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U(s) 80 60 40 20 0 -20 -40 10-7 − G(s) G (s) Wua(s) U(s) ∆(s) Gn(s) + + Y(s) Wua(s) U(s) ∆(s) Gn(s) Fig. 3. Additive uncertainty model. + + Y(s) We(s) E(s) R(s)+ E(s) C(s) U(s) 10-6 10-5 10-4 10-3 Frequency (rad/s) 10-2 S (s) = E(s) = n R(s) 1 . (16) 1 + C(s)Gn(s) Additive Uncertainty means of the previous functions defined as it is shown in Fig. 5. As can be noticed, the new blocks correspond to the con- troller C(s) and We(s) that defines the system performance. The controller C(s) can be designed in order to get a trade off between performance and stability, which can be established by means of the || · ||∞. 1) Performance condition: The performance of the system can be established in terms of the track error between the output voltage and the desired one, as well as the settling time. These parameters can be set using the sensitivity function of a system Sn(s): Fig. 4. Second order Wua(s) function (red) of the model uncertainty. In order to obtain Wua(s), the following procedure has been done: 1) Select a certain number of plants to perform the uncer- tainty. In this work, 2000 plants have been selected. 2) Determine the frequency range where to model the uncertainty. Looking Fig. 2 the bound selected is [10−7, 10−2] rad/s. 3) Define a number of points logarithmically distributed along the bound defined in 2). 4) Define the set of plants with the frequency defined in 3). 5) Calculate the error (Eplant) of each new plant with respect to the nominal ones. 6) Obtain a function that bounds all errors of step 5). Fig. 4 shows the corresponding Wua(s) function that has been obtained following the procedure presented. C. Controller design Once the uncertainty has been modeled allowing the spec- ification of all operating conditions, next step correspond to the design of the controller. The proposed controller follows a feedback configuration, that can be easily implemented by Function Sn(s) must be bounded selecting the proper perfor- mance parameters. A typical weighting function for Sn(s) is [22]: Fig. 5. Feedback controller for the uncertainty model. Magnitude (dB) We(s) = 1 s + Mswb (17) Ms s+εwb where 1/Ms is the distance to the critical point (-1,0), and usually Ms takes a value between 1 and 2, parameters wb is the bandwidth used to place the desired poles (and therefore the settling time) and ε is the desired steady-state error. Thus, in order to obtain the desired performance, the fol- lowing condition for the sensitivity function must be fulfilled: |Sn(jω)| < 1 , ∀ω (18) |We(jω)| Taking use of the || · ||∞, it can be rewritten as: ∥We(s)Sn(s)∥∞ < 1. (19) 2) Stability condition: Once the performance condition has been formulated, next step corresponf to ensure the stability of the controlled system in closed-loop. Considering Fig. 5, the stability condition can be formulated using the || · ||∞ as: ∥Wua(s)C(s)Sn(s)∥∞ < 1. (20)PDF Image | H control of a redox flow battery overpotentials
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