Showing papers on "Riccati equation published in 1994"

Observer design for a class of nonlinear systems

Abstract: A viable design methodology to construct observers for a class of nonlinear systems is developed. The proposed technique is based on the off-line solution of a Riccati equation, and can be solved using commercially available software packages. For globally valid results, the class of systems considered is characterized by globally Lipschitz nonlinearities. Local results relax this assumption to only a local requirement. For a more general description of nonlinear systems, the methodology yields approximate observers, locally. The proposed theory is used to design an observer for a single-link flexible joint robot. This observer estimates the robot link variables based on the joint measurements.

Numerical solution of the discrete-time periodic Riccati equation

Abstract: In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with. >

Solution of physical problems by decomposition

Abstract: Recent generalizations are discussed and results are presented for the theory and applications of the decomposition method. Application is made to the Duffing equation with an error of 0.0001% in only four terms and less than 10−16 in thirteen terms of the decomposition series. Application is also made to a dissipative wave equation, a matrix Riccati equation, and advection-diffusion nonlinear transport.

Characterization of the ${\cal L}_2$-Induced Norm for Linear Systems with Jumps with Applications to Sampled-Data Systems

Abstract: This paper considers a continuous-time linear system with finite jumps at discrete instants of time. An iterative method to compute the ${\cal L}_2$-induced norm of a linear system with jumps is presented. Each iteration requires solving an algebraic Riccati equation. It is also shown that a linear feedback interconnection of a continuous-time finite-dimensional linear time-invariant (FDLTI) plant and a discrete-time finite-dimensional linear shift-invariant (FDLSI) controller can be represented as a linear system with jumps. This leads to an iterative method to compute the ${\cal L}_2$-induced norm of a sampled-data system.

Multiscale systems, Kalman filters, and Riccati equations

Abstract: An algorithm analogous to the Rauch-Tung-Striebel algorithm/spl minus/consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step/spl minus/was developed previously by the authors (ibid. vol.39, no.3, p.464-78 (1994)). In this paper they present a detailed system-theoretic analysis of this filter and of the new scale-recursive Riccati equation associated with it. While this analysis is similar in spirit to that for standard Kalman filters, the structure of the dyadic tree leads to several significant differences. In particular, the structure of the Kalman filter error dynamics leads to the formulation of an ML version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the Hamiltonian for the smoothing problem. In addition, the notion of stability for dynamics requires some care as do the concepts of reachability and observability. Using these system-theoretic constructs, the stability and steady-state behavior of the fine-to-coarse Kalman filter and its Riccati equation are analysed. >

Solution and asymptotic behavior of coupled Riccati equations in jump linear systems

Abstract: A new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled Riccati equations occurring in jump linear systems is derived. By verifying a Riccati inequality it is shown that such a solution exists; in addition two numerical algorithms are given to compute it. An example is given to illustrate the proposed method. >

Missile longitudinal autopilot design using the state-dependent Riccati equation method

Abstract: A missile output feedback pitch autopilot is designed using the state-dependent Riccati equation (SDRE) approach presented in. The particular SDRE design methodology chosen for this paper is referred to as SDRE H{sub 2}. The SDRE H{sub 2} design structure is the same as that of linear H{sub 2}, except that the two Riccati equations are state-dependent. Hence, SDRE H{sub 2} design is a nonlinear extension of linear H{sub 2} design. The procedural steps in the SDRE H{sub 2} design process are presented along with design results.

The Riccati Equation for Optimal Control Problems with Mixed State-Control Constraints: Necessity and Sufficiency

Abstract: The goal of this paper is to conduct a complete study of second-order conditions for the optimal control problem with mixed state-control constraints. The conjugate point theory is presented and a necessary condition in terms of the corresponding Riccati equation is obtained. Sufficiency criteria are developed in terms of strengthened necessary conditions, including the Riccati equation. The results generalize the known ones for pure control constraints as well as for the mixed state-control constraints.

Extensions of mixed-µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory†

Abstract: In this paper we make explicit connections between classical absolute stability theory and modern mixed-μ analysis and synthesis. Specifically, using the parameter-dependent Lyapunov function framework of Haddad and Bernstein and the frequency dependent off-axis circle interpretation of How and Hall, we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty. An immediate application of this framework is the generalization and reformulation of mixed-μ analysis and synthesis in terms of Lyapunov functions and Riccati equations. This observation is exploited to provide robust, reduced-order controller synthesis while avoiding the standard D, N - K iteration and curve-fitting procedures.

The discrete-time Riccati equation related to the H/sub /spl infin// control problem

Abstract: The H/sub /spl infin// control problem has been solved recently with the use of discrete-time algebraic Riccati equations. In this paper, we investigate this Riccati equation. We derive recursive methods to find the stabilizing solution of this Riccati equation. Moreover, we derive several properties of the class of positive semi-definite solutions of this equation. >

Positive definiteness in the numerical solution of Riccati differential equations

Abstract: In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization.

Nonlinear regulation and nonlinear H{sub {infinity}} control via the state-dependent Riccati equation technique: Part 3, examples

Abstract: A little known technique for systematically designing nonlinear regulators is analyzed. The technique consists of first using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficients (SDC). A state-dependent Riccati equation (SDRE) is then solved at each point x along the trajectory to obtain a nonlinear feedback controller of the form u = -R{sup -1}(x)B{sup T}(x)P(x)x, where P(x) is the solution of the SDRE. In the case of scalar x, it is shown that the SDRE approach yields a control solution which satisfies all of the necessary conditions for optimality even when the state and control weightings are functions of the state. It is also shown that the solution is globally asymptotically stable. In the multivariable case, the optimality, suboptimality and stability properties of the SDRE method are investigated. Under various mild assumptions of controllability and observability, the following is shown: (a) concerning the necessary conditions for optimality, where H is the Hamiltonian of the system, H{sub u} = 0 is always satisfied and, under stability, {lambda} = -H{sub x} is asymptotically satisfied at a quadratic rate as the states are driven toward the origin, (b) if it exists, a parameter-dependent SDC parameterization can bemore » computed such that the multivariable SDRE closed loop solution satisfies all of the necessary conditions for optimality for a given initial condition, and (c) the method is locally asymptotically stable. A general nonlinear minimum-energy (nonlinear H{sub {infinity}}) problem is then posed. For this problem, the SDRF, method involves the solution of two coupled state-dependent Riccati equations at each point x along the trajectory. In the case of full state information, again under mild assumptions of controllability and observability, it is shown that the SDRE non-linear H{sub {infinity}} controller is internally locally asymptotically stable.« less

Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems

Abstract: An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solved in this paper. The distributed parameter system is modeled by an evolution equation with an infinitesimal generator for an analytic semigroup. Since there is boundary or point control, the linear transformation for the control in the state equation is also an unbounded operator. The unknown parameters in the model appear affinely in both the infinitesimal generator of the semigroup and the linear transformation of the control. Strong consistency is verified for a family of least squares estimates of the unknown parameters. An Ito formula is established for smooth functions of the solution of this linear stochastic distributed parameter system with boundary or point control. The certainty equivalence adaptive control is shown to be self-tuning by using the continuity of the solution of a stationary Riccati equation as a function of parameters in a uniform operator topology. For a quadratic cost functional of the state and the control, the certainty equivalence control is shown to be self-optimizing; that is, the family of average costs converges to the optimal ergodic cost. Some examples of stochastic parabolic problems with boundary control and a structurally damped plate with random loading and point control are described that satisfy the assumptions for the adaptive control problem solved in this paper.

Iterative matrix bounds and computational solutions to the discrete algebraic Riccati equation

Abstract: Bilateral matrix bounds for the solution of the discrete algebraic Riccati equation (DARE) are presented. They are new or tighter than the existing bound. Computational algorithms to solve the DARE follow. >

Robust linear quadratic designs with real parameter uncertainty

Abstract: This note derives a linear quadratic regulator which is robust to real parametric uncertainty, by using the overbounding method of Petersen and Hollot (1986). The resulting controller is determined from the solution of a single modified Riccati equation. This controller has the same guaranteed robustness properties as standard linear quadratic designs for known systems. It is proven that when applied to a structural system, the controller achieves its robustness by minimizing the potential energy of uncertain stiffness elements, and minimizing the rate of dissipation of energy by the uncertain damping elements. >

Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

Abstract: We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain Ω ź źn, arbitraryn, with boundaryL2-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual "Finite Cost Condition" and "Detectability Condition," the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameterź, here calledźc (criticalź),źcź0, such that a complete theory is available forź >źc, while the maximization problem does not have a finite solution if 0 <ź <źc. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theHź-robust stabilization problem with partial observation.

Time-Varying Discrete Linear Systems: Input-Output Operators, Riccati Equations, Disturbance Attenuation

Abstract: 0. General motivation.- 1. Evolutions and related basic notions.- 2. Nodes.- 3. Riccati equations and nodes.- 4. Disturbance Attenuation.- Appendix A. Discrete-time stochastic control.- 1. Discrete-time Riccati equation of stochastic control.- 2. Optimal compensator under independent random disturbances.- Notes and References.- Appendix B. Almost periodic discrete-time systems.- 1. Standeard theory of almost periodic sequences.- 2. A new Bochner theory for almost periodic sequences.- 3. Almost periodic evolution.- 4. Evolutions under Besicovitch sequences.- Notes and References.- References.

The discrete algebraic Riccati equation and linear matrix inequality

Abstract: In this paper we study the discrete time algebraic Riccati equation and its connection to the discrete time linear matrix inequality. We show that in general only a subset of the set of rank-minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic Riccati equation, and study under what conditions these sets are equal. In this process we also derive very weak assumptions under which a Riccati equation has a solution. >

Convergence of the time-invariant riccati differential-equation and lq-problem - mechanisms of attraction.

Abstract: The nature of the attraction of the solution of the time-invariant matrix Riccati differential equation towards the stabilizing solution of the algebraic Riccati equation is studied. This is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis. Various aspects of this convergence are analysed by displaying explicit mechanisms of attraction, and connections are made with the literature. The analysis ultimately shows the exponential nature of the convergence of the solution of the Riccati differential equation and of the related finite horizon LQ-optimal state and control trajectories as the horizon recedes. Computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite-horizon LQ problem by the solution of an infinite-horizon LQ problem.

Consideration of plant uncertainties in the optimum structural-control design

Abstract: An integrated design approach for the optimum design of a controlled structure is presented. The control design method takes into consideration both structured and unstructured uncertainties by improving a bound on the H∞ norm of the closed-loop system and satisfying a constraint on the linear quadratic Gaussian performance index. The controller is designed by solving three Riccati equations. The weight of the structure is specified as an objective function with constraints on the structural frequencies, the closed-loop damping, and the performance index. The numerical results are presented for a three-dimensional truss structure to illustrate the application of the integrated approach

Transformation and Symbolic Calculations in Filtering and Control

Abstract: This thesis deals with several aspects of optimal control and filtering problems. A fundamental problem in the optimal control theory is the design of a regulatorfora linear system which minimizes a quadratic cost function characterizing the control effort and the deviations of the system from the ideal operation. On the other hand, a basic problem in signal analysis is the optimal estimation of a useful signal from observations corrupted by additive noise. The solutions of both problems depend in a crucial way on solutions of the matrix Riccati differential equation.This work is an effort to unify the theoretical analysis and the numerical or symbolic calculation of solutions of the Riccati differential equation (RDE) as well as the algebraic Riccati equation (ARE) via a matrix transformation. It is shown that the most important issues evolving around the Riccati equation solution can be completely chara:cterized by submatrices of a transformation matrix. Not only the necessary and sufficient conditions for the existence of a solution but also an explicit expression of the solution are obtained from this computable matrix transformation. The transformation matrix can be calculated whether the solution exists or not. It can clearly be seen from this transformation matrix that the solution of the ARE can be explicitly expressed via the submatrices even if the system is not stabilizable (in the optimal filtering context). Furthermore, the Riccati differential equation can also be solved analytically in terms of these submatrices. The criterion, which ensures a solution of the Riccati differential equation to converge to the stabilizing solution, or the strong solution, are established via the same transformation matrix and with a more relaxed requirement on the initial condition than existing results. The criterion is proven to be sufficient and necessary, thus extending existing convergence results.The technique of the matrix transformation is shown to be very useful for exploring the Riccati equation associated with linear time varying systems. We show that an important class of linear time varying systems can be transformed using an appropriate time varying matrix transformation to a linear time invariant form. Hence, instead of attempting to solve the RDE with time varying coefficients, its time invariant correspondence may be solved symbolically.In applying the theoretical analysis for practical systems, a central question is how well-established theories for linear systems can be applied to practical nonlinear systems. As an effort to bridge the gap, we extend linear optimal filtering to some nonlinear problems. The technique proposed in the thesis is applied to solve the problem of transformer saturation in protective relaying for power systems. Simulation results are provided to illustrate the accuracy of the method.

Robust stabilization of uncertain linear dynamical systems with time-varying delay

Abstract: In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh − Some analytical methods are employed to investigate these stability conditions Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper

Lectures on Discrete Time Filtering

Abstract: This text is based on a course given at the University of Southern California, at the University of Nice, and at Cheng Kung University in Taiwan. It discusses linear and nonlinear sequential filtering theory: that is, the problem of estimating the process underlying a stochastic signal. For the linear coloured-noise problem, the theory is due to Kalman, and in the case of white noise it is the continuous Kalman-Bucy theory. The techniques considered have applications in fields as diverse as economics (prediction of the money supply), geophysics (processing of sonar signals), electrical engineering (detection of radar signals), and numerical analysis (in integration packages). The nonlinear theory is treated thoroughly, along with some novel synthesis methods for this computationally demanding problem. The author also discusses the Burg technique, and gives a detailed analysis of the matrix Riccati equation.