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%% Explicit MPC Control of an Aircraft with Unstable Poles
% This example shows how to control an unstable aircraft with saturating
% actuators using Explicit MPC.
%
% Reference:
%
% [1] P. Kapasouris, M. Athans and G. Stein, "Design of feedback control
% systems for unstable plants with saturating actuators", Proc. IFAC Symp.
% on Nonlinear Control System Design, Pergamon Press, pp.302--307, 1990
%
% [2] A. Bemporad, A. Casavola, and E. Mosca, "Nonlinear control of
% constrained linear systems via predictive reference management", IEEE(R)
% Trans. Automatic Control, vol. AC-42, no. 3, pp. 340-349, 1997.
%
% See also MPCAIRCRAFT.
% Copyright 1990-2014 The MathWorks, Inc.
%% Define Aircraft Model
% The linear open-loop dynamic model is as follows:
A = [-0.0151 -60.5651 0 -32.174;
-0.0001 -1.3411 0.9929 0;
0.00018 43.2541 -0.86939 0;
0 0 1 0];
B = [-2.516 -13.136;
-0.1689 -0.2514;
-17.251 -1.5766;
0 0];
C = [0 1 0 0;
0 0 0 1];
D = [0 0;
0 0];
plant = ss(A,B,C,D);
x0 = zeros(4,1);
%%
% The manipulated variables are the elevator and flaperon angles, the
% attack and pitch angles are measured outputs to be regulated.
%
% The open-loop response of the system is unstable.
pole(plant)
%% Design MPC Controller
% To obtain an Explicit MPC controller, you must first design a traditional
% MPC (also referred as Implicit MPC) that is able to achieves your control
% objectives.
% *MV Constraints*
%%
% Both manipulated variables are constrained between +/- 25 degrees. Since
% the plant inputs and outputs are of different orders of magnitude, you
% also use scale factors to faciliate MPC tuning. Typical choices of scale
% factor are the upper/lower limit or the operating range.
MV = struct('Min',{-25,-25},'Max',{25,25},'ScaleFactor',{50,50});
%%
% *OV Constraints*
%%
% Both plant outputs have constraints to limit undershoots at the first
% prediction horizon. You also specify scale factors for outputs.
OV = struct('Min',{[-0.5;-Inf],[-100;-Inf]},'Max',{[0.5;Inf],[100;Inf]},'ScaleFactor',{1,200});
%%
% *Weights*
%%
%%
% The control task is to get zero offset for piecewise-constant references,
% while avoiding instability due to input saturation. Because both MV and
% OV variables are already scaled in MPC controller, MPC weights are
% dimensionless and applied to the scaled MV and OV values. In this
% example, you penalize the two outputs equally with the same OV weights.
Weights = struct('MV',[0 0],'MVRate',[0.1 0.1],'OV',[10 10]);
%%
% *Construct the traditional MPC controller*
%%
% Create an MPC controller with plant model, sample time and horizons.
Ts = 0.05; % Sampling time
p = 10; % Prediction horizon
m = 2; % Control horizon
mpcobj = mpc(plant,Ts,p,m,Weights,MV,OV);
%% Generate Explicit MPC Controller
% Explicit MPC executes the equivalent explicit piecewise affine version of
% the MPC control law defined by the traditional MPC. To generate an
% Explicit MPC from a traditional MPC, you must specify range for each
% controller state, reference signal, manipulated variable and measured
% disturbance so that the multi-parametric quadratic programming problem is
% solved in the parameter space defined by these ranges.
%%
% *Obtain a range structure for initialization*
%%
% Use |generateExplicitRange| command to obtain a range structure where you
% can specify range for each parameter afterwards.
range = generateExplicitRange(mpcobj);
%%
% *Specify ranges for controller states*
%%
% MPC controller states include states from plant model, disturbance model
% and noise model in that order. Setting the range of a state variable is
% sometimes difficult when the state does not correspond to a physical
% parameter. In that case, multiple runs of open-loop plant simulation
% with typical reference and disturbance signals are recommended in order
% to collect data that reflect the ranges of states.
range.State.Min(:) = -10000;
range.State.Max(:) = 10000;
%%
% *Specify ranges for reference signals*
%%
% Usually you know the practical range of the reference signals being used
% at the nominal operating point in the plant. The ranges used to generate
% Explicit MPC must be at least as large as the practical range.
range.Reference.Min = [-1;-11];
range.Reference.Max = [1;11];
%%
% *Specify ranges for manipulated variables*
%%
% If manipulated variables are constrained, the ranges used to generate
% Explicit MPC must be at least as large as these limits.
range.ManipulatedVariable.Min = [MV(1).Min; MV(2).Min] - 1;
range.ManipulatedVariable.Max = [MV(1).Max; MV(2).Max] + 1;
%%
% *Construct the Explicit MPC controller*
%%
% Use |generateExplicitMPC| command to obtain the Explicit MPC controller
% with the parameter ranges previously specified.
mpcobjExplicit = generateExplicitMPC(mpcobj, range);
display(mpcobjExplicit);
%%
% Use |simplify| command with the "exact" method to join pairs of regions
% whose corresponding gains are the same and whose union is a convex set.
% This practice can reduce memory footprint of the Explicit MPC controller
% without sacrifice any performance.
mpcobjExplicitSimplified = simplify(mpcobjExplicit, 'exact');
display(mpcobjExplicitSimplified);
%%
% The number of piecewise affine region has been reduced.
%% Plot Piecewise Affine Partition
% You can review any 2-D section of the piecewise affine partition defined
% by the Explicit MPC control law.
%%
% *Obtain a plot parameter structure for initialization*
%%
% Use |generatePlotParameters| command to obtain a parameter structure
% where you can specify which 2-D section to plot afterwards.
params = generatePlotParameters(mpcobjExplicitSimplified);
%%
% *Specify parameters for a 2-D plot*
%%
% In this example, you plot the pitch angle (the 4th state variable) vs.
% its reference (the 2nd reference signal). All the other parameters must
% be fixed at a value within its range.
%%
% Fix other state variables
params.State.Index = [1 2 3 5 6];
params.State.Value = [0 0 0 0 0];
%%
% Fix other reference signals
params.Reference.Index = 1;
params.Reference.Value = 0;
%%
% Fix manipulated variables
params.ManipulatedVariable.Index = [1 2];
params.ManipulatedVariable.Value = [0 0];
%%
% *Plot the 2-D section*
%%
% Use |plotSection| command to plot the 2-D section defined previously.
plotSection(mpcobjExplicitSimplified, params);
axis([-10 10 -10 10]);
grid;
xlabel('Pitch angle (x_4)');
ylabel('Reference on pitch angle (r_2)');
%% Simulate Using Simulink(R)
% To run this example, Simulink(R) is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
%%
% Simulate closed-loop control of the linear plant model in Simulink, using
% the Explicit MPC Controller block. Controller "mpcobjExplicitSimplified"
% is specified in the block dialog.
mdl = 'empc_aircraft';
open_system(mdl)
sim(mdl)
%%
% The closed-loop response is identical to the traditional MPC controller
% designed in the "mpcaircraft" example.
%%
bdclose(mdl)