Tracking with prescribed transient behaviour

Achim Ilchmann; E. P. Ryan; C. J. Sangwin

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 7, page 471-493
  • ISSN: 1292-8119

Abstract

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Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary ℝ M -valued reference signal r of class W1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ϕ ( t ) ∥ e ( t ) ∥ < 1 for all t ≥ 0, where φ a prescribed real-valued function of class W1,∞ with the property that φ(s) > 0 for all s > 0 and lim inf s → ∞ ϕ ( s ) > 0 . A simple (neither adaptive nor dynamic) error feedback control of the form u ( t ) = - α ( ϕ ( t ) ∥ e ( t ) ∥ ) e ( t ) is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain α ( ϕ ( · ) ∥ e ( · ) ∥ ) .

How to cite

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Ilchmann, Achim, Ryan, E. P., and Sangwin, C. J.. "Tracking with prescribed transient behaviour." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 471-493. <http://eudml.org/doc/90632>.

@article{Ilchmann2010,
abstract = { Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary $\mathbb\{R\}^M$-valued reference signal r of class W1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that $\{\varphi\}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of class W1,∞ with the property that φ(s) > 0 for all s > 0 and $\liminf_\{s\rightarrow\infty\}\{\varphi\}(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha (\{\varphi\}(t)\|e(t)\|)e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha (\{\varphi\}(\cdot )\|e(\cdot )\|)$. },
author = {Ilchmann, Achim, Ryan, E. P., Sangwin, C. J.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear systems; functional differential equations; feedback control; tracking; transient behaviour.; nonlinear systems; functional differential equations; time-varying feedback control; tracking funnel control; transient behaviour; reference signals; performance funnel; nonlinear delay system; systems with hysteresis; relay hysteresis; backlash hysteresis; asymptotic stabilization; simulations; constant gain feedback; adaptive -tracking; universal tracking; memory; disturbance; nonlinear causal operator},
language = {eng},
month = {3},
pages = {471-493},
publisher = {EDP Sciences},
title = {Tracking with prescribed transient behaviour},
url = {http://eudml.org/doc/90632},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Ilchmann, Achim
AU - Ryan, E. P.
AU - Sangwin, C. J.
TI - Tracking with prescribed transient behaviour
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 471
EP - 493
AB - Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary $\mathbb{R}^M$-valued reference signal r of class W1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ${\varphi}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of class W1,∞ with the property that φ(s) > 0 for all s > 0 and $\liminf_{s\rightarrow\infty}{\varphi}(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha ({\varphi}(\cdot )\|e(\cdot )\|)$.
LA - eng
KW - Nonlinear systems; functional differential equations; feedback control; tracking; transient behaviour.; nonlinear systems; functional differential equations; time-varying feedback control; tracking funnel control; transient behaviour; reference signals; performance funnel; nonlinear delay system; systems with hysteresis; relay hysteresis; backlash hysteresis; asymptotic stabilization; simulations; constant gain feedback; adaptive -tracking; universal tracking; memory; disturbance; nonlinear causal operator
UR - http://eudml.org/doc/90632
ER -

References

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  2. A. Ilchmann, E.P. Ryan and C.J. Sangwin, Systems of controlled functional differential equations and adaptive tracking. SIAM J. Control Optim.40 (2002) 1746-1764.  
  3. H. Logemann and A.D. Mawby, Low-gain integral control of infinite dimensional regular linear systems subject to input hysteresis, in Advances in Mathematical Systems Theory, edited by F. Colonius, U. Helmke, D. Prätzel-Wolters and F. Wirth. Birkhäuser Verlag, Boston, Basel, Berlin (2000) 255-293.  
  4. D.E. Miller and E.J. Davison, An adaptive controller which provides an arbitrarily good transient and steady-state response. IEEE Trans. Automat. Control36 (1991) 68-81.  
  5. E.P. Ryan and C.J. Sangwin, Controlled functional differential equations and adaptive stabilization. Int. J. Control74 (2001) 77-90.  
  6. E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control34 (1989) 435-443.  
  7. C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange attractors. Springer-Verlag, New York (1982).  
  8. G. Weiss, Transfer functions of regular linear systems, Part 1: Characterization of regularity. Trans. Amer. Math. Soc.342 (1994) 827-854.  

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