Mixed 𝐻_𝟐/𝐻_∞ Performance Analysis and State-Feedback Control Design for Networked Systems with Fading Communication Channels

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 𝐻₂/𝐻_∞-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 𝐻₂/𝐻_∞ 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., [1–3] 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 [9–11], 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 𝜇-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., [13–15] and their references). In [8], an 𝐻_∞-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 𝐻_∞ 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 𝐻₂/𝐻_∞-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 𝐻₂/𝐻_∞ 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 𝐻₂/𝐻_∞ 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 𝐻₂/𝐻_∞ 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: 𝜉̇=−𝐾𝛿_𝑑̇=𝑌𝜉,𝛿−𝐾_𝑥𝜉−𝑥̇,(2.1) where 𝛿∈ℝ³ denotes the deviation of the aircraft with respect to its desired position and 𝜉∈ℝ³ stands for the deviation of its state 𝑥=[𝑉𝜓𝛾]^𝑇 (𝑉 representing the airspeed, 𝜓 the heading angle, and 𝛾 the flight path angle) with respect to some specified value 𝑥. The input vector 𝑥̇‌ includes the desired derivatives of 𝑥 and plays the role of a reference signal in the model (2.1). The constant matrix 𝑌 has the diagonal form 𝑌=diag(1,𝑉₀,𝑉₀), and the state-feedback control gains 𝐾_𝑑 and 𝐾_𝑥 are diagonal, too. In the present paper, the case when the reference signal 𝑥̇‌ 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 𝐿th-order Rice model was adopted. This model is frequently used in wireless mobile links, and it is given by the discrete-time equation:𝑟(𝑖)=^𝐿_𝑘=0𝑎_𝑘(𝑖)𝑣(𝑖−𝑘)+𝑛(𝑖),(2.2) where 𝑖 denotes the moment of time, 𝑣(⋅) denotes the transmitted information, 𝑟(⋅) is the received information, 𝑛 is a Gaussian white noise with zero mean and unit variance, and 𝑎_𝑘(𝑖), 𝑘=0,…,𝐿 are independent random variables with known mean 𝑎_𝑘 and variance 𝜎^2𝑘. In the case of the application considered in this paper, 𝑣(⋅) is just the transmitted reference signal 𝑥̇‌. A state-space representation of (2.2) is 𝑟(𝑖)=_𝐿𝑝(𝑖+1)=𝑀𝑝(𝑖)+𝑁𝑣(𝑖),_𝑘=1𝑎_𝑘(𝑖)𝑃_𝑘𝑝(𝑖)+𝑎₀(𝑖)𝑣(𝑖)+𝑛(𝑖),𝑖=0,1,…,(2.3) where, by definition, 𝑀=⎡⎢⎢⎣⋮𝐼00⋱⋯00⋯⋯𝐼000⎤⎥⎥⎦,𝑁=⎡⎢⎢⎣⋮𝐼00⎤⎥⎥⎦,𝑃_𝑘=0⋯0𝐼0⋯0,(2.4) and 𝑝(𝑖)∈ℝ^{𝐿⋅𝑛𝑣×1} stands for the state vector of the communication channel, 𝑛_𝑣 denoting the dimension of the transmitted information vector 𝑣(⋅). In (2.4), the identity and the zero matrices have the size 𝑛_𝑣×𝑛_𝑣, and the identity matrix in 𝑃_𝑘 is on the 𝑘th 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: 𝑥𝑖ℎ₊𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑤(𝑡),𝑡≠𝑖ℎ,=𝐴_𝑑𝑥(𝑖ℎ)+𝐵_𝑑𝑤_𝑑(𝑖),𝑖=0,1,…,(2.5) in which ℎ>0 denotes the sampling period, the state 𝑥(𝑡) is left continuous and right discontinuous at the sampling moments 𝑡=𝑖ℎ, 𝑖=0,1,…, and 𝑤(𝑡), 𝑡≠𝑖ℎ, and 𝑤_𝑑(𝑖), 𝑖=0,1,…, are the continuous-time and the discrete-time inputs, respectively, of the system (see, e.g., [19, 20]). Since the received information 𝑟 is constant between the sampling moments, it can be represented as 𝑟𝑖ℎ₊𝑟̇(𝑡)=0,𝑡≠𝑖ℎ,=^𝐿_𝑘=1𝑎_𝑘(𝑖)𝑃_𝑘𝑝(𝑖ℎ)+𝑎₀(𝑖)𝑣(𝑖)+𝑛(𝑖),𝑖=0,1,…,(2.6) where the state 𝑝 of the communication system is given by 𝑝𝑖ℎ₊𝑝̇(𝑡)=0,𝑡≠𝑖ℎ,=𝑀𝑝(𝑖ℎ)+𝑁𝑣(𝑖),𝑖=0,1,….(2.7) A similar model can be adopted for the continuous-time control system (2.1) which can be represented as 𝑞̇(𝑡)=𝐴𝑞(𝑡)+𝐵𝑟(𝑡),𝑡≥0,(2.8) where 𝑞(𝑡)∶=𝛿^𝑇(𝑡)𝜉^𝑇(𝑡)^𝑇,𝐴∶=−𝐾0_𝑑−𝑌𝐾_𝑥,𝐵∶=−0𝐼.(2.9) In (2.8) the state 𝑞(𝑡) is continuous-time, and therefore, 𝑞𝑖ℎ⁺=𝑞(𝑖ℎ),(2.10) and 𝑟(𝑡) is the received perturbed reference signal described by (2.6). From (2.6)–(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 𝑥=[𝑞^𝑇𝑟^𝑇𝑝^𝑇]^𝑇. Moreover, since in (2.6) the coefficients 𝑎_𝑘(𝑖), 𝑘=0,…,𝐿 and 𝑖=0,1,…, are random variables, one may consider the following stochastic version of (2.5) which includes both multiplicative noise components and additive white noise terms: 𝑥𝑖ℎ₊𝑑𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝑤(𝑡)𝑑𝑡+𝐴₁𝑥(𝑡)+𝐵₁𝑤(𝑡)𝑑𝜈(𝑡)+𝐺𝑑𝜂(𝑡),𝑡≠𝑖ℎ,=𝐴_0𝑑𝑥(𝑖ℎ)+𝐵_0𝑑𝑤_𝑑𝑦(𝑖)+𝑦(𝐴𝑡)_1𝑑=𝑥𝐶(𝑖𝑥ℎ()𝑡+),𝐵_1𝑑𝑡𝑤≠_𝑑𝑖ℎ(𝑖,_𝑑)𝜈_𝑑(𝑖)+𝐺_𝑑𝜂_𝑑(𝑖),𝑖=0,1,…,(𝑖)=𝐶_𝑑𝑥(𝑖ℎ),𝑖=0,1,…,(2.11) where the random variables 𝜂(𝑡)∈ℝ,𝑡≥0, and 𝜈(𝑡)∈ℝ^𝑟, 𝑡≥0, are such that the pair (𝜂(𝑡),𝜈(𝑡)) is an 𝑟+1-dimensional standard Wiener process, and 𝜈_𝑑(𝑖)∈ℝ and 𝜂_𝑑(𝑖)∈ℝ^𝑟𝑑, 𝑖=0,1,… are sequences of independent random variables on a probability space (Ω,𝒫,ℱ). It is assumed that 𝜈(𝑡),𝜂(𝑡), 𝑡≥0, 𝜈_𝑑(𝑖),𝜂_𝑑(𝑖), 𝑖=0,1,…, are independent stochastic processes with zero mean and unitary second moments. The outputs 𝑦(𝑡) and 𝑦_𝑑(𝑖) 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 ℱ_𝑡-adapted solution for any initial condition 𝑥(0), ℱ_𝑡 denoting the 𝜎-algebra generated by the random vectors 𝜈(𝑠),𝜂(𝑠),𝜈_𝑑(𝑖), and 𝜂_𝑑(𝑖), 0≤𝑠≤𝑡,0≤𝑖ℎ≤𝑡. This solution is almost surely left continuous.

Figure 1 

Control configuration of networked systems with fading communication channel.

A mixed 𝐻₂/𝐻_∞ problem for this class of stochastic systems with jumps will be treated in the next section.

3. Mixed 𝐻₂/𝐻_∞-Type Norm for Systems with Jumps Corrupted with Multiplicative Noise

Before defining and computing the mixed 𝐻₂/𝐻_∞ 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 𝑤 and 𝑤_𝑑 denote continuous-time and discrete-time energy bounded inputs, respectively. It means that 𝑤∈𝐿²[0,∞), where 𝐿²[0,∞) denotes the space of the functions 𝑓(𝑡),𝑡≥0 for which ‖𝑓‖^2𝐿₂∶=∫^∞0|𝑓(𝑡)|²𝑑𝑡<∞, and 𝑤_𝑑∈ℓ² where ℓ² is the space of the discrete-time vectors 𝑔(𝑖),𝑖=0,1,… with the property ‖𝑔‖^2ℓ₂∶=∑^∞𝑖=0|𝑔(𝑖)|²<∞, where |⋅| stands for the Euclidian norm.

Definition 3.1. The stochastic system with jumps and with multiplicative noise 𝑥𝑖ℎ𝑑₊𝑥(𝑡)=𝐴₀𝑥(𝑡)𝑑𝑡+𝐴₁𝑥(𝑡)𝑑𝜈(𝑡),𝑡≠𝑖ℎ,=𝐴_0𝑑𝑥(𝑖ℎ)+𝐴_1𝑑𝑥(𝑖ℎ)𝜈_𝑑(𝑖)𝑖=0,1,…(3.1) is exponentially stable in mean square (ESMS) if there exist 𝛼>0 and 𝛽≥1 such that 𝐸[|𝑥(𝑡)|²]≤𝛽𝑒^−𝛼𝑡|𝑥(0)|² for any initial condition 𝑥(0) and for all 𝑡≥0, 𝐸[⋅] denoting the mean of the random variable and 𝑥(𝑡) representing the solution of (3.1) with the initial condition 𝑥(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 𝑋−(𝑖𝑋ℎ₋̇(𝑡)=𝐴^𝑇0𝑋(𝑡)+𝑋(𝑡)𝐴₀+𝐴^𝑇1𝑋(𝑡)𝐴₁,𝑡≠𝑖ℎ,)=𝐴^𝑇0𝑑𝑋(𝑖ℎ)𝐴_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑,𝑖=0,1,…(3.2) has a unique symmetric solution 𝑋(𝑡)≥0,𝑡≥0, right continuous and ℎ-periodic.

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

Proposition 3.3. Let 𝑣(𝑡,𝑥) be a continuous function with respect to (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑛. If 𝑥(𝑡) is a solution of the stochastic differential equation 𝑑𝑥(𝑡)=𝑎(𝑡)𝑑𝑡+𝑏(𝑡)𝑑𝛽(𝑡),(3.3) then 𝑑𝑣(𝑡,𝑥(𝑡))=𝜕𝑣𝜕𝑡(𝑡,𝑥(𝑡))+𝜕𝜕𝑥𝑣(𝑡,𝑥(𝑡))^𝑇𝑎(𝑡)+12Tr𝑏^𝑇(𝑡)𝜕²𝑣(𝑡)𝜕𝑥²+𝜕𝑣(𝑡,𝑥(𝑡))𝑏(𝑡)𝑑𝑡𝜕𝑥(𝑡,𝑥(𝑡))^𝑇𝑏(𝑡)𝑑𝛽(𝑡),(3.4) where Tr(⋅) denotes the trace of the matrix (⋅).

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 𝑑𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝑢(𝑡)𝑑𝑡+𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡)𝑑𝜈(𝑡)(3.5) and the cost function 𝐽𝑡₀,𝜏,𝑢=𝐸^𝜏𝑡₀𝑥^𝑇(𝑡)𝑢^𝑇(𝑡)𝐿𝑀𝐿_𝑇𝑅𝑥𝑢((𝑡𝑡))𝑑𝑡,(3.6) then 𝐽𝑡₀,𝜏,𝑢=+𝑥^𝑇0𝐸𝑋𝑡_0𝜏𝑡0𝑥₀−𝐸𝑥^𝑇(𝜏)𝑋(𝜏)𝑥(𝜏)𝑢(𝑡)−𝐹(𝑡)(𝑥(𝑡))^𝑇𝑅+𝐵^𝑇1𝑋(𝑡)𝐵₁𝑢(𝑡)−𝐹(𝑡)𝑥(𝑡)𝑑𝑡,(3.7) where 𝑋(𝑡) verifies the equation −Ẋ(t)=×𝐴^𝑇0𝑅𝑋+(𝑡)𝐵+_𝑇1𝑋(𝑡)𝐴₀+𝐴^𝑇1𝑋(𝑡)𝐴₁−𝑋(𝑡)𝐵₀+𝐴^𝑇1𝑋(𝑡)𝐵₁𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁+𝑀(3.8) and where 𝐹(𝑡)=−(𝑅+𝐵^𝑇1𝑋(𝑡)𝐵₁)⁻¹(𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁+𝐿^𝑇).

3.2. The Mixed 𝐻₂/𝐻_∞ Norm of the System (2.11)

Assume that the system (2.11) which will be denoted below by 𝒢 is ESMS and that 𝑥(0)=0. As in the deterministic case, 𝐻₂ and 𝐻_∞ norms can be defined as follows.

(i) For 𝑤(𝑡)≡0 and 𝑤_𝑑(𝑖)≡0, the impulse-to-energy gain induced from (𝜂,𝜂_𝑑) to (𝑦,𝑦_𝑑) stands for the 𝐻₂-type norm of the system (2.11). The 𝐻₂-type norm of (2.11) denoted by ‖𝒢‖₂ can be determined as ‖𝒢‖²²=Tr𝐺^𝑇𝑑𝑄(𝑖ℎ)𝐺_𝑑+1ℎ^ℎ0Tr𝐺^𝑇𝑄(𝑡)𝐺𝑑𝑡,(3.9) where 𝑄(𝑡),𝑡≥0 is the solution of the Lyapunov-type system 𝑄(𝑖−ℎ𝑄₋̇(𝑡)=𝐴^𝑇0𝑄(𝑡)+𝑄(𝑡)𝐴₀+𝐴^𝑇1𝑄(𝑡)𝐴₁+𝐶^𝑇𝐶,𝑡≠𝑖ℎ,)=𝐴^𝑇0𝑑𝑄(𝑖ℎ)𝐴_0𝑑+𝐴^𝑇1𝑑𝑄(𝑖ℎ)𝐴_1𝑑+𝐶^𝑇𝑑𝐶_𝑑,𝑖=0,1,…,(3.10) (see also [18]).

(ii) For 𝜂(𝑡)≡0 and 𝜂_𝑑(𝑖)≡0, the energy-to-energy gain induced from (𝑤,𝑤_𝑑) to (𝑦,𝑦_𝑑) stands for the 𝐻_∞-type norm of the system (2.11), denoted by ‖𝒢‖_∞. It represents the smallest 𝛾>0 for which the following system of coupled Riccati equations −𝑋̇(𝑡)=×𝐴^𝑇0𝛾𝑋₂(𝑡)+𝑋(𝑡)𝐴₀+𝐴^𝑇1𝑋(𝑡)𝐴₁+𝐶^𝑇𝐶+𝑋(𝑡)𝐵₀+𝐴^𝑇1𝑋(𝑡)𝐵₁𝑋(𝑖ℎ₋𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁,𝑡≠𝑖ℎ,)=×𝐴^𝑇0𝑑𝛾𝑋₂(𝑖ℎ)𝐴_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑+𝐶^𝑇𝑑𝐶_𝑑+𝐴^𝑇0𝑑𝑋(𝑖ℎ)𝐵_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑⁻¹𝐵^𝑇0𝑑𝑋(𝑖ℎ)𝐴_0𝑑+𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑,𝑖=0,1,….(3.11) has a stabilizing solution 𝑋(𝑡)≥0,𝑡≥0. Recall that a symmetric right continuous, ℎ-periodic function 𝑋(𝑡) verifying (3.11) is called a stabilizing solution of (3.11) if 𝛾𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁>0,𝑡≠𝑖ℎ,₂𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑>0,𝑖=0,1…,(3.12) and the system with jumps 𝑥𝑖ℎ₊𝑑𝑥(𝑡)=𝐴₀+𝐵₀𝐹(𝑡)𝑥(𝑡)𝑑𝑡+𝐴₁+𝐵₁𝐹(𝑡)𝑥(𝑡)𝑑𝜈(𝑡),𝑡≠𝑖ℎ,=𝐴_0𝑑𝑥(𝑖ℎ)+𝐵_0𝑑𝐹(𝑖ℎ)𝑥(𝑖ℎ)+𝐴_1𝑑+𝐵_1𝑑𝐹(𝑖ℎ)𝑥(𝑖ℎ)𝜈_𝑑(𝑖),𝑖=0,1,…(3.13) is ESMS, where, by definition, 𝐹(𝑖ℎ)=𝐹𝛾₂(𝑡)=𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁,𝑡≠𝑖ℎ,𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑⁻¹𝐵^𝑇0𝑑𝑋(𝑖ℎ)𝐴_0𝑑+𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑,𝑖=0,1,….(3.14) Similarly with the deterministic case (see, e.g., [22, 23]), a mixed 𝐻₂/𝐻_∞-type norm of (2.11) can be defined solving the optimization problem: 𝐽₀=sup_{𝑤,𝑤𝑑,𝜂,𝜂𝑑}𝐸‖𝑦‖^2𝐿₂+‖‖𝑦_𝑑‖‖^2ℓ₂−𝛾²‖𝑤‖^2𝐿2+‖‖𝑤_𝑑‖‖^2ℓ₂,(3.15) where (𝑤,𝑤_𝑑)∈𝐿²[0,∞)×ℓ², the white-noise-type random inputs 𝜂 and 𝜂_𝑑 are as in previous subsection and 𝛾>‖𝐺‖_∞ with ‖𝒢‖_∞. Notice that, if 𝑤 and 𝑤_𝑑 are null in (3.15), then 𝐽₀ gives the square of the 𝐻₂-type norm induced by the random inputs 𝜂 and 𝜂_𝑑. The main result of this subsection is the following theorem.

Theorem 3.5. The optimum 𝐽₀ defined in (3.15) is given by 𝐽₀=Tr𝐺^𝑇𝑑𝑋(ℎ)𝐺_𝑑+1ℎ^ℎ0Tr𝐺^𝑇𝑋(𝑡)𝐺𝑑𝑡,(3.16) where 𝑋(𝑡) is the stabilizing solution of the system of coupled Riccati equations (3.11).

Proof. The proof follows applying Itô’s formula (Proposition 3.3) for the function 𝑣(𝑡,𝑥)=𝑥^𝑇(𝑡)𝑋(𝑡)𝑥(𝑡) with 𝑥(𝑡) being the solution of (2.11) and with 𝑋(𝑡) the stabilizing solution to the system (3.11). Thus, by direct computations, one obtains 𝑑𝑥(𝑡)^𝑇𝑋(𝑡)𝑥(𝑡)=+−2𝒫𝑥_𝑐𝑇(𝑡)−𝑦^𝑇(𝑡)𝑦(𝑡)+𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)+Tr𝐺^𝑇𝑋(𝑡)𝐺𝑑𝑡(𝑡)𝑋(𝑡)𝐺𝑑𝜂(𝑡)+2𝑥^𝑇(𝑡)𝑋(𝑡)𝐴₁𝑥(𝑡)+𝐵₁𝑤(𝑡)𝑑𝜈(𝑡),(3.17) where, by definition, 𝒫_𝑐(𝑡)∶=×𝑥^𝑇𝛾(𝑡)𝑋(𝑡)𝐵₀+𝐴^𝑇1𝑋(𝑡)𝐵₁−𝑤^𝑇(𝑡)𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁₂𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁𝑥(𝑡)−𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁𝑤(𝑡)≥0.(3.18) On the other hand using the second equations of (2.11) and of (3.11), it follows that 𝐸^{(𝑖+1)ℎ𝑖ℎ}₊𝑑𝑥^𝑇𝑋𝑥==𝐸𝐸𝑥𝑥^𝑇_𝑇((𝑖+1)ℎ)𝑋((𝑖+1)ℎ)𝑥((𝑖+1)ℎ)−𝑥^𝑇𝑖ℎ⁺𝑋𝑖ℎ⁺𝑥𝑖ℎ⁺−𝛾(₂(𝑖+1)ℎ)𝑋((𝑖+1)ℎ)𝑥((𝑖+1)ℎ)−𝑥^𝑇(𝑖ℎ)𝑋(𝑖ℎ⁻)𝑥(𝑖ℎ)+𝑦^𝑇𝑑(𝑖)𝑦_𝑑(𝑖)𝑤^𝑇𝑑(𝑖)𝑤_𝑑(𝑖)+𝒫_𝑑(𝑖)−Tr𝐺^𝑇𝑑𝑋(𝑖ℎ)𝐺_𝑑,(3.19) where 𝒫=_𝑑(𝑖𝑥)∶_𝑇(𝑖ℎ)𝐴^𝑇0𝑑𝑋(𝑖ℎ)𝐵_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑−𝑤^𝑇(𝑖)𝛾²𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑𝛾²𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑⁻¹×𝑥^𝑇(𝑖ℎ)𝐴^𝑇0𝑑𝑋(𝑖ℎ)𝐵_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑−𝑤^𝑇(𝑖)𝛾²𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑^𝑇≥0.(3.20) Integrating (3.17) from 𝑡=0 to ∞ and equalizing it with (3.19) summed up from 𝑖=0 to ∞, based on the fact that 𝒫_𝑐(𝑡)≥0 and 𝒫_𝑑(𝑖)≥0 and that 𝑋(𝑡) is ℎ-periodic, one obtains (3.16).

4. State-Feedback Mixed 𝐻₂/𝐻_∞ Control Design

Consider the following linear stochastic system with multiplicative noise and finite jumps: 𝑥𝑑𝑖𝑥ℎ(₊𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝑤(𝑡)+𝐵₂𝑢(𝑡)𝑑𝑡+𝐴₁𝑥(𝑡)+𝐵₁𝑤(𝑡)𝑑𝜈(𝑡)+𝐺𝑑𝜂(𝑡),𝑡≠𝑖ℎ,=𝐴_0𝑑𝑥(𝑖ℎ)+𝐵_0𝑑𝑤_𝑑𝑦(𝑖)+𝐴_1𝑑𝑥(𝑖ℎ)+𝐵_1𝑑𝑤_𝑑(𝑖)𝜈_𝑑(𝑖)+𝐺_𝑑𝜂_𝑑(𝑖),𝑖=0,1,…,₁(𝑡)𝑦=𝐶𝑥(𝑡)+𝐷𝑢(𝑡),𝑡≠𝑖ℎ,₂𝑦(𝑡)=𝑥(𝑡),𝑡≠𝑖ℎ,_𝑑(𝑖)=𝐶_𝑑𝑥(𝑖ℎ),𝑖=0,1,…,(4.1) where 𝑤(𝑡)∈𝐿²[0,∞) is an exogenous input, 𝑢(𝑡) denotes the control variable, 𝑦₁(𝑡) stands for the regulated output, and 𝑦₂(𝑡) is the measured output. For the simplicity of the computations, the following orthogonality assumption is made:𝐷^𝑇𝐶𝐷=0𝐼.(4.2) As seen from the above system, the state vector 𝑥(𝑡) 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, [24–26].

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 𝐹(𝑡),𝑡≠𝑖ℎ, such that the resulting system obtained with 𝑢(𝑡)=𝐹(𝑡)𝑥(𝑡),𝑡≠𝑖ℎ, satisfies the following conditions.(i)It is ESMS.(ii)The 𝐻_∞-type norm of the stochastic system with jumps obtained by ignoring the noises 𝜂(𝑡) and 𝜂_𝑑(𝑖) is less than a given 𝛾>0.(iii)The performance index (3.16) is minimized, where 𝑋(𝑡) in (3.16) denotes the stabilizing solution of the norm-type Riccati system (3.11) corresponding to the resulting system obtained with 𝑢(𝑡)=𝐹(𝑡)𝑥(𝑡),𝑡≠𝑖ℎ, namely, replacing 𝐴₀ by 𝐴₀+𝐵₂𝐹(𝑡).

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

Theorem 4.1. The solution of the state-feedback mixed 𝐻₂/𝐻_∞ control problem considered above is given by 𝐹(𝑡)=−𝐵^𝑇2𝑋(𝑡),𝑡≠𝑖ℎ,(4.3) where 𝑋(𝑡) denotes the ℎ-periodic stabilizing solution of the game-theoretic Riccati type system with jumps −𝑋̇(𝑡)=×𝐴^𝑇0𝛾𝑋₂(𝑡)+𝑋(𝑡)𝐴₀+𝐴^𝑇1𝑋(𝑡)𝐴₁+𝐶^𝑇𝐶+𝑋(𝑡)𝐵₀+𝐴^𝑇1𝑋(𝑡)𝐵₁𝑋(𝑖ℎ₋𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁−𝑋(𝑡)𝐵₂𝐵^𝑇2𝑋(𝑡),𝑡≠𝑖ℎ,)=×𝐴^𝑇0𝑑𝛾𝑋₂(𝑖ℎ)𝐴_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑+𝐶^𝑇𝑑𝐶_𝑑+𝐴^𝑇0𝑑𝑋(𝑖ℎ)𝐵_0𝑑+𝐴^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑𝐼−𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐵_1𝑑⁻¹𝐵^𝑇0𝑑𝑋(𝑖ℎ)𝐴_0𝑑+𝐵^𝑇1𝑑𝑋(𝑖ℎ)𝐴_1𝑑,𝑖=0,1….(4.4)

Proof. Consider the cost function 𝐽𝑥₀,𝜏,𝑤,𝑢=𝐸^𝜏0||𝑦¹(𝑡)||²−𝛾²||𝑤(𝑡)||²𝑑𝑡(4.5) associated with the system 𝑑𝑥𝑥(𝑡)𝑖ℎ=₊𝐴₀𝑥(𝑡)+𝐵₀𝑤(𝑡)+𝐵₂𝑢(𝑡)𝑑𝑡+𝐴₁𝑥(𝑡)+𝐵₁𝑤(𝑡)𝑑𝜈(𝑡),𝑡≠𝑖ℎ,=𝐴_0𝑑𝑥(𝑖ℎ)+𝑦𝐵_0𝑑𝑤_𝑑(𝑖)+𝐴_1𝑑𝑥(𝑖ℎ)+𝐵_1𝑑𝑤_𝑑(𝑖),𝑖=0,1,…,₁(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢(𝑡),𝑡≠𝑖ℎ,(4.6) with the initial condition 𝑥(0)=𝑥₀. Applying Proposition 3.4 for 𝑢₁=𝑤 and 𝑢₂=𝑢, one obtains that 𝐽𝑥₀,𝜏,𝑤,𝑢=+𝑥^𝑇0𝐸𝑋_𝜏0(0)𝑥₀−𝐸𝑥^𝑇(𝜏)𝑋(𝜏)𝑥(𝜏)𝑢(𝑡)+𝐵^𝑇2𝑋(𝑡)𝑥(𝑡)^𝑇𝑢(𝑡)+𝐵^𝑇2𝑋(𝑡)𝑥(𝑡)−𝒫(𝑤(𝑡),𝑥(𝑡),𝑋(𝑡))𝑑𝑡,(4.7) where the following notation has been introduced 𝒫(𝑤(𝑡),𝑥(𝑡),𝑋(𝑡))∶=𝑤(𝑡)−𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵₁𝑋(𝑡)𝐴₁𝑥(𝑡)^𝑇××𝑤𝛾²𝐼(𝑡)−−𝐵^𝑇1𝛾𝑋₂(𝑡)𝐵₁𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵₁𝑋(𝑡)𝐴₁𝑥(𝑡),(4.8)𝑋(𝑡),𝑡≠𝑖ℎ, denoting the stabilizing solution of the Riccati-type system (4.4).
Equation (4.7) shows that the minimum of 𝐽 with respect to the control input 𝑢 is obtained for 𝑢(𝑡)=−𝐵^𝑇2𝑋(𝑡)𝑥(𝑡).
Further, consider a stabilizing state-feedback control 𝑢(̂𝑡)=𝐹(𝑡)𝑥̂(𝑡),𝑡≠𝑖ℎ, for which the 𝐻_∞ norm of the resulting system without additive white noise (see the requirement (ii) above) 𝑑𝑥̂(𝑡)=𝐴₀+𝐵₂𝐹(𝑡)𝑦(𝑥𝑡̂)(𝑡=)+(𝐶𝐵+₀𝑤𝐷(𝐹𝑡)(𝑡𝑑)𝑡)𝑥+̂(𝑡)𝐴₁𝑥̂(𝑡)+𝐵₁𝑤(𝑡)𝑑𝜈(𝑡),(4.9) is less than 𝛾.
Using again Proposition 3.4 for the system (4.9), direct computations give 𝐽𝑥₀,𝜏,𝑤,𝑢̂=𝑥^𝑇0𝑋(0)𝑥₀−𝐸𝑥̂^𝑇(𝜏)𝑋(𝜏)𝑥̂(𝜏)−𝐸^𝜏0𝒫𝑤(𝑡),𝑥̂(𝑡),𝑋(𝑡)𝑑𝑡,(4.10) where 𝒫(⋅,⋅,⋅) is defined by (4.8) and 𝑋(𝑡) is the stabilizing solution of the Riccati system of form (3.11) corresponding to (4.9). Then, defining 𝑢(̃𝑡)∶=𝐹(𝑡)𝑥(𝑡) and 𝑤(𝑡)∶=𝛾²𝐼−𝐵^𝑇1𝑋(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋(𝑡)+𝐵^𝑇1𝑋(𝑡)𝐴₁𝑥(𝑡),(4.11) one obtains that 𝐽(𝑥₀,𝜏,𝑤,𝑢)̃≤𝐽(𝑥₀,𝜏,𝑤,𝑢)̂; namely, 𝑥−^𝑇0𝐸𝑋(0_𝜏0)𝑥₀−𝐸𝑥^𝑇(𝜏)𝑋(𝜏)𝑥(𝜏)≤𝑥^𝑇0𝑋(0)𝑥₀−𝐸𝑥̂^𝑇(𝜏)𝑋(𝜏)𝑥̂(𝜏),𝒫𝑤(𝑡),𝑥̂(𝑡),𝑋(𝑡)𝑑𝑡≤𝑥^𝑇0𝑋(0)𝑥₀−𝐸𝑥̂^𝑇(𝜏)𝑋(𝜏)𝑥̂(𝜏).(4.12) Since 𝑋(𝑡) and 𝑋(𝑡) are stabilizing solutions of the Riccati systems, it follows that lim_𝜏→∞𝑥(𝜏)=lim_𝜏→∞𝑥̂(𝜏)=0. Therefore, making 𝜏→∞ in (4.12), one obtains that 𝑋(0)≤𝑋(0).
Further, using a similar reasoning for the cost function 𝐽_𝑡𝑥₀,𝜏,𝑤,𝑢=𝐸^𝜏𝑡||𝑦¹(𝑡)||²−𝛾²||𝑤(𝑡)||²𝑑𝑡(4.13) with 𝑡∈(0,ℎ), one obtains that 𝑋(𝑡)≤𝑋(𝑡), and; thus, one concludes that the minimum of (3.16) is obtained for the stabilizing solution 𝑋(𝑡) 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 𝐽₀ with the expression given in the statement of Theorem 3.5, the stabilizing solution 𝑋(𝑡),𝑡≥0, 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 𝑋(𝑡) is ℎ-periodic and right continuous. The proposed iterative procedure is the following: 𝑋𝑋̇_𝑘+1(𝑡)+𝐴₀+𝐵₀𝐹_𝑘(𝑡)^𝑇𝑋_𝑘+1(𝑡)+𝑋_𝑘+1(𝑡)𝐴₀+𝐵₀𝐹_𝑘(𝑡)+𝑀_𝑘(𝑡)=0,𝑡≠𝑖ℎ,_𝑘+1(𝑖ℎ⁻)=𝐴_0𝑑+𝐵_0𝑑𝐹_𝑑,𝑘(𝑖)^𝑇𝑋_𝑘+1(𝑖ℎ)𝐴_0𝑑+𝐵_0𝑑𝐹_𝑑,𝑘(𝑖)+𝑁_𝑘(𝑖),𝑖=0,1,…,(5.1) where 𝑀𝐹_𝑘𝑘(𝑡)=𝛾²𝐼−𝐵^𝑇1𝑋_𝑘(𝑡)𝐵₁⁻¹𝐵^𝑇0𝑋_𝑘(𝑡)+𝐵^𝑇1𝑋_𝑘(𝑡)𝐴₁,(𝑡)=𝐴^𝑇1𝑋_𝑘(𝑡)𝐵₁𝛾²𝐼−𝐵^𝑇1𝑋_𝑘(𝑡)𝐵₁⁻¹𝐵^𝑇1𝑋_𝑘(𝑡)𝐴₁𝐹−𝑋_𝑘(𝑡)𝐵₀𝛾²𝐼−𝐵^𝑇1𝑋_𝑘(𝑡)𝐵₁⁻¹𝐵^𝑇𝑜𝑋_𝑘(𝑡)+𝐴^𝑇1𝑋_𝑘(𝑡)𝐴₁+𝐶^𝑇𝐶,_𝑑,𝑘𝑁_𝑘(𝑖)=𝛾²𝐼−𝐵^𝑇1𝑑𝑋_𝑘(𝑖ℎ)𝐵_1𝑑⁻¹𝐵^𝑇0𝑑𝑋_𝑘(𝑖ℎ)𝐴_0𝑑+𝐵^𝑇1𝑑𝑋_𝑘(𝑖ℎ)𝐴_1𝑑,(𝑖)=+−𝐹𝐴^{𝑇𝑑,𝑘}𝐵^𝑇0𝑑𝑋_𝑘(𝑖ℎ)𝐴_0𝑑+𝐵_0𝑑𝐹_𝑘,𝑑(𝑖)−𝐴^𝑇0𝑑𝑋_𝑘(𝑖ℎ)𝐵_0𝑑𝐹_𝑘,𝑑(𝑖)_1𝑑+𝐵_1𝑑𝐹_𝑘,𝑑(𝑖)^𝑇𝑋_𝑘(𝑖ℎ)𝐴_1𝑑+𝐵_1𝑑𝐹_𝑘,𝑑(𝑖)+𝐶^𝑇𝑑𝐶_𝑑.(5.2) For the initial step of the above iterative procedure, one takes 𝑋(0)=0,𝑡∈(0,ℎ), and 𝐹₀(𝑡) and 𝐹_0,𝑑(𝑖) stabilizing (5.1). In order to solve (5.1) at each iteration, one solves the first equation (5.1) obtaining 𝑋_𝑘+1(𝑖ℎ)=𝑒^{(𝐴0+𝐵0𝐹𝑘)𝑇ℎ}𝑋_𝑘+1(𝑖ℎ⁻)𝑒^{(𝐴0+𝐵0𝐹𝑘)ℎ}+^ℎ0𝑒^{(𝐴0+𝐵0𝐹𝑘)𝑇𝜏}𝑀_𝑘(𝜏)𝑒^{(𝐴0+𝐵0𝐹𝑘)𝜏}𝑑𝜏,(5.3) which is substituted then in the second equation (5.1) obtaining, thus, a Lyapunov-type equation with the unknown variable 𝑋_𝑘+1(𝑖ℎ⁻). Then, by backward integration on the interval [(𝑖−1)ℎ,𝑖ℎ⁻) with the initial condition 𝑋_𝑘+1(𝑖ℎ⁻), one obtains 𝑋_𝑘+1(𝑡) for 𝑡∈[(𝑖−1)ℎ,𝑖ℎ⁻).

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 𝐾_𝑑=diag(0.7,0.05,0.05) and 𝐾_𝑥=diag(7,5,5), for 𝑉₀=150 m/s (see Section 2). One determined the performance index 𝐻₂/𝐻_∞ performance computing the value of the index 𝐽₀ 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 𝛾_min 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 𝛾_min 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.

Figure 2 

Variation of 𝐽0 with respect to the sampling period ℎ.

Figure 3 

Variation of 𝛾min with respect to ℎ and 𝜎.

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 𝐻₂/𝐻_∞-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.