Stochastic LQ control under asymptotic tracking for discrete systems over multiple lossy channels

Abstract : This paper addresses the asymptotic tracking problem subjected to linear quadratic constraints for linear discrete-time systems, where packet dropout occurs in actuating channels. In order to solve this objective control problem, the controller-coding co-design approach is adopted, i.e., the controller, encoder and decoder are designed for taking full advantage of the network resource collaboratively, thereby achieving better transmission of control signals. A stabilizability condition in the mean square sense that reveals the fundamental limitation among the H 2 norm of the plant, data arrival rates and coding matrices is first derived. Then, a solvability condition is conducted to handle the additional stochastic LQ control objective by a modified discrete-time algebraic Riccati equation, and an iterative algorithm is also given for designing the corresponding state feedback gain and coding matrices. Relied on such design, the asymptotic tracking constraint is further fulfilled through solving a Sylvester equation, and the feedforward gain related to tracking is parameterized. Finally, a simulation with the implementation of the design method on two cooperative robots is included to show the effectiveness of the current results. 1 Introduction Unlike classical digital control systems, networked control systems (NCSs) are closed-loop systems where the plants, the sensors, the actuators and controllers are coordinated through certain form of communication network. The main features of NCSs are low cost, high reliability, ease of maintenance and expansion [1-3]. Meanwhile numerous special issues on NCSs have been concerned by many researchers, inspired by wide applications of NCSs in cooperative vehicles [4, 5], sensor networks [6, 7], multi-agent systems [8, 9] and so on. In NCSs, the imperfect communication channels can introduce various constraints and uncertainties, for instance, packet dropouts [10, 11], time delays [12, 13], fading [14, 15], limited data rates [16, 17] and quantization [18, 19], etc. The stabilization of NCSs with networked constraints and uncertainties is a wildly attentional research field and numerous results have been reported in the literatures. For instance, the Lyapunov function approach is applied in [20] to reach the coarsest quan-tization density required for the stabilization of the single-input discrete-time linear time-invariant system with logarithmic quan-tized state feedback. Authors in [21] consider a stabilization problem with actuating channel subjecting to packet dropout, and necessary and sufficient conditions are determined in terms of the packet dropout probability and the spectral radius of the system matrix. [22] solves the stabilization problem for single-input system with both packet dropout and logarithmic quantization and the above-mentioned problem for single-input single-output system is solved in [23] by computing two algebraic Riccati equations and an algebraic Riccati inequality, where the tradeoff between the robust stability and the robust performance is revealed. Furthermore, [24] gives the solution for the multi-input multi-output case. The output regulation problem is considered in a cooperative and distributed scheme with constraints as switching network topology in [25]. The regulation problem is alternatively tackled by using input/output weighting filters, and the classical internal model principle is extended to the co-called comprehensive admissibility in [26]. Moreover, in order to find the least total channel capacity achieving stabilization for multi-input NCSs, the technique of channel resource allocation is proposed in [27] and it is appealed to deal with the constraints as signal-to-noise ratio (SNR), logarithmic quantization and fading [28, 29]. For channel resource allocation, the designers are endowed with the freedom to allocate the capacities among different input channels with the total capacity of the communication network being given. When it comes to the case where sub-channels' capacities are assumed to be fixed static, a coding and controller co-design method is put forward in [30] to obtain the stabilizability condition for multi-input systems under signal-to-noise ratio constraint. Linear quadratic (LQ) control has been possessing an indispensable role in systems and control theory. Such problem for determin-istic systems has been extensively investigated, and reaches a mature state in the 1970s [31]. The optimal control problem for systems subjected to stochastic perturbations and network-induced constraints has equally received considerable attention from the control community. For example, [32] addresses the LQ control problem for networked control systems subjected to data rate constraints, where a state feedback control scheme is employed in order to achieve the minimum data rate for mean square stabilization of the system. To addresses the LQ problem for Itô-type stochastic systems with input delays, Wang and Zhang introduce a forward-backward stochastic differential equation based approach for obtaining the optimal controller [33]. [34] is concerned with the stochastic LQ control for Markovian jumping systems by using the averaging approach to aggregate states according to their jump rates. [35] studies the LQ performance of systems where control signals are subject to packet dropout, and both zero-control and hold strategies are discussed. Authors in [36] study the LQ control problem for discrete-time linear systems over single packet-dropping link and a controller-coding co-design method is proposed in order to deal with the lost packet, where estimates are used to replace the missing measurements. To the best of authors' knowledge, the difficulties in designing NCSs mainly focus on the following aspects. First, analytical solution to controller synthesis of multi-input NCSs turns out to be an essential µ-synthesis problem [24, 28]. Second, controller synthesis problem with multiple coupled factors becomes significantly complicated and harder to cope with [22, 26]. Finally, it is usually a considerable and difficult problem to combine the proposed results with actual application requirements. Motivated by the above
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Rongyao Ling, Yu Feng, Fabien Claveau, Philippe Chevrel. Stochastic LQ control under asymptotic tracking for discrete systems over multiple lossy channels. IET Control Theory and Applications, Institution of Engineering and Technology, 2019, 13 (18), pp.3107-3116. ⟨10.1049/iet-cta.2018.6234⟩. ⟨hal-02378967⟩

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