Designing a Nonlinear PID Neural Controller of Differential Braking System for Vehicle Model Based on Particle Swarm Optimization

This paper presents a nonlinear PID neural controller for the 2-DOF vehicle model in order to improve stability and performances of vehicle lateral dynamics by achieving required yaw rate and reducing lateral velocity in a short period of time to prevent vehicle from sliding out the curvature. The scheme of the discrete-time PID control structure is based on neural network and tuned the parameters of the nonlinear PID neural controller by using a particle swarm optimization PSO technique as a simple and fast training algorithm. The differential braking system and front wheel steering angle are the outputs of the nonlinear PID neural controller that has automatically controlled the vehicle lateral motion when the vehicle rotates the curvatures. Simulation results show the effectiveness of the proposed control algorithm in terms of the best transient state outputs of the system and minimum tracking errors as well as smoothness control signals obtained with bounded external disturbances.


INTRODUCTION
ecently, the automotive electronic technology has led to the rapid development of modem automotive system technology such as Vehicle Dynamics Control (VDC), Anti-lock Brake System (ABS), Acceleration Slip Regulation (ASR) System, Electronic Stabilization Program (ESP) and Automatic Guidance Control (AGC) [1 and 2] where they are played an important role on improving the vehicle active safety and stability in order to prevent the wheel from slipping on the braking or accelerating as well as to avoid vehicle from sliding out the curvature.
The purpose of these systems is to actively control the vehicle under emergency situations where the vehicle is at the physical limit of adhesion between the tires and the road.These emergency situations are those that the normal driver usually cannot handle, and often loses control of the vehicle [3].
A system, which automatically intervenes in such situations, allows the driver to keep control of the vehicle and enhances the chance of avoiding an accident.In fact, there are three points that an active safety system must be addressed [4].
1) A vehicle must provide good controllability by responding quickly and accurately (i.e. with the right amount of change) to the driver's operational inputs.
2) A vehicle must provide good stability, with little change in behavior in relation to changes in driving conditions.
3) There must be an effective control loop between the driver and the vehicle for conveying operational inputs and the vehicle response in order to ensure that the driver can easily recognize present operating conditions and also predict vehicle behavior.
There are various ways to address these control issues such as in [5] it is used independent front and rear wheel drive to control the vehicle to track a desired lateral velocity by using fuzzy logic controller with variable gain structures.In [6], it is an analysis of vehicle lateral dynamics and the basic look-ahead control law is suggested for controller design to investigate the characteristics of vehicle dynamics and the performance requirements of steering controllers in vehicle-following collisions.
Also a robust PI controller is presented in [7] for the tracking of predefined vehicle lateral dynamics and using a full nonlinear vehicle simulator and four wheels steering control signal with the presence of a wide variety of disturbances.The model control is optimized based on the model predictive control theory as explained in [8] through adjusting the tire parameters on-line, the vehicle stability control performance is well, but the robustness was poor.In addition to that, the vehicle stability controller is designed based on the Linear Quadratic Regulator (LQR) and fuzzy control theory as proposed in [1] where the yaw moment is produced by differential braking and the target of control vehicle stability is achieved.And in [9] it is used neural controller based genetic algorithm to control the vehicle to track a desired lateral velocity from front and rear steering angles.A genetic neural fuzzy antilock brake system ABS controller is applied that consists of a non-derivative neural optimizer and fuzzy-logic components (FLC) as presented by Y.Z.[10].It is used (ABS) senses when the wheel lockup is to occur, releases the brakes momentarily, and then reapplies the brakes when the wheel spins up again.The remainder of this paper is organized as follows: Section two is a description of the 2DOF vehicle mathematical model.In section three, the proposed of nonlinear PID neural network controller approach and PSO tuning algorithm are derived.Simulation results of the proposed robust PID neural control algorithm are presented in section four and the conclusions are drawn in section five.

Vehicle Dynamics Model 2-DOF
There are many variables that effects on the dynamics of vehicle lateral motion such as vehicle speed, vehicle mass and tires state on road.The independent control of lateral and yaw motion requires at least one additional control input, which is independent of the front steering angle.There are three possible solutions for these inputs as follows: four wheel steering system; braking forces; and torque driving wheel [5 and 9].
In this paper, the focus is on the vehicle yaw rate and lateral velocity as the desired and the differential braking and front steering angle are the control action variables.
Figure (1) shows the two DOF vehicle model and it is widely used for lateral control design and has been shown to provide accurate response characteristics compared to more complex models for conditions up to 0.3g lateral acceleration [6, 7, 11, 12 and 13].The linear dynamics model of vehicle lateral motion [13 and 14] with interaction in multi-input multi-output system are expressed as the state space equations as follows: where y 1 =V and y 2 = r Equation ( 1) represents linear mathematical model of vehicle lateral motion with interaction in multiple-input-multiple-output system (see appendix 1) where V is lateral velocity, r is yaw rate and both are system variable states.f δ is the front steering angle and Bs F is brake steer force and they are inputs to the system.where the brake steering force can be described as in equation ( 1) from figure (1) as follows [13 and 14]: BS M is brake steer moment.XR F and XL F are front and rear longitudinal tire forces.

Nonlinear PID Neural Controller
The control of linear MIMO dynamics system is considered in this section.The approach used to control on the lateral velocity and yaw rate of the system that depends on the information available about the system model and the control objectives.The feedback nonlinear PID neural controller is necessary to stabilize the tracking error dynamics of the system when the output variable states of the system are drifted from the desired inputs by using two control signals front steering angle and brake steering force.The general structure of the nonlinear PID neural controller type can be given in the form of the block diagram shown in figure (2).The nonlinear PID neural controller for MIMO system can be shown in figure (3).It has the characteristics of control agility, strong adaptability, good dynamic characteristic and robustness because it is based on that of a conventional PID controller that consists of three terms: proportional, integral and derivative.The standard form of a PID controller is given in the s-domain as equation ( 5) [15].
where K p , K i and K d are called the proportional gain, the integral gain and the derivative gain respectively.The proposed nonlinear PID neural controller scheme is like neural network PID controller structure as the discrete-time equation ( 6) [16].
Therefore, the tuning PID input vector consists of denote the input error signals and the PID output respectively.Particle swarm optimization algorithm technique is used to adjust the parameters of the nonlinear PID neural controller.

2-DOF Vehicle Model
Nonlinear PID

Neural Controller
Particle Swarm Optimization Algorithm

Bounded External Disturbances
∆ is defined as a delay mapping.The proposed control law of the feedback front steering angle and brake steer force ( 1 u and 2 u ) respectively can be proposed as follows: Assume w 1 and w 2 are constant positive weights. 1 o and 2 o are the outputs of the neural networks that can be obtained from sigmoid function has nonlinear relationship as presented in the following function:  Kd of the nonlinear PID neural controller are adjusted using the particle swarm optimization techniques.Scaling functions have to be added at the neural network terminals to convert the scaled values to actual values and vice versa; therefore, the final feedback control actions δ for next sample can be calculated as the proposed law with scale factors δ S for the front steering angle and F S for the brake steer force.Each particle has its own position and velocity to move around the search space.The particles are evaluated using a fitness function to see how close they are to the optimal solution [17, 18 and 19].
V , is the velocity of the r th particle at k iteration.k m r y , is the position of the r th particle at k iteration.The previous best value is called as pbest.Thus, pbest is related only to a particular particle.It also has another value called gbest, which is the best value of all the particles pbest in the swarm.The nonlinear PID neural controller with six weights parameters and the matrix is rewritten as an array to form a particle.Particles are then initialized randomly and updated afterwards according to equations (17, 18, 19, 20, 21 and 22) in order to tune the PID parameters: where pop is number of particles.
γ is the weight of particle m at k iteration.c 1 and c 2 are the acceleration constants with positive values equal to 2. r 1 and r 2 are random numbers between 0 and 1. m pbest , γ is best previous weight of m th particle.gbest is best particle among all the particle in the population.The number of dimension in particle swarm optimization is equal to six because there are two nonlinear PID and each one has three parameters.The mean square error function is chosen as criterion for estimating the model performance as equation ( 23) [16]: The steps of PSO for neural network like self-tuning PID controller can be described as follows: • Step1 Initial searching points searching point is set to pbest for each particle.The best-evaluated value of pbest is set to gbest and the particle number with the best value is stored.

• Step2
The objective function value is calculated for each particle by using equation ( 23).If the value is better than the current pbest of the particle, the pbest value is replaced by the current value.If the best value of pbest is better than the current gbest, gbest is replaced by the best value and the particle number with the best value is stored.

• Step4
The current iteration number reaches the predetermined maximum iteration number, then exit.Otherwise, return to step 2.

Simulation Results
The proposed controller is verified by means of computer simulation using Matlab/Simulink.The dynamic model of the 2-DOF vehicle described in section two is used.The vehicle parameters as given in appendix ( 2) is taken to clarify the features of the nonlinear PID neural controller explained in section three.Scaling function has to be added at the neural network terminals to convert the scaled values to actual values "where the differential braking range is N 7000 ± and front steering angle is 1 .0 ± rad" and vice versa in order to overcome a numerical problem that is involved within real values.Therefore the signals entering to the network have been normalized to lie within (-1 and +1).The proposed control scheme is applied to the vehicle model and it is used the proposed learning algorithm steps of PSO for tuning nonlinear PID neural controller's parameters.The PSO algorithm is set to the following parameters: Population of particle is equal to 30.Number of weight in each particle is 6 because there are two nonlinear PID neural controllers.Scaling functions have to be added at the neural network terminals to convert the scaled values to actual values and vice versa.The performance of the proposed controller is evaluated using the closed-loop step lateral velocity and yaw rate responses for linear dynamic system.The desired lateral velocity must be zero to over come the vehicle may rotate around itself at high vehicle velocity.And desired yaw rate must be verified: where R is curvature radius.After training with sampling time is equal to 0.1 sec, it can be observed that the actual yaw rate output of the system are following the desired trajectory and the yaw rate error can be shown as the figures (8-a and b) respectively.Figure (8-a) is the yaw rate response and its fast smoothness response without overshoot.Steady-state error is equal to zero while at the transient time, the error is approximately equal to 0.075rad/sec when the vehicle velocity is change as (35, 25, 15, 20, 30 and 40) m/sec with fixed curvature radius equal to 100m as shown in figure (8-b).The differential braking and the front steering angle can be shown in figures (13-a and b) while the error between the desired yaw rate and actual output for disturbance case can be shown in figure (14-a and b).

Conclusions
The nonlinear PID neural network controller with particle swarm optimization algorithm technique for MIMO linear dynamic vehicle motion model has been presented in this paper.The state outputs of the 2DOF vehicle model are yaw rate and lateral velocity and they are followed the desired inputs because there are two control actions front steer angle and differential brake that are generated from the proposed controller with PSO algorithm that is used to tune the nonlinear PID neural controller with minimum time and more stability of the controller and no oscillation with best

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Eng. & Tech.Journal , Vol.32,Part (A), No.1, 2014 Designing a Nonlinear PID Neural Controller of Differential Braking System for Vehicle Model Based on Particle Swarm Optimization 199

Figure ( 2 )
Figure (2): The general proposed structure of nonlinear PID Neural feedback controller for 2DOF vehicle model.
des des Eng.& Tech.Journal , Vol.32,Part (A), No.1, 2014 Designing a Nonlinear PID Neural Controller of Differential Braking System for Vehicle Model Based on Particle Swarm Optimization 202

2 net
are calculated from these equations

2 …( 14 )
Particle Swarm optimization (PSO) is a kind of algorithm to search for the best solution by simulating the movement and flocking of birds.PSO algorithms use a population of individual (called particles) "flies" over the solution space in search for the optimal solution.

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usually generated randomly within the allowable range.Note that the dimension of search space is consists of all the parameters used in the nonlinear PID neural controller as shown in figure(3).The current Designing a Nonlinear PID Neural Controller of Differential Braking System for Vehicle Model Based on Particle Swarm Optimization 204