place

For this example, consider a simple second-order system with the following state-space matrices:

Input the matrices and create the state-space system.

A = [-1,-2;1,0];
B = [2;0];
C = [0,1];
D = 0;
sys = ss(A,B,C,D);

Compute the open-loop poles and check the step response of the open-loop system.

Pol  = pole(sys)

Pol = 2×1 complex

  -0.5000 + 1.3229i
  -0.5000 - 1.3229i

figure(1)
step(sys)
hold on;

Notice that the resultant system is underdamped. Hence, choose real poles in the left half of the complex-plane to remove oscillations.

p = [-1,-2];

Find the gain matrix K using pole placement and check the closed-loop poles of syscl.

K = place(A,B,p);
Acl = A-B*K;
syscl = ss(Acl,B,C,D);
Pcl = pole(syscl)

Pcl = 2×1

   -2.0000
   -1.0000

Now, compare the step response of the closed-loop system.

figure(1)
step(syscl)

Hence, the closed-loop system obtained using pole placement is stable with good steady-state response.

Note that choosing poles that are further away from the imaginary axis achieves faster response time but lowers the steady-state gain of the system. For instance, consider using the poles [-2,-3] for the above system.

p = [-2, -3];
K2 = place(A,B,p);
syscl2 = ss(A-B*K2,B,C,D);
figure(1);
step(syscl2);

stepinfo(syscl)

ans = struct with fields:
         RiseTime: 2.5901
    TransientTime: 4.6002
     SettlingTime: 4.6002
      SettlingMin: 0.9023
      SettlingMax: 0.9992
        Overshoot: 0
       Undershoot: 0
             Peak: 0.9992
         PeakTime: 7.7827

stepinfo(syscl2)

ans = struct with fields:
         RiseTime: 1.4130
    TransientTime: 2.4766
     SettlingTime: 2.4766
      SettlingMin: 0.3003
      SettlingMax: 0.3331
        Overshoot: 0
       Undershoot: 0
             Peak: 0.3331
         PeakTime: 4.1216

For this example, consider the pole locations [-2e-13,-3e-4,-3e-3]. Compute the precision of the actual poles.

A = [4,2,1;0,-1,2;0,1e-8,1];
B = [1,2;3,1;1e-6,0];
p = [-2e-13,-3e-4,3e-3];
[~,prec] = place(A,B,p)

prec = 
2

A precision value of 2 is obtained indicating that the actual pole locations are precise up to 2 decimal places.

For this example, consider the following transfer function with complex-conjugate poles at $-2\pm 2\mathit{i}$:

Input the transfer function model. Then, convert it to state-space form since place uses the A and B matrices as input arguments.

s = tf('s');
systf = 8/(s^2+4*s+2);
sys = ss(systf);

Next, compute the gain matrix K using the complex-conjugate poles.

p = [-2+2i,-2-2i];
K = place(sys.A,sys.B,p)

K = 1×2

         0    1.5000

The values of the gain matrix are real since the poles are self-conjugate. The values of K would be complex if p did not contain self-conjugate poles.

Now, verify the step response of the closed-loop system.

syscl = ss(sys.A-sys.B*K,sys.B,sys.C,sys.D);
step(syscl)

For this example, consider the following SISO state-space model:

Create the SISO state-space model defined by the following state-space matrices:

A = [-1,-0.75;1,0];
B = [1;0];
C = [1,1];
D = 0;
Plant = ss(A,B,C,D);

Now, provide a pulse to the plant and simulate it using lsim. Plot the output.

N = 250;
t = linspace(0,25,N);
u = [ones(N/2,1); zeros(N/2,1)];
x0 = [1;2];
[y,t,x] = lsim(Plant,u,t,x0);

figure
plot(t,y);
title('Output');

For this example, assume that all the state variables cannot be measured and only the output is measured. Hence, design an observer with this measurement. Use place to compute the estimator gain by transposing the A matrix and substituting C' for matrix B. For this instance, select the desired pole locations at -2 and -3.

L = place(A',C',[-2,-3])';

Use the estimator gain to substitute the state matrices using the principle of duality/separation and create the estimated state-space model.

At = A-L*C;
Bt = [B,L];
Ct = [C;eye(2)];
sysObserver = ss(At,Bt,Ct,0);

Simulate the time response of the system using the same pulse input.

[observerOutput,t] = lsim(sysObserver,[u,y],t);
yHat = observerOutput(:,1);
xHat = observerOutput(:,[2 3]);

Compare the response of the actual system and the estimated system.

figure;
plot(t,x);
hold on;
plot(t,xHat,'--');
legend('x_1','x_2','xHat_1','xHat_2')
title('Comparison - Actual vs. Estimated');