State Feedback Observer Design Example

pdf version NJUPT <Modern Control Theory> P131

Question5.7

Consider the following transfer function:

            z=[1 2];
          
            p=[-1 2 -3];
          
            G1=zpk(z,p,1)
          
                G1 = (s-1) (s-2) ----------------- (s+1) (s+3) (s-2)
                Continuous-time zero/pole/gain model.
              
            Gt=zpk(1,[-2 -3],1)
          
                Gt = (s-1) ----------- (s+2) (s+3) Continuous-time
                zero/pole/gain model.

Is it possible to transform G1 to Gt using state feedback control?

solution

step1. Controlable?

            [A,B,C,D]=zp2ss(z,p,1);
          
            Qc=ctrb(A,B);
          
            Qo=obsv(A,C);
          
            if rank(Qc)>=size(A,2)
          
             disp('Controlable, so can design state feedback control')
          
            else 
          
             disp('Uncontrolable')
          
            end
          
                Controlable, so can design state feedback control
              
            if rank(Qo)>=size(A,2)
          
             disp('Observable,so can design output feedback control');
          
            else
          
             disp('Unobservable');
          
            end
          
                Unobservable

step2. pole placement

            P=[2 -2 -3];
          
            K=acker(A,B,P)
          
                    K = 1×3
                  
                    1.0000 -3.0000 -5.1962

step3. state feedback observer system

            A2=A-B*K;
          
            % G2ss=ss(A2,B,C,0);
          
            [z2,p2]=ss2zp(A2,B,C,0);
          
            G2=zpk(z2,p2,1)
          
                G2 = (s-2) (s-1) ----------------- (s-2) (s+2) (s+3)
                Continuous-time zero/pole/gain model.