Approximate computation of storage functions for discrete-time systems using sum-of-squares techniques

Learning control for polynomial systems using sum of squares relaxations

“…In this case, the matrix X 1,T contains the derivatives of the states at the sampling times when the measurements are taken (see [6,Remark 2]). In this paper, we will focus on continuous-time polynomial systems because this allows us to adopt the tools from [7], [10], [11], while for discrete-time polynomial systems less results are available [12], [13].…”

Verification of Dissipativity and Evaluation of Storage Function in Economic Nonlinear MPC using Q-Learning

“…Note that using SOS method in this method has less complexity and more accuracy in the storage function rather than solving (7) directly using SOS method. Indeed, to solve dissipativity (7) using SOS one needs to approximate the exact stage cost and dynamics by polynomials and the obtained storage function is based of these approximations and may have a bias with the exact storage function (Pirkelmann et al (2019)). However, to solve (31), one only needs to provide a SOS MPC stage cost ˆ θ .…”

“…The dissipativity property allows one to equalize the ENMPC with a tracking MPC which has a well-established stability conditions. Unfortunately, it is not trivial to prove that a given problem is dissipative (Pirkelmann et al (2019)). In order to prove it, one has to find a storage function that satisfies the dissipation inequality.…”

“…A similar method is used by Koch et al (2020) to verify dissipativity properties based on noisy data for linear systems. Furthermore, the Sum-of-Squares (SOS) method is used by Pirkelmann et al (2019) for polynomials dynamics and stage cost. Zanon et al (2016) showed how tracking MPC schemes can approximate generic economic NMPC locally and provided conditions for the local existence of a storage function using a Semi-Definite Program (SDP).…”

Homothetic tube-based robust offset-free economic Model Predictive Control

“…Remark 3. The authors in [24] propose a computational method for automatic verification of dissipativity for discrete time systems with polynomial dynamics and stage cost. Assumption 2 requires that inequality (20) holds for all γ ∈ [0, 1], whereas in the standard definition of dissipativity γ = 0 and the supply rate is based on the original cost function , rather than˜ .…”