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Periodic event-based asynchronous filtering of switched systems
Accepted Manuscript
Periodic event-based asynchronous filtering of switched systems Guoqi Ma, Xinghua Liu, Prabhakar R. Pagilla, Shuzhi Sam Ge PII: DOI: Reference:
S0016-0032(19)30257-1 https://doi.org/10.1016/j.jfranklin.2019.04.010 FI 3884
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
3 July 2018 2 February 2019 9 April 2019
Please cite this article as: Guoqi Ma, Xinghua Liu, Prabhakar R. Pagilla, Shuzhi Sam Ge, Periodic event-based asynchronous filtering of switched systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.04.010
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ACCEPTED MANUSCRIPT
Periodic event-based asynchronous filtering of switched systems✩
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Guoqi Maa , Xinghua Liub , Prabhakar R. Pagillaa,∗, Shuzhi Sam Gec a J.
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Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, United States b Department of Electrical Engineering, School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China c Department of Electrical and Computer Engineering, National University of Singapore, 117583, Singapore
Abstract
This paper provides a periodic event-based asynchronous H∞ filter design for switched systems subject to limited communication resources. The governing equations for the physical system are combined with those of the proposed event-based asynchronous filter and the resulting equations are formulated as
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a switched filtering error system with time-varying delay. For the achieved switched delay filtering error system, sufficient conditions in terms of linear
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matrix inequalities are obtained to achieve exponential stability and a prescribed H∞ disturbance attenuation level. Tools and techniques from delay-dependent Lyapunov theory, free-weighting matrices, and average dwell time for switched
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systems are used to obtain the main results. Further, a synthesis procedure for the filter gains, the event-triggering condition parameters, and a lower bound on
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the average dwell time of the switching signal is provided. A numerical example on a switched RLC circuit is finally provided to evaluate the proposed design and results.
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Keywords: Periodic event-detection, Switched systems, Average dwell time, Exponential stability, Asynchronous H∞ filtering.
✩ The
material in this paper was not present at any conference. author. Tel.: +1 (979) 458-4829 E-mail addresses: [email protected] (Guoqi Ma), [email protected] (Xinghua Liu), [email protected] (Prabhakar R. Pagilla), [email protected] (Shuzhi Sam Ge) ∗ Corresponding
Preprint submitted to Journal of The Franklin Institute
April 25, 2019
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1. Introduction As an important class of hybrid dynamical systems that exhibit interactive continuous and discrete dynamics, switched systems are composed of finite sub-
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systems (or modes) described by a group of differential or difference equations
and a switching signal that takes values from a finite integer set to orchestrate the switching between subsystems. In the past several decades, switched systems
have received considerable attention due to their applicability and effectiveness in describing a wide range of practical engineering systems, such as boost con-
verter circuits Ren et al. (2018), flight control systems Lian et al. (2017), power
systems Cardim et al. (2009), networked control systems Donkers et al. (2011),
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chemical processes Niu et al. (2015), and vehicle platoons Malloci et al. (2012). Due to the unavailability of state information that could be encountered in most practical control systems, it is important to estimate the system states in the presence of noise; state estimation can be be achieved by the Kalman filter Kalman (1960) if the external noise satisfies a normal probability distri-
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bution. When such a priori information on the external noise is not known, some new filters, e.g. H∞ filter, have been proposed. In recent years, state es-
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timation or filtering problems for switched systems have received attention; see for example, Mahmoud and Shi (2012), Lian et al. (2013), Wang et al. (2016), Zhang et al. (2017a), Xiao et al. (2017), Zhang et al. (2017b, 2018), Wang et al.
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(2018) and references therein. However, in most existing work, the output measurements are continuously transmitted to the filter regardless of whether it is
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really necessary or not. This not only creates problems for practical implementation of the filter but also can result in unnecessary usage of communication
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resources, which can render the filter inapplicable in systems subject to limited
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communication resources. Therefore, it is necessary to investigate the problem of alleviating communication burden in the filtering system, while achieving a satisfactory filtering performance; this problem is addressed in this paper. Recently, there has been a substantial increase in research related to event-
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based communication mechanisms Abdelrahim et al. (2018), Hu et al. (2018),
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He et al. (2018), Li et al. (2017), Zou et al. (2017), Khashooei et al. (2017), Liu et al. (2017a), Meng and Chen (2014). Compared with sampled-data (also called time-triggered) communication mechanisms, in which the signal is sam-
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pled and transmitted at a fixed rate without paying much attention to the system status, the main advantages of event-based communication mechanisms
are twofold: (1) reduction in communication frequency within the system loop resulting in relieving communication bandwidth and (2) reduction in the update
frequency of the actuator, which can alleviate actuator wear. These are achieved by incorporating an event-detecting mechanism in the feedback channel to con-
tinuously or discretely detect the “event” which is characterized by whether or
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not the designed error signal has violated a prescribed threshold. Only when the “event” occurs, a new signal is triggered and transmitted, otherwise, the signal is kept unchanged via a zero-order hold (ZOH). The application of eventbased communication mechanisms can be found in linear systems Tanwani et al. 45
(2016), stochastic systems Wu et al. (2017), fuzzy systems Liu et al. (2017b),
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multi-agent systems Viel et al. (2017), networked control systems (NCSs) Yin et al. (2016). However, results on the application of event-based communication
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mechanisms for switched systems is sparse, except for some recent work found in Xiao et al. (2017), Li and Fu (2017), Qi et al. (2017), Ma et al. (2016); these 50
existing event-based schemes rely on a continuous event-detection mechanism,
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which is not only difficult to implement in practical applications but also can lead to Zeno behavior Tabuada (2007).
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In this paper, the problem of periodic event-based asynchronous filtering for switched systems with limited communication capacity is considered. By
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utilizing only the event-based output measurement input to the filter, one can
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reduce the communication load between the output and the filter while achieving similar or even better filtering performance. In addition, to further relieve the communication burden, a periodic event-detecting mechanism instead of a continuous one is synthesized between the system measured output and the fil-
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ter to periodically detect the event-triggering condition. This condition is an inequality constructed via the output measurement and its error signal to char3
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acterize the variation of the output measurement from the last event-triggering instant. Only when the event-triggering condition is satisfied, i.e., the inequality is violated, an “event” occurs and the current output measurement is transmitted to the filter via a ZOH. Besides, by taking into account the switching delay
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between the filter and the subsystem, an asynchronous filter is considered. The
overall periodic event-based asynchronous switched filter system is modeled as
a switched system with time-varying delays. Based on this model, sufficient conditions are obtained for the overall system to be exponentially stable with a 70
prescribed H∞ disturbance attenuation level. Furthermore, a co-design proce-
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dure to synthesize the asynchronous filter gains and event-triggering condition parameters is provided and a lower bound on the average dwell time that will facilitate the switching signal design is obtained. Compared with existing results, the main contributions of this paper are as follows:
(i) A periodic event-based mode-dependent filter is designed for switched sys-
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tems subject to limited communication resources. The event detector samples the event-triggering condition periodically rather than continuously. (ii) The switching lag between the filter and the subsystem is also taken into
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consideration, and thus an asynchronous filter is designed. (iii) By constructing a novel Lyapunov function, the filter gains, event-triggering
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condition parameters, and the switching signal are co-designed in a unified framework, without unnecessary constraints on the switching signal. The remainder of the paper is organized as follows. Section 2 presents the
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system description and problem statement. Section 3 gives the description of
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the periodic event-triggering mechanism (PETM) and the asynchronous filter.
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Section 4 contains some mathematical preliminaries necessary for deriving the main results. Section 5 provides the main theoretical results. A numerical example is performed in Section 6, followed by remarks concluded in Section 7.
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The following notation is employed throughout the paper. N, R+ , and R+
denote the sets of natural numbers, positive real numbers, and nonnegative real numbers, respectively; L2 [0, ∞) represents the space of square integrable infi4
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nite sequences; λ(A), λmax (A), and λmin (A) denote the eigenvalues, maximum eigenvalue, and minimum eigenvalue of a matrix A ∈ Rn×n , respectively. The superscripts ‘>’ and ‘−1’ are used for matrix transpose and matrix inverse, respectively; A 0(≺ 0) is used to indicate that the matrix A is positive (negative)
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definite; sym(A) represents A + A> . A(i,j) denotes the block matrix element at
the i-th row group and j-th column group of a block matrix A partitioned into m row groups and n column groups, denoted as [A]m×n . The terms induced by
symmetry in a matrix are denoted by ‘?’. Compatible dimensions are assumed for matrices whose dimensions are not explicitly stated.
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2. System description and problem statement
Consider a class of switched systems of the form ˙ = Aσ(t) x(t) + Bσ(t) ω(t), x(t) y(t) = Cσ(t) x(t), z(t) = D x(t),
(1)
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σ(t)
where x(t) ∈ Rn is the state vector, y(t) ∈ Rp is the measured output, z(t) ∈ Rq
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is the signal to be estimated, ω(t) ∈ Rl denotes the external disturbance and is assumed to belong to L2 [0, ∞). Aσ(t) , Bσ(t) , Cσ(t) , Dσ(t) are system matrices of compatible dimensions pertaining to subsystems indicated by the switching
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signal σ(t). Let N = {1, 2, · · · , N } be a finite index set. The switching signal σ(t) : [0, ∞) → N is a right-continuous, piecewise constant function, which
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orchestrates the switching among the subsystems, that is, corresponding to the
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switching signal σ(t) there exists a switching sequence {(i0 , t0 ), (i1 , t1 ), · · · , (ik , tk ), · · · |ik ∈ N , k ∈ N} ,
(2)
where tk ∈ R+ denotes switching instants, which means that the subsystem ik
is activated at tk and remains active for all t ∈ [tk , tk+1 ). Based on Duan and Wu (2014), a switching signal σ(t) is said to have a dwell time τD if tk+1 − tk ≥
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τD , ∀k ∈ N, and is said to have an average dwell time τa Liberzon (2012) if 5
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Nσ (T1 , T2 ) ≤ N0 +
T2 −T1 τa
for any 0 ≤ T1 ≤ T2 , where Nσ (T1 , T2 ) denotes the
number of switchings of σ(t) over (T1 , T2 ). Given the above switched system, the problem is to co-design a periodic
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event-triggering mechanism and an asynchronous filter to estimate the signal z(t). The goal of the periodic event-triggering mechanism is to provide the measured output as the input to the filter only when the change of the output is large enough as characterized by an event-triggering condition. We will formalize this in the next section.
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3. Periodic event-triggering mechanism and asynchronous filter
The proposed design strategy for the overall system is illustrated in Fig. 1.
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Besides the switched system, it mainly consists of two modules: (i) the periodic event-detector; and (ii) the filter system. The event-detector, with a period of T , samples the measured system output and evaluates the event-triggering
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condition to determine whether the input to the filter needs to be updated or not. The governing equations for the periodic event detector and the filter are
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given in the following.
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(t )
(t )
Switched System
Sensor
T
z (t )
y (t )
Event Detector
Clock
z f (t )
yˆ(t )
y (tˆn )
Filter ZOH Figure 1: Overall system configuration.
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3.1. Periodic event-detector Let the event-triggering instants be given by tˆn = ˆin T , where ˆin , n ∈ N,
and {ˆi0 , ˆi1 , ˆi2 , · · · } ⊂ {0, 1, 2, 3, · · · } with ˆi0 = 0 and ˆin < ˆin+1 . Note that for
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∀n ∈ N we have tˆn+1 − tˆn ≥ T . Define ln,j = (ˆin + j)T , j = 0, 1, 2, · · · dn , where Sd n dn = ˆin+1 − ˆin − 1. Then, tˆn , tˆn+1 = j=0 [ln,j , ln,j+1 ). Further, denote the
input to the filter as yˆ(t), then when t ∈ [ln,j , ln,j+1 ), n ∈ N, j = 0, 1, 2, · · · dn , then an error signal on the measured output is defined as ey (t) = yˆ(t) − y(ln,j ).
(3)
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Based on ey (t) defined in (3), we consider the following mode-dependent eventtriggering condition:
> e> y (t)Ω1,σ(t) ey (t) ≤ φy (ln,j )Ω2,σ(t) y(ln,j ), t ∈ [ln,j , ln,j+1 ) ,
(4)
where φ ∈ R+ is an adjustable design parameter, and Ω1,σ(t) , Ω2,σ(t) 0 are
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mode-dependent positive definite matrices, which are to be designed. Remark 3.1. It should be pointed that when the inequality (4) is violated,
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> i.e., e> y (t)Ω1,σ(t) ey (t) > φy (ln,j )Ω2,σ(t) y(ln,j ), we say the event-triggering con-
dition is satisfied. Nonetheless, at the event-triggering instants, the error signal ey (t) will be reset to zero, together with the fact that ey (t) is a piecewise con-
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stant and continuous from the right signal, then, the inequality (4) is held for all time.
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To model the overall periodic event-based filter system in terms of the time variable t, motivated by Zhang and Feng (2014), define τ (t) = t − ln,j , when t ∈ [ln,j , ln,j+1 ), thus, τ (t) can be considered as a time-varying delay satisfying
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τ (t) ∈ [0, T ) and τ˙ (t) = 1. With this definition of τ (t), y(ln,j ) can be rewritten
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as y (t − τ (t)) and the input to the filter as yˆ(t) = ey (t) + y (t − τ (t)). We will use this later when we formulate the filtering error system.
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3.2. Filter system model
f
f,σf (t) f
(5)
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The asynchronous filter is described by x˙ (t) = A ˆ(t), f f,σf (t) xf (t) + Bf,σf (t) y z (t) = C x (t),
where xf (t) denotes the filter state vector, zf (t) is the estimate of z(t), and Af,σf (t) , Bf,σf (t) , Cf,σf (t) are filter gains corresponding to the filter subsystems specified by the filter switching signal σf (t). Ideally, one can choose σf (t) to be
the same as σ(t), which means that one can instantaneously detect the switching of the subsystems of the switched system and match the corresponding filter
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subsystem. However, in reality, there will inevitably be a lag between the two, and we define it as d ∈ R+ . Hence, we model σf (t) in the following manner: σf (t) = σ(t − d).
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The switching lag d is assumed to satisfy 0 < d < τD , in which τD is the dwell
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time of σ(t).
3.3. Modeling of the filtering error system
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When t ∈ [tk , tk+1 ), ∀k ∈ N, assume σ(tk ) = i, σf (tk ) = j, i, j ∈ N , i 6= j. h i> Moreover, define ε(t) = z(t) − zf (t) as the filtering error, ζ(t) = x> (t), x> (t) f as the augmented state vector. Then, we can obtain the following filtering error
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system equations: ˙ = Aij ζ(t) + Bi ω(t) + Cj ey (t) + Dj y (t − τ (t)) , ζ(t) t ∈ [tk , tk + d) ε(t) = Eij ζ(t), ˙ = Aii ζ(t) + Bi ω(t) + Ci ey (t) + Di y (t − τ (t)) , ζ(t) t ∈ [tk + d, tk+1 ) ε(t) = Eii ζ(t),
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(7)
h Ai 0 Bi 0 , Bi = , Cj = Dj = , Eij = Ei where Aij = 0 Af,j 0 Bf,j h i Ai 0 0 , Ci = Di = , Eii = Ei −Cf,i . Aii = 0 Af,i Bf,i 8
i −Cf,j ,
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4. Preliminaries Before proceeding with the main results, we first present the following definitions and lemmas, which play a key role in the proof of the main results.
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Definition 4.1. Wu et al. (2009) The equilibrium ξ ∗ = 0 of the filtering error system (7) with ω(t) = 0 is said to be exponentially stable under σ(t) if its solution ζ(t) satisfies kζ(t)k ≤ ηkζt0 kC 1 e−λ(t−t0 ) ,
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n o ˙ + θ)k . for η ≥ 1 and λ > 0, and kζt kC 1 = sup−d≤θ≤0 kζ(t + θ)k, kζ(t
Definition 4.2. Xiao et al. (2017) For a given scalar > 0, the filtering error system (7) is said to be exponentially stable with a weighted H∞ performance index (, γ) if the following two conditions hold:
(i) The filtering error system (7) is exponentially stable when ω(t) = 0;
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(ii) Under zero initial conditions and for any nonzero ω(t) ∈ L2 [0, ∞], the following inequality is satisfied:
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Z∞ 0
e
−s >
ε (s)ε(s)ds ≤ γ
2
Z∞
ω > (s)ω(s)ds.
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Lemma 4.1. For any real vectors u, v with appropriate dimensions and symmetric positive definite matrix Q with compatible dimension, the following
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inequality holds:
u> v + v > u ≤ u> Qu + v > Q−1 v.
Lemma 4.2. Let W ∈ Rn×n be a symmetric matrix, and let x ∈ Rn , then
the following inequality holds: λmin (W )x> x ≤ x> W x ≤ λmax (W )x> x.
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5. Main results In this section, we will first determine the sufficient conditions under which the filtering error system (7) is exponentially stable when the disturbance ω(t) =
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0. Then, we will derive conditions under which the H∞ attenuation performance is achieved for any nonzero ω(t) ∈ L2 [0, ∞]. 5.1. Exponential stability analysis
We will use tools from delay-dependent Lyapunov functions, free-weighting
matrices, and average dwell time techniques, to derive sufficient conditions which
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can ensure the filtering error system (7) is exponentially stable in the following theorem.
Theorem 5.1. Given parameters α ∈ R+ , β ∈ R+ , φ ∈ R+ , µ ≥ 1, d ∈ R+ ,
and T ∈ R+ , the filtering error system (7) with ω(t) = 0 is exponentially stable
under any switching signal σ(t) with average dwell time τa satisfying τa > τa∗ =
if for ∀σ(t) = i ∈ N , there exist positive definite matrices
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2 ln µ+(α+β)(d+T ) , α
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Pi 0, Qi 0, Ri 0, and arbitrary compatible matrices Mi , Ni such that Ni Mi 0, [Ξi ] Λi = 4×4 ≺ 0, ? e−αT Ri (1,1)
where Ξi is symmetric and its block matrix elements are given by Ξi
=
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(1,2) (1,3) (1,4) > sym {Pi Ai } + αPi + T A> = 0, Ξi = 0, Ξi = Ci> Mi , i Ci Ri Ci Ai , Ξi (2,2) (2,3) (2,4) (3,3) Ξi = sym {Qi Af,i + αQi }, Ξi = Qi Bf,i , Ξi = Qi Bf,i , Ξi = −Ω1,i , (3,4) (4,4) Ξi = 0, Ξi = −sym {Mi } + φΩ2,i + τ (t)Ni .
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And for ∀i ∈ N , j ∈ N , i 6= j, Nj Mj 0, [Θij ] Πij = 4×4 ≺ 0, ? Rj
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(1,1)
where Θij is symmetric and its block matrix elements are given by Θij (1,2)
> sym {Pj Ai } − βPj + T A> i Ci Rj Ci Ai , Θij (2,2)
Θij
(3,4) Θij
(2,3)
= sym {Qj Af,j }−βQj , Θij = 0,
(4,4) Θij
(1,3)
= 0, Θij (2,4)
= Qj Bf,j , Θij
= −sym {Mi } + τ (t)Ni + φΩ2,i . 10
(1,4)
= 0, Θij
= Ci> Mj ,
(3,3)
= Qj Bf,j , Θij
=
= −Ω1,i ,
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Proof 5.1. Choose a piecewise Lyapunov-Krasovskii functional candidate of the form
where k ∈ N and
V1,σ(t) (t) = ζ (t)diag Pσ(t) , Qσ(t) ζ(t) + >
>
(8)
V2,σf (t) (t), t ∈ [tk , tk + d) , Z0 Zt
eα(s−t) y˙ > (s)Rσ(t) y(s)dsdθ, ˙
−T t+θ
Z0 Zt
eβ(t−s) y˙ > (s)Rσf (t) y(s)dsdθ. ˙
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V2,σf (t) (t) = ζ (t)diag Pσf (t) , Qσf (t) ζ(t) +
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V (t) =
V1,σ(t) (t), t ∈ [tk + d, tk+1 ) ,
−T t+θ
First, for any positive real numbers α, β, it is held that Z0 Zt
−T t+θ
e
α(s−t) >
y˙ (s)Ri y(s)dsdθ ˙ ≤
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≤
Z0 Zt
y˙ > (s)Ri y(s)dsdθ ˙
Z0 Zt
eβ(t−s) y˙ > (s)Ri y(s)dsdθ. ˙
−T t+θ
(9)
−T t+θ
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Second, assume for ∀i, j ∈ N = {1, 2, · · · , N }: Pi ≤ µPj , Qi ≤ µQj , Ri ≤ µRj , µ ≥ 1.
(10)
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Then, combining conditions (9) and (10), at time instant tk + d, we obtain V1,σ(t) (tk + d) ≤ µV2,σf (t) (tk + d)− .
(11)
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On the other hand, at the time instant tk , we can obtain V2,σf (t) (tk ) ≤ µe(α+β)T V1,σ(t) (t− k ).
(12)
Consider t ∈ [tk , tk + d), and assume σ(t) = i ∈ N , σf (t) = j ∈ N , i 6= j. Taking the derivative of V2,σf (t) (t) with respect to t along the trajectories of the
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filtering error system (7) yields V˙ 2,σf (t) (t) = 2x> (t)Pj x(t) ˙ + 2x> ˙ f (t) + β f (t)Qj x
−T t+θ
e−βθ y˙ > (t + θ)Rj y(t ˙ + θ)dθ
−T
≤ 2x> (t)Pj x(t) ˙ + 2x> ˙ f (t) + β f (t)Qj x
Z0 Zt
eβ(t−s) y˙ > (s)Rj y(s)dsdθ ˙
−T t+θ
Zt
y˙ > (s)Rj y(s)ds. ˙
(13)
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+ T y˙ > (t)Rj y(t) ˙ −
eβ(t−s) y˙ > (s)Rj y(s)dsdθ ˙
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+ T y˙ > (t)Rj y(t) ˙ −
Z0
Z0 Zt
t−τ (t)
Then, we can get
V˙ 2,σf (t) (t) − βV2,σf (t) (t) > ≤ x> (t) sym {Pj Ai } − βPj + T A> i Ci Rj Ci Ai x(t)
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> + x> f (t) [sym {Qj Af,j } − βQj ] xf (t) + 2xf (t)Qj Bf,j ey (t)
+ 2x> f (t)Qj Bf,j y(t − τ (t)) −
Zt
y˙ > (s)Rj y(s)ds. ˙
(14)
t−τ (t)
two identities:
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For arbitrary compatible matrices Mi and Ni , one can obtain the the following
>
h
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2y (t − τ (t))Mi y(t) − y(t − τ (t)) −
Zt
t−τ (t)
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>
τ (t)y (t − τ (t))Ni y(t − τ (t)) −
Zt
i y(s)ds ˙ = 0.
y > (t − τ (t))Ni y(t − τ (t))ds = 0.
(15)
(16)
t−τ (t)
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In addition, the event-triggering condition (4) implies > −e> y (t)Ω1,i ey (t) + φy (t − τ (t))Ω2,i y(t − τ (t)) ≥ 0.
(17)
Then, based on (14), (15), (16), and (17), we obtain V˙ 2,σf (t) (t) − βV2,σf (t) (t) ≤ ξ (t)Θij ξ(t) − >
Zt
t−τ (t)
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%> (t, s)Πij %(t, s)ds,
(18)
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h i> > > > where ξ(t) = x> (t), x> , %(t, s) = y > (t − τ (t)), y˙ > (s) . f (t), ey (t), y (t − τ (t)) Therefore, if Πij 0 and Θij ≺ 0 are satisfied, it can be got from (18) that V˙ 2,σf (t) (t) − βV2,σf (t) ≤ 0.
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(19)
Integrating (19) from tk to t, t ∈ [tk , tk + d), together with (11) and (12), it can be deduced that V (t) ≤ eβ(t−tk ) V2 (t+ k)
≤ µeβd e(α+β)T e−α(tk −tk−1 −d) V1 (tk−1 + d)
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≤ µ2 e2βd e(α+β)T e−α(tk −tk−1 −d) V2 (t+ k−1 )
≤ µ3 e2βd e2(α+β)T e−α(tk −tk−2 −2d) V1 (tk−2 + d)+ .. .
≤ e(α+β)d e[
2 ln µ+(α+β)(d+T ) −α τa
](t−t0 ) V (t ). 0
(20)
Moreover, according to the definition of V (t) in (8), together with Lemma 4.2,
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one can obtain
(21)
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V (t) ≥ c1 kζ(t)k2 , n o where c1 = min min λ (Pi ) , min λ (Qi ) . i∈N
i∈N
On the other hand, it can be deduced that
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V (t0 ) ≤ λmax (Pi )x> (t0 )x(t0 ) + λmax (Qi )x> f (t0 )xf (t0 )
CE
+ T eβT λmax (Ri )λmax (Ci> Ci )ke x(t0 )k2
e 0 )k2 , ≤ c2 kζ(t
where x e(t0 ) = max
−T ≤θ≤0
(22)
e 0 ) = max kζ(θ)k, kζ(θ)k ˙ kx(θ)k ˙ , ζ(t , −T ≤θ≤0
c2 = max{λmax (Pi ), λmax (Qi ), T eβT λmax (Ri )λmax (Ci> Ci )}.
AC
Then, by combining (20), (21), and (22), we get r ) c2 (α+β)d [ 2 ln µ+(α+β)(d+T −α e 0 )k. 2τa 2 ](t−t0 ) kζ(t kζ(t)k ≤ e 2 e c1
(23)
The same result be achieved when t ∈ [tk + d, tk+1 ), if Λi 0 and Ξi ≺ 0 are
170
satisfied. Therefore, according to Definition 4.1, the filtering error system (7) is exponentially stable when ω(t) = 0. This completes the proof. 13
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Remark 5.1. Note that in (7), if we write y (t − τ (t)) as Cσ(t−τ (t)) x(t − τ (t)) + Dσ(t−τ (t)) ν(t − τ (t)), then σ(t − τ (t)) is difficult to handle in the stability analysis. Although a switching signal merging technique Vu and Morgansen (2010) is utilized in Xiao et al. (2017), where at most one subsystem switching
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175
is allowed within any two consecutive event-triggering instants, which places
restrictions on the switching signal σ(t). A constraint on the switching signal σ(t) was made in Ren et al. (2019) as well when describing the delayed term of
the measured output. The method in this paper does not place such restrictions. 5.2. H∞ performance analysis and co-design procedure
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180
In this subsection, we will investigate the H∞ performance of the filtering error system (7) with any nonzero ω(t) ∈ L2 [0, ∞]. We will provide sufficient conditions that can guarantee a prescribed H∞ performance for the filtering error system (7) and a co-design procedure for the filter gains, the event-triggering 185
condition parameters, and the switching signal are provided by the following
M
theorem.
Theorem 5.2. Given parameters α ∈ R+ , β ∈ R+ , φ ∈ R+ , µ ≥ 1, γ ∈ R,
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d ∈ R+ , and T ∈ R+ , the filtering error system (7) has a prescribed H∞ disturbance attenuation level for any nonzero ω(t) ∈ L2 [0, ∞] under any switching signal σ(t) with average dwell time τa satisfying τa > τa∗ =
2 ln µ+(α+β)(d+T ) , α
PT
if for ∀i ∈ N , there exist positive definite matrices Pi 0, Qi 0, Ri 0, Ω1,i 0, Ω2,i 0, and arbitrary compatible matrices Mi , Ni , Hi , W1,i , W2,i ,
AC
CE
W3,i such that Λi 0 hold, and Υi =
Υ1i
Υ2i
?
−I
≺ 0,
(24)
where [Υ1i ]6×6 is symmetric block matrix, [Υ2i ]6×2 is a block matrix, and the (1,1)
block elements in Υ1i and Υ2i are given by Υ1i (1,2)
Υ1i
190
(1,3)
= −Ei> W3,i , Υ1i
Pi B i ,
(2,2) Υ1i
(2,6) Υ1i
= 0,
(1,4)
= 0, Υ1i
= sym {Pi Ai } + αPi + Ei> Ei , (1,5)
= Ci> Mi , Υ1i
(2,3) (2,4) = sym {W1,i } + αQi , Υ1i = W2,i , Υ1i = W2,i , (3,3) (3,4) (3,5) (3,6) Υ1i = −Ω1,i , Υ1i = 0, Υ1i = 0, Υ1i =
14
(1,6)
=
(2,5) Υ1i = (4,4) 0, Υ1i
0,
> = A> i Ci Hi , Υ1i
=
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(4,5)
= 0, Υ1i
2
(2,1) Υ2i
−sym {Mi } + T Ni + φΩ2,i , Υ1i
(5,6) Υ1i
= Hi C i B i ,
(6,6) Υ1i
= −γ I,
(4,6)
=
(5,5)
= 0, Υ1i
> W3,i ,
= −sym {Hi } + T Ri ,
the other elements in Υ2i are
195
And for ∀i, j ∈ N , i 6= j, Πij 0, and S1ij S2ij ≺ 0, Sij = ? −I
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all zero blocks.
(25)
where [S1ij ]6×6 is symmetric block matrix, [S2ij ]6×2 is a block matrix, and the (1,1)
block elements in S1ij and S2ij are given by S1ij = sym {Pj Ai }−βPj +Ei> Ei , (1,2)
S1ij
(1,3)
= −Ei> W3,j , S1ij
(1,4)
= 0, S1ij
200
(1,6)
> = A> i Ci Hj , S1ij
=
(2,3) (2,4) (2,5) P j Bi , = sym {W1,j } − βQj , S1ij = W2,j , S1ij = W2,j , S1ij = 0, (2,6) (3,3) (3,4) (3,5) (3,6) (4,4) S1ij = 0, S1ij = −Ω1,i , S1ij = 0, S1ij = 0, S1ij = 0, S1ij = (4,5) (4,6) (5,5) −sym {Mj } + φΩ2,i + T Nj , S1ij = 0, S1ij = 0, S1ij = −sym {Hj } + T Rj , (5,6) (6,6) (2,1) S1ij = Hj Ci Bi , S1ij = −γ 2 I, S2ij = Cf,j , the other elements in S2ij are
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(2,2) S1ij
(1,5)
= Ci> Mj , S1ij
zero blocks.
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In addition, the filter gains are given by
−1 Af,i = Q−1 i W1,i , Bf,i = Qi W2,i , Cf,i = W3,i .
(26)
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Proof 5.2. Choose the same piecewise Lyapunov-Krasovskii functional candidate as in (8), and define z(t) = ε> (t)ε(t) − γ 2 ω > (t)ω(t). Consider the asynchronous period, i.e. t ∈ [tk , tk + d), assume that σ(t) = i ∈ N , σf (t) = j ∈ N .
PT
Taking the derivative of V (t) with respect to t along the trajectories of (7) and adding the term −βV2 (t) + z(t) on both sides, we obtain
AC
CE
V˙ 2 (t) − βV2 (t) + z(t) h i ≤ x> (t) sym {Pj Ai } − βPj + Ei> Ei x(t) h i > + x> (t) sym {Q A } − βQ + C C xf (t) j f,j j f,j f f,j + 2x> (t)Pj Bi ω(t) − 2x> (t)Ei> Cf,j xf (t)
> + 2x> f (t)Qj Bf,j ey (t) + 2xf (t)Qj Bf,j y(t − τ (t)) >
2
>
+ T y˙ (t)Rj y(t) ˙ − γ ω (t)ω(t) −
Zt
t−τ (t)
15
y˙ > (s)Rj y(s)ds. ˙
(27)
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Based on the system equation (1), we obtain y(t) ˙ = Ci Ai x(t) + Ci Bi ω(t).
(28)
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Using the above equation, we have
2y˙ > (t)Hi Ci Ai x(t) + Ci Bi ω(t) − y(t) ˙ = 0,
(29)
where Hi is an arbitrary matrix of compatible dimension.
Then, combining (15), (16), (17), (27), and (29), we have Zt
%> (t, s)Πij %(t, s)ds,
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V˙ 2 (t) − βV2 (t) + z(t) ≤ χ> (t)Xij χ(t) −
(30)
t−τ (t)
h i> > > > > where χ(t) = x> (t), x> (t), e (t), y (t − τ (t)), y ˙ (t), ω (t) , [Xij ]6×6 is symy f (1,1)
metric block matrix, whose elements are given by Xij Di> Di , (1,6)
(1,2) Xij
(2,2)
= 0,
(1,4) Xij
= sym {Pj Ai } − βPj + (1,5)
= Ci> Mj , Xij
> = A> i C i Hj ,
(2,3)
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> = sym {Qj Af,j } − βQj + Cf,j Cf,j , Xij
= Qj Bf,j ,
(2,5) (2,6) (3,3) (3,4) (3,5) = Qj Bf,j , Xij = 0, Xij = 0, Xij = −Ω1,i , Xij = 0, Xij = (4,4) (4,5) (4,6) = 0, Xij = −sym {Mj } + φΩ2,i + τ (t)Nj , Xij = 0, Xij = (5,6) (6,6) 2 = −sym {Hj } + T Rj , Xij = Hj Ci Bi , Xij = −γ I, (4,4) that there exists a time-varying term in Xij , hence we decompose Xij
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follows:
ED
(2,4) Xij (3,6) Xij (5,5) Xij
Note
= −Di> Cf,j ,
= Pj Bi , Xij
Xij
(1,3) Xij
Xij =
CE
where
T − τ (t) b τ (t) e Xij + Xij , T T
0, 0,
as
(31)
b ij = Xij −diag {0, 0, 0, τ (t)Nj , 0, 0} , X e ij = Xij +diag {0, 0, 0, (T − τ (t))Nj , 0, 0} , X
AC
from which, one can obtain
e ij = X b ij + diag {0, 0, 0, T Nj , 0, 0} , X
e ij ≺ 0 ⇒ X b ij ≺ 0, then, if X e ij ≺ 0, we have which implies X Xij =
τ (t) e T − τ (t) b Xij + Xij ≺ 0. T T 16
(32)
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e ij , using Schur Complement Lemma Note that, there are nonlinear terms in X
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e ij converts X e ij into its equivalent form as two times on X Ψ11,ij Ψ12,ij Ψ13,ij (33) Ψij = ? −I 0 , ? ? −I n o e ij − diag 0, C > Cf,j , 0, 0, 0, 0 , Ψ12,ij = [0, Cf,j , 0, 0, 0, 0]> , where Ψ11,ij = X f,j >
Ψ13,ij = [0, 0, 0, 0, Hj Di , 0] .
Then, by letting Qj Af,j = W1,j , Qj Bf,j = W2,j , Cf,j = W3,j in Ψij , we can
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obtain (25). Therefore, if conditions Πij 0 and Sij ≺ 0 given in Theorem 5.2 are satisfied, we can obtain
V˙ 2,σf (t) (t) − βV2,σf (t) (t) + z(t) < 0.
(34)
Integrating (34) from tk to t, t ∈ [tk , tk + d), it can be derived that
≤ µe
tk
V1 (t− k)
eβ(t−s) z(s)ds
−
Zt
e
β(t−s)
ED
(α+β)T +βd
Zt
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V2 (t) ≤ eβ(t−tk ) V2 (tk ) −
tk
2 (α+β)T +2βd−α(tk −tk−1 −d)
PT
≤µ e
eβ(t−s) z(s)ds
tk
Ztk
(α+β)T +βd
V2 (tk−1 ) − µe
e−α(tk −s) z(s)ds
tk−1 +d
− µ2 e(α+β)T +2βd−α(tk −tk−1 −d)
CE
z(s)ds −
Zt
tk−1 Z +d tk−1
eβ(tk −s) z(s)ds −
Zt
eβ(t−s) z(s)ds
tk
AC
.. .
≤ e(α+β)d e[
where Φ2 (t) =
2 ln µ+(α+β)(d+T ) −α](t−t0 ) τa
k P
Rtj
j=1 tj−1 +d
e[Nσ (s,t)−1]αd z(s)ds +
V (t0 ) − Φ2 (t),
e[2Nσ (s,t)−1] ln µ−α(tk −s)+Nσ (s,t)(α+β)T +Nσ (s,t)βd ×
k P
tj−1 R +d
j=1 tj−1
e2Nσ (s,t) ln µ−αtk +(α+β)tj−1 +β[Nσ (s,t)+1]d ×
eαNσ (s,t)d+Nσ (s,t)(α+β)T e−βs z(s)ds +
Rt
tk
eβ(t−s) z(s)ds.
17
(35)
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Under zero initial condition, we have Φ2 (t) ≤ 0.
(36)
of the above inequality yields e 2 (t) ≤ Φ b 2 (t), Φ
where
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Then, multiplying e−2Nσ (t0 ,t) ln µ−Nσ (t0 ,t)(α+β)(d+T )+ln µαd−αt+αtk on both sides
(37)
tj−1 Z +d k X e 2 (t) = Φ e−2Nσ (0,s) ln µ+ln µ−αt+(α+β)tj−1 −(α+β)[Nσ (0,s)−1]d−βs ×
e
−Nσ (0,s)(α+β)T >
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j=1 t j−1
ε (s)ε(s)ds +
Zt
eβ(t−s)+ln µ+αd−αt+αtk ε> (s)ε(s)ds
tk
+
k X
Ztj
e−2Nσ (0,s) ln µ−α(t−s)−Nσ (0,s)(α+β)(d+T ) ε> (s)ε(s)ds,
j=1t j−1 +d
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tj−1 Z +d k X e−2Nσ (0,s) ln µ+ln µ−αt+(α+β)tj−1 −(α+β)[Nσ (0,s)−1]d−βs j=1 t j−1
ED
b 2 (t) =γ 2 Φ
e−Nσ (0,s)(α+β)T ω > (s)ω(s)ds + γ 2
eβ(t−s)+ln µ+αd−αt+αtk ω > (s)ω(s)ds
tk
e−2Nσ (0,s) ln µ−α(t−s)−Nσ (0,s)(α+β)(d+T ) ω > (s)ω(s)ds.
PT
+ γ2
Ztj k X
Zt
j=1t j−1 +d
CE
It is obvious that
eβ(t−s)+ln µ+αd−αt+αtk ε> (s)ε(s)ds ≥ e−αt ε> (s)ε(s),
AC
and
eβ(t−s)+ln µ+αd−αt+αtk ω > (s)ω(s)ds ≤ e−α(t−s)+ln µ+(α+β)d ω > (s)ω(s)ds.
Then, we get Z∞
e
−αs >
ε (s)ε(s)ds < γ
t0
2
Z∞
t0
18
ω > (s)ω(s)ds,
(38)
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where γ = e
ln µ+(α+β)d 2
γ. The same result can also be guaranteed for t ∈ [tk + d, tk+1 )
if conditions Λi 0 and Υi ≺ 0 given in Theorem 5.2 hold. Then according to Definition 4.2, the H∞ disturbance attenuation performance is ensured for the filtering error system (7). This completes the proof.
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205
Remark 5.2. It should be pointed out if the LMI conditions (24) and (25)
210
given in Theorem 5.2 are solvable, then the adjustable design parameter φ in n the event-triggering condition (4) has an upper bound min λmin sym{Mi } − −1 o T Ni Ω−1 , λ sym{M } − T N Ω2,i , which can be derived as follows: min j j 2,i (4,4)
is a diagonal element, we have −sym{Mi } + T Ni + φΩ2,i ≺ 0, which is equivalent to sym{Mi } − T Ni Ω−1 2,i − φI 0, then we can obtain there exist invertible matrices Pi and Qi such that Pi−1 sym{Mi } − T Ni Ω−1 2,i − φI Pi = diag{σ1 , σ2 , · · · , σp }, i.e. Pi−1 sym{Mi } − T Ni Ω−1 2,i Pi = diag{σ1 + −1 −1 φ, σ2 + φ, · · · , σp + φ} and Q sym{Mi } − T Ni Ω2,i Qi = diag{λ1 , λ2 , · · · , λp }, therefore, it can be deduced that φ < min{λ1 , λ2 , · · · , λp }, i.e., φ < λmin sym{Mi }− −1 T Ni Ω−1 ; similarly, from (25), we can get φ < λ sym{M } − T N min j j Ω2,i . 2,i
M
215
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First, as Υ1i
Thus an upper bound of φ can be obtained by taking the smaller value between
220
ED
the above two bounds achieved from (24) and (25), respectively.
6. Simulations
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In this section, a switched electrical circuit system Shen et al. (2014) is provided to illustrate the effectiveness of the proposed filter design and the
CE
results in Section 5. A sketch of the switched circuit is provided in Fig. 2. >
Choose x(t) = [iL (t), uC (t)] as state vector, and the voltage of the resistor
AC
as the measured output y(t), ω(t) as the disturbance input. Then this switched
19
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R
(t )
Subsystem 2
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Subsystem 1
iL (t ) L1
L2
uC (t ) C1 !
C2
!
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Figure 2: A switched RLC series circuit.
ED
M
circuit can be modeled as the form of (1) with two subsystems as follows: R 1 1 − − Li Li Li ω(t), x(t) ˙ = x(t) + 1 0 0 Ci | {z } | {z } Ai Bi h i (39) y(t) = R 0 x(t), | {z } Ci z(t) = D x(t). i The values of the resistor, capacity, and inductor are chosen as R = 5Ω,
L1 = 0.1H, L2 = 0.2H, C1 = 10F, C2 = 8F. Moreover, choose z(t) = 0.6uC (t)
PT
225
for subsystem 1 and z(t) = 0.5uC (t) for subsystem 2. The parameters of the
AC
CE
two subsystems are given as follows: Subsystem 1: h i h i −50 −10 10 , B1 = , C1 = 5 0 , D1 = 0 0.6 . A1 = 0.1 0 0 Subsystem 2:
−25
A2 = 0.125
h i h i 5 , B2 = , C2 = 5 0 , D2 = 0 0.5 . 0 0
−5
20
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We choose α = 0.2, β = 0.1, µ = 1.1, d = 0.2, T = 1, φ = 0.05, then we obtain τa∗ = 2.7531, we choose τa = 3, choose γ = 2.5, then γ = e
ln µ+(α+β)d 2
γ=
Af,2
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2.7019. From Theorem 5.2, we obtain the filter gains as follows: h i> h i −0.60 0 , Bf,1 = 0.43 0.73 , Cf,1 = −0.15 0.45 ; Af,1 = 0 −0.60 h i> h i −0.99 0 , Bf,2 = 0.12 0.16 , Cf,2 = −0.13 0.35 . = 0 −0.99
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The event-triggering condition parameters are obtained as
Ω1,1 = Ω2,1 = 25.93; Ω1,2 = Ω2,2 = 62.09.
The switching signals σ(t) and σf (t) are shown in Fig. 3. The initial values of >
>
system states and filter states are chosen as x(0) = [2 − 2] , xf (0) = [5 − 3] . 230
The response of system states x(t) and filter states xf (t) are illustrated in Fig. 4. z(t), zf (t), and the filtering error ε(t) are given in Fig. 5. The event-triggering
M
intervals and the event-triggered input to the filter are plotted in Fig. 6 and Fig. 7, respectively.
ED
235
>
>
In addition, under zero initial conditions, i.e., x(0) = [0 0] , xf (0) = [0 0] , R∞ choose the external disturbance input as ω(t) = e−0.05t with ω 2 (t)dt = 10 < 0
∞, so ω(t) ∈ L2 [0, ∞). Besides, in order to examine the robustness of the filter
PT
to noise sensitivity, a Gaussian white noise is added to ω(t) with a signal-to-noise ratio (SNR) 10, and denote the synthetic noise signal as ω ˆ (t). The trajectories
CE
of ω ˆ (t), ω(t), and
240
Rt
e−αs ε> (s)ε(s)ds
0
Rt 0
, ω ˆ > (s)ˆ ω (s)ds
Rt
e−αs ε> (s)ε(s)ds
0
Rt
are shown in Fig. 8, and ω > (s)ω(s)ds
0
since γ = 2.7019, the H∞ disturbance attenuation performance is achieved for
AC
both ω ˆ (t) and ω(t). From the simulation results, it can be seen that the filtering error system is
stable and a prescribed H∞ disturbance attenuation level is achieved. Moreover, the output measurements are only transmitted to the filter when needed, which
245
has significantly reduced the communication frequency between the measured output and the filter. 21
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Figure 3: The switching signals σ(t) and σf (t).
5
0
20
40
60
80
100
40
60
80
100
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-5
M
0
PT
5
0
0
AC
CE
-5
20
Figure 4: The responses of system states x(t) and filter states xf (t).
7. Conclusions The problem of periodic event-based asynchronous H∞ filtering has been
addressed for switched systems. The overall system has been transformed to a 22
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1
-1 -2 0
50
0
50
100
1
-1 -2
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0
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0
100
150
150
250
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Figure 5: z(t), zf (t) and the filtering error ε(t).
switched delay filtering error system by utilizing the input-delay approach. Us-
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ing tools and techniques from average dwell time, multiple Lyapunov functional, and free-weighting matrices, sufficient conditions under which the switched filtering error system is exponentially stable with a prescribed H∞ disturbance 255
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attenuation performance have been derived in terms of linear matrix inequalities. A numerical example and simulation results are provided to demonstrate
CE
the effectiveness of the proposed design scenario.
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Figure 6: The event-triggering inter-intervals.
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Periodic event-based asynchronous filtering of switched systems Guoqi Ma, Xinghua Liu, Prabhakar R. Pagilla, Shuzhi Sam Ge PII: DOI: Reference:
S0016-0032(19)30257-1 https://doi.org/10.1016/j.jfranklin.2019.04.010 FI 3884
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
3 July 2018 2 February 2019 9 April 2019
Please cite this article as: Guoqi Ma, Xinghua Liu, Prabhakar R. Pagilla, Shuzhi Sam Ge, Periodic event-based asynchronous filtering of switched systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.04.010
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Periodic event-based asynchronous filtering of switched systems✩
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Guoqi Maa , Xinghua Liub , Prabhakar R. Pagillaa,∗, Shuzhi Sam Gec a J.
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Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, United States b Department of Electrical Engineering, School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China c Department of Electrical and Computer Engineering, National University of Singapore, 117583, Singapore
Abstract
This paper provides a periodic event-based asynchronous H∞ filter design for switched systems subject to limited communication resources. The governing equations for the physical system are combined with those of the proposed event-based asynchronous filter and the resulting equations are formulated as
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a switched filtering error system with time-varying delay. For the achieved switched delay filtering error system, sufficient conditions in terms of linear
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matrix inequalities are obtained to achieve exponential stability and a prescribed H∞ disturbance attenuation level. Tools and techniques from delay-dependent Lyapunov theory, free-weighting matrices, and average dwell time for switched
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systems are used to obtain the main results. Further, a synthesis procedure for the filter gains, the event-triggering condition parameters, and a lower bound on
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the average dwell time of the switching signal is provided. A numerical example on a switched RLC circuit is finally provided to evaluate the proposed design and results.
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Keywords: Periodic event-detection, Switched systems, Average dwell time, Exponential stability, Asynchronous H∞ filtering.
✩ The
material in this paper was not present at any conference. author. Tel.: +1 (979) 458-4829 E-mail addresses: [email protected] (Guoqi Ma), [email protected] (Xinghua Liu), [email protected] (Prabhakar R. Pagilla), [email protected] (Shuzhi Sam Ge) ∗ Corresponding
Preprint submitted to Journal of The Franklin Institute
April 25, 2019
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1. Introduction As an important class of hybrid dynamical systems that exhibit interactive continuous and discrete dynamics, switched systems are composed of finite sub-
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systems (or modes) described by a group of differential or difference equations
and a switching signal that takes values from a finite integer set to orchestrate the switching between subsystems. In the past several decades, switched systems
have received considerable attention due to their applicability and effectiveness in describing a wide range of practical engineering systems, such as boost con-
verter circuits Ren et al. (2018), flight control systems Lian et al. (2017), power
systems Cardim et al. (2009), networked control systems Donkers et al. (2011),
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chemical processes Niu et al. (2015), and vehicle platoons Malloci et al. (2012). Due to the unavailability of state information that could be encountered in most practical control systems, it is important to estimate the system states in the presence of noise; state estimation can be be achieved by the Kalman filter Kalman (1960) if the external noise satisfies a normal probability distri-
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bution. When such a priori information on the external noise is not known, some new filters, e.g. H∞ filter, have been proposed. In recent years, state es-
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timation or filtering problems for switched systems have received attention; see for example, Mahmoud and Shi (2012), Lian et al. (2013), Wang et al. (2016), Zhang et al. (2017a), Xiao et al. (2017), Zhang et al. (2017b, 2018), Wang et al.
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(2018) and references therein. However, in most existing work, the output measurements are continuously transmitted to the filter regardless of whether it is
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really necessary or not. This not only creates problems for practical implementation of the filter but also can result in unnecessary usage of communication
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resources, which can render the filter inapplicable in systems subject to limited
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communication resources. Therefore, it is necessary to investigate the problem of alleviating communication burden in the filtering system, while achieving a satisfactory filtering performance; this problem is addressed in this paper. Recently, there has been a substantial increase in research related to event-
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based communication mechanisms Abdelrahim et al. (2018), Hu et al. (2018),
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He et al. (2018), Li et al. (2017), Zou et al. (2017), Khashooei et al. (2017), Liu et al. (2017a), Meng and Chen (2014). Compared with sampled-data (also called time-triggered) communication mechanisms, in which the signal is sam-
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pled and transmitted at a fixed rate without paying much attention to the system status, the main advantages of event-based communication mechanisms
are twofold: (1) reduction in communication frequency within the system loop resulting in relieving communication bandwidth and (2) reduction in the update
frequency of the actuator, which can alleviate actuator wear. These are achieved by incorporating an event-detecting mechanism in the feedback channel to con-
tinuously or discretely detect the “event” which is characterized by whether or
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not the designed error signal has violated a prescribed threshold. Only when the “event” occurs, a new signal is triggered and transmitted, otherwise, the signal is kept unchanged via a zero-order hold (ZOH). The application of eventbased communication mechanisms can be found in linear systems Tanwani et al. 45
(2016), stochastic systems Wu et al. (2017), fuzzy systems Liu et al. (2017b),
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multi-agent systems Viel et al. (2017), networked control systems (NCSs) Yin et al. (2016). However, results on the application of event-based communication
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mechanisms for switched systems is sparse, except for some recent work found in Xiao et al. (2017), Li and Fu (2017), Qi et al. (2017), Ma et al. (2016); these 50
existing event-based schemes rely on a continuous event-detection mechanism,
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which is not only difficult to implement in practical applications but also can lead to Zeno behavior Tabuada (2007).
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In this paper, the problem of periodic event-based asynchronous filtering for switched systems with limited communication capacity is considered. By
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utilizing only the event-based output measurement input to the filter, one can
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reduce the communication load between the output and the filter while achieving similar or even better filtering performance. In addition, to further relieve the communication burden, a periodic event-detecting mechanism instead of a continuous one is synthesized between the system measured output and the fil-
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ter to periodically detect the event-triggering condition. This condition is an inequality constructed via the output measurement and its error signal to char3
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acterize the variation of the output measurement from the last event-triggering instant. Only when the event-triggering condition is satisfied, i.e., the inequality is violated, an “event” occurs and the current output measurement is transmitted to the filter via a ZOH. Besides, by taking into account the switching delay
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between the filter and the subsystem, an asynchronous filter is considered. The
overall periodic event-based asynchronous switched filter system is modeled as
a switched system with time-varying delays. Based on this model, sufficient conditions are obtained for the overall system to be exponentially stable with a 70
prescribed H∞ disturbance attenuation level. Furthermore, a co-design proce-
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dure to synthesize the asynchronous filter gains and event-triggering condition parameters is provided and a lower bound on the average dwell time that will facilitate the switching signal design is obtained. Compared with existing results, the main contributions of this paper are as follows:
(i) A periodic event-based mode-dependent filter is designed for switched sys-
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tems subject to limited communication resources. The event detector samples the event-triggering condition periodically rather than continuously. (ii) The switching lag between the filter and the subsystem is also taken into
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consideration, and thus an asynchronous filter is designed. (iii) By constructing a novel Lyapunov function, the filter gains, event-triggering
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condition parameters, and the switching signal are co-designed in a unified framework, without unnecessary constraints on the switching signal. The remainder of the paper is organized as follows. Section 2 presents the
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system description and problem statement. Section 3 gives the description of
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the periodic event-triggering mechanism (PETM) and the asynchronous filter.
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Section 4 contains some mathematical preliminaries necessary for deriving the main results. Section 5 provides the main theoretical results. A numerical example is performed in Section 6, followed by remarks concluded in Section 7.
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The following notation is employed throughout the paper. N, R+ , and R+
denote the sets of natural numbers, positive real numbers, and nonnegative real numbers, respectively; L2 [0, ∞) represents the space of square integrable infi4
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nite sequences; λ(A), λmax (A), and λmin (A) denote the eigenvalues, maximum eigenvalue, and minimum eigenvalue of a matrix A ∈ Rn×n , respectively. The superscripts ‘>’ and ‘−1’ are used for matrix transpose and matrix inverse, respectively; A 0(≺ 0) is used to indicate that the matrix A is positive (negative)
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definite; sym(A) represents A + A> . A(i,j) denotes the block matrix element at
the i-th row group and j-th column group of a block matrix A partitioned into m row groups and n column groups, denoted as [A]m×n . The terms induced by
symmetry in a matrix are denoted by ‘?’. Compatible dimensions are assumed for matrices whose dimensions are not explicitly stated.
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2. System description and problem statement
Consider a class of switched systems of the form ˙ = Aσ(t) x(t) + Bσ(t) ω(t), x(t) y(t) = Cσ(t) x(t), z(t) = D x(t),
(1)
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σ(t)
where x(t) ∈ Rn is the state vector, y(t) ∈ Rp is the measured output, z(t) ∈ Rq
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is the signal to be estimated, ω(t) ∈ Rl denotes the external disturbance and is assumed to belong to L2 [0, ∞). Aσ(t) , Bσ(t) , Cσ(t) , Dσ(t) are system matrices of compatible dimensions pertaining to subsystems indicated by the switching
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signal σ(t). Let N = {1, 2, · · · , N } be a finite index set. The switching signal σ(t) : [0, ∞) → N is a right-continuous, piecewise constant function, which
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orchestrates the switching among the subsystems, that is, corresponding to the
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switching signal σ(t) there exists a switching sequence {(i0 , t0 ), (i1 , t1 ), · · · , (ik , tk ), · · · |ik ∈ N , k ∈ N} ,
(2)
where tk ∈ R+ denotes switching instants, which means that the subsystem ik
is activated at tk and remains active for all t ∈ [tk , tk+1 ). Based on Duan and Wu (2014), a switching signal σ(t) is said to have a dwell time τD if tk+1 − tk ≥
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τD , ∀k ∈ N, and is said to have an average dwell time τa Liberzon (2012) if 5
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Nσ (T1 , T2 ) ≤ N0 +
T2 −T1 τa
for any 0 ≤ T1 ≤ T2 , where Nσ (T1 , T2 ) denotes the
number of switchings of σ(t) over (T1 , T2 ). Given the above switched system, the problem is to co-design a periodic
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event-triggering mechanism and an asynchronous filter to estimate the signal z(t). The goal of the periodic event-triggering mechanism is to provide the measured output as the input to the filter only when the change of the output is large enough as characterized by an event-triggering condition. We will formalize this in the next section.
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3. Periodic event-triggering mechanism and asynchronous filter
The proposed design strategy for the overall system is illustrated in Fig. 1.
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Besides the switched system, it mainly consists of two modules: (i) the periodic event-detector; and (ii) the filter system. The event-detector, with a period of T , samples the measured system output and evaluates the event-triggering
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condition to determine whether the input to the filter needs to be updated or not. The governing equations for the periodic event detector and the filter are
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given in the following.
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(t )
(t )
Switched System
Sensor
T
z (t )
y (t )
Event Detector
Clock
z f (t )
yˆ(t )
y (tˆn )
Filter ZOH Figure 1: Overall system configuration.
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3.1. Periodic event-detector Let the event-triggering instants be given by tˆn = ˆin T , where ˆin , n ∈ N,
and {ˆi0 , ˆi1 , ˆi2 , · · · } ⊂ {0, 1, 2, 3, · · · } with ˆi0 = 0 and ˆin < ˆin+1 . Note that for
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∀n ∈ N we have tˆn+1 − tˆn ≥ T . Define ln,j = (ˆin + j)T , j = 0, 1, 2, · · · dn , where Sd n dn = ˆin+1 − ˆin − 1. Then, tˆn , tˆn+1 = j=0 [ln,j , ln,j+1 ). Further, denote the
input to the filter as yˆ(t), then when t ∈ [ln,j , ln,j+1 ), n ∈ N, j = 0, 1, 2, · · · dn , then an error signal on the measured output is defined as ey (t) = yˆ(t) − y(ln,j ).
(3)
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Based on ey (t) defined in (3), we consider the following mode-dependent eventtriggering condition:
> e> y (t)Ω1,σ(t) ey (t) ≤ φy (ln,j )Ω2,σ(t) y(ln,j ), t ∈ [ln,j , ln,j+1 ) ,
(4)
where φ ∈ R+ is an adjustable design parameter, and Ω1,σ(t) , Ω2,σ(t) 0 are
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mode-dependent positive definite matrices, which are to be designed. Remark 3.1. It should be pointed that when the inequality (4) is violated,
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> i.e., e> y (t)Ω1,σ(t) ey (t) > φy (ln,j )Ω2,σ(t) y(ln,j ), we say the event-triggering con-
dition is satisfied. Nonetheless, at the event-triggering instants, the error signal ey (t) will be reset to zero, together with the fact that ey (t) is a piecewise con-
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stant and continuous from the right signal, then, the inequality (4) is held for all time.
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To model the overall periodic event-based filter system in terms of the time variable t, motivated by Zhang and Feng (2014), define τ (t) = t − ln,j , when t ∈ [ln,j , ln,j+1 ), thus, τ (t) can be considered as a time-varying delay satisfying
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τ (t) ∈ [0, T ) and τ˙ (t) = 1. With this definition of τ (t), y(ln,j ) can be rewritten
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as y (t − τ (t)) and the input to the filter as yˆ(t) = ey (t) + y (t − τ (t)). We will use this later when we formulate the filtering error system.
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3.2. Filter system model
f
f,σf (t) f
(5)
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The asynchronous filter is described by x˙ (t) = A ˆ(t), f f,σf (t) xf (t) + Bf,σf (t) y z (t) = C x (t),
where xf (t) denotes the filter state vector, zf (t) is the estimate of z(t), and Af,σf (t) , Bf,σf (t) , Cf,σf (t) are filter gains corresponding to the filter subsystems specified by the filter switching signal σf (t). Ideally, one can choose σf (t) to be
the same as σ(t), which means that one can instantaneously detect the switching of the subsystems of the switched system and match the corresponding filter
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subsystem. However, in reality, there will inevitably be a lag between the two, and we define it as d ∈ R+ . Hence, we model σf (t) in the following manner: σf (t) = σ(t − d).
(6)
The switching lag d is assumed to satisfy 0 < d < τD , in which τD is the dwell
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time of σ(t).
3.3. Modeling of the filtering error system
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When t ∈ [tk , tk+1 ), ∀k ∈ N, assume σ(tk ) = i, σf (tk ) = j, i, j ∈ N , i 6= j. h i> Moreover, define ε(t) = z(t) − zf (t) as the filtering error, ζ(t) = x> (t), x> (t) f as the augmented state vector. Then, we can obtain the following filtering error
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system equations: ˙ = Aij ζ(t) + Bi ω(t) + Cj ey (t) + Dj y (t − τ (t)) , ζ(t) t ∈ [tk , tk + d) ε(t) = Eij ζ(t), ˙ = Aii ζ(t) + Bi ω(t) + Ci ey (t) + Di y (t − τ (t)) , ζ(t) t ∈ [tk + d, tk+1 ) ε(t) = Eii ζ(t),
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(7)
h Ai 0 Bi 0 , Bi = , Cj = Dj = , Eij = Ei where Aij = 0 Af,j 0 Bf,j h i Ai 0 0 , Ci = Di = , Eii = Ei −Cf,i . Aii = 0 Af,i Bf,i 8
i −Cf,j ,
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4. Preliminaries Before proceeding with the main results, we first present the following definitions and lemmas, which play a key role in the proof of the main results.
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Definition 4.1. Wu et al. (2009) The equilibrium ξ ∗ = 0 of the filtering error system (7) with ω(t) = 0 is said to be exponentially stable under σ(t) if its solution ζ(t) satisfies kζ(t)k ≤ ηkζt0 kC 1 e−λ(t−t0 ) ,
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n o ˙ + θ)k . for η ≥ 1 and λ > 0, and kζt kC 1 = sup−d≤θ≤0 kζ(t + θ)k, kζ(t
Definition 4.2. Xiao et al. (2017) For a given scalar > 0, the filtering error system (7) is said to be exponentially stable with a weighted H∞ performance index (, γ) if the following two conditions hold:
(i) The filtering error system (7) is exponentially stable when ω(t) = 0;
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(ii) Under zero initial conditions and for any nonzero ω(t) ∈ L2 [0, ∞], the following inequality is satisfied:
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Z∞ 0
e
−s >
ε (s)ε(s)ds ≤ γ
2
Z∞
ω > (s)ω(s)ds.
0
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Lemma 4.1. For any real vectors u, v with appropriate dimensions and symmetric positive definite matrix Q with compatible dimension, the following
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inequality holds:
u> v + v > u ≤ u> Qu + v > Q−1 v.
Lemma 4.2. Let W ∈ Rn×n be a symmetric matrix, and let x ∈ Rn , then
the following inequality holds: λmin (W )x> x ≤ x> W x ≤ λmax (W )x> x.
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5. Main results In this section, we will first determine the sufficient conditions under which the filtering error system (7) is exponentially stable when the disturbance ω(t) =
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0. Then, we will derive conditions under which the H∞ attenuation performance is achieved for any nonzero ω(t) ∈ L2 [0, ∞]. 5.1. Exponential stability analysis
We will use tools from delay-dependent Lyapunov functions, free-weighting
matrices, and average dwell time techniques, to derive sufficient conditions which
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can ensure the filtering error system (7) is exponentially stable in the following theorem.
Theorem 5.1. Given parameters α ∈ R+ , β ∈ R+ , φ ∈ R+ , µ ≥ 1, d ∈ R+ ,
and T ∈ R+ , the filtering error system (7) with ω(t) = 0 is exponentially stable
under any switching signal σ(t) with average dwell time τa satisfying τa > τa∗ =
if for ∀σ(t) = i ∈ N , there exist positive definite matrices
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2 ln µ+(α+β)(d+T ) , α
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Pi 0, Qi 0, Ri 0, and arbitrary compatible matrices Mi , Ni such that Ni Mi 0, [Ξi ] Λi = 4×4 ≺ 0, ? e−αT Ri (1,1)
where Ξi is symmetric and its block matrix elements are given by Ξi
=
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(1,2) (1,3) (1,4) > sym {Pi Ai } + αPi + T A> = 0, Ξi = 0, Ξi = Ci> Mi , i Ci Ri Ci Ai , Ξi (2,2) (2,3) (2,4) (3,3) Ξi = sym {Qi Af,i + αQi }, Ξi = Qi Bf,i , Ξi = Qi Bf,i , Ξi = −Ω1,i , (3,4) (4,4) Ξi = 0, Ξi = −sym {Mi } + φΩ2,i + τ (t)Ni .
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And for ∀i ∈ N , j ∈ N , i 6= j, Nj Mj 0, [Θij ] Πij = 4×4 ≺ 0, ? Rj
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(1,1)
where Θij is symmetric and its block matrix elements are given by Θij (1,2)
> sym {Pj Ai } − βPj + T A> i Ci Rj Ci Ai , Θij (2,2)
Θij
(3,4) Θij
(2,3)
= sym {Qj Af,j }−βQj , Θij = 0,
(4,4) Θij
(1,3)
= 0, Θij (2,4)
= Qj Bf,j , Θij
= −sym {Mi } + τ (t)Ni + φΩ2,i . 10
(1,4)
= 0, Θij
= Ci> Mj ,
(3,3)
= Qj Bf,j , Θij
=
= −Ω1,i ,
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Proof 5.1. Choose a piecewise Lyapunov-Krasovskii functional candidate of the form
where k ∈ N and
V1,σ(t) (t) = ζ (t)diag Pσ(t) , Qσ(t) ζ(t) + >
>
(8)
V2,σf (t) (t), t ∈ [tk , tk + d) , Z0 Zt
eα(s−t) y˙ > (s)Rσ(t) y(s)dsdθ, ˙
−T t+θ
Z0 Zt
eβ(t−s) y˙ > (s)Rσf (t) y(s)dsdθ. ˙
AN US
V2,σf (t) (t) = ζ (t)diag Pσf (t) , Qσf (t) ζ(t) +
CR IP T
V (t) =
V1,σ(t) (t), t ∈ [tk + d, tk+1 ) ,
−T t+θ
First, for any positive real numbers α, β, it is held that Z0 Zt
−T t+θ
e
α(s−t) >
y˙ (s)Ri y(s)dsdθ ˙ ≤
M
≤
Z0 Zt
y˙ > (s)Ri y(s)dsdθ ˙
Z0 Zt
eβ(t−s) y˙ > (s)Ri y(s)dsdθ. ˙
−T t+θ
(9)
−T t+θ
ED
Second, assume for ∀i, j ∈ N = {1, 2, · · · , N }: Pi ≤ µPj , Qi ≤ µQj , Ri ≤ µRj , µ ≥ 1.
(10)
CE
PT
Then, combining conditions (9) and (10), at time instant tk + d, we obtain V1,σ(t) (tk + d) ≤ µV2,σf (t) (tk + d)− .
(11)
AC
On the other hand, at the time instant tk , we can obtain V2,σf (t) (tk ) ≤ µe(α+β)T V1,σ(t) (t− k ).
(12)
Consider t ∈ [tk , tk + d), and assume σ(t) = i ∈ N , σf (t) = j ∈ N , i 6= j. Taking the derivative of V2,σf (t) (t) with respect to t along the trajectories of the
11
ACCEPTED MANUSCRIPT
filtering error system (7) yields V˙ 2,σf (t) (t) = 2x> (t)Pj x(t) ˙ + 2x> ˙ f (t) + β f (t)Qj x
−T t+θ
e−βθ y˙ > (t + θ)Rj y(t ˙ + θ)dθ
−T
≤ 2x> (t)Pj x(t) ˙ + 2x> ˙ f (t) + β f (t)Qj x
Z0 Zt
eβ(t−s) y˙ > (s)Rj y(s)dsdθ ˙
−T t+θ
Zt
y˙ > (s)Rj y(s)ds. ˙
(13)
AN US
+ T y˙ > (t)Rj y(t) ˙ −
eβ(t−s) y˙ > (s)Rj y(s)dsdθ ˙
CR IP T
+ T y˙ > (t)Rj y(t) ˙ −
Z0
Z0 Zt
t−τ (t)
Then, we can get
V˙ 2,σf (t) (t) − βV2,σf (t) (t) > ≤ x> (t) sym {Pj Ai } − βPj + T A> i Ci Rj Ci Ai x(t)
M
> + x> f (t) [sym {Qj Af,j } − βQj ] xf (t) + 2xf (t)Qj Bf,j ey (t)
+ 2x> f (t)Qj Bf,j y(t − τ (t)) −
Zt
y˙ > (s)Rj y(s)ds. ˙
(14)
t−τ (t)
two identities:
ED
For arbitrary compatible matrices Mi and Ni , one can obtain the the following
>
h
PT
2y (t − τ (t))Mi y(t) − y(t − τ (t)) −
Zt
t−τ (t)
CE
>
τ (t)y (t − τ (t))Ni y(t − τ (t)) −
Zt
i y(s)ds ˙ = 0.
y > (t − τ (t))Ni y(t − τ (t))ds = 0.
(15)
(16)
t−τ (t)
AC
In addition, the event-triggering condition (4) implies > −e> y (t)Ω1,i ey (t) + φy (t − τ (t))Ω2,i y(t − τ (t)) ≥ 0.
(17)
Then, based on (14), (15), (16), and (17), we obtain V˙ 2,σf (t) (t) − βV2,σf (t) (t) ≤ ξ (t)Θij ξ(t) − >
Zt
t−τ (t)
12
%> (t, s)Πij %(t, s)ds,
(18)
ACCEPTED MANUSCRIPT
h i> > > > where ξ(t) = x> (t), x> , %(t, s) = y > (t − τ (t)), y˙ > (s) . f (t), ey (t), y (t − τ (t)) Therefore, if Πij 0 and Θij ≺ 0 are satisfied, it can be got from (18) that V˙ 2,σf (t) (t) − βV2,σf (t) ≤ 0.
CR IP T
(19)
Integrating (19) from tk to t, t ∈ [tk , tk + d), together with (11) and (12), it can be deduced that V (t) ≤ eβ(t−tk ) V2 (t+ k)
≤ µeβd e(α+β)T e−α(tk −tk−1 −d) V1 (tk−1 + d)
AN US
≤ µ2 e2βd e(α+β)T e−α(tk −tk−1 −d) V2 (t+ k−1 )
≤ µ3 e2βd e2(α+β)T e−α(tk −tk−2 −2d) V1 (tk−2 + d)+ .. .
≤ e(α+β)d e[
2 ln µ+(α+β)(d+T ) −α τa
](t−t0 ) V (t ). 0
(20)
Moreover, according to the definition of V (t) in (8), together with Lemma 4.2,
M
one can obtain
(21)
ED
V (t) ≥ c1 kζ(t)k2 , n o where c1 = min min λ (Pi ) , min λ (Qi ) . i∈N
i∈N
On the other hand, it can be deduced that
PT
V (t0 ) ≤ λmax (Pi )x> (t0 )x(t0 ) + λmax (Qi )x> f (t0 )xf (t0 )
CE
+ T eβT λmax (Ri )λmax (Ci> Ci )ke x(t0 )k2
e 0 )k2 , ≤ c2 kζ(t
where x e(t0 ) = max
−T ≤θ≤0
(22)
e 0 ) = max kζ(θ)k, kζ(θ)k ˙ kx(θ)k ˙ , ζ(t , −T ≤θ≤0
c2 = max{λmax (Pi ), λmax (Qi ), T eβT λmax (Ri )λmax (Ci> Ci )}.
AC
Then, by combining (20), (21), and (22), we get r ) c2 (α+β)d [ 2 ln µ+(α+β)(d+T −α e 0 )k. 2τa 2 ](t−t0 ) kζ(t kζ(t)k ≤ e 2 e c1
(23)
The same result be achieved when t ∈ [tk + d, tk+1 ), if Λi 0 and Ξi ≺ 0 are
170
satisfied. Therefore, according to Definition 4.1, the filtering error system (7) is exponentially stable when ω(t) = 0. This completes the proof. 13
ACCEPTED MANUSCRIPT
Remark 5.1. Note that in (7), if we write y (t − τ (t)) as Cσ(t−τ (t)) x(t − τ (t)) + Dσ(t−τ (t)) ν(t − τ (t)), then σ(t − τ (t)) is difficult to handle in the stability analysis. Although a switching signal merging technique Vu and Morgansen (2010) is utilized in Xiao et al. (2017), where at most one subsystem switching
CR IP T
175
is allowed within any two consecutive event-triggering instants, which places
restrictions on the switching signal σ(t). A constraint on the switching signal σ(t) was made in Ren et al. (2019) as well when describing the delayed term of
the measured output. The method in this paper does not place such restrictions. 5.2. H∞ performance analysis and co-design procedure
AN US
180
In this subsection, we will investigate the H∞ performance of the filtering error system (7) with any nonzero ω(t) ∈ L2 [0, ∞]. We will provide sufficient conditions that can guarantee a prescribed H∞ performance for the filtering error system (7) and a co-design procedure for the filter gains, the event-triggering 185
condition parameters, and the switching signal are provided by the following
M
theorem.
Theorem 5.2. Given parameters α ∈ R+ , β ∈ R+ , φ ∈ R+ , µ ≥ 1, γ ∈ R,
ED
d ∈ R+ , and T ∈ R+ , the filtering error system (7) has a prescribed H∞ disturbance attenuation level for any nonzero ω(t) ∈ L2 [0, ∞] under any switching signal σ(t) with average dwell time τa satisfying τa > τa∗ =
2 ln µ+(α+β)(d+T ) , α
PT
if for ∀i ∈ N , there exist positive definite matrices Pi 0, Qi 0, Ri 0, Ω1,i 0, Ω2,i 0, and arbitrary compatible matrices Mi , Ni , Hi , W1,i , W2,i ,
AC
CE
W3,i such that Λi 0 hold, and Υi =
Υ1i
Υ2i
?
−I
≺ 0,
(24)
where [Υ1i ]6×6 is symmetric block matrix, [Υ2i ]6×2 is a block matrix, and the (1,1)
block elements in Υ1i and Υ2i are given by Υ1i (1,2)
Υ1i
190
(1,3)
= −Ei> W3,i , Υ1i
Pi B i ,
(2,2) Υ1i
(2,6) Υ1i
= 0,
(1,4)
= 0, Υ1i
= sym {Pi Ai } + αPi + Ei> Ei , (1,5)
= Ci> Mi , Υ1i
(2,3) (2,4) = sym {W1,i } + αQi , Υ1i = W2,i , Υ1i = W2,i , (3,3) (3,4) (3,5) (3,6) Υ1i = −Ω1,i , Υ1i = 0, Υ1i = 0, Υ1i =
14
(1,6)
=
(2,5) Υ1i = (4,4) 0, Υ1i
0,
> = A> i Ci Hi , Υ1i
=
ACCEPTED MANUSCRIPT
(4,5)
= 0, Υ1i
2
(2,1) Υ2i
−sym {Mi } + T Ni + φΩ2,i , Υ1i
(5,6) Υ1i
= Hi C i B i ,
(6,6) Υ1i
= −γ I,
(4,6)
=
(5,5)
= 0, Υ1i
> W3,i ,
= −sym {Hi } + T Ri ,
the other elements in Υ2i are
195
And for ∀i, j ∈ N , i 6= j, Πij 0, and S1ij S2ij ≺ 0, Sij = ? −I
CR IP T
all zero blocks.
(25)
where [S1ij ]6×6 is symmetric block matrix, [S2ij ]6×2 is a block matrix, and the (1,1)
block elements in S1ij and S2ij are given by S1ij = sym {Pj Ai }−βPj +Ei> Ei , (1,2)
S1ij
(1,3)
= −Ei> W3,j , S1ij
(1,4)
= 0, S1ij
200
(1,6)
> = A> i Ci Hj , S1ij
=
(2,3) (2,4) (2,5) P j Bi , = sym {W1,j } − βQj , S1ij = W2,j , S1ij = W2,j , S1ij = 0, (2,6) (3,3) (3,4) (3,5) (3,6) (4,4) S1ij = 0, S1ij = −Ω1,i , S1ij = 0, S1ij = 0, S1ij = 0, S1ij = (4,5) (4,6) (5,5) −sym {Mj } + φΩ2,i + T Nj , S1ij = 0, S1ij = 0, S1ij = −sym {Hj } + T Rj , (5,6) (6,6) (2,1) S1ij = Hj Ci Bi , S1ij = −γ 2 I, S2ij = Cf,j , the other elements in S2ij are
AN US
(2,2) S1ij
(1,5)
= Ci> Mj , S1ij
zero blocks.
M
In addition, the filter gains are given by
−1 Af,i = Q−1 i W1,i , Bf,i = Qi W2,i , Cf,i = W3,i .
(26)
ED
Proof 5.2. Choose the same piecewise Lyapunov-Krasovskii functional candidate as in (8), and define z(t) = ε> (t)ε(t) − γ 2 ω > (t)ω(t). Consider the asynchronous period, i.e. t ∈ [tk , tk + d), assume that σ(t) = i ∈ N , σf (t) = j ∈ N .
PT
Taking the derivative of V (t) with respect to t along the trajectories of (7) and adding the term −βV2 (t) + z(t) on both sides, we obtain
AC
CE
V˙ 2 (t) − βV2 (t) + z(t) h i ≤ x> (t) sym {Pj Ai } − βPj + Ei> Ei x(t) h i > + x> (t) sym {Q A } − βQ + C C xf (t) j f,j j f,j f f,j + 2x> (t)Pj Bi ω(t) − 2x> (t)Ei> Cf,j xf (t)
> + 2x> f (t)Qj Bf,j ey (t) + 2xf (t)Qj Bf,j y(t − τ (t)) >
2
>
+ T y˙ (t)Rj y(t) ˙ − γ ω (t)ω(t) −
Zt
t−τ (t)
15
y˙ > (s)Rj y(s)ds. ˙
(27)
ACCEPTED MANUSCRIPT
Based on the system equation (1), we obtain y(t) ˙ = Ci Ai x(t) + Ci Bi ω(t).
(28)
CR IP T
Using the above equation, we have
2y˙ > (t)Hi Ci Ai x(t) + Ci Bi ω(t) − y(t) ˙ = 0,
(29)
where Hi is an arbitrary matrix of compatible dimension.
Then, combining (15), (16), (17), (27), and (29), we have Zt
%> (t, s)Πij %(t, s)ds,
AN US
V˙ 2 (t) − βV2 (t) + z(t) ≤ χ> (t)Xij χ(t) −
(30)
t−τ (t)
h i> > > > > where χ(t) = x> (t), x> (t), e (t), y (t − τ (t)), y ˙ (t), ω (t) , [Xij ]6×6 is symy f (1,1)
metric block matrix, whose elements are given by Xij Di> Di , (1,6)
(1,2) Xij
(2,2)
= 0,
(1,4) Xij
= sym {Pj Ai } − βPj + (1,5)
= Ci> Mj , Xij
> = A> i C i Hj ,
(2,3)
M
> = sym {Qj Af,j } − βQj + Cf,j Cf,j , Xij
= Qj Bf,j ,
(2,5) (2,6) (3,3) (3,4) (3,5) = Qj Bf,j , Xij = 0, Xij = 0, Xij = −Ω1,i , Xij = 0, Xij = (4,4) (4,5) (4,6) = 0, Xij = −sym {Mj } + φΩ2,i + τ (t)Nj , Xij = 0, Xij = (5,6) (6,6) 2 = −sym {Hj } + T Rj , Xij = Hj Ci Bi , Xij = −γ I, (4,4) that there exists a time-varying term in Xij , hence we decompose Xij
PT
follows:
ED
(2,4) Xij (3,6) Xij (5,5) Xij
Note
= −Di> Cf,j ,
= Pj Bi , Xij
Xij
(1,3) Xij
Xij =
CE
where
T − τ (t) b τ (t) e Xij + Xij , T T
0, 0,
as
(31)
b ij = Xij −diag {0, 0, 0, τ (t)Nj , 0, 0} , X e ij = Xij +diag {0, 0, 0, (T − τ (t))Nj , 0, 0} , X
AC
from which, one can obtain
e ij = X b ij + diag {0, 0, 0, T Nj , 0, 0} , X
e ij ≺ 0 ⇒ X b ij ≺ 0, then, if X e ij ≺ 0, we have which implies X Xij =
τ (t) e T − τ (t) b Xij + Xij ≺ 0. T T 16
(32)
ACCEPTED MANUSCRIPT
e ij , using Schur Complement Lemma Note that, there are nonlinear terms in X
CR IP T
e ij converts X e ij into its equivalent form as two times on X Ψ11,ij Ψ12,ij Ψ13,ij (33) Ψij = ? −I 0 , ? ? −I n o e ij − diag 0, C > Cf,j , 0, 0, 0, 0 , Ψ12,ij = [0, Cf,j , 0, 0, 0, 0]> , where Ψ11,ij = X f,j >
Ψ13,ij = [0, 0, 0, 0, Hj Di , 0] .
Then, by letting Qj Af,j = W1,j , Qj Bf,j = W2,j , Cf,j = W3,j in Ψij , we can
AN US
obtain (25). Therefore, if conditions Πij 0 and Sij ≺ 0 given in Theorem 5.2 are satisfied, we can obtain
V˙ 2,σf (t) (t) − βV2,σf (t) (t) + z(t) < 0.
(34)
Integrating (34) from tk to t, t ∈ [tk , tk + d), it can be derived that
≤ µe
tk
V1 (t− k)
eβ(t−s) z(s)ds
−
Zt
e
β(t−s)
ED
(α+β)T +βd
Zt
M
V2 (t) ≤ eβ(t−tk ) V2 (tk ) −
tk
2 (α+β)T +2βd−α(tk −tk−1 −d)
PT
≤µ e
eβ(t−s) z(s)ds
tk
Ztk
(α+β)T +βd
V2 (tk−1 ) − µe
e−α(tk −s) z(s)ds
tk−1 +d
− µ2 e(α+β)T +2βd−α(tk −tk−1 −d)
CE
z(s)ds −
Zt
tk−1 Z +d tk−1
eβ(tk −s) z(s)ds −
Zt
eβ(t−s) z(s)ds
tk
AC
.. .
≤ e(α+β)d e[
where Φ2 (t) =
2 ln µ+(α+β)(d+T ) −α](t−t0 ) τa
k P
Rtj
j=1 tj−1 +d
e[Nσ (s,t)−1]αd z(s)ds +
V (t0 ) − Φ2 (t),
e[2Nσ (s,t)−1] ln µ−α(tk −s)+Nσ (s,t)(α+β)T +Nσ (s,t)βd ×
k P
tj−1 R +d
j=1 tj−1
e2Nσ (s,t) ln µ−αtk +(α+β)tj−1 +β[Nσ (s,t)+1]d ×
eαNσ (s,t)d+Nσ (s,t)(α+β)T e−βs z(s)ds +
Rt
tk
eβ(t−s) z(s)ds.
17
(35)
ACCEPTED MANUSCRIPT
Under zero initial condition, we have Φ2 (t) ≤ 0.
(36)
of the above inequality yields e 2 (t) ≤ Φ b 2 (t), Φ
where
CR IP T
Then, multiplying e−2Nσ (t0 ,t) ln µ−Nσ (t0 ,t)(α+β)(d+T )+ln µαd−αt+αtk on both sides
(37)
tj−1 Z +d k X e 2 (t) = Φ e−2Nσ (0,s) ln µ+ln µ−αt+(α+β)tj−1 −(α+β)[Nσ (0,s)−1]d−βs ×
e
−Nσ (0,s)(α+β)T >
AN US
j=1 t j−1
ε (s)ε(s)ds +
Zt
eβ(t−s)+ln µ+αd−αt+αtk ε> (s)ε(s)ds
tk
+
k X
Ztj
e−2Nσ (0,s) ln µ−α(t−s)−Nσ (0,s)(α+β)(d+T ) ε> (s)ε(s)ds,
j=1t j−1 +d
M
tj−1 Z +d k X e−2Nσ (0,s) ln µ+ln µ−αt+(α+β)tj−1 −(α+β)[Nσ (0,s)−1]d−βs j=1 t j−1
ED
b 2 (t) =γ 2 Φ
e−Nσ (0,s)(α+β)T ω > (s)ω(s)ds + γ 2
eβ(t−s)+ln µ+αd−αt+αtk ω > (s)ω(s)ds
tk
e−2Nσ (0,s) ln µ−α(t−s)−Nσ (0,s)(α+β)(d+T ) ω > (s)ω(s)ds.
PT
+ γ2
Ztj k X
Zt
j=1t j−1 +d
CE
It is obvious that
eβ(t−s)+ln µ+αd−αt+αtk ε> (s)ε(s)ds ≥ e−αt ε> (s)ε(s),
AC
and
eβ(t−s)+ln µ+αd−αt+αtk ω > (s)ω(s)ds ≤ e−α(t−s)+ln µ+(α+β)d ω > (s)ω(s)ds.
Then, we get Z∞
e
−αs >
ε (s)ε(s)ds < γ
t0
2
Z∞
t0
18
ω > (s)ω(s)ds,
(38)
ACCEPTED MANUSCRIPT
where γ = e
ln µ+(α+β)d 2
γ. The same result can also be guaranteed for t ∈ [tk + d, tk+1 )
if conditions Λi 0 and Υi ≺ 0 given in Theorem 5.2 hold. Then according to Definition 4.2, the H∞ disturbance attenuation performance is ensured for the filtering error system (7). This completes the proof.
CR IP T
205
Remark 5.2. It should be pointed out if the LMI conditions (24) and (25)
210
given in Theorem 5.2 are solvable, then the adjustable design parameter φ in n the event-triggering condition (4) has an upper bound min λmin sym{Mi } − −1 o T Ni Ω−1 , λ sym{M } − T N Ω2,i , which can be derived as follows: min j j 2,i (4,4)
is a diagonal element, we have −sym{Mi } + T Ni + φΩ2,i ≺ 0, which is equivalent to sym{Mi } − T Ni Ω−1 2,i − φI 0, then we can obtain there exist invertible matrices Pi and Qi such that Pi−1 sym{Mi } − T Ni Ω−1 2,i − φI Pi = diag{σ1 , σ2 , · · · , σp }, i.e. Pi−1 sym{Mi } − T Ni Ω−1 2,i Pi = diag{σ1 + −1 −1 φ, σ2 + φ, · · · , σp + φ} and Q sym{Mi } − T Ni Ω2,i Qi = diag{λ1 , λ2 , · · · , λp }, therefore, it can be deduced that φ < min{λ1 , λ2 , · · · , λp }, i.e., φ < λmin sym{Mi }− −1 T Ni Ω−1 ; similarly, from (25), we can get φ < λ sym{M } − T N min j j Ω2,i . 2,i
M
215
AN US
First, as Υ1i
Thus an upper bound of φ can be obtained by taking the smaller value between
220
ED
the above two bounds achieved from (24) and (25), respectively.
6. Simulations
PT
In this section, a switched electrical circuit system Shen et al. (2014) is provided to illustrate the effectiveness of the proposed filter design and the
CE
results in Section 5. A sketch of the switched circuit is provided in Fig. 2. >
Choose x(t) = [iL (t), uC (t)] as state vector, and the voltage of the resistor
AC
as the measured output y(t), ω(t) as the disturbance input. Then this switched
19
ACCEPTED MANUSCRIPT
R
(t )
Subsystem 2
CR IP T
Subsystem 1
iL (t ) L1
L2
uC (t ) C1 !
C2
!
AN US
Figure 2: A switched RLC series circuit.
ED
M
circuit can be modeled as the form of (1) with two subsystems as follows: R 1 1 − − Li Li Li ω(t), x(t) ˙ = x(t) + 1 0 0 Ci | {z } | {z } Ai Bi h i (39) y(t) = R 0 x(t), | {z } Ci z(t) = D x(t). i The values of the resistor, capacity, and inductor are chosen as R = 5Ω,
L1 = 0.1H, L2 = 0.2H, C1 = 10F, C2 = 8F. Moreover, choose z(t) = 0.6uC (t)
PT
225
for subsystem 1 and z(t) = 0.5uC (t) for subsystem 2. The parameters of the
AC
CE
two subsystems are given as follows: Subsystem 1: h i h i −50 −10 10 , B1 = , C1 = 5 0 , D1 = 0 0.6 . A1 = 0.1 0 0 Subsystem 2:
−25
A2 = 0.125
h i h i 5 , B2 = , C2 = 5 0 , D2 = 0 0.5 . 0 0
−5
20
ACCEPTED MANUSCRIPT
We choose α = 0.2, β = 0.1, µ = 1.1, d = 0.2, T = 1, φ = 0.05, then we obtain τa∗ = 2.7531, we choose τa = 3, choose γ = 2.5, then γ = e
ln µ+(α+β)d 2
γ=
Af,2
CR IP T
2.7019. From Theorem 5.2, we obtain the filter gains as follows: h i> h i −0.60 0 , Bf,1 = 0.43 0.73 , Cf,1 = −0.15 0.45 ; Af,1 = 0 −0.60 h i> h i −0.99 0 , Bf,2 = 0.12 0.16 , Cf,2 = −0.13 0.35 . = 0 −0.99
AN US
The event-triggering condition parameters are obtained as
Ω1,1 = Ω2,1 = 25.93; Ω1,2 = Ω2,2 = 62.09.
The switching signals σ(t) and σf (t) are shown in Fig. 3. The initial values of >
>
system states and filter states are chosen as x(0) = [2 − 2] , xf (0) = [5 − 3] . 230
The response of system states x(t) and filter states xf (t) are illustrated in Fig. 4. z(t), zf (t), and the filtering error ε(t) are given in Fig. 5. The event-triggering
M
intervals and the event-triggered input to the filter are plotted in Fig. 6 and Fig. 7, respectively.
ED
235
>
>
In addition, under zero initial conditions, i.e., x(0) = [0 0] , xf (0) = [0 0] , R∞ choose the external disturbance input as ω(t) = e−0.05t with ω 2 (t)dt = 10 < 0
∞, so ω(t) ∈ L2 [0, ∞). Besides, in order to examine the robustness of the filter
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to noise sensitivity, a Gaussian white noise is added to ω(t) with a signal-to-noise ratio (SNR) 10, and denote the synthetic noise signal as ω ˆ (t). The trajectories
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of ω ˆ (t), ω(t), and
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Rt
e−αs ε> (s)ε(s)ds
0
Rt 0
, ω ˆ > (s)ˆ ω (s)ds
Rt
e−αs ε> (s)ε(s)ds
0
Rt
are shown in Fig. 8, and ω > (s)ω(s)ds
0
since γ = 2.7019, the H∞ disturbance attenuation performance is achieved for
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both ω ˆ (t) and ω(t). From the simulation results, it can be seen that the filtering error system is
stable and a prescribed H∞ disturbance attenuation level is achieved. Moreover, the output measurements are only transmitted to the filter when needed, which
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has significantly reduced the communication frequency between the measured output and the filter. 21
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Figure 3: The switching signals σ(t) and σf (t).
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Figure 4: The responses of system states x(t) and filter states xf (t).
7. Conclusions The problem of periodic event-based asynchronous H∞ filtering has been
addressed for switched systems. The overall system has been transformed to a 22
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Figure 5: z(t), zf (t) and the filtering error ε(t).
switched delay filtering error system by utilizing the input-delay approach. Us-
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ing tools and techniques from average dwell time, multiple Lyapunov functional, and free-weighting matrices, sufficient conditions under which the switched filtering error system is exponentially stable with a prescribed H∞ disturbance 255
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attenuation performance have been derived in terms of linear matrix inequalities. A numerical example and simulation results are provided to demonstrate
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the effectiveness of the proposed design scenario.
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