Iterative algorithms for discrete periodic Riccati matrix equations

In this paper, two iterative algorithms are constructed to obtain the positive definite solutions of the discrete periodic algebraic Riccati matrix equations. In these two algorithms, the estimation of the unknown matrices are updated by using the available estimation information at the current iteration step. The convergence properties of the proposed algorithms are also given. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithms.     

Controllability, pole placement, and stabilizability for homogenous polynomial systems

In linear system theory the concepts of controllability, pole assignment, and stabilizability are very familiar. These topics are now addressed for the class of homogeneous polynomial systems. First, general results on necessary conditions for the state controllability are derived using results from linear algebra and algebraic geometry. Then pole placement and stabilizability results are developed for the two-dimensional case. Finally, practical examples illustrate the results.     

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.     

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.     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Boundary problems and periodic Riccati equations

We consider the resolution problem of a periodic Riccati equation from the point of view of a generalized boundary problem that lets us reduce the differential problem to an algebraic equation of Riccati type. Sufficient conditions for the existence and calculus of special types of solutions are given.     

Controllability and stabilizability of a networked control system with periodic communication constraints

This paper proposes a new model for a networked control system and considers its controllability and stabilizability. To control a linear time-invariant discrete-time plant via a bus with limited capacity, we introduce a hold device and a communication sequence which follows a given ω-periodic pattern. Incorporating the communication sequences and hold device into the original plant amounts to extending the original time-invariant state equation to a ω-periodic one which has the higher order. We assume that the communication sequence is admissible. It is shown that controllability and stabilizability of the plant are preserved in the periodic extended system under the assumption that the zeros of the communication sequence characteristic polynomial do not coincide with the eigenvalues of the plant.     

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.     

周期性系统的能控性理论

主要探讨周期性系统(包括连续和离散两种情形)的能控性问题，对文献中有关周期为To的连续周期性系统的能控性结论进行了完善，推导出系统能控等价于单周期(0，To)内能控的结论，并将这些结论直接推广到离散周期性系统；然后讨论了离散前后周期性系统能控性保持不变的可能性，证明在采用不等间距采样方式时，离散前后周期性系统的能控性保持不变。     

Multiscale systems, Kalman filters, and Riccati equations

An algorithm analogous to the Rauch-Tung-Striebel algorithm-consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step-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     

Controllability and stabilizability of switched linear-systems

In this paper, we study the conjecture made in (IEEE Trans. Automat. Control 47 (8) (2002) 1401) concerning the existence of basic switching sequence for switched systems, and give it an affirmative answer and a construction method. This paper proves that for switched linear system, there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system. Hence, the controllability can be realized by using only one switching sequence. Then, the result is extended to the systems with time-delay in controls. Furthermore, we define the stabilizability of switched linear systems, and based on certain transformation of state coordinate, we give a necessary and sufficient condition for the stabilizability. Two examples are presented to illustrate the obtained results.     

New Riccati equations for well-posed linear systems

We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative self-adjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the infinite-dimensional generalization of this Riccati equation is not always well-defined. We show that X always satisfies alternative Riccati equations, which are more suitable for algebraic and numerical computations.     

Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations

In this paper, we propose lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations. We give comparisons between the parallel estimates. Finally, we give examples showing that our bounds can be better than the previous results for some cases.     

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.     

Controllability, stabilizability, and continuous-time Markovianjump linear quadratic control

Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem     

Linear matrix inequalities, Riccati equations, and indefinitestochastic linear quadratic controls

This paper deals with an optimal stochastic linear-quadratic (LQ) control problem in infinite time horizon, where the diffusion term in dynamics depends on both the state and the control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem, which may still be well posed due to the deep nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE), which is, however, fundamentally different from the classical algebraic Riccati equation as a result of the indefinite nature of the LQ problem. To analyze the SARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the SARE. Moreover, we develop a computational approach to the SARE via a semi-definite programming associated with the LMIs. Finally, numerical experiments are reported to illustrate the proposed approach     

Controllability and observability of periodic hybrid interval systems

This paper derives necessary and sufficient conditions for periodic hybrid interval systems where the A, B and C system matrices are interval matrices to be controllable and observable. The results are easily extended to impulsive periodic hybrid interval systems and linear discrete-time systems.