Nonlinear model predictive control (NMPC) allows one to explicitly treat nonlinear dynamics and constraints. To apply NMPC in real time on embedded hardware, online algorithms as well as efficient code implementations are crucial. A tutorial-style approach is adopted in this article to present such algorithmic ideas and to show how they can efficiently be implemented based on the ACADO Toolkit from MATLAB (MathWorks, Natick, MA, USA). Using its code generation tool, one can export tailored Runge-Kutta methods-explicit and implicit ones-with efficient propagation of their sensitivities. The article summarizes recent research results on autogenerated integrators for NMPC and shows how they allow to formulate and solve practically relevant problems in only a few tens of microseconds. Several common NMPC formulations can be treated by these methods, including those with stiff ordinary differential equations, fully implicit differential algebraic equations, linear input and output models, and continuous output independent of the integration grid. One of the new algorithmic contributions is an efficient implementation of infinite horizon closed-loop costing. As a guiding example, a full swing-up of an inverted pendulum is considered. explicit system of ordinary differential equations (ODEs), but this will be further generalized to implicit systems of differential algebraic equations (DAEs). The optimization problem depends on the parameter N x 0 2 R n x through the initial value constraint of (1b) and can also be time dependent. Hence, the control trajectory obtained by solving problem (1) provides a feedback strategy u .t 0 ; N x 0 /, which depends on the current state and time. The OCP is in practice often solved by a direct approach where one first discretizes the problem to obtain a structured nonlinear program (NLP), which is generally nonconvex. A Newton-type algorithm is able to find a locally optimal solution by solving the Karush-Kuhn-Tucker conditions. Two popular Newton-type techniques are interior point (IP) methods and sequential quadratic programming (SQP) [1]. IP methods for nonconvex problems treat the inequality constraints therein by the use of a smoothening technique [2]. SQP instead consists in sequentially approximating the NLP by convex quadratic program (QP) subproblems.Recent algorithmic progress [3,4] allowed to reduce computational delays between receiving the new state estimate and applying the next control input to the process [3]. This made it possible to apply NMPC also to fast dynamic systems with sampling times in the millisecond or even microsecond range. The real-time iteration (RTI) scheme [5] is an SQP-type online algorithm. The resulting sequence of sparse QPs can either be solved directly using a structure exploiting convex solver such as FORCES [6] or qpDUNES [7] or by reducing the size of the QP subproblems with a condensing technique [8,9] and using a dense linear algebra solver such as qpOASES [10]. A discussion on these algorithmic aspects can be found in [11]. It is important...
This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuoustime dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robust MPC scheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system.
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