- Instantaneous Control
The idea behind instantaneous control in this context is two-fold - to construct a feedback law which matches the desired state, but at considerable control costs. The method works as follows. The uncontrolled dynamical system is discretized with respect to time. Then, at selected time slices an instantaneous version of the cost functional is approximately minimized subject to a stationary system, whose structure depends on the chosen discretization method. The control obtained is used to steer the system to the next time slice, where the procedure is repeated. Instantaneous control is closely related to receding horizon control (RHC) or model predictive control (MPC) with finite time horizon [9,10,11]. Its difference to (RHC) and (MPC) can best be explained in terms of a chess analogy: (RHC) and (MPC) compute the next n, say, optimal moves and apply the first one to steer the system to the next time slice. Instantaneous control computes an approximation to the next optimal move by applying exactly one step of the gradient method to the solution of the instantaneous control problem and applies the approximate control to advance to the next time slice. Clearly, it can not be expected that the instantaneous control technique is a candidate to stabilize arbitrary dynamical systems, compare the citations above. However, it was shown at hand of the Burgers equation, it is capable of enhancing the stability properties of dissipative systems. In its nature instantaneous control is a discrete-in-time feedback control approach and, as shall be shown, can be interpreted as the stable time discretization of a closed-loop control law [12]. Recent developments in the field of suboptmal control strategies for time dependent incompressible Navier-Stokes equations were summarized in [13]. Reduced-order modeling as well as instantaneous control strategies were described and illustrated by numerical examples for the laminar backward facing step flow. In [14] suboptimal distributed as well as boundary control strategies for fluid flows governed by the Navier-Stokes equations together with aspects of the numerical and practical implementation of the underlying algorithm were considerd. Choi, Hinze and Kunisch investigated suboptimal boundary control strategies for the time-dependent, incompressible flow over the backward-facing step in [15]. The objective consists in the reduction of the recirculation bubble of the flow behind the step. Several cost functionals were suggeted and a frame for the derivation of the optimality system for a general class of cost functionals was presented. Numerical examples were also included. Instantaneous control is applied to the control of the Burgers equation in [16]. This control technique is closely related to receding horizon control and allows for an interpretation as suboptimal closed loop controller, whose parameters may be adjusted so as to stabilize the nonlinear equation under consideration. Besides convergence analysis for the distributed control case several numerical examples for the continuous and discrete-in-time control laws are presented, including boundary control. Instantaneous control has also been applied to viscoelastic fluid flows [17]. Several results are given, including the influence of the design and numerical algorithm parameters on the resulting computed control.
References:
[9] Garcia, C. E., Prett, D. M. and Morari, M.: Model predictive control: theory and practice - a survey. Automatica, 25(3):335-348, 1989.
[10] Nevistic, V., and Primbs, J. A.: Finite receding horizon control: A general framework for stability and performance analysis. Automatic Control Laboratory, ETH Zürich, Preprint, 1997.
[11] Rawlings, J. B., and Muske, K. R.: The stability of constrained receding horizon control. IEEE Transactions on Automatic Control, 38(10):1512-1516, 1993.
[12] Hinze, M., and Kauffmann, A.: A new class of feedback control laws for dynamical sytems. Fachbereich Mathematik, Technische Universität Berlin, Preprint No. 602/1998, 1998.
[13] Hinze, M., and Kunisch, K.: On suboptimal control strategies for the Navier-Stokes equations. ESAIM: Proceedings, Vol. 4, 1998, 181-198.
[14] Hinze, M., and Kunisch, K.: Suboptimal control strategies for backward facing step flows. Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 110, Mai 1997.
[15] Choi, H., Hinze, M., and Kunisch, K.: Instantaneous control of backward-facing step flows. Applied Numerical Mathematics 31 (1999), 133-158.
[16] Hinze, M., and Volkwein, S.: Instantaneous control for the instationary Burgers equation - convergence analysis & numerical implementation-. Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 170, September 1999.
[17] Kunisch, K., and Marduel, X.: Sub-optimal control of transient non-isothermal viscoelastic fluid flows. Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 155, Juni 1999.
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