Structure of Expansion-Contraction Matrices in the Inclusion Principle for Dynamic Systems

Design of guaranteed cost overlapping controllers for a class of uncertain state-delay systems

“…These transformations involve a set of socalled complementary matrices. The influence of the choice of these matrices on properties like stability, controllability or observability has been illustrated in previous works [1], [2], [3], [4].…”

Sequential design of multi-overlapping controllers for longitudinal multi-overlapping systems

“…Definition 1 (Inclusion Principle) A systemS is an expansion of the system S if there exist transformations (U, V, R, S) as in (2) such that, for any initial state x 0 ∈R n and any input u(t)∈R m , ifx 0 =V x 0 andũ(t)=Ru(t) then x(t; x 0 , u)=Ux(t; V x 0 , Ru) for all t≥0.…”

“…Definition 2 (Contractibility) Suppose thatS is an expansion of the system S. Then, a control lawũ(t)=−Kx(t) forS is contractible to the control law u(t)=−Kx(t) for S if there exist transformations as in (2) such that, for any initial state x 0 ∈R n and any input u(t)∈R m , ifx 0 =V x 0 andũ(t)=Ru(t) then Kx(t; x 0 , u)=QKx(t; V x 0 , Ru) for all t≥0.…”

“…Now, we proceed as described in Section 2 and we use the expansion matrices V (2) , R (2) , defined by the numbers in (40) to compute the decoupled expanded pairs contains no overlapping blocks, and we choose the quadratic index…”

“…We select the quadratic index J (1) 1 with weighting matrices [Q (1) 1 ] * =I n (1) and [R (1) 1 ] * =10 −15 I n (1) ×m (2) , n (1) =n…”