bilin - Multivariable bilinear transform of frequency (s or z)

Syntax

GT = bilin(G,VERS,METHOD,AUG)

Description

bilin computes the effect on a system of the frequency-variable substitution,

The variable VERS denotes the transformation direction:

VERS= 1, forward transform (s→z) or .

VERS=-1, reverse transform (z→s) or .

This transformation maps lines and circles to circles and lines in the complex plane. People often use this transformation to do sampled-data control system design [1] or, in general, to do shifting of jω modes [2], [3], [4].

Bilin computes several state-space bilinear transformations such as backward rectangular, etc., based on the METHOD you select

Bilinear Transform Types

Examples

Example 1. Tustin continuous s-plane to discrete z-plane transforms

Consider the following continuous-time plant (sampled at 20 Hz):

Following is an example of four common "continuous to discrete" bilin transformations for the sampled plant:

A= [-1 1; 0 -2]; B=[1 0; 1 1];  
C= [1 0; 0 1];   D=[0 0; 0 0]; 
sys = ss(A,B,C,D);                     % ANALOG
Ts=0.05;  % sampling time
[syst] = c2d(sys,Ts,'tustin');         % Tustin 
[sysp] = c2d(sys,Ts,'prewarp',40);        % Pre-warped Tustin 
[sysb] = bilin(sys,1,'BwdRec',Ts);     % Backward Rectangular
[sysf] = bilin(sys,1,'FwdRec',Ts);     % Forward Rectangular
w = logspace(-2,3,50); % frequencies to plot
sigma(sys,syst,sysp,sysb,sysf,w); 

Comparison of Four Bilinear Transforms from Example 1

Example 2. Bilinear continuous to continuous pole-shifting 'S_ftjw'

Design an H mixed-sensitivity controller for the ACC Benchmark plant

such that all closed-loop poles lie inside a circle in the left half of the s-plane whose diameter lies on between points [p1,p2]=[–12,–2]:

p1=-12; p2=-2; s=zpk('s');
G=ss(1/(s^2*(s^2+2)));          % original unshifted plant
Gt=bilin(G,1,'Sft_jw',[p1 p2]); % bilinear pole shifted plant Gt
Kt=mixsyn(Gt,1,[],1);           % bilinear pole shifted controller
K =bilin(Kt,-1,'Sft_jw',[p1 p2]); % final controller K

As shown in the following figure, closed-loop poles are placed in the left circle [p1 p2]. The shifted plant, which has its non-stable poles shifted to the inside the right circle, is

'S_ftjw' final closed-loop poles are inside the left [p1,p2] circle

Algorithms

bilin employs the state-space formulae in [3]:

References

[1] Franklin, G.F., and J.D. Powell, Digital Control of Dynamics System, Addison-Wesley, 1980.

[2] Safonov, M.G., R.Y. Chiang, and H. Flashner, "H_∞ Control Synthesis for a Large Space Structure," AIAA J. Guidance, Control and Dynamics, 14, 3, p. 513-520, May/June 1991.

[3] Safonov, M.G., "Imaginary-Axis Zeros in Multivariable H_∞ Optimal Control", in R.F. Curtain (editor), Modelling, Robustness and Sensitivity Reduction in Control Systems, p. 71-81, Springer-Varlet, Berlin, 1987.

[4] Chiang, R.Y., and M.G. Safonov, "H_∞ Synthesis using a Bilinear Pole Shifting Transform," AIAA, J. Guidance, Control and Dynamics, vol. 15, no. 5, p. 1111-1117, September-October 1992.

See Also

c2d | d2c | sectf

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