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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-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.</p>

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

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  • 发表日期：

    2026年2月13日

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  • Vladimir Turetsky

摘要

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 \(L_2\) L 2 -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.