This chapter discusses a number of methods available for designing feedback controllers for multivariable systems. A system is a mechanism or a collection of entities, physical or otherwise, that transforms a set of clearly distinguishable quantities called inputs to a set of outputs. The analysis problem includes such things as stability and evaluation of certain 'structural' properties of the system. When the number of inputs and/or outputs is more than one, we call it a multivariable system.

This chapter presents modal control. The conditions for modal controllability are identical to those of state controllability. A number of algorithms are developed for pole placement in single and multi-input systems. The extra freedom available in the multi-input case was shown to be available for eigenvector placement with certain restrictions.

This chapter shows that an estimate of the states x can be obtained using an asymptotic observer whenever the system is observable. We also showed that an observer of order (n-m) is adequate to estimate all the states. We also defined a minimal order observer to estimate linear functions of states. When only a single (scalar) linear function of states is desired, an observer of order (r-1), where r is the observability index of the system, is possible.

In this chapter we defined return difference and return ratio matrices and showed that the closed loop stability is related to det N (if all the open loop poles are stable). We also found methods for checking closed loop stability of multivariable systems. By defining a system matrix P(s), we formulated various ideas related to multivariable poles and zeros. We shall develop and use these ideas in controller design in the next three chapters. Note that the formulation is such that a dynamic compensator is anticipated and methods for accommodating it are already in place.

In this chapter we have introduced the concept of characteristic transfer functions and seen that these may not be ratios of polynomials in s. No difficulty, however, is encountered in evaluating these functions along s =jω to find the characteristic loci. It has further been shown how the characteristic loci can be used to determine the stability of a multivariable system and how one approach to the design of a controller is basically an exercise in shaping characteristic loci.

This section summarised certain basic results in stochastic control theory. Using the basic results on estimation theory in Section 11.3, a number of identification algorithms can be derived. Modification and extension of these results will lead to self tuning control. Some of the other topics in stochastic control not considered here are systems with stochastic parameters and nonlinear feedback systems with stochastic disturbances.