This paper demonstrates a symbolic tool that generates C code for nonlinear model predictive controllers. The optimality conditions are derived in a quick tutorial on optimal control. A model based workflow using MapleSim for modeling and simulation, and Maple for analysis and code generation is then explained. In this paper, we assume to have a control model of a nonlinear plant in MapleSim. The first step of the workflow is to get the equations of the control model from MapleSim. These equations are usually in the form of differential algebraic equations. After converting the equations to ordinary differential equations, the C code for the model predictive controller is generated using a tool created in Maple. The resulting C code can be used to simulate the control algorithm and program the hardware controller. The proposed tool for automatic code generation for model predictive controllers is open and can be employed by users to create their own customized code generation tool.

Model Predictive Control (MPC) design methods are becoming popular among automotive control researchers because they explicitly address an important challenge faced by today’s control designers: How does one realize the full performance potential of complex multi-input, multi-output automotive systems while satisfying critical output, state and actuator constraints? Nonlinear MPC (NMPC) offers the potential to further improve performance and streamline the development for those systems in which the dynamics are strongly nonlinear. These benefits are achieved in the MPC framework by using an on-line model of the controlled system to generate the control sequence that is the solution of a constrained optimization problem over a receding horizon. Motivated by the application of NMPC to the Diesel engine air path control problem, we present a control design environment that leverages Maple’s symbolic computation engine to facilitate NMPC problem formulation, solution, and C code-generation. Given the limited on-line computational resources available for automotive control implementation and the dependence of effective NMPC problem formulation and solution on the application at hand, the designer needs to be able to fully explore the NMPC formulation / solution / computation cost design space. Thus our Symbolic Computing Design Environment (SCDE) for NMPC is constructed so that the designer can rapidly evaluate the performance and computation cost of several implementation options. In particular, we show by example how SCDE can be used to choose between numeric and symbolic solutions approaches and reduce NMPC computation cost by generating functions from the controller’s equations that re-use the sub-expressions common to different aspects of the solution.

This paper introduces a numerical method to estimate the region of attraction for polynomial nonlinear systems using sum of squares programming. This method computes a local Lyapunov function and an invariant set around a locally asymptotically stable equilibrium point. The invariant set is an estimation of the region of attraction for the equilibrium point. In order to enlarge the estimation, a subset of the invariant set defined by a shape factor is enlarged by solving a sum of squares optimization problem. In this paper, a new algorithm is proposed to select the shape factor based on the linearized dynamic model of the system. The shape factor is updated in each iteration using the computed local Lyapunov function from the previous iteration. The efficiency of the proposed method is shown by a few numerical examples.

The main objective of this paper is to present a unified dissipativity approach for stability analysis of piecewise smooth (PWS) systems with continuous and discontinuous vector fields. The Filippov definition is considered for the solution of these systems. Using the concept of generalized gradients for nonsmooth functions, sufficient conditions for the stability of a PWS system are formulated based on Lyapunov theory. The importance of the proposed approach is that it does not need any a-priori information about attractive sliding modes on switching surfaces, which is in general difficult to obtain. A section on application of the main results to piecewise affine (PWA) systems followed by a section with extensive examples clearly show the usefulness of the proposed unified methodology. In particular, we present an example with a stable sliding mode where the proposed method works and previously suggested methods fail.

An approach to estimate vehicle state and tire road friction forces using an extended Kalman filter (EKF) is presented. A numerically stable algorithm is used to implement the EKF. This approach does not require knowledge of the tire model and road friction coefficient. This is an advantage, because although many tire models have been developed so far, there is still a significant difference between these models and the real behavior of the tire-road interface. The main advantages of the proposed method are numerical stability, computational efficiency and to use vehicle mounted sensors. The effectiveness of the presented method is confirmed by simulation of a lane-change and an ABS braking maneuver for a full vehicle. In these simulations, a seven DOF vehicle model, a Pacejka tire model and a nonlinear model for a hydraulic brake system are used. The results show that the EKF has good performance in presence of significant sensor noise in both scenarios.