Tracking Error/Control Effort Trade-Off in the Presence of Disturbance

Abstract A bi-objective control problem for a linear system in the presence of a disturbance is considered. The first cost functional is a generalized tracking error defined as a Lebesgue-Stieltjes discrepancy integral comprising both continuous and discrete discrepancies. The second cost is the control effort. The relaxed Pareto control, which guarantees a balance between two costs, is defined for an unknown disturbance and controls from $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -bounded sets. The solution is constructed based on the auxiliary generalized linear-quadratic differential game formulated in open-loop controls. It is shown that by a proper choice of the cost penalty coefficients, the game-optimal control strategy solves the bi-objective control problem. As a by-product, a novel solvability condition for the game in open-loop controls is derived. Illustrative examples are presented.