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.

Chapter Contents:

• 5.1 Three-time-scale discrete systems
• 5.1.1 State-space model
• 5.1.2 Analysis
• 5.1.3 Three-time-scale property
• 5.2 Open-loop optimal control problem
• 5.3 Singular perturbation method
• 5.3.1 Outer series
• 5.3.2 Initial boundary-layer correction series
• 5.3.3 Final boundary-layer correction series
• 5.3.4 Total series solution
• 5.4 Conclusions and discussion

Inspec keywords: state-space methods; optimal control; open loop systems; singularly perturbed systems; boundary-value problems; discrete systems; series (mathematics)

Other keywords: quadratic cost function; singularly perturbed discrete systems; singularly perturbed form; open loop optimal control; three-time scale property; state-space model; two-point boundary-value problem; correction series solution; TPBVP; costate equation; boundary layer behaviour; discrete maximum principle; outer series solution; perturbation method

Subjects: Discrete control systems