Adaptive Back-Stepping Control of Automotive Electronic Control Throttle

Transcription

1 Journal of Software Engineering an Applications, 07, 0, ISSN Online: ISSN Print: Aaptive Back-Stepping Control of Automotive Electronic Control Throttle Nobuo Kurihara, Hiroyuki Yamaguchi Department of System an Information Engineering, Hachinohe Institute of Technology, Hachinohe, Japan How to cite this paper: Kurihara, N. an Yamaguchi, H. (07) Aaptive Back- Stepping Control of Automotive Electronic Control Throttle. Journal of Software Engineering an Applications, 0, Receive: December 6, 06 Accepte: January 0, 07 Publishe: January 3, 07 Copyright 07 by authors an Scientific Research Publishing Inc. This work is license uner the Creative Commons Attribution International License (CC BY 4.0). Open Access Abstract Back-stepping control (BSC), which is eeme effective for a non-holonomic system, is applie to improving both responsiveness an resolution performance of an electronic control throttle (ECT) use in automotive engines. This paper is characterize by the use of a two-step type BSC in a manner that achieves an improvement in responsiveness with the ETC operate in a fully opene state by aing a erivative term in Step an the improvement in resolution performance with the ETC operate in a minutely opene state by aing an aaptive feature in the form of an integral term using the control eviation in Step. This paper presents an ECT control expresse as a secon-orer system incluing nonlinearities such as backlash of gear train an static friction in sliing area, a BSC system esigne base on Lyapunov stability, an a etermination metho for control parameters. Also, a two-step type BSC system is formulate using Matlab/Simulink with a physics moel as a control object. As a result of simulation analyses, it becomes clear that the BSC system can achieve quicker response because the erivative term works effectively an finer resolution because the aaptive control absorbs the error margin of the nonlinear compensation than conventional PID control. Keywors Back-Stepping Control, Aaptive Control, Electronic Throttle, Automotive Engine. Introuction The electronic control throttle in automobile engine control systems is inispensable to increasing fuel economy an curbing exhaust emissions. It is the main actuator for torque control in gasoline engines an emission control in iesel engines, an its responsiveness an resolution have to meet strict requirements. It is anticipate that the electronic control throttle will not only provie a solution to relevant mechanical problems, but also eliver improvements DOI: 0.436/jsea January 3, 07

2 through its metho of control. Research on nonlinear control base on feeback control enhance by the Lyapunov esign metho has been in progress since the early 990 s. Krstić et al. propose this type of control as Back-stepping Design [], an Zhou et al. then systematize it as Aaptive Back-stepping Control []. This back-stepping control (BSC) is a promising metho applicable to electronic throttle control as a means of solving nonlinear problems, such as the backlash of the gear train an frictional characteristics of rotational sliing. Pan et al. propose a relevant approach [3], but this focuses on state observers, an oes not take avantage of the benefit obtaine from BSC. The author et al. has achieve high responsiveness with the ai of sliing-moe control [4] [5]. However, nonlinear compensation for fine control is insufficient. This paper escribes a metho for esigning back-stepping control (BSC) of a low-orer control object with the nonlinear characteristics expecte for an automobile electronic control throttle. The control use is two-step type BSC base on a quaratic linear moel so that esign can be formulate in a manner similar to that for conventional PID control. We ae a erivative control function to Step in orer to improve responsiveness, an an aaptive control function to Step in orer to absorb moel errors in nonlinear compensation. The effects of these functions were examine by simulation using Matlab/Simulink. As control objects, we built a physical moel base on the structure of the electronic control throttle, a backlash moel for examining the rive torque, an a Stribeck moel for consieration of frictional characteristics [6] [7]. The erivative term in Step serves to improve the starting characteristics that the electronic control throttle exhibits in response to suen acceleration. Minute opening control on the electronic control throttle may cause sticking ue to static friction, overshooting after a motion, an a steay-state eviation ue to backlash. The results of simulations suggest that such problems with resolution egraation ue to response lag an nonlinear characteristics can be solve by improving Steps an of the two-step type BSC. We also achieve further improvement by referring to a reference linear moel.. Basic BSC Design This paper systematizes BSC esign by assuming that the control object is an (N N) MISO system that combines the linear an nonlinear characteristics of a state variable. The control object is assume to be an object escribe by Equation (). N x = a, jxj + ψ ( x) j= N x i= ai, jxj+ ψ i( x) () j= N x N= an, jxj+ ψ N( x) + bu N j= 4

3 We esigne an N-step control system that fees back all the components of the state variable. The control law meeting the conitions for Lyapunov stability in each step is given by Equations () to (4). Derivation of these equations is omitte. Step : Step i: Step N: where α N( j ) = z cz ajxj ψ ( ) a x () j= N( j + i ) = cz a, z a, x ( ) a x (3) ii, + j= α α ψ i i i i i i i i j j i N u = αn cnzn an, NzN anjxj ψ N ( x) (4) bn j= z is the esire value, an is the control error in Step i. 3. Moeling an Electronic Control Throttle 3.. Linear Moel Figure illustrates the structure of an electronic control throttle use for automobile engines. Its main elements are a DC motor, gear train (pinion gear, intermeiate gear, an sector gear), butterfly valve, an springs. For these elements, a DC motor circuit equation, DC motor rotational motion equation, gear-train rotational motion equation, an valve rotational motion equation are erive. These physics are shown from Equations (5) to (8). Table shows the parameter names an their meanings use in the throttle valve s mathematical moels. In aition, Equations (5) to (8) can be reuce to Equations (9) to (0) by consiering the gear ratios. i θm Lm + Ri + Km = V (5) Figure. Structure of ECT. 43

4 Table. Parameters of ECT. Parameters Physical meanings Units θ v Valve opening eg θ m Motor rotary angle eg 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 D m, D g, D v Viscous friction N m s E g, E v Transmission efficiency - R Motor internal resistance Ω where θm θm Jm + D m = Kmi T (6) θg θg Jg + D g = NgET g T (7) θv θv Jv + D v = NvET v Tsp ( θ ) (8) θv θv J + D + T sp ( θv ) = K I (9) i Lm + Ri + Ke N g Nv θv = V (0) J = J N N E E + J N E + J m g v g v g v v v D = D N N E E + D N E + D m g v g v g v v v K = K N E N E m g g v v 3.. Nonlinear Moel We consier the backlash characteristics escribe by Figure an Equation () an the friction characteristics escribe by Figure 3 an Equation () as the nonlinear characteristics of the electronic control throttle. The friction characteristics are known as the Stribeck moel an are obtaine as a composite of static, Coulomb, an viscous friction. Since the viscous friction is alreay containe as the by Equation () is applie. Dθ v term of Equation (9), it is exclue when st ( v ) ( ) ( ) ( ) if ( ) 0 an ( ) ( ) or if ( ) 0 an ( ) ( ) T θ expresse mt v t Tv t Tv t = m Tv t Br T v( t) = T v t < Tv t = m Tv t Bl () 0 otherwise 44

5 Figure. Backlash characteristics of gear train. Figure 3. Stribeck friction nonlinearity. T st ( θv ) if θv θth ( Tc + ( Tbrk Tc ) exp( c s θv )) sign( θv ) + D vθv = if θv < θth (( Tc + ( Tbrk Tc ) exp( c s θv )) + D vθ v ) θ v θth () 3.3. Electronic Control Throttle The block iagram in Figure 4 is obtaine by combining the linear moel in 3. an nonlinear moel in 3.. As the physical moel of an electronic control throttle, it is use as a control object in the creation of a control moel or for confirming the performance of a control system Linear Control Moel A control moel built irectly from Equations (9) to (0) can be applie as a compensation moel integrate into BSC. However, this paper erives a qua- 45

6 ratic linear moel in orer to ensure consistency with conventional PID control systems. Assuming a control moel expresse by Equation (3), we ientifie the step response of the electronic control moel of Figure 4 with the nonlinear moel exclue. We thereby obtaine a secon-orer elay moel that is in goo agreement with the physical moel, as can be seen in Figure 5. x = x (3) x = ax + ax + bu Figure 4. Block iagram of electronic control throttle. Time [s] Figure 5. Fitting using n orer elay moel of electronic control throttle. 46

7 4. Detaile Design of BSC When esigning the BSC control system for the electronic control throttle, we change the control object from Equation () to Equation (3) by reucing its orer. This control object is a quaratic linear moel suite to the control moel erive in the preceing section with a friction an backlash moel ae. In the Equation (4), x is the throttle opening, ψ ( x ) the friction torque, an B( ) the backlash function. x = x x = aixi + ψ ( x) + bu (4) i= u = B( v) The BSC is a two-step type of control consisting of Step for the x feeback an Step for the x loop. 4.. Aing a Derivative Term to Step In this stuy, we ae a erivative term to Step in orer to improve responsiveness (particularly the starting characteristics). The following emonstration of stability omits escription of nonlinear characteristics. Equation (5) efines the control law of Step using a linear combination of a control eviation z an its change rate z. α = z cz + z c > 0 0 (5) ( ) ( ) For the Lyapunov caniate function V efine by Equation (5), we obtain: V = z (6) z = x z (7) z = x z = x z (8) ( ) ( α ) ( α ) V = zz = z x z = z x cz z = cz zz + z x = cz V + zz = + + ( cz zz ) With Step esigne such that the intermeiate control eviation z = 0, the following inequality hols. V 0 (0) Since V is a Lyapunov function, the control eviation z converges to zero. For Step, the function V of the intermeiate control eviation z efine by Equation () is selecte as the Lyapunov caniate function. (9) 47

8 V = V + z ( + ) () By ifferentiating both sies, we obtain the following. zz V = V + = cz cz 0 () ( ) ( ) ( ) + + Since V is a Lyapunov function, this inequality shows that Step is stable, with the intermeiate control eviation z converging to zero. Equation () is erive as follows. ( ) ( α ) + V = cz + zz + zax+ zax + zbu z (3) Assuming that the control law u is given by Equation (4), Equation (5) is obtaine by substituting Equation (4) into Equation (). ( ) ( cz ) cz + u = ( cz ax ax z) ( c 0) b α > (4) V = cz + zz ( ) + zax + zax z α+ zb cz z ax ax + α ( + ) b (5) = 0 A stable control system can therefore be achieve by using Equation (4) as the control law for the BSC. 4.. Aing an Integral Term to Step Heaings, or heas, are organizational evices that guie the reaer through your paper. There are two types: Component heas an text heas. As shown by Equations () an (3), BSC has the avantage of allowing nonlinear compensation to be easily achieve. When the friction function ψ ( x) is known, Equation (5) can be use as the control law for Step. u = ( cz ax ax z) ( c 0) b α ψ > (6) In actual applications, however, nonlinear characteristics contain errors or are unknown. In aition, the backlash function B( ) an other characteristics nee to be offset not by subtraction but with an inverse moel. In this stuy, those issues were solve by aing an aaptive control term f ( z ) that integrates the control eviation z in Step uner the conitions for Lyapunov stability. Equation (6) or (7) can be applie as the aaptive control term f z. a ( ) a ( ) ς ( ) f z z z z t (7) a ( ) ( ) η ( ) f z sign z z z t (8) In these equations, z is the control eviation with respect to the reference linear moel, but it is not always necessary within the allowable range of steay a 48

9 eviation. If an aaptive control term is ae, the control law is given by Equation (9). u= ax cz z + f b i i ψ α a (9) i= In this case, since Equation (7) can be expresse as Equation (30), stable control is guarantee. ( ) ( ) + V = cz cz z f a 0 (30) 4.3. Aaptive BSC System Figure 6 shows a block iagram of the aaptive back-stepping control system obtaine from the above esign. The backlash function is isregare base on the following results of paragraph 5.3. The reference linear moel in Figure 6 is the BSC system assume without nonlinear control moel an without parameter error margin of linear moel an is not inispensable. 5. Simulations an Discussion 5.. Improvement in Response We examine 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 use Matlab/Simulink an built the BSC system illustrate in Figure 6. The erivative term ae to Step improve the starting characteristics but elaye convergence near the esire value. To solve this Figure 6. Two-step BSC system with aaptive control. 49

10 problem, we attache an imperfect ea zone expresse by Equation (3) to the subsequent stage of the erivative term (also inclue in Figure 6). This ea zone makes the control eviation z minute an graually reuces the erivative term gain. The Step parameters a, a, b were etermine by fitting the control object to Equation (). z ( ) ( ) ( ) ( e ) i t L zo t = zi t + L (3) Figure 7 shows the results of the simulation. Response becomes higher with increasing erivative term gain in Step. This effect graually becomes smaller, an the response oes not excee critical amping, so that no oscillation is observe. 5.. Setting Control Parameters The control parameters to be ajuste in esigning the BSC are, in Step, the proportional term gain c of the control eviation z an the erivative term gain, an, in Step, the proportional term gain c of the intermeiate control eviation z an the parameters a, a, b of the linear control moel expresse in Equation (3). The parameters a, a, b can be ajuste by fitting them to the step response of the control object. The proportional term gains c an c, which are unique to the BSC, are mutually complementary, as shown in Table which lists the simulation results. Hence, cc is etermine by consiering factors such as the constraint on the operation quantity u( t ), an supu ( t), the constraint on the controlle quantity x, an avoiance of overshoot. Next, c c is etermine from the response spee of the controlle quantity x, for example, a 95% response in 80 ms. The remaining erivative term gain in Step is etermine by varying it from = 0 such that the target responsiveness is ensure uner the above-mentione constraints. Figure 7. Effect in step response at erivative term. 50

11 5.3. Compensating for Backlash Characteristics Backlash is inclue as Equation () in the control object. As shown in Equation (4), the relationship between the input u an output v of the control moel is expresse as u = B( v). This suggests that backlash compensation requires an inverse moel. In this stuy, we assume that the backlash was unknown an examine how effectively the aaptive control term fa ( z ) absorbe it in Step. We implemente the simulation using Equation (3), which was obtaine by omitting z from Equation (7). f z z ς z t (3) a ( ) Figure 8 shows the simulation results. As the backlash motion is repeate, the effect of the aaptive control term causes the motion to graually approach that without backlash (z). The results show that this tenency also iffers epening on the BSC control parameter c c. It can be seen that, when c c is sufficiently large relative to the wih of the backlash ea zone, goo responsiveness is obtaine even without the aaptive control term Compensating for Friction Characteristics Since the electronic control throttle nees to perform fine air-intake ajustment, the responsiveness to be achieve when the target egree of opening is change Table. Ajustment of control parameters. Case c c c /c c c Re s. at 80 ms Figure 8. Behavior of aaptive control changing by esign parameters. 5

12 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 use the fric- ψ x expresse by Equation (33). tion compensation moel ( ) ψ ( x ) ξψ ( x ) = (33) The friction characteristics are known when ξ =, contain an error when > ξ > 0, an are unknown when ξ = 0. The simulation was implemente with a step response of 0.05, an inclues the effect of aing the aaptive control term ( ) Figure 9 an Figure 0. f z. The results are given in Figure 9 shows the results corresponing to ξ = 0.7. It can be seen that the throttle oes not operate without compensation for the static friction being applie. In aition, a large control eviation remains when compensation is im- ψ x. In contrast, the plemente using only the friction compensation moel ( ) a response is in close agreement with that corresponing to the no-friction state when the aaptive control term ( ) f z is ae. a Figure 0 shows the results for ξ = 0, in which no friction compensation moel is use since the friction characteristics are unknown. Although a steay eviation remains when only the aaptive control term ( ) f z is ae, it can be reuce using the control eviation z of Step of the reference linear moel Evaluating Electronic Throttle Control Figure shows the results of simulating operation of the electronic control throttle from the fully close to fully open position (WOT: Wie Open Throttle), an compares it with operation using conventional PID control. All of the a Time [s] Figure 9. Responses when friction function contains error margin ( ξ = 0.7 ). 5

13 Figure 0. Responses when friction function is unknown ( ξ = 0 ). Figure. Test of Wie-Open-Throttle. operations meet the 95% response in 80 ms require as the target specification. It can be seen that the starting characteristics are improve by aing a erivative term to Step (D-BSC). Responsiveness can be ajuste by PID control, but too large a erivative term gain will cause overshooting. Critical amping is not exceee with BSC. Figure shows the results of examining the resolution of the electronic control throttle. When its target egree of opening was change on a continuous basis by 0.05 intervals, the setting value for PID control was affecte by backlash 53

14 Figure. Test of Narrow-Open-Throttle. an friction characteristics, making it inefinite. However, a resolution of 0.05 is require as the target specification of the equipment. In contrast, the effect of the compensation expresse by Equation (6) of Step allows BSC to keep up with minute changes in the esire value. 6. Summary Back-stepping control (BSC) can compensate separately for linear an nonlinear characteristics in the system esign, an provie a stable response even when moel errors are present. This stuy applie BSC to an electronic control throttle for automobiles, an thereby achieve improvements in both responsiveness an resolution performance. We expresse a general-purpose esign metho for BSC as Equations () to (4), an etermine the following through the use of simulations. () Two-step type BSC is suitable for electronic control throttles. () We presente Equation (3) as the control law for a erivative term ae to Step of the two-step type BSC. This improves starting characteristics. (3) We presente Equations (3) an (4) as control laws for an integral term ae to Step of the two-step type BSC (aaptive BSC). This absorbs errors that occur in nonlinear control moels. (4) Consieration of backlash compensation becomes unnecessary if the control gain ( c c ) of the BSC is increase. (5) Aaptive BSC allows the esire value to be followe even when friction characteristics are unclear ( ξ = 0.7 ) or unknown ( ξ = 0 ). The above results inicate that electronic control throttles using the two-step type BSC that we have propose can achieve a higher egree of responsiveness 54

15 an resolution performance than conventional PID control. Acknowlegements We will express our gratitue for Hiroshi Hayashi, Masahiro Miura, an Hiroki Toa that avances the simulation examination in this research. References [] Krstić, M., Kanellakopoulos, I. an Kokotović, P. (995) Nonlinear an Aaptive Control Design. Wiley-Interscience, New York. [] Zhou, J. an Wen, C.Y. (008) Aaptive Backstepping Control of Uncertain Systems. Springer Publication, Heielberg. [3] Pan, Y.D., Ozguner, U. an Dagci, O.H. (008) Variable-Structure Control of Electronic Trottle Valve. IEEE Transactions on Inustrial Electronics, 55, [4] Zhang, Y. an Kurihara, N. (0) An Integral Variable Structure Compensation Metho for Electronic Throttle Control with Input Constraint. Proceeings of 0 3r International Conference on avance Computer Control (ICACC), 8-0 January 0, [5] Zhang, Y. an Kurihara, N. (0) A Stuy of Integral Sliing Moe Control with Input Constraint for Engine Iling-Spee Control. IEEJ Transactions on Electrical an Electronic Engineering, 7, 4-9. [6] Brian, A.-H., Dupont, P. an De Wit, C.C. (994) A Survey of Moels, Analysis Tools an Compensation Methos for the Control of Machines with Friction. Automatica, 30, [7] Iwasaki, M., Kitoh, Y. an Matsui, N. (996) Analysis an Performance Improvement of Motor Spee Control System with Nonlinear Friction. IEEJ Transactions on Electronics, Information an Systems, 6, Submit or recommen next manuscript to SCIRP an we will provie best service for you: Accepting pre-submission inquiries through , Facebook, LinkeIn, Twitter, etc. A wie selection of journals (inclusive of 9 subjects, more than 00 journals) Proviing 4-hour high-quality service User-frienly online submission system Fair an swift peer-review system Efficient typesetting an proofreaing proceure Display of the result of ownloas an visits, as well as the number of cite articles Maximum issemination of your research work Submit your manuscript at: Or contact jsea@scirp.org 55