Robust nonlinear feedback control for uncertain linear systems with nonquadratic performance criteria

In this paper we develop a unified framework to address the problem of optimal nonlinear robust control for linear uncertain systems. Specifically, we transform a given robust control problem into an optimal control problem by properly modifying the cost functional to account for the system uncertainty. As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. The overall framework generalizes the classical Hamilton–Jacobi–Bellman conditions to address the design of robust nonlinear optimal controllers for uncertain linear systems. © 1998 Elsevier Science B.V.     

Optimal controllers for spatially invariant systems with delayed communication requirements

We consider the problem of optimal design of semi-decentralized controllers for a special class of spatially distributed systems. This class includes spatially invariant and distributed discrete-time systems with an inherent temporal delay in the interaction of neighboring sites. We consider the problem of optimal design of distributed controllers that have the same information passing delay structure as the plant. We show how for stable plants, the YJBK parameterization of such stabilizing controllers yields a convex parameterization for this class. We then show how the optimal problem can be solved.     

Design of the l1-suboptimal Robust Controller for the Linear Object with Nonstructured Uncertainty

Design of the l ₁-suboptimal robust controller for the linear discrete scalar object with constrained external perturbation and nonstructured operator perturbation in input and control was considered. The worst value of the l -norm of object output in the class of admissible perturbations was regarded as the performance index of the control system. The problem of designing the optimal robust controller was interpreted in the geometric terms, and an algorithm to design the suboptimal controller was proposed. A numerical example was used to demonstrate effciency of the algorithm, the ways to improving it, and the differences between the suboptimal robust controllers with structured and nonstructured operator perturbations.     

Hybrid state-space fuzzy model-based controller with dual-ratesampling for digital control of chaotic systems

We develop a hybrid state-space fuzzy model-based controller with dual-rate sampling for digital control of chaotic systems. A Takagi-Sugeno (TS) fuzzy model is used to model the chaotic dynamic system and the extended parallel-distributed compensation technique is proposed and formulated for designing the fuzzy model-based controller under stability conditions. The optimal regional-pole assignment technique is also adopted in the design of the local feedback controllers for the multiple TS linear state-space models. The proposed design procedure is as follows: an equivalent fast-rate discrete-time state-space model of the continuous-time system is first constructed by using fuzzy inference systems. To obtain the continuous-time optimal state-feedback gains, the constructed discrete-time fuzzy system is then converted into a continuous-time system. The developed optimal continuous-time control law is finally converted into an equivalent slow-rate digital control law using the proposed intelligent digital redesign method. The main contribution of the paper is the development of a systematic and effective framework for fuzzy model-based controller design with dual-rate sampling for digital control of complex such as chaotic systems. The effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic Chua circuit     

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.     

Robust output-feedback control of linear discrete-time systems

The problem of designing H_∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H_∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.     

Multicriteria Robust Generalized <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and γ<Subscript>0</Subscript> Controllers with Application to Stabilization of a Rotor in Electromagnetic Bearings

For linear plants with unstructured or structured uncertainty of bounded norm, this paper designs Pareto optimal robust controllers in terms of linear matrix inequalities in multicriteria control problems with the generalized H₂ or γ₀ norms. The controller design procedure is based on optimization of a scalar objective function (Germeier convolution) and semi-definite programming. The developed theory is used to design multicriteria robust controllers in the stabilization problem for a rotor in electromagnetic bearings.     

Non‐smooth structured control design with application to PID loop‐shaping of a process

Feedback controllers with specific structure arise frequently in applications because they are easily apprehended by design engineers and facilitate on‐board implementations and re‐tuning. This work is dedicated to H_∞ synthesis with structured controllers. In this context, straightforward application of traditional synthesis techniques fails, which explains why only a few ad hoc methods have been developed over the years. In response, we propose a more systematic way to design H_∞ optimal controllers with fixed structure using local optimization techniques. Our approach addresses in principle all those controller structures which can be built into mathematical programming constraints. We apply non‐smooth optimization techniques to compute locally optimal solutions, and provide practical tests for descent and optimality. In the experimental part we apply our technique to H_∞ loop‐shaping proportional integral derivative (PID) controllers for MIMO systems and demonstrate its use for PID control of a chemical process. Copyright © 2007 John Wiley & Sons, Ltd.     

A Constructive Approach to Approximate Linear Periodic Systems

In this note, the problem on approximating a linear periodic system with period N by a periodic model with period M(0infin norms. Under this framework, both finite impulse response (FIR) and infinite impulse response (IIR) periodic systems can be efficiently approximated by using a constructive linear matrix inequality approach. Finally, some numerical examples are given to illustrate the effectiveness of the proposed approach     

Stabilization of sets with application to multi-vehicle coordinated motion

In this paper, we develop stability and control design framework for time-varying and time-invariant sets of nonlinear dynamical systems using vector Lyapunov functions. Several Lyapunov functions arise naturally in multi-agent systems, where each agent can be associated with a generalized energy function which further becomes a component of a vector Lyapunov function. We apply the developed control framework to the problem of multi-vehicle coordinated motion to design distributed controllers for individual vehicles moving in a specified formation. The main idea of our approach is that a moving formation of vehicles can be characterized by a time-varying set in the state space, and hence, the problem of distributed control design for multi-vehicle coordinated motion is equivalent to the design of stabilizing controllers for time-varying sets of nonlinear dynamical systems. The control framework is shown to ensure global exponential stabilization of multi-vehicle formations. Finally, we implement the feedback stabilizing controllers for time-invariant sets to achieve global exponential stabilization of static formations of multiple vehicles.     

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

We consider the problem of designing controllers for spatially-varying interconnected systems distributed in one spatial dimension. The matrix structure of such systems can be exploited to allow fast analysis and design of centralized controllers with simple distributed implementations. Iterative algorithms are provided for stability analysis, $H_{infty}$ analysis and sub-optimal controller synthesis. For practical implementation of the algorithms, approximations can be used, and the computational efficiency and accuracy are demonstrated on an example.      

Design of full state feedback finite‐precision controllers

In this paper, the Generalized L2 Synthesis framework is brought to bear on the problem of control design of full state feedback finite‐precision controllers. In particular, we investigate the problem of designing full state feedback controllers that achieve guaranteed H‐infinity performance objectives, subject to finite precision constraints on the controller. It is shown that by adopting the Generalized L2 Synthesis framework, the errors in the controller implementation can be captured as full structured uncertainty, and computationally tractable linear matrix inequality techniques used for analysis and synthesis. Copyright © 2002 John Wiley & Sons, Ltd.     

Distributed hierarchical control design of coupled heterogeneous linear systems under switching networks

In this paper, the distributed hierarchical control design is investigated for cooperations of heterogeneous linear systems subjected to the switching networks, where each subgraph of switching networks is allowed to be disconnected. Each heterogeneous system contains uncontrollable states and evolves in different dimension with different dynamics. The hierarchical control framework proposed in this paper consists of distributed hierarchical controllers, which contain the upper‐layers dealing with the communication topologies and the lower‐layers dealing with the heterogeneity of the systems. Under the framework, the cooperation problem of heterogeneous linear systems is decoupled into a cooperation problem of homogeneous virtual systems in the upper‐layer and tracking problems of each single systems in the lower‐layer. These two layers are designed based on the optimal control method and the output regulation method, respectively. Under the hierarchical control framework, the leader‐following consensus of heterogeneous linear systems is achieved and the result is extended to the formation control problem. Finally, numerical examples are provided to demonstrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal Process Control Using Neural Networks

Infinite time optimal controllers have been designed for a dispersion type tubular reactor model by using the framework of adaptive critic optimal control design. For the reactor control problem, which is governed by two coupled nonlinear partial differential equations, an optimal controller synthesis is presented through two sets of neural networks. One set of neural networks captures the relationship between the states and the control, whereas the other set of networks captures the relationship between the states and the costates. This innovative approach embeds the solutions to the optimal control problem for a large number of initial conditions in the domain of interest. Although the main aim of this paper is to solve a process control problem, the methodology presented here can be viewed as a practical computational tool for many problems associated with nonlinear distributed parameter systems. Numerical results demonstrate the viability of the proposed method.     

Robust non-fragile LQ controllers: The static state feedback case

This paper describes the synthesis of non-fragile or resilient regulators for linear systems. A general framework for fragility is described using state-space methodologies, and the LQ/H₂ static state-feedback problem is examined in detail. We discuss the multiplicative structured uncertainties case, and propose remedies of the fragility problem using an optimization programming framework via matrix inequalities. A special case that leads to a convex optimization framework via linear matrix inequalities (LMIs) will be considered. The benchmark problem is taken as an example to show how special controller gain variations can affect the performance of the closed-loop system.     

Design method of gain-scheduled controllers not depending on derivatives of parameters

In general, designing Gain-Scheduled (GS) controllers for Linear Parameter-Varying (LPV) systems using Parameter-Dependent Lyapunov Functions (PDLFs) yields controllers that depend on the derivatives of parameters. This makes their implementation impractical because ideal parameter derivatives are not available in the real world. In contrast to this, for open-loop system design, such as filter design and inverse system design, we propose a GS controller design method using PDLFs that gives controllers which do not depend on parameter derivatives. Although the proposed method uses structured PDLFs, we show that it is as conservative as existing methods. Numerical examples of designing GS filters and GS right inverse systems for LPV systems are included to demonstrate our conclusion.     

Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

线性离散时滞系统的鲁棒耗散控制

考虑线性离散时滞系统的二次型耗散控制问题.对于确定系统,给出渐近稳定且严格二次型耗散的条件和动态输出反馈控制器使闭环系统渐近稳定且严格二次型耗散.对于不确定系统,考虑不确定性具有耗散特性的情形,讨论鲁棒耗散性分析和动态输出反馈鲁棒耗散控制问题.通过构造增广系统,将不确定系统的鲁棒严格二次型耗散分析和设计转化为确定系统的情况.所得结果为离散时滞系统的无源控制和H∞控制提供了统一框架,且为离散时滞系统的分析和设计提供了一种更灵活、保守性更小的方法.     

System level synthesis

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty – such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.