Problem in matrix partition for uncertain system: how to find matrices A, B1, B2, C1, C2, D11, D12, D21, D22?

Hi everybody!

I have obtained my uncertain system using this code:

a1 = ureal('a1',p_p2, 'Range' , [p_p2-0.0070 p_p2+0.0070]);
a2 = ureal('a2',p_p3, 'Range' , [p_p3-0.0128 p_p3+0.0128]);
b0 = ureal('b0',p_z1, 'Range' , [p_z1-0.0019 p_z1+0.0019]);
b1 = ureal('b1',p_z2, 'Range' , [p_z2-0.0145 p_z2+0.0145]);

A   = [0 1; -a2 -a1];
B   = [0; 1];
C   = [b1 b0];

sys = ss(A,B,C,0);
sys.InputGroup.ActualIn   = 1;
sys.OutputGroup.ActualOut = 1;
[Msys,Delta] = lftdata(sys);

(Of course p_p2, p_p3, p_z1, p_z2 have a value, but I am omitting the equations for the sake of simplicity)

So Msys has the form:

But I am searching a way to partition A, B, C, D to get A, B1, B2, C1, C2, D11, D12, D21, D22, as in this form:

What I know is that A should be the same and D22 should be 0, but how to implement this partitioning?

Thanks to anyone can help me.

Francesco

[Paul]

It's unclear what the goal is. Are you starting from A,B,C,D and trying to figure out what the partitions are based on know dimensions of w, u, z, and y? Or are you starting from the partitions and trying to form an ss object that represents the partitioned system? It's hard to tell from the example becuase sys is SISO, and so there really isn't any partitioning to speak of.

[Francesco]

Thank you Paul for your reply, I’m sorry, I will try to make it clearer. Msys is already the system with uncertainty that I have built using the functions ureal and then ss, but it is formed by matrices A, B, C, D. These matrices are formed by the matrices I am searching, so for example B1 and B2 are obtained by splitting matrix B into parts, and the same is happening for C and D. What I’m asking is the rule on how to split A, B, C and D that compose Msys to get A, B1, B2, C1, C2, D11, D12, D21, D22 (or a function eventually, for the job), because these are the matrices I need for my project for H infinite control.

Thank you in advance!

Answer 1

[Paul]

Suppose you have a model

xdot = A*x + B1*w + B2*u

z = C1*x + D11*w + D12*u

y = C2*x + D21*w + D22*u

This system can be rewritten as:

xdot = A*x + [B1 B2]*[w;u] = A*x + B*[w;u]

[z;y]= [C1;C2]*x + [D11 D12;D21 D22]*[w;u] = C*x + D*[w;u];

Assume you have n states, and w is a vector of nw inputs, and u is vector of nu inputs, and z is vector of nz outputs, y is vector of ny outputs. Then we must have

B1 = B(:,1:nw);
B2 = B(:,nw+1:nw+nu); % or B(:,nw+1:end)
C1 = C(1:nz,:);
C2 = C(nz+1:nz+ny,:); % or C(nz+1:end,:);
D11 = D(1:nz,1:nw);
D12 = D(1:nz,nw+1:nw+nu);  % or D(1:nz,nw+1:end)
D21 = D(nz+1:nz+ny,1:nw);  % or D(nz+1:end,1:nw)
D22 = D(nz+1:nz+ny,nw+1:nw+nu); % or D(nz+1:end,nw+1:end)

The dimensions of w, u, z, and y must be known to determine the partitions of A,B,C,D of compatible dimensions.

[Francesco]

Perfect, thank you very much Paul!

[Mohammed Yakoob]

Hello dear @Paul and sorry to bother you! I'm also working with your method that explained it above but has given me a wrong results firstly, my system has this form x_dot= A*x+B*u+d*w and Y= C*x and D=0 the dimensions of A(2*2), B(2*1),d(2*1) and C(1*2) in matlab I have made new B=[B d] and the Incorrect results in dimensions of C2(0*2),D11(1),D12(1),D21(0*1) and D22(0*1) also, so please what I will to solve this problem because the dimensions is very important in H_inf that solving by LMI?

[Paul]

Is there also an equation for the output variable z?

[Mohammed Yakoob]

Thank you so much for your answer!! The first thing I have this state space model as below x_dot= A*x+B*u+d*w and Y= C*x and D=0 the dimensions of A(2*2), B(2*1),d(2*1) and C(1*2) After that that system have discrete form as below X(m+1)= A*x(m) +B*w and y(m) =C*x(m) +D*w Then they have used another form (LTI) that solved H-infinty with LMI as below x(m) = A*x(m) +B1*w +B2*u Z(m) = C1*x(m) +D11*w+D12*u Y(m) =C2*x(m) + D21*w+D22*u So, I have done this partition as mentioned below but as I told you there were some incorrect matrices!!

[Paul]

Are you sarting with pthe partitions and trying to form the full system matrices or starting with the full system matrices and tyring to find the partitions? Can you post some code that shows what you're trying to do and the error that results?

[Mohammed Yakoob]

clc

close all

clear

%% Plant Paramiters

Ts= 0.0005;

f= 60;

Wo= 2*pi*f;

Vdc= 300;

C_st= ureal('C_st',15*10^-6,'Percentage',[-40,40]);

L_t= ureal('L_t',2*10^-3,'Percentage',[-30,30] );

R_l= 40;

%% Implement a Single Phase Microgrid in a contious time form

% x_dot= AX+BU+dW ; Y= CX+DU.

% X=[il Vg]' ; U=[Vsw] ; d= [ig]; Y=[Vg].

Aop=[0 1/L_t; 1/C_st 0];

Bop= [1/L_t; 0];

d=[0; -1/C_st];

Cop=[0 1];

Dop= [0 0];

% B= [Bop d];

sys1= ss(Aop,[Bop d] ,Cop,Dop);

sys2= ss(Aop,[Bop d] ,Cop,Dop,Ts);

Ad= sys2.A;

Bd= sys2.B;

Cd=sys2.C;

Dd= sys2.D;

% % Determine sizes

ns = size(Aop,1); % number of states

nu = size(Bop,2); % number of actuator inputs

ny = size(Cop,1); % number of measured outputs

nq = size(Cop,1); % number of regulated output

nw = nu; % number of disturbances

% % 9 Matrix Representation

B1 = Bd(:,1:nw);

B2 = Bd(:,nw+1:nw+nu);

C1 = Cd(1:nq,:);

C2 = Cd(nq+1:nq+ny,:);

D11 = Dd(1:nq,1:nw);

D12 = Dd(1:nq,nw+1:nw+nu);

D21 = 0; %Dd(nq+1:end,1:nw);

D22 = Dd(nq+1:nq+ny,nw+1:nw+nu);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% lsim(sys1,step);

%% LMI by using YALMIP

% eta=1.5;

% Settings

opt = sdpsettings('verbose',0,'solver','sedumi');

% Define Variables

Y= sdpvar(2,2);

P= sdpvar(1,2);

H= sdpvar(2,2);

y= sdpvar(2,2);

F = sdpvar(2,1);

Q = sdpvar(2,2);

R = sdpvar(1,1);

S = sdpvar(2,2);

J = sdpvar(2,2);

Lo = sdpvar(1,2);

L= sdpvar(2,2);

eta = sdpvar(1,1);

m11= L;

m12= J;

m13= Ad*Y+B2*P;

m14= Ad+B2*R*C2;

m15= B1+B2*R*D21;

m16= zeros(2,1);

m22= H;

m23= Q;

m24= y*Ad+F*C2;

m25= y*B2+F*D21;

m26= zeros(2,1);

m33= Y+Y'-L;

m34= eye(2)+S'-J;

m35= zeros(2,1);

m36= Y'*C1'+Lo'*D12';

m44= y+y'-H;

m45=zeros(2,1);

m46= C1'+C2'*R'*D12';

m55= eye(1);

m56= D11'+D21'*R'*D12';

m66= eta*eye(1);

mat2= [m11 m12 m13 m14 m15 m16;...

m12' m22 m23 m24 m25 m26;...

m13' m23' m33 m34 m35 m36;...

m14' m24' m34' m44 m45 m46;...

m15' m25' m35' m45' m55 m56;...

m16' m26' m36' m46' m56' m66];

F = [ mat2>=0];

% Optimization Problem

% optimize(F,eta,opt);

opt = sdpsettings('solver','lmilab','verbose',2); %(better SDP solver recommended)

% opt.lmilab.maxiter = 1000;

% opt.lmilab.feasradius = 1e20;

solvesdp(F,[],opt);

eta= double(eta);

L = double(L);

F = double(F);

R = double(R);

Lo = double(Lo);

[Mohammed Yakoob]

Hello dear again

the above is my codes but not the final one , Im dealing with uncertian system and this also add some defctes to my code , I want to detrmine he H_inf to design a controller

Answer 2

[Paul]

Edit: This answer thread was intended to be a response to this comment.

%% Plant Paramiters 
Ts= 0.0005;
f= 60;
Wo= 2*pi*f;
Vdc= 300;
C_st= ureal('C_st',15*10^-6,'Percentage',[-40,40]);
L_t= ureal('L_t',2*10^-3,'Percentage',[-30,30] );
R_l= 40;
%% Implement a Single Phase Microgrid in a contious time form

Based on the code as written, the output equation for Y is really Y = CX + DU + 0w

% x_dot= AX+BU+dW ; Y= CX+DU.
% X=[il Vg]' ; U=[Vsw] ; d= [ig]; Y=[Vg].
Aop=[0 1/L_t; 1/C_st 0];
Bop= [1/L_t; 0];
d=[0; -1/C_st];
Cop=[0 1];
Dop= [0 0];
% B= [Bop d];

For H-inf control, the disturbance input, w, typically is "before" the control input, u.

%sys1= ss(Aop,[Bop d] ,Cop,Dop);
sys1= ss(Aop,[d Bop],Cop,Dop);

At this point sys1 has two inputs (w,u) and only one output, y.

size(sys1)

Uncertain state-space model with 1 outputs, 2 inputs, 2 states, and 2 blocks.

I don't understand the next line. If sys2 is supposed to the discrete approximation to sys1, then it should be using c2d(). Becasue the actual definition of sys2 isn't really relevant to the question of partitioning, I'm going to just set sys2 to sys1 so that there aren't any problems with subsequent references to sys2.

% sys2= ss(Aop,[Bop d] ,Cop,Dop,Ts);
sys2 = sys1;
Ad= sys2.A;
Bd= sys2.B;
Cd=sys2.C;
Dd= sys2.D;
% Determine sizes 
ns = size(Aop,1);   % number of states
nu = size(Bop,2);  % number of actuator inputs
ny = size(Cop,1);  % number of measured outputs

So far so good. Now the following line indicates that there should be as many regulated outputs, z, as there are measured ouputs, y (why use nq, and not nz?). But nowhere is there an equation that defines the output equation for z. in terms of x, w, and u. And z is not included as an output in sys1.

nq = size(Cop,1);  % number of regulated output 
nw = nu;     % number of disturbances

Because I don't know what z is supposed to be, I'm just going to assume that z = 2*y for purposes of illustration. Under this assumption, the output equation for z is z = 2*Cop*x + 2*Dop*[w;u]. Recreate sys1 with z as the first output and y as the second output, and the repeat all the subsequent code.

sys1= ss(Aop,[d Bop],[2*Cop; Cop],[2*Dop; Dop]);
size(sys1)

Uncertain state-space model with 2 outputs, 2 inputs, 2 states, and 2 blocks.

sys2 = sys1;
Ad   = sys2.A;
Bd   = sys2.B;
Cd   = sys2.C;
Dd   = sys2.D;
% Determine sizes 
ns = size(Aop,1);  % number of states
nu = size(Bop,2);  % number of actuator inputs
ny = size(Cop,1);  % number of measured outputs
nq = ny;          % number of regulated output 
nw = nu;          % number of disturbances

Now get the partitions

% % 9 Matrix Representation
A  = Ad;
B1 = Bd(:,1:nw);
B2 = Bd(:,nw+1:nw+nu);
C1 = Cd(1:nq,:);
C2 = Cd(nq+1:nq+ny,:);
D11 = Dd(1:nq,1:nw);
D12 = Dd(1:nq,nw+1:nw+nu);
D21 = Dd(nq+1:end,1:nw);
D22 = Dd(nq+1:nq+ny,nw+1:nw+nu);
disp('complete')

complete

[Mohammed Yakoob]

Hello My Dear and thank you so much for your help !!

1. I have used this way (sys2= ss(Aop,[Bop d] ,Cop,Dop,Ts);) because I deal with uncertainity and (c2d) not accepted with it.
2. (sys1= ss(Aop,[d Bop],[2*Cop; Cop],[2*Dop; Dop]);) in this command I have the H_infinity from W to Z equal to 1.5 so, the gain of the controlled outpt will be (1.5*y)!! is that okay?
3. after that , I want to design the H_infinity controller with LMI but I deal with the uncertainity so I get this error (Error using * (line 80)

Invalid operand. Variables of type "sdpvar" cannot be combined with other models.)

please if you can, help me ??

thank you so much a gain and good luck!!

[Paul]

If you have the partitioned model set up correctly, I'm afraid I can't be of any more help. Good luck.

[Mohammed Yakoob]

Thank you so much dear and good luck with my respect