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Quantized stabilization of switched systems with switching delays and packet loss
Accepted Manuscript
Quantized stabilization of switched systems with switching delays and packet loss Jingjing Yan, Yuanqing Xia, Chenglin Wen PII: DOI: Reference:
S0016-0032(18)30321-1 10.1016/j.jfranklin.2018.05.009 FI 3443
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
5 May 2017 5 March 2018 4 May 2018
Please cite this article as: Jingjing Yan, Yuanqing Xia, Chenglin Wen, Quantized stabilization of switched systems with switching delays and packet loss, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.05.009
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Quantized stabilization of switched systems with switching delays and packet loss Jingjing Yan, Yuanqing Xia, and Chenglin Wen
Abstract
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This paper is concerned with the problem of designing an observer-based quantized feedback controller for the continuous-time switched linear systems, in which the transmission of switching signal is subject to unbounded delays and packet loss. To deal with the unbounded switching delays, we design a constant d¯ to determine that the switching signal received by controller is ignored or not. Based on that, if the signal is timestamped, the controller’s mode is uniquely determined. Moreover, we adjust the quantizer parameters in real time depending on the actual transmission situations to ensure the unsaturation of quantizer and thus the boundness of quantization error. Within this setup, we derive a maximum allowable packet loss rate ensuring the mean square stability of the closed-loop switched systems. An illustrative example is given to show the usefulness of the proposed framework for the quantized stabilization of some classes of switched systems. Index Terms
Switched systems, quantization, unbounded delays, packet loss.
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I. I NTRODUCTION As an important class of hybrid dynamical systems, switched systems have received a lot of attention in recent years [1–6], since they have been successfully applied to model different practical systems, such as highly complex systems, chemical process, power electronic systems, manipulator robots, networked control systems, flight control systems, servomechanism systems and so on. With the advancement of network process, data in switched systems is transmitted over the network frequently. While bringing convenience, it should have a significant impact on the system performance, especially for not timely and imprecise data transmission. Not timely data transfer is deduced by network-induced time-delay and packet loss [7–10]. So, among the control and communication communities, many researchers make a great effort to study the stability and controller design for switched systems affected by time-delay [11–18] or packet loss [19–21]. The global exponential stability of the nonlinear time-varying switched control systems is obtained in [11]. The stability and L2 -gain analysis problem for a class of switched linear systems is studied in [12]. Observer-based finite-time stabilization and H∞ control problems for switched neutral systems with mixed time-varying delays is discussed in [13]. Under the influence of non-linear disturbances, an asynchronous dynamic output feedback controller is given in [14]. The dynamic properties of switched nonlinear systems with perturbations and delays is addressed in [15]. Literature [16] discusses the stability analysis problem of switched positive T-S fuzzy systems with time-varying delays. Stabilization controller is designed in [17] for two-dimensional discrete-time switched systems. For discrete-time hybrid switched nonlinear systems, the reduced-order model approximation problem is addressed in [18]. For the switched systems with packet loss, literature [19] obtains some conditions ensuring the stability of the closed-loop systems in case of an unstable open loop system. Robust stability and stabilization problem of uncertain switched time-delay systems is studied in [20]. Exponential stabilization for sampled-data Takagi-Sugeno (T-S) fuzzy control systems with packet dropouts is solved by a switched system approach in [21]. Inaccurate data transfer is often caused by data quantization. Thus, quantized control for switched systems has also received increasing attention recently. For Markovian jump systems, the problem of Jingjing Yan and Chenglin Wen are with the College of Electrical Engineering, Henan University of Technology, Zhengzhou 450052, China. Yuanqing Xia is with the Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China. Email: [email protected]; xia [email protected]; [email protected].
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quantized state feedback are studied in [22–25]. For sampled-data systems, switched linear controller is designed by using sampled and quantized measurements in [26, 27]. The quantized stabilization for affine systems is discussed in [28]. Moreover, quantized and switching control methods are adopted by many papers to stabilize planar systems [29], discontinuous nonlinear systems [30], centralized [1] and decentralized [2] networked control systems etc. As we can see, switched systems affected just by time-delay, packet loss or quantization have gotten plentiful results, but the relating research for the systems influenced by these three network-induced factors simultaneously is lacking, which is the issue studied in this paper. In fact, literature [31] has already involved the discrete-time switched systems with time-delay, packet loss and logarithmic quantization simultaneously. There are three differences between our paper and literature [31]. First, the continuoustime switched systems is studied here. Second, the switching delays deduced by communication delays is considered in this paper, but time-delay in [31] is modeled as delay-system. Third, uniform quantizer rather than logarithmic quantizer is used in our paper to quantize the data transmitted. Moreover, switching signals are key factors in a switched system. If the controller can not receive switching signals timely and accurately, it will not be able to follow the plant mode and thus cause the system divergent. Existing literatures always assume that the controller knows exactly that one switch has occurred, but the case that it dose not know exactly where is more challenging as pointed out in [26]. Up to now, few papers address quantized control for switched systems whose switches are detected with bounded delays [32], let alone unbounded delays and packet loss. In [32], the authors study the phenomenon of switching delays which lead to mode mismatches between plant and controller. By restraining the upper bound of delays, the authors set the maximum length of the mismatched time interval. Thus it ensures that the closed-loop system states load in the designed invariant sets. Within this setup, a sufficient condition, which is characterized by maximum delay and dwell time of the switching signal, is given for the existence of a quantizer that guarantees asymptotic stability of the closed-loop system. Nevertheless, if unbounded delays and even packet loss occur, it is obvious that the sufficient condition based on maximum delay is invalid. In this paper, we additionally address unbounded switching delays and packet loss. Due to this, the upper bound of the mismatches between plant and controller can not be restrained. Since the states of the closed-loop system are monotonically increasing in the case of mismatch, it is difficult to design the invariant sets such that the system states are located in which after certain moments. To solve this problem, we first design a constant d¯ to deal with the unbounded delays. If the controller receives switching information and the corresponding delay is less than d,¯ then its mode switches accordingly. Otherwise, the controller ignores switching information received and keeps the current mode. Next, we adjust the quantizer parameters in real time according to the actual situations of switching delays and packet loss, and design invariant sets to guarantee the boundedness of the closed-loop system states and the unsaturation of the quantizer. Based on this, we ensure the mean square stability of the closed-loop system when the packet loss rate is less than a given upper bound. In fact, the key difficulties of this paper lie in how to design quantizer parameter µ based on transmission situations and how to illustrate that the states are always located in the invariant sets under the designed parameters. The main contributions of this paper are third aspects with respect to earlier literature. First, the stabilization problem of the continuous-time switched systems with uniform quantization, communication delay and packet loss is studied here. Second, switching signals considered here are affected by unbounded delays and packet loss. Last, a constant d¯ is designed to certain the controller mode. On the basis of that, quantizer parameter µ is adjusted according to the actual transmission situations which can ensure the unsaturation of the quantizer when delay or packet loss occurs. The remainder of this paper is organized as follows. We describe the problem and main result of this paper in Section II. Section III gives the detailed proof of the mean square stability. Finally, we show the stability results using a well-known benchmark example in Section IV and draw some conclusions in Section V. Notation: Rn denotes the n-dimensional Euclidean space. R+ and N denote the set of positive real numbers and positive integers, respectively. We denote by k · k the standard Euclidean norm in Rn and the
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corresponding induced matrix norm in Rn×n . λmax (P) and λmin (P) represent the maximum and minimum eigenvalue of matrix P, respectively. diag(A1 , . . . , An ) stands for a block-diagonal matrix with the entries A1 , . . . , An on the diagonal and A> ∈ Rm×n means the transposed of matrix A ∈ Rn×m . The signal byc indicates the largest integer not greater than y. II. P ROBLEM FORMULATION
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A. Switched system A class of continuous-time switched linear systems under consideration is described by the following model: x(t) ˙ = Aσ (t) x(t) + Bσ (t) u(t) (1) y(t) = Cσ (t) x(t)
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where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input and y(t) ∈ Rl is the output. For a finite index set P, {(A p , B p ,C p ) : p ∈ P} is a set defining the individual control systems of the switched system. The symbol σ : [0, ∞) → P called as “switching signal” or “switches” is a right-continuous, piecewise constant function. We call the discontinuities of σ as switching times and denote Nσ (t, s) as the number of discontinuities on the interval (s,t]. Assume that the output y(t) of system (1) is transmitted to the controller over the network, then the controller side can only receive the quantized value of which. For technical simplicity, a continuous-time quantizer is considered here similar to [32]. Thus the observer-based controller can be constructed as: ξ˙ (t) = (Aσc (t) + Lσc (t)Cσc (t) )ξ (t) + Bσc (t) u(t) − Lσc (t) qµ(t) (y(t)) (2) u(t) = Kσc (t) ξ (t)
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where ξ (t) ∈ Rn is the observer state, σc : [0, ∞) → P is a right-continuous, piecewise constant function, which is called the switching signal of controller. Kσc (t) ∈ Rn×m and Lσc (t) ∈ Rl×n are feedback gain and observer gain, respectively. qµ(t) (·) is a quantizer defined as in [33]: y(t) qµ (y(t)) = µ(t) q µ(t)
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where µ(t) ∈ R+ is an adjustable parameter. Assume that the following conditions on qµ (·) are satisfied: I. If ky(t)k ≤ Mµ(t), then kqµ (y(t)) − y(t)k ≤ 4µ(t); II. If ky(t)k > Mµ(t), then kqµ (y(t))k > Mµ(t) − 4µ(t). where M is the saturation value and 4 the sensitivity. We consider the situation that the transmission of switching information from plant to controller may be under the influence of time-delay and packet loss, thus the modes of them might be mismatched. Let the switching times of the plant are kτd , k ∈ N with switching period τd > 0. Due to the influence of time-delay and packet loss, the controller either not received switching information or received switching information at kτd + dk , where dk denotes the transmission delay corresponding to the kth switching information. For the time-delay dk , k ∈ N, here we just assume that it is greater than or equal to 0, and without any constrains on its upper bound. In order to avoid the occurrence of disordering, we will design a positive constant d¯ < τd . If the controller receives switching information at kτd + dk and ¯ ∀k ∈ N, then its mode switches according to this information. Otherwise, if dk ≥ d,¯ the controller dk < d, ignores switching information and keeps the current mode, that is, the corresponding information is seen as lost. In fact, if switching information is with timestamp, the controller is always able to determine its own mode under above protocol. In the following, we call the event that packet loss occurs or time-delay is larger than d¯ as packet loss for brevity. Assume that the controller knows the plant mode at the initial moment, we can define σc (t) as: ( σ (0), ∀t ∈ [0, k˜ 1 τd + dk˜ 1 ) σc (t) = σ (k˜ i τd ), ∀t ∈ [k˜ i τd + d ˜ , k˜ i+1 τd + d ˜ ) ki
ki+1
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where k˜ i τd + dk˜ i , i, k˜ i ∈ N denotes the moment when the controller receives switching information for the ith time. It is worth mentioning that if time-delay is larger than or equal to d,¯ then the controller is regarded as receiving no plant mode information, thus dk˜ i < d.¯ Remark 1: Comparing with [32], the main innovations of this paper are removing the constraint on the upper bound of time-delay and considering the packet loss phenomenon simultaneously, which is more realistic and challenging.
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B. Closed-loop system Assume that plant mode σ and controller mode σc satisfy σ (t) = p and σc (t) = r at time t, then the system (1) with the controller (2) can be rewritten as z˙(t) = Fp,r z(t) + Lˆ r (qµ (y(t)) − y(t))
where z :=
x x−ξ
, Fp,r :=
"
(1,1)
(1,2)
Fp,r Fp,r (2,2) (2,1) Fp,r Fp,r
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, Lˆ r :=
with (1,1)
:= A p + B p Kr
(1,2) Fp,r (2,1) Fp,r (2,2) Fp,r
:= −B p Kr
0 Lr
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Fp,r
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:= A p,r + B p,r Kr + LrC p,r
:= Ar − B p,r Kr + LrCr
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and A p,r := A p − Ar , B p,r := B p − Br ,C p,r := C p −Cr . If p = r, then Fp := Fp,p defined in (4) is as: A p + B p Kp −B p K p Fp = 0 A p + L pC p
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Assumption 1: Matrix pairs (C p , A p ) is observable for every p ∈ P. Assumption 2: It exists a positive-definite matrix P such that Fp> P + PFp < 0, ∀p ∈ P. Remark 2: Assumptions above are very standard. Assumption 1 is to ensure the nonsingularity of observability Gramian matrix. Under Assumption 2, we can always define a Lyapunov function V (z) = zT Pz such that V˙ (z) < 0, that is, the closed-loop individual systems of the switched system are stable without time-delay and packet loss.
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C. Main purpose ˆ c, Define the packet loss rate α as α = (aˆ + b)/ ˆ where a, ˆ bˆ and cˆ denote the actual number of packet ¯ lost, the number of delays which are greater than d and the total number of transmissions, respectively. The objective of this paper is to design a positive number d¯ and the quantizer parameter µ(t) ensuring the mean square stability of the system (3) if α is small enough. The main result of this paper is shown as the following theorem, which will be verified in Section III. Theorem 1: If Assumption 1 and Assumption 2 hold, we can always select suitable positive constants > P + PF ≤ β¯ P, ∀p, r ∈ P. Let ω ∈ (0, 1) and β and β¯ such that −Fp P − PFp ≥ β P and Fp,r p,r Θ :=
2 max p∈P kPLˆ p k kPLˆ p k 2 . · max , ω¯ := ω p∈P λmin (Q p ) β¯ Θλmin (P)
Select M large enough satisfying
s λmax (P) ¯ γ¯d/2 M > max 24, e Cmax Θ4 λmin (P)
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with Cmax := max p∈P kC p k. For the designed constant d¯ satisfying ¯ β¯ , if the packet loss rate α satisfies and γ¯ := (1 + ω) ¯
γ γ+γ¯ τd
≥ d¯ ≥ 0, where γ := (1 − ω)β
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1 − e(γ¯d−γ(τd −d)) (7) ¯ ¯ ¯ d − e(γ¯d−γ(τ d −d)) eγτ then we can design suitable quantizer parameter µ such that the closed-loop system (3) is mean square stability, that is, limt→∞ E{kz(t)k2 } = 0. ˆ c, Remark 3: If α defined above is changed as α = b/ ˆ then the transmission of the switching information is only influenced by unbounded time-delay. On the other hand, if α is as α = a/ ˆ c, ˆ then it is the packet loss rate in common sense. Remark 4: The selection of the positive constant d¯ can ensure that the upper bound of α belongs to the interval (0, 1].
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III. P ROOF OF MEAN SQUARE STABILITY To prove theorem 1, we first illustrate a useful lemma, on the basis of which, zoom strategy including zooming-out and zooming-in is adopted to show the establishment of the mean square stability of the closed-loop system (3). ¯ γ, γ¯ and Lemma 1: [32] Assume Assumption 1 and Assumption 2 hold. For the parameters β , β¯ , ω, ω, Θ defined in Theorem 1, if the state of the closed-loop system (3) meets Θ4µ ≤ |z| ≤
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then Lyapunov function V (z) = zT Pz satisfies −γV (z) ˙ V (z) ≤ ¯ (z) γV
Mµ Cmax
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if σ (t) = σc (t) if σ (t) 6= σc (t)
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where σ and σc denote the switching signals of the plant and the controller, respectively. Remark 5: The detailed proof of Lemma 1 can be seen in [32], in which the bounded time-delay on switching information is discussed. Here we discuss the influence of unbounded time-delay and packet loss on the switched systems performance, which may lead to the increases of mode mismatched interval between controller and plant, but it does not affect the increase (decrease) rate of a common Lyapunov function for the mismatched (matched) case.
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A. Stage 1: zooming-out The purpose of this stage is to determine a moment, and the estimator state at this moment such that the state of the closed-loop system (3) is bounded. Similar to [32] and [33], for all t ∈ [kη, (k + 1)η), we set µ(0) = 0 and µ(t) = eνΓkη with Γ := max p∈P kA p k and arbitrary constants ν > 1, η > 0. Consider that the growth rate of µ(t) is larger than 0 that of ky(t)k, it follows from M > 24 that there exists a time t0 ≥ 0 satisfying 0
ky(t)k ≤ Mµ(t) − 24µ(t), ∀t ≥ t0
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Thus kqµ(t) (y(t))k ≤ Mµ(t)−4µ(t) and kqµ(t) (y(t))−y(t)k ≤ 4µ(t) hold for all t ≥ t0 by the conditions of quantizer. To construct the estimator state and the bound of closed-loop system state at a certain moment, we define signals as follows and show them in Fig. 1 for ease of understanding. ˜ d : if t 0 /τd is a positive integer, then kτ ˜ d := t 0 , otherwise, kτ ˜ d denotes a moment at which a switch kτ 0 0 0 occurs for the first time after t0 . 0 s: a moment at which the controller receives switching information for the first time after t0 . It is ˆ d + d ˆ for a fixed integer k, ˆ and thus s ∈ [kτ ˆ d , (kˆ + 1)τd ). Consider obvious that s can be denoted as kτ k
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kWd
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sW s sW0 sW s W
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sˆ The packet is actually lost
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The packet is successfully transmitted but time-delay is larger than d The packet is successfully transmitted and time-delay is less than d Maybe s is at the position of sˆ
Fig. 1.
The illustration of the times defined below.
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˜ Moreover, the controller may receive the packet loss, we know that kˆ is not necessarily equal to k. 0 ˜ d ) due to the effect of time-delay. In this case, kˆ is equal to k˜ − 1, that is, switching information in [t0 , kτ 0 ˜ ˜ ˜ s ∈ [t0 , kτd ) ⊂ [(k − 1)τd , kτd ). ¯ τd ]. τ: a given constant satisfying τ ∈ (d, ¯ τ0 : which is equal to τ − d. 0 0 τ : if a switch occurs during the interval (s + τ0 , s + τ], we define the switching time as s + τ . ¯ if the controller receives switching information in (s + τ0 , s + τ], then the received time is denoted τ: ¯ as s + τ. ˜ which is defined as τ˜ = min{τ, τ}. ¯ τ: Lemma 2: For the variables defined above, it is assumed σc (s) = p and σ ((kˆ +1)τd ) = r. If Assumption 1 and Assumption 2 hold, by defining
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ξ (s) := Wp (τ0 )
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eA p (τ0 −t)C> p qµ(s+t) (y(s + t))dt
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Es := τ0 kWp (τ0 )−1 k · max kC p eA pt k · 4µ(s + τ0 )
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˜ ˜ Es+τ˜ := max max keAr (τ−t) eA pt kEs + max max keAr (τ−t) eA pt − eA p τ˜ k · kξ (s)k
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>
A pt eA p t C> p C p e dt
˜ := eA p τ˜ ξ (s) and Es+τ˜ := kξ (s + τ)k ˜ + 2Es+τ˜ , we get kz(s + τ)k ˜ ≤ Es+τ˜ . and letting ξ (s + τ) Proof: Let us first explain that a switch does not appear during the interval (s, s + τ0 ]. Based on definitions of d¯ and τ, it sees dkˆ + τ < d¯ + τd denoting dkˆ + τ0 < τd . Then we have s + τ0 = ˆ d + d ˆ + τ0 < (kˆ + 1)τd which means that a switch does not appear in (s, s + τ0 ], and thus x(s) can be kτ k illustrated as Z τ0 > −1 x(s) = Wp (τ0 ) eA p (τ0 −t)C> p y(s + t)dt 0
0
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Using (10a) and the fact that the quantization errors are bounded in (s, s + τ0 ] due to s ≥ t0 , we can also derive 0 |x(s) − ξ (s)| ≤ Es Next, we declare that a switch may appear during the interval (s + τ0 , s + τ].
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0
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˜ ) Apτ ˜ = eAr (τ−τ x(s + τ) e x(s)
shows 0
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0 In fact, if τ > τd − d,¯ then it may be (kˆ + 1)τd ∈ (s + τ0 , s + τ], where (kˆ + 1)τd := s + τ is a switching time. Moreover, there must not exist two switching times within the interval (s + τ0 , s + τ] by τ − τ0 = d¯ < τd . The third step describes that the controller may receive the switching information during (s + τ0 , s + τ]. If a switch occurs during (s + τ0 , s + τ] and the controller receives this switching information before ¯ such that z(s + τ) ¯ s + τ, that is, (kˆ + 1)τd + dk+1 := s + τ¯ ≤ s + τ, then we will design the value of ξ (s + τ) ˆ has an upper bound. Otherwise, if the controller receives no switching information during (s + τ0 , s + τ], ξ (s + τ) will be designed. 0 ¯ ξ (s + τ) ˜ := eA p τ˜ ξ (s) and Es+τ˜ := kξ (s + τ)k ˜ + 2Es+τ˜ , we finally prove kz(s + τ)k ˜ ≤ Let τ˜ := min{τ, τ}, Es+τ˜ . ˜ := eA p τ˜ ξ (s), in conjunction with By setting ξ (s + τ)
0
˜ ˜ ) Apτ ) Apτ ˜ − ξ (s + τ)k ˜ ≤ keAr (τ−τ ˜ ≤ Es+τ˜ kx(s + τ) e (x(s) − ξ (s))k + keAr (τ−τ e ξ (s) − ξ (s + τ)k 0
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If there dose not exist a switch in (s + τ0 , s + τ], then the above inequality also holds just replaced Ar by A p . Thus the bound of the closed-loop system state can be derived as ˜ ≤ kξ (s + τ)k ˜ + 2kx(s + τ) ˜ − ξ (s + τ)k ˜ ≤ Es+τ˜ kz(s + τ)k
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This completes the proof. Above Lemma tells us that the state of the closed-loop system (3) is bounded at a fixed time s + τ˜ by designing the estimator state suitably. Remark 6: Compared with [32], the main innovation of above Lemma is the introduction of variable ˜ the advantages of which are twofold. τ, i) The completion time of the zooming-out stage can be advanced. In fact, if a switching signal is received during (s + τ0 , s + τ), then τ˜ < τ, and thus the boundedness of the closed-loop system state is ˜ ≤ Es+τ˜ . Hence the analysis of zooming-out ensured at s + τ˜ which is less than s + τ based on kz(s + τ)k stage is completed. ii) The design method of the controller is more in line with its own switching rule. In literature [32], the observer state at s + τ is set as ξ (s + τ) = eA p τ ξ (s) whether the controller receives switching signal in (s + τ0 , s + τ) or not. However, following the switching rule, controller mode should be updated at time instant s + τ˜ ∈ (s + τ0 , s + τ) if at which the controller receives the switching signal. It means that ˜ A p τ˜ ξ (s + τ) should be defined as ξ (s + τ) = eAr (τ−τ) e ξ (s) for more reasonable. However, by the fact ˜ under which situation that the boundedness of the closed-loop system state can be guaranteed at s + τ, ˜ we the controller need not to update its mode due to no switching signals are received during (s, s + τ), ˜ A τ p ˜ = e ξ (s) meets the switching rule. claim that the selection of ξ (s + τ) B. Stage 2: zooming-in ˜ zoom strategy transfers from zooming-out stage to zooming-in stage. If At the moment T0 := s + τ, parameters µ(T0 ), κ, M and 4 are selected satisfying s λmax (P) Cmax ET0 µ(T0 ) ≥ λmin (P) M and ¯
Ω := eγ¯d/2
s
λmax (P) (1 + κ)Θ4Cmax <1 λmin (P) M
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where the second inequality can be guaranteed by (6), then we can define the level sets R1 (µ) and R2 (µ) by λmin (P)M 2 µ 2 2n R1 (µ) := z ∈ R : V (z) ≤ 2 Cmax 2n 2 2 R2 (µ) := z ∈ R : V (z) ≤ λmax (P)((1 + κ)Θ4) µ ¯
we can get T1 as
T1 =
(kˆ + 1)τd (kˆ + 2)τd
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satisfying z(T0 ) ∈ R1 (µ(T0 )) and R2 (eγ¯d/2 µ(t)) ⊂ R1 (µ(t)), ∀t ∈ R+ . Let T1 be the moment at which a switch occurs for the first time after T0 . Consider ˆ d , (kˆ + 1)τd ) [kτ if no switch occurs during (s + τ0 , s + τ] ˘ ˘ T0 ∈ [kτd , (k + 1)τd ) = ˆ ˆ [(k + 1)τd , (k + 2)τd ) if a switch occurs during (s + τ0 , s + τ] if no switch occurs during (s + τ0 , s + τ] if a switch occurs during (s + τ0 , s + τ]
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For any k ∈ N, let Tk+1 := Tk + τd and define φ0 and φk , respectively, as ˘ d , (k˘ + 1)τd ) 0 if packet loss occurs during [kτ φ0 = 1 otherwise 0 if packet loss occurs during [Tk , Tk+1 ) φk = 1 otherwise
¯ ¯ (1 − φk )eγt/2 + φk eγ¯d/2 µ(Tk ) ¯ ¯ ¯ d /2 (1 − φk )eγτ + φk e(γ¯d−γ(τd −d))/2 µ(Tk )
if t ∈ (0, τd ) if t = τd
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µ(Tk + t) =
(
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it holds Pr(φk = 1) = 1 − α and Pr(φk = 0) = α, ∀k ∈ N ∪ {0}. Lemma 3: If quantizer parameter µ(t) is designed as ¯ + φ0 µ(T0 ) if t ∈ (0, T1 − T0 ) (1 − φ0 )eγt/2 µ(T0 + t) = ¯ d /2 γτ (1 − φ0 )e + φ0 µ(T0 ) if t = T1 − T0
(11a)
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then we have z(t) ∈ R1 (µ(t)), ∀t ≥ T0 . Proof: The lemma is proved by inductive method. Step 1: z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. ˆ d , (kˆ + 1)τd ), it is obvious that φ0 ≡ 1 and s is the moment when the switching signal is If T0 ∈ [kτ ˆ d , (kˆ + 1)τd ). From T0 = s + τ˜ > s, we know that the modes of the plant and the received during [kτ controller must be matched in [T0 , (kˆ + 1)τd ). Thus Lyapunov function is monotone decreasing in the looped region R1 (µ(T0 ))\R2 (µ(T0 )). Then z(t) ∈ R1 (µ(T0 )), ∀t ∈ [T0 , T1 ] can be guaranteed by z(T0 ) ∈ R1 (µ(T0 )). Taking the definition of µ(t) and (11a) into consideration, we obtain z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If T0 ∈ [(kˆ + 1)τd , (kˆ + 2)τd ) and a switch is received in this interval, that is, φ0 = 1, then T0 is a moment when the information is received. Similar analysis results in z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If there is no switching information received by the controller, then the length of the total mismatched interval is T1 − T0 ≤ τd in [T0 , T1 ]. In the following, two different cases are discussed. Case 1: By assuming z(T0 ) ∈ R1 (µ(T0 ))\R2 (µ(T0 )), the increasing rate of the bound of R1 (µ(t)), ∀t ∈ (T0 , T1 ], is γ¯ based on (11a). Due to the increasing rate of V (z(t)) is less than γ¯ in R1 (µ(t))\R2 (µ(t)) based on Lemma 1, there may be a time t˜1 ∈ [T0 , T1 ] satisfying z(t) ∈ R1 (µ(t))\R2 (µ(t)), ∀t ∈ [T0 , t˜1 ), z(t˜1 ) ∈ R2 (µ(t˜1 ))
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Obviously, if t˜1 does not exist, then the increasing rate of V (z(t)) is less than γ¯ throughout the interval (T0 , T1 ], and thus z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If z(t) ∈ R2 (µ(t)) ⊂ R1 (µ(t)),t ∈ [t˜1 , T2 ), then z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. Otherwise, if there exists a time t˜2 such that z(t) ∈ R2 (µ(t)),t ∈ [t˜1 , t˜2 ), z(t˜2 ) ∈ R1 (µ(t˜2 ))\R2 (µ(t˜2 ))
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it guarantees z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ] by considering the fact that the increasing rate of V (z(t)) in ¯ R1 (µ(t))\R2 (µ(t)) is less than γ¯ but the one of the bound of R1 (µ(t)) is γ. Case 2: When z(T0 ) ∈ R2 (µ(T0 )) and z(t) ∈ R2 (µ(t)), ∀t ∈ (T0 , T1 ], it holds z(t) ∈ R1 (µ(t)), ∀t ∈ (T0 , T1 ] based on R2 (µ(t)) ⊂ R1 (µ(t)). If there is a time t˜3 such that z(t) ∈ R2 (µ(t)),t ∈ [T0 , t˜3 ), z(t˜3 ) ∈ / R2 (µ(t˜3 ))
the fact that the increasing rate of V (z(t)) in R1 (µ(t))\R2 (µ(t)) is less than γ¯ can ensure z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. Step 2: Assume that z(Tk ) ∈ R1 (µ(Tk )), then z(Tk +t) ∈ R1 (µ(Tk +t)), ∀t ∈ (0, τd ] must be established. Case 1: If z(Tk ) ∈ R1 (µ(Tk )) and φk = 1, then the upper bound of the mismatched time range is d¯ and the lower bound of the matched time range is τd − d¯ in the interval [Tk , Tk+1 ). If there exists a time t˜4 ∈ [Tk , Tk+1 ) such that ¯
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¯ ¯ ¯ z(Tk+1 ) ∈ R1 (e(γ¯d−γ(τd −d))/2 µ(Tk )), z(t) ∈ R1 (eγ¯d/2 µ(Tk )), ∀t ∈ [t˜4 , Tk+1 ] Hence (11b) gives z(t) ∈ R1 (µ(t)), ∀t ∈ [t˜4 , Tk+1 ]. Combining with z(t) ∈ R1 (µ(t)),t ∈ [Tk , t˜4 ), tells us z(t) ∈ R1 (µ(t)),t ∈ [Tk , Tk+1 ]. Case 2: If z(Tk ) ∈ R1 (µ(Tk )) and φk = 0, then the length of the total mismatched interval is τd in [Tk , Tk+1 ). Consider that µ(Tk + t),t ∈ (0, τd ] and µ(t),t ∈ [T0 , T1 ] have the same definitions, similar analysis of Case 1 and Case 2 in Step 1 results in z(t) ∈ R1 (µ(t)),t ∈ [Tk , Tk+1 ]. Based on above analysis and the inductive method, it holds z(t) ∈ R1 (µ(t)), ∀t ≥ T0 . The proof of Lemma 3 is completed. From Lemma 3, we see that Lyapunov function V (z(t)), ∀t ≥ T0 satisfies λmin (P)M 2 µ 2 (t) V (z(t)) = z> (t)Pz(t) ≤ 2 Cmax 2
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IV. S IMULATION In this section, a well-known benchmark example is adopted to illustrate the effectiveness of the main results. The linearized batch reactor is given by x(t) ˙ = Ax(t) + Bu(t) (12) y(t) = Cx(t)
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Step 6: If quantizer parameter µ(t) is designed by (11a) and (11b), we can ensure E{|z(t)|2 } ≤ and the mean square stability of the closed-loop system (12), which is shown in Fig. 3. It is worth mentioning that the curve shown in Fig. 3 is not fixed due to the impact of packet loss. Comparison: If the switching information is only influenced by bounded time-delay rather than unbounded time-delay and packet loss, the state of the closed-loop system (12) and the upper bound of which are shown in Fig. 4. It means that the control strategy proposed in this paper is still useful to deal with the stability problem discussed in literature [32]. However, if the parameters are selected as M = 100, 4 = 0.001 and x0 = [10, 0, 10, 20]> same as in [32], the state trajectories of the closed-loop system (12) is shown as Fig. 5. Compared with Fig. 2 in [32], we know that the convergence rate under our control strategy is slower. Summarized above, it sees that our algorithm has a wider range of applicability but lower convergence speed compared with [32]. Discussion: From step 4, we see that d¯ should be selected as 0.001 to meet γ+γ γ¯ τd ≥ d.¯ It means that switching signals will be neglected if the transmission delay of which is larger than 0.001. Although this approach can ensure the system stability, it will inevitably reduce system performance such as convergence speed. The small value of the upper bound of d¯ is the main conservativeness of Theorem 1.
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V. C ONCLUSION When switching signals are affected by unbounded delays and packet loss, we propose design methods of a constant d¯ and the quantizer parameter µ, under which sufficient conditions ensuring the mean square stability of the closed-loop switched system are obtained if the packet loss rate is less than a given upper bound. As pointed in discussion, how to increase the value of d¯ to improve the system performance is one of our future research directions. Moreover, this paper is only relating to fixed-length switching period, future work will also involve designing d¯ and a quantizer for arbitrary switchings case. VI. ACKNOWLEDGEMENTS The work was supported by the National Natural Science Foundation of China (61773154, 61403125, 61720106010, 61333005). Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2015RCJH15), Natural Science Foundation of Henan Province Education Department (15A413012).
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R EFERENCES [1] M. C. F. Donkers, W. P. M. H. Heemels, N. van de Wouw, and L. Hetel, Stability analysis of networked control systems using a switched linear systems approach, IEEE Transactions on Automatic Control 56 (9) (2011) 2101-2115. [2] N. W. Bauer, M. C. F. Donkers, N. van de Wouw, and W. P. M. H. Heemels, Decentralized observerbased control via networked communication, Automatica, 49 (2013) 2074-2086. [3] X. Su, P. Shi, L. Wu and Y.-D. Song, Fault detection filtering for nonlinear switched stochastic systems, IEEE Transactions on Automatic Control, 61 (5) (2016) 1310-1315. [4] Y. Yuan, H. Yuan, Z. Wang, L. Guo, and H. Yang, Optimal control for networked control systems with disturbances: a delta operator approach, IET Control Theory & Applications, 11 (9) (2017) 1325-1332. [5] X. Su, L. Wu, P. Shi, and C. L. P. Chen. Model approximation for fuzzy switched systems with stochastic disturbance, IEEE Transactions on Fuzzy Systems, 23 (5) (2015) 1458-1473. [6] X. Su, X. Liu, P. Shi, and R. Yang, Sliding mode control of discrete-time switched systems with repeated scalar nonlinearity, IEEE Transactions on Automatic Control, 62 (9) (2017) 4604-4610. [7] T. Wang, H. Gao, and J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control, IEEE Transactions on Neural Networks and Learning Systems 27 (2) (2016) 416-425. [8] L. Qiu, Y. Shi, F. Yao, G. Xu, and B. Xu, Network-based robust H2 /H∞ control for linear systems with two-channel random packet dropouts and time delays, IEEE Transactions on Cybernetics 45 (8) (2015) 1450-1462. [9] Q. Zhu, K. Lu, Y. Zhu and G. Xie, Modeling and state feedback control of multi-rate networked control systems with both short time delay and packet dropout, International Journal of Innovative Computing, Information and Control 12 (3) (2016) 779-793. [10] X. Su, P. Shi, L. Wu, and Y.-D. Song, A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays, IEEE Transactions on Fuzzy Systems 21 (4) (2013) 655-671, 2013. [11] Y.-E Wang, X.-M. Sun, and F. Mazenc, Stability of switched nonlinear systems with delay and disturbance, Automatica 69 (2016) 78-86. [12] Y.-E. Wang, B. Wu, and C. Wu, Stability and L2 -gain analysis of switched inputdelay systems with unstable modes under asynchronous switching, Journal of the Franklin Institute 354 (11) (2017) 4481-4497. [13] Y. Dong, W. Liu, T. Li, and S. Liang, Finite-time boundedness analysis and H∞ control for switched neutral systems with mixed time-varying delays, Journal of the Franklin Institute 354 (2) (2017) 787811. [14] X. Wang, G. Zong, and H. Sun, Asynchronous finite-time dynamic output feedback control for switched time-delay systems with non-linear disturbances, IET Control Theory and Applications 10 (10) (2016) 1142-1150. [15] X. Liu, S. Zhong, and Q. Zhao, Dynamics of delayed switched nonlinear systems with applications to cascade systems, Automatica 87 (2018) 251-257. [16] S. Du, J. Qiao, Stability analysis and L1 -gain controller synthesis of switched positive T-S fuzzy systems with time-varying delays, Neurocomputing, (2017) doi.org/10.1016/j.neucom.2017.11.026. [17] L. Wu, R. Yang, P. Shi, X. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica 59 (2015) 206-215. [18] X. Su, X. Liu, Y.-D. Song, H. K. Lam, and L. Wang, Reduced-order model approximation of fuzzy switched systems with pre-specified performance, Information Sciences 370-371 (2016) 538-550. [19] J. Nygrenand and K. Pelckmans, A closed loop stability condition of switched systems applied to NCSs with packet loss, IFAC-Papers On Line 48 (22) (2015) 8-13. [20] L. Zhang, P. Shi, and M. Basin, Robust stability and stabilisation of uncertain switched linear discrete time-delay systems, IET Control Theory & Applications 2 (7) (2008) 606-614. [21] M. Wang, J. Qiu, M. Chadli, and M. Wang, A switched system approach to exponential stabilization
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Quantized stabilization of switched systems with switching delays and packet loss Jingjing Yan, Yuanqing Xia, Chenglin Wen PII: DOI: Reference:
S0016-0032(18)30321-1 10.1016/j.jfranklin.2018.05.009 FI 3443
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
5 May 2017 5 March 2018 4 May 2018
Please cite this article as: Jingjing Yan, Yuanqing Xia, Chenglin Wen, Quantized stabilization of switched systems with switching delays and packet loss, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.05.009
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Quantized stabilization of switched systems with switching delays and packet loss Jingjing Yan, Yuanqing Xia, and Chenglin Wen
Abstract
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This paper is concerned with the problem of designing an observer-based quantized feedback controller for the continuous-time switched linear systems, in which the transmission of switching signal is subject to unbounded delays and packet loss. To deal with the unbounded switching delays, we design a constant d¯ to determine that the switching signal received by controller is ignored or not. Based on that, if the signal is timestamped, the controller’s mode is uniquely determined. Moreover, we adjust the quantizer parameters in real time depending on the actual transmission situations to ensure the unsaturation of quantizer and thus the boundness of quantization error. Within this setup, we derive a maximum allowable packet loss rate ensuring the mean square stability of the closed-loop switched systems. An illustrative example is given to show the usefulness of the proposed framework for the quantized stabilization of some classes of switched systems. Index Terms
Switched systems, quantization, unbounded delays, packet loss.
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I. I NTRODUCTION As an important class of hybrid dynamical systems, switched systems have received a lot of attention in recent years [1–6], since they have been successfully applied to model different practical systems, such as highly complex systems, chemical process, power electronic systems, manipulator robots, networked control systems, flight control systems, servomechanism systems and so on. With the advancement of network process, data in switched systems is transmitted over the network frequently. While bringing convenience, it should have a significant impact on the system performance, especially for not timely and imprecise data transmission. Not timely data transfer is deduced by network-induced time-delay and packet loss [7–10]. So, among the control and communication communities, many researchers make a great effort to study the stability and controller design for switched systems affected by time-delay [11–18] or packet loss [19–21]. The global exponential stability of the nonlinear time-varying switched control systems is obtained in [11]. The stability and L2 -gain analysis problem for a class of switched linear systems is studied in [12]. Observer-based finite-time stabilization and H∞ control problems for switched neutral systems with mixed time-varying delays is discussed in [13]. Under the influence of non-linear disturbances, an asynchronous dynamic output feedback controller is given in [14]. The dynamic properties of switched nonlinear systems with perturbations and delays is addressed in [15]. Literature [16] discusses the stability analysis problem of switched positive T-S fuzzy systems with time-varying delays. Stabilization controller is designed in [17] for two-dimensional discrete-time switched systems. For discrete-time hybrid switched nonlinear systems, the reduced-order model approximation problem is addressed in [18]. For the switched systems with packet loss, literature [19] obtains some conditions ensuring the stability of the closed-loop systems in case of an unstable open loop system. Robust stability and stabilization problem of uncertain switched time-delay systems is studied in [20]. Exponential stabilization for sampled-data Takagi-Sugeno (T-S) fuzzy control systems with packet dropouts is solved by a switched system approach in [21]. Inaccurate data transfer is often caused by data quantization. Thus, quantized control for switched systems has also received increasing attention recently. For Markovian jump systems, the problem of Jingjing Yan and Chenglin Wen are with the College of Electrical Engineering, Henan University of Technology, Zhengzhou 450052, China. Yuanqing Xia is with the Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China. Email: [email protected]; xia [email protected]; [email protected].
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quantized state feedback are studied in [22–25]. For sampled-data systems, switched linear controller is designed by using sampled and quantized measurements in [26, 27]. The quantized stabilization for affine systems is discussed in [28]. Moreover, quantized and switching control methods are adopted by many papers to stabilize planar systems [29], discontinuous nonlinear systems [30], centralized [1] and decentralized [2] networked control systems etc. As we can see, switched systems affected just by time-delay, packet loss or quantization have gotten plentiful results, but the relating research for the systems influenced by these three network-induced factors simultaneously is lacking, which is the issue studied in this paper. In fact, literature [31] has already involved the discrete-time switched systems with time-delay, packet loss and logarithmic quantization simultaneously. There are three differences between our paper and literature [31]. First, the continuoustime switched systems is studied here. Second, the switching delays deduced by communication delays is considered in this paper, but time-delay in [31] is modeled as delay-system. Third, uniform quantizer rather than logarithmic quantizer is used in our paper to quantize the data transmitted. Moreover, switching signals are key factors in a switched system. If the controller can not receive switching signals timely and accurately, it will not be able to follow the plant mode and thus cause the system divergent. Existing literatures always assume that the controller knows exactly that one switch has occurred, but the case that it dose not know exactly where is more challenging as pointed out in [26]. Up to now, few papers address quantized control for switched systems whose switches are detected with bounded delays [32], let alone unbounded delays and packet loss. In [32], the authors study the phenomenon of switching delays which lead to mode mismatches between plant and controller. By restraining the upper bound of delays, the authors set the maximum length of the mismatched time interval. Thus it ensures that the closed-loop system states load in the designed invariant sets. Within this setup, a sufficient condition, which is characterized by maximum delay and dwell time of the switching signal, is given for the existence of a quantizer that guarantees asymptotic stability of the closed-loop system. Nevertheless, if unbounded delays and even packet loss occur, it is obvious that the sufficient condition based on maximum delay is invalid. In this paper, we additionally address unbounded switching delays and packet loss. Due to this, the upper bound of the mismatches between plant and controller can not be restrained. Since the states of the closed-loop system are monotonically increasing in the case of mismatch, it is difficult to design the invariant sets such that the system states are located in which after certain moments. To solve this problem, we first design a constant d¯ to deal with the unbounded delays. If the controller receives switching information and the corresponding delay is less than d,¯ then its mode switches accordingly. Otherwise, the controller ignores switching information received and keeps the current mode. Next, we adjust the quantizer parameters in real time according to the actual situations of switching delays and packet loss, and design invariant sets to guarantee the boundedness of the closed-loop system states and the unsaturation of the quantizer. Based on this, we ensure the mean square stability of the closed-loop system when the packet loss rate is less than a given upper bound. In fact, the key difficulties of this paper lie in how to design quantizer parameter µ based on transmission situations and how to illustrate that the states are always located in the invariant sets under the designed parameters. The main contributions of this paper are third aspects with respect to earlier literature. First, the stabilization problem of the continuous-time switched systems with uniform quantization, communication delay and packet loss is studied here. Second, switching signals considered here are affected by unbounded delays and packet loss. Last, a constant d¯ is designed to certain the controller mode. On the basis of that, quantizer parameter µ is adjusted according to the actual transmission situations which can ensure the unsaturation of the quantizer when delay or packet loss occurs. The remainder of this paper is organized as follows. We describe the problem and main result of this paper in Section II. Section III gives the detailed proof of the mean square stability. Finally, we show the stability results using a well-known benchmark example in Section IV and draw some conclusions in Section V. Notation: Rn denotes the n-dimensional Euclidean space. R+ and N denote the set of positive real numbers and positive integers, respectively. We denote by k · k the standard Euclidean norm in Rn and the
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corresponding induced matrix norm in Rn×n . λmax (P) and λmin (P) represent the maximum and minimum eigenvalue of matrix P, respectively. diag(A1 , . . . , An ) stands for a block-diagonal matrix with the entries A1 , . . . , An on the diagonal and A> ∈ Rm×n means the transposed of matrix A ∈ Rn×m . The signal byc indicates the largest integer not greater than y. II. P ROBLEM FORMULATION
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A. Switched system A class of continuous-time switched linear systems under consideration is described by the following model: x(t) ˙ = Aσ (t) x(t) + Bσ (t) u(t) (1) y(t) = Cσ (t) x(t)
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where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input and y(t) ∈ Rl is the output. For a finite index set P, {(A p , B p ,C p ) : p ∈ P} is a set defining the individual control systems of the switched system. The symbol σ : [0, ∞) → P called as “switching signal” or “switches” is a right-continuous, piecewise constant function. We call the discontinuities of σ as switching times and denote Nσ (t, s) as the number of discontinuities on the interval (s,t]. Assume that the output y(t) of system (1) is transmitted to the controller over the network, then the controller side can only receive the quantized value of which. For technical simplicity, a continuous-time quantizer is considered here similar to [32]. Thus the observer-based controller can be constructed as: ξ˙ (t) = (Aσc (t) + Lσc (t)Cσc (t) )ξ (t) + Bσc (t) u(t) − Lσc (t) qµ(t) (y(t)) (2) u(t) = Kσc (t) ξ (t)
ED
M
where ξ (t) ∈ Rn is the observer state, σc : [0, ∞) → P is a right-continuous, piecewise constant function, which is called the switching signal of controller. Kσc (t) ∈ Rn×m and Lσc (t) ∈ Rl×n are feedback gain and observer gain, respectively. qµ(t) (·) is a quantizer defined as in [33]: y(t) qµ (y(t)) = µ(t) q µ(t)
AC
CE
PT
where µ(t) ∈ R+ is an adjustable parameter. Assume that the following conditions on qµ (·) are satisfied: I. If ky(t)k ≤ Mµ(t), then kqµ (y(t)) − y(t)k ≤ 4µ(t); II. If ky(t)k > Mµ(t), then kqµ (y(t))k > Mµ(t) − 4µ(t). where M is the saturation value and 4 the sensitivity. We consider the situation that the transmission of switching information from plant to controller may be under the influence of time-delay and packet loss, thus the modes of them might be mismatched. Let the switching times of the plant are kτd , k ∈ N with switching period τd > 0. Due to the influence of time-delay and packet loss, the controller either not received switching information or received switching information at kτd + dk , where dk denotes the transmission delay corresponding to the kth switching information. For the time-delay dk , k ∈ N, here we just assume that it is greater than or equal to 0, and without any constrains on its upper bound. In order to avoid the occurrence of disordering, we will design a positive constant d¯ < τd . If the controller receives switching information at kτd + dk and ¯ ∀k ∈ N, then its mode switches according to this information. Otherwise, if dk ≥ d,¯ the controller dk < d, ignores switching information and keeps the current mode, that is, the corresponding information is seen as lost. In fact, if switching information is with timestamp, the controller is always able to determine its own mode under above protocol. In the following, we call the event that packet loss occurs or time-delay is larger than d¯ as packet loss for brevity. Assume that the controller knows the plant mode at the initial moment, we can define σc (t) as: ( σ (0), ∀t ∈ [0, k˜ 1 τd + dk˜ 1 ) σc (t) = σ (k˜ i τd ), ∀t ∈ [k˜ i τd + d ˜ , k˜ i+1 τd + d ˜ ) ki
ki+1
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where k˜ i τd + dk˜ i , i, k˜ i ∈ N denotes the moment when the controller receives switching information for the ith time. It is worth mentioning that if time-delay is larger than or equal to d,¯ then the controller is regarded as receiving no plant mode information, thus dk˜ i < d.¯ Remark 1: Comparing with [32], the main innovations of this paper are removing the constraint on the upper bound of time-delay and considering the packet loss phenomenon simultaneously, which is more realistic and challenging.
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B. Closed-loop system Assume that plant mode σ and controller mode σc satisfy σ (t) = p and σc (t) = r at time t, then the system (1) with the controller (2) can be rewritten as z˙(t) = Fp,r z(t) + Lˆ r (qµ (y(t)) − y(t))
where z :=
x x−ξ
, Fp,r :=
"
(1,1)
(1,2)
Fp,r Fp,r (2,2) (2,1) Fp,r Fp,r
#
, Lˆ r :=
with (1,1)
:= A p + B p Kr
(1,2) Fp,r (2,1) Fp,r (2,2) Fp,r
:= −B p Kr
0 Lr
(4)
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Fp,r
(3)
:= A p,r + B p,r Kr + LrC p,r
:= Ar − B p,r Kr + LrCr
(5)
ED
M
and A p,r := A p − Ar , B p,r := B p − Br ,C p,r := C p −Cr . If p = r, then Fp := Fp,p defined in (4) is as: A p + B p Kp −B p K p Fp = 0 A p + L pC p
CE
PT
Assumption 1: Matrix pairs (C p , A p ) is observable for every p ∈ P. Assumption 2: It exists a positive-definite matrix P such that Fp> P + PFp < 0, ∀p ∈ P. Remark 2: Assumptions above are very standard. Assumption 1 is to ensure the nonsingularity of observability Gramian matrix. Under Assumption 2, we can always define a Lyapunov function V (z) = zT Pz such that V˙ (z) < 0, that is, the closed-loop individual systems of the switched system are stable without time-delay and packet loss.
AC
C. Main purpose ˆ c, Define the packet loss rate α as α = (aˆ + b)/ ˆ where a, ˆ bˆ and cˆ denote the actual number of packet ¯ lost, the number of delays which are greater than d and the total number of transmissions, respectively. The objective of this paper is to design a positive number d¯ and the quantizer parameter µ(t) ensuring the mean square stability of the system (3) if α is small enough. The main result of this paper is shown as the following theorem, which will be verified in Section III. Theorem 1: If Assumption 1 and Assumption 2 hold, we can always select suitable positive constants > P + PF ≤ β¯ P, ∀p, r ∈ P. Let ω ∈ (0, 1) and β and β¯ such that −Fp P − PFp ≥ β P and Fp,r p,r Θ :=
2 max p∈P kPLˆ p k kPLˆ p k 2 . · max , ω¯ := ω p∈P λmin (Q p ) β¯ Θλmin (P)
Select M large enough satisfying
s λmax (P) ¯ γ¯d/2 M > max 24, e Cmax Θ4 λmin (P)
(6)
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with Cmax := max p∈P kC p k. For the designed constant d¯ satisfying ¯ β¯ , if the packet loss rate α satisfies and γ¯ := (1 + ω) ¯
γ γ+γ¯ τd
≥ d¯ ≥ 0, where γ := (1 − ω)β
¯
1 − e(γ¯d−γ(τd −d)) (7) ¯ ¯ ¯ d − e(γ¯d−γ(τ d −d)) eγτ then we can design suitable quantizer parameter µ such that the closed-loop system (3) is mean square stability, that is, limt→∞ E{kz(t)k2 } = 0. ˆ c, Remark 3: If α defined above is changed as α = b/ ˆ then the transmission of the switching information is only influenced by unbounded time-delay. On the other hand, if α is as α = a/ ˆ c, ˆ then it is the packet loss rate in common sense. Remark 4: The selection of the positive constant d¯ can ensure that the upper bound of α belongs to the interval (0, 1].
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α<
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III. P ROOF OF MEAN SQUARE STABILITY To prove theorem 1, we first illustrate a useful lemma, on the basis of which, zoom strategy including zooming-out and zooming-in is adopted to show the establishment of the mean square stability of the closed-loop system (3). ¯ γ, γ¯ and Lemma 1: [32] Assume Assumption 1 and Assumption 2 hold. For the parameters β , β¯ , ω, ω, Θ defined in Theorem 1, if the state of the closed-loop system (3) meets Θ4µ ≤ |z| ≤
M
then Lyapunov function V (z) = zT Pz satisfies −γV (z) ˙ V (z) ≤ ¯ (z) γV
Mµ Cmax
(8)
if σ (t) = σc (t) if σ (t) 6= σc (t)
(9)
PT
ED
where σ and σc denote the switching signals of the plant and the controller, respectively. Remark 5: The detailed proof of Lemma 1 can be seen in [32], in which the bounded time-delay on switching information is discussed. Here we discuss the influence of unbounded time-delay and packet loss on the switched systems performance, which may lead to the increases of mode mismatched interval between controller and plant, but it does not affect the increase (decrease) rate of a common Lyapunov function for the mismatched (matched) case.
AC
CE
A. Stage 1: zooming-out The purpose of this stage is to determine a moment, and the estimator state at this moment such that the state of the closed-loop system (3) is bounded. Similar to [32] and [33], for all t ∈ [kη, (k + 1)η), we set µ(0) = 0 and µ(t) = eνΓkη with Γ := max p∈P kA p k and arbitrary constants ν > 1, η > 0. Consider that the growth rate of µ(t) is larger than 0 that of ky(t)k, it follows from M > 24 that there exists a time t0 ≥ 0 satisfying 0
ky(t)k ≤ Mµ(t) − 24µ(t), ∀t ≥ t0
0
Thus kqµ(t) (y(t))k ≤ Mµ(t)−4µ(t) and kqµ(t) (y(t))−y(t)k ≤ 4µ(t) hold for all t ≥ t0 by the conditions of quantizer. To construct the estimator state and the bound of closed-loop system state at a certain moment, we define signals as follows and show them in Fig. 1 for ease of understanding. ˜ d : if t 0 /τd is a positive integer, then kτ ˜ d := t 0 , otherwise, kτ ˜ d denotes a moment at which a switch kτ 0 0 0 occurs for the first time after t0 . 0 s: a moment at which the controller receives switching information for the first time after t0 . It is ˆ d + d ˆ for a fixed integer k, ˆ and thus s ∈ [kτ ˆ d , (kˆ + 1)τd ). Consider obvious that s can be denoted as kτ k
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kWd
t0'
Plant
kˆWd
(kˆ 1)Wd
sW s sW0 sW s W
Controller
'
sˆ The packet is actually lost
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The packet is successfully transmitted but time-delay is larger than d The packet is successfully transmitted and time-delay is less than d Maybe s is at the position of sˆ
Fig. 1.
The illustration of the times defined below.
M
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˜ Moreover, the controller may receive the packet loss, we know that kˆ is not necessarily equal to k. 0 ˜ d ) due to the effect of time-delay. In this case, kˆ is equal to k˜ − 1, that is, switching information in [t0 , kτ 0 ˜ ˜ ˜ s ∈ [t0 , kτd ) ⊂ [(k − 1)τd , kτd ). ¯ τd ]. τ: a given constant satisfying τ ∈ (d, ¯ τ0 : which is equal to τ − d. 0 0 τ : if a switch occurs during the interval (s + τ0 , s + τ], we define the switching time as s + τ . ¯ if the controller receives switching information in (s + τ0 , s + τ], then the received time is denoted τ: ¯ as s + τ. ˜ which is defined as τ˜ = min{τ, τ}. ¯ τ: Lemma 2: For the variables defined above, it is assumed σc (s) = p and σ ((kˆ +1)τd ) = r. If Assumption 1 and Assumption 2 hold, by defining
0
Z τ0
ED
ξ (s) := Wp (τ0 )
−1
0
>
eA p (τ0 −t)C> p qµ(s+t) (y(s + t))dt
(10a)
Es := τ0 kWp (τ0 )−1 k · max kC p eA pt k · 4µ(s + τ0 )
(10b)
˜ ˜ Es+τ˜ := max max keAr (τ−t) eA pt kEs + max max keAr (τ−t) eA pt − eA p τ˜ k · kξ (s)k
(10c)
PT
0
0≤t≤τ0
0
CE
r∈P t∈[τ0 ,τ] ˜
r∈P t∈[τ0 ,τ] ˜
AC
with observability Gramian Wp (τ0 ) shown as Wp (τ0 ) :=
Z τ0 0
>
A pt eA p t C> p C p e dt
˜ := eA p τ˜ ξ (s) and Es+τ˜ := kξ (s + τ)k ˜ + 2Es+τ˜ , we get kz(s + τ)k ˜ ≤ Es+τ˜ . and letting ξ (s + τ) Proof: Let us first explain that a switch does not appear during the interval (s, s + τ0 ]. Based on definitions of d¯ and τ, it sees dkˆ + τ < d¯ + τd denoting dkˆ + τ0 < τd . Then we have s + τ0 = ˆ d + d ˆ + τ0 < (kˆ + 1)τd which means that a switch does not appear in (s, s + τ0 ], and thus x(s) can be kτ k illustrated as Z τ0 > −1 x(s) = Wp (τ0 ) eA p (τ0 −t)C> p y(s + t)dt 0
0
0
Using (10a) and the fact that the quantization errors are bounded in (s, s + τ0 ] due to s ≥ t0 , we can also derive 0 |x(s) − ξ (s)| ≤ Es Next, we declare that a switch may appear during the interval (s + τ0 , s + τ].
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0
0
˜ ) Apτ ˜ = eAr (τ−τ x(s + τ) e x(s)
shows 0
0
0
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0 In fact, if τ > τd − d,¯ then it may be (kˆ + 1)τd ∈ (s + τ0 , s + τ], where (kˆ + 1)τd := s + τ is a switching time. Moreover, there must not exist two switching times within the interval (s + τ0 , s + τ] by τ − τ0 = d¯ < τd . The third step describes that the controller may receive the switching information during (s + τ0 , s + τ]. If a switch occurs during (s + τ0 , s + τ] and the controller receives this switching information before ¯ such that z(s + τ) ¯ s + τ, that is, (kˆ + 1)τd + dk+1 := s + τ¯ ≤ s + τ, then we will design the value of ξ (s + τ) ˆ has an upper bound. Otherwise, if the controller receives no switching information during (s + τ0 , s + τ], ξ (s + τ) will be designed. 0 ¯ ξ (s + τ) ˜ := eA p τ˜ ξ (s) and Es+τ˜ := kξ (s + τ)k ˜ + 2Es+τ˜ , we finally prove kz(s + τ)k ˜ ≤ Let τ˜ := min{τ, τ}, Es+τ˜ . ˜ := eA p τ˜ ξ (s), in conjunction with By setting ξ (s + τ)
0
˜ ˜ ) Apτ ) Apτ ˜ − ξ (s + τ)k ˜ ≤ keAr (τ−τ ˜ ≤ Es+τ˜ kx(s + τ) e (x(s) − ξ (s))k + keAr (τ−τ e ξ (s) − ξ (s + τ)k 0
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If there dose not exist a switch in (s + τ0 , s + τ], then the above inequality also holds just replaced Ar by A p . Thus the bound of the closed-loop system state can be derived as ˜ ≤ kξ (s + τ)k ˜ + 2kx(s + τ) ˜ − ξ (s + τ)k ˜ ≤ Es+τ˜ kz(s + τ)k
AC
CE
PT
ED
M
This completes the proof. Above Lemma tells us that the state of the closed-loop system (3) is bounded at a fixed time s + τ˜ by designing the estimator state suitably. Remark 6: Compared with [32], the main innovation of above Lemma is the introduction of variable ˜ the advantages of which are twofold. τ, i) The completion time of the zooming-out stage can be advanced. In fact, if a switching signal is received during (s + τ0 , s + τ), then τ˜ < τ, and thus the boundedness of the closed-loop system state is ˜ ≤ Es+τ˜ . Hence the analysis of zooming-out ensured at s + τ˜ which is less than s + τ based on kz(s + τ)k stage is completed. ii) The design method of the controller is more in line with its own switching rule. In literature [32], the observer state at s + τ is set as ξ (s + τ) = eA p τ ξ (s) whether the controller receives switching signal in (s + τ0 , s + τ) or not. However, following the switching rule, controller mode should be updated at time instant s + τ˜ ∈ (s + τ0 , s + τ) if at which the controller receives the switching signal. It means that ˜ A p τ˜ ξ (s + τ) should be defined as ξ (s + τ) = eAr (τ−τ) e ξ (s) for more reasonable. However, by the fact ˜ under which situation that the boundedness of the closed-loop system state can be guaranteed at s + τ, ˜ we the controller need not to update its mode due to no switching signals are received during (s, s + τ), ˜ A τ p ˜ = e ξ (s) meets the switching rule. claim that the selection of ξ (s + τ) B. Stage 2: zooming-in ˜ zoom strategy transfers from zooming-out stage to zooming-in stage. If At the moment T0 := s + τ, parameters µ(T0 ), κ, M and 4 are selected satisfying s λmax (P) Cmax ET0 µ(T0 ) ≥ λmin (P) M and ¯
Ω := eγ¯d/2
s
λmax (P) (1 + κ)Θ4Cmax <1 λmin (P) M
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where the second inequality can be guaranteed by (6), then we can define the level sets R1 (µ) and R2 (µ) by λmin (P)M 2 µ 2 2n R1 (µ) := z ∈ R : V (z) ≤ 2 Cmax 2n 2 2 R2 (µ) := z ∈ R : V (z) ≤ λmax (P)((1 + κ)Θ4) µ ¯
we can get T1 as
T1 =
(kˆ + 1)τd (kˆ + 2)τd
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satisfying z(T0 ) ∈ R1 (µ(T0 )) and R2 (eγ¯d/2 µ(t)) ⊂ R1 (µ(t)), ∀t ∈ R+ . Let T1 be the moment at which a switch occurs for the first time after T0 . Consider ˆ d , (kˆ + 1)τd ) [kτ if no switch occurs during (s + τ0 , s + τ] ˘ ˘ T0 ∈ [kτd , (k + 1)τd ) = ˆ ˆ [(k + 1)τd , (k + 2)τd ) if a switch occurs during (s + τ0 , s + τ] if no switch occurs during (s + τ0 , s + τ] if a switch occurs during (s + τ0 , s + τ]
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For any k ∈ N, let Tk+1 := Tk + τd and define φ0 and φk , respectively, as ˘ d , (k˘ + 1)τd ) 0 if packet loss occurs during [kτ φ0 = 1 otherwise 0 if packet loss occurs during [Tk , Tk+1 ) φk = 1 otherwise
¯ ¯ (1 − φk )eγt/2 + φk eγ¯d/2 µ(Tk ) ¯ ¯ ¯ d /2 (1 − φk )eγτ + φk e(γ¯d−γ(τd −d))/2 µ(Tk )
if t ∈ (0, τd ) if t = τd
PT
µ(Tk + t) =
(
ED
M
it holds Pr(φk = 1) = 1 − α and Pr(φk = 0) = α, ∀k ∈ N ∪ {0}. Lemma 3: If quantizer parameter µ(t) is designed as ¯ + φ0 µ(T0 ) if t ∈ (0, T1 − T0 ) (1 − φ0 )eγt/2 µ(T0 + t) = ¯ d /2 γτ (1 − φ0 )e + φ0 µ(T0 ) if t = T1 − T0
(11a)
(11b)
AC
CE
then we have z(t) ∈ R1 (µ(t)), ∀t ≥ T0 . Proof: The lemma is proved by inductive method. Step 1: z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. ˆ d , (kˆ + 1)τd ), it is obvious that φ0 ≡ 1 and s is the moment when the switching signal is If T0 ∈ [kτ ˆ d , (kˆ + 1)τd ). From T0 = s + τ˜ > s, we know that the modes of the plant and the received during [kτ controller must be matched in [T0 , (kˆ + 1)τd ). Thus Lyapunov function is monotone decreasing in the looped region R1 (µ(T0 ))\R2 (µ(T0 )). Then z(t) ∈ R1 (µ(T0 )), ∀t ∈ [T0 , T1 ] can be guaranteed by z(T0 ) ∈ R1 (µ(T0 )). Taking the definition of µ(t) and (11a) into consideration, we obtain z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If T0 ∈ [(kˆ + 1)τd , (kˆ + 2)τd ) and a switch is received in this interval, that is, φ0 = 1, then T0 is a moment when the information is received. Similar analysis results in z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If there is no switching information received by the controller, then the length of the total mismatched interval is T1 − T0 ≤ τd in [T0 , T1 ]. In the following, two different cases are discussed. Case 1: By assuming z(T0 ) ∈ R1 (µ(T0 ))\R2 (µ(T0 )), the increasing rate of the bound of R1 (µ(t)), ∀t ∈ (T0 , T1 ], is γ¯ based on (11a). Due to the increasing rate of V (z(t)) is less than γ¯ in R1 (µ(t))\R2 (µ(t)) based on Lemma 1, there may be a time t˜1 ∈ [T0 , T1 ] satisfying z(t) ∈ R1 (µ(t))\R2 (µ(t)), ∀t ∈ [T0 , t˜1 ), z(t˜1 ) ∈ R2 (µ(t˜1 ))
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Obviously, if t˜1 does not exist, then the increasing rate of V (z(t)) is less than γ¯ throughout the interval (T0 , T1 ], and thus z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. If z(t) ∈ R2 (µ(t)) ⊂ R1 (µ(t)),t ∈ [t˜1 , T2 ), then z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. Otherwise, if there exists a time t˜2 such that z(t) ∈ R2 (µ(t)),t ∈ [t˜1 , t˜2 ), z(t˜2 ) ∈ R1 (µ(t˜2 ))\R2 (µ(t˜2 ))
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it guarantees z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ] by considering the fact that the increasing rate of V (z(t)) in ¯ R1 (µ(t))\R2 (µ(t)) is less than γ¯ but the one of the bound of R1 (µ(t)) is γ. Case 2: When z(T0 ) ∈ R2 (µ(T0 )) and z(t) ∈ R2 (µ(t)), ∀t ∈ (T0 , T1 ], it holds z(t) ∈ R1 (µ(t)), ∀t ∈ (T0 , T1 ] based on R2 (µ(t)) ⊂ R1 (µ(t)). If there is a time t˜3 such that z(t) ∈ R2 (µ(t)),t ∈ [T0 , t˜3 ), z(t˜3 ) ∈ / R2 (µ(t˜3 ))
the fact that the increasing rate of V (z(t)) in R1 (µ(t))\R2 (µ(t)) is less than γ¯ can ensure z(t) ∈ R1 (µ(t)), ∀t ∈ [T0 , T1 ]. Step 2: Assume that z(Tk ) ∈ R1 (µ(Tk )), then z(Tk +t) ∈ R1 (µ(Tk +t)), ∀t ∈ (0, τd ] must be established. Case 1: If z(Tk ) ∈ R1 (µ(Tk )) and φk = 1, then the upper bound of the mismatched time range is d¯ and the lower bound of the matched time range is τd − d¯ in the interval [Tk , Tk+1 ). If there exists a time t˜4 ∈ [Tk , Tk+1 ) such that ¯
AN US
¯
z(t) ∈ R1 (µ(Tk )) ⊂ R1 (eγ¯d/2 µ(Tk )) = R1 (µ(t)),t ∈ [Tk , t˜4 )
and z(t˜4 ) ∈ R1 (eγ¯d/2 µ(Tk ))\R1 (µ(Tk )), then the modes of the plant and the controller must be mismatched at time t˜4 , that is, t˜4 ∈ [Tk , Tk + dk ). Otherwise, if t˜4 dose not exist, then z(t) ∈ R1 (µ(Tk )) ⊂ R1 (µ(t)). ¯ Next, we discuss the upper bound of z(t),t ∈ [t˜4 , Tk+1 ]. Since z(t˜4 ) ∈ R1 (eγ¯d/2 µ(Tk ))\R1 (µ(Tk )) and ¯ ¯ R1 (µ(Tk )) ⊃ R2 (eγ¯d/2 µ(Tk )) from Ω < 1, we have z(t˜4 ) ∈ / R2 (eγ¯d/2 µ(Tk )). By (11b) we know z(t˜4 ) ∈ R1 (µ(t˜4 ))\R2 (µ(t˜4 )). Recall that the growth rate of Lyapunov function is bounded by (9) if z(t) ∈ R1 (µ(t))\R2 (µ(t)), we have
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¯ ¯ ¯ z(Tk+1 ) ∈ R1 (e(γ¯d−γ(τd −d))/2 µ(Tk )), z(t) ∈ R1 (eγ¯d/2 µ(Tk )), ∀t ∈ [t˜4 , Tk+1 ] Hence (11b) gives z(t) ∈ R1 (µ(t)), ∀t ∈ [t˜4 , Tk+1 ]. Combining with z(t) ∈ R1 (µ(t)),t ∈ [Tk , t˜4 ), tells us z(t) ∈ R1 (µ(t)),t ∈ [Tk , Tk+1 ]. Case 2: If z(Tk ) ∈ R1 (µ(Tk )) and φk = 0, then the length of the total mismatched interval is τd in [Tk , Tk+1 ). Consider that µ(Tk + t),t ∈ (0, τd ] and µ(t),t ∈ [T0 , T1 ] have the same definitions, similar analysis of Case 1 and Case 2 in Step 1 results in z(t) ∈ R1 (µ(t)),t ∈ [Tk , Tk+1 ]. Based on above analysis and the inductive method, it holds z(t) ∈ R1 (µ(t)), ∀t ≥ T0 . The proof of Lemma 3 is completed. From Lemma 3, we see that Lyapunov function V (z(t)), ∀t ≥ T0 satisfies λmin (P)M 2 µ 2 (t) V (z(t)) = z> (t)Pz(t) ≤ 2 Cmax 2
resulting in kz(t)k2 ≤ CM2 µ 2 (t). Taking the mean value of the both sides gives
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We conclude from
E{kz(t)k2 } ≤
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2 ¯ ¯ ¯ d /2 E{µ 2 (Tk+1 )} = E{ (1 − φk )eγτ + φk e(γ¯d−γ(τd −d))/2 }E{µ 2 (Tk )}
that limk→∞ E{µ 2 (Tk )} = 0 if α satisfies
¯
¯
1 − e(γ¯d−γ(τd −d)) α< ¯ ¯ ¯ d −d)) eγτd − e(γ¯d−γ(τ Moreover, µ(Tk + t) defined by (11b) results in
¯ d lim E{µ 2 (Tk + t)} ≤ eγτ lim E{µ 2 (Tk )} = 0
k→∞
for any t ∈ (0, τd ). Hence, we get limt→∞
k→∞ 2 E{kz(t)k } = 0,
which completes the proof of Theorem 1.
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IV. S IMULATION In this section, a well-known benchmark example is adopted to illustrate the effectiveness of the main results. The linearized batch reactor is given by x(t) ˙ = Ax(t) + Bu(t) (12) y(t) = Cx(t)
where
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0 0 5.679 0 1 0 1 −1 B := , C := 1.136 −3.146 0 1 0 0 1.136 0
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M
If the output y(t) in system (12) is transmitted to the controller over the network, then qµ(t) (y(t)) is received by the controller side. Moreover, switching signals σ (t) may be delayed or lost when they are transmitted by the influence of external factors. Under this circumstance, it is suitable to adopt the control strategy designed in this paper to ensure the mean square stability of the closed-loop system (12). To show the usage of the control method proposed here, the detailed simulation steps are listed as follows. Step 1: Design a quantizer as i (t) 4 +√ , µ(t) yµ(t) l M √ µ(t)sgn{yi (t)},
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(
qµ (yi (t)) =
l
if ky(t)k ≤ Mµ(t) if ky(t)k > Mµ(t)
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where l is the dimension of y(t), and yi (t) the ith component of y(t). Obviously, the above quantizer satisfies conditions I and II. Step 2: For pairs (A, B p ) and (A> ,C> p ), p ∈ {1, 2}, we design feedback gain K p and observer gain L p by linear quadratic regulator with state weighting matrix diag(1, 1, 1, 1) and input weighting matrix diag(1, 1) such that the matrices A p + B p K p and A p + L pC p , and thus Fp are stable. Then we can pursue a suitable positive-definite matrix P satisfying Fp> P + PFp < 0, p ∈ {1, 2}. Step 3: For the selected matrices K p and L p , p ∈ {1, 2}, the stability of the matrix F1,2 and the unstability of F2,1 can be testified, which means that a mode mismatch due to switching delays and packet loss makes the closed-loop system unstable. > P+PF ≤ β¯ P, respectively. Step 4: Let β = 0.5 and β¯ = 7.47 which satisfy −Fp P−PFp ≥ β P and Fp,r p,r Select ω = 0.5, direct calculations give Θ = 59.0726, ω¯ = 0.0669, γ = 0.25 and γ¯ = 7.97. Assume τd = 0.1, choose a constant d¯ = 0.001 such that γ+γ γ¯ τd ≥ d¯ ≥ 0. Let 4 = 0.001, then the saturation value can be selected as M = 5 according to (6). Up to now, we can get the upper bound of α is 0.0135. Step 5: Let x0 = [40, 0, 10, 20]> , ν = 1.01 and η = 0.1. If we design µ(0) = 0 and µ(t) = eνΓkη ,t ∈ 0 [kη, (k + 1)η), then Fig. 2 can be illustrated. Now we get t0 = 0.3 and k˜ = 3. Assume that the switching ˜ d is lost but the ones at (k˜ + 1)τd and (k˜ + 2)τd are transmitted successfully, then we get signal at kτ kˆ = 4. For ease of analysis, we set dkˆ = 0.001, dk+1 = 0.0002, τ = 0.0998, and thus s = 0.401, k˘ = 5, ˆ 0 τ = 0.0990 and τ¯ = 0.0992. So far, we know that the zooming-out stage finishes at T0 = 0.5002 which is less than s + τ.
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500 ||y(t)|| (M-2∆)||µ(t)|| 400
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Selection of t0 .
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Mean square stability of closed-loop system.
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Step 6: If quantizer parameter µ(t) is designed by (11a) and (11b), we can ensure E{|z(t)|2 } ≤ and the mean square stability of the closed-loop system (12), which is shown in Fig. 3. It is worth mentioning that the curve shown in Fig. 3 is not fixed due to the impact of packet loss. Comparison: If the switching information is only influenced by bounded time-delay rather than unbounded time-delay and packet loss, the state of the closed-loop system (12) and the upper bound of which are shown in Fig. 4. It means that the control strategy proposed in this paper is still useful to deal with the stability problem discussed in literature [32]. However, if the parameters are selected as M = 100, 4 = 0.001 and x0 = [10, 0, 10, 20]> same as in [32], the state trajectories of the closed-loop system (12) is shown as Fig. 5. Compared with Fig. 2 in [32], we know that the convergence rate under our control strategy is slower. Summarized above, it sees that our algorithm has a wider range of applicability but lower convergence speed compared with [32]. Discussion: From step 4, we see that d¯ should be selected as 0.001 to meet γ+γ γ¯ τd ≥ d.¯ It means that switching signals will be neglected if the transmission delay of which is larger than 0.001. Although this approach can ensure the system stability, it will inevitably reduce system performance such as convergence speed. The small value of the upper bound of d¯ is the main conservativeness of Theorem 1.
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State trajectories of closed-loop system under bounded time-delay.
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State trajectories of closed-loop system with parameters same as in [32].
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V. C ONCLUSION When switching signals are affected by unbounded delays and packet loss, we propose design methods of a constant d¯ and the quantizer parameter µ, under which sufficient conditions ensuring the mean square stability of the closed-loop switched system are obtained if the packet loss rate is less than a given upper bound. As pointed in discussion, how to increase the value of d¯ to improve the system performance is one of our future research directions. Moreover, this paper is only relating to fixed-length switching period, future work will also involve designing d¯ and a quantizer for arbitrary switchings case. VI. ACKNOWLEDGEMENTS The work was supported by the National Natural Science Foundation of China (61773154, 61403125, 61720106010, 61333005). Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2015RCJH15), Natural Science Foundation of Henan Province Education Department (15A413012).
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