st_ility

Arguments

Sl

syslin list (linear system)

ns

integer (dimension of stabilizable subspace)

nc

integer (dimension of controllable subspace nc <= ns)

U

basis such that its ns (resp. nc) first components span the stabilizable (resp. controllable) subspace

Slo

a linear system (syslin list)

tol

threshold for controllability detection (see contr)

Description

Slo=( U'*A*U, U'*B, C*U, D, U'*x0 ) (syslin list) displays the stabilizable form of Sl. Stabilizability means ns=nx (dim. of A matrix).

[*,*,*]            [*]
U'*A*U = [0,*,*]     U'*B = [0]
[0,0,*]            [0]

where (A11,B1) (dim(A11)= nc) is controllable and A22 (dim(A22)=ns-nc) is stable. "Stable" means real part of eigenvalues negative for a continuous linear system, and magnitude of eigenvalues lower than one for a discrete-time system (as defined by syslin).

Examples

A=diag([0.9,-2,3]);B=[0;0;1];Sl=syslin('c',A,B,[]);
[ns,nc,U]=st_ility(Sl);
U'*A*U
U'*B
[ns,nc,U]=st_ility(syslin('d',A,B,[]));
U'*A*U
U'*B

See also

• dt_ility — detectability test
• contr — controllability, controllable subspace, staircase
• stabil — stabilization
• ssrand — random system generator