How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

Active state estimation of nonlinear systems

In this paper, state observers for control systems with nonlinear outputs are studied. For such systems, the observability does not only depend on the initial conditions, but also on the exciting control used. Thus, for such systems, design of active control is an integral part of the design for state observers. Here some sufficient conditions are given for the convergence of an observer. It is also discussed, via a camera example, how to actively excite a system in order to improve the observability.     

Stochastic observability in network state estimation and control

A hidden Markov model for the traffic congestion control problem in transmission control protocol (TCP) networks is developed, and the question of observability of this system is posed. Of specific interest are the dependence of observability on the congestion control law and the interaction between observability ideas and the effectiveness of feedback control. Analysis proceeds with a survey of observability concepts and an extension of some available definitions for linear and nonlinear stochastic systems. The key idea is to link the improvement of state estimator performance to the conditioning on the output data sequence. The observability development proceeds from linear deterministic systems to linear Gaussian systems, nonlinear systems, etc., with backwards compatibility to deterministic ideas. The principal concepts relate to the entropy decrease of scalar functions of the state, which in the linear case are describable in terms of covariance matrices. A feature of nonlinear systems is that the estimator properties may affect the closed-loop control performance. Results are derived linking stochastic reconstructibility to strict improvement of the optimal closed-loop control performance over open-loop control for the hidden Markov model. The entropy provides a means to quantify and thus order simulation results for a simplified TCP network. Motivated by the link between feedback control and reconstructibility, the entropy formulation is also explored as a means to discriminate between different control strategies for improving estimator performance. This approach has connections to dual-adaptive control ideas, where the control has the simultaneous and opposing goals of regulating the system and of exciting the system to prevent estimator divergence.     

Connections between finite-gain and asymptotic stability

The relationship between input-output and Lyapunov stability properties for nonlinear systems is studied. Well-known definitions for the input-output properties of finite-gain and passivity, even with quite reasonable minimality assumptions on a state-space representation, do not necessarily imply any form of stability for the state. Attention is given to the precise versions of input-output and observability properties which guarantee asymptotic stability. Particular emphasis is given to the possibility of multiple equilibria for the dynamical system.     

非线性切换系统的集合稳定性分析

将输出对状态稳定和状态模可观测等概念进行推广.提出了输出对V(x)稳定、小时间y(z)可观测、大时间V(x)可观测的定义.利用上述定义,给出了非线性切换系统的V(x)零值集稳定的充分条件.分别利用统一Lyapunov函数和多Lyapunov函数证明了所提出的结论,并详细讨论了输出对V(x)稳定与输出对状态稳定及其他相关定义之间的关系.数值例子验证了所提出结论的正确性.     

A state observer for nonlinear systems

A state observer design methodology for a class of nonlinear systems is proposed. The nonlinear observer design methodology is, in fact, an extension of linear observer design techniques using transformations related to linear observability matrices     

Energy functions for dissipativity-based balancing of discrete-time nonlinear systems

Most of the energy functions used in nonlinear balancing theory can be expressed as storage functions in the framework of dissipativity theory. By defining a framework of discrete-time dissipative systems, this paper presents existence conditions for their discrete-time energy functions along with algorithms to find them based on dynamic optimization problems. Furthermore, the important case of the nonlinear discrete-time versions of the controllability and observability functions, its properties and algorithms to find them are presented. These algorithms are illustrated with linear and nonlinear examples.     

Stochastic balancing and approximation-stability and minimality

A new method of stochastic model reduction has recently been introduced by Desai et al. [1], [2]. The stability of the reduced order model has not previously, been considered. In this paper, we show that the stability of the reduced order model follows directly from the results of Pernebo and Silverman [3]. It is also shown that the reduced order model is minimal, in the controllability/observability sense. The relevance of this notion of minimality to stochastic minimality is made clear.     

The robustness of controllability and observability of linear time-varying systems

Fixed point methods from nonlinear analysis are used to establish conditions under which the uniform complete controllability, of linear time-varying systems is preserved under nonlinear perturbations in the state dynamics and the zero-input uniform complete observability of linear time-varying systems is preserved under nonlinear perturbation in the state dynamics and output read-out map. Robustness of partial controllability., observability, and a specific kind of nonzero input observability are also proven.     

Abstract bilinear systems: The forward shift approach

A bilinear realization theory for a Volterra series input-output map is given. The approach involves a special transform representation for a Volterra series and certain shift operators on a Fock space. The approach yields in a very simple manner a theory of span reachability, observability and minimality for bilinear systems.     

Computation of reachable and observable realizations of spatial filters

The input-output behaviour of a two-dimensional linear filter is defined by a formal power series in two variables- If the power series is rational, the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given.

In general reachability and observability do not imply the minimality of the realization. Nevertheless the dimension of a minimal realization is the least rank in a family of Hankel matrices.     

Design of state observers from a drift-observability property

In this paper, the problem of state observation with exponential error decay for nonlinear systems affine in the input is considered and an observer is proposed. For such an observer, the drift-observability property of the system (i.e., observability for zero input), together with an assumption on the input amplitude or on the observation relative degree, are sufficient conditions for exponential convergence of the observation error. The existence of an exponential observer is correlated to the existence of a solution for an H_∞ Riccati-like inequality. Global and semiglobal convergence results are presented     

Hamiltonian realizations of nonlinear adjoint operators

This paper addresses the issue of state-space realizations for nonlinear adjoint operators. In particular, the relationships between nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are established. Then, characterizations of the adjoints of controllability, observability and Hankel operators are derived from this analysis. The state-space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory. Finally, a duality between the controllability and observability energy functions is proved.     

Discrete-time model reference adaptive control for continuous-timesystems using generalized sampled-data hold functions

The generalized sampled-data hold functions approach of P.T. Kabamba (1987) is extended to the control of linear time-invariant systems with unknown parameters. The idea of Kabamba's approach is to periodically sample the plant output and define the control as the sampled output plus a discrete-time reference, each multiplied by an individual modulating function. Such a control makes it possible to assign an arbitrary discrete-time transfer function for the sampled closed-loop system and does not make assumptions on the plant other than controllability and observability. The authors propose an indirect adaptive controller which is based on this approach and estimates the modulating functions online. In particular, the control is modified so that persistent excitation of the continuous-time plant is ensured without making an assumption on the reference signal, and discrete-time asymptotic model-following is nevertheless obtained. The only assumptions on the plant are minimality, for the continuous and sampled plant, and known order     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Local observability of nonlinear differential-algebraic equations(DAEs) from the linearization along a trajectory

The main results of this note establish sufficient rank conditions for local observability of nonlinear differential-algebraic equations (DAE) systems near a known trajectory associated with a given control. Verification of the observability rank condition is addressed, and the relationship with previous work on observability of DAE is indicated     

Observability of nonlinear control systems on time scales

The problem of observability of the family of nonlinear control systems defined on any time scale and parameterised by time instant t ₀ is developed. Necessary and sufficient conditions for observability are given. Because time scale is a rich model of time that includes continuous- and discrete-time models as standard cases, the obtained results extend classical observability rank conditions given for continuous-time systems and discrete-time systems.     

An integral-invariance principle for nonlinear systems

In this paper we present an integral-invariance principle generalizing LaSalle's invariance principle for nonlinear systems. The principal new ingredients are the use of observation functions and certain integrability conditions, which are particularly well suited for dynamical systems involving control and observations. The integral-invariance principle leads to the development of a series of results relating stability, observability, and the converse theorems of Lyapunov theory. Corollaries include apparently diverse stabilizability results for adaptive control, nonlinear control, and passive circuits and systems     

非线性DEDS的标准结构

非线性DEDS是指由极大极小函数描述的系统, 常见于计算机科学、控制论、运筹学等领域, 考虑非自治非线性DEDS的结构问题, 通过引入白色图和凝白色图, 得到了系统能达和能观的两个充要条件以及系统的标准结构, 同时还给出了它们的矩阵表示.     

Optimal selection of sensors for state estimation in a reactive distillation process

The success of state estimation in a high dimensional system like multicomponent reactive distillation depends on the rigorous evaluation of the observability and appropriate selection of measurements that adequately characterize the process behavior. In this work, the dynamic state sensitive measurement information extracted from the nonlinear reactive distillation process is employed to configure the gramian covariance matrices which are then subjected to various scalar quantification measures to find the degree of observability in order to select the temperature sensors for state estimation in the process. These optimally configured process measurements are then incorporated in a process model based composition estimation scheme. The validity of the sensors that are selected by the gramian quantification measures are further ascertained through the evaluation of the estimator performance for various nonoptimal measurement combinations. The results on application to a metathesis reactive distillation column exhibit the usefulness of the empirical observability gramian based sensor selection strategy for inferential state estimation of reactive distillation process.