An algebraic approach to dynamic optimisation of nonlinear systems: a survey and some new results

Abstract

Dynamic optimisation, with a particular focus on optimal control and nonzero-sum differential games, is considered. For nonlinear systems solutions sought via the dynamic programming strategy are inevitably characterised by partial differential equations (PDEs) which are often difficult to solve. A detailed overview of a control design framework which enables the systematic construction of approximate solutions for optimal control problems and differential games without requiring the explicit solution of any PDE is provided along with a novel design of a nonlinear control gain aimed at improving the 'level of approximation' achieved. Multi-agent systems are considered as a possible application of the theory.

Disturbance Attenuation by Measurement Feedback in Nonlinear Systems via Immersion and Algebraic Conditions

“…The algebraic conditions (H2) and (H4) in Theorem 1 essentially replace the partial differential equations (13) and (15), respectively, in Theorem IV.1 of [3]. Note that, despite the nonlinear dependence on the variable x in (6) and (11), the former are much milder conditions than the latter. In fact, in the latter, roughly speaking, one must still solve an algebraic expression -with respect to the co-vector describing the gradient of the function sought for -together with the additional requirement that such solution is an exact differential.…”

“…with Ψ e (x e , ξ e ) = ∇ ξe (P e (ξ e )x e ), and recalling (11) with M e given in (21) above, D e = Ψ(x e , ξ e ) −R e and q e (x e ) = x e Q e (x e )x e . Note that Ψ e (0, ξ e ) = 0 and, as a result, Z e = I 0 spans the kernel of D e xe=0 .…”

“…To demonstrate local asymptotic stability, consider the candidate Lyapunov function W (x, ξ, x e , ξ e ) = V (x, ξ) + V e (x e , ξ e ) , and notice that by (6) and (11) and by the definition ofP andP e , W > 0 at least in a neighbourhood of the origin. The two previous steps imply that there exists a non-empty neigbourhood N = W ∩ W e such that for any k >k and any k e >k ė…”

“…Therefore, any solution to the PDEs in [3] is also a solution to the algebraic equations in Theorem 1, while the converse does not hold. Hence, the set of solutions to equations (6), (11) contains the solutions to the PDEs in [3]. Moreover, algebraic Riccati-like equations similar to (6) and (11) have been extensively studied and efficient strategies have been proposed in the literature to to compute their solution, see for instance [19].…”

“…In this paper, we follow the approach of [3] and provide an alternative solution for the problem of local disturbance attenuation with internal stability which is characterised by algebraic equations in place of PDEs and which relies on the immersion of the underlying nonlinear dynamics into an auxiliary (extended) system. These algebraic equations are similar in spirit to the so-called alge-braicP solutions introduced in [9]- [11] and utilised in [12], [13] in the context of optimal control and differential games. The approach enables the systematic construction of dynamic output feedbacks which solve -without approximation -the problem of local disturbance attenuation with internal stability.…”