Discontinuous Systems

This book is intended to develop analysis tools and the control synthesis of discontinuous systems operating under uncertainty conditions. Throughout this book, discontinuous systems are regarded as the ones consisting of a family of continuoustime subsystems, equipped with the rule of switching between them. Conceptually, these systems are described as follows. Let the continuous state space, say an Euclidean space or a Hilbert space, be partitioned into an infinite number of operating domains by means of a family of switching surfaces. Inside an operating domain, such a system is governed by an ordinary differential equation with a differentiable right-hand side. Whenever the system trajectory hits a switching surface, the continuous state makes a jump, specified by a restitution rule. Such a rule is a map whose domain and range are the union of the switching surfaces and the entire state space, respectively. The state jumps are typically referred to as impulse effects. A special case is when impulse effects are absent. Just in case, the state trajectory is always continuous, but in general it is not differentiable when it hits a switching surface. After hitting a switching surface, the trajectory can either cross it or evolve along the surface on a finite time interval. In the latter case, sliding motions occur in the system. To summarize, a discontinuous system is specified by i) the family of switching surfaces determining the operating domains, ii) the family of continuous time subsystems, well-posed for each domain, iii) the restitution rule, determining potential jumps of the system, and iv) the sliding mode dynamics, governing potential state motions on the switching surfaces. We introduce here some basic models of discontinuous systems to preview issues that will arise in our study.

The present section develops a consistent modeling methodology which gives rise to mathematical models of nonlinear dynamical systems with nonlinear impulse responses. Modeling accommodates the details of the realization of the impulse input, and admits the impulse response to depend upon the manner in which the impulse is implemented. The set of all possible impulse responses is found from a certain auxiliary system with integrable inputs. Necessary and sufficient conditions are additionally obtained for the impulse response to be unique and independent of the impulse realization. The proposed modeling methodology is subsequently used to derive filtering equations over sampled-datameasurements and to synthesize impulsive controllers.

The analysis of discontinuous dynamic systems has attracted a considerable research interest in the last decades. Although the existing literature on this subject includes numerous books and papers such as [30, 52, 77, 108, 126, 132, 139, 212, 227], to name a few, these systems are far from being fully understood. In the present chapter, the analysis tools of such systems are developed within the framework of the second Lyapunov method with a focus on the use of nonsmooth Lyapunov functions, possessing non-positive time derivatives along the system trajectories.

Until recently, the finite-time stability of asymptotically stable homogeneous systems has been well-recognized for only continuous vector fields [26, 100, 153]. Extending this result to switched systems has required proceeding differently [128, 169] because a smooth homogeneous Lyapunov function, whose existence was proven in [195] for continuous asymptotically stable homogeneous vector fields, can no longer be brought into play. The aforementioned work [169] forms a basis of the present chapter, where the finite-time stability property is established for homogeneous asymptotically stable discontinuous systems whose homogeneity degree is negative. Exemplified with a second-order system, the finite-time stability is then additionally demonstrated to remain in force regardless of inhomogeneous perturbations.

Quasihomogeneity-based control synthesis is presently developed to stabilize uncertain minimum phase systems of uniform m-vector relative degree (2, . . . ,2) ^T . The proposed synthesis does not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts.

The sliding mode control technique is long recognized as a powerful controlmethod to counteract non-vanishing external disturbances and unmodeled dynamics. The standard sliding mode control is synthesized to steer the system to a submanifold in finite time, after that the system stays in this submanifold forever. Typically [227], there are several switching points as one coordinate after the other hits the discontinuity manifold, and in order to design such a controller, one needs to additionally follow the hierarchy of the control components that ensures the existence of the sliding motion.

ℋ_∞-disturbance attenuation theory is fully understood when the underlying system is linear. In the state-space formulation, the problem of minimizing the ℋ_∞-norm of a linear control system is viewed as a differential game of two antagonistic persons and a solution of the problem relates to certain solutions of the Riccati equations arising in linear quadratic differential game theory (see, e.g., [18, 62] for details).

The unit feedback synthesis is developed for a class of linear infinite-dimensional systems with a finite-dimensional unstable part using finite-dimensional sensing and actuation. The present development is essentially from [180]. Initially, the class of infinite-dimensional systems is precisely formulated. Modal decomposition is then used to decompose the original infinite-dimensional system into an interconnection of a finite-dimensional (possibly unstable) system and an infinite-dimensional stable system. A stabilizing unit state feedback controller is constructed on the basis of the finite-dimensional system. Subsequently, an infinite-dimensional Luenberger state observer, which utilizes a finite number of measurements, is constructed to provide estimates of the state of the infinite-dimensional system. Finally, an output feedback controller design is completed by coupling the infinite-dimensional Luenberger state observer and the unit state feedback controller. In order to obtain the fully practical finite-dimensional framework for controller synthesis, a finite-dimensional approximation of the Luenberger observer as well as a continuous approximation of the unit feedback controller are carried out at the implementation stage.

In the present section, the unit control approach is developed for minimum phase semilinear infinite-dimensional systems driven in a Hilbert space. Control algorithms presented ensure asymptotic stability, global or local according as state feedback or output feedback is available. The desired robustness properties of the closedloop system against external disturbances with a priori known norm bounds make the algorithms extremely suited for stabilization of the underlying system operating under uncertainty conditions. It is, in particular, shown that discontinuous feedback stabilization is constructively available in the case where complex nonlinear dynamics of the uncertain system does not admit factoring out a destabilizing nonlinear gain, and thus the destabilizing gain can not be handled through nonlinear damping. The theory is applied to the stabilization of chemical processes around pre-specified steady-state temperature and concentration profiles corresponding to a desired coolant temperature. Two specific cases, a plug flow reactor and an axial dispersion reactor, governed by hyperbolic and parabolic partial differential equations of the first order and of the second order, respectively, are under consideration. To achieve a regional temperature feedback stabilization around the desired profiles, with the region of attraction, containing a prescribed set of interest, a component concentration observer is constructed and included into the closed-loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice. Performance issues of the unit feedback design are illustrated in a simulation study of the plug flow reactor.

To this end, the unit feedback synthesis is developed for a class of uncertain timedelay systems with nonlinear disturbances and unknown delay values whose unperturbed dynamics are linear. Being inspired from [179], the present synthesis is based on the delay-dependent stability criterion, which is derived within the framework developed in Sect. 3.8. The controller constructed proves to be robust against sufficiently small delay variations and weak external (possibly, unmatched) disturbances. It is worth noticing that allowing unmatched disturbances is a step beyond a standard sliding mode control treatment. Specifically, the critical delay value when the closed-loop system, corresponding to this value, becomes asymptotically unstable, is explicitly calculated as a function of linear growth constants of the unmatched disturbances. Performance issues of the controller are illustrated in a simulation study.

The robust control of mechanical systems has attracted considerable research interest (see, e.g., related surveys compiled in [10] and [199]). The existing controllers from [19, 20, 44, 68, 225], which were derived via the nonlinear ℋ_∞-control approach coupled to the feedback linearization techniques, as well as passivity-based controllers from [187], are widely used in practice due to their robustness and simplicity of implementation. However, the frictional influence and backlash effects, as well as the fact of the incompleteness of the state measurements, were ignored in these studies, that severely limited the achievable performance and their practical utility. The nonsmooth ℋ_∞-synthesis of Chap. 7 is, in principle, capable of accounting for hard-to-model friction forces and backlash effects. It is the latter approach that is applied in the present chapter to regulation problems for a frictional mechanical manipulator and for a servomechanism with backlash, both with incomplete measurements. Due to the nature of the approach, the resulting regulators yield the desired robustness properties against the discrepancy between the real friction/backlash phenomenon and that described in the model.

Quasihomogeneity-based synthesis appears to be extremely suited for fully actuated systems with dry friction. This claim is supported by several arguments. Firstly, mechanical systems with relatively strong Coulomb friction require discontinuous controllers for adequate regulation. Secondly, external disturbances, affecting these systems, meet the matching condition so that their influence on the underlying system is not simply attenuated as it would be the case under ℋ_∞-synthesis, but it is also rejected under the quasihomogeneous synthesis. Thirdly, the global position regulation becomes possible provided that an upper bound on the magnitude of the external disturbances is known a priori. These features are subsequently illustrated by means of the orbitally stabilizing synthesis of a simple inverted pendulum and by means of the global position regulation of a multi-link robot manipulator.

The stabilization of underactuated systems, forced by fewer actuators than DOFs, is more complex than that of fully actuated systems [67, 186]. As well known (see, e.g., [23, 249]), these systems possess nonholonomic properties, caused by nonintegrable differential constraints, and therefore, they cannot be stabilized by means of smooth feedback. Due to this, the present investigation is based on hybrid synthesis, which is recognized as an effective tool of controlling underactuated mechanical systems [250, 251]. Representative papers analyzing some control problems for underactuated systems include the study of accessibility [191], the stabilization of equilibria through passivity techniques [186] and energy shaping [27], stabilization and tracking via backstepping control [206], the use of virtual constraints to produce stable oscillations [209], path planning [37], and the control of mechanical systems with an unactuated cyclic variable [86], among others.