www.gusucode.com > robust_featured 案例源码程序 matlab代码 > robust_featured/RobustControlExample.m
%% First-Cut Robust Design
% This example shows how to use the Robust Control Toolbox(TM) commands |usample|,
% |ucover| and |dksyn| to design a robust controller with standard
% performance objectives. It can serve as a template for more complex
% robust control design tasks.
% Copyright 1986-2012 The MathWorks, Inc.
%% Introduction
% The plant model consists of a first-order system with uncertain gain and
% time constant in series with a mildly underdamped resonance and
% significant unmodeled dynamics. The uncertain variables are specified
% using |ureal| and |ultidyn| and the uncertain plant model |P| is constructed
% as a product of simple transfer functions:
gamma = ureal('gamma',2,'Perc',30); % uncertain gain
tau = ureal('tau',1,'Perc',30); % uncertain time-constant
wn = 50; xi = 0.25;
P = tf(gamma,[tau 1]) * tf(wn^2,[1 2*xi*wn wn^2]);
% Add unmodeled dynamics
delta = ultidyn('delta',[1 1],'SampleStateDim',5,'Bound',1);
W = makeweight(0.1,20,10);
P = P * (1+W*delta);
%%
% A collection of step responses for randomly sampled uncertainty values
% illustrates the plant variability.
step(P,5)
%% Covering the Uncertain Model
% The uncertain plant model |P| contains 3 uncertain elements.
% For feedback design purposes, it is often desirable to simplify the
% uncertainty model while approximately retaining its overall variability.
% This is one use of the command |ucover|. This command
% takes an array of LTI responses |Pa| and a nominal response |Pn| and
% models the difference |Pa-Pn| as multiplicative uncertainty in the system
% dynamics (|ultidyn|).
%
% To use |ucover|, first map the uncertain model |P| into a family of LTI
% responses using |usample|. This command samples the uncertain elements in
% an uncertain system. It returns an array of LTI models where each model
% representing one possible behavior of the uncertain system.
% In this example, generate 60 sample values of |P|:
rng('default'); % the random number generator is seeded for repeatability
Parray = usample(P,60);
%%
% Next, use |ucover| to cover all behaviors in |Parray| by a simple
% uncertain model of the form
%
% Usys = Pn * (1 + Wt * Delta)
%
% where all the uncertainty is concentrated in the "unmodeled dynamics"
% component |Delta| (a |ultidyn| object). Choose the nominal value of |P| as
% center |Pn| of the cover, and use a 2nd-order shaping filter |Wt| to capture
% how the relative gap between |Parray| and |Pn| varies with frequency.
Pn = P.NominalValue;
orderWt = 2;
Parrayg = frd(Parray,logspace(-3,3,60));
[Usys,Info] = ucover(Parrayg,Pn,orderWt,'in');
%%
% Verify that the filter magnitude (in red) "covers" the
% relative variations of the plant frequency response (in blue).
Wt = Info.W1;
bodemag((Pn-Parray)/Pn,'b--',Wt,'r')
%% Creating the Open-Loop Design Model
% To design a robust controller for the uncertain plant |P|,
% choose a target closed-loop bandwidth |desBW| and perform a
% sensitivity-minimization design using the simplified uncertainty
% model |Usys|. The control structure is shown in Figure 1.
% The main signals are the disturbance |d|, the measurement noise |n|,
% the control signal |u|, and the plant output |y|.
% The filters |Wperf| and |Wnoise| reflect
% the frequency content of the disturbance and noise signals, or
% equivalently, the frequency bands where good disturbance
% and noise rejection properties are needed.
%
% Our goal is to keep |y| close to zero by rejecting the disturbance |d|
% and minimizing the impact of the measurement noise |n|. Equivalently,
% we want to design a controller that keeps the gain from |[d;n]| to |y|
% "small." Note that
%
% y = Wperf * 1/(1+PC) * d + Wnoise * PC/(1+PC) * n
%
% so the transfer function of interest consists of performance- and
% noise-weighted versions of the sensitivity function
% 1/(1+PC) and complementary sensitivity function PC/(1+PC).
%
% <<../designdemoIC.png>>
%
% *Figure 1*: Control Structure.
%%
% Choose the performance weighting function |Wperf| as
% a first-order low-pass filter with magnitude greater
% than 1 at frequencies below the desired closed-loop bandwidth:
desBW = 0.4;
Wperf = makeweight(500,desBW,0.5);
%%
% To limit the controller bandwidth and induce
% roll off beyond the desired bandwidth, use a sensor noise
% model |Wnoise| with magnitude greater than 1 at frequencies greater
% than |10*desBW|:
Wnoise = 0.0025 * tf([25 7 1],[2.5e-5 .007 1]);
%%
% Plot the magnitude profiles of |Wperf| and |Wnoise|:
bodemag(Wperf,'b',Wnoise,'r'), grid
title('Performance weight and sensor noise model')
legend('Wperf','Wnoise','Location','SouthEast')
%%
% Next build the open-loop interconnection of Figure 1:
Usys.InputName = 'u'; Usys.OutputName = 'yp';
Wperf.InputName = 'd'; Wperf.OutputName = 'yd';
Wnoise.InputName = 'n'; Wnoise.OutputName = 'yn';
sumy = sumblk('y = yp + yd');
sume = sumblk('e = -y - yn');
M = connect(Usys,Wperf,Wnoise,sumy,sume,{'d','n','u'},{'y','e'});
%% First Design: Low Bandwidth Requirement
% The controller design is carried out with the automated robust design
% command |dksyn|. The uncertain open-loop model is given by |M|.
[ny,nu] = size(Usys);
[K1,ClosedLoop,muBound] = dksyn(M,ny,nu);
muBound
%%
% The |muBound| is a positive scalar. If it is near 1, then the design
% is successful and the desired and effective closed-loop bandwidths match
% closely. As a rule of thumb, if |muBound| is less than 0.85, then the
% achievable performance can be improved. When |muBound| is greater than
% 1.2, then the desired closed-loop bandwidth is not achievable for the
% given amount of plant uncertainty.
%
% Since, here, |muBound| is approximately 0.82, the objectives are met, but
% could ultimately be improved upon. For validation purposes, create
% Bode plots of the open-loop response for different values of the uncertainty
% and note the typical zero-dB crossover frequency and phase margin:
opt = bodeoptions;
opt.PhaseMatching = 'on';
opt.Grid = 'on';
bodeplot(Parray*K1,{1e-2,1e2},opt);
%%
% Randomized closed-loop Bode plots confirm a
% closed-loop disturbance rejection bandwidth of approximately 0.4 rad/s.
S1 = feedback(1,Parray*K1); % sensitivity to output disturbance
bodemag(S1,{1e-2,1e3}), grid
%%
% Finally, compute and plot the closed-loop responses to a step disturbance at the
% plant output. These are consistent with the desired
% closed-loop bandwidth of 0.4, with settling times approximately 7 seconds.
step(S1,8);
%%
% In this naive design strategy, we have correlated the noise bandwidth
% with the desired closed-loop bandwidth. This simply helps limit
% the controller bandwidth. A fair perspective is
% that this approach focuses on output disturbance attenuation in
% the face of plant model uncertainty. Sensor noise is not truly
% addressed. Problems with considerable amounts of
% sensor noise would be dealt with in a different manner.
%% Second Design: Higher Bandwidth Requirement
% Let's redo the design for a higher target bandwidth and adjusting the noise
% bandwidth as well.
desBW = 2;
Wperf = makeweight(500,desBW,0.5);
Wperf.InputName = 'd'; Wperf.OutputName = 'yd';
Wnoise = 0.0025 * tf([1 1.4 1],[1e-6 0.0014 1]);
Wnoise.InputName = 'n'; Wnoise.OutputName = 'yn';
M = connect(Usys,Wperf,Wnoise,sumy,sume,{'d','n','u'},{'y','e'});
[K2,ClosedLoop2,muBound2] = dksyn(M,ny,nu);
muBound2
%%
% With |muBound2| about 0.99, this design achieves a good tradeoff between
% performance goals and plant uncertainty. Open-loop Bode plots confirm a
% fairly robust design with decent phase margins, but not as good as the lower
% bandwidth design.
bodeplot(Parray*K2,{1e-2,1e2},opt)
%%
% Randomized closed-loop Bode plots confirm a
% closed-loop bandwidth of approximately 2 rad/s. The frequency response
% has a bit more peaking than was seen in the lower bandwidth design, due
% to the increased uncertainty in the model at this frequency. Since the
% Robust Performance mu-value was 0.99, we expected some degradation in the
% robustness of the performance objectives over the lower bandwidth
% design.
S2 = feedback(1,Parray*K2);
bodemag(S2,{1e-2,1e3}), grid
%%
% Closed-loop step disturbance responses further illustrate the
% higher bandwidth response, with reasonable robustness across the
% plant model variability.
step(S2,8);
%% Third Design: Very Aggressive Bandwidth Requirement
% Redo the design once more with an extremely optimistic
% closed-loop bandwidth goal of 15 rad/s.
desBW = 15;
Wperf = makeweight(500,desBW,0.5);
Wperf.InputName = 'd'; Wperf.OutputName = 'yd';
Wnoise = 0.0025 * tf([0.018 0.19 1],[0.018e-6 0.19e-3 1]);
Wnoise.InputName = 'n'; Wnoise.OutputName = 'yn';
M = connect(Usys,Wperf,Wnoise,sumy,sume,{'d','n','u'},{'y','e'});
[K3,ClosedLoop3,muBound3] = dksyn(M,ny,nu);
muBound3
%%
% Since |muBound3| is greater than 1.5, the closed-loop performance goals
% are not achieved under plant uncertainties. The frequency responses of the
% closed-loop system have higher peaks indicating the poor performance of
% the designed controller.
S3 = feedback(1,Parray*K3);
bodemag(S3,{1e-2,1e3}), grid
%%
% Similarly, step responses under uncertainties illustrate the poor
% closed-loop performance.
step(S3,1);
%% Robust Stability Calculations
% The Bode and Step response plots shown above are generated from
% samples of the uncertain plant model |P|. We can use the
% uncertain model directly, and assess the robust stability
% of the 3 closed-loop systems.
ropt = robOptions('Display','on','MussvOptions','sm5');
stabmarg1 = robstab(feedback(P,K1),ropt);
%%
stabmarg2 = robstab(feedback(P,K2),ropt);
%%
stabmarg3 = robstab(feedback(P,K3),ropt);
%%
% The robustness analysis reports confirm what we have observed by sampling the
% closed-loop time and frequency responses. The second design is a good
% compromise between performance and robustness, and the third design is
% too aggressive and lacks robustness.