Extended Ho–Kalman algorithm for systems represented in generalized orthonormal bases

This paper considers the construction of minimal state space models of linear time-invariant systems on the basis of system representations in terms of generalized orthogonal basis function expansions. Starting from the classical Ho–Kalman algorithm that solves the problem using Markov parameter expansions, a generalization is obtained by analysing the matrix representations of the Hankel operators in generalized orthonormal bases. Using the so-called Hambo-domain techniques an efficient algorithm is given to implement the proposed method.     

Continuous state representations for AR Systems

In this paper we consider in a behavioral setting the subclass of dis-crete-time, linear, finite-dimensional systems, which can be represented by autoregressive (AR) equations. It will be shown that, with respect to the convergence of all coefficients in an AR representation, there exist continuously dependent input-state-output (i/s/o) representations, under the condition that some specified degree remains constant. This continuous i/s/o representation is given by the Fuhrmann realization.     

Representations and structural properties of periodic systems

We consider periodic behavioral systems as introduced in [Kuijper, M., & Willems, J. C. (1997). A behavioral framework for periodically time-varying systems. In Proceedings of the 36th conference on decision & control (Vol. 3, pp. 2013-2016). San Diego, California, USA, 10-12 December 1997] and analyze two main issues: behavioral representation/controllability and autonomy. More concretely, we study the equivalence and the minimality of kernel representations, and introduce latent variable (and, in particular, image) representations. Moreover we relate the controllability of a periodic system with the controllability of an associated time-invariant system known as lifted system, and derive a controllability test. Further, we prove the existence of an autonomous/controllable decomposition similar to the time-invariant case. Finally, we introduce a new concept of free variables and inputs, which can be regarded as a generalization of the one adopted for time-invariant systems, but appears to be more adequate for the periodic case.     

Feedback norm minimisation with regional pole placement

This article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time-invariant multi-input systems. The inherent non-convexity in this optimisation is resolved by a combination of two separate approaches: (1) an inner convex approximation of the polynomial matrix stability region due to Henrion and (2) a novel convex parameterisation of column reduced matrix fraction system representations. Using a sequence of approximations enabled by the above two methods, it is shown that the constraints on closed-loop poles (both pre-specified exact locations and regional placement) define linear matrix inequalities. Finally, the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Detection filter design for LPV systems—a geometric approach

This paper investigates fault detection and isolation of linear parameter-varying (LPV) systems by using parameter-varying (C,A)-invariant subspace and parameter-varying unobservability subspaces. The so called “detection filter” approach, formulated as the fundamental problem of residual generation (FPRG) for linear time-invariant (LTI) systems, is extended for a class of LPV systems. The question of stability is addressed in the terms of Lyapunov quadratic stability by using linear matrix inequalities. The results are applied to the model of a generic small commercial aircraft.     

Local identifiability of time-invariant linear systems by periodic test signals

This paper is concerned with the parameter local identifiability of dynamic systems operating in periodic regimes. The systems dealt with are time-invariant and discrete-time linear systems, with an output white gaussian disturbance. Sets of frequencies are characterized such that the parameter vector is identifiable if and only if the spectrum of the periodic test signal does not vanish over one of these sets. The analysis is developed by taking also into consideration measurement schedule sets which are scattered along the time axis.     

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.     

Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models

In this paper, new results on the open-loop balanced representation of continuous time linear time-invariant systems are reported. More particularly, the effect of lossless positive real transformations on open-loop balanced representations is investigated with specific attention to the problem of model order reduction. The properties of systems where a lossless positive real transformation has been applied are discussed showing that, if the original system is open-loop balanced, the resulting transformed system is still open-loop balanced. Furthermore, the singular values of the transformed system are related to those of the original one. These results allow to derive a model order reduction strategy for this class of systems that leads to a consistent decrease of the numerical complexity. The proposed approach reveals to be of particular interest for the design of reduced-order systems with specific amplitude responses, including analog multiband filters.     

On tracking control for affine nonlinear systems by sliding mode

The stability problem of output tracking of a bounded time-varying reference by decouplable affine nonlinear systems using sliding mode control is investigated. It is shown that when the error dynamics on the sliding surface is chosen to be linear time-invariant, closed loop stability of systems under the presented sliding mode control can be guaranteed only if the systems are minimum phase.     

On the controllability of switching linear systems

This note presents a necessary and sufficient condition for small time controllability of a linear switching system (that is, a collection of linear time-invariant control systems, where a trajectory is any concatenation of trajectories of the individual systems). This result extends the controllability condition recently obtained for unconstrained linear switching systems to the case of control which is constrained in a cone.     

Addendum to ‘(Singular) state models and (singular) LTID systems’

In this note, a construction of state representations of singular linear time-invariant differential systems is described directly in terms of trajectories.     

On controllability,near-controllability,controllable subspaces,and nearly-controllable subspaces of a class of discrete-time multi-input bilinear systems

This paper considers a class of discrete-time multi-input inhomogeneous bilinear systems. The structure of such systems is most close to linear time-invariant systems’ but they own a strong property. That is, if the systems are uncontrollable, they can still be nearly controllable. Necessary and sufficient conditions for controllability and near-controllability of the systems are established by using a classical decomposition. Furthermore, a geometric characterization is given for the systems such that controllable subspaces and nearly-controllable subspaces are derived and characterized. Similar results on controllability are also obtained for the continuous-time counterparts of the systems. Finally, examples are provided to demonstrate the conceptions and results of this paper.     

基于定常线性迭代法的PSO算法收敛性分析

PSO算法本身是线性时变离散系统,现有的PSO算法收敛性条件的研究都是通过一定的假设将其转化为线性定常离散系统,线性定常离散系统的数学模型与求解线性方程组的单步定常线性迭代法的数学模型完全一致,这样对线性定常离散系统的稳定性分析就转化为对单步定常线性迭代格式的收敛性分析,为PSO算法的收敛性研究提供了一种新的思路和方法。     

Parametrization method for calculating exact stability bounds of stochastic linear systems with multiplicative noise

For deterministic time-invariant linear systems, stability results are quite simple. For stochastic systems, however, even for linear ones, they are rather complicated. In this paper, some results on second mean stability (mean square stability) of time-invariant linear systems with multiplicative noise are summarized and the parametrization method of getting an exact bound for pth mean stability (p ≥ 2) via second mean stability is stated. Moreover, relations between pth mean stabilities for various values of p are given. On the basis of these relations, a simpler method for getting sufficient conditions for pth mean stability is also given, though the resulting sufficient bound is, of course, more conservative. Comparative studies of various conditions are made by using examples.     

On the quantitative characterization of approximate decentralized fixed modes using transmission zeros

The spectrum of a linear time-invariant multivariable system, using decentralized linear time-invariant controllers, can only be assigned to a symmetric set of complex numbers that include the decentralized fixed modes (DFM). Hence only systems with stable DFM can be stabilized. Although the concept of DFM characterizes when a decentralized controller can stabilize a system, it gives no indication of howhard it is to effect such a stabilization. A system is considered hard to stabilize if large controller gains are required. Modes that are hard to shift are termedapproximate decentralized fixed modes. In this paper two new assignability measures which quantify the difficulty of shifting a mode are derived. The first is coordinate invariant and is based on the distance between a mode and a set of transmission zeros. The second is coordinate dependent and is based on the minimum singular value of a set of transmission zero matrices. This work has been supported by the Natural Sciences are Engineering Research Council of Canada under Grant No. A4396.     

Reliable stabilization of linear plants using a two-controller configuration

A reliable controller design method is developed for linear, time-invariant, multi-input multi-output control systems; two controllers are designed to stabilize the closed-loop system when acting together and acting independently if one fails. All reliable controllers which achieve closed-loop stability are characterized for strongly stabilizable plants using a factorization approach.     

Identifiability of linear stochastic systems operating under linear feedback

The identifiability of multiple input-multiple output stochastic systems operating in closed loop is considered for the case where the plant and the regulator are both linear and time-invariant. Two basic identification methods have been proposed for such systems: the joint input-output method, in which the input and output processes are modelled jointly as the output of a white noise driven system; and the direct method, in which a prediction error method is used on the input-output data as if the system were in open loop. Previously obtained identifiability results for the joint input-output method are extended to a number of new situations, including but extending beyond the identifiability results obtained with the direct method.