Stabilization of Nonlinear Systems with Semi-Quadratic Cost
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Stabilization of Nonlinear Systems with Semi-Quadratic Cost Sergey Nikitin
Received: 23 January 2008 / Accepted: 11 July 2008 / Published online: 19 August 2008 © Springer Science+Business Media B.V. 2008
Abstract The paper addresses the stabilization of nonlinear systems with semi-quadratic cost: quadratic with respect to controls and nonlinear for state variables. Paper presents the effective new feedback synthesis procedure. The novel feedback design procedure is based on the ideas borrowed from nonlinear optics and the theory of semi-classical asymptotics developed in quantum mechanics. Keywords Nonlinear system · Stabilization · Semi-classical asymptotic · Maslov canonical operator · Lagrange manifold
1 Introduction Stabilization is one of the central topics of control theory. Moreover, in numerous applications of control theory one needs not only to stabilize a nonlinear system but also to minimize a certain cost function (like, for example, energy cost). These type of problems is well-known in linear system theory and the solution can be found in the class of linear feedbacks [12]. It is also closely related to H∞ -control theory [13]. The nonlinear generalization of H∞ theory is presented in [11]. This paper in its spirit is similar to [11]. However, the main topic of this publication is different and devoted to the more narrow subject of the new synthesis of feedback stabilizers that minimize the semi-quadratic cost functional defined as ∞ ε(x) + u, Q(x)udt. 0
This paper follows the same ideology as publications [7–9]. That means, we assume that the problem of optimal stabilization is solved locally in some small neighborhood of the target equilibrium. In the majority of real world applications that local solution is obtained in the framework of classical linear system theory [4]. Then we follow the ideas outlined S. Nikitin () Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, USA e-mail: [email protected]
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in [9] and extend the optimal feedback outside the small neighborhood. This process can be interpreted in optical terms: the stabilization problem in the reverse time can be treated as a problem from the nonlinear optic with sources of “light” located on the level set of the “local” Lyapunov function. The propagation of “light” is described by projection into x-space the bicharacteristics of the Lagrangian manifold associated with the semi-quadratic functional. This scenario efficiently solves the problem of optimal stabilization. However, on this path we encounter the following pitfalls. The first is related to caustic phenomena. Our “light” propagates in non-uniform “media” (distortion is due to function ε(x)) and we face scattering phenomena that leads to caustics. The second problem is related to the caustics. However, its true nature is not “scattering of light” but the singularity of optimizers connected, in particular, to those semi-quadratic functionals where the matrix Q(x) is degenerate. Discussion of this problem for linear system
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