How to create a state space model with constant term and do feedback.

I have the discrete time system -

x(k+1) = [A]*x(k) + [B]*u(k) + c

y(k) = [C]*x(k)

I have read online that I can merge the constant term into B by doing [B 1] * [u1(k) ; u2(k)] with u2(k) = c ( I have no control over this input). ie the system becomes

x(k+1) = [A]*x(k) + [B 1]*[u1(k) ; c]

y(k) = [C]*x(k).

However, when I do state feed back, how can I ensure that u2(k) = c?

[Sam Chak]

Hi @Marcus

When the constant c is zero, how would you design and implement the state feedback? Can you demo a bit with your MATLAB code?

[Marcus]

See my recent response to Aquatris. Just represent closed loop dyanamics with ss(A-B*K,B*G,C,D,Ts) due to the control law u_k = G*r_k - K*x_k.

[Sam Chak]

You appear to have used the continuous-time linear system to place the discrete-time closed-loop pole inside the unit circle (stable region of discrete-time linear system), which is considered incorrect.

Answer 1

[Marcus]

The solution that I found to work for my system is by representing

as the follwing uncontrollable system

with initial conditions .

State feedback and steady state tracking involve placing the poles of A and B of the origianl system and not the poles of the uncontrollable system.

ie the close loop system is -

Acl = [A-B*K eye(size(A));
    0 eye(size(A))]
Bcl = [B;0]
C = [C,eye(size(A))]
Dcl = 0

sys_cl = ss(Acl,Bcl,Ccl,Dcl)

Answer 2

[Aquatris]

You can define which inputs and outputs are connected to your controller. Checkout the feedback function page and look at the 'Specify Input and Output Connections in a Feedback Loop' section.

First try it yourself to figure it out, cause it is a nice exercise to understand documentation and how to search for things. If you get stuck while doing it, provide what you have done and we can guide you.

[Marcus]

I am interested in doing state feedback. its seems that this documentation is for output feedback?

[Aquatris]

It is a general representation. State feedback essentially means your observation matrix C is identity matrix with a size of nxn where n is the number of states.

So in addition to your actual output, you can create another output for your feedback controller that would have the state information, something like:

Then you need to connect the y_states as an input to your feedback controller and connect u1 to the output of your feedback controller

[Marcus]

I seem to still be having trouble. This is what I've done so far.

A = [0.999979515574799];
B = [0.00001070633417008353561593759356585 1];
C = [1; 1];
D = [0];
c = 0.0046089956808409359920175596414538;
Ts = 1;
sys1 = ss(A,B,C,D,Ts)

sys1 has 2 inputs / 2 outputs / 1 state

I want to use the following control law where for steady state tracking and K is my gains matrix.

K = place(A,B,0.5);
G = inv(C*inv(1-(A-B*K))*B);

I am not sure how to represent this controller in state space form. I want to connect y_states to u1 with feedback(sys1, sys2, [1], [2], -1]), but I am not sure how to formulate sys2. How should I do this?

Typically, if c=0, I would use ss(A-B*K,B*G,C,D) to represend my closed loop dynamics, but this already incoroprates feedback.

Also, nowhere have I specified that u2 = c, is this done by setting my reference input to c (ie r2_k = c)?

Edit: Seems that G does not exist since a left inverse doenst exist. Anyway how would this be done with u = r - Kx?

Thanks

Answer 3

[Pavl M.]

There are 2 simple precise ways to do it:

Just make your

Baug = [B 1]

and regard your c = u2(k) as constant control input to your plant(environment) with u1(k) as the control from the controller(agent) you are designing.

then

sys1 = ss(A,Baug,C,D,Ts)
K = place(A,Baug,0.5);
G = inv(C*inv(1-(A-Baug*K))*Baug);
cldyn = ss(A-Baug*K,Baug*G,C,D,Ts)

or treat the c as a noise(disturbance), then don't augment much your B, leave it as B ( I worked on it ) and use disturbance rejection PID or LQR next controller:

https://fr.mathworks.com/help/control/ref/lti.lqr.html

with defining state cost, control cost and cross control-state cost matrices as per your problem objections and econometrics,

use lqgtrack or lqg or Kalman:

%State cost matrix:
Q1 = gain*eye(A_size_row+aug_level+1);
Q2 = gain*eye(A_size_row+aug_level);
%Control cost matrix:
%symmetric positive semi-definite state cost matrix, and R (k × k) is a symmetric positive definite control cost matrix.
R = eye(n_states,n_states); %eye(size(A));
%Cross Control-state cost:
N = 1; %eye(A_size_row,length(C));
S = eye(n_states,n_outputs);
E = eye(n_states,n_states);
[~,p] = chol(R)
CM = [Q2;N;N';R]
##Function File: [G, X, L] = dlqr (SYS, Q, R)
## -- Function File: [G, X, L] = dlqr (SYS, Q, R, S)
## -- Function File: [G, X, L] = dlqr (A, B, Q, R)
## -- Function File: [G, X, L] = dlqr (A, B, Q, R, S)
## -- Function File: [G, X, L] = dlqr (A, B, Q, R, [], E)
## -- Function File: [G, X, L] = dlqr (A, B, Q, R, S, E)
##     Linear-quadratic regulator for discrete-time systems.
##
##     *Inputs*
##     SYS
##          Continuous or discrete-time LTI model (p-by-m, n states).
##     A
##          State transition matrix of discrete-time system (n-by-n).
##     B
##          Input matrix of discrete-time system (n-by-m).
##     Q
##          State weighting matrix (n-by-n).
##     R
##          Input weighting matrix (m-by-m).
##     S
##          Optional cross term matrix (n-by-m).  If S is not specified, a
##          zero matrix is assumed.
##     E
##          Optional descriptor matrix (n-by-n).  If E is not specified,
##          an identity matrix is assumed.
##
##     *Outputs*
##     G
##          State feedback matrix (m-by-n).
##     X
##          Unique stabilizing solution of the discrete-time Riccati
##          equation (n-by-n).
##     L
##          Closed-loop poles (n-by-1).
##
##     *Equations*
##          x[k+1] = A x[k] + B u[k],   x[0] = x0
##
##                  inf
##          J(x0) = SUM (x' Q x  +  u' R u  +  2 x' S u)
##                  k=0
##
##          L = eig (A - B*G)
[K1,XG1,L1] = lqi(sys,Q1,R)
[G, X, L] = dlqr (A, B, Q2, R,S)
[Fr,Pr,Er]=lqrpid(sys,Q,R)
%dare provides similar to dlqr results
%QXU =
%QWV =
%QI =
%https://www.mathworks.com/help/control/getstart/design-an-lqg-servo-controller.html
%reg = lqg(sys,QXU,QWV,QI)
K_i = -min(X3)
K_x = -X3
%Hamiltonian:
H = A - B*G;
p = 0.5;
%poles placement control:
Ka=acker(A, B, p)
A2 = A - Ka*B

Hope this clears up full solution to your conundrum. For complete solution help me financially if you wanna I do it for you. I have had been developing 3/4 of the disturbance rejection, contact me more for exact codes if you are interested.