Fixed‐Structure ℋ∞ Controller Design for Systems with Polytopic Uncertainty: An LMI Solution

This note provides a new method for fixed‐structure H_∞ controller design for discrete‐time single input single output (SISO) systems with polytopic uncertainty. New conditions are derived based on the concept of robust strict positive realness (SPRness) of an uncertain polynomial with respect to a parameter‐dependent polynomial. The quality of this approximation depends on this SPR‐maker. A procedure is proposed for choosing the parameter‐dependent SPR‐maker that guarantees the improvement of the performance for the designed controller in comparison with the traditional approaches employed fixed SPR‐makers. The proposed conditions are given in terms of solutions to a set of linear matrix inequalities. The effectiveness of the proposed approach is demonstrated by comparing with the existing results.     

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.     

Robust Performance Controller Synthesis Based on Discrete‐Time Noncausal Linear Periodically Time‐Varying Scaling

This paper is concerned with the technique called discrete‐time noncausal linear periodically time‐varying (LPTV) scaling for robust stability analysis and synthesis. It is defined through the lifting treatment of discrete‐time systems, and naturally leads to a sort of noncausal operation of signals. In the robust stability analysis of linear time‐invariant (LTI) systems, it has been shown that even static noncausal LPTV scaling induces some frequency‐dependent scaling when it is interpreted in the context of lifting‐free treatment. This paper first discusses in detail different aspects of the effectiveness of noncausal LPTV scaling, with the aim of showing its effectiveness in controller synthesis. More precisely, we study the robust performance controller synthesis problem, where we allow the controllers to be LPTV. As in the LTI robust performance controller synthesis problem, we tackle our problem with an iterative method without guaranteed convergence to a globally optimal controller. Despite such a design procedure, the closed‐loop H_∞ performance is expected to improve as the period of the controller is increased, and we discuss how the frequency‐domain properties of noncausal LPTV scaling could contribute to such improvement. We demonstrate with a numerical example that an effective LPTV controller can be designed for a class of uncertainties for which the well‐known μ‐synthesis fails to derive even a robust stabilization controller.     

Dissipative control for a class of uncertain nonlinear control systems

A general dissipative controller is proposed to achieve robust tracking control performance for a class of uncertain single‐input single‐output (SISO) nonlinear systems. The feedback linearization technique is employed to transform the nonlinear system into an assignable inner linear system with a differential control input so that the relationship of the external (input) power and the stored energy of system can be shown clearly. Then, a dissipative controller with an assignable attenuation level is proposed to make the system energy dynamics fit a required dissipative inequality. The unstable factors of the system can then be attenuated accordingly. The system stability is guaranteed even if the system has permanent unavoidable uncertainties. The proposed design can be achieved without the use of traditional means, i.e. optimal control, which requires solution of a Hamilton inequality (or Riccati equation). The Lyapunov stable condition is assured in our approach when the system uncertainties belong to L _∞. Moreover, due to the compatibility of the proposed controller, the controller can be embedded into the designs of other controllers. In those designs, knowledge of the system functions is not required. Basically, the proposed dissipative controller is independent of the system functions. The use of the bounds of the system function is considered to prove the system stability only. Two simulations are given to illustrate the effectiveness of the proposed controller. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A two‐degree‐of‐freedom ℋ︁∞ control design method for robust model matching

We propose an ??_∞ controller design method which achieves a closed‐loop transfer function equal or otherwise sensibly close to a desired transfer function, viz. a model reference design. The proposed controller design method inherits the model reference feature of the internal model control design method and incorporates the weighting scheme of the ??_∞ loop‐shaping. It utilizes Youla–Kucera parameterization in a two‐degree‐of‐freedom scheme to achieve robust model reference and high performance design while ensuring a sensible robust stability margin, and can be readily applied to the generic class of LTI systems (SISO, MIMO, stable, unstable). Copyright © 2006 John Wiley & Sons, Ltd.     

Robust control of multivariable critical systems

Based on the method of inequalities and H _∞-optimization method, this paper develops an approach to robust control design of multivariable critical systems with external and internal uncertainties. In this approach the formulation of the robust control design of these systems is expressed by a set of inequalities which includes output performance criteria in the time domain and a robust performance criterion in the frequency domain of the system. Some relationships between an input space, a modelling error space, a controller, output performance and robust performance are established by inequalities for SISO and MIMO critical systems so that the robust control design problem of these systems is largely simplified.     

Bode 理想传递函数在分数阶控制中的应用

本文将Bode 理想传递函数应用于分数阶控制器的设计和分数阶PID 控制器参数整定中．所得控制器 可以在满足系统要求的截止频率和相角裕度的前提下,使补偿后系统Bode 图的相频特性曲线在截止频率附近有一 个水平区域,即闭环系统对增益的变化具有鲁棒性．它不仅适合于分数阶对象,也适用于整数阶对象,并能够提高 系统的控制品质．仿真结果证明了上述方法的有效性．     

NARMAX modelling and robust control of internal combustion engines

Detailed in this paper is a SISO non-linear modelling and robust controller design methodology experimentally verified on an internal combustion engine. The methodology begins with the identification of a NARMAX model that captures the non-linear dynamics relating the input to the output of a system. This model is converted to a describing function representation for the purpose of robust feedback controller design. The ideology for the describing function recovery is developed in the form of an algorithm which can be extended to other NARMAX model structures not considered here. The controller design is executed in the frequency domain where the output performance specification is |y(t)|≤β?t>0 and the actuator saturation constraint is |u(t)|≤K?t>0. For the engine idle speed control application of this study, a SISO NARMAX model of the engine is developed between the by-pass idle air valve (BPAV) and engine speed. The performance objective for the controller design is the time domain tolerance of |Δ rpm| ≤ 100 rpm on idle speed perturbations despite a non-measurable 20 N m external torque disturbance. The controller is validated through numerical simulations as well as experimental verification.     

Controller reduction with closed loop H infinity performance constraints: A QFT perspective

Assume that an initial stabilizing controller K₀₍s₎, which satisfies various closed loop frequency domain specifications, has been a priori synthesized using, e.g. H infinity control, mu synthesis techniques or closed loop convex synthesis. Remembering that the order of K₀₍s₎ is typical y at least equal to the order of the plant to be controlled, the aim of this paper is to find a stabilizing reduced order controller, which also satisfies the performance specifications and which moreover minimizes the open loop bandwidth. To this aim, the structured singular value mu is used to translate at each frequency closed loop frequency domain specifications into requirements on the frequency response of the controller. The principle of our reduction method is thus very close to the original idea of the SISO QFT design approach, except that the problem of translating closed loop frequency domain specifications into open loop ones is much more complex in the MIMO case. The problem reduces to the issue of finding a controller, whose frequency response belongs at each frequency to a template. The convexity of these templates greatly facilitates the practical realization of the controller. As a final point, the method is successfully applied to a standard H problem, which is extracted from the mu Analysis and Synthesis Toolbox of Matlab, namely the synthesis of an autopilot for the space shuttle.     

Loop shaping and a zero-placement technique as applied to the benchmark problem

A classical and a ‘quasi-modern’ approach are taken to develop robust control laws for the 1992 ACC benchmark problem of Wie and Bernstein. In both approaches, copious use of loop-shaping à la Bode design is employed. The quasi-modern approach reported herein is a zero-placement technique, which is a variant on LQG/LTR. This approach allows for the direct selection of the zeros of the loop transfer function, leading to the desired loop shape. Therefore, undesirable pole-zero cancellations may be avoided. The modern and quasi-modern approaches are closely related, and can lead to similar control laws. Though the problem addressed here is SISO, the zero-placement approach can be used on MIMO systems. However, in the case discussed herein, the classically designed controller has superior performance and stability robustness.     

Output‐Feedback Control for Continuous‐time Interval Positive Systems under L1 Performance

This paper is concerned with the design of an L₁‐induced output‐feedback controller for continuous‐time positive systems with interval uncertainties. A necessary and sufficient condition for stability and an L₁‐induced performance of interval positive linear systems is proposed in terms of linear inequalities. Based on this, conditions for the existence of robust static output‐feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions. For special single‐input‐multiple‐output (SIMO) positive systems, the problem of controller synthesis is completely solved with the help of an analytical formula for the L₁‐induced norm. An illustrative example is provided to show the effectiveness and applicability of the theoretical results.     

A Parameter Dependent Controller Design Approach for Delayed LPV System

This paper investigates the memory state feedback controller synthesis problem for linear parameter‐varying (LPV) systems with time‐varying input delay. A novel state transformation is proposed to reduce the influence of time delay, and two types of Lyapunov‐Krasovskii functional are employed to guarantee the stability of the closed‐loop system and induced L₂ norm performance. The controller gain is formulated to parameter dependent, and design method is then led to a convex optimization problem and solved by parameter gridding technique. A numerical example is provided to validate the effectiveness of the proposed approach.     

Robust Fault‐Tolerant Control of Launch Vehicle Via GPI Observer and Integral Sliding Mode Control

A robust fault‐tolerant attitude control scheme is proposed for a launch vehicle (LV) in the presence of unknown external disturbances, mismodeling dynamics, actuator faults, and actuator's constraints. The input‐output representation is employed to describe the rotational dynamics of LV rendering three independently decoupled second order single‐input‐single‐output (SISO) systems. In the differential algebraic framework, general proportional integral (GPI) observers are used for the estimations of the states and of the generalized disturbances, which include internal perturbations, external disturbances, and unknown actuator failures. In order to avoid the defects of the conventional sliding surface, a new nonlinear integral sliding manifold is introduced for the robust fault‐tolerant sliding mode controller design. The stability of the GPI observer and that of the closed‐loop system are guaranteed by Lyapunov's indirect and direct methods, respectively. The convincing numerical simulation results demonstrate the proposed control scheme is with high attitude tracking performance in the presence of various disturbances, actuator faults, and actuator constraints.     

具有最优动态性能的鲁棒镇定控制器设计

针对ＳＩＳＯ线性离散系统,利用线性规划方法设计具有指定最优动态性能的鲁棒稳定控制器。当线性离散模型的零,极点已知时,将最优动态性能指标在指定输入信号下直接转化为线性规划问题。从而解出最优响应输出序列。最优动态性能指标与鲁棒稳定性的统一使该控制器的设计方法具备了工业应用条件。仿真实例验证了结果的正确性。     

Robust PID‐PSD Controller Design: BMI Approach

A new robust proportional‐integral‐derivative (PID)–proportional‐sum‐derivative (PSD) controller design method based on linear (bilinear) matrix inequalities (LMI, BMI) is proposed for uncertain affine linear system. The design procedure guarantees the parameter dependent quadratic stability, and guaranteed cost control with a new quadratic cost function (LQRS) including the derivative term for the state vector as a tool to influence the overshoot and response rate. The second approach to the PSD controller design procedure is based on a Lyapunov function with a special term corresponding to the time‐delay part of the control algorithm. The results obtained are illustrated on three examples to show the robust PID, PSD control design procedure and the influence of the choice of matrix S in the extended cost function.     

Relay feedback tuning of robust PID controllers with iso-damping property.

A new tuning method for proportional-integral-derivative (PID) controller design is proposed for a class of unknown, stable, and minimum phase plants. We are able to design a PID controller to ensure that the phase Bode plot is flat, i.e., the phase derivative w.r.t. the frequency is zero, at a given frequency called the "tangent frequency" so that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. At the "tangent frequency," the Nyquist curve tangentially touches the sensitivity circle. Several relay feedback tests are used to identify the plant gain and phase at the tangent frequency in an iterative way. The identified plant gain and phase at the desired tangent frequency are used to estimate the derivatives of amplitude and phase of the plant with respect to frequency at the same frequency point by Bode's integral relationship. Then, these derivatives are used to design a PID controller for slope adjustment of the Nyquist plot to achieve the robustness of the system to gain variations. No plant model is assumed during the PID controller design. Only several relay tests are needed. Simulation examples illustrate the effectiveness and the simplicity of the proposed method for robust PID controller design with an iso-damping property.     

A robust integrated controller/diagnosis aircraft application

In this paper, an application of the robust integrated control/diagnosis approach using ??_∞‐optimization techniques to the nonlinear longitudinal dynamics of a Boeing 747‐100/200 aircraft is presented. The integrated approach allows to address directly the trade‐off between the conflicting controller and fault diagnosis objectives. The integrated design formulation (interconnection and weight selection) is defined using five LTI plants obtained through out the Up‐and‐Away flight envelope. Linear and nonlinear closed‐loop time simulations are carried out under a realistic turbulence and noise environment. A comparison drawn with the non‐integrated design of a controller and a diagnosis filter with the same objectives shows that the integrated case results in similar diagnosis characteristics but improved fault tolerant performance and ease of design. Copyright © 2005 John Wiley & Sons, Ltd.     

Plasma multivariable robust control system design and simulation for a thermonuclear tokamak-reactor

The paper reports results on the design and analysis of the multivariable feedback H_infin; robust system for plasma current, position and shape control in the fusion energy advanced tokamak (FEAT) developed in the International Thermonuclear Experimental Reactor (ITER) project. The system contains the fast loop with the SISO plasma vertical speed robust controller and the slow loop with the MIMO plasma current and shape robust controller. The goal is to study the resources of the system robustness to achieve a higher degree of the FEAT operation reliability. Two H_infin; block diagonal controllers {K _SISO, K _MIMO} were designed by a mixed sensitivity approach in the framework of the disturbance rejection configuration. These controllers were compared with block diagonal decoupling, PI and LQG controllers at the set of FEAT key scenario points according to the multiple-criterion: nominal performance at minor disruptions, robust stability and robust performance. The H_infin; controllers showed larger multivariable stability margin and better nominal performance.     

H ∞ control of a Wiener-type system

A Wiener system is a system which can be modelled as a linear dynamic followed by a static gain. The goal of this paper is to develop a robust H _∞ compensator for controlling an SISO Wiener system. The controller also takes the form of a Wiener model. The design approach consists of the approximation of the non-linear gain using a piecewise linear (PWL) function and in using a linear controller for each sector obtained from this approximation. Therefore, the general controller structure can be stated as a linear dynamic compensator in series with a PWL static gain.

As an illustrative case, a neutralization pH reaction between a strong acid and a strong base in the presence of a buffer agent is dealt with. Computer simulations are developed for showing the performance of the proposed controller.     

Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

In this paper, a robust fractional‐order PID (FOPID) controller design method for fractional‐order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D‐decomposition method. Then, the optimal fractional orders λ and μ of FOPID controller are achieved at the biggest three‐dimensional PSR, which means the best robustness. Given the optimal λ and μ, the other FOPID controller parameters k_p, k_i, k_d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI). The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm.