Abstract
This paper proposes a model predictive control (MPC) approach for non-linear systems where the non-linear dynamics are embedded inside a linear parameter-varying (LPV) representation. The non-linear MPC problem is therefore replaced by an LPV MPC problem, without using linearization. Compared to general non-linear MPC, advantages of this approach are that it allows for the tractable construction of a terminal set and cost, and that only a single convex program must be solved online. The key idea that enables proving recursive feasibility and stability, is to restrict the state evolution of the non-linear system to a time-varying sequence of state constraint sets. Because in LPV embeddings, there exists a relationship between the scheduling and state variables, these state constraints are used to construct a corresponding future scheduling tube. Compared to non-time-varying state constraints, tighter bounds on the future scheduling trajectories are obtained. Computing a scheduling tube in this setting requires applying a non-linear function to the sequence of constraint sets. Outer approximations of this non-linear projection-based scheduling tube can be found, e.g., via interval analysis. The computational properties of the approach are demonstrated on numerical examples.
Highlights
Controlling a non-linear system using model predictive control (MPC) typically requires solving a non-convex optimization problem online, which can be computationally demanding
The paper [15] presents the construction of a terminal set and cost, based on the concept of finite-step contraction, to guarantee stability of a tube model predictive control (TMPC) scheme for linear parameter-varying (LPV) systems
This paper considers constrained non-linear systems (1) that can be embedded in a constrained LPV state-space (LPV-SS) representation of the form x(k + 1) = A θ(k) x(k) + Bu(k), (4a) θ(k) = μ(x(k)), (4b) where the matrix A(⋅) is an affine function given as
Summary
Controlling a non-linear system using MPC typically requires solving a non-convex optimization problem online, which can be computationally demanding. The resulting optimal state and input trajectories are used to generate a new predicted scheduling trajectory, and the procedure is iterated until the predicted trajectories converge This is computationally highly efficient because at each time instant, only the solution of a sequence of LTV MPC sub-problems is required. Where [10] solves a sequence of LTV MPC problems at each time instant, the tube-based method that this paper proposes solves a single LP or QP which is, more complex than the problems solved in [10] Both approaches provide a different trade-off between computational efficiency and a priori stability guarantees. The paper [15] presents the construction of a terminal set and cost, based on the concept of finite-step contraction, to guarantee stability of a tube model predictive control (TMPC) scheme for LPV systems.
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