Controlled functional differential equations : approximate and exact asymptotic tracking with prescribed transient performance

Ryan, Eugene P.; Sangwin, Chris J.; Townsend, Philip

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 745-762

A tracking problem is considered in the context of a class $𝒮$ of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, $m$-input, $m$-output, minimum-phase systems with sign-definite “high-frequency gain”. The first control objective is tracking of reference signals $r$ by the output $y$ of any system in $𝒮$: given $\lambda \ge 0$, construct a feedback strategy which ensures that, for every $r$ (assumed bounded with essentially bounded derivative) and every system of class $𝒮$, the tracking error $e=y-r$ is such that, in the case $\lambda >0$, ${lim\; sup}_{t\to \infty }\parallel e\left(t\right)\parallel <\lambda$ or, in the case $\lambda =0$, ${lim}_{t\to \infty }\parallel e\left(t\right)\parallel =0$. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ${ℱ}_{\varphi }$ (determined by a function $\varphi$). For suitably chosen functions $\alpha$, $\nu$ and $\theta$, both objectives are achieved via a control structure of the form $u\left(t\right)=-\nu \left(k\right(t\left)\right)\theta \left(e\right(t\left)\right)$ with $k\left(t\right)=\alpha \left(\varphi \right(t)\parallel e(t)\parallel )$, whilst maintaining boundedness of the control and gain functions $u$ and $k$. In the case $\lambda =0$, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case $\lambda \ge 0$.

Classification : 93D15, 93C30, 34K20, 34A60
Keywords: functional differential inclusions, transient behaviour, approximate tracking, asymptotic tracking

[1] J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984). | Zbl | MR

[2] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer, New York (1996). | Zbl | MR

[3] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). | Zbl | MR

[4] A. Ilchmann, Non-Identifier-Based High-Gain Adaptive Control. Springer-Verlag, London (1993). | Zbl | MR

[5] A. Ilchmann, E.P. Ryan and C.J. Sangwin, Systems of controlled functional differential equations and adaptive tracking. SIAM J. Contr. Opt. 40 (2002) 1746-1764. | Zbl | MR

[6] A. Ilchmann, E.P. Ryan and C.J. Sangwin, Tracking with prescribed transient behaviour. ESAIM: COCV 7 (2002) 471-493. | Zbl | MR | Numdam

[7] A. Ilchmann, E.P. Ryan and P.N. Townsend, Tracking control with prescribed transient behaviour for systems of known relative degree. Systems Control Lett. 55 (2006) 396-406. | Zbl | MR

[8] A. Ilchmann, E.P. Ryan and P.N. Townsend, Tracking with prescribed transient behaviour for nonlinear systems of known relative degree. SIAM J. Contr. Opt. 46 (2007) 210-230. | Zbl | MR

[9] P.D. Loewen, Optimal Control via Nonsmooth Analysis, CRM Proc. & Lecture Notes 2. AMS, Providence RI (1993). | Zbl | MR

[10] 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, F. Colonius, U. Helmke, D. Prätzel-Wolters and F. Wirth Eds., Birkhäuser Verlag, Boston (2001) 255-293. | MR

[11] R.D. Nussbaum, Some remarks on a conjecture in parameter adaptive control. Systems Control Lett. 3 (1983) 243-246. | Zbl | MR

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

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

[14] E.D. Sontag, Adaptation and regulation with signal detection implies internal model. Systems Control Lett. 50 (2003) 119-126. | Zbl | MR

[15] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer-Verlag, New York (1982). | Zbl | MR

[16] M. Vidyasagar, Control System Synthesis: A Factorization Approach. MIT Press, Cambridge (1985). | Zbl | MR

[17] W.M. Wonham, Linear Multivariable Control: A Geometric Approach. 2nd edn., Springer-Verlag, New York (1979). | Zbl | MR