CHAPTER 4 A CONVENIENT SOLVER FOR SOLVING OPTIMAL CONTROL
4.7 Illustrative Examples
Two types of optimal control problems mentioned in the literature have been used as test problems to evaluate the performance of the proposed method. In the AOCP method, both the acceptable violation of constraints for feasible designs and the acceptable tolerance for the convergence parameter are 10-3. The numerical results for all example problems were obtained on a Pentium 4 Celeron 1.2 GHz computer with 384 MB of RAM.
4.7.1 The van der Pol Oscillator Problem
The van der Pol oscillator problem was given and solved by Bullock and Franklin (1967) using a second variation method. The problem was also used by Jaddu and Shimemura (1999) to verify their computational method. In this dissertation, it is further used to evaluate the performance and capabilities of the proposed method and the OCP solver. The van der Pol oscillator problem can be formulated by the following minimization
dt
subject to
Based on this problem, Jaddu and Shimemura considered three cases that can also be solved by the OCP solver: the unconstrained problem, the terminal state constrained problem, and the terminal states and control constrained problem.
Case I: Free end point and no control constraints
The optimal solution for this problem found by Bullock and Franklin (1967) using a second variation method was J0* = 1.433508, while that found by Jaddu and Shimemura (1999) using a ninth-order Chebyshev series to approximate x1(t) was J0* = 1.4334872.
Using the OCP solver, in which the control variable u is discretized into 21 grid points, the optimal value is J0* = 1.4334723, smaller than both earlier reported results. The numerical parameters for MOST are listed in Table 4.2, and the optimal control and state trajectories are shown in Figure 4.12.
Case II: Terminal state constraint
( x ( ) t
f) 1 x
2( ) t
fx
1( ) t
fϕ
= − + = 0
(4.3)For this problem, Bullock and Franklin (1967), again using the second variation method, found an optimal value of J0* = 1.6905756, while Jaddu and Shimemura (1999), also using a ninth-order Chebyshev series to approximate x1(t), found an optimal value of J0* = 1.6857113. In this study, the terminal state constraint is treated as an equality constraint and the other number parameters, the same as in case I. With the OCP package, the value obtained is J0* = 1.6856957. Figure 4.13 shows the optimal control and state trajectories for the proposed OCP solver.
Case III: Terminal state constraints and saturation constraints on control
The terminal state constraints and the saturation constraints on control are described in the following equation:
When Bashein and Enns (1972) solved the problem, they obtained J0* = 2.1439039, while Jaddu and Shimemura (1999), this time using a twelfth-order Chebyshev series to approximate x1(τ), found an optimal value of J0* = 2.1443893. The solution produced by the OCP solver is an optimal value of J0* = 2.1375360. The optimal control and state trajectories for the OCP solver are shown in Figure 4.14.
4.7.2 Time-optimal Control Problem: Overhead Crane System
Overhead cranes are widely used in factories and workplaces to transport objects. An overhead crane system, like that sketched in Figure 4.15, is a high-order nonlinear system that consists of a cart with a point load suspended by cables. The control problem is to transfer the load from an arbitrary point A to point B in minimal time subject to the requirement of zero residual vibration at point B. The control inputs are the horizontal acceleration of the cart and the hoisting acceleration of the cable. Hu et al. (2002) proposed this problem and solved it using an enhanced DCNLP method. In this dissertation, this problem will be used to demonstrate the ability of the proposed method to solve a high-order time-optimal control problem.
Given x1 =z, x2 =z,
x
3 =θ
,x
4 =θ
=ω
,x
5 =l
,x
6 =l
as the state variables, and u1 = z, as the control inputs, the OCP formulation of the overhead crane system can be minimized as followsl
u
2 =J0 = tf (4.5) 4, 0]T where g is the gravitational acceleration.
The state and control constraints are as follows:
tf
Using the admissible control formulation delineated in the previous chapter, the control variables are converted into design variables that can then be treated as design variable boundaries. Furthermore, the state constraints are transferred into standard constraint form as follows:
In this problem, both the state and the control variables are divided into 101 grid points.
The minimum time J* = tf = 12.0004 is solved in the OCP solver by applying a cubic piecewise interpolation scheme to the control function. Two local optimal solutions are obtained by the OCP solver with different initial points. The trajectories of the rope angle and angular velocity are shown in Figure 4.16, in which the solid line represents the results obtained by the DCNLP method (Hu. et al., 2002) and the dashed line represents a second
optimal solution obtained by the OCP solver. As the figure illustrates, the solid line totally matches the results obtained by Hu et al. (2002), meaning that one of the optimal solutions found by the OCP solver tallies exactly with the trajectories obtained by the DCNLP method (Hu. et al., 2002). In addition, the performance index (terminal time, tf) obtained by the OCP solver (tf ≅ 12.00) is very close to the result using the DCNLP method (tf ≅ 12.00). Moreover, according to the trajectories shown in Figure 4.16, the amplitudes of rope angle and angular velocity for the second optimal solution obtained by the OCP solver, as represented by the dashed line, are smaller than the others. Figures 4.17 and Figure 4.18 depict the corresponding inputs and states with local optimal solutions, respectively. In Figure 4.17, the trajectories of the control inputs conform to the dynamic control constraints given in Eq. (4.8). According to the state trajectories in Figure 4.18, the initial conditions, xT(t0) = [0, 0, 0, 0, 4, 0]T ,and the terminal conditions, xT(tf) = [10, 0, 0, 0, 4, 0]T are satisfied. Obviously, all constraints are fulfilled, thereby proving the correctness of the solutions. In other word, both solutions solved by the OCP solver are local optimal solutions. In practice, small amplitudes of rope angle and angular velocity for an overhead crane will be adopted because they benefit operational safety.
As the numerical results show, both examples convert successfully into NLP problems using the admissible control formulation and can then be solved using the AOCP method.
Moreover, the results of the numerical schemes of the proposed method are quite accurate.
With the OCP solver, users need not spend a vast amount of effort on programming to obtain solutions. Rather, once the problems are formulated, the solver can be implemented and the problems solved easily. In addition, rapidly advancing computer capabilities will ensure that computing time for the OCP solver will decrease. Thus, users will be able to obtain optimal results more quickly than before.