Optimal control of a bilinear system with a quadratic cost functional

Abstract

Various control systems in engineering problemsare modelled with linear differential equations where a linearcontrol is used. On the other hand, linear models are notcapable of representing many systems where the control isapplied in a multiplicative ways. These multiplicative controlsyield bilinear systems (BLS). Products of state and control takepart in BLS, which means that state and control are linearseparately but not jointly. In this paper, optimal control ofbilinear systems with a quadratic cost functional is studied. Adistributed parameter system is considered and a bilinear controlis applied to the system. The control problem is turned intoa modal control problem by way of reduced order modelling.Performance index (cost functional) is defined as a measure ofthe dynamic response and a penalty term on control energy.Pontryagins maximum principle is used to obtain the optimalcontrol function that leads to a nonlinear two-point boundaryvalue problem. Optimal control and optimal trajectory of thesystem are determined by solving this two-point boundary valueproblem using steepest descent method. The programming is donein MATLAB platform. Numerical results are given in graphicsfor showing the effectiveness and applicability of the introducedbilinear control for a parabolic distributed parameter system.