i am trying to find the lqr controller system Transfer function, but getting error as "Arrays have incompatible sizes for this operation."

% Define system matrices
A = [0 20.95; -709.22 -106.85];
B = [3771.21; 12765.96];
C = [1 0];
D = 0;

% Define the LQR weighting matrices
Q = [0.0057312 0; 0 1];  % Weighting matrix for states
R = 1;                    % Weighting matrix for control inputs

% Compute LQR gain matrix
K = lqr(A, B, Q, R);

% Augment the system
[n, ~] = size(A);
AA = [A, zeros(n, 1); -C, 0];
BB = [B; 0];

% Construct the closed-loop system
Ac = AA - BB * K;

Arrays have incompatible sizes for this operation.

Bc = BB;
Cc = C;
Dc = D;

% Create state-space model of the closed-loop system
sys_cl = ss(Ac, Bc, Cc, Dc);

% Convert state-space model to transfer function
sys_tf = tf(sys_cl);

% Display the transfer function
disp('Transfer Function of the Closed-Loop System with LQR Controller:');
disp(sys_tf);

Answer 1

[Sam Chak]

Hi @Senthilnathan A

I have marked the lines that require fixing. The step response characteristics have also been displayed. The selected Q and R weights of the LQR design resulted in a settling time of 0.0105 seconds and a percent overshoot of over 16%.

Are these the desired control performance requirements? If not, then pole placement, also known as eigenvalue assignment, would be a better design option for a second-order system.

% Define system matrices
A = [0 20.95; -709.22 -106.85];
B = [3771.21; 12765.96];
C = [1 0];
D = 0;

% Define the LQR weighting matrices
Q = [0.0057312 0; 0 1];         % Weighting matrix for states
R = 1;                          % Weighting matrix for control inputs

% Compute LQR gain matrix
K = lqr(A, B, Q, R);

% % Augment the system          % no reason to augment it unless LQR not what you want
% [n, ~] = size(A);
% AA = [A, zeros(n, 1); -C, 0];
% BB = [B; 0];

% Construct the closed-loop system
Ac = A - B*K;                   % <-- fix it here
Bc = B;                         % <-- fix it here
Cc = C;
Dc = D;

% Create state-space model of the closed-loop system
sys_cl  = ss(Ac, Bc, Cc, Dc);
op      = findop(sys_cl, y=1);  % <-- add it here

% Convert state-space model to transfer function
sys_tf = tf(op.u*sys_cl);       % <-- fix it here

% Display the transfer function
disp('Transfer Function of the Closed-Loop System with LQR Controller:');

Transfer Function of the Closed-Loop System with LQR Controller:

sys_tf

sys_tf =
 
     1.505e04 s + 2.675e06
  ---------------------------
  s^2 + 1.298e04 s + 2.675e06
 
Continuous-time transfer function.

%% plot step response
S   = stepinfo(sys_tf)          % <-- add it here

S = struct with fields:
         RiseTime: 1.0662e-04
    TransientTime: 0.0105
     SettlingTime: 0.0105
      SettlingMin: 0.9067
      SettlingMax: 1.1617
        Overshoot: 16.1749
       Undershoot: 0
             Peak: 1.1617
         PeakTime: 4.7615e-04

step(sys_tf), grid on           % <-- add it here

[Mr. Pavl M.]

Why did you do system matrices augmentation after calling to lqr function, while in theory they do augmentation for lqr and lqi before calling to lqr/lqi function, what is the sense of doing so?

Let me sharpen thinking, improve understanding to catch apprehend as in nature all the subleties.

How will such controller be possible to implement in reality in real hardware embedded C MCU, like A-B*K, the gain is just amplifier, do you need Kalman state estimator to add to it, how the blockscheme of the closed loop system will look like (without integrator and with just Kalman or other State Estimator)?

Answer 2

[Sam Chak]

Hi @Senthilnathan A

If fast-rising spike and the overshoot are undesired and you would like to eliminate it, I suggest adding a prefilter at the reference input path.

% Define system matrices
A = [0 20.95; -709.22 -106.85];
B = [3771.21; 12765.96];
C = [1 0];
D = 0;

% Define the LQR weighting matrices
Q = [0.0057312 0; 0 1];         % Weighting matrix for states
R = 1;                          % Weighting matrix for control inputs

% Compute LQR gain matrix
K = lqr(A, B, Q, R);

% % Augment the system          % no reason to augment it unless LQR not what you want
% [n, ~] = size(A);
% AA = [A, zeros(n, 1); -C, 0];
% BB = [B; 0];

% Construct the closed-loop system
Ac = A - B*K;                   % <-- fix it here
Bc = B;                         % <-- fix it here
Cc = C;
Dc = D;

% Create state-space model of the closed-loop system
sys_cl  = ss(Ac, Bc, Cc, Dc);

% Convert state-space model to transfer function
sys_tf  = tf(sys_cl);           % <-- fix it here
[n, d]  = tfdata(sys_tf, 'v');  % <-- add it here

%% Pre-filter
Gf      = tf(d(end), n)         % <-- add it here

Gf =
 
      2.675e06
  -----------------
  3771 s + 6.704e05
 
Continuous-time transfer function.

%% Display the transfer function of the Command Compensated System
disp('Transfer Function of the Command Compensated System:');

Transfer Function of the Command Compensated System:

Gcc = minreal(Gf*sys_tf)        % <-- fix it here

Gcc =
 
           2.675e06
  ---------------------------
  s^2 + 1.298e04 s + 2.675e06
 
Continuous-time transfer function.

%% plot step response
S   = stepinfo(Gcc)             % <-- add it here

S = struct with fields:
         RiseTime: 0.0105
    TransientTime: 0.0187
     SettlingTime: 0.0187
      SettlingMin: 0.9029
      SettlingMax: 0.9993
        Overshoot: 0
       Undershoot: 0
             Peak: 0.9993
         PeakTime: 0.0349

step(Gcc), grid on              % <-- add it here