LQG controller design with LQR

I am using lqg command and got a b c and d value for the controller. And from lqr I have k value. But how to incorporate them in my system together? I mean what is my final optimal systems A, B, C and D value that can be obtained from lqg controller and LQR gain?

Answer 1

[Pavl M.]

clc
clear all
close all

n_states = 5;
n_controls = 5;
n_outputs = 5;

A = [-0.4628, 0.3217, 0.5789, -0.3823, -0.0507;
     0.7593, 1.0396, -0.2008, -0.2438, -0.0678;
     0.2106, 0.6490, 0.3973, 0.8735, 0.2070;
     0.3128, 0.3380, 0.1524, 0.0914, -0.1088;
     0.0353, 0.0419, -0.0355, 0.0682, 0.7576];

B = [0.0026;
     0.0088;
    -0.0036;
     0.0490;
    -0.1443];

C = [1.1099, 1.3983, 0.8999, -0.0136, 0.0652];

D = 0;

dt = 0.02;
aug_level = 0;
gain = 4;
sys = ss(A, B, C, D, dt,'InputName','u','OutputName',{'y'});

% define your control, cross and state cost matrices:
R = 1;
N = 1;
Q = [1 0 0 0 0 0;0 2 0 0 0 0;0 0 5 4 0 0;0 0 0 5 0 0; 0 1 0 0 5 1;0 0 0 0 0 1];
Q1 = (Q+Q')/2

Q1 = 6×6

1.0000 0 0 0 0 0 0 2.0000 0 0 0.5000 0 0 0 5.0000 2.0000 0 0 0 0 2.0000 5.0000 0 0 0 0.5000 0 0 5.0000 0.5000 0 0 0 0 0.5000 1.0000

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A = rand(n_states+1, n_states+1);
A_symmetric = tril(A) + triu(A', 1);
Q2 = A_symmetric

Q2 = 6×6

0.6629 0.6854 0.6830 0.7862 0.5655 0.1663 0.6854 0.7813 0.6157 0.7265 0.0305 0.1561 0.6830 0.6157 0.6874 0.8049 0.1267 0.4960 0.7862 0.7265 0.8049 0.8787 0.5216 0.8728 0.5655 0.0305 0.1267 0.5216 0.7036 0.1557 0.1663 0.1561 0.4960 0.8728 0.1557 0.8840

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QXU = blkdiag(0.1*eye(n_states),sqrt(Q))

QXU = 11×11

0.1000 0 0 0 0 0 0 0 0 0 0 0 0.1000 0 0 0 0 0 0 0 0 0 0 0 0.1000 0 0 0 0 0 0 0 0 0 0 0 0.1000 0 0 0 0 0 0 0 0 0 0 0 0.1000 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 1.4142 0 0 0 0 0 0 0 0 0 0 0 2.2361 2.0000 0 0 0 0 0 0 0 0 0 0 2.2361 0 0 0 0 0 0 0 0 1.0000 0 0 2.2361 1.0000

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QWV = blkdiag(Q,0.01)

QWV = 7×7

1.0000 0 0 0 0 0 0 0 2.0000 0 0 0 0 0 0 0 5.0000 4.0000 0 0 0 0 0 0 5.0000 0 0 0 0 1.0000 0 0 5.0000 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0.0100

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QI = 1 ; %eye(ny)

KLQG1 = lqg(sys,Q1,Q,QI,'1dof')

Warning: QN is not symmetric and was replaced by (QN+QN')/2.

KLQG1 =
 
  A = 
              x1_e       x2_e       x3_e       x4_e       x5_e        xi1
   x1_e    -0.6969    0.02017      0.394    -0.3729   -0.06068   0.002294
   x2_e     0.2617     0.3901    -0.5879    -0.2157   -0.08427   0.007763
   x3_e    -0.3262   -0.01802   -0.04461     0.8711     0.1702  -0.003176
   x4_e    -0.0487     -0.243   -0.04954     0.2184   -0.05898    0.04323
   x5_e     0.2911      0.734   -0.09655    -0.2959     0.5634    -0.1273
   xi1           0          0          0          0          0          1
 
  B = 
              e1
   x1_e  -0.2057
   x2_e  -0.4308
   x3_e  -0.4908
   x4_e  -0.2278
   x5_e  -0.0579
   xi1      0.02
 
  C = 
         x1_e     x2_e     x3_e     x4_e     x5_e      xi1
   u   -2.218   -5.357  0.06196    2.529     1.32   0.8822
 
  D = 
      e1
   u   0
 
Input groups:        
    Name     Channels
    Error       1    
                     
Output groups:          
      Name      Channels
    Controls       1    
                        
Sample time: 0.02 seconds
Discrete-time state-space model.

KLQG2 = lqg(sys,Q1,Q,QI,'2dof')

Warning: QN is not symmetric and was replaced by (QN+QN')/2.

KLQG2 =
 
  A = 
              x1_e       x2_e       x3_e       x4_e       x5_e        xi1
   x1_e    -0.6969    0.02017      0.394    -0.3729   -0.06068   0.002294
   x2_e     0.2617     0.3901    -0.5879    -0.2157   -0.08427   0.007763
   x3_e    -0.3262   -0.01802   -0.04461     0.8711     0.1702  -0.003176
   x4_e    -0.0487     -0.243   -0.04954     0.2184   -0.05898    0.04323
   x5_e     0.2911      0.734   -0.09655    -0.2959     0.5634    -0.1273
   xi1           0          0          0          0          0          1
 
  B = 
             r1       y
   x1_e       0  0.2057
   x2_e       0  0.4308
   x3_e       0  0.4908
   x4_e       0  0.2278
   x5_e       0  0.0579
   xi1     0.02   -0.02
 
  C = 
         x1_e     x2_e     x3_e     x4_e     x5_e      xi1
   u   -2.218   -5.357  0.06196    2.529     1.32   0.8822
 
  D = 
      r1   y
   u   0   0
 
Input groups:              
       Name        Channels
     Setpoint         1    
    Measurement       2    
                           
Output groups:          
      Name      Channels
    Controls       1    
                        
Sample time: 0.02 seconds
Discrete-time state-space model.

%csys = ss(A-B*K,B,C,D)

figure
step(KLQG1)
title('1DOF Servo Controller designed')

figure
step(KLQG2)
title('2DOF Servo Controller designed')

%Constructing closed-loop system after designing:
%Ther are many similar questions this answer will fit(answer) to:
%https://www.mathworks.com/matlabcentral/answers/484846-how-to-use-the-generated-lqg-controller
C=KLQG1;
P = sys;
set(C,'InputName','e')
set(C,'OutputName','u')
set(P,'OutputName','y')
set(P,'InputName','u')
Sum_Error = sumblk('e = r - y');
clsys = tf(connect(C,P,Sum_Error,'r','y'))

clsys =
 
  From input "r" to output "y":
         0.003938 z^9 + 0.00479 z^8 - 0.01414 z^7 - 0.04427 z^6 + 0.1179 z^5 - 0.08028 z^4 - 0.003101 z^3 + 0.01298 z^2 + 0.007207 z - 0.004974
  ----------------------------------------------------------------------------------------------------------------------------------------------------
  z^11 - 3.254 z^10 + 3.293 z^9 - 0.3241 z^8 - 1.017 z^7 - 0.02715 z^6 + 0.4166 z^5 - 0.0324 z^4 - 0.05791 z^3 - 0.002832 z^2 + 0.005646 z - 4.509e-05
 
Sample time: 0.02 seconds
Discrete-time transfer function.

clsys_m = minreal(clsys)

clsys_m =
 
  From input "r" to output "y":
         0.003938 z^9 + 0.00479 z^8 - 0.01414 z^7 - 0.04427 z^6 + 0.1179 z^5 - 0.08028 z^4 - 0.003101 z^3 + 0.01298 z^2 + 0.007207 z - 0.004974
  ----------------------------------------------------------------------------------------------------------------------------------------------------
  z^11 - 3.254 z^10 + 3.293 z^9 - 0.3241 z^8 - 1.017 z^7 - 0.02715 z^6 + 0.4166 z^5 - 0.0324 z^4 - 0.05791 z^3 - 0.002832 z^2 + 0.005646 z - 4.509e-05
 
Sample time: 0.02 seconds
Discrete-time transfer function.

figure
step(clsys_m)
title('Constructed closed loop system response')

%Constructed from needing help code by:
    %https://join.skype.com/invite/oXnJhbgys7oW
    %https://independent.academia.edu/PMazniker
    %[phone]
    %https://diag.net/u/u6r3ondjie0w0l8138bafm095b
    %https://github.com/goodengineer
    %https://orcid.org/0000-0001-8184-8166
    %https://willwork781147312.wordpress.com/portfolio/cp/
    %https://www.youtube.com/channel/UCC__7jMOAHak0MVkUFtmO-w
    %https://nanohub.org/members/130066
    %https://pangian.com/user/hiretoserve/
    %https://substack.com/profile/191772642-paul-m