Performance issues in the individual channel design of 2-input 2-output systems. Part 2. Robustness issues

In this second part of a three part series of papers examining the performance issues raised by the introductory paper (O'Reilly and Leithead 1991) on individual channel design for multivariable control, attention is focused on robustness: that is, on the meeting of performance specifications—in particular closed-loop system stability—in the face of plant uncertainty. The main results are that the use of phase margins and gain margins to assess robustness of 2-input 2-output multi-variable systems is justified. Further, it is shown that controllers for uncertain 2-input 2-output multivariable systems can be designed by classical SISO Nyquist-Bode local loop shaping using stable minimum phase controllers. Several additional results are also described in the paper.     

Satisfaction of gain and phase margin constraints using proportional controllers

Gain and phase margins (GM and PM) are frequently used as robustness indicators for linear time invariant systems. In many cases, the design problem is reduced to determination of a loop gain to satisfy several design criteria including gain and PM specifications. Therefore finding the set of fixed order compensators that satisfy given gain and PM specifications attracted the attention of a considerable part of the scientific community recently. It is possible to show that it may not be possible to satisfy the required gain and/or PM specifications using only a proportional controller. As a result, finding the limits of the proportional controllers for a given plant (i.e. finding the maximum achievable gain and PMs, MAGM and MAPM) is also important. After providing alternative methods (to those that already exist in the literature) for determining all stabilising proportional controllers that satisfy gain and PM constraints, a new method for calculating MAGM and MAPM using proportional controllers is given in this article. A formulation to calculate maximum gain that results in MAGM is also provided.     

Evolutionary design and implementation of a hard disk drive servo control system

The ever increasing demand for higher storage capacity and smaller magnetic hard disk drives have driven the need of developing a high performance head positioning servo control system. To meet the challenge, this paper presents the design and real-time implementation of a robust two-degree-of-freedom servo system for physical 3.5-in. hard disk drive with single voice-coil-motor actuator using a multi-objective evolutionary algorithm toolbox. Besides the simplicity in controller structure, such an evolutionary servo control system is capable of meeting various performance specifications of hard disk drives in both the time and frequency domains. It is shown that the servo system optimally moves the magnetic head onto the desired track with minimal control effort, and keeps it on the track robustly against plant uncertainties or runout disturbances. Validation results of the evolutionary servo control system are compared with classical PID and RPT controllers, which show excellent closed-loop response and robustness in the face of practical perturbations in HDD.     

混合频小增益动态输出反馈控制综合

本文主要研究线性连续时间系统经动态输出反馈的混合频小增益控制综合问题. 提出了一种使相应闭环系统渐近稳定并满足有限频范围和全频范围小增益条件要求的动态输出反馈控制器的新设计方法. 设计条件以一系列线性矩阵不等式的解的形式给出. 最后, 给出一个数值例子来说明设计程序以及这种方法对比于现有方法的优势.     

Design of low‐order controllers satisfying sensitivity constraints for unstructured uncertain plants

This paper presents a simple analytically based algorithm for the design of reduced‐order controllers satisfying frequency‐dependent sensitivity specifications for SISO plants having unstructured uncertainty. The uncertainties can be additive as well as multiplicative, and can take the form of circles, polygons or sectors located around a nominal plant. Moreover, the circle radius and polygon and sector sizes may depend on the frequency. The proposed method is applicable to both continuous and discrete designs. Copyright © 2004 John Wiley & Sons, Ltd.     

A convex parametrization of risk-adjusted stabilizing controllers

The focal point of this paper is the synthesis of controllers under risk-specifications. In recent years there has been a growing interest in the development of techniques for controller design where, instead of requiring that the performance specifications are met for every possible value of admissible uncertainty, it is required that the risk of performance violation is below a small well-defined risk level. In contrast to previous work, where the search for the controller gains is done using randomized algorithms, the results in this paper show that for a class of uncertain linear time invariant systems, the search for the “risk-adjusted” controller can be done efficiently using deterministic algorithms. More precisely, for the case when the characteristic polynomial of the closed loop system depends affinely on the uncertainty, we provide a convex parametrization of “risk-adjusted” stabilizing controllers.     

Controller synthesis free of analytical model: fixed-order controllers

This paper extends the existing results on model-free approach for three-term controllers to fixed-order controllers. It is shown that knowing the frequency response of a plant is sufficient to calculate a subset of stabilising fixed-order controllers using a set of linear inequalities. The main feature of the proposed approach is that the possible lowest order stabilising controllers corresponding to any stable, unstable, minimum and non-minimum phase plants can be obtained. Also, it is shown that the performance criterion can be transformed to simultaneously stabilising problem for a family of real and complex plants derived from the nominal plant. The usefulness of the proposed approach is illustrated by some examples.     

Design of controllers for a class of nonlinear control systems

A design method is developed for a class of controllers which bound a nonlinear control systems response within prescribed specifications for a set of initial conditions. The design method is based upon the quantitative properties of a class of Lyapunov functions and a time domain ratio criterion. A digital computer algorithm which permits the automation of a large portion of the design method is presented.     

A DIALOG-ORIENTED AND GRADIENT-BASED STABILITY MARGIN IN UNCERTAIN SYSTEMS

ABSTRACT

Gradients and matricial gradients for optimally increasing the stability margin and the admissible uncertainty of a dynamic system are the targets of this presented article. To design a dynamic system, the gradients are used in a dialog between a system scientist and gradient-based computer support. The stability margin is derived for output state controllers, including regular state controllers. The resonant frequency and the damping factors are investigated as a direct function of the maximum admissible uncertainty. The resulting gradients are extended to observer-assisted controllers, to minimum-order observers, and to dynamic-output state controllers.     

Multi-rate digital redesign of cascaded and dynamic output feedback systems

In this paper, a new indirect digital redesign method is presented for multi-rate sampled-data control systems with cascaded and dynamic output feedback controllers. These analogue controllers are often pre-designed based on desirable frequency specifications, such as bandwidth, natural angular frequency, etc. To take advantage of the digital controller over the analogue controller, digital implementation of these analogue controllers are often desirable. As only measured input-output signals are available, an ideal state reconstructing algorithm is utilised to obtain the multi-rate discrete-time states of the original continuous-time system. Based on the Chebyshev quadrature method, the gains of the multi-rate cascaded and the output feedback digital controllers are determined from their continuous-time counterparts according to the different sampling rates employed in the different parts of the closed-loop system. As a result, the respective analogue controllers with the high-frequency and low-frequency characteristics can be implemented using the respective fast-rate sampling and slow-rate sampling digital controllers. Unlike the classical direct bilinear transform method, which is an open-loop direct digital redesign method, the proposed digital controllers take into account the state-matching of the original continuous-time closed-loop system and the digitally redesigned sampled-data closed-loop system. To further improve the state-matching performance, an improved digital redesign approach is also developed to construct the multi-rate cascaded and dynamic output feedback digital controllers. Illustrative examples are given to demonstrate the effectiveness of the developed methods.     

On the design of robust controllers for arbitrary uncertainty structures

The focal point of this paper is the design of robust controllers for linear time-invariant uncertain systems. Given bounds on performance (defined by a convex performance evaluator) the algorithm converges to a controller that robustly satisfies the specifications. The procedure introduced has its basis on stochastic gradient algorithms and it is proven that the probability of performance violation tends to zero with probability one. Moreover, this algorithm can be applied to any uncertain plant, independently of the uncertainty structure. As an example of application of this new approach, we demonstrate its usefulness in the design of robust H/sub 2/ controllers.     

The design of suboptimal asymptotic stabilising controllers for nonlinear slowly varying systems

The design of asymptotic stabilising controllers for slowly varying nonlinear systems is considered in this paper. The designed control law is based on finding a slowly varying control Lyapunov function. Also, consideration of the Hamilton–Jacobi–Bellman equation showed that the proposed controller is a suboptimal controller and the response of the system may be very close to its optimal solution. The maximum admissible rate of changes of the system dynamic is also evaluated. This technique is first applied to a created example and then to a practical example (optimal autopilot design for an air vehicle). The air vehicle is modelled as a nonlinear slowly varying system and the efficiency of the designed autopilot in terms of transient responses, control signals and the values of cost function are shown by numerical simulations.     

Generalized KYP lemma: unified frequency domain inequalities with design applications

The celebrated Kalman-Yakubovic/spl caron/-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an S-procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects-the frequency range and the class of systems-and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers.     

Bi-C eigenvector sequences and the design of causal commutative controllers

The use of Laurent power series expansions of the eigenvector matrix of a linear multivariable transfer function matrix G(z) holds the key to the physical realization of commutative controllers. In general, however, such controllers would be anti-causal. It is the purpose of this paper to show that there are enough degrees of freedom in the choice of the controller eigenfunctions both to effect gain/phase compensation of the frequency response of the eigenfunctions of G(z), and to force the resulting control law to be causal. The results of the paper are shown to be superior to those possible through the use of approximate commutative controllers proposed earlier.     

An interactive parameter space method for robust performance inmixed sensitivity problems

This paper presents an interactive graphical method to determine the set of fixed-order stabilizing controllers achieving robust performance, in the mixed sensitivity framework. The method is limited to single-input/single-output (SISO) systems but offers significant advantages over traditional loop gain shaping methods such as H^∞ and μ-synthesis. It can handle pure time delays in an exact manner and the weighting functions need not be rational. The technique translates frequency-domain weighting functions and stability constraints into the parameter space and thus gives the user more insights into the design than conventional methods. By virtue of producing the required parameter space region for the frequency response criteria, subsequent optimization of secondary objectives is possible. The controllers obtained are of lower order for comparable performance than those produced by current H^∞ and μ-synthesis techniques. The method is particularly well-suited to robust control problems where frequency-domain constraints emerge from the analysis of nonparametric uncertainties in the system and also to control problems where the frequency domain loop shaping is used to achieve time-domain specifications     

A Measurement‐Based Approach for Designing Fixed‐Order Controllers for Unknown Closed‐Loop Architecture

This paper presents a new technique to design fixed‐structure controllers for linear unknown systems using a set of measurements. In model‐based approaches, the measured data are used to identify a model of the plant for which a suitable controller can be designed. Due to the fact that real processes cannot be described perfectly by mathematical models, designing controllers using such models to guarantee some desired closed‐loop performance is a challenging task. Hence, a possible alternative to model‐based methods is to directly utilize the measured data in the design process. We propose an approach to designing structured controllers using a set of closed‐loop frequency‐domain data. The principle of such an approach is based on computing the parameters of a fixed‐order controller for which the closed‐loop frequency response fits a desired frequency response that describes some desired performance indices. This problem is formulated as an error minimization problem, which can be solved to find suitable values of the controller parameters. The main feature of the proposed control methodology is that it can be applied to stable and unstable plants. Additionally, the design process depends on a pre‐selected controller structure, which allows for the selection of low‐order controllers. An application of the proposed method to a DC servomotor system is presented to experimentally validate and demonstrate its efficacy.     

可重构飞行控制律设计的混合特征结构配置方法研究

将特征结构配置与模糊控制理论相结合对飞机侧向飞行重构控制律进行设计.首先介绍了重构控制律中特征结构配置的原理和反馈增益矩阵的算法实现,在此基础上结合带优化修正参数的无量化模糊控制方法对闭环系统进行鲁棒控制器设计.该控制策略以特征结构配置作为故障后系统的内环控制器,模糊控制器作为外环控制器,使系统获得较好的动态性能和较强的鲁棒性.仿真结果表明系统有效抑制了内部参数摄动对飞行任务的不良影响.     

Controller Synthesis Free of Analytical Models: Three Term Controllers

The main focus of this paper is on direct data driven synthesis and design of controllers. We show that the complete set of stabilizing proportional–integral–derivative (PID) and first-order controllers for a finite dimensional linear time-invariant plant, possibly cascaded with a delay, can be calculated directly from the frequency response (Nyquist/Bode) data $P(jmath omega )$ for $omega in [0, infty )$ without the need of producing an identified analytical model. It is also shown that complete sets achieving guaranteed levels of performance measures such as gain margin, phase margin, and $H_{infty} $ norms can likewise be calculated directly from only Nyquist/Bode data. The solutions have important new features. For example it is not necessary to know the order of the plant or even the number of left half plane or right half plane poles or zeros. The solution also identifies, in the case of PID controllers an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate. These constitute important new guidelines for identification when the latter is to be used for control design.      

Robust IMC–PID tuning for cascade control systems with gain and phase margin specifications

In this article, an internal model control plus proportional-integral-derivative (IMC–PID) tuning procedure for cascade control systems is proposed based on the gain and phase margin specifications of the inner and outer loop. The internal model control parameters are adjusted according to the desired frequency response of each loop with a minimum interaction between the inner and outer PID controllers, obtaining a fine tuning and the desired gain and phase margins specifications due to an appropriate selection of the PID controller gains and constants. Given the design specifications for the inner and outer loop, this tuning procedure adjusts the IMC parameter of each controller independently, with no interference between the inner and outer loop obtaining a robust method for cascade controllers with better performance than sequential tuning or other frequency domain-based methods. This technique is accurate and simple, providing a convenient technique for the PID tuning of cascade control systems in different applications such as mechanical, electrical or chemical systems. The proposed tuning method explained in this article provides a flexible tuning procedure in comparison with other tuning procedures because each loop is tuned simultaneously without modifying the robustness characteristics of the inner and outer loop. Several experiments are shown to compare and validate the effectiveness of the proposed tuning procedure over other sequential or cascade tuning methods; some experiments under different conditions are done to test the performance of the proposed tuning technique. For these reasons, a robustness analysis based on sensitivity is shown in this article to analyze the disturbance rejection properties and the relations of the IMC parameters.