System with Uncertain Parameters
As an example of a closed-loop system with uncertain parameters, consider the two-cart "ACC Benchmark" system [13] consisting of two frictionless carts connected by a spring shown as follows.
ACC Benchmark Problem
The system has the block diagram model shown below, where the individual carts have the respective transfer functions.
The parameters m1, m2, and k are uncertain, equal to one plus or minus 20%:
m1 = 1 ± 0.2 m2 = 1 ± 0.2 k = 1 ± 0.2
"ACC Benchmark" Two-Cart System Block Diagram y1 = P(s) u1
The upper dashed-line block has transfer function matrix F(s):
This code builds the uncertain system model P shown above:
m1 = ureal('m1',1,'percent',20); m2 = ureal('m2',1,'percent',20); k = ureal('k',1,'percent',20); s = zpk('s'); G1 = ss(1/s^2)/m1; G2 = ss(1/s^2)/m2; F = [0;G1]*[1 -1]+[1;-1]*[0,G2]; P = lft(F,k);
The variable P is a SISO uncertain state-space (USS) object with four states and three uncertain parameters,
m1, m2, and k. You can
recover the nominal plant with the command:
zpk(P.nominal)
ans =
1
-------------
s^2 (s^2 + 2)
Continuous-time zero/pole/gain model.
If the uncertain model P(s) has LTI negative feedback controller
then you can form the controller and the closed-loop system
y1 =
T(s) u1
and view the closed-loop system's step response on the time interval from
t=0 to t=0.1 for a Monte Carlo random sample of five combinations of the three uncertain
parameters k, m1, and m2 using
this code:
C=100*ss((s+1)/(.001*s+1))^3; % LTI controller T=feedback(P*C,1); % closed-loop uncertain system step(usample(T,5),.1);
