A multiscale approximation scheme for explicit model predictive control with stability, feasibility, and performance guarantees

In this paper, an algorithm is introduced based on classical wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolating wavelets. It is proven that the resulting interpolation is everywhere feasible. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a gridded hierarchy, the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. A simple example is provided which both illustrates the approach and motivates further research in this direction

Fast Explicit Nonlinear Model Predictive Control Via Multiresolution Function Approximation with Guaranteed Stability Nonlinear Control Systems

In this paper an algorithm for nonlinear explicit model predictive control is introduced based on multiresolution function approximation that returns a low complexity approximate receding horizon control law built on a hierarchy of second order interpolets. Feasibility and stability guarantees for the approximate control law are given using reachability analysis, where interval methods are used to construct a capture basin (feasible region). A constructive algorithm is provided that combines adaptive function approximation with interval methods to build a receding horizon control law that is suboptimal, yet with a region of guaranteed feasibility and stability. The resulting control law is built on a grid hierarchy that is fast to evaluate in real-time systems

A multiresolution approximation method for fast explicit model predictive control

A model predictive control law is given by the solution to a parametric optimization problem that can be pre- computed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

In this chapter an algorithm for nonlinear explicit model predictive control is presented. A low complexity receding horizon control law is obtained by approximating the optimal control law using multiscale basis function approximation. Simultaneously, feasibility and stability of the approximate control law is ensured through the computation of a capture basin (region of attraction) for the closed-loop system. In a previous work, interval methods were used to construct the capture basin (feasible region), yet this approach suffered due to slow computation times and high grid complexity. In this chapter, we suggest an alternative to interval analysis based on zonotopes. The suggested method significantly reduces the complexity of the combined function approximation and verification procedure through the use of DC (difference of convex) programming, and recursive splitting. The result is a multiscale function approximation method with improved computational efficiency for fast nonlinear explicit model predictive control with guaranteed stability and constraint satisfaction

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

Abstract Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit MPC, will often be prohibitively complex for fast systems. In this work we study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs (SDPs) or linear programs (LPs), in case of 2-norm or 1, ∞-norm based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, we provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results. The authors are with the Hybrid an

Shifting strategy for efficient block-based nonlinear model predictive control using real-time iterations

Nonlinear Model Predictive Control (NMPC) requires the use of efficient solutions and strategies for its implementation in fast/real-time systems. A popular approach for this is the Real Time Iteration (RTI) Scheme which uses a shifting strategy, namely the Initial Value Embedding (IVE), that shifts the solution from one sampling time to the next. However, this strategy together with other efficient strategies such as Move Blocking, present a recursive feasibility problem. This paper proposes a novel modified shifting strategy which preserve both recursive feasibility and stability properties, as well as achieves a significant reduction in the computational burden associated with the optimisation. The proposed approach is validated through a simulation of an inverted pendulum where it clearly outperforms other standard solutions in terms of performance and recursive feasibility properties. Additionally, the approach was tested on two computing platforms: a laptop with an i7 processor and a Beaglebone Blue Linux-based computer for robotic systems, where computational gains compared to existing approaches are shown to be as high as 100 times faster

Learning a feasible and stabilizing explicit model predictive control law by robust optimization

Fast model predictive control on embedded sys- tems has been successfully applied to plants with microsecond sampling times employing a precomputed state-to-input map. However, the complexity of this so-called explicit MPC can be prohibitive even for low-dimensional systems. In this pa- per, we introduce a new synthesis method for low-complexity suboptimal MPC controllers based on function approximation from randomly chosen point-wise sample values. In addition to standard machine learning algorithms formulated as convex programs, we provide sufficient conditions on the learning algo- rithm in the form of tractable convex constraints that guarantee input and state constraint satisfaction, recursive feasibility and stability of the closed loop system. The resulting control law can be fully parallelized, which renders the approach particularly suitable for highly concurrent embedded platforms such as FPGAs. A numerical example shows the effectiveness of the proposed method