Class Of Discontinuous Dynamical Systems Research Articles

A Small-Gain Approach to Incremental Input-to-State Stability Analysis of Hybrid Integrator-Gain Systems

Incremental input-to-state stability plays an important role in the analysis of nonlinear systems, as it opens up the possibility for accurate performance characterizations beyond classical approaches. In this paper, we are interested in deriving conditions for incremental stability of a specific class of discontinuous dynamical systems containing a so-called hybrid integrator. Recently, it was shown that hybrid integrators have the potential for overcoming fundamental performance limitations of linear time-invariant control, thereby making them interesting for use in, e.g., high-precision motion control applications. The main contribution of this paper is to show that these hybrid integrators have incremental input-to-state stability properties, and that, under an incremental small-gain condition, the feedback interconnection of a hybrid integrator and a linear time-invariant plant is incrementally input-to-state stable.

Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization

Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vec...

Optimal control approach for discontinuous dynamical systems

SummaryIn this work, we consider a wide class of discontinuous dynamical systems, discontinuity of which is based on the sign (for short sgn) function. We propose a smooth optimal control problem to solve the main discontinuous system. By solving some numerical examples in mechanical engineering, we show the efficiency of our approach with respect to 2 smoothing methods for discontinuous systems.

OGY method for a class of discontinuous dynamical systems

In this paper, we prove that the OGY method to control unstable periodic orbits (UPOs) of continuous-time systems can be applied to a class of systems discontinuous with respect the state variable, by using a generalized derivative. Because the discontinuous problem may have not classical solutions, the initial value problem is transformed into a set-valued problem via Filippov regularization. The existence of the ingredients necessary to apply OGY method (UPO, Poincare map and stable and unstable directions) is proved and the numerically implementation is explained. Another possible way analyzed in this paper is the continuous approximation of the underlying initial value problem, via Cellina’s theorem for differential inclusions. Thus, the problem is approximated by a continuous initial value problem, and the OGY method can be applied as usual.

Chaos suppression via periodic change of variables in a class of discontinuous dynamical systems of fractional order

By introducing a suitable change of variable theorem for a class of fractional discontinuous equations, we study the possibility to use a periodic perturbation algorithm to stabilize chaotic trajectories. For this purpose, some new issues of fractional differential inclusions and results on Filippov systems are used. The algorithm, which periodically changes the system variables, has been used so far to stabilize discrete, continuous and discontinuous systems of integer order. As an example, a piece-wise continuous variant of the Chen system is utilized.

APPROACH OF A CLASS OF DISCONTINUOUS DYNAMICAL SYSTEMS OF FRACTIONAL ORDER: EXISTENCE OF SOLUTIONS

In this paper, we are concerned with the possible approach to the existence of solutions for a class of discontinuous dynamical systems of fractional order. To this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a Péano-like theorem for ordinary differential equations of fractional order.

Hybrid automata: an insight into the discrete abstraction of discontinuous systems

We develop a novel computational–dynamical framework for the modelling of a class of discontinuous dynamical systems (DDSs). In particular, what is referred to as the DDS hybrid automaton with inputs and outputs is proposed. This is a general hybrid automaton that provides a suitable mathematical model for DDSs with discontinuous state derivatives and sliding motions. The chief characteristic of this model is that, following the computational divide-and-conquer principle, a system with multiple discontinuous elements can be represented by the composition of several DDS hybrid automata. Although discontinuous, non-smooth or switched dynamical systems have been well-investigated within different frameworks, it is still a challenge to give satisfactory solutions for specifying the transitions between the different modes of operation of these systems. We propose a new way of solving this problem, which is especially effective for systems with multiple switching elements. An example is used to illustrate these ideas. Several simulations are presented. The simulations results are obtained with Stateflow® and Modelica®.

Integral Input-to-Output Stability for one Class of Discontinuous Dynamical Systems

It is well known that integral input-to-output stability(IIOS) is weaker than input-to -output stability(IOS).In this paper,IIOS problems for one class of discontinuous dynamical systems are considered.Piecewise smooth IIOS-Lyapunov functions are adopted.Furthermore,interconnecti ons of discontinuous dynamical systems are studied.The results for a class of discontinuous dynam ical systems and interconnection of discontinuous dynamical systems' IIOS character are shown,res pectively.

Numerical approximation of a class of discontinuous systems of fractional order

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams–Bashforth–Moulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.

Chaotic behavior of a class of discontinuous dynamical systems of fractional-order

In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that end, the initial value problem is first transformed, by using the Filippov regularization (Filippov in Differential Equations with Discontinuous Right-Hand Sides, 1988), into a set-valued problem of fractional-order, then by Cellina’s approximate selection theorem (Aubin and Cellina in Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska in Set-valued Analysis, 1990). The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical scheme proposed by Diethelm et al. (Nonlinear Dyn. 29:3–22, 2002). Two typical examples of systems belonging to this class are analyzed and simulated.

On the uniqueness of solutions to a class of discontinuous dynamical systems

The study of uniqueness of solutions of discontinuous dynamical systems has an important implication: multiple solutions to the initial value problem could not be found in real dynamical systems; also the (attracting or repulsive) sliding mode is inherently linked to the uniqueness of solutions. In this paper a strengthened Lipschitz-like condition for differential inclusions and a geometrical approach for the uniqueness of solutions for a class of Filippov dynamical systems are introduced as tools for uniqueness. Several theoretical and practical examples are discussed.

Hybrid-automaton models for simulating systems with sliding motion: still a challenge

Abstract There is a gap between the modelling and the simulation of discontinuous dynamical systems (DDS) exhibiting sliding-mode behaviour. Several challenges arise. Mainly, the definition of unique solutions on the discontinuity surfaces and at their crossings. There are also numerical issues in the detection and location of the trajectory entering and going out of the discontinuity surface, as well as when it remains on the discontinuity surface (chattering phenomenon). The aim of this paper is establishing a semantics for DDS so that a deterministic dynamical behaviour can be defined for simulation purposes. Particularly, a class of DDS is modelled by using two different hybrid automata. One of them is called the DDS hybrid automaton , proposed by the author previously. The other is obtained by modifying the DDS hybrid automaton , and is inspired in computer-simulation-oriented friction models. The use of a computational model is an elegant way for specifying the multiple transitions in DDS. The modelling framework is specially useful for specifying transitions when multiple discontinuity surfaces are present. A system with discontinuous friction and stick-slip oscillations is used to validate the models. In the example, it is concluded that discontinuous systems can be appropriately simulated by using hybrid-automaton-based models without the problem of chattering. On the other hand, the system dynamical behaviour can change depending on the step size of the numerical integration method and on the hybrid-automaton representation used. The hybrid models are simulated by means of Simulink/Stateflow ® package.

Hybrid modelling of a discontinuous dynamical system including switching control

Although discontinuous, non-smooth or switched dynamical systems have been broadly studied, there still exist dynamical behaviours which are not completely understood. In particular, those related to transitions between state-space regions and to crossings through discontinuity surfaces. The problem of specifying system transitions becomes crucial when multiple discontinuity surfaces or switching elements are present. In this case, a great deal of care has to be taken in order to simulate the system and to verify its properties. Obtaining a computational model appears to be an elegant way for the specification of the transitions and event-triggered phenomena involved in the dynamical behaviour of discontinuous systems. This is the goal of this paper. A class of discontinuous dynamical systems including control inputs and outputs is reinterpreted within the hybrid-automata framework, and what is referred as to discontinuous-dynamical-systemhybrid automaton is proposed. An example is used, which corresponds to a 2-degrees-of-freedom electromechanical system including discontinuous friction and sliding-mode control. This system exhibits several discontinuity surfaces which give rise to different types of transitions and dynamical behaviours. New computational-kind phenomena may arise when the system is reinterpreted as a hybrid dynamical system. Nevertheless, this paper only deals with the modelling and specification of the system.

Stability of discontinuous Cauchy problems in Banach space

We present Lyapunov stability results, including Converse Theorems, for a class of discontinuous dynamical systems (DDS) determined by differential equations in Banach space or Cauchy problems on abstract spaces. We demonstrate the applicability of our results in the analysis of several important classes of DDS, including systems determined by functional differential equations, Volterra integro-differential equations and partial differential equations.

Stability of discontinuous retarded functional differential equations with applications

We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations.

ASYMPTOTIC STABILITY OF DISCONTINUOUS CAUCHY PROBLEMS IN BANACH SPACE WITH APPLICATIONS

We present an asymptotic stability result for a class of discontinuous dynamical systems (DDS) determined by differential equations in Banach space (resp., Cauchy problems on abstract spaces). We demonstrate the applicability of our result in the analysis of several important classes of DDS, including systems determined by functional differential equations and partial differential equations.

Sliding Motion and Stability of a Class of Discontinuous Dynamical Systems

A closed container with gas and liquid inflows and one outlet for outflow, described by a nonlinear hydraulic model, is shown to reach a state of sliding mode. Through a Lyapunov function, the sliding motion is shown to be stable for all positive exponents in the flow model. This property is independent of factors such as valve opening, valve coefficients, friction factor and initial states, making this device a suitable one to study two phase flow. The sliding solution of this system, with square root hydraulic model, is found using Filippov's equivalent dynamic approach. The discontinuous system is also solved using an accurate and efficient method that accounts for persistent discontinuity sticking. These two solutions are shown to be in exact agreement.

CHAOTIFYING DISCONTINUOUS DYNAMICAL SYSTEMS VIA TIME-DELAY FEEDBACK ALGORITHM

In this paper the anti-control technique of chaos to systems continuous with respect to the state variable using a time-delay feedback technique introduced by Wang, Chen and Yu, is adapted to a class of dynamical systems discontinuous with respect to the state variable. The considered discontinuous initial value problem is transformed into a differential inclusion using the Filippov regularization. Then, certain results on existence and uniqueness of solutions to differential inclusions are used to define our class of discontinuous dynamical systems. Afterwards, introducing an adequate concept of derivative for the considered discontinuous functions, we show that the algorithm for continuous dynamical systems can be adapted to our class of discontinuous problems. Three examples are given.

An Accurate and Efficient Integration Algorithm for a Class of Discontinuous Dynamical Systems in Sliding Motion

The concept of correctness of the numerical solution of a gas–liquid system with persistent discontinuity sticking in the gas model is proposed. An algorithm that handles discontinuity while in the liquid model alone is shown to be correct. This technique is applied to the gas–liquid system with linear hydraulics and compared with the analytical solution. For a system with squareroot hydraulics, the results of this approach are compared with those obtained from the numerical solution of equivalent dynamics. The proposed technique is one to two orders of magnitude more efficient than a previously reported, and a less accurate, method.

Controlling chaos in discontinuous dynamical systems

In this paper we consider the possibility to implement the technique of changes in the system variables to control the chaos introduced by Güemez and Matías for continuous dynamical systems to a class of discontinuous dynamical systems. The approach is realized via differential inclusions following the Filippov theory. Three practical examples are considered.