Creating an uncertain model based on a family of multiple inputs multiple outputs models (MIMOs) - matlab

Question

Objective
Currently I am trying to create an uncertain system based on a family of statespace models using ucover. For this I am basing my script on the documentation "Modeling a Family of Responses as an Uncertain System" which shows the technique for creating an uncertain system based on a single-input-single-output system (SISO) explicitly but makes it clear that this is fully useable for MIMO systems as well.
Technical details
Specifically it is stated with the documentation of ucover that it supports MIMO systems:
USYS = ucover(PARRAY,PNOM,ORD1,ORD2,UTYPE) returns an uncertain
system USYS with nominal value PNOM and whose range of behaviors
includes all LTI responses in the LTI array PARRAY. PNOM and PARRAY
can be SS, TF, ZPK, or FRD models. USYS is of class UFRD if PNOM
is an FRD model and of class USS otherwise.
ORD1 and ORD2 specify the order (number of states) of each diagonal
entry of W1 and W2. If PNOM has NU inputs and NY outputs, ORD1 and ORD2
should be vectors of length:
UTYPE ORD1 ORD2
InputMult NU-by-1 NU-by-1
OutputMult NY-by-1 NY-by-1
Additive NY-by-1 NU-by-1
In my case I am using both 2 inputs and 2 outputs so both ORD1 adn ORD2 should be 2 by 1. I am using 8 as the number of states used by W1 and W2 (just because, I will try adjusting that once this issue is sorted).
The Attempt
Based on the SISO example I have attempted to create a MIMO example, this is shown below
noInputs=2;
noOutputs=2;
noOfStates=4;
Anom=rand(noOfStates,noOfStates);
Bnom=rand(noOfStates,noInputs);
Cnom=rand(noOutputs,noOfStates);
Dnom=rand(noOutputs,noInputs);
Pnom=ss(Anom, Bnom, Cnom, Dnom);
p1 = Pnom*tf(1,[.06 1]); % extra lag
p2 = Pnom*tf([-.02 1],[.02 1]); % time delay
p3 = Pnom*tf(50^2,[1 2*.1*50 50^2]);
Parray = stack(1,p1,p2,p3);
Parrayg = frd(Parray,logspace(-1,3,60));
[P,Info] = ucover(Parrayg,Pnom,[8 8]',[8 8]','InputMult');
Wt = Info.W1;
bodemag((Pnom-Parray)/Pnom,'b--',Wt,'r'); grid
title('Relative Gaps vs. Magnitude of Wt')
The problem
Unlike the image in the documentation my uncertain model (when put through a bode plot) only shows a response on the lead diagonal. See the screenshot for what I mean:
Where blue is the individual models and red is the uncertain model
Question
How can I create an uncertain system based on a family of MIMO statespace models that correctly covers responses between all inputs and outputs?

Answer 1

If you use [8,8]' as your uncertainty order structure ord1,ord2, matlab will try to have two diagonal blocks in your uncertainty block each.
However matlab only supports diagonal weighting functions (due to some complications about nonconvex search) and what you are plotting is the diagonal weighting that will multiply the 2x2 full block LTI dynamic uncertainty. W1 affects the rows and W2 affects the columns of the uncertainty.
Hence you should check the samples of that uncertainty multiplied by the weights and then the plant. Then you can compare it with the uncertain model stack. Notice that your off-diagonal entries are practically zero (<1e-10) hence almost decoupled. But W1, W2 search looks for the H-infinity norm hence you don't get to see perfect covering at each block of the Bode plot. It combines the rows/columns of the required minimum uncertainty amount (see the examples on the help file). That's why you see 1 plot per each weight in the demos.
If you would like to model the each uncertainty affecting each block separately then you need to form a new augmented LFT such that the uncertainty is four 1x1(scalar) LTI dynamic uncertainty on the diagonal then you can have four entries in ord1 and ord2.

Answer 2

Since this is a MIMO system, you shouldn't compare things element-by-element. You are using the input-multiplicative form, so the uncertain system being created is of the form
Pnom*(I + W1*Delta*W2), where Delta is any stable (2-by-2, in this case) system, with ||Delta|| <= 1. So, to verify that the produced uncertain model "covers" your array of system, you should think of the equation
Parray = Pnom*(I + W1*Delta*W2)
and solve for Delta. Plot it (with SIGMA, say), and you will see that it is less than 1 in magnitude, for all frequencies. The Matlab code would be (multiply everything listed below, in order - my mulitplication symbol is not showing up in the posted answer...)
sigma(inv(W1)*inv(Pnom)*(Parrayg-Pnom)*inv(W2))
Now, using the syntax you specified, you are using weights W1 and W2 of the following form:
W1 = [W1_11 0;
0 W1_22]
and
W2 = [W2_11 0;
0 W2_22]
where you've specified 8th-order fits for all nonzero entries. Certainly for your example, this is overkill (although on a richer problem, it might be fine).
I would try much simpler, like
ucover(Parrag,Pnom,3,[],'InputMult')
That syntax will make an uncertain model of the form
Pnom*(I + w1*Delta)
where w1 is a scalar, 3rd order system. You could still see the covering by plotting SIGMA(Delta), namely
sigma((1/w1)*inv(Pnom)*(Parrayg-Pnom))
I hope that helps.

Answer 3

In order to create discrete or continuous time uncertain systems you can use uss associated with ureal.
Quick example
Define an uncertain propeller radius
% Propeller radius (m)
rp = ureal('rp',13.4e-2,'Range',[0.08 0.16]);
Define uncertain continuous time system
tenzo_unc = uss(A,Bw,Clocal,D,'statename',states,'inputname',inputs,'outputname',outputsLocal);
Simulate step response:
N = 5;
% Prende alcuni campioni del sistema incerto e calcola bound su incertezze
for i=1:1:N
sys{i} = usample(tenzo_unc);
step(sys{i})
hold on
cprintf('text','.');
end
Complete example
Quadcopter uncertain linearized model control with LQR. Code is available here
Step response
Closed Loop Step response
<script src="https://gist.github.com/GiovanniBalestrieri/f90a20780eb2496e730c8b74cf49dd0f.js"></script>
NB:
If you don't have the utility cprintf, include this script in your folder and use it.