×

Control of flexible structures: Continuous theory and approximation issues. (English) Zbl 0991.93052

Rodellar, J. (ed.) et al., Advances in structural control. Barcelona: CIMNE. 31-50 (1999).
The author reviews some important recent developments in the control theory of structures modeled by systems of partial differential equations, in other words of systems with distributed parameters. He observes that usually the starting point is acceptance of some mathematical model originating with continuum mechanics. Of course, it is much easier to deal with a finite-dimensional system derived by lumping the parameters. The two approaches consist of either discretizing the system and then deriving some control results, or vice-versa first developing control for the distributed parameter system and then discretizing it for implementation of these results. Reasons for unsatisfactory results in the “first discretize-then control” methodology, which is most commonly practiced, relate to the continuum mechanical properties, such as propagation of different kinds of waves in solids, which are entirely lost or badly distorted in the discrete scheme. It has been known for some time that the controllability and observability results lose much if the system is a priori discretized. The instability of boundary controls caused by mesh size has been demonstrated by Infante and Zuazua. The author concludes that a continuous model produces refined results and offers better predictions. Moreover the discrete model may hide some elegant results, such as the bang-bang solutions predicted by Pontryagin’s Hamiltonian.
The author considers a classical elastodynamic problem: \(\rho\partial_{tt} y-\text{div} Ae(y)=0\), where \(y\) is a “small” displacement, \(e\) is the strain tensor, \(A\) is a Hooke-type matrix. Thus \(Ae\) represents the stress tensor. The control \(u=y\) is applied on part of the boundary region \(\Omega\), otherwise \(u=0\). All this can be rewritten in the first order PDE form: \(\partial_t z=Az+Bu\), with “usual” initial conditions, and with \(z(0)=z_0\), where \(z\) denotes \(\{y(t),y_t(t)\}\). The author introduces the adjoint system with zero boundary conditions, whose adjoint state solution is given by the vector \(\varphi\), with initial state \((\varphi^0, \varphi^1)\).
The Hilbert Uniqueness Method (HUM), reinterpreted by Lions and popularized by Lagnese, is defined here in terms of a strong observability and the exact controllability inequality, which was first observed by D. L. Russell in the early 1970’s. The author comments that while the HUM offers an immediate algorithm for computing the optimal control, a lot of hard analysis is required to make certain that this result is physically meaningful. For example, the \(P\) and \(S\) waves travel with finite speed, and there is a problem of reaching a controlled region of an elastic body within the allocated time. If this condition is violated, we may be dealing with approximate controllability instead of exact controllability, which in this interpretation means that some (hopefully small) controlled regions are not reached for a given initial state. What is worse, an arbitrarily small perturbation of an exactly controllable system may result in the appearance of this weaker form of controllability.
Following these introductory remarks, the author briefly touches upon several topics. The relation between the control Grammian and energy in the adjoint state unexpectedly resembles that of the original state of the system. The author calls this queer, because this demands exact linear momentum conservation in the adjoint state. But the usual Galerkin or finite element discretization does not respect this demand. This apparent lack of robustness of strong observability was noted by Glowinski, and others.
Discussing a closed-loop feedback, the author observes that control problems become easier to tackle, since exact controllability is a necessary condition for the existence of a feedback law. Here the Komornik study of wave equations shows that the state and its time derivative decay exponentially.
The remainder of the article discusses computational issues. The pioneering work of Glowinski, Lions, and others used a conjugate gradient technique. High frequency vibrations gave trouble, but Tikhonov’s regularization helped to filter out some of these inconvenient phenomena. The instability with respect to grid size produced the appearance of really non-existing oscillations. The author offers a discussion of various approaches to discretization, but admits that this cannot be done in a short review. He ends with a remark on terminology and techniques used by mathematicians and structural engineers, saying that there is much these groups can learn from each other, perhaps giving rise to new innovative ideas.
For the entire collection see [Zbl 0959.00020].

MSC:

93C20 Control/observation systems governed by partial differential equations
74M05 Control, switches and devices (“smart materials”) in solid mechanics
93B40 Computational methods in systems theory (MSC2010)
93B05 Controllability
93B07 Observability
93D15 Stabilization of systems by feedback
PDFBibTeX XMLCite