Showing papers on "Riccati equation published in 1996"
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01 Dec 1996TL;DR: In this paper, the authors present a generalization of the Hamilton-Jacobi theory for systems on Lie groups and homogenous spaces, including linear and polynomial control systems with quadratic costs.
Abstract: Introduction Acknowledgments Part I. Reachable Sets and Controllability: 1. Basic formalism and typical problems 2. Orbits of families of vector fields 3. Reachable sets of Lie-determined systems 4. Control affine systems 5. Linear and polynomial control systems 6. Systems on Lie groups and homogenous spaces Part II. Optimal Control Theory: 7. Linear systems with quadratic costs 8. The Riccati equation and quadratic systems 9. Singular linear quadratic problems 10. Time-optimal problems and Fuller's phenomenon 11. The maximum principle 12. Optimal problems on Lie groups 13. Symmetry, integrability and the Hamilton-Jacobi theory 14. Integrable Hamiltonian systems on Lie groups: the elastic problem, its non-Euclidean analogues and the rolling-sphere problem References Index.
1,066 citations
TL;DR: In this article, the problem of reliable computation of H"~ controllers given a solution (R, S) of the characteristic system of linear matrix inequalities (LMI) is addressed.
434 citations
TL;DR: An algorithm for iterative learning control is developed on the basis of an optimization principle which has been used previously to derive gradient-type algorithms and has numerous benefits which include realization in terms of Riccati feedback and feedforward components.
Abstract: An algorithm for iterative learning control is developed on the basis of an optimization principle which has been used previously to derive gradient-type algorithms. The new algorithm has numerous benefits which include realization in terms of Riccati feedback and feedforward components. This realization also has the advantage of implicitly ensuring automatic step size selection and hence guaranteeing convergence without the need for empirical choice of parameters. The algorithm is expressed as a very general norm optimization problem in a Hilbert space setting and hence, in principle, can be used for both continuous and discrete time systems. A basic relationship with almost singular optimal control is outlined. The theoretical results are illustrated by simulation studies which highlight the dependence of the speed of convergence on parameters chosen to represent the norm of the signals appearing in the optimization problem.
308 citations
TL;DR: In this paper, the authors take a new approach to the problem of peak-to-peak gain minimization by minimizing the * -norm, the best upper bound on the induced L∞ norm obtainable by bounding the reachable set with inescapable ellipsoids.
Abstract: In this paper we take a new approach to the problem of peak-to-peak gain minimization (the L1 or induced L∞ problem). This is done in an effort to circumvent the complexity problems of other approaches. Instead of minimizing the induced L∞ norm, we minimize the * -norm, the best upper bound on the induced L∞ norm obtainable by bounding the reachable set with inescapable ellipsoids. Controller and filter synthesis for * -norm minimization reduces to minimizing a continuous function of a single real variable. This function can be evaluated, in the most complicated case, by solving a Riccati equation followed by an LMI eigenvalue problem. We contend that synthesis is practical now, but a key computational question-is the function to be minimized convex?—remains open. The filters and controllers that result from this approach are at most the same order as the plant, as in the case of LQG and H∞ design.
284 citations
TL;DR: An alternative formulation of a randomized optimal controller which depends on a solution of a functional equation with a simpler structure than general dynamic programming equations is presented which leads to a simpler form of design equations.
191 citations
TL;DR: In this paper, the authors considered the propagation of acoustic waves in waveguides with variable cross section using multimodal decomposition and derived an infinite matricial Riccati equation for the impedance matrix.
Abstract: The propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition. The approach adopted is to construct two infinite first‐order differential equations for the components of the pressure and the velocity projected over the normal modes. From these an infinite matricial Riccati equation is derived for the impedance matrix. These equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes. The stiffness of the pressure‐velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide. The method is checked using different examples where the solutions of the plane‐wave approximation or the finite element method are known. Results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare.
178 citations
TL;DR: This work considers coupled Riccati equations that arise in the optimal control of jump linear systems and shows how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs).
Abstract: We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations that arise, e.g., in the optimal control of linear systems subject to state-dependent multiplicative noise. Some nonstandard Riccati equations (such as those connected to linear systems subject to both state- and control-dependent multiplicative noise) are not amenable to the method. We show that we can still use LMI optimization to compute the optimal control law for the underlying control problem without solving the Riccati equation.
161 citations
TL;DR: In this article, asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior are presented. But their results are not applicable to non-Gaussian initial conditions.
Abstract: Consider the problem of estimation of a diffusion signal observed in additive white noise. If the solution to the filtering equations, initialized with an incorrect prior distribution, approaches the true conditional distribution asymptotically in time, then the filter is said to be asymptotically stable with respect to perturbations of the initial condition. This paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior. For the linear case, stability of the Riccati equation of Kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non-Gaussian initial conditions. In the nonlinear case, asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution.
138 citations
11 Dec 1996
TL;DR: In this paper, a finite difference approximation scheme is proposed to solve the nonlinear matrix inequalities (NLMIs) and the state-dependent Riccati equation for analysis and synthesis.
Abstract: The methods of frozen Riccati equations and nonlinear matrix inequalities (NLMIs) offer certain computational advantages over Hamilton-Jacobi equations (HJE), but may fail to be optimal. The frozen Riccati method uses a non-unique state-dependent linear representation to reduce the HJE to a state-dependent Riccati equation. While there usually exists some choices that recover the optimality, it may be difficult to find. The NLMI computation for analysis and synthesis is shown to be as hard as Lyapunov stability analysis; a finite difference approximation scheme is proposed to solve the NLMIs.
137 citations
TL;DR: In this article, the authors show how an ordinary differential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduced equation via a pair of quadratures and a linear second order equation.
133 citations
TL;DR: Using a Razumikhin approach for the stability of functional differential equations, upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable, in the case of constrained input, via memoryless state feedback control laws.
Abstract: A class of uncertain time-delay systems containing a saturating actuator is considered. These systems are characterized by delayed state equations (including a saturating actuator) with norm-bounded parameter uncertainty (possibly time varying) in the state and input matrices. The delay is assumed to be constant bounded but unknown. Using a Razumikhin approach for the stability of functional differential equations, upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable, in the case of constrained input, via memoryless state feedback control laws. These bounds are given in terms of solutions of appropriate finite dimensional Riccati equations.
TL;DR: In this article, the authors consider the matrix equation X = Q + NX −1 N∗, where the Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil.
15 Sep 1996
TL;DR: In this article, a new nonlinear filter referred to as the state-dependent Riccati equation filter (SDthis article) is presented, which is derived by constructing the dual of a little known nonlinear regulator control design technique which involves the solution of a state-dependent RICE (SDRE) and which has been appropriately called the SDRE control method.
Abstract: A new nonlinear filter referred to as the state-dependent Riccati equation filter (SDREF) is presented. The SDREF is derived by constructing the dual of a little known nonlinear regulator control design technique which involves the solution of a state-dependent Riccati equation (SDRE) and which has been appropriately called the SDRE control method. The resulting SDREF has the same structure as the continuous steady-state linear Kalman filter. In contrast to the linearized Kalman filter (LKF) and the extended Kalman filter (EKF) which are based on linearization, the SDREF is based on a parameterization that brings the nonlinear system to a linear structure having state-dependent coefficients (SDC). In a deterministic setting, before stochastic uncertainties are introduced, the SDC parameterization fully captures the nonlinearities of the system, It was shown in Cloutier et al. (1996) that, in the multivariable case, the SDC parameterization is not unique and that the SDC parameterization itself can be parameterized. This latter parameterization creates extra degrees of freedom that are not available in traditional filtering methods. These additional degrees of freedom can be used to either enhance filter performance, avoid singularities, or avoid loss of observability. The main intent of this paper is to introduce the new nonlinear filter and to illustrate the behaviorial differences and similarities between the new filter, the LKF, and the EKF using a simple pendulum problem.
TL;DR: Comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H/sub 2//H/sub /spl infin//-type problems are presented.
Abstract: Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H/sub 2//H/sub /spl infin//-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples.
TL;DR: A robust control design method for parameter uncertain systems that have delay in both state and control input is presented and a state feedback controller is obtained that guarantees a stability for parameter uncertainties and delays.
01 Sep 1996
TL;DR: In this paper, a robust decentralised load-frequency controller based on the Riccati-equation approach is proposed for multi-area power systems with parametric uncertainties, which consists of N local robust loadfrequency controllers.
Abstract: A robust decentralised load-frequency controller, based on the Riccati-equation approach, is proposed for multi-area power systems with parametric uncertainties. It comprises N local robust load-frequency controllers for a N-area power system. N interlinked Riccati equations are produced initially which are separated via a decoupling technique. Bounds of system parametric uncertainties are included in these Riccati equations to improve the robustness of the intended controller. One local robust load-frequency relationship is obtained by solving the corresponding decoupled Riccati equation. It operates on its local measurements; feedback from other areas is not needed. The overall system is asymptotically stable, for all admissible system parametric uncertainties, when all the local load-frequency controllers are working together. Good performance is reported from the studied three-area power system even in the presence of the generation-rate constraint.
TL;DR: It is shown that a linear discrete-time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open-loop poles are located inside or on the unit circle.
Abstract: We revisit the problem of semiglobal stabilization of linear discrete-time systems subject to input saturation and give an algebraic Riccati equation (ARE)-based approach to the proof of a fact established earlier (Lin and Saberi, 1995), i.e. a linear discrete-time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open-loop poles are located inside or on the unit circle. Moreover, we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work.
TL;DR: In this paper, the relations of a number of bounds for the solutions of the algebraic Riccati and Lyapunov equations that have been reported during the last two decades are investigated.
Abstract: This paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic Riccati and Lyapunov equations that have been reported during the last two decades. Also presented are bounds for the unified Riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete Riccati equations can be unified by them.
TL;DR: In this article, the General Dynamic Equation for aerosol evolution is converted into a set of ordinary differential equations for the moments Mm by multiplying by vm and integrating over particle volume, v. Closure of these equations is achieved by assuming a functional form for the moment, instead of the usual assumption of a functional shape for the size distribution itself.
TL;DR: In this paper, the authors provide results concerning the optimal feedback control of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for their work.
Abstract: In this paper, we provide results concerning the optimal feedback control of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for our work. This system consists of two coupled PDEs exhibiting hyperbolic and parabolic characteristics, respectively, with the control action being modeled by a highly unbounded operator. We rigorously justify an optimal control theory for this class of problems and further characterize the optimal control through a suitable Riccati equation. This is achieved in part by exploiting recent techniques in the area of optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analysis which considers the propagation of singularities and optimal trace behavior of the solutions.
TL;DR: In this article, the Laguerre-Hahn forms are defined as those whose formal Stieltjes function satisfies a Riccati equation, and the definition of the class of a lgHahn form is given.
Abstract: We study the Laguerre-Hahn forms, that is to say those whose formal Stieltjes function satisfies a Riccati equation. We give the definition of the class of a Laguerre-Hahn form. We carry out the complete description of the symmetric forms of class s=1. Essentially, three canonical cases appear. Some particular cases are exhibited which refer to well-known orthogonal sequences. In general, the problem of representing these forms is open.
TL;DR: In this article, the authors present a more natural extension of the Weiss singular manifold method, which makes use of only one singular manifold but is capable of dealing with such PDEs, and they allow the possibility that the DT might in fact correspond to an infinite Painleve expansion, for a certain choice of the arbitrary coefficients.
Abstract: For many completely integrable partial differential equations (PDEs) the singular manifold method of Weiss allows the recovery of the Lax pair and Darboux transformation (DT), and so also the Backlund transformation, from a truncated Painleve expansion. Recently the so‐called ‘‘two‐singular manifold method’’ has been proposed in order to handle PDEs such as the modified Korteweg–de Vries (MKdV) equation. Here we present a more natural extension of the Weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such PDEs. In this approach we allow the possibility that the DT might in fact correspond to an infinite Painleve expansion, for a certain choice of the arbitrary coefficients. This then leads us to a new and more consistent definition of ‘‘singular manifold equation’’ (SME); this can give SMEs different from those usually presented. The summation of infinite Painleve expansions is effected by seeking a truncation in a new Riccati variable Z. The use o...
TL;DR: In this paper, a sufficient condition is derived such that the closed-loop state delayed system is stable and guarantees a prescribed H ∞ norm bound of the transfer matrix from the disturbance to the controlled output.
Abstract: A sufficient condition is derived such that the closed-loop state delayed system is stable and guarantees a prescribed H ∞ norm bound of the closed-loop transfer matrix from the disturbance to the controlled output. Based on this derivation, a full-order observer-based H ∞ controller for the stace delayed linear systems is constructed by solving two modified algebraic Riccati equations. The state feedback H ∞ controller for the state delayed linear systems is also obtained by solving a modified algebraic Riccati equation. An illustrative example is given to show the applicability of the proposed approach.
TL;DR: In this paper, a nonlinear H∞ filter is designed such that the induced l 2 norm of the mapping from the noise signal to the estimation error is within a specified bound.
Abstract: This paper deals with the H∞ filtering problem for a class of discrete-time nonlinear systems with or without real time-varying parameter uncertainty and unknown initial state. For the case when there is no parametric uncertainty in the system, we are concerned with designing a nonlinear H∞ filter such that the induced l2 norm of the mapping from the noise signal to the estimation error is within a specified bound. It is shown that this problem can be solved via one Riccati equation. We also consider the design of nonlinear filters which guarantee a prescribed H∞ performance in the presence of parametric uncertainties. In this situation, a solution is obtained in terms of two Riccati equations.
TL;DR: A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented, expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions.
Abstract: In this paper, the problem of pole assignment in a disk by output feedback for continuous or discrete-time uncertain systems is addressed. A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented. This condition is expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions. An output d stabilization algorithm is derived and a controller formula given. Some related problems are also discussed.
TL;DR: In this article, generalized Riccati differential and difference equations obtained from standard RICCati equations by adding a semidefinite perturbation term were investigated and results on the monotonic dependence of the solutions on the coefficients and initial values as well as results on convergence of solutions were given.
TL;DR: In this article, the Riccati equation for the Dirichlet-to-Neumann map is derived from the exact operator factorization of the two-dimensional variable coefficient Helmholtz equation.
Abstract: The operator Riccati equation for the Dirichlet‐to‐Neumann map is derived from the exact operator factorization of the two‐dimensional variable coefficient Helmholtz equation. Numerical schemes are developed for the operator Riccati equation and its variant using a local eigenfunction expansion. This leads to a practical computational method for acoustic wave propagation over large range distances, since the boundary value problem of the Helmholtz equation is reduced to ‘‘initial’’ value problems that are solved by marching in the range. The efficiency and accuracy of the method is demonstrated by numerical experiments including the plane‐parallel range‐dependent waveguide benchmark problem proposed by the Acoustical Society of America.
TL;DR: In this article, the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control, which yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.
Abstract: In this paper the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control. This yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation. The same approach allows to investigate as well smoothness of the value function.
TL;DR: In this article, the size of the set of all H ∞ filters is considered on the basis of the monotonicity of the above Riccati solution, and it is shown that under a certain condition, the degree of freedom of the H- ∞ filter reduces a 1 the optimal H-∞ norm bound.
Abstract: This paper is concerned with the algebraic Riccati equations (AREs) related to the H ∞ filtering problem. A necessary and sufficient condition for the H ∞ problem to be solvable is that the H ∞ ARE has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a stabilizing solution is a monotonically non-increasing convex function of the prescribed H ∞ norm bound γ. This property of the H ∞ ARE is very important for the analysis of the performance of the H ∞ filter. In this paper, the size of the set of all H ∞ filters is considered on the basis of the monotonicity of the above Riccati solution. It turns out that, under a certain condition, the degree of freedom of the H ∞ filter reduces a1 the optimal H ∞ norm bound. These results provide a guideline for selecting the value of γ Some numerical examples are included.