Self-tuning Control Based on Discrete Sliding Mode

Transcription

1 Int. J. Mech. Eng. Autom. Volume 1, Number 6, 2014, pp Received: July 18, 2014; Published: December 25, 2014 International Journal of Mechanical Engineering and Automation Akira Ohata 1, Akihiko Sugiki 2 and Katsuhisa Furuta 3 1. Toyota Motor Corporation, Higashifuji Technical Center, Shizuoka , Japan 2. Nikki Denso Co., Ltd., Chiba , Japan 3. Tokyo Denki University, Tokyo , Japan Abstract: This paper presents a multivariable self-tuning control based on the idea of discrete-time sliding mode. The previous stability proof based on the Lyapunov function is corrected in the MIMO framework that plant input and output have the same dimension. The performance of the proposed method is evaluated in the simulation environment designed for MBD (model based development) of automotive systems. An application to the engine-start control of an automotive V6 engine model is given. Its model order is over 50, and the control objective is to regulate the engine speed at 650 ± 50 rpm within 1.5 s just after the engine starts. This control requirement is achieved by controlling the engine speed and intake pressure with using the proposed multivariable adaptive scheme. Key words: Self-tuning control, discrete-time sliding mode, Lyapunov function, multi-input multi-output systems, automotive engine control. 1. Introduction Adaptive control has gathered great attentions from automotive industry because the production variance and aging effect deteriorates the performance of mass produced automobiles. Adaptive control is a potential solution to mitigate the issue and to reduce the calibration effort in the development of engine control systems, but there have been proposed few MIMO (multi-input multi-output) adaptive control methods practically applicable. The idea of combining the LSE (least squares estimation) with feedback control to obtain the adaptive capability goes back to Kalman [1]. In Astrom and Wittenmark [2], self-tuning control was originated for discrete-time systems, which is named istc (implicit self-tuning control) later. They showed that the control with LSE converges to a minimum-variance regulator if the estimates converge. Corresponding author: Akira Ohata, senior general manager, research fields: automotive control and control theory. akira_ohata@mail.toyota.co.jp. Goodwin, et al. [3] studied the global stability of minimum phase discrete-time MIMO noise-free linear systems in a seminal paper, in which they showed that the tracking error converges to zero if the input and output are bounded. Clarke, et al. [4, 5] and Gawthrop [6] studied the convergence of STC for non-minimum phase system, incorporating the input, output, and set point variations into the cost function. Clarke [7] also studied the practical features of STC using the recursive LSE together with the minimization of an auxiliary output for non-minimum phase systems. Slotine and Li [8] combined sliding mode control and adaptive control to assure the robustness when uncertainty exceeds the bound assumed. This idea has not been succeeded in applying to discrete-time systems. Patete, et al. [9, 10] showed that the idea is generalized to discrete-time systems combining LSE and discrete-time sliding mode, and the stability is proved based on the Lyapunov function. The authors pointed out in the presentation of the IFAC congress [11] that the stability was not completely proved and then corrected the proof in Ref. [12]. A purpose of this

2 368 paper is to show the corrected proof also in a MIMO framework. This paper is organized as follows: Section 2 presents a MIMO self-tuning control design; Section 3 gives an application to the cycle by cycle model of automotive V6 gasoline engine proposed in Refs. [13, 14] and shows that the proposed MIMO istc improves the performance of non-adaptive control; Section 4 concludes the paper. 2. Implicit Self-tuning Control for MIMO System Consider a -input -output system described by ( ) =( ) ( ) =+ + + ( ) = where =[,,,, ], =[,,,, ],, are its input, output, backward-shift operator, and delay time, respectively. and are polynomial matrices, and is assumed nonsingular. Define the vector as =( )( )+( ) (1) ( ) =+ + + ( ) =diag{ (1 ),, (1 )} where =[,,,, ], is an matrix of the Schur polynomials, =[,,,, ] is a constant reference vector, and (, ) are designed so that the following system is stable: = Define the polynomial matrices ( ) =+ + + () ( ) = () which satisfy =+, then (1) yields =(( )( )+ ( )) ( ) +( ) =(( )( )+( )) +( ) ( ) (2) Denote = +. Letting =0, which is considered a kind of discrete-time sliding mode, gives the control: =( ) (( ) ( ) ) Theorem 1: (MIMO self-tuning control) As, the control law = ( ) (( ) ( ) ) (3) makes 0 where =[, ] and =[, ] (, =1,,) denote the estimates of controller parameters at time. The parameters, =[,,,,,,,,, ( )=,,,,,,,,,, +,, ( )=, are estimated by the algorithm,,, ] (4) + +, () +, + +, (), =, +, (, + ( ), ) (5), =, ( +, ), = {,, ( +, ), } (6) =[,,, (),,,,,, (),,,, (),,,,,, (), ] where is a forgetting factor and is the i-th row of. Proof Define the variable, ( =1,,) by Eq. (7), which is introduced to complete the proof:, =, + ( ), =, (7) where, =, is the parameters error vector:, =[,,,,,,,,,,,,,, Consider the Lyapunov function,,,,,, ] =, = , = 1 2, + 1 2,,, where =diag,,,,,, > 0 ( = 1,,), =[,,,,], and = [,,,, ]. Adding the terms 0=,,, 1 2,,,

3 369 0= 1 2, 1 2,,,,, + 1 2,,, to, =,, yields, =,,,, 1 2 (,, ), (,, ) + 1 2, (,, ), +,, (,, +,, ) (8) Since Eq. (6) gives, =(, + ), the 3rd term in the right hand side of (8) vanishes. Using, =, given by (2), (5) is written as (,, )+,, =0 (9) where the following equality has been used:, ( +, ) = ( +, ), Eq. (9) implies that the 4th term in Eq. (8) vanishes. Thus, 0. =, 0 means 0 as. This yields, 0 because,, = (,, ),, 0 We complete the proof. Notice that =(,, +), [( ) ( ),, () ] is used alternatively to avoid the singularity in Eq. (3) where ɛ is a sufficiently small positive constant and ( ) =, +, + +, (). 3. Engine Speed Control Idle reduction is regarded as a useful method to avoid the waste of fuel and reduce exhaust gas emissions. But, it highly requires the smooth engine restart. It is well known that almost HC, CO, NO x emissions are exhausted in the short period from the engine start to the activation of the catalyst and the smooth engine start reduces the exhaust gas emissions considerably. Hybrid vehicles also adopt idle reduction because electrical motors easily realize to restart engines. Therefore, the engine start control has received a great attention. However, the engine start control to keep lean air-fuel ratio is the one of the most difficult portions of engine control although almost all people may take it for granted that their engines can start easily. Research committee on Advanced Powertrain Control Theory of SICE provided the engine start benchmark problem and the engine model. Fig. 1 shows the 3 litters, V6, port injection gasoline engine model by using Simulink with two-way connections of which a line connected between two blocks shows two direction signals. It is an in-cylinder model calculating each cylinder pressure and the engine speed fluctuation during an engine cycle. It is constructed by the first principles and the system order is over 50. The manipulation signals are the throttle angle, the spark advance and the amount of fuel injection mass of each cylinder. The purpose of the benchmark problem is to start the engine and to regulate the engine speed at 650 ± 50 rpm within 1.5 s. Actually, the overshoot reaches around 2,000 rpm without the throttle and the spark advance controls as shown in Fig. 2 and the following engine speed drop yields the uncontrollable rich air-fuel ratio that causes much HC exhaust gas emission. Thus, the suppression of the engine speed overshoot just after the engine start is additionally required. In this study, the model is modified to remove the fuel dynamics in the intake port because the purpose of this study is to show an effectiveness of the proposed istc. When applying istc to the engine start control, the engine speed and the intake pressure are controlled with the throttle angle and the spark advances which are treated as a single input representing

4 370 Fig. 1 V6 gasoline engine model provided from SICE. where ω is the engine speed (rad s 1 ), p is the intake pressure (kpa), is the throttle angle, is the spark advance (deg), the suffix means the value of convergence, and is the number of sampling. The error system is represented by Fig. 2 Engine start simulation without the throttle angle and the spark advance controls. all spark advances of the V6 engine. To feed the constant throttle angle of 6 deg and the spark advances of 10 deg CA into the engine model gave the stable engine speed of 650 rpm and the intake pressure of 28 kpa. It took 40 s to obtain the stable engine condition. The error system from the stable condition was identified by using simulation data as follows:, = 1 ( ), = 1 ( ), = 1 ( ), = 1 ( ) ( + ), =,, (10), The square waves of which the amplitudes are the throttle angle of 0.5 (deg) and the spark advance of 0.5 (deg CA) respectively are fed to the engine model. The identification results of Eq. (10) from the measured data are = , = = (11) and the comparison between the simulation data and the identification results is shown in Fig. 3. According to the identified parameters (11), ( )= ( )= are obtained with the given matrices of ( )= ( )=

5 371 Fig. 3 Comparison of the engine model and the identification result. Fig. 4 Top layer of closed-loop simulation. Fig. 4 shows the top layer of the model provided from SICE. The proposed control is implemented in the controller block as shown in the upper portion of Fig. 4. The air flow rate was fed to the original controller block but switched by the intake pressure when istc was designed. The controller expressed by (3) was designed by using the identified model parameters and the above matrices and. The above and were used as the initial estimation of (4). For the estimation of and, the forgetting factors = are applied. Fig. 5 shows a simulation result of the proposed istc. We succeeded to start the engine model and to regulate the engine speed at 650 rpm within 1.5 s. An interesting thing of this result is that the throttle is closed when the engine starts. The motion of the throttle valve is effective to suppress the engine speed overshoot because closing the throttle valve reduces the intake pressure rapidly and makes the first ignition weak. The proposed control gives the motion by detecting the higher intake pressure than the target value during the engine start. Next, to investigate the robustness of the proposed istc, the plenum chamber volume was changed from 6.0 L to 25.0 L. Fig. 6 shows the comparison of the control results with istc and DLQR (discrete-time linear quadratic regulator). The criterion of DLQR is Fig. 5 Fig. 6 where Result of multi-variable self-tuning control. Comparison of istc and DLQR. =,, +,,

6 372 = , = istc was designed such as the initial closed loop poles are equal to the ones of DLQR. DLQR failed to start the engine as shown by the dotted line but istc succeeded to do it although the fluctuation appears in the top of Fig. 6. The fluctuation may be caused by the saturation of the throttle angle. This result indicates that the proposed control is well adapted to the change of the plenum chamber volume. 4. Conclusions In the paper, a multivariable self-tuning control design was presented. Its stability proof is a corrected version of the previous result, which was done by additionally introducing one intermediate vector in showing the convergence of discrete-time sliding mode type variables. The control result of V6 gasoline engine model was given for validation. The proposed control successfully regulates the engine speed at 650 ± 50 rpm within 1.5 s by simultaneously controlling the engine speed and intake pressure with the throttle angle and spark advance. This control result contributes to the idle reduction which is effective for saving fuel and engine restart. From another perspective, the proposed multivariable self-tuner is useful to refine the MBD environment. Since complicated mechanical systems like automotive engines are varying its dynamics by changing parameters under different operating conditions. The possible controlled performances of such systems are evaluated efficiently by the multivariable adaptive control. Acknowledgment This research has been done by the support of Grant-in-Aid for Scientific Research (B) Grant Number References [1] R.E. Kalman, Design of a self-optimizing control system, Transactions of the ASME 80 (2) (1958) [2] K.J. Astrom, B. Wittenmark, On self-tuning regulators, Automatica 9 (1973) [3] G.C. Goodwin, P.J. Ramadge, P.E. Caines, Discrete-time multivariable adaptive control, IEEE Transactions on Automatic Control 25 (3) (1980) [4] D.W. Clarke, D. Phil, P.J. Gawthrop, Self-tuning controller, IEE Proceedings 122 (9) (1975) [5] D.W. Clarke, D. Phil, P.J. Gawthrop, Self-tuning control, IEE Proceedings 126 (6) (1979) [6] P.J. Gawthrop, On the stability and convergence of a self-tuning controller, International Journal of Control 31 (5) (1980) [7] D.W. Clarke, Self-tuning control of non-minimum phase systems, Automatica 20(5) (1984) [8] J-J.E. Slotine, W. Li, On the adaptive control of robot manipulators, The International Journal of Robotics Research 6 (3) (1987) [9] A. Patete, K. Furuta, M. Tomizuka, Stability of self-tuning control based on Lyapunov function, International Journal of Adaptive Control and Signal Processing 22 (2008) [10] A. Patete, K. Furuta, M. Tomizuka, Self-tuning control based on generalized minimum-variance criterion for auto-regressive models, Automatica 44 (2008) [11] K. Furuta, A. Ohata, A. Sugiki, Self-tuning control based on discrete-time sliding mode with applications, in: Proceedings of the 18th IFAC World Congress 2011, MoA22.5. [12] A. Sugiki, K. Furuta, A. Ohata, H. Nita, Nonlinear variable structure adaptive control, in: Proceedings of the American Control Conference 2014, pp [13] A. Ohata, S. Shen, K. Ito, Introduction to the benchmark challenge on SICE engine start control problem, in: Proceedings of the 17th IFAC World Congress 2008, Mo28.5. [14] A. Ohata, H. Ito, S. Gopalswamy, K. Furuta, Plant modeling environment based on conservation laws and projection method for automotive control system, The SICE Journal of Control, Measurement, and System Integration 2 (3) (2008)