A scheduling algorithm for networked control systems
AA SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS
ATREYEE KUNDUAbstract. This paper deals with the design of scheduling logics for Networked ControlSystems (NCSs) whose shared communication networks have limited capacity. We assumethat among π plants, only π ( < π ) plants can communicate with their controllers at anytime instant. We present an algorithm to allocate the network to the plants periodically suchthat stability of each plant is preserved. The main apparatus for our analysis is a switchedsystems representation of the individual plants in an NCS. We rely on multiple Lyapunov-like functions and graph-theoretic arguments to design our scheduling logics. The set ofresults presented in this paper is a continuous-time counterpart of the results proposed in[15]. We present a set of numerical experiments to demonstrate the performance of ourtechniques.
1. IntroductionNetworked Control Systems (NCSs) are spatially distributed control systems in whichthe communication between plants and their controllers occurs through shared communi-cation networks. NCSs ο¬nd wide applications in sensor networks, remote surgery, hapticscollaboration over the internet, automated highway systems, unmanned aerial vehicles, etc.[6]. While the use of shared communication networks in NCSs oο¬ers ο¬exible architecturesand reduced installation and maintenance costs, the exchange of information between theplants and their controllers often suο¬ers from network induced limitations and uncertainties.In this paper we deal with NCSs whose communication networks have limited bandwidth.While NCSs applications typically involve a large number of plants, the bandwidth ofthe shared network is often limited. Examples of communication networks with limitedbandwidth include wireless networks (an important component of smart home, smarttransportation, smart city, remote surgery, platoons of autonomous vehicles, etc.) andunderwater acoustic communication systems. The scenario in which the number of plantssharing a communication network is higher than the capacity of the network is called medium access constraint . This constraint motivates a need for allocating the networkto the plants in a manner so that good qualitative properties of each plant in the NCS arepreserved. This task of eο¬cient allocation of a shared communication network is commonlyreferred to as a scheduling problem and the corresponding allocation scheme is called a scheduling logic . We are interested in algorithmic design of scheduling logics for NCSs.Scheduling logics can be classiο¬ed broadly into two categories: static and dynamic . Incase of the former, a ο¬nite length allocation scheme of the network is determined oο¬ineand is applied eternally in a periodic manner, while in case of the latter, the allocation ofthe shared network is determined based on some information about the plant (e.g., states,outputs, access status of sensors and actuators, etc.). For NCSs with continuous-time linearplants, static scheduling logics that preserve stability of all plants are characterized usingcommon Lyapunov functions in [7] and piecewise Lyapunov-like functions with averagedwell time switching in [16]. A more general case of co-designing a static scheduling logicand control action is addressed using combinatorial optimization with periodic controltheory in [24] and Linear Matrix Inequalities (LMIs) optimization with average dwell time
Date : January 5, 2021.The author is with the Department of Electrical Engineering, Indian Institute of Science Bangalore, Bengaluru- 560012, Karnataka, India, E-mail: [email protected] . a r X i v : . [ ee ss . S Y ] J a n ATREYEE KUNDU technique in [4]. In the discrete-time setting, a blend of multiple Lyapunov-like functionsand graph theory was employed to design stability preserving periodic scheduling logicsrecently in [15]. The authors of [27] characterize static switching logics that ensurereachability and observability of the plants under limited communication, and design anobserver-based feedback controller for these logics. The corresponding techniques werelater extended to the case of constant transmission delays [8] and Linear Quadratic Gaussian(LQG) control problem [9]. Event-triggered scheduling logics that preserve stability of allplants under communication delays are proposed in [1]. In [18] the authors propose amechanism to allocate network resources by ο¬nding optimal node that minimizes a certaincost function in every network time instant. The design of dynamic scheduling logicsfor stability of each plant under both communication uncertainties and computationallimitations is studied in [25]. In [5] a class of distributed control-aware random networkaccess logics for the sensors such that all control loops are stabilizable, is presented. Adynamic scheduling logic based on predictions of both control performance and channelquality at run-time, is proposed recently in [20].In this paper we consider an NCS consisting of multiple continuous-time linear plantswhose feedback loops are closed through a shared communication network. A blockdiagram of such an NCS is shown in Figure 1. We assume that the plants are unstable
Controller 1 Plant 1Controller 2 Plant 2Controller N Plant N
Communication network ...
Figure 1. Block diagram of NCSin open-loop and asymptotically stable in closed-loop. Due to a limited communicationcapacity of the network, only a few plants can exchange information with their controllersat any instant of time. Consequently, the remaining plants operate in open-loop at everytime instant. We will design periodic scheduling logics that preserve global asymptoticstability (GAS) of each plant in the NCS. The set of results presented in this paper is acontinuous-time counterpart of the results in [15].We employ switched systems and graph theory as the main apparatuses for our analysis.We model the individual (open-loop unstable) plants of an NCS as switched systems, wherethe switching is between their open-loop (unstable mode) and closed-loop (stable mode)operations. In this setting no switched system can operate in stable mode for all time asthat will destabilize some of the plants in the NCS. The search for a stabilizing schedulinglogic then becomes the problem of ο¬nding switching logics that obey the limitations of theshared network and preserve stability. We assume that the exchange of information betweena plant and its controller is not aο¬ected by communication uncertainties. We associate aweighted directed graph with the NCS that captures the communication limitation of theshared network, and design stabilizing switching logics for each plant in the NCS. MultipleLyapunov-like functions are employed for analyzing stability of the switched systems.Towards designing a stabilizing periodic scheduling logic, we combine stabilizing switching
SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 3 logics in terms of a class of cycles on the underlying weighted directed graph of the NCSthat satisο¬es appropriate contractivity properties. We also discuss algorithmic constructionof these cycles. It is known that periodic scheduling logics are easier to implement, oftennear optimal, and guarantee activation of each sensor and actuator, see [23, 19, 10] fordetailed discussions. They are preferred for safety-critical control systems [19, Β§2.5.1]. Itis also observed in [22, 23] that periodic phenomenon appears in non-periodic schedules.We demonstrate the techniques proposed in this paper on a numerical example.The remainder of this paper is organized as follows: in Β§2 we formulate the problemunder consideration. The apparatuses for our design of scheduling logics and analysis ofstability are described in Β§3. Our results appear in Β§4. We present a numerical example inΒ§5 and conclude in Β§6 with a brief discussion of future research directions.We employ standard notations throughout the paper. N is the set of natural numbers, N = N βͺ { } , and R is the set of real numbers. (cid:107)Β·(cid:107) denotes the Euclidean norm (resp.,induced matrix norm) of a vector (resp., a matrix). For a ο¬nite set π΄ , we employ | π΄ | todenote its cardinality, i.e., the number of elements in π΄ . For a matrix π β R π Γ π , π max ( π ) and π min ( π ) denote the maximum and minimum eigenvalues of π , respectively. For a scalar π , possibly complex, Re ( π ) denotes the real part of π .2. Problem statementWe consider an NCS with π plants whose dynamics are given by (cid:164) π₯ π ( π‘ ) = π΄ π π₯ π ( π‘ ) + π΅ π π’ π ( π‘ ) , π₯ π ( ) = π₯ π , π‘ β [ , +β[ , (1)where π₯ π ( π‘ ) β R π and π’ π ( π‘ ) β R π are the vectors of states and inputs of the π -th plant attime π‘ , respectively, π = , , . . . , π . All plants communicate with their remotely locatedstate-feedback controllers π’ π ( π‘ ) = πΎ π π₯ π ( π‘ ) , π = , , . . . , π (2)through a shared communication network. The matrices π΄ π β R π Γ π , π΅ π β R π Γ π and πΎ π β R π Γ π , π = , , . . . , π are known.We will operate under the following set of assumptions: Assumption 1.
The shared communication network has a limited communication capacityin the sense that at any time instant, only π plants ( < π < π ) can access the network.Consequently, π β π plants operate in open-loop at every time instant. Assumption 2.
The open-loop dynamics of each plant is unstable and each controller isstabilizing. More speciο¬cally, the matrices π΄ π + π΅ π πΎ π , π = , , . . . , π are stable (Hurwitz)and the matrices π΄ π , π = , , . . . , π are unstable. Assumption 3.
The shared communication network is ideal in the sense that the exchangeof information between the plants and their controllers is not aο¬ected by communicationuncertainties.In view of Assumptions 1 and 2, each plant in (1) operates in two modes: (a) stablemode when the plant has access to the shared communication network and (b) unstablemode when the plant does not have access to the network. Let us denote the stable andunstable modes of the π -th plant as π π and π π’ , respectively, π΄ π π = π΄ π + π΅ π πΎ π and π΄ π π’ = π΄ π , π = , , . . . , π . In this paper we are interested in ο¬nding a mechanism to allocate the sharedcommunication network to the plants such that stability of each plant in (1) is preserved.Let S : = { π β { , , . . . , π } π | all elements of π are distinct } Recall that a matrix π΄ β R π Γ π is Hurwitz if every eigenvalue of π΄ has strictly negative real part. We call π΄ unstable if it is not Hurwitz. ATREYEE KUNDU be the set of vectors that consist of π distinct elements from { , , . . . , π } . We call afunction πΎ : [ , +β[β S a scheduling logic . There exists a diverging sequence of times0 = : π < π < π < Β· Β· Β· and a sequence of indices π , π , π , . . . with π π β S , π = , , , . . . such that πΎ ( π‘ ) = π π for all π‘ β [ π π : π π + [ , π = , , , . . . . In other words, πΎ speciο¬es, atevery time π‘ , π plants of the NCS which access the communication network at that time.The remaining π β π plants operate in open-loop, i.e., with π’ π ( π‘ ) = Deο¬nition 1. [12, Lemma 4.4] The π -th plant in (1) is globally asymptotically stable (GAS)for a given scheduling logic πΎ , if there exists a class KL function π½ π such that the followinginequality holds: (cid:107) π₯ π ( π‘ )(cid:107) (cid:54) π½ π ((cid:107) π₯ π ( )(cid:107) , π‘ ) for all π₯ π ( ) β R π and π‘ β [ , +β[ . (3)We will solve the following problem: Problem 1.
Given the matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and a number π ( < π ) , designa scheduling logic πΎ that preserves GAS of each plant in (1) . We will model each plant in (1) as a switched system and associate a labelled andweighted directed graph with the NCS under consideration. Scheduling logics πΎ thatpreserve GAS of each plant π in (1) will be designed by concatenating cycles on thisdirected graph that satisfy certain contractivity properties. In the sequel we will refer tosuch scheduling logics as stabilizing scheduling logics. Prior to presenting our solution toProblem 1, we catalog a set of preliminaries required for our analysis.3. Preliminaries3.1. Individual plants and switched systems.
We model the dynamics of the π -th plantin (1) as a switched system (cid:164) π₯ π ( π‘ ) = π΄ π π ( π‘ ) π₯ π ( π‘ ) , π₯ π ( ) = π₯ π , π π ( π‘ ) β { π π , π π’ } , π‘ β [ , +β[ , (4)where the subsystems are { π΄ π π , π΄ π π’ } and a switching logic π π : N β { π π , π π’ } satisο¬es: π π ( π‘ ) = (cid:40) π π , if π is an element of πΎ ( π‘ ) ,π π’ , otherwise . Clearly, a switching logic π π , π = , , . . . , π is a function of the scheduling logic πΎ . Inorder to ensure GAS of the individual plants, it therefore, suο¬ces to design a πΎ that renderseach π π stabilizing in the following sense: π π guarantees GAS of the switched system (4)for each π = , , . . . , π . Lemma 1.
For each π = , , . . . , π , there exist pairs ( π π , π π ) , π β { π π , π π’ } , where π π β R π Γ π are symmetric and positive deο¬nite matrices, and π π π > π π π’ (cid:54)
0, such thatwith R π (cid:51) π β¦β π π ( π ) : = π (cid:62) π π π β [ , +β[ (5)we have π π ( π§ π ( π‘ )) (cid:54) exp (β π π π‘ ) π π ( π§ π ( )) , π‘ β [ , +β[ , (6)and π§ π (Β·) solves the π -th recursion in (4), π β { π π , π π’ } . Proof.
Fix π β { , , . . . , π } . Since π΄ π π is Hurwitz, the matrix π π π can be selected as asymmetric and positive deο¬nite solution to the Lyapunov equation π΄ (cid:62) π π π π π + π π π π΄ π π + π π π = π for some pre-selected symmetric and positive deο¬nite matrix π π π β R π Γ π [2, Corollary11.9.1]. A straightforward calculation gives πππ‘ (cid:18) π π π (cid:0) π§ π π ( π‘ ) (cid:1)(cid:19) = β π§ π π ( π‘ ) (cid:62) π π π π§ π π ( π‘ ) , SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 5 where π§ π π (Β·) solves the π π -th system dynamics in (4). Recall that for any symmetric matrix π β R π Γ π , we have [2, Lemma 8.4.3] π min ( π ) (cid:107) π§ (cid:107) (cid:54) π§ (cid:62) π π§ (cid:54) π max ( π ) (cid:107) π§ (cid:107) for all π§ β R π . (7)It, therefore, follows that β π§ π π ( π‘ ) (cid:62) π π π π§ π π ( π‘ ) (cid:54) β π min ( π π π ) π max ( π π π ) π§ π π ( π‘ ) (cid:62) π π π π§ π π ( π‘ ) . Deο¬ning π π π = π min ( π π π ) π max ( π π π ) , we arrive at πππ‘ (cid:18) π π π (cid:0) π§ π π ( π‘ ) (cid:1)(cid:19) (cid:54) β π π π π π π (cid:0) π§ π π ( π‘ ) (cid:1) , which gives (6) with π π π > π΄ π π’ unstable, there exists π π π’ > π΄ π π’ β π π π’ πΌ π is Hurwitz. Fixa symmetric and positive deο¬nite matrix π π π’ , and let π π π’ be the symmetric and positivedeο¬nite solution to the Lyapunov equation ( π΄ π π’ β π π π’ πΌ π ) (cid:62) π π π’ + π π π’ ( π΄ π π’ β π π π’ πΌ π ) + π π π’ = π . It follows that πππ‘ (cid:18) π π π’ (cid:0) π§ π π’ ( π‘ ) (cid:1)(cid:19) = β π§ π π’ ( π‘ ) (cid:62) π π π’ π§ π π’ ( π‘ ) , where π§ π π’ (Β·) solves the π π’ -th system dynamics in (4). Applying (7), we arrive at β π§ π π’ ( π‘ ) (cid:62) π π π’ π§ π π’ ( π‘ ) (cid:54) β (cid:18) π π π’ β π min ( π π π’ ) π max ( π π π’ ) (cid:19) π§ π π’ ( π‘ ) (cid:62) π π π’ π§ π π’ ( π‘ ) . Deο¬ning π π π’ = π π π’ β π min ( π π π’ ) π max ( π π π’ ) , we arrive at πππ‘ (cid:18) π π π’ (cid:0) π§ π π’ ( π‘ ) (cid:1)(cid:19) (cid:54) β π π π’ π π (cid:0) π§ π π’ ( π‘ ) (cid:1) . Notice that any scalar larger than Re (cid:0) π max ( π΄ π π’ ) (cid:1) is a valid choice of π π π’ . One, therefore,needs to choose π π π’ and π π π’ such that 2 π π π’ β π min ( π π π’ ) π max ( π π π’ ) (cid:54) (cid:3) Lemma 2.
For each π = , , . . . , π , there exist π ππ (cid:62) π π ( π ) (cid:54) π ππ π π ( π ) for all π β R π and π, π, β { π π , π π’ } . (8) Proof.
Linear comparability of π π βs is clear from the deο¬nition of π π , π β { π π , π π’ } in (5).The assertion of Fact 2 follows at once. (cid:3) The function π π , π β { π π , π π’ } , π = , , . . . , π are called Lyapunov-like functions. Thescalars π π , π β { π π , π π’ } give quantitative measures of (in)stability associated to (un)stablemodes of operation of the π -th plant. A tight estimate of the scalars π ππ , π, π β { π π , π π’ } isgiven by π max ( π π π β π ) [14, Proposition 4]. Facts 1 and 2 will be employed in our design of πΎ . ATREYEE KUNDU
NCS and directed graphs.
We associate a directed graph πΊ ( π, πΈ ) with the NCSunder consideration. The vertex set π contains (cid:0) ππ (cid:1) vertices that are labelled distinctly. Thelabel associated to a vertex π£ β π is given by πΏ ( π£ ) = { β π£ ( ) , β π£ ( ) , . . . , β π£ ( π )} , where β π£ ( π ) = π π for any π elements of { , , . . . , π } and β π£ ( π ) = π π’ for the remaining π β π elements. The edge set πΈ contains directed edges ( π’, π£ ) for every π’, π£ β π , π’ β π£ .Let the functions π€ ( π£ ) = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) π€ ( π£ ) π€ ( π£ ) ...π€ π ( π£ ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , π£ β π with π€ π ( π£ ) = (cid:40) β (cid:12)(cid:12) π π π (cid:12)(cid:12) , if β π£ ( π ) = π π , (cid:12)(cid:12) π π π’ (cid:12)(cid:12) , if β π£ ( π ) = π π’ , π = , , . . . , π, (9)and π€ ( π’, π£ ) = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) π€ ( π’, π£ ) π€ ( π’, π£ ) ...π€ π ( π’, π£ ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , ( π’, π£ ) β πΈ with π€ π ( π’, π£ ) =  ln π π π π π’ , if β π’ ( π ) = π π and β π£ ( π ) = π π’ , ln π π π’ π π , if β π’ ( π ) = π π’ and β π£ ( π ) = π π , , otherwise , π = , . . . , π, (10)be the weight associated to a vertex π£ β π and the weight associated to an edge ( π’, π£ ) β πΈ ,respectively. Here, the scalars π π , π β { π π , π π’ } and π ππ , π, π β { π π , π π’ } , π = , , . . . , π , areas described in Fact 1 and Fact 2, respectively. Remark 1.
The association of a directed graph with an NCS was ο¬rst proposed in [15].Here we employ a natural extension of this association to the continuous-time setting. Thelabel πΏ ( π£ ) of a vertex π£ β π gives a combination of π plants operating in stable mode andthe remaining π β π plants operating in unstable mode. Since π contains (cid:0) ππ (cid:1) verticesand the label of each vertex is distinct, it follows that the set of vertex labels consists ofall possible combinations of π plants accessing the communication network and π β π plants operating in open-loop. A directed edge ( π’, π£ ) from a vertex π’ to a vertex π£ ( β π’ ) corresponds to a transition from a set of π plants accessing the communication network(as speciο¬ed by πΏ ( π£ ) ). The vertex (subsystem) weights of πΊ ( π, πΈ ) capture the rate ofincrease/decrease of the Lyapunov-like functions π π , π β { π π , π π’ } and the edge weights of πΊ ( π, πΈ ) capture the βjumpβ between Lyapunov-like functions π π and π π , π, π β { π π , π π’ } .This choice of weights is employed with the objective to compensate the increase in π π , π β { π π , π π’ } caused by activation of unstable mode π π’ and switches between stable andunstable modes ( π π to π π’ and π π’ to π π ) by the decrease in π π , π β { π π , π π’ } caused by theactivation of the stable modes π π , π = , , . . . , π , as will be useful in our analysis for GASof the individual plants.Recall that [3, p. 4] a cycle on the directed graph πΊ ( π, πΈ ) is an alternating (ο¬nite) se-quence of vertices and edges that begin and end on the same vertex, e.g., π = Λ π£ , ( Λ π£ , Λ π£ ) , Λ π£ , . . . , Λ π£ π , ( Λ π£ π , Λ π£ ) , Λ π£ . The number of edges that appear in the sequence is called the length of the cycle. Here the length of π is π . We will employ the following class of cycles in ourdesign of stabilizing scheduling logics: Deο¬nition 2.
A cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ on πΊ ( π, πΈ ) is called π -contractive if there exist R (cid:51) π π£ π > π = , , . . . , π β
1, 2 (cid:54) π (cid:54) | π | , such that the SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 7 following set of inequalities is satisο¬ed: Ξ π ( π ) : = π β βοΈ π = π€ π ( π£ π ) π π£ π + π β βοΈ π = π£ π : = π£ π€ π ( π£ π , π£ π + ) < π = , , . . . , π , where π is the length of π , π€ ( π£ π ) is the weight associated tovertex π£ π , π€ π ( π£ π ) is the π -th element of π€ ( π£ π ) , and π€ ( π£ π , π£ π + ) is the weight associatedto the edge ( π£ π , π£ π + ) , π€ π ( π£ π , π£ π + ) is the π -th element of π€ ( π£ π , π£ π + ) , π = , , . . . , π , π = , , . . . , π β Remark 2.
Deο¬nition 2 is a natural extension of [15, Deο¬nition 2] to the continuous-timesetting. The scalars π π£ π , π = , , . . . , π β π£ π , π = , , . . . , π β π . This time duration will determinehow long a set of π plants can access the shared communication network while preservingGAS of all plants in the NCS under consideration. Naturally, π π£ π βs are real numbers for thecontinuous-time case, while they are integers for the discrete-time setting. In the sequel wewill call the scalars π π£ π , π = , , . . . , π β π -factor of vertex π£ π , π = , , . . . , π β Remark 3.
The concept of contractive cycles and its variants have appeared in the contextof designing switching signals that preserve stability of continuous-time switched systemsearlier in the literature, see e.g., [13]. In this paper we will use the notion of π -contractivecycles to address a harder problem of simultaneously preserving GAS of π switchedsystems.We now move on to our solution to Problem 1.4. ResultsWe will solve Problem 1 in two steps: β¦ First, we present an algorithm that constructs periodic scheduling logics πΎ by employinga π -contractive cycle on the underlying directed graph πΊ ( π, πΈ ) of an NCS and itscorresponding π -factors. β¦ Second, we show that scheduling logics obtained from our algorithm preserve GAS ofeach plant in (1).We will also present an algorithm to design π -contractive cycles on πΊ ( π, πΈ ) .Given the matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and a number π , Algorithm 1 designs ascheduling logic πΎ , that speciο¬es, at every time, π plants that access the shared communi-cation network at that time. Algorithm 1 is a continuous-time counterpart of [15, Algorithm1]. The key ingredient of Algorithm 1 is a π -contractive cycle on the underlying directedgraph πΊ ( π, πΈ ) of the NCS under consideration. In Step I, corresponding to each vertex π£ π , π = , , . . . , π β
1, a vector π π , π = , , . . . , π β π is created with the elements π β { , , . . . , π } for which β π£ π ( π ) = π π . In Step II, a scheduling logic πΎ is constructed fromthe vectors π π , π = , , . . . , π β π -factors π π£ π , π = , , . . . , π β π plants corresponding to the elements in π π access the shared communicationnetwork for π π£ π duration of time, π = , , . . . , π β
1, and the process is repeated. Clearly, ascheduling logic πΎ constructed as above, is periodic with period π β βοΈ π = π π£ π . Theorem 1 assertsthat a scheduling logic obtained from Algorithm 1 is stabilizing. Theorem 1.
Consider an NCS described in Β§2. Let the matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and a number π ( < π ) be given. Then each plant π in (1) is GAS under a scheduling logic πΎ obtained from Algorithm 1.Proof. (Sketch) Consider the NCS described in Β§2 and its underlying directed graph πΊ ( π, πΈ ) . Let π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ be a π -contractive cycle on ATREYEE KUNDU
Algorithm 1
Construction of periodic scheduling logics
Input: a π -contractive cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ and the corre-sponding π -factors π , π , . . . , π π β . Output: a periodic scheduling logic πΎ Step I: For each vertex π£ π , π = , , . . . , π β , pick the elements π with label β π£ π ( π ) = π π , π = , , . . . , π , and construct π - dimensional vectors π π , π = , , . . . , π β . for π = , , . . . , π β do Set π = for π = , , . . . , π do if β π£ π ( π ) = π π then Set π = π + π’ π ( π ) = π . end if end for end for Step II: Construct a scheduling logic using the vectors π π , π = , , . . . , π β obtained inStep I and the π -factors π π£ π , π = , , . . . , π β Set π = π = for π = ππ, ππ + , . . . , ( π + ) π β do Set πΎ ( π π ) = π π β ππ and π π + = π π + π π£ π β ππ . Output π π and πΎ ( π π ) . end for Set π = π + πΊ ( π, πΈ ) . Consider a scheduling logic πΎ obtained from Algorithm 1 constructed by em-ploying π . We will show that πΎ preserves GAS of each plant in (1).Fix an π β { , , . . . , π } arbitrary. It suο¬ces to show that the switched system (4) isGAS under the switching signal π π corresponding to πΎ .Fix a time π‘ >
0. Recall that 0 = : π < π < Β· Β· Β· are the points in time where πΎ changesvalues. Let π πΎπ‘ denote the total number of times πΎ has changed its values on ] , π‘ ] .In view of (6), we have π π π ( π‘ ) ( π₯ π ( π‘ )) (cid:54) exp (cid:18) β π π π ( π ππΎπ‘ ) (cid:0) π‘ β π π πΎπ‘ (cid:1)(cid:19) π π π ( π‘ ) ( π₯ π ( π π πΎπ‘ )) . (12)A straightforward iteration of (12) applying (6) and (8) leads to π π π ( π‘ ) ( π₯ π ( π‘ )) (cid:54) exp (cid:18) β π πΎπ‘ βοΈ π = π ππΎπ‘ + : = π‘ π π π ( π π ) ( π π + β π π ) (cid:19) Γ π πΎπ‘ β (cid:214) π = π π π ( π π ) π π ( π π + ) π π π ( ) ( π₯ π ( )) . (13)Now, π πΎπ‘ β (cid:214) π = π π π ( π π ) π π ( π π + ) = exp ln (cid:32) π πΎπ‘ β (cid:214) π = π π π ( π π ) π π ( π π + ) (cid:33) = exp (cid:32) π πΎπ‘ β βοΈ π = ln π π π ( π π ) π π ( π π + ) (cid:33) = exp (cid:32) βοΈ π β{ π π ,π π’ } π πΎπ‘ β βοΈ π = βοΈ π β π : π β{ π π ,π π’ } ,π π ( π π ) = π,π π ( π π + ) = π ln π ππ (cid:33) . SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 9
Let π ππ ( π , π‘ ) denote the total number of transitions from subsystem (mode) π to subsystem(mode) π , π, π β { π π , π π’ } on ] π , π‘ ] . Then the RHS of the above expression is equal toexp (cid:18) ln π π π π π’ π π π π π’ ( , π‘ ) + ln π π π’ π π π π π’ π π ( , π‘ ) (cid:19) , (14)since ln π π π π π = ln π π π’ π π’ =
0. Moreover,exp (cid:32) β π πΎπ‘ βοΈ π = π ππΎπ‘ + : = π‘ π π π ( π π ) ( π π + β π π ) (cid:33) = exp (cid:32) β π πΎπ‘ βοΈ π = π π ππ‘ + : = π‘ (cid:18) βοΈ π β{ π π ,π π’ } ( π π ( π π ) = π ) π π ( π π + β π π ) (cid:19)(cid:33) . Let π· π ( π , π‘ ) and π· π’ ( π , π‘ ) denote the total durations of activation of the stable and unstablemodes of π on ] π , π‘ ] , respectively. Consequently, the above expression is equal toexp (cid:18) β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( , π‘ ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( , π‘ ) (cid:19) . (15)Substituting (14) and (15) in (13), we obtain π π π ( π‘ ) ( π₯ π ( π‘ )) (cid:54) π π ( π‘ ) π π π ( ) ( π₯ π ( )) , (16)where N (cid:51) π‘ β¦β π π ( π‘ ) : = exp (cid:18) β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( , π‘ ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( , π‘ )+ ln π π π π π’ π π π π π’ ( , π‘ ) + ln π π π’ π π π π π’ π π ( , π‘ ) (cid:19) . (17)From the deο¬nition of π π , π β { π π , π π’ } in (5) and properties of positive deο¬nite matrices[2, Lemma 8.4.3], it follows that (cid:107) π₯ π ( π‘ )(cid:107) (cid:54) ππ π ( π‘ ) (cid:107)((cid:107) π₯ π ( )) for all π‘ β [ , +β[ , (18)where π = βοΈ max π β{ π π ,π π’ } π max ( π π ) min π β{ π π ,π π’ } π min ( π π ) . In order to establish GAS of (4), we require to showthat π (cid:107) π₯ π ( )(cid:107) π π ( π‘ ) can be bounded above by a class KL function. It is immediate that π (cid:107) π₯ π ( )(cid:107) is a class K β function. It remains to show that π π ( π‘ ) is bounded above by afunction belonging to class L .Recall that πΎ is constructed by employing a π -contractive cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ on πΊ , and π π£ π , π = , , . . . , π β π -factors associated to vertices π£ π , π = , , . . . , π β
1. Let π π : = π β βοΈ π = π π£ π , π‘ (cid:62) ππ π , π β N , and Ξ π ( π ) = β π π , π π > Ξ π ( π ) is as deο¬ned in (11). By construction of πΎ , we have π π ( π‘ ) = exp (cid:32) β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( , π‘ ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( , π‘ ) + ln π π π π π’ π π π π π’ ( , π‘ ) + ln π π π’ π π π π π’ π π ( , π‘ ) (cid:33) = β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( , ππ π ) β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( ππ π , π‘ ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( , ππ π ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( ππ π , π‘ )+ ln π π π π π’ π π π π π’ ( , ππ π ) + ln π π π π π’ π π π π π’ ( ππ π , π‘ )+ ln π π π’ π π π π π’ π π ( , ππ π ) + ln π π π’ π π π π π’ π π ( ππ π , π‘ ) . (19)Notice that β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( , ππ π ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( , ππ π ) + ln π π π π π’ π π π π π’ ( , ππ π ) + ln π π π’ π π π π π’ π π ( , ππ π ) = β (cid:12)(cid:12) π π π (cid:12)(cid:12) π βοΈ π : β π£π ( π ) = π π π = , ,...,π β π π£ π + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π βοΈ π : β π£π ( π ) = π π’ π = , ,...,π β π π£ π + ln π π π π π’ π ( π π β π π’ ) π + ln π π π’ π π π ( π π’ β π π ) π , where ( π β π ) π denotes the number of times a transition from a vertex π£ π to a vertex π£ π + has occurred in π such that β π£ π ( π ) = π and β π£ π + ( π ) = π , π, π β { π π , π π’ } , π β π . Theright-hand side of the above equality can be rewritten as π (cid:32) β (cid:12)(cid:12) π π π (cid:12)(cid:12) βοΈ π : β π£π ( π ) = π π π = , ,...,π β π π£ π + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) βοΈ π : β π£π ( π ) = π π’ π = , ,...,π β π π£ π + ln π π π π π’ ( π π β π π’ ) π + ln π π π’ π π ( π π’ β π π ) π (cid:33) . (20)From the deο¬nition of weights associated to vertices and edges of πΊ , we have that the aboveexpression is equal to β ππ π . Also, β (cid:12)(cid:12) π π π (cid:12)(cid:12) π· π ( ππ π , π‘ ) + (cid:12)(cid:12) π π π’ (cid:12)(cid:12) π· π’ ( ππ π , π‘ )+ ln π π π π π’ π π π π π’ ( ππ π , π‘ ) + ln π π π’ π π π π π’ π π ( ππ π , π‘ ) (cid:54) (cid:12)(cid:12) π π π’ (cid:12)(cid:12) ( π‘ β ππ π ) + ππ ( ln π π π π π’ + ln π π π’ π π ) : = π (say) . (21)From (20) and (21), we obtain that the right-hand side of (19) is bounded above byexp (cid:0) β ππ π + π (cid:1) .Let π π : [ , π‘ ] β R be a function connecting ( , exp ( π )+ π π ) , ( ππ π , exp (β( π β ) π π + π )) , ( π‘, exp (β ππ π + π )) , π = , , . . . , π , with straight line segments. By construction, π π isan upper envelope of π β¦β π π ( π ) on [ ,π‘ ] , is continuous, decreasing, and tends to 0 as π‘ β +β . Hence, π π β L .Since π β { , , . . . , π } was selected arbitrarily, the assertion of Theorem 1 follows. (cid:3) A next natural question is: given the matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and the number π , how do we design a π -contractive cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ onthe underlying directed graph πΊ ( π, πΈ ) of the NCS under consideration? In the remainderof this section we address this question. Deο¬nition 3. [15, Deο¬nition 3] We call a cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ on πΊ ( π, πΈ ) candidate contractive , if for each π = , , . . . , π , there exists at least one π£ π , π β { , , . . . , π β } such that β π£ π ( π ) = π π .Let πΆ πΊ denote the set of all candidate contractive cycles on πΊ ( π, πΈ ) . We next providean algorithm to design a π -contractive cycle on πΊ ( π, πΈ ) .Given the matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and the number π , Algorithm 2 designsa π -contractive cycle π on πΊ ( π, πΈ ) . Recall the computations of the pairs ( π π , π π ) , π β { π π , π π’ } from our proof of Fact 1. We perform line searches over the intervals [ π βππ π , π π’ππ π ] and [ π βππ π’ , π π’ππ π’ ] with step sizes β π and β π’ , respectively, and solve the feasibility problem(22) for the pairs ( π π , π π ) , π β { π π , π π’ } , π = , , . . . , π . The condition π πΌ (cid:22) π π π , π π π’ limitsthe condition numbers of π π π and π π π’ to π β , and the condition π π π , π π π’ (cid:22) πΌ π guaranteesthat the set of feasible π π π , π π π’ is bounded. If (22) admits solutions for all π = , , . . . , π ,then we compute the scalars π ππ , π, π β { π π , π π’ } , π = , , . . . , π by using the estimate of[14, Proposition 4], and check if any of the candidate contractive cycles on πΊ ( π, πΈ ) is π -contractive. If a solution is found, then Algorithm 2 outputs the cycle and its corresponding π -factors, and terminates. Otherwise, the values of π π π and π π π’ are updated, and the searchcontinues. Remark 4.
The design of π -contractive cycles on πΊ ( π, πΈ ) in this paper diο¬ers from [15,Algorithm 2] in terms of designing the pairs ( π π , π π ) , π β { π π , π π’ } , π = , , . . . , π . In [15, SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 11
Algorithm 2
Design of a π -contractive cycle on πΊ ( π, πΈ ) Input : The matrices π΄ π , π΅ π , πΎ π , π = , , . . . , π and the set πΆ πΊ . Output : A π -contractive cycle π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ on πΊ ( π, πΈ ) and the corresponding π -factors π , π , . . . , π π β . Step I: Compute the matrices π΄ π π and π΄ π π’ , π = , , . . . , π for π = , , . . . , π do Set π΄ π π = π΄ π + π΅ π πΎ π and π΄ π π’ = π΄ π end for Step II: Fix ranges of values for π π , π β { π π , π π’ } , π = , , . . . , π and step-sizes β π and β π’ Fix π βππ > π π’ππ > π βππ . Fix β π > π π such that π π is the maximum integersatisfying π βππ + π π β π (cid:54) π π’ππ . Fix π π’ππ’ = π βππ’ < π π’ππ’ . Fix β π’ > π π’ such that π π’ is the largest integer satisfying π βππ’ + π π’ β π’ (cid:54) π π’ππ’ . Step III: Check for pairs ( π π , π π ) , π β { π π , π π’ } , π = , , . . . , π under which πΊ ( π, πΈ ) admitsa π -contractive cycle for π π = π βππ , π βππ + β π , π βππ + β π , . . . , π βππ + π π β π do for π π’ = π βππ’ , π βππ’ + β π’ , π βππ’ + β π’ , . . . , π βππ’ + π π’ β π’ do for π = , , . . . , π do Solve for ( π π , π π ) , π β { π π , π π’ } :minimize 1subject to  π΄ (cid:62) π π π π π + π π π π΄ π π (cid:22) β π π π π π π ,π΄ (cid:62) π π’ π π π’ + π π π’ π΄ π π’ (cid:22) β π π π’ π π π’ ,π π π , π π π’ (cid:31) ,π πΌ (cid:22) π π π , π π π’ (cid:22) πΌ π , π > ,π small enough . (22) end for if there is a solution to (22) for all π = , , . . . , π then Compute π π π π π’ = π max ( π π π’ π β π π ) and π π π’ π π = π max ( π π π π β π π’ ) end if Solve for π π£ π , π = , , . . . , π β π β πΆ πΊ (cid:40) π is π -contractive ,π π£ π > , π = , , . . . , π β . (23) if there is a solution to (23) then Output π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ and the corresponding π -factors π , π , . . . , π π β , and halt. end if end for end for Algorithm 2] the authors employed designed Lyapunov-like functions in the discrete-timesetting, while in the current paper we cater to their continuous-time counterparts.Algorithm 2, however, provides only a partial solution to the problem of designing π -contractive cycles on the underlying directed graph of an NCS. Indeed, even if the step-sizes β π and β π’ are chosen to be very small, only a ο¬nite number of possibilities for the pair ( π π , π π ) , π β { π π , π π’ } , π = , , . . . , π , are explored. Consequently, if no solution to thefeasibility problem (23) is found, we cannot conclude that there does not exist choice ofscalars π π , π β { π π , π π’ } and π ππ , π, π β { π π , π π’ } , π = , , . . . , π , under which πΊ ( π, πΈ ) admits a π -contractive cycle. Remark 5.
Switched systems have appeared before in NCSs literature, see e.g., [11, 26,17, 28], and average dwell time switching logic is proven to be a useful tool. In the presenceof unstable systems, stabilizing average dwell time switching involves two conditions onevery interval of time [21]: i) an upper bound on the number of switches and ii) a lowerbound on the ratio of durations of activation of stable to unstable subsystems. In contrast,our design of a stabilizing scheduling logic involves design of a π -contractive cycle onthe underlying weighted directed graph of the NCS. To design these cycles, we solve thefeasibility problems (22) and (23). We do not impose restrictions on the behaviour of ascheduling logic on every interval of time, thereby leading to numerically tractable stabilityconditions. Remark 6.
Lyapunov-like functions and graph-theory have been employed to study stabilityof continuous-time switched systems earlier in the literature, see e.g., [13]. In particular,stabilizing switching signals are constructed by concatenating cycles on the underlyingdirected graph of a switched system that satisfy certain properties. The design of thesecycles requires βcoβ-designing the Lyapunov-like functions and cycles on the directed graphunder consideration. This problem is known to be numerically hard. As a natural choice, theexisting literature considers the Lyapunov-like functions and a set of scalars correspondingto these functions to be βgivenβ, and designs cycles on the underlying directed graph ofa switched system that satisfy the desired conditions. However, non-existence of such acycle with the given choice of functions does not conclude non-existence of a class offunctions for which the underlying directed graph of a switched system admits a favourablecycle. Algorithm 2 extends the literature on stability of continuous-time switched systemsby providing a partial solution to the problem of βcoβ-designing Lyapunov-like functionsand suitable cycles on the underlying directed graph of a switched system.We now present two numerical experiments to test our technique for designing stabilizingscheduling logics. 5. Numerical experiments
Experiment 1.
We consider an NCS with two plants. At any point in time only oneplant can communicate with its controller over the shared communication network. Morespeciο¬cally, π = π =
1. The numerical values of the matrices π΄ π , π΅ π , πΎ π , π = , π΄ = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) . . . . (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ,π΅ = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ,πΎ = (cid:18) β . . . β . . β . β . . (cid:19) , SCHEDULING ALGORITHM FOR NETWORKED CONTROL SYSTEMS 13 and π΄ = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) . . .
05 0 00 0 0 β (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ,π΅ = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ,πΎ = . π + β (cid:18) β . . β . β . β . . β . β . (cid:19) . The eigenvalues of π΄ and π΄ + π΅ πΎ are 0 . , . , . , . β . β . , β . β . π΄ and π΄ + π΅ πΎ are β , . , . , . β . , β . , β . β . πΊ ( π, πΈ ) of the NCS under consideration has 2 verticesand a directed edge between every pair of vertices. The labels of the vertices are: πΏ ( π£ ) = (cid:18) π π’ (cid:19) and πΏ ( π£ ) = (cid:18) π’ π (cid:19) , π£ π β π, π = , .
2) We obtain the set of all candidate contractive cycles, πΆ πΊ has two components: π£ , ( π£ , π£ ) , π£ , ( π£ , π£ ) , π£ and π£ , ( π£ , π£ ) , π£ , ( π£ , π£ ) , π£ .3) We input the matrices π΄ π , π΅ π , πΎ π , π = , πΆ πΊ to Algorithm 2and obtain a π -contractive cycle π = π£ , ( π£ , π£ ) , π£ , ( π£ , π£ ) , π£ with its corresponding π -factors π π£ = .
25 and π π£ = .
36. The algorithm is implemented in MATLABR2020a. The following choices of scalars are used: π βππ = . π π’ππ’ = β π = . π βππ’ = π π’ππ’ = β π’ = .
01, and π = . π -contractive cycle obtained in Step 3) to Algorithm 1 that designs ascheduling logic πΎ . We have that πΎ is periodic with period π π£ + π π£ = .
61 units oftime.5) We choose 10 diο¬erent initial conditions from the interval [β , ] uniformly atrandom, and plot (cid:107) π₯ π ( π‘ )(cid:107) versus π‘ under πΎ until π‘ =
150 units of time, π = ,
2. SeeFigures 2 and 3. GAS of each plant is observed. t || x ( t ) || Figure 2. State trajectory for plant 1 under πΎ Experiment 2.
We now test the performance of our techniques in large scale settings, i.e.,when the number of plants in the NCS is large. First, we generate π unstable matrices π΄ π β R Γ and vectors π΅ π β R Γ with entries from the interval [β , ] and the set { , } , respectively, chosen uniformly at random, and ensuring that each pair ( π΄ π , π΅ π ) , π = , , . . . , π , is controllable. The linear quadratic regulators πΎ π are computed with π π = π = πΌ Γ and π π = π =
1. Second, the underlying directed graph πΊ ( π, πΈ ) of theNCS is considered, and Algorithm 2 is employed to design a π -contractive cycle. Third, t || x ( t ) || Figure 3. State trajectory for plant 2 under πΎ this cycle is employed to design stabilizing scheduling logics. In Table 1 we list sizes of πΊ and lengths of candidate contractive cycles π = π£ , ( π£ , π£ ) , π£ , . . . , π£ π β , ( π£ π β , π£ ) , π£ for various values of π , and the (rounded-oο¬) computation times. π | π | π Computation time (sec)100 1 . Γ
64 5000200 2 . Γ
101 9000500 2 . Γ
327 37000700 7 . Γ
521 740001000 2 . Γ
803 90000Table 1. Graph and cycle data6. Concluding remarksIn this paper we addressed the design of scheduling logics for NCSs whose communi-cation networks have limited bandwidth. We presented an algorithm that designs purelytime-dependent periodic scheduling logics under which GAS of each plant is preserved. Ablend of multiple Lyapunov-like functions and graph theory was employed as the main appa-ratus for our analysis. The results presented in this paper are a continuous-time counterpartof the techniques proposed in [15].We identify the following two directions for our future work: First, the extension of ourtechniques to the design of scheduling logics when the communication networks are alsoprone to uncertainties like delays, data losses, etc. Second, the co-design of controllers πΎ π , π = , , . . . , π and a scheduling logic πΎ such that good qualitative properties of each plantin an NCS are preserved. The above topics are currently under investigation and will bereported elsewhere. References [1] S. Al-Areqi, D. GΓΆrges, and S. Liu, Event-based control and scheduling codesign: stochastic and robustapproaches , IEEE Transactions on Automatic Control, 60 (2015), pp. 1291β1303.[2] D. S. Bernstein,
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