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On the Stabilization of Linear Time Invariant Systems with Positive Controls

Klein Schaarsberg, F.L.H. (2019) On the Stabilization of Linear Time Invariant Systems with Positive Controls.

Abstract:This report studies the stabilization of control systems for which the control input has a positivity constraint, that is, one or more of the control inputs is either positive or zero. Typical examples of such systems are systems for which the control can only add (for example energy or substances), but cannot extract. A classic example is that of a heating system in buildings. Radiators can only release heat into the rooms, but it cannot extract the heat when the temperature overshoots the setpoint. This report is build up out of three main parts. The first part concerns a review on mathematical research on systems with positive control, in particular for linear time invariant systems. Such systems may be represented by the state-space representation x' = Ax + Bu, with state x and control input u. Two representative papers are studied in more detail. One approaches the positive control problem from the perspective of optimal control. The other investigates positive state feedback stabilization of linear time invariant systems with at most one pair of unstable complex conjugate poles. This latter paper forms the basis for this project. The second and third part focus on linear time invariant systems in general. It extends known results to systems with more than one pair of unstable complex conjugate poles, where the positive control input is scalar. Two approaches are considered. The first approach uses Lyapunov’s stability theory as a base for a asymptotically stabilizing positive control law. Formal proofs of stability are given for stable (but not asymptotically) oscillatory systems. The feasibility of the control law for unstable oscillatory systems is investigated through simulations. The second approach concerns techniques from singular perturbations for ordinary differential equations. The viability of the application of known techniques to the positive control problem is investigated and substantiated with various simulations.
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics MSc (60348)
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