Tian Qi

Optimal tracking design and performance analysis for LTI systems with quantization effects

This paper studies the tracking problem for linear time-invariant multi-input single-output (MISO) discrete-time systems with quantization effects. Logarithmic quantization laws are adopted in the systems. The tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Our goals are to design an optimal controller for the tracking problem and to find an explicit formula of the minimum tracking cost. It turns out that the optimal state feedback law can be obtained by solving a modified discrete-time Riccati equation associated with the state space model of the plant and the features of the quantization law. Furthermore, from the unique positive solution of the modified Riccati equation, we obtain an analytic expression for the minimum tracking cost in terms of the nonminimum phase zeros and the bound of quantization error. When the quantization error approaches zero, the minimum tracking cost degrades to the minimum tracking cost of the system without quantization effects, which is presented some existing works. The results obtained in this work explicitly show how is the optimal tracking performance limited by the quantization error.

Optimal Tracking Design of an MIMO Linear System with Quantization Effects

Abstract This paper studies optimal design for a linear time-invariant (LTI) MIMO discrete-time networked feedback system in tracking a step signal. It is assumed that the outputs of the controller are quantized by logarithm quantization laws, respectively, and then transmitted through a communication network to the remote plant in the feedback system, whereas the quantization errors in all quantized signals are modeled as a product of a white noise with zero mean and the source signal respectively, the variances of the white noises are determined by the accuracies of the quantization laws. The tracking performance of the system we interested in is defined as the averaged energy of the error between the output of the plant and the reference input. Three problems are studied for the system: 1) For a set of given logarithm laws, how to design an optimal stabilizing controller for the closed-loop system in mean-square stability sense? 2) What is a minimal communication load to stabilize the networked feedback system in terms of the characteristics of the logarithm quantization laws? 3) For a set of given logarithm laws, how to design an optimal controller to achieve minimal tracking cost? We find that the problems 1 and 3 have a unique solution, respectively, and obtain an analytic solution for problem 2 when the plant is a minimum phase system.

Tracking performance limitation of a linear MISO unstable system with quantized control signals

This paper studies the tracking performance for linear time-invariant multi-input single-output (MISO) discrete-time feedback systems with quantized control signals. The specific problem under consideration is that the quantized control signals transmitted to a given plant by two different communication channels which satisfy some constraint. The tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Logarithmic quantization law is used in the quantizers. To obtain the best attainable tracking performance of the feedback system, two parameter controller is adopted. By using stochastic dynamic programming approach, the best attainable tracking performance is in terms of the state equation of the plant and the largest relative quantization errors of the logarithmic quantization laws. Furthermore, a sufficient condition for asymptotical tracking is given for the feedback system in an average sense.

Small-gain stability conditions for linear systems with time-varying delays ☆

Sufficient stability conditions for linear systems subject to time-varying delays are developed in this paper, which are in the form of scaled small gain conditions and can be checked readily by solving corresponding linear matrix inequality problems. From a robust control perspective, our development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. Examples show significant improvement of our results over the existing criteria.

Fundamental Limits on Uncertain Delays: When is a Delay System Stabilizable by LTI Controllers?

This paper concerns the stabilization of linear time-invariant (LTI) systems subject to uncertain, possibly time-varying delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we derive fundamental bounds on the delay margin, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. Our contribution is threefold. First, for single-input single-output (SISO) systems with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output (MIMO) systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by finding bounds on the radius of delay variations. Third, we show that these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a LTI controller.

Control under Stochastic Multiplicative Uncertainties: Part I, Fundamental Conditions of Stabilizability

In this two-part paper we study stabilization and optimal control of linear time-invariant systems with stochastic multiplicative uncertainties. We consider structured multiplicative perturbations, which, unlike in robust control theory, consist of static, zero-mean stochastic processes, and we assess the stability and performance of such systems using mean-square measures. While Part 2 of this paper tackles and solves optimal control problems under the mean-square criterion, Part 1 is devoted to the stabilizability problem. We develop fundamental conditions of mean-square stabilizability which ensure that an open-loop unstable system can be stabilized by output feedback in the mean-square sense. For single-input single-output systems, a general, explicit stabilizability condition is obtained. This condition, both necessary and sufficient, provides a fundamental limit imposed by the systems unstable poles, nonminimum phase zeros and time delay. For multi-input multi-output systems, we provide a complete, computationally efficient solution for minimum phase systems possibly containing time delays, in the form of a generalized eigenvalue problem readily solvable by means of linear matrix inequality optimization. Limiting cases and nonminimum phase plants are analyzed in depth for conceptual insights, revealing, among other things, how the directions of unstable poles and nonminimum phase zeros may affect mean-square stabilizability in MIMO systems. Other than their independent interest, stochastic multiplicative uncertainties have found utilities in modeling networked control systems pertaining to, e.g., packet drops, network delays, and fading. Our results herein lend solutions applicable to networked control problems addressing these issues.

On delay radii and bounds of MIMO systems

Abstract The delay margin of a time-delay system constitutes the fundamental limit beyond which no single controller may exist to robustly stabilize an unstable delay plant for a range of delay values. For single-input single-output (SISO) systems with a linear time-invariant (LTI) controller, this margin is known to be finite for an unstable plant, and bounds on the delay margin are available. This paper extends the existing results to multi-input multi-output (MIMO) systems. We derive upper bounds on a generalized notion called delay radius. Our results show that for a delay whose direction is orthogonal to that of an unstable pole, no constraint is imposed by the pole on that delay, while if the delay direction is parallel to that of a nonminimum phase zero, its allowable range will be further restricted by the nonminimum phase zero.

Consensus over directed graph: Output feedback and topological constraints

In this paper, the discrete-time multi-agent systems consensus problem over a directed, fixed network communication graph is studied. A distributed dynamic output feedback control protocol is employed. Drawing upon concepts and techniques from robust control, notably those concerning gain-phase margin optimization and analytic interpolation, we obtained explicit sufficient conditions for general linear agents to achieve consensus. The results display an explicit dependence of the consensus condition on the agents unstable poles, non-minimum phase zeros and their relative degree.

Stability for networked control system with quantization constraints

This paper studies thestability of linear time-invariant (LTI) single-input single-output (SISO) discrete-time networked feedback systems. Only quantization effect is considered and the quantizer is logarithmic. Second order stochastic input-output stability is studied. A stabilizing output feedback controller which is realized by Youla parameterization is adopted. We get a necessary condition to guarantee the closed-loop system stability in mean-square sense. This result also provides a method to design an output feedback controller to stabilize a linear system with multiplicative noise.

Consensus of Multi-agent Systems Under Delay

This chapter concerns the consensus problem for continuous-time multi-agent systems (MAS). The network topology is assumed to be fixed, which can be undirected and directed. We assume that the agents’ input is subject to a constant, albeit possibly unknown time delay, and is generated by a distributed dynamic output feedback control protocol. Drawing upon concepts and techniques from robust control theory, notably those concerning gain margin and gain-phase margin optimizations and analytic interpolation, we derive explicit, closed-form conditions for general linear agents to achieve consensus. The results display an explicit dependence of the consensus conditions on the delay value as well as on the agent’s unstable poles and nonminimum phase zeros, showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology. We also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method.