Figure 1 illustrates the structure of an electronic control throttle used for automobile engines. Its main elements are a DC motor, gear train (pinion gear, intermediate gear, and sector gear), butterfly valve, and springs. For these elements, a DC motor circuit equation, DC motor rotational motion equation, gear-train rotational motion equation, and valve rotational motion equation are derived. These physics are shown from Equations (5) to (8).

Table 1 shows the parameter names and their meanings used in the throttle valve’s mathematical models. In addition, Equations (5) to (8) can be reduced to Equations (9) to (10) by considering the gear ratios.

where

We consider the backlash characteristics described by Figure 2 and Equation (11) and the friction characteristics described by Figure 3 and Equation (12) as the nonlinear characteristics of the electronic control throttle. The friction characteristics are known as the Stribeck model and are obtained as a composite of static, Coulomb, and viscous friction. Since the viscous friction is already contained as the term of Equation (9), it is excluded when expressed by Equation (12) is applied.

The block diagram in Figure 4 is obtained by combining the linear model in 3.1 and nonlinear model in 3.2. As the physical model of an electronic control throttle, it is used as a control object in the creation of a control model or for confirming the performance of a control system.

A control model built directly from Equations (9) to (10) can be applied as a compensation model integrated into BSC. However, this paper derives a quadratic linear model in order to ensure consistency with conventional PID control systems. Assuming a control model expressed by Equation (13), we identified the step response of the electronic control model of Figure 4 with the nonlinear model excluded. We thereby obtained a second-order delay model that is in good agreement with the physical model, as can be seen in Figure 5.

In this study, we added a derivative term to Step 1 in order to improve responsiveness (particularly the starting characteristics). The following demonstration of stability omits description of nonlinear characteristics.

Equation (15) defines the control law of Step 1 using a linear combination of a control deviation and its change rate.

For the Lyapunov candidate function defined by Equation (15), we obtain:

With Step 2 designed such that the intermediate control deviation, the following inequality holds.

Since is a Lyapunov function, the control deviation converges to zero. For Step 2, the function of the intermediate control deviation defined by Equation (21) is selected as the Lyapunov candidate function.

By differentiating both sides, we obtain the following.

Since is a Lyapunov function, this inequality shows that Step 2 is stable, with the intermediate control deviation converging to zero.

Equation (22) is derived as follows.

Assuming that the control law u is given by Equation (24), Equation (25) is obtained by substituting Equation (24) into Equation (21).

A stable control system can therefore be achieved by using Equation (24) as the control law for the BSC.

Headings, or heads, are organizational devices that guide the reader through your paper. There are two types: Component heads and text heads.

As shown by Equations (2) and (3), BSC has the advantage of allowing nonlinear compensation to be easily achieved. When the friction function is known, Equation (25) can be used as the control law for Step 2.

In actual applications, however, nonlinear characteristics contain errors or are unknown. In addition, the backlash function and other characteristics need to be offset not by subtraction but with an inverse model. In this study, those issues were solved by adding an adaptive control term that integrates the control deviation in Step 2 under the conditions for Lyapunov stability. Equation (26) or (27) can be applied as the adaptive control term.

In these equations, is the control deviation with respect to the reference linear model, but it is not always necessary within the allowable range of steady deviation. If an adaptive control term is added, the control law is given by Equation (29).

In this case, since Equation (27) can be expressed as Equation (30), stable control is guaranteed.

Figure 6 shows a block diagram of the adaptive back-stepping control system obtained from the above design. The backlash function is disregarded based on the following results of paragraph 5.3. The reference linear model in Figure 6 is the BSC system assumed without nonlinear control model and without parameter error margin of linear model and is not indispensable.

We examined the control performance of the BSC system by measuring the step response. Assuming that the control object has the physical characteristics shown in Figure 4, we used Matlab/Simulink and built the BSC system illustrated in Figure 6. The derivative term added to Step 1 improved the starting characteristics but delayed convergence near the desired value. To solve this

problem, we attached an imperfect dead zone expressed by Equation (31) to the subsequent stage of the derivative term (also included in Figure 6). This dead zone makes the control deviation minute and gradually reduces the derivative term gain. The Step 2 parameters were determined by fitting the control object to Equation (11).

Figure 7 shows the results of the simulation. Response becomes higher with increasing derivative term gain in Step 1. This effect gradually becomes smaller, and the response does not exceed critical damping, so that no oscillation is observed.

The control parameters to be adjusted in designing the BSC are, in Step 1, the proportional term gain of the control deviation and the derivative term gain, and, in Step 2, the proportional term gain of the intermediate control deviation and the parameters of the linear control model expressed in Equation (13). The parameters can be adjusted by fitting them to the step response of the control object. The proportional term gains and, which are unique to the BSC, are mutually complementary, as shown in Table 2 which lists the simulation results. Hence, is determined by considering factors such as the constraint on the operation quantity, and, the constraint on the controlled quantity, and avoidance of overshoot. Next, is determined from the response speed of the controlled quantity, for example, a 95% response in 80 ms. The remaining derivative term gain in Step 1 is determined by varying it from such that the target responsiveness is ensured under the above-mentioned constraints.

Backlash is included as Equation (11) in the control object. As shown in Equation (14), the relationship between the input and output of the control model is expressed as. This suggests that backlash compensation requires an inverse model. In this study, we assumed that the backlash was unknown and examined how effectively the adaptive control term absorbed it in Step 2. We implemented the simulation using Equation (32), which was obtained by omitting from Equation (27).

Figure 8 shows the simulation results. As the backlash motion is repeated, the effect of the adaptive control term causes the motion to gradually approach that without backlash (zd). The results show that this tendency also differs depending on the BSC control parameter ×. It can be seen that, when ×is sufficiently large relative to the width of the backlash dead zone, good responsiveness is obtained even without the adaptive control term.

Since the electronic control throttle needs to perform fine air-intake adjustment, the responsiveness to be achieved when the target degree of opening is changed

on a continuous basis by 0.05˚ intervals is given as the target specification. In the operation of such minute openings (NOT: Narrow Open Throttle), friction characteristics have a relatively large influence. In this simulation, we used the friction compensation model expressed by Equation (33).

The friction characteristics are known when, contain an error when, and are unknown when.

The simulation was implemented with a step response of 0.025˚, and includes the effect of adding the adaptive control term. The results are given in Figure 9 and Figure 10.

Figure 9 shows the results corresponding to. It can be seen that the throttle does not operate without compensation for the static friction being applied. In addition, a large control deviation remains when compensation is implemented using only the friction compensation model. In contrast, the response is in close agreement with that corresponding to the no-friction state when the adaptive control term is added.

Figure 10 shows the results for ξ = 0, in which no friction compensation model is used since the friction characteristics are unknown. Although a steady deviation remains when only the adaptive control term is added, it can be reduced using the control deviation of Step 2 of the reference linear model.

Figure 11 shows the results of simulating operation of the electronic control throttle from the fully closed to fully open position (WOT: Wide Open Throttle), and compares it with operation using conventional PID control. All of the

operations meet the 95% response in 80 ms required as the target specification. It can be seen that the starting characteristics are improved by adding a derivative term to Step 1 (D-BSC). Responsiveness can be adjusted by PID control, but too large a derivative term gain will cause overshooting. Critical damping is not exceeded with BSC.

Figure 12 shows the results of examining the resolution of the electronic control throttle. When its target degree of opening was changed on a continuous basis by 0.025˚ intervals, the setting value for PID control was affected by backlash

and friction characteristics, making it indefinite. However, a resolution of 0.05˚ is required as the target specification of the equipment. In contrast, the effect of the compensation expressed by Equation (26) of Step 2 allows BSC to keep up with minute changes in the desired value.