Coprime factorizations and stabilizability of infinite-dimensional linear systems Kalle M. Mikkola Helsinki University of Technology Finland Kalle.Mikkola@hut.fi http://www.math.hut.fi/˜kmikkola/ 13th December 2005 CDC-ECC’05 MTNS06, 13th of December 2005
Main Theorem The following are equivalent for a holomorphic function P : (i) P has a dynamic stabilizing controller . (ii) P has a right coprime factorization . [Smith89] [M05d] (iii) P has a stabilizable and detectable realization . [Staffans98] [CurOpm05] [M05c] We work in discrete time, but essentially the same results hold in continuous time too. Part of the results are new even in the scalar-valued case. As corollaries, one obtains analogous results for exponential (power) stabilization. MTNS06, 13th of December 2005 1
Notation U , X , Y : complex Hilbert spaces of arbitrary dimensions. � � | z | < 1 } . D : the unit disc { z ∈ C B ( U , Y ) : bounded linear maps U → Y . H ∞ ( U , Y ) : the set of bounded holomorphic functions D → B ( U , Y ) . I : the identity operator, e.g., I = I U ∈ B ( U , U ) , or the corresponding constant function, e.g., I = I U ∈ H ∞ ( U , U ) . proper function = holomorphic (operator-valued) function defined near the origin; strictly proper = P is proper and P ( 0 ) = 0 ; stable = H ∞ (a restriction of a H ∞ function is identified with the H ∞ function). Motivation: P ∈ H ∞ ( U , Y ) = ⇒ P is bounded (stable) multiplication operator H 2 ( U ) → H 2 ( Y ) . MTNS06, 13th of December 2005 2
Dynamic (output-feedback) stabilization y in + + � ✲ ✛ P ❡ u = u in + Qy y ✻ u ❄ y = y in + Pu u in + + ✲ ❡ ✛ Q Figure 1: Controller Q for the transfer function P y ] is stable ( H ∞ ). stabilizing controller = [ u in y in ] �→ [ u A proper B ( Y , U ) -valued function Q is called a (dynamic output feedback) proper stabilizing controller for a proper B ( U , Y ) -valued function P if the “input-to-error” y ] in Figure 1 is stable ( E ∈ H ∞ ).The map E is obviously given by map E : [ u in y in ] �→ [ u � � − 1 � � ( I − QP ) − 1 Q ( I − PQ ) − 1 − Q I E : = = (1) . P ( I − QP ) − 1 ( I − PQ ) − 1 − P I (Observe that then P is also a proper stabilizing controller for Q .) MTNS06, 13th of December 2005 3
Right coprime The following are equivalent for a proper holomorphic function P : � I � − 1 ∈ H ∞ ). − Q (i) P has a proper stabilizing controller Q (i.e., − P I (ii) P has a right coprime factorization . (iii) P has a stabilizable and detectable realization . Two functions M , N ∈ H ∞ are called (B´ ezout) right coprime if [ M N ] is left-invertible in Y ∈ H ∞ satisfying the B´ H ∞ , i.e., if there exist ˜ X , ˜ ezout identity XM − ˜ ˜ YN ≡ I (on D ) . (2) We call the factorization P = NM − 1 a right coprime factorization of P if N ∈ H ∞ ( U , Y ) and M ∈ H ∞ ( U ) are right coprime, M ( 0 ) is invertible and P = NM − 1 . X − 1 exists). X − 1 ˜ Then Q = ˜ Y is a stabilizing controller for P (if ˜ MTNS06, 13th of December 2005 4
All stabilizing controllers Let P be B ( U , Y ) -valued and have a right coprime factorization P = NM − 1 . Then N ] ∈ H ∞ ( U , U × Y ) can be extended to an invertible element of H ∞ ( U × Y ) , say [ M Y [ M N X ] . (This is called a doubly coprime factorization of P .) [Tolokonnikov81] [Treil04] [M05d] All stabilizing controllers for P are given by the Youla(–Bongiorno) parameterization Q = ( Y + MV )( X + NV ) − 1 (3) where V ∈ H ∞ ( Y , U ) is arbitrary (the controller is proper iff ( X + NV ) − 1 is proper). [CuWeWe01] [M05d] If P is strictly proper ( P ( 0 ) = 0 ), then all these controllers are proper. MTNS06, 13th of December 2005 5
Matrix-valued case Let P be a proper C n × m -valued function. Then also the following are equivalent to the existence of a proper stabilizing controller: (i*) P has a stable ( Q ∈ H ∞ ( C n , C m ) ) stabilizing controller. [Treil92] [Quadrat04] (ii*) P = NM − 1 , where N , M ∈ H ∞ , N ∗ N + M ∗ M ≥ ε I on D , ε > 0 and det M �≡ 0 . [Carleson62] [Fuhrman68] (The corona condition in (ii’) is not sufficient for coprimeness in the operator-valued case [Treil89]. It is not known whether (i’) is necessary in general.) MTNS06, 13th of December 2005 6
Controllers with internal loop Also the following is equivalent to the existence of a proper stabilizing controller of P : (i”) P has a stabilizing controller with internal loop. [CuWeWe01] [M05d] � R 11 R 12 � We call R a stabilizing controller with internal loop for P if R = is a R 21 R 22 proper B ( Y × Ξ , U × Ξ ) -valued function for some Hilbert space Ξ and the combined � u in � � u � in Figure 2 becomes stable ( H ∞ ). map y in �→ y ξ ξ in y in + + P ✲ ✛ y = y in + Pu ❢ � � � � � � y ✻ u u in y u ❄ = + R u in ξ ξ in ξ + + ✲ ❢ ✛ R 11 R 12 ξ in + + R 21 R 22 ✲ ❢ ✛ ξ ✻ Figure 2: Controller R with internal loop for P If I − R 22 ( 0 ) is invertible, then R corresponds to the proper stabilizing controller Q = R 11 + R 12 ( I − R 22 ) − 1 R 21 . MTNS06, 13th of December 2005 7
Main Theorem (ver. 3) The following are equivalent for a proper function P : � I � − 1 ∈ H ∞ ). − Q (i) P has a proper stabilizing controller Q (i.e., − P I (i’) P has a strictly proper stabilizing controller. (i”) P has a stabilizing controller with internal loop. (ii) P has a right coprime factorization P = NM − 1 . M − 1 ˜ (ii’) P has a left coprime factorization P = ˜ N . N X ] − 1 ∈ H ∞ ( U × Y ) . (ii”) P has a doubly coprime factorization P = NM − 1 , [ M Y N X ] , [ M Y (iii) P has a stabilizable and detectable realization . MTNS06, 13th of December 2005 8
Discrete-time system ( A B D ) ∈ B ( X × U , X × Y ) C Given input u ∈ ℓ 2 ( N ; U ) and initial state x 0 ∈ X , we associate the state trajectory x : N → X and output y : N → Y through � x k + 1 = Ax k + Bu k , k ∈ N . (4) y k = Cx k + Du k , � A � The transfer function P ( z ) : = D + C ( z − 1 − A ) − 1 B of B is proper. C D � A � B We call a realization of P . C D u ( z ) : = ∑ n z n u n . The Z -transform � u of u : N → U is defined by � For x 0 = 0 , we have � y = P � u . MTNS06, 13th of December 2005 9
State feedback u k = Fx k State feedback means that we feed the state back to the input through some state-feedback operator F ∈ B ( X , U ) : u k : = Fx k +( u in ) k ( k ∈ N ) , (5) where u in denotes an exogenous input (or disturbation), as in Figure 3. x τ − 1 ❄ ✛ x · + 1 ✻ x · + 1 = Ax + Bu A B ✛ y y = Cx + Du C D ✛ Fx F 0 t ✻ + u = Fx + u in u in + ❄ ✲ ✲ ❢ t Figure 3: State-feedback connection � A + BF � � x k + 1 � � � B x k ⇒ x k + 1 = ( A + BF ) x k + B ( u in ) k ⇒ �→ . : y k C D ( u in ) k u k F I MTNS06, 13th of December 2005 10
Closed-loop system � � A + BF B x k + 1 � � � � x k : . C + DF �→ (6) D y k ( u in ) k F I u k The transfer function of the closed-loop system (6) is obviously given by � � � � � � N ( z ) C + DF D ( z − 1 − A − BF ) − 1 B . = + (7) M ( z ) I F �� � y y is given by P = NM − 1 . Because [ N M ] maps � u in �→ , a factorization of P : � u �→ � � u Finite Cost Condition (FCC): For each x 0 ∈ X , some u ∈ ℓ 2 makes y ∈ ℓ 2 . If(f) the FCC holds, then there exists F ∈ B ( X , U ) that minimizes ∑ ∞ k = 0 ( � y k � 2 Y + � u k � 2 U ) (LQR cost) for every x 0 . The resulting factorization P = NM − 1 is weakly coprime [M05a]. � A ∗ � , then P = NM − 1 is right coprime [CO05]. C ∗ If the FCC holds for B ∗ D ∗ MTNS06, 13th of December 2005 11
State-feedback stabilization of ( A B D ) C � A � output stable = y ∈ ℓ 2 whenever x 0 ∈ X and u = 0 ; B C D i.e., � CA · x 0 � 2 ≤ K � x 0 � X ( x 0 ∈ X ) . � A � stable = y ∈ ℓ 2 and x is bounded whenever x 0 ∈ X and u ∈ ℓ 2 ( N ; U ) ; i.e., B C D ( n ≥ 0 , x 0 ∈ X , u ∈ ℓ 2 ( N ; U )) . � x n � X + � y � 2 ≤ K ( � x 0 � X + � u � 2 ) (8) � A + BF � � A � B B [output-]stabilizable = [output-]stable for some F . C D C D F I � A � � A ∗ � C ∗ B [input-]detectable = [output-]stabilizable. B ∗ D ∗ C D (iii) P has a stabilizable and detectable realization . (iii’) P has an output-stabilizable and input-detectable realization . Theorem Output-stabilizability ⇔ Finite Cost Condition. [M05a] � A � � A � � A ∗ � C ∗ B B (iii”) P has a realization such that and satisfy the Finite Cost B ∗ D ∗ C D C D Condition. MTNS06, 13th of December 2005 12
Recommend
More recommend
