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## Über dieses Buch

Linear, Time-varying Approximations to Nonlinear Dynamical Systems introduces a new technique for analysing and controlling nonlinear systems. This method is general and requires only very mild conditions on the system nonlinearities, setting it apart from other techniques such as those – well-known – based on differential geometry. The authors cover many aspects of nonlinear systems including stability theory, control design and extensions to distributed parameter systems.

Many of the classical and modern control design methods which can be applied to linear, time-varying systems can be extended to nonlinear systems by this technique. The implementation of the control is therefore simple and can be done with well-established classical methods. Many aspects of nonlinear systems, such as spectral theory which is important for the generalisation of frequency domain methods, can be approached by this method.

## Inhaltsverzeichnis

### Introduction to Nonlinear Systems

Overview
In this book we shall present a new way of approaching nonlinear systems by regarding them as limits of linear, time-varying ones. In order to explain the method, consider an unforced nonlinear dynamical system defined by the differential equation
$$\dot {x} = f(x,t) , x(0)=x_{0}. (1.1)$$
Suppose that f is differentiable at x = 0, for all t, and
$$f(0,t)=0 , \textrm{for all } t.$$
María Tomás-Rodríguez, Stephen P. Banks

### Linear Approximations to Nonlinear Dynamical Systems

Introduction
In this chapter the iteration approach to nonlinear systems under study is explained in detail. This technique is based on the replacement of the original nonlinear system by a sequence of linear time-varying systems, whose solutions will converge to the solution of the nonlinear problem. The only condition required for its application is a mild Lipschitz condition which must be satisfied by a matrix associated with the nonlinear system.
María Tomás-Rodríguez, Stephen P. Banks

### The Structure and Stability of Linear, Time-varying Systems

Introduction
In view of the basic approximation theory in Chapter 2, nonlinear dynamical systems can be approximated uniformly on compact intervals by linear, time-varying systems. It is therefore important to study the general questions of existence, uniqueness, etc. for dynamical systems of this type. In this chapter we shall consider the general theory of linear time-varying dynamical systems first from the point of view of existence and uniqueness, and then we shall determine a number of explicit solutions, based on the theory of Lie algebras.
The remainder of the chapter is concerned, essentially with stability theory. After considering the classical theory, we shall introduce the ideas of Lyapunov numbers and describe Oseledec’s theorem on decomposition of the state-space into invariant subbundles, which generalises the hyperbolic splitting of the state-space for timeinvariant systems. Finally we shall consider the theory of exponential dichotomies and its generalisation to invariant subbundles.
María Tomás-Rodríguez, Stephen P. Banks

### General Spectral Theory of Nonlinear Systems

Introduction
The spectral theory of linear (time-invariant) systems is, of course, the most important aspect of control theory in classical feedback design, and historically this was the approach taken by control engineers until the introduction of state-space theory. The techniques developed in the past include Nyquist and Bode diagrams, pole assignment and root locus methods. Frequency domain methods are therefore extremely important, especially for the suppression of resonant vibrations in mechanical systems.
In this chapter we shall outline a general spectral theory for nonlinear systems. First, a generalised transform theory is developed which can generate Volterra series kernels directly using Schwartz’ kernel theorem, without the need of a (somewhat arbitrary) definition of multi-dimensional Laplace transforms. This theory can then be directly applied, by using the iteration scheme developed in this book to derive a general spectral theory for nonlinear systems. We shall then briefly show how to apply the same ideas to generalise exponential dichotomies and the Sacker-Sell spectrum to nonlinear systems. We shall assume a basic knowledge of functional analysis and distribution theory.
María Tomás-Rodríguez, Stephen P. Banks

### Spectral Assignment in Linear, Time-varying Systems

Introduction
Pole placement is a well-known tool in designing control for linear systems. It belongs to the class of so-called feedback stabilisation methods. Basically, the aim of this approach is to design a controller so that the closed-loop poles of the plant are assigned to some desired locations chosen according to some specific stability and performance criteria.
María Tomás-Rodríguez, Stephen P. Banks

### Optimal Control

Introduction
Optimal control is one of the main techniques of modern control design, as it has been for many years. The linear-quadratic theory of optimal control design is well established and has various forms including the receding horizon approach for a robust, easily implementable variation of the theory. It is also useful in H  ∞  control in the well-known state-space game theoretic formulation. Obtaining the ‘best’ controller in any given circumstance is clearly important, but for general nonlinear systems, one is led to the solution of an extremely difficult (in general, non-smooth) partial differential equation. This makes the existing general nonlinear theory very difficult to apply.
In this chapter we shall show how to use the iteration technique developed above to solve nonlinear optimal control problems. In the next section we shall outline the classical linear quadratic regulator theory and derive the optimal feedback control in terms of the solution of a Riccati equation.We shall also indicate the modifications necessary to solve the linear tracking problem. The generalisation to nonlinear systems will be given in Section 6.3 and some examples will be presented in Section 6.4. Some comments on viscosity solutions of the Hamilton-Jacobi-Bellman (HJB) equation and the optimality of the solution will be given in Section 6.5.
María Tomás-Rodríguez, Stephen P. Banks

### Sliding Mode Control for Nonlinear Systems

Introduction
In this chapter a method of sliding mode control for nonlinear systems will be presented. Sliding mode techniques are a different approach to solve control problems and are an area of increasing interest. It is well-known that in most of the cases, in the formulation of any control problem, there will appear some differences between the actual plant and the mathematical model developed for the control design. These discrepancies may be due to any number of factors such as unmodelled dynamics, variation in system parameters or the approximation of complex plant behaviour by a simpler model. It is the engineer’s responsibility to guarantee some level of performance in spite of the existence of plant/model mismatches. This has led to the development of the so-called robust methods.
María Tomás-Rodríguez, Stephen P. Banks

### Fixed Point Theory and Induction

Introduction
In this chapter we shall show that we can obtain results on various aspects of systems of the form
$$\dot{x} = A(x)x, \: x(0)=x_{0} ~(8.1)$$
by using a sequence of approximations
$$\dot{x}^{[i]}(t) = A(x^{[i-1]}(t))x^{[i]}(t), \: x^{[0]}(0)=x_{0} ~(8.2)$$
as before and a combination of fixed point theorems and induction. The induction will proceed in the following way: suppose we want to prove some property P of Equation 8.1, and assume we can find a function x [0](t) which has this property. Suppose also that if x [i − 1](t) has the property, then the solution x [i](t) of Equation 8.2 also has the property. Then if the sequence { x [i](t) } converges on some interval [0,T], it follows by induction that the nonlinear system (8.1) (or the solutions thereof) also have the property P.
We shall see that this can be applied to stability of nonlinear systems and the existence of periodic solutions. The same idea can, however, be applied to many other situations.
María Tomás-Rodríguez, Stephen P. Banks

### Nonlinear Partial Differential Equations

Introduction
In this chapter we shall show how to generalise the results of the previous chapters on finite-dimensional nonlinear systems to partial differential equations. Rather than try to cover any significant part of this vast field, we shall concentrate on two problems, since the ideas then apply to many other nonlinear distributed systems. These two problems are concerned with moving boundaries in heat flow and the motion of solitary nonlinear waves (solitons).
María Tomás-Rodríguez, Stephen P. Banks

### Lie Algebraic Methods

Introduction
In this chapter we shall consider systems of the form
$$\dot{x} = A(x)x ~(10.1)$$
where $$A:\mathbb{R}^{n} \rightarrow \mathfrak{g}$$ and $$\mathfrak{g}$$ is the Lie algebra of a Lie group G. The classical structure theory of Lie groups and Lie algebras (see Appendix B and [1,2]) will be used to decompose the system (10.1) into simpler subsystems in a way which generalises the classical Jordan decomposition of single matrices.
María Tomás-Rodríguez, Stephen P. Banks

### Global Analysis on Manifolds

Introduction
In this chapter we shall consider the non-local theory of systems – i.e. the theory of sections of the tangent bundle of a differentiable manifold which are called vector fields. Most control systems are described in terms of local operating points, i.e. they are linearised about some equilibrium point and then local feedback control is applied to hold this system ‘near’ this point. (Such is the case, for example, with aircraft systems, where the operating point is called a ‘trim condition’.)
María Tomás-Rodríguez, Stephen P. Banks

### Summary, Conclusions and Prospects for Development

Introduction
In this book we have presented a theory which provides a general approach to nonlinear problems in systems theory. The method consists of writing a nonlinear system as the limit of a sequence of an approximating sequence of linear, time-varying ones and applying linear theory to each of the approximating systems. We have proved general convergence theorems and given applications to frequency-domain theory of nonlinear systems, optimal control, nonlinear sliding control and to nonlinear partial differential equations.
In this final chapter we shall show that there are many more potential applications of the method by using two illustrative examples which are now in the process of development. One is the application of the method to the problem of travelling waves in nonlinear partial differential equations and the other is to the separation theorem for nonlinear stochastic systems.
María Tomás-Rodríguez, Stephen P. Banks

### Backmatter

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