compute the controllability and observability grammians
Grammians( sys, opts )
System(ss); state-space System object
(optional) equation(s) of the form option = value; specify options for the Grammians command
checkstability = truefalse
True means check whether the system is stable; if it is not stable, an error occurs. False means skip the check. The default is true.
output = C or O or list of these names
Specifies the returned values. If equal to C, then the controllability grammian is returned. If equal to O, then the observability grammian is returned. If a list of these names, then the output is a list/sequence (see returnlist) with each C replaced with the controllability grammian and each O replaced with the observability grammian. The default is [C,O].
returnlist = truefalse
True means return a list; false means return an expression sequence. The default is false.
The Grammians command computes the grammians of sys, a state-space system.
Depending on the value of the output option, either the controllability grammian, the observability grammian, or both, is computed.
For a grammian to exist, the system must be stable. For a continuous-time system, all eigenvalues, λ, of A must lie in the open left-half plane: ℜ⁡λ<0. For a discrete-time system, all eigenvalues, λ, of A must lie in the open unit-circle: λ<1. If sys is not stable, an error occurs, unless the option checkstability is false.
A grammian is the positive-definite matrix X that solves the appropriate Lyapunov equation.
For a continuous system, the Lyapunov equation is A.X+X.AT=−B.BT.
For a discrete system, the Lyapunov equation is A.X.AT−X=−B.BT.
For a continuous system, the Lyapunov equation is AT.X+X.A=−CT.C.
For a discrete system, the Lyapunov equation is AT.X.A−X=−CT.C.
Assign a state-space system.
aSys ≔ StateSpace⁡−5,3|3,−4,2,3,1,0|0,1,0,0:
Compute its controllability grammian.
Cg ≔ Grammians⁡aSys,output=C
Cg ≔ 1.681818181818182.136363636363642.136363636363642.72727272727273
Verify that Cg meets the Lyapunov equation (b^+ is the transpose of b, see LinearAlgebra[Transpose]).
Download Help Document
What kind of issue would you like to report? (Optional)
Thank you for submitting feedback on this help document. Your feedback will be used
to improve Maple's help in the future.