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.     

Internal model based robust inversion feedforward and feedback 2DOF control for LPV system with disturbance

To reduce the adverse effects on the control performance and disturbance rejection caused by system uncertainty, a novel internal model based robust inversion feedforward and feedback 2DOF control approach was proposed for LPV system with disturbance. The proposed control approach combines the internal model control and robust inversion based 2DOF control, it utilizes internal model based control to reject external disturbance, utilizes robust inversion 2DOF control to enhance the control resolution and guarantee the system control performance. At first, a LMI synthesis approach for LPV system model identification and a disturbance compensator optimization design method which could minimize H_∞ norm of output error caused by disturbance are presented. Then, combined with internal loop for disturbance compensation, a robust inversion feedforward controller is designed by robust inversion approach and the feedback controller which could render the requirements of reference signal tracking performance and robustness satisfied is obtained by the H_∞ mixed sensitivity synthesis approach. Finally, atomic force microscopy (AFM) vertical positioning simulation experiments are conducted and the experiment results showed that the proposed control approach could achieve better output performance and disturbance rejection compared with conventional internal model based control and robust inversion based 2DOF control approach.     

Robust dynamic ship positioning control system design and applications

This paper describes the robust control system design for a ship dynamic positioning system. The control design is based on an approximate linear model derived from the nonlinear hydrodynamic equations governing the horizontal motions of the ship. The nonlinear models of the ship, seawaves, current, wind and thrusters are derived and simulated for control design verification. The H_∞ control design technique is employed to design the controller. The control problem is formulated in state‐space form and the design specifications are translated into requirements on the weighting functions of the error signal and the thrusters input. A tuning procedure is proposed based on the wind and wave disturbances. The controller is initially tested on the nonlinear ship model and simulation results are presented to demonstrate the robustness of the H_∞ controller. Tank tests results are then presented to assess the controller performance. Copyright © 2001 John Wiley & Sons, Ltd.     

Robust nonlinear ship course‐keeping control by H∞ I/O linearization and μ‐synthesis

In this paper, the H_∞ input/output (I/O) linearization formulation is applied to design an inner‐loop nonlinear controller for a nonlinear ship course‐keeping control problem. Due to the ship motion dynamics are non‐minimum phase, it is impossible to use the ordinary feedback I/O linearization to resolve. Hence, the technique of H_∞ I/O linearization is proposed to obtain a nonlinear H_∞ controller such that the compensated nonlinear system approximates the linear reference model in I/O behaviour. Then a μ‐synthesis method is employed to design an outer‐loop robust controller to address tracking, regulation, and robustness issues. The time responses of the tracking signals for the closed‐loop system reveal that the overall robust nonlinear controller is able to provide robust stability and robust performance for the plant uncertainties and state measurement errors. Copyright © 2002 John Wiley & Sons, Ltd.     

Backstepping Robust H ∞ Control for a Class of Uncertain Switched Nonlinear Systems Under Arbitrary Switchings

This paper addresses global robust H _∞ control for a class of switched nonlinear systems with uncertainty under arbitrary switchings. Each subsystem is in lower triangular form. The uncertainties are assumed to be in a known compact set. The backstepping design technique is used to design a smooth state feedback controller that renders the associated closed‐loop switched system globally robustly asymptotically stable and imposes a pre‐specified upper bound to the L ₂‐gain under arbitrary switchings. An example is provided to demonstrate the efficacy of the design approach.     

Robust control design of power system stabilizers using multivariable frequency domain techniques

H_∞ and structured singular value optimization techniques are used to design robust power system stabilizers (PSS) for a single-machine and a two-machine system with varying operating conditions. Realistic uncertainty models to represent the possible operating conditions as perturbations from a nominal operating condition are developed. System experience is used to select weighting functions to provide adequate damping and shape the controller frequency response. Computer simulations show that the PSS designed using the proposed technique provides improved damping compared to a conventional PSS.     

Robust H ∞ control for a class of uncertain mechanical systems

In this article, the problem of H _∞ control is investigated for a class of mechanical systems with input delay and parameter uncertainties which appear in all the mass, damping and stiffness matrices. Two approaches, norm-bounded and linear fractional transformation (LFT) uncertainty formulations, are considered. By using a new Lyapunov–Krasovskii functional approach, combined with the advanced techniques for achieving delay dependence, improved robust H _∞ state-feedback controller design methods are developed. The existence condition for admissible controllers is formulated in the form of linear matrix inequalities (LMIs), and the controller design is cast into a convex optimisation problem subject to LMI constraints. If the optimisation problem is solvable, a desired controller can be readily constructed. The result for the norm-bounded uncertainty case improves the existing ones in terms of design conservatism, and that for the LFT uncertainty case represents the first attempt in this direction. An illustrative example is provided to show the effectiveness and advantage of the proposed controller design methodologies.     

Discrete-time H∞ control of linear parameter-varying systems

We introduce new conditions for the H_∞ synthesis of discrete-time linear parameter-varying systems in the form of linear matrix inequalities. A distinctive feature of the proposed conditions is the ability to handle variation in both the dynamics and the input matrices without resorting to dynamic augmentation or iterative procedures. We show that this new condition contains the existing poly-quadratic H_∞ synthesis result as a particular case. We also derive a corollary which shows improvement even in the stronger case of quadratic H_∞ synthesis. Additionally, we show that, surprisingly, a dynamic gain-scheduled quadratic H_∞ controller can result in inferior performance compared to a static robust controller. Numerical examples illustrate our results.     

Robust sliding mode-<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control approach for a class of nonlinear systems affected by unmatched uncertainties using a poly-quadratic Lyapunov function

This paper proposes a robust sliding mode-H _∞ control design methodology for a class of nonlinear systems with unmatched parametric uncertainty and external disturbance. The design procedure combines the high robustness of the sliding mode control (SMC) with the H _∞ norm performance. First, based on linear matrix inequalities (LMI) technique and multiple Lyapunov functions approach, the sliding surface design problem is formulated as a H _∞ state-feedback control for a reduced uncertain nonlinear system with polytopic representation. Then, a sliding mode controller that drives the system states to the sliding surface in finite time and maintains a sliding mode is constructed. Finally, a comparative study is done to prove the effectiveness of the results.     

Robust observer–controller compensator design using the loop shaping design procedure and the algebraic method

This paper investigates robust observer‐controller compensator design using Vidyasagar's structure (VS). VS has a unit matrix parameter H similar to the Q parameter for the Youla–Kucera parameterization. VS can be designed based on the left coprimeness of the central controller in the H_∞‐loop shaping design procedure (H_∞‐LSDP) and therefore can preserve the intrinsic properties of the H_∞‐LSDP. This paper introduces algebraic methods to simplify the design of H in the VS controller by solving specific algebraic equations. In particular, the algebraic design of H can achieve two things. First, a dynamic H adjusts the tracking performance and yields the integral action. Second, a dynamic H rejects the input and output sinusoidal disturbances with known frequencies. These attributes are indications of the flexibility of the proposed method since the output‐feedback controller design of the H_∞‐LSDP cannot easily deal with such conditions. This paper discusses the achieved loop and the closed‐loop behavior of the system with VS, and also gives two numerical examples. The first example shows that the proposed method results in a better design in many aspects than the resulting from H_∞‐LSDP. The second example shows the application of the proposed method to rejecting input and output step disturbances, and input and output multiple sinusoidal disturbances, for which the H_∞‐LSDP can hardly be used. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust discrete-time control design with time-varying parametric uncertainties

A novel multiplier approach for robust controller design in discrete-time systems with real, time-varying parametric uncertainty is presented. An important feature of our approach is that bounds on the rate of variation of the uncertain parameters are assumed and, unlike in most related approaches, dynamic multipliers are obtained that utilize this information. A convex minimization procedure formulated as an LMI problem is presented to obtain multipliers that satisfy the robustness conditions derived. Such conditions are transformed to an equivalent scaled H_∞ norm condition and a μ/k_m-synthesis approach is proposed to design robust controllers. Copyright © 1999 John Wiley & Sons, Ltd.     

GH ∞ self-tuning control for multivariable systems

A H _∞ self-tuning controller which is based on the generalized H _∞ cost-function is described that includes robust design features. The GH _∞ control law is simple to compute and is suitable for multivariable self-tuning control applications. The simplifications achieved in the controller structure which are necessary for use in self-tuning, depend upon the assumptions made on system and cost-weighting models. The cost-function weights are assumed to ensure a system matrix can be factorized into a particular form. The control law is coupled to an extended least squares identification algorithm. The GH _∞ control law may be designed to be insensitive to unmodelled dynamics and the self-tuning action provides adaption to parametric uncertainty.     

Optimisation of the weighting functions of an H ∞ controller using genetic algorithms and structured genetic algorithms

In this article, the optimisation of the weighting functions for an H _∞ controller using genetic algorithms and structured genetic algorithms is considered. The choice of the weighting functions is one of the key steps in the design of an H _∞ controller. The performance of the controller depends on these weighting functions since poorly chosen weighting functions will provide a poor controller. One approach that can solve this problem is the use of evolutionary techniques to tune the weighting parameters. The article presents the improved performance of structured genetic algorithms over conventional genetic algorithms and how this technique can assist with the identification of appropriate weighting functions’ orders.     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust global stabilization of linear systems with input saturation via gain scheduling

The problem of robust global stabilization of linear systems subject to input saturation and input‐additive uncertainties is revisited in this paper. By taking advantages of the recently developed parametric Lyapunov equation‐based low gain feedback design method and an existing dynamic gain scheduling technique, a new gain scheduling controller is proposed to solve the problem. In comparison with the existing ?₂‐type gain scheduling controller, which requires the online solution of a state‐dependent nonlinear optimization problem and a state‐dependent ?₂ algebraic Riccati equation (ARE), all the parameters in the proposed controller are determined a priori. In the absence of the input‐additive uncertainties, the proposed controller also partially recovers Teel's ?_∞‐type scheduling approach by solving the problem of global stabilization of linear systems with actuator saturation. The ?_∞‐type scheduling approach achieves robustness not only with non‐input‐additive uncertainties but also requires the closed‐form solution to an ?_∞ ARE. Thus, the proposed scheduling method also addresses the implementation issues of the ?_∞‐type scheduling approach in the absence of non‐input‐additive uncertainties. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic output feedback control of a flexible air-breathing hypersonic vehicle via T–S fuzzy approach

By utilising Takagi–Sugeno (T–S) fuzzy set approach, this paper addresses the robust H_∞ dynamic output feedback control for the non-linear longitudinal model of flexible air-breathing hypersonic vehicles (FAHVs). The flight control of FAHVs is highly challenging due to the unique dynamic characteristics, and the intricate couplings between the engine and fight dynamics and external disturbance. Because of the dynamics’ enormous complexity, currently, only the longitudinal dynamics models of FAHVs have been used for controller design. In this work, T–S fuzzy modelling technique is utilised to approach the non-linear dynamics of FAHVs, then a fuzzy model is developed for the output tracking problem of FAHVs. The fuzzy model contains parameter uncertainties and disturbance, which can approach the non-linear dynamics of FAHVs more exactly. The flexible models of FAHVs are difficult to measure because of the complex dynamics and the strong couplings, thus a full-order dynamic output feedback controller is designed for the fuzzy model. A robust H_∞ controller is designed for the obtained closed-loop system. By utilising the Lyapunov functional approach, sufficient solvability conditions for such controllers are established in terms of linear matrix inequalities. Finally, the effectiveness of the proposed T–S fuzzy dynamic output feedback control method is demonstrated by numerical simulations.     

Decentralised indirect adaptive output feedback fuzzy H ∞ tracking design for a class of large-scale nonlinear systems

A novel decentralised indirect adaptive output feedback fuzzy controller with a compensation controller and an H _∞ tracking controller is presented for a class of uncertain large-scale nonlinear systems in this article. The compensator adaptively compensates for interconnections between subsystems as well as mismatched errors, while the H _∞ controller suppresses the effect of external disturbances. Based upon the combination of fuzzy inference systems, a state observer, H _∞ tracking technique and the strictly positive real condition, the proposed overall observer-based decentralised algorithm guarantees not only asymptotical tracking of reference trajectories but also an arbitrary small attenuation level of the unmodelled error dynamics including the disturbances on the tracking control. Simulation results substantiate the effectiveness of the proposed scheme.     

New robust stability criterion and robust controller synthesis

In this paper, a new robust stability criterion for linear systems is proposed by combining the passivity and small gain theorems in different frequency bands. A controller synthesis method based on the new criterion is also developed. The controller can achieve both good performance and robustness in the same frequency band, if the uncertainty in that frequency band is passive or near passive. For processes with near passive lumped uncertainties with large gain in the frequency region in which good performance is required, the proposed controller can have better performance than that of H_∞ control for the same robustness specification. © 1998 John Wiley & Sons, Ltd.