# 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

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topIlchmann, 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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- 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. Zbl1015.93027
- 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.
- 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. Zbl0725.93074
- E.P. Ryan and C.J. Sangwin, Controlled functional differential equations and adaptive stabilization. Int. J. Control74 (2001) 77-90. Zbl1009.93068
- E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control34 (1989) 435-443. Zbl0682.93045
- C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange attractors. Springer-Verlag, New York (1982). Zbl0504.58001
- G. Weiss, Transfer functions of regular linear systems, Part 1: Characterization of regularity. Trans. Amer. Math. Soc.342 (1994) 827-854. Zbl0798.93036

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