Linh Vu

Brief paper: 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.

Common Lyapunov functions for families of commuting nonlinear systems

Abstract We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based on an iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individual systems. Our results extend a previously available one which relies on exponential stability of the vector fields.

Stability of Time-Delay Feedback Switched Linear Systems

We address stability of state feedback switched linear systems in which delays are present in both the feedback state and the switching signal of the switched controller. For switched systems with average dwell-time switching signals, we provide a condition, in terms of upper bounds on the delays and in terms of a lower bound on the average dwell-time, to guarantee asymptotic stability of the closed loop. The condition also implies that, in general, feedback switched linear systems are robust with respect to both small state delays and small switching delays. Our approach combines existing multiple Lyapunov function techniques with the merging switching signal technique, which gives relationships between the average dwell times of two mismatched switching signals and their mismatched times. A methodology for numerical solution based on linear matrix inequality is also included.

Supervisory Control of Uncertain Linear Time-Varying Systems

We consider the problem of adaptively stabilizing linear plants with unknown time-varying parameters in the presence of noise, disturbances, and unmodeled dynamics using the supervisory control framework, which employs multiple candidate controllers and an estimator based switching logic to select the active controller at every instant of time. Time-varying uncertain linear plants can be stabilized by supervisory control, provided that the plants parameter varies slowly enough in terms of mixed dwell-time switching and average dwell-time switching, the noise and disturbances are bounded and small enough in terms of L-infinity norms, and the unmodeled dynamics are small enough in the input-to-state stability sense. This work extends previously reported works on supervisory control of linear time-invariant systems with constant unknown parameters to the case of linear time-varying uncertain systems. A numerical example is included, and limitations of the approach are discussed.

ISS of Switched Systems and Applications to Switching Adaptive Control

In this paper we prove that a switched nonlinear system has several useful 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. Illustrative examples are included.

Stabilizing uncertain systems with dynamic quantization

We consider state feedback stabilization of uncertain linear systems with quantization. The plant uncertainty is dealt with by the supervisory control framework, which employs switching among a finite family of candidate controllers. For a static quantizer, we quantify a relationship between the quantization range and the quantization error bound to guarantees closed loop stability. Using a dynamic quantizer which can vary the quantization parameters in real time, we show that the closed loop can be asymptotically stabilized, provided that additional conditions on the quantization range and the quantization error bound are satisfied. Our results extend previous results on stabilization of known systems with quantization to the case of uncertain systems.

Modeling and analysis of dynamic decision making in sequential two-choice tasks

We present the construction and analysis of a dynamical system model for human decision making in sequential two-choice tasks, in which a human subject makes a series of interrelated decisions between two choices in order to obtain the maximum reward. For a nominal decision making policy inspired by behavioral aspects of humans, we show asymptotic behavior of such decision making process in sequential two-choice tasks for various types of reward structures. Our work gives a control theory oriented perspective to the experiments carried out by cognitive scientists.

Stability of feedback switched systems with state and switching delays

We study stability of state feedback switched systems in which time delays are present in both the feedback state and the switching signal of the controller. For switched linear systems with average dwell-time switching signals, we provide a condition in terms of upper bounds on the delays and a lower bound on the average dwell-time to guarantee asymptotic stability of the closed loop. Our approach employs multiple Lyapunov functions and the merging switching signal technique. We then apply our stability results in switched systems to consensus networks with asymmetric time-varying delays and switching topologies.

On Invertibility of Switched Linear Systems

We address a new problem - the invertibility problem for continuous-time switched linear systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the subsystems should be invertible and there should be no 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

Deterministic Modeling and Evaluation of Decision-Making Dynamics in Sequential Two-Alternative Forced Choice Tasks

The focus of the work in this paper is a systems-theoretic construction, analysis, and evaluation of a deterministic model of human decision making relative to experimental data. In sequential two-alternative forced choice decision tasks, a human subject is presented with two choices at every time step, is given finite time to select one of the choices, and receives a reward after a choice is made (presented as a number on a computer screen). The goal for the human is to obtain the maximal reward while not knowing the underlying reward assignment process. In this work, we present a parameterized deterministic model for human decision making in this context and analyze optimality and stability using a finite state machine approach. This model is then evaluated relative to experimental data from human subjects performing each of six tasks.