sectf :: Functions (Robust Control Toolbox™)

sectf

Purpose

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_T2and y_T2, respectively--see Figure 6-16.

Figure 6-16: 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]

[0,Inf] or [0;Inf]

[A,B] or [A;B]

[a,b] or [a;b]

S

S

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

Output Arguments
Description

G
Transformed plant G(s)=lftf(T,F)

T
LFT sector transform, maps conic sector SECF into conic sector SECG

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

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

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

Examples

The statement G(j) inside sector[-1, 1] is equivalent to the H inequality

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 Figure 6-18

Figure 6-17: Sector Transform Block Diagram

Here is a simple example of the sector transform.

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 Figure 6-18, Example of Sector Transform.. The condition P₁(s) inside [0, ] implies that P₁(s) is stable and P₁(j) is positive real, i.e.,

Figure 6-18: Example of Sector Transform.

Algorithm

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

is solved for the two-port transfer function T(s) from to. Finally, the function lftf is used to compute G(s) as G=lftf(T,F).

Limitations

A well-posed conic sector must have or .
Also, you must have

since sectors are only defined for square systems.

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.