Observer-Based Feedback Stabilization of Networked Control Systems with Random Packet Dropouts

Abstract

This paper is concerned with observer-based feedback stabilization of networked control systems (NCSs) with random packet dropouts. Both sensor-to-controller (S/C) and controller-to-actuator (C/A) packet dropouts are considered, and their behavior is assumed to obey the Bernoulli random binary distribution. The hold-input strategy is adopted, in which the previous packet is used if the packet is lost. An observer-based feedback controller is designed, and sufficient conditions for stochastic stability are derived in the form of linear matrix inequalities (LMIs). A numerical example illustrates the effectiveness of the results.

1. Introduction

In the past few years, dramatic progress has been made for network analysis. Unparticular, many important results have been obtained in engineering, such as the stability analysis and controller design for networked control systems (NCSs) [1–3] and consensus for networked multiagent systems [4, 5]. While NCSs have recently been receiving increasing attention due to their advantages over classical feedback control systems, the insertion of the communication network gives rise to new challenges. Among these challenges, time delays and packet losses are two important factors that can severely degrade the performance of NCSs. In the past few years, much attention has been paid to the time delay problem of NCSs, see [6–10], to name a few. Moreover, some existing control methods for time delays [11] can be used to NCSs with time delay. On the other hand, the control problem of NCSs with packet losses has also attracted considerable research interests (see, e.g., [12, 13] and references therein). This paper focuses on the impact of packet losses on the controller design for NCSs.

An arguably popular approach for modeling the packet loss phenomenon is to view the packet loss as a binary switching sequence according to a Bernoulli process which takes on values of zero and one with certain probability. Recently, some results have been obtained on such model [15–17]. For example, in [15], the Kalman filtering problem is presented in the setting of intermittent observations and how the expected estimation error covariance depends on the tradeoff between loss probability and the system dynamics is showed. In [16], the problem of robust finite-horizon filtering is investigated for a class of uncertain systems with missing measurements. Moreover, in [17], the feedback stabilization schemes for discrete-time control systems with packet dropping network link are studied, and the feedback strategy presented is considerably simpler to implement.

It is noticed that, based on the Bernoulli distributed model, almost all the stability conditions and controller designs given in the aforementioned references are derived in terms of the assumption that the packet dropout exists only in the sensor-to-controller (S/C) side. The effect of controller-to-actuator (C/A) packet dropouts is neglected due to the complicated NCS modeling. Lately, there have appeared some research results which simultaneously consider S/C and C/A packet losses. In [14], the robust H∞ control problem is considered for NCSs with both S/C and C/A random communication packet losses. By modeling the random packet loss as a linear function of the stochastic variable satisfying Bernoulli binary distribution, stability analysis and controller synthesis problems are investigated. In [18], the observer-based H∞ control problem is studied for discrete-time mixed delay systems with random packet dropouts and multiplicative noises. By modeling the packet-loss phenomenon as Bernoulli distributed white sequences, the packet losses from S/C and from C/A are simultaneously considered. Furthermore, in [19], the similar problem is investigated for a class of networked nonlinear systems with global Lipschitz nonlinearities and random communication packet losses. In the above works, the zero-input strategy is adopted, in which, without considering the disturbance, the actuator (controller) input is set to zero when the C/A  (S/C) packet is lost. However, for systems whose state and input signals change little from one time step to the next, for example, process control systems, the strategy may not perform well. In these situations, the hold-input strategy, that the latest packet stored in the buffer is used when the C/A or S/C packet is lost, gives a better performance [20]. To the best of the authors’ knowledge, based on the hold-input strategy, the problem of observer-based feedback stabilization for NCSs with random S/C and C/A packet dropouts has not been investigated to date, which motivates the present study.

In this paper, the observer-based feedback stabilization problem for NCSs with both random S/C and C/A packet losses is considered. If the packet is lost, the hold-input strategy is adopted. Sufficient conditions for stochastic stability are given, and corresponding controller design steps are provided. An example is finally given to show the effectiveness of the control scheme proposed.

2. Problem Formulation

Consider the following NCS with random data packet dropouts shown in Figure 1, where sensors, controllers, and actuators are clock-driven: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢_𝑐(𝑘),𝑦_𝑐(𝑘)=1−𝛼_𝑘𝐶𝑥(𝑘)+𝛼_𝑘𝑦_𝑐(),𝑢𝑘−1_𝑐(𝑘)=1−𝛽_𝑘𝑢(𝑘)+𝛽_𝑘𝑢_𝑐(𝑘−1), (1)(2)(3) where 𝑥(𝑘)∈𝑅^𝑛 is the state vector, 𝑢_𝑐(𝑘)∈𝑅^𝑚 is the control input to the actuator, 𝑢(𝑘)∈𝑅^𝑚 is the desired control input computed by the controller, and 𝑦_𝑐(𝑘)∈𝑅^𝑝 is the measurement vector with transmission missing. 𝐴,𝐵,and𝐶 are known real constant matrices with appropriate dimensions. The stochastic variable 𝛼_𝑘 models the S/C packet loss: if the measurement packet is lost, then 𝛼_𝑘=1. Otherwise 𝛼_𝑘=0. Similarly, the stochastic variable 𝛽_𝑘 models the C/A packet loss: 𝛽_𝑘=1 implies that the control packet is lost, and 𝛽_𝑘=0 implies that the control packet is correctly delivered. (𝛼_𝑘,𝛽_𝑘) are independent identically distributed (i.i.d.) Bernoulli random variables with prob𝛼_𝑘=1=𝐸𝛼_𝑘=𝛽𝛼,prob_𝑘=1=𝐸𝛽_𝑘=𝛽. (4) The buffers store only one packet, and they will be updated if a new packet arrives. By such a mechanism, the buffers always store the most recent packet and are used for the purpose of packet dropout compensation. For buffer 1, if the measurement packet is correctly delivered, then 𝛼_𝑘=0, that is, 𝑦_𝑐(𝑘)=𝐶𝑥(𝑘), while if the packet is lost, 𝛼_𝑘=1, that is, 𝑦_𝑐(𝑘)=𝑦_𝑐(𝑘−1). This observation model is summarized by (2). For buffer 2, if the packet is correctly delivered, then 𝛽_𝑘=0, that is, 𝑢_𝑐(𝑘)=𝑢(𝑘). Otherwise, the actuator will employ the previous control value, that is, 𝛽_𝑘=1, 𝑢_𝑐(𝑘)=𝑢_𝑐(𝑘−1), as suggested in [21]. This compensation scheme is summarized by (3).

Figure 1 

Structure of an NCS with random data packet dropouts.

The dynamic observer-based control scheme for the system is described as follows: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢_𝑐(𝑘)+𝐿𝑦_𝑐(𝑘)−(1−𝛼)𝐶𝑥(𝑘)−𝛼𝑦_𝑐(,𝑢(𝑘)𝑥(𝑘),𝑘−1)=𝐾 (5) where 𝑥(𝑘)∈𝑅^𝑛 is the state estimation and 𝐿∈𝑅^𝑛×𝑝 and 𝐾∈𝑅^𝑚×𝑛 are the observer and controller gains, respectively.

Let the estimation error be 𝑒(𝑘)=𝑥(𝑘)−𝑥(𝑘), (6) and by substituting (2), (3), and (5) to (1) and (6), we have 𝑥(𝑘+1)=𝐴+1−𝛽𝑥(𝑘)−𝐵𝐾1−𝛽𝐵𝐾𝑒(𝑘)+𝛽𝐵𝑢_𝑐()−𝛽𝑘−1_𝑘−𝛽𝐵𝐾𝑥(𝑘)+𝛽_𝑘−𝛽(𝑘)+𝛽𝐵𝐾𝑒_𝑘−𝛽𝐵𝑢_𝑐(),𝑒()=[(𝑘−1𝑘+1𝐴−1−𝛼)]𝑒(𝑘)+𝛼𝐿𝐶_𝑘−𝛼𝐿𝐶𝑥(𝑘)−𝛼_𝑘−𝛼𝐿𝑦_𝑐(𝑘−1). (7) Moreover, we can rewrite (2) and (3) as 𝑦_𝑐(𝑘)=(1−𝛼)𝐶𝑥(𝑘)+𝛼𝑦_𝑐()−𝛼𝑘−1_𝑘−𝛼𝐶𝑥(𝑘)+𝛼_𝑘−𝛼𝑦_𝑐(),𝑢𝑘−1_𝑐(𝑘)=1−𝛽𝐾𝑥(𝑘)−1−𝛽𝐾𝑒(𝑘)+𝛽𝑢_𝑐()−𝛽𝑘−1_𝑘−𝛽𝐾𝑥(𝑘)+𝛽_𝑘−𝛽(𝑘)+𝛽𝐾𝑒_𝑘−𝛽𝑢_𝑐(𝑘−1). (8)

By defining 𝑧(𝑘)=𝑥^𝑇(𝑘)𝑒^𝑇(𝑘)𝑦^𝑇_𝑐(𝑘−1)𝑢^𝑇_𝑐(𝑘−1)^𝑇, (9) (7)-(8) can be rewritten in a compact form as follows: 𝑧(𝑘+1)=Φ₁+Φ₂𝑧(𝑘), (10) where Φ₁=𝐴+1−𝛽𝐵𝐾−1−𝛽𝐵𝐾0(𝛽𝐵0𝐴−1−𝛼)(𝐿𝐶001−𝛼)𝐶0𝛼01−𝛽𝐾−1−𝛽𝐾0𝛽,Φ₂=−𝛽_𝑘−𝛽𝐵𝐾𝛽_𝑘−𝛽𝐵𝐾0𝛽_𝑘−𝛽𝐵𝛼_𝑘−𝛼𝐿𝐶0−𝛼_𝑘−𝛼−𝛼𝐿0_𝑘−𝛼𝐶0𝛼_𝑘−−𝛽𝛼0_𝑘−𝛽𝐾𝛽_𝑘−𝛽𝐾0𝛽_𝑘−𝛽. (11) Since 𝛼_𝑘 and 𝛽_𝑘 are stochastic variables, we need to introduce the following definition before proceeding further.

Definition 1. System (1) is said to be mean square stable if for any 𝜀>0, there is a 𝛿(𝜀)>0 such that 𝐸{|𝑥(𝑘)|²}<𝜀,𝑘>0 when 𝐸{|𝑥(0)|²}<𝛿(𝜀). In addition, if lim_𝑘→∞𝐸{|𝑥(𝑘)|²}=0 for any initial conditions, the system (1) is said to be globally mean-square asymptotically stable (GMSAS).

3. Main Results

In this section, we shall discuss the observer-based feedback controller design problem for (1). Without loss of generality, we make the following assumption.

Assumption 2. The matrix 𝐵∈𝑅^𝑛×𝑚 is of full-column rank; that is, Rank(𝐵)=𝑚.

For the matrix 𝐵∈𝑅^𝑛×𝑚 being of full-column rank, there always exist two orthogonal matrices 𝑈∈𝑅^𝑛×𝑛 and 𝑉∈𝑅^𝑚×𝑚, such that 𝑈𝑈𝐵𝑉=¹𝑈₂Λ0𝐵𝑉=, (12) where 𝑈₁∈𝑅^𝑚×𝑛,𝑈₂∈𝑅^{(𝑛−𝑚)×𝑛} and Λ=diag𝜆₁𝜆₂⋯𝜆_𝑚, in which 𝜆_𝑖(𝑖=1,2,…,𝑚) are nonzero singular values of 𝐵.

Furthermore, in order to derive the main result, the following lemma will be needed.

Lemma 3 (see [22] ). Let the matrix 𝐵∈𝑅^𝑛×𝑚 be of full-column rank. If matrix 𝑃∈𝑅^𝑛×𝑛 is of the following structure 𝑃=𝑈^𝑇𝑃₁00𝑃₂𝑈=𝑈^𝑇₁𝑃₁𝑈₁+𝑈^𝑇₂𝑃₂𝑈₂, (13) where 𝑃₁∈𝑅^𝑚×𝑚>0, 𝑃₂∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}>0, and 𝑈₁,𝑈₂ are defined in (12); then there exists a nonsingular matrix 𝑃∈𝑅^𝑚×𝑚 such that 𝐵𝑃=𝑃𝐵.

Theorem 4. Consider the networked control problem in Figure 1. Given the dynamic observer-based control scheme (5), system (1) is GMSAS for any S/C and C/A random data packet dropouts if there exist matrices 𝑃>0,𝑆>0,𝑀>0,𝑁>0,𝐿, and 𝐾 satisfying Ω₁Π₁Π₂∗Ω₁⁻¹0∗∗Ω₂⁻¹<0, (14) where Ω₁,Π=diag−𝑃−𝑆−𝑀−𝑁₁=Φ^T₁,Π₂=𝐵𝐾−𝐵𝐾0−𝐵𝐿𝐶0−𝐿0𝐶0−𝐼0𝐾−𝐾0−𝐼^𝑇,Ω₂=diag−𝛼²₁𝑃−𝛼²₂𝑆−𝛼²₂𝑀−𝛼²₁𝑁, (15) in which 𝛼₁=[𝛽(1−𝛽)]^1/2,𝛼₂=[𝛼(1−𝛼)]^1/2.

Proof. To prove the theorem, we introduce the following functions: 𝑉(𝑧(𝑘),𝑘)=𝑉₁+𝑉₂+𝑉₃+𝑉₄,𝑉₁=𝑥^𝑇(𝑘)(𝑘),𝑉𝑃𝑥₂=𝑒^𝑇(𝑘)(𝑘),𝑉𝑆𝑒₃=𝑦^𝑇_𝑐(𝑘−1)𝑀𝑦_𝑐(),𝑉𝑘−1₄=𝑢^𝑇_𝑐(𝑘−1)𝑁𝑢_𝑐(𝑘−1), (16) where 𝑃,𝑆,𝑀, and 𝑁 are positive definite matrices to be determined. Then, we have 𝐸{(𝑘)(0{𝑉(𝑧())(𝑘)(0(𝑧(𝑘))Δ𝑉∣𝑧,…,𝑧)}=𝐸𝑘+1,𝑘+1∣𝑧,…,𝑧)}−𝑉,𝑘=𝐸Δ𝑉₁∣𝑧(𝑘),…,𝑧(0)+𝐸Δ𝑉₂∣𝑧(𝑘),…,𝑧(0)+𝐸Δ𝑉₃∣𝑧(𝑘),…,𝑧(0)+𝐸Δ𝑉₄∣𝑧(𝑘),…,𝑧(0), (17) where 𝐸Δ𝑉₁(𝑘)(0=∣𝑧,…,𝑧)𝐴+1−𝛽𝐵𝐾𝑥(𝑘)−1−𝛽(𝑘)+𝐵𝐾𝑒𝛽𝐵𝑢_𝑐(𝑘−1)^𝑇𝑃×𝐴+1−𝛽𝐵𝐾𝑥(𝑘)−1−𝛽(𝑘)+𝐵𝐾𝑒𝛽𝐵𝑢_𝑐(𝛽𝑘−1)+𝐸_𝑘−𝛽²𝐵𝐾𝑥(𝑘)−𝐵𝐾𝑒(𝑘)−𝐵𝑢_𝑐(𝑘−1)^𝑇𝑃×𝐵𝐾𝑥(𝑘)−𝐵𝐾𝑒(𝑘)−𝐵𝑢_𝑐(𝑘−1)−𝑥^𝑇(𝑘)(𝑘),𝐸𝑃𝑥Δ𝑉₂(𝑘)(0=(∣𝑧,…,𝑧){[𝐴−1−𝛼)𝐿𝐶]𝑒(𝑘)}^𝑇𝑆{[𝐴−(1−𝛼)]𝑒(𝑘𝛼𝐿𝐶)}+𝐸_𝑘−𝛼²𝐿𝐶𝑥(𝑘)−𝐿𝑦_𝑐(𝑘−1)^𝑇×𝑆𝐿𝐶𝑥(𝑘)−𝐿𝑦_𝑐(𝑘−1)−𝑒^𝑇(𝑘)(𝑘),𝐸𝑆𝑒Δ𝑉₃(𝑘)(0=∣𝑧,…,𝑧)(1−𝛼)𝐶𝑥(𝑘)+𝛼𝑦_𝑐(𝑘−1)^𝑇×𝑀(1−𝛼)𝐶𝑥(𝑘)+𝛼𝑦_𝑐(𝛼𝑘−1)+𝐸_𝑘−𝛼²𝐶𝑥(𝑘)−𝑦_𝑐(𝑘−1)^𝑇×𝑀𝐶𝑥(𝑘)−𝑦_𝑐(𝑘−1)−𝑦^𝑇_𝑐(𝑘−1)𝑀𝑦_𝑐(),𝐸𝑘−1Δ𝑉₄(𝑘)(0=∣𝑧,…,𝑧)1−𝛽𝐾𝑥(𝑘)−1−𝛽𝐾𝑒(𝑘)+𝛽𝑢_𝑐(𝑘−1)^𝑇𝑁×1−𝛽𝐾𝑥(𝑘)−1−𝛽𝐾𝑒(𝑘)+𝛽𝑢_𝑐(𝛽𝑘−1)+𝐸_𝑘−𝛽²𝐾𝑥(𝑘)−𝐾𝑒(𝑘)−𝑢_𝑐(𝑘−1)^𝑇𝑁×𝐾𝑥(𝑘)−𝐾𝑒(𝑘)−𝑢_𝑐(𝑘−1)−𝑢^𝑇_𝑐(𝑘−1)𝑁𝑢_𝑐(𝑘−1). (18)
Noting that 𝐸{(𝛼_𝑘−𝛼)²}=𝛼(1−𝛼) and 𝐸{(𝛽_𝑘−𝛽)²}=𝛽(1−𝛽), we obtain 𝐸{Δ𝑉∣𝑧(𝑘),…,𝑧(0)}=𝑧^𝑇(𝐾)Σ𝑧(𝑘), (19) in which Σ=Ω₁−Π₁Ω₁Π^𝑇₁−Π₂Ω₂Π^𝑇₂. (20)
By the Schur complement, (14) guarantees Σ<0. Thus, for all 𝑧(𝑘)≠0, it is easy to know that 𝐸{𝑉(𝑧(𝑘+1),𝑘+1)∣𝑧(𝑘),…,𝑧(0)}−𝑉(𝑧(𝑘),𝑘)<0. (21) That is, there exists 0<𝛾<1 satisfying 𝐸{𝑉(𝑧(𝑘+1),𝑘+1)∣𝑧(𝑘),…,𝑧(0)}≤𝛾𝑉(𝑧(𝑘),𝑘). (22) Using the smoothing property 𝐸{𝑉(𝑧(𝑘))()(0{𝐸{𝑉(𝑧(𝑘))()(0∣𝑧()(0,𝑘∣𝑧𝑘−2,…,𝑧)}=𝐸,𝑘∣𝑧𝑘−1,…,𝑧)}𝑘−2,…,𝑧)} (23) in (22) after taking the conditional expectation, we can conclude 𝐸{𝑉(𝑧(𝑘))()(0{𝑉(𝑧())()(0,𝑘∣𝑧𝑘−2,…,𝑧)}≤𝛾𝐸𝑘−1,𝑘−1∣𝑧𝑘−2,…,𝑧)}≤𝛾²𝑉(𝑧(𝑘−2),𝑘−2). (24) Continuing this process, we get 𝐸{𝑉(𝑧(𝑘),𝑘)}≤𝛾^𝑘𝑉(𝑧(0),0), (25) which implies that 𝐸^𝑁_𝑘=0𝑉(𝑧(𝑘),𝑘)≤1+𝛾+⋯+𝛾^𝑁𝑉(𝑧(0))=,01−𝛾^𝑁+1𝑉(𝑧(0)).1−𝛾,0 (26) Since 𝑆,𝑀,and𝑁 are positive definite matrices, it is easy to conclude that lim_𝑁→∞𝐸^𝑁_𝑘=0𝑥^𝑇(𝑘)𝑥(𝑘≤1)1−𝛾𝜆_min(𝑃)𝑉(𝑧(0),0), (27) which implies lim_𝑁→∞𝐸{|𝑥(𝑘)|²}=0, and the proof is completed.

Note that the condition (14) is not an LMI, hence, cannot be solved by MATLAB LMI Toolbox. In the following, we will deal with the controller design problem and derive the explicit expression of the controller parameters in terms of LMI.

Theorem 5. Consider the networked control problem in Figure 1. The dynamic observer-based control scheme (5) exists such that system (1) is GMSAS for any 𝑆/𝐶 and 𝐶/𝐴 random data packet dropouts if there exist matrices 𝑃₁>0, 𝑃₂>0, 𝑆>0, 𝑀>0, 𝐺, and 𝐻, satisfying Ω₁Ξ₁Ξ₂∗Ω₁0∗∗Ω₁<0, (28) where Ξ₁=𝑃𝐴+1−𝛽𝐵𝐺−1−𝛽𝐵𝐺0(𝛽𝑃𝐵0𝑆𝐴−1−𝛼)(𝐻𝐶001−𝛼)𝑀𝐶0𝛼𝑀01−𝛽𝑉Λ²𝑉^𝑇𝐺−1−𝛽𝑉Λ²𝑉^𝑇𝐺0𝛽𝑁^𝑇,Ξ₂=𝛼₁𝐵𝐺−𝛼₁𝐵𝐺0−𝛼₁𝛼𝑃𝐵₂𝐻𝐶0−𝛼₂𝛼𝐻0₂𝑀𝐶0−𝛼₂𝛼𝑀0₁𝑉Λ²𝑉^𝑇𝐺−𝛼₁𝑉Λ²𝑉^𝑇𝐺0−𝛼₁𝑁^𝑇, (29) in which 𝑃=𝑈^𝑇₁𝑃₁𝑈₁+𝑈^𝑇₂𝑃₂𝑈₂,𝑁=𝑉Λ𝑃₁Λ𝑉^𝑇, and 𝑈₁,𝑈₂,𝑉, and Λ come from (12). Moreover, the controller parameters are given by 𝐾=𝑉Λ⁻¹𝑃₁⁻¹Λ𝑉^𝑇𝐺,𝐿=𝑆⁻¹𝐻. (30)

Proof. Since there exist 𝑃₁>0, and 𝑃₂>0, such that 𝑃=𝑈^𝑇₁𝑃₁𝑈₁+𝑈^𝑇₂𝑃₂𝑈₂, where 𝑈₁, and 𝑈₂ come from (12). It follows from Lemma 3 that there exists a nonsingular matrix 𝑃=𝑉Λ⁻¹𝑃₁Λ𝑉^𝑇 such that 𝐵𝑃=𝑃𝐵. Let 𝐵𝑃=𝑃𝐵, 𝐺=𝑃𝐾, and 𝐻=𝑆𝐿., we can conclude that (28) is equivalent to (14).

4. Numerical Example

In this section, an example is used to demonstrate that the controller designed in this paper is effective. We compare our strategy with zero-input strategy [14], which shows that, for systems whose state and input signals change little from one time step to the next, the hold-input strategy adopted in the paper has a better performance.

Consider the following discrete-time unstable system: ,,.𝐴=0.70.201.12𝐵=−0.03−1𝐶=10 (31) It is assumed that 𝛼=𝛽=0.1.

By zero-input strategy in [14], the controller gain and the observer gain are 𝐾=[0.22350.4312] and 𝐿=[0.90641.5446]^𝑇, respectively. Suppose that the initial conditions are 𝑥(0)=[11]^𝑇, 𝑥(0)=[11]^𝑇, and 𝑢_𝑐(−1)=𝑦_𝑐(−1)=0. We have Figures 2(a) and 2(b). Then, by Theorem 5 of our paper, we obtain 𝐾=[0.17800.3916] and 𝐿=[0.57701.1488]^𝑇. With the same initial conditions, we have Figures 3(a) and 3(b). From Figures 2 and 3 we know that the system can be stabilized by both controllers; however, by comparison, it is easy to know that our controller gives a better performance.

(a)

(b)

Figure 2 

State response with controller in [14]. (a) First component of the state vector 𝑥(𝑘). (b) Second component of the state vector 𝑥(𝑘).

(a)

(b)

Figure 3 

State response with controller proposed in this paper. (a) First component of the state vector 𝑥(𝑘). (b) Second component of the state vector 𝑥(𝑘).

Remark 6. From the system in this example, we can observe that the state, measurement, and input signals change slowly. Moreover, since the latest packet stored in the buffer is used when the C/A or S/C packet is lost, our controller provides a smoother sequence of inputs. These are the reason why hold-input strategy adopted in the paper gives a better performance.

5. Conclusion

In this paper, the problem of observer-based feedback stabilization is considered for NCSs with random packet dropouts. The hold-input strategy is adopted, and sufficient conditions for stochastic stability are derived in the form of linear matrix. Furthermore, the dynamic observer-based control scheme is designed. An example shows that, for systems whose state and input change little from one time step to the next, the hold-input strategy adopted performs better than the zero-input strategy.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61202478 and the Fundamental Research Funds for the Central Universities under Grant 2012QNB18.