5.7: Chapter References

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8. Z. Lou, Q. Deng, S. Wen, Y. Zhang, M. Yu, M. Sun, and G. Zhu, “Progress in camless variable valve actuation with two-spring pendulum and electrohydraulic latching,” SAE Int. J. of Engines, SAE 2013-01-0590, vol. 6, no. 1, 2013, doi:10.4271/2013-01-0590.
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19. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Princeton, NJ, USA: Princeton Univ. Press, 1973, pp.72–73, ISBN: 978-0-6911-7779-3.
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23. K. S. Peterson, A. G. Stefanopoulou, and J. Freudenberg, “Current versus flux in the control of electromechanical valve actuators,” in Proc. 2005 Amer. Control Conf., Portland, OR, USA, Jun. 8-10, 2005.
24. J. Ma, G. Zhu, and H. Schock, “A dynamic model of an electropneumatic valve actuator for internal combustion engines,” ASME J. of Dyn. Syst., Meas., and Control, vol. 132, no. 2, p. 021007, Mar. 2010, doi: 10.1115/1.4000816.
25. Y. Wang, A. Stefanopoulou, M. Haghgooie, I. Kolmanovsky, and M. Hammoud, “Modeling of an electromechanical valve actuator for a camless engine,” in Proc. 2000 (5th) Int. Symp. of Adv. Veh. Control, no. 93, Ann Arbor, MI, Aug. 22–24, 2000.
26. C. Tai and T. Tsao, “Control of an electromechanical camless valve actuator,” in Proc. 2002 Amer. Control Conf., Anchorage, AK, USA, May 8–10, 2002.
27. Y. Wang, T. Megli, M. Haghgooie, K. Peterson, and A. Stefanopoulou, “Modeling and control of electromechanical valve actuator,” SAE 2002 Trans. J. of Engines, SAE 2002-01-1106, 2002.
28. S. Butzmann, J. Melbert, and A. Kock, “Sensor-less control of electromagnetic actuators for variable valve train,” SAE Technical Paper 2000-01-1225, 2000.
29. M. Montanari, F. Ronchi, C. Rossi, and A. Tonielli, “Control of a camless engine electromechanical actuator: Position reconstruction and dynamic performance analysis,” IEEE Trans. on Ind. Electron., vol. 51, no. 2, pp. 299–311, Apr. 2004.
30. K. Peterson, A. Stefanopoulou, Y. Wang, and T. Megli, “Output observer-based feedback for soft landing of electromechanical camless valvetrain actuator,” in Proc. 2002 Amer. Control Conf., Anchorage, AK, USA, May 8–10, 2002, pp. 1413–1418.
31. K. Peterson and A. Stefanopoulou, “Rendering the electromechanical valve actuator globally asymptotically stable,” in Proc. 2003 IEEE Conf. on Decis. Control, Maui, HI, USA, Dec. 2003, pp. 1753–1758.
32. L. Mianzo, “A unified Hamiltonian framework for \({\mathcal{H}}_2\) and \({\mathcal{H}}_{\infty }\) preview control algorithms with application to a variable valve timing engine,” Ph.D. dissertation, University of Michigan, MI, USA, 2002.
33. Z. Lou, “Camless variable valve actuation designs with two-spring pendulum and electrohydraulic latching,” SAE Tech. Paper 2007-01-1295, 2007.
34. J. P. Watson and R. J. Wakeman, “Simulation of a pneumatic valve actuation system for internal combustion engine,” SAE Technical Paper 2005-01- 0771, 2005.
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37. R. Zope, J. Mohammadpour, K. Grigoriadis, and M. Franchek, “Robust fueling strategy for an SI engine modeled as a linear parameter-varying time-delayed system,” in Proc. 2010 Amer. Control Conf., Baltimore, MD, USA, Jun. 30–Jul. 2, 2010.
38. J. Partington, “Some frequency-domain approaches to the model reduction of delay systems,” Annu. Rev. of Control, vol. 28, no. 1, pp. 65–73, Apr. 2004.
39. F. White, Fluid Mechanics, 7th ed., New York, NY, USA: McGraw-Hill Education, 2010, ch. 6, ISBN-13: 978-0-0735-2934-9.
40. A. White, Z. Ren, G. Zhu, and J. Choi, “Mixed \({\mathcal{H}}_2\)/\({\mathcal{H}}_{\infty }\) observer-based LPV control of a hydraulic engine cam phasing actuator," IEEE Trans. on Control Syst. Tech., vol. 21, no. 1, pp. 229–238, Jan. 2013.
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43. F. Wu and M. G. Karolos, “LPV systems with parameter-varying time delays: analysis and control,” Automatica, vol. 37, no. 2, pp. 221–229, Feb. 2001.
44. A. White, G. Zhu, and J. Choi, Linear Parameter Varying Control for Engineering Applications, London, UK: Springer-Verlag, 2013, ch. 2, ISBN-13: 978-1-4471-5039-8.
45. H. Li, Y. Huang, G. Zhu, and L. Zheng, “Profile tracking for an electrohydraulic variable valve actuator using receding horizon LQT,” IEEE/ASME Trans. on Mechatron., vol. 24, no. 1, pp. 338–349, Feb. 2019, doi: 10.1109/TMECH.2019.2892038.
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Solutions to Practice Problems

1. \[SA=\dfrac{360{}^\circ }{2\times 2\times 6}=15{}^\circ\]
   1.  

   The solution for part b) is as below.

   \[d_l=G_{r2l}\times θ_r=0.007056\times 900=0.635\ mm\]

   1. The control sequence for the desired movement is shown in the following table.

   Moving Steps	Energized Solenoids	Resulting Polarities
   Step #0	Solenoid #5 & #7	N (#5) & S (#7)
   Step #1	Solenoid #4 & #6	N (#4) & S (#6)
   Step #2	Solenoid #3 & #5	N (#3) & S (#5)
   Step #3	Solenoid #2 & #4	N (#2) & S (#4)

   1.  
   2. The three-torque curves plus the motor output torque curve can be plotted as below.

   From the red curve, the first peak occurs at \(θ_{max}=π/6\) and the first valley at \(θ_{max}=π/3\). The maximum and minimal torque can be calculated as follows.

   \[T_{max}=2{\mathrm{sin} \dfrac{π}{6}\ }+2{\mathrm{sin} \left(\dfrac{π}{6}+\dfrac{π}{3}\right)+\ }2{\mathrm{sin} \left(\dfrac{π}{6}+\dfrac{2π}{3}\right)\ }=1+2+1=4\]

   \[T_{min}=2{\mathrm{sin} \dfrac{π}{3}\ }+2{\mathrm{sin} \left(\dfrac{π}{3}+\dfrac{π}{3}\right)+\ }2{\mathrm{sin} \left(\dfrac{π}{3}+\dfrac{2π}{3}\right)\ }=\sqrt{3}+\sqrt{3}+0=2\sqrt{3}\]

   and the percentage motor torque ripple is shown below.

   \[R_{\%}=\dfrac{T_{max}-T_{min}}{T_{max}}\times 100=\dfrac{4-2\sqrt{3}}{4}\times 100=13.4\%\]

   This indicates that as the number of poles increases, the torque ripple reduces. For most DC motors in production, the number of poles is often greater than or equal to six.

   1.  

   The no-load speed can be calculated using the equation below.

   \[ω_{no-load}=\dfrac{e_a}{K_m}=\dfrac{10}{0.5}=20\ rad/s\]

   Therefore, the calculated stall torque is \(50\ Nm\) and the calculated no-load speed \(20\ rad/s\).

   1.  

   The no-load speed of the gearbox output shaft can be calculated using the equation below.

   \[\dfrac{ω}{ω_o}=0.5\ \ \to \ ω_o=\dfrac{ω}{0.5}=\dfrac{8}{0.5}=16\ rad/s\]

   Therefore, the calculated gear ratio \(η=0.5\) and the calculated associated no-load speed \(16\ rad/s\).

   1.  

   The time constant can be calculated by the following equation.

   \[T=\dfrac{JR_a}{R_ab_m+K_τK_m}=\dfrac{0.4\times 0.1}{0.1\times 2+{0.5}^2}=0.08889\ s\]

   Therefore, the calculated \(K=1.1111\) \((rad/s)/A\) and the calculated \(T=0.08889\) second. The steady state speed in this case can be calculated by the following equation.

   \[ω_{ss}=K\times e_a=1.1111\times 5=5.555\ rad/s\]

   1.  

   Therefore, the reference voltage \(v_{ref}\) is \(1.5\ V\).

   1.  

   With the 20% duty-cycle, the averaged motor current is shown below.

   \[i_a=i_{a-on}\times 20\%=60\times 0.2=12\ A\]

   1. The other braking regeneration operational mode, when the motor rotates in the anticlockwise direction, can be implemented by turning on only the upper-right switch, as illustrated below.
   2. Because the PWM conversion block has a saturation effect to the analog control signal generated by the PID controller, the controller shown in Figure 5.29 is the fastest one.
   3. The answer is c).
   4. For the equivalent circuits of DC motor and EMVA (lower solenoid), a) the lower solenoid inductance, \(L_{low}\left(i_{low},y\right)\), is related to DC-motor armature inductance \(L_a\); b) the lower solenoid back-EMF coefficient \(K_{low}\left(i_{low},y\right)\) is related to the back-EMF coefficient of DC-motor \(K_m\), and c) EMVA linear motion speed \(\dot{y}\) is related to the DC-motor rotational speed \(\dot{θ}.\)
   1.  

   Note that the torque generated by the DC motor is a linear function of armature current \(i_a\) (see DC-motor modeling in Chapter 2). The magnetic force generated by the EMVA lower solenoid is a quadratic function of its current \(i_{low}\); see equation \ref{5.41}.

   1. To delay the valve closing starting time from \(15\ ms\) to \(20\ ms\), we need to adjust the upper solenoid control input timing shown in the following table. Note that the upper solenoid time input value changes in the bottom line of the table.

   Output voltage	\(380\)	\(380\)	12\(0\)	\(12\)	\(12\)	0	0	\(0\)
   Input time (\(ms\))	\(0\)	\(1.5\)	\(2.2\)	\(6\)	\(19.95\)	25	25	\(40\)
   Output voltage	\(0\)	\(0\)	\(380\)	\(380\)	\(120\)	12	12	\(12\)
   Input time (\(ms\))	\(0\)	\(14.95\)	\(20\)	21.2	\(22.5\)	26	30	40

   1. The corresponding displacement, speed, and acceleration responses (similar to Figure 5.39) are plotted below.
   2. The system matrix \(A\) described by equation \ref{5.63} is affine because there is a constant term in equation \ref{5.64}. Control input matrix \(B\) defined by equation \ref{5.64} is linear because there is no constant part in matrix \(B\).
   3. The MATLAB/Simulink diagram of the LPV EHVA system is shown below.
   4. Part a) is true. Part b) is false; the correct statement is: The speed of an induction motor is slightly below its stator current frequency.
   1. The pseudo inverse of \(\ T_{clark}\) can be calculated by conducting SVD first as below for \(\ T_{clark}\).

   \[T_{clark}=U\cdot S\cdot V^*=\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0.8165 & 0 \end{array} \right]\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -0.707 & 0.707 \\ 0 & 0.707 & 0.707 \end{array} \right]\]

   and

   \[T^+_{clark}=\left[ \begin{array}{cc} 1 & 0 \\ 0 & -0.707 \\ 0 & 0.707 \end{array} \right]\left[ \begin{array}{cc} 1 & 0 \\ 0 & {0.8165}^{-1} \end{array} \right]\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0.866 \\ 0 & -0.866 \end{array} \right]=\left[ \begin{array}{cc} 1 & 0 \\ 0 & \dfrac{\sqrt{3}}{2} \\ 0 & -\dfrac{\sqrt{3}}{2} \end{array} \right]\]

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