Showing papers on "Riccati equation published in 1995"
Book•
01 Jan 1995
TL;DR: Geometric theory: the complex case 8.
Abstract: 1. Preliminaries from the theory of matrices 2. Indefinite scalar products 3. Skew-symmetric scalar products 4. Matrix theory and control 5. Linear matrix equations 6. Rational matrix functions 7. Geometric theory: the complex case 8. Geometric theory: the real case 9. Constructive existence and comparison theorems 10. Hermitian solutions and factorizations of rational matrix functions 11. Perturbation theory 12. Geometric theory for the discrete algebraic Riccati equation 13. Constructive existence and comparison theorems 14. Perturbation theory for discrete algebraic Riccati equations 15. Discrete algebraic Riccati equations and matrix pencils 16. Linear-quadratic regulator problems 17. The discrete Kalman filter 18. The total least squares technique 19. Canonical factorization 20. Hoo control problems 21. Contractive rational matrix functions 22. The matrix sign function 23. Structured stability radius Bibliography List of notations Index
1,465 citations
TL;DR: An extension of a Lyapunov equation result is derived for the countably infinite Markov state-space case and guarantees existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy.
Abstract: Optimal control problems for discrete-time linear systems subject to Markovian jumps in the parameters are considered for the case in which the Markov chain takes values in a countably infinite set. Two situations are considered: the noiseless case and the case in which an additive noise is appended to the model. The solution for these problems relies, in part, on the study of a countably infinite set of coupled algebraic Riccati equations (ICARE). Conditions for existence and uniqueness of a positive semidefinite solution to the ICARE are obtained via the extended concepts of stochastic stabilizability (SS) and stochastic detectability (SD), which turn out to be equivalent to the spectral radius of certain infinite dimensional linear operators in a Banach space being less than one. For the long-run average cost, SS and SD guarantee existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy. Furthermore, an extension of a Lyapunov equation result is derived for the countably infinite Markov state-space case.
218 citations
TL;DR: In this paper, a generalized Lyapunov theorem for continuous-time descriptor systems is presented, which is applied to the infinite-horizon descriptor LQ regulator problem, and the result is extended to continuous time descriptor systems.
209 citations
TL;DR: A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions.
Abstract: This paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system, the uncertainty being norm bounded. For this the "quadratic d stabilizability" concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived. This condition expressed as a parameter dependent discrete Riccati equation enables one to design the control gain matrix by solving iteratively a discrete Riccati equation. >
187 citations
TL;DR: In this paper, a minimax control problem for an uncertain system containing structured uncertainties is considered, where the uncertainties in this system are assumed to satisfy a certain integral quadratic constraint, and the minimax optimal controller is constructed by solving a parameter-dependent Riccati equation of the game type.
Abstract: This paper considers a minimax control problem for an uncertain system containing structured uncertainties. The uncertainties in this system are assumed to satisfy a certain integral quadratic constraint. For a given initial condition, the minimax optimal controller is constructed by solving a parameter-dependent Riccati equation of the game type. This controller leads to a closed-loop uncertain system which is absolutely stable.
120 citations
TL;DR: In this article, sufficient conditions for the existence of an invariant distribution, for all values of the delay, were given in the form of some positive definite matrices satisfying certain Riccati-type equations, and the covariance and correlation matrix function of the resulting stationary process were completely characterized by a Lyapunov-type equation.
108 citations
TL;DR: In this article, the second-order sufficient conditions for local minima of optimal control problems with state and control constraints were extended to include general boundary conditions and a direct sufficiency criterion based on a quadratic function that satisfies a Hamilton-Jacobi inequality.
Abstract: References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.
107 citations
TL;DR: It is shown that a positive-semidefinite solution of the coupled algebraic discrete-time Riccati-like equation occurring in Markovian jump control problems exists and can be obtained as a limit of a monotonic sequence.
95 citations
TL;DR: In this article, the validity question for the Newell-Whitehead equation is treated, i.e., estimates between the exact solutions of the original problem and the associated approximations are given.
Abstract: Modulation equations play an essential role in the description of systems which exhibit patterns of nearly periodic nature, e.g. in Benard's problem. The so called Newell-Whithead equation is derived to describe the envelope of modulated roll-solutions in systems with two large extended or unbounded space directions. Here the validity question for the Newell-Whitehead equation is treated, i.e., estimates between the exact solutions of the original problem and the associated approximations are given. The Newell-Whitehead equation is a good example how important this validity question is. Although the modulation equation can formally be derived, the solutions of the original system can behave in some situations in a completely different manner than predicted by the modulation equation. This happens on a time scale which is very small compared to the natural one.
92 citations
TL;DR: In this article, an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates.
Abstract: In this paper we first present an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates. We then consider a certainty equivalence adaptive linear-quadratic Gaussian feedback control law using the estimates generated by the nonlinear filter and demonstrate the global existence of solutions to the resulting closed-loop system. A stochastic Lyapunov analysis establishes that the certainty equivalence law stabilizes the Markov jump parameter linear system in the mean square average sense. The conditions for this result are that certain products of (i) the parameter process jump rate and (ii) the solution of the control Riccati equation and its second derivatives should be less than certain given bounds. An example is given where the controlled linear system has state dimension 2. Finally, the stabilizing properties of certainty equivalence laws which depend on (i) the maximum likelihood estimate of the parameter value and (ii) a modified version of this estimate are established under certain conditions.
73 citations
TL;DR: In this paper, it was shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter p of a Banach space, and the proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC.
Abstract: This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameter p of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parametric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corre- sponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC. Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.
01 Jan 1995
TL;DR: In this article, the symmetric coupled algebraic Riccati equations corresponding to the steady state Nash strategies are studied and the Lyapunov iterations are constructed such that the proposed algorithm converges to the nonnegative (positive) definite stabilizing solution of the coupled algebraIC Riccaci equations.
Abstract: In this paper we study the symmetric coupled algebraic Riccati equations corresponding to the steady state Nash strategies. Under control-oriented assumptions, imposed on the problem matrices, the Lyapunov iterations are constructed such that the proposed algorithm converges to the nonnegative (positive) definite stabilizing solution of the coupled algebraic Riccati equations. In addition, the problem order reduction is achieved since the obtained Lyapunov equations are of the reduced-order and can be solved independently. As a matter of fact a parallel synchronous algorithm is obtained. A high-order numerical example is included in order to demonstrate the efficiency of the proposed algorithm. In the second part of this paper we have proposed an algorithm, in terms of the Lyapunov iterations, for finding the positive semidefinite stabilizing solution of the algebraic Riccati equation of the zero-sum differential games. The similar algebraic Riccati type equations appear in the H ∞ optimal control and related problems.
TL;DR: A sequence of Lyapunov algebraic equations, whose solutions converge to the solutions of the coupled algebraic Riccati equations of the optimal control problem for jump linear systems are constructed.
Abstract: In this paper we construct a sequence of Lyapunov algebraic equations,whose solutions converge to the solutions of the coupled algebraic Riccati equations of the optimal control problem for jump linear systems. The obtained solutions are positive semidefinite, stabilizing, and unique. The proposed algorithm is extremely efficient from the numerical point of view since it operates only on the reduced-order decoupled Lyapunov equations, Several examples are included to demonstrate the procedure. >
TL;DR: In this paper, the authors present a design strategy in terms of ordinary differential equations which creates chaotic attractors with an increasing number of positive Lyapunov exponents as the (finite) dimension of the system is increased.
Abstract: We present a design strategy in terms of ordinary differential equations which creates chaotic attractors with an increasing number of positive Lyapunov exponents as the (finite) dimension of the system is increased. First, we introduce the most simple abstract equation containing only one nonlinearity. Second, we suggest a piecewise linear version of the abstract equation. Third, we propose a set of chemical reactions and demonstrate that the corresponding rate equations produce hyperchaotic behavior equivalent to the abstract system.
TL;DR: This note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant, which yields for the closed-loop transfer matrix a specified H/sub /spl infin norm bound.
Abstract: This note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant, which yields for the closed-loop transfer matrix a specified H/sub /spl infin norm bound. The algebraic results of covariance control are applied to the H/sub /spl infin control problem to yield a parameterization in terms of the Lyapunov matrix, which carries many system properties (such as H/sub 2/ performance, covariance bounds, system entropy at infinity, etc.). All low-order H/sub /spl infin controllers are shown to have observer-based structure for "reduced-order models" of the plant and are characterized by two Riccati equations with a coupling condition. >
TL;DR: In this paper, the Conley index theory is used to describe the dynamics on the global attractors of bistable gradient-like evolution equations via a semiconjugacy onto the dynamics of a simple system of ordinary differential equations.
Abstract: The dynamics on the global attractors of bistable gradient-like evolution equations are described via a semiconjugacy onto the dynamics of a simple system of ordinary differential equations. The fact that the semiconjugacy is “onto” implies that, given any solution to the ordinary differential equation, there exists a corresponding orbit on the attractor of the evolution equation. It is also shown that these results apply to the damped wave equation, a viscoelastic beam equation, the Fitz–Hugh–Nagumo equations, the Cahn–Hilliard equation, and the phase-field equations. The proofs are based on the Conley index theory.
01 Mar 1995
TL;DR: In this article, a Riccati equation approach to the output feedback quadratic stabilisation of an uncertain system is presented, which involves the minimisation of a certain bound on an LQG cost function.
Abstract: The paper presents a Riccati equation approach to the output feedback quadratic stabilisation of an uncertain system. The uncertain systems under consideration depend on a norm-bounded time-varying matrix of uncertain parameters. A feature of the approach taken is that it involves the minimisation of a certain bound on an LQG cost function. The results were obtained by combining results on an optimal guaranteed cost linear quadratic regulator problem for uncertain systems and an optimal guaranteed cost state estimation problem for uncertain systems.
TL;DR: In this article, the duality between the filtering and optimal control equations is not straightforward and a generalization of the concepts of detectability and stabilizability, as well as a discussion of how the dualness between these concepts should be understood, is presented Conditions for existence and uniqueness of a positive semi-definite solution to a set of coupled Riccati equations in a generic form are provided in terms of these concepts.
TL;DR: Multigrid or, more appropriately, multilevel techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations and Smith’s method for solving matrix LyAPunov equations are considered.
Abstract: We consider multigrid or, more appropriately, multilevel techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations. The Riccati equation, which is quadratic, plays an essential role in the solution of linear-quadratic optimal control problems. The linear Lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the Newton–Kleinman (NK) algorithm for the solution of algebraic Riccati equations. Both equations are operator equations when the underlying linear system is infinite dimensional. In this case, finite-dimensional discretization is required. However, as the level of discretization increases, the convergence rate of the standard iterative techniques for solving high order matrix Lyapunov and Riccati equations decreases. To deal with this, multileveling is introduced into the iterative NK method for solving the algebraic Riccati equation and Smith’s method for solving matrix Lyapunov equations. Theoretical result...
TL;DR: This work considers the infinite horizon quadratic cost minimization problem for a linear system with finitely many inputs and outputs, and reduces the problem to a symmetric Wiener-Hopf problem, that can be solved by means of a canonical factorization of the symbol.
Abstract: We consider the infinite horizon quadratic cost minimization problem for a linear system with finitely many inputs and outputs. A common approach to treat a problem of this type is to construct a semigroup in an abstract state space, and to use infinite-dimensional control theory. However, this approach is less appealing in the case where there are discrete time delays in the impulse response, because such time delays force both the control operator and the observation operator to be unbounded at the same time. In order to be able to include this case we take an alternative approach. We work in an input-output framework, and reduce the problem to a symmetric Wiener-Hopf problem, that can be solved by means of a canonical factorization of the symbol. In a standard shift semigroup realization this amounts to factorizations of the Riccati operator and the feedback operator into convolution operators and projections. Our approach leads to a new significant discovery: in the case where the impulse response of the system contains discrete time delays, the standard Riccati equation is incorrect; to get the correct Riccati equation the feed-through matrix of the system must be partially replaced by the feed-through matrix of the spectral factor. This means that, before it is even possible to write down the correct Riccati equation, a spectral factorization problem must first be solved to find one of the weighting matrices in this equation.
TL;DR: In this paper, a linear dynamic output feedback controller that robustly stabilizes the uncertain system and achieves the extended strict positive realness property for a given closed-loop transfer function is presented.
TL;DR: In this article, a method for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas, is proposed for the root-finding problem.
Abstract: A method is proposed for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas. This equation has singularities which make the application of standard root-finding methods difficult. In order to solve the equation, a class of transformations of variables is considered, which transform the equation into one for which Newton's method converges from any point in a certain given interval. In addition, the form of the transformed equation suggests a convergence accelerating modification of Newton's method. The same ideas are applied to the secant method and numerical results are presented.
TL;DR: In this article, it is shown how the zero dynamics of spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic Riccati inequality.
Abstract: In this paper it is shown how the zero dynamics of (not necessarily square) spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic Riccati inequality. The notion of output-induced subspace of a minimal Markovian splitting subspace, which is the stochastic analogue of the supremal output-nulling subspace in geometric control theory, is introduced. Through this concept, the analysis can be made coordinate-free and straightforward geometric methods can be applied. It is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal Markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces. Finally, the well-known characterization of the solutions of the algebraic Riccati equation is generalized in terms of Lagrangian subspaces invariant under the corresponding Hamiltonian to the larger solution set of the algebraic Riccati inequality.
TL;DR: In this paper, a necessary and sufficient condition for the solution of the time-invariant Riccati differential equation to converge towards the strong solution of corresponding algebraic Riccaci equation, without assuming that the Hamiltonian matrix has no eigenvalues on the imaginary axis (or equivalently that there are no critical unobservable modes).
TL;DR: In this paper, the authors consider a general class of systems subject to two types of uncertainty: a continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain.
TL;DR: An ℓ2-approach permits the optimal control problem to be reduced to a norm minimization one in Hilbert spaces and the singularities of the 2D Riccati equation are examined to characterize suboptimal control laws that apply whenever the solvability conditions are not satisfied.
Abstract: The infinite horizon optimal control problem is solved for 2D systems described by the Fornasini-Marchesini model. An l2-approach permits us to reduce the optimal control problem to a norm minimization one in Hilbert spaces. Both necessary and sufficient conditions for solvability and the structure of the solution are established. Moreover, a comparison with known results is presented, and the singularities of the 2D Riccati equation are examined in order to characterize suboptimal control laws that apply whenever the solvability conditions are not satisfied.
TL;DR: A bound for the degree of the extension C ⊃ Q is given for the second-order differential equation y ″ + ay ′ + by = 0 with a, b ϵ Q ( x).
TL;DR: In this article, an ansatz motivated by the classical form of el phi, where phi is the angle variable, was used to construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator.
Abstract: Using an ansatz motivated by the classical form of el phi , where phi is the angle variable, we construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator. The matrix elements of these operators can be expressed in terms of entire functions in the position complex plane. These functions provide solutions of the Ricatti equation associated with the time-independent Schrodinger equation. We relate the normalizability of the eigenstates to the global properties of the flows of this equation. These exact results yield approximations which complement the WKB approximation and allow an arbitrarily precise determination of the energy levels. We give numerical results for the first 10 levels with 30 digits. We address the question of the quantum integrability of the system.
TL;DR: A procedure of designing a linear (decentralized) feedback control law which will stabilize a given symmetric composite system with uncertainty is presented by using the Riccati equation approach.
01 Jan 1995