Discrete-time Algebraic Riccati Equation Research Articles

Bauer’s spectral factorization method for low order multiwavelet filter design

Para-Hermitian polynomial matrices obtained by matrix spectral factorization lead to functions useful in control theory systems, basis functions in numerical methods or multiscaling functions used in signal processing. We introduce a fast algorithm for matrix spectral factorization based on Bauer’s method. We convert Bauer’s method into a nonlinear matrix equation (NME). The NME is solved by two different numerical algorithms (Fixed Point Iteration and Newton’s Method) which produce approximate scalar or matrix factors, as well as a symbolic algorithm which produces exact factors in closed form for some low-order scalar or matrix polynomial matrices, respectively. Convergence rates of the two numerical algorithms are investigated for a number of singular and nonsingular scalar and matrix polynomials taken from different areas. In particular, one of the singular examples leads to new orthogonal multiscaling and multiwavelet filters. Since the NME can also be solved as a Generalized Discrete Time Algebraic Riccati Equation (GDARE), numerical results using built-in routines in Maple 17.0 and six Matlab versions are presented.

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.

Finding the LQR Weights to Ensure the Associated Riccati Equations Admit a Common Solution

This paper addresses the problem of finding the linear quadratic regulator (LQR) weights such that the associated discrete-time algebraic Riccati equations admit a common optimal stabilising solution. Solving such a problem is key to designing LQR controllers to stabilise discrete-time switched linear systems under arbitrary switching, or stabilise polytopic systems (e.g., Takagi-Sugeno fuzzy systems and linear parameter varying systems) in the entire operating region. To ensure problem tractability and reduce the searching space, this paper proposes an efficient framework of finding only the state weights based on the given input weights. Linear matrix inequality conditions are derived to conveniently check feasibility of the problem. An iterative algorithm with quadratic convergence and low computational complexity is developed to solve the problem. Efficacy of the proposed method is illustrated through numerical simulations of systems with various sizes.

Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems.

Observer-Based Dynamic Event-Triggered Bipartite Consensus of Discrete-Time Multi-Agent Systems

In this brief, the dynamic event-triggered bipartite consensus is explored for discrete-time multi-agent systems (MASs) under competitive network. Different from the existing works on dynamic event-triggered consensus of discrete-time MASs, observer based protocol is proposed by using the related output information. With the help of Lyapunov function method, some sufficient conditions are obtained and the control gains can be determined by solving discrete-time algebraic Riccati equations. Finally, a simulation example is presented to effectively verify the theoretical results.

Structured Doubling Algorithm for a Class of Large-Scale Discrete-Time Algebraic Riccati Equations with High-Ranked Constant Term

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared to the existing doubling algorithm of O(2kn) flops at the k-th iteration, the newly developed version merely needs O(n) flops for preprocessing and O((k+1)3m3) flopsfor iterations and is more proper for large-scale computations when m≪n. The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments indicate that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of computing the solution efficiently for large-scale DAREs.

Distributed Fault-Tolerant Bipartite Output Synchronization of Discrete-Time Linear Multiagent Systems.

This article studies the distributed fault-tolerant bipartite output synchronization problem of discrete-time linear multiagent systems (MASs) with process faults under a general directed signed graph. The reference signal is generated by an autonomous exosystem, which can also be seen as a leader. All followers are divided into two subgroups with antagonistic interactions, and the followers in each subgroup are cooperative. We aim to solve the bipartite fault-tolerant control (FTC) problem via the output regulation theory such that bipartite output synchronization can be achieved in the presence of process faults, that is, the outputs of followers with different subgroups can approach the output of exosystem with the same magnitude and the opposite sign regardless of process faults. To estimate the states and the faults of each follower, a simultaneous state and fault estimator based on the neighboring signed output estimation error and the standard discrete-time algebraic Riccati equation (ARE) is designed. Besides, a new exosystem observer with two classes of convergence conditions relying on the respective solutions of standard and modified AREs is provided. All eigenvalues of the exosystem matrix can lie completely outside the unit circle. Based on these estimations, we present a distributed fault-tolerant output feedback controller, which can overcome the no-loops constraint. Finally, simulation results are given to demonstrate the analytic results.

Fault-tolerant control of a hydraulic servo actuator via adaptive dynamic programming

<abstract><p>The fault-tolerant control problem of a hydraulic servo actuator in the presence of actuator faults is studied utilizing adaptive dynamic programming. This task is challenging because of unknown system dynamics, uncertain disturbances or unmeasurable system states of such highly nonlinear systems in real applications. The aim is to achieve asymptotic tracking and actuator faults compensation by minimizing some predefined performance index. The discrete-time algebraic Riccati equation is iteratively solved by the adaptive dynamic programming approach. For practical reasons, adaptive dynamic programming techniques and fault compensation are integrated to iteratively compute an approximated optimal fault-tolerant control using real-time input/output data without any a priori knowledge of the system dynamics and unmeasurable states. As a result, a fault-tolerant control of hydraulic servo actuator is then designed based on adaptive dynamic programming via output feedback. Also, the convergence analysis of a data-driven fault-tolerant control is theoretically shown as well. Finally, intensive simulation results are given to prove the validity and merits of the developed data-driven fault-tolerant control strategy.</p></abstract>

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.

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.

Optimal output tracking control of linear discrete-time systems with unknown dynamics by adaptive dynamic programming and output feedback

This paper discusses the output-feedback-based model-free optimal output tracking control problem of discrete-time systems with completely unknown system models under mild assumptions. The only information that allows utilisation is the system output and the reference output. To overcome these challenges, the paper aims to solve a model-free optimal output regulation problem for achieving optimal output tracking control; solving an optimal output regulation problem is equivalent to solving a linear quadratic regulation (LQR) problem and a constrained static optimisation problem. The state reconstruction is first given to represent the system state in terms of the input and output sequences. A data-driven value iteration (VI) algorithm is then proposed to iteratively approximate the solution to the discrete-time algebraic Riccati equation (ARE) of the corresponding LQR problem on the basis of input and output data. Next, based on the iterative solutions with respect to the ARE, a model-free solution is provided for the corresponding constrained static optimisation problem. Finally, an alternative reference system equivalent to the original reference system is established to avoid the requirement of having the knowledge of the reference system dynamics and the reference state. A numerical example is employed to demonstrate the effectiveness of the proposed control scheme.

Consensus-based distributed filtering with fusion step analysis

For consensus on measurement-based distributed filtering (CMDF), through infinite consensus fusion operations during each sampling interval, each node in the sensor network can achieve optimal filtering performance with centralized filtering. However, due to the limited communication resources in physical systems, the number of fusion steps cannot be infinite. To deal with this issue, the present paper analyzes the performance of CMDF with finite consensus fusion operations. First, by introducing a modified discrete-time algebraic Riccati equation and several novel techniques, the convergence of the estimation error covariance matrix of each sensor is guaranteed under a collective observability condition. In particular, the steady-state covariance matrix can be simplified as the solution to a discrete-time Lyapunov equation. Moreover, the performance degradation induced by reduced fusion frequency is obtained in closed form, which establishes an analytical relation between the performance of the CMDF with finite fusion steps and that of centralized filtering. Meanwhile, it provides a trade-off between the filtering performance and the communication cost. Furthermore, it is shown that the steady-state estimation error covariance matrix exponentially converges to the centralized optimal steady-state covariance matrix with fusion operations tending to infinity during each sampling interval. Finally, the theoretical results are verified with illustrative numerical experiments.

Dynamic Mode Decomposition: A New Approach for Computing the DMD Modes and Eigenvalues

In this paper we study a special case of linearly perturbed discrete-time algebraic Riccati equation. We give some sufficient conditions forthe existence of a positive definite solution of the considered equation.We propose a basic fixed point iteration and its inversion free variantfor finding a positive definite solution. Moreover, by specially choosingthe initial value in the basic fixed point iteration we prove that it con-verges to the largest solution. The theoretical results are illustratedby numerical examples.

The convergence analysis of an accelerated iteration for solving algebraic Riccati equations

The discrete-time algebraic Riccati equation (DARE) have extensive applications in optimal control problems. We provide new theoretical supports to the stability properties of solutions to the DARE and reduce the convergence conditions under which the accelerated fixed-point iteration (AFPI) can be applied to compute the numerical solutions of DARE. In particular, we verify that the convergence of AFPI is R-superlinear when the spectral radius of the closed-loop matrix is greater than 1, which is shown by mild assumption and only using primary matrix theories. Numerical examples are shown to illustrate the consistency and effectiveness of our theoretical results.

On the lack of monotonicity of Newton–Hewer updates for Riccati equations

We provide a set of counterexamples for the monotonicity of the Newton–Hewer method (Hewer, 1971) for solving the discrete-time algebraic Riccati equation in dynamic settings, drawing a contrast with the Riccati difference equation (Caines and Mayne, 1970).

Disturbance Observer-Based Robust Formation-Containment of Discrete-Time Multiagent Systems with Exogenous Disturbances

This paper investigates robust formation-containment control of discrete-time multiagent systems (MASs) with exogenous disturbances. Based on the discrete-time disturbance observer method, both state feedback and output feedback control protocols are proposed. Formation-containment conditions are obtained and convergency analysis is given according to Lyapunov stability theory. And, the corresponding control gains are obtained by solving some discrete-time algebraic Riccati equations. Numerical simulations are presented to illustrate the theoretical findings.

Modified Algebraic Riccati Equation Closed-Form Stabilizing Solution

A modified discrete-time algebraic Riccati equation (MARE) is a discrete-time algebraic Riccati equation (DARE) for which the quadratic term is weighted by a modifying parameter $\alpha $ . The MARE is known to arise for example when conducting estimation or stabilization of single-input single-output (SISO) systems subject to packet losses in networked control systems (NCSs). In the present paper we characterize the solution to the MARE in closed form that, to the best of the author’s knowledge, is a completely novel result. We then verify the already known critical value of the modifying parameter $\alpha _{c}$ for which the MARE is solvable and propose closed-form expressions for the optimal state feedback gain matrix. We finally present examples to illustrate the obtained contributions.

POSITIVE DEFINITE SOLUTIONS OF A LINEARLY PERTURBED MATRIX EQUATION

In this paper we study a special case of linearly perturbed discrete-time algebraic Riccati equation. We give some sufficient conditions forthe existence of a positive definite solution of the considered equation.We propose a basic fixed point iteration and its inversion free variantfor finding a positive definite solution. Moreover, by specially choosingthe initial value in the basic fixed point iteration we prove that it con-verges to the largest solution. The theoretical results are illustratedby numerical examples.

Event-Triggered Adaptive Optimal Control With Output Feedback: An Adaptive Dynamic Programming Approach.

This article presents an event-triggered output-feedback adaptive optimal control method for continuous-time linear systems. First, it is shown that the unmeasurable states can be reconstructed by using the measured input and output data. An event-based feedback strategy is then proposed to reduce the number of controller updates and save communication resources. The discrete-time algebraic Riccati equation is iteratively solved through event-triggered adaptive dynamic programming based on both policy iteration (PI) and value iteration (VI) methods. The convergence of the proposed algorithm and the closed-loop stability is carried out by using the Lyapunov techniques. Two numerical examples are employed to verify the effectiveness of the design methodology.

Multi-Objective Model Predictive Control for Vehicle Active Suspension System

This paper puts forward a multi-objective model predictive control scheme in order to address the conflicting control objectives of a vehicle active suspension system. The key problem in designing an active suspension controller is that the controller has to realize a feasible control input that can satisfy the competing control requirements such as ride comfort, suspension travel and road handling. Hence, in this work, these constraints are integrated into an optimal control framework and a finite horizon model predictive controller is used to solve the multi-objective cost function. The key advantage of the proposed scheme is that the model predictive control design finds the optimal control input by solving the discrete time algebraic Riccati equation. This guarantees not only a robust closed loop system but also a realizable control effort, without violating the hard constraints of the active suspension system. The proposed model predictive control design is experimentally validated on a laboratory scale quarter car suspension system using hardware-in-loop testing. The performance of the model predictive control scheme is compared with the one of the unconstrained linear quadratic regulator and tested for four realistic road profiles. The experimental results substantiate that the suspension system controlled by the model predictive controller offers better ride comfort and road handling features when compared to the conventional linear quadratic regulator.