state feedback control and observer

Can you explain how to design a state feedback control and an observer for a DC motor with the following specifications? If you could provide instructions for both MATLAB and Simulink, I would greatly appreciate it.

Rs = 0.47;
Ld = 1.23e-3;
B = 0.0002;
Kb = 0.42;
Kt = 0.73;
J = 6.5e-4;

% Second-order State Matrix
A1 = [-B/J Kt/J;
    -Kb/Lq -Rs/Lq];
B1 = [0;
    1/Lq];
C1 = [1 0];
D1 = 0;

Answer 1

[Oguz Kaan Hancioglu]

You can follow this tutorial,

https://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed&section=ControlStateSpace

Answer 2

[Sam Chak]

Hi @영탁 한

Since no requirement is provided, we follow the design criteria given in the example:

• Settling time less than 2 seconds
• Overshoot less than 5%
• Steady-stage error less than 1%

Like many collaborative robot arms and autonoumous vehicles, it is necessary to design the desired reference trajectory so that the control systems can follow. In this exercise, a reference model (transfer function) is designed to satisfy the design criteria.

From the reference model, one can obtain the desired closed-loop pole and the pole placement technique (solving algebraic equations) can be applied. The rest of the controller and observer design are procedural steps. So, the key is the reference model design!

% Parameters
Rs   = 0.47;
Lq   = 1.23e-3;
B    = 0.0002;
Kb   = 0.42;
Kt   = 0.73;
J    = 6.5e-4;

% State-space Matrices of the Uncompensated Plant
A    = [-B/J   Kt/J;
       -Kb/Lq -Rs/Lq];
B    = [0;
        1/Lq];
C    = [1 0];
D    = 0;

% Reference trajectory design
wr   = 3;                   % reference omega, ω
zr   = 0.7;                 % reference zeta,  ζ
nr   = wr^2;
dr   = [1 2*zr*wr wr^2];    % x" + 2·ζ·ω·x' + ω²·x
Gr   = tf(nr, dr)           % reference model

Gr =
 
         9
  ---------------
  s^2 + 4.2 s + 9
 
Continuous-time transfer function.

stepinfo(Gr)

ans = struct with fields:
         RiseTime: 0.7089
    TransientTime: 1.9930
     SettlingTime: 1.9930
      SettlingMin: 0.9001
      SettlingMax: 1.0460
        Overshoot: 4.5986
       Undershoot: 0
             Peak: 1.0460
         PeakTime: 1.4693

% Controller design
cp   = pole(Gr)             % controller poles

cp = 
  -2.1000 + 2.1424i
  -2.1000 - 2.1424i

K    = place(A, B, cp)      % controller gains

K = 1×2

-0.4200 -0.4652

% Closed-loop control system without observer
sys  = ss(A-B*K, B, C, D);
Nbar = 1/dcgain(sys);       % Normalizer (to rescale the reference input)
cls  = ss(A-B*K, B*Nbar, C, D)

cls =
 
  A = 
              x1         x2
   x1    -0.3077       1123
   x2  -0.006947     -3.892
 
  B = 
             u1
   x1         0
   x2  0.008014
 
  C = 
       x1  x2
   y1   1   0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

stepinfo(cls)

ans = struct with fields:
         RiseTime: 0.7089
    TransientTime: 1.9930
     SettlingTime: 1.9930
      SettlingMin: 0.9001
      SettlingMax: 1.0460
        Overshoot: 4.5986
       Undershoot: 0
             Peak: 1.0460
         PeakTime: 1.4693

% Observer design
or   = 10*real(cp(1));      % make the observer response 10 times faster
op   = [or+1 or-1]          % observer poles (make op ≪ cp but no repeated poles)

op = 1×2

-20.0000 -22.0000

L    = place(A', C', op)'   % observer gains

L = 2×1

-340.4215 -225.3519

% Combine state-feedback controller and observer
Aco  = [A-B*K B*K;
        zeros(size(A)) A-L*C];
Bco  = [B*Nbar;
       zeros(size(B))];
Cco  = [C zeros(size(C))];
Dco  = 0;

% Closed-loop observer-based control system
clco = ss(Aco, Bco, Cco, Dco)

clco =
 
  A = 
              x1         x2         x3         x4
   x1    -0.3077       1123          0          0
   x2  -0.006947     -3.892     -341.5     -378.2
   x3          0          0      340.1       1123
   x4          0          0     -116.1     -382.1
 
  B = 
             u1
   x1         0
   x2  0.008014
   x3         0
   x4         0
 
  C = 
       x1  x2  x3  x4
   y1   1   0   0   0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

step(clco, 6), grid on

stepinfo(clco)

ans = struct with fields:
         RiseTime: 0.7089
    TransientTime: 1.9930
     SettlingTime: 1.9930
      SettlingMin: 0.9001
      SettlingMax: 1.0460
        Overshoot: 4.5986
       Undershoot: 0
             Peak: 1.0460
         PeakTime: 1.4693

sse = fix(1 - dcgain(clco))      % steady-state error

sse = 0

As shown, all three design criteria are satisfied.