Discrete-time Riccati Equation Research Articles

Inheritance properties of the conjugate discrete-time algebraic Riccati equation

In this paper we consider a class of conjugate discrete-time Riccati equations, arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Under mild and reasonable assumptions, the existence of the maximal solution to the conjugate discrete-time Riccati equation, in which the control weighting matrix is nonsingular and its constant term is Hermitian, will be inherited to a transformed discrete-time algebraic Riccati equation. Based on this inheritance property, an accelerated fixed-point iteration is proposed for finding the maximal solution via the transformed Riccati equation. Numerical examples are shown to illustrate the correctness of our theoretical results and the feasibility of the proposed algorithm.

Leader-following almost output consensus for discrete-time heterogeneous multi-agent systems in the presence of external disturbances

In this paper, we consider the problem of leader-following almost output consensus for discrete-time multi-agent systems with heterogeneous follower agents. Conditions on agent dynamics, the communication topology and how the disturbance enters the agent dynamics are established under which leader-following almost output consensus can be achieved, that is, the leader-following output consensus can be reached with any pre-specified level of accuracy and the effect of the disturbance on the leader-following consensus can be attenuated to any pre-specified level. A family of state feedback consensus protocols are constructed by using the low gain feedback design technique and a modified discrete-time Riccati equation. These consensus protocols are shown to achieve leader-following almost output consensus. Additional conditions are identified under which output feedback results are established. Simulation results demonstrate the effectiveness of the proposed consensus protocols.

On the maximal solution of the conjugate discrete-time algebraic Riccati equation

In this paper we consider a class of conjugate discrete-time Riccati equations, arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Under some mild assumptions and the framework of the fixed-point iteration, a constructive proof is given for the existence of the maximal solution to the conjugate discrete-time Riccati equation, in which the control weighting matrix is nonsingular and its constant term is Hermitian. Moreover, starting with a suitable initial matrix, we also show that the nonincreasing sequence generated by the fixed-point iteration converges at least linearly to the maximal solution of the Riccati equation. An example is given to demonstrate the correctness of our main theorem and provide considerable insights into the study of another meaningful solutions.

Observer-based discrete-time sliding mode control for systems with unmatched uncertainties

This paper addresses an observer-based sliding mode control (SMC) approach for discrete-time systems with unmatched uncertainties. A modified sliding surface based on disturbance estimation and a sliding mode controller are designed to counteract with the unmatched disturbance. The proposed method exhibits the following three features. First, the hyperplane matrix is designed in a simple way based on the discrete-time Riccati equation. Second, a chattering-free SMC method is utilized. Third, the proposed approach retains the nominal performance of the system. The stability of the overall system is achieved and simulation results are presented to verify the effectiveness of the proposed method.

Game Theory-Based Control Strategy For Trajectory Following of Four-Wheel Independently Actuated Autonomous Vehicles

Trajectory Following Control (TFC) and lateral stability control are always significant for Autonomous Ground Vehicles (AGVs). However, sometimes it is quite a challenge to achieve excellent tracking performance with less stability loss. In this paper, a novel two-agent non-cooperative game framework is built to explore the coordination approach of the integrated TFC and Direct Yaw Control (DYC) issue. Then the game theory-based control strategies are proposed and two information patterns are investigated. In the open-loop case, the Nash equilibrium of each control cycle is determined by solving discrete-time Riccati equations. When it comes to the closed-loop pattern, the game complexity increases because each player can revise his own game strategy according to the other players’ strategy evolution, and thus the dynamic programming method based on the Bellman's principle is applied to derive the control strategies. Besides, a Linear Parameter-Varying (LPV) method is adopted to facilitate the process of discretizing the state-space model in every control cycle. Finally, simulation tests based on Carsim-Simulink joint platform and experimental tests are carried out to verify the effectiveness of the proposed approach.

Stability Analysis for Input Saturated Discrete-Time Switched Systems With Average Dwell-Time

This paper studies the stability analysis of the input saturated discrete-time switched systems with average dwell-time based on the parametric discrete-time Riccati equation. The state feedback controller and the observer-based output feedback controller are designed to guarantee the exponential stability of the closed-loop system. The proposed method is simple and easy to operate in practice. The designed controllers can be computed easily by solving the parametric discrete-time Lyapunov equation. The main advantages of this paper are that the stability analysis is based on the properties of the saturation function and limited structure information of the actuator saturation and that the designed controllers have good robustness to the structure uncertainty of the input saturation. The simulation results illustrate the effectiveness of the proposed methods.

Hyperplane design for discrete‐time sliding mode control with event‐trigger strategy and disturbance observer

In this study, a simple hyperplane design method based on the discrete-time Riccati equation is proposed, and the inter-relations are investigated for the basic requirement on sliding surface selection, the assumption of stabilisability, the Riccati inequality and the standard Riccati equation. A state and disturbance observer is embedded in discrete-time sliding mode control (DSMC) to achieve its applicability when only partial system states can be measured. In order to improve the performance of the system in control updating times, a novel event-triggered DSMC with a state and disturbance observer is proposed. It is shown that the proposed method achieves a quasi sliding mode with a small boundary layer. Simulation examples are presented to show the effectiveness and advantages of the proposed design schemes.

Consensus of Linear Discrete-Time Multi-Agent Systems: A Low-Gain Distributed Impulsive Strategy

This paper aims to investigate the semiglobal consensus problem of a class of linear discrete-time multi-agent systems with distributed impulsive control. First, a novel distributed impulsive strategy is presented for achieving semiglobal consensus by considering low-gain feedback control, in which the magnitude of the impulsive protocol converges to zero when the low-gain parameter tends to zero. By utilizing the Lyapunov function and low-gain theory, a parametric discrete-time Riccati equation is considered for calculating impulsive control gain matrix. Furthermore, based on the low-and-high-gain approach, another distributed impulsive strategy is proposed. Moreover, two algorithms are then presented to derive the control gain matrices of distributed impulsive protocols for meeting different performance requirements. Subsequently, the applicability of proposed strategies is validated through two examples.

A Tradeoff Approach for Optimal Fault Detection of Networked Control Systems With Event-Triggered Scheme

An optimal fault detection (FD) approach for a class of networked control systems (NCSs) is concerned in this work. To improve the accuracy of the FD, a new event-triggered scheme (ETS) is addressed, where the output measurement transmitted or not is determined by a set of conditions instead of a single condition. Moreover, output measurements in these conditions are not only the last transmitted data, but also the data in the past period of time. The structure of fault detection and isolation (FDI) consisting of the residual generation and residual evaluation can detect faults as accurately as possible, and show robustness to disturbances in the meantime. Optimal observer gains can be obtained by utilizing the discrete-time Riccati equation (DTRE), and the time-varying threshold can be specified by employing the linear matrix inequality (LMI). Finally, a numerical simulation and the application on closed-loop continuous stirred-tank reactor (CSTR) are adopted to prove the superiority of the proposed approach.

An efficient algorithm for periodic Riccati equation with periodically time-varying input matrix

This paper proposes an efficient algorithm for the periodic discrete-time Riccati equation arising from a linear periodic time-varying system (Ak,Bk). Although the algorithm can be used for general problem, it is particularly efficient when Ak is time-invariant and Bk is periodic, a special property associated with the problem of spacecraft attitude control using magnetic torques. We will compare the proposed algorithm, by using analysis and simulation, to some well-established algorithm to confirm our claim.

Variational Integrators for Structure-Preserving Filtering

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discrete-time approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, and constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefits from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

The Maximal Solution and Comparison Theorems for the Periodic Discrete-Time Riccati Equation

AbstractProperties and comparison theorems for the maximal solution of the periodic discrete-time Riccati equation are supplemented by an extension of some earlier results and analysis, for the discrete-time Riccati equation to the periodic case.

The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

ABSTRACTIn this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.

Min–max piecewise constant optimal control for multi-model linear systems

The present work addresses a finite-horizon linear-quadratic optimal control problem for uncertain systems driven by piecewise constant controls. The precise values of the system parameters are unknown, but assumed to belong to a finite set (i.e., there exist only finitely many possible models for the plant). Uncertainty is dealt with using a min-max approach (i.e., we seek the best control for the worst possible plant). The optimal control is derived using a multi-model version of Lagrange's multipliers method, which specifies the control in terms of a discrete-time Riccati equation and an optimization problem over a simplex. A numerical algorithm for computing the optimal control is proposed and tested by simulation.

Composite control of a class of nonlinear singularly perturbed discrete-time systems via D-SDRE

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.

Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.

A Recursive Elimination Method for Finite-Horizon Optimal Control Problems of Discrete-Time Rational Systems

In this technical note, a method of solving finite-horizon optimal control problems involving discrete-time rational systems is proposed. Sequences of algebraic equations for the control input and costate at each time are constructed backward, starting from the terminal condition, by the recursive elimination of variables in the optimality conditions. This recursive elimination can be viewed as a generalization of the classical backward sweep method to obtain the discrete-time Riccati equation for finite-horizon linear quadratic control. Sufficient conditions are given for the existence and uniqueness of locally optimal state feedback laws in the form of algebraic functions of the state.

New forms of Riccati equations and the further results of the optimal control for linear discrete-time systems

For linear discrete-time systems, the traditional finite horizon optimal controller is proved to render the closed-loop systems asymptotically stable under some assumptions in literature. In this paper, a new form of finite horizon discrete-time Riccati equation is proposed. It is proved that the new form of finite horizon discrete-time Riccati equation is equivalent to the other three old ones. Based on this new form of finite horizon discrete-time Riccati equation, the finite horizon optimal controller of linear discrete time systems is proved to render the closed-loop system exponentially stable without any assumptions. At the same time, a new form of infinite horizon discrete-time Riccati equation is proposed when the discrete system is controllable or stabilizable. It is proved that the new form of infinite horizon discrete-time Riccati equation is equivalent to the other three old ones too. Based on this new form of infinite horizon discrete-time Riccati equation, the infinite horizon optimal controller of linear discrete-time systems is proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally an unstable batch reactor and an unstable inverted pendulum are used to verify the theory results of this paper.

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.

Optimal state estimation using randomly delayed measurements without time stamping

SUMMARY This paper studies an optimal state estimation (Kalman filtering) problem under the assumption that output measurements are subject to random time delays caused by network transmissions without time stamping. We first propose a random time delay model which mimics many practical digital network systems. We then study the so-called unbiased, uniformly bounded linear state estimators and show that the estimator structure is given based on the average of all received measurements at each time for different maximum time delays. The estimator gains can be derived by solving a set of recursive discrete-time Riccati equations. The estimator is guaranteed to be optimal in the sense that it is unbiased with uniformly bounded estimation error covariance. A simulation example shows the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.