Synthesis of H∞ PID controllers: A parametric approach

This paper considers the problem of synthesizing proportional-integral-derivative (PID) controllers for which the closed-loop system is internally stable and the H_∞-norm of a related transfer function is less than a prescribed level for a given single-input single-output plant. It is shown that the problem to be solved can be translated into simultaneous stabilization of the closed-loop characteristic polynomial and a family of complex polynomials. It calls for a generalization of the Hermite-Biehler theorem applicable to complex polynomials. It is shown that the earlier PID stabilization results are a special case of the results developed here. Then a linear programming characterization of all admissible H_∞ PID controllers for a given plant is obtained. This characterization besides being computationally efficient reveals important structural properties of H_∞ PID controllers. For example, it is shown that for a fixed proportional gain, the set of admissible integral and derivative gains lie in a union of convex sets.     

Design of digital PID controllers using the parameter space approach

In this paper, a parameter space approach is taken for designing digital PID controllers. The stability domains of the coefficients of the controllers are computed. The existing continuous-time results are extended to the case of discrete-time systems. In this approach, the stability region is obtained in the plane of two auxiliary controller coefficients by assuming a fixed value for a third auxiliary controller coefficient. The stability region is defined by several line segments or equivalently by several linear equalities and inequalities. Then, through mapping from the auxiliary coefficient space to the original controller coefficient space, exact stability domain in the (K_P ???K_I ???K_D ) space is obtained. The method is also extended for locating the closed-loop poles of PID control systems inside the circles with arbitrary radii, centred at the origin of the z-plane. The results can be used in the design of dead-beat control systems.     

A new approach to robust and non-fragile H∞ control for uncertain fuzzy systems

This paper investigates the robust H_∞ control and non-fragile control problems for Takagi-Sugeno (T-S) fuzzy systems with linear fractional parametric uncertainties. The robust H_∞ control problem is to design a state feedback controller such that the robust stability and a prescribed H_∞ performance of the resulting closed-loop system is ensured. And the non-fragile H_∞ control problem is to design a state feedback controller with parameter uncertainties. Based on the linear matrix inequality (LMI) approach, new sufficient conditions for the solvability of the two problems are obtained. It is shown that the desired state feedback fuzzy controller can be constructed by solving a set of LMIs. Numerical examples are also provided to demonstrate the effectiveness of the proposed design method.     

Robust PID controller tuning based on the constrained particle swarm optimization

This paper proposes a novel tuning strategy for robust proportional-integral-derivative (PID) controllers based on the augmented Lagrangian particle swarm optimization (ALPSO). First, the problem of PID controller tuning satisfying multiple H_∞ performance criteria is considered, which is known to suffer from computational intractability and conservatism when any existing method is adopted. In order to give some remedy to such a design problem without using any complicated manipulations, the ALPSO based robust gain tuning scheme for PID controllers is introduced. It does not need any conservative assumption unlike the conventional methods, and often enables us to find the desired PID gains just by solving the constrained optimization problem in a straightforward way. However, it is difficult to guarantee its effectiveness in a theoretical way, because PSO is essentially a stochastic approach. Therefore, it is evaluated by several simulation examples, which demonstrate that the proposed approach works well to obtain PID controller parameters satisfying the multiple H_∞ performance criteria.     

H<Subscript>2</Subscript>-optimal tuning algorithms for fixed structure controllers

An H₂-method of optimal tuning is proposed for a fixed order controller. The SISO plant model is considered in state space. The H₂-method of tuning parameter design is based on the minimization of a transient process closeness criterion for appropriate open-loop and closed-loop control systems and their reference models. The controller tuning algorithms use the plant parameter estimations obtained during the plant parameter identification. The analytical expressions are obtained for the square of H₂-norm of a stable dynamic system. The following theorem is proven: the minimum necessary conditions for the functionals of transfer function H₂-norm of open-loop and closed-loop systems are the same as the minimum necessary conditions for the Frobenius norm of the controller parameter tuning polynomial.     

Multiobjective robust control of LTI systems subject to unstructured perturbations

We consider a multiobjective robust controller synthesis problem for an LTI system subject to unstructured perturbations. Our design specifications include robust stability, robust performance (H₂-norm) bounds and time-domain bounds (output and command input peak). We derive sufficient conditions, based on a single quadratic Lyapunov function, for the existence of an LTI controller such that the closed-loop system satisfies all specifications simultaneously. These conditions can be numerically checked using finite-dimensional, convex optimization over LMIs, associated with a two-dimensional search. When considering only a subset of the specifications, we recover previous results from the literature, such as those obtained in mixed H₂/H_∞ control.     

<Emphasis Type="BoldItalic">H</Emphasis><Subscript>2</Subscript>-Control and the Separation Principle for Discrete-Time Markovian Jump Linear Systems

In this paper we consider the H₂-control problem of discrete-time Markovian jump linear systems. We assume that only an output and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed-loop system is mean square stable and minimizes the H₂-norm of the system. As in the case with no jumps, we show that an optimal controller can be obtained from two sets of coupled algebraic Riccati equations, one associated with the optimal control problem when the state variable is available, and the other associated with the optimal filtering problem. This is the principle of separation for discrete-time Markovian jump linear systems. When there is only one mode of operation our results coincide with the traditional separation principle for the H₂-control of discrete-time linear systems. Date received: June 1, 2001. Date revised: October 13, 2003.     

Robust control of distillation columns: μ- vs H∞-synthesis

High-purity distillation columns are known to be difficult to control due to their ill-conditioned and non-linear behaviour. Two approaches for the design of robust controllers yielding high performance are presented. For the first approach, first principles are used to develop an uncertainty model describing the nonlinear dynamics within the entire operating range of an industrial distillation column. This structured uncertainty model is used for μ-synthesis. In a second approach which is based on loop shaping ideas, an H_∞-controller is designed. This controller performs as well as the μ-controller. The H_∞-approach offers the advantage that the burden for uncertainty modelling and computation is greatly reduced. However, the GS/T augmentation scheme, which is developed in this paper, must be used for the design of the H_∞-controller to avoid the inversion of the plant in the controller. The paper concludes with a comparison of the H_∞- and μ-synthesis methods.     

Decentralised H ∞ control for singular systems

In this article, considering the design problem of decentralised H _∞ controller of singular systems, the two cases of controllers via measurement feedback are designed: one is precise controller, and the other is additive controller gain variation. The design procedures of the two cases of controllers are presented in terms of the solutions to generalised algebraic Riccati inequalities. The designed controllers in each case guarantee that closed-loop singular systems are admissible and with H _∞-norm bound on disturbance attenuation. Finally, a numerical example to demonstrate the validity of the proposed approach is given.     

Robust Controller Design And Pid Tuning For Multivariable Processes

In this paper, a robust controller design method is first formulated to deal with both performance and robust stability specifications for multivariable processes. The optimum problem is then dealt with using a loop‐shaping H_∞ approach, which gives a sub‐optimal solution. Then a PID approximation method is proposed to reduce a high‐order controller. The whole procedure involves selecting several parameters and the computation is simple, so it serves as a PID tuning method for multivariable processes. Examples show that the method is easy to use and the resulting PID settings have good time‐domain performance and robustness.     

Stability and performance recovery within discretized non-linear control systems

In this paper, a closed-loop fast sampling analysis of a discretized (emulated) continuous-time controller in a sampled-data environment is presented. The analysis involves a general weighted version of the L_p-norm. This allows a qualitative and quantitative stability and performance analysis for an emulated controller. Several examples are used to show the relevance of these results for the analysis of sampled-data implementations and the computation of quantitative upper limits on the sampling period to achieve recovery of stability and performance.     

Robust H∞ control for uncertain discrete singular systems with pole placement in a disk

This paper addresses the design of robust H_∞ controllers for uncertain discrete singular systems with time-invariant uncertainty in both the state and measurement matrices. The singular system to be controlled is not assumed to be regular. A regular dynamic output feedback controller is designed such that a prescribed H_∞ performance condition is satisfied and the closed-loop poles are placed in a specified disk while the regularity, causality and stability of the closed-loop system can be guaranteed for all admissible uncertainties. The desired controller can be obtained by solving a set of matrix inequalities. A numerical example is given to demonstrate the application of the proposed method.     

An LMI approach to vibration control of base-isolated building structures with delayed measurements

In this article, we address a convex optimisation approach to the problem of state-feedback H _∞ control design for vibration reduction of base-isolated building structures with delayed measurements, where the delays are time-varying and bounded. An appropriate Lyapunov–Krasovskii functional and some free-weighting matrices are utilised to establish some delay-range-dependent sufficient conditions for the design of desired controllers in terms of linear matrix inequalities. The controller, which guarantees asymptotic stability and an H _∞ performance, simultaneously, for the closed-loop system of the structure, is then developed. The performance of the controller is evaluated by means of simulations in MATLAB/Simulink.     

output feedback control for uncertain stochastic systems with time-varying delays

This paper deals with the problem of H_∞ output feedback control for uncertain stochastic systems with time-varying delays. The parameter uncertainties are assumed to be time-varying norm-bounded. The aim is the design of a full-order dynamic output feedback controller ensuring robust exponential mean-square stability and a prescribed H_∞ performance level for the resulting closed-loop system, irrespective of the uncertainties. A sufficient condition for the existence of such an output feedback controller is obtained and the expression of desired controllers is given.     

H ∞ PID controller design for Lur’e systems and its application to a ball and wheel apparatus

In this paper, we consider the problem of synthesizing proportional-integral-derivative (PID) controllers for a given Lur’e system in the presence of exogenous energy-bounded disturbance. Based on the circle criterion, sufficient conditions for guaranteeing closed-loop absolute stability and achieving disturbance attenuation are given in terms of the multi-objective H _∞ specifications. The results from the earlier work are then used to solve the resulting multi-objective H _∞ PID design problem. It is shown that for a fixed proportional gain, and by sweeping over a variable, the set of admissible integral and derivative gain values can be determined constructively using linear-programming techniques. The proposed synthesis method is used to design a PID controller for the ball and wheel system and the experimental results are presented to show effectiveness of the proposed method.     

H ∞ control for 2-D singular delayed systems

This article considers the problem of H _∞ control for two-dimensional (2-D) singular delayed systems in Roesser models. The problem to be addressed is the design of a state feedback controller such that the acceptability, internal stability and causality of the resulting closed-loop system is guaranteed and a prescribed H _∞ performance level is ensured. In terms of a linear matrix inequality (LMI), a sufficient condition for the solvability of the problem is obtained. A desired state feedback controller can be designed by solving a certain LMI. A numerical example is provided to demonstrate the application of the proposed method.     

不确定多智能体互联动态系统分布式鲁棒H∞协同控制

研究一类具有时延和切换拓扑的不确定多智能体互联系统的分布式协同控制问题,提出一类分布式鲁棒H∞协同控制器.该控制器能够在满足期望的H∞性能指标的前提下,使得所有智能体鲁棒地跟踪虚拟Leader.针对所提出的分布式H∞协同控制器,借助Lyapunov-Krasovskii泛函,利用线性矩阵不等式,推导出一系列充分(必要)条件,并且在通信拓扑连通的前提条件下,给出整个闭环系统的稳定性证明.仿真实例表明了所提出方法的有效性.     

Robust optimal pole-clustering in a vertical strip and disturbance rejection for uncertain Lagrange's systems

This paper presents an approach to design a state-feedback robust control law for uncertain Lagrange's systems such that the designed closed-loop systems have the properties of robust pole-clustering within a vertical strip and disturbance rejection with anH -norm constraint. This approach is based on solving an algebraic Riccati equation with the adjustable scalars and prespecified parameters. The uncertainties considered include both unstructured and structured uncertainties in the system and the input matrices. Also, a constraint is established to verify that the proposed robust LQRs preserveH ₂ optimality with respect to a specific quadratic cost function.     

A mixed H2/H∞ adaptive tracking control for constrained non-holonomic systems

In this study, a PID type controller incorporating an adaptive control scheme for the mixed H₂/H_∞ tracking performance is developed for constrained non-holonomic mechanical systems under unknown or uncertain plant parameters and external disturbances. By virtue of the skew-symmetric property of the non-holonomic mechanical systems and an adequate choice of a state variable transformation, sufficient conditions are developed for the adaptive mixed H₂/H_∞ tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled non-linear differential equations. The coupled algebraic equations can be solved analytically.     

Distributed H_∞ PID Feedback for Improving Consensus Performance of Arbitrary-delayed Multi-agent System

The H∞ proportional-integral-differential(PID) feedback for arbitrary-order delayed multi-agent system is investigated to improve the system performance. The closed-loop multi-input multi-output(MIMO) control framework with the distributed PID controller is firstly described for the multi-agent system in a unified way. Then, by using the matrix theory, the prescribed H∞performance criterion of the multi-agent system is shown to be equivalent to several independent H∞ performance constraints of the single-input single-output(SISO) subsystem with respect to the eigenvalues of the Laplacian matrix. Subsequently, for each subsystem,the set of the PID controllers satisfying the required H∞ performance constraint is analytically characterized based on the extended Hermite-Biehler theorem. Finally, the three-dimensional set of the decentralized H∞ PID control parameters is derived by finding the intersection of the H∞ PID regions for all the decomposed subsystems. The simulation results reveal the effectiveness of the proposed method.