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Über dieses Buch

The first German edition of this book appeared in 1972, and in Polish translation in 1976. It covered the analysis and synthesis of sampled-data systems. The second German edition of 1983 ex­ tended the scope to design, in particular design for robustness of control system properties with respect to uncertainty of plant parameters. This book is a revised translation of the second Ger­ man edition. The revisions concern primarily a new treatment of the finite effect sequences and the use of nice numerical proper­ ties of Hessenberg forms. The introduction describes examples of sampled-data systems, in particular digital controllers, and analyzes the sampler and hold; also some design aspects are introduced. Chapter 2 reviews the modelling and analysis of continuous systems. Pole shifting is formulated as an affine mapping, here some n~w material on fixing some eigenvalues or some gains in a design step is included. Chapter 3 treats the analysis of sampled-data systems by state­ space and z-transform methods. This includes sections on inter­ sampling behavior, time-delay systems, absolute stability and non­ synchronous sampling. Chapter 4 treats controllability and reach­ ability of discrete-time systems, controllability regions for con­ strained inputs and the choice of the sampling interval primarily under controllability aspects. Chapter 5 deals with observability and constructability both from the discrete and continuous plant output. Full and reduced order observers are treated as well as disturbance observers.



1. Introduction

If you are driving a car and want to know your speed, you do not have to watch the speedometer continuously. It is sufficient that you monitor the continuously indicated velocity by an occasional glance, and that is “Sampling”. Likewise the continuously changing voter opinion of political parties is sampled every few years by an election and the composition of the congress is then fixed for the following term of office. Stock market quotations are set every working day, and the temperature of a sick person is checked several times per day.
Jürgen Ackermann

2. Continuous Systems

There are several reasons to include this chapter on continuous systems into a book devoted to sampled-data systems.
Jürgen Ackermann

3. Modelling and Analysis of Sampled-Data Systems

Let the plant be a continuous system which can be described by the vector differential equation
$${\rm{\dot x}}({\rm{t}}){\rm{ = }}\underline {{\rm{Fx}}} \left( t \right) + u\left( t \right)$$
and the measurement equation
$${\rm{y}}({\rm{t}}){\rm{ = }}'\underline x \left( t \right)$$
Jürgen Ackermann

4. Controllability, Choice of Sampling Period and Pole Assignment

In this chapter the properties of controllability and reachability will be introduced for discrete systems. Also their relation with the corresponding properties of the continuous subsystem will be presented.
Jürgen Ackermann

5. Observability and Observers

In chapter 4 a state-vector feedback u = -kx was assumed for the controller structure. In most control systems not all components of the state vector are measured. Some variables may be difficult to measure; for others the sensors may be expensive or unreliable. If a Vector \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y} ' = [y_1 \,y_2 ...\,y_s ]\) of s linearly independent outputs is measured where
$$\underline y = \underline {Cx} + \underline d u, rank \underline C = s < n$$
then an output-vector feedback
$$u = \underline x y'\underline y = - \underline k y'\underline {Cx} - \underline k y'\underline {du}$$
$$u = - \frac{1}{{1 + \underline k y'\underline d }} x \underline k y'\underline {Cx}$$
can be implemented. Neglecting the scalar factor \(1/\left( {1 + \underline k y'\underline d } \right)\) the connection with state-vector feedback is
$$\underline {k'} = \underline k y'\underline C$$
or, writing in full
$${\left[ {{k_1} \ldots {k_n}} \right]^\prime } = {\left[ {{k_{y1}} \ldots {k_{ys}}} \right]^\prime }\left[ {\begin{array}{*{20}{l}} {{c_{11}} \ldots {c_{1n}}}\\ \vdots \\ {{c_{s1}} \ldots {c_{sn}}} \end{array}} \right] $$
Jürgen Ackermann

6. Control Loop Synthesis

In the chapters 1 to 5 several aspects of control system design have been discussed separately; we will now put these elements together.
Jürgen Ackermann

7. Geometric Stability Investigation and Pole Region Assignment

The most important requirement on a control system is the stability of the closed loop. In section 3.2 it was indicated that a necessary and sufficient condition for the asymptotic stability of a discrete rational system is that all eigenvalues of its system matrix have an absolute value of less than one.
Jürgen Ackermann

8. Design of Robust Control Systems

For most plants the mathematical model is not known exactly or it is too complicated for the controller design (for example it may be nonlinear). The usual procedure is to design the controller with a simplified model and nominal values of the plant parameters. An important design goal is therefore to reduce the influence of parameter uncertainty, neglected dynamics, and nonlinearity on the dynamics of the closed loop. A survey on the origin of uncertainty in modelling and model simplifications and on design of robust, adaptive and intelligent control systems is given in [85.1].
Jürgen Ackermann

9. Multivariable Systems

In the last two decades many papers and books on linear systems with several inputs and several outputs, i.e. multivariable systems, have been published. More and more abstract notations and definitions have been introduced in both frequency domain and state space descriptions and other newly developed formalisms. It is not the aim of this last chapter to review this vast amount of literature. The main motivation for this chapter is the observation that some practically important concepts can be made clear and applicable in a simple language if we restrict ourselves to the discrete-time case. This applies in particular to the theory of finite effect sequences (FES) which is developed and applied in this chapter. Therefore this topic seems to be an appropriate conclusion for a book on sampled-data systems.
Jürgen Ackermann


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