System Identification and Model Predictive Control using CVXGEN for Electro-Hydraulic Actuator

Hydraulics have been widely used in heavy industries for decades. The demand for intelligent hydraulic control system has been increasing as tough robotic researches are getting more popular. Despite the high power to weight ratio delivery, the hydraulic actuator suffers from nonlinearity properties that cause difficulties in applying precise position control.  In this paper we proposed Model Predictive Control (MPC) to control an Electro-Hydraulic Actuator (EHA) where its dynamic characteristics is obtained through system identification method.  Control signal generation optimisation and constraint handling are seldom included in the conventional control system design process. Therefore we introduce CVXGEN, a Code Generator for Embedded Convex Optimization that utilises the Quadratic Programming (QP) interior-point solver for MPC optimisation problem. Predictive Functional Control (PFC) is used to validate the CVXGEN-MPC and both algorithms are implemented in simulation and experiment of EHA position control to highlight the optimisation and constraint handling problem. Control performance, control effort, constraint handling and disturbance handling of both methods are discussed

Approximate Dynamic Programming via Sum of Squares Programming

We describe an approximate dynamic programming method for stochastic control problems on infinite state and input spaces. The optimal value function is approximated by a linear combination of basis functions with coefficients as decision variables. By relaxing the Bellman equation to an inequality, one obtains a linear program in the basis coefficients with an infinite set of constraints. We show that a recently introduced method, which obtains convex quadratic value function approximations, can be extended to higher order polynomial approximations via sum of squares programming techniques. An approximate value function can then be computed offline by solving a semidefinite program, without having to sample the infinite constraint. The policy is evaluated online by solving a polynomial optimization problem, which also turns out to be convex in some cases. We experimentally validate the method on an autonomous helicopter testbed using a 10-dimensional helicopter model.Comment: 7 pages, 5 figures. Submitted to the 2013 European Control Conference, Zurich, Switzerlan

Embedded Code Generation with CVXPY

We introduce CVXPYgen, a tool for generating custom C code, suitable for embedded applications, that solves a parametrized class of convex optimization problems. CVXPYgen is based on CVXPY, a Python-embedded domain-specific language that supports a natural syntax (that follows the mathematical description) for specifying convex optimization problems. Along with the C implementation of a custom solver, CVXPYgen creates a Python wrapper for prototyping and desktop (non-embedded) applications. We give two examples, position control of a quadcopter and back-testing a portfolio optimization model. CVXPYgen outperforms a state-of-the-art code generation tool in terms of problem size it can handle, binary code size, and solve times. CVXPYgen and the generated solvers are open-source

Actuators coordination of heavy vehicles using model predictive control allocation

This report proposes the use of a novel method called Model Predictive Control Allocation (MPCA) in order to conveniently coordinate the different actuators present on a heavy vehicle. The actuators analysed in this report are disc brakes, powertrain and rear active steering. All these actuators can technically be controlled by an external electronic device and their utilization has an impact on the planar dynamics of the vehicle. The actuators are designed so that, if the driver wants to modify the vehicle behaviour, there are several ways of using the actuators that lead to the same requested behaviour. This property identifies the vehicle as an over-actuated system. Considering the nature of all the actuators and their effects on the vehicle is essential for the designated method to coordinate the actuators. The method used for the coordination merges the characteristics of two different types of controllers: Model Predictive Control (MPC) and Control Allocation (CA). The potential of a model predictive control method resides in its ability to explicitly take into account the nature of the actuators for a certain time horizon ahead before deciding the control action to be applied to the system. The control allocation, on the other hand, is a suitable method to decide how to combine the actuators in order to modify the behaviour of the vehicle. The peculiarity of these controllers lies in the way they compute the control input to the system. Unlike a classical PID controller, in fact, they use a cost function, which has to be iteratively minimized, in order to find out the best input for the system. Common issues related to this class of controllers are the robustness and speed of the algorithms used to solve the problem. The problem defined by the MPCA controller belongs to the class of Quadratic Programming (QP) problems for which several methods have been developed. A primal-dual interior-point method with Mehrotra’s predictor-corrector is used by the solver selected to deal with the QP problem. In order to evaluate the performance of the controller, three test scenarios have been analysed: split-� braking, split-� acceleration and brake blending. In each one of the scenarios there is a need to precisely coordinate the actuators in order to improve the vehicle’s dynamics. The expected behaviour of the controller when facing the three different situations has firstly been analysed and explained. Later, the controller has been validated using simulations and tests on a real vehicle. Both simulations and tests have shown promising results. The controller is able to effectively deal with each one of the situations leading to a satisfactory enhancement of the vehicle dynamics. The controller has also been compared with other methods, a Control Allocation formulation with rate limits and a vehicle without rear active steering (RAS). In general, better performances can be observed during the transitions when using the MPCA formulation rather than the CA formulation and improved stability can be achieved on the vehicle when the RAS is introduced. The different behaviours of the vehicle for every different scenario have been presented and explained

Field programmable gate array based predictive control system for spacecraft rendezvous in elliptical orbits

A field programmable gate array (FPGA)-based model predictive controller (MPC) for two phases of spacecraft rendezvous is presented. Linear time varying prediction models are used to accommodate elliptical orbits, and a variable prediction horizon is used to facilitate finite time completion of the longer-range man{\oe}uvres, whilst a fixed and receding prediction horizon is used for fine-grained tracking at close range. The resulting constrained optimisation problems are solved using a primal dual interior point algorithm. The majority of the computational demand is in solving a system of simultaneous linear equations at each iteration of this algorithm. To accelerate these operations, a custom circuit is implemented, using a combination of Mathworks HDL Coder and Xilinx System Generator for DSP, and used as a peripheral to a MicroBlaze soft core processor on the FPGA, on which the remainder of the system is implemented. Certain logic that can be hard-coded for fixed sized problems is implemented to be configurable online, in order to accommodate the varying problem sizes associated with the variable prediction horizon. The system is demonstrated in closed loop by linking the FPGA with a simulation of the spacecraft dynamics running in Simulink on a PC, using Ethernet. Timing comparisons indicate that the custom implementation is substantially faster than pure embedded software-based interior point methods running on the same MicroBlaze, and could be competitive with a pure custom hardware implementation.This work was supported by the Engineering and Physical Sciences Research Council Grant Number [EP/G030308/1] as well as industrial support from Xilinx, Mathworks, and the European Space Agency

Multi-parametric Programming for Model Predictive Control

Model predictive control (MPC) solves a quadratic optimization problem to generate control law in each step. The usual methods of solution for quadratic optimization problem are interior point method, active set method etc. But most of the techniques are computationally heavy to perform the job in small amount of time. So a method is required where on-line computation is less. In multi-parametric quadratic programming (mp-QP) method an off-line computation is done a prior and a binary search tree is prepared. The on-line computation mainly involves a search through the binary-tree. The mp-QP is suitable for the class of optimization problem, where the objective function is to minimize or maximize a performance criterion subject to a given set of constraints where some of the parameter vary between lower and upper bounds. Also mp-QP is suitable for multi-objective optimization, where multi criteria problems can be reformulated as multi-parametric programming problems and a parametrized optimal solution is obtained. Multi-parametric programming is a technique for obtaining: (i) the objective and optimization variable as functions of the varying parameters and (ii) the regions in the space of the parameters where these functions are valid. The newly developed convex optimization solver CVXGEN is utilized successfully for off-line calculations which involves of dividing the parameter space into different polyhedral regions.In each one, the objective function has a constant value. The process involves another kind of optimization problem. For CVXGEN, worst case solving time is in milliseconds, even for a large problem.Thus, the use of CVXGEN minimizes the off-line calculation in mp-QP technique. In this work, an input constraint MPC problem is chosen from existing literature. The problem is solved for both two step prediction and three step prediction cases.The control input and states are ploted for both the MPC problems, and the results are compared

Dominant speed factors of active set methods for fast MPC

The paper presents a review of active set (AS) algorithms that have been deployed for implementation of fast model predictive control (MPC). The main purpose of the survey is to identify the dominant features of the algorithms that contribute to fast execution of online MPC and to study their influence on the speed. The simulation study is conducted on two benchmark examples where the algorithms are analyzed in the number of iterations and in the workload per iteration. The obtained results suggest directions for potential improvement in the speed of existing AS algorithms

Model Predictive Control for Micro Aerial Vehicles: A Survey

This paper presents a review of the design and application of model predictive control strategies for Micro Aerial Vehicles and specifically multirotor configurations such as quadrotors. The diverse set of works in the domain is organized based on the control law being optimized over linear or nonlinear dynamics, the integration of state and input constraints, possible fault-tolerant design, if reinforcement learning methods have been utilized and if the controller refers to free-flight or other tasks such as physical interaction or load transportation. A selected set of comparison results are also presented and serve to provide insight for the selection between linear and nonlinear schemes, the tuning of the prediction horizon, the importance of disturbance observer-based offset-free tracking and the intrinsic robustness of such methods to parameter uncertainty. Furthermore, an overview of recent research trends on the combined application of modern deep reinforcement learning techniques and model predictive control for multirotor vehicles is presented. Finally, this review concludes with explicit discussion regarding selected open-source software packages that deliver off-the-shelf model predictive control functionality applicable to a wide variety of Micro Aerial Vehicle configurations