# Stabilization of the Magnetic Levitation System
## 4 Multiple Real Dominant Pole Controller Design
### PD Controller Design by the Triple Real Dominant Pole Method  
The considered control structure is shown in Figure 5. In this case, the stabilizing PD controller  
\[C_{0}(s)=K_{p}+K_{d}s \tag{17}\]  
will be combined with a second order low-pass filter  
\[Q_{2}(s)=\frac{1}{\left(T_{f}s+1\right)^{2}}. \tag{18}\]  
Then  
\[C(s)=C_{0}(s)Q_{2}(s) \tag{19}\]  
Figure 4: Output and control signal of the controller based on the plant model (2), stabilizing controller (3)–(6), and the feedforward PI controller (13).  
In the controller tuning, its two time constants \(T_{f}\) will be considered by the modified half-rule method [33; 42], as an amendment to the total dead-time by an equivalent filter dead-time \(T_{e}\)  
\[T_{d}=T_{m}+T_{e};\ T_{e}=2T_{f}/2=T_{f}. \tag{20}\]  
Thus, it can be omitted from further derivations. The control structure segments with the transfer functions \(C_{1}(s)\), \(C_{2}(s)\), and \(C_{3}(s)\) will be omitted by choosing \(C_{1}(s)=1\) and \(C_{2}(s)=C_{3}(s)=0\).  
The PD control combined with a DIPDT plant yields a closed loop transfer function  
\[F_{yy}(s)=\frac{Y(s)}{W(s)}=\frac{K_{s}(K_{d}s+K_{p})}{e^{T_{d}s}s^{2}+K_{s}(K_ {p}+K_{d}s)} \tag{21}\]  
As shown, e.g., in [31; 32], for the characteristic polynomial  
\[A_{PD}(s)=e^{T_{d}s}s^{2}+K_{s}(K_{p}+K_{d}s), \tag{22}\]  
a triple-real-dominant-pole (TRDP) \(s_{o}\) corresponds to a tuning, satisfying conditions:  
\[\left[A_{PD}(s);\ \frac{dA_{PD}(s)}{ds};\ \frac{dA_{PD}^{2}(s)}{ds^{2}}\right]_ {s=s_{o}}=0 \tag{23}\]  
From  
\[\frac{dA_{PD}^{2}(s)}{ds^{2}}=(T_{d}^{2}s^{2}+4T_{d}s+2)e^{T_{d}s}=0 \tag{24}\]  
follows the TRDP as  
\[s_{o}=E/T_{d};\ E=\sqrt{2}-2 \tag{25}\]  
where appropriate, the triple closed-loop time constant  
\[T_{o}=-1/s_{o}=1.707T_{d} \tag{26}\]  
can be used to characterize the speed of the responses. The corresponding optimal PD controller tuning can then be specified by the gains  
\[K_{p0}=\frac{(10E+6)e^{E}}{K_{s}T_{d}^{2}}=\frac{0.079}{K_{s}T_{d}^{2}};\ K_{d0}=-\frac{2(E+1)e^{E}}{K_{s}T_{d}}= \frac{0.461}{K_{s}T_{d}} \tag{27}\]  
To compensate the dominant input disturbance caused by the gravitation force, and leading to a significant steady-state error, a simple static feedforward can be used based on the inverse IOSSCH, given as \(u_{ff}=f^{-1}(r)\), where \(y_{ss}=f(u_{ss})\) is IOSSCH. In the simplest case, it may be approximated by the slope \(K_{1}\) and the offset \(K_{2}\). These values can be found  
Figure 5: General control structure for MRDP design.  
after approximating IOSSCH by straight-line dependence and determining its slope (\(K_{1}\)) and y-intercept (\(K_{2}\)).  
Another alternative is to use static feedforward via inversion of the measured IOSSCH approximated by a lookup table in MATLAB/Simulink. In such a case, in the control structure in Figure 5, the block \(K_{2}\) is omitted and the block \(K_{1}\) replaced by the inverse IOSSCH (see Figure 6).  
The corresponding step responses in Figures 7 and 8 show much faster responses than with the first method. The steady-state control error can be eliminated by a more accurate inverse IOSSCH approximation for lower reference signal values, extended possibly by gain-scheduling, modifying the \(K_{m}\) value, depending on the reference setpoint \(r\). However, when the circuit properties change (e.g., by increased coil temperature), the used control structure is not able to change its properties, and a permanent error may reappear again.  
Figure 6: Modified control structure with lookup table.  
Figure 7: PD controller with IOSSCH, approximated by two linear segments with \(K_{1}=0.65\) and \(K_{2}=1\).