Universal characteristic decomposition of radical differential ideals

We call a differential ideal universally characterizable, if it is characterizable w.r.t. any ranking on partial derivatives. We propose a factorization-free algorithm that represents a radical differential ideal as a finite intersection of universally characterizable ideals. The algorithm also constructs a universal characteristic set for each universally characterizable component, i.e., a finite set of differential polynomials that contains a characterizing set of the ideal w.r.t. any ranking. As a part of the proposed algorithm, the following problem of satisfiability by a ranking is efficiently solved: given a finite set of differential polynomials with a derivative selected in each polynomial, determine whether there exists a ranking w.r.t. which the selected derivatives are leading derivatives and, if so, construct such a ranking.     

Invertibility of switched nonlinear systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.     

Full state constraints-based adaptive control for switched nonlinear pure-feedback systems

This paper investigates the problem of full state constraints-based adaptive control for a class of switched nonlinear pure-feedback systems under arbitrary switchings. First, the switched pure-feedback system is transformed into a switched strict-feedback system with non-affine terms based on the mean value theorem. Then, by exploiting the common Lyapunov function (CLF) method, the Barrier Lyapunov function method and backstepping, state feedback controllers of individual subsystems and a common Barrier Lyapunov function (CBLF) are constructed, which guarantee that all signals in the closed-loop system are global uniformly bounded under arbitrary switchings, and full state constraints are not violated. Furthermore, the tracking error can converge to a bounded compact set. Two examples, which include a single-link robot as a practical example, are provided to demonstrate the effectiveness of the proposed design method.     

An algorithm for the robust exponential stabilization of a class of switched systems

In this paper, we consider continuous‐time switched systems whose subsystems are linear, or, more generally, homogeneous of degree one. For that class of systems, we present a control algorithm that under certain conditions generates switching signals that globally exponentially stabilizes the switched system, even in the case in which there are model uncertainties and/or measurement errors, provided that the bounds of that uncertainties and errors depend linearly on the norm of the state of the system and are small enough in a suitable sense. We also show that in the case in which the measurement errors and the model uncertainties are bounded, the algorithm globally exponentially stabilizes the system in a practical sense, with a final error which depends linearly on the bounds of both the model uncertainties and the measurement errors. In other words, the closed‐loop system is exponentially input‐to‐state‐stable if one considers the perturbations and output measurements bounds as inputs. For switched linear systems, under mild observability conditions, we design an observer whose state‐estimation drives the control algorithm to exponentially stabilize the system in absence of perturbations and to stabilize it in an ultimately bounded way when the perturbations and the output measurement errors are bounded. Finally, we illustrate the behavior of the algorithm by means of simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

A DESCRIPTION OF A SET OF HYPOTHESES FOR SYSTEM IDENTIFICATION PROBLEM*

This paper concerns the system identification process which is a specific form of the hypothetico-deductive process. More specifically, this paper deals with the inductive inference, i.e., with the process of generating a set of hypotheses σ(E) that explain a given finite set of input-output experiments performed on a finite sequential system being identified. It is shown that Sεσ(E) if and only if there exists a homomorphism of a basic hypothesis explaining E, into S. Next, a set of hypotheses σ¹(E). defined as follows: Sεσ¹(E) if E is structure-complete w.r.t. S, is considered. Then it is proved that Sεσ¹(E) if and only if there exists a full homomorphism of a basic hypothesis onto S. Some important methodological consequences of obtained results are derived. Finally, relationship linking the properties of a system identification algorithm is investigated.     

Controller switching based on output prediction errors

We consider a switched nonlinear feedback control strategy for controlling a plant with unknown parameters so that the output asymptotically tracks a reference signal. The controller is selected online from a given set of controllers according to a switching rule based on output prediction errors. For control problems requiring asymptotic tracking of a reference input we provide sufficient conditions under which the switched closed-loop control system is exponentially stable and asymptotically achieves good control even if the switching does not stop, Our results are illustrated with three examples     

Model predictive control of switched nonlinear systems under average dwell-time

In this paper, we propose a model predictive control (MPC) algorithm for switched nonlinear systems under average dwell-time switching signals. Assuming that a stabilizing MPC controller exists for each of the subsystems, we show that recursive feasibility of the repeatedly solved optimal control problem and asymptotic stability of the closed-loop switched system can be established if a certain average dwell-time condition is satisfied. If the switching times are unknown a priori and cannot be detected instantly, the MPC controller calculated for the previously active subsystem might not be stabilizing for the newly activated subsystem, thus possibly leading to an unstable closed-loop system. It is shown that if the switches can be detected fast enough, then still ultimate boundedness in an arbitrarily small region around the origin can be ensured, i.e., the proposed MPC algorithm is inherently practically robust with respect to small errors in the detection of the switching times. Finally, we apply our results to a continuous stirred-tank reactor (CSTR) with two different modes of operation.     

Stabilization of switched linear systems with bounded disturbances and unobservable switchings

This paper studies the stabilization problem of switched linear systems with bounded disturbances. It is assumed that the system switches among an infinite set of uniformly controllable linear systems, and that the switching signals are not observable, but the switching duration has a lower bound. It will be shown that by combining on-line adaptive estimation and control in the controller design, a feedback control law can be constructed which makes the switched linear system globally stable.     

Input-to-state stability of switched systems and switching adaptive control

In this paper we prove that a switched nonlinear system has several useful input-to-state stable (ISS)-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.     

First-order logic characterization of program properties

A program is first-order reducible (FO-reducible) w.r.t. a set IC of integrity constraints if there exists a first-order theory T such that the set of models for T is exactly the set of intended models for the program w.r.t. all possible EDBs. In this case, we say that P is FO-reducible to T w.r.t. IC. For FO-reducible programs, it is possible to characterize, using first-order logic implications, properties of programs that are related to all possible EDBs as in the database context. These properties include, among others, containment of programs, independence of updates w.r.t. queries and integrity constraints, and characterization and implication of integrity constraints in programs, all of which have no known proof procedures. Therefore, many important problems formalized in a nonstandard logic can be dealt with by using the rich reservoir of first-order theorem-proving tools, provided that the program is FO-reducible. The following classes of programs are shown to be FO-reducible: (1) a stratified acyclic program P is FO-reducible to comp(P)∪IC w.r.t. IC for any set IC of constraints; (2) a general chained program P is FO-reducible to comp(P')∪IC w.r.t. certain acyclicity constraints IC; and (3) a bounded program P is FO-reducible to comp(P')∪IC w.r.t. any set IC of constraints, where P' is a nonrecursive program equivalent to P. Some heuristics for constructing FO-reducible programs are described     

Funnel‐like prescribed tracking control for uncertain nonlinear stochastic switched systems

This paper deals with the funnel‐like prescribed tracking control problem for a class of uncertain nonlinear stochastic switched systems. An improved performance technique is developed to restrain the fluctuation at the moment of switches and a new algorithm is proposed to address the funnel‐like prescribed tracking problem. First, a dynamic gain‐based switched K‐filter is constructed to estimate the unmeasured state information of the switched system. Subsequently, the performance technique is applied to prescribe output tracking error and restrain fluctuations of the system. Thereafter, the dynamic output feedback switched controller is designed by the use of the backstepping method. Moreover, based on the Lyapunov stability theory, it is proved strictly that all signals of the resulting closed‐loop system are bounded in probability if the switching signal satisfies the average dwell time. Finally, a numerical simulation is presented to illustrate the effectiveness of the proposed theoretical results.     

Links between different stabilities of switched homogeneous systems with delays and uncertainties

This paper addresses the links between three stabilities (attractivity, asymptotic stability, and exponential stability) of switched homogeneous systems with delays and uncertainties. A system has a certain property over a given set of switching signals if the property holds for all switching signals in . It is shown that a switched homogeneous system of degree one is exponentially stable over a given set of switching signals if it is attractive or asymptotically stable over the same set. The result is then applied to switched linear systems with delays and uncertainties. Finally, an example follows to show that ‘being over a given set of switching signals’ is necessary to guarantee the equivalence between different stabilities. Copyright © 2015 John Wiley & Sons, Ltd.     

Input/output-to-state stability and state-norm estimators for switched nonlinear systems

In this paper, the concepts of input/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that when the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for.     

Gröbner fan and universal characteristic sets of prime differential ideals

The concepts of Gröbner cone, Gröbner fan, and universal Gröbner basis are generalized to the case of characteristic sets of prime differential ideals. It is shown that for each cone there exists a set of polynomials which is characteristic for every ranking from this cone; this set is called a strong characteristic set, and an algorithm for its construction is given. Next, it is shown that the set of all differential Gröbner cones is finite for any differential ideal. A subset of the ideal is called its universal characteristic set, if it contains a characteristic set of the ideal w.r.t. any ranking. It is shown that every prime differential ideal has a finite universal characteristic set, and an algorithm for its construction is given. The question of minimality of this set is addressed in an example. The example also suggests that construction of a universal characteristic set can help in solving a system of nonlinear PDE’s, as well as maybe providing a means for more efficient parallel computation of characteristic sets.     

Stabilization of a class of cascade nonlinear switched systems with application to chaotic systems

In this study, the problem of finite‐time practical control of a class of nonlinear switched systems in the presence of input nonlinearities is investigated. The subsystems of the switched system are considered as complex nonlinear systems with a cascade structure. Each subsystem is fluctuated by lumped uncertainties. Moreover, some parts of the system's dynamics are considered to be unknown in advance. This paper sets no restrictive assumption on the switching logic of the system. Therefore, the aim is to propose a controller to work under any arbitrary switching signals. After providing a smooth sliding manifold, a simple adaptive control input is developed such that the system trajectories approach the prescribed sliding mode dynamics in finite‐time sense. The adopted control signal does not use the upper bounds of the lumped uncertainties, and it is robust against unknown nonlinear parts of the subsystems. It is proved that the origin is the (practical) finite‐time stable equilibrium point of the overall closed‐loop system. Subsequently, the proposed control rule is modified to handle the same switched system with no input nonlinearities. Computer simulations via 2 chaotic electric direct current machines demonstrate the robust performance of the derived variable structure control algorithm against system fluctuations and nonlinear inputs.     

Reachability analysis of constrained switched linear systems

In this note, we investigate the reachability of switched linear systems with switching/input constraints. We prove that, under a mild assumption of the feasible switching signals, the reachability set is the reachable subspace of the unconstrained system. We also address the local reachability for switched linear systems with input constraints and present a complete criterion for a general class of switched linear systems.     

Boolean-controlled systems via receding horizon and linear programing

We consider dynamic systems controlled by boolean signals or decisions. We show that in a number of cases, the receding horizon formulation of the control problem can be solved via linear programing by relaxing the binary constraints on the control. The idea behind our approach is conceptually easy: a feasible control can be forced by imposing that the boolean signal is set to one at least one time over the horizon. We translate this idea into constraints on the controls and analyze the polyhedron of all feasible controls. We specialize the approach to the stabilizability of switched and impulsively controlled systems. This work was supported by MURST-PRIN 2007ZMZK5T “Decisional model for the design and the management of logistics networks characterized by high interoperability and information integration”.     

The gain scheduling and the robust state feedback stabilization problems

We consider dynamical uncertain systems of the form x/spl dot/=a(x, w)+b(x, w)u where w(t)/spl isin/W is an unknown but bounded time-varying parameter. We consider the robust state feedback stabilization problem, in which we consider a control of the form u=/spl Phi/(x), and the gain-scheduling stabilization problem in which a control of the form u=/spl Phi/(x, w) (full information control) is admitted. We show that for convex processes, those systems in which for fixed x the set of all [a(x, w)b(x, w)] is convex, the two problems are equivalent; i.e., if there exists a (locally Lipschitz) gain scheduling stabilizing control then there exists a robustly stabilizing control (continuous everywhere except possibly at the origin). Thus, for convex process stabilization, knowledge of w(t) is not an advantage for the compensator. Then we consider the special class of polytopic LPV systems, and show that there is no loss of regularity as in the general case, if we pass from a gain-scheduling controller to a state feedback controller. In particular, no discontinuity at the origin may occur. We show that the existence of a dynamic controller always implies the existence of a static one. Finally we show that we can always find a linear gain-scheduling controller for a stabilizable system. This means that a possible advantage of the online measurement of the parameter w(t) is that this always allows for linear compensators, whose implementation can be easier than that of nonlinear ones.     

Convergence rate of nonlinear switched systems

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems.We first consider a switched system which is asymptotically stable for a class of switching signals but not for all switching signals. We show that solutions corresponding to that class of switching signals converge arbitrarily slowly to the origin.Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for switching signals with a fixed dwell-time.     

Improving performance in model predictive control: Switching cost functionals under average dwell-time

In this paper, we propose a novel switched model predictive control (MPC) algorithm for nonlinear continuous-time systems, where we switch between different cost functionals in order to enhance performance. Thus, different performance criteria can be taken into account. In order to ensure stability of the resulting closed-loop system, we consider switching signals which exhibit a certain average dwell-time. When considering switching signals of this type, certain assumptions are common in the switched systems literature in order to ensure stability, like a matching condition for the different Lyapunov functions and the possibility to find Lyapunov functions with an exponential decay rate. In this paper, we show how these assumptions can be satisfied in the MPC context and thus stability of the proposed switched MPC algorithm can be established.