stabilization
F=stabil(A,B,alfa)
K=stabil(Sys,alfa,beta)
:A square real matrix ( nx x nx) : :B real matrix ( nx x nu) : :alfa, beta real or complex vector (in conjugate pairs) or real
number.
: :F real matrix ( nx x nu) : :Sys linear system ( syslin list) ( m inputs, p outputs). : :K linear system ( p inputs, m outputs) :
F=stabil(A,B,alfa) returns a gain matrix F such that A+B*F is stable if pair (A,B) is stabilizable. Assignable poles are set to alfa(1),alfa(2),.... If (A,B) is not stabilizable a warning is given and assignable poles are set to alfa(1),alfa(2),.... If alfa is a number all eigenvalues are set to this alfa (default value is alfa=-1).
K=stabil(Sys,alfa,beta) returns K, a compensator for Sys such that (A,B)-controllable eigenvalues are set to alfa and (C,A)-observable eigenvalues are set to beta.
All assignable closed loop poles (which are given by the eigenvalues of Aclosed=h_cl(Sys,K) are set to alfa(i)‘s and beta(j)‘s.
// Gain:
Sys=`ssrand`_(0,2,5,`list`_('st',2,3,3));
A=Sys('A');B=Sys('B');F=stabil(A,B);
`spec`_(A) //2 controllable modes 2 unstable uncontrollable modes
//and one stable uncontrollable mode
`spec`_(A+B*F) //the two controllable modes are set to -1.
// Compensator:
Sys=`ssrand`_(3,2,5,`list`_('st',2,3,3)); //3 outputs, 2 inputs, 5 states
//2 controllables modes, 3 controllable or stabilizable modes.
K=stabil(Sys,-2,-3); //Compensator for Sys.
`spec`_(Sys('A'))
`spec`_(`h_cl`_(Sys,K)) //K Stabilizes what can be stabilized.