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The aim of this chapter is to introduce you to some ideas about modelling non-linear systems from first principles, about their behaviour and how it can be analyzed, and finally, about how they might be made to behave in a desirable way. As a vehicle I have chosen what I hope you will find to be an interesting problem which is outside the mainstream of technology-based control but is, I believe, part of an area of increasing importance for the near future. The topic is the application of control theory in a biological setting - the optimal enhancement of the immune system. In what follows I will first outline how models of systems of this type can be constructed largely through deduction. They belong to a class of models called Lotka-Volterra Models which are widely used to describe the dynamics of interacting populations such as predator-prey systems, epidemics and, in our case, the concentrations of various biological quantities in the immune system. Next we will look briefly at the behaviour of this class of dynamics. Finally we look at a difficult problem - how control theory can be used to enhance the response of the immune system by allowing us to compute an optimal policy that will prevent organ failure in conditions where, left to itself, organ damage or death would naturally occur. The case study builds on a series of recent papers by Stengel and colleagues [1-3] with the introduction of a newly developed approach to non-linear optimal control [4] which has not yet become well known but appears to have a number of advantages over conventional solutions. Throughout I will try to maintain a "systems viewpoint" and avoid too much biology. I will also avoid too much mathematics other than stating the key equations and providing references to more detail. Those of you versed in linear quadratic optimal control will recognize the form of the non-linear generalization. [ABSTRACT FROM AUTHOR]

This chapter considers the design of a robust servo systemof hard disk drive(HDD). Three robust design methods are applied, namely, the μ-synthesis, $\mathcal{H}_\infty$ optimal design and $\mathcal{H}_\infty$ loop-shaping design procedure (LSDP). After a description of the HDD servo system dynamics in the first section, it is shown in details (in Section 12.2) how to derive the plant model which involves several uncertain parameters. Then we consider the synthesis of continuoustime controllers using the available methods for robust control design. These controllers are compared in aspects of robustness of closed-loop system stability and of performance in the frequency domain and in the time-domain. The design of discrete-time controller is included as well, for two sampling frequencies. Finally, we present the simulation results of the nonlinear continuous-time and discrete-time closed-loop systems, followed by conclusions at the end of the chapter. The prevalent trend in hard disks design is towards smaller disks with increasing larger capacities. This implies that the track width has to be smaller leading to lower error tolerance in the positioning of the read/write heads. The controller for track following has to achieve tighter regulation in the presence of parameter variations, nonlinearities and noises. Hence it is appropriate to use advanced design methods like μ synthesis and $\mathcal{H}_\infty$ optimization in order to achieve robust stability and robust performance of the closed-loop servo system. [ABSTRACT FROM AUTHOR]

Engineers use feedback, both positive and negative, to perform a wide array of signaling functions. Biological systems are also faced with many of the same requirements In this tutorial we examine examples from different cellular signaling systems to show how biology also uses feedback paths to perform many of the same tasks. [ABSTRACT FROM AUTHOR]

This chapter uses a railway vehicle as an example of a mechanical dynamic system to which control can be applied in a manner that yields significant benefits from an engineering and operational viewpoint. The first part describes the fundamentals of railway vehicles and their dynamics: the normal configuration, the suspension requirements, how they are modelled and an overview of the types of control concept that are currently applied or under consideration. The second part provides a case study of controller design issues. [ABSTRACT FROM AUTHOR]

In this Chapter we present the design of a robust control system for a high-purity distillation column. The original nonlinear model of the column is of high order and it includes parametric gain and time delay uncertainty. A low order linearized distillation column model is used in design of a two-degree-of freedom (2DOF) $\mathcal{H}_\infty$ loop shaping controller and a μ-controller. Both controllers ensure robust stability of the closed loop system and fulfillment of a mixture of time domain and frequency domain specifications. A reduced order μ-controller is then found which preserves the robust stability and robust performance of the closed loop system. The simulation of the closed loop system with the nonlinear distillation column model shows very good performance for different reference and disturbance signals as well as for different values of the uncertain parameters. [ABSTRACT FROM AUTHOR]

This chapter demonstrates the use of anti-windup compensation in the control loop of a micro-actuator which is nominally controlled by a linear, discrete robust controller. The micro-actuator is part of a hard disk drive dual-stage servo-control system for positioning of the read/write head. The actuator inputs are constrained to retain the micro-actuator's displacement range of less than 0.4 μm for mechanical protection. In the first part of the chapter, the anti-windup compensation scheme exemplifies an approach suggested by Weston & Postlethwaite [29]. Here, the scheme is posed as a discrete full-order compensator and the closed loop analysis uses a generalized circle criterion approach. The design of the compensator is posed in LMI-form. In the second part of the chapter, it is shown how the linear micro-actuator control loop with anti-windup compensation is incorporated into the non-linear servo-control scheme for positioning of the read/write head in a hard disk drive. [ABSTRACT FROM AUTHOR]

The main features of sliding-mode control (SMC) and the associated feedback control law of Variable Structure Control (VSC) systems will be summarized in this chapter. SMC is a well-known solution to the problem of the deterministic control of uncertain systems, since it yields invariance to a class of parameter variations [4,7,33-36,43]. The characterizing feature is sliding motion, which occurs when the system state repeatedly crosses certain subspaces, or sliding hyperplanes, in the state space. A sliding controller may comprise nonlinear and linear parts, and has been well studied in the literature. [ABSTRACT FROM AUTHOR]

The focus of much of the research in the area of control systems theory during the seventies and eighties addressed the issue of robustness - i.e., designing controllers with the ability to maintain stability and performance in the presence of discrepancies between the plant and model. One nonlinear approach to robust controller design which emerged during this period is the Variable Structure Control Systems methodology. Variable Structure Control Systems evolved from the pioneering work in Russia of Emel'yanov and Barbashin in the early 1960's. The ideas did not appear outside the Soviet Union until the mid 1970's when a book by Itkis [20] and a survey paper by Utkin [29] were published in English. Variable structure systems concepts have subsequently been utilized in the design of robust regulators, model-reference systems, adaptive schemes, tracking systems and state observers. The ideas have successfully been applied to problems as diverse as automatic flight control, control of electrical motors, chemical processes, helicopter stability augmentation, space systems and robotics [12, 23, 28, 30]. [ABSTRACT FROM AUTHOR]

To motivate a study of asymptotic behaviour of nonlinear systems modelled by ordinary differential equations and differential inclusions, we indicate how such equations/inclusions arise naturally in control of dynamical process by feedback. The concept of control pertains to modifying the behaviour of the process, by manipulation of inputs to the process, in order to achieve some prescribed goal. Fundamental to this is the notion of feedback: a strategy in which the inputs to the process are determined on the basis of concurrent observations on (or outputs from) the process. [ABSTRACT FROM AUTHOR]

This chapter summarises the basic ideas behind $\mathcal{H}_\infty$ control theory. [ABSTRACT FROM AUTHOR]

This chapter introduces the problem of input saturation in systems which are otherwise linear. It is demonstrated how input saturation may cause such systems to exhibit undesirable behaviour. The notion of anti-windup compensators is introduced, followed by a detailed description of a typical anti-windup problem and an effective design procedure. The chapter also emphasizes the role of various nonlinear stability and performance concepts such as Lyapunov stability, sector-boundedness and $\mathcal{L}_2$ gain. [ABSTRACT FROM AUTHOR]

This chapter gives an introduction to the use of linear matrix inequalities (LMIs) in control. LMI problems are defined and tools described for transforming matrix inequality problems into a suitable LMI-format for solution. Several examples explain the use of these fundamental tools. [ABSTRACT FROM AUTHOR]

The approximation of high-order plant and controller models by models of lower-order is an integral part of control system design and analysis. Until relatively recently model reduction was often based on physical intuition. For example, chemical engineers often assume that mixing is instantaneous and that packed distillation columns may be modelled using discrete trays. Electrical engineers represent transmission lines and the eddy currents in the rotor cage of induction motors by lumped circuits. Mechanical engineers remove high-frequency vibration modes from models of aircraft wings, turbine shafts and flexible structures. It may also be possible to replace high-order controllers by low-order approximations with little sacrifice in performance. [ABSTRACT FROM AUTHOR]

This paper assumes familiarity with the basic Control and Dynamics, as covered in undergraduate courses. It introduces the different alternative system representations for linear systems and provides a quick review of the fundamental mathematical tools, which are essential for the treatment of the more advanced notions in Linear Systems. The paper focuses on some fundamental concepts underpinning the study of linear systems and dynamics and which play a crucial role in the analysis and design of control systems; thus, the paper deals with notions such as those of Controllability, Observability, Stability, poles and zeros and re lated dynamics and their properties under different compensation schemes. There is a concrete flavour in the current approach which runs through this presentation and this is that of the underlying algebraic structure. The term "structure" refers to aspects of the state space/transfer function description, which remain invariant under a variety of transformations. The set of transformations considered here are of the compensation type and include state feedback and output injection, dynamic compensation, as well as of the representation type transformations that include state, input, and output co-ordinate transformations. This structure stems from the system description and defines the nature of the dynamics and the related geometric proper ties and these in turn define what it is possible to achieve under feedback; such an approach is known as a structural approach. Central to our analysis are the notions of poles and zeros. The poles of a system are crucial characteristics of the internal system dynamics, characterise system free response, stability and general aspects of the performance of a system. The poles of a system are affected by the different compensation schemes and their assignment is the subject of many design methodologies aiming at shaping the internal system dynamics under different compensation schemes. The notion of zeros is more complex, since they express the interaction between internal dynamics and the effort to control and observe the system and they are thus products of overall system design, that apart from process synthesis involves selection of actuation and measurement schemes for the system. The significance of zeros is mainly due to that they remain invariant under a large set of compensation schemes, as well as that they define limits of what can be achieved under compensation. This makes zeros crucial for design, since they are part of those factors characterising the potential of a given system to achieve certain design objectives under compensation. The invariance of zeros implies that their design is an issue that has to be addressed outside the traditional control design; this requires understanding of the zero formation process and involves early design stages mechanisms such as process instrumentation. Poles and zeros are conceptually inverse concepts (resonances, antiresonances) and such mechanisms are highlighted throughout the paper. The role of system structure in characterising different system properties is central to this paper and it is defined by a set of discrete and continuous invariants; these invariants characterise a variety of key system properties and their type/values define the structure of canonical forms and determine somehow the potential of a given system for compensation. Invariants and canonical forms under the general transformation group are linked to compensation theory, whereas those associated with representation transformations play a key role in system identification. The emphasis in this article is to provide an overview of the fundamentals concepts, the back ground mathematical tools, explain their dynamic significance and link them to problems of control and systems design. [ABSTRACT FROM AUTHOR]

This chapter introduces a Linear Matrix Inequality formulation of $\mathcal{H}_\infty$ output feedback controller synthesis. Within this framework, the design procedure of McFarlane and Glover is detailed. Existence conditions for full-order and static output loop-shaping controllers are given in terms of matrix inequalities. The approach is extended to the class of polytopic systems. The effectiveness of the various controller synthesis algorithms is demonstrated on two aerospace control design examples. [ABSTRACT FROM AUTHOR]

This chapter describes the development of evolutionary and deterministic global optimisation methods for the clearance of nonlinear flight control laws for highly augmented aircraft. The algorithms are applied to the problem of evaluating a nonlinear handling qualities clearance criterion for the ADMIRE benchmark model. An optimisation-based approach for computing worst-case pilot input demands is also presented. Hybrid versions of the global algorithms, incorporating local gradient-based optimisation, are shown to significantly reduce computational complexity while at the same time improving global convergence properties. The proposed approach to flight clearance is shown to have significant potential for improving both the reliability and efficiency of the current industrial flight clearance process. [ABSTRACT FROM AUTHOR]

In this chapter, the challenging problem of aircraft on-ground modelling for control design and analysis is dealt with using a novel NDI-based force identification approach and a general nonlinear symbolic LFT modelling framework. It is shown how the modelling framework is applicable to derive an exact LFT model of the aircraft-on-ground as well as some simplified design-oriented versions. An important contribution is the proposed force identification method, which allows nonlinear ground forces to be replaced with saturation-type nonlinearities. As a result, the simplified model boils down to a reduced-order LFT plant where the Δ block only contains time-varying or constant (but uncertain) parameters on the one hand, and saturation-type non-linearities on the other hand. Such a model is then very useful for applying modern analysis and synthesis techniques. [ABSTRACT FROM AUTHOR]

In this chapter, a general nonlinear symbolic LFT modelling framework and its supporting LFT tools are presented. The modelling approach developed combines the natural modularity and clarity of presentation from LFT modelling with the ease of manipulation from symbolic algebra. It results in an exact nonlinear symbolic LFT that represents an ideal starting point to apply subsequent assumptions and simplifications to finally transform the model into an approximated symbolic LFT ready for design and analysis. The development of the framework is supported by a novel algebraic algorithm for symbolic matrix decomposition and two new LFT operations: nested substitution and symbolic differentiation. [ABSTRACT FROM AUTHOR]

The main objective of the Action Group was to investigate the potential benefits of nonlinear design and analysis methods for control law development in aerospace vehicles. To guarantee the industrial relevance of the project, two highly realistic simulation models were developed, together with demanding design/analysis challenges. These benchmarks in themselves represent significant achievements of the project, since there are still very few industrially relevent aircraft models, with realistic design and analysis specifications, available in the open literature on which control theoreticians can test and validate new techniques and algorithms. An additional benefit of the on-ground transport aircraft benchmark developed by Airbus for the project is that it represents a non-standard control application (at least in the context of aerospace control!) and thus adds another new and challenging set of problems to those traditionally addressed by flight control law designers. [ABSTRACT FROM AUTHOR]