International Journal of Vehicle Mechanics and ...

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Optimal Preview Control of Rear Suspension Using Nonlinear Neural Networks A. MORAN & M. NAGAI

a

a

Tokyo University of Agriculture and Technology , Koganei-shi Naka-machi 2-24-16, Tokyo, 184, Japan Published online: 27 Jul 2007.

To cite this article: A. MORAN & M. NAGAI (1993) Optimal Preview Control of Rear Suspension Using Nonlinear Neural Networks, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 22:5-6, 321-334, DOI: 10.1080/00423119308969034 To link to this article: http://dx.doi.org/10.1080/00423119308969034

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Vehicle System Dynamics. 22 (1993). pp. 321-334

0042-31 14/93R205-0321525.00

O Society of Automotive Engineers of Japan, inc.

Optimal Preview Control of Rear Suspension Using Nonlinear Neural Networks

Downloaded by [Antonio Moran] at 15:55 12 January 2015

A. MORAN and M. NAGAI'

SUMMARY The performance of neural networks to be used for identification and optimal control of nonlinear vehicle suspensions is analyzed. It is shown that neuro-vehicle models can be efficiently trained to identify the dynamical characteristics of actual vehicle suspensions. After trained, this neuro-vehicle is used to train both front and rear suspension neuro-controllers under a nonlinear rear preview control scheme. To do that, a neuro-observer is trained to identify the inversedynamics of the front suspension so that front road disturbances can be identified and used to improbe the response of the rear suspension. The performance of the vehicle with neuro-control and with LQ control are compared.

1. INTRODUCTION

Artificial Neural Networks (A.N.N.) have recently attracted a great deal of attention owing to their ability to learn most classes of nonlinear continuous functions with bounded inputs and outputs to arbitrary precision [I]. This learning and versatile input-output mapping capabilities together with parallel and collective processing abilities arise expectations for applying A.N.N. to the identification and control of nonlinear dynamical systems [2] which include ground vehicles as a potential field of application. The mapping capabilities of neural networks constitute potential means for controlling nonlinear systems which can not be well controlled by conventional linear feedback controllers. The learning and collective processing abilities can be useful for controllers self-tuning, adaptation to changing hardware and varying objective-cost function. Also these abilities can be used for reducing human effort in designing controllers and suggests a potential for discovering new and better control strategies than the presently known. So far the most of studies dealing with the analysis and design of active suspensions consider a linear suspension model: the suspension components are linearized around the operational point and control laws are derived using linear model-based control methods such as optimal LQR [3] or H- robust control [4]. Although these control methods are of simple implementation they do not allow Tokyo University of Agriculture and Technology, Koganei-shi Naka-machi 2-24-16, Tokyo 184, Japan.


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A. MORAN AND M. NAGAI

a full exploitation of the active control resources when they are applied to nonlinear active suspensions. In order to improve the performance of vehicles with nonlinear active suspensions and improve the use of control resources, this paper proposes the use of A.N.N. for the design of optimal nonlinear controllers. This paper will analyze the performance and dynamical behavior of A.N.N. for forward and inverse dynamics identification and control of vehicle active suspensions with nonlinear characteristics. The dynamical behavior of a half vehicle equipped with front and rear nonlinear active suspensions is identified by a neuro-vehicle model which is posteriorly used for training both front and rear suspension neuro-controllers under a nonlinear rear preview control scheme. The front neuro-controller generates control signals after processing information coming from the whole vehicle dynamics. The rear neuro-controller process information from the vehicle and from road disturbances placed between front and rear suspensions. These disturbances are identified by a neuro-observer representing the inverse dynamics of the front suspension. The performance of the vehicle with nonlinear neuro-active suspensions is compared with that of a vehicle with optimal linear controllers (LQ) designed after linearizing the suspension components around the equilibrium point and having the same rear preview control action.

2. VEHICLE NONLINEAR MODEL

Figure 1 shows the longitudinal view of a four degree-of-freedom half car model with rigid heave and pitch motions. The car body is supported by front and rear suspensions compossed by nonlinear spring and damper and independently controlled actuators. Tires are represented by a nonlinear spring and damper. Suspensions and tires equivalent springs are assumed to have the following nonlinear behavior [ 5 ] :

fs = ks sgn (Ax) I Ax I " where fs is the spring dynamical force, ks is the equivalent stiffness, Ax is the relative motion of both extremes of the spring and n is an exponent representing the spring nonlinearity and usually ranges between 1 and 1.4 when Ax is in cm. Figure 2(a) shows the nonlinear relationship between fs and Ax. Front and rear suspensions nonlinear dampers are governed by the equation

PI:

where fd is the damper force and c, and cc are damping coefficients for tension


OF'TIMAL PREVIEW CONTROL 0E;REAK SUSPENSION

REAR

323

FRONT

Fig. 1. Half car model.

Fig. 2.

(a) Nonlinear characteristics of suspension spring. (b) Nonlinear characteristics of asymmetrical damper. C

and compression, respectively. The ratio usually ranges between 1 and 1.5. cc Figure 2(b) shows the asymmetrical relationship between fd and A i . Tire damping is assumed to be constant and relatively low. The equations describing the vehicle vertical motion can be expressed as: Car body (motion at center of gravity) heave motion:

m 2 x 2 + f V + f2, = 0

pitch motion:

I0

+ f 2 t f - f2rlr = 0

324


where m2 and I are the car body mass and moment of inertia, x2 and 8 are the car body vertical displacement at center of gravity and pitch angle, fZf and f,, are the front and rear suspension forces, respectively. Front tire (unsprung mass)


Rear tire (unsprung mass)

The front suspension and tire forces fZf and f,f are:

where f2sf and f,$ are the front suspension and tire equivalent nonlinear spring forces expressed by equation (I), f2df is the suspension nonlinear damper force expressed by equation (2), fldf is the tire equivalent damper force and u is the f. controlled force provided by the front actuator whose dynamics is not considered here. Rear suspension and tire forces can be expressed in similar way as equations (7) and (8). Defining the vehicle state variable vector as :

the control vector u and disturbance vector w as:

where wf-- xOf and wr = x,, the discrete-time state space equation of the suspension system can be written as:

where is a vectorial nonlinear function determinable from the vehicle vertical motion equations described above and k represents the discrete time. The symbol A in equation (9) represents relative displacements, i.e., AxZlf= xZf- IT

3. REAR PREVIEW CONTROL In order to improve the dynamical performance of the vehicle by active control,


front and rear suspensions are equipped with controlled actuators. The front actuator generates control signals after processing information coming from the front and rear suspension state variables. The rear actuator process information no only from both suspensions state variables but also information coming from the road disturbances which have previously excited the front suspension. While the front actuator works as a feedback controller, the rear actuator provides both feedback and preview control actions. Front actuator:

Rear actuator:

where xf and xr represent the state variables vectois of the front and rear suspenare front and rear controllers nonlinear functions and p is a sions, yf and vector containing the velocities of the road disturbances placed between front and rear suspensions. At time k, vector p is: .

.

where d is thei number of 'measured.' road disturbances placed between front and rear suspensions. The objective of the control action is to improve the vehicle dynamical performance expressed in terms of ride comfort, running stability and traction characteristics. Since ride comfort is related to car bod? acceleration and running stability is related to tire-road contact force and suspension stroke, the control objective can be expressed as the minimization of the following cost function:

where AF,o represents the variations in the tire-road contact force, p,, ... pa are weighting coefficients and N is a large number. Several methods have been proposed to solve this nonlinear optimal control problem [6,7] but the most of them are cumbersome, memory demanding and:difficult for practical on-line implementation. In this paper this control problem will be solved using neural networks. The mathematical validation of the use of neural networks to solve nonlinear optimal control problems has been derived in reference [a] using calculus of variations theory.

4. VEHICLE NEURAL IDENTIFICATION AND CONTROL Figure 3 shows the structure of the closed loop vehicle system analyzed in this paper. There are 4 neural networks: one neuro-vehicle model representing the


A. MORAN AND M.NAGAI

Fig. 3.

Vehicle 'closed loop' structure

vehicle forward dynamics, two neuro-controllers (front and rear) generating control signals in a nonlinear scheme and one neuro-observer identifying the inverse dynamics of the front suspension so that it can determine -observe- the road disturbances that excited this suspension.

4.1. Vehicle Neuro-Identification The objective of vehicles neuro-identification is to represent the vehicle dynamics, expressed by equation (1 l), using neural networks. Neuro-identification is an optimization problem for which the synaptic weighting coefficients of the neural network (network parameters) are determined so that the following identification error E (network training error) is minimized:

where x and 2 are the neuro-vehicle model and actual vehicle outputs, respectively. The structure of the neuro-identification process is shown in Figure 4. The most common method for neuro-identification of dynamical systems is the error back propagation method [9] which can be implemented considering two types of neural networks: forward networks and recurrent networks as it is shown in Figure 4. The basic difference between both networks structures is the dynamical relationship between network inputs and outputs. While in forward networks inputs and outputs do not have a temporal relationship, in recurrent networks outputs will become inputs in the next time step. Forward networks only performs a static mapping between inputs and outputs. Recurrent networks possess memory, have dynamics and can identify the sequential characteristics of dynamical systems. Also the feedback connections of recurrent networks provide them with the capability of isolating the effect of measurement noise (vector

OPTIMAL PREVIEW CONTROL OF REAR SUSPENSION

u (k)

-

VEHICLE

NEURO VEHICLE X (k+l)


-.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 4.

Block diagram of neuro-vehicle training. A: Forward neuro-vehicle. B: Recurrent neuro-vehicle.

v(k) in Figure 4). Recurrent networks can be trained using the back propagation through time [9] or the dynamic back propagation [lo] methods in off-line or online implementations, respectively.

4.2. Vehicle Neuro-Control Since the task of controllers is to drive a system (vehicle) according to a desired response, neuro-controllers design can be seen as an identification problem where the objective is to train the neuro-controller so that the closed loop system follows the behavior of a reference model with known good performance. The same training methods presented for vehicles neural identification can be used to train neuro-controllers. In this case, the neuro-vehicle model of Figure 4 is replaced by an augmented network compossed of a neuro-controller and a neuro-vehicle model as it is shown in Figure 5(a). When training the neuro-controller, the neuro-vehicle model (already trained) is maintained fixed and used only to back propagate the training errors from the vehicle output to the neuro-controller. However, when training a neuro-controller only to identify the dynamics of a reference model, all the nonlinear capabilities of neural networks are not fully exploited. Therefore, after the neuro-controller has identified the reference model, it is trained to minimize, in a nonlinear scheme, the cost function of equation (15). The structure of the process to train neuro-controllers for cost function minimization is shown in Figure 5(b). The error back propagation algorithm is used to calculate the derivatives of the cost function with respect to the neurocontroller weighting coefficients (optimizing procedure) [2]. In this paper, the structure of the neuro-controllers for feedback and preview control are determined considering the Linear Structure Preserving Principle which states that once the LQ equivalent structure has been found, this linear structure is maintained but the controller gain matrices are replaced by nonlinear mappings (neural networks). The structure of LQ preview controllers can be found in references [7] or [ l l ] .

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ANTONIO MORAN AND MASAO NAGAI

REFERENCE +a

1-

+

ACTUAL SYSTEM

-

A

X (k+l)

AUGMENTED NETWORK


...........................................

(a)

I

I

Model reference controller

VEHICLE

AUGMENTED NETWORK

NEURO CONTROLLER r......

(b)

Fig. 5.

I

I

-.-?

;&+I)

I

j

NEURO

.......L

Cost function optimizing controller

Block diagram of neuro-controller training process. A: Forward neuro-vehicle.B: Recurrent neuro-vehicle.

4.3. Inverse Dynamics Neuro-Model Neural networks are capable of identifying not only the forward dynamics of systems but also the inverse dynamics of them [a]. The system whose inverse dynamics will be identified can be nonlinear or no bijective linear where conventional linear techniques can not be applied. In this paper, the inverse dynamics of the front suspension will be identified by a neuro-observer in order to estimate the front road disturbances w,from the variations in the state variable vector Ax, and control variable ufof the front suspension as it is shown in Figure 6.


* +

AXf("

INVERSE

Wf(k)

NEURO


-. Fig. 6.

1

329

FORWARD X (k+l NEURO MODEL

FRONT

Block diagram of neuro-observer training.

The method used for inverse dynamics neuro-identification is similar to the neuro-controller training method presented above: the forward neuro-model of the system (front suspension) is previously identified and after it is used to train its inverse neuro-model as it is shown in Figure 6 . This method has good convergence properties even in cases when the inputs and outputs of the system have not a one-to-one relationship [a].

5. SIMULATION RESULTS In order to verify the capabilities of neural networks for forward and inverse identification and control of nonlinear vehicle active suspensions, simulations have been carried out considering a 13 ton passenger bus running at 110 kmlh. Neuro-vehicle model, front and rear suspension neuro-controllers and the neuroobserver for identifying road disturbances are multilayered networks designed having one input and one output layers with linear neurons and two hidden layers with nonlinear neurons (sigmoid neurons). Forward and recurrent networks have been used to represent both the neuro-vehicle model and the neuro-observer. 'The neuro-controllers are forward networks trained considering the recurrent neurovehicle model. 5.1. Vehicle Neuro-Identification In order to compare the performance of forward and recurrent networks for vehicle suspensions identification under the presence of measurement noise, the teaching signals (actual suspension outputs) generated from the actual suspension nonlinear dynamics given by equation (1 1) have been contaminated by random noise sequences. The training process has been performed considering a set of 250 input patterns of random plant exogenous inputs (control signals u,and ur and disturbance signals wfand wr) and specified initial state variables. Since



Fig. 7.

-2 1

I

0

1

I

I

2 Time [sec]

Neuro-vehicle response. Actual vehicle: - Recurrent neuro-vehicle: - - - - Forward neuro-vehicle: -.-.-.-

the neuro-vehicle model has 12 inputs (8 state variables and 4 exogenous inputs) and 8 outputs (8 state variables), the network has been sized with 20 neurons in the first hidden layer and 15 neurons in the second. After learning, the neurovehicle model has been tested by exciting it with random and deterministic exogenous inputs and initial conditions different from those used for network training. The responses obtained for the car body vertical acceleration at C.G.1, and for the front suspension stroke Ax,,, for sinusoidal road input are shown in Figure 7(a) and (b), respectively. These results show how the neuro-vehicle model can identify the nonlinear dynamics of the actual vehicle (solid line) and predict its future behavior. From these results it is clear that the recurrent neurovehicle model (broken line) approximates the actual vehicle response better than the forward neuro-vehicle model (broken-dotted line). This difference is in part owed to the presence of noise in the measured outputs of the vehicle (teaching signals): while in recurrent networks these noisy signals are used only as teaching signals, in forward networks they are used as teaching signals and network inputs.

5.2. Front Suspension Inverse Dynamics As it has been stated previously, the road disturbance exciting the front suspension has been determined by a neuro-observer working as a soft sensor and

OF'TIMAL PREVIEW CONTROL OF REAR SUSPENSION


0 Fig. 8.

1

2 Time [sec] 3

Neuro-observer response. Actual road disturbance: -Neuro-observer: - - - - -

trained to identify the inverse dynamics of this suspension. Since the neuroobserver has 5 inputs and 1 output, the network has been sized having 10 neurons in the first hidden layer and 5 neurons in the second. Figure 8 compares the time history of the actual road disturbances (solid line) and the output of the neuroobserver (broken line). From this result it is clear that the neuro-observer has identified the inverse dynamics of the front suspension and is capable of estimating the road disturbances affecting this suspension. The present analysis considers a weak dynamical coupling between front and rear suspensions so that the rear suspension motion almost no affect the front suspension [4]. It explains why rear suspension state variables were not included as inputs of the neuro-observer. 5.3. Suspension Neuro-Controllers The training of both front and rear suspension neuro-controllers have been performed in two steps: (1) Identification of the dynamics of a reference closed loop vehicle model with front and rear linear optimal (LQ) suspensions designed after linearizing the tire and suspension components around the equilibrium point and considering a rear preview control strategy. (2) Minimization of the cost function of equation (15) considering the nonlinear characteristics of the suspension as expressed by equation (1 1). The structure of the training process for steps (1) and (2) are shown in Figures 5(a) and (b), respectively. In these figures, w represents the disturbance vector given by equation (10) and p represents the 'previewed' road disturbances vector of equation (14) whose components have been determined by the neuro-observer. Considering a sampling control time of 20 msec., a wheel base of 7 m. and a running speed of 110 kmlh, the number of components of vector p is 11. Since the front neuro-controller has 8 inputs and 1 output sinals, it has been sized having 15 inputs in the first hidden layer and 5 in the second. The rear neurocontroller having 19 inputs (8 state variables and 11 previewed disturbances) and 1 output signals has been sized with 25 neurons in the first hidden layer and 5 in the second.


332


Step Response. Rear preview Effect In order to examine the effect of the rear preview control action, the step response of the nonlinear vehicle with rear preview neuro-control has been compared with the response obtained for the case of LQ suspension controllers designed considering a linearized vehicle model but without rear preview control action. Figure 9(a) and (b) show the absolute displacements of the car body at ~ step road input. The effect of the rear preview rear position x2r and rear tire x , for control action is clearly noted: the rear suspension begins moving before the step input reaches it. The rear actuator generates control forces just after the step input excited the front suspension. Given this preview action, the dynamical response (overshoot, rising and settling times) of rear suspension variables are improved. Random Response. Neuro-control Effect The response of the vehicle with rear preview neuro-control has been compared with the response of the same vehicle but with controllers designed according to LQ control minimizing the same cost function and with the same rear preview action. In both cases the road disturbances placed between front and rear suspensions are determined by the neuro-observer. Figure 10 shows the time history of car body vertical acceleration at rear suspension position and rear suspension

xzr

-

Fig. 9.

0

0.2

0.4

Time [sec] 0.8

Response for step road input. Rear preview neuro-control: - - - - - LQ control (no preview action): -



333

A

z m

-?

L

0

0

L-

4 -5 0

1

2 Time [sec]

Fig. 10. Response for random road input. Rear preview neuro-control: - - - - - Rear preview LQ control: -

tirelroad contact force variations AFlo, when the suspension is excited by random road irregularities different to those used for neuro-controllers training. The neuro-controller response for xZr and AF,ol are better than those of the LQ preview controller. The RMS values of i21 and AF,o, are 22%and 15% lower for the neuro-controller than for the LQ controller. Although the RMS values of the front and rear control signals are 2% higher for the neuro-controllers and the RMS values of the suspension stroke are similar for both controllers, the value of the cost function is 30% lower for the neuro-controller. Several simulations have been performed to examine the performance of the neuro-controllers. As it is the case of most of nonlinear systems, the performance depends on the input disturbances: road disturbances with the same spectral characteristics but differing in amplitude yield vehicle responses which do not keep the proportional characteristics of linear systems responses. For all the types of road disturbances analyzed the vehicle with neuro-control presents better performance than the vehicle with LQ controllers especially for random road disturbances with ,low-pass spectral characteristics. For some types of random road disturbances the value of the cost function for the vehicle with neurocontrol is 40% lower than that of the vehicle with LQ control. The improved performance of the neuro-control can be explained not only by the fact that the controllers design process takes into account the nonlinear dynamics of the vehi-

334


cle but also by the nonlinear characteristcs of the neuro-controllers which allows a nonlinear strategy for further minimization of the cost function of equation (15). The difference between the responses of the vehicle with neuro-control and with LQ control will increase as the suspension nonlinearities are more pronounced.


6. CONCLUSIONS This paper has shown how neural networks can be used for the forward and inverse identification as well as the optimal control of nonlinear active suspensions. Well trained neuro-vehicle models can identify the dynamics of actual vehicle suspensions and predict their future behavior. Well trained neuro-observers can identify the inverse dynamics of nonlinear vehicles and are capable of determining the exogenous inputs exciting the vehicle from the dynamical response of it. Well trained neuro-controllers can improve the performance of active suspensions under a nonlinear control scheme and without the requirement of suspension linearization. Questions regarding the internal structure of neural networks, optimum number of hidden layers and neurons and robustness and sensitivity properties remain as the topics for further research.

7. REFERENCES I . Lippmann. R., 1987. "An Introduction to Computing with Neural Nets", IEEEASSP Magazine, Vol. 4, NO. 2, pp. 4-22. 2. Narendra, K., and Parthasarathy, K., 1990. "Identification and Control of Dynamical Systems Using Neural Networks", IEEE Trans. on Neural Networks. Vol. 1, No. 1. pp. 4-27. 3. Thompson, A.G., 1976, "An Active Suspension with Optimal Linear State Feedback", Vehicle System Dynamics. Vol. 5. pp. 187-203. 4. Moran, A,, and Nagai, M., 1992. "Analysis and Design of Active Suspensions by H-infiniry Robust Control Theory", JSME Int. Journal, Series 111, Vol. 35, No. 3, pp. 427-437. 5. Dixon, J.C., 1991. "Tires, Suspension and Handling". Cambridge University Press. U.K. 6. Nedevkovic, N.. 1981. "New Algorithms for Unconstrained Nonlinear Optimal Control Problems". IEEE Trans. on Automatic Control, Vol. AC-26. No. 4, pp. 868-876. 7. Bryson, A.E., and Ho, Y.C., 1975. "Applied Optimal Control. Optimization. Estimation and Control", Hemisphere Publishing, New York. 8. Moran, A,, and Nagai, M., 1992. "Optimal Active Control of Nonlinear Vehicle Suspensions Using Neural Networks". To appear in JSME Int. Journal. 9. Rumelhart, D.E., and McClelland. J.L., 1986. "Parallel Distributed Processing: Explorations in the Microstructure of Cognition". Vol. 1 . MIT Press, Cambridge, MA. 10. Moran, A,, and Nagai, M., 1993. "Efficient On-line Training of Recurrent Networks for Identification and Optimal Control of Nonlinear Systems", Submitted to Int. Joint Con& of Neural Networks IJCNN'93, Nagoya, Japan. I I . Tomizuka. M., and Whitney, D.E., 1975. "Optimal Discrete Finite Preview Problems (Why and How is Future Information Important?)", ASME Trans., Journal of Dynamics Systems, Measurement, and Control, Vol. 97, No. 4, pp. 319-325.