Showing papers on "Riccati equation published in 1997"

State-dependent Riccati equation techniques: an overview

Abstract: State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design methods which provide a systematic and effective means of designing nonlinear controllers, observers, and filters. This paper provides an overview of several SDRE design techniques including SDRE nonlinear regulation, SDRE nonlinear H/sub /spl infin//, SDRE nonlinear H/sub 2/, and SDRE nonlinear filtering.

H ∞ -control for Markovian jumping linear systems with parametric uncertainty

Abstract: This paper studies the problem of H∞-control for linear systems with Markovian jumping parameters The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H∞-performance are achieved We also deal with the robust H∞-control problem for linear systems with both Markovian jumping parameters and parameter uncertainties The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix Both the finite-horizon and infinite-horizon cases are analyzed We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case Particularly, robust H∞-controllers are also designed when the jumping rates have parameter uncertainties

Game theory approach to discrete H/sub /spl infin// filter design

Abstract: A finite-horizon discrete H/sub /spl infin// filter design with a linear quadratic (LQ) game approach is presented. The exogenous inputs composed of the "hostile" noise signals and system initial condition are assumed to be finite energy signals with unknown statistics. The design criterion is to minimize the worst possible amplification of the estimation error signals in terms of the exogenous inputs, which is different from the classical minimum variance estimation error criterion for the modified Wiener or Kalman filter design. The approach can show how far the estimation error can be reduced under an existence condition on the solution to a corresponding Riccati equation. A numerical example is given to compare the performance of the H/sub /spl infin// filter with that of the conventional Kalman filter.

The Generalized Emden-Fowler Equation

Abstract: We give the description of nonlinear nonautonomous ordinary differential equations of order n with a so-called reducible linear part. The group classification of generalized Emden-Fowler equations of the mentioned class is done. We have found such laws of the variation of f (x) that the equation admits one, two, or tree one-parameter Lie groups.

Optimal quadratic guaranteed cost control of a class of uncertain time-delay systems

Abstract: The design of robust state feedback controllers for a class of uncertain linear time-delay systems with norm-bounded uncertainty is presented. The state feedback results extend previous results on quadratic guaranteed cost control to the case of uncertain time-delay systems. This is done by the authors' definition of quadratic stability for uncertain time-delay systems with norm bounded uncertainty. It is shown that the state feedback controller can be constructed via the solution of a parameter dependent Riccati equation.

Bounded real lemma and H/sub /spl infin// control for descriptor systems

Abstract: The purpose of this paper is to give a necessary and sufficient condition under which for a given plant of descriptor system model there exists a normal, internally stabilizing controller that has an upper limit on its order and that satisfies a closed-loop H/sub /spl infin// norm bound. The approach used in this paper is based on a generalized version of bounded real lemma, thus the proofs are simple.

Stabilization and estimation for perturbed discrete time-delay large-scale systems

Abstract: Based on a derived algebraic inequality, a criterion to guarantee the robust stabilization and the state estimation for perturbed discrete time-delay large-scale systems is proposed. This criterion is independent of time delay and does not need the solution of a Lyapunov equation or Riccati equation. Furthermore, the same problem for the continuous-time systems is also solved by a much simpler derivation.

Optimal control of stable weakly regular linear systems

Abstract: The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.

On the partial stochastic realization problem

Abstract: We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

General matrix pencil techniques for the solution of algebraic Riccati equations: a unified approach

Abstract: We present a unified theory of matrix pencil techniques for solving both continuous and discrete-time algebraic Riccati equations (AREs) under fairly general conditions on the coefficient matrices. The theory applies to a large class of AREs and Riccati-like equations arising from the singular H/sup /spl infin//- and H/sup 2/-control problems, singular linear quadratic control, the 4-block Nehari problem, or from singular J-spectral factorizations. The underlying concept is the so-called proper deflating subspace of a (possibly singular) matrix pencil in terms of which necessary and sufficient conditions for the solvability of Riccati equations are given. It is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm.

${\cal H}_\infty$ Control and Estimation Problems with Delayed Measurements: State-Space Solutions

Abstract: Most physical processes exhibit transport delay in the measured output, and it is well known that this can have disastrous effects on system stability and performance if it is not accounted for. In this paper, we give necessary and sufficient conditions for existence of estimators and controllers that achieve the desired ${\cal H}_\infty$ performance criterion when such a measurement delay is present. We also give the complete characterization of all controllers and estimators that achieve the desired performance criterion. The necessary and sufficient conditions are easy to check and are given in terms of the familiar pair of algebraic Riccati equations that appear in the nondelay versions of the corresponding ${\cal H}_\infty$ problems, along with an additional Riccati differential equation. Explicit state-space formulas for the controllers and estimators are also obtained. They have a linear periodic structure and are easily implementable. To obtain these results, we first obtain state-space results for a "modified" Nehari problem, which may be of independent interest (see Problem 5 in section 2).

Quadratic optimal control of stable well-posed linear systems

Abstract: We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

Observer design and detection for nonlinear descriptor systems

Abstract: A nonlinear observer is considered for a class of continuous nonlinear descriptor systems subject to unknown inputs and faults. This class is partly characterized by globally Lipschitz nonlinearities, and a member system may be singular and possibly non-causal. The observer structure chosen makes it useful for both state estimation for feedback controls and residual generation for fault detection. Results on the existence of solutions are given and some useful bounds are derived which are important in the observer design, which is based on a transformed system and on the solution of a Riccati equation. The existence, convergence properties and robustness of the observer are investigated by the use of a quadratic Lyapunov function. Conditions are given for a bound to hold on the norm of the transfer function relating the residual error to the disturbance (linear case). A design algorithm is given and applied to the estimation of the states of a flexible joint robot.

Full Envelope Missile Longitudinal Autopilot Design Using the State-Dependent Riccati Equation Method

Abstract: A full envelope missile output feedback pitch autopilot is designed using the state-dependent Riccati equation (SDRE) approach presented in [1]. The particular SDRE design methodology chosen for this paper is referred to as SDRE HZ- The SDRE HI design structure is the same as that of linear HZ, except that the two Riccati equations are state-dependent. Hence, SDRE HZ design is a nonlinear extension of linear HZ design. A full envelope missile model using the same aerodynamics as an earlier work [5] is used to demonstrate the usefulness of the SDRE method for full envelope design. In this paper the SDRE Hz method is breifly discussed. The full envelope model is described and the state dependent coefficient (SDC) parameterization is presented. Finally the SDRE Hz design and design results are presented.

Attitude control of a satellite using the SDRE method

Abstract: Examines the use of nonlinear regulation for attitude control of a satellite controlled by internal momentum rotors. Specifically, the nonlinear method using the state dependent Riccati equation (SDRE) developed by Cloutier, D'Souza, and Mracek (1996) is applied and examined under numerical simulations.

The linear quadratic optimal control problem for linear descriptor systems with variable coefficients

Abstract: We study linear quadratic optimal control problems for linear variable coefficient descriptor systems. Generalization from the case of standard control problems leads to several difficulties. We discuss a behavioral approach that solves some of these difficulties. Furthermore, an analysis of general rectangular systems is given and generalized Euler-Lagrange equations and Riccati differential algebraic equations are discussed.

Sufficient Optimality Conditions for Optimal Control Subject to State Constraints

Abstract: Strong second-order sufficient optimality conditions (SSC) are derived for optimal control problems of systems described by nonlinear ODEs subject to mixed control-state and pure state constraints. The obtained SSC are expressed in terms of a modified Legendre--Clebsch condition and the associated Riccati equation. The role of SSC in stability analysis of solutions to parametric optimal control problems is briefly discussed.

Delay-independent stability of linear neutral systems: A Riccati equation approach

Abstract: This note focuses on the problem of asymptotic stability of a class of linear neutral systems described by differential equations with delayed state. The delay is assumed unknown, but constant. Sufficient conditions for delay-independent asymptotic stability are given in terms of the existence of symmetric and positive definite solutions of a continuous Riccati algebraic matrix equation coupled with a discrete Lyapunov equation. The approach adopted here is based on a Lyapunov-Krasovskii functional technique.

When the telegrapher's equation furnishes a better approximation to the transport equation than the diffusion approximation

Abstract: It has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher’s equation @A.J. Ishimaru, Appl. Opt. 28, 2210 ~1989!#. We show that in one dimension the telegrapher’s equation furnishes an exact solution to the transport equation. In two dimensions, we show that, since the solution can become negative, the telegrapher’s equation will not furnish a usable approximation. A comparison between simulated data in three dimensions indicates that the solution to the telegrapher’s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation. @S1063-651X~97!04205-0#

Constrained Regular LQ-Control Problems

Abstract: In this paper we give a complete description of how the Jacobi theory of conjugate points can be extended to a regular linear-quadratic control problem where the state end-points are jointly constrained to belong to a subspace of ${\bf R}^n \times {\bf R}^n$ and there is a linear pointwise state-control constraint. We introduce also the definition of semiconjugate point which describes a distinctive feature of these problems and state the corresponding necessary and sufficient conditions for the quadratic form to be nonnegative or coercive. In the case in which the constraints and the costs act separately on the initial and final points we give equivalent characterizations of the coercivity of the quadratic form by means of the solutions of an associated Riccati equation in both the controllable and uncontrollable case.

Benchmarks for the numerical solution of algebraic Riccati equations

Abstract: Two collections of benchmark examples are presented for the numerical solution of continuous-time and discrete-time algebraic Riccati equations. These collections may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

Minimum entropy static output-feedback control with an H/sub /spl infin//-norm performance bound

Abstract: The problem of designing static output-feedback controllers is considered. The controller is required to minimize the closed-loop entropy (at s=/spl infin/) and to satisfy a prescribed H/sub /spl infin//-norm bound. The design equations consist of a modified Riccati equation and a Lyapunov equation that are coupled via a projection. These equations are solved using a homotopy algorithm. The design procedure is illustrated by two examples. The first one is a classical example that was solved in the literature using convex optimization techniques. The second example is one of designing a gain-scheduled normal acceleration loop for an air launched unmanned air vehicle.

The sampled-data H problem: A unified framework for discretizationbased methods and Riccati equation solution

Abstract: H A new discretization-based solution to the sampled-data control problem is given. In contrast to previous solution procedures, the method is not based on the lifting technique. Instead, an equivalent finite-dimensional discrete problem representation is derived directly from a description of the sampled-data system. This is achieved via a closed-loop expression of the worst-case intersample disturbance and an associated variable transformation. In this way the solution is obtained completely in terms of classical linear-quadratic optimal control theory. The discretization procedure described here is closely related to both the liftingH based technique and a two-Riccati equation solution of the sampled-data problem, in which the solution is obtained in terms of two coupled Riccati differential equations with jumps. In this way the method makes the connections between the various solution procedures transparent. In particular, it can be shown that the lifting approach and the two-Riccati equation solution...

Receding horizon H tracking control for time-varying discrete linear systems

Abstract: In this paper, a new receding horizon tracking control law based on the H control concept, is proposed for time-varying discrete linear systems. The proposed controller is constructed using a dynamic game approach minimizing a worst case finite horizon performance index with the finite terminal weighting matrix. The conditions on the terminal weighting matrix are proposed under which the proposed control law guarantees closed loop stability and, simultaneously, the infinite horizon H norm bound. It is shown that the proposed stability condition on the terminal weighting matrices can be converted to a Linear Matrix Inequality (LMI). In order to prove closed-loop stability, the monotonicity property of the cost is utilized instead of the monotonicity property of the Riccati equation. It is also shown that the proposed stability condition on both terminal weighting matrices and the cost horizon size are more general than the conventional results. Some examples are included to illustrate the proposed results.

Control of nonlinear systems via state feedback state-dependent Riccati equation techniques

Abstract: : Nonlinear regulation and nonlinear H-infinity control via state-dependent Riccati equation (SDRE) techniques are considered. Relationships between SDREs and Hamilton-Jacobi/Bellman inequalities/equations are examined, and a necessary condition for existence of solutions involving nonlinear stabilizability is derived. A single additional necessary criterion is given for the SDRE methods to yield the optimal control or guaranteed induced L2 gain properties. Pointwise stabilizability and detectability of factorizations prove necessary and sufficient, respectively, for well-posedness of standard numerical implementations of suboptimal SDRE regulators, but neither proves necessary if analytical solutions are allowed. For scalar analytic systems or those with full rank constant control input matrices, stabilizability and nonsingularity of the state weighting matrix function result in local and global asymptotic stability, respectively, due to equivalence between nonlinear and factored controllability in these cases. A proof of asymptotic stability for sampled data analytic SDRE controllers is also given, but restrictive assumptions make the main utility of these results guidance in choosing appropriate system factorizations. Conditions for exponential stability are also derived. All results are extendable to SDRE nonlinear H-infinity control with additional assumptions. The SDRE theory is illustrated by application to momentum control of a dual-spin satellite and comparison with other current methods.

Further results about quadratic stability of continuous-time fuzzy control systems

Abstract: Further results about the quadratic stability of continuous-time fuzzy control systems are presented in this paper. We first point out that the boundary conditions of the results of Cao et al, (1996a) should be explicitly indicated in the theorem. Then, several new results about quadratic stability of continuous-time fuzzy control systems are given. All of these results address the boundary conditions.