code-snippet shows how this can be done:

% populate run time parameters struct
params.all_parameters = repmat(2, model.N, 1);
params.xinit = zeros(model.neq, 1); % initial condition to ODE
params.x0 = repmat([1.2;zeros(2,1)], model.N, 1); % initial guess
params.reinitialize = 0;

% Solve optimization problem
[output, exitflag, info] = FORCESPROSolver(params);

The FORCESPROSolver is expected to return an exitflag equal to \(1\), which corresponds to a successful solve. However, note that the QP could become infeasible in some cases. In this case, one should expect an exitflag equal to \(−8\).

8.11.1.6. Results

The control objective is to track an angular speed of 2 and -2 respectively. As can be seen in Figure 8.54 the controller completes this task. The control (\(u\)) is plotted in Figure 8.50, the first state (\(x_1\)) is plotted in Figure 8.51 and second state (\(x_2\)) in Figure 8.52. As a measure of control quality, the closed loop objective value is plotted in Figure 8.53.

../../_images/dc_motor_control.png

Figure 8.50 The control (\(u\), in blue) as a function of simulation time (s). The control obeys its constraints (red) throughout the simulation.

../../_images/dc_motor_first_state.png

Figure 8.51 The first state (\(x_1\), in blue) as a function of simulation time. It obeys its constraints (red) throughout the simulation.

../../_images/dc_motor_second_state.png

Figure 8.52 The second state (\(x_2\), in blue) as a function of simulation time. It obeys its constraints (red) throughout the simulation.

../../_images/dc_motor_obj.png

Figure 8.53 Closed-loop objective value as a function of time

../../_images/dc_motor_reference.png

Figure 8.54 Angular speed (blue) and tracked reference (red) value as a function of time.

BerUnb

S. Daniel-Berhe; H. Unbehauen: Experimental physical parameter estimation of a thyristor driven DC-motor using the HMF-method. Control Engineering Practice, 6:615–626, 1998