Jeff S. Shamma

Survey Research on gain scheduling

Current research on gain scheduling is clarifying customary practices as well as devising new approaches and methods for the design of nonlinear control systems.

Consensus Filters for Sensor Networks and Distributed Sensor Fusion

Consensus algorithms for networked dynamic systems provide scalable algorithms for sensor fusion in sensor networks. This paper introduces a distributed filter that allows the nodes of a sensor network to track the average of n sensor measurements using an average consensus based distributed filter called consensus filter. This consensus filter plays a crucial role in solving a data fusion problem that allows implementation of a scheme for distributed Kalman filtering in sensor networks. The analysis of the convergence, noise propagation reduction, and ability to track fast signals are provided for consensus filters. As a byproduct, a novel critical phenomenon is found that relates the size of a sensor network to its tracking and sensor fusion capabilities. We characterize this performance limitation as a tracking uncertainty principle. This answers a fundamental question regarding how large a sensor network must be for effective sensor fusion. Moreover, regular networks emerge as efficient topologies for distributed fusion of noisy information. Though, arbitrary overlay networks can be used. Simulation results are provided that demonstrate the effectiveness of consensus filters for distributed sensor fusion.

Analysis of gain scheduled control for nonlinear plants

Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the robustness, performance, or even nominal stability of the overall gain-scheduled design. An analysis is presented for two types of nonlinear gain-scheduled control systems: (1) scheduling on a reference trajectory, and (2) scheduling on the plant output. Conditions which guarantee stability, robustness, and performance properties of the global gain schedule designs are given. These conditions confirm and formalize popular notions regarding gain scheduled designs, such as that the scheduling variable should vary slowly, and capture the plants nonlinearities. >

Guaranteed properties of gain scheduled control for linear parameter-varying plants

Abstract Gain scheduling has proven to be a successful design methodology in many engineering applications. However in the absence of a sound theoretical analysis, these designs come with no guarantees on the robustness, performance, or even nominal stability of the overall gain scheduled design. This paper presents such an analysis for one type of gain scheduled system, namely, a linear parameter-varying plant scheduling on its exogenous parameters. Conditions are given which guarantee that the stability, robustness, and performance properties of the fixed operating point designs carry over to the global gain scheduled design. These conditions confirm and formalize popular notions regarding gain scheduled design, such as the scheduling variable should “vary slowly.”

Cooperative Control and Potential Games

We present a view of cooperative control using the language of learning in games. We review the game-theoretic concepts of potential and weakly acyclic games, and demonstrate how several cooperative control problems, such as consensus and dynamic sensor coverage, can be formulated in these settings. Motivated by this connection, we build upon game-theoretic concepts to better accommodate a broader class of cooperative control problems. In particular, we extend existing learning algorithms to accommodate restricted action sets caused by the limitations of agent capabilities and group based decision making. Furthermore, we also introduce a new class of games called sometimes weakly acyclic games for time-varying objective functions and action sets, and provide distributed algorithms for convergence to an equilibrium.

Robust stability with time-varying structured uncertainty

Considers the problem of assessing robust stability in the presence of block-diagonally structured time-varying dynamic uncertainty. It is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition. The notion of stability here is finite-gain stability over finite-energy signals. In sharp contrast to the case of time-invariant dynamic uncertainty, this result is not limited by the number uncertainty blocks. These results parallel previous results regarding finite-gain stability over persistent bounded signals. >

Payoff-Based Dynamics for Multiplayer Weakly Acyclic Games

We consider repeated multiplayer games in which players repeatedly and simultaneously choose strategies from a finite set of available strategies according to some strategy adjustment process. We focus on the specific class of weakly acyclic games, which is particularly relevant for multiagent cooperative control problems. A strategy adjustment process determines how players select their strategies at any stage as a function of the information gathered over previous stages. Of particular interest are “payoff-based” processes in which, at any stage, players know only their own actions and (noise corrupted) payoffs from previous stages. In particular, players do not know the actions taken by other players and do not know the structural form of payoff functions. We introduce three different payoff-based processes for increasingly general scenarios and prove that, after a sufficiently large number of stages, player actions constitute a Nash equilibrium at any stage with arbitrarily high probability. We also show how to modify player utility functions through tolls and incentives in so-called congestion games, a special class of weakly acyclic games, to guarantee that a centralized objective can be realized as a Nash equilibrium. We illustrate the methods with a simulation of distributed routing over a network.

Set-valued observers and optimal disturbance rejection

A set-valued observer (also called guaranteed state estimator) produces a set of possible states based on output measurements and models of exogenous signals. We consider the guaranteed state estimation problem for linear time-varying systems with a priori magnitude bounds on exogenous signals. We provide an algorithm to propagate the set of possible states based on output measurements and show that the centers of these sets provide optimal estimates in an l/sup /spl infin//-induced norm sense. We then consider the utility of set-valued observers for disturbance rejection with output feedback and derive the following general separation structure. An optimal controller can consist of a set-valued observer followed by a static nonlinear function on the observed set of possible states. A general construction of this function is provided in the scalar control case. Furthermore, in the special case of full-control, i.e., the number of control inputs equals the number of states, optimal output feedback controllers can take the form of an optimal estimate of the full-state feedback controller.

An Overview of LPV Systems

The framework of Linear Parameter Varying (LPV) systems concerns linear dynamical systems whose state-space representations depend on exogenous nonstationary parameters. Since its introduction by Shamma and Athans in 1988 to model gain-scheduling, the LPV paradigm has become a standard formalism in systems and controls, with many papers devoted to analysis, controller synthesis, and system identification of LPV models. This chapter reviews basic concepts and presents a representative selection of analytical approaches for LPV systems.

Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization

It is shown that time-varying compensation does not improve the optimal rejection of persistent bounded disturbances. This result is obtained by exploiting a key observation that any time-varying compensator which yields a given degree of disturbance rejection must do so uniformly over time, thereby removing any advantage of time-variation. This key observation is exploited to show that time-varying compensation does not improve the optimal rejection of disturbances, regardless of the norm used to measure the disturbances. Thus, absolutely summable, finite-energy, or persistent bounded disturbances may be treated in the same manner. It is shown that time-varying compensation does not help in the bounded-input bounded-output robust stabilization of time-invariant plants with unstructured uncertainty. In doing so, it is also shown that the small-gain theorem is both necessary and sufficient for the bounded-input bounded-output stability of certain linear time-varying plants subject to unstructured linear time-varying perturbations. >