Banded Null Basis and ADMM for Embedded MPC

© 2017 In this paper, we propose an improved QP solver for embedded implementations of MPC controllers. We adopt a “reduced Hessian” approach for handling the equality constraints that arise in the well-known “banded” formulation of MPC (in which the predicted states are not eliminated). Our key observation is that a banded basis exists for the null space of the banded equality-constraint matrix, and that this leads to a QP of the same size as the “condensed” formulation of MPC problems, which is considerably smaller than the “banded” formulation. We use the Alternating Direction Method of Multipliers (ADMM) - which is known to be particularly suitable for embedded implementations - to solve this smaller QP problem. Our C implementation results for a particular MPC example (a 9-state, 3-input quadrotor) show that our proposed algorithm is about 4 times faster than an existing well-performing ADMM variant (“indirect indicator” ADMM or “iiADMM”) and 3 times faster than the well-known QP solver CVXGEN. The convergence rate and code size of the proposed ADMM variant is also comparable with iiADMM.National Research Foundation, Singapore

Model predictive control with prioritised actuators

This paper deals with the control of systems for which there is a clear distinction between preferred and auxiliary actuators, the latter to be used only when the control error is large. Explicit MPC and exact penalty functions are used to show how ℓasso-MPC can implement this idea. Two ℓasso-MPC versions are reviewed, that allow the designer to impose a certain nominal operations zone, namely, a neighbourhood of the set-point in which the auxiliary actuators are never used. For the sake of brevity, the required procedures are shown only for version 1, but it is also discussed how they can be extended to version 2. Limitations due to the presence of constraints are also formalised. The ℓasso-MPC version 1 can be used to embed an existing linear quadratic MPC, while ℓasso-MPC version 2 can be used to obtain multiple levels of priority. The paradigm is demonstrated for version 1 through the control of the linearised lateral dynamics of a Boeing 747. In particular, the approach uses the spoilers only when the control error is larger than a desired threshold.Research supported by the EPSRC grant “Control for Energy and Sustainability”, EP/G066477/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ECC.2015.733059

Custom optimization algorithms for efficient hardware implementation

The focus is on real-time optimal decision making with application in advanced control systems. These computationally intensive schemes, which involve the repeated solution of (convex) optimization problems within a sampling interval, require more efficient computational methods than currently available for extending their application to highly dynamical systems and setups with resource-constrained embedded computing platforms. A range of techniques are proposed to exploit synergies between digital hardware, numerical analysis and algorithm design. These techniques build on top of parameterisable hardware code generation tools that generate VHDL code describing custom computing architectures for interior-point methods and a range of first-order constrained optimization methods. Since memory limitations are often important in embedded implementations we develop a custom storage scheme for KKT matrices arising in interior-point methods for control, which reduces memory requirements significantly and prevents I/O bandwidth limitations from affecting the performance in our implementations. To take advantage of the trend towards parallel computing architectures and to exploit the special characteristics of our custom architectures we propose several high-level parallel optimal control schemes that can reduce computation time. A novel optimization formulation was devised for reducing the computational effort in solving certain problems independent of the computing platform used. In order to be able to solve optimization problems in fixed-point arithmetic, which is significantly more resource-efficient than floating-point, tailored linear algebra algorithms were developed for solving the linear systems that form the computational bottleneck in many optimization methods. These methods come with guarantees for reliable operation. We also provide finite-precision error analysis for fixed-point implementations of first-order methods that can be used to minimize the use of resources while meeting accuracy specifications. The suggested techniques are demonstrated on several practical examples, including a hardware-in-the-loop setup for optimization-based control of a large airliner.Open Acces

Predictive Control for Alleviation of Gust Loads on Very Flexible Aircraft

In this work the dynamics of very flexible aircraft are described by a set of non-linear, multi-disciplinary equations of motion. Primary structural components are represented by a geometrically-exact composite beam model which captures the large dynamic deformations of the aircraft and the interaction between rigid-body and elastic degrees-of-freedom. In addition, an implementation of the unsteady vortex-lattice method capable of handling arbitrary kinematics is used to capture the unsteady, three-dimensional flow-eld around the aircraft as it deforms. Linearization of this coupled nonlinear description, which can in general be about a nonlinear reference state, is performed to yield relatively high-order linear time-invariant state-space models. Subsequent reduction of these models using standard balanced truncation results in low-order models suitable for the synthesis of online, optimization-based control schemes that incorporate actuator constraints. Predictive controllers are synthesized using these reduced-order models and applied to nonlinear simulations of the plant dynamics where they are shown to be superior to equivalent optimal linear controllers (LQR) for problems in which constraints are active

Coordinate-Descent Augmented Lagrangian Methods for Interpretative and Adaptive Model Predictive Control

Model predictive control (MPC) of nonlinear systems suffers a trade-off between model accuracy and real-time compu- tational burden. This thesis presents an interpretative and adaptive MPC (IA-MPC) framework for nonlinear systems, which is related to the widely used approximation method based on successive linearization MPC and Extended Kalman Filtering (SL-MPC-EKF). First, we introduce a solution algo- rithm for linear MPC that is based on the combination of Co- ordinate Descent and Augmented Lagrangian (CDAL) ideas. The CDAL algorithm enjoys three features: (i) it is construction-free, in that it avoids explicitly constructing the quadratic pro-gramming (QP) problem associated with MPC; (ii) is matrix-free, as it avoids multiplications and factorizations of matri-ces; and (iii) is library-free, as it can be simply coded without any library dependency, 90-lines of C-code in our implemen-tation. We specialize the algorithm for both state-space for-mulations of MPC and formulations based on AutoRegres-sive with eXogenous terms models (CDAL-ARX). The thesis also presents a rapid-prototype MPC tool based on the gPROMS platform, in which the qpOASES and CDAL algorithm was integrated. In addition, based on an equivalence between SS-based and ARX-based MPC problems we show,we investigate the relation between the proposed IA-MPC and the classical SL-MPC-EKF method. Finally, we test and show the effectiveness of the proposed IA-MPC frameworkon four typical nonlinear MPC benchmark examples

Certification of the proximal gradient method under fixed-point arithmetic for box-constrained QP problems

In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of optimization algorithms. However, a typical underlying assumption is that the operations performed by the algorithm are exact, i.e., that there is no numerical error during the mathematical operations, which is hardly a valid assumption in a real hardware implementation. This is particularly true in the case of fixed-point hardware, where computational inaccuracies are not uncommon. This article presents a certification procedure for the proximal gradient method for box-constrained QP problems implemented in fixed-point arithmetic. The procedure provides a method to select the minimal fractional precision required to obtain a certain suboptimality bound, indicating the maximum number of iterations of the optimization method required to obtain it. The procedure makes use of formal verification methods to provide arbitrarily tight bounds on the suboptimality guarantee. We apply the proposed certification procedure on the implementation of a non-trivial model predictive controller on $32$-bit fixed-point hardware.Comment: 8 page

Dictionary-free Koopman model predictive control with nonlinear input transformation

This paper introduces a method for data-driven control based on the Koopman operator model predictive control. Unlike exiting approaches, the method does not require a dictionary and incorporates a nonlinear input transformation, thereby allowing for more accurate predictions with less ad hoc tuning. In addition to this, the method allows for input quantization and exploits symmetries, thereby reducing computational cost, both offline and online. Importantly, the method retains convexity of the optimization problem solved within the model predictive control online. Numerical examples demonstrate superior performance compared to existing methods as well as the capacity to learn discontinuous lifting functions

Nonlinear model predictive low-level control

This dissertation focuses on the development, formalization, and systematic evaluation of a novel nonlinear model predictive control (MPC) concept with derivative-free optimization. Motivated by a real industrial application, namely the position control of a directional control valve, this control concept enables straightforward implementation from scratch, robust numerical optimization with a deterministic upper computation time bound, intuitive controller design, and offers extensions to ensure recursive feasibility and asymptotic stability by design. These beneficial controller properties result from combining adaptive input domain discretization, extreme input move-blocking, and the integration with common stabilizing terminal ingredients. The adaptive discretization of the input domain is translated into time-varying finite control sets and ensures smooth and stabilizing closed-loop control. By severely reducing the degrees of freedom in control to a single degree of freedom, the exhaustive search algorithm qualifies as an ideal optimizer. Because of the exponentially increasing combinatorial complexity, the novel control concept is suitable for systems with small input dimensions, especially single-input systems, small- to mid-sized state dimensions, and simple box-constraints. Mechatronic subsystems such as electromagnetic actuators represent this special group of nonlinear systems and contribute significantly to the overall performance of complex machinery. A major part of this dissertation addresses the step-by-step implementation and realization of the new control concept for numerical benchmark and real mechatronic systems. This dissertation investigates and elaborates on the beneficial properties of the derivative-free MPC approach and then narrows the scope of application. Since combinatorial optimization enables the straightforward inclusion of a non-smooth exact penalty function, the new control approach features a numerically robust real-time operation even when state constraint violations occur. The real-time closed-loop control performance is evaluated using the example of a directional control valve and a servomotor and shows promising results after manual controller design. Since the common theoretical closed-loop properties of MPC do not hold with input moveblocking, this dissertation provides a new approach for general input move-blocked MPC with arbitrary blocking patterns. The main idea is to integrate input move-blocking with the framework of suboptimal MPC by defining the restrictive input parameterization as a source of suboptimality. Finally, this dissertation extends the proposed derivative-free MPC approach by stabilizing warm-starts according to the suboptimal MPC formulation. The extended horizon scheme divides the receding horizon into two parts, where only the first part of variable length is subject to extreme move-blocking. A stabilizing local controller then completes the second part of the prediction. The approach involves a tailored and straightforward combinatorial optimization algorithm that searches efficiently for suboptimal horizon partitions while always reproducing the stabilizing warm-start control sequences in the nominal setup