Abstract
The present work deals with the design problem of a robust observer-based controller for a motorcycle system using LPV approach. The designed model is specifically uncertain and disturbed one, whose uncertainties are related to variations of both the cornering stiffness and the longitudinal velocity. The nonlinear motorcycle model is firstly transformed on an uncertain LPV model with two vertices; then an observer-based robust controller is designed. Both the controller and observer gain matrices are computed by solving a unified convex optimization problem under LMI constraints using YALMIP solver. Numerical simulation results are given to illustrate the effectiveness of the designed method.
1. Introduction
In recent years, more and more people are moving towards fast, light, less expensive means of transport and small jigs. Among these means of transport, motorized two-wheeled vehicles are increasingly popular because of the great freedom of driving and the possibility of avoiding traffic congestion and parking difficulties.
At the present time, scooters and motorcycles have become necessary means of transport in our society and the last statics of sales of these vehicles reinforce this statement. The increase in the number of motorcycles and scooters is, unfortunately, followed by an increase in accidents on the road.
In these circumstances, safety in two-way transport has become a central concern for public bodies and car manufacturers. Several preventive and repressive measures have been put in place to warn drivers of the risks associated with certain behavior on the steering wheel and the handlebars. On the other hand, several research projects have been initiated in order to provide solutions in the field of prevention and driver assistance systems.
In the last years, many motorcycle equipment manufacturers focus on active and semiactive safety systems. The first work to stabilize a motorcycle has been developed in [1], by considering a simple linear model to describe the motorcycle behavior. The control of motorcycles has been investigated in [2, 3] and the problem of observation and estimation of unmeasurable state variables have been studied in [4–7]. Overall, motorcycle safety system design is still an open problem and few results exist in the literature. To the best of our knowledge, there are no studies that deal with both the motorcycle stability and estimation in presence of disturbance, except the work in [7], which has studied the two problems separately, without considering external disturbance.
Overall, safety system design for motorcycle is still an open problem and few results exist in the literature. In the modelling aspect, we develop an uncertain polytope that can cover the time-varying longitudinal velocity range and has few vertices. Moreover, the nonlinear tyre model is reformulated into a Linear Parameter-Varying (LPV) model with norm-bounded uncertainties. LPV models have been studied in many works like [8].
Motivated by these observations, a robust observer-based controller will be investigated. The observer will be used to estimate the unmeasured variables of the motorcycle like the yaw rate, the roll rate, and the steering rate and the controller ensures the robust stability despite the variation of the road conditions and external disturbances. The less conservative LMI-based design conditions of both the observer and the controller are proposed and can be solved in one step.
The rest of the paper is structured as follows: the analysis of the motorcycle described by an LPV model is introduced in Section 3. Hereafter, in Section 4, the analysis and the design problems of the robust observer-based controller are studied. To show the performance of the proposed observer in different cases, some simulations are given in Section 5. Finally, conclusions are given in Section 6.
2. Notations and Preliminaries
The following lemmas are used to prove our results.
Lemma 1 (see [9]). Given a positive scalar and two matrices and , the following inequality holds:
Lemma 2 (see [10]). Considering a negative definite matrix , a given matrix , and a scalar , the following inequality holds:
3. Motorcycle Model Analysis
3.1. Nonlinear Model of the Motorcycle
The model used in this work describes the motorcycle’s lateral and roll dynamics, which are obtained by considering Sharp’s motorcycle model [11] (see Figure 1). The dynamics of a motorcycle can be represented by a model with four equations [12, 13] describing the lateral motion mainly caused by lateral forces and the yaw and roll motions under rider’s steering actions.
The movements that correspond to lateral, roll, yaw, and steering motions, respectively, are expressed by the following equations:Using the moment principle, the following expressions can be obtained: where coefficients , , , and are given in the Appendix.
Lateral and roll forces have the following expressions: where
Substituting (4) and (5) in model (3), we obtain the following form: where motorcycle parameters are defined in Nomenclature and parameters and are given in the Appendix.
Let us consider state vector ; system (7) can be written as follows:whereMatrix is a nonsingular matrix, so system (8) can be expressed as follows: whereTo simplify the development, is written as .
In the following, we mention that the yaw control is only tripped at a nonzero of the longitudinal velocity. For this, we propose to accommodate the variation of this speed.
3.2. Motorcycle Uncertain LPV Model Using Polytopic Approach and Reducing Vertices
Let us consider that the longitudinal speed is time-varying in interval . Then, variable is time-varying in . For pair , we can use a rectangular polytope [14] to describe it.
Since the choices for the parameter set only occur on the solid line and most of the area inside the rectangle is not achievable, the description using a rectangular polytope could be conservative. Moreover, if the number of the polytope vertices increases, the computational load and the complexity also increase. In the following, we propose to use the idea proposed by [15] to reduce the number of the polytope vertices and also to simplify the modelling complexity.
Based on [15] and considering matrix defined in Section 3.1, for the set of the variables , we consider a rectangular polytope given in Figure 2.
As is shown in Figure 2, the rectangular polytope obtained by the four vertices illustrates the variation of the set of variables . A straight line crosses and ; then, shifting the straight line until the tangent to the hyperbola , we obtain another straight line . crosses the straight line in point and, on the other hand, in point . Then, the line crosses the middle of lines and via and . The sweep above and below along the two segments and can achieve all possibilities of most of the area inside the polytope.
The two newly obtained vertices are defined by the following coordinates:Since is less than one, the uncertain straight line segment crossing points and can cover the whole parallelogram . Then, based on the work of [16], each point inside the parallelogram can be represented by a linear combination of the new uncertain vertices and as follows:where and and .
The description of the two parameters and , in expression (13), has only two vertices and it can overlay all the possibilities for the set of and .
To take into account the road condition variations, the cornering stiffness is given by the following equations:Considering the representation in (10) and (13), the obtained model isWe define the state matrices by the first term containing nominal parameters. So, the state matrices of the multiple models are written as The uncertain ones are given bywherewherewith For more details, see [16].
4. Analysis and Synthesis of the Robust Observer-Based Controller
The objective is to design a robust observer-based controller such that the closed-loop system is globally stable and the performances are guaranteed.
Based on LPV model (15), the observer and the controller are defined as where is the estimated state and are the controller gains to be determined, are the observer gains to be determined, as the yaw rate, the roll rate, and the steering rate can be generally measured, and is an adequate matrix which is given byDefine the estimation error as The derivative of the error estimation is given byBy using (15), (21), and (24), we can express the augmented system in the following form: whereTo improve motorcycle’s lateral dynamics stability, the objective in controller design is to possess robustness against external disturbance (steering angle) and uncertainties (cornering stiffness and the longitudinal velocity), which guarantees a given attenuation level of disturbance rejection attenuation.
The objective of this work is to design an observer-based controller such that the following requirements are satisfied:(i)Ensure the stability for closed-loop system (25) with the controller when (ii)Ensure disturbance rejection for system (25) such that The stability conditions of augmented system (25) are given in terms of LMIs in the following theorem and can be solved in one step.
Theorem 3. For given scalars and , system (25) is asymptotically stable via the robust observer-based controller (21), if there exist positive definite matrices and and matrices , , and , linear variables , , and , and a positive scalar , which guarantees the performance, such that the following LMIs are verified:where And the gains of the controller and the observer are, respectively,
Proof. The Lyapunov function is chosen aswhere the Lyapunov matrix is chosen as
The derivation of the Lyapunov function gives Then, objective (28) is guaranteed by ensuring the following inequality: Therefore, we haveThen, inequality (36) holds if the following condition is satisfied: By substituting the expressions of and defined in (26) and (27), inequality (38) is equivalent towhereIn the following step, we expand matrix (39) in three matrices; that is, with and According to expressions of and and using Lemma 1, inequality (39) holds if there exist real scalars , , , and satisfying with
Prepost multiplying (44) by the matrix with and , we obtain with Matrices , , and are given byThen, is rewritten as follows:Using Lemma 2, there exists a positive scalar such that matrices can be rewritten: Using Schur’s complement, (50) can be written as which is equivalent to Using Schur’s complement, sufficient conditions in Theorem 3 are established.
5. Simulation Results
To show the performance and the effectiveness of the proposed observer-based controller law, we have considered both the variation of the longitudinal velocity as given in Figure 3 and the cornering stiffness. The steering angle is given by Figure 4.
Longitudinal velocity is varying between m·s−1 and m·s−1. The reduced model using two-vertex approach is firstly parameterized. Taking into account the uncertainties of the cornering stiffness which are 12% of the nominal values given in Table 1, the observer-based robust state feedback controller is then designed. Figures 5–9 show the performance of the designed observer-based controller by considering fixed parameters and and the following initial conditions and .
and controller gains are given by
Matrix is given by
Parameters , , and are obtained asFigures 5–9 show the results of a driving test with varying longitudinal velocity, Figures 5–8 present the comparison between one of the states of the model and its estimate from the robust observer-based controller. These figures show the good estimation, stabilization, and the robustness with respect to uncertainties. It can be seen that there is a close approximation of the estimated and measured lateral velocity, yaw rate, roll rate, and roll angle of the motorcycle. The yaw rate, however, shows a deviation that is related to steering angle of the motorcycle. We can also see that the designed robust observer-based controller improves the stability and safety under some unmeasured states and cornering manoeuver.
6. Conclusion
In this paper, the problem of the design of the observer-based robust control of the lateral dynamics of the motorcycle is studied. Using LPV approach and some geometrical transformations, the nonlinear motorcycle model is rewritten in uncertain LPV model form. Then, a robust observer-based controller has been designed using LMI approach. To prove the performance of the controller, simulations using Matlab-SIMULINK toolbox have been presented. The comparisons of the estimated and measured states evolution and then the stabilization and the robustness under some unmeasured states and cornering manoeuver with respect to parameters uncertainties of the LPV representation of lateral dynamic motorcycle model are only treated in this manuscript.
Appendix
Coefficients and Parameters
Nomenclature
Illustration of Parameters of Motorcycle| , , : | Total weight of the motorcycle, mass of the rear, and front portions |
| : | Distance from the gravity center of the rear frame to the front wheel contact |
| : | Distance from the gravity center of the rear frame to the rear wheel contact |
| , , , : | Cornering front and rear stiffness |
| , : | The rear and front tire relaxation lengths |
| : | Steering head angle, curvature of the road, and road curvature radius |
| : | Pneumatic trail |
| : | Coefficient of friction of the road and height of the bike’s center of gravity |
| : | Distance from the center of gravity of the front frame to the ground |
| : | Distance between the gravity centers of the rear and front frames |
| , : | Front wheel load and the gravity force |
| : | Distance from the center of gravity of the front frame to the fork |
| : | Yaw angle, steering angle, and roll angle |
| , : | Longitudinal and lateral velocities |
| , : | Inertia’s polar moments of the front and rear wheel |
| : | Polar moment of inertia of the camber inertia of the rear wheel. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.