Aggregation of a class of interconnected, linear dynamical systems

In this paper, we introduce the notion of a “meaningful” average of a collection of dynamical systems as distinct from an “ensemble” average. Such a notion is useful for the study of a variety of dynamical systems such as traffic flow, power systems, and econometric systems. We also address the associated issue of the existence and computation of such an average for a class of interconnected, linear, time invariant dynamical systems. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the average behavior of the interconnected dynamical systems. The problem of analysis and control of heirarchical, large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system.     

Analysis and design for discrete-time linear systems subject to actuator saturation

We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback which makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.     

Controlled invariance of ellipsoids: linear vs. nonlinear feedback

Several equivalent conditions or statements for set invariance were obtained for systems with one saturating actuator in a recent paper. In particular, it was shown that the existence of a nonlinear feedback that makes an ellipsoid invariant is equivalent to the existence of a feedback linear inside the ellipsoid that makes it invariant. In this paper, we will show that this equivalence property holds conditionally for systems with multiple saturating actuators. We will provide a criterion to check if the largest ellipsoid made invariant by nonlinear feedback can also be made invariant by a feedback linear inside the ellipsoid. Numerical examples reveal that this criterion is usually satisfied. The equivalence of other set invariance conditions will also be investigated.     

Invariant cones for non-smooth dynamical systems

In this paper we study the dynamics of piecewise linear systems with regard to generalisations of the familiar notion of Hopf bifurcation for smooth systems. It is shown that the concept of invariant planes with a change from a stable focus to an unstable focus through a center should be replaced in the case of non-smooth systems by an invariant cone with similar dynamical behaviour. In that way results for planar systems are extended into higher dimensions. To determine the invariant cone and its stability properties various extended systems are derived.     

离散非线性时变凸多面体系统族的鲁棒正不变集

动态系统的状态约束和控制约束等问题可归结为状态空间中某些集合的正不变性.利用 混合单调分解方法研究离散非线性、时变凸多面体系统族的线性状态约束集合的鲁棒正不变性． 对由矩阵凸多面体和加性区间扰动描述的线性时变离散系统族,得到了鲁棒正不变集的充分必要 条件;对非线性系统族则得到有关充分条件.这些条件均由系统族的顶点表述,易于检验,同时给 出示例.     

Some remarks on duality over a commutative ring

This paper addresses the duality problem for dynamical, linear, time-invariant systems defined over a ring. The duality principle is at the core of theoretical results for systems defined over a field, but this principle can not be applied for systems over a ring. From the definition of controlled and conditioned invariant submodules in the geometric approach, we analyze the relationships among various notions of invariance, using the concept of orthogonal submodule. These logical relations are summarized in two non-equivalent schemes.     

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.     

分数阶动力学的分析力学方法及其应用

综述我们在分数阶动力学分析力学方法的研究进展,包括：分数阶动力学系统的分析力学表示,构造分数阶动力学模型的分析力学方法,构造分数阶动力学模型团簇的分析力学方法,三类分数阶Lie群无限小变换方法,分数阶动力学系统的对称性、对称性摄动和共形不变性的分析力学方法,分数阶动力学系统的代数结构与Poisson积分的分析力学方法,构造分数阶动力学系统积分不变量的分析力学方法,分数阶动力学系统梯度表示的分析力学方法,分数阶动力学系统稳定性的分析力学方法,分数阶微分方程的分析力学方法等,介绍了对于物理学、力学、生物学、非线性科学等领域的10多种分数阶动力学模型的应用,并指出了若干进一步研究的问题.     

任意初始状态下的学习控制

本文对线性时不变系统和一类非线性系统提出了一种新的学习控制方案,此方案包括两 种学习,即同时进行系统的输入和初始状态的学习.因此在学习开始时对系统的初始状态无 要求.仿真结果表明了这种方法的可行性.     

Linear cone-invariant control systems and their equivalence

In this paper, we study invariant control systems that generalise positive systems. A characterisation of linear control systems invariant on polyhedral cones (corner regions) in the state-space, called cone-invariant linear control systems, is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone. The problem of equivalence between control systems invariant on corner regions is introduced. For cone-invariant linear control systems, we study invariance-preserving state-equivalence and invariance-preserving feedback-equivalence and present characterisations of both notions of equivalence.     

Construction of invariant polytopic sets with specified complexity

The problem of construction of controlled invariant polytopic sets with specified complexity, for linear systems subject to linear state and control constraints, is investigated. First, geometric conditions for the enlargement of a polytopic set by adding a new vertex, in order to produce a polytopic set of specified complexity, are established. Next, conditions for such an enlargement of controlled invariant sets to preserve the controlled invariance property are presented. The established theoretical results are used to develop methods for the construction of admissible controlled invariant sets with specified complexity. Two numerical examples show how these results can be used for the computation of monotonic sequences of admissible controlled invariant sets of specified complexity.     

The contact problem for linear continuous-time dynamical systems: ageometric approach

In this paper linear time-invariant dynamical systems described by a combination of differential equalities and static inequalities in state-space formulation are investigated. Of special interest is the contact problem: the effect of the boundary of the constraint set on the behavior of the system. This effect is studied by dividing the state-space in a number of disjunct subsets. It is shown that these subsets are invariant under linear state feedback. In our framework, a specific place is reserved for modeling the laws of collision, i.e., physical modeling, which are regarded as external factors. Our main results are a system theoretical framework in which we describe what happens upon contact and a definition of the constrained state-space system in terms of its restricted behavior. The results presented here can be considered as an extension for restricted linear systems of the classic positive invariance theory for linear systems     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Distributed control of spatially invariant systems

We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), H₂ and H_∞, optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed     

Set Invariance Conditions for Singular Linear Systems Subject to Actuator Saturation

In this note, we establish a set of conditions under which an ellipsoid is contractively invariant with respect to a singular linear system under a saturated linear feedback. These conditions can be expressed in terms of linear matrix inequalities, and the largest contractively invariant ellipsoid can be determined by solving an optimization problem with LMI constraints. With the feedback gain viewed as an additional variable, this optimization problem can be readily adapted for the design of feedback gain that results in the largest contractively invariant ellipsoid. Moreover, in the degenerate case where the singular linear system reduces to a regular system, our set invariance conditions reduce to the existing set invariance conditions for normal linear systems.     

Positive invariance, monotonicity and comparison of nonlinear systems

In this paper, the interrelation between positive invariance, monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied. First, necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v(x)≤w with respect to nonlinear systems are established. Then, various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established. Finally, necessary and sufficient conditions for a, not necessarily monotone, nonlinear system to be a comparison system are established. No specific algebraic structure is necessary for developing the main results of this work. Numerical examples illustrating the applicability of all these results to the case of discrete-time dynamical systems are also given.     

Stability Analysis of Linear Shift‐Invariant Dynamics in Honeycomb Structure

This paper deals with the stability of linear shift‐invariant multidimensional dynamical systems defined on honeycomb structure. Two different honeycomb structures are discussed. The local dynamical states are assumed to be distributed to honeycomb cells in the first consideration, and they are assumed to be distributed to the nodes of honeycomb mesh in the second consideration. In each honeycomb structure, the fundamental linear shift‐invariant dynamics is introduced and then the stability criterion is presented.     

Stability theory for hybrid dynamical systems

We first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples     

A unified method for the design of nonovershooting linear multivariable state-feedback tracking controllers

We consider the use of linear multivariable feedback control to achieve a nonovershooting step response. A method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot and arbitrarily small rise time, under some mild assumptions. We present a unified design method that can be applied to continuous and discrete time systems, square and non-square systems, minimum and nonminimum phase systems, and also strictly proper and nonstrictly proper systems.     

On maximizing the convergence rate for linear systems with input saturation

In this note, we consider a few important issues related to the maximization of the convergence rate inside a given ellipsoid for linear systems with input saturation. For continuous-time systems, the control that maximizes the convergence rate is simply a bang-bang control. Through studying the system under the maximal convergence control, we reveal several fundamental results on set invariance. An important consequence of maximizing the convergence rate is that the maximal invariant ellipsoid is produced. We provide a simple method for finding the maximal invariant ellipsoid, and we also study the dependence of the maximal convergence rate on the Lyapunov function.