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4.5.2 Theory 4 –

4.5.2 Theory 4 – FUNDAMENTALS The first step in control theory is to build a physical model of the control path and then to describe it mathematically. For simplifying the physical model of an electromechan- ical system, the mechanical and electrical part shall be considered separately. Their models are sketched in figure 4.10, whereof the applicable formulas can be deduced. x k m c Figure 4.10: Models of the FH as a spring mass system (left) and the VCA as a RL circuit (right). The mathematical model of the mechanics and its Laplace transform follow to UR RC m¨x + k ˙x + cx = FV CA(x, t) (4.22) ¢ s 2 mX(s) + skX(s) + cX(s) = FV CA(s). (4.23) Thereof, the transfer function for the mechanical part GF H(s) can be described as GF H(s) = X(s) FV CA(s) = IC UC UL 1 s2 . (4.24) m + sk + c Analogously follows for the mathematical model of the electric circuit and its Laplace transform the equations uR + uL − uC = 0 (4.25) RCiC + L diC dt = uC (4.26) ¢ RCIC(s) + sLIC(s) = UC(s). (4.27) L 28

4 – FUNDAMENTALS Thus, the transfer function of the electromagnetic part GV CA(s) results in GV CA(s) = IC(s) UC(s) = 1 . (4.28) sL + RC Since the VCA drives the FH, the mechancial part can be considered as connected in series to the electromagnetical part. Thus, a multiplication of the transfer functions results in the frequency domain. As interconnecting part, the force factor KF derived in (4.16) must be taken into account, whereof its Laplace transformation can be denoted by KF figure 4.11. KF (s) = FV CA(s)/IC(s). The extracted control path is shown in Note that this description must be considered as a first approximation, what corre- sponds to the general path of modeling in control theory. Obviously, no distrubance variable appears in figure 4.11. But when the MPO once arrives in Mercury’s orbit and the MSTS starts working, no disturbances should occur anymore. UC GVCA(s) IC FVCA x KF(s) GFH(s) Figure 4.11: Model of the control path consisting of a serial connection of the transfer functions GV CA(s), KF (s) and GF H(s). 29

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