This book presents the twin topics of singular perturbation methods and time scale analysis to problems in systems and control. The heart of the book is the singularly perturbed optimal control systems, which are notorious for demanding excessive computational costs. The book addresses both continuous control systems (described by differential equations) and discrete control systems (characterised by difference equations).
Inspec keywords: open loop systems; continuous systems; optimal control; discrete systems; closed loop systems
Other keywords: open loop optimal control; singular perturbation method; discrete systems; continuous system; closed loop optimal control; time-scale analysis
Subjects: Optimal control; Control system analysis and synthesis methods; Discrete control systems
A fundamental problem in the theory of systems and control is the mathematical modelling of a physical system. The realistic representation of many systems calls for high-order dynamic equations. The presence of some 'parasitic' parameters such as small time constants, masses, moments of inertia, resistances, inductances and capacitances is often the cause for the increased order and 'stiffness' of these systems. The stiffness, attributed to the simultaneous occurrence of 'slow' and 'fast' phenomena, gives rise to time scales. The systems in which the suppression of a small parameter is responsible for the degeneration of dimension are labelled as 'singularly perturbed' systems, which are a special representation of the general class of time-scale systems.
In this chapter we present the singular perturbation method for continuous and discrete control systems. The boundary-layer method is also discussed where the approximate solution is given by the outer series and a boundary-layer correction which is equivalent to the difference between the inner and inter mediate series. The boundary-value problem is treated and the structural properties-stability, controllability and observability-are discussed. We conclude by bringing out various characteristic features of singular perturbations in differential and difference equations describing continuous and discrete control systems, respectively.
In this chapter, we focus our attention on the time-scale analysis of continuous and discrete systems. Section 3.1 describes a block diagonalisation procedure to decouple a continuous system having the two-time property into a slow and a fast subsystem. The aspects of permutation and scaling are discussed in Section 3.2. Section 3.3 provides an interesting fact that singularly perturbed continuous systems can be viewed as two-time-scale systems. The feedback design for eigenvalue placement is the content of Section 3.4. Sections 3.5-3.7 discuss the above aspects of block diagonalisation and feedback design with reference to discrete systems. The last Section summarises the Chapter with a brief discussion.
In this chapter, our main intention is to describe the singluar perturbation method in order to obtain asymptotic power-series expansions for the singularly perturbed TPBVP arising in the open-loop optimal control of linear and nonlinear continuous systems. Thus in Section 4.1 we consider a linear, time-invariant singularly perturbed system along with minimisation of a standard quadratic performance index as a free-end-point problem. Using the necessary conditions for optimisation, we arrive at the singularly perturbed TPBVP in terms of the state and co-state variables with appropriate boundary conditions.
In this chapter, we present a perturbation method for the TPBVP arising in the open-loop optimal control of singularly perturbed discrete systems. A state-space model with a three-time-scale property exhibiting boundary layer behaviour at the initial and final points is formulated in Section 5.1. The solution of the model is obtained as the sum of an outer series solution and two correction series solutions for the initial and final boundary layer. In Section 5.2, the optimal control problem with a quadratic cost function is then considered. Using the discrete maximum principle, the state and co state equations are obtained and cast in the singularly perturbed form which exhibits the three-time-scale property. In Section 5.3, a method is described to solve the resulting two-point boundary-value problem.
In this chapter, we intend to use Vasileva's method (Vasileva, 1963; Wasow, 1965) where the approximate solution is made up of an outer series, inner series and intermediate series. It should be noted that, as shown in Chapter 2, the boundary-layer correction is equivalent to the difference between the inner and intermediate series. However, in working out a large number of control problems by Vasileva's method and the boundary-layer method (Naidu, 1977), it has been found that the former has computational advantages, especially in the case of nonlinear equations as will be apparent at the end of this Chapter. Thus, the contents of this Chapter depend mainly on results from Naidu (1977), Naidu and Rajagopalan (1980) and Rajagopalan and Naidu (19806).
In this chapter, first in Section 7.1, a method is described to analyse the singularly perturbed nonlinear difference equations for initial and boundary value problems. The approximate solution is obtained in the form of an outer series and a correction series. It is seen that considerable care has to be taken in formulating the equations for the boundary-layer correction series in the case of nonlinear equations. Then, in Section 7.2, the closed-loop optimal control problem is formulated, resulting in the singularly perturbed nonlinear matrix Riccati difference equation. It is seen that the degeneration (the process of suppressing a small parameter) affects some of the final conditions of the Riccati equation. In Section 7.3, a method is given to obtain approximate solutions in terms of an outer series and a terminal boundary-layer correction series. A method is also discussed in Section 7.4 for the important case of the steady-state solution of the matrix Riccati equation. The time-scale analysis of the regulator problem is also given. It is found that these methods, with the special feature of order reduction, offer considerable computational simpli city in evaluating the inverse of a matrix associated with the solution of the Riccati equation. Examples are given to illustrate these methods.
