Zeros of Continuous-time Linear Periodic Systems

Zeros of continuous-time linear periodic systems are defined and their properties investigated. Under the assumption that the system has uniform relative degree, the zero-dynamics of the system is characterized and a closed-form expression of the blocking inputs is derived. This leads to the definition of zeros as unobservable characteristic exponents of a suitably defined periodic pair. The zeros of periodic linear systems satisfy blocking properties that generalize the well-known time-invariant case. Finally, an efficient computational scheme is provided that essentially amounts to solving an eigenvalue problem.     

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.     

一类线性时变不确定周期奇异系统的鲁棒镇定

讨论了一类线性时变不确定周期广义系统的鲁棒镇定问题．基于线性时变不确定周期广义系统的鲁棒稳定的概念, 提出鲁棒稳定的充分必要条件, 并基于对偶系统的等价性得到鲁棒镇定的充分必要条件. 通过引入自由矩阵, 所得结果表示为线性矩阵不等式, 验证过程更简单、可靠.     

Performance limitations in the robust servomechanism problem for discrete time periodic systems

Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for discrete time periodic systems. In studying this problem, the RSP for a multi-input/multi-output discrete time system is considered; application of these results is then made to the “periodic system robust servomechanism problem”, and explicit expressions for the limiting costs for error tracking regulation are obtained. These limitations can be characterized completely by the number and location of the non-minimum phase transmission zeros of the system's associated lifted system, together with a term which depends on the order of the system, the number of inputs, the number of transmission zeros and the system's period.     

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     

Linear SISO systems with extremely sensitive zero structure

If a system with regular system pencil and relative degree greater than one is perturbed, the relative degree will typically decrease, and new finite zeros will appear. These new zeros are singularly perturbed. This paper applies a new canonical parameterization to systems with singular system pencils. Such systems have undefined relative degree. In singular systems, new zeros also appear under small perturbation, but they are not necessarily singularly perturbed. Rather, these zeros may appear at any frequency     

Computation of zeros of linear multivariable systems

Several algorithms have been proposed in the literature for the computation of the zeros of a linear system described by a state-space model {A, B, C, D}. In this paper we discuss the numerical properties of a new algorithm and compare it with some earlier techniques of computing zeros. The method is a modified version of Silverman's structure algorithm and is shown to be backward stable in a rigorous sense. The approach is shown to handle both nonsquare and/or degenerate systems. Several numerical examples are also provided.