Stability of feedback systems with periodic parameters†

This paper deals with a frequency domain technique for stability analysis of linear feedback systems with periodically time-varying parameters. The approximate method of signal balance used by Clark et al. (1064) is generalized to yield better results in the case of a wide variety of systems including some TVP-coupled systems. For systems with TVP coupling, a novel and simple method of ‘pivotal points’ is suggested for quickly sketching the stability boundaries making use of the known stability of subsystems. The results of perturbation analysis (Rao 1969) are found to support the pivotal paint method.     

The role of connection in the nonlinear behavior of locomotion systems with symmetry

Using a geometric framework, the role of connection in nonlinear behavior of locomotion systems with symmetry is investigated in this paper. It is shown that the covariant derivative of connection plays a fundamental role in describing the nonlinear behavior of locomotion systems with Lie group symmetries. In particular, by placing prescribed closed paths properly in the shape space, it is argued that the system will exhibit nonlinear behaviors such as limit cycles and bifurcation in the fiber. The idea is used to investigate nonlinear behavior of a three link fish-like articulated body in perfect fluid. Numerical results are presented showing limit cycles and other coherent gaits resulting from various locations of shape circle.     

H∞ control of nonlinear continuous-time systems based on dynamical fuzzy models

This paper presents a solution of the H_∞ control problem for a class of continuous-time nonlinear systems. The method is based on a fuzzy dynamical model of the nonlinear system. A suitable piecewise differentiate quadratic (PDQ) Lyapunov function is used to establish asymptotic stability of the closed-loop system. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of suitable algebraic Riccati equations. An example is given to illustrate the application of the method     

Spectral necessary and sufficient conditions for sampling-period-independent stabilisation of periodic and aperiodic sampled-data systems using a class of generalised sampled-data hold functions

Sampling-period-independent (SPI) stabilisation of both periodic and aperiodic sampled-data systems using a class of generalised sampled-data hold functions is addressed. It is proved that for this specific class of hold functions, a continuous-time linear system with rank-minimal input matrix and with non-defective eigenvalues on the imaginary axis is SPI-stabilisable if and only if the spectrum of the system is contained in the closed left-half plane. A systematic procedure for constructing suitable state-feedback controllers is also provided. Several examples are finally discussed for illustration.     

Modeling of nonlinear dynamical systems based on deterministic learning and structural stability基于确定学习和结构稳定理论的非线性动态系统建模

Recently, a deterministic learning (DL) theory was proposed for accurate identification of system dynamics for nonlinear dynamical systems. In this paper, we further investigate the problem of modeling or identification of the partial derivative of dynamics for dynamical systems. Firstly, based on the locally accurate identification of the unknown system dynamics via deterministic learning, the modeling of its partial derivative of dynamics along the periodic or periodic-like trajectory is obtained by using the mathematical concept of directional derivative. Then, with accurately identified system dynamics and the partial derivative of dynamics, a C¹-norm modeling approach is proposed from the perspective of structural stability, which can be used for quantitatively measuring the topological similarities between different dynamical systems. This provides more incentives for further applications in the classification of dynamical systems and patterns, as well as the prediction of bifurcation and chaos. Simulation studies are included to demonstrate the effectiveness of this modeling approach.     

Energetically consistent simulation of simultaneous impacts and contacts in multibody systems with friction

This paper presents a methodology for treating energy consistency when considering simultaneous impacts and contacts with friction in the simulation of systems of interconnected bodies. Hard impact and contact is considered where deformation of the impacting surfaces is negligible. The proposed approach uses a discrete algebraic model of impact in conjunction with moment and tangential coefficients of restitution (CORs) to develop a general impact law for determining post-impact velocities. This process depends on impulse–momentum theory, the complementarity conditions, a principle of maximum dissipation, and the determination of contact forces and post-impact accelerations. The proposed methodology also uses an energy-modifying COR to directly control the system’s energy profile over time. The key result is that different energy profiles yield different results and thus energy consistency should be considered carefully in the development of dynamic simulations. The approach is illustrated on a double pendulum, considered to be a benchmark case, and a bicycle structure.     

Adaptive output feedback stabilisation for planar nonlinear systems with unknown control coefficients

The paper is concerned with the global adaptive stabilisation via output feedback for a class of uncertain planar nonlinear systems. Remarkably, the unknowns in the systems are rather serious: the control coefficients are unknown constants which do not belong to any known interval, and the growth of the systems heavily depends on the unmeasured states and has the rate of unknown polynomial of output. First, a delicate state transformation is introduced to collect the unknown control coefficients, and subsequently, a suitable state observer is successfully designed with two different dynamic gains. Then, an adaptive output feedback controller is proposed by flexibly combining the universal control idea and the backstepping technique. Meanwhile, an appropriate estimation law is constructed to overcome the negative effect caused by the unknown control coefficients. It is shown that, with the appropriate choice of the design parameters, all the states of the resulting closed-loop system are globally bounded, and furthermore, the states of the original system converge to zero.     

Formal verification of real-time systems with preemptive scheduling

In this paper, we propose a method for the verification of timed properties for real-time systems featuring a preemptive scheduling policy: the system, modeled as a scheduling time Petri net, is first translated into a linear hybrid automaton to which it is time-bisimilar. Timed properties can then be verified using HyTech. The efficiency of this approach leans on two major points: first, the translation features a minimization of the number of variables (clocks) of the resulting automaton, which is a critical parameter for the efficiency of the ensuing verification. Second, the translation is performed by an over-approximating algorithm, which is based on Difference Bound Matrix and therefore efficient, that nonetheless produces a time-bisimilar automaton despite the over-approximation. The proposed modeling and verification method are generic enough to account for many scheduling policies. In this paper, we specifically show how to deal with Fixed Priority and Earliest Deadline First policies, with the possibility of using Round-Robin for tasks with the same priority. We have implemented the method and give some experimental results illustrating its efficiency.

H∞control of interconnected nonlinear sampled-data systems with parametric uncertainty

This paper studies the problem of robust control design for a class of interconnected uncertain systems under sampled measurements. The class of system under consideration is described by a state space model containing unknown cone bounded nonlinear interaction and time-varying norm-bounded parameter uncertainties in both state and output equations. Our attention is focused on the design of linear dynamic output feedback controllers using sampled measurements. We address the problem of robust H_∞ control in which both robust stability and a prescribed H_∞ performance are required to be achieved irrespective of the uncertainties and nonlinearities. The H_∞ performance measure involves both continuous-time and discrete-time signals. It has been shown that the above problems can be recast into H_∞ syntheses for related N decoupled linear sampled-data systems without parameter uncertainties and unknown nonlinearities, which can be solved in terms of Riccati differential equations with finite discrete jumps. A numerical example is given to show the potential of the proposed technique.     

Permanence and almost periodic solution of two-species delayed Lotka–Volterra cooperative systems with impulsive perturbations

By applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a class of two-species Lotka–Volterra cooperative systems with impulses. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.     

On global controllability of affine nonlinear systems with a triangular-like structure

In this paper, we investigate a class of affine nonlinear systems with a triangular-like structure and present its necessary and sufficient condition for global controllability, by using the techniques developed by Sun Yimin and Guo Lei recently. Furthermore, we will give two examples to illustrate its application.     

Satisfactory control of discrete-time linear periodic systems

In this paper satisfactory control for discrete-time linear periodic systems is studied. Based on a suitable time-invariant state sampled reformulation, periodic state feedback controller has been designed such that desired requirements of steady state covariance, H-infinity rejection bound and regional pole assignment for the periodic system are met simultaneously. By using satisfactory control theory, the problem of satisfactory periodic controller can be transformed into a linear programming problem subject to a set of linear matrix inequalities （LMIs）, and a feasible designing approach is presented via LMI technique. Numeric example validates the obtained conclusion.     

Optimal higher order learning adaptive control approach for a class of SISO nonlinear systems

1IntroductionMany dynamic systems to be controlled have constant orslowly_varying uncertain parameters .Adaptive control is apopular approachtothe control ofsuchsystems [1] .Inthepast two decades ,significant progress has been made in theresearch and design of adaptive control systems [2,3] .Fairly complete and comprehensive guidelines are nowavailable for both design and implementation of adaptivecontrollers inthe cases where the systemunder control canbe adequately modeled as a linear dynami…     

Robust adaptive tracking control of MIMO nonlinear systems in the presence of actuator hysteresis

Adaptive tracking control of a class of MIMO nonlinear system preceded by unknown hysteresis is investigated. Based on dynamic surface control, an adaptive robust control law is developed and compensators are designed to mitigate the influences of both the unknown bounded external uncertainties and the unknown Prandtl–Islinskii hysteresis. By adopting the low-pass filters, the explosion of complexity caused by tedious computation of the time derivatives of the virtual control laws is overcome. With the proposed control scheme, the closed-loop system is proved to be semi-globally ultimately bounded by the Lyapunov stability theory, and the output of the controlled system can track the desired trajectories with an arbitrarily small error. Finally, numerical simulations are given to verify the effectiveness of the proposed approach.     

Analysis of multibody systems with flexible plates using variational graph-theoretic methods

This work treats the problem of modelling multibody systems with structural flexibility. By combining linear graph theory with the principle of virtual work and finite elements, a dynamic formulation is obtained that extends graph-theoretic (GT) modelling methods to the analysis of thin flexible plates for multibody systems. The system is represented by a linear graph, in which nodes represent reference frames on flexible plates, and edges represent components that connect these frames. To generate the equations of motion with elastic deformations, the flexible plates are discretized using a triangular thin shell finite element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as satellite panels, flatbed trailers, and mechanisms with plates. Three flexible systems with plates are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for flexible multibody systems (FMS) without additional user input.     

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems concerned are of （generalized） switched Byrnes-Isidori canonical form, which has all switched models in （generalized） Byrnes- Isidori canonical form. First, a stability result of switched systems is obtained. Then it is used to solve the stabilization problem of the switched nonlinear control systems. In addition, necessary and sufficient conditions are obtained for a switched affine nonlinear system to be feedback equivalent to （generalized） switched Byrnes-Isidori canonical systems are presented. Finally, as an application the stability of switched lorenz systems is investigated.     

Robust H ∞ fuzzy control of nonlinear systems with multiple time delays

A robustness design of fuzzy control via model-based approach is proposed in this article to overcome the effect of approximation error between multiple time-delay nonlinear systems and Takagi--Sugeno (T-S) fuzzy models. A stability criterion is derived based on Lyapunov's direct method to ensure the stability of nonlinear multiple time-delay systems especially for the resonant and chaotic systems. Positive definite matrices P and R_k of the criterion are obtained by using linear matrix inequality (LMI) optimization algorithms to solve the robust fuzzy control problem. In terms of the control scheme and this criterion, a fuzzy controller is then designed via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay system and the H ^∞ control performance is achieved at the same time. Finally, two numerical examples of the chaotic and resonant systems are demonstrated to show the concepts of the proposed approach.