Numerical Examples - Vehicle Suspension Design Problem

CHAPTER 6 ENGINEERING APPLICATIONS

6.2 Vehicle Suspension Design Problem

6.2.2 Numerical Examples

This paper uses the numerical data from Haug and Arora (1979) to validate the model specified in Section 6.2.1. The following parameters in the vehicle system equations are fixed during the calculations; m1

g = 290 lb, m

g = 4500 lb, m

g = m

g = 96.6 lb, I = 41,000

lb-in-sec², H = 20 in, L = 120 in, k4 = k5 = 1500 lb / in, vallow = 1056 in / sec (60 mph), and c4

= c5 = 5 lb-sec / in. The coefficients of the suspension system are selected as design variables,

b = [k

1, k2, k3, c1, c2, c3]^T. The lower and upper bounds on b are [50, 200, 200, 2, 5, 5]^T and [500, 1000, 1000, 50, 80, 80]^T, respectively. The maximum allowable values for the state variable constraints in Eqs. (6.23) – (6.29) are selected to be [400, 2, 5, 5, 2, 2, 1056]^T. The units of z1, z2, z4, z5 and z6 are inches and those of z3 are radians.

Model validation

The physical phenomenon of rigid body motion can be used to confirm the correctness of the vehicle model. Therefore, cases of braking and accelerating while traveling straight ahead are considered here to validate the longitudinal vehicle dynamics. For convenience of observation, the vehicle is assumed to travel along a straight path such that R1(y) = R2(y) = 0.

In cases of acceleration, the control problem is to determine a feasible acceleration trajectory along which a vehicle with various initial speeds can arrive at a destination in minimal time.

Hence, one additional terminal constraint is imposed:

6( )_f _t

z t

y

(6.33)

where yt is the destination. Similarly, one additional terminal constraint is included in cases of braking

12( ) 0_f

z t

= (6.34)

All the acceleration and braking test cases are transformed into time-optimal control

problems and solved by applying the proposed NLP method. Figure 6.6 and Figure 6.7 show the velocity trajectories of the vehicle with various starting speeds. As Figure 6.6 illustrates, the vehicle accelerates at the maximum allowable acceleration until the speed constraint defined in Eq. (6.29) becomes pertinent, from which point the speed is maintained. In contrast to the cases of acceleration, the vehicle decelerates with maximal allowable deceleration until it stops. Figure 6.8 shows the driver’s seat acceleration trajectory for the case of straight-ahead braking. According to these results, the vehicle motion is consistent with the motion of a rigid body, meaning that the longitudinal vehicle dynamics of the proposed model are validated.

Optimal design of the vehicle suspension system

The vertical displacement functions and system equations specified in Section 3 can be used to define an optimal suspension design problem. The driver is to be made as comfortable as possible over a range of road conditions and traveling speeds. Thus, the design objective is to minimize the maximum absolute acceleration of the driver’s seat by adjusting the vehicular suspension properties subject to the constraints that certain relative displacements do not exceed imposed limits. The objective function is therefore

0 max[0, ] 1( )

Two design cases considered by Haug and Arora (1979) are used here to examine the correctness of the proposed method. Figure 6.9 represents the road displacement profiles in the test cases. In case 1, the road surface profile includes a cavity. Case 2 involves two road displacement profiles, presented in Figure 6.9(b) and (c). The speed of the vehicle in case 1 is 450 in/sec and that in case 2 is 960 in/sec. Table 6.2 gives the optimal solutions. A comparison with the results present in the research sources (Hsieh and Arora, 1984; Haug and Arora, 1979)

shows that the results obtained by the proposed method are quite accurate.

Vehicle speed control problem

In most emergency situations, drivers must stop the vehicle quickly and safely. Changing the speed of the vehicle according to the conditions of the road and the distance from the current position to the site of accident is a vehicle speed control problem that the vehicle model and system equations derived in Section 3 can be used to solve. In this case, the initial speed of the vehicle is 880 in / sec (50 mph) and the road surface profile is as plotted in Figure 6.10. According to the definition in Section 2.2, the emergency braking problem is transformed into a time-optimal control problem that is then solved using the proposed NLP method. The minimum time, tf = 3.4 seconds, and the terminal displacement, z6 = 1585.7 inches, are obtained using the OCP solver. Figure 6.11 shows the trajectories of the vehicle speed and acceleration. Figure 6.12 plots the trajectories of the acceleration and pitch angle of the passenger seat, which are of interest to vehicle designers. The solid bold curves at the bottom of Figure 6.11 and Figure 6.12 represent the corresponding road profile. The numerical results indicate that all the constraints are satisfied and the optimal control law that solves the emergency braking problem is determined.

6.3 Summary

In this chapter, two practical applications, the flight level control problem and the vehicle suspension design problem – both highly nonlinear optimal control problems – have been formulated following the procedure suggested in this dissertation and solved by the proposed OCP solver. In the case of the flight level control problem, two common types of optimal control problem, the tracking problem and the minimum time problem, were derived to simulate practical situations. The vehicle suspension design problem provided a useful example of dynamic system design. After the problem has been formulated and the proper constraints imposed, users can solve their dynamic optimization problems by applying the

proposed method.

Because the proposed solver provides a convenient tool for solving dynamic optimization problems, proper modeling and formulating the physical problem become the major decisive factors in whether the solution is meaningful or not. Moreover, the constraints must meet actual environmental conditions or the solution will make no sense. Overconstraining the problem will considerably increase the computational efforts and make obtaining the solution harder. In contrast, loosely constraining the problem will provide no practically applicable solution.

Table 6.1 User subroutines for the flight level tracking problem.

//---Program parameters --- //B: Discrete design parameters of design variable vector. (INPUT) // U: Admissible control function vector. (INPUT)

// Z: State variable vector. (INPUT) // T: Given time grid point. (INPUT)

// G: First term of performance index or functional constraint or // dynamic constraint. (OUTPUT)

// NV: Number of design variables for optimizer (INPUT) // NU: Number of control functions. (INPUT)

// NEQ: Number of state equations (INPUT)

// N: Index of current number of function evaluation. (INPUT) //

// ---FFN()--- // Routine to calculate the integral term of the performance index // or functional constraint

void ffn(double *B, double *U, double *Z, double *T, double *F, int NV, int NU, int NEQ, int N, int NBJ) // Routine to calculate the first term of the performance index or // functional constraint or dynamic constraint

void gfn(double *B, double *U, double *Z, double *T, double *G, int NV, int NU, int NEQ, int N, int NBJ)

Table 6.1 (cont.) User subroutines for the flight level tracking problem.

//--- HFN() --- //Routine to calculate the state trajectory.

void hfn(double *B, double *U, double *Z, double *DZ, double *T, int NV, int NU, int NEQ )

{

DZ[0] = -1*((aD*Z[0]*Z[0]/m) + (g*sin(Z[1]*PI/180.0))) + U[0]

*10000 / m;

DZ[1] = (aL*Z[0]*(1-c*Z[1])/m) - (g*cos(Z[1]*PI/180.0)/Z[0]) + aL*c*Z[0]*U[1]/m;

DZ[2] = Z[0]*sin(Z[1]*PI/180.0);

}

//--- Z0FN() --- // Routine to calculate the initial state vector.

void z0fn(double *B, double *ZINT, int NV, int NEQ ) {

ZINT[0] = 100.0;

ZINT[1] = 20.0;

ZINT[2] = -120.0;

}

Table 6.2 Optimal solutions for vehicle suspension.

(a) Case 1

125.50 125.60

Figure 6.1 Aircraft model (Lygeros, 2003).

0 2 4 6 8 X2: Flight Path Angle (deg.) X3: Altitude distance(m)

x₂ x₁

x₃

(a) State trajectories.

0 2 4 6 8 u2: the pitch angle (deg.)

u₂ u₁

(b) The thrust trajectory.

Figure 6.2 Numerical results for the tracking problem.

0 20 40 60

(a) Control input trajectories.

0 20 40 60 X2: Flight Path Angle (deg.) X3: Altitude distance(m)

(b) State trajectories.

Figure 6.3 Trajectories for the minimum time problem.

Figure 6.4 Six-degrees-of-freedom vehicle model

Figure 6.5 Sinusoidal displacement functions.

0 5 10 15 20 25 Time (sec.)

0 200 400 600 800 1000 1200

Velocity (in / sec)

V = 0

V = 528 in/sec (30 mph) V = 880 in/sec (50 mph)

0 0 0

Figure 6.6 Vehicle speed histories for straight-ahead accelerating.

0 0.5 1 1.5 2 2.5 3 3 Time (sec.)

.5 0

200 400 600 800 1000 1200

Velocity (in / sec)

V = 528 in/sec ( 30 mph) V = 880 in/sec ( 50 mph) V = 1056 in/sec ( 60 mph)

0 0 0

Figure 6.7 Vehicle speed histories for straight-ahead braking.

0 0.5 1 1.5 2 2.5 3 Time

3.5 (sec.)

-40 -20 0 20 40

Seat acceleration ( in / sec2 )

V = 30 mph ( Tf = 1.51 sec.) V = 50 mph ( Tf = 2.51 sec.) V = 60 mph ( Tf = 3.02 sec.)

0 0 0

Figure 6.8 Driver’s seat acceleration for straight-ahead braking.

(a) Case I

(b) Case II – road profile 1.

Figure 6.9 Road displacement profiles for model validation (Haug and Arora 1979).

Figure 6.10 Road displacement profiles for emergency braking.

0 200 400 600 800 1000 1200 1400 1600 Displacement (in.)

0 200 400 600 800 1000

Vehicle Speed ( in. / s)

Vehicle speed Vehicle acceleration

-400 -300 -200 -100 0 100 200

Vehicle acceleration( in / s2 )

Figure 6.11 Vehicle speed and acceleration histories for emergency braking.

0 200 400 600 800 1000 1200 1400 1600 Displacement (in)

-40 -30 -20 -10 0 10 20 30 40

Seat acceleration ( in / sec2 )

Seat acceleration Vehicle pitch angle

-6 -4 -2 0 2

Vehicle pitch angle (Deg.)

Road Profile

Figure 6.12 Seat acceleration and pitch angle histories for emergency braking.

CHAPTER 7 CONCLUSIONS AND FUTURE STUDY

7.1 Concluding Remarks

This study has introduced and compared two typical methods – the indirect and direct approaches – usually used to solve optimal control problems. Because most of practical control problems are described by strongly nonlinear differential equations that are difficult to solve by indirect methods, direct methods have been widely studied in the recent literature and are also adopted in this dissertation. In spite of extensive use of direct and indirect methods to solve optimal control problems, engineers still expend much effort on reformulating problems and implementing corresponding programs for different control problems. Therefore, the first objective of this dissertation was to develop a convenient solver and provide a systematic computational procedure capable of assisting engineers and students easily solve their dynamic system design problems.

To this end, a computational AOCP method, one kind of sequential direct method, has been developed and presented herein. Subsequently, it was implemented and coupled with many robust numerical schemes to develop a general OCP solver. Besides outlining the theoretical and numerical foundations involved in the proposed solver, the discussion detailed the OCP solver implementation, including the dynamic constraint treatments, ODE solver, sensitivity analysis, and so forth. The van der Pol oscillator problem with three different terminal conditions and a highly nonlinear time-optimal control problem were used to illustrate and verify the stability and capability of the proposed solver. In these examples, different numerical schemes and different time intervals were applied to investigate the numerical schemes’ effect on the validity of the solution and computational efficiency. The results indicate that the OCP solver coupled with the systematic procedure suggested in this study can truly facilitate the solving of engineering control problems in a systematic and

efficient way.

The other objective of this dissertation was to develop a novel method for solving discrete-valued optimal control problems. Most traditional methods focus on the continuous optimal control problems and fail when applied to a discrete-valued optimal control problem.

One common type of such problems is the bang-bang type control problem arising from time-optimal control problems. When the controls are assumed to be of the bang-bang type, the time-optimal control problem becomes one of determining the TOCP switching times.

Several methods for such determination have been studied extensively in the literature;

however, these methods require that the number of switching times be known before their algorithms can be applied. As a result, they cannot meet practical situations in which the number of switching times is usually unknown before the control problem is solved.

Therefore, to solve discrete-valued optimal control problems, this dissertation has focused on developing a computational method consisting of two phases: (a) the calculation of switching times using existing optimal control methods and (b) the use of the information obtained in the first phase to compute the discrete-valued control strategy. The proposed algorithm combines the proposed OCP solver with an enhanced branch-and-bound method. To demonstrate the proposed computational scheme, the study applied third-order and fourth-order systems and an F-8 fighter aircraft control problem considered in several pioneering studies. Comparing the results of this study with the results from the literature indicates that the proposed method provides a better solution and the accuracy of the terminal constraints is acceptable. Finally, the proposed solver and procedure were applied to two engineering applications: the flight level control problem and the vehicle suspension design problems.

7.2 Future Study

Future study will focus on two topics: one is to develop a web-based dynamic

optimization solver and another one is to improve the global convergence of proposed method.

In terms of the former, although a convenient solver for dynamic optimization has been developed in this dissertation, constructing a turnkey system for solving dynamic system design problems is still a major problem for inexperienced engineers and students. In addition, because of the resource limitation including finances and laboratory facilities, it is difficult to provide each engineer with a turnkey system to solve his/her dynamic system design problem.

The advent of the Internet and distributed computing technologies suggest that a Web-based optimization tool may provide a potential partial solution to this problem (Chu, 1999). A Web-based dynamic optimization solver would also be useful to engineering curriculums because students could share limited resource via the Internet. Many Internet techniques such as Web service, Simple Object Access Protocol (SOAP), DataSocket, and XML, could be applied to the development of such a solver, for which a conceptual flowchart is given in Figure 7.1. For this suggested model, the proposed OCP solver would serve as a numerical engine.

As regards improving global convergence, the convergency of the OCP solver depends on good initial guesses that speed up optimization problem convergence and produce high-precision solutions. However, it is difficult for the inexperienced to provide good initial guesses that lie within the convergence domain. Therefore, a module that assists with estimation of the optimal solution will be developed to help the novice making the proper initial guess.

It is expected that the contents of Chapters 2 through 5 will be the basis for addressing these two topics, which, because of modular programming techniques, can be effected by adding external modules into the proposed solver or replacing the original modules with new ones. This feature allows the proposed solver to be easily updated by state-of-the-art algorithms.

Figure 7.1 Conceptual flow chart of a web-based dynamic optimization solver.

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