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RegulationProblem

UI.RegulationProblem History

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November 09, 2013, at 08:57 PM by mk -
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** [[#mpexplicit|Explicit solution]]
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** [[#mpcexplicit|Explicit solution]]
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November 09, 2013, at 08:56 PM by mk -
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!! Regulation problem
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! Regulation problem
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!!! Finite horizon MPC
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!! Finite horizon MPC
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!!! Online solution
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!!! Explicit solution
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!!! Finite horizon MPC  with terminal penalty and terminal set constraints
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!! Finite horizon MPC  with terminal penalty and terminal set constraints
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!!! Regulation to a nonzero steady state
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!! Regulation to a nonzero steady state
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November 09, 2013, at 08:55 PM by mk -
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** [[#mpconline|Online solution]]
** [[#mpexplicit|Explicit solution]]
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[[#mpconline]]
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[[#mpcexplicit]]
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November 06, 2013, at 01:51 PM by mk -
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As an example, let us 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]@@:
to:
As an example, let us consider an LTI prediction model, represented by the state-update equation {$x^+ = A x + B u$} with {$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$} and {$ B = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix} $}
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* Add state constraints @@ [-5; -5] <= x <= [5; 5]@@:
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* Add state constraints {$ \begin{bmatrix} -5 \\ -5 \end{bmatrix} \le x \le \begin{bmatrix} 5 \\ 5 \end{bmatrix} $}
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* Add input constraints @@-1 <= u <= 1@@:
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* Add input constraints {$-1 \le u \le 1 $}:
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* Add quadratic state penalty, i.e., penalize @@x_k^T Q x_k@@ with @@Q = [1 0; 0 1]@@:
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* Add quadratic state penalty, i.e., penalize {$ x_k^T Q x_k $} with {$ Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $}
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* Use quadratic penalization of inputs, i.e., @@u_k^T R u_k@@ with @@R=1@@:
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* Use quadratic penalization of inputs, i.e., {$ u_k^T R u_k $} with {$ R=1 $}:
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Finally, we can synthesize an MPC controller with prediction horizon say @@N=5@@:
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Finally, we can synthesize an MPC controller with prediction horizon say {$ N=5 $}:
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August 29, 2013, at 07:24 AM by Michal Kvasnica -
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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]@@:
to:
As an example, let us 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]@@:
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August 07, 2013, at 09:26 AM by Martin Herceg -
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model.u.penalty = OneFunction(R);
to:
model.u.penalty = QuadFunction(R);
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August 05, 2013, at 09:45 AM by Martin Herceg -
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* [[#mpcss|Regulation to nonzero steady state]]

to:
* [[#mpcss|Regulation to a nonzero steady state]]

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!!! Regulation to nonzero steady state
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!!! Regulation to a nonzero steady state
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August 05, 2013, at 09:44 AM by Martin Herceg -
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%width=500% Attach:aircraft_nss.jpg %%
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August 05, 2013, at 09:34 AM by Martin Herceg -
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* [[#mpcss|Regulation to nonzero steady state]]

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[[#mpcss]]
!!! Regulation to nonzero steady state
The next example demonstrates how to achieve regulation to a nonzero steady state. Consider a model of an aircraft discretized with 0.2s sampling time
(:source lang=MATLAB -getcode:) [@
A = [0.99 0.01 0.18 -0.09 0; 0 0.94 0 0.29 0; 0 0.14 0.81 -0.9 0; 0 -0.2 0 0.95 0; 0 0.09 0 0 0.9];
B = [0.01  -0.02; -0.14 0; 0.05 -0.2; 0.02 0; -0.01 0];
C = [0 1 0 0 -1; 0 0 1 0 0; 0 0 0 1 0; 1 0 0 0 0];
model = LTISystem('A', A, 'B', B, 'C', C, 'Ts', 0.2);
@]
Given the dynamics of the model, one can compute the new steady state values for a particular value of the input, e.g.
(:source lang=MATLAB -getcode:) [@
us = [0.8; -0.3];
ys = C*( (eye(5)-A)\B*us );
@]
The computed values will be used as references for the desired steady state which can be added using "reference" filter
(:source lang=MATLAB -getcode:) [@
model.u.with('reference');
model.u.reference = us;
model.y.with('reference');
model.y.reference = ys;
@]
To fomulate MPC optimization problem one shall provide constraints and penalties on the system signals. In this example, only the input constraints are considered
(:source lang=MATLAB -getcode:) [@
model.u.min = [-5; -6];
model.u.max = [5; 6];
@]
The penalties on outputs and inputs are provided as quadratic functions
(:source lang=MATLAB -getcode:) [@
model.u.penalty = QuadFunction( diag([3, 2]) );
model.y.penalty = QuadFunction( diag([10, 10, 10, 10]) );
@]
which yields a quadratic optimization problem to be solved in MPC. The online MPC controller object is constructed with a horizon 6
(:source lang=MATLAB -getcode:) [@
ctrl = MPCController(model, 6)
@]
and simulated for zero initial conditions over 30 samples
(:source lang=MATLAB -getcode:) [@
loop = ClosedLoop(ctrl, model);
x0 = [0; 0; 0; 0; 0];
Nsim = 30;
data = loop.simulate(x0, Nsim);
@]
The simulation results can be plotted to inspect that the convergence to the new steady state has been achieved
(:source lang=MATLAB -getcode:) [@
subplot(2,1,1)
plot(1:Nsim, data.Y);
hold on;
plot(1:Nsim, ys*ones(1, Nsim), 'k--')
title('outputs')
subplot(2,1,2)
plot(1:Nsim, data.U);
hold on;
plot(1:Nsim, us*ones(1, Nsim), 'k--')
title('inputs')
@]

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August 03, 2013, at 04:55 PM by Martin Herceg -
Added lines 3-10:
* [[#mpc|Finite horizon MPC]]
* [[#mpctset|Finite horizon MPC with terminal penalty and terminal set constraints]]

----


[[#mpc]]
!!! Finite horizon MPC
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h1. Adding terminal set and terminal penalty
to:
[[#mpctset]]
!!! Finite horizon MPC  with terminal penalty and terminal set constraints
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August 03, 2013, at 04:49 PM by Martin Herceg -
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@]
to:
@]

Back to [[UI.Control|MPC synthesis]] overview.
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August 03, 2013, at 04:47 PM by Martin Herceg -
Added lines 1-143:
!! Regulation problem

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 ([[UI.Systems|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:) [@
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@@:
(:source lang=MATLAB -getcode:) [@
R = 1;
model.u.penalty = OneFunction(R);
@]

A number of other constraints and penalties can be added using the [[UI.Filters|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 [[UI.ClosedLoop|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 = QuadFunction(eye(2));
model.u.penalty = QuadFunction(1);
@]

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);
@]
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