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Back-stepping control (BSC), which is deemed effective for a non-holonomic system, is applied to improving both responsiveness and resolution performance of an electronic control throttle (ECT) used in automotive engines. This paper is characterized by the use of a two-step type BSC in a manner that achieves an improvement in responsiveness with the ETC operated in a fully opened state by adding a derivative term in Step 1 and the improvement in resolution performance with the ETC operated in a minutely opened state by adding an adaptive feature in the form of an integral term using the control deviation in Step 2. This paper presents an ECT control expressed as a second-order system including nonlinearities such as backlash of gear train and static friction in sliding area, a BSC system designed based on Lyapunov stability, and a determination method for control parameters. Also, a two-step type BSC system is formulated using Matlab/Simulink with a physics model as a control object. As a result of simulation analyses, it becomes clear that the BSC system can achieve quicker response because the derivative term works effectively and finer resolution because the adaptive control absorbs the error margin of the nonlinear compensation than conventional PID control.

The electronic control throttle in automobile engine control systems is indispensable to increasing fuel economy and curbing exhaust emissions. It is the main actuator for torque control in gasoline engines and emission control in diesel engines, and its responsiveness and resolution have to meet strict requirements. It is anticipated that the electronic control throttle will not only provide a solution to relevant mechanical problems, but also deliver improvements through its method of control.

Research on nonlinear control based on feedback control enhanced by the Lyapunov design method has been in progress since the early 1990’s. Krstić et al. proposed this type of control as “Back-stepping Design” [

This paper describes a method for designing back-stepping control (BSC) of a low-order control object with the nonlinear characteristics expected for an automobile electronic control throttle. The control used is two-step type BSC based on a quadratic linear model so that design can be formulated in a manner similar to that for conventional PID control. We added a derivative control function to Step 1 in order to improve responsiveness, and an adaptive control function to Step 2 in order to absorb model errors in nonlinear compensation. The effects of these functions were examined by simulation using Matlab/Simulink. As control objects, we built a physical model based on the structure of the electronic control throttle, a backlash model for examining the drive torque, and a Stribeck model for consideration of frictional characteristics [

This paper systematizes BSC design by assuming that the control object is an (N × N) MISO system that combines the linear and nonlinear characteristics of a state variable. The control object is assumed to be an object described by Equation (1).

We designed an N-step control system that feeds back all the components of the state variable. The control law meeting the conditions for Lyapunov stability in each step is given by Equations (2) to (4). Derivation of these equations is omitted.

Step 1:

Step i:

Step N:

where

Parameters | Physical meanings | Units |
---|---|---|

θ_{v} | Valve opening | deg |

θ_{m} | Motor rotary angle | deg |

V | Motor voltage | V |

T_{sp}, T_{spl} | Spring torque | - |

N_{g}, N_{v} | Gear ratio | N∙m |

J_{m}, J_{g}, J_{v} | Moment of inertia | N∙m∙s^{2} |

D_{m}, D_{g}, D_{v} | Viscous friction | N∙m∙s |

E_{g}, E_{v} | Transmission efficiency | - |

R | Motor internal resistance | Ω |

where

We consider the backlash characteristics described by

The block diagram in

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

When designing the BSC control system for the electronic control throttle, we changed the control object from Equation (1) to Equation (13) by reducing its order. This control object is a quadratic linear model suited to the control model derived in the preceding section with a friction and backlash model added. In the Equation (14),

The BSC is a two-step type of control consisting of Step 1 for the

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

For the Lyapunov candidate function

With Step 2 designed such that the intermediate control deviation

Since

By differentiating both sides, we obtain the following.

Since

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

In actual applications, however, nonlinear characteristics contain errors or are unknown. In addition, the backlash function

In these equations,

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

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

problem, we attached an imperfect dead zone expressed by Equation (31) to the subsequent stage of the derivative term (also included in

The control parameters to be adjusted in designing the BSC are, in Step 1, the proportional term gain

Backlash is included as Equation (11) in the control object. As shown in Equation (14), the relationship between the input

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

Case | c_{1} | c_{2} | c_{1}/c_{2} | c_{1}c_{2} | Re s. at 80 ms |
---|---|---|---|---|---|

1 | 60 | 60 | 1.00 | 3600 | 95 |

2 | 45 | 80 | 0.56 | 3600 | 94 |

3 | 36 | 100 | 0.44 | 3600 | 93 |

4 | 30 | 120 | 0.36 | 3600 | 91 |

5 | 40 | 90 | 0.25 | 3600 | 88 |

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

The friction characteristics are known when

The simulation was implemented with a step response of 0.025˚, and includes the effect of adding the adaptive control term

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.

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.

Back-stepping control (BSC) can compensate separately for linear and nonlinear characteristics in the system design, and provide a stable response even when model errors are present. This study applied BSC to an electronic control throttle for automobiles, and thereby achieved improvements in both responsiveness and resolution performance. We expressed a general-purpose design method for BSC as Equations (1) to (4), and determined the following through the use of simulations.

(1) Two-step type BSC is suitable for electronic control throttles.

(2) We presented Equation (13) as the control law for a derivative term added to Step 1 of the two-step type BSC. This improves starting characteristics.

(3) We presented Equations (23) and (24) as control laws for an integral term added to Step 2 of the two-step type BSC (adaptive BSC). This absorbs errors that occur in nonlinear control models.

(4) Consideration of backlash compensation becomes unnecessary if the control gain (

(5) Adaptive BSC allows the desired value to be followed even when friction characteristics are unclear (

The above results indicate that electronic control throttles using the two-step type BSC that we have proposed can achieve a higher degree of responsiveness and resolution performance than conventional PID control.

We will express our gratitude for Hiroshi Hayashi, Masahiro Miura, and Hiroki Toda that advances the simulation examination in this research.

Kurihara, N. and Yamaguchi, H. (2017) Adaptive Back- Stepping Control of Automotive Electronic Control Throttle. Journal of Software Engineering and Applications, 10, 41-55. http://dx.doi.org/10.4236/jsea.2017.101003