# Recent Posts

### Product Design Space

My current mind map

Is MBRL=MPC?

### Convex Optimization: A Practical Guide

Python scripts included

# Projects

#### Computational Tools for Piecewise Affine Systems

PWATOOLS is a set of tools for the analysis and design of piecewise affine (PWA) systems.

#### Model based automatic code generation for nonlinear model predictive control

Symbolic computation methods are employed to generate optimized C code for nonlinear model predictive controllers.

# Selected Publications

### Model based automatic code generation for nonlinear model predictive control

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.
International Workshop on Numerical Software Verification, 2016.

### Design environment for nonlinear model predictive control

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.
SAE Technical Paper, 2016.

### Estimation of region of attraction for polynomial nonlinear systems: A numerical method

A numerical method is proposed to enlarge the estimation of the region of attraction for locally asymptotically stable equilibrium points. The proposed algorithm uses an iteratively updated shape factor based on the dynamic model of the system. The problem is formulated as an iterative sum of squares programming. The effectiveness of the proposed method compared to existing methods is shown using numerical examples.
ISA transactions, 2014.

### A unified dissipativity approach for stability analysis of piecewise smooth systems

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.
Automatica, 2011.

### Real-time estimation of vehicle state and tire-road friction forces

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.
American Control Conference (ACC), 2001.

# Recent Publications

. Model based automatic code generation for nonlinear model predictive control. International Workshop on Numerical Software Verification, 2016.

. Design environment for nonlinear model predictive control. SAE Technical Paper, 2016.

. Piecewise-affine approximations for a powertrain control verification benchmark. ARCH14-15. 1st and 2nd International Workshop on Applied Verification for Continuous and Hybrid Systems, 2015.

. A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems. International Journal of Robust and Nonlinear Control, 2014.

. Estimation of region of attraction for polynomial nonlinear systems: A numerical method. ISA transactions, 2014.

. Active front-steering control of a sport utility vehicle using a robust linear quadratic regulator method, with emphasis on the roll dynamics. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2013.

. A convex formulation of controller synthesis for piecewise-affine slab systems based on invariant sets. 2012 IEEE 51st Annual Conference on Decision and Control (CDC), 2012.

. PWATOOLS: A MATLAB toolbox for piecewise-affine controller synthesis. American Control Conference (ACC), 2012.

. Controller design for vehicle stability improvement using optimal distribution of tire forces. 2nd International Conference on Control, Instrumentation and Automation (ICCIA), 2011.

. Handling safety improvement for steer-by-wire vehicle using fuzzy controller. 2nd International Conference on Control, Instrumentation and Automation (ICCIA), 2011.

. Stability of sampled-data piecewise affine systems: A time-delay approach. Automatica, 2009.

. Backstepping controller synthesis for piecewise polynomial systems: a sum of squares approach. American Control Conference (ACC), 2009.

. Extension of local linear controllers to global piecewise affine controllers for uncertain non-linear systems. International Journal of Systems Science, 2008.

. Sampled-data piecewise affine slab systems: A time-delay approach. American Control Conference (ACC), 2008.

. Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach. 46th IEEE Conference on Decision and Control, 2007.

. Backstepping controller synthesis for piecewise affine systems: A sum of squares approach. IEEE International Conference on Systems, Man and Cybernetics, 2007.

. Extension of a local linear controller to a stabilizing semi-global piecewise affine controller. 7th Portuguese Conference on Automatic Control, 2006.

. Fuzzy based stability enhancement system for a four-motor-wheel electric vehicle. SAE Transactions, 2002.

. Improving vehicle handling by differential braking. Sharif Scientific Research Quarterly, 2002.

. Real-time estimation of vehicle state and tire-road friction forces. American Control Conference (ACC), 2001.

. Estimation of vehicle speed and tire-road friction forces. Amirkabir Journal of Science and Technology, 2001.

# Talks

Controller Synthesis for Nonholonomic Robots
Jul 25, 2013 1:00 PM
Model Based Code Generation for Nonlinear Model Predictive Control
Jul 17, 2016 2:00 PM
Modeling, Control, and Optimization for Aerospace Systems
Jun 26, 2012 1:00 PM
Nonlinear Model Predictive Control Webinar
Dec 19, 2014 1:00 PM
Nonlinear Model Predictive Control for Autonomous Race Cars
Nov 8, 2017 2:00 PM
PWATOOLS: A MATLAB Toolbox for Piecewise-Affine Controller Synthesis
Jun 28, 2012 5:20 PM
Vehicle Control Systems
Jul 25, 2001 1:00 PM
Virtual Prototyping
Jan 27, 2012 10:20 AM
Virtual Prototyping
Apr 9, 2014 10:20 AM