arXiv subjects

Huanshui Zhang

Publications and source records attributed to Huanshui Zhang.

63 records · Page 4Linked to original sources

arXiv2018

Optimal Control and Stabilization Problem for Discrete-time Markov Jump Systems with Indefinite Weight Costs

It is well known that stability is the most fundamental nature with regard to a control system, in view of this, the stabilization becomes an inevitable control problem. This article mainly discusses the optimal control and stabilization problem for discrete-time systems involving Markov jump and multiplicative noise. The state and control weighting matrices in the cost function are allowed to be indefinite. By solving the forward-backward stochastic difference equations with Markov jump (FBSDEs-MJ) derived from the maximum principle, we conclude that the necessary and sufficient conditions of the solvability of indefinite optimal control problem in finite-horizon, whose method is different from most previous works [13], etc. Furthermore, necessary and sufficient conditions that stabilize the Markov jump discrete- time systems in the mean square sense with indefinite weighting matrices in the cost are first developed under the basic assumption of exactly observable, which is different from the previous works [12], [14] where an additional assumption of stabilization of systems is made. The key points of this article can be summed up as that an analytic solution to FBSDEs- MJ which makes the optimal controller to be explicitly expressed and the method of trans- formation, i.e., the stabilization problem of indefinite case is boiled down to a definite one whose stabilization is expressed by defining Lyapunov function via the optimal cost subject to a new algebraic Riccati equation involving Markov jump (NGARE-MJ).

math.OC↗

arXiv2018

Linear Quadratic Optimal Control and Stabilization for Discrete-time Markov Jump Linear Systems

This paper mainly investigates the optimal control and stabilization problems for linear discrete-time Markov jump systems. The general case for the finite-horizon optimal controller is considered, where the input weighting matrix in the performance index is just required to be positive semi-definite. The necessary and sufficient condition for the existence of the optimal controller in finite-horizon is given explicitly from a set of coupled difference Riccati equations (CDRE). One of the key techniques is to solve the forward and backward stochastic difference equation (FDSDE) which is obtained by the maximum principle. As to the infinite-horizon case, we establish the necessary and sufficient condition to stabilize the Markov jump linear system in the mean square sense. It is shown that the Markov jump linear system is stabilizable under the optimal controller if and only if the associated couple algebraic Riccati equation (CARE) has a unique positive solution. Meanwhile, the optimal controller and optimal cost function in infinite-horizon case are expressed explicitly.

math.OC↗

arXiv2018

Distributed Control and Stabilization for Discrete-time Large Scale Systems With Imposed Constraints

This paper is concerned with the distributed control and stabilization problems for linear discrete-time large scale systems with imposed constraints. The main contributions of this paper are: Firstly, by using the maximum principle (necessary condition) for the finite horizon optimal control developed in this paper, the optimal distributed controller is thus derived, which can be easily calculated; Secondly, by defining the Lyapunov function with the optimal cost function, we show that the systems with imposed constraints can be stabilized by the optimal distributed controller for the infinite horizon case. The main techniques adopted in this paper are the maximum principle and the solution to the forward and backward difference equations (FBDE), which are the basis in solving distributed control and stabilization problems for linear systems with constraints.

math.OC↗

arXiv2017

On Irregular Linear Quadratic Control: Stochastic Case

As it is popular known, Riccati equation is the key basic tool for optimal control in the modern control theory. The solvability conditions of optimal control, stabilization conditions and controller design are all based on the Riccati equation. However, these results highly depends on a strictly assumption that the Riccati equation is regular. If the Riccati equation is irregular, the controller could not be derived from the equilibrium condition. This paper is concerned with the general stochastic linear quadratic (LQ) with irregular Riccati equation. Different from the classical control theory for regular LQ problems, a new approach of `multi-layer optimization' is proposed. With the approach, we show that different controller entries of irregular-LQ controller need to be derived from different equilibrium conditions and a specified terminal constraint condition in different layers, which is much different from the classical regular LQ control where all the controller entries can be obtained from equilibrium condition in one layer based on regular Riccati equation. The presented results clarify the differences of open-loop control from closed-loop control in the aspects of solvability and controller design and also explores in essentially the differences of regular control from irregular control. Several examples are presented to show the effectiveness of the proposed approach.

math.OC↗

arXiv2017

Delayed Optimal Control of Stochastic LQ Problem

A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper, which does not require any definiteness constraint on the cost weighting matrices. From some abstract representations of the system and cost functional, the solvability of this LQ problem is characterized by some conditions with operator form. Based on these, necessary and sufficient conditions are derived for the case with a fixed time-state initial pair and the general case with all the time-state initial pairs. For both cases, a set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. In particular, for the general case with all the initial pairs, the existence of the delayed optimal control is equivalent to the solvability of the Riccati-like equations with some algebraic constraints, and both of them are also equivalent to the solvability of a set of coupled linear matrix equality-inequalities. Note that both the constrained Riccati-like equations and the linear matrix equality-inequalities are introduced for the first time in the literature for the proposed LQ problem. Furthermore, the convexity and the {uniform convexity} of the cost functional are fully characterized via certain properties of the solution of the Riccati-like equations.

math.OC↗

arXiv2017

Stabilization Control for Linear Continuous-time Mean-field Systems

This paper investigates the stabilization and control problems for linear continuous-time mean-field systems (MFS). Under standard assumptions, necessary and sufficient conditions to stabilize the mean-field systems in the mean square sense are explored for the first time. It is shown that, under the assumption of exact detectability (exact observability), the mean-field system is stabilizable if and only if a coupled algebraic Riccati equation (ARE) admits a unique positive semi-definite solution (positive definite solution), which coincides with the classical stabilization results for standard deterministic systems and stochastic systems. One of the key techniques in the paper is the obtained solution to the forward and backward stochastic differential equation (FBSDE) associated with the maximum principle for an optimal control problem. Actually, with the analytical FBSDE solution, a necessary and sufficient solvability condition of the optimal control, under mild conditions, is derived. Accordingly, the stabilization condition is presented by defining an Lyaponuv functional via the solution to the FBSDE and the optimal cost function. It is worth of pointing out that the presented results are different from the previous works for stabilization and also different from the works on optimal control.

math.OC↗

arXiv2016

A Complete Solution to Optimal Control and Stabilization for Mean-field Systems: Part I, Discrete-time Case

Different from most of the previous works, this paper provides a thorough solution to the fundamental problems of linear-quadratic (LQ) control and stabilization for discrete-time mean-field systems under basic assumptions. Firstly, the sufficient and necessary condition for the solvability of mean-field LQ control problem is firstly presented in analytic expression based on the maximum principle developed in this paper, which is compared with the results obtained in literatures where only operator type solvability conditions were given. The optimal controller is given in terms of a coupled Riccati equation which is derived from the solution to forward and backward stochastic difference equation (FBSDE). Secondly, the sufficient and necessary stabilization conditions are explored. It is shown that, under exactly observability assumption, the mean-field system is stabilizable in mean square sense if and only if a coupled algebraic Riccati equation (ARE) has a unique solution $P$ and $\bar{P}$ satisfying $P>0$ and $P+\bar{P}>0$. Furthermore, under the exactly detectability assumption, which is a weaker assumption than exactly observability, we show that the mean-field system is stabilizable in mean square sense if and only if the coupled ARE has a unique solution $P$ and $\bar{P}$ satisfying $P\geq 0$ and $P+\bar{P}\geq 0$. The key techniques adopted in this paper are the maximum principle and the solution to the FBSDE obtained in this paper. The derived results in this paper forms the basis to solve the mean-field control problem for continuous-time systems and other related problems.

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arXiv2014

Stochastic Recursive Optimal Control Problem with Time Delay and Applications

This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.

math.OC↗

arXiv2013

A New Distributed Localization Method for Sensor Networks

This paper studies the problem of determining the sensor locations in a large sensor network using relative distance (range) measurements only. Our work follows from a seminal paper by Khan et al. [1] where a distributed algorithm, known as DILOC, for sensor localization is given using the barycentric coordinate. A main limitation of the DILOC algorithm is that all sensor nodes must be inside the convex hull of the anchor nodes. In this paper, we consider a general sensor network without the convex hull assumption, which incurs challenges in determining the sign pattern of the barycentric coordinate. A criterion is developed to address this issue based on available distance measurements. Also, a new distributed algorithm is proposed to guarantee the asymptotic localization of all localizable sensor nodes.

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