Developing a Rollover Stability Control System Using Model-Based Design

The trend of integrating active safety mechanisms in automotive electronics is increasing, forcing automakers to integrate rollover prevention into traditional vehicle chassis control systems. For example, antilock braking systems and traction control systems have now been enhanced to include rollover prevention. The National Highway Traffic Safety Administration (NHTSA) has furthered this trend by mandating that all 2011 model year vehicles and newer vehicles be equipped with rollover prevention controllers. This requirement is based on NHTSA’s analysis of accident data on rollover crashes. For example, according to data from NHTSA’s National Center for Statistics and Analysis, in 2001, 10,138 people died in rollover crashes, accounting for 32% of all deaths in that year. Implementing active safety mechanisms can reduce the risk of vehicle rollovers, thereby reducing potential casualties. One way to reduce the risk of rollovers is to implement electronic stability control (ESC) that applies differential braking based on measured and predicted vehicle states. This article describes the use of Model-Based Design to develop and automatically optimize ESC for a sport utility vehicle (SUV).

Car and controller models

In Model-Based Design, the core concept is an executable specification or model that describes the dynamic behavior of the system. You can significantly reduce the development cost and time associated with controller design by leveraging a validated vehicle model (in this case, a high-fidelity SUV model). Digital simulations of the model can be used to study how the vehicle responds to different steering maneuvers, and such tests can be easily repeated with different parameters such as road surfaces, tire types, and vehicle attributes. In addition, the model can be used in the development and validation of embedded control systems.

The vehicle used in this article is a typical mid-size SUV. The vehicle model is available in CarSim®, a commercial off-the-shelf vehicle dynamics simulation tool. The performance of the vehicle model is validated against test data and is suitable for simulating the vehicle's response to severe roll motions. The vehicle model has two independent front suspensions and a solid rear axle to support the sprung masses. The nonlinear mathematical model provides degrees of freedom for the sprung masses, axles, wheels, steering system, and braking system. The vehicle model can be customized with different vehicle parameters as well as road and environmental conditions.

Figure 1: Setting vehicle parameters using the CarSim user interface.

Figure 1 shows the CarSim user interface and some of the physical vehicle parameters used to build the vehicle model. These parameters can be modified individually from the controller parameters to test the behavior of the controller under different vehicle conditions, such as one passenger, multiple passengers, and high center of gravity. The vehicle model used in this article applies steering inputs consistent with the NHTSA fishhook maneuver test, a standard test for evaluating dynamic vehicle stability. This test is designed to simulate the actions a driver might take when avoiding an obstacle that suddenly appears on the road. For the digital simulation, we set the steering input for the SUV model and verified that the vehicle would roll over without ESC.

Controller development and optimization

[page]

The ESC implemented in this article avoids unsafe vehicle roll and slip maneuvers caused by driver input. It applies differential braking to the wheels to adjust the vehicle roll and slip rate while minimizing the vehicle speed reduction caused by the electronic brakes automatically applied by the controller. Our ESC implementation switches between three control modes. The control modes are activated based on the three possible causes of the vehicle entering a wheel slip state: loss of traction, excessive roll, and excessive slip. The mode switching logic controls a set of proportional-integral-derivative (PID) compensators that adjust the brake pressure applied by the driver to the wheels based on measured and estimated parameters. The controller design implemented in Simulink® has six PID gains that can be changed to optimize ESC performance.

In this model, we can look at wheel speed, brake pressure, body roll, sideslip, and slip. Some vehicle states are predicted from available sensor data, just like in a real vehicle controller, while others are predicted from mathematical relationships between measured and predicted parameters. Vehicle speed is predicted from the average wheel speed of the unbraked wheels. A low-pass filter is used to simulate the effect of vehicle inertia on the measured wheel speeds to avoid uncertainty in the vehicle speed measurements when brake pressure is applied to the four wheels.

Body slip is a parameter that is difficult to measure directly without the use of expensive sensors. Our ESC implementation predicts body slip from the measured body slip. Body roll angle is predicted by a transfer function that relates lateral acceleration to body roll angle. This transfer function is valid when the body roll angle is within specified design limits. By ensuring that the optimization algorithm will impose strict actions on the controller when the predicted body roll angle exceeds the design limits, we show that we do not need a prediction algorithm that can accurately predict the body roll angle outside the design range. As a result, we can significantly simplify the body roll angle prediction algorithm under normal vehicle operating conditions.

Once the controller structure is specified, the next task is to tune the controller gains to meet the design requirements. Without models that can be experimented with in a systematic way, engineers typically rely on knowledge gained from past vehicle programs or invest a lot of time in trying to tune the parameter values ​​of the PID compensator through road experiments. Model-Based Design takes the hassle out of hardware and instead uses models to explore the design space. By combining these models with automated optimization-based methods, engineers can significantly reduce the need for tedious testing through prototypes or simulations to obtain optimal controller gains.

For this application, the optimization algorithm first sets the controller gains to zero and finds the optimal controller gains that keep the system within the design limits, requiring about 100 iterations and about 4 minutes of computation time. Iterative trial-and-error methods require intensive manual testing, and even if the testing is fully repeatable and the rollover during the tuning process does not cause any damage to the vehicle, it would take more than 4 hours to run the same number of test cases. A 10-second NHTSA fishhook maneuver can be digitally simulated in less than 3 seconds on a modern PC and can be repeated indefinitely without the overhead associated with road testing.

In this model, we are looking for the optimal controller gains for the PID compensator in the ESC to keep the vehicle's roll angle, slip ratio, and slip angle within certain design limits while minimizing speed losses due to differential braking. Six tunable gains provide a nearly infinite number of controller gain combinations, making exhaustive testing nearly impossible. Simulink® Response Optimization™ allows you to graphically set system requirements to limit rollover and vehicle slip while minimizing energy losses from ESC braking. Once performance criteria are specified, an optimization-based routine automatically adjusts parameters to enable the vehicle to perform a fishhook maneuver without rolling over.

We provide the signal to be constrained to the Signal Constraint block and graphically set its design constraints, as shown by the horizontal solid lines in Figure 2. We selected the following requirements (constraints) to meet the design goals:

The vehicle rollover angle is limited to +/-11.5 degrees.

[page]

· Vehicle slip angle is limited to +/-11.5 degrees.

· The maximum slip rate is set to +/-37.25 degrees/second.

· The minimum vehicle speed at the end of the Fishhook manipulation experiment was set to 10 mph.

· The simulation end time is set to 10 seconds.

To avoid premature termination of the vehicle rollover simulation due to an incorrect set of controller gain values, a simulation time limit needs to be specified.

Figure 2: Signals provided to the Signal Constraint module (left) and the evolution of the rollover and slip signals during the optimization process (right). The yellow area represents the impermissible signal value range.

Each signal constraint defines a piecewise linear upper and lower limit on the signal. During the optimization process, the controller gains are adjusted and the simulation is repeated in an iterative loop until the simulated signal meets the specified bounds or the optimization routine fails to solve the problem. Figure 2 shows the evolution of the rollover and slip signals as the optimization algorithm iterates to a solution. When solving such a feasibility problem, the optimization algorithm calculates the maximum signed distance between the constrained signal and each piecewise linear bound. Typically, a negative value indicates that the corresponding constraint is met.

[page]

The optimization algorithm uses the signed distance to each bound to update the controller parameters. The optimization algorithm uses a method independent of the numerical solution used to compute the system state when formulating the optimization problem. Gradient-based or non-gradient-based methods, such as genetic algorithms, can be used. In this case, given the switching nature of the controller and the subsequent non-smooth behavior, it is difficult for a gradient-based solution to yield a global solution. Therefore, a pattern search algorithm is used. In practice, we recommend switching between multiple types of optimization methods to ensure that the optimization algorithm can find the global extremum and exclude convergence to a local minimum of the cost function.
Controller Validation and Performance Validation

Figure 3: Visualization of the behavior of SUVs with and without ESC during a fishhook maneuver at 50 mph. The blue SUV is equipped with optimized ESC, and the red SUV is without ESC.

Figure 3 shows the performance of the optimized ESC in preventing vehicle rollovers. The red car was not equipped with a controller and rolled over, while the blue car was equipped with an optimized controller. Through such simulations, we can demonstrate the controller design that can prevent SUV rollovers, greatly reducing the number of road tunings and avoiding reliance on actual vehicle testing.

Next Steps and Closing Remarks

The next step in the design work usually involves converting the control algorithm from the Simulink model into code implemented on the chassis controller. To perform design verification before the vehicle goes into production, the code can be tested on the road using a prototype vehicle equipped with measurement instruments using integrated rapid prototyping and hardware-in-the-loop (HIL) simulation tools. The algorithm can be implemented using production code generation tools to obtain code implemented on the prototype vehicle, which can minimize errors in the conversion process and further accelerate the vehicle development process. In addition, using this model, engineers can test the controller under different vehicle configurations, enabling rapid modifications and maximizing the reuse of the controller design in multiple vehicle programs.

This article highlights the use of Model-Based Design in developing an ESC algorithm to address the rollover problem. It also demonstrates a method to automatically tune the ESC based on design requirements.