Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2012 / Article
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Mathematical Problems for Complex Networks

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Volume 2012 |Article ID 291714 | 16 pages | https://doi.org/10.1155/2012/291714

Mixed ๐ป๐Ÿ/๐ปโˆž Performance Analysis and State-Feedback Control Design for Networked Systems with Fading Communication Channels

Academic Editor: Zidong Wang
Received01 Jul 2011
Accepted02 Sep 2011
Published14 Dec 2011

Abstract

The purpose of the present paper is to provide a performance analysis approach of networked systems with fading communication channels. For a Ricean model of the fading communication channel, it is shown that the resulting system has a hybrid structure including the continuous-time dynamics of the networked systems and a discrete-time dynamics of the communication channels. Moreover, this resulting hybrid system has both multiplicative and additive noise terms. The performance analysis naturally leads to an ๐ป2/๐ปโˆž-type norm evaluation for systems with finite jumps and multiplicative noise. It is proved that this norm depends on the stabilizing solution of a specific system of coupled Riccati's equations with jumps. A state-feedback design problem to accomplish a mixed ๐ป2/๐ปโˆž performance is also considered. A numerical iterative procedure allowing to compute the stabilizing solution of the Riccati-type system with jumps is presented. The theoretical results are illustrated by numerical results concerning the tracking performances of a flight formation with fading communication channel. The paper ends with some concluding remarks.

1. Introduction

The analysis and synthesis of networked control systems have received a major attention over the last decade due to their wide area of applications (see, e.g., [1โ€“3] and their references). These applications include aerial and terrestrial surveillance [3], formation atmospheric flight [4], terrain mapping, and satellites formations for space science missions [5]. In all these applications, the formation members are autonomous vehicles from which the human pilots have been removed in order to avoid their participation at dangerous and repetitive tasks. The specific feature of such networked systems is that the control loop is closed through a communication channel shared by all autonomous vehicles. This communication network is required since the control law of each formation member usually depends on the measurements from all other vehicles. Even in the distributed control architectures when each vehicle uses only the information about its neighboring vehicles, a communication network is very useful. Indeed, in [6] it is proved that in a predecessor following approach, the relative positioning error between vehicles is amplified if the members of the formation have no information about the leader position. This conclusion motivates the use of a predecessor and leader following method in which both information about the predecessor and about the leader position are available for each formation member. In [6] it is also proved that in this case the relative spacing errors can be attenuated. These interesting results emphasize the importance of the communication in networked control systems. Most of the communication systems are based on wireless networks which, in contrast with the wired systems, are much more sensitive to information transmission errors.

The main goal of this paper is to analyze the interaction between the control and the communication system with fading. To this end, a model of the fading communication channel is required. There are many such models developed in the recent literature (see, e.g., [7, 8]). Deterministic models of communication networks with time-varying delays are considered in [9โ€“11], in which the maximum admissible delays are determined using the Lyapunov stability theory. In other deterministic models of fading communication channels, the transmission errors are represented as uncertain parameters, and, the control system is designed via specific robust synthesis procedures including linear quadratic Gaussian (LQG) and ๐œ‡-synthesis. Another class of representations of fading communication channels is based on stochastic models either with Markovian jumps or with white noise [7, 12]. Many useful results concerning the stability, control, and disturbance attenuation of such systems are available in the control literature (see, e.g., [13โ€“15] and their references). In [8], an ๐ปโˆž-type design is used to determine a controller for a system with fading communication channel represented as a Markovian system. A Markovian representation of the network status is also used to solve robust fault detection problems by ๐ปโˆž techniques for communication systems which may be found in [16]. An extended version for the case of random measurement delays and stochastic data-missing phenomenon is treated in [17]. In the present paper, a discrete-time Ricean model of the communication channel is considered. The stochastic Rice models are often used for models of wireless links [7]. They include both additive and multiplicative white noise terms. The problem analyzed in this paper is the influence of the fading communication channel over the tracking performance of a flight formation. The control system of the flight formation is the one derived in [4]. Since the exogenous inputs in the networked system are both deterministic (the reference signals for the formation control) and stochastic (the white noise terms in the fading communication channel model), a mixed ๐ป2/๐ปโˆž-type approach is appropriate for this analysis. Moreover, the networked system has a hybrid structure due to its continuous-time component represented by the vehicles dynamics and a discrete-time one corresponding to the communication channel. The above mentioned considerations lead to a mixed ๐ป2/๐ปโˆž analysis problem for stochastic systems with finite jumps. The systems with finite jumps are used to represent dynamic systems with continuous-time and discrete-time components. Useful results and developments concerning these systems may be found, for instance, in [18, 19]. The paper provides an analysis and an optimization approach of the mixed ๐ป2/๐ปโˆž performance for a hybrid model of networked systems with fading communication channels. This model is derived in Section 2 of the paper. In Section 3, the expression of the ๐ป2/๐ปโˆž performance is determined in terms of the stabilizing solution of a specific system of coupled Riccati equations with finite jumps. A state-feedback design procedure to optimize the mixed performance is presented in Section 4. Numerical aspects concerning the computation of the stabilizing solution are given in Section 5. The theoretical results are illustrated by a numerical example concerning the tracking performance of an aircraft formation. The paper ends with some final remarks and future work.

2. A Model of Networked Systems over Fading Communication Channel

In this section a control problem of a formation of unmanned air vehicles (UAVs) will be briefly presented. Such problems have been intensively analyzed over the last fifteen years (see, e.g., [3]) both for their wide area of applications and for the challenges addressed to the control engineer. In [4], a dynamic inversion-type approach is used to linearize the nonlinear dynamic and kinematic equations of the UAV motion. A simplified linearized model of a flight formation member has the form: ฬ‡ฬ‡๐›ฟ=๐‘Œ๐œ‰,๐œ‰=โˆ’๐พ๐‘‘๐›ฟโˆ’๐พ๐‘ฅฬ‡๐œ‰โˆ’๐‘ฅ,(2.1) where ๐›ฟโˆˆโ„3 denotes the deviation of the aircraft with respect to its desired position and ๐œ‰โˆˆโ„3 stands for the deviation of its state ๐‘ฅ=[๐‘‰๐œ“๐›พ]๐‘‡ (๐‘‰ representing the airspeed, ๐œ“ the heading angle, and ๐›พ the flight path angle) with respect to some specified value ๐‘ฅ. The input vector ฬ‡โ€Œ๐‘ฅ includes the desired derivatives of ๐‘ฅ and plays the role of a reference signal in the model (2.1). The constant matrix ๐‘Œ has the diagonal form ๐‘Œ=diag(1,๐‘‰0,๐‘‰0), and the state-feedback control gains ๐พ๐‘‘ and ๐พ๐‘ฅ are diagonal, too. In the present paper, the case when the reference signal ฬ‡โ€Œ๐‘ฅ is transmitted from the ground station or from the formation leader using a fading communication channel is considered. The aim is to analyze how the tracking performances of the flight formation are altered due to the communication system. To this end, a model of the fading communication channel is required. In this paper, the ๐ฟth-order Rice model was adopted. This model is frequently used in wireless mobile links, and it is given by the discrete-time equation:๐‘Ÿ(๐‘–)=๐ฟ๎“๐‘˜=0๐‘Ž๐‘˜(๐‘–)๐‘ฃ(๐‘–โˆ’๐‘˜)+๐‘›(๐‘–),(2.2) where ๐‘– denotes the moment of time, ๐‘ฃ(โ‹…) denotes the transmitted information, ๐‘Ÿ(โ‹…) is the received information, ๐‘› is a Gaussian white noise with zero mean and unit variance, and ๐‘Ž๐‘˜(๐‘–), ๐‘˜=0,โ€ฆ,๐ฟ are independent random variables with known mean ๐‘Ž๐‘˜ and variance ๐œŽ2๐‘˜. In the case of the application considered in this paper, ๐‘ฃ(โ‹…) is just the transmitted reference signal ฬ‡โ€Œ๐‘ฅ. A state-space representation of (2.2) is ๐‘(๐‘–+1)=๐‘€๐‘(๐‘–)+๐‘๐‘ฃ(๐‘–),๐‘Ÿ(๐‘–)=๐ฟ๎“๐‘˜=1๐‘Ž๐‘˜(๐‘–)๐‘ƒ๐‘˜๐‘(๐‘–)+๐‘Ž0(๐‘–)๐‘ฃ(๐‘–)+๐‘›(๐‘–),๐‘–=0,1,โ€ฆ,(2.3) where, by definition, โŽกโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽฃ๐ผ0โ‹ฎ0โŽคโŽฅโŽฅโŽฆ๐‘€=00โ‹ฏ0๐ผ0โ‹ฏ0โ‹ฎโ‹ฑ0โ‹ฏ๐ผ0,๐‘=,๐‘ƒ๐‘˜=๎€บ๎€ป,0โ‹ฏ0๐ผ0โ‹ฏ0(2.4) and ๐‘(๐‘–)โˆˆโ„๐ฟโ‹…๐‘›๐‘ฃร—1 stands for the state vector of the communication channel, ๐‘›๐‘ฃ denoting the dimension of the transmitted information vector ๐‘ฃ(โ‹…). In (2.4), the identity and the zero matrices have the size ๐‘›๐‘ฃร—๐‘›๐‘ฃ, and the identity matrix in ๐‘ƒ๐‘˜ is on the ๐‘˜th position. The configuration of the communication system (2.3) coupled with the system (2.1) is illustrated in Figure 1. The resulting system from Figure 1 is in fact a hybrid system since the dynamics of (2.1) is a continuous-time one, and (2.3) is a discrete-time system. A state-space realization of such hybrid system can be given using systems with finite jumps of the general form: ๐‘ฅ๎€ทฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๐ต๐‘ค(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ด๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต๐‘‘๐‘ค๐‘‘(๐‘–),๐‘–=0,1,โ€ฆ,(2.5) in which โ„Ž>0 denotes the sampling period, the state ๐‘ฅ(๐‘ก) is left continuous and right discontinuous at the sampling moments ๐‘ก=๐‘–โ„Ž, ๐‘–=0,1,โ€ฆ, and ๐‘ค(๐‘ก), ๐‘กโ‰ ๐‘–โ„Ž, and ๐‘ค๐‘‘(๐‘–), ๐‘–=0,1,โ€ฆ, are the continuous-time and the discrete-time inputs, respectively, of the system (see, e.g., [19, 20]). Since the received information ๐‘Ÿ is constant between the sampling moments, it can be represented as ๐‘Ÿ๎€ทฬ‡๐‘Ÿ(๐‘ก)=0,๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ฟ๎“๐‘˜=1๐‘Ž๐‘˜(๐‘–)๐‘ƒ๐‘˜๐‘(๐‘–โ„Ž)+๐‘Ž0(๐‘–)๐‘ฃ(๐‘–)+๐‘›(๐‘–),๐‘–=0,1,โ€ฆ,(2.6) where the state ๐‘ of the communication system is given by ๐‘๎€ทฬ‡๐‘(๐‘ก)=0,๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐‘€๐‘(๐‘–โ„Ž)+๐‘๐‘ฃ(๐‘–),๐‘–=0,1,โ€ฆ.(2.7) A similar model can be adopted for the continuous-time control system (2.1) which can be represented as ฬ‡๐‘ž(๐‘ก)=๐ด๐‘ž(๐‘ก)+๐ต๐‘Ÿ(๐‘ก),๐‘กโ‰ฅ0,(2.8) where ๎€บ๐›ฟ๐‘ž(๐‘ก)โˆถ=๐‘‡(๐‘ก)๐œ‰๐‘‡๎€ป(๐‘ก)๐‘‡,๎‚ƒ๐ดโˆถ=0๐‘Œโˆ’๐พ๐‘‘โˆ’๐พ๐‘ฅ๎‚„,๎‚ƒ0๎‚„.๐ตโˆถ=โˆ’๐ผ(2.9) In (2.8) the state ๐‘ž(๐‘ก) is continuous-time, and therefore, ๐‘ž๎€ท๐‘–โ„Ž+๎€ธ=๐‘ž(๐‘–โ„Ž),(2.10) and ๐‘Ÿ(๐‘ก) is the received perturbed reference signal described by (2.6). From (2.6)โ€“(2.10), it follows that the hybrid networked system with fading communication channels can be represented using a model with finite jumps of the form (2.5) where ๐‘ฅ=[๐‘ž๐‘‡๐‘Ÿ๐‘‡๐‘๐‘‡]๐‘‡. Moreover, since in (2.6) the coefficients ๐‘Ž๐‘˜(๐‘–), ๐‘˜=0,โ€ฆ,๐ฟ and ๐‘–=0,1,โ€ฆ, are random variables, one may consider the following stochastic version of (2.5) which includes both multiplicative noise components and additive white noise terms: ๎€ท๐ด๐‘‘๐‘ฅ(๐‘ก)=0๐‘ฅ(๐‘ก)+๐ต0๐‘ค๎€ธ๎€ท๐ด(๐‘ก)๐‘‘๐‘ก+1๐‘ฅ(๐‘ก)+๐ต1๐‘ค๎€ธ๐‘ฅ๎€ท(๐‘ก)๐‘‘๐œˆ(๐‘ก)+๐บ๐‘‘๐œ‚(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ด0๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต0๐‘‘๐‘ค๐‘‘(๎€ท๐ด๐‘–)+1๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต1๐‘‘๐‘ค๐‘‘(๎€ธ๐œˆ๐‘–)๐‘‘(๐‘–)+๐บ๐‘‘๐œ‚๐‘‘(๐‘ฆ๐‘–),๐‘–=0,1,โ€ฆ,๐‘ฆ(๐‘ก)=๐ถ๐‘ฅ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘‘(๐‘–)=๐ถ๐‘‘๐‘ฅ(๐‘–โ„Ž),๐‘–=0,1,โ€ฆ,(2.11) where the random variables ๐œ‚(๐‘ก)โˆˆโ„,๐‘กโ‰ฅ0, and ๐œˆ(๐‘ก)โˆˆโ„๐‘Ÿ, ๐‘กโ‰ฅ0, are such that the pair (๐œ‚(๐‘ก),๐œˆ(๐‘ก)) is an ๐‘Ÿ+1-dimensional standard Wiener process, and ๐œˆ๐‘‘(๐‘–)โˆˆโ„ and ๐œ‚๐‘‘(๐‘–)โˆˆโ„๐‘Ÿ๐‘‘, ๐‘–=0,1,โ€ฆ are sequences of independent random variables on a probability space (ฮฉ,๐’ซ,โ„ฑ). It is assumed that ๐œˆ(๐‘ก),๐œ‚(๐‘ก), ๐‘กโ‰ฅ0, ๐œˆ๐‘‘(๐‘–),๐œ‚๐‘‘(๐‘–), ๐‘–=0,1,โ€ฆ, are independent stochastic processes with zero mean and unitary second moments. The outputs ๐‘ฆ(๐‘ก) and ๐‘ฆ๐‘‘(๐‘–) denote the continuous-time and the discrete-time outputs, respectively. By virtue of standard results from the theory of stochastic differential equations (see, e.g., [21]), the system (2.11) has a unique โ„ฑ๐‘ก-adapted solution for any initial condition ๐‘ฅ(0), โ„ฑ๐‘ก denoting the ๐œŽ-algebra generated by the random vectors ๐œˆ(๐‘ ),๐œ‚(๐‘ ),๐œˆ๐‘‘(๐‘–), and ๐œ‚๐‘‘(๐‘–), 0โ‰ค๐‘ โ‰ค๐‘ก,0โ‰ค๐‘–โ„Žโ‰ค๐‘ก. This solution is almost surely left continuous.

A mixed ๐ป2/๐ปโˆž problem for this class of stochastic systems with jumps will be treated in the next section.

3. Mixed ๐ป2/๐ปโˆž-Type Norm for Systems with Jumps Corrupted with Multiplicative Noise

Before defining and computing the mixed ๐ป2/๐ปโˆž norm for systems of the form (2.11), some useful definitions and preliminary results will be briefly presented.

3.1. Notations, Definitions, and Some Useful Results

Consider the stochastic system with jumps (2.11) in which ๐‘ค and ๐‘ค๐‘‘ denote continuous-time and discrete-time energy bounded inputs, respectively. It means that ๐‘คโˆˆ๐ฟ2[0,โˆž), where ๐ฟ2[0,โˆž) denotes the space of the functions ๐‘“(๐‘ก),๐‘กโ‰ฅ0 for which โ€–๐‘“โ€–2๐ฟ2โˆซโˆถ=โˆž0|๐‘“(๐‘ก)|2๐‘‘๐‘ก<โˆž, and ๐‘ค๐‘‘โˆˆโ„“2 where โ„“2 is the space of the discrete-time vectors ๐‘”(๐‘–),๐‘–=0,1,โ€ฆ with the property โ€–๐‘”โ€–2โ„“2โˆ‘โˆถ=โˆž๐‘–=0|๐‘”(๐‘–)|2<โˆž, where |โ‹…| stands for the Euclidian norm.

Definition 3.1. The stochastic system with jumps and with multiplicative noise ๐‘‘๐‘ฅ(๐‘ก)=๐ด0๐‘ฅ(๐‘ก)๐‘‘๐‘ก+๐ด1๐‘ฅ๎€ท๐‘ฅ(๐‘ก)๐‘‘๐œˆ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ด0๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ด1๐‘‘๐‘ฅ(๐‘–โ„Ž)๐œˆ๐‘‘(๐‘–)๐‘–=0,1,โ€ฆ(3.1) is exponentially stable in mean square (ESMS) if there exist ๐›ผ>0 and ๐›ฝโ‰ฅ1 such that ๐ธ[|๐‘ฅ(๐‘ก)|2]โ‰ค๐›ฝ๐‘’โˆ’๐›ผ๐‘ก|๐‘ฅ(0)|2 for any initial condition ๐‘ฅ(0) and for all ๐‘กโ‰ฅ0, ๐ธ[โ‹…] denoting the mean of the random variable and ๐‘ฅ(๐‘ก) representing the solution of (3.1) with the initial condition ๐‘ฅ(0).
The following result gives necessary and sufficient conditions in which the system with finite jumps (3.1) is ESMS, and its proof may be found in [13].

Proposition 3.2. The system (3.1) is ESMS if and only if the system of coupled Lyapunov equations โˆ’ฬ‡๐‘‹(๐‘ก)=๐ด๐‘‡0๐‘‹(๐‘ก)+๐‘‹(๐‘ก)๐ด0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ด1,๐‘กโ‰ ๐‘–โ„Ž,๐‘‹(๐‘–โ„Žโˆ’)=๐ด๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘,๐‘–=0,1,โ€ฆ(3.2) has a unique symmetric solution ๐‘‹(๐‘ก)โ‰ฅ0,๐‘กโ‰ฅ0, right continuous and โ„Ž-periodic.

Another useful result is the differentiation rule of functions of solutions to stochastic differential equations, well known in the literature as Itรดโ€™s formula [21].

Proposition 3.3. Let ๐‘ฃ(๐‘ก,๐‘ฅ) be a continuous function with respect to (๐‘ก,๐‘ฅ)โˆˆ[0,๐‘‡]ร—โ„๐‘›. If ๐‘ฅ(๐‘ก) is a solution of the stochastic differential equation ๐‘‘๐‘ฅ(๐‘ก)=๐‘Ž(๐‘ก)๐‘‘๐‘ก+๐‘(๐‘ก)๐‘‘๐›ฝ(๐‘ก),(3.3) then ๎‚ธ๐‘‘๐‘ฃ(๐‘ก,๐‘ฅ(๐‘ก))=๐œ•๐‘ฃ๎‚€๐œ•๐‘ก(๐‘ก,๐‘ฅ(๐‘ก))+๐œ•๐‘ฃ๎‚๐œ•๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก))๐‘‡1๐‘Ž(๐‘ก)+2Tr๐‘๐‘‡๐œ•(๐‘ก)2๐‘ฃ(๐‘ก)๐œ•๐‘ฅ2๎‚น+๎‚€(๐‘ก,๐‘ฅ(๐‘ก))๐‘(๐‘ก)๐‘‘๐‘ก๐œ•๐‘ฃ๎‚๐œ•๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก))๐‘‡๐‘(๐‘ก)๐‘‘๐›ฝ(๐‘ก),(3.4) where Tr(โ‹…) denotes the trace of the matrix (โ‹…).

The next result will be used in the following sections, and its proof may be found in [13, page 162].

Proposition 3.4. Consider the stochastic system with multiplicative noise ๎€ท๐ด๐‘‘๐‘ฅ(๐‘ก)=0๐‘ฅ(๐‘ก)+๐ต0๐‘ข๎€ธ๎€ท๐ด(๐‘ก)๐‘‘๐‘ก+1๐‘ฅ(๐‘ก)+๐ต1๐‘ข๎€ธ(๐‘ก)๐‘‘๐œˆ(๐‘ก)(3.5) and the cost function ๐ฝ๎€ท๐‘ก0๎€ธ๎‚ธ๎€œ,๐œ,๐‘ข=๐ธ๐œ๐‘ก0๎€บ๐‘ฅ๐‘‡(๐‘ก)๐‘ข๐‘‡๎€ป๎‚ธ๐ฟ(๐‘ก)๐‘€๐ฟ๐‘‡๐‘…๎‚น๎‚ƒ๎‚„๎‚น,๐‘ฅ(๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก(3.6) then ๐ฝ๎€ท๐‘ก0๎€ธ,๐œ,๐‘ข=๐‘ฅ๐‘‡0๐‘‹๎€ท๐‘ก0๎€ธ๐‘ฅ0๎€บ๐‘ฅโˆ’๐ธ๐‘‡๎€ป๎‚ธ๎€œ(๐œ)๐‘‹(๐œ)๐‘ฅ(๐œ)+๐ธ๐œ๐‘ก0๎‚€๎‚๎‚๐‘ข(๐‘ก)โˆ’๐น(๐‘ก)(๐‘ฅ(๐‘ก))๐‘‡๎€ท๐‘…+๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธ๎‚€๎‚๎‚๎‚น,๐‘ข(๐‘ก)โˆ’๐น(๐‘ก)๐‘ฅ(๐‘ก)๐‘‘๐‘ก(3.7) where ๐‘‹(๐‘ก) verifies the equation โˆ’ฬ‡X(t)=๐ด๐‘‡0๐‘‹(๐‘ก)+๐‘‹(๐‘ก)๐ด0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ด1โˆ’๎€ท๐‘‹(๐‘ก)๐ต0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธร—๎€ท๐‘…+๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธ+๐‘€(3.8) and where ๎‚๐น(๐‘ก)=โˆ’(๐‘…+๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1)โˆ’1(๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1+๐ฟ๐‘‡).

3.2. The Mixed ๐ป2/๐ปโˆž Norm of the System (2.11)

Assume that the system (2.11) which will be denoted below by ๐’ข is ESMS and that ๐‘ฅ(0)=0. As in the deterministic case, ๐ป2 and ๐ปโˆž norms can be defined as follows.

(i) For ๐‘ค(๐‘ก)โ‰ก0 and ๐‘ค๐‘‘(๐‘–)โ‰ก0, the impulse-to-energy gain induced from (๐œ‚,๐œ‚๐‘‘) to (๐‘ฆ,๐‘ฆ๐‘‘) stands for the ๐ป2-type norm of the system (2.11). The ๐ป2-type norm of (2.11) denoted by โ€–๐’ขโ€–2 can be determined as โ€–๐’ขโ€–22๎€ท๐บ=Tr๐‘‡๐‘‘๐‘„(๐‘–โ„Ž)๐บ๐‘‘๎€ธ+1โ„Ž๎€œโ„Ž0๎€ท๐บTr๐‘‡๐‘„๎€ธ(๐‘ก)๐บ๐‘‘๐‘ก,(3.9) where ๐‘„(๐‘ก),๐‘กโ‰ฅ0 is the solution of the Lyapunov-type system โˆ’ฬ‡๐‘„(๐‘ก)=๐ด๐‘‡0๐‘„(๐‘ก)+๐‘„(๐‘ก)๐ด0+๐ด๐‘‡1๐‘„(๐‘ก)๐ด1+๐ถ๐‘‡๐ถ,๐‘กโ‰ ๐‘–โ„Ž,๐‘„(๐‘–โ„Žโˆ’)=๐ด๐‘‡0๐‘‘๐‘„(๐‘–โ„Ž)๐ด0๐‘‘+๐ด๐‘‡1๐‘‘๐‘„(๐‘–โ„Ž)๐ด1๐‘‘+๐ถ๐‘‡๐‘‘๐ถ๐‘‘,๐‘–=0,1,โ€ฆ,(3.10) (see also [18]).

(ii) For ๐œ‚(๐‘ก)โ‰ก0 and ๐œ‚๐‘‘(๐‘–)โ‰ก0, the energy-to-energy gain induced from (๐‘ค,๐‘ค๐‘‘) to (๐‘ฆ,๐‘ฆ๐‘‘) stands for the ๐ปโˆž-type norm of the system (2.11), denoted by โ€–๐’ขโ€–โˆž. It represents the smallest ๐›พ>0 for which the following system of coupled Riccati equations โˆ’ฬ‡๐‘‹(๐‘ก)=๐ด๐‘‡0๐‘‹(๐‘ก)+๐‘‹(๐‘ก)๐ด0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ด1+๐ถ๐‘‡๎€ท๐‘‹๐ถ+(๐‘ก)๐ต0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธร—๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธ,๐‘กโ‰ ๐‘–โ„Ž,๐‘‹(๐‘–โ„Žโˆ’)=๐ด๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘+๐ถ๐‘‡๐‘‘๐ถ๐‘‘+๎€ท๐ด๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ต0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธร—๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘๎€ธ,๐‘–=0,1,โ€ฆ.(3.11) has a stabilizing solution ๐‘‹(๐‘ก)โ‰ฅ0,๐‘กโ‰ฅ0. Recall that a symmetric right continuous, โ„Ž-periodic function ๐‘‹(๐‘ก) verifying (3.11) is called a stabilizing solution of (3.11) if ๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๐›พ>0,๐‘กโ‰ ๐‘–โ„Ž,2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘>0,๐‘–=0,1โ€ฆ,(3.12) and the system with jumps ๎€ท๐ด๐‘‘๐‘ฅ(๐‘ก)=0+๐ต0๐น๎€ธ๐‘ฅ๎€ท๐ด(๐‘ก)(๐‘ก)๐‘‘๐‘ก+1+๐ต1๐น๎€ธ๐‘ฅ๐‘ฅ๎€ท(๐‘ก)(๐‘ก)๐‘‘๐œˆ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๎€ท๐ด0๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต0๐‘‘๎€ธ๎€ท๐ด๐น(๐‘–โ„Ž)๐‘ฅ(๐‘–โ„Ž)+1๐‘‘+๐ต1๐‘‘๎€ธ๐น(๐‘–โ„Ž)๐‘ฅ(๐‘–โ„Ž)๐œˆ๐‘‘(๐‘–),๐‘–=0,1,โ€ฆ(3.13) is ESMS, where, by definition, ๎€ท๐›พ๐น(๐‘ก)=2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธ๎€ท๐›พ,๐‘กโ‰ ๐‘–โ„Ž,๐น(๐‘–โ„Ž)=2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘๎€ธ,๐‘–=0,1,โ€ฆ.(3.14) Similarly with the deterministic case (see, e.g., [22, 23]), a mixed ๐ป2/๐ปโˆž-type norm of (2.11) can be defined solving the optimization problem: ๐ฝ0=sup๎€ท๐‘ค,๐‘ค๐‘‘,๐œ‚,๐œ‚๐‘‘๎€ธ๐ธ๎‚ƒโ€–๐‘ฆโ€–2๐ฟ2+โ€–โ€–๐‘ฆ๐‘‘โ€–โ€–2โ„“2โˆ’๐›พ2๎‚€โ€–๐‘คโ€–2๐ฟ2+โ€–โ€–๐‘ค๐‘‘โ€–โ€–2โ„“2,๎‚๎‚„(3.15) where (๐‘ค,๐‘ค๐‘‘)โˆˆ๐ฟ2[0,โˆž)ร—โ„“2, the white-noise-type random inputs ๐œ‚ and ๐œ‚๐‘‘ are as in previous subsection and ๐›พ>โ€–๐บโ€–โˆž with โ€–๐’ขโ€–โˆž. Notice that, if ๐‘ค and ๐‘ค๐‘‘ are null in (3.15), then ๐ฝ0 gives the square of the ๐ป2-type norm induced by the random inputs ๐œ‚ and ๐œ‚๐‘‘. The main result of this subsection is the following theorem.

Theorem 3.5. The optimum ๐ฝ0 defined in (3.15) is given by ๐ฝ0๎€ท๐บ=Tr๐‘‡๐‘‘๐‘‹(โ„Ž)๐บ๐‘‘๎€ธ+1โ„Ž๎€œโ„Ž0๎€ท๐บTr๐‘‡๐‘‹๎€ธ(๐‘ก)๐บ๐‘‘๐‘ก,(3.16) where ๐‘‹(๐‘ก) is the stabilizing solution of the system of coupled Riccati equations (3.11).

Proof. The proof follows applying Itรดโ€™s formula (Proposition 3.3) for the function ๐‘ฃ(๐‘ก,๐‘ฅ)=๐‘ฅ๐‘‡(๐‘ก)๐‘‹(๐‘ก)๐‘ฅ(๐‘ก) with ๐‘ฅ(๐‘ก) being the solution of (2.11) and with ๐‘‹(๐‘ก) the stabilizing solution to the system (3.11). Thus, by direct computations, one obtains ๐‘‘๎€ท๐‘ฅ(๐‘ก)๐‘‡๎€ธ=๎€บ๐‘‹(๐‘ก)๐‘ฅ(๐‘ก)โˆ’๐’ซ๐‘(๐‘ก)โˆ’๐‘ฆ๐‘‡(๐‘ก)๐‘ฆ(๐‘ก)+๐›พ2๐‘ค๐‘‡๎€ท๐บ(๐‘ก)๐‘ค(๐‘ก)+Tr๐‘‡๐‘‹(๐‘ก)๐บ๎€ธ๎€ป๐‘‘๐‘ก+2๐‘ฅ๐‘‡(๐‘ก)๐‘‹(๐‘ก)๐บ๐‘‘๐œ‚(๐‘ก)+2๐‘ฅ๐‘‡๎€ท๐ด(๐‘ก)๐‘‹(๐‘ก)1๐‘ฅ(๐‘ก)+๐ต1๎€ธ๐‘ค(๐‘ก)๐‘‘๐œˆ(๐‘ก),(3.17) where, by definition, ๐’ซ๐‘๎€บ๐‘ฅ(๐‘ก)โˆถ=๐‘‡๎€ท(๐‘ก)๐‘‹(๐‘ก)๐ต0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’๐‘ค๐‘‡๎€ท๐›พ(๐‘ก)2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1ร—๎€ท๐›พ๎€ธ๎€ป2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๐ต๎€บ๎€ท๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธ๎€ท๐›พ๐‘ฅ(๐‘ก)โˆ’2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธ๎€ป๐‘ค(๐‘ก)โ‰ฅ0.(3.18) On the other hand using the second equations of (2.11) and of (3.11), it follows that ๐ธ๎‚ธ๎€œ(๐‘–+1)โ„Ž๐‘–โ„Ž+๐‘‘๎€ท๐‘ฅ๐‘‡๎€ธ๎‚น๎€บ๐‘ฅ๐‘‹๐‘ฅ=๐ธ๐‘‡((๐‘–+1)โ„Ž)๐‘‹((๐‘–+1)โ„Ž)๐‘ฅ((๐‘–+1)โ„Ž)โˆ’๐‘ฅ๐‘‡๎€ท๐‘–โ„Ž+๎€ธ๐‘‹๎€ท๐‘–โ„Ž+๎€ธ๐‘ฅ๎€ท๐‘–โ„Ž+๎€บ๐‘ฅ๎€ธ๎€ป=๐ธ๐‘‡((๐‘–+1)โ„Ž)๐‘‹((๐‘–+1)โ„Ž)๐‘ฅ((๐‘–+1)โ„Ž)โˆ’๐‘ฅ๐‘‡(๐‘–โ„Ž)๐‘‹(๐‘–โ„Žโˆ’)๐‘ฅ(๐‘–โ„Ž)+๐‘ฆ๐‘‡๐‘‘(๐‘–)๐‘ฆ๐‘‘(๐‘–)โˆ’๐›พ2๐‘ค๐‘‡๐‘‘(๐‘–)๐‘ค๐‘‘(๐‘–)+๐’ซ๐‘‘๎€ป๎€ท๐บ(๐‘–)โˆ’Tr๐‘‡๐‘‘๐‘‹(๐‘–โ„Ž)๐บ๐‘‘๎€ธ,(3.19) where ๐’ซ๐‘‘=๎€บ๐‘ฅ(๐‘–)โˆถ๐‘‡๎€ท๐ด(๐‘–โ„Ž)๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ต0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’๐‘ค๐‘‡๎€ท๐›พ(๐‘–)2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๐›พ๎€ธ๎€ป๎€ท2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’1ร—๎€บ๐‘ฅ๐‘‡๎€ท๐ด(๐‘–โ„Ž)๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ต0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’๐‘ค๐‘‡๎€ท๐›พ(๐‘–)2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธ๎€ป๐‘‡โ‰ฅ0.(3.20) Integrating (3.17) from ๐‘ก=0 to โˆž and equalizing it with (3.19) summed up from ๐‘–=0 to โˆž, based on the fact that ๐’ซ๐‘(๐‘ก)โ‰ฅ0 and ๐’ซ๐‘‘(๐‘–)โ‰ฅ0 and that ๐‘‹(๐‘ก) is โ„Ž-periodic, one obtains (3.16).

4. State-Feedback Mixed ๐ป2/๐ปโˆž Control Design

Consider the following linear stochastic system with multiplicative noise and finite jumps: ๎€ท๐ด๐‘‘๐‘ฅ(๐‘ก)=0๐‘ฅ(๐‘ก)+๐ต0๐‘ค(๐‘ก)+๐ต2๐‘ข๎€ธ๎€ท๐ด(๐‘ก)๐‘‘๐‘ก+1๐‘ฅ(๐‘ก)+๐ต1๐‘ค๎€ธ๐‘ฅ๎€ท(๐‘ก)๐‘‘๐œˆ(๐‘ก)+๐บ๐‘‘๐œ‚(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ด0๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต0๐‘‘๐‘ค๐‘‘(๎€ท๐ด๐‘–)+1๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต1๐‘‘๐‘ค๐‘‘(๎€ธ๐œˆ๐‘–)๐‘‘(๐‘–)+๐บ๐‘‘๐œ‚๐‘‘(๐‘ฆ๐‘–),๐‘–=0,1,โ€ฆ,1๐‘ฆ(๐‘ก)=๐ถ๐‘ฅ(๐‘ก)+๐ท๐‘ข(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,2๐‘ฆ(๐‘ก)=๐‘ฅ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘‘(๐‘–)=๐ถ๐‘‘๐‘ฅ(๐‘–โ„Ž),๐‘–=0,1,โ€ฆ,(4.1) where ๐‘ค(๐‘ก)โˆˆ๐ฟ2[0,โˆž) is an exogenous input, ๐‘ข(๐‘ก) denotes the control variable, ๐‘ฆ1(๐‘ก) stands for the regulated output, and ๐‘ฆ2(๐‘ก) is the measured output. For the simplicity of the computations, the following orthogonality assumption is made:๐ท๐‘‡๎€บ๎€ป=๎€บ๎€ป.๐ถ๐ท0๐ผ(4.2) As seen from the above system, the state vector ๐‘ฅ(๐‘ก) is assumed measurable. It is not the purpose of the present paper to analyze the case when the state variables must be estimated. For some results concerning the discrete-time filtering methods associated to networked systems, see, for instance, [24โ€“26].

The second equation of the system (4.1) does not include a discrete-time control input since, in the application presented in the previous section, the control law has only a continuous-time component.

The problem analyzed in this section consists in finding a state-feedback gain ๐น(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž, such that the resulting system obtained with ๐‘ข(๐‘ก)=๐น(๐‘ก)๐‘ฅ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž, satisfies the following conditions.(i)It is ESMS.(ii)The ๐ปโˆž-type norm of the stochastic system with jumps obtained by ignoring the noises ๐œ‚(๐‘ก) and ๐œ‚๐‘‘(๐‘–) is less than a given ๐›พ>0.(iii)The performance index (3.16) is minimized, where ๐‘‹(๐‘ก) in (3.16) denotes the stabilizing solution of the norm-type Riccati system (3.11) corresponding to the resulting system obtained with ๐‘ข(๐‘ก)=๐น(๐‘ก)๐‘ฅ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž, namely, replacing ๐ด0 by ๐ด0+๐ต2๐น(๐‘ก).

The solution of this problem is given by the following result.

Theorem 4.1. The solution of the state-feedback mixed ๐ป2/๐ปโˆž control problem considered above is given by ๐น(๐‘ก)=โˆ’๐ต๐‘‡2๐‘‹(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,(4.3) where ๐‘‹(๐‘ก) denotes the โ„Ž-periodic stabilizing solution of the game-theoretic Riccati type system with jumps โˆ’ฬ‡๐‘‹(๐‘ก)=๐ด๐‘‡0๐‘‹(๐‘ก)+๐‘‹(๐‘ก)๐ด0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ด1+๐ถ๐‘‡๎€ท๐‘‹๐ถ+(๐‘ก)๐ต0+๐ด๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธร—๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธโˆ’๐‘‹(๐‘ก)๐ต2๐ต๐‘‡2๐‘‹(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘‹(๐‘–โ„Žโˆ’)=๐ด๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘+๐ถ๐‘‡๐‘‘๐ถ๐‘‘+๎€ท๐ด๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ต0๐‘‘+๐ด๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธร—๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‘๐‘‹(๐‘–โ„Ž)๐ด0๐‘‘+๐ต๐‘‡1๐‘‘๐‘‹(๐‘–โ„Ž)๐ด1๐‘‘๎€ธ,๐‘–=0,1โ€ฆ.(4.4)

Proof. Consider the cost function ๐ฝ๎€ท๐‘ฅ0๎€ธ๎‚ธ๎€œ,๐œ,๐‘ค,๐‘ข=๐ธ๐œ0๎‚€||๐‘ฆ1||(๐‘ก)2โˆ’๐›พ2||||๐‘ค(๐‘ก)2๎‚๎‚น๐‘‘๐‘ก(4.5) associated with the system ๎€ท๐ด๐‘‘๐‘ฅ(๐‘ก)=0๐‘ฅ(๐‘ก)+๐ต0๐‘ค(๐‘ก)+๐ต2๐‘ข๎€ธ๎€ท๐ด(๐‘ก)๐‘‘๐‘ก+1๐‘ฅ(๐‘ก)+๐ต1๐‘ค๎€ธ๐‘ฅ๎€ท(๐‘ก)๐‘‘๐œˆ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,๐‘–โ„Ž+๎€ธ=๐ด0๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต0๐‘‘๐‘ค๐‘‘(๎€ท๐ด๐‘–)+1๐‘‘๐‘ฅ(๐‘–โ„Ž)+๐ต1๐‘‘๐‘ค๐‘‘(๎€ธ๐‘ฆ๐‘–),๐‘–=0,1,โ€ฆ,1(๐‘ก)=๐ถ๐‘ฅ(๐‘ก)+๐ท๐‘ข(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž,(4.6) with the initial condition ๐‘ฅ(0)=๐‘ฅ0. Applying Proposition 3.4 for ๐‘ข1=๐‘ค and ๐‘ข2=๐‘ข, one obtains that ๐ฝ๎€ท๐‘ฅ0๎€ธ,๐œ,๐‘ค,๐‘ข=๐‘ฅ๐‘‡0๐‘‹(0)๐‘ฅ0๎€บ๐‘ฅโˆ’๐ธ๐‘‡๎€ป๎€œ(๐œ)๐‘‹(๐œ)๐‘ฅ(๐œ)+๐ธ๐œ0๎‚ƒ๎€ท๐‘ข(๐‘ก)+๐ต๐‘‡2๎€ธ๐‘‹(๐‘ก)๐‘ฅ(๐‘ก)๐‘‡๎€ท๐‘ข(๐‘ก)+๐ต๐‘‡2๎€ธโˆ’๎‚๎‚„๐‘‹(๐‘ก)๐‘ฅ(๐‘ก)๐’ซ(๐‘ค(๐‘ก),๐‘ฅ(๐‘ก),๐‘‹(๐‘ก))๐‘‘๐‘ก,(4.7) where the following notation has been introduced ๎‚๎‚ƒ๎€ท๐›พ๐’ซ(๐‘ค(๐‘ก),๐‘ฅ(๐‘ก),๐‘‹(๐‘ก))โˆถ=๐‘ค(๐‘ก)โˆ’2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต1๐‘‹(๐‘ก)๐ด1๎€ธ๎‚„๐‘ฅ(๐‘ก)๐‘‡ร—๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธร—๎‚ƒ๐‘ค๎€ท๐›พ(๐‘ก)โˆ’2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต1๐‘‹(๐‘ก)๐ด1๎€ธ๐‘ฅ๎‚„,(๐‘ก)(4.8)๐‘‹(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž, denoting the stabilizing solution of the Riccati-type system (4.4).
Equation (4.7) shows that the minimum of ๐ฝ with respect to the control input ๐‘ข is obtained for ๐‘ข(๐‘ก)=โˆ’๐ต๐‘‡2๐‘‹(๐‘ก)๐‘ฅ(๐‘ก).
Further, consider a stabilizing state-feedback control ๎ฬ‚๐‘ข(๐‘ก)=๐น(๐‘ก)ฬ‚๐‘ฅ(๐‘ก),๐‘กโ‰ ๐‘–โ„Ž, for which the ๐ปโˆž norm of the resulting system without additive white noise (see the requirement (ii) above) ๐ด๐‘‘ฬ‚๐‘ฅ(๐‘ก)=๎‚ƒ๎‚€0+๐ต2๎๎‚๐น(๐‘ก)ฬ‚๐‘ฅ(๐‘ก)+๐ต0๎‚„๎€ท๐ด๐‘ค(๐‘ก)๐‘‘๐‘ก+1ฬ‚๐‘ฅ(๐‘ก)+๐ต1๎€ธ๐‘ค(๐‘ก)๐‘‘๐œˆ(๐‘ก),๐‘ฆ(๐‘ก)=(๐ถ+๐ท๐น(๐‘ก))ฬ‚๐‘ฅ(๐‘ก)(4.9) is less than ๐›พ.
Using again Proposition 3.4 for the system (4.9), direct computations give ๐ฝ๎€ท๐‘ฅ0๎€ธ,๐œ,๐‘ค,ฬ‚๐‘ข=๐‘ฅ๐‘‡0๎๐‘‹(0)๐‘ฅ0๎‚ƒโˆ’๐ธฬ‚๐‘ฅ๐‘‡๎๎‚„๎‚ธ๎€œ(๐œ)๐‘‹(๐œ)ฬ‚๐‘ฅ(๐œ)โˆ’๐ธ๐œ0๎‚๐’ซ๎‚€๎๎‚๎‚น,๐‘ค(๐‘ก),ฬ‚๐‘ฅ(๐‘ก),๐‘‹(๐‘ก)๐‘‘๐‘ก(4.10) where ๎‚๐’ซ(โ‹…,โ‹…,โ‹…) is defined by (4.8) and ๎๐‘‹(๐‘ก) is the stabilizing solution of the Riccati system of form (3.11) corresponding to (4.9). Then, defining ฬƒ๐‘ข(๐‘ก)โˆถ=๐น(๐‘ก)๐‘ฅ(๐‘ก) and ๎‚๎€ท๐›พ๐‘ค(๐‘ก)โˆถ=2๐ผโˆ’๐ต๐‘‡1๐‘‹(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹(๐‘ก)+๐ต๐‘‡1๐‘‹(๐‘ก)๐ด1๎€ธ๐‘ฅ(๐‘ก),(4.11) one obtains that ๐ฝ(๐‘ฅ0๎‚,๐œ,๐‘ค,ฬƒ๐‘ข)โ‰ค๐ฝ(๐‘ฅ0๎‚,๐œ,๐‘ค,ฬ‚๐‘ข); namely, ๐‘ฅ๐‘‡0๐‘‹(0)๐‘ฅ0๎€บ๐‘ฅโˆ’๐ธ๐‘‡๎€ป(๐œ)๐‘‹(๐œ)๐‘ฅ(๐œ)โ‰ค๐‘ฅ๐‘‡0๎๐‘‹(0)๐‘ฅ0๎‚ƒโˆ’๐ธฬ‚๐‘ฅ๐‘‡๎๎‚„,๎‚ธ๎€œ(๐œ)๐‘‹(๐œ)ฬ‚๐‘ฅ(๐œ)โˆ’๐ธ๐œ0๎‚๐’ซ๎‚€๎๎๎‚๎‚น๐‘ค(๐‘ก),ฬ‚๐‘ฅ(๐‘ก),๐‘‹(๐‘ก)๐‘‘๐‘กโ‰ค๐‘ฅ๐‘‡0๎๐‘‹(0)๐‘ฅ0๎‚ƒโˆ’๐ธฬ‚๐‘ฅ๐‘‡๎๎‚„.(๐œ)๐‘‹(๐œ)ฬ‚๐‘ฅ(๐œ)(4.12) Since ๐‘‹(๐‘ก) and ๎๐‘‹(๐‘ก) are stabilizing solutions of the Riccati systems, it follows that lim๐œโ†’โˆž๐‘ฅ(๐œ)=lim๐œโ†’โˆžฬ‚๐‘ฅ(๐œ)=0. Therefore, making ๐œโ†’โˆž in (4.12), one obtains that ๎๐‘‹(0)โ‰ค๐‘‹(0).
Further, using a similar reasoning for the cost function ๐ฝ๐‘ก๎€ท๐‘ฅ0๎€ธ๎‚ธ๎€œ,๐œ,๐‘ค,๐‘ข=๐ธ๐œ๐‘ก๎‚€||๐‘ฆ1||(๐‘ก)2โˆ’๐›พ2||||๐‘ค(๐‘ก)2๎‚๎‚น๐‘‘๐‘ก(4.13) with ๐‘กโˆˆ(0,โ„Ž), one obtains that ๎๐‘‹(๐‘ก)โ‰ค๐‘‹(๐‘ก), and; thus, one concludes that the minimum of (3.16) is obtained for the stabilizing solution ๐‘‹(๐‘ก) of the Riccati-type system (4.4).

5. A Numerical Procedure to Compute the Stabilizing Solution of the Riccati System with Jumps

In order to determine ๐ฝ0 with the expression given in the statement of Theorem 3.5, the stabilizing solution ๐‘‹(๐‘ก),๐‘กโ‰ฅ0, of the Riccati-type system (3.11) must be determined. Since the two Riccati equations of this system are coupled, an iterative procedure will be used. The proposed iterative method is similar with the iterative numerical methods used to solve Riccati equations of norm in the deterministic continuous-time and discrete-time cases (see, for instance, [27, 28]). These Newton-type iterative procedures are adapted to the particularities of the Riccati systems with jumps derived in the previous sections, and a detailed proof of the convergence towards the stabilizing solution is not the purpose of the present paper. Roughly speaking, the proof based follows showing that the solutions obtained at each iteration determine a monotonic and bounded sequence. An important particular feature of the Riccati systems with jumps, already mentioned above, is that their solution ๐‘‹(๐‘ก) is โ„Ž-periodic and right continuous. The proposed iterative procedure is the following: ฬ‡๐‘‹๐‘˜+1๎€ท๐ด(๐‘ก)+0+๐ต0๐น๐‘˜๎€ธ(๐‘ก)๐‘‡๐‘‹๐‘˜+1(๐‘ก)+๐‘‹๐‘˜+1๎€ท๐ด(๐‘ก)0+๐ต0๐น๐‘˜๎€ธ(๐‘ก)+๐‘€๐‘˜๐‘‹(๐‘ก)=0,๐‘กโ‰ ๐‘–โ„Ž,๐‘˜+1(๐‘–โ„Žโˆ’๎€ท๐ด)=0๐‘‘+๐ต0๐‘‘๐น๐‘‘,๐‘˜๎€ธ(๐‘–)๐‘‡๐‘‹๐‘˜+1๎€ท๐ด(๐‘–โ„Ž)0๐‘‘+๐ต0๐‘‘๐น๐‘‘,๐‘˜๎€ธ(๐‘–)+๐‘๐‘˜(๐‘–),๐‘–=0,1,โ€ฆ,(5.1) where ๐น๐‘˜๎€ท๐›พ(๐‘ก)=2๐ผโˆ’๐ต๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ต1๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‹๐‘˜(๐‘ก)+๐ต๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ด1๎€ธ,๐‘€๐‘˜(๐‘ก)=๐ด๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ต1๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ต1๎€ธโˆ’1๐ต๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ด1โˆ’๐‘‹๐‘˜(๐‘ก)๐ต0๎€ท๐›พ2๐ผโˆ’๐ต๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ต1๎€ธโˆ’1๐ต๐‘‡๐‘œ๐‘‹๐‘˜(๐‘ก)+๐ด๐‘‡1๐‘‹๐‘˜(๐‘ก)๐ด1+๐ถ๐‘‡๐น๐ถ,๐‘‘,๐‘˜๎€ท๐›พ(๐‘–)=2๐ผโˆ’๐ต๐‘‡1๐‘‘๐‘‹๐‘˜(๐‘–โ„Ž)๐ต1๐‘‘๎€ธโˆ’1๎€ท๐ต๐‘‡0๐‘‘๐‘‹๐‘˜(๐‘–โ„Ž)๐ด0๐‘‘+๐ต๐‘‡1๐‘‘๐‘‹๐‘˜(๐‘–โ„Ž)๐ด1๐‘‘๎€ธ,๐‘๐‘˜(๐‘–)=โˆ’๐น๐‘‡๐‘‘,๐‘˜๐ต๐‘‡0๐‘‘๐‘‹๐‘˜๎€ท๐ด(๐‘–โ„Ž)0๐‘‘+๐ต0๐‘‘๐น๐‘˜,๐‘‘๎€ธ(๐‘–)โˆ’๐ด๐‘‡0๐‘‘๐‘‹๐‘˜(๐‘–โ„Ž)๐ต0๐‘‘๐น๐‘˜,๐‘‘+๎€ท๐ด(๐‘–)1๐‘‘+๐ต1๐‘‘๐น๐‘˜,๐‘‘๎€ธ(๐‘–)๐‘‡๐‘‹๐‘˜๎€ท๐ด(๐‘–โ„Ž)1๐‘‘+๐ต1๐‘‘๐น๐‘˜,๐‘‘๎€ธ(๐‘–)+๐ถ๐‘‡๐‘‘๐ถ๐‘‘.(5.2) For the initial step of the above iterative procedure, one takes ๐‘‹(0)=0,๐‘กโˆˆ(0,โ„Ž), and ๐น0(๐‘ก) and ๐น0,๐‘‘(๐‘–) stabilizing (5.1). In order to solve (5.1) at each iteration, one solves the first equation (5.1) obtaining ๐‘‹๐‘˜+1(๐‘–โ„Ž)=๐‘’(๐ด0+๐ต0๐น๐‘˜)๐‘‡โ„Ž๐‘‹๐‘˜+1(๐‘–โ„Žโˆ’)๐‘’(๐ด0+๐ต0๐น๐‘˜)โ„Ž+๎€œโ„Ž0๐‘’(๐ด0+๐ต0๐น๐‘˜)๐‘‡๐œ๐‘€๐‘˜(๐œ)๐‘’(๐ด0+๐ต0๐น๐‘˜)๐œ๐‘‘๐œ,(5.3) which is substituted then in the second equation (5.1) obtaining, thus, a Lyapunov-type equation with the unknown variable ๐‘‹๐‘˜+1(๐‘–โ„Žโˆ’). Then, by backward integration on the interval [(๐‘–โˆ’1)โ„Ž,๐‘–โ„Žโˆ’) with the initial condition ๐‘‹๐‘˜+1(๐‘–โ„Žโˆ’), one obtains ๐‘‹๐‘˜+1(๐‘ก) for ๐‘กโˆˆ[(๐‘–โˆ’1)โ„Ž,๐‘–โ„Žโˆ’).

In the final part of this section, some of the above theoretical results will be used to analyze the mixed performance of the UAVs formation networked with fading communication channel considered in Section 2. The values of the gains considered in this example are ๐พ๐‘‘=diag(0.7,0.05,0.05) and ๐พ๐‘ฅ=diag(7,5,5), for ๐‘‰0=150โ€‰m/s (see Section 2). One determined the performance index ๐ป2/๐ปโˆž performance computing the value of the index ๐ฝ0 defined by (3.15). The results are illustrated in Figure 2. One can see the the tracking performances of the flight formation are severely deteriorated when the sampling period of the transmission in the communication channel increases. In Figure 3, the variation of ๐›พmin with respect to the sampling period and the variance of the multiplicative noise, in the absence of the additive white noise, are shown in Figure 3. It can be seen that ๐›พmin is not very much influenced by the multiplicative noise at small sampling periods, but it becomes very sensitive with respect to this noise when the sampling period increases.

6. Conclusions

The purpose of the paper was to provide an appropriate methodology to evaluate the performance of networked systems interconnected via fading communication channels. The main difficulty arises from the hybrid structure of the resulting system which includes a continuous-time component specific to the network individual members and a discrete-time component given by the communication system. It is shown that such a hybrid configuration can be analyzed from the point of view of stability and disturbance attenuation performances using dynamic models with finite jumps. In the actual stage of the research, a method to compute a mixed ๐ป2/๐ปโˆž-type performance has been developed, and a state-feedback control law to optimize it has been designed. Further research will be focused on the state estimation problems arising in the implementation of such control laws.

Acknowledgments

The author thanks the anonymous reviewers for their useful comments and suggestions. This work was sponsored by the National University Research Council (CNCS) under Grant 1721.

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Copyright ยฉ 2012 Adrian-Mihail Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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