Generalized Sampled State Hold Functions Applied to Feedback Control of Non Periodic Linear Systems

CRONE control of continuous linear time periodic systems: Application to a testing bench

In this note, the novel representation is proposed for a linear periodic continuous-time system with T-periodic real-valued coefficients. We prove that a T-periodic real-valued factor and two real-valued matrix exponential functions can be extracted from a state transition matrix, while, in the well-known Floquet representation theorem, a 2T-periodic real-valued factor and a real-valued matrix exponential function are extracted from the state transition matrix. Then we also proved that any T-periodic system can be transformed to a system with T-periodic real-valued trigonometric coefficients using a T-periodic real-valued coordinate transformation, while, in the well-known Lyapunov reducibility theorem, a 2T-periodic real-valued coordinate transformation is utilized to transform the given periodic system into a time-invariant system with real coefficients. This new information can be useful for designing a T-periodic control law.

For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here, a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists. It is determined by the spectral subspaces of the homogeneous part which is a periodic linear differential equation. Using the Poincaré sphere, one obtains a compactification of the state space allowing us to describe the behavior “near infinity” of the original control system. Furthermore, an application to quasi-affine systems yields a unique control set with nonvoid interior.

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. >

Guaranteed cost control of periodic piecewise linear time-delay systems

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

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.

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.

Robust time-weighted guaranteed cost control of uncertain periodic piecewise linear systems

In this article stabilization of fractional-order periodic discrete-time linear system by a complete nonlinear parametric approach for pole assignment with respect to stability and reachability of system via state feedback is proposed. To modify the dynamic response of linear system we should change the poles of state feedback matrices. Nonlinear parametric feedback matrix makes some freedoms in conditions such as minimum norm of feedback matrix. But it makes large monodromy matrix and changing all of eigenvalues makes some problems. Reassigning a part of bad spectrums, leaving the rest of the spectrums invariant, we have lower-ordered matrix to modify the dynamic response of linear system and we can make nonlinear parametric for this new lower-ordered matrix by less expenses and better stability conditions. Numerical examples illustrate the effectiveness of the proposed approaches.

We consider a linear periodic control system with zero mean of the coefficient matrix and with linear state feedback control periodic with the same period. We obtain necessary and sufficient conditions for the solvability of the frequency spectrum control problem with a given goal set for strongly irregular periodic vibrations. In this problem, one should find a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.

A linear control system with a periodic matrix of coefficients and program control is considered. The matrix under control is constant, rectangular (the number of columns does not exceed the number of rows) and its rank is not maximum, i. e. less than the number of columns. It is assumed that the control is periodic, and the module of its frequencies, i. e. the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. The following task is posed: to construct such a control from an admissible set that switches the system to asynchronous mode, i. e. the system must have periodic solutions such that the intersection of the frequency moduli of the solution and the coefficient matrix is trivial. The problem posed is called the problem of synthesis of asynchronous mode. The solution to the formulated problem significantly depends on the structure of the average value of the coefficient matrix. In particular, its solution is known for systems with zero average. In addition, solvability conditions were obtained in the case when the matrix under control has zero rows, the averaging of the coefficient matrix is reduced to the form with upper left diagonal block and with zero remaining blocks. In this paper we consider a more general case with a nontrivial left lower block. Assuming an incomplete column rank of the matrix function composed from the rows of oscillation path of the coefficient matrix, we construct the control explicitly. This control switches the system to asynchronous mode.

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 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.

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.