On the use of reachability Gramians for the stabilization of linear periodic systems

This note is concerned with stabilization of continuous-time periodic linear (CPL) systems with state feedback. The design is based on solutions to a class of parametric periodic Lyapunov differential equations (PLDEs) resulting from the problem of minimal energy control with guaranteed convergence rate. By carefully studying the properties of the PLDEs and their solutions, a continuous periodic state feedback is designed. The PLDE based approach is effective in designing stabilizing controller for CPL systems as the designers need only to solve a linear differential equation whose solution can be obtained analytically if the system is relative simple and can be computed numerically by integration in general cases. Necessary and sufficient conditions on the free parameter in the PLDE are proposed to guarantee the stability of the closed-loop system whose characteristic multiplier set can even be exactly computed accordingly. A numerical example is worked out to show the effectiveness of the proposed approach.

show abstract

“…It is known from [18] that the !-periodic system (1) is controllable at t 0 if and only if W (t 0 ; t 0 + n!) > 0, where…”

Section: Preliminary Results On Cpl Systemsmentioning

confidence: 99%

“…Moreover, periodic controller can also significantly improve control performance of closed-loop systems (see, for example, [12]). For these reasons, modeling, analysis and control of periodic systems have witnessed a renewed interest in recent years (see [3], [6], [10], [13], [14], [16], [18], [20]- [22] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Periodic Lyapunov Equation Based Approaches to the Stabilization of Continuous-Time Periodic Linear Systems

Zhou

Duan

2012

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

show abstract

“…A similar approach can be developed to solve the adjoint PDLE (2). The solution X(t) at time moments t and t − ∆ are related as [1] …”

Section: B Multi-shot Approachmentioning

confidence: 99%

“…1) Periodic Lyapunov differential equations (PLDE) either in the direct forṁ X(t) = A(t)X(t) + X(t)A T (t) + Q(t) (1) or in the adjoint form −Ẋ(t) = A T (t)X(t) + X(t)A(t) + Q(t) (2) where Q(t) = Q T (t), Q(t) = Q T (t) and A(t), Q(t), and Q(t) are n × n T -periodic matrices (i.e., ∀t A(t+T ) = A(t), Q(t+T ) = Q(t), Q(t+T ) = Q(t)). These equations play an important role in the analysis of controllability/observability of linear continuous-time periodic systems [1], in solving periodic stabilization problems [2], computing Hankel-and H 2 -norms of periodic systems [3], or in solving periodic differential Riccati equations by employing Newtons' method [4].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On solving periodic differential matrix equations with applications to periodic system norms computation

Varga

Proceedings of the 44th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

Periodic Lyapunov, Sylvester and Riccati differential equations have many important applications in the analysis and design of linear periodic control systems. For the numerical solution of these equations efficient numerically reliable algorithms based on the periodic Schur decomposition are proposed. The new multi-shot type algorithms compute periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to traditionally used oneshot periodic generator methods, the multi-shot type methods have the advantage to be able to address problems with large periods and/or unstable dynamics. Applications of the proposed techniques to compute several system norms are presented.

show abstract

On solving periodic Riccati equations

Varga

2008

Numerical Linear Algebra App

View full text Add to dashboard Cite

Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, the multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represent extensions of the periodic QZ algorithm to non-square periodic pairs, whereas the third method represents an extension of a quotient-product swapping and collapsing 'fast' algorithm. All proposed approaches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting matrices.Provided a non-negative N -periodic stabilizing solution X k of the DPRE (5) is known, the periodic state-feedback matrix F k in the optimal control law u * k = F k x k , which minimizes the performance index (7), results asAlternatively, it is possible to determine the periodic generator X 0 := X (0) by solving the algebraic Riccati equation [3]: 21 + 22 X 0 − X 0 11 − X 0 12 X 0 = 0

show abstract

On the use of reachability Gramians for the stabilization of linear periodic systems

Cited by 11 publications

References 17 publications

Periodic Lyapunov Equation Based Approaches to the Stabilization of Continuous-Time Periodic Linear Systems

Periodic Lyapunov Equation Based Approaches to the Stabilization of Continuous-Time Periodic Linear Systems

On solving periodic differential matrix equations with applications to periodic system norms computation

On solving periodic Riccati equations

Contact Info

Product

Resources

About