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.

Chapter Contents:

• 4.1 Formulation of the linear problem
• 4.2 Singular perturbation method
• 4.2.1 Outer expansion
• 4.2.2 Initial boundary-layer correction
• 4.2.3 Final boundary-layer correction
• 4.2.4 Asymptotic expansions for the optimal control and cost
• 4.2.5 Total series solution
• 4.3 Fixed-end-point problem
• 4.4 Nonlinear systems
• 4.4.1 Formulation of the problem
• 4.4.2 Asymptotic expansions
• 4.5 Conclusions and discussion

Inspec keywords: nonlinear control systems; optimisation; performance index; boundary-value problems; linear systems; continuous time systems; optimal control; singularly perturbed systems

Other keywords: free end point problem; linear singularly perturbed system; standard quadratic performance index; open loop optimal control; asymptotic power series expansion; costate variable; singularly perturbed TPBVP; time invariant singularly perturbed system; nonlinear continuous system; linear continuous system; boundary condition

Subjects: Optimal control