Discrete-time Algebraic Riccati Equation Research Articles

A geometric approach to optimal state‐space solutions with minimal realization for standard discrete‐time H 2 control problem

This paper shows that the controllable and unobservable subspaces of the discrete‐time H 2 optimal controller can be characterized by the image and kernel spaces of two matrices Z 2 and W 2, where Z 2 and W 2 are positive semi‐definite solutions of two pertinent Lyapunov equations whose coefficients involve the stabilizing solutions of two celebrated discrete‐time algebraic Riccati equations (DAREs) used in solving the H 2 optimal control problem. By suitably choosing the bases adapted to Z 2 and W 2, a minimal order state‐space realization of an H 2 optimal controller is then given via an elegant geometric approach. In terms of geometric language, all the results and proofs given are clear and simple.

Stability analysis and optimal tuning of EWMA controllers – Gain adaptation vs. intercept adaptation

Exponentially weighted moving average (EWMA) controllers are the most commonly used run-to-run controllers in semiconductor manufacturing industry. An EWMA controller can be implemented in two different ways. One way is to keep the process gain as its off-line estimate and update the intercept term at each run, which is termed EWMA with intercept adaptation; the other is to keep the intercept term as its off-line estimate and update the process gain at each run, which is termed EWMA with gain adaptation. Despite the fact that gain variation and adaptation is typical in semiconductor industry, most EWMA formulations are for intercept adaptation and few results exist on the stability and sensitivity of EWMA with gain adaptation. In this paper, we propose a general formulation to analyze the stability of both EWMA controllers. The proposed state-space representation not only reveals the similarities and differences between two types of EWMA controllers, but also explains why the stability conditions for both types of EWMA controllers are independent of process disturbances. In addition, we propose a general framework that unifies the analysis of the optimal control performance for both types of EWMA controllers. The proposed framework is different from existing approaches in that it decouples the state estimation from the control law, and derives the optimal weighting based on the state estimation performance. The proposed framework significantly simplifies the analysis procedure, especially for EWMA with gain adaptation. Using this framework, we derive the optimal EWMA weighting through solving the discrete-time algebraic Riccati equation (DARE) for various process disturbances that are encountered in semiconductor manufacturing industry. Simulation examples are given to illustrate the optimality of the EWMA weighting derived using the framework. Some practical aspects of controller tuning are also discussed based on the simulation results.

Model-free multiobjective approximate dynamic programming for discrete-time nonlinear systems with general performance index functions

In this paper, a forward-in-time optimal control method for a class of discrete-time nonlinear systems with general multiobjective performance indices is proposed with unknown system dynamics. The proposed approximate dynamic programming (ADP) method aims to find out the increments of both the controls and states instead of computing the controls and states directly. Using the technique of dimension augment, the vector-valued performance indices are transformed into additive quadratic form which satisfies the corresponding discrete-time algebraic Riccati equation (DTARE). Both the action and critic networks can be adaptively tuned by adaptive critic methods without the information of the system model. The convergence property is guaranteed by a rigorous mathematical proof and finally the simulation results show the effectiveness of the method.

Adaptive State Feedback Nash Strategies for Linear Quadratic Discrete-Time Games

A substantial effort has been devoted to various adaptive techniques of systems. Most of these concepts work in the control domain, where every system only has one controller. Yet, for the multi-controller counterpart — dynamic games, adaptations are usually considered from a perspective of systems, for an example, evolutionary games. In this paper, we propose a new adaptive approach for linear quadratic discrete-time games with scalar inputs and state feedback Nash strategies. We consider the effort of adaptation under a Fictitious Play (FP) framework with learning algorithms derived from conventional adaptive control methods. Convergence to Nash strategies is proved with the condition that there exists a unique state feedback strategy, which implies that the associated coupled discrete-time algebraic Riccati equations (DAREs) have a unique positive semi-definite solution. The requirement of Persistency of Excitation (PE) is satisfied by proper reference signals to be tracked.

Bounded Real Lemma for Linear Discrete-Time Descriptor Systems

Under some rank condition, a new version of bounded real lemma, which is expressed in terms of an admissible solution of a generalized discrete-time algebraic Riccati equation (GDARE) rather than inequality, is presented for linear discrete-time descriptor systems. When a linear discrete-time descriptor system is admissible, with the H∞-norm of its transfer matrix less than a prescribed positive number γ, a constructive procedure is also given to obtain an admissible solution of the above-mentioned GDARE.

Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property ( P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.

Optimal periodic disturbance reduction for active noise cancelation

The design of an optimal internal model-based (IMB) controller by extending standard discrete time optimal control theory for IMB controllers is described. The optimal observer and state feedback gains of the IMB controller are given via the solution of discrete time algebraic Riccati equations. The design method is applied to an acoustic system that is subjected to disturbances from a server fan. Periodic disturbances from the server fan appear as harmonics of the fundamental frequency of the fan. Parametric models for the plant and non-periodic part of the disturbance are identified from experimental data. An internal model is designed in discrete time and the internal model principle is used to design a feedback controller that rejects periodic disturbances in the acoustic system. The controller is implemented in real-time and successfully attenuates the first four harmonics of the fan noise.

On the Solution of the Rational Matrix Equation

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation, where is symmetric positive definite and is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.

Observer-Based Fast Rate Fault Detection for a Class of Multirate Sampled-Data Systems

This note deals with the problem of observer-based fast rate fault detection for a class of multirate sampled-data (MSD) systems. Applying a lifting technique, a linear time-invariant (LTI) representation with slow sampling period is firstly obtained for the MSD systems and, based on this, an observer-based fault detection filter is considered as a residual generator. Then an optimization fault detection approach for LTI systems is modified to the residual generation for the MSD systems and, by solving a discrete-time Algebraic Riccati equation, a family of optimal solutions with causality constraint can be obtained. An inverse lifting operation on the generated residual implements its fast rate. The residual evaluation problem is also considered. A numerical example is finally given to illustrate the effectiveness of the proposed design techniques

Order reduction of discrete-time algebraic Riccati equations with singular closed loop matrix

We study the general discrete-time algebraic Riccati equation and deal with the case where the closed loop matrix corresponding to an arbitrary solution is singular. In this case the extended symplectic pencil associated with the DARE has 0 as a characteristic root and the corresponding spectral deflating subspace gives rise to a subspace where all solutions of the DARE coincide. This allows for a reduction of the original DARE to an equation of smaller size.

MATLAB TOOLS FOR SOLVING PERIODIC EIGENVALUE PROBLEMS

Software for computing eigenvalues and invariant subspaces of general matrix products is proposed. The implemented algorithms are based on orthogonal transformations of the original data and thus attain numerical backward stability, which enables good accuracy even for small eigenvalues. The prospective toolbox combines the efficiency and robustness of library-style Fortran subroutines based on state-of-the-art algorithms with the convenience of Matlab interfaces. It will be demonstrated that this toolbox can be used to address a number of tasks in systems and control theory, such as the solution of periodic discrete-time algebraic Riccati equations.

A characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

FAST RATE FAULT DETECTION FOR MULTIRATE SAMPLED-DATA SYSTEMS WITH TIME-DELAYS

This paper deals with the fast rate residual generation problem for a class of multirate sampled-data (MSD) systems with multiple data transfer time-delays. By using the l2-induced gain to measure the robustness of residual with respect to unknown input and sensitivity to fault, the design of residual generator for such MSD systems is formulated as an H∞ optimization problem. A solution of the optimization problem is then presented via a discrete-time algebraic Riccati equation by applying some recent results of fault detection of single rate SD system. The fast rate of residual generator can be implemented when its causality is guaranteed. The main contribution of this paper lies in the treatment of residual generation problem for such an MSD system as that of linear time-invariant system without time-delays and, further, the implement of its fast rate. An illustrative design example is employed to demonstrate the effectiveness of the proposed approach.

A Parametrization of Solutions of the Discrete-time Algebraic Riccati Equation Based on Pairs of Opposite Unmixed Solutions

The paper describes the set of solutions of the discrete-time algebraic Riccati equation. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution. The results of the paper provide an extended counterpart of the parametrization theory of continuous-time algebraic Riccati equations by Willems, Coppel, and Shayman.

A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations

In Chu et al. (2004), an efficient structure-preserving doubling algorithm (SDA) was proposed for the solution of discrete-time algebraic Riccati equations (DAREs). In this paper, we generalize the SDA to the G-SDA, for the generalized DARE: E T XE = A T XA − (A T XB…+C TS )(R + B T XB)−1(B T XA + S TC ) + C T QC. Using Cayley transformation twice, we transform the generalized DARE to a DARE in a standard symplectic form without any explicit inversions of (possibly ill-conditioned) R and E. The SDA can then be applied. Selected numerical examples illustrate that the G-SDA is efficient, out-performing other algorithms.

A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations

Continuous-time algebraic Riccati equations (CAREs) can be transformed, à la Cayley, to discrete-time algebraic Riccati equations (DAREs). The efficient structure-preserving doubling algorithm (SDA) for DAREs, from [E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-28, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003; E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, C.-S. Wang, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-18, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003], can then be applied. In this paper, we develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to CAREs. A modified version of the SDA, developed for DAREs with a “doubly symmetric” structure, is also presented. Extensive numerical results show that our approach is efficient and competitive.

A Parametrization of the Set of Solutions of the Discrete-Time Algebraic Riccati Equation

The set of solutions of the discrete-time algebraic Riccati equation is described. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix associated to an unmixed solution. The results of the paper are a counterpart of the parametrization theory of continuous-time algebraic Riccati equations due to Willems, Coppel and Shayman.

On the Structure of the Solutions of Discrete-Time Algebraic Riccati Equation With Singular Closed-Loop Matrix

The classical discrete-time algebraic Riccati equation (DARE) is considered in the case when the corresponding closed-loop matrix is singular. It is shown that in this case all the symmetric solutions of the DARE coincide along some directions. A parametrization of the set of solutions in terms of a reduced-order DARE is then obtained. This parametrization provides an algorithm (that appears to be computationally very attractive when the multiplicity of the eigenvalue /spl lambda/=0 of the closed-loop matrix is large) for the computation of the solutions of the DARE. The same issue for the generalized DARE is also addressed.

Structure-Preserving Algorithms for Periodic Discrete-Time Algebraic Riccati Equations

In this paper we investigate structure-preserving algorithms for computing the symmetric positive semi-definite solutions to the periodic discrete-time algebraic Riccati equations (P-DAREs). Using a structure-preserving swap and collapse procedure, a single symplectic matrix pair in standard symplectic form is obtained. The P-DAREs can then be solved via a single DARE, using a structure-preserving doubling algorithm. We develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability. With several numerical results, the algorithm is shown to be efficient, out-performing other algorithms on a large set of benchmark problems.

Backward Perturbation Analysis of the Periodic Discrete-Time Algebraic Riccati Equation

Normwise backward errors and residual bounds for an approximate Hermitian positive semidefinite solution set to the periodic discrete-time algebraic Riccati equation are obtained. The results are illustrated by using simple numerical examples.