Evaluation of Modeling Error

It is necessary to evaluate modeling error of the nominal plant for Gyu(s) to guarantee closed-loop stability.
To do that, additive uncertainty model is introduced, and the weight which represents the gain characteristic of the additive uncertainty, is specified in this stage.

The following is the rule of thumb which describes relations between modeling error estimation and the resultant system performance:

too much estimation of modeling error causes poor system performance.
reduction of the estimation causes better performance.
however, too short estimation of modeling error might break stability of the system.

Type the following command on Matlab:

>> weight

Then, gain characteristic of the modeling error for Gyu(s) and weight function are plotted.

Example:

Gain characteristic of modeling error (blue points) and weight function (red curve)

The weight function is defined in the file weight.m.
Tune the parameters of the weight function and/or change the weight function by editing and executing weight.m repetitively.
Some examples of weight functions are listed bellow.
The blank ??? should be filled by proper value.
Use the text editor xemacs to edit the file weight.m.
The lines which starts by % are comment.

Constant weight function:
```
W = ???;
	
```

Frequency dependent weight function (1st order)

%             s + w1     w2
% W(s) = w0 * ------ * ------
%               w1     s + w2

w0 = ???;
w1 = 2*pi*???;
w2 = 2*pi*???;
W = nd2sys(w0*w2*[1, w1], w1*[1, w2]);

where the dimension of w1 and w2 is [rad/sec].

Frequency dependent weight function (2nd order, notch filter)

%             s^2 + 2*a*w3*s + w3^2
% W(s) = w0 * ---------------------
%             s^2 + 2*b*w3*s + w3^2	

w0 = ???;
w3 = 2*pi*???;
a = ???;
b = ???;
W = nd2sys([1, 2*a*w3, w3*w3], [1, 2*b*w3, w3*w3]);

Frequency dependent weight function (series combination of the above weight functions)

%             s + w1     w2     s^2 + 2*a*w3*s + w3^2
% W(s) = w0 * ------ * ------ * ---------------------
%               w1     s + w2   s^2 + 2*b*w3*s + w3^2

% 1st order weight function
w0 = ???;
w1 = 2*pi*???;
w2 = 2*pi*???;
W1 = nd2sys(w0*w2*[1, w1], w1*[1, w2]);

% 2nd order weight function (notch filter)
w3 = 2*pi*???;
a = ???;
b = ???;
W2 = nd2sys(w3*[1, 2*a*w3, w3*w3], [1, 2*b*w3, w3*w3]);

% series combination
W = mmult(W1, W2);

where `nd2sys' is a Matlab function which makes a transfer function from polynomial coefficients of numerator (1st argument) and that of denominator (2nd argument), and `mmult' is a Matlab function which multiply the transfer functions given as arguments and returns the resultant multiplied transfer function.

Yasuhide Kobayashi

Last modified: Thu Jun 3 09:51:00 JST 2004