errors with musyn -- Plant must have at least <> inputs and <> outputs

Taking a very simple inverted pendulum found from these dynamics: https://towardsdatascience.com/comparing-optimal-control-and-reinforcement-learning-using-the-cart-pole-swing-up-openai-gym-772636bc48f4

4 states, 1 input, 4 outputs.

Attempting to find a robust controller using musyn, but coming across this error message

```

Error using uss/musyn (line 74)

P must have at least 2 inputs and 5 outputs.

```

Code below:

% PLANT MODEL
g = ureal('g', 9.8, 'percentage', 80);
mcart = ureal('mcart', 1.0, 'percentage', 80);
mpole = ureal('mpole', 0.1, 'percentage', 80);
l = ureal('l', 2*0.5, 'percentage', 80);
M = mcart + mpole;

a = g /(l*(4.0/3 - mpole/(mpole+mcart)));
A = [0 1 0 0; 0 0 a 0; 0 0 0 1; 0 0 a 0];

b = -1/(l*(4.0/3 - mpole/(mpole+mcart)));
B = [0;1/mcart; 0; b];
C = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1];
D = [0; 0; 0; 0];

% MAKE SS
P = ss(A, B, C, D);
P.StateName = {'cart pos (m)';'car vel (m/s)';...
               'pole angle (\theta)';'pole angular vel  (\theta/s)'};
P.InputName = {'force'};
P.OutputName = {'x'; 'xdot'; 'theta'; 'thetadot'};

% ROBUST ANALYSIS
nY =  4;
nU = 1;
[Krob, ~, gamma_inf] = musyn(P, nY, nU)

Answer 1

[Paul]

The plant model has 4 outputs and 1 input. If you look at

doc musyn

you'll see that the second (nY) and third inputs (nU) are the number of measurements (y) from and control inputs (u) to the plant. But the plant has additional inputs (w) and outputs (z) that define the the closed loop transfer function for optimization. So the plant has to have at least 2 inputs and 5 outputs altogether, which is what the error message is saying.

Having said all that, are you sure musyn is the correct function to use? I'm asking because

a) the comment says "Robust Analysis" and musyn is used for robust design, and

b) the outputs from the call to musyn don't seem to match any of the signatures on doc page.

[Jonathan Salfity]

thank you for the comment !

The only uncertainty I intend is the physics parameters, g, mcart, mpole, l. How do I add in the extra input , w, and output, z ?

a) you're right, I incorrectly titled my comment, it should be ROBUST CONTROLLER SYNTHESIS.

b) the code should probably look like this: [Krob, G_uncCL_perf, info] = musyn(G_unc, nmeas, ncont)

[Paul]

Defining w and z is the art of mu synthesis. Have you looked at any of the examples in the documenation? There appear to be several.

[Jonathan Salfity]

@Paul -- thanks for the follow up. I searched around and found some good examples, i looked at these two and modified my code like below:

I think I'm getting what I want, except I didnt realize Krob is a ss, is there a K matrix that gets output for use in state feedback ?

https://www.mathworks.com/help/robust/ug/robust-controller-for-spinning-satellite.html
https://www.mathworks.com/help/robust/gs/using-mixsyn-for-h-infinity-loop-shaping.html

% To achieve good reference-tracking and disturbance-rejection performance, 
% shape S to be small inside the control bandwidth, 
% which means choosing W1 large at low frequency, rolling off at high frequency. 
% For this example, specify W1 with:
wS = makeweight(33, 5, 0.5);  % sensitivity      

% For robustness and noise attenuation, 
% shape T to be small outside the control bandwidth, 
% which means choosing W3 large at high frequency.
wT = makeweight(0.5,20,20);  % complementary sensitivity function

wKS = []; % w to u, control effort 

P = augw(G_unc, wS, wKS, wT);

nmeas =  4;
ncont = 1;
[Krob, G_uncCL_Perf, info] = musyn(P, nmeas, ncont)

[Paul]

I don't think you can get a state feedback solution, but it's been a long time since I did any mu synthesis stuff and so could very well be wrong.