Predictive control for polynomial systems subject to constraints using sum of squares

Offline NMPC for continuous-time systems using sum of squares

“…Part (a): The important steps are first applying Positivstellensatz (Lemma 1) to (5e) ensuring local positive definiteness on the domain D × Γ (represented by h D and p k (θ)) 1 , and in a second step to exploit Schur complement of Φ(x, θ) (cf. [6]). Then, we use the fact that if a matrix Λ ∈ S n + ⇒Z T ΛZ ≥ 0, 1 Using the slack variables S h 0 (x, θ) and Sp k (x, θ).…”

“…In [5], such an offline NMPC scheme has been derived for a continuous-time setting. For polynomial control systems as treated in [6], an offline NMPC approach is described in [10], however in a discrete-time setting and resulting in a non-convex optimization problem. Polynomial control systems have received increasing attention since the development of the SOS relaxation [11].…”

“…Although their variation rates are known, these are not used since they may contain non-polynomial nonlinearities and their incorporation would increase the computational effort significantly. Using PPV systems, the class of nonlinear systems to which the NMPC approach [6] can be applied is enlarged. Additionally, a computationally attractive offline formulation of the NMPC approach is developed using the idea of nested invariant sets.…”

“…This may lead to computational issues when the optimization problem cannot be solved fast enough. To overcome these problems, an idea to perform a repeated online calculation of a state feedback has been developed for linear systems [4] which has been extended to Lur'e systems [5] and polynomial control systems [6]. In [7]- [9], the basic idea of [4] is modified in order to solve the optimization problem offline to reduce online computational effort.…”

Long-time average cost control of polynomial systems: A sum of squares approach

“…Recent results on SOS decomposition have transformed the verification of non-negativity of polynomials into SDP, hence providing promising algorithmic procedures for stability analysis of polynomial systems. However, using SOS techniques for optimal control, as for example in [13], [14], [15], is subject to a generic difficulty: while the problem of optimizing the candidate Lyapunov function certifying the stability for a closed-loop system for a given controller and the problem of optimizing the controller for a given candidate Lyapunov function are reducible to an SDP and thus, are tractable, the problem of optimizing both the control and the Lyapuniov function is non-convex. Naturally, iterative procedures are attempted for overcoming this difficulty.…”