Weizhou Su

Fundamental performance limitations in tracking sinusoidal signals

This paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies. The tracking performance is measured by an integral square error between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the attainable tracking performance, under any control structure and parameters, in terms of the characteristics and structural parameters of the given plant, as well as those of the reference signal under consideration. It is shown that this fundamental limit depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency-dependent directional information.

Attainability of the minimum data rate for stabilization of linear systems via logarithmic quantization

This paper investigates the attainability of the minimum average data rate for stabilization of linear systems via logarithmic quantization. It is shown that a finite-level logarithmic quantizer suffices to approach the well-known minimum average data rate for stabilizing an unstable linear discrete-time system under two basic network configurations. In particular, we derive explicit finite-level logarithmic quantizers and the corresponding controllers to approach the minimum average data rate.

Robust control of nonlinear feedback passive systems

In this paper we consider a class of nonlinear systems with uncertain parameters which enter the system nonlinearly. We assume that the uncertain nonlinear system is minimum phase and the uncertain parameters are from a bounded compact set. The problem under consideration is the design of a nonlinear static state feedback controller such that the closed-loop system is passive for all admissible uncertainties.

Optimal tracking performance of discrete-time systems over an additive white noise channel

This paper studies the optimal tracking performance of multiple-input multiple-output (MIMO), finite dimensional, linear time-invariant discrete-time systems with a power-constrained additive white noise (AWN) channel in the feedback path. We adopt the tracking error power as a measure of the performance and examine the best achievable performance by all two-parameter stabilizing controllers. In the due process, a scaling scheme is introduced as a means of integrating controller and channel design, and is optimized to better the tracking performance. In contrast to the standard setting where tracking of a step reference signal is conducted with no communication constraint, in which the tracking error can be made as zero for minimum phase plants, it is shown explicitly herein that the tracking performance will be additionally constrained by the plant unstable poles, as a consequence of noisy, power-constrained channels in the feedback loop.

Dual-Stage Actuator Control Design Using a Doubly Coprime Factorization Approach

This paper first reveals that the tracking and disturbance rejection problems can be decoupled into two independent optimization problems under the 2-DOF control framework. This result is then used for the design of a 2-DOF controller for a dual-stage actuator (DSA) system to provide desired performance of disturbance rejection and step tracking. The 2-DOF controller is designed based on the doubly coprime factorization approach, with which the closed-loop transfer function is expressed explicitly in terms of design parameters. This greatly simplifies the optimization of design parameters in meeting desired specifications. We further study how to use the design parameters to deal with specific problems in the DSA, i.e., control allocation and trajectory planning. For step tracking beyond the secondary actuator range, a nonlinear controller is also used for the primary actuator to complete the task. Experimental results demonstrate the practical implementation of the DSA control system and verify its effectiveness for step tracking and disturbance rejection and its robust performance under load changes.

On performance limitation in tracking a sinusoid

This note studies the performance limitation of a feedback system with a given linear time-invariant (LTI) plant in tracking a sinusoidal signal. It continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal. In this note, we consider the more realistic (and more difficult) situation where the controller only accesses the current and past values of the reference. An explicit formula of the best attainable performance is obtained for a single-input-single-output (SISO) system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid. Compared to the previously studied case when the future of the reference is available, this formula clearly shows the extra effort one has to pay due to the lack of the reference information. A partial result for a multiple-input-multiple-output (MIMO) system is also given

Fundamental limit of discrete-time systems in tracking multi-tone sinusoidal signals

This paper studies the tracking performance of linear time-invariant multi-variable discrete-time systems. The specific problem under consideration is to track a multi-tone sinusoidal reference signal consisting of linear combinations of a step and several sinusoidal signals, whereas the tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the best attainable performance, under any control structures and parameters, and we seek to determine this limit analytically in terms of the given plant and reference characteristics. Both the full-information and partial-information tracking schemes are formulated and investigated to address these goals, which are concerned with whether or not the reference information is fully available for tracking. Analytical expressions are developed in full generality under full-information tracking, and for a more specialized case under partial-information scheme. In addition, an optimal cheap control design is constructed to show that the performance limit can be attained asymptotically in the full-information case. The results show that in general plant nonminimum phase zeros and reference modes can interact to fundamentally constrain a systems tracking ability. They also show that absence of full reference information can degrade the tracking performance, thus demonstrating an intrinsic trade-off between the tracking objective and the availability of the reference information.

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.