Optimal dual-rate digital redesign with closed-loop order reduction

This paper presents a new method for the design of control laws for systems with on-off nonlinear actuators. The new methodology improves the design of control laws for pulse-modulated systems by allowing the use of continuous-time multi-inputlmultioutput design procedures. Discrete-time state feedback control gains are developed from the of continuous-time feedback gains, based on a geometric series approximation, to closely match the states of the closed-loop hybrid system to those of the original designed closed-loop continuous-time system at each sampling instant. A delay is then incorporated into a pulse-width modulator design to closely match the states of the modulated system to those of the discrete-time pulse-amplitude modulated system. The delay time is improved from previous work to provide a close match between the states for pulse durations of all sizes, in effect matching the states of the pulse-width modulated system to those of the closed-loop continuous-time system. The new control law design is applied to the problem of attitude control of the Space Shuttle Orbiter with the Hubble Space Telescope deployed on its remote manipulator. * Senior Member Technical Staff. Member A I M . t Professor, Electrical Engineering. Member AIAA. tt Aerospace Engineer. Member AIAA. Copyright O 1995 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Many paradigms are available for the synthesis of control laws for continuous-time systems, and though many methods also exist for developing discrete-time or digital controllers, it is often desirable to design a control law for the continuous-time plant. It is often more direct to determine the control specifications in the continuous-time or frequency domain. Also, direct digital control law design has difficulty in predetermining the hybrid control specifications and ignores inter sampling effects. Yet, these continuoustime control laws are generally implemented via discrete-time computer control, requiring that a discrete-time control, equivalent to the continuous-time control, be developed. The process of developing discrete-time control laws that provide closed-loop performance of the hybrid system equivalent to the performance of the designed closed-loop continuoustime system is known as redesign. Several techniques have been developed to digitally redesign continuous-time feedback controllers[l] ,[2] ,[3]. Unfortunately, the principal applied for the design of discrete-time control laws, i.e., that the amplitude of the control signal can take any finite value, which is why discrete-time control is often referred to as pulse-amplitude control (PAM), is not valid for all computer controlled systems. Systems that utilize onoff control devices, such as the reaction jet or electronic relay, can only provide a single finite value of control. To provide control with these types of actuator, either complex non-linear algorithms[4] must be developed or the control must be applied by modulating the pulse width, the pulse frequency or both the pulse width and frequency. These modulation techniques, in particular pulse-width modulation (PWM), allow the design of linear control laws to control non-linear systems. Although methods exist for the design of linear control laws for pulse width modulation systems[5], they still

Stability and sensitivity of nonlinear sampled data systems

A new method for digital model conversion of a continuous-time uncertain system and a new digital redesign method for robust control of a sampled-data uncertain system are presented. The concept of the principle of equivalent areas together with interval arithmetic is utilised for finding the discrete-time uncertain model and the digital robust control law from the continuous-time uncertain state equation and the analogue robust control law, respectively. Using the newly digitally redesigned controllers, the resulting dynamic states of the digitally controlled sampled-data uncertain systems are able to closely match those of the original analogously controlled continuous-time uncertain systems for a relatively longer sampling period.

ABSTRACTIn this paper, the digital redesign of a continuous suboptimal tracker for the two‐dimensional (2‐D) systems is proposed. This paper presents a new optimal digital redesign technique of 2‐D systems for finding a dynamic digital control law from the given continuous‐time 2‐D systems by minimizing a quadratic cost function (performance index). We directly convert the original continuous‐time 2‐D quadratic cost function into the discretized form and solve the optimization problem in the discrete‐time domain. The developed optimal digital redesign control law enables the output of the digitally controlled closed‐loop systems to closely match the reference signal for 2‐D systems, and it can be easily implemented using microcomputers. An illustrative example is presented to demonstrate the effectiveness of the proposed procedure.

The discrete-time state feedback control systems share many similarities with their continuous-time counterpart. This chapter deals with the introduction of state feedback control in discrete-time. It discusses for a control system with multiple inputs or multiple outputs, the discrete-time linear quadratic regulator (DLQR) provides both optimal controller and observer design. The chapter presents DLQR with a prescribed degree of stability. The state feedback control systems in discrete-time share many characteristics, properties, and design methods with the continuous-time state feedback control systems, which includes the same characteristic equation expression. The concept of observability is associated with an observer design. Closed-loop performance with the discrete-time LQR is determined by the selection of weighting matrices. The chapter provides case studies to illustrate vibration control using linear quadratic regulator.

Using the bilinear and tuning bilinear approximation together with the interval arithmetic operation, the discrete-time uncertain model and robust digital redesign of sampled-data uncertain system with input time delay are presented. The system matrices characterizing the stale-space representation of the original uncertain system are assumed to be interval matrices. The developed discrete-time interval model tightly encloses the exactly discretized continuous-time input time-delay uncertain system, and it can be utilized for digital simulation. Based on bilinear approximation, a tuning bilinear approximation is presented to find the digitally redesign controller of the digitally controlled sampled-data uncertain system with input time delay. The digitally redesigned predictor controller is able to closely match the states of the digitally redesigned uncertain sampled-data system with those of the original continuous-time predictor controlled uncertain system.

STABILITY AND SENSITIVITY OF NONLINEAR SAMPLED DATA SYSTEMS

본 논문에서는 불확실성을 포함한 연속시간 비선형 동적 시스템에 대한 퍼지 모델 기반 제어기의 지능형 디지털 재설계를 위한 조직적인 방법을 제안한다. 불확실 비선형 시스템을 표현하기 위해 연속시간 불확실 TS 퍼지 모델이 구성되었다. 안정화와 추적을 위한 퍼지 모델 기반 제어기 설계를 위해 EPDC 기법이 이용되었다. 설계된 연속시간 제어기는 지능형 디지털 재설계 기법을 이용함으로써 이산시간 제어기로 전환되었다. 이 새로운 설계 기법은 퍼지 모델 기반 제어 이론과 불확실성을 가진 비선형 동적 시스템에 대한 진보된 디지털 재설계 기법의 통합을 위한 조직적이고, 효과적인 틀을 제공한다. 마지막으로, 단일 링크 유연 로봇 시스템의 모의 실험을 이용해 개발된 설계 방법의 효용성과 실행 가능성을 입증한다. In this paper, we propose a systematic method for intelligent digital redesign of a fuzzy-model-based controller for continuous-time nonlinear dynamical systems which may also contain uncertainties. The continuous-time uncertain TS fuzzy model is first constructed to represent the uncertain nonlinear systems. An extended parallel distributed compensation(EPDC) technique is then used to design a fuzzy-model-based controller for both stabilization and tracking. The designed continuous-time controller is then converted to an equivalent discrete-time controller by using an integrated intelligent digital redesign method. This new design technique provides a systematic and effective framework for integration of the fuzzy-model-based control theory and the advanced digital redesign technique for nonlinear dynamical systems with uncertainties. Finally, The single link flexible-joint robot arm is used as an illustrative example to show the effectiveness and the feasibility of the developed design method.

Basing on lifting technology, this paper researches a design method for multirate sampled-data control system in the background of flight control system design, which consummates the theory of multirate sampled-date control system design for further steps, and provides a method to design multirate sampled-data controller. As the reason of uncertainty in the aircraft model and the need of different sampling frequencies in different control loops, H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is considered. First, multirate sampled-data control system is outlined, and then the signal description are presented. Because of multirate sampled-data system is time-variant, the direct digital controller design becomes difficult. The equivalent discrete time-invariant systems can be converted from the linear time-variant systems through lifting technology. Then the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> discretization of multirate sampled-data systems is researched. After the sampled-data control system was lifted, the equivalent L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> [0, h) -induced norm is proven. Base on the above, the method of H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is summarized. A height control is simulated using a simplified aircraft model to illustrate the design. The simulation shows that the multirate Sampled-data control system can achieve the satisfactory objective accurately and optimally.

An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach

Many of nonlinear control systems are sampled-data system, i.e. the continuous-time nonlinear plants are controlled by digital controllers. So it is important to investigate that if the solution of the discrete-time output regulation problem is effective to sampled-data nonlinear control systems. Recently a feedforward neural network based approach to solving the discrete regulator equations has been presented. This approach leads to an effective way to practically solve the discrete nonlinear output regulation problem. In this paper the approach is used to sampled-data nonlinear control system. The simulation of the sampled-data system shows that the control law designed by the proposed approach performs much better than the linear control law does.

The robust stability criterion for sampled-data control systems with a sector nonlinearity was presented in our previous papers (1996, 1998). Although it is applicable only to the sampled-data control system of a certain class, a usual discrete-time control system belongs to this type of class. The criterion contains the robust stability condition in regard to the nonlinear allowable band (sector) for continuous-time systems and the linear allowable gain band (interval set) for discrete-time systems. The paper focuses primarily on the concept of robust stability in order to analyze a nonlinear element by the linear gain band, in other words, the validity of Aizerman's conjecture extended into discrete-time systems.

In this paper, the stability problem for the digital redesign of discrete-time switched systems using H∞ linear matrix inequality (LMI) is investigated. We propose the switching time approach for digital redesign between controller work and failure, and this switching time will limit the system output within the system capacity. When the controller fails, the overall system will be unstable. Therefore, if the digital redesign controller is not restored in a certain period of time, the system output will exceed the system capacity. To solve this problem, we propose a switching law to determine the switching time between the stable mode (controller work) and the unstable (controller failure) mode; this will limit the overall system states in the unstable mode. In addition, the digital redesign controller has the advantage of faster tracking. After we propose a discrete-time switching system with stable and unstable modes, we use H∞ linear matrix inequality (LMI) and Lyapunov functions to prove the stability in detail. Finally, the numerical example illustrates the feasibility of the proposed approach.

The authors consider sampled-data control systems which consist of an interconnection of a continuous-time nonlinear plant and a digital controller which has quantizers (but is otherwise linear), along with the required interface elements (A/D and D/A converters). In the present paper the authors study the qualitative effects of the quantizers and the plant nonlinearities of these systems.

&lt;div class="htmlview paragraph"&gt;In this work we study the stability of digital controls of flexible/Vibratory aerospace/automobile systems by the graph norm technique, occurring in sampled-data control systems due to sampling period variations. To do so, this work tries to establish regions (graphs) of stability and instability in a Banach Space, the distances (norms) between them and a given design to detect analytically and/or numerically its margins of stability or conditions of instability. Based on that, we sketch the first steps for a design methodology of stable digital controllers of flexible/vibratory systems embedded in a sampled-data system with adjustable sampling periods of A/D and D/A converters. A short tutorial about the graph norm technique is also given and some theoretical results as well numerical results are shown. This work was done in two folds to unmask the stability secrets hidden in a general sampled-data control system, until today not revelated. The first part we have used the graph norm and at the second part we made use of the Liapunov stability theory.&lt;/div&gt;