Robust controller design by linear programming with application to a double-axis positioning system

A linear programming approach is proposed to tune fixed-order linearly parameterized controllers for stable LTI plants. The method is based on the shaping of the open-loop transfer function in the Nyquist diagram. A lower bound on the crossover frequency and a new linear stability margin which guarantees lower bounds for the classical robustness margins are defined. Two optimization problems are proposed and solved by linear programming. In the first one the robustness margin is maximized for a given lower bound on the crossover frequency, whereas in the second one the closed-loop performance in terms of the load disturbance rejection is optimized with constraints on the new stability margin. The method can directly consider multi-model as well as frequency-domain uncertainties. An application to a high-precision double-axis positioning system illustrates the effectiveness of the proposed approach.     

Finite Horizon Minimax Optimal Control of Nonlinear Continuous Time Systems with Stochastic Uncertainty

This paper develops a general continuous-time stochastic framework for robustness analysis and robust control synthesis. We consider a stochastic minimax optimization problem for general stochastic uncertain systems. A general method is presented for converting problems of performance analysis or controller synthesis into unconstrained optimization problems.     

An LMI approach to dencentralized H8 control

We consider the design of a decentralized controller for a linear time invariant (LTI) system. This system is modelled as an interconnection of subsystems. For every subsystem, a linear time invariant controller is sought such that the overall closed loop system is stable and achieves a given H performance level. The main idea is to design every local controller such that the corresponding closed loop subsystem has a certain input-output (dissipative) property. This local property is constrained to be consistent with the overall objective of stability and performance. The local controllers are designed simultaneously, avoiding the traditional iterative process: both objectives (the local one and the global one) are achieved in one shot. Applying this idea leads us to solving the following new problem: given an LTI system, parameterize all the dissipative properties which can be achieved by feedback. The proposed approach leads to solving convex optimization problems that involve linear matrix inequality constraints.     

Normalized H∞ controller reduction with a priorierror bounds

This paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized H_∞ problem. It is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented. When the approximation is performed on the controller generator, guaranteed a priori stability and performance bounds are obtained in terms of the synthesis Riccati equation solutions of the normalized H_∞ control problems     

Robust control of linear time-invariant plants using periodic compensation

This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.     

Evolving an interval type-2 fuzzy PID controller for the redundant robotic manipulator

Robotic manipulators are a multi-input multi-output, dynamically coupled, highly time-varying, complex and highly nonlinear systems wherein the external disturbances, parameter variations, and random noise adversely affects the performance of the robotic system. Therefore, in order to deal with such complexities, however, an intriguing task for control researchers, these systems require an efficient and robust controller. In this paper, a novel application of genetic algorithms (GA) optimization approach to optimize the scaling factors of interval type-2 fuzzy proportional derivative plus integral (IT2FPD+I) controllers is proposed for 5-DOF redundant robot manipulator for trajectory tracking task. All five controllers' parameters are optimized simultaneously. Further, a procedure for selecting appropriate initial search space is also demonstrated. In order to make a fair comparison between different controllers, the tuning of each of the controllers' parameters is done with GA. This optimization technique uses the time domain optimal tuning while minimizing the fitness function as the sum of integral of multiplication of time with square error (ITSE) for each joint. To ascertain the effectiveness of IT2FPID controller, it is compared against type-1 fuzzy PID (T1FPID) and conventional PID controllers. Furthermore, robustness testing of developed IT2FPID controller for external disturbances, parameter variations, and random noise rejection is also investigated. Finally, the experimental study leads us to claim that our proposed controller can not only assure best trajectory tracking in joint and Cartesian space, but also improves the robustness of the systems for external disturbances, parameter variations, and random noise.     

On the stabilization and stability robustness against small delaysof some damped wave equations

In this paper we consider a system which can be modeled by two different one-dimensional damped wave equations in a bounded domain, both parameterized by a nonnegative damping constant. We assume that the system is axed at one end and is controlled by a boundary controller at the other end. We consider two problems, namely the stabilization and the stability robustness of the closed-loop system against arbitrary small time delays in the feedback loop. We propose a class of dynamic boundary controllers and show that these controllers solve the stabilization problem when the damping coefficient is nonnegative and the stability robustness problem when the damping coefficient is strictly positive     

Nonsmooth multi-objective synthesis with applications

Nonsmooth optimization is used to design feedback controllers subject to closed-loop performance specifications both in time and frequency-domains. In time domain the nonlinear plant is submitted to a set of test input signals and the closed-loop responses so generated are called scenarios. A design technique is proposed which computes a controller with a prescribed structure that satisfies performance specifications for a given set of scenarios in tandem with robustness constraints in the frequency-domain, and is locally optimal among other controllers with these properties.     

具有预期闭环极点区域的稳定控制器的H∞优化设计方法

本文从工程应用角度出发,提出了一种将闭环极点设置于预期稳定区域的稳定控制器的 H∞优化设计方法,即实参数优化求解控制器的方法.这种方法综合考虑了闭环动态性能、抗 干扰、鲁棒性以及控制器本身的稳定性.实例表明,与现有的H∞设计方法相比,用本文方法 设计的控制系统在不显著增大抗干扰和鲁棒性指标情况下,具有闭环动态性能良好及控制器 稳定等优点,因此具有工程应用价值.     

Variable structure model-reference adaptive control (VS-MRAC) usingonly input and output measurements: the general case

A globally asymptotically stable VS-MRAC for the general case of plants with arbitrary relative degree is presented. It is shown that the VS approach leads to general adaptive controllers with outstanding performance. The controllers use only I/O measurements and are free of differentiators. The results indicate that several of the usual drawbacks of conventional MRACs (e.g. unsatisfactory transient behavior, lack of robustness, difficulties in designing controllers for time-varying or nonlinear plants) can be circumvented     

Uniting Local and Global Controllers with Robustness to Vanishing Noise

We consider control systems for which we know two stabilizing controllers. One is globally asymptotically stabilizing, the other one is only locally asymptotically stabilizing but for some reason we insist on using it in a neighborhood of the origin. We look for a uniting control law being equal to the local feedback on a neighborhood of the origin, equal to the global one outside of a larger neighborhood and being a globally stabilizing controller. We study several solutions based on continuous, discontinuous, hybrid, time-varying controllers. One criterion of the selection of a controller is the robustness of the stability to vanishing noise. This leads us in particular to consider a kind of generalization of Krasovskii trajectories for hybrid systems. Date received: November 29, 1999. Date revised: August 7, 2000.     

An evolutionary approach to design practical μ synthesis controllers

This paper proposes an evolutionary approach to solve μ synthesis problem. The goal is to achieve low order, practical μ synthesis controllers without any order reduction. In the proposed approach µ synthesis problem is solved as a constraint optimization problem in which robust stability and robust performance based on μ analysis are considered as the constraint and the cost function respectively. In order to solve the optimization problem an improved particle swarm optimization (PSO) is chosen to find the required coefficients of a structure-specified controller. The performance and robustness of the proposed controller are investigated by an uncertain mass-damper-spring system and is compared with the D-K iteration controller (the conventional solution to μ synthesis problem). Simulation results demonstrate the advantages of the proposed controller in terms of simple structure and robustness against plant perturbations and disturbances in comparison with D-K iteration controller.     

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.     

Model-reference robust tuning of 2DoF PI controllers for first- and second-order plus dead-time controlled processes

The aim of this paper is to present a robust tuning method for two-degree-of-freedom (2DoF) proportional integral (PI) controllers. This is based on the use of a model reference optimization procedure with servo and regulatory target closed-loop transfer functions for first- and second-order plus dead-time (FOPDT, SOPDT) controlled process models. The designer is allowed to deal with the performance/robustness trade-off of the closed-loop control system by specifying the desired robustness level by selecting a maximum sensitivity in the range from 1.4 to 2.0. In addition, a smooth servo/regulatory performance combination is obtained by forcing both closed-loop transfer functions to perform as closely as possible to non-oscillatory dynamic targets. A unified set of controller tuning equations is provided for FOPDT and SOPDT models with normalized dead-times from 0.1 to 2.0 that guarantees the achievement of the design robustness level. The robustness of the control system is analyzed as well as the robustness–fragility and performance–fragility of the optimized controllers. Comparative examples show the effectiveness of the proposed tuning method. The exact achievement of the control system robustness target for all the controlled process models considered (first- and second-order) is one of the distinctive characteristics of the proposed model reference robust tuning (MoReRT) method.     

A comparative experimental study of type-1/type-2 fuzzy cascade controller based on genetic algorithms and particle swarm optimization

In this study, we introduce the design methodology of an optimized fuzzy controller with the aid of particle swarm optimization (PSO) for ball and beam system.The ball and beam system is a well-known control engineering experimental setup which consists of servo motor, beam and ball. This system exhibits a number of interesting and challenging properties when being considered from the control perspective. The ball and beam system determines the position of ball through the control of a servo motor. The displacement change of the position of ball leads to the change of the angle of the beam which determines the position angle of a servo motor.The fixed membership function design of type-1 based fuzzy logic controller (FLC) leads to the difficulty of rule-based control design when representing linguistic nature of knowledge. In type-2 FLC as the expanded type of type-1 FL, we can effectively improve the control characteristic by using the footprint of uncertainty (FOU) of the membership functions. Type-2 FLC exhibits some robustness when compared with type-1 FLC.Through computer simulation as well as real-world experiment, we apply optimized type-2 fuzzy cascade controllers based on PSO to ball and beam system. To evaluate performance of each controller, we consider controller characteristic parameters such as maximum overshoot, delay time, rise time, settling time, and a steady-state error. In the sequel, the optimized fuzzy cascade controller is realized and also experimented with through running two detailed comparative studies including type-1/type-2 fuzzy controller and genetic algorithms/particle swarm optimization.     

Best achievable decentralized performance

In this paper, a novel parameterization of all decentralized stabilizing controllers is employed in mathematically formulating the best achievable decentralized performance problem as an infinite dimensional optimization problem, Finite dimensional optimization problems are then constructed that have values arbitrarily close to this infinite dimensional problem. An algorithm which identifies the best achievable performance over all linear time-invariant decentralized controllers is then presented. It employs a global optimization approach to the solution of these finite dimensional approximating problems     

Hybrid multi-objective shape design optimization using Taguchi’s method and genetic algorithm

This research is based on a new hybrid approach, which deals with the improvement of shape optimization process. The objective is to contribute to the development of more efficient shape optimization approaches in an integrated optimal topology and shape optimization area with the help of genetic algorithms and robustness issues. An improved genetic algorithm is introduced to solve multi-objective shape design optimization problems. The specific issue of this research is to overcome the limitations caused by larger population of solutions in the pure multi-objective genetic algorithm. The combination of genetic algorithm with robust parameter design through a smaller population of individuals results in a solution that leads to better parameter values for design optimization problems. The effectiveness of the proposed hybrid approach is illustrated and evaluated with test problems taken from literature. It is also shown that the proposed approach can be used as first stage in other multi-objective genetic algorithms to enhance the performance of genetic algorithms. Finally, the shape optimization of a vehicle component is presented to illustrate how the present approach can be applied for solving multi-objective shape design optimization problems.     

Simultaneous closed-loop tuning of cascade controllers based directly on set-point step-response data

This study presents a novel closed-loop tuning method for cascade control systems, in which both primary and secondary controllers are tuned simultaneously by directly using set-point step-response data without resorting to process models. The tuning method can be applied on-line to improve the performance of existing underperforming cascade controllers by retuning controller parameters, using routine operating data. The goal of the proposed design is to obtain the parameters of two proportional-integral-derivative (PID)-type controllers, so that the resulting inner and outer loops behave as similarly as possible to the appropriately specified reference models. The tuning rule and optimization problem related to the proposed design are derived. Based on the rationale behind cascade control, the secondary controller is designed based on disturbance rejection to quickly attenuate disturbances. The primary controller is designed to accurately account for the inner-loop dynamics, without requiring an additional test. In addition, robustness considerations are included in the proposed tuning method, which enable the designer to explicitly address the trade-off between performance and robustness for inner and outer loops independently. Simulation examples show that the proposed method exhibits superior control performance compared with the previous (model-based) tuning methods, confirming the effectiveness of this novel tuning method for cascade control systems.     

Influence of meta-heuristic optimization on the performance of adaptive interval type2-fuzzy traffic signal controllers

Intelligent traffic control systems optimized using meta-heuristic algorithms can greatly alleviate traffic congestions in urban areas. Meta-heuristics are broadly used as efficient approaches for complex optimization problems. Comparing the performance of optimization methods on different applications is a way to evaluate their effectiveness. The current literature lacks studies on how performance of traffic signal controllers is affected by utilized optimization algorithms. This paper evaluates the performance of three meta-heuristic optimization methods on an advanced interval type-2 adaptive neuro-fuzzy inference system (IT2ANFIS)-based controller for complex road networks. Simulated annealing (SA), genetic algorithm (GA), and the cuckoo search (CS) are applied for optimal tuning of IT2ANFIS controller. Optimizations methods adjust the parameters in a way to reduce the total travel time of vehicles in the road network. Paramics is used to design and simulate urban traffic network models and implement proposed timing controllers. Comprehensive simulation and performance evaluation are done for both single and multi-intersection traffic networks. Obtained results reveal significant superiority of IT2ANFIS trained using CS method over other controllers. The average performance of the CS-IT2ANFIS is about 31% better than the benchmark fixed-time controllers. This is 17% and only 3% for GA-IT2ANFIS and SA-IT2ANFIS controllers respectively.     

A review on the design and optimization of interval type-2 fuzzy controllers

A review of the methods used in the design of interval type-2 fuzzy controllers has been considered in this work. The fundamental focus of the work is based on the basic reasons for optimizing type-2 fuzzy controllers for different areas of application. Recently, bio-inspired methods have emerged as powerful optimization algorithms for solving complex problems. In the case of designing type-2 fuzzy controllers for particular applications, the use of bio-inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems. In this review, we consider the application of genetic algorithms, particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type-2 fuzzy controllers. We also mention alternative approaches to designing type-2 fuzzy controllers without optimization techniques. We also provide a comparison of the different optimization methods for the case of designing type-2 fuzzy controllers.