# Tracking with prescribed transient behaviour

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

ESAIM: Control, Optimisation and Calculus of Variations (2002)

- 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 (2002): 471-493. <http://eudml.org/doc/245146>.

@article{Ilchmann2002,

abstract = {Universal tracking control is investigated in the context of a class $\{\mathcal \{S\}\}$ 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 $W^\{1,\infty \}$ (absolutely continuous and bounded with essentially bounded derivative) and every system of class $\{\mathcal \{S\}\}$, the tracking error $e$ between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that $\{\varphi \}(t)\Vert e(t)\Vert < 1$ for all $t\ge 0$, where $\{\varphi \}$ a prescribed real-valued function of class $W^\{1,\infty \}$ with the property that $\{\varphi \}(s)>0$ for all $s > 0$ and $\liminf _\{s\rightarrow \infty \}\{\varphi \}(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha (\{\varphi \}(t)\Vert e(t)\Vert )e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha (\{\varphi \}(\cdot )\Vert e(\cdot )\Vert )$.},

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; time-varying feedback control; tracking funnel control; 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},

pages = {471-493},

publisher = {EDP-Sciences},

title = {Tracking with prescribed transient behaviour},

url = {http://eudml.org/doc/245146},

volume = {7},

year = {2002},

}

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

PY - 2002

PB - EDP-Sciences

VL - 7

SP - 471

EP - 493

AB - Universal tracking control is investigated in the context of a class ${\mathcal {S}}$ 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 $W^{1,\infty }$ (absolutely continuous and bounded with essentially bounded derivative) and every system of class ${\mathcal {S}}$, the tracking error $e$ between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ${\varphi }(t)\Vert e(t)\Vert < 1$ for all $t\ge 0$, where ${\varphi }$ a prescribed real-valued function of class $W^{1,\infty }$ with the property that ${\varphi }(s)>0$ for all $s > 0$ and $\liminf _{s\rightarrow \infty }{\varphi }(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha ({\varphi }(t)\Vert e(t)\Vert )e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha ({\varphi }(\cdot )\Vert e(\cdot )\Vert )$.

LA - eng

KW - nonlinear systems; functional differential equations; feedback control; tracking; transient behaviour; time-varying feedback control; tracking funnel control; 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/245146

ER -

## References

top- [1] C.I. Byrnes and J.C. Willems, Adaptive stabilization of multivariable linear systems, in Proc. 23rd Conf. on Decision and Control. Las Vegas (1984) 1574-1577.
- [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. Zbl1015.93027MR1897194
- [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. Control 36 (1991) 68-81. Zbl0725.93074MR1084247
- [5] E.P. Ryan and C.J. Sangwin, Controlled functional differential equations and adaptive stabilization. Int. J. Control 74 (2001) 77-90. Zbl1009.93068MR1824464
- [6] E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34 (1989) 435-443. Zbl0682.93045MR987806
- [7] C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange attractors. Springer-Verlag, New York (1982). Zbl0504.58001MR681294
- [8] G. Weiss, Transfer functions of regular linear systems, Part 1: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. Zbl0798.93036MR1179402

## Citations in EuDML Documents

top- Eugene P. Ryan, Chris J. Sangwin, Philip Townsend, Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance
- Eugene P. Ryan, Chris J. Sangwin, Philip Townsend, Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance

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