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in Section 3.2) and using the results of [14] (details and discussion therein), provided a transfor- mation V exists2 such that −1Aj−BjKxl −BjKσl max V V (42) j,l∈{1,...,N } −1 Sζ:= ζ:|V ζ|≤(I−Λ) Λ := is a Schur matrix, then the trajectories of the closed-loop system (41) are bounded and the set −τCj 1 system can then be suitably bounded as 0 τr |ζ| ≤ ζ := |V |(I − Λ) −1 −1Ej−BjKwl 0w −1 −1Ej−BjKwl 0w max V , (43) j,l∈{1,...,N } is an attractive invariant set for the closed-loop system dynamics, where |w| ≤ w ̄, |r| ≤ r ̄. The max V . (44) j,l∈{1,...,N } 0 τr The existence of the transformation V such that the matrix Λ (42) is Schur, ensures closed-loop system stability (in terms of boundedness) of the overall system using the tracking controller with integral action (25) or (39). Note that tighter invariant sets for the system dynamics, where 125 necessary (e.g. when used in fault detection mechanisms), can be obtained by considering a “central system” (see, e.g., [16], which considers the constant offset of the reference, r, as the centre of its containing set). Furthermore, the above stability analysis can also be extended to consider the tracking error dynamics or when additional bounds are known with respect to measurement and process noise. 130 4. Implementation Results In this section, the simulation and experimental results of the proposed LPV based control scheme (see Section 3), are presented. Details of the system parameters are given in Table 1. c 2The existence of a transformation matrix, say Vc with Λc = V−1AcVc being Schur (where Ac denotes the closed-loop system), implies the existence of a common Lyapunov function (see [15]) and hence the stability of the closed-loop system. 15PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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