Continuous Time Controller Design
State space analysis of systems. Modal control. Quadratic optimal control. Design of observers. Other selected design methods in the state space. Frequency domain analysis of multivariable systems. The Inverse Nyquist array method. The characteristic locus method. Frequency domain design by factorisation methods. Selected stochastic problems.
Inspec keywords: state-space methods; stochastic systems; continuous time systems; frequency-domain analysis; multivariable control systems; linear systems; control system synthesis
Other keywords: characteristic locus design; state space design methods; frequency domain design methods; linear multivariate systems; control engineering; graduate courses; stochastic control problems; control systems; continuous time controllers design
Subjects: Control system analysis and synthesis methods; Time-varying control systems; Mathematical analysis; Multivariable control systems
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.
The state space representation of dynamic systems is a representation via a first order matrix differential equation. We shall introduce this by means of some examples.
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.
In this chapter we have introduced and solved the linear quadratic optimal control problem. The infinite time problem is solved by solving a matrix quadratic equation called the Riccati equation. The truly surprising thing is that the optimal control turned out to be linear state feedback control. One of the main reasons for this is that the control is 'independent' of xο. For a different performance index, the optimal control may have turned out to be a non-feedback control or nonlinear feedback control.
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.
We have seen that a linear state feedback controller can be designed to place the closed loop poles in specified locations or to minimise a quadratic performance index. Researchers, over the last two decades, have also looked into the question of: 'What more can be accomplished with linear state feedback?'. For example in Section 3.7.1 we found that, in the multi-input case, eigenvectors with certain restrictions can also be specified along with the eigenvalues, and a linear state feedback controller can be designed to accomplish this. In this chapter we first consider two design schemes, employing 'essentially' state feedback, to meet a prescribed specification: in the first scheme we design a 'decoupling controller' and in the second 'a model following' controller. The last part of the chapter is concerned with the fundamental problem of designing a controller to follow a prescribed reference at the same time as rejecting a disturbance.
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.
The inverse Nyquist array method (INA method) is one of the earliest frequency domain methods developed for multivariable systems design. The basic philosophy of this method is to approach multivariable system design in two stages. In the first stage, the m-input/m -output multivariable system is 'decomposed' into m SISO systems by a preliminary controller. In the second stage a suitable controller is designed for each of the m 'more or less individual systems'. The decoupling controller of Section 5.2, in fact, at tempted to accomplish this by decoupling the multivariable system to m SISO systems via state variable feedback.
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.
In this chapter we have discussed a design technique for multivariable feedback systems based on coprime factorisation of G(s) by way of matrices defined over a suitable ring. Most of the state space design results can be rederived using this approach. Using the theoretical results developed in Sections 10.2 -10.5, one should be able to do this without much difficulty.
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.
