Linear systems theory revisited

This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.     

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.     

Minimality and local state decompositions of a nonlinear state space realization using energy functions

In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state-space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies minimality. Using the nonlinear analogue of the Kalman decomposition, connections are then established between minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.     

Minimal realisations of odd transfer functions for first-degree nD systems

ABSTRACT

Minimal realisation problems of odd transfer functions for first-degree (multi-linear) nD single-input single-output discrete systems have been studied, but it has not been well solved. This paper provides a new, different method to solve absolutely minimal realisation problems. By methods of limits and algebraic techniques, without using the symbolic approach by Gröbner basis, the requirements of absolutely minimal realisation are transformed into a system of equations represented by the determinants. Since the equations for first-degree 2D systems are solvable by quadratic equations and the conditions for higher-dimensional realisations can be expressed by the results of 2D systems, the absolutely minimal realisations for nD systems can be found by using the realisations of n(n ? 1)/2 2D systems. Furthermore, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived from the Pfaffian function of the skew-symmetric matrix. Finally, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.     

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.     

On minimality and stable minimality of time-varying linear systems with well-posed boundary conditions

For time-varying linear systems with well-posed boundary conditions the concept of stable minimality is introduced. A complete characterization is given of stably minimal systems with analytic coefficients.     

Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems

Using the minimality property of finite-horizon time-varying compensators, established in this paper, and the Moore-Penrose pseudo-inverse instead of the standard inverse, strengthened discrete-time optimal projection equations (SDOPE) and associated boundary conditions are derived for finite-horizon fixed-order LQG compensation. They constitute a two-point boundary value problem explicit in the LQG problem parameters which is equivalent to first-order necessary optimality conditions and which is suitable for numerical solution. The minimality property implies that minimal compensators have time-varying dimensions and that the finite-horizon optimal full-order compensator is not minimal. The use of the Moore-Penrose pseudo-inverse is further exploited to reveal that the optimal projection approach can be generalised, but only to partially include non-minimal compensators. Furthermore, the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent. Max-min compensator dimensions are introduced and their significance in solving numerically the two-point boundary value problem is explained. The numerical solution is presented in a recently published companion paper, which relies on the results of this paper.     

Necessary and sufficient conditions for controllability and observability of switched impulsive control systems

Many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This note first introduces the impulsive behavior into switched linear systems and studies the controllability and observability of such systems. Necessary and sufficient criteria for reachability and controllability are established. It is proved that reachability is equivalent to controllability for such systems. Then, necessary and sufficient criteria for observability and determinability of such systems are established by duality. It is also proved that observability is equivalent to determinability. Our criteria are of geometric type, they can be transformed into algebraic form conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.     

Uniform exponential stability of linear time-varying systems: revisited

Some classical results known in the adaptive control literature are often used as analysis tools for nonlinear systems by evaluating the nonlinear differential equations along trajectories. While this technique is widely used, as we remark through examples, one must take special care in the consideration of the initial conditions in order to conclude uniform convergence. One way of taking care explicitly of the initial conditions is to study parameterized linear time-varying systems. This paper re-establishes known results for linear time-varying systems via new techniques while stressing the importance of imposing that the formulated sufficient and necessary conditions must hold uniformly in the parameter. Our proofs are based on modern tools which can be interpreted as an “integral” version of Lyapunov theorems; rather than on the concept of uniform complete observability which is most common in the literature.     

基于H表示的时变随机Markov 跳跃系统的能观性

研究时变连续和离散随机Markov 跳跃系统(SMJSs) 的能观性问题. 基于H-表示方法将时变SMJSs 转化为等价的时变线性系统, 根据线性系统理论得到时变连续和离SMJSs 的能观性Gramian 矩阵判据. 数值仿真表明了所得结论的正确性.     

On stochastic observability and controllability

New definitions of stochastic observability, detectability, controllability and reachability are proposed. For both continuous and discrete time linear systems easily verifiable necessary and sufficient conditions are obtained, which are compatible with the usual ones for the corresponding (degenerate) deterministic systems. Furthermore, for stochastically observable, linear systems explicit expressions for the LUMCE of the state at time $\text{t≧0}$, with no information of the initial state, are obtained.     

Realisation of linear time-varying systems

In this paper, the realisation problem of linear multi-input multi-output, time-varying systems is studied. The approach, based on the theory of non-commutative polynomial rings, yields explicit and simple formulas for computation of the state coordinates as well as for state equations in observable canonical form. The formulas are based on (left) Euclidean polynomial division.     

线性时变广义系统的能控性与能观性问题

讨论了线性时变系统和线性时变广义系统的两个基本问题, 得到了两种判定时变系统能控性与能观性的必要条件, 该判定条件只依赖于系统矩阵A(t)和输入矩阵B(t), 不必计算系统的系统状态转移矩阵, 使得判别时变系统能控性与能观性易于实现. 说明了本文结论是线性定常系统相应结论的自然扩展. 对进一步深入研究时变系统和时变广义系统具有实际启发作用.     

Duality Properties of Linear Impulsive Systems

Reachability and observability properties of linear switched, impulsive, and switched/impulsive systems have received a great deal of attention over the past decade. The purpose of this paper is to rigorously establish a duality between reachability and observability for linear impulsive systems both in direct system‐theoretic terms as well as in geometric terms. These connections are established using an appropriately defined adjoint system associated with the original impulsive system. The duality between reachability and observability is then used to leverage recent reachability results in order to tighten the relationship between the set of unobservable states and a certain invariant subspace.     

Reachability, observability and strict equivalence of networks of linear systems

This paper aims at a better understanding of reachability and observability properties of heterogeneous networks of linear systems. It extends prior work by Hara et al. (SICE J Control Meas Syst Integr 2:299–306, 2009), who characterized reachability for homogeneous networks of identical linear SISO systems. Our approach is based on extending the classical notion of strict system equivalence to networks of linear systems. We survey and extend known characterizations for reachability and observability for arbitrary interconnected linear MIMO systems. Both static and dynamic interconnection laws are considered, and various applications to classes of homogeneous and heterogeneous networks are derived.     

Contraction analysis of non-linear distributed systems

Contraction theory is a comparatively recent dynamic analysis and non-linear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of non-linear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton–Jacobi dynamics, and optimal controllers and observers. By contrast with stability proofs based on energy dissipation, stability and convergence can be determined for energy-based systems excited by time-varying inputs.

The Hamilton–Jacobi–Bellman controller and a similar optimal non-linear observer design are studied based on explicitly computable conditions on the convexity of the cost function. These stability conditions extend the well-known conditions on controllability and observability Grammians for linear time-varying systems, without requiring the unknown transition matrix of the underlying differential dynamics.     

A necessary and sufficient minimality condition for uncertainsystems

A necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations (LFTs) on structured operator sets. This condition is based on the existence of a rank-deficient solution to either of a pair of linear matrix inequalities which generalize Lyapunov equations; the notion of Gramians is thus also generalized to uncertain systems, as well as Kalman-like decomposition structures. A related minimality condition, the converse of the reducibility condition, may then be inferred from these results and the equivalence class of all minimal LFT realizations defined. These results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems. Subsequent results, such as the definition of and rank tests on structured controllability and observability matrices are also given. The minimality results described are applicable to multidimensional system realizations as well as to uncertain systems; connections to formal powers series representations also exist     

Balancing related methods for minimal realization of periodic systems

We propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time-varying dimensions. The first method belongs to the family of square-root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations. An alternative balancing-free square-root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems. The key numerical computation in both methods is the solution of nonnegative periodic Lyapunov equations directly for the Cholesky factors of the solutions. For this purpose, a numerically reliable computational algorithm is proposed to solve nonnegative periodic Lyapunov equations with time-varying dimensions.     

Synthesis of linear digital controllers using dead-beat observers for systems with inaccessible states†

A procedure is given to synthesize linear digital controllers for uniformly observable linear time-varying discrete data systems where some of the state variables are not available for feedback. Dead-beat observer theory is used and the controller generates the exact optimal control law in at most q?1 sampling periods, where q is the observability index of the system.     

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.