A Canonical State-Space Model for m-Dimensional Discrete Systems

Multivariable Control

“…Hidden Modes. This grd is equal to a grd of (E.cD,N) since (40) which implies, in view of N(DcD)-l PE.c-l , that the unobservable 97 eigenvalues ~ exactly. They correspond to closed loop eigenvalues which are uncontrollable and/or unobservable.…”

“…Consequently equality obtains in (39) if and only if the mXm minor formed from the first m columns of the left-hand matrix in (42) is nonsingular. Hence by (41) equality obtains in (39) if and only if (36) holds, which completes the proof.…”

“…Making use of (5), (8), (10), (16), (19), (22) and (46) we can compute matrices F, A, Q, T, Au' Su' E and then (39) in closed analytical form. Making use of (5), (8), (10), (16), (19), (22) and (46) we can compute matrices F, A, Q, T, Au' Su' E and then (39) in closed analytical form.…”

“…Making use of (5), (8), (10), (16), (19), (22) and (46) we can compute matrices F, A, Q, T, Au' Su' E and then (39) in closed analytical form. Finally, since Theorem 2 the solution of (1) is given by (23) with N explicitely computed by means of (40) and (32). Finally, since Theorem 2 the solution of (1) is given by (23) with N explicitely computed by means of (40) and (32).…”

“…compoyield (39) n-q range for the observer is that given by r where ---<r<n-q q iff{M,F} and{A,C} form observable pairs. In the illustrative example that follows n=4, q=2, thus an observer of dimension 4-2 as low as --2-=1 can be used to implement the control whereas if we first estimated x(k) we would require an observer of dimension 4-2=2.…”