This paper presents the stabilization approach for linear time-varying continuous-time systems using proportional-derivative (PD) state feedback control. The solvability conditions for the problem are considered. The general analytical expressions for the PD controller gains are derived, which describe the available degrees of freedom offered by PD state feedback. The non-uniqueness of the controller gains is utilized to obtain closed-loop systems with small gain elements. Two numerical examples are introduced to demonstrate the effectiveness of the proposed approach.
AbdelazizTHS (2010) Optimal control using derivative feedback for linear systems. Journal of Systems and Control Engineering224(2): 185–202.
2.
AbdelazizTHS (2012) Parametric eigenstructure assignment using state-derivative feedback for linear systems. Journal of Vibration & Control18(12): 1809–1827.
3.
AbdelazizTHS (2015a) Pole placement for single-input linear system by proportional-derivative state feedback. ASME Journal Dynamic Systems, Measurement & Control137(4): 041015–1–10.
4.
AbdelazizTHS (2015b) Stabilization of single-input LTI systems by proportional-derivative feedback. Asian Journal of Control17(6): 2165–2174.
5.
AbdelazizTHSValašekM (2005) Direct algorithm for pole placement by state-derivative feedback for multi-input linear systems – nonsingular case. Kybernetika41(5): 637–660.
6.
CaiGHuCDuanG (2011) Eigenstructure assignment for linear parameter-varying systems with applications, Mathematical & Computer Modelling53(5): 861–870.
7.
CaverlyRJZlotnikDEBridgemanLJForbesJR (2014) Saturated proportional derivative control of flexible-joint manipulators. Robotics & Computer-Integrated Manufacturing30(6): 658–666.
8.
D’AngeloH (1970) Linear Time-Varying Systems: Analysis and Synthesis. Boston: Allyn & Bacon.
9.
GolnaraghiFKuoBC (2010) Automatic Control Systems. 9th Edn. New York, USA: John Wiley & Sons. Inc.
10.
Gonzalez-VazquezSMoreno-ValenzuelaJ (2013) Time-scale separation of a class of robust PD-type tracking controllers for robot manipulators. ISA Transactions52(3): 418–428.
11.
HuQXiaoB (2012) Intelligent proportional-derivative control for flexible spacecraft attitude stabilization with unknown input saturation. Aerospace Science & Technology23(1): 63–74.
12.
KamenEW (1995) Fundamentals of linear time-varying systems. The Control Handbook. CRC Press, pp. 451–468.
13.
KemmotsuMAkutsuWAwatsuLMutohY (2015) Trajectory tracking control of manipulator using time-varying observer based pole placement technique. In: The 54th Annual Conference Society of Instrument & Control Engineers of Japan (SICE), 28–30 July 2015, pp. 771–777.
14.
LeeHCChoiJW (2005) Ackermann-like eigenvalue assignment formulae for linear time-varying systems. IET Control Theory & Applications152(4): 427–434.
15.
LinCYChangCM (2012) Hybrid proportional derivative/repetitive control for active vibration control of smart piezoelectric structures. Journal of Vibration & Control19(7): 992–1003.
16.
LuoJZouXCaoC (2012) Eigenvalue assignment for linear time-varying systems with disturbances. IET Control Theory & Applications6(3): 365–374.
17.
MutohY (2011) A new design procedure of the pole-placement and the state observer for linear time-varying discrete systems. In: Informatics in Control, Automation & Robotics, Lecture Notes in Electrical Engineering, vol. 89, pp. 321–333. Springer International Publishing.
18.
MutohYKemmotsuMAwatsuL (2016) Pole placement of linear time-varying discrete systems and its application to trajectory tracking control of nonlinear systems. In: Informatics in Control, Automation & Robotics, Lecture Notes in Electrical Engineering, vol. 383, pp. 337–354. Springer International Publishing.
19.
NguyenCC (1986) Canonical transformation for a class of time-varying multivariable systems. International Journal of Control43(4): 1061–1074.
20.
NguyenCC (1987) Arbitrary eigenvalue assignments for linear time-varying multivariable control systems. International Journal of Control45(3): 1051–1057.
21.
O’BrienRTIglesiasPA (2001) On the poles and zeros of linear, time-varying systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory & Applications48(5): 565–577.
22.
ShiehLSGanesanSNavarroJM (1987) Transformations of a class of time-varying multivariable control systems to block companion forms. Computers & Mathematics with Applications14(6): 471–477.
23.
SilvermanLMAndersonBDO (1968) Controllability, observability and stability of linear systems. SIAM Journal on Control6(1): 121–130.
24.
SuYZhengC (2014) Velocity-free saturated PD controller for asymptotic stabilization of spacecraft. Aerospace Science & Technology39(1): 6–12.
25.
SunMYangRWangZChenZ (2013) Stability margin-based PD attitude control tuning for unstable flight vehicle. International Journal of System Science44(2): 240–251.
26.
TsakalisKSIoannouPA (1993) Linear Time-Varying Systems: Control and Adaptation, Advances in Industrial Control. Englewood Cliffs, NJ: Prentice-Hall.
27.
ValášekMOlgacN (1995) Efficient eigenvalue assignments for general linear MIMO systems. Automatica31(11): 1605–1617.
28.
WolovichWA (1968) On the stabilization of controllable systems. IEEE Transactions on Automatic Control13(5): 569–572.