Robust Control Toolbox™

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 or .

VERS=-1, reverse transform 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 j modes [2], [3], [4].

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

Table 5-1: Bilinear transform types.

Method	Type of bilinear transform
'BwdRec'	backward rectangular: AUG = T, the sampling period.
'FwdRec'	forward rectangular: AUG = T, the sampling period.
'S_Tust'	shifted Tustin: AUG = [T h], is the "shift" coefficient.
'S_ftjw'	shifted j-axis, bilinear pole-shifting, continuous-time to continuous-time: AUG = [p2 p1].
'G_Bilin'	METHOD = 'G_Bilin', general bilinear, continuous-time to continuos-time: AUG = .

Example

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); 
  
  .
  
  
  
  

Figure 5-2: . Comparison of 4 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 Figure 5-3, 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

Figure 5-3: 'S_ftjw' final closed-loop poles are inside the left [p1,p2] circle.

Algorithm

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         Convert from continous- to discrete-time

d2c         Convert from continous- to discrete-time

sectf       Sector transformation

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