On flat systems behaviors and observable image representations

In this short paper we propose a definition of flatness for systems not necessarily given in input/state/output representation. A flat system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping, and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order. For linear differential systems, flatness is equivalent to controllability. We will generalize the main theorem of Levine and Nguyen (Systems Control Lett. 48 (2003) 69) to general linear differential systems.     

On the stability of qualitative models

This paper considers a discrete-time continuous-variable autonomous system for which only a quantised state measurement [x(k)] is available. The qualitative model of this system is a nondeterministic automaton thay describes, for a given quantised initial state [x(0)], the sequence of quantised states [x(k)]. The paper deals with the question for which stable linear systems is the qualitative model also stable. A necessary and sufficient condition on the qualitative model is proved under which the continuous-variable system is stable. This condition refers to the eventual boundedness of the nondeterministic automation. An example demonstrates that this stability criterion can be used only if a qualitative model of the system is available.     

A simple graph theoretic characterization of reachability for positive linear systems

In this paper we consider discrete-time linear positive systems, that is systems defined by a pair (A,B) of non-negative matrices. We study the reachability of such systems which in this case amounts to the freedom of steering the state in the positive orthant by using non-negative control sequences. This problem was solved recently [Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl., 310 (2000) 49]. However we derive here necessary and sufficient conditions for reachability in a simpler and more compact form. These conditions are expressed in terms of particular paths in the graph which is naturally associated with the system.     

On perturbations of linear controllable infinite-dimensional time-varying discrete-time systems

It is shown that for a broad class of linear (possibly time-varying and infinite-dimensional) discrete-time systems x_k+1 = A_kx_k + B_ku_k the property of being uniformly equicontrollable is preserved under small perturbations of system parameters. The problem of controllability of asymptotically time-invariant systems is also studied.     

A robust least squares based approach to min‐max model predictive control

This article deals with the model predictive control (MPC) of linear, time‐invariant discrete‐time polytopic (LTIDP) systems. The 2‐fold aim is to simplify the treatment of complex issues like stability and feasibility analysis of MPC in the presence of parametric uncertainty as well as to reduce the complexity of the relative optimization procedure. The new approach is based on a two degrees of freedom (2DOF) control scheme, where the output r(k) of the feedforward input estimator (IE) is used as input forcing the closed‐loop system ∑_f. ∑_f is the feedback connection of an LTIDP plant ∑_p with an LTI feedback controller ∑_g. Both cases of plants with measurable and unmeasurable state are considered. The task of ∑_g is to guarantee the quadratic stability of ∑_f, as well as the fulfillment of hard constraints on some physical variables for any input r(k) satisfying an “a priori” determined admissibility condition. The input r(k) is computed by the feedforward IE through the on‐line minimization of a worst‐case finite‐horizon quadratic cost functional and is applied to ∑_f according to the usual receding horizon strategy. The on‐line constrained optimization problem is here simplified, reducing the number of the involved constraints and decision variables. This is obtained modeling r(k) as a B‐spline function, which is known to admit a parsimonious parametric representation. This allows us to reformulate the minimization of the worst‐case cost functional as a box‐constrained robust least squares estimation problem, which can be efficiently solved using second‐order cone programming.     

Lyapunov stability of n-D strongly autonomous systems

In this article we look into stability properties of strongly autonomous n-D systems, i.e. systems having finite-dimensional behaviour. These systems are known to have a first-order representation akin to 1-D state-space representation; we consider our systems to be already in this form throughout. We first define restriction of an n-D system to a 1-D subspace. Using this we define stability with respect to a given half-line, and then stability with respect to collections of such half-lines: proper cones. Then we show how stability with respect to a half-line, for the strongly autonomous case, reduces to a linear combination of the state representation matrices being Hurwitz. We first relate the eigenvalues of this linear combination with those of the individual matrices. With this we give an equivalent geometric criterion in terms of the real part of the characteristic variety of the system for half-line stability. Then we extend this geometric criterion to the case of stability with respect to a proper cone. Finally, we look into a Lyapunov theory of stability with respect to a proper cone for strongly autonomous systems. Each non-zero vector in the given proper cone gives rise to a linear combination of the system matrices. Each of these linear combinations gives a corresponding Lyapunov inequality. We show that the system is stable with respect to the proper cone if and only if there exists a common solution to all of these Lyapunov inequalities.     

Stability independent of delay using rational functions

This paper is concerned with the problem of assessing the stability of linear systems with a single time-delay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer k and a positive real number T has only stable roots for any finite T≥0 and any k≥2. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number δ in the open interval (0,1). The paper is closed by showing counterexamples where stability of the roots of the rational function when k=1 is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.     

Constrained MPC for uncertain linear systems with ellipsoidal target sets

Robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain/time-varying linear systems with input constraints under fixed state feedback control laws. With the introduction of extra degrees of freedom in the form of perturbed control laws, these stabilizable regions can be enlarged. This was done in Lee and Kouvaritakis (Automatica 36 (2000) 1497–1504) in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets. We also extend the analysis to take into account both polytopic and unstructured bounded disturbances, as well as unstructured uncertainties.     

A Review of:“ORTHOGONAL POLYNOMIALS FOR ENGINEERS AND PHYSICISTS”, by Petr Beckmann. The Golem Press,Boulder, Colorado, 1973. 280 pp.

This paper explores the problems that are associated with building a model (representation) of two systems which are central to the field of policy analysis. The systems are the Lockean and Dialectical Inquiring Systems (ISs). Given two or more belief systems, a Lockean IS will attempt to secure maximum agreement between them, whereas a Dialectical IS will attempt to secure maximum disagreement as the basis for forming a policy.

It is shown that a representation of these two systems necessitates a notion of probability assignment in which the law of conditional probabilities, p(A)p(B\A) = p(B)p(A\B), does not hold. Cross impact analysis provides such a notion of probability and hence is necessary to the representation of what happens as a decision-maker goes from a Lockean state to a Dialectical state. The movement from a Lockean to a Dialectical state is termed Conflict Production or Generation whereas the movement from a Dialectical to a Lockean state is termed Conflict Resolution or Reduction. It is argued that the representation of these two systems is basic to the foundation of a mathematical approach to policy analysis.     

State feedback for non-linear control systems

A theory of static state feedback for non-linear discrete-time systems is developed. The theory applies to non-linear systems possessing a recursive representation of the form x _k+1 =?(x _k , u_k ), where ? is a continuous function, and it deals with the construction of continuous state feedback functions that internally stabilize a given system. The theory yields an explicit method for the computation of stabilizing feedback functions, and several examples of the computation of such functions are provided.     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Deep Teleology in Artificial Systems

Teleological variations of non-deterministic processes are defined. The immediate past of a system defines the state from which the ordinary (non-teleological) dynamical law governing the system derives different possible present states. For every possible present state, again a number of possible states for the next time step can be defined, and so on. After k time steps, a selection criterion is applied. The present state leading to the selected state after k time steps is taken to be the effective present state. Hence, the present state of a system is defined by its past in the sense that the past determines the possible states that are to be considered, and by its future in the sense that the selection of a possible future state determines the effective present state. A system that obeys this type of teleological dynamics may have significantly better performance than its non-teleological counterpart. The basic reason is that evolutions that are less optimal for the present time step, but which lead to a higher optimality after k time steps, may be preferred. This abstract concept of teleology is implemented for two concrete systems. First, it is applied to a general method for function approximation and classification problems. The method at issue treats all problems handled by conventional connectionism, and is suited for information with inner structure also. Second, it is applied to a dynamics in which forms of maximal homogeneity have to be produced. The relevance of the latter dynamics for generative art is illustrated. The teleology is `deep' in the sense that it is situated at the cellular level, in contradistinction with the teleology that is usually met in cognitive contexts, and which refers to macroscopic processes such as making plans. It is conjectured that deep level teleology is useful for machines, even though the issue if natural systems use this teleology is left open.     

Reliability evaluation of generalised multi-state k-out-of-n systems based on FMCI approach

Most studies on k-out-of-n systems are in the binary context. The k-out-of-n system has failed if and only if at least k components have failed. The generalised multi-state k-out-of-n: G and F system models are defined by Huang et al. [Huang, J., Zuo, M.J., and Wu, Y.H. (2000), ‘Generalized Multi-state k-out-of-n: G Systems’, IEEE Transactions on reliability, 49, 105–111] and Zuo and Tian [Zuo, M.J., and Tian, Z.G. (2006), ‘Performance Evaluation of Generalized Multi-state k-out-of-n Systems’, IEEE Transactions on Reliability, 55, 319–327], respectively. In this article, by using the finite Markov chain imbedding (FMCI) approach, we present a unified formula with the product of matrices for evaluating the system state distribution for generalised multi-state k-out-of-n: F systems which include the decreasing multi-state F system, the increasing multi-state F system and the non-monotonic multi-state F system. Our results can be extended to the generalised multi-state k-out-of-n: G system. Three numerical examples are presented to illustrate the results.     

Maximal controllability of input constrained unstable systems by the addition of implicit constraints

The null controllable set of a system is the largest set of states that can be controlled to the origin. Control systems that have a region of attraction equal to the null controllable set are said to be maximally controllable closed-loop systems. In the case of open-loop unstable plants with amplitude constrained control it is well known that the null controllable set does not cover the entire state-space. Further the combination of input constraints and unstable system dynamics results in a set of state constraints which we call implicit constraints. It is shown that the simple inclusion of implicit constraints in a controller formulation results in a controller that achieves maximal controllability for a class of open-loop unstable systems.     

给定环境下稳定开放量子系统的哈密顿量方法

针对耗散已知情况下Lindblad主方程描述的开放量子系统,本文通过哈密顿量的设计实现了系统对于目标平衡态的稳定化.借助相干矢量体系,将矩阵形式下的原始系统模型转换为了一个矢量形式的线性系统,并证明了变换前后系统稳定属性的等价性.通过保证矢量化线性系统模型的稳定性,并使系统的唯一平衡态等于期望的目标态,得到了系统哈密顿量的设计框架.特别地,本文讨论了这两类条件下系统哈密顿量各元素的范围,并指出根据它们的交集即可构成所设计的系统哈密顿量.最后,在一个两能级系统上进行了数值仿真实验,验证了本文哈密顿量稳定化方案的有效性.     

Design of reduced-order optimal estimators directly in the frequency domain

The frequency-domain design of optimal reduced-order estimators ((Caiman filters) for continuous-time linear time-invariant systems containing 0 ≤ k ≤ m noise-free output measurements is considered. In a direct frequency-domain approach to state feedback and estimation the ‘states’ of a system do not appear explicitly, and therefore we here focus our attention mainly on the frequency-domain characteristics of the optimal filter. By suitably modifying known time-domain results, a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization. The equivalence between the time- and the frequency-domain results is established with the aid of a non-minimal representation of reduced-order observers.     

Velocity observers for linear mechanical systems subject to single non-smooth impacts

In this paper, we consider the problem of the velocity estimation from position measurement for linear mechanical systems (with multiple degrees of freedom) subject to single non-smooth impacts, both elastic and inelastic (i.e., with coefficient of restitution e=1 and e(0,1), respectively). Through a simple example, it is shown that a classical Luenberger observer is not able to reproduce instantaneously the jumps in the mass velocities, since it recovers the error induced by such jumps only asymptotically: an infinite sequence of impacts can prevent the estimation error to asymptotically go to zero. A new observer structure is proposed for linear mechanical systems subject to single unilateral constraints, that guarantees that the corresponding error dynamics are exponentially stable, also in presence of an infinite sequence of non-smooth impacts. The observer that we propose switches at the impact times, that can be recognized by position measurements only. To validate the proposed observer, both simulation and experimental tests have been carried out and are briefly reported, pointing out the drawbacks and advantages of the observer.     

The fundamental closed-form solution of control-related states of kth order S3PR system with left-side non-sharing resource places of Petri nets

Due to the state explosion problem, it has been unimaginable to enumerate reachable states for Petri nets. Chao broke the barrier earlier by developing the very first closed-form solution of the number of reachable and other states for marked graphs and the kth order system. Instead of using first-met bad marking, we propose ‘the moment to launch resource allocation’ (MLR) as a partial deadlock avoidance policy for a large, real-time dynamic resource allocation system. Presently, we can use the future deadlock ratio of the current state as the indicator of MLR due to which the ratio can be obtained real-time by a closed-form formula. This paper progresses the application of an MLR concept one step further on Gen-Left kth order systems (one non-sharing resource place in any position of the left-side process), which is also the most fundamental asymmetric net structure, by the construction of the system's closed-form solution of the control-related states (reachable, forbidden, live and deadlock states) with a formula depending on the parameters of k and the location of the non-sharing resource. Here, we kick off a new era of real-time, dynamic resource allocation decisions by constructing a generalisation formula of kth order systems (Gen-Left) with r^* on the left side but at arbitrary locations.