How to incorporate equations in the connect function for control design?

I would like to synthesize a h-infinity controller and need to connect a few blocks. How do I go about integrating the equation

y=x^2+a^2 

in the following manner with the connect function?

It is no issue in Simulink but I feel like I am forgetting some basics with MATLAB.

[Sam Chak]

@Jiten Parmar, more info are needed. I'd advise you to show the basic model in Simulink.

Is 'a' the output signal from the so-called controller?

Is the Plant a linear or nonlinear system?

The y signal is a nonlinear non-negative function , which has function values equal to or greater than zero.

[Jiten Parmar]

@Sam Chak Firstly, I apologize for using the word "integrating". I did not mean it in the math sense.

However, the question that I have posted is a simplified version of the actual issue because I just need to extrapolate the method of implementing it with the "connect" function. Anyways, here is a picture of the model I am working with. It is a model of a wind turbine. Lambda is a simple equation of the form (ab/c). Same with Ta. However, Cp has a for-loop inside it. I need to implement this into matlab so I can use the hinfsyn function to make a controller. The output from the controller would go into the Tg input on the mux.

[Sam Chak]

Hi @Jiten Parmar

Correct me if I'm wrong. If the model looks like the following:

where is the disturbance torque as a function of the wind speed, and the rotor speed, then the plant is LTI, and you can use the connect() function. Consult @Paul on this matter if you wish to connect to the controller.

The reason I used the Double Integrator in my Example (Answer below) because the simple rotational motion model resembles it.

Coming from the radical thinking perspective, if the Wind speed is measurable, then it is possible to build the feedforward control model to estimate , so that you can reliably counter it in .

If the estimation is pretty accurate, such that the residual error is small , then you will have more flexibilities (less conservative) in designing in

.

[Jiten Parmar]

That is not the model that I am utilizing. The wind turbine model is as follows:

However. I believe I have found a viable approach based on your double integrator example. I need to linearize the model above with respect to Ta and then use the augmentation approach.

[Sam Chak]

@Jiten Parmar, Good to hear that you have found the approach.

Answer 1

[Sam Chak]

Hi @Jiten Parmar

Thanks for clarifying that. For more info, you can find out how use 'connect' here. But I have seen people using 'augw' to augment the LTI plant with the weighting functions for general control design.

Here is an example for the control of a Double Integrator:

s   = zpk('s');

% Augmenting plant with the weighting functions
Gp  = 1/(s^2);      % Double Integrator plant
W1  = 0.1*(s + 1e2)/(1e2*s + 1); 
W2  = 0.1; 
W3  = [];
P   = augw(Gp, W1, W2, W3);

% H-infinity control design
[K, CL, gamma] = hinfsyn(P, 1, 1)

K =
 
  A = 
            x1       x2       x3
   x1    -0.01        0        0
   x2    4.007   -1.699  0.06469
   x3    328.4   -139.2   -78.46
 
  B = 
               u1
   x1      0.3125
   x2    -0.01633
   x3  -1.254e-06
 
  C = 
           x1      x2      x3
   y1   328.4  -139.2  -78.46
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

CL =
 
  A = 
              x1         x2         x3         x4         x5         x6
   x1      -0.01    -0.3125          0          0          0          0
   x2          0          0          1          0          0          0
   x3          0          0          0      328.4     -139.2     -78.46
   x4          0    -0.3125          0      -0.01          0          0
   x5          0    0.01633          0      4.007     -1.699    0.06469
   x6          0  1.254e-06          0      328.4     -139.2     -78.46
 
  B = 
               u1
   x1      0.3125
   x2           0
   x3           0
   x4      0.3125
   x5    -0.01633
   x6  -1.254e-06
 
  C = 
           x1      x2      x3      x4      x5      x6
   y1    0.32  -0.001       0       0       0       0
   y2       0       0       0   32.84  -13.92  -7.846
 
  D = 
          u1
   y1  0.001
   y2      0
 
Input groups:       
    Name    Channels
     U1        1    
                    
Output groups:      
    Name    Channels
     Y1       1,2   
                    
Continuous-time state-space model.

gamma = 0.1339

Gc  = tf(K)         % transfer function form of H-inf ctrl

Gc =
 
  104.9 s^2 + 0.01633 s + 1.254e-06
  ---------------------------------
  s^3 + 80.17 s^2 + 143.1 s + 1.423
 
Continuous-time transfer function.

margin(Gc*Gp)

% Closed-loop control system
Gcl = tf(minreal(feedback(Gc*Gp, 1)))

Gcl =
 
                 104.9 s^2 + 0.01633 s + 1.254e-06
  ---------------------------------------------------------------
  s^5 + 80.17 s^4 + 143.1 s^3 + 106.3 s^2 + 0.01633 s + 1.254e-06
 
Continuous-time transfer function.

step(Gcl)

S   = stepinfo(Gcl)

S = struct with fields:
         RiseTime: 2.1066
    TransientTime: 3.2529
     SettlingTime: 3.2529
      SettlingMin: 0.9061
      SettlingMax: 1.0074
        Overshoot: 0.7396
       Undershoot: 0
             Peak: 1.0074
         PeakTime: 4.2777

ssy = dcgain(Gcl)   % steady-state output

ssy = 1.0000

Answer 2

[Paul]

connect only works with lti system objects. Nonlinear elemens have to be linearized around an equilibrium point and the linearized models can be connected, etc. for LTI system analysis.