Tracking with prescribed transient behaviour

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

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 471-493.

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 $W^{1,\infty }$ (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 $\varphi \left(t\right)\parallel e\left(t\right)\parallel <1$ for all $t\ge 0$, where $\varphi$ a prescribed real-valued function of class $W^{1,\infty }$ with the property that $\varphi \left(s\right)>0$ for all $s>0$ and ${lim\; inf}_{s\to \infty }\varphi \left(s\right)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u\left(t\right)=-\alpha \left(\varphi \right(t)\parallel e(t)\parallel )e\left(t\right)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha \left(\varphi \right(·)\parallel e(·)\parallel )$.

[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. | MR | Zbl

[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. | MR | Zbl

[5] E.P. Ryan and C.J. Sangwin, Controlled functional differential equations and adaptive stabilization. Int. J. Control 74 (2001) 77-90. | MR | Zbl

[6] E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34 (1989) 435-443. | MR | Zbl

[7] C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange attractors. Springer-Verlag, New York (1982). | MR | Zbl

[8] G. Weiss, Transfer functions of regular linear systems, Part 1: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR | Zbl