Said Hadd and Qing-Chang Zhong Dept. of Electrical Eng

A PPROXIMATE C ONTROLLABILITY OF N EUTRAL S YSTEMS WITH S TATE AND I NPUT D ELAYS IN BANACH S PACES Said Hadd and Qing-Chang Zhong Dept. of Electrical Eng. & Electronics The University of Liverpool Liverpool, L69 3GJ United Kingdom [email protected]

Outline Quick overview of recent activities Motivation of the research Problem statement Methodology Main results Special cases

S. H ADD

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Q.-C. Z HONG : A PPROXIMATE

CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN

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Recent research activities Control engineering Power electronics: grid connection etc Renewable energy: wind power Automotive electronics: hybrid electric vehicles Control theory Robust control: J-spectral factorisation, algebraic Riccati equations etc Time-delay systems: a series of problems Infinite-dimensional systems: feedback stabilizability, controllability S. H ADD

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Key publications

One research monograph IEEE Trans. Automatic Control: 7 papers Automatica: 4 papers other IEEE Transactions: 3 papers

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Current funding EPSRC: EP/C005953/1, £126k, to expire in 09/2008 EPSRC: EP/E055877/1, £88k EPSRC: one DTA studentship, £50k EPSRC and Add2 Ltd: Dorothy Hodgkin Postgraduate Award, £90k ESPRC and Nheolis, France: Dorothy Hodgkin Postgraduate Award, £90k

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Motivation Controllability is one of the most important properties of a control system. Basically, it tells us if a system is controllable, in other words, whether a point in the state space is reachable. The most widely known formula is the Kalman rank condition Rank( B AB · · · An−1 B ) = n.

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Controllability: Analysis and design Normally, controllability is used to analyse whether a system is controllable. It can also be used for design. Many flexible modes to be suppressed The number of actuators to be minimised Many locations for placing actuators The question is where to put them and how many? S. H ADD

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Infinite-dimensional case For finite-dimensional systems, there are different ways to define controllability. However, they all end up with the same concept. However, this is no longer true for infinite-dimensional case. Different concepts need to be used. Approximate or exact controllability in finite or infinite time etc. (Nine different concepts are defined in Stafani’s book)

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Approximate controllability (AC) For a given arbitrary ε > 0, it is possible to steer the state from the origin to the neighbourhood, with a radius ε, of all points in the state space.

This is weaker than the exact controllability, where ε = 0.

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Some notation X, U, Y are Banach spaces, p > 1 is a real number and (A, D(A)) is the generator of a C0 -semigroup (T (t))t≥0 on X. The type of T (t) is defined as ω0 (A) := inf{t−1 log kT (t)k : t > 0}. The domain D(A), which is a Banach space, is endowed with the graph norm kxkA := kxk + kAxk, x ∈ D(A). Denote the resolvent set of A by ρ(A) and the resolvent operator of A by R(µ, A) := (µ − A)−1 for µ ∈ ρ(A). S. H ADD

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The completion of X with respect to the norm kxk−1 = kR(µ, A)xk for x ∈ X and some µ ∈ ρ(A) is a Banach space denoted by X−1 , which is called the extrapolation space of A. For any Banach spaces E and F , denote the Banach space of all linear bounded operators from E to F by L(E, F ) with L(E) := L(E, E). The history function of z : [−r, ∞) → Z is the function zt : [−r, 0] → Z defined by zt (θ) = z(t + θ) for t ≥ 0 and θ ∈ [−r, 0] with delay r > 0. For p > 1, Lp ([−r, 0], Z) is the Banach space of all pintegrable functions f : [−r, 0] → Z and W 1,p ([−r, 0], Z) is its associated Sobolev space. S. H ADD

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Problem statement To investigate the approximate controllability of  d   x(t) − Dxt − K0 u(t) − K1 ut = B0 u(t) + B1 ut   dt    +A x(t) − Dxt − K0 u(t) − K1 ut + Lxt , t ≥ 0,   lim x(t) − Dxt − K0 u(t) − K1 ut = z0 ,   t→0   x = ϕ, u = ψ. 0 0

(1)

Here x : [−r, ∞) → X, u : [−r, ∞) → U , the operators D, L : W 1,p ([−r, 0], X) → X, K1 , B1 : W 1,p ([−r, 0], U ) → X and B0 , K0 : U → X are linear bounded, and A is the generator of a C0 -semigroup T := (T (t))t≥0 on X. S. H ADD

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How general it is? Distributed delays in the state AND the input, could be multiple discrete delays Delays in the derivative: neutral K0 6= 0 and K1 6= 0.

 d   x(t) − Dxt − K0 u(t) − K1 ut = B0 u(t) + B1 ut    dt +A x(t) − Dxt − K0 u(t) − K1 ut + Lxt , t ≥ 0,  lim x(t) − Dxt − K0 u(t) − K1 ut = z0 ,     t→0 x0 = ϕ, u0 = ψ. S. H ADD

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Background This area had been very active during 70s and 80s, focusing on retarded systems with state or input delays. Recent works include: 1) Diblik et al: Using a representation of solutions with the aid of a discrete matrix delayed exponential for linear discrete-time systems with state delays; 2) Sun et al: Using the matrix Lambert W function for linear (continuous-time) systems with state delays; 3) Rabah et al: Using the moment problem approach for neutral systems with distributed state delays. The closest one is the last one, where K0 = 0, K1 = 0, A = 0 and B1 = 0. Also, the operator D is more general here. S. H ADD

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Key steps Transform the problem into a perturbed boundary control problem

Solve the approximate controllability for the boundary control problem

Then transfer the results back to the original problem

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More notation Define QX ˙ m ϕ = ϕ, U ˙ Q ψ = ψ, m

1,p D(QX ) = W ([−r, 0], X), m

=W

1,p

D(Q ) = {ϕ ∈ W

1,p

U D(Qm )

([−r, 0], U ),

and X

Q ϕ = ϕ, ˙ ˙ QU ψ = ψ,

X

([−r, 0], X) : ϕ(0) = 0},

D(QU ) = {ψ ∈ W 1,p ([−r, 0], U ) : ψ(0) = 0}.

It is well known that QX and QU generate the left shift semigroups on Lp ([−r, 0], X) and Lp ([−r, 0], U ), respectively. S. H ADD

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U X U Denote by QX and Q the extension of Q and Q , −1 −1 respectively, and define the operators, for µ ∈ C,

β X := (µ − QX −1 )eµ ,

β U := (µ − QU−1 )eµ

(2)

where eµ : Z → Lp ([−r, 0], Z),

(eµ z)(θ) = eµθ z

(3)

is defined for µ ∈ C, z ∈ Z, and θ ∈ [−r, 0]. Denote by J Z (Z = X or U ) the following continuous injection Z

Z

J : D(Q ) → W S. H ADD

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1,p

([−r, 0], Z).


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Transforming into a boundary control problem Introduce the space X := X × Lp ([−r, 0], X) × Lp ([−r, 0], U ), and the matrix operator   A L B1 Am :=  0 QX 0 , m 0 0 QUm

D(Am ) := D(A) × W 1,p ([−r, 0], X) × W 1,p ([−r, 0], U ),

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Moreover, define the boundary operators N , M : D(Am ) → U := X × U as 0 δ0 0 I D K1 , N = , M= 0 0 δ0 0 0 0 and, for any u ∈ U , the operators B u 0 Bu = 0 , Ku = Ku0 u 0

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Then the system (1) can be reformulated as w(t) ˙ = Am w(t) + Bu(t),

t ≥ 0,

(5)

N w(t) = Mw(t) + Ku(t).

by setting w(t) = (z(t), xt , ut )⊤ , t ≥ 0 with w(0) = ̟ = (z0 , ϕ, ψ)⊤ , where z(t) = x(t) − Dxt − K0 u(t) − K1 ut with limz(t) = z0 . Here, Am : Xm → X and t→0

N , M : Xm → U are linear operators with Banach spaces X , U and Xm being a dense domain of X endowed with a norm | · |, which is finer than the norm k · k of X such that (Xm , | · |) is complete. This is a boundary control problem. S. H ADD

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Verification of conditions Standard conditions for boundary control problems: (H1) A = Am with domain D(A) := KerN generates a C0 -semi group (T (t))t≥0 on X , (H2) ImN = U. These assumptions show that the boundary value problem with M = 0 and B = 0 is well-posed in the sense that it can be reformulated as a well-posed open loop system on the state space X and control space U. Clearly N satisfies the condition (H2).

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In order to verify condition (H1), assume (A1) (QX , β X , LJ X ) generates a regular linear system on Lp ([−r, 0], X), X, X, (A2) (QU , β U , B1 J U ) generates a regular linear system on Lp ([−r, 0], U ), U, X. Define the operator 



A L B1 A :=  0 QX 0  , 0 0 QU D(A) := KerN = D(A) × D(QX ) × D(QU ).

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It satisfies A = A0 + L with   A 0 0 A0 :=  0 QX 0  , 0 0 QU D(A0 ) := KerN .,





0 L B1 L := 0 0 0  0 0 0 D(L) := KerN .

Clearly, A0 generates a diagonal C0 -semigroup on X . It is not difficult to see that if L and B1 are admissible observation operators for QX and QU (in particular if (A1) and (A2) holds), respectively, then L is a Miyadera–Voigt perturbation for A0 . Then A generates a C0 -semigroup T := (T (t))t≥0 on X . Hence N satisfies the condition (H1). S. H ADD

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Admissible observation operator Definition 1. C ∈ L(D(A), Y ) is called an admissible observation operator for A if, for constants τ ≥ 0 and γ := γ(τ ) > 0, Z τ kCT (t)xkp dt ≤ γ p kxkp , ∀x ∈ D(A), (6) 0

The map Ψ∞ x := CT (·)x, defined on D(A), extends to a bounded operator Ψ∞ : X → Lploc (R+ , Y ). For any x ∈ X we set Ψ(t)x := Ψ∞ x on [0, t] and call (T (t), Ψ(t))t≥0 an observation system.

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Admissible control operator Definition 2. B ∈ L(U, X−1 ) is called an admissible control operator for (A, D(A)), the generator of a C0 semigroup (T (t))t≥0 on X, if for all t > 0 and u ∈ Lp ([−r, 0], U ) the control map Z t Φ(t)u := T (t − s)Bu(s) ds (7) 0

takes values in X. The pair (T, Φ) is called a control system represented by the operator B.

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Formal definition of AC Definition 3. Let (T, Φ) be the control system represented by the operator B. Define the reachability space R := ∪t≥0 RanΦ(t). Then, (A, B) is said to be approximately controllable if R is dense in X. This means that, for a given arbitrary ε > 0, it is possible to steer the state from the origin to the neighbourhood, with a radius ε, of all points in the state space.

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Further transformation It can be proved that the boundary control system (5) is equivalent to the distributed-parameter system w(t) ˙ = Aw(t) + (B + BK)u(t),

w(0) = ̟,

(8)

for t ≥ 0, with A and B defined as Ax = Am x, D(A) = x ∈ D(Am ) : N x = Mx . B := (µ − A−1 )Dµ ,

µ ∈ ρ(A).

where Dµ is the Dirichlet operator −1 Dµ := N |Ker(µ−Am ) : U → Ker(µ − Am ). (9) S. H ADD

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AC for the boundary control problem When the conditions (H1) and (H2) are satisfied, Dµ exists and is bounded. For µ ∈ ρ(A), there is (Am − A−1 ) = (µ − A−1 )Dµ N

on

D(Am ).

Definition 4. Assume that (H1)–(H2) are satisfied. The boundary value problem (5) is said to be approximately controllable if the open-loop system (A, B+BK) is (in the sense of Definition 3).

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Theorem Theorem 5. Assume that (H1)–(H2) are satisfied. The boundary value problem (5) is approximately controllable if and only if, for µ ∈ ρ(A) ∩ ρ(A) and ϕ ∈ X ′ , the fact that h(I − Dµ M)−1 (Dµ K + R(µ, A)B)u, ϕi = 0, for ∀u ∈ U, (10) implies that ϕ = 0.

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Going back to the neutral system ...

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Dirichlet operator associated with N Note that ρ(A) = ρ(A). For µ ∈ ρ(A), one can see that the Dirichlet operator associated with N , as defined in (9), is given by   R(µ, A)(Le u + B e v) µ 1 µ u u , Dµ v =  eµ u v ∈ U. eµ v (11)

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More assumptions Let J : D(A) → D(A) × W 1,p ([−r, 0], X) × W 1,p ([−r, 0], U ) be the continuous injection, and set C := MJ ∈ L(D(A), U). Then C is an admissible observation operator for A under the following assumptions: (A3) (QX , β X , DJ X ) generates a regular linear system on Lp ([−r, 0], X), X, X with IX as an admissible feedback, (A4) (QU , β U , K1 J U ) generates a regular linear system on Lp ([−r, 0], U ), U, X. S. H ADD

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∆(µ) Define ∆(µ) := I − Deµ − R(µ, A)Leµ ,

µ ∈ ρ(A).

By using (11), for µ ∈ ρ(A), we have ∆(µ) −(K1 eµ + R(µ, A)B1 eµ ) . IU − MDµ = 0 IU (12) Thus, ∆(µ) is invertible if and only if µ ∈ ρ(A)∩ρ(A).

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AC: the transformed system Definition 6. Assume that the conditions (A1) to (A4) are satisfied. Define Z t R(t)u := (TI )−1 (t − τ )(BK + B)u(τ ) dτ 0

p Lloc (R+ , U ).

for t ≥ 0 and u ∈ The system (5) is said to be Υ-approximately controllable if Cl( ∪ PΥ (RanR(t))) = Υ, t≥0

where Υ can be any of X , X, X × Lp ([−r, 0], X), Lp ([−r, 0], X) and Lp ([−r, 0], U ), and PΥ is the projection operator from X to Υ. In particular, PX = I. S. H ADD

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



z(t) w(t) =  xt  ut X-approximately controllable: the state z is approximately controllable; X × Lp ([−r, 0], X)-approximately controllable: states z and xt are approximately controllable; Lp ([−r, 0], U )-approximately controllable: state ut is approximately controllable. Apparently, it is always approximately controllable. S. H ADD

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Lp ([−r, 0], U )-


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AC: Transformed system Theorem 7. Assume that (A1) to (A4) are satisfied. Then the following conditions are equivalent: (i) the system (5) is X -approximately controllable, (ii) it is X × Lp ([−r, 0], X)-approximately controllable, (iii) for µ ∈ ρ(A) ∩ ρ(A), ϕ ∈ Lq ([−r, 0], X) with 1p + 1q = 1 and x ∈ X ′ , the fact that, for ∀u ∈ U,

−1 (Ω(µ) + Γ(µ)Λ(µ))u, x + (eµ K0 + eµ ∆(µ) Λ(µ))u, ϕ = 0 implies that x = 0 and ϕ = 0. Here,

Γ(µ) := R(µ, A)Leµ ∆(µ)−1 , Ω(µ) := R(µ, A)(LeµK0 + B1 eµ + B0 ), Λ(µ) : = Ω(µ) + Deµ K0 + K1 eµ . S. H ADD

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Outline of proof (I − Dµ M)

−1



 R(µ, A) =  



I + Γ(µ)  −1 =  eµ ∆(µ) 0

Γ(µ)D I + eµ

Γ(µ)K1

∆(µ)−1 D



eµ ∆(µ)−1 K1  

0

I

R(µ, A)

R(µ, A)LR(µ, QX )

0

R(µ, QX )

0

0

0

R(µ, QU ) 



R(µ, A)B1

R(µ, QU )

.



 . 



Ω(µ)u    Dµ Ku + R(µ, A)Bu =  eµ K0 u . eµ u Hence, 

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

Ω(µ) + Γ(µ)Λ(µ)   −1  (I − Dµ M)−1 (Dµ Ku + R(µ, A)Bu) =   eµ K0 + eµ ∆(µ) Λ(µ) u eµ

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State z(t) Assume that the conditions (A1) to (A4) are satisfied. Then the system (5) is X-approximately controllable iff, for µ ∈ ρ(A) ∩ ρ(A) and x ∈ X ′ , the fact that

(Ω(µ) + Γ(µ)Λ(µ))u, x = 0, for ∀u ∈ U, implies that x = 0.

This characterises the condition when z(t), partial state of system (5), can reach all points in X. In general, this is irrelevant to the approximate controllability of system (1).

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State xt Assume that the conditions (A1) to (A4) are satisfied. Then the system (5) is Lp ([−r, 0], X)-approximately controllable iff, for µ ∈ ρ(A) ∩ ρ(A) and ϕ ∈ Lq ([−r, 0], X) with 1p + 1q = 1, the fact that

−1 eµ (K0 + ∆(µ) Λ(µ))u, ϕ = 0, for ∀u ∈ U, (13) implies that ϕ = 0. This actually describes the approximate controllability of system (1).

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AC: the neutral system Theorem 8. Assume that the conditions (A1) to (A4) are satisfied. Then the general neutral system (1) is approximately controllable iff, for µ ∈ ρ(A) ∩ ρ(A) and x ∈ X, the fact that

−1 (K0 + ∆(µ) Λ(µ))u, x = 0, for ∀u ∈ U, implies that x = 0.

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Special case I: Difference equations with state and input delays Consider the difference equation x(t) = Dxt + K0 u(t) + K1 ut ,

t ≥ 0,

(14)

with x0 = ϕ, u0 = ψ. Note that the equation (14) is a special case of (1), with A = 0, L = 0, B0 = 0, B1 = 0. In this case, ∆(µ) = I − Deµ, Γ(µ) = 0,

Ω(µ) = 0,

Λ(µ) = DeµK0 + K1 eµ . S. H ADD

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Define the operator QI ϕ = ϕ, ˙ n o D(QI ) = ϕ ∈ W 1,p ([−r, 0], X) : ϕ(0) = Dϕ . ∆(µ) is invertible if and only if µ ∈ ρ(QI ). Assume that the conditions (A3) to (A4) are satisfied. Then the system (14) is approximately controllable iff, for µ ∈ ρ(QI ) and x ∈ X, the fact that

−1 (I − Deµ) (K0 + K1 eµ )u, x = 0, for ∀u ∈ U implies that x = 0. S. H ADD

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Special case II: Retarded delay systems with state and input delays Consider the following retarded delay system with finite-dimensional delay-free dynamics, which widely exist in engineering: t ≥ 0,

x(t) ˙ = Ax(t)+Lr x(t−r)+B0 u(t)+Br u(t−r),

where x ∈ Rn , A ∈ Rn×n , Lr ∈ Rn×n , B0 ∈ Rn×m , Br ∈ Rn×m and u ∈ Rm . The variants of this system have been targets of most of the papers in this area.

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This system can be obtained from the system (1) by taking D = 0, K0 = 0, K1 = 0, L = Lr δ−r and B1 = Br δ−r , where δ−r is the Dirac operator. These operators satisfy the conditions (A1)–(A4). Then, ∆(µ) = I − e−rµR(µ, A)Lr ,

µ ∈ ρ(A).

Define ρ∆ = {µ : ∆(µ) is invertible}. Then for µ ∈ ρ(A) ∩ ρ∆ , Γ(µ) = e−rµ R(µ, A)Lr ∆(µ)−1 = ∆(µ)−1 − I, Λ(µ) = Ω(µ) = R(µ, A)[e

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−rµ

Br + B0 ].


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Now Theorem 8 shows that the system is approximately controllable iff, for µ ∈ ρ(A) ∩ ρ∆ and x ∈ Rn , the fact that

−1 ∆(µ) Λ(µ)u, x = 0, for ∀u ∈ U, (15)

implies that x = 0. It can be found that the equality in (15) is equivalent to that

−rµ −1 −rµ (µI − A − e Lr ) (e Br + B0 )u, x = 0.

When Br = 0, this condition is simpler than the one for systems with state delays obtained by Manitius et al in 70s. S. H ADD

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A rank condition Assume Lr = 0. The left-hand side of the equality in (15), assuming µ 6= 0 for the moment, is u⊤ (e−rµ Br + B0 )⊤ (µI − A)−⊤ x 1 ⊤ −rµ ⊤ n−1 = u (e Br + B0 ) Σk=0 fk ( )(Ak )⊤ x µ 1 −rµ 1 n−1 ⊤ = Σk=0 u (fk ( )e Br + fk ( )B0 )⊤ (Ak )⊤ x µ µ where (µI − A)−1 is replaced with (µI − A)

−1

1 −1 1 1 k n−1 = (I − A) = Σk=0 fk ( )A µ µ µ

with fk ( µ1 ), k = 0, 1 · · · , n − 1, as polynomials in µ1 . S. H ADD

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This can be written as ⊤   ⊤ uf0 ( µ1 )e−rµ B r   ⊤ ⊤  uf1 ( µ1 )e−rµ   B  r A    .. ..    . .     uf ( 1 )e−rµ   ⊤ ⊤ n−1   Br (A )  n−1 µ    1 B0⊤ uf0 ( µ )       ⊤ ⊤ 1   B A ) uf (  0 1 µ    .. ..    . .   ⊤ ⊤ n−1 1 B (A ) ufn−1 ( µ ) 0 S. H ADD

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             

x = 0.

2n·m×n


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Hence, the system is approximately controllable if and only if Rank

Br ABr · · · An−1 Br B0 AB0 · · · An−1 B0

= n.

The condition µ ∈ ρ(A) ∩ ρ∆ and µ 6= 0 are now irrelevant. This recovers the results for systems with input delays by Sebakhy (1971). If Br = 0, then Rank B0 AB0 · · · An−1 B0 = n. We have not lost ourselves! S. H ADD

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A rank condition when Lr 6= 0 can be derived similarly. Instead of using Ak in the series expansion, (A + e−rµLr )k , k = 0, 1 · · · , n − 1, should be used. However, it is difficult to express the condition for the general case in a compact form and hence omitted.

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Summary The approximate controllability of neutral systems is discussed via converting the problem into that for a perturbed control problem. After solving the approximate controllability for the perturbed control problem, the conditions for the approximate controllability of neutral systems are obtained. Some special cases are discussed.

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Said Hadd and Qing-Chang Zhong Dept. of Electrical Eng

Said Hadd and Qing-Chang Zhong Dept. of Electrical Eng

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