Infinite horizon multiobjective optimal control of stochastic cooperative linear-quadratic dynamic difference games

This article is concerned with the infinite horizon stochastic cooperative linear-quadratic (LQ) dynamic difference game in both the regular and the indefinite cases. Firstly, due to the constraints imposed on the weighting matrices and the linearity of the dynamic system, the costs are shown to be convex spontaneously for the regular stochastic cooperative LQ difference game, which yields the equivalence between the minimization of the weighted sum of costs and the Pareto optimal control. Secondly, the Pareto optimal control is derived for the regular game on the ground of the solution to the weighted algebraic Riccati equation (WARE) under exact observability, and then Pareto solutions are identified via the optimal feedback gain matrices and the solution to the weighted algebraic Lyapunov equation (WALE). Moreover, a new criterion which is also necessary and sufficient is developed to guarantee the costs to be convex for the indefinite case, and the Pareto optimality is investigated based on the solutions to the weighted generalized algebraic Riccati equation (WGARE) and the weighted generalized algebraic Lyapunov equation (WGALE) combining with the semidefinite programming (SDP). Finally, the fishery management game in the economy is presented to illustrate the obtained results.     

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.     

Finite and infinite horizon indefinite linear quadratic optimal control for discrete-time singular Markov jump systems

This paper concerns the indefinite linear quadratic (LQ) optimal control problem for discrete-time singular Markov jump systems (MJSs) with finite and infinite horizon, where the weight matrices for state and control of cost function are all indefinite. Firstly, the indefinite LQ problem for singular MJSs is equivalently transformed into indefinite LQ problem for MJSs under a series of equivalent transformations. Then, the sufficient and necessary condition is proposed for the solvability of finite horizon case, the optimal control and optimal cost value are given, and the resulting optimal closed-loop system is regular, casual. Next, some sufficient and necessary conditions are obtained to ensure the transformed equivalent LQ problem for MJSs to be definite one, which can guarantee the generalized algebraic Riccati equation with Markov jump has a unique semi-positive definite solution. Meanwhile, the optimal control and nonnegative optimal cost value in infinite horizon are acquired, and the resulting optimal closed-loop system is stochastically admissible. Finally, three examples are presented to illustrate the theoretical results.     

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.     

Output regulation for matrix second order singular systems via measurement output feedback

This paper is concerned with the output regulation problem of matrix second order singular systems via measurement output feedback. It is shown that the solvability of output regulation problem of matrix second order singular systems is equivalent to the solvability of a set of matrix equations, which are called regulator equations. The solvability condition and solutions of the regulator equations are given, which can be used to solve the output regulation problem. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

对纯交换帕累托最优条件的再思考——兼与 Andreu Mas-Colell, Michael D. Whinston 和 Jerry R. Green 商榷

本文认为现有的纯交换帕累托最优条件存在问题,一则帕累托最优均衡点不是唯一的,原因是条件不充分;二则虽然使用提供曲线解决了均衡点的唯一性问题,但提供曲线本身是有缺陷的。本文认为,在信息完全的条件下,通过博奕分析方法,可得出纯交换帕累托最优均衡是唯一的,并对此完善了纯交换的帕累托最优条件。     

行延拓矩阵的矩阵方程组的最佳逼近解

主要讨论行延拓矩阵的线性约束矩阵方程组的最佳逼近;介绍了延拓矩阵的概念;利用矩阵奇异值分解得到了行延拓矩阵的线性约束矩阵方程组有解的充要条件、通解表达式;最后讨论了相应问题的最佳逼近解的表达式。     

Delay-dependent H∞ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities

This paper is devoted to the investigation of the delay-dependent H_∞ filtering problem for a class of discrete-time singular Markov jump systems with Wiener process and partly unknown transition probabilities. The class of stochastic singular model under consideration is more general and covers the stochastic singular Markov jump time-varying delay systems with completely known and completely unknown transition probabilities as two special cases. Firstly, based on a stochastic Lyapunov–Krasovskii candidate function and an auxiliary vector function, by employing some appropriate free-weighting matrices, the discretized Jensen inequality and combining them with the structural characteristics of the filtering error system, a set of delay-dependent sufficient conditions are established, which ensure that the filtering error system is stochastically admissible. And then, a singular filter is designed such that the filtering error system is not only regular, causal and stochastically stable, but also satisfy a prescribed H_∞ performance for all time-varying delays no larger than a given upper bound. Furthermore, the sufficient conditions for the solvability of the H_∞ filtering problem are obtained in terms of a new type of Lyapunov–Krasovskii candidate function and a set of linear matrix inequalities. Finally, simulation examples are presented to illustrate the effectiveness of the proposed method in the paper.     

Feedback Stackelberg strategies for the discrete-time mean-field stochastic systems in infinite horizon

This paper deals with the feedback Stackelberg strategies for the discrete-time mean-field stochastic systems in infinite horizon. The optimal control problem of the follower is first studied. Employing the discrete-time linear quadratic (LQ) mean-field stochastic optimal control theory, the sufficient conditions for the solvability of the optimization of the follower are presented and the optimal control is obtained based on the stabilizing solutions of two coupled generalized algebraic Riccati equations (GAREs). Then, the optimization of the leader is transformed into a constrained optimal control problem. Applying the Karush-Kuhn-Tucker (KKT) conditions, the necessary conditions for the existence and uniqueness of the Stackelberg strategies are derived and the Stackelberg strategies are expressed as linear feedback forms involving the state and its mean based on the solutions $(K_i,{\widehat{K}}_i),$ $i=1,2$ of a set of cross-coupled stochastic algebraic equations (CSAEs). An iterative algorithm is put forward to calculate efficiently the solutions of the CSAEs. Finally, an example is solved to show the effectiveness of the proposed algorithm.     

Optimal control of singular boolean control networks via Ledley solution method

This paper investigates the optimal control problem for a class of Boolean control networks, called singular Boolean control networks (SBCNs), which consist of two parts: difference equations and algebraic equations. By constructing the truth matrix of Ledley solution, necessary and sufficient conditions are provided for the solvability of SBNs (or SBCNs). Then an effective algorithm is presented to design an optimal control sequence by using the controllability matrix of normalized Boolean control networks.     

Distributed semistable LQR control for discrete-time dynamically coupled systems

A distributed linear-quadratic-regulator (LQR) semistability theory for discrete-time systems is developed for designing optimal semistable controllers for discrete-time coupled systems. Unlike the standard LQR control problem, a unique feature of the proposed optimal control problem is that the closed-loop generalized discrete-time semistable Lyapunov equation can admit multiple solutions. Necessary and sufficient conditions for the existence of solutions to the generalized discrete-time semistable Lyapunov equation are derived and an optimization-based design framework for distributed optimal controllers is presented.     

Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems

This paper deals with the problem of non-fragile guaranteed cost control for a class of uncertain stochastic nonlinear time-delay systems. The parametric uncertainties are assumed to be time-varying and norm bounded. The time-delay factors are unknown and time-varying with known bounds. The aim of this paper is to design a memoryless non-fragile state feedback control law such that the closed-loop system is stochastically asymptotically stable in the mean square for all admissible parameter uncertainties and the closed-loop cost function value is not more than a specified upper bound. A new sufficient condition for the existence of such controllers is presented based on the linear matrix inequality (LMI) approach. Then, a convex optimization problem is formulated to select the optimal guaranteed cost controller which minimizes the upper bound of the closed-loop cost function. Numerical example is given to illustrate the effectiveness of the developed techniques.     

Regional eigenvalue-clustering robustness of linear uncertain multivariable output feedback PID control systems

This paper proposes a time domain approach to deal with the regional eigenvalue-clustering robustness analysis problem of linear uncertain multivariable output feedback proportional-integral-derivative (PID) control systems. The robust regional eigenvalue-clustering analysis problem of linear uncertain multivariable output feedback PID control systems is converted to the regional eigenvalue-clustering robustness analysis problem of linear uncertain singular systems with static output feedback controller. Based on some essential properties of matrix measures, a new sufficient condition is proposed for ensuring that the closed-loop singular system with both structured and mixed quadratically-coupled parameter uncertainties is regular and impulse-free, and has all its finite eigenvalues retained inside the same specified region as the nominal closed-loop singular system does. Two numerical examples are given to illustrate the application of the presented sufficient condition.     

Stabilization of discrete-time switched singular time-delay systems under asynchronous switching

This paper is concerned with the problem of state feedback stabilization of a class of discrete-time switched singular systems with time-varying state delay under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the subsystems. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence of a class of stabilizing switching laws is first derived to guarantee the closed-loop system to be regular, causal and exponentially stable in the presence of asynchronous switching. The stabilizing switching laws are characterized by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.     

LQ preview state feedback with output regulation constraint

This paper addresses the output regulation problem for a class of preview control systems, and derives a state feedback law which suppresses the steady-state error caused by the excitation from polynomial or sinusoidal exogenous inputs. Recently, the output regulation condition for the broader class of distributed parameter systems is characterized via the operator regulator equation. We show that a solution of the operator regulator equation specialized to the preview control system is obtained by solving the matrix regulator equation, and provide the state feedback law which attenuates the transient error optimally with respect to an LQ (Linear Quadratic) performance index.     

State feedback control for stochastic Markovian jump delay systems based on LaSalle-type theorem

This paper investigates the problem of stochastic stability and stabilization of stochastic Markovian jump delay systems (SMJDSs) based on LaSalle theorem. The time delays are assumed to be time-varying and numerous stochastic disturbances are considered. Attention is focused on the design of the mode-dependent state feedback controller for SMJDSs based on LaSalle theorem such that the closed-loop SMJDSs are almost surely asymptotically stable. The sufficient conditions for the solvability of the state feedback control problem are obtained in terms of linear matrix inequalities (LMIs). When the LMIs are feasible, the desired state feedback controller is also given. Two numerical examples including the vertical take-off and landing (VTOL) helicopter system are employed to demonstrate the effectiveness and usefulness of the method proposed in this paper     

Robust sliding mode control for stochastic singular Markovian jump systems with unmatched one-sided Lipschitz nonlinearities

This paper investigates sliding mode control of stochastic singular Markovian jump systems with nonlinearity. The unmatched nonlinearity satisfies one-sided Lipschitz condition and quadratically inner-boundedness. In term of a new technical variable transformation, sufficient conditions are developed for nonlinear stochastic singular Markovian jump systems constrained on sliding manifold to guarantee stochastic admissibility and uniqueness of solution based on implicit function theorem. The sliding mode control law by which the trajectories of system can be compelled to the predefined sliding surface in finite time no matter what initial state value is, is synthesized. The derivative singular matrix is fully considered in the whole design process such that the derived conditions can be checked easily.The technical treatment of the nonlinear matrix term avoids the classification discussion of sliding mode controller design. Convex optimization problems subject to linear matrix inequalities are formulated to optimize the desired indexes of interest. Finally, the effectiveness of the proposed approach is illustrated by a numerical example and a practical example.     

Observer-based control for singular nonhomogeneous Markov jump systems with packet losses

This paper is concerned with the observer-based H_∞ control for a class of singular Markov jump systems over a finite-time interval, where the transition probability (TP) is time-varying and is limited to a convex hull. Due to the limited capacity of network medium, packet losses are presented in the underlying systems. Firstly, using a stochastic Lyapunov functional, a sufficient condition on singular stochastic H_∞ finite-time boundedness for the corresponding closed-loop error systems is provided. Subsequently, a linear matrix inequality (LMI) condition on the existence of the H_∞ observer-based controller is developed from a new perspective. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed controller design method, wherein it is shown that the proposed method yields less conservative results than those in the literature.