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Doubling algorithm for continuous-time algebraic Riccati equation
Abstract
A second-order convergent algorithm is presented which gives an approximation to the unique positive definite solution of the continuous-time algebraic Riccati equation (CARE). First the CARE is transformed into the discrete-time algebraic Riccati equation (DARE) using the transformation given by Hitz and Anderson (1972). Then the discrete-time doubling algorithm, whose initial values are expressed in forms suitable for computation, is applied to the DARE. Next, it is shown that this algorithm is convergent under the condition that the CARE has the unique positive definite solution. Finally an ‘inverse’ of the Hitz and Anderson (1972) transformation is presented, which transforms the DARE into the CARE. It is proved that the ‘inverse’ transformation preserves the stabilizability, controllability, detectability and observability of the DARE.
... In this paper we investigate a structure-preserving doubling algorithm [24,37] for the computation of the symmetric positive semi-definite (s.p.s.d.) solution X (i.e. X 0) to the continuous-time algebraic Riccati equation (CARE): ...
... In this section we propose a structure-preserving doubling algorithm (SDA) for solving the CARE (1) based on the doubling algorithms in [24,37]. In addition, the well-known structure-preserving matrix sign function methods [7,[11][12][13][14]20,21,27,33,46] are also reviewed from the point of view of preserving Hamiltonian structure. ...
... solution to the above DARE as well as the CARE (1). Moreover, in Theorems 1 and 2 of [37], the pairs ( A, B) and ( A, C) are proven to be stabilizable and detectable, respectively, where the matrices G = B B T and H = C T C are full rank decompositions (FRD). Using (9)-(11) to transform the CARE (1) to an equivalent DARE (12) with the associated symplectic matrix pair ( N, L) in SSF, the SDA in [24] can then be modified to the following algorithm for CAREs: (with Im denoting the imaginary axis). ...
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.
... As noted by Anderson (1978) and Kimura (1989), a doubling algorithm for a discretetime symplectic system can be used to solve a c o n tinuous-time Hamiltonian system. Recall that in our discussion of solving control problems via a matrix sign algorithm, we s h o wed how t o c o vert a discrete-time symplectic system into a continuous-time Hamiltonian system. ...
... See [1,[3][4][5] and the references therein. Many numerical methods have been proposed, such as invariant subspace methods [6], Schur method [7], doubling algorithm [8], and structure-preserving doubling algorithm [9,10]. At the same time the perturbation theory was developed in [11][12][13][14][15], as well as the unified methods for the discrete-time and continuous-time algebraic Riccati equations [16,17]. ...
The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or
unique HPD solution is designed and tested by numerical experiments.
... When H in (1.3) has no purely imaginary eigenvalues, a strongly stable method has been proposed by [10] for computing the Hamiltonian Schur form of H, and therefore, the H-stable Lagrangian subspace. Efficient structured doubling algorithms (incorporating an appropriate Cayley transform) [11] [27] and the matrix sign function methods [5] [8] [14] [16] have been developed to compute the unique positive semidefinite solution of CARE (1.1). When H in (1.3) satisfies Assumption (A1), an eigenvector deflation technique proposed by [13] guarantees that the eigenvalues appear with the correct pairing. ...
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.
... There have also been unconventional methods for the DREs arising in optimal control, e.g. [5], [6], [9], [13], [14] and [15]. These cover various analytical solutions, and doubling algorithms for time-invariant problem. ...
Using the tools of semiconvex duality and max-plus algebra, this work derives a new fundamental solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. Such a fundamental solution, is the counterpart of the state transition matrix in linear time-varying differential equations, and can solve the DRE analytically starting with any initial condition. By parametrizing the exit cost of the underlying optimal control problem using an additional variable, a bivariate DRE is derived. Any particular solution of such bivariate time-varying DRE, can generate the fundamental solution, and hence the general solution, analytically. The fundamental solution is equivalently represented by three matrices, and the solution for any initial condition is obtained by a few matrix operations on the initial condition. It covers the special case of time invariant DRE, and derives the kernel matching conditions for transforming the DRE into the semiconvex dual DRE. As a special case, this dual DRE can be made linear, and is thus solvable analytically. Using this, the paper rederives the analytical solutions previously obtained by Leipnik [4] and Rusnak [6]. It also suggests a modification to the exponentially fast doubling algorithm described in [1], used to solve the time invariant DRE, and makes it more stable and accurate numerically for the propagation at small time step. This work is inspired from the previous work by McEneaney and Fleming [1],[2].
... As noted by Anderson (1978) and Kimura (1989), a doubling algorithm for a discretetime symplectic system can be used to solve a continuous-time Hamiltonian system. Recall that in our discussion of solving control problems via a matrix sign algorithm, we showed how to covert a discrete-time symplectic system into a continuous-time Hamiltonian system. ...
this paper use routines from the FORTRAN packages LAPACK, LINPACK and RICPACK. All of these packages can be obtained by anonymous ftp from netlib.att.com and various mirrors. MATLAB is a commercial matrix algebra package available from The MathWorks, Inc. All of our FORTRAN routines are implemented as MATLAB MEX-files.
This paper describes the recent advances for rapidly and accurately solving matrix Riccati and Sylvester equations and applies them to devise efficient computational methods for solving and estimating dynamic linear economies. The chapter explores the most promising solution methods available and compares their speed and accuracy for some particular economic examples. Except for the simplest dynamic linear models, it is necessary to compute solutions numerically. In estimation contexts, computation speed is important because climbing a likelihood function can require that a model be solved many times. Methods that are faster than direct iterations on the Riccati equation and are more reliable than solutions based on eigenvalue–eigenvector decompositions of the state–costate evolution equation are discussed in the chapter. Two generalizations are presented in the chapter: The first generalization introduces forcing sequences or “uncontrollable states” into the deterministic regulator problem, while the second generalization introduces, among other things, discounting and uncertainty into the augmented regulator problem.
This paper catalogues formulas that are useful for estimating dynamic linear economic models. We describe algorithms for computing equilibria of an economic model and for recursively computing a Gaussian likelihood function and its gradient with respect to parameters. We apply these methods to several example economies.
The paper describes an iterative algorithm for computing the limiting, or steady-state, solution of the matrix Riccati differential equation associated with quadratic minimisation problems in linear systems. It is shown that the positive-definite solution of the algebraic equation PF + F¿P¿PGR¿1G¿P + S = 0, provided that it exists and is unique, can be obtained as the limiting solution of a quadratic matrix difference equation that converges from any nonnegative definite initial condition. The algorithm is simple, and, at least for moderate dimensions of the solution matrix, competitive in computational effort with other current techniques for obtaining the limiting solution of the Riccati equation.
This paper describes a technique for obtaining error bounds for certain characteristic subspaces associated with the algebraic eigenvalue problem, the generalized eigenvalue problem, and the singular value decomposition. The method also gives perturbation bounds for isolated eigenvalues and useful information about clusters of eigenvalues. The bounds are obtained from an iterative process for generating the subspaces in question, and one or more steps of the iteration can be used to construct perturbation estimates whose error can be bounded.
If the doubling algorithm (DA) for the discrete-time algebraic Riccati equation converges, the speed of convergence is high. However, its convergence has not yet been examined exactly. Firstly, a certain matrix appearing in the DA is shown to be non-singular and therefore the algorithm is well defined. Secondly, it is found that the loss of significant digits hardly occurs in the DA. Finally, it is proved that if the time-invariant discrete-time linear system, whose state is estimated by the steady-state Kalman filter, is reachable and detectable, or stabilizable and observable, then all three matrix sequences in the DA converge.
The problem of optimal linear regulation considered is the return of a stablo linear constant coefficient dynamical systom to its equilibrium position with minimum value of a prescribed functional of system variables and source outputs. Pontryagin's Maximum Principle is used to set up the equations governing optimally regulated motion. A solution to these equations is obtained in terms of certain eigenvalues and eigenvectors of a matrix obtained from the equations defining optimal motion. It is shown that an equivalent linear constant coefficient dynamical system exists whose free motion is identical to the optimally regulated motion of the given system. Explicit expressions are obtained for the optimally regulated motion, optimal vnluo of performance functional and optimal source outputs.
Algebraic matrix Ricatti equation solution via Newton-Raphson iteration algorithm, noting application to quadratic optimal controller and least square state estimator
In this paper we shall present two new algorithms for solution of the diserete-time algebraic Riccati equation. These algorithms are related to Potter's and to Laub's methods, but are based on the solution of a generalized rather than an ordinary eigenvalue problem. The key feature of the new algorithms is that the system transition matrix need not be inverted. Thus, the numerical problems associated with an ill-conditioned transition matrix do not arise and, moreover, the algorithm is directly applicable to problems with a singular transition matrix. Such problems arise commonly in practice when a continuous-time system with time delays is sampled.
In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors, thereby gaining substantial numerical advantages. Considerable discussion is devoted to a number of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor.
Starting with certain identifies obtained by Reid [6] and Redheffer [11] for general matrix Riccati equations (RE's), we give various algorithms for the case of constant coefficients. The algorithms are based on two ideas-first, relate the RE solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant, the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration-free, superlinearly convergent iterative solution to the algebraic RE. Preliminary numerical experiments show that our algorithms, arranged in square-root form, provide a method that is numerically stable and appears to be competitive with other methods of solving the algebraic RE.
In this paper, using the ‘partitioning’ approach to estimation, exceptionally robust and fast computational algorithms for the effective solution of continuous Riccati equations are presented. The algorithms have essentially a decomposed or ‘partitioned’ structure which is both theoretically interesting as well as computationally attractive. Specifically, the ‘partitioned’ solution is given exactly in terms of a set of elemental solutions which are both simple as well as completely decoupled from each other, and as such computable in either a parallel or serial processing mode. Moreover, the overall solution is given by a simple recursive operation of the elemental solution. Extensive computer simulation has shown that the ‘partitioned’ algorithm is numerically very effective and robust, especially in the case of ill-conditioned Riccati solutions, e.g. for ill-conditioned initial conditions, or for stiff system matrices. Further, the ‘partitioned’ algorithm is very fast, ranging up to several orders of magnitude faster than the corresponding Runge-Kutta algorithm.
A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.
This paper considers computational problems arising in the solution of the linear optimal regulator problem. The proposed solution for the constant feedback gain matrix is an adaption of the eigenvector solution proposed by many authors. Techniques are given which are numerically stable and do not require complex arithmetic. These techniques offer considerable savings in computation time and eliminate many of the problems that plague more conventional methods. Copyright © 1970 by The Institute of Eiectrical and Electronics Engineers, Inc.
First Page of the Article
- ANDERSON B. D. O.