In this chapter we have presented a realtime parameter estimation algorithm using a multi- ple model switching approach incorporating simple linear models. Based on the simulation results, we demonstrated the accuracy of the suggested technique as compared to the tra- ditional least squares identification approach, which shows significant benefits. We also presented preliminary tests of the algorithm with off-line measurement data taken from an undisclosed test vehicle, and results were promising. The results showed that the algorithm can also work in cases where the signals are corrupted by noise and bias. Moreover, the load
0 5 10 15 20 0.4 0.5 0.6 0.7 0.8 0.9 t [sec] Estimated CG height, h
[m] with initial candidate modelswith refined models
Figure 2.41: Adaptive CG height estimation.
condition estimator example demonstrated that a simple version of the suggested algorithm can easily be integrated into current rollover or lateral dynamics controllers to enhance their performance. In the last part of the chapter we conducted an analysis of the cost function, and also introduced a simple adaptation scheme to improve estimations based on multiple model estimation method. With simple numerical examples we showed that the suggested adaptation method can provide good estimation results while utilizing a smaller number of identification models as compared to estimations with fixed models alone. One important observation in our analysis was that the multiple model algorithm employing only fixed models required too many models to produce the desired estimation accuracy and perfor- mance (as apparent from numerical simulations, where we had 240 models for CG height estimation based on roll dynamics). Our adaptation scheme can be used to circumvent this problem, which employs only a small number of models initially and are updated and re- parameterized in fixed time intervals. In our numerical simulations we managed to get a good CG height estimation using only 96 models in conjunction with the adaptive estima- tion method, which shows efficacy of the suggested algorithm.
0 5 10 15 20 2 2.5 3 3.5 4x 10 4 Spring stiffness, k [Nm] 0 5 10 15 20 3000 4000 5000 6000 t [sec] Damping coefficient, c [Nms]
c, with initial cand. models c, with refined models k, with initial cand. models k, with refined models
A Methodology for the Design of
Robust Rollover Prevention
Controllers for Automotive Vehicles
In this chapter we present a robust controller design methodology for vehicle rollover prevention utilizing active steering and differential braking actuators. Control design is based on keeping the magnitude of the vehicle performance outputs, including load transfer ratio (LTR), below a certain level in the pres- ence of driver steering inputs; we also develop an exact expression for cal- culating LTR. The proposed controllers have a proportional-integral structure whose gain matrices are obtained by solving a set of LMIs, which provide controllers to robustly guarantee that the peak magnitudes of the performance outputs do not exceed certain values. We show that using the design method the controllers can be designed to be robust with respect to unknown vehicle parameters such as speed and center of gravity height. We also provide a switching rule for controller activation based on the potential for rollover.
3.1
Chapter contributions
The scientific contribution of this chapter over the literature is mainly in the area of vehicle dynamics control; particularly in the area of automotive rollover prevention. Our control design was formulated as a bounded input bounded output (BIBO) disturbance rejection problem. We viewed the automotive vehicle as an uncertain dynamical system with dis- turbance inputs, and our controllers guarantee that the performance outputs of the system relevant to rollover are bounded. In doing so, we suggested using a dynamic version of the load transfer ratio (LTR) as a criterion for rollover occurrence. Our suggested robust control design method is unique in the sense that it gives way to a quantification of robustness of the controllers. We also considered vehicle parameter uncertainty in our control designs given that the uncertainty satisfies certain conditions.
The work contained in this chapter has resulted in the following publications:
(i) Solmaz S., Corless M., Shorten R., “A methodology for the design of robust rollover prevention controllers for automotive vehicles: Part 1-Differential Braking”, 45th IEEE Conference on Decision and Control, San Diego, Dec 13-15, 2006.
(ii) Solmaz S., Corless M., Shorten R., “A methodology for the design of robust rollover prevention controllers for automotive vehicles: Part 2-Active steering”, HYCON- CEMaCS Joint Workshop on Automotive Systems & Control, Lund, June 1-2, 2006.
(iii) Solmaz S., Corless M., Shorten R., “A methodology for the design of robust rollover prevention controllers for automotive vehicles: Part 2-Active steering”, American Control Conference, New York, July 11-13, 2007.
(iv) Solmaz S., Corless M., Shorten R., “A methodology for the design of robust rollover prevention controllers for automotive vehicles with active steering”, International Journal of Control, Special Issue on Automotive Systems and Control, Vol. 80, No. 11, pages 1763-1779, November 2007.
3.2
Introduction
It should be clear from the preceding chapter that the vehicle center of gravity position directly affects vehicle accident behavior. Particularly, it is well known that vehicles with a high center of gravity such as vans, trucks and the highly popular SUVs (Sport Utility Vehicles) are more prone to rollover accidents, which are, by far, the most dangerous type of accidents. As evident from to the 2004 accident data [1] compiled in the USA, light trucks (pickups, vans and SUVs) were involved in nearly 70% of all the rollover accidents, with SUVs alone responsible for almost 35% of this total. The fact that the composition of the current automotive fleet in the U.S. consists of nearly 36% pickups, vans and SUVs [22], along with the recent increase in the popularity of SUVs worldwide, makes rollover an important vehicle safety problem.
There are two distinct types of vehicle rollover: tripped and un-tripped. Tripped rollover is usually caused by impact of the vehicle with something else (e.g. obstacles, curb etc.) resulting in the rollover incident. For example, a tripped rollover commonly occurs when a vehicle slides sideways and digs its tires into soft soil or strikes an object such as a curb or guardrail. Driver induced un-tripped rollover can occur during typical driving situations and poses a real threat for top-heavy vehicles. Examples are excessive speed during cornering, obstacle avoidance and severe lane change maneuvers, where rollover occurs as a direct result of the wheel forces induced during these maneuvers. In recent years, rollover has been the subject of intensive research, especially by the major automobile manufacturers; see, for example, [28, 27]. That research is geared towards the development of rollover prediction schemes and occupant protection devices. It is however, possible to prevent such a rollover incident by monitoring the car dynamics and applying appropriate control effort ahead of time. Therefore there is a need to develop driver assistance technologies which would be transparent to the driver during normal driving conditions, but which act when needed to recover handling of the vehicle during extreme maneuvers [22].
In this chapter we present a robust rollover prevention controller design methodology, which represents the first of the two available approaches (i.e., robust and adaptive) towards the feedback design for systems with parameter uncertainties. Although most of the controller designs for automotive applications are in this category, our robust design method is unique in the sense that, unlike the traditional approaches, it quantifies the robustness of the atten- uation from the actuator inputs to the performance outputs, which can be used as structured way of tuning the controllers. The robust controller design described in the sequel is based on two separate type of actuators: active steering and differential braking. Also, as an ac- curate indicator of performance related to rollover, we consider the vehicle Load Transfer Ratio (LTR) in the feedback design. This measure of performance is related to tire lift-off and it can be considered as an early indicator of impending vehicle rollover. Vehicle wheel lift off occurs when the magnitude of this variable reaches one. We develop an exact ex- pression for this variable taking the vehicle roll dynamics fully into account. To distinguish our expression from previous (static) approximations of LTR in the literature, we denote it by LT Rd. We emphasize that although vehicle rollover is a dynamical process, the static
approximations of LTR ignore the roll dynamics; thus, they are not fully capable of deter- mining the onset of rollover.
Our proposed controllers based on differential braking have a P (proportional) structure with a fixed gain matrix KP, while active steering based controllers have a PI (proportional-
integral) structure with two fixed gain matrices KPand KI. By utilizing the integral action in
the latter, we ensure that the steady state steering response of the vehicle is as expected by the driver. The gain matrices are chosen to reduce the magnitude of LT Rd during transient
behavior.
The design of the controller gain matrices is based on recent results in [92] where they consider uncertain systems with performance outputs and subject to a bounded disturbance input. For each performance output zj they introduce a performance measure γj which
the magnitude of the disturbance. They present a controller design procedure which can be used to minimize the performance level for one main output while keeping the performance levels for the other outputs below some prespecified levels. In addition, the controllers in [92] are robust in the sense that they ensure performance in the presence of any allowable uncertainty which was taken into account in the control design. In applying these results to rollover problem, we consider the driver steering input as a disturbance input. Since we wish to keep the magnitude of LT Rd less than one, we view this as the main performance
output. To limit the amount of control effort and to accommodate actuator constraints, we choose the control input as an additional performance output in the feedback design. We note that many robust control designs in the literature are based on keeping the root mean square (or Euclidian norm1) of a performance output (i.e., kz
j(t)k2) small. However, for
this problem we consider it to be more important to utilize a controller which is designed to keep the peak magnitude (infinity norm or maximum norm2) of outputs (i.e.,kzj(t)k∞) to
be small rather than their rms value; this choice is motivated by the fact thatkLT Rdk ≥ 1
implies rollover, where LT Rd is the main performance output for this problem.
We initially consider control design for fixed vehicle parameters and illustrate the efficacy of our approach with some numerical simulations using typical data for a compact car. We then design a fixed robust controller which is effective for a range of vehicle speeds and vehicle CG (center of gravity) heights. The efficacy of this controller is illustrated by simulating the vehicle with different CG heights and with varying speeds. Finally, we propose a modification to our controllers so that they only activate when the potential for rollover is significant. This modification prevents the controllers from activating in non- critical situations and possibly annoying the driver.
1for a vector y∈ Rnwith y= (y
1, . . . , yn)T, the Euclidian norm is given bykyk2=
q
y2
1+ . . . + y2n.
2for a vector y∈ Rn, the infinity (or maximum) norm is given bykyk