mpc = MPCController(sys)

model = LTISystem('A', [1 1; 0 1], 'B', [1; 0.5]);

model.x.min = [-5; -5];
model.x.max = [5; 5];

model.u.min = -1;
model.u.max = 1;

Q = [1 0; 0 1];
model.x.penalty = QuadFunction(Q);

R = 1;
model.u.penalty = OneFunction(R);

N = 5;
mpc = MPCController(model, N)

x0 = [4; 0];
u = mpc.evaluate(x0)

u =

    -1

[u, feasible, openloop] = mpc.evaluate(x0)

openloop = 

    cost: 32.4898
       U: [-1 -1 0.1393 0.3361 -5.2042e-16]
       X: [2x6 double]
       Y: [0x5 double]

expmpc = mpc.toExplicit();

[u, feasible, openloop] = expmpc.evaluate(x0)

openloop = 

    cost: 32.4898
       U: [-1 -1 0.1393 0.3361 -5.2042e-16]
       X: [2x6 double]
       Y: [0x5 double]

A = [ 0.5403, -0.8415; 0.8415,  0.5403];
B = [ -0.4597; 0.8415];
C = [1 0];
D = 0;

sys = ss(A,B,C,D,1);
model = LTISystem(sys);

model.y.min = -10;
model.y.max = 10;

model.u.min = -1;
model.u.max = 1;

model.x.penalty = QuadFunction(eye(2));
model.u.penalty = QuadFunction(1);

Tset = model.LQRSet;

PN = model.LQRPenalty;

model.x.with('terminalSet');
model.x.terminalSet = Tset;
model.x.with('terminalPenalty');
model.x.terminalPenalty = PN;

ctrl = MPCController(model,5);

• Regulation problem
• Tracking problem

model.x.penalty = QuadFunction(eye(2)); model.u.penalty = QuadFunction(1);

Q = [1 0; 0 1]; model.x.penalty = QuadFunction(Q);

• Use quadratic penalization of inputs, i.e., u_k^T R u_k with R=1:

R = 1; model.u.penalty = OneFunction(R);

MPC synthesis in MPT3

Synthesis of MPC controllers in MPT3 is based on two principles:

1. Constraints and penalties are specified directly in the prediction model
2. Every MPC controller in MPT3 starts as an on-line MPC regulator. Explicit controllers are only synthesized on demand.

To create an MPC controller in MPT3, use the MPCController constructor:

(:source lang=MATLAB -getcode:)

mpc = MPCController(sys)

where sys represents the prediction model, created by LTISystem, PWASystem or MLDSystem constructors (see here).

As an example, let use consider an LTI prediction model, represented by the state-update equation x^+ = A x + B u with A = [1 1; 0 1] and B = [1; 0.5]:

(:source lang=MATLAB -getcode:)

model = LTISystem('A', [1 1; 0 1], 'B', [1; 0.5]);

Next we specify more properties of the model:

• Add state constraints [-5; -5] <= x <= [5; 5]:

(:source lang=MATLAB -getcode:)

model.x.min = [-5; -5];
model.x.max = [5; 5];

• Add input constraints -1 <= u <= 1:

(:source lang=MATLAB -getcode:)

model.u.min = -1;
model.u.max = 1;

• Add quadratic state penalty, i.e., penalize x_k^T Q x_k with Q = [1 0; 0 1]:

(:source lang=MATLAB -getcode:)

model.x.penalty = Penalty([1 0; 0 1], 2);  % 2 = quadratic penalty

• Use quadratic penalization of inputs, i.e., u_k^T Q u_k with Q = 1:

(:source lang=MATLAB -getcode:)

model.u.penalty = Penalty(1, 2);

A number of other constraints and penalties can be added using the concept of filters.

Finally, we can synthesize an MPC controller with prediction horizon say N=5:

(:source lang=MATLAB -getcode:)

N = 5;
mpc = MPCController(model, N)

Now the mpc object is ready to be used as an MPC controller. For instance, we can ask it for the optimal control input for a given value of the initial state:

(:source lang=MATLAB -getcode:)

x0 = [4; 0];
u = mpc.evaluate(x0)

u =

    -1

You can also retrieve the full open-loop predictions:

(:source lang=MATLAB -getcode:)

[u, feasible, openloop] = mpc.evaluate(x0)

openloop = 

    cost: 32.4898
       U: [-1 -1 0.1393 0.3361 -5.2042e-16]
       X: [2x6 double]
       Y: [0x5 double]

Please consult this page to learn how to perform closed-loop simulations in MPT3.

Once you are satisfied with the controller's performance, you can convert it to an explicit form by calling the toExplicit() method:

(:source lang=MATLAB -getcode:)

expmpc = mpc.toExplicit();

This will generate an instance of the EMPCController class, which represents all explicit MPC controllers in MPT3. Explicit controllers can be used for evaluation/simulation in exactly the same way, i.e.,

(:source lang=MATLAB -getcode:)

[u, feasible, openloop] = expmpc.evaluate(x0)

openloop = 

    cost: 32.4898
       U: [-1 -1 0.1393 0.3361 -5.2042e-16]
       X: [2x6 double]
       Y: [0x5 double]

h1. Adding terminal set and terminal penalty

Consider an oscillator model defined in 2D.

(:source lang=MATLAB -getcode:)

A = [ 0.5403, -0.8415; 0.8415,  0.5403];
B = [ -0.4597; 0.8415];
C = [1 0];
D = 0;

Linear discrete-time model with sample time 1

(:source lang=MATLAB -getcode:)

sys = ss(A,B,C,D,1);
model = LTISystem(sys);

Set constraints on output

(:source lang=MATLAB -getcode:)

model.y.min = -10;
model.y.max = 10;

Set constraints on input

(:source lang=MATLAB -getcode:)

model.u.min = -1;
model.u.max = 1;

Include weights on states/inputs

(:source lang=MATLAB -getcode:)

model.x.penalty = Penalty(eye(2),2);
model.u.penalty = Penalty(1,2);

Compute terminal set

(:source lang=MATLAB -getcode:)

Tset = model.LQRSet;

Compute terminal weight

(:source lang=MATLAB -getcode:)

PN = model.LQRPenalty;

Add terminal set and terminal penalty

(:source lang=MATLAB -getcode:)

model.x.with('terminalSet');
model.x.terminalSet = Tset;
model.x.with('terminalPenalty');
model.x.terminalPenalty = PN;

Formulate finite horizon MPC problem

(:source lang=MATLAB -getcode:)

ctrl = MPCController(model,5);