A decomposition approach to suboptimal control of discrete-time systems

A suboptimal controller for a class of discrete-time systems is presented. The controller is derived by first solving ‘off-line’ a simplified optimal control problem obtained by neglecting part of the system state and by considering a larger time step, then by solving ‘on-line’ at each time step an optimization problem based on the results of the previously solved ‘off-line’ problem. A simple numerical example is presented to illustrate the control scheme.     

On linear quadratic optimal control of discrete-time complex-valued linear systems

We study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete-time complex-valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete-time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex-valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti-Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete-time time-delay system with one-step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods.     

Delay‐dependent robust passive control for a class of nonlinear systems with time‐varying delays

In this paper, the problem of robust passive control for a class of nonlinear systems with time‐varying delays is considered. The uncertainties investigated in this paper are norm bounded and time varying, and they enter all system matrices. Based on the Lyapunov–Krasovskii functionals approach, a new robust passive control criterion is proposed in terms of linear matrix inequalities, which is dependent on the size of time delay. We also design a state feedback controller that guarantees a robust asymptotically stable and strictly passive closed‐loop system for all admissible uncertainties. Finally, two numerical examples are given to illustrate the effectiveness of the developed techniques. Copyright © 2007 John Wiley & Sons, Ltd.     

A robust and accurate disturbance damping control design for nonlinear dynamical systems

The principle result of this paper is the following disturbance rejection control scheme for a class of nonlinear dynamical systems. By using the internal model principle, the problem of disturbance damping control is converted into a nonlinear quadratic regulator (NQR) problem for an undisturbed augmented system. Then, an iterative technique is designed to solve this NQR problem effectively. The proposed iterative method is also extended through the use of a nonlinear model predictive control in an offline framework. In this case and in the presence of unknown disturbances, the Lyapunov stability of the closed‐loop system is guaranteed. Numerical simulations and comparative results verify the effectiveness of the proposed approach.     

A novel neural network discrete-time optimal control design for nonlinear time-delay systems using adaptive critic designs

In this article, a novel neural network (NN) optimal control approach using adaptive critic designs is developed for nonlinear discrete-time (DT) systems with time delays. First, to eliminate the delay term of control input, a time-delay matrix function is developed by designing a M network. Furthermore, the cost function is approximated by the critic NN, and the control signal can be obtained directly by using the information of critic NN according to the equilibrium condition. In addition, to shorten the learning time and reduce the computational burden in the control process, a novel control strategy with less adjustable parameters for the time-delay DT nonlinear systems is proposed in this article, in which the norm of the weight estimations of critic NN is updated to generate a novel long-term performance function. The proposed control algorithm using adaptive critic designs has the advantage of reducing adaptive learning parameters and lessening calculative burden. The Lyapunov stability analysis shows that the time-delay DT controlled systems can be uniformly ultimately bounded stable. Finally, three simulations are presented to demonstrate the control performance of the developed method.     

Reachable set estimation for uncertain nonlinear systems with time delay

This article studies reachable set estimation for parameter uncertain systems with time delay. A new reachable set estimation method is proposed for nonlinear discrete-time systems with parameter uncertainty and unknown but bounded disturbance. By analyzing Lyapunov-Krasovskii functional, reachable set estimation is formulated to an optimization problem in terms of linear matrix inequalities. The proposed method not only can handle nonzero initial condition but also provide real-time reachable set estimation. The effectiveness and superiority of the proposed method are illustrated by numerical simulations.     

Adaptive fuzzy finite-time optimal control for switched nonlinear systems

This article addresses the adaptive fuzzy finite-time control problem for a class of switched nonlinear systems whose powers are positive odd rational numbers and vary with the switching signal. The fuzzy logic systems (FLSs) are used to approximate the unknown nonlinearities of the controlled systems, and then by combining backstepping control algorithm with adding a power integrator technique, an adaptive finite-time controller is designed. By modifying and optimizing the relevant design parameters of the proposed adaptive finite-time controller, a novel adaptive finite-time optimal control approach is developed. The proposed two adaptive finite-time optimal control schemes can guarantee semi-globally practically finite-time stability of the closed-loop system. Moreover, the adaptive finite-time optimal controller can also achieve optimized performance in relation to the cost functional. Finally, a simulation example is given to illustrate the effectiveness of the proposed two adaptive finite-time control strategies.     

Robust stabilization of hybrid uncertain stochastic systems by discrete‐time feedback control

This paper aims to stabilize hybrid stochastic differential equations with norm‐bounded uncertainties by feedback controls based on the discrete‐time observations of both state and mode. The control structure appears only in the drift part (the deterministic part) of a stochastic differential equations, and the controlled system will be robustly exponentially stable in mean square. Our stabilization criteria are in terms of linear matrix inequalities whence the feedback controls can be designed more easily in practice. An example is given to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Decentralized output‐feedback stabilization for interconnected discrete systems with unknown delays

The decentralized feedback stabilization problem of a class of nonlinear interconnected discrete‐time systems is considered. This class of systems has unknown‐but‐bounded state‐delay and uncertain nonlinear perturbations satisfying quadratic constraints that are functions of the overall state and delayed state vectors. A decentralized output feedback scheme is proposed and analyzed such that the overall closed‐loop system guarantees global delay‐dependent stability condition, derived in terms of local subsystem variables. Incorporating feedback gain perturbations, new resilient decentralized feedback scheme is subsequently developed. The proposed approach is formulated within the framework of convex optimization over linear matrix inequalities. Simulation results illustrate the effectiveness of the proposed decentralized output‐feedback controllers. Copyright © 2009 John Wiley & Sons, Ltd.     

Distributed RHC stabilization for discrete-time large-scale systems with imposed constraints

This paper is concerned with finite horizon distributed stabilization of linear discrete-time large-scale systems with imposed constraints. Firstly, by constructing the performance index with two terminal weighted matrices, the necessary and sufficient conditions for stabilizing the system are derived. It is shown that distributed receding horizon control (RHC) can stabilize the discrete time large system with constraints if and only if two novel Lyapunov inequalities on the terminal weighting matrix are satisfied. Secondly, the finite horizon optimal control problem is solved by using the maximum principle (necessary condition), and the Riccati equation satisfying the corresponding distributed control system is derived. Based on the Riccati equation, the explicit optimal distributed RHC controller is obtained. Finally, simulation examples are used to verify the feasibility of distributed RHC to stabilize the system.     

LMI-based cooperative distributed model predictive control for Lipschitz nonlinear systems

In this paper, a distributed model predictive control is proposed to control Lipschitz nonlinear systems. The cooperative distributed scheme is considered where a global infinite horizon objective function is optimized for each subsystem, exploiting the state and input information of other subsystems. Thus, each control law is obtained separately as a state feedback of all system's states by solving a set of linear matrix inequalities. Due to convexity of the design, convergence properties at each iteration are established. Additionally, the proposed algorithm is modified to optimize only one control input at a time, which leads to a further reduction in the computation load. Finally, two application cases are studied to show the effectiveness of the proposed method.     

Guaranteed-cost stabilization of non-linear stochastic systems

The control of uncertain non-linear discrete-time systems having stochastic cone-bounded non-linearities is considered. First, a quadratic performance bound and a guaranteed-cost optimal state feedback controller are derived. Then, an auxiliary system is introduced. It is shown that the quadratic optimal control for this auxiliary system is the same as the guaranteed-cost control for the original system, and, therefore, the existence of the infinite-horizon guaranteed-cost controller can be based on the stabilizability and observability properties of the auxiliary system. Finally, the stochastic boundedness of the controlled uncertain system is proved based on the properties of the auxiliary system. © 1998 John Wiley & Sons, Ltd.     

Optimal control of discrete-time nonlinear heterogeneous multi-agent systems via a distributed DISOPE algorithm

A distributed offline DISOPE algorithm for optimal state synchronization of leader-follower systems with nonlinear discrete-time dynamics is considered, which integrates the model optimization idea and parameter estimation technique together. It can be seen that the convergent solutions of modified linear optimal control problems satisfy the optimality conditions of the original nonlinear optimization problem with non-LQ performance indices. The heterogeneous agents can cooperate and exchange information via network communication. Based on DISOPE algorithm, a distributed optimal control policy is obtained to assure state synchronization and minimize performance indices in finite time horizon. Finally, a simulation example is provided to illustrate the effectiveness of the distributed DISOPE algorithm.     

A hierarchical approach to optimized control of water distribution systems: Part I decomposition

Consideration is given to the problem of evaluating optimized control schedules for common types of water distribution systems. For these systems the major operating costs are associated with electricity charges for pumped source supplies. This paper provides a rigorous formulation for system operations. Proposals are then made for hierarchical decomposition into these levels: an upper level for dynamic optimization of reservoir storage, an intermediate level for static optimization of source extraction and a tower level for static optimization of individual sources.     

Robust quasi-time-dependent control for switched time-delay systems with performance guarantee

In this paper, guaranteed cost control for a class of switched linear systems subject to time delays is investigated, where a nonweighted quadratic performance index is considered. In particular, a novel quasi-time-dependent Lyapunov-Krasovskii function based on a time-varying positive definite matrix is proposed that is decreasing during the switching intervals and at the switching instants. By exploiting this interesting property, a family of stability condition based on mode-dependent dwell time is developed to guarantee asymptotic stability of the considered switched linear system. Moreover, design condition for the quasi-time-dependent state feedback controller is formulated by extending the stability conditions, guaranteeing asymptotic stability of the closed-loop system and an initial-condition-dependent upper bound of the performance index. A computationally traceable optimization problem is presented to minimize the proposed upper bound as well. A numerical example is considered to show the effectiveness of the proposed control method.     

Inverse optimal control for discrete‐time nonlinear systems via passivation

This paper presents an inverse optimal control approach for stabilization and trajectory tracking of discrete‐time nonlinear systems, avoiding to solve the associated Hamilton–Jacobi–Bellman equation, and minimizing a meaningful cost functional. The proposed controller is based on a discrete‐time control Lyapunov function and passivity theory; its applicability is illustrated via simulations for an unstable nonlinear system and a planar robot. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal fixed‐levels control for nonlinear systems with quadratic cost‐functionals

This paper is devoted to general optimal control problems (OCPs) associated with a family of nonlinear continuous‐time switched systems in the presence of some specific control constraints. The stepwise (fixed‐level type) control restrictions we consider constitute a common class of admissible controls in many real‐world engineering systems. Moreover, these control restrictions can also be interpreted as a result of a quantization procedure appglied to the inputs of a conventional dynamic system. We study control systems with a priori given time‐driven switching mechanism in the presence of a quadratic cost functional. Our aim is to develop a practically implementable control algorithm that makes it possible to calculate approximating solutions for the class of OCPs under consideration. The paper presents a newly elaborated linear quadratic‐type optimal control scheme and also contains illustrative numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust H∞ control for switched singular linear systems with uncertainties in the derivative matrices: an improved average dwell time approach

This paper addresses the robust H_∞ control problem for continuous‐time switched singular systems with uncertainties in the state, input, and derivative matrices. A two‐phase method is proposed in this paper to optimize H_∞ performance and to minimize the upper bound of the average dwell time. First, sufficient conditions for finding a state feedback controller guaranteeing the minimum H_∞ disturbance attenuation are proposed. Besides, the average dwell time is obtained by solving linear matrices inequalities introduced in this paper. Second, to minimize the upper bound of the average dwell time, an improve average dwell time approach is put forward by introducing a new technique. And based on the result in the first phase, a new feedback controller, which optimizes H_∞ performance and minimizes the upper bound of the average dwell time is obtained by using the improved average dwell time approach. The point is that when the upper bound of the average dwell time is decreasing, H_∞ disturbance attenuation will increase and vice versa. Therefore, a trade‐off should be built between H_∞ disturbance attenuation and average dwell time in practice. Finally, several numerical examples are presented to illustrate the effectiveness of the methods proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

Inverse optimal control for asymptotic trajectory tracking of discrete‐time stochastic nonlinear systems in block controllable form

This paper concerns an inverse optimal control–based trajectory tracking of discrete‐time stochastic nonlinear systems. It is assumed that the nonlinear system can be transformed to the so called nonlinear block controllable form. Additionally, the synthesized control law minimizes a cost functional, which is posteriori determined. In contrast to the optimal control technique, this scheme avoids to solve the stochastic Hamilton‐Jacobi‐Bellman equation, which is not an easy task. Based on a discrete‐time stochastic control Lyapunov function, the proposed optimal controller is analyzed. The proposed approach is applied successfully to the two degrees‐of‐freedom helicopter with uncertainties in real time.