Multivariable gain-phase and sensitivity integral relations anddesign trade-offs

We present several extensions of the classical Bode integral relations to multivariable systems. The main contributions are an extended gain-phase integral formula and an extended sensitivity integral relation. These results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems. In particular, they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single-input/single-output systems. Both results show that design limitations in multivariable systems depend strongly upon certain directionality properties     

Logarithmic integrals, interpolation bounds, and performancelimitations in MIMO feedback systems

We study performance limitation issues found in linear multivariable feedback systems. Our main contributions include Bode and Poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions. These results characterize and quantify explicitly how open-loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance. The role of time delay is also studied in this context. Most notably, we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros, and in particular, on how pole and zero directions are aligned. The latter is characterized by angles measuring the mutual orientation between zero and pole directions, and it is shown to play a crucial role in multivariable system design     

Control of LTI systems subject to unanticipated extreme perturbations using self-tuning 3-term switching controllers

In the design of conventional control systems for a multivariable system, using robust/adaptive control techniques, the motivation is to design a controller which "works satisfactorily" in the presence of plant uncertainty. Unfortunately, however, if large unanticipated structural changes subsequently occur in the system, severe limitations in practical performance may occur, since such conventional control schemes usually do not have the ability to control systems which are subject to unplanned extreme changes. Moreover, for the realistic situation when control input constraints exist, few results for continuous time multivariable systems are available. In this paper, a new class of self-tuning proportional-integral-derivative switching controllers, which is an extension of the self-tuning integral controller of Miller and Davison, is described, and has the property that it is robust to unplanned extreme changes in the plant and satisfies any feasible control signal input constraints. Results of this self-tuning controller when applied to an experimental multivariable system also are described.     

Fundamental limitations and intrinsic limits of feedback: An overview in an information age

This paper presents a review on the fundamental performance limitations and design tradeoffs of feedback control systems, ranging from classical performance tradeoff issues to the more recent information-theoretic analysis, and from conventional feedback systems to networked control systems, with an attempt to document some of the key achievements in more than seventy years of intellectual inquiries into control performance limitation studies, as so embodied by the timeless contributions of Bode known as the Bode integral relations.     

Properties of sensitivity and complementary sensitivity functions in single-input single-output digital control systems

Sensitivity and complementary sensitivity functions play a key role in feedback control system design. This paper is concerned with the relation between the sampling period and the properties of these functions in digital control systems. Some integral constraints and the lower bounds of the H ^∞-norm are derived, which show that the feedback performance for the unstable plant with the stable digital compensator can be improved as the sampling period goes to zero.     

Right half plane poles and zeros and design tradeoffs in feedback systems

This paper expresses limitations imposed by right half plane poles and zeros of the open-loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed-loop system. The limitations are determined by integral relationships which must be satisfied by these functions. The integral relationships are interpreted in the context of feedback design.     

控制系统的脆弱性分析

脆弱性是指一闭环系统的稳定性对参数摄动极端敏感.这类系统的开环频率特性经过距临界点(-1,j0)的近旁,这种与(-1,j0)点之间脆弱的相对关系遇到任何可能的摄动就会遭到破坏.文中分析指出,控制系统的脆弱性可以用Bode积分来进行定量分析.对于一个非最小相位的不稳定对象的控制来说,控制器的不稳定极点与对象的不稳定极点加到一起会大大增加系统灵敏度函数的峰值.故这类系统的设计必然是脆弱的.对于不稳定的多变量对象的控制来说,如果其中一个等价的输出反馈回路中有一个较大的不稳定极点,那么其灵敏度函数也会出现大的峰值,会导致脆弱性,而利用Bode积分则可以避免设计的脆弱性.     

Minimum sensitivity design of linear multivariable feedback control systems by matrix spectral factorization

A scalar measure of system sensitivity to plant parameter variations is employed in the design of linear lumped stationary multivariable feedback control systems. The plant parameters are treated as random variables, and design formulas are derived which lead to systems with the smallest expected value for the chosen scalar sensitivity measure. The design formulas give physically realizable feedback and tandem compensation network transfer function matrices provided the overall system transfer function matrix is properly specified. The solution of the minimum sensitivity design problem is obtained by first solving the multivariable semi-free-configuration Wiener problem.     

基于特征结构配置的最小灵敏度控制器设计

本文描述了一种多变量控制系统的最小灵敏度控制器的设计方法.该方法除将闭环系统 的特征值配到给定点外,还利用状态反馈所剩余的自由度,对闭环特征向量实行部分配置,以 满足其它设计指标,如低参数灵敏度和良好的响应特性等.设计举例和仿真表明,采用这种最 小灵敏度控制器,被控系统既能获得满意的瞬态响应,又能降低状态轨迹对参数扰动的灵敏 度.本设计采用的算法是以解矩阵Sylvester方程为主.     

Feedback properties of multivariable systems: The role and use of the return difference matrix

For linear time-invariant multivariable feedback systems, the feedback properties of plant disturbance attenuation, sensor noise response, stability margins, and sensitivity to plant and sensor variation are quantitatively related to the Bode magnitude versus frequency plots of the singular values of the return difference matrixI + Land of the associated inverse-return difference matrixI + L^{-1}. Implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed. The penalty function in the stochastic linear quadratic Gaussian (LQG) optimal control problem is found to be a weighted-sum of the singular values, with the weights determined by the quadratic cost and noise intensity matrices. This enables systematic "tuning" of LQG cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties. The theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft.     

New roles for feedforward in multivariable control by individual channel design

Considerable though the benefits of feedback control are, they do not extend to ameliorating adverse multivariable plant dynamical characteristics that are of a structural nature. Feedforward control on the other hand, is shown in this paper to complement multivariable feedback control in a simple but powerful way by providing a strikingly direct resolution of such structural difficulties as non-minimum phase plant behaviour, plant excessive structure sensitivity (uncertainty in the number of channel RHPZs), plant excessive phase sensitivity, and decoupling without increasing system uncertainty. This use of feedforward is believed to be new and is completely different from the well-known application of feedforward control to anticipate and counteract the effect of some known plant disturbance before it affects the output, as widely used in the process industries. The paper ends with a complete statement of the philosophy and methodology of multivariable control, feedforward and feedback, within the context of the individual channel design of 2-input 2-output systems (O'Reilly and Leithead 1991). In particular, it is pointed out that it is immaterial which multivariable technique is used to determine the feedback controller for the plant amended by feedforward, although diagonal controller matrices are preferred (Leithead and O'Reilly 1992b) otherwise, if the feedback controller is non-diagonal, the extent to which the sensitivity to system uncertainty is increased should be checked (Leithead and O'Reilly 1992b, O'Reilly and Leithead 1993).     

Fundamental design limitations of the general control configuration

The theory of fundamental design limitations is well understood for the case that the performance variable is measured for feedback. In the present paper, we extend the theory to systems for which the performance variable is not measured. We consider only the special case for which the performance and measured outputs and the control and exogenous inputs are all scalar signals. The results of the paper depend on the control architecture, specifically, on the location of the sensor relative to the performance output, and the actuator relative to the exogenous input. We show that there may exist a tradeoff between disturbance attenuation and stability robustness that is in addition to the tradeoffs that exist when the performance output is measured. We also develop a set of interpolation constraints that must be satisfied by the disturbance response at certain closed right half plane poles and zeros, and translate these constraints into generalizations of the Bode and Poisson sensitivity integrals. In the absence of problematic interpolation constraints we show that there exists a stabilizing control law that achieves arbitrarily small disturbance response. Depending on the system architecture, this control law will either be high gain feedback or a finite gain controller that depends explicitly on the plant model. We illustrate the results of this paper with the problem of active noise control in an acoustic duct.     

A fundamental multivariable robustness theorem for robusteigenvalue assignment

A fundamental robustness theorem for robust eigenvalue assignment of multivariable feedback systems is derived. The theorem determines whether a perturbed multivariable feedback system has its characteristic polynomial zeros located in the same regions as the nominal system does. It can be applied to continuous systems as well as to discrete systems. The theorem can handle nonsquare transfer matrices as well as dynamic output feedback. Conditions are discussed under which the proposed theorem reduces to a fundamental theorem for stability robustness analysis     

Sensitivity in sampled-data systems

The sensitivity of sampled-data systems to plant variations and load disturbance is examined with reference to two fundamental configurations. It is shown that the sensitivity function of a sampled-data system containing no continuous feedback is theoretically limited if the plant possesses low-pass response characteristics. Since the limitation imposed by sampling can be removed by continuous feedback, consideration is given to a sampled-data system containing a continuous minor-loop feedback path. A sensitivity function is derived which measures the individual effectiveness of the inner and outer loops in desensitizing the system, thereby providing the basis for a realistic design approach. This sensitivity function is applied to a design example.     

Integral constraints and performance limits on complementary sensitivity: discrete-time systems

We study performance limitation issues for multivariable discrete-time feedback systems. The complementary sensitivity function is employed as a performance measure, and Bode and Poisson-type integral inequalities and -type performance limits are derived. The results exhibit frequency-dependent constraints as well as best achievable limits on the complementary sensitivity function, which are shown to be determined by nonminimum phase zeros, unstable poles, and time delays. In particular, the directions of such zeros and poles are seen to play a central role to this effect.     

Design of minimum sensitivity systems

A method for the design of linear time-invariant multivariable minimum sensitivity systems is presented. The method utilizes a quadratic form in the parameter-induced output errors as a performance index to be minimized, with the constraint that a prescribed nominal transfer function matrix be obtained. An essential ingredient in the procedure is the use of a comparison sensitivity matrix. Two advantages that follow from the use of the sensitivity matrix are: 1) Physical realization constraints on the compensators may be included in the design. 2) The computational aspects of the problem are relatively simple and may be carried out routinely using any parameter optimization algorithm. A nontrivial multivariable example illustrates the procedure. The design method may be viewed as the second part in a two-part procedure, where the first part is the determination of a desired nominal transfer function. A two-degree-of-freedom feedback structure is used to realize an optimum or desired nominal closed-loop transfer matrix, as well as a minimum in a sensitivity index.     

Generalization of integral constraints on sensitivity to time-delaysystems

Extends results on integral constraints to time-delay systems. Key technical issues in this extension include properties of nonminimum phase zeros, the convergence of Blaschke products, and the Poisson integral. The results have implications for scalar and multivariable control system design as well as in filtering problems     

Frequency domain design trade-offs for linear periodic feedback systems*

The purpose of this paper is to analyse inherent design limitations associated with systems that are linear and periodically time-varying. The contributions of the paper are (i) to relate frequency domain raising methods from signal processing literature to time-domain lifting used in control literature, and (ii) to develop extensions of the Poisson sensitivity and complementary sensitivity integral constraints. In particular, it is shown that there is generally an additional cost associated with having a time invariant target closed loop for a periodic open loop plant. It is also shown that design limitations due to unstable poles and/or non-minimum phase zeros of a discrete linear time-invariant plant remain even if a periodic time-varying controller is used. As a consequence, the utility of periodic control in circumventing design limitations is questioned.     

The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.     

多输入输出反馈系统虚轴的零点对跟踪问题的约束

反馈系统的设计会受到闭环系统特性的制约. 本文给出了反馈控制系统接近虚轴或虚轴上的稳定零点对跟踪误差的时间域积分约束. 这一时间域积分约束是任何一个线性时不变系统为保证其闭环系统的稳定性, 其跟踪误差必须满足的. 接近虚轴或虚轴上的稳定零的存在表明跟踪误差与调节时间之间存在某种折中. 对固定的调节时间, 本文给出了在虚轴上存在零点情形下, 其跟踪误差的无穷范数下界的一个有效估计. 此估计表明, 零点的绝对值越小, 其无穷范数的下界越大. 这些约束由一个例子加以解释.