A Floquet-like factorization for linear periodic systems

Sample and hold functions have been described for periodic linear system control applications where they are used to assign the eigenvalues of the system monodromy matrix. If the system is controllable in one period it was shown that the complete monodromy matrix can be assigned. In this paper we apply these methods to non periodic continuous time linear systems. When the system is controllable across each sampling interval, the state transition matrix associated with each interval can be completely assigned with full state feedback. Thus the resulting closed loop system resembles a closed loop periodic system whose monodromy matrix is completely specified.

In the former part of this paper, a trace formula is established for the H 2 norms of a class of finite-dimensional linear continuous-time periodic (FDLCP) systems based on the solution of the harmonic Lyapunov equations. This trace formula is quite similar in form to what we have for the H 2 norm of an LTI continuous-time system apart from the fact that infinite-dimensional matrices are involved in the FDLCP setting. Based on this formula, in the latter part of this paper some trace formulas are developed via the approximate modelling and truncating approach, which are numerically implementable in most practical FDLCP systems. There are numerical examples to illustrate the efficacy of the trace formulas suggested.

The relation between the classical G. Floquet characteristic exponents (1879) and entities called ED (essential D) eigenvalues and PED (primary essential D) eigenvalues for specified scalar periodic linear systems is established. In particular, some important results on the existence and uniqueness of periodic ED-eigenvalues and periodic PED-eigenvalues for these systems are obtained, and it is shown that the mean value of periodic PED-eigenvalues for these systems are their classical Floquet characteristic exponents. Using these results, the authors derive two necessary and sufficient stability criteria for F-nonderogatory and F-derogatory scalar periodic linear systems. The derived stability criteria not only serve to unify the stability theory for the general scalar time-varying linear system, but can also be used to develop novel techniques for evaluating and/or approximating the Floquet characteristic exponents for scalar periodic linear systems. >

This article deals with the problem of distributed filtering of periodic systems over sensor networks under event-triggered communications, based on a piecewise impulsive method. The continuous-time periodic system is approximated by several linear time-invariant subsystems through a piecewise constant method. The sensor network deployed to the periodic system is taken into account for the actual needs of monitoring or estimation. A dynamic event-triggering mechanism is designed to reduce the unnecessary data transmission over the wireless network from the sensor to the filter as well as between each filter. Distributed peak-to-peak filters with periodic time-varying parameters are designed under the event-triggering mechanism to estimate the networked continuous-time periodic systems with the bounded and local differentiable disturbance. Then, a periodic piecewise filtering error system is modeled as an impulsive periodic piecewise system, for which sufficient conditions of a guaranteed peak-to-peak performance are proposed. The gains of the filters are obtained by the method of convex linearization. The validity of the proposed results is demonstrated by a pendulum tracking system.

ABSTRACTIn this paper, the finite-time stability, stabilisation, L2-gain and H∞ control problems for a class of continuous-time periodic piecewise linear systems are addressed. By employing a time-varying control scheme in which the time interval of each subsystem constitutes a number of basic time segments, the finite-time controllers can be developed with periodically time-varying control gains. Based on a piecewise time-varying Lyapunov-like function, a sufficient condition of finite-time stability and the relevant time-varying controller are proposed. Considering the finite-time boundedness of the closed-loop periodic system, the L2-gain criterion with continuous time-varying Lyapunov-like matrix function is studied. A finite-time H∞ controller is proposed based on the L2-gain analysis. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed criteria.

Guaranteed cost control of periodic piecewise linear time-delay systems

Rotational Motion Control by Feedback with Minimum L1-Norm

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.

The problem of intermittent fault (IF) detection is studied for a class of linear discrete-time stochastic systems with unknown disturbance under the influence of a class of networked phenomena by using the moving horizon estimation (MHE). Aiming to normalise the transmission of data in network systems, the Round-Robin (RR) protocol is introduced for linear discrete-time stochastic systems, so the system is converted to a linear periodic system. By using the lifting technique and superposition principle, the linear periodic discrete-time system is transformed into a linear discrete time-invariant system. An MHE strategy is developed to cope with unknown disturbance by introducing provisional estimates of external unknown disturbances. By utilizing the moving horizon estimator, a scalar residual decoupled from the unknown disturbance is designed. Finally, the effectiveness of the proposed method is verified by a simulation example.

Constrained stabilization of periodic discrete-time systems via periodic Lyapunov functions

Stability of Periodic Linear Systems and the Geometry of Lie Groups

Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

In this paper, the classical notion of a conditioned invariant subspace is extended to the case of linear periodic discrete-time systems, properly characterized and suitably specialized to the notion of an outer reconstructible subspace. These ideas are used in order to find necessary and sufficient conditions for the existence of a linear function dead-beat observer for such systems under unknown and unmeasurable disturbances and a synthesis procedure for such an observer.

In this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete-time systems, and for general classes of disturbance functions and reference signals, consisting in free responses of periodic processes. The necessary and sufficient conditions for the existence of a solution, which involve the periodic notion of invariant zero, are found. A constructive design procedure of the periodic compensator is given; it is based on the internal model principle.

The classical row-by-row decoupling problem with stability by the use of a dynamic compensator is studied for linear periodic discrete-time systems. The necessary and sufficient conditions for the existence of a solution are found, and a design procedure of the dynamic compensator (including a feedback subcompensator and a pre-subcompensator) is provided by the sufficiency proof, also for the case of a dead-beat requirement for the state free response.