This article attempts to clarify the question of what the essential difference between the optimal and “clipped-optimal” control is that the Hamilton–Jacobi–Bellman or Hamilton–Jacobi–Isaacs partial differential equation is linear or nonlinear. An adaptive optimal control based on policy iteration to the constrained semi-active vehicle suspension system with a magnetorheological damper is presented. The problem of improving the optimal performance of semi-active control suspension system is converted to L2-gain optimal control problem. A two-player policy iteration scheme is employed to solve the Hamilton–Jacobi–Isaacs equation by use of neural networks to approximate optimal policies and value functions in the admissible global region. Simulation results demonstrate that the adaptive optimal controlled semi-active vehicle suspension system outperforms greater than that of the clipped-LQR controlled.
Abu-KhalafMLewisFL (2005) Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach. Automatica41(5): 779–791.
2.
Abu-KhalafMLewisFLHuangJ (2006) Policy iterations on the Hamilton–Jacobi–Isaacs equation for H∞ state feedback control with input saturation. IEEE Transactions on Automatic Control51(12): 1989–1995.
3.
Abu-KhalafMLewisFLHuangJ (2008) Neurodynamic programming and zero-sum games for constrained control systems. IEEE Transactions on Neural Networks19(7): 1243–1252.
4.
Al-TamimiAAbu-KhalafMLewisFL (2007a) Adaptive critic designs for discrete-time zero-sum games with application to control. IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics)37(1): 240–247.
5.
Al-TamimiALewisFLAbu-KhalafM (2007b) Model-free Q-learning designs for linear discrete-time zero-sum games with application to H-infinity control. Automatica43(3): 473–481.
6.
BalakrishnanSNBiegaV (1996) Adaptive-critic-based neural networks for aircraft optimal control. Journal of Guidance, Control, and Dynamics19(4): 893–898.
7.
BeardRWSaridisGNWenJT (1997) Galerkin approximations of the generalized Hamilton–Jacobi–Bellman equation. Automatica33(12): 2159–2177.
8.
BertsekasDPTsitsiklisJN (1995) Neuro-dynamic programming: an overview. In: Proceedings of the 34th IEEE conference on decision and control, vol. 1, New Orleans, LA, 13–15 December 1995, pp. 560–564. New York: IEEE.
ChoiS-BSeongM-SHaS-H (2009) Vibration control of an MR vehicle suspension system considering both hysteretic behavior and parameter variation. Smart Materials and Structures18(12): 125010.
11.
DierksTJagannathanS (2010) Optimal control of affine nonlinear continuous-time systems using an online Hamilton–Jacobi–Isaacs formulation. In: Proceedings of the 2010 49th IEEE conference on decision and control (CDC), Atlanta, GA, 15–17 December 2010, pp. 3048–3053. New York: IEEE.
12.
DuttaSChoiS-MChoiS-B (2016) A new adaptive sliding mode control for Macpherson strut suspension system with magneto-rheological damper. Journal of Intelligent Material Systems and Structures27(20): 2795–2809.
13.
DykeSJ (1996) Acceleration feedback control strategies for active and semi-active control systems: modeling, algorithm development, and experimental verification (directed by BF Spence Jr). Department of Civil Engineering and Geological Sciences, University of Notre Dame. Available at: https://engineering.purdue.edu/IISL/Publications/DSc_Dissertations/Shirley_Dyke.pdf
14.
GiorgettiNBemporadATsengHE. (2006) Hybrid model predictive control application towards optimal semi-active suspension. International Journal of Control79(5): 521–533.
15.
HarveyPSJrGavinHPScruggsJT (2012) Optimal performance of constrained control systems. Smart Materials and Structures21(8): 085001.
16.
HarveyPSJrGavinHPScruggsJT. (2014) Determining the physical limits on semi-active control performance: a tutorial. Structural Control and Health Monitoring21(5): 803–816.
17.
HeydariABalakrishnanSN (2013) Finite-horizon control-constrained nonlinear optimal control using single network adaptive critics. IEEE Transactions on Neural Networks and Learning Systems24(1): 145–157.
18.
HeydariABalakrishnanSN (2014a) Fixed-final-time optimal tracking control of input-affine nonlinear systems. Neurocomputing129: 528–539.
19.
HeydariABalakrishnanSN (2014b) Global optimality of approximate dynamic programming and its use in non-convex function minimization. Applied Soft Computing24: 291–303.
20.
HuangJLinC-F (1995) Numerical approach to computing nonlinear H-infinity control laws. Journal of Guidance, Control, and Dynamics18(5): 989–994.
21.
HudhaKJamaluddinHSaminPM. (2005) Effects of control techniques and damper constraint on the performance of a semi-active magnetorheological damper. International Journal of Vehicle Autonomous Systems3(2–4): 230–252.
22.
JansenLMDykeSJ (2000) Semi-active control strategies for MR dampers: a comparative study. Journal of Engineering Mechanics126: 795–803.
23.
JohnsonMBhasinSDixonWE (2011) Nonlinear two-player zero-sum game approximate solution using a policy iteration algorithm. In: Proceedings of the 2011 50th IEEE conference on decision and control and European control conference (CDC-ECC), Orlando, FL, 12–15 December 2011, pp. 142–147. New York: IEEE.
24.
KarkoubMAZribiM (2006) Active/semi-active suspension control using magnetorheological actuators. International Journal of Systems Science37(1): 35–44.
25.
ModaresHLewisFLNaghibi-SistaniMB (2013) Adaptive optimal control of unknown constrained-input systems using policy iteration and neural networks. IEEE Transactions on Neural Networks and Learning Systems24(10): 1513–1525.
26.
NguyenLHHongK-SParkS (2010) Road-frequency adaptive control for semi-active suspension systems. International Journal of Control, Automation and Systems8(5): 1029–1038.
ShinDKPhuDXChoiS-M. (2015) An adaptive fuzzy sliding mode control of magneto-rheological seat suspension with human body model. Journal of Intelligent Material Systems and Structures27(7): 925–934.
29.
SungK-GHanY-MChoJ-W. (2008) Vibration control of vehicle ER suspension system using fuzzy moving sliding mode controller. Journal of Sound and Vibration311(3): 1004–1019.
30.
TsengHEHedrickJK (1994) Semi-active control laws—optimal and sub-optimal. Vehicle System Dynamics23(23): 545–569.
31.
TsengHEHrovatD (2015) State of the art survey: active and semi-active suspension control. Vehicle System Dynamics53(7): 1034–1062.
32.
VrabieDPastravanuOAbu-KhalafM. (2009) Adaptive optimal control for continuous-time linear systems based on policy iteration. Automatica45(2): 477–484.
33.
WeiQSongRYanPF (2016) Data-driven zero-sum neuro-optimal control for a class of continuous-time unknown nonlinear systems with disturbance using ADP. IEEE Transactions on Neural Networks and Learning Systems27(2): 444–458.
34.
WerbosPJ (1992) Approximate dynamic programming for real-time control and neural modeling. In: WhiteDASofgeDA (eds) Handbook of Intelligent Control: Neural, Fuzzy and Adaptive Approaches (Chapter 13). New York: Van Nostrand, pp. 493–525.
35.
WuHNLuoB (2012) Neural network based online simultaneous policy update algorithm for solving the HJI equation in nonlinear H-infinity control. IEEE Transactions on Neural Networks and Learning Systems23(12): 1884–1895.
36.
YangXLiuDRMaHW. (2016) Online approximate solution of HJI equation for unknown constrained-input nonlinear continuous-time systems. Information Sciences328: 435–454.
37.
YasiniSNaghibi SistaniMBKarimpourA (2014) Approximate dynamic programming for two-player zero-sum game related to H∞ control of unknown nonlinear continuous-time systems. International Journal of Control, Automation and Systems13(1): 99–109.
38.
ZarehSHSarrafanAKhayyatAAA. (2012) Intelligent semi-active vibration control of eleven degrees of freedom suspension system using magnetorheological dampers. Journal of Mechanical Science and Technology26(2): 323–334.
39.
ZongLHGongXLGuoCY. (2012) Inverse neuro-fuzzy MR damper model and its application in vibration control of vehicle suspension system. Vehicle System Dynamics50(7): 1025–1041.
