In this paper, we solve the output tracking and disturbance rejection problem for a system described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞[0, ∞) and L∞[0, ∞), respectively. Generally, these signals cannot be generated from an exosystem. We explore an approach based on proportional control. It is shown that a proportional gain controller can achieve exponentially the output tracking while rejecting disturbance. Our method consists of three steps: first, we convert the original system without disturbance into two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. Second, based on our recent result on disturbance estimator, we apply the estimation/cancellion strategy to cancel to the external disturbance and to track the reference asymptotically. Third, we design a controller using a state observer. Since disturbance does not appear in the observer explicitly (the disturbance is exactly compensated), the controlled output signal is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals.
Accepté le :
DOI : 10.1051/cocv/2018049
Keywords: Output tracking, disturbance rejection, wave equation, anti-damping, exponential stabilization
Zhou, Hua-Cheng 1
@article{COCV_2019__25__A69_0,
author = {Zhou, Hua-Cheng},
title = {Output tracking and disturbance rejection for {1-D} anti-stable wave equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018049},
mrnumber = {4031687},
zbl = {1441.93258},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018049/}
}
TY - JOUR AU - Zhou, Hua-Cheng TI - Output tracking and disturbance rejection for 1-D anti-stable wave equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018049/ DO - 10.1051/cocv/2018049 LA - en ID - COCV_2019__25__A69_0 ER -
%0 Journal Article %A Zhou, Hua-Cheng %T Output tracking and disturbance rejection for 1-D anti-stable wave equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018049/ %R 10.1051/cocv/2018049 %G en %F COCV_2019__25__A69_0
Zhou, Hua-Cheng. Output tracking and disturbance rejection for 1-D anti-stable wave equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 69. doi: 10.1051/cocv/2018049
[1] , , and , Output regulation for linear distributed parameter systems. IEEE Trans. Automat. Control 45 (2000) 2236–2252. | MR | Zbl | DOI
[2] and , Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica 50 (2014) 1407–1415. | MR | Zbl | DOI
[3] and , An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995). | MR | Zbl | DOI
[4] , The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Automat. Control 21 (1976) 25–34. | MR | Zbl | DOI
[5] , A backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica 57 (2015) 56–64. | MR | Zbl | DOI
[6] , Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs. IEEE Trans. Automat. Control 61 (2016) 2288–2294. | MR | Zbl | DOI
[7] and , Backstepping design of robust state feedback regulators for second order hyperbolic PIDEs. IFAC-PapersOnLine 49 (2016) 80–85. | MR | DOI
[8] , Introduction to Time-Delay Systems. Birkhäuser/Springer, Cham (2014). | MR | Zbl | DOI
[9] and , The internal model principle of control theory. Automatica 12 (1976) 457–465. | MR | Zbl | DOI
[10] and , A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. IEEE Trans. Automat. Control 62 (2017) 3774–3787. | MR | Zbl | DOI
[11] and , Performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance. Automatica 68 (2016) 194–202. | MR | Zbl | DOI
[12] , and , Performance output tracking and disturbance rejection for one-dimensional wave equation with boundary disturbance. IEEE, 54th, Annual Conference on Decision and Control, Osaka, Japan (2015).
[13] , and , Adaptive rejection of harmonic disturbance anticollocated with control in 1D wave equation. Automatica 79 (2017) 17–26. | MR | Zbl | DOI
[14] , and , Adaptive error feedback output regulation for 1d wave equation. Int. J. Robust Nonlinear Control 28 (2018) 4309–4329. | MR | Zbl | DOI
[15] and , Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance. IEEE Trans. Automat. Control 60 (2015) 824–830. | MR | Zbl | DOI
[16] and , The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Trans. Automat. Control 52 (2007) 371–377. | MR | Zbl | DOI
[17] and , The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance. IEEE Trans. Automat. Control 60 (2015) 143–157. | MR | Zbl | DOI
[18] and , Output regulation of periodic signals for DPS: an infinite-dimensional signal generator. IEEE Trans. Automat. Control 50 (2005) 1799–1804. | MR | Zbl | DOI
[19] and , Feedback and feedforward output regulation of bounded uniformly continuous signals for infinite-dimensional systems. SIAM J. Control Optim. 45 (2006) 1714–1735. | MR | Zbl | DOI
[20] , Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhäuser, Boston (2009). | MR | Zbl | DOI
[21] and , Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking. Syst. Control Lett. 86 (2015) 24–33. | MR | Zbl | DOI
[22] , and , Output tracking for one-dimensional Schrödinger equation subject to boundary disturbance. Asian J. Control 20 (2018) 659–668. | MR | Zbl | DOI
[23] and , Tracking control for boundary controlled parabolic PDEs with varying parameters: combining backstepping and differential flatness. Automatica 45 (2009) 1182–1194. | MR | Zbl | DOI
[24] , and , The state feedback regulator problem for regular linear systems. IEEE Trans. Automat. Control 59 (2014) 2708–2723. | MR | Zbl | DOI
[25] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl | DOI
[26] and , The internal model principle for systems with unbounded control and observation. SIAM J. Control Optim. 52 (2014) 3967–4000. | MR | Zbl | DOI
[27] and , Internal model based tracking and disturbance rejection for stable well-posed systems. Automatica 39 (2003) 1555–1569. | MR | Zbl | DOI
[28] and , Boundary control of an anti-stable wave equation with antidamping on the uncontrolled boundary. Syst. Control Lett. 58 (2009) 617–623. | MR | Zbl | DOI
[29] and , Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Basel (2009). | MR | Zbl
[30] , Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527–545. | MR | Zbl | DOI
[31] and , Integral control of stable nonlinear systems. Preprint (2016). | arXiv
[32] and , Multivariable boundary PI control and regulation of a fluid flow system. Math. Control Relat. Fields 4 (2014) 501–520. | MR | Zbl | DOI
[33] , and , Adaptive stabilization for a class of PDE-ODE cascade systems with uncertain harmonic disturbances. ESAIM: COCV 23 (2017) 497–515. | MR | Zbl | Numdam
[34] and , Performance output tracking for one-dimensional wave equation subject to unmatched general disturbance and non-collocated control. Eur. J. Control 39 (2018) 39–52. | MR | Zbl | DOI
[35] and , The regulation problem for the one-dimensional Schrödinger equation via the backstepping approach. Proc. of the International Conference on the Science of Electrical Engineering (ICSEE), Eilat, Israel (2016).
[36] and , Output feedback exponential stabilization for one-dimensional unstable wave equations with boundary control matched disturbance. SIAM J. Control Optim. 56 (2018) 4098–4129. | MR | Zbl | DOI
[37] and , Output feedback exponential stabilization of a nonlinear 1-D wave equation with boundary input. Proc. of the IFAC World Congress, Toulouse, France (2017).
[38] and , Output tracking and disturbance rejection for a one-dimensional anti-stable wave equation. IEEE 56th Annual Conference on Decision and Control, Melbourne, Australia (2017). | Numdam | Zbl | MR
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