Hybrid Persistency of Excitation in Adaptive Estimation for Hybrid Systems

We propose a framework to analyze stability for a class of linear non-autonomous hybrid systems, where the continuous evolution of solutions is governed by an ordinary differential equation and the instantaneous changes are governed by a difference equation. Furthermore, the jumps are triggered by the influence of an external hybrid signal. The proposed framework builds upon a generalization of the well-known persistency of excitation (PE) and uniform observability (UO) notions to the realm of hybrid systems. That is, we establish conditions, under which, hybrid PE implies hybrid UO and, in turn, uniform exponential stability (UES) and input-to-state stability (ISS). Our proofs rely on an original statement for hybrid systems, expressed in terms of Lp bounds on the solutions. We demonstrate the utility of our results on generic adaptive estimation problems. The first one concerns the so-called gradient systems, reminiscent of the popular gradient-descent algorithm. The second one pertains to designing adaptive observers/identifiers for a class of hybrid systems that are nonlinear in the input and the output, and linear in the unknown parameters. In both cases, we illustrate through examples that the proposed hybrid framework succeeds when the classic purely continuous- or discrete-time counterparts fail.Comment: 30 pages, 6 figures. This is v2. Some corrections were added, relative to v

Observer-based Synchronization of Multi-agent Systems Using Intermittent Output Measurements

The problem of synchronizing multiple continuous-time linear time-invariant systems connected over a complex network, with intermittently available measurements of their outputs, is considered. To solve this problem, we propose a distributed observer-based feedback controller that utilizes a local hybrid observer to estimate neighboring states only from output measurements at such potentially nonperiodic isolated event times. Due to the inherent continuous and discrete dynamics emerging from coupling the impulsive measurement updates and the interconnected networked systems, we use hybrid systems to model and analyze the resulting closed-loop system. The problem of synchronization and state estimation is then recast as a set stabilization problem, and, utilizing a Lyapunov-based analysis for hybrid systems, we provide sufficient conditions for global exponential stability of the synchronization and zero estimation error set. A numerical example is provided to illustrate the results

Performance Analysis and Validation of a Recoverable Flight Control System in a Simulated Neutron Environment

This paper introduces a class of stochastic hybrid models for the analysis of closed-loop control systems implemented with NASA\u27s Recoverable Computer System. Such Recoverable Computer Systems have been proposed to insure reliable control performance in harsh environments. The stochastic hybrid models consist of either a stochastic finite-state automaton or a finite-state machine driven by a Markov input, which in turn drives a switched linear discrete-time dynamical system. Their stability and output tracking performance are analyzed using an extension of the existing theory for Markov jump-linear systems. For illustration, a stochastic hybrid model is used to calculate the tracking error performance of a Boeing 737 at cruising altitude and in closed-loop with a Recoverable Computer System subject to neutron-induced single-event upsets. The upsets are modeled with a Markov process. The results are validated using experimental data obtained from a simulated neutron environment in NASA\u27s SAFBTI Laboratory. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved

Follow the bouncing ball: global results on tracking and state estimation with impacts

peer reviewedIn this paper we formulate tracking and state- estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous trajectories arising from the impacts, we use hybrid systems stability analysis tools to establish the results. Using a novel concept of mirrored images of the target mass we prove that 1) a tracking control algorithm, and 2) an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards, including rectangles. Moreover, we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking. The results are illustrated via simulations

Qualitative Studies of Nonlinear Hybrid Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results

Interval Observer Approach to Output Stabilization of Linear Impulsive Systems

International audienceThe problem of output stabilization is studied for a class of linear hybrid systems subject to signal uncertainties: linear impulsive systems under dwell-time constraints. Two problems are considered. First, an interval observer estimating the set of admissible values for the state is designed. Next, an output stabilizing feedback design problem is studied where the stability is checked using linear matrix inequalities (LMIs). To the best of our knowledge, interval observer approach has never been proposed for the stabilization of this class of hybrid systems. Efficiency of the proposed approach is demonstrated by computer experiments for Fault Detection and Isolation (FDI) and Fault-Tolerant Control (FTC) of a power split device with clutch for heavy-duty military vehicles

High Performance Control of a Transmission Based Servo Actuator System

High performance actuation is a key factor in the industrial robot area. The transmission based servo actuator system (TBA) is a new type of robot actuator with a brushless DC servo motor and a three speed discrete variable transmission (DVT). The proposed TBA design can match the performance of a typical hydraulic actuator with compact size and weight. The TBA is a typical hybrid dynamic system consisting of three continuous dynamic systems and a discrete state controller. This dissertation addresses the fundamental problems associated with the TBA system control from a hybrid system point of view. A detailed dynamic model of the TBA is developed. Due to the complexity of the TBA system, an exact model is unwieldy for control design and analysis purposes. In this research, the TBA system is simplified into a hybrid system with three second order linear time invariant systems, on which all the controls are developed.Dynamic stability of the TBA is critical for its function as a servoactuator. For a hybrid system, the stability problem has much broader range of issues than a purely continuous system. In general, the plant stability and the subsystem stability are independent. For example, a hybrid system with stable subsystems can be unstable for certain switch sequences; on the other hand, a hybrid system with unstable subsystems can be stabilized by proper switch signals. In this dissertation, a sufficient condition is established for stability of the TBA system. It is proven that the hybrid system is stable under asynchronous switching if there exists a common Lyapunov function for all subsystems. It is proven that the TBA subsystems can have a common Lypunov function by designing appropriate feedback controller. The feedback controller to stabilize the TBA can be transformed into a PID equivalent controller because the subsystems are second order linear time invariant systems (LTI). The PID controller was then implemented and high performance in terms of position error and transient suppression has been achieved. The discrete state controller should be stable, which means that its output should be consistent if the hybrid system is subjected to disturbances. A common phenomenon is that the state changes back and forth very frequently near the switch boundary, which is referred to as transition instability. This research proposes a switch strategy consisting of two boundaries to achieve the transition stability, and it is proved that the proposed switch strategy is transition stable. An optimal controller is designed and difficulties associated with implementation are generated. Based on the proposed control methods, a multithread real time control software has been developed to achieve a deterministic control loop sampling. The control software is developed in C/C++ under Real Time Application Interface (RTAI), which provides a real time programming environment in a normal Linux operating system. With the proposed controller and a prototype TBA test system, TBA stability and control performance was demonstrated and evaluated. The following results were observed: Steady state error of 0.005 degrees at the emulated robust manipulator shoulder pitch joint Control loop sampling period of 1 millisecond with negligible delay Transient disturbances associated with the gear shifting of ~20% in most cases. The methods and applications used in this dissertation can be extended to a large range of hybrid dynamic systems in terms of control system design, analysis and implementation. This research contributes to the literature and research knowledge base in the following ways: Exploration and solution of the control problems of TBA’s in the hybrid system control context. Expansion of the fundamental understanding of the practical control issues of TBA’s. Analysis, design, and implementation of a real time TBA control system, and identification of the most suitable control strategy for the TBA. The development of analysis and control methods that can be extended to a much broader range of hybrid dynamic systems