Global aspects of the matrix Riccati equation

Matrix Riccati equations are interpreted as differential equations on Grassman manifolds. Necessary conditions for the Riccati equation to be a Morse-Smale system are given in the autonomous and periodic cases. Under this condition, the equation is structurally stable and has a unique asymptotically stable equilibrium point or periodic solution.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Nonsymmetric Riccati equations: Partitioned algorithms

Generalized partitioned solutions (GPS) of nonsymmetric matric Riccati equations are presented in terms of forward and backward time differential equations that are of theoretical interest and also are computationally powerful. The GPS are the natural framework for the effective change of initial conditions, and the transformation of backward Riccati equation to forward Riccati equation and vice versa.Based on the GPS, computationally effective algorithms are obtained for the numerical solution of Riccati equations. These partitioned numerical algorithms have a decomposed or “partitioned” structure. They are given exactly in terms of a set of elemental solutions which are completely decoupled, and as such computable in either a parallel or serial processing mode. The overall solution is given exactly in terms of a simple recursive operation on the elemental solutions. Except for a subinterval of the total computation interval, the partitioned numerical algorithms are integration-free for the Riccati equation with constant or periodic matrices.Most importantly based on the GPS, a computationally attractive numerical algorithm is obtained for the computation of the steady-state solution of time-invariant Riccati equations. By making use of the GPS and some simple iterative operations, the Riccati solution is obtained in an interval which is twice as long as the previous interval requiring integration only in the initial subinterval.     

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.     

Numerical solution of the discrete-time periodic Riccati equation

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     

New explicit exact solutions for the generalized coupled Hirota–Satsuma KdV system

In this paper, we study the generalized coupled Hirota–Satsuma KdV system by using the two new improved projective Riccati equations method. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and combined formal solitary wave solutions and combined formal periodic wave solutions are obtained.     

Difference games with periodic information structures with application to the worst case design of regulators†

Nash equilibrium strategies of general linear-quadratic two-player difference games with two kinds of periodic information structures are considered. Solution algorithms are developed for problems where the players' information is of periodic open-loop or periodic open-closed type. In the former case the players receive measurements of the state periodically only at the beginning of each interval and in the latter case one of the players has a perfect memory information of the state within each period. The solutions are obtained by recursive algorithms where a series of coupled difference equations of Riccati type are solved repeatedly and where the boundary values of these equations are determined by similar difference equations. A new game theoretic worst case design method based on games with periodic open-closed information structure is then proposed and applied to the design of a state regulator for a pilot process. The results obtained in the example suggest that this new approach can be successfully employed in practical design problems.     

Controllability, stabilizability, and matrix Riccati equations for periodic systems

In this note, controllability, stabilizahility, and related concepts for periodically time-varying systems are discussed. Especially, it is proved that the definition of controllability employed in connection with the existence of periodic solutions of periodic matrix Riccati equations is equivalent to Kalman's original definition.     

Dampening controllers via a Riccati equation approach

An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed-loop system dynamics. In other words, a feedback gain matrix is computed such that the eigenvalues of the closed-loop state matrix are within the region of the left half-plane where the magnitude of the real part of each eigenvalue is greater than that of the imaginary part, This may be accomplished by solving a damped algebraic Riccati equation and a degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms, Damped Riccati equations are unusual in that they may be formulated as an invariant subspace problem of a related periodic Hamiltonian system. This periodic Hamiltonian system induces two damped Riccati equations: one with a symmetric solution and another with a skew symmetric solution. These two solutions result in two different state feedbacks, both of which dampen the system dynamics, but produce different closed-loop eigenvalues, thus giving the controller designer greater freedom in choosing a desired feedback     

Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations,II

In this paper, elementary techniques from linear algebra and elementary properties of the Grassmann manifolds are used to prove the existence of periodic orbits and to study the equilibrium structure of Riccati differential equations.Supported in part by NASA Grants #2384 and NAG-82 and DOE Contract #DE-AC01-80RA-5256Supported in part by NASA Grant #NSG-2402, ARMY Grant #ILIG1102RHN7-05 and the National Science Foundation.     

Periodic solutions of periodic Riccati equations

For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are shown to be sufficient for the existence of a unique positive definite periodic solution.     

Periodic Riccati difference equations: approaching equilibria byimplicit systems

A parameterization of solutions to the periodic Riccati difference equation (RDE) is given. By using behavioral tools the moving equilibria of the periodic RDE are encompassed among the trajectories of the implicit AR model associated to the symplectic pencil of the corresponding Hamiltonian equations. This allows a parameterization of solutions of periodic RDEs even in the case the dynamic matrix is singular     

Periodic strong solution for the optimal filtering problem oflinear discrete-time periodic systems

The periodic Riccati difference equation (PRDE) for the optimal filtering problem of linear periodic discrete-time systems is addressed. Specifically, the author provides a number of results on the existence, uniqueness, and stability properties of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation in the case of nonreversible and nonstabilizable periodic systems. The convergence of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation is also analyzed. The results have been established under weaker assumptions and include both necessary and sufficient conditions. The existence and properties of symmetric periodic nonnegative-definite solutions of the PRDE are established directly from the PRDE     

Cost translation and a lifting approach to the multirate LQGproblem

It is shown how to translate an instance of a multirate sampled-data LQG problem into an equivalent, modified, single-rate, shift-invariant problem via a lifting isomorphism approach. Using this approach, one can solve the multirate LQG problem without using periodic system theory or solving periodic Riccati equations and without suffering any increases in state dimension. This translation procedure shows the correct way to translate RMS noise specification to the lifted domain for a multirate Q-design computer-aided-design package     

The standard H∞ problem in systems with a singleinput delay

The standard H_∞ problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Lyapunov and Riccati equations: Periodic inertia theorems

The Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered. Under the assumption of detectability of the underlying periodic system, two inertia theorems are provided linking the inertia of the solution to the one of the so-called monodromy matrix.     

Remarks on the periodic orbits of the matrix Riccati equation

We investigate certain questions concerning the periodic structure of the matrix Riccati differential equation with constant coefficients. A closed-form expression for the periodic solutions is obtained for both the cases involving distinct or repeated eigenvalues in the associated linear hamiltonian system. Previous results are extended by establishing that periodic solutions are bounded if and only if the span of their range does not intersect the orthogonal complement of the controllable subspace of the associated linear system.     

F-reduction of the operator Riccati equation for hereditary differential systems

A number of independent treatments of the linear quadratic optimal control problem for retarded systems are available in the literature. Namely (a) using abstract theory of evolution equations in a Hilbert space, (b) via the theory of Fredholm integral equations, (c) via the Bellman-Hamilton-Jacobi equation. These treatments result in different characterizations of the optimal system via complicated Riccati equations. It is shown that by introducing a certain ‘hereditary operator’ F one can characterize more precisely the structure of the solution of the operator Riccati equation. This in turn provides a missing link between the existing theories, and results in a simplification and in some reduction in the Riccati equation.     

State-space solution to the periodic multirate H∞control problem: a lifting approach

A two-Riccati equation solution to the H_∞ control problem for periodic multirate systems is derived via the lifting method. The solution is expressed in terms of two algebraic Riccati equations. The causality constraints are represented by a set of positive definiteness conditions and coupling criteria. As a by-product, the study shows that there is a close connection between the solution of the causally constrained lifted problem and the solution obtained by solving the problem directly as a periodic H_∞ problem defined for the multirate system