Generalized Nyquist and generalized Root-Locus diagrams are algebraic curves derived from the spectral analysis of appropriate matrix-valued functions of a complex variable. Their key properties can be comprehensively analysed in terms of a state-space model of the feedback loop which is being studied Such an analysis is essentially geometrical in nature and is concerned with the way in which certain subspaces, defined via the various operators involved, sit with respect to one another, and with ways of assigning complex frequencies to these subspaces. The current status of work on the' design of feedback systems using these results is briefly discussed, and possible future developments are noted.

A 3-input multivariable control system is developed for the regulation of the product concentration and temperature of a 2-bed exothermic catalytic reactor. Two of the inputs are the flow rate and temperature of a quench stream injected between the beds; the third input is the feed temperature. Through the use of the characteristic locus method of control system analysis, a cascade of dynamic compensators is developed by which interaction among variables is suppressed and the effects of feed concentration disturbances are con trolled. In this development, an analysis is made of the potential improvements in performance accruing from the use of internal bed temperature measurements. The system also has the feature of resetting the quench flow rate to its nominal value upon sustained system disturbance.

The steam raising plant of a British designed nuclear power station for which the installed control scheme has given rise to performance and stability problems is considered. The lecture is based on studies carried out for the plant which illustrate the way in which multivariable frequency response methods can be used to analyse and identify the source of control problems and further enable alternative control schemes, having improved performance, to be designed.

Control design requires the choice both of structure of the controller and parameters for a given structure; the first task is guided by system theory; the latter is aided by mathematical programming. For a given control structure design objectives can often be specified as satisfying a set of inequalities or minimizing a function subject to these in equalities. It is shown that many objectives can be expressed as infinite dimensional constraints (φ(z,α) ≤ 0 for all α ∈ A). Algorithms for the resultant semi-infinite programming problems are briefly outlined.

In this chapter, pole assignment is discussed. The problem of pole assignment using state feedback or output feedback is examined. An algorithm for the design of full rank minimum degree output-feedback compensators is presented and results in a unification of previously obtained results. Pole assignment is simply concerned with moving the poles (or eigenvalues) of a given time-invariant linear system to a specified set of locations in the s-plane (subject to complex pairing) by means of state or output feedback. The state-feedback approach is well established, however the output-feedback case is still being explored as a research area.

As a result of the enormous impact of microprocessors, electronic engineers, with sometimes only a cursory back ground in control theory, are being involved in direct digital-control (D.D.C.) system design. There appears to be a real need for an easily understood and simply imple mented comprehensive design technique for single-input d.d.c. systems. The proposed design technique provides, first of all, a simple calculation that ensures that the data sampling rate is consistent with the control system's accuracy specification or the fatigue life of its actuators. Pulsed transfer-function design for a plant controller is based on two simple rules and a few standard frequency response curves, which are easily computed once and for all time. Structural resonances are eliminated by digital notch filters, the pole-zero locations of which are directly re lated to the frequency and bandwidth of an oscillatory mode; this is exactly as with analogue networks. In addition a computationally simple formula gives an upper bound on the amplitude of the control error (deviation) component due to multiplicative rounding effects in the digital computer; this thereby enables the selection of a suitable machine wordlength or machine. A distinct advantage of the proposed design technique is that its implementation does not necess arily involve a complex computer-aided-design facility.

Much of control theory is concerned with systems which are modelled by ordinary differential equations, so-called lumped parameter systems. In this chapter it is shown how the concepts of controllability, observability, optimal control and estimation may be investigated for system models based upon partial differential equations, so-called distributed parameter systems. Such a system with a single input and a single output is used to illustrate the theory presented.