Showing papers in "Mathematics of Control, Signals, and Systems in 2012"
TL;DR: A family of time-varying hyperbolic systems of balance laws is considered and a strictification approach is taken to obtain a time-Varying strict Lyapunov function, which allows to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of input-to-state stability (ISS) type.
Abstract: A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach to obtain a time-varying strict Lyapunov function. It allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of input-to-state stability (ISS) type. Two examples illustrate the results.
158 citations
TL;DR: For large natural frequencies, it is shown that scalar proportional mean-field feedback is able to induce desynchronization, and how it can be induced in a network of originally synchronized oscillators.
Abstract: Motivated by neuroscience applications, and in particular by the deep brain stimulation treatment for Parkinson’s disease, we have recently derived a simplified model of an interconnected neuronal population under the effect of its mean-field proportional feedback. In this paper, we rely on that model to propose conditions under which proportional mean-field feedback achieves either oscillation inhibition or desynchronization. More precisely, we show that for small natural frequencies, this scalar control signal induces an inhibition of the collective oscillation. For the closed-loop system, this situation corresponds to a fixed point which is shown to be almost globally asymptotically stable in the fictitious case of zero natural frequencies and all-to-all coupling and feedback. In the case of an odd number of oscillators, this property is shown to be robust to small natural frequencies and heterogencities in both the coupling and feedback topology. On the contrary, for large natural frequencies, we show that scalar proportional mean-field feedback is able to induce desynchronization. After having recalled a formal definition for desynchronization, we show how it can be induced in a network of originally synchronized oscillators.
42 citations
TL;DR: The overall system is composed of switched subsystems each of which has the nonlinear MIMO generalized triangular form, which (in contrast to strict-feedback form) has non-invertible input–output maps.
Abstract: We solve the problem of global uniform input-to-state stabilization with respect to external disturbance signals for a class of large-scale interconnected nonlinear switched systems. The overall system is composed of switched subsystems each of which has the nonlinear MIMO generalized triangular form, which (in contrast to strict-feedback form) has non-invertible input–output maps. The switching signal is an arbitrary unknown piecewise constant function and the feedback constructed does not depend on the switching signal.
39 citations
TL;DR: It is shown that the ‘funnel controller’ achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel, which determines the transient behaviour.
Abstract: We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ‘strict and non-positive relative degree’. We present two main results: first, a so-called ‘zero dynamics form’ is derived; this form is—within the class of system equivalence—a simple form of the DAE; it is a counterpart to the well-known Byrnes–Isidori form for ODE systems with strictly proper transfer function. The ‘zero dynamics form’ is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ‘funnel controller’ (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
29 citations
TL;DR: It is shown that, under some realistic assumptions, any n-dimensional dynamically perturbed strict-feedback nonlinear system can be globally practically stabilized by a quantized control law using 2n three-level dynamic quantizers.
Abstract: This paper proposes a new tool for quantized nonlinear control design of dynamic systems transformable into the dynamically perturbed strict-feedback form. To address the technical challenges arising from measurement and actuator quantization, a new approach based on set-valued maps is developed to transform the closed-loop quantized system into a large-scale system composed of input-to-state stable (ISS) subsystems. For each ISS subsystem, the inputs consist of quantization errors and interacting states, and moreover, the ISS gains can be assigned arbitrarily. Then, the recently developed cyclic-small-gain theorem is employed to guarantee input-to-state stability with respect to quantization errors and to construct an ISS-Lyapunov function for the closed-loop quantized system. Interestingly, it is shown that, under some realistic assumptions, any n-dimensional dynamically perturbed strict-feedback nonlinear system can be globally practically stabilized by a quantized control law using 2n three-level dynamic quantizers.
25 citations
TL;DR: A scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay and it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds.
Abstract: Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.
22 citations
TL;DR: Input-to-state stability (ISS) properties for a class of time-varying hybrid dynamical systems via averaging method through averaging method can be used in a framework for a systematic design of hybrid feedbacks for pulse-width modulated control systems.
Abstract: Input-to-state stability (ISS) properties for a class of time-varying hybrid dynamical systems via averaging method are considered. Two definitions of averages, strong average and weak average, are used to approximate the time-varying hybrid systems with time-invariant hybrid systems. Closeness of solutions between the time-varying system and solutions of its weak or strong average on compact time domains is given under the assumption of forward completeness for the average system. We also show that ISS of the strong average implies semi-global practical (SGP)-ISS of the actual system. In a similar fashion, ISS of the weak average implies semi-global practical derivative ISS (SGP-DISS) of the actual system. Through a power converter example, we show that the main results can be used in a framework for a systematic design of hybrid feedbacks for pulse-width modulated control systems.
17 citations
TL;DR: This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach that avoids any reference to the related notion of probability measures induced by a PFSA.
Abstract: Probabilistic finite state automata (PFSA) have found their applications in diverse systems. This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach. The vector space is constructed in a stationary setting, which eliminates the need for an initial state in the specification of PFSA. This algebraic model formulation avoids any reference to the related notion of probability measures induced by a PFSA. A formal language-theoretic and symbolic modeling approach is adopted. Specifically, semantic models are constructed in the symbolic domain in an algebraic setting. Applicability of the theoretical formulation has been demonstrated on experimental data for robot motion recognition in a laboratory environment.
17 citations
TL;DR: The question of how many non-ISS subsystems are allowed in the network, and the question of the necessity of small-gain-type properties are given, and a concise representation of the iISS small- gain criteria for an arbitrary number of subsystems forming a cycle in networks is proposed.
Abstract: In this paper, necessary conditions are investigated for global asymptotic stability of dynamical networks consisting of integral input-to-state stable (iISS) subsystems. Employing the dissipation formulation for subsystems, this paper naturally covers input-to-state stable (ISS) subsystems, which constitute a strict subset of the iISS. The number n of subsystems composing a network is allowed to be arbitrary. This paper gives answers to the question of how many non-ISS subsystems are allowed in the network, and the question of the necessity of small-gain-type properties. All the developments in this paper are natural, but non-trivial extensions of the results obtained previously for n = 2. This paper also proposes a concise representation of the iISS small-gain criteria for an arbitrary number of subsystems forming a cycle in networks.
16 citations
TL;DR: In this article, a homotopy algorithm was proposed to compute a decay point of a monotone operator. But the decay point is strictly smaller than the point itself, i.e., a point whose image under the monotonous operator is not larger than the one itself.
Abstract: In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system, and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone operator. For this purpose we use a fixed-point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage over an earlier algorithm is demonstrated. Furthermore, an example is given which shows how to analyze a given perturbed interconnected system.
16 citations
TL;DR: It is shown that the solvability of the output regulation problem can be characterized by the solvation of a certain constrained infinite-dimensional Sylvester differential equation.
Abstract: In this paper the output regulation of a linear distributed parameter system with a non-autonomous periodic exosystem is considered. It is shown that the solvability of the output regulation problem can be characterized by the solvability of a certain constrained infinite-dimensional Sylvester differential equation. Conditions are given for the existence of feedforward and feedback controllers solving the regulation problem along with a method for their construction. The theoretical results are applied to output regulation of a controlled delay equation.
TL;DR: A Kalman rank condition in the one-dimensional case of linear parabolic systems with constant coefficients is proved and related results such as a Carleman estimate are established.
Abstract: In this work, we study the null-controllability properties of linear parabolic systems with constant coefficients in the case where several controls are acting on different distributed subdomains and/or on the boundary. We prove a Kalman rank condition in the one-dimensional case. In the case where only distributed controls are considered, we also establish related results such as a Carleman estimate.
TL;DR: A general approach to the problem of optimizing the long-run average cost per unit time that consists of first determining an optimal steady state (periodic) behavior and then to design a feedback scheduling protocol ensuring convergence to this behavior as time progresses is considered.
Abstract: The paper considers standard fluid models of multi-product multiple-server production systems where setup times are incurred whenever a server changes product. We consider a general approach to the problem of optimizing the long-run average cost per unit time that consists of first determining an optimal steady state (periodic) behavior and then to design a feedback scheduling protocol ensuring convergence to this behavior as time progresses. In this paper, we focus on the latter part and introduce a systematic approach. This approach gives rise to protocols that are cyclic and distributed: the servers do not need information about the entire system state. Each of them proceeds basically from the local data concerning only the currently served queue, although a fixed finite number of one-bit notification signals should be exchanged between the servers during every cycle. The approach is illustrated by simple instructive examples concerning polling systems, single server systems with processor sharing scheme, and the re-entrant two-server manufacturing network with non-negligible setup times introduced by Kumar and Seidman. For the last network considered in the analytical form, some cases of optimal steady-state (periodic) behavior are first recalled. For all examples, based on the desired steady state behavior and using the presented theory, we designed simple distributed feedback switching control laws. These laws not only give rise to the required behaviors but also make them globally attractive, irrespective of the system parameters and initial state.
TL;DR: This paper considers discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph, and proves general results and criteria relating controllability to the combinatorial and topological properties of the walk.
Abstract: In this paper, we consider discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied In particular, the set of states that one can achieve can be described by studying controllability Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra Controllability is verified when any unitary operation between two states can be implemented as a result of the evolution of the quantum walk We prove general results and criteria relating controllability to the combinatorial and topological properties of the walk In particular, controllability is verified if and only if the underlying graph is not a bipartite graph and therefore it depends only on the graph and not on the particular quantum walk defined on it We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms
TL;DR: The solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the corresponding duality problem in how far the solutions of the corresponding homogeneous systems determine the equations are investigated.
Abstract: In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the corresponding duality problem in how far the solutions of the corresponding homogeneous systems determine the equations. For ordinary delay-differential (DD) equations these behavioral problems were investigated in a seminal paper by GlusingLurssen(1997)andinlaterpapersbyHabets, Glusing-Lurssen, VettoriandZampieri. In these papers the delay-differential operators are considered as distributions with compact support which act on smooth functions or on arbitrary distributions via convolution. The entire analytic Laplace transforms of the distributions with compact support play an important part in the quoted papers. In our approach the partial differential-difference operators belong to various topological operator rings A of holomorphic functions on subsets of C n and are thus studied in the frequency domain, the arguments of these (operator) functions being interpreted as generalized
TL;DR: A generalized interpolation problem, which relaxes the usual assumptions on disjointness of the interpolation nodes and the poles of theInterpolating conditions, is formulated and solved.
Abstract: We investigate here the interpolation conditions connected to an interpolating function \(Q\) obtained as a Linear Fractional Transformation of another function \(S\). In general, the degree of \(Q\) is equal to the number of interpolating conditions plus the degree of \(S\). We show that, if the degree of \(Q\) is strictly less that this quantity, there is a number of complementary interpolating conditions which has to be satisfied by \(S\). This induces a partitioning of the interpolating conditions in two sets. We consider here the case where these two sets are not necessarily disjoint. The reasoning can also be reversed (i.e. from \(S\) to \(Q\)). To derive the above results, a generalized interpolation problem, which relaxes the usual assumptions on disjointness of the interpolation nodes and the poles of the interpolant, is formulated and solved.
TL;DR: In this paper, the controllability of an artificial advection-diffusion system through the boundary was studied and suitable Carleman estimates gave us the observability of the adjoint system in the one dimensional case.
Abstract: In this paper we study the controllability of an artificial advection–diffusion system through the boundary. Suitable Carleman estimates give us the observability of the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity.
TL;DR: Interest in linear large-scale systems analysis and synthesis has been revived by new developments in nonlinear systems and control, which include but are not limited to dissipative systems theory, input-to-state stability, and various Lyapunov and smallgain design schemes.
Abstract: In a broad sense, large-scale systems theory is concerned with the analysis and design of often very high-dimensional systems that admit some kind of partition into smaller subsystems. Research in linear large-scale systems analysis and synthesis can be traced back to at least the 1970s. Recently, interest in this subject has been revived by new developments in nonlinear systems and control, which include but are not limited to dissipative systems theory, input-to-state stability, and various Lyapunov and smallgain design schemes. Systems and control theory itself has seen many important advances since the 1970s, including, e.g., concepts like control Lyapunov functions, effective implementations of nonlinear model predictive control, or entire emerging areas like hybrid systems and cooperative multi-agent systems. Naturally, these developments also leap into the area of large-scale systems. It is hence to be expected that the interest in this area will continue to increase, ultimately also due to, e.g., current research efforts in distributed model predictive control, next generation smart electricity grids, incorporating green and renewable energy sources, or bioand neuro-engineering. Many questions are associated with the theory of large-scale systems. A quite central one is about sufficient conditions on interconnections of a number of stable systems such that the composite system is also stable. Another one is how individual