This paper is concerned with the boundary error feedback regulation for a one-dimensional anti-stable wave equation with distributed disturbance generated by a finite-dimensional exogenous system. Transport equation and regulator equation are introduced first to deal with the anti-damping on boundary and the distributed disturbance of the original system. Then, the tracking error and its derivative are measured to design an observer for both exosystem and auxiliary partial differential equation (PDE) system to recover the state. After proving the well-posedness of the regulator equations, we propose an observer-based controller to regulate the tracking error to zero exponentially and keep the states of all the internal loop uniformly bounded. Finally, some numerical simulations are presented to validate the effectiveness of the proposed controller.
Bresch-PietriDKristicM (2014) Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica50: 1407–1415.
2.
DavisonEJ (1976) The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Transactions on Automatic Control21(1): 25–34.
3.
DeutscherJ (2011) Output regulation for linear distributed-parameter systems using finite-dimensional dual observers. Automatica47(11): 2468–2473.
4.
DeutscherJ (2015) A backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica57(4): 56–64.
5.
FengHGuoBZ (2017a) A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. IEEE Transactions on Automatic Control62(8): 3774–3787.
6.
FengHGuoBZ (2017b) Observer design and exponential stabilization for wave equation in energy space by boundary displacement measurement only. IEEE Transactions on Automatic Control62(3): 1438–1444.
7.
FengHGuoBZWuXH (2020) Trajectory planning approach to output tracking for a 1-D wave equation. IEEE Transactions on Automatic Control65(5): 1841–1854.
8.
FrancisBAWonhamWM (1976) The internal model principle of control theory. Automatica12(5): 457–465.
9.
GuJJWangJMGuoYP (2018) Output regulation of anti-stable coupled wave equations via the backstepping technique. IET Control Theory and Applications12(4): 431–445.
10.
GuJJWeiCQWangJM (2019) Backstepping-based output regulation of ordinary differential equations cascaded by wave equation with in-domain anti-damping. Transactions of the Institute of Measurement and Control41(1): 246–262.
11.
GuoBZKangW (2014) The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance. International Journal of Robust and Nonlinear Control24(1): 54–69.
12.
GuoBZMengTT (2020) Robust error based non-collocated output tracking control for a heat equation. Automatica114: 108818.
GuoWJinFF (2020b) Adaptive error feedback regulator design for 1D heat equation with unknown harmonic disturbance anti-collocated with control. IEEE Transactions on Automatic Control65(2): 824–830.
15.
GuoWShaoZCKrsticM (2017) Adaptive rejection of harmonic disturbance anti-collocated with control in 1D wave equation. Automatica79(1): 17–26.
16.
GuoWZhouHCKrsticM (2018) Adaptive error feedback regulation problem for 1D wave equation. International Journal of Robust and Nonlinear Control28(6): 4309–4329.
17.
IlchmannALogemannH (1998) Adaptive λ-tracking for a class of infinite-dimensional systems. Systems and Control Letters34(1–2): 11–21.
18.
IsidoriAByrnesCI (1990) Output regulation of nonlinear systems. IEEE Transactions on Automatic Control35(2): 131–140.
19.
JinFFGuoBZ (2019) Performance boundary output tracking for a wave equation with control unmatched disturbance. European Journal of Control50(11): 30–40.
20.
KalaithT (1980) Linear Systems. NJ: Prentice Hall.
21.
KobayashiTOyaM (2002) Adaptive servomechanism design for boundary control system. IMA Journal of Mathematical Control and Information19(3): 279–295.
22.
KrsticM (2009) Adaptive control of an anti-stable wave PDE. In: 2009 American control conference, St. Louis, MO, 10–12 June 2009.
23.
KrsticMSmyshlyaevA (2008) Boundary Control of PDES: A Course on Backstepping Designs. Philadelphia, PA: SIAM.
24.
KrsticMGuoBZBaloghA, et al. (2008) Output-feedback stabilization of an unstable wave equation. Automatica44: 63–74.
25.
LiuWWGuoWWangJM (2020) Backstepping-based adaptive error feedback regulator design for one dimensional reaction-diffusion equation. Journal of Mathematical Analysis and Applications484(1): 123666.
26.
LogemannHIlchmannA (1994) An adaptive servomechanism for a class of infinite-dimensional systems. SIAM Journal on Control and Optimization32(4): 917–936.
27.
PaunonenL (2016) Controller design for robust output regulation of regular linear systems. IEEE Transactions on Automatica Control61(10): 2974–2986.
28.
PisanoAOrlovYUsaiE (2011) Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization49(2): 363–382.
29.
RichterHWarnerH (2020) Motion optimization for musculoskeletal dynamics: A flatness-based polynomial approach. IEEE Transactions on Automatic Control66: 3289–3295.
30.
SaidBSaidHRachidS (2016) Error feedback regulation problem for regular linear systems. IMA Journal of Mathematical Control and Information35: 1–25.
31.
SmyshlyaevAKrsticM (2009) Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Systems and Control Letters58(8): 617–623.
32.
SmyshlyaevACerpaEKrsticM (2009) Boundary stabilization of an anti-stable wave equation with in-domain anti-damping. In: 48th IEEE conference on decision and control and 28th Chinese control conference, Shanghai, P.R. China, 16–18 December 2009.
33.
WeissG (1989) Admissibility of unbounded control operators. SIAM Journal on Control and Optimization27(3): 527–545.
34.
XuXDDubljevicS (2017) Output and error feedback regulator designs for linear infinite-dimensional systems. Automatica83(6): 170–178.
35.
ZhangXYFengHChaiSG (2016) Performance output exponential tracking for a wave equation with a general disturbance and non-collocated control. Systems and Control Letters98(12): 79–85.
36.
ZhouHCGuoBZ (2018) Performance output tracking for one-dimensional wave equation subject to unmatched general disturbance and non-collocated control. European Journal of Control39(1): 39–52.
37.
ZhouHCWeissG (2017) Output tracking and disturbance rejection for a one-dimensional anti-stable wave equation. In: 56th annual conference on decision and control, Melbourne, VIC, Australia, 13 December 2017.
