On the synthesis of robust PID controllers for plants with structured and unstructured uncertainty

In this paper, we consider the problems of synthesizing PID controllers for robust stability and performance for a given linear time‐invariant plant subject to both parametric and H_∞‐norm‐bounded perturbations. Using results from the area of parametric robust control, synthesis problems are converted into simultaneous stabilization of a family of complex segment polynomials. The results on H_∞ PID synthesis are then used to devise a design procedure for determining the admissible PID gain values. One of the important features of the proposed method is that it constructively characterizes the approximated set of all admissible PID controllers. This characterization can facilitate the optimal design of any additional design requirements. Copyright © 2005 John Wiley & Sons, Ltd.     

一类模糊P I D 控制器的鲁棒优化设计

研究一类模糊 PID控制器的鲁棒设计。以小增益定理分析得到该模糊 PID控制系统稳定性条件。针对参数摄动系统的“最坏点”,用该稳定性条件作为约束 ,采用遗传算法对标称系统的性能进行优化 ,求得优化鲁棒控制器。以倒立摆为例进行鲁棒模糊 PID控制器的设计 ,实验结果表明了该方法的有效性     

Synthesis of globally optimal controllers for robust performance tounstructured uncertainty

This paper solves the problem of synthesis of controllers achieving globally optimal robust performance against unstructured time-varying and/or nonlinear uncertainty. The performance measure considered is the infinity-to-infinity induced norm of a system's transfer function. The solution utilizes sensitivity analysis of linear programming and the theory of parametric programming     

A noniterative method for the design of linear robust controllers

A procedure for determining a linear control law which guarantees asymptotic stability for an uncertain system is derived. Using an elementary matrix identity, an alternative proof of the existence of stabilizing control law for the matched case is provided. Bounds for the rates of decay are established, and by using these bounds it is shown that every trajectory of the closed-loop system can be made to decay at a prespecified exponential rate. The technique is simpler to use than existing techniques since it does not require a numerical procedure, but allows the control law to be obtained from some simple formulas     

Independent design of robust partially decentralized controllers

An independent design method for robust partially decentralized controllers is developed. In the proposed design method, the partially decentralized control system is first expanded to the nonsquare decentralized structure. Using the Internal Model Control parametrization, the independent design procedure for robust nonsquare decentralized controllers can then be applied directly to the expanded system. Two examples, including a nonlinear stirred mixing tank, are used to illustrate the developed design method and a comparison to a decentralized controller is made.     

Representations of controllers that achieve robust performance for systems with real parameter uncertainty

Consider a family of single input single output plants described by transfer functions that involve real parameter uncertainty. Parameter values are known to lie in a hypercube. Assume that a class of available controllers has been prescribed, along with a bound for the sensitivity transfer function to ensure tracking. It is of interest to determine whether a controller from the given class exists that guarantees robust stability and robust asymptotic tracking. In this paper we present a problem formulation and then provide a solution based on it. Not only do we address the existence question but also give representation of controllers from the class that meet the robustness requirements.     

Relations between uncertainty structures in identification for robust control

Various techniques of system identification exist that provide a nominal model and an uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. It is shown that an amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and ν-gap uncertainty. As a result, the choice of structure does not matter provided that the identification methods deliver optimal uncertainty sets rather than an uncertainty bound around a prefixed nominal model. Frequency-dependent closed-loop performance functions based on the uncertainty sets are again bounded by circles in the frequency domain, allowing for analytical expressions for worst-case performance and for the evaluation of the consequences of uncertainty for robust design. The results can be used to tune optimal experimental conditions in view of robust control design and in the further development of experiment-based robust control design methods.     

The design of robust feedback controllers in the time domain

In this new approach to designing multivariable feedback controllers in the time domain the model uncertainties are dealt with in the sense of unknown-but-bounded uncertainties. They are described by multi-input multi-output comparison systems. Thus, the character of the plant as a multi variable dynamical system is preserved even in the considerations of the uncertainties. The controller parameters are selected by means of well-known design algorithms. The robustness of the controller is investigated on the basis of a stability condition and an estimate of the input-output behaviour of the incompletely known closed-loop original system. The results are illustrated by a practical design example.     

Combined design of structures and controllers for optimal maneuverability

This paper treats the problem of the combined design of structure/control systems for achieving optimal maneuverability. A maneuverability index which directly reflects the time required to perform a given maneuver or set of maneuvers is introduced. By designing the flexible appendages of a spacecraft, its maneuverability is optimized under the constraints of structural properties, and of the postmaneuver spill-over being within a specified bound. The spillover reduction is achieved by making use of an appropriate control design model. The distributed parameter design problem is approached using assumed shape functions and finite element analysis with dynamic reduction. Characteristics of the problem and problem solving procedures have been investigated. Adaptive approximate design methods have been developed to overcome computational difficulties. It is shown that the global optimal design may be obtained by tuning the natural frequencies of the spacecraft to satisfy specific constraints. We quantify the difference between a lower bound to the objective function associated with the original problem and the estimate obtained from the modified problem as the index for the adaptive refinement procedure. Numerical examples show that the results of the optimal design can provide substantial improvement.     

On robust design optimization of truss structures with bounded uncertainties

This paper investigates robust design optimization of truss structures with uncertain-but-bounded parameters and loads. The variations of the cross-sectional areas, Young’s moduli and applied loads are treated with non-probabilistic ellipsoid convex models. A robustness index for quantitatively measuring the maximal allowable magnitude of system variations is presented, and the design problem is then formulated as to maximize the minimum of the robustness indices for all the concerned design requirements under a given material volume constraint. For circumventing the difficulty associated with the max-min type problem, an aggregate function technique is employed to construct a smooth objective function. The computational scheme for the sensitivity of the robustness index is derived on the basis of optimum sensitivity analysis. The optimization problem is then solved by using the GCMMA optimizer. Numerical examples illustrate the validity and effectiveness of the present formulation and solution techniques.     

On analytical design of controllers

A method to design controllers of linear plants under bounded sectionally continuous external disturbances was proposed.     

Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design

Robust design is an effective approach to design under uncertainty. Many works exist on mitigating the influence of parametric uncertainty associated with design or noise variables. However, simulation models are often computationally expensive and need to be replaced by metamodels created using limited samples. This introduces the so-called metamodeling uncertainty. Previous metamodel-based robust designs often treat a metamodel as the real model and ignore the influence of metamodeling uncertainty. In this study, we introduce a new uncertainty quantification method to evaluate the compound effect of both parametric uncertainty and metamodeling uncertainty. Then the new uncertainty quantification method is used for robust design. Simplified expressions of the response mean and variance is derived for a Kriging metamodel. Furthermore, the concept of robust design is extended for metamodel-based robust design accounting for both sources of uncertainty. To validate the benefits of our method, two mathematical examples without constraints are first illustrated. Results show that a robust design solution can be misleading without considering the metamodeling uncertainty. The proposed uncertainty quantification method for robust design is shown to be effective in mitigating the effect of metamodeling uncertainty, and the obtained solution is found to be more “robust” compared to the conventional approach. An automotive crashworthiness example, a highly expensive and non-linear problem, is used to illustrate the benefits of considering both sources of uncertainty in robust design with constraints. Results indicate that the proposed method can reduce the risk of constraint violation due to metamodel uncertainty and results in a “safer” robust solution.     

Iterative design of robust controllers under unknown bounded perturbation norms

A suboptimal robust controller for a multidimensional linear stationary nominal system with output- and control-operator perturbations and a bounded external perturbation is designed. Perturbation norms (weights) are assumed to be unknown and estimated from measurement data. The quality index of the closed-loop control system is taken to be the worst -norm for the output of the system in the class of admissible perturbations. The quality index for systems in the class of linear stationary stabilizing controllers is a fractional linear function of induced norms of transfer matrices of the closed-loop system and perturbation norms. The classical design of suboptimal robust controllers under known perturbation norms is reduced to a standard ₁-optimization problem, and optimal estimation of unknown perturbation norms is reduced to a linear programming problem. Therefore, the iterative sequential method of controller design and perturbation norm estimation is used as a heuristic method for designing suboptimal robust controllers under unknown perturbation norms. Modeling results corroborate the effectiveness of this method.Translated from Avtomatika i Telemekhanika, No. 4, 2005, pp. 110–126.Original Russian Text Copyright © 2005 by Sokolov.This work was supported by the Russian Foundation for Basic Research, project no. 02-01-00474.This paper was recommended for publication by A.P. Kurdyukov, a member of the Editorial Board     

The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods

The problem of designing realistic multivariable controllers to solve the servomechanism problem is considered in this paper. Specifically, it is desired to find a controller for a plant to solve the robust servomechanism problem, so that closed-loop stability and asymptotic regulation occur, and also so that other desirable properties of the controlled system, such as fast response, low-interaction, integrity, tolerance to plant variations, etc., occur. The method of design is based on using state space methods via a two-stage process: 1) using theory, determine the existence of a solution and control structure required to solve the problem, and 2) using nonlinear programming methods, determine the unknown controller parameters so as to minimize a performance index for the system subject to certain constraint requirements. Numerous examples, varying from a single-input/single-output to a four-input/four-output system, are given to illustrate the design method, and the results obtained are compared with the results obtained by using other alternate design methods. In all cases, the controllers obtained have been highly competitive with controllers obtained by alternate design methods.     

On the design of stabilizing controllers for singularly perturbedsystems

The robustness of output feedback control designs for singularly perturbed systems based on the reduced systems is discussed. A frequency-domain approach is presented to determine the conditions for the stability of linear time-invariant systems subject to neglected high-frequency dynamics. In contrast with the qualitative analyses for robust stability that have appeared in the literature, this approach gives an explicit, computable bound on unmodeled dynamics which does not destabilize the systems. Stability conditions for various feedback control schemes are presented. It is noted that the approach, extended to systems with mixed singular-regular perturbations, can be used to derive the robust stability condition     

Generalized linear transformations on the design of robust dynamic output feedback controllers

The problem of designing stable, robust closed-loop systems is considered in this paper. A new bound of robustness measure is introduced. Free design parameters are exploited in the achievement of desirable performance. Designs of minimal and reduced-order dynamic output controllers are carried out using special algorithms and an illustrative example of the design method is included.     

Robust design of pole placement controllers in the presence of structured uncertainty

This paper addresses the robust design of pole placement controllers in the presence of structured parametric uncertainty in a plant. A design algorithm is proposed to minimize pole placement error so as to achieve the desired closed-loop performances better. It is shown that the proposed design algorithm not only has a fast rate convergence in the search of maximum tolerance with regard to pole placement error but also is easy to understand and convenient to implement. A numerical example is given to illustrate the application of the design algorithm.     

Universal controllers for robust control problems

In this paper we discuss the construction of “universal” controllers for a class of robust stabilization problems. We give a general theorem on the construction of these controllers, which requires that a certain nonlinear inequality is solvablepointwisely or, equivalently, that arobust control Lyapunov function does exist. The constructive procedure producesalmost smooth controllers. The robust control Lyapunov functions extend to uncertain systems the concept of control Lyapunov functions. If such a robust control Lyapunov function also satisfies a small control property, the resulting stabilizing controller is also continuous in the origin of the state space. Applications of our results range from optimal to robust control.     

Nonlinear time-varying controllers for robust stabilization

In this paper we consider the problem of robust stabilizallon for a certain class of plants with unstructured, infinite-dimensional uncertainty. We demonstrate that for the problem of robustly stabilizing this class of plants, linear time-invariant controllers perform as well as nonlinear time-varying (NLTV) ones. Ihis, in particular, implies that adaptive control laws offer no advantage as far as the problem of robust stabilization of this class of plants is concerned. As a corollary we demonstrate that the small-gain theorem is both necessary and sufficient for a certain class of NLTV operators.