On nonlinear controllability of homogeneous systems linear in control

This paper considers small-time local controllability (STLC) of single- and multiple-input systems, x/spl dot/=f/sub 0/(x)+/spl Sigma//sub i=1//sup m/f/sub i/u/sub i/ where f/sub 0/(x) contains homogeneous polynomials and f/sub 1/,...,f/sub m/ are constant vector fields. For single-input systems, it is shown that even-degree homogeneity precludes STLC if the state dimension is larger than one. This, along with the obvious result that for odd-degree homogeneous systems STLC is equivalent to accessibility, provides a complete characterization of STLC for this class of systems. In the multiple-input case, transformations on the input space are applied to homogeneous systems of degree two, an example of this type of system being motion of a rigid-body in a plane. Such input transformations are related via consideration of a tensor on the tangent space to congruence transformation of a matrix to one with zeros on the diagonal. Conditions are given for successful neutralization of bad type (1,2) brackets via congruence transformations.     

Local stabilization and controllability of a class of non-triangular nonlinear systems

This paper considers the problems of local asymptotic stabilization using continuous static state feedback (LCFS) and small time local controllability (STLC) of single-input nonlinear systems. A specific class of nontriangular systems, called the essentially triangular form, is introduced. For this class of systems sufficient conditions for LCFS and necessary and sufficient conditions for STLC are obtained. In particular, it is shown that STLC implies LCFS. A motivating physical example of the ETF system is presented.     

Exponential stabilization of an underactuated autonomous surface vessel

This paper studies the problem of controlling the planar position and orientation of an autonomous surface vessel using two independent thrusters. It is first shown that although the system is not asymptotically stabilizable to a given configuration using a time-invariant continuous feedback, it is strongly accessible and small-time locally controllable at any equilibrium. Time-invariant discontinuous feedback laws are then constructed to asymptotically stabilize the system to the desired configuration with exponential convergence rates. A simulation example is included to demonstrate the results.     

On the relation between small-time local controllability and normal self-reachability

We introduce a local property of nonlinear systems called the nontangency property and we show that, in the presence of this nontangency property, small-time local controllability by measurable controls implies small-time local controllability by piecewise-constant controls; furthermore, the initial state is normally reachable from itself in arbitrarily small time. The class of systems that are small-time locally controllable and satisfy the nontangency property is shown to contain all real-analytic systems, all smooth systems with the Lie-algebra rank condition, and all locally boundedC ¹ systems. Some consequences of small-time normal self-reachability are also discussed.     

切换系统不变集的输出反馈镇定

利用一个非负函数V (x), 推广了小时间状态模可观测和大时间状态模可观测的定义, 提出了小时间V (x)可观测和大时间V (x)可观测的概念. 利用V (x)可观测的概念, 讨论了子系统无源的切换系统不变集的有界输出反馈镇定及动态输出反馈镇定. 利用多Lyapunov函数证明了闭环系统渐近稳定. 仿真例子验证了文中结论正确性.     

Equilibrium analysis of multibody dynamic systems using genetic algorithm in comparison with constrained and unconstrained optimization techniques

The present paper describes a set of procedures for the solution of nonlinear equilibrium problems in complex multibody systems. To find the equilibrium position of the system, six different optimization algorithms are used to minimize the total potential energy (TPE) of the system and compared with respect to accuracy and efficiency. A computer program is developed to evaluate the equality constraints and objective function of a general multibody dynamic system to find the equilibrium condition. It is seen that the indirect methods have better results and converge faster. Also it is shown that the genetic algorithm (GA) results in a global optimum while the other methods converge to a local optimum.     

至少有一个增长无源的子系统切换非线性系统的输出跟踪控制

本文利用平均驻留时间方法解决至少有一个增长无源的子系统的切换非线性系统的输出跟踪问题.对于给定的无源率,设计子系统控制器使得平均驻留时间变小.所得到的平均驻留时间依赖于增长无源子系统的增长指数小时间范数可观性.最后,通过数值例子验证所提出方法的有效性.     

Pareto Effectiveness of Equilibria in Active Systems with the Distributed Control

Active systems with the distributed control (one active element and a few control centers) are considered. Questions for the existence of the Nash equilibrium of the game of centers in pure strategies are studied. It is shown that at any distribution of the incomes from the work of an active system among a few centers and the existence of at least one Pareto-ineffective Nash equilibrium, any Pareto-effective outcome can be implemented as a Nash equilibrium. It is proved that in the system consisting of two centers there always exists a Pareto-effective Nash equilibrium in pure strategies.     

基于有理平方根逼近的非高斯随机分布系统的故障诊断和容错控制

随机分布系统指的是输入为常规向量而输出为系统输出的概率密度函数所描述的一类随机系统．该类系统控制算法的目标是选择一个控制输入使得系统的实际输出概率密度函数尽可能跟踪一个事先给定的概率密度函数．本文对采用有理平方根B样条逼近其输出概率密度函数的非高斯动态随机分布系统,提出了一种基于非线性自适应观测器的故障诊断方法．该方法可快速有效地诊断出非高斯随机分布系统故障．通过对故障系统的重组,使故障后系统的输出概率密度函数仍能跟踪给定的分布,实现了该随机系统的容错控制,提高了随机系统的可靠性．     

Mode identification of bilinear systems

A method is presented for the identification of the eigenvalues of systems characterized by bilinear differential equation models. The identification procedure is based on the existence of a zeroth harmonic, or DC shift, in the output of non-linear systems perturbed by a periodic input. The magnitude of the DC shift is a function of system properties and of the input waveform and frequency. After the appropriate equations based on a truncated Volterra series representation of a general bilinear system are presented, the regression problem associated with the identification procedure is discussed in detail. It is shown that in many cases good approximations of the system eigenvalues are obtained. Furthermore, a minimal order for any proposed system model is obtained. The method is robust in the sense that it employs system inputs which are relatively easy to generate. Also, the method requires only time averages of the system output, thereby minimizing the effects of measurement noise.     

Local bifurcations and feasibility regions indifferential-algebraic systems

The dynamics of a large class of physical systems such as the general power system can be represented by parameter-dependent differential-algebraic models of the form x˙=f and 0=g. Typically, such constrained models have singularities. This paper analyzes the generic local bifurcations including those which are directly related to the singularity. The notion of a feasibility region is introduced and analyzed. It consists of all equilibrium states that can be reached quasistatically from the current operating point without loss of local stability. It is shown that generically loss of stability at the feasibility boundary is caused by one of three different local bifurcations, namely the saddle-node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation which is analyzed precisely here for the first time. The latter results when an equilibrium point is at the singular surface. Under certain transversality conditions, the change in the eigenstructure of the system Jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed     

Synchronizing linear systems via partial-state coupling

A basic result in the synchronization of linear systems via output coupling is presented. For identical discrete-time linear systems that are detectable from their outputs and neutrally stable, it is shown that a linear output feedback law exists under which the coupled systems globally asymptotically synchronize for all fixed connected (asymmetrical) network topologies. An algorithm is provided to compute such a feedback law based on individual system parameters.     

On Asymptotic Stabilizability of Linear Systems With Delayed Input

This paper examines the asymptotic stabilizability of linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open-loop system is not exponentially unstable (i.e., all the open-loop poles are on the closed left-half plane). A simple example shows that such results would not be true if the open-loop system is exponentially unstable. It is further shown that such systems, when subject to actuator saturation, are semiglobally asymptotically stabilizable by linear state or output feedback.     

Threshold superposition in morphological image analysis systems

It is shown that four composite morphological systems, namely morphological edge detection, peak/valley extraction, skeletonization, and shape-size distributions obey a weak linear superposition, called threshold-linear superposition. The output image signal or measurement from each system is shown to be the sum of outputs due to input binary images that result from thresholding the input gray-level image at all levels. These results are generalized to a vector space formulation, e.g. to any finite linear combination of simple morphological systems. Thus many such systems processing gray-level images are reduced to corresponding binary image processing systems, which are easier to analyze and implement     

对称循环组合系统的几个控制问题

研究了对称循环组合系统的输出调节问题和不确定对称循环组合的二次稳定性和二次稳定化问题。结果表明,由于对称循环组合系统结构的特殊性,这的输出调节问题可以转化为一些低价系数的输出调节问题。而不确定对称循环组合系统的二次稳定性可以通过考察若干个低价系数的二次稳定性来判断,且其稳定比化控制律也可以通过低价系数的稳定化控制律来构造。     

A local feedback synthesis of time-optimal stabilizing controls in dimension three

Under generic conditions a local feedback synthesis for the problem of time-optimally stabilizing an equilibrium point in dimension three is constructed. There exist two surfaces which are glued together along a singular are on which the optimal control is singular. Away from these surfaces the optimal controls are piecewise constant with at most two switchings. Bang-bang trajectories with two switchings but different switching orders intersect in a nontrivial cut-locus and optimality of trajectories ceases at this cut-locus. The construction is based on an earlier result by Krener and Schättler which gives the precise structure of the small-time reachable set for an associated system to which time has been added as an extra coordinate.     

Nonlinear influence in the frequency domain: Alternating series

The nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have Volterra series expansion. It is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities. The magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series. Sufficient (and necessary) conditions in which the output spectrum can be cast into an alternating series are studied. These results reveal a novel frequency-domain insight into the nonlinear influence on a system, and provide a new method for the analysis and design of nonlinear systems in the frequency domain. Examples are given to illustrate the results.     

Stochastic Distribution Control of Singular Systems: Output PDF Shaping

This paper presents a new algorithm designed to control the shape of the output probability density function (PDF) of singular systems subjected to non-Gaussian input. The aim is to select a control input uk such that the output PDF is made as close as possible to a given PDF. Based on the B-spline neural network approximation of the output PDF, the control algorithm is formulated by extending the developed PDF control strategies of non-singular systems to singular systems. It has been shown that under certain conditions the stability of the closed-loop system can be guaranteed. Simulation examples are given to show the effectiveness of the proposed control algorithm.     

State‐limiting PID controller for a class of nonlinear systems with constant uncertainties

Proportional Integral Derivative (PID) controllers still represent the core control method for achieving output regulation of either linear or nonlinear systems in the majority of industrial applications. However, conventional PID control cannot guarantee specific state constraint requirements for the plant, when the system introduces uncertainties. In this paper, a novel nonlinear PID control that achieves output regulation and guarantees a desired state limitation below a given value for a wide class of nonlinear systems with constant uncertainties is proposed. Using nonlinear ultimate boundedness theory, it is shown that the proposed state‐limiting PID (sl‐PID) control maintains a given bound for the desired system states at all times, ie, even during transients, whereas an analytic method for selecting the controller gains is also presented to ensure closed‐loop system stability and convergence at the desired equilibrium. Two nonlinear engineering examples that include an electric motor and a dc/dc converter are investigated using the conventional PID and the proposed sl‐PID to validate the superiority of the proposed controller in achieving the desired output regulation with a given bounded state requirement.     

Nonlinear output regulation for invertible nonlinear MIMO systems

In this paper, we address the problem of output regulation for a broad class of multi‐input multi‐output (MIMO) nonlinear systems. Specifically, we consider input–affine systems, which are invertible and input–output linearizable. This class includes, as a trivial special case, the class of MIMO systems which possess a well‐defined vector relative degree. It is shown that if a system in this class is strongly minimum phase, in a sense specified in the paper, the problem of output regulation can be solved via partial‐state feedback or via (dynamic) output feedback. The result substantially broadens the class of nonlinear MIMO systems for which the problem in question is known to be possible. Copyright © 2015 John Wiley & Sons, Ltd.