Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment

Systems & Control Letters 69 (2014) 47–52

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment F. Cacace a,∗ , A. Germani a,b , C. Manes b a

Università Campus Bio-Medico, Via Àlvaro del Portillo, 21, 00128 Roma, Italy

b

Dip. di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, 67100 Coppito (AQ), Italy

article

info

Article history: Received 6 May 2013 Received in revised form 14 February 2014 Accepted 22 April 2014

Keywords: Feedback stabilization Delay compensation Linear time-delay systems

abstract We consider the problem of controlling a linear system when the state is available with a known timevarying delay (delayed-state feedback control) or the actuator is affected by a delay. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system to guarantee the exponential zero-state stability with a prescribed decay rate by means of a finite-dimensional control law. A non conservative bound on the maximum allowed delay for the prescribed decay rate is presented, which holds for both cases of constant and time-varying delays. An advantage over recent and similar approaches is that differentiability or continuity of the delay function is not required. We compare the performance of our approach, in terms of delay bound and input signal, with another recent approach. © 2014 Published by Elsevier B.V.

1. Introduction Since the birth of automatic control the problem of stabilizing systems with delays in the state and/or input variables has been investigated by a multitude of researchers (see e.g. [1–6] and the references therein). For the general case of linear systems with delays in the state equations several control approaches have been developed, including Finite Spectrum Assignment [7,8], Continuous Pole Placement [9], optimization based pole assignment [10], Infinite Spectrum Assignment [11], Lyapunov–Krasovskii functional based methods [12,13], parametric Lyapunov equations [14], and Matrix Lambert Function [15,16]. Predictor-based approaches for controlling time-delay systems have long been used for systems with delayed input, and have received renewed interest in recent years, even if most approaches only consider the case of constant delay [17,18]. In [19] a sequence of predictors is used to overcome the delay limitation in the constant delay case. Predictorbased state feedback and output feedback controllers have been proposed in [20,21] for nonlinear systems with time-varying input delays. State predictors for nonlinear systems with output delays have been studied in [22,23]. Traditional predictor-based controllers use infinite-dimensional feedback laws, whose accurate implementation has been widely

∗

Corresponding author. Tel.: +39 3496570566. E-mail addresses: [email protected], [email protected] (F. Cacace), [email protected] (A. Germani), [email protected] (C. Manes). http://dx.doi.org/10.1016/j.sysconle.2014.04.007 0167-6911/© 2014 Published by Elsevier B.V.

discussed in the literature (see for example [24,5,25]). A feedback law that only involves finite dimensional static state feedback is simpler to implement, because it does not require to store the previous values of the state and it does not need discretization. An approach of this kind, named Truncated Prediction Feedback, has been used in [26] for linear systems with constant delays that are not exponentially unstable and extended in [27] to the case of time-varying input delays and in [28] to exponentially unstable systems. In this paper we use a finite-dimensional method for linear systems with no restriction on the position of the poles of the open-loop system, when the state is available with a possibly timevarying delay. This is equivalent to solving a delayed input stabilization problem when the delay function is known in advance (see [27] for details). It can be seen as an extension to the variable delay case of the general structure of predictors outlined in [29], and it improves the recent results cited above in several regard. In particular, continuity and differentiability of the delay function are not needed, and the delay bound for a prescribed exponential rate of the controlled system is the same in the constant and variable delay case. Sufficient and, under some assumptions, necessary conditions on the delay bound are easy to check. The work here presented extends the preliminary results in [30], where only the case of single-input systems has been investigated. This paper is organized as follows. In Section 2 the control problem is formally stated, and preliminary definitions are given. In Section 3 the proposed control law and its features are presented. Section 4 investigates stability conditions for time-varying delays.

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F. Cacace et al. / Systems & Control Letters 69 (2014) 47–52

We compare the features of the proposed method with the recent approach presented in [28] in Section 5, and a numerical comparison is provided in Section 6. Notation. Given a real number α , the symbol C>α (C≥α ) denotes the set of all complex numbers s such that ℜ(s) > α (ℜ(s) ≥ α ). SC denotes the set of all the countable subsets of C that are symmetric w.r.t. the real axis (i.e., if U ∈ SC , then z ∈ U ⇒ z ∗ ∈ U). σ (A) ∈ SC denotes the spectrum of a real square matrix A. Given a positive real number δ¯ and an integer n, the symbol Cδ¯n denotes

¯ 0] in Rn , with the space of continuous functions that map [−δ, the uniform convergence norm, denoted ∥ · ∥∞ . For a set L ∈ SC , µ(L) = maxλi ∈L {ℜ(λi )}. Abbreviations: LTI: Linear Time Invariant; LTDS: Linear Time Delay System; DDE: delay-differential equation.

Consider a linear system whose state is available with a known ¯ : time-delay δt , possibly time-varying in a given interval [0, δ]

ξ (t ) = x(t − δt ),

t ≥ 0,

¯ δt : R+ → [0, δ],

(1)

where x(t ) ∈ Rn is the state, u(t ) ∈ Rp is the input. The pair (ξ (t ), δt ) is the measurement available at time t. Under the assumption that the pair (A, B) is controllable, we consider the problem of constructing a stabilizing feedback control law with delaydependent gain matrix: u(t ) = −K (δt ) ξ (t ),

¯ → Rp×n . K : [0, δ]

(2)

We assume that the control law (2) starts operating at time t = 0. Thus, the closed-loop system is a time-delay system with the following structure: x˙ (t ) = Ax(t ) − BK (δt ) x t − δt ,



x(t ) = φ(t ),



t ≥ 0,

¯ 0], δt : R+ → [0, δ], ¯ t ∈ [−δ,

(3)

where φ ∈ Cδ¯n is the so called preshape function. Of course, the gain function K (·) and the delay function δt must be such to ensure that the solution of (3) exists and is unique. For this reason we assume K (·) continuous in [0, δ¯ ] and δt measurable in R+ . Note that the control problem described above is equivalent to the problem of controlling a linear system when the state is available without delay and the actuator is affected by a known time-varying delay. We are interested in designing control laws that ensure exponential decay of the state with a prescribed rate. The following definition is useful for our purposes. Definition 1 (α -exp Stability). For a given real number α > 0, the system (3) is said to be α -exp stable if there exists γ > 0 such that

∥x(t )∥ ≤ e−αt γ ∥φ∥∞ ,

Of course, the set of modal numbers M A, −BK (δt ) extends the concept of spectrum to time-varying systems, and it can be an empty set if K (·) is not properly chosen. Like the spectrum, if A, B and K (·) are real and number, then also λ∗ is  λ is a modal  a modal number, i.e. M A, −BK (δt ) ∈ SC . For δt = 0 we have





M A, −BK (0) = σ A−BK (0) , and for constant delay δt = δ > 0,   the set M A, −BK (δ) is made of the countably infinite roots of the









characteristic function

2. Problem statement

x˙ (t ) = Ax(t ) + Bu(t ),

Definition 3. A normal mode of the LTDS (3), where the gain ¯ are given, is a function K (·) and the delay function δt ∈ [0, δ] solution of (3) of the type x(t ) = eλ t v , with λ ∈ C and v ∈ Cn. If such a solution exists, λ is called a modal number of (3).  M A, −BK (δt ) ⊂ C will denote the set of modal numbers of the system (3).

∀t ≥ 0, ∀φ ∈ Cδ¯n .

(4)

Definition 2. Consider the system (1) with a given control law of the type (2). For a given α > 0 the maximal delay for α -exp stability, denoted ∆α , is the supremum among all δ¯ > 0 such that the closed ¯ . If the loop system (3) is α -exp stable for any measurable δt ∈ [0, δ] system (3) is α -exp stable for any δt ∈ [0, ∞), then ∆α = ∞. ∆0 denotes the maximal delay for asymptotic stability. Definition 2 of ∆α implies that if δ¯ > ∆α , then there exists at ¯ such that the system (3) is not least one delay function δt ∈ [0, δ] α -exp stable. Note that if the delay-dependent gain K (δt ) is properly chosen the system (3), although time-varying, may admit solutions of the type eλt v , for some constant λ ∈ C and v ∈ Cn . Thus, we can give the following:

  νδ (s) = sIn − A + BK (δ) e−sδ , (5)   i.e. λ ∈ M A, −BK (δ) ⇐⇒ νδ (λ) = 0. Let us define σ¯ δ =   max{ℜ(λ), λ ∈ M A, −BK (δ) }. It is known that σ¯ δ exists and σ¯ δ < ∞ (Lemma 1.4.1 in [31]). Moreover, if σ¯ δ < 0, then the system (3) is α -exp stable with α ∈ (0, |σ¯ δ |) (Thm. 1.6.2 in [31]). These results can be summarized in the following Proposition: Proposition 1. Consider the system (3), with a constant delay δt = δ . For a given real number α > 0 the system is α -exp stable if νδ (s) ̸= 0 ∀s ∈ C≥−α , and only if νδ (s) ̸= 0 ∀s ∈ C>−α . Control problem formulation: Partial Spectrum Assignment with α exp stability. Let L = {λ1 , . . . , λn } ∈ SC denote a set of n real or complex numbers. The problem of Partial Spectrum Assignment (PSA) consists in finding a time-dependent feedback gain K : [0, δ¯ ] → Rp×n , such that with the feedback law (2) the set L is included in ¯ , i.e. L ⊂ the set of modal  numbers for any delay function δt ∈ [0, δ] ¯ . The PSA problem with α -exp stability M A, −BK (δt ) , ∀δt ∈ [0, δ] (PSAα ) requires that, in addition, the system (3) is α -exp stable. Of course, a necessary condition for having α -exp stability is that µ(L) ≤ −α . Note that when δt = 0, if the pair (A, B) is controllable, then for any choice of L ∈ SC , there exists K ∈ Rp×n such that σ (A − B K ) = L. 3. The feedback law This section summarizes the main results of the paper. For the sake of clarity, their formal statement and corresponding proof is postponed to Section 4 or to Appendices A and B as detailed in the sequel. Given a set L ∈ SC of n desired eigenvalues for the system (1), where the pair (A, B) is assumed controllable, consider the following delay-dependent feedback law u(t ) = −K (δt )ξ (t ),

with K (δt ) = K e Aδt ,

(6)

where A = A − B K,

with K ∈ Rp×n : σ ( A) = L.

(7)

We will prove in Theorem 2 that the feedback the  gain (6) solves  PSA problem for the system (3), i.e. L ⊂ M A, −BK (δt ) for any delay function δt . Moreover, Theorem 1 states that for a given desired decay rate α > 0, such that µ(L) < −α , the feedback gain ¯ as (6) achieves α -exp stability for the system (3) for any δt ∈ [0, δ] long as δ¯ is such that: δ¯

 0

∥ K e At B∥eαt dt < 1.

(8)

F. Cacace et al. / Systems & Control Letters 69 (2014) 47–52

The sufficient condition (8) of α -exp stability allows an easy computation of a lower bound for ∆α (the maximal delay for α exp stability of Definition 2) or, in some cases, the exact value of ∆α . Let δ¯α ∈ R+ be such that δ¯α



∥q(t )∥eαt dt = 1,

(9)

∥q(t )∥eαt dt ≤ 1. Then, condition (8) implies that 0 ¯δα ≤ ∆α , i.e. δ¯α is a lower bound for ∆α . ∞

α

We will conclude that the control gain (6) solves the PSA ¯ , provided that δ¯ < δ¯α given by (9). problem for any δt ∈ [0, δ] Further interesting results can be proved for single-input systems, for which q(t ) defined in (9) is a scalar function. In this case, Theorem 4 in Appendix A states that δ¯ α only depends on the two sets σ (A) and L, and not on B (provided that (A, B) is controllable). Theorem 1 in Section 4 states that if K e At B > 0 for t ∈ [0, δ¯ α ], then δ¯ α = ∆α . This implies the remarkable consequence that the delay bound in (8) is a necessary and sufficient delay condition of α -exp stability for both constant and variable delays. Remark 1. Note that the structure of the gain in (6) is seemingly similar to the one in [26] and in [27]. However, in (6) the gain K (δt ) depends on the exponential of the closed-loop stable matrix A, while in [26,27] the gain depends on the exponential of the open-loop, possibly unstable, matrix A. Some implications of this difference will be discussed in Section 5. 4. Results for time-varying delays

Theorem 1. Consider the system (1), a set L ∈ SC of n desired eigenvalues, and the control law (6), with time-varying delay δt ∈ ¯ , and δ¯ given. For a given α > 0 such that µ(L) < −α , consider [0, δ] δ¯α as defined in (9). Then: (i) If δ¯ α = ∞, then the system (3) is α -exp stable for any value of δ¯ ; (ii) If δ¯ α < ∞, then the system (3) is α -exp stable if δ¯ < δ¯ α (i.e., δ¯ α is a lower bound of ∆α ); (iii) In the case of scalar input (p = 1), if δ¯ α < ∞ and q(t ) > 0 ∀t ∈ [0, δ¯α ], then the system (3) is α -exp stable if and only if δ¯ < δ¯α (i.e., δ¯α = ∆α ). α

αt

Proof. Defining the α -perturbed state x (t ) = e x(t ), for t ∈ ¯ ∞), the proof of α -exp stability under condition (i) or (ii) is [−δ, achieved by showing that there exists x¯ > 0 such that ∥xα (t )∥ ≤ x¯ , ∀t ≥ 0, for any φ ∈ Cδ¯n . Computing the derivative x˙ α (t ) = α xα (t ) + eα t x˙ (t ), in which x˙ (t ) is replaced by the expression in ¯ 0], (3), with K (δt ) as in (6), and defining φ α (t ) = eα t φ(t ), t ∈ [−δ, we have α

x˙ (t ) = Aα x (t ) − B K e xα (t ) = φ α (t ),

Aα δ t α

x (t − δt ),

xα (t ) = e Aα (t −δt ) xα (t − δt ) +

t ≥ 0,

¯ 0], t ∈ [−δ,



t

e Aα (t −τ ) B K x˜ α (τ ) dτ .

(13)

t −δt

x˜ α (t ) =  φ α (t ) +

t



e Aα (t −τ ) B K x˜ α (τ ) dτ ,

(10)

t ∈ [0, t0 ),

0

x˜ α (t ) =



t

e Aα (t −τ ) B K x˜ α (τ ) dτ ,

t ≥ t0

(14)

t −δt

where

 φ α (t ) = e Aα t φ α (0) − e Aα δt φ α (t − δt ),

t ∈ [0, t0 ).

(15)

˜α

α

Defining the function c (t ) = K x (t ), t ≥ 0, Eq. (11) is rewritten as x˙ α (t ) = Aα xα (t ) + Bc α (t ), c α (t ) = K xα (t ) − e Aα δt xα (t − δt ) ,





(16)

t ≥ 0.

From this, recalling that xα (0) = φ(0) we have xα (t ) = e Aα t φ(0) +

t



e Aα (t −τ ) Bc α (τ )dτ .

(17)

0

Aα is Hurwitz stable, so that ∥e Aα t ∥ ≤ ρ , ∀t ≥ 0, for some ρ > 0. Noting also that ∥φ(0)∥ ≤ ∥φ∥∞ , from (17) we get t



∥xα (t )∥ ≤ ρ ∥φ∥∞ +

We start by proving that δ¯ α defined in (9) is a lower bound of ∆α , and in some cases it is exactly ∆α .

α

while for t ≥ t0

From these we have

where q(t ) = K e At B,

0

or δ¯ α = ∞, if

49

ρ ∥B∥ ∥c α (τ )∥ dτ .

(18)

0

In the following we will prove that if δ¯ α = ∞ (i.e., if

 δ¯

αt

∞ 0

∥q(t )∥

αt

e dt ≤ 1), or if δ¯ < δ¯ α < ∞, (i.e., if 0 ∥q(t )∥ e dt < 1), then there exist κ > 0 and β > 0 such that ∥c α (t )∥ ≤ e−β t κ∥φ∥∞ , t ≥ 0. Defining qα (t ) = eα t q(t ) = K e Aα t B, and recalling that c α (t ) = K x˜ α (t ), from (14) we have c α (t ) = K  φ α (t ) +

t



qα (t − τ ) c α (τ ) dτ ,

t ∈ [0, t0 ),

0

c α (t ) =



t

qα (t − τ ) c α (τ ) dτ ,

t ≥ t0 .

(19)

t −δt

The bound ∥c α (t )∥ ≤ e−β t κ∥φ∥∞ can be obtained by defining the ¯ (k + 1)δ) ¯ , for k = 0, 1, 2, . . . , and sequence of intervals Ik = [kδ, the nonnegative sequence c¯kα = sup ∥c α (t )∥,

k = 0, 1, 2, . . . .

t ∈Ik

Defining q¯ α = we have

∥c α (t )∥ ≤ q¯ α

 δ¯ 0

(20)

∥qα (t )∥dt, and recalling that t0 ≤ δ¯ , from (19)

sup ¯ t) τ ∈[t −δ,

∥c α (τ )∥,

¯ t ≥ δ.

(21)

where Aα = A + α In and Aα = A + α In (note that Aα is stable because µ(L) < −α by assumption). Adding and subtracting B K xα (t ) in (10), and noting that Aα = Aα − B K , we get

Taking the sup of ∥c α (t )∥ over Ik+1 , after some computations we get, for k ≥ 0,

x˙ α (t ) = Aα xα (t ) + B K x˜ α (t ),

c¯kα+1 ≤ sup

t ≥ 0. x˜ α (t ) = xα (t ) − e Aα δt xα (t − δt ),

(11)

¯ be such that t0 − δt0 = 0 (so that t ≥ t0 , ⇐⇒ Let t0 ∈ [0, δ] t − δt ≥ 0). Then, from (11), for t ≥ 0 xα (t ) = e Aα t xα (0) +

t

 0

e Aα (t −τ ) B K x˜ α (τ ) dτ ,

(12)

t ∈Ik+1



q¯ α

sup ¯ t) τ ∈[t −δ,

 ∥c α (τ )∥ = q¯ α max(¯ckα , c¯kα+1 ).

(22)

Note that under the assumptions (i) (δ¯ α = ∞) or (ii) (δ¯ < δ¯ α < ∞) we have q¯ α ∈ (0, 1). It follows that the only consistent solution of (22) is c¯kα+1 ≤ q¯ α c¯kα , for k = 0, 1, 2, . . . , and therefore c¯kα ≤ (¯qα )k c¯0α ,

k = 0, 1, 2, . . . .

(23)

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F. Cacace et al. / Systems & Control Letters 69 (2014) 47–52

¯ Defining β = − log(¯qα )/δ¯ > 0, so that e−β δ = q¯ α , we easily get the following:

∥c α (t )∥ ≤ e−β t (¯qα )−1 c¯0α ,

t ≥ 0.

(24)

Further computations (reported in Appendix B) show that

∥c α (t )∥ ≤ e−β t κ∥φ∥∞ ,

t ≥ 0,

(25)

for κ = 2(¯qα )−1 ∥ K ∥ρ e2q¯ α (see Eq. (B.1)). Substitution of (25) into (18) gives α

∥x (t )∥ ≤ ρ ∥φ∥∞ + ρ ∥B∥ κ∥φ∥∞

t



e−β t dτ ,

(26)

0

and from this

∥B∥ κ ∥x (t )∥ ≤ ρ 1 + β α





∥φ∥∞ ,

t ≥ 0.

(27)

As discussed before, this implies the α -exp stability of the system (3), and concludes the proofs of the points (i) and (ii) of the theorem. The point (iii) states that the condition δ¯ < δ¯ α , which has been proved to be sufficient for α − exp stability, is also necessary when the input is scalar and q(t ) > 0. To prove necessity of course we can limit ourselves to the case of constant delay equal to δ¯ , which ¯ . Such a result is a particular case of time-varying delay δt ∈ [0, δ] is proved in Proposition 4 in Appendix A.  Remark 2. In the previous result δ¯ α provided by (9) is proved to be a lower bound of ∆α under the assumption that α < −µ(L) because for any α > −µ(L) (3) cannot be α -exp stable. From (9), it follows that δ¯ α is a decreasing function of α . Thus, defining α¯ = −µ(L), it is that δ¯α > δ¯α¯ , for α ∈ [0, α) ¯ , and therefore δ¯α¯ is a ¯ , where δ¯ < δ¯α¯ , then lower bound of ∆α . It follows that if δt ∈ [0, δ] the closed-loop system (3) is α -exp stable for α < α¯ . Moreover, if the input is scalar and q(t ) > 0 for t ∈ [0, δ¯ α ] the lower bound δ¯ α¯ is tight, in the sense that for any δ¯ > δ¯ α¯ there exists an α < α¯ such that the closed-loop is not α -exp stable when δt = δ¯ . The interesting point about Theorem 1 is the statement that under the control law (6) it is possible to derive a delay bound that is a sufficient (possibly necessary) condition of α -exp stability not only for the constant delay case but also for uniformly bounded time-varying delays. Here time-varying delays belong to the broad class of measurable but not necessarily continuous delay functions. Point (i) of the Theorem seems to correspond to the results in [26,27], where no exponentially unstable open loop eigenvalue is allowed and the feedback law guarantees asymptotic convergence for arbitrarily large delays. Point (iii) of the Theorem states that for δ¯ > δ¯ α = ∆α , there exist delay functions δt ≤ δ¯ that are not α -exp stable. In other words, the bound δ¯ α is on sets of uniformly bounded delay functions. A specific δt may satisfy α -exp stability even if δt > δ¯ α . When the input is scalar the bound δ¯ α depends only on σ (A), α and L, the set of ‘‘desired’’ modal numbers (see Theorem 4 in Appendix A). Since there are infinite choices of L that satisfy the constraint α < −µ(L) an interesting optimization problem is to find L, given σ (A) and α , to have the largest δ¯ α . We now show that the delay-dependent gain K (δt ) defined in (6) solves the PSA problem, i.e. is such to include a set L ∈ SC of n desired eigenvalues in the set of modal numbers of the closed-loop system (3). Theorem 2. Consider the system (1), a set L ∈ SC of n desired eigenvalues, and the control law (6). ¯ , with Then, for any bounded time-varying delay δt ∈ [0, δ] arbitrary δ¯ > 0, the set L belongs to the set of modal numbers of the   closed loop system (3) (i.e., L ⊂ M A, −BK (δt ) ).

Proof. Recall that K in the control law (6) is such that σ ( A) = L. Thus, the proof consists in showing that for any λi ∈ σ ( A) there ¯ ∞) exists vi ∈ Cn such that the function x(t ) = eλi t vi , for t ∈ [−δ, satisfies the DDE  (3) for t ∈ [0, ∞), so that we can conclude that λi ∈ M A, −BK (δt ) (see Definition 3). To this aim, let vi be the eigenvector of A associated to λi and replace in (3) x(t ) with eλi t vi , x˙ (t ) with λi eλi t vi , and x(t − δt ) with eλi (t −δt ) vi . With these substitutions, the identity (3) is readily verified by taking into account that A = A − K B, Avi = λi vi , e Aδt vi = eλi δt vi .  Corollary 2. Under the same conditions and assumptions of Theorem 2, the set L is the intersection of all the sets of modal numbers of (3) for all bounded delay functions δt . Proof. This result follows from Theorem 2, considering that L ⊂ M (A, −BK (δt )) for all δt ∈ [0, δ¯ ], and in particular for δt = 0 L = M A, −BK (0) = σ (A − B K ).  The combination of Theorems 1 and 2 allows to conclude that the control law (6) solves the PSAα problem when δ¯ satisfies the α -exp stability conditions of Theorem 1. 5. Comparison with similar approaches In this section we compare our method to the one presented in [28], since this is, to our knowledge, the only approach that deals with the control problem of possibly unstable linear systems with time-varying input delays and static finite dimensional feedback. The control law in [28] is u(t ) = −KeAδt x(t − δt ),

(28)

where K = −BT P and the semi-definite positive matrix P is the solution of a parametric algebraic Riccati equation (ARE). Note that in (28) the exponential of the dynamic matrix A of (1) replaces the closed-loop matrix A of the control law (6). The essential differences can be summarized as follows. 1. In [28] the delay function δt is assumed to be continuously differentiable with δ˙ t < 1. This limitation may descend from the proof technique employed rather than from a limitation of the method, but it is not necessary in our approach. Non continuous piecewise constant delay functions are rather common in the important area of networked control systems, where data are buffered and sent into packets. Another important case of non continuous delay functions is sampling. 2. Our method allows to choose any desired rate α for the controlled system, while the design parameter γ in [28] belongs to an interval and it is only indirectly related to the convergence rate, through the parametric ARE. The former fact implies that the convergence rate has an upper limit. The possibility of choosing α arbitrarily large (at the expense of a lower delay bound) is an advantage, especially in the presence of nonlinear disturbances to the nominal systems. 3. In [28] the time-varying feedback gain increases when the delay is large, due to the exponential of the unstable matrix A. A larger gain can be a problem when the delayed state measurement is affected by additive noise. Although we do not work out this analysis, it is well known that a high feedback gain amplifies the effects of noise, and therefore a scheme that provides a lower gain is preferable. Moreover, a smaller gain yields a smaller u(t ). 4. The delay bound of our method at a given convergence rate appears to be larger, as illustrated by the example in the next section. This may be difficult to prove in general, since the dependence of the bound from the convergence rate is different in the two cases. Similarly, the input signal u(t ) consistently appears to be smaller.

F. Cacace et al. / Systems & Control Letters 69 (2014) 47–52

51

Fig. 1. Plots of x(t ) (left) and u(t ) (right) for the delay function δt = 3.5(1 + 0.5 cos(t )), δ¯ = 5.25.

6. Example In this section we consider the delayed double oscillator system presented in Section 6 of [28] with a positive real pole. The state space matrices are



p 0  A = 0 0 0

1 0

−ω 0 0

0

ω 0 0 0

0 0 1 0

−ω



0 0  0 ,

ω 0

  0

0   B = 0 . 0

(29)

1

Note that σ (A) = {0.1, ±j} with p = 0.1 and ω = 1. The maximum delay bound for the TPF controller of [28] at γ = 0.09 (close to the fastest attainable convergence to 0) is δmax,1 = 0.458 (see [28], Sect. 6.4). Since this is a sufficient condition, the authors have found, through simulation, the actual value of the bound by using the specific delay function δt = d(1 + 0.5 cos(t )) at increasing d. The actual delay bound reported in [28] is δmax,2 = 2.55 (that is, d = 1.7). The difference between δmax,1 and δmax,2 may depend on the fact that the former refers to a generic δt and the latter to a specific δt , as well as on the conservatism of the conditions. To compare PSA with TPF we have chosen σ ( A) = {−0.1, −0.1 ± j}. By setting (the convergence rate α = 0.02 the convergence rate of TPF at γ = 0.09 is not specified but it appears to be less than 0.01, see Fig. 7 in [28]), we obtain δ¯ α = 5.395, more than 10 times larger than the corresponding TPF limit for a generic δt and about twice the actual TPF bound on a specific δt . This limit is not strict, since in this case the input is scalar but q(t ) is not positive in the interval (0, δ¯ α ). We have found through simulation that for δt = d(1 + 0.5 cos(t )) the actual PSA delay bound is about δ¯ 0.02 = 8.25. Fig. 1 (left) shows the state response of the closed-loop system for α = 0.02 and the delay function δt = 3.5(1 + 0.5 cos(t )), that corresponds to δ¯ = 5.25. Fig. 1 (right) shows the input signal u(t ). Comparing these plots with the ones in Figs. 7 and 8 of [28], obtained at δ¯ = 2.55, it may be appreciated that the convergence rate is faster for twice the delay and with an input signal which is an order of magnitude smaller. The plot of the function ∥x(t )∥e0.02t (not reported) confirms that the controlled system is 0.02-exp stable. 7. Conclusions In this paper a feedback law is proposed for the exponential stabilization of linear systems when the state is available with a known time-delay, possibly time-varying. The proposed technique is computationally very simple because it is a simple extension of the classical eigenvalue assignment method with an exponential term that depends on the delay. The delay-dependent delayedstate feedback proposed in this paper is such to include a given set

of eigenvalues in the set of modal numbers (spectrum, for constant delays) of the closed-loop system. We showed a method for computing lower bounds of the maximum delay that guarantees exponential stability with a prescribed rate of convergence. Non restrictive and easy to check conditions have been provided that show the trade-off between the exponential stability rate and such a maximum delay. Such conditions hold for both cases of constant and time-varying delay. A problem left for future work is the optimal choice of the set L of desired modal numbers that achieves the desired convergence rate α and obtains the largest maximal delay ∆α . Another future research theme is the solution of the PSAα problem when delayed output measurements are available (i.e. the whole state is not available). Appendix A. Results for the constant delay case δt = δ Theorem 3. Consider the system (1), a set L ∈ SC of n desired eigenvalues, and the control law (6), with constant delay δt = δ . Let Qδ (s) =

δ

 0

q(t )e−st dt =

δ



K e At Be−st dt .

(A.1)

0

Then, for any α > 0 such that µ(L) < −α , the closed loop system (3) is α -exp stable if |Ip − Qδ (s)| ̸= 0, ∀s ∈ C>−α , and only if |Ip − Qδ (s)| ̸= 0, ∀s ∈ C≥−α . Corollary 3 (Scalar Input). Consider the system (1) with p = 1, a set L ∈ SC of n desired eigenvalues, and the control law (6), with constant delay δt = δ . Moreover, assume that q(t ) > 0 ∀t ∈ [0, δ]. Then, for any α > 0 such that µ(L) < −α , the closed loop system (3) is α -exp stable if Qδ (−α) < 1 and only if Qδ (−α) ≤ 1. Proposition 4 (Scalar Input). In the same hypotheses of Corollary 3, for a given α > 0 such that µ(L) < −α , let δ¯ α be defined as in (9). If q(t ) > 0 ∀t ∈ [0, δ¯ α ], then δ¯ α = ∆α . Proof. The case δ¯ α = ∞ is trivial, because it means that the closed-loop system is α -exp stable for any delay, i.e. ∆α = δ¯ α = ∞. Then, let us consider the case δ¯α < ∞. By definition δ¯α ≤ ∆α . Thus we only have to show that ∆α cannot be larger than δ¯ α . Assume, by contradiction, that ∆α > δ¯ α . Let δˆ ∈ (δ¯ α , ∆α ) be such that ˆ (δˆ exists thanks to continuity arguments, q(t ) > 0 for t ∈ [0, δ) because q(δα ) > 0). By definition of ∆α , the closed loop system is ˆ ⊂ [0, ∆α ). Being Qδ (−α) monotone α -exp stable for any δ ∈ [0, δ] ˆ , and being Qδ¯ (−α) = 1, it follows strictly increasing for δ ∈ [0, δ] α

ˆ . Thus, the necessary condition that Qδ (−α) > 1 for any δ ∈ (δ¯ α , δ] Qδ (−α) ≤ 1 for the α -exp stability of Corollary 3 is violated, contradicting the assumption that ∆α > δ¯ α . It follows that ∆α cannot be larger than δα , so that δ¯ α = ∆α . 

52

F. Cacace et al. / Systems & Control Letters 69 (2014) 47–52

Theorem 4 (Scalar Input). In the same hypotheses of Corollary 3, the scalar function q(t ) = K e At B and δ¯ α defined in (9) depend only on the two sets σ (A) and L. Appendix B. A bound needed in the proof of Theorem 1 In this Appendix the following bound is proved

∥c α (t )∥ ≤ e−β t κ∥φ∥∞ , t ≥ 0, with κ = 2(¯qα )−1 ∥ K ∥ρ e2q¯ α ,

(B.1)

where ρ > 0 is a bound on ∥e Aα t ∥. This result is achieved by finding α an upper bound for c¯0α (defined as supt ∈[0,δ] ¯ ∥c (t )∥) to be replaced into the inequality (24). From (19), defining φ¯ α = supt ∈[0,δ] φα ∥ ¯ ∥ we have for t ∈ [0, t0 )

∥c α (t )∥ ≤ ∥ K ∥ φ¯ α +

t



∥qα (t − τ )∥ ∥c α (τ )∥ dτ ,

0

≤ ∥ K ∥ φ¯ α e

t 0

∥qα (t −τ )∥ dτ

,

t ∈ [0, t0 )

(B.2)

where the Gronwall–Bellman inequality has been used. From these, recalling that q¯ α =

∥c α (t )∥ ≤ ∥ K ∥ φ¯ α eq¯ α ,

 δ¯ 0

∥qα (t )∥dt and δ¯ ≥ t0 ,

t ∈ [0, t0 ).

(B.3)

Now considering (19) for t ∈ [t0 , δ¯ ] we have

∥c α (t )∥ ≤

t



∥qα (t − τ )c α (τ )∥ dτ

t −δt t

 ≤

∥qα (t − τ )c α (τ )∥ dτ .

(B.4)

0

Putting (B.3) and (B.4) together we have for t ∈ [0, δ¯ ]

∥c α (t )∥ ≤ ∥ K ∥ φ¯ α eq¯ α +

t



∥qα (t − τ )∥ ∥c α (τ )∥ dτ .

(B.5)

0

Again, by the Gronwall–Bellman inequality t

∥c α (t )∥ ≤ ∥ K ∥ φ¯ α eq¯ α e

0

∥qα (t −τ )∥ dτ

≤ ∥ K ∥ φ¯ α e2q¯ α ,

(B.6)

α

from which we get the following bound on c¯0 : c¯0α ≤ ∥ K ∥ φ¯ α e2q¯ α .

(B.7)

A bound on φ¯ α is obtained by recalling the definition φ¯ α = supt ∈[0,t0 ] ∥ φ α (t )∥, where  φ α (t ) = e Aα t φ α (0) − e Aα δt φ α (t − δt ), as in Eq. (15). Recalling that ∥e Aα t ∥ ≤ ρ , it easily follows

φ¯ α ≤ 2ρ max ∥φ α (t )∥ ≤ 2ρ∥φ∥∞ .

(B.8)

¯ 0] t ∈[−δ,

Replacing this in (B.7) the following bound on c¯0α is obtained c¯0α ≤ 2∥ K ∥ρ e2q¯ α ∥φ∥∞ .

(B.9)

Using this as an upper bound on (24) we get for t ≥ 0

∥c α (t )∥ ≤ e−β t 2(¯qα )−1 ∥ K ∥ρ e2q¯ α ∥φ∥∞ , which is exactly the bound (B.1).

t ≥ 0,

(B.10)

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