sectf

State-space sector bilinear transformation

Syntax

[G,T] = sectf(F,SECF,SECG)

Description

[G,T] = sectf(F,SECF,SECG) computes a linear fractional transform T such that the system lft(F,K) is in sector SECF if and only if the system lft(G,K) is in sector SECG where

G=lft(T,F,NU,NY)

where NU and NY are the dimensions of u_T2 and y_T2, respectively—see the following figure.

Sector transform G=lft(T,F,NU,NY).

sectf are used to transform general conic-sector control system performance specifications into equivalent H_∞-norm performance specifications.

Input Arguments
F	LTI state-space plant
SECG, SECF:	Conic Sector:
	[-1,1] or [-1;1]	$${\Vert y\Vert }^2\le {\Vert u\Vert }^2$$
	[0,Inf] or [0;Inf]	$$0\le \mathrm{Re}\left[y\ast u\right]$$
	[A,B] or [A;B]	$$0\ge \mathrm{Re}\left[\left(y-Au\right)\ast \left(y-Bu\right)\right]$$
	[a,b] or [a;b]	$$0\ge \mathrm{Re}\left[\left(y-diag(a)u\right)\ast \left(y-diag(b)u\right)\right]$$
	S	$$0\ge \mathrm{Re}\left[\left(S_{11}u+S_{12}y\right)\ast \left(S_{21}u+S_{22}y\right)\right]$$
	S	$$0\ge \mathrm{Re}\left[\left(S_{11}u+S_{12}y\right)\ast \left(S_{21}u+S_{22}y\right)\right]$$

where A,B are scalars in [–_∞, _∞] or square matrices; a,b are vectors; S=[S11 S12;S21,S22] is a square matrix whose blocks S11,S12,S21,S22 are either scalars or square matrices; S is a two-port system S=mksys(a,b1,b2,...,'tss') with transfer function

$$S(s)=\left[\begin{array}{cc}{S}_{11}(s)& S_{12}(s)\\ S_{21}(s)& S_{22}(s)\end{array}\right]$$

Output Arguments	Description
G	Transformed plant G(s)=lftf(T,F)
T	LFT sector transform, maps conic sector SECF into conic sector SECG

Output Variables
G	The transformed plant G(s) = lftf(T,F):
T	The linear fractional transformation T(s) = T

Examples

The statement G(jω) inside sector[–1, 1] is equivalent to the H_∞ inequality

$$\underset{\omega }{\sup }\overline{\sigma }\left(G(j\omega )\right)={\Vert G\Vert }_{\infty }\le 1$$

Given a two-port open-loop plant P(s) := P, the command P1 = sectf(P,[0,Inf],[-1,1])computes a transformed P₁(s):= P1 such that if lft(G,K) is inside sector[–1, 1] if and only if lft(F,K) is inside sector[0, _∞]. In other words, norm(lft(G,K), inf)<1 if and only if lft(F,K) is strictly positive real. See Example of Sector Transform.

Sector Transform Block Diagram

Here is a simple example of the sector transform.

$$P(s)=\frac{1}{s+1}\in \text{sector}\left[-1,1\right]\to P_1(s)=\frac{s+2}{2}\in \text{sector}\left[0,\infty \right].$$

You can compute this by simply executing the following commands:

P = ss(tf(1,[1 1])); 
P1 = sectf(P,[-1,1],[0,Inf]);

The Nyquist plots for this transformation are depicted in Example of Sector Transform. The condition P₁(s) inside [0, _∞] implies that P₁(s) is stable and P₁(jω) is positive real, i.e.,

$$P_1^{\ast }(j\omega )+P_1(j\omega )\ge 0\forall \omega$$

Example of Sector Transform

Limitations

A well-posed conic sector must have det(B–A)≠ 0 or

$$\det \left(\left[\begin{array}{cc}{s}_{11}& s_{12}\\ s_{21}& s_{22}\end{array}\right]\right)\ne 0.$$

Also, you must have $$\dim (u_{F1})=\dim (y_{F1})$$ since sectors are only defined for square systems.

Algorithms

sectf uses the generalization of the sector concept of [3] described by [1]. First the sector input data Sf= SECF and Sg=SECG is converted to two-port state-space form; non-dynamical sectors are handled with empty a, b1, b2, c1, c2 matrices. Next the equation

$$S_{g(}s)\left[\begin{array}{c}{u}_{g_1}\\ y_{g_1}\end{array}\right]=S_f(s)\left[\begin{array}{c}{u}_{f_1}\\ y_{f_1}\end{array}\right]$$

is solved for the two-port transfer function T(s) from $$u_g{}_{{}_1}{y}_{f_1}$$ to $$u_f{}_{{}_1}{y}_{g_1}$$. Finally, the function lftf is used to compute G(s) as G = lftf(T,F).

References

[1] Safonov, M.G., Stability and Robustness of Multivariable Feedback Systems. Cambridge, MA: MIT Press, 1980.

[2] Safonov, M.G., E.A. Jonckheere, M. Verma and D.J.N. Limebeer, “Synthesis of Positive Real Multivariable Feedback Systems,” Int. J. Control, vol. 45, no. 3, pp. 817-842, 1987.

[3] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems ≥— Part I: Conditions Using Concepts of Loop Gain, Conicity, and Positivity,” IEEE Trans. on Automat. Contr., AC-11, pp. 228-238, 1966.

Version History

Introduced before R2006a