A robust adaptive model predictive control framework for nonlinear uncertain systems

Abstract

Summary In this article, we present a tube‐based framework for robust adaptive model predictive control (RAMPC) for nonlinear systems subject to parametric uncertainty and additive disturbances. Set‐membership estimation is used to provide accurate bounds on the parametric uncertainty, which are employed for the construction of the tube in a robust MPC scheme. The resulting RAMPC framework ensures robust recursive feasibility and robust constraint satisfaction, while allowing for less conservative operation compared with robust MPC schemes without model/parameter adaptation. Furthermore, by using an additional mean‐squared point estimate in the objective function the framework ensures finite‐gain ℒ2 stability w.r.t. additive disturbances. As a first contribution we derive suitable monotonicity and nonincreasing properties on general parameter estimation algorithms and tube/set‐based RAMPC schemes that ensure robust recursive feasibility and robust constraint satisfaction under recursive model updates. Then, as the main contribution of this article, we provide similar conditions for a tube‐based formulation that is parametrized using an incremental Lyapunov function, a scalar contraction rate and a function bounding the uncertainty. With this result, we can provide simple constructive designs for different RAMPC schemes with varying computational complexity and conservatism. As a corollary, we can demonstrate that state of the art formulations for nonlinear RAMPC are a special case of the proposed framework. We provide a numerical example that demonstrates the flexibility of the proposed framework and showcase improvements compared with state of the art approaches.

Learning‐Based Model Predictive Control for Multi‐Agent Systems using Gaussian Processes

“…It was shown for an academic example of an omnidirectional robot with limited traction that the natural inclusion of nonlinear rate constraints in the investigated robust nonlinear MPC framework [1] is possible. In fact it is straight forward to extend the latter to more general nonlinear systems.…”

“…The variable open-loop tube size around a nominal trajectory is implicitly specified at every time step via level sets of a local Lyapunov function using a scalar parameter s k . The interested reader is referred to [1]. Here, we use a disturbed discrete time single integrator system with a sampling time of T to model the mobile robot.…”

“…The robust tube-based MPC framework used here was introduced in [1]. It is based on the idea of local incremental stabilizability of nonlinear systems using a tube-based approach with an ancillary controller κ.…”

Adaptive model predictive control design using multiple model second level adaptation for parameter estimation of two‐degree freedom of helicopter model

“…In Reference 17, authors have proposed an adaptive dual MPC for a nonlinear SISO system to get quadratic stability under mismatched uncertainty. For handling the parametric uncertainty and additive disturbances, a tube‐based adaptive MPC was proposed in Reference 18, where the set‐membership estimation technique is used to provide accurate bounds on the parametric uncertainty. Furthermore, in Reference 19, authors have used a combination of parameter estimation and MPC for max‐plus linear discrete‐time systems where the control covers only a specific class of systems.…”

An adaptive feedback linearized model predictive controller design for a nonlinear multi‐input multi‐output system

“…Later in Reference 9, the author has proposed adaptive NMPC for automated vehicles in combination with a state and parameter estimator. A tube‐based adaptive MPC was proposed in Reference 10, where set‐membership estimation is used to handle the parametric uncertainty. Recently, a learning‐based adaptive MPC has been developed by a researcher in References 11,12, where a modified extended Kalman filter (EKF) is used to perform the joint state and parameter estimation.…”