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Scheduling-and-Control Codesign for a Collection of Networked Control Systems With Uncertain Delays


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Scheduling-and-Control Codesign for a Collection of Networked Control Systems With Uncer

tain Delays
Shi-Lu Dai, Student Member, IEEE, Hai Lin, Member, IEEE, and Shuzhi Sam Ge, Fellow, IEEE
Abstract—This paper is concerned with the simultaneous stabilization of a collection of continuous-time linear time-invariant (LTI) plants whose feedback-control loops are closed via a shared digital communication network. Because of the limitation of communication capacity, only a limited number of controller-plant connections can be accommodated at any time instant. Therefore, it is necessary to carefully determine a scheduling policy so as to achieve a simultaneous stabilization for all these control loops. A suf?cient condition on the existence of such a scheduling policy is presented for a collection of networked LTI systems with sampled-data controllers and uncertain network-induced delays. The proof for the schedulability condition is in a constructive way, which can also serve as a systematic method to design a scheduling policy. Finally, a scheduling-and-feedback-control codesign procedure is proposed for the simultaneous stabilization of the collection of networked LTI systems, and the effectiveness of the proposed codesign procedure is demonstrated with simulation results. Index Terms—Average dwell time, codesign, networked control systems (NCSs), scheduling, switched systems.

I. INTRODUCTION

N

ETWORKED control systems (NCSs) are feedback-control systems in which the communication between spatially distributed system components like sensors, actuators, and controllers occurs through shared band-limited digital communication networks. Compared with conventional point-to-point interconnected control systems, NCSs possess many attractive features due to the inclusion of a communication network. These advantages include higher system testability and resource utilization, as well as lower cost, reduced weight and power, and

Manuscript received May 20, 2008; revised September 03, 2008. Manuscript received in ?nal form November 25, 2008. First published April 21, 2009; current version published December 23, 2009. Recommended by Associate Editor C.-Y. Su. This work was supported by the Singapore Ministry of Education’s AcRF Tier 1. S.-L. Dai is with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education of China, Northeastern University, Shenyang 110004, China, and also with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: dslwm@yahoo.com.cn). H. Lin is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: elelh@nus.edu.sg). S. S. Ge is with the Social Robotics Laboratory, Interactive Digital Media Institute and the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: samge@nus. edu.sg). Color versions of one or more of the ?gures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identi?er 10.1109/TCST.2008.2010459

simpler installation and maintenance [1], [2], which make the use of networks in control systems connecting sensors/actuators to controllers more and more popular in many applications, including traf?c control, satellite clusters, and mobile robots [3]–[5]. Consequently, considerable attention has been paid to the study of NCSs recently, see for example, the survey papers [4], [6], and [7], the recent special issues [8] and [9], and the references therein. In NCSs, it is common that many spatially distributed system components, like sensors, controllers, and actuators, share a common communication network. This is very typical when using the base station to control and coordinate a group of mobile robots through a wireless network. Many speci?c application setups can be found in the real-world control systems through networks, such as controller area networks (CANs) [10] and Fieldbus [11]. For example, cart–pendulums are distributively controlled over CAN, which is taken from [10], as shown in Fig. 1. In this paper, we consider a class of NCSs consisting of a collection of continuous-time linear time-invariant (LTI) plants whose open-loop dynamics might be unstable. Each plant communicates with its remotely located controller over a shared network link, as shown in Fig. 2. Because of limited communication capacities, it is assumed that controller–plant communication is restricted, and not all the control loops in the NCSs can be addressed at the same time. That is, only a few controller–plant connections can be granted at any one time, while the other feedback-control loops are assumed to be open loop. It is clear that, if some connections monopolize the network, the other plants will not be stabilizable. In order to guarantee the stabilization of each plant, it is necessary to design a scheduling algorithm to schedule the NCSs. It should be pointed out that similar problems have been studied in [12]–[16]. In [12], the rate-monotonic scheduling algorithm was applied to schedule a set of NCSs. A schedulability condition was presented in [13]–[15] for the simultaneous stability of a group of continuous-time linear systems by a common Lyapunov function, and a time-division-based scheduling policy was developed in [16] by employing the average dwell-time technique [17], [18] incorporated with piecewise Lyapunov-like functions. So far, these studies are carried out in the continuous-time domain and under the assumption that no time delay happens. The network-induced delay is one of the basic problems in NCSs [1], and the characteristics of delay can be constant, bounded, or even random, which can usually degrade a system’s performance and even cause system instability. For instance, a typical motion-control example was presented in [19], in which the time variation of the network-induced delays results

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Fig. 1. Cart–pendulums are distributively controlled over CAN.

in an unstable system. In view of discretizing continuous-time systems affected by time-varying delays [19]–[23], these systems can be described by uncertain systems for whose robust control methods can be applied. Additionally, other effective approaches, such as the maximum allowable delay-bound approach combined with Lyapunov–Krasovskii functionals [22]–[24], the stochastic system approach [25], [26], the predictive control approach [27], [28], and the switched system approach [29]–[32] have been developed to study the modeling, analysis, and synthesis of NCSs with constant, random, or time-varying delays. On the other hand, the problem of feedback-control-and-scheduling codesign has recently received increasing attention in the NCS literature, since NCS performances not only depend on the design of control algorithms but also on the scheduling of the shared network resources [12]. In [33], this problem was addressed to improve control performances for a linear system over limited bandwidth networks. A codesign scheme was presented in [34] for stabilizing an LTI system, in which only limited sensors/actuators can exchange information with a remote controller via a shared common communication network. In [35], a predictive control-and-scheduling-codesign approach was proposed to deal with the communication constraints for a set of NCSs with network-induced delays. In this paper, we address the problem of feedback-control-and-scheduling codesign in a sampled-data control framework and explicitly consider uncertain network-induced delays in the control loops. The motivation comes from the popularity of digital control and unavoidable transmission delays in communication networks. The main contributions of this paper are that: 1) a constructive proof method is used to show that the schedulability condition only depends on the convergence rate of the closed-loop system and the divergence rate of the open-loop plant; 2) simultaneous stability conditions are presented by employing a parameter-dependent Lyapunov function method combined with average-dwell-time technique; and 3) a scheduling-and-feedback-control codesign procedure is proposed for the simultaneous stabilization of the collection of NCSs under consideration. The rest of this paper is organized as follows. In Section II, a group of continuous-time LTI plants affected by network-induced delays are modeled as a collection of discrete-time polytopic uncertain systems with one step delay. In Section III, we

give suf?cient stability conditions for a single-control system and a schedulability condition for the considered NCSs, and then, simultaneous stability conditions are proposed by a parameter-dependent Lyapunov function method. In Section IV, a scheduling-and-feedback-control codesign procedure is developed for the simultaneous stabilization of the NCSs. Simulation studies are performed to demonstrate the effectiveness of the proposed codesign procedure in Section V. Finally, conclusions are included in Section VI. II. NCS MODEL In this paper, we consider the NCSs consisting of a collection of continuous-time LTI plants whose feedback-control loops are closed via a shared network link, as shown in Fig. 2. The th , is given by plant, (1) where are the system states and are the control inputs. The pair is assumed to be stabilizable, might be unstable matrices. but In this paper, the following assumptions are made. Assumption 1 [13], [15]: Because of the limitation of communication capacities, not all the control loops in the NCSs can be addressed at the same time. At any time instant, only of the plants ( ) can communicate with their reof the mote controllers while others must wait, i.e., only plants can close their feedback loops at any time instant while the other control loops are assumed to be open loop. Assumption 2 [13], [15]: When a plant fails to communicate with its corresponding controller, the open-loop system is unstable; otherwise, the plant communicates with its controller, and the resulting closed-loop system is stable. Assumption 3 [23]: The following setup is considered in this paper: a clock-driven sensor, which periodically samples the plant outputs, an event-driven controller, which calculates the control signal as soon as the sensor data arrive, and an eventdriven actuator, which updates the plant inputs as soon as the controller data arrive. The data are transmitted in a single packet at each time step. Remark 1: To make the problem nontrivial, in Assumption 2, the open-loop systems are assumed to be unstable. In this paper, we only consider the worst case of unstable open-loop systems,

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Fig. 2. Communication of controllers and plants through a shared network link.

and it should be mentioned that stable open-loop systems will be special cases in our discussion. This is, the results of this paper will still hold when the open-loop systems are stable, while the results may be conservative for the special cases. In order to design a computer-based control, a sampled model of the continuous-time plant is used, which is based on [36]. Assume that the constant sampling period of the th plant is ; the equivalent discrete representation of plant (1) is given by (2) where , , and (nonnegative integers). , , and Throughout this paper, we use to denote , , and , respectively, for convenience. A more realistic discrete representation should consider transmission delays in a data network. There are two sources of delays in the network [23]: the sensor-to-controller delay and the controller-to-actuator delay . For static control law, the sensor-to-controller and controller-to-actuator . delays can be lumped together as Now, we assume that is unknown time varying and satis?es (3) and are constant positive scalars representing the where lower and upper bounds of delays, respectively. Sampling system (1) with period gives the following discrete representation [36]: (4) where and (5) (6) . with and are dependent on the unNote that , and thus, system (4) is an unknown time-varying delay certain linear system with time-varying uncertainties. Note that

cannot take any value; they belong to the convex hulls [20], [21], [23] (7) where denotes the convex hull and constant matrices represent the vertices of the hulls. Remark 2: There are some approaches which can be used to obtain vertices of the convex hull (7) in the NCS literature, e.g., [20], [21]. Using the Taylor series expansion, in [20], a discretized system with a time-varying network-induced delay can be approximated by a polytopic uncertain system. A method for calculating the vertices of polytopic systems was developed in [21] by the real Jordan form for the representation of NCS with uncertain delay. In simulation studies, we will apply one . of them to derive the convex hull vertices Consider system (4) with (5), (6), and (7) and static state; thus, the closed-loop system is feedback laws given by (8) where

It is clear that system (8) is a discrete-time polytopic uncertain system with one step delay. The discretizing continuous-time systems affected by uncertain network-induced delays are represented as system (8). It should be pointed out that this modeling method might be conservative, particularly in the case where the delay has very large values, but this does not occur frequently. From the control point of view, the digital control synthesis of continuous-time systems affected by time-varying delays is a challenge problem [20]. In this paper, we try to solve the control synthesis problem for the system with uncertain network-induced delays from the robust control point of view. The objective of this paper is to design a scheduling policy such that all and state-feedback-control laws systems in (8) are exponentially stabilized.

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III. STABILITY FOR NCSS A. Single-Control Loop Stability Analysis In this section, the subscript will be dropped for short. We to denote the time interval , where is the use sampling period of the th plant. Throughout this paper, the subscripts “ ” and “ ” stand for closed-loop and open-loop systems, respectively. This section focuses on the stability of a single plant with closed/open feedback-control loop. As we know, a switched system may be unstable even if all its subsystems are stable, and it is still possible to be stabilized when all its subsystems are unstable by properly designed switching laws [17], [18], [37], [38]. Average-dwell-time technique [17] is an effective tool for designing such switching laws. As shown in [39], if the average dwell time is chosen suf?ciently large and the total activation time of unstable subsystems is relatively small compared with that of stable subsystems, then exponential stability with a desired decay rate of the switched system is guaranteed. Motivated by the results, it is possible for a plant to be exponentially stabilized if the plant closes its feedback control for suf?ciently large average dwell times and the total activation time of its open loop is relatively small compared with that of its close loop. Here, we are interested in solving the following problem: how often and how long one should close the control loop such that the control system is stable. It is supposed that the single-control system is an open loop for some time because the shared network link is occupied by another network user. The single-control system can be described by a switched system, which is composed of the open-loop subsystem (9) and the closed-loop subsystem

The following result gives the exponential stability of a single-control system. Lemma 1: Consider the single-control system composed of the unstable open-loop subsystem (9) and the stable closedloop subsystem (10). The single-control system is exponentially if the following conditions stable with decay rate hold. and 1) The positive de?nite quadratic functions in (11) satisfy (12) where and . 2) There exists a constant scalar such that (13) for any . 3) The attention rate satis?es (14) 4) The attention frequency satis?es

(15) where and are said to be the average dwell time and , and the chatter bound, respectively, . Proof: See Appendix A. Remark 3: It is worth making a few remarks about this lemma. First of all, condition 1) of Lemma 1 implies that in (11) along the state trajectory of the the function closed-loop subsystem (10) has an exponential decay property: , where and is the initial time step. Moveover, means that , which gives an exponential along the state trajectory of the open-loop increase of subsystem (9). Condition 2) ?rst appeared in [40] and has almost become a standard in applying the average-dwell-time technique to design switching laws for switched systems [17], [39], [41], [42]. This condition restricts the class of applicable Lyapunov-like functions by requiring the existence of a maximal global constant ratio among the functions. Quadratic functions are universally considered in linear systems, and in this case, the existence of a global constant is automatically guaranteed [42]. To ensure the exponential stability of the system, condition 3) implies that the attention rate of the th plant is required to be suf?ciently large, while the attention frequency is restricted with condition 4). B. Scheduling Policy for NCSs In accordance with Assumptions 1 and 2, if some plants and their controllers monopolize the network, then the other plants might not be stabilizable. In order to achieve simultaneous stabilization for all the control systems, it is necessary to carefully schedule the communication tasks for the collection of NCSs.

(10) and and are de?ned in (8). where For stability analysis, the following de?nitions play crucial roles in the sequel. with De?nition 1: The system is said to be exponentially stable with decay rate if holds for a constant . , let denote the total De?nition 2 [16]: For any number of sampling periods of the th plant being a closed loop , and the ratio (attended by the controller) during is said to be the attention rate of the th plant, and let denote the total number of switching for the th plant between open- and closed-loop statuses, which is said to be the attention frequency. For the single-control system composed of subsystems (9) and (10), choose a piecewise quadratic Lyapunov-like function candidate if closed loop if open loop. (11)

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This section concentrates on ?nding a scheduling policy for establishing and terminating communication between each system and its controller in a way that stabilizes all systems. Lemma 2: Under Assumptions 1 and 2, consider the collection of NCSs with communication constraints as shown in Fig. 2. Suppose that conditions 1) and 2) in Lemma 1 hold for any single-control system, and the following condition holds: (16) , , , and denotes where the maximum number of plants which can communicate with their remote controllers at any time instant. Then, there exists a scheduling policy which guarantees the exponential stabilization of each system (8). Proof: The proof is inspired by the continuous-time one in [16]. From (16), it is clear that there exists a positive scalar such that the following inequality:

Now, we show that, under the aforementioned scheduling policy, the NCSs are exponentially stable with decay rate . , it can be 2) Stability veri?cation: For any written as , , where is a nonnegative integer and is a real number. , the following two cases need to be considered. For , then 1) If and . Then, we obtain

and . 2) If and , then

. Thus, it follows that

. Then, we have

holds for

. Let . Then, we have and . If set which , then we have . yields From the aformentioned discussions and the de?nition of in (18), it follows that

Therefore, we obtain (17) which holds for all . Note that verify that . Let and and (17); then, it is easy to for . Let (18) and . Then, we have Next, we propose a periodic scheduling policy which guarantees the asymptotic stability of the NCSs. It should be mentioned that the proposed discrete-time scheduling policy is motivated by the continuous-time one in [16]. 1) Scheduling policy: , where is a positive 1) Choose integer suf?ciently large to satisfy the average-dwell-time is the sampling condition in (15) for the th plant and period of the th plant. For example, we may set , where is the lower bound of the average dwell and denotes the upper integer bound. time 2) Close control loops for their plants at any time instant. Activate the control loops from 1 to in order, and let the th control loop work for a time interval of length for .

which satis?es condition 3) in Lemma 1. According to the sein scheduling policy 1), conditions 3) and 4) in lections for , under the proposed schedLemma 1 are both satis?ed for uling policy. , by shifting the initial time to the beginning of For the ?rst closed-loop sampling period of the th plant and adcorrespondingly, it reduces to the justing the initial state case , and conditions 3) and 4) in Lemma 1 are both satis?ed for the shifted th plant. It is straightforward to show the exponential stability between the time-shifted (by a ?nite constant) control system and the original system. Therefore, all the systems are exponentially stable with a speci?ed decay rate under the scheduling policy, which completes the proof. Remark 4: The form of schedulability condition (16) is similar to the continuous-time one in [14] and [16]. According to Lemma 2, it is shown that the schedulability condition (16) only depends on the convergence rate of the th closed-loop of the th open-loop plant. system and the divergence rate and To obtain a large value of , it is desirable that are small. Therefore, we should focus on designing state-feedare minimized in the sequel. back-control laws such that Remark 5: The proof of Lemma 2 employs a constructive method, which provides a systematic way to design a scheduling policy to guarantee the simultaneous stability of all systems. The scheduling policy is a static periodic scheduling policy, which is quite simple and can be easily implemented in engineering practice [16].

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C. Simultaneous Stability In this section, simultaneous stability conditions are presented for the collection of networked LTI systems. For the polytopic system (8) with closed/open control loops, we choose the following piecewise Lyapunov-like function candidate as if closed loop if open loop where (19)

with

, , , , and . and Remark 6: In (19), the Lyapunov function matrices are similar to the form of (10), which are time variant and . dependent on uncertain parameters and When , in (19) shrinks to the single Lyapunov function . The use of the single Lyapunov function usually renders conservative stability conditions as shown in [43]. To improve the single Lyapunov function-based results, we therefore employ the parameter-dependent Lyapunov function (19) for the analysis and synthesis of the NCSs. Theorem 1: Under Assumptions 1 and 2, consider the collection of NCSs with communication constraints as shown in be state-feedback laws. Given a Fig. 2 and let and constants , , positive integer , suppose that there exist positive-de?nite matrices and , , , , , and , and matrices satisfying (20)–(23), as shown at the bottom of the page. Then, there exists a scheduling policy which guarantees the exponential stabilization of each system (8), and the state decay estimations are given by

,

, and denote the minimum and maximum eigenvalues of a symmetric matrix, . respectively, and Proof: See Appendix B. Although exponential stabilization conditions for each discrete-time system (8) have been given in Theorem 1, we still cannot conclude that the each original continuous-time system (1) is asymptotically stable. The following result shows that the exponential stabilization of each system (8) implies simultaneous stabilization of the original collection of continuous-time linear plants. Proposition 1: If conditions (20), (21), (22), and (23) in Theorem 1 hold, then each system (8) is exponentially stabilized, and its original continuous-time system (1) is asymptotically stable. Proof: See Appendix C. IV. CONTROL-AND-SCHEDULING CODESIGN We shall devote this section to the design of state-feedbackcontrol laws and scheduling policy for the NCSs. and constants are given, then If the gain matrices inequality (20) is a linear matrix inequality (LMI) over the , , , and , and matrices . matrix variables However, since our purpose is to determine the gain matrices , inequality (20) is actually a nonlinear matrix inequality. , and Note that inequality (20) implies is obviously nonsingular. By performing congruence trans, de?ning formation to (20) by and , and using the Schur complement formula [44], we can thus conclude that inequality (20) is equivalent to (24) and (25), as shown at the bottom of the next page. , we need to solve To obtain the controller gain matrices the feasibility solution to (24) and (25). Note that (25) is an equality constraint, and this problem is a nonconvex feasibility problem. Several approaches have been proposed to solve such nonconvex feasibility problem. One ef?cient way to solve the nonconvex problem is the cone complementarity linearization (CCL) algorithm proposed in [45], which has been used to solve the controller design problem, e.g., [24]. Now, using the CCL algorithm [45], the controller gain matrices can be obtained by solving the following nonlinear minimization problem with LMI constrains: minimize

where decay rates

, constants

, , ,

(20) (21) (22) (23)

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subject to (24) and (26) From Theorem 1 and the aforementioned minimization problem, we can outline a procedure for scheduling-and-feedback-control codesign for the NCSs. Algorithm 1: 1) For any ?xed , minimize and solve the th state-feedback gain . The minimization problem can be solved by the following iterative algorithm. such that Step 1) Choose a suf?ciently large initial there exists a feasible solution to (24) and (26). . Set Step 2) Find a feasible solution set to (24) and (26). Set Step 3) Solve the following LMI problem: minimize subject to (24) and (26) Set , , , , , , and .

Remark 7: Algorithm 1 can also be applied in the design of , where static output-feedback laws is the output of the th plant. We can determine the static output-feedback gains by replacing in (24) with . Remark 8: For Algorithm 1, we can give a conservative method for the estimates of constants

(27) , which satis?es (22). However, this estimate for of may result in a large average-dwell-time lower bound from (15). In fact, there is another method, which will be used for simulation studies in Section V, for ?nding a proper . Solving Steps 1–4 in Algorithm 1 gives the parameterized maand . Choose two small initial and , then trices solve LMIs (21) and (22); if there is no feasible solution to (21) or to some extent and solve LMIs (21) and (22), increase and (22) again, until (21) and (22) are feasible. V. SIMULATION STUDIES To show the proposed codesign procedure and validate its effectiveness, the control network setup in Fig. 1 is used for simulation studies, in which cart–pendulums are distributively controlled over CAN. Consider a cart–pendulum system, whose dynamics, taken from [46] and [47], can be described as

. Step 4) If inequality (20) is satis?ed, then set and return to Step 2 after decreasing to some extent. If inequality (20) is not satis?ed within a speci?ed number of iterations, then exit. Otherand go to Step 3. wise, set subject to (21). 2) Minimize satisfying (22). 3) Find constants 4) For given a positive integer , calculate the maximal value of subject to (23). 5) Design the following periodic static scheduling policy for NCSs. , where a) Choose with being the lower bound of the average dwell . It is clear that satis?es the average-dwelltime time condition for the th plant. b) Close control loops for their plants at any time instant. Activate the control loops from 1 to in order and let the th control loop work for a time interval of for . length Conclusion: The simultaneous stabilization of the collection of NCSs with network-induced delay (3) and communication constraints, as shown in Fig. 2, can be obtained.

where a force is applied to the cart for keeping the pendulum balanced upright, is the pendulum angular position, is the angle of the pendulum, and are the masses of the cart and the pendulum, respectively, is the length of the pendulum, and is the acceleration due to gravity. Selecting the state variables and linearizing the aforementioned model at the equilibrium point (i.e., =0) yield the following state-space model [47]:

Let

kg, kg, m, and m/s . The simpli?ed model of the cart–pendulum

(24) (25)

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TABLE I CALCULATE THE MINIMUM VALUES OF  FOR DIFFERENT VALUES OF d (h : s AND d )

= 0 04

=0

TABLE II CALCULATE THE MINIMUM VALUES OF  FOR DIFFERENT VALUES OF d (h : s AND d )

= 0 06

=0

process is given by

which results in and satisfying (23). Therefore, it can be concluded from Theorem 1 that such two identical systems with network-induced delays and sampling period can share a common network link, which only takes care of one control loop at a time. It follows by analogous analysis that three alike control loops can be scheduled under the assumption that ( ). For sampling period and , two identical plants can be simultaneously stabilized via a shared common network link. Then, we will focus on designing state-feedback laws and a scheduling policy such that the following two control systems are exponentially stabilized. s A sampling system (28) with different periods ( s) yields the following discrete representation: and

(28) where and .

(29) (30) where , , and are de?ned in (4), respectively. Assume the network-induced delay . Applying Algorithm 1 gives the following parameters:

The eigenvalues of matrix are , which is an unstable matrix. Now, consider the effect of means that sattransmission delays in the networks and suppose that is?es (3). We apply the procedure proposed in [21] to calculate the vertices of the convex hull (7), which are given by

for system (29) and

for system (30). Hence, we have Assuming that the lower delay bound of is , we are interested in the relationship between the minimum value and the upper delay bound . With different sampling of and , Tables I and II list the minimum periods values of for different by applying Steps 1–4 in Algorithm 1. grows as increases from It is clear that the value of Tables I and II. Moreover, when (i.e., no networkedapproaches zero; hence, tends to in?nity induced delay), (for from the schedulability condition (16). When ) or (for ), (20) is unfeasible. Next, ?nd constants and such that (21) and (22) hold. By Remark 8, there exists a feasible solution to (21) and (22) for a and ( ). Select constants given and ( ), (21) and (22) are feasible. For simplicity, assume from now on that , i.e., only one plant can close its feedback loop at any time instant. and into (23) gives Substituting

which means that systems (29) and (30) can share a common network link. Let the desired exponential decay rates . According to the proof of Lemma 2, select the following , , , parameters: , , , and . It follows from 5) of Algorithm 1 that , , s, s, and s. Therefore, the static period scheduling policy is determined. for the time interval Close (30) ?rst and activate controller of length s, then close (29) and let its controller work s, then, again, close (30) for the time for the time interval s. Hence, the time period of the schedinterval of length uling policy is s. Let the initial states and the net. work-induced delays Under the aforementioned period scheduling policy, the state trajectories of system (29) and the corresponding control input trajectory are shown in Figs. 3 and 4. Figs. 5 and 6 show the state and control input trajectories of system (30). Simulation

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Fig. 3. State trajectories ( h

= 0:04 s).

Fig. 5. State trajectories ( h

= 0:06 s).

Fig. 4. Control input trajectory ( h

= 0:04 s).

Fig. 6. Control input trajectory ( h

= 0:06 s).

results demonstrate that the unstable open-loop systems are successfully stabilized by the proposed control laws and scheduling policy. VI. CONCLUSION In this paper, a scheduling-and-feedback-control codesign procedure has been proposed for the simultaneous stabilization of a collection of the plants whose feedback-control loops are closed via a shared communication network. The presence of feedback-based communication constraints and the effect of networked-induced delays have been taken into account in the communication network. Using average-dwell-time technique, stability conditions have been addressed for a single plant whose feedback-control loop is closed/open. Based on these stability conditions, a schedulability condition has been shown by a constructive proof which provides a systematic way to design a scheduling policy. The schedulability condition only depends on the convergence rate of the closed-loop system and the divergence rate of the open-loop plant. The effectiveness of the proposed codesign procedure has been illustrated through the application to the inverted pendulum control.

Controllability and reachability are two fundamental concepts in the analysis and design of control systems. The design of the communication sequences that preserve the controllability and reachability of a group of NCSs with communication constraints constitutes a challenging opportunity for future work. The ideas and algorithms presented in [48]–[50] for ?nding switching sequences to achieve the controllability and reachability of switched systems inspire us to design such communication sequences for the NCSs. APPENDIX A PROOF OF LEMMA 1 Without loss of generality, we assume that the controller and that the plant is an open loop works during , , where . Choose the during piecewise quadratic Lyapunov-like function candidate as (11). , it holds from (12) that For any if if where and . (31)

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Therefore, if De?nition 2 that

, it follows from (13) and

(32) and are de?ned in De?nition 2 and where denotes the time instant that is immediately before . for Similarly, we have . From (14), we have

NCSs. It is obvious that (23) is the same as (16), and we shall prove that conditions 1) and 2) in Lemma 1 hold if (20), (21), and (22) hold for any ?xed . First of all, it can be veri?ed that holds according to (20) as follows. and Let . . From (19), the forward De?ne difference for along the state trajectory of system (10) is given by

(38) where and is as

which is equivalent to (33) From (15), we have (39) From (20) and (39), we have (34) Combining (32), (33), and (34) yields (35) If a quadratic form is considered in the piecewise Lyapunovlike function (11), then there exist , , , and such that (40) where , , and are de?ned in (8) and (9), respectively. Performing congruence transformation to (40) by yields On the other hand, note that (20) implies and that is obviously nonsingular. Since , we have , which is equivalent to

which leads to hold, which implies (36) where , Euclidean norm. From (35) and (36), we have , and denotes the (41)

(37) which means that the single-control system is exponentially stable with decay rate , where .

Let

By pre- and postmultiplying (41) by and its transpose, respectively, it follows that (42)

where APPENDIX B PROOF OF THEOREM 1 The subscript will be dropped for brevity in the proof. From Lemma 2, we know that there exists a scheduling policy that exponentially stabilizes all systems if conditions 1) and 2) in Lemma 1 and (16) hold for every single-control system of the

is de?ned in (38). From (38) and (42), we have (43)

Therefore, we can conclude that holds. Second, we shall verify that (21) results in .

if (20)

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 1, JANUARY 2010

From (19) and (21), the forward difference for the state trajectory of system (9) is given by

along

and are bounded. Therefore, it is clear that is also bounded. Moreover, the stability of each system (8) means , which results in that its states converge to zero as that converges to zero as . Hence, the original continuous-time system (1) is asymptotically stable. ACKNOWLEDGMENT

which means that

Therefore, we can conclude that condition 1) in Lemma 1 is satis?ed if (20) and (21) hold. Third, it follows from (19) and (22) that

The authors would like to thank the Associate Editor and the anonymous reviewers for their helpful and insightful comments that further improve the quality and presentation of this paper. S.-L. Dai would like to thank Prof. J. Zhao from Northeastern University, Shenyang, China, Dr. X.-M. Sun from Dalian University of Technology, Dalian, China, and X. Li from National University of Singapore, Singapore, for their helpful discussions and constructive suggestions. REFERENCES
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which implies that and . Lastly, for the considered Lyapunov-like function (19), it follows from (36) and (37) that

where

,

, , and . This completes the proof.

APPENDIX C PROOF OF PROPOSITION 1 To study the intersample behavior of the th original continuous-time LTI system (1), the solution of the th system (1) over is given by the interval

Let we have

. According to Lemma 5.1 [19],

for

and

for . If conditions (20)–(23) in Theorem 1 hold, then each system (8) is exponentially stabilized, which implies that states

DAI et al.: SCHEDULING-AND-CONTROL CODESIGN FOR A COLLECTION OF NCSs WITH UNCERTAIN DELAYS

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[41] X.-M. Sun, J. Zhao, and D. J. Hill, “Stability and L -gain analysis for switched delay systems: A delay-dependent method,” Automatica, vol. 42, no. 10, pp. 1769–1774, Oct. 2006. [42] D. Chatterjee and D. Liberzon, “On stability of randomly switched nonlinear systems,” IEEE Trans. Autom. Control, vol. 52, no. 12, pp. 2390–2394, Dec. 2007. [43] P. Gahinet, P. Apkarian, and M. Chilali, “Af?ne parameter-dependent Lyapunov functions and real parametric uncertainty,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 436–442, Mar. 1996. [44] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [45] L. E. Ghaoui, F. Oustry, and M. AitRami, “A cone complementarity linearization algorithm for static output-feedback and related problems,” IEEE Trans. Autom. Control, vol. 42, no. 8, pp. 1171–1176, Aug. 1997. [46] J. Zhao and M. W. Spong, “Hybrid control for global stabilization of the cart–pendulum system,” Automatica, vol. 37, no. 12, pp. 1941–1951, Dec. 2001. [47] H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” IEEE Trans. Autom. Control, vol. 52, no. 2, pp. 328–334, Feb. 2007. [48] S. S. Ge, Z. Sun, and T. H. Lee, “Reachability and controllability of switched linear discrete-time systems,” IEEE Trans. Autom. Control, vol. 46, no. 9, pp. 1437–1441, Sep. 2001. [49] Z. Sun, S. S. Ge, and T. H. Lee, “Controllability and reachability criteria for switched linear systems,” Automatica, vol. 38, no. 5, pp. 775–786, May 2002. [50] S. S. Ge and Z. Sun, “Switched controllability via bumpless transfer input and constrained switching,” IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1702–1706, Aug. 2008.

Shi-Lu Dai (S’09) received the B.Eng. and M.Eng. degrees from Northeastern University, Shenyang, China, in 2002 and 2006, respectively, where he is currently working toward the Ph.D. degree in the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education of China. He has been a visiting student in the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, since November 2007. His current research interests lie in switched systems, networked embedded systems, and control applications. Mr. Dai was a recipient of the Chinese Government Scholarship from the China Scholarship Council in 2007–2009.

Hai Lin (M’04) received the B.S. degree from the University of Science and Technology Beijing, Beijing, China, in 1997, the M.Eng. degree from the Chinese Academy of Science, Beijing, in 2000, and the Ph.D. degree from the University of Notre Dame, Notre Dame, IN, in 2005. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore. His research interests include the multidisciplinary study of the problems at the intersection of control, communication, computation, and life sciences. He is particularly interested in hybrid systems, networked embedded systems, and systems biology.

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 1, JANUARY 2010

Shuzhi Sam Ge (S’90–M’92–SM’99–F’06) received the B.Sc. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1986, and the Ph.D. degree and the Diploma of Imperial College from the Imperial College of Science, Technology, and Medicine, University of London, London, U.K., in 1993. He is the founding Director of the Social Robotics Laboratory, Interactive Digital Media Institute, National University of Singapore, Singapore, where he is also a Professor with the Department of Electrical and Computer Engineering. He has (co)authored three books entitled Adaptive Neural Network Control of Robotic Manipulators (World Scienti?c, 1998), Stable Adaptive Neural Network Control (Kluwer, 2001), and Switched Linear Systems: Control and Design (Springer-Verlag, 2005) and edited the book Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications (Taylor & Francis, 2006) and over 300 international journal and conference pa-

pers. He provides technical consultancy to industrial and government agencies. His current research interests include social robotics, multimedia fusion, adaptive control, intelligent systems, and arti?cial intelligence. Dr. Ge was the recipient of the Changjiang Guest Professor, Ministry of Education, China, 2008, the Outstanding Overseas Young Research Award, National Science Foundation, China, 2004, the Inaugural Temasek Young Investigator Award, Singapore, 2002, and the Outstanding Young Researcher Award, National University of Singapore, 2001. He has served/been serving as an Associate Editor for a number of ?agship journals including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, and IEEE TRANSACTIONS ON NEURAL NETWORKS. He is the Editor-in-Chief of the International Journal of Social Robotics (Springer) and the Associate Editor of Automatica. He has been serving/served as the Vice President of Technical Activities, 2009–2010, an elected member of the Board of Governors, 2007–2009, and the Chair of the Technical Committee on Intelligent Control, 2005–2008, for the IEEE Control Systems Society.


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