A complete stability analysis for planar delta operator systems subject to state saturation

In this paper, stability analysis is investigated for planar delta operator systems subject to state saturation. General properties of limit trajectories are characterized for the planar delta operator systems subject to state saturation. A nontrivial limit trajectory only intersects with two pairs of opposite sides but not neighboring sides of a unit square. Hence, a relation is established between the present intersection of a trajectory with lines x₂ = ±1 and the next intersection. Necessary and sufficient conditions on global asymptotic stability are obtained for the planar delta operator systems with full and partial saturated state. Simulation results are given to show the effectiveness of the proposed methods.     

Globally periodic behavior of switched flow networks with a cyclic switching policy

The paper considers the problem of the qualitative analysis of switched flow networks. Such networks can be used to model various flexible manufacturing, communication, and computer systems. We prove that for any network from a quite general class, there exists a limit cycle such that all the trajectories of the system converge to this cycle.     

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACT

In this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.     

Switched systems with multiple invariant sets

This paper explores dwell time constraints on switched systems with multiple, possibly disparate invariant limit sets. We show that, under suitable conditions, trajectories globally converge to a superset of the limit sets and then remain in a second, larger superset. We show the effectiveness of the dwell-time conditions by using examples of switching limit cycles commonly found in robotic locomotion and flapping flight.     

Lifting of trajectories of control systems related by smooth mappings

Given a pair of control systems and whose state spaces are differentiable manifolds M and N, respectively, we focus on situations where there exists a mapping that induces some type of correspondence between the trajectories of the systems. An important instance of this situation occurs when f is a complex system (for example, one having many states), h is a subsystem of f (presumably, with fewer states and a simpler structure than f), and Φ is a mapping from the full state space M to a reduced state space N. Some recent research has concentrated on the concept of Φ-related systems, where it is required that Φ send trajectories of f onto trajectories of h. Here we approach the problem from a different direction and ask under what conditions trajectories of the subsystem h can be lifted to trajectories of the full system f. We provide computable sufficient conditions for the local and global lifting of trajectories from N to M. Connections between the lifting problem and the problem of global (f,g)-invariance are also discussed briefly.     

Stability theory for nonnegative and compartmental dynamical systems with time delay

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments. These models are widespread in biological and physical sciences and play a key role in understanding these processes. A key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. In this paper, we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear Lyapunov–Krasovskii functionals.     

On the design and synthesis of limit cycles using switching linear systems

This article deals with the design and synthesis of limit cycles for a class of switched linear systems. This work is motivated by an application in the context of electrical networks, but the methodologies can be applied in other engineering fields as well. Several methods for the design and synthesis of limit cycles are presented. Based on a monodromy matrix associated with a periodically switched system, a design method and two feedback strategies are developed. The first strategy is based on linear feedback design using pole placement. The second strategy is based on the observation that certain state-dependent switching strategies can be implemented by means of a simple nonlinear output feedback controller. Advantages of this latter strategy are not only the ease with which the switching strategy can be implemented, but also the fact that classical techniques may also be used to ascertain the stability of the resulting limit cycle. Our next contribution is the development of a novel frequency domain-based approach to limit cycle design. This approach is based on the observation that the existence of certain limit cycles can be deduced from an infinity of circle criteria generated by a family of periodic systems. By making use of recent results, this observation can be used to develop a one-parameter spectral search to deduce the approximate frequencies of feasible limit cycles. Having selected a frequency of oscillation, one may then make use of the aforementioned nonlinear elements to realise a switching strategy that generates a stable limit cycle with given frequency and amplitude. Examples are given to illustrate the effectiveness of the approaches presented.     

Multicanonical sampling of rare trajectories in chaotic dynamical systems

In chaotic dynamical systems, a number of rare trajectories with low level of chaoticity are embedded in chaotic sea, while extraordinary unstable trajectories can exist even in weakly chaotic regions. In this study, a quantitative method for searching these rare trajectories is developed; the method is based on multicanonical Monte Carlo and can estimate the probability of initial conditions that lead to trajectory fragments of a given level of chaoticity. The proposed method is successfully tested with four-dimensional coupled standard maps, where probabilities as small as ¹⁰−14 are estimated.     

A comparison between LQR control for a long string of SISO systems and LQR control of the infinite spatially invariant version

In this paper we consider a long string of SISO systems which in the limit becomes a scalar infinite spatially invariant system. We compare the LQR control for long-but-finite strings with the LQR control for the corresponding infinite strings. We give analytical and numerical examples where these are quite different and we investigate the cause. In addition, we obtain sufficient conditions for the LQR solutions to be similar as the length of the string increases.     

Computation of the phase response curve: a direct numerical approach

Neurons are often modeled by dynamical systems--parameterized systems of differential equations. A typical behavioral pattern of neurons is periodic spiking; this corresponds to the presence of stable limit cycles in the dynamical systems model. The phase resetting and phase response curves (PRCs) describe the reaction of the spiking neuron to an input pulse at each point of the cycle. We develop a new method for computing these curves as a by-product of the solution of the boundary value problem for the stable limit cycle. The method is mathematically equivalent to the adjoint method, but our implementation is computationally much faster and more robust than any existing method. In fact, it can compute PRCs even where the limit cycle can hardly be found by time integration, for example, because it is close to another stable limit cycle. In addition, we obtain the discretized phase response curve in a form that is ideally suited for most applications. We present several examples and provide the implementation in a freely available Matlab code.     

A Transient-State Analysis of Tyson's Model for the Cell Division Cycle by Means of KCC-Theory

The transient-state stability analysis for the trajectories of Tyson's equations for the cell-division cycle is given by the so-called KCC-Theory. This is the differential geometric theory of the variational equations for deviation of whole trajectories to nearby ones. The relationship between Lyapunov stability of steady-states and limit cycles is throughly examined. We show that the region of stability (where, in engineering parlance, the system is hunting) encloses the Tyson limit cycle, while outside this region the trajectories exhibit aperiodic behaviour.     

Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems

This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.     

Control of constrained robots including effects of joint flexibility and actuator dynamics

We consider mathematical representations of constrained robot systems in which the effects of joint flexibility and actuator dynamics are significant. The objective is to design a feedback control law so that the position output variables and the force output variables of the robot follows the desired position and the desired force trajectories respectively despite the presence of joint flexibility and actuator dynamics. A systematic procedure is developed for designing a feedback control law which ensures that the position variables track the desired position trajectories exponentially, and the force variables track the desired force trajectories exponentially. The development of the control law is based on the model of a constrained robot system which includes the effects of actuator dynamics and joint flexibility. Thus using the force/position control law developed in this paper one can achieve better tracking performance in cases where such effects are significant.     

Global bounded consensus of multi-agent systems with non-identical nodes and communication time-delay topology

This article investigates the global bounded consensus problem of networked multi-agent systems exhibiting nonlinear, non-identical node dynamics with communication time-delays. Globally bounded consensus conditions for both delay-independent and delay-dependent conditions based on the Lypunov–Krasovskii functional method are derived. The proposed consensus criteria ensures that all agents eventually move along the desired trajectories in the sense of boundedness. The proposed consensus criteria generalises the case of identical agent dynamics to the case of non-identical agent dynamics, and many related results in this area can be viewed as special cases of the above results. We finally demonstrate the effectiveness of the theoretical results by means of a numerical simulation.     

Optimal fixed‐structure control for linear non‐negative dynamical systems

In this paper, we develop optimal output feedback controllers for set‐point regulation of linear non‐negative dynamical systems. Specifically, using a constrained fixed‐structure control framework we develop optimal output feedback control laws that guarantee that the trajectories of the closed‐loop system remain in the non‐negative orthant of the state space for non‐negative initial conditions. In addition, we characterize domains of attraction predicated on closed and open Lyapunov level surfaces contained in the non‐negative orthant for unconstrained optimal linear‐quadratic output feedback controllers. Output feedback controllers for compartmental systems with non‐negative inputs are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

An embedding approach for the design of state‐feedback tracking controllers for references with jumps

We study the problem of designing state‐feedback controllers to track time‐varying state trajectories that may exhibit jumps. Both plants and controllers considered are modeled as hybrid dynamical systems, which are systems with both continuous and discrete dynamics, given in terms of a flow set, a flow map, a jump set, and a jump map. Using recently developed tools for the study of stability in hybrid systems, we recast the tracking problem as the task of asymptotically stabilizing a set, the tracking set, and derive conditions for the design of state‐feedback tracking controllers with the property that the jump times of the plant coincide with those of the given reference trajectories. The resulting tracking controllers guarantee that solutions of the plant starting close to the reference trajectory stay close to it and that the difference between each solution of the controlled plant and the reference trajectory converges to zero asymptotically. Constructive conditions for tracking control design in terms of LMIs are proposed for a class of hybrid systems with linear maps and input‐triggered jumps. The results are illustrated by various examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalisation of Euler–Lagrange equations to find min–max optimal solution of uncertain systems

A major problem in optimal control theory is existing uncertainty in initial state, cost function, or in state equations. In this paper, a generalised Euler–Lagrange approach to find min–max optimal solution of uncertain systems with uncertain or certain cost function using calculus of variation is considered. The work is based on an admissible system path or a trajectory, which minimises the maximum value of the functional over all uncertainty. First of all, a new form of Euler–Lagrange conditions for uncertain systems is presented. Then several cases are indicated where final condition can be specified or free. Also necessary conditions are introduced to existence of min–max optimal solution of the uncertain systems. Also the method is generalised when the uncertainties are bounded with using Pontryagin's minimum principle. Finally, efficiency of the proposed methods is verified through some examples.     

Computational networks and systems – homogenization of variational problems on micro-architectured networks and devices

Networked materials and micro-architectured systems gain increasingly importance in multi-scale physics and engineering sciences. Typically, computational intractable microscopic models have to be applied to capture the physical processes and numerous transmission conditions at singularities, interfaces and borders. The topology of the periodic microstructure governs the effective behaviour of such networked systems. A mathematical concept for the analysis of microscopic models on extremely large periodic networks is developed. We consider microscopic models for diffusion–advection–reaction systems in variational form on periodic manifolds. The global characteristics are identified by a homogenization approach for singularly perturbed networks with a periodic topology. We prove that the solutions of the variational models on varying networks converge to a two-scale limit function. In addition, the corresponding tangential gradients converge to a two-scale limit function for vanishing lengths of branches. We identify the variational homogenized model. Complex network models, previously considered as completely intractable, can now be solved by standard PDE-solvers in nearly no time. Furthermore, the homogenized coefficients provide an effective characterization of the global behaviour of the variational system.     

Technology diffusion: analysing the diffusion of agent technologies

Despite several examples of deployed agent systems, there remain barriers to the large-scale adoption of agent technologies. In order to understand these barriers, this paper considers aspects of marketing theory which deal with diffusion of innovations and their relevance to the agents domain and the current state of diffusion of agent technologies. In particular, the paper examines the role of standards in the adoption of new technologies, describes the agent standards landscape, and compares the development and diffusion of agent technologies with that of object-oriented programming. The paper also reports on a simulation model developed in order to consider different trajectories for the adoption of agent technologies, with trajectories based on various assumptions regarding industry structure and the existence of competing technology standards. We present details of the simulation model and its assumptions, along with the results of the simulation exercises.     

Stability and performance analysis of time‐delay LPV systems with brief instability

Linear parameter‐varying (LPV) systems provide a systematic framework for the study of nonlinear systems by considering a representative family of linear time‐invariant systems parameterized by system parameters residing in a compact set. The brief instability concept in such systems allows the linear system to be unstable for some trajectories of the LPV parameter set, so that instability occurs only for short periods of time. In the present paper, we extend the notion of brief instability to LPV systems with time delay in their dynamics. The results provide tools for the stability and performance analysis of such systems, where performance is evaluated in terms of induced ??₂‐gain (or so‐called ??_∞ norm). The main results of this paper illustrate that stability and performance conditions can be evaluated by examining the feasibility of parameterized sets of linear matrix inequalities (LMIs). Using the results of this paper, we then investigate analysis conditions to guarantee the asymptotic stability and ??_∞ performance of fault‐tolerant control (FTC) systems, in which instability may take place for a short period of time due to the false identification of the fault signals provided by a fault detection and isolation (FDI) module. The numerical examples are used to illustrate the qualification of the proposed analysis and synthesis results for addressing brief instability in time‐delay systems. Copyright © 2010 John Wiley & Sons, Ltd.