Adaptive Output Feedback Control System Design for Slip Control of a Lock-Up Clutch Time-Delay System

11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France

ThS5T2.2

Adaptive Output Feedback Control System Design for Slip Control of a Lock-Up Clutch Time-Delay System Ikuro Mizumoto ∗ Taro Takagi ∗ Kenshi Yamanaka ∗∗ ∗

Department of Intelligent Mechanical Systems, Kumamoto University, Kumamoto 860-8555, Japan (e-mail: [email protected]). ∗∗ Department of Mechanical Systems Engineering, Kumamoto University, Kumamoto 860-8555, Japan Abstract: In this paper, we consider designing an adaptive output feedback control system with a simple structure for the slip control of a lock-up clutch system with a time delay. A design scheme of a parallel feedforward compensator (PFC) for systems with a time-delay for rendering ASPR augmented system will be proposed in order to realize a stable adaptive output feedback control to a slip control system of lock-up clutch with a time-delay. The effectiveness of the proposed method is confirmed through numerical simulations for a slip control of lock-up clutch system. Keywords: adaptive control, output feedback control, almost strictly positive real, time-delay system 1. INTRODUCTION The lock-up clutch is a system which improves the transmission efficiency of torque converters. However, in the situation of clutch where the engine and the drive system were directory connected, it often happens the vibration and thick sound due to engine torque pulsation. Therefore, it is required to make an efficient slip speed of clutch and to control it effectively. The slip control system of lock-up clutch in the torque converter is generally modelled as a system with a time-delay and the parameters of the system may often vary according to the engine speed. Thus, it might be very difficult to control the slip revolution speed accurately by a fixed controller. In this paper, we consider designing an adaptive output feedback control system with a simple structure for the slip control of a lock-up clutch system with a time delay. In order to design a simple and stable adaptive output feedback control system, the almost strictly positive real-ness (ASPR-ness) of the controlled system would be required (Kaufman et al., 1997; Fradkov, 1994; Mizumoto and Iwai, 1996). The system is said to be ASPR if there exists a static output feedback such that the resulting closedloop system is strictly positive real (SPR). Although the ASPR-based adaptive control scheme is one of the powerful control strategies for controlling uncertain systems, unfortunately however, the ASPR-ness of the system is a severe restriction for the practical systems so that the application of the ASPR-based adaptive control schemes would be restricted. As a countermeasure to this problem, the introduction of a parallel feedforward compensator (PFC) has been proposed (Barkana, 1987; Iwai and Mizumoto, 1994; Mizumoto and Iwai, 1996) and the ASPR restriction for practical systems has been alleviated by 978-3-902823-37-3/2013 © IFAC

Torque converter

Lock-up clutch

472

Hydraulic pressure Circuit of of Hydraulic Input Hydraulic pressure pressure

Input of revolution speed

Engine speed

Fig. 1. Lock-up clutch system designing the ASPR-based adaptive control system for the ASPR augmented system with the PFC (Kaufman et al., 1997; Fradkov, 1996; Mizumoto et al., 2009). However, for systems with a time-delay, no effective design scheme of the PFC so as to make an ASPR augmented controlled system have been proposed. This paper deals with a design problem of a parallel feedforward compensator (PFC) for systems with a time-delay for designing a stable adaptive output feedback control to a slip control system of lock-up clutch with a time-delay. A concrete PFC design method based on the idea of Smith predictor will be shown for the slip control of lock-up clutch system with time-delay in order to design an adaptive output feedback control of the slip system of lock-up clutch. The effectiveness of the proposed method is confirmed through numerical simulations for a slip control of lock-up clutch system. 2. LOCK-UP CLUTCH SYSTEM Consider a lock -up clutch system in a torque converter shown in Fig. 1. A simple model of the slip system of the lock-up clutch can be obtained as shown in Fig. 2 by 10.3182/20130703-3-FR-4038.00067

11th IFAC ALCOSP July 3-5, 2013. Caen, France

TEG rLup

G p (s)

e −Ts

TLup

+

TCV

−

r +

P (s)

e

+

ω SLP

u

C

+

−

P(s )

G ( s)e −Ts

y

F ( s)e −Ts

~ yf

~ ya

+ +

Fig. 2. Block diagram of a lock-up clutch system u

G ( s )e

− Ts

F ( s )e

− Ts

y

Fig. 4. Control system for the virtual augmented timedelay system

~ ya

r

~ yf

+

e

C

u

G ( s)e −Ts

y

~ G a ( s ) e − Ts

denoting the transfer function from the hydraulic pressure input signal τLup of the lock-up to the lock-up torque TLup by GP (s) and the transfer function from the converter torque TCV to slip revolution speed ωSLP by P (s). Thus, by considering the engine torque TEG as a disturbance and considering a time-delay T from input torque signal to actual torque action, the slip system of the lock-up clutch can be modelled as follows: GLup (s) = GP (s)P (s)e−T s = G(s)e−T s (1) In general, a simple approximated model of GP (s) can be given as second order system and P (s) can be approximated by a first order system as follows: ωnp (2) GP (s) = 2 2 s + 2ζωnp + ωnp K P (s) = (3) Ts + 1 and the parameters of these model will vary according to the engine speed. The objective of this paper is to design an adaptive output feedback control for the slip speed control of the lock-up clutch system with a time-delay. 3. ADAPTIVE OUTPUT FEEDBACK CONTROL SYSTEM DESIGN

ya

yf

H (s)

Fig. 3. Virtual augmented controlled system with timedelay

+ +

−

Fig. 5. Equivalent block diagram of the closed-loop system To this end, we first consider introducing a virtual PFC: F (s)e−T s with the same time-delay of the controlled system: G(s)e−T s as shown in Fig. 3. The resulting virtual augmented system can be represented as ˜ a (s)e−T s [u(t)] y˜a (t) = y(t) + y˜f (t) = G (6) where y˜a (t) is the output of the virtual augmented system: ˜ a (s)e−T s with G ˜ a (s) = G(s) + F (s) G (7) and y˜f (t): y˜f (t) = F (s)e−T s [u(t)] is the output of the virtual PFC: F (s)e

−T s

(8) .

For this augmented time-delay system, consider designing a control system with a controller C by using the Smith predictor: P (s) as shown in Fig. 4. In this case, the resulting closed-loop system from r(t) to y˜a (t) can be represented as y˜a (t) = =

1

C −T s ˜ 1−CP (s) Ga (s)e [r(t)] C −T s ˜ + 1−CP (s) Ga (s)e

˜ a (s)e−T s CG [r(t)]. ˜ a (s)e−T s 1 − CP (s) + C G

(9)

Thus designing the Smith predictor by

3.1 PFC design for making ASPR augmented system As shown in the previous section, the considered lock-up clutch system can be modelled as the following time-delay system: y(t) = G(s)[u(t − T )] = G(s)e−T s [u(t)], (4) where T is an input time-delay and the notation W (s)[u(t)] implies the output of the system expressed by a transfer function of W (s) with input u(t). That is, y(t) given in (4) is the output of the system G(s)e−T s with input u(t). Here, we will present a PFC design scheme so as to render the resulting augmented system with a PFC H(s): Ga (s) = G(s)e−T s + H(s) (5) ASPR in order to design a stable adaptive output feedback control system for the considered lock-up clutch system with a time-delay. 473

˜ a (s)e−T s − G ˜ a (s), P (s) = G

(10)

the following closed-loop system from r(t) to y˜a (t) is obtained. ˜ a (s) CG e−T s [r(t)]. (11) y˜a (t) = ˜ a (s) 1 + CG ˜ a (s) given in (7) is designed to be ASPR, by Then, if G designing the controller C = k ∗ as a static feedback gain, one can obtain a stable control system easily by setting a ˜ a (s) k∗ G sufficient large k ∗ so as to make 1+k ∗G ˜ (s) SPR. a

˜ a (s) is ASPR, then there It should be noted that if G necessarily exists a feedback gain k0 such that the closed˜ a (s) k∗ G ∗ loop system 1+k ∗G ˜ (s) should be SPR for all k ≥ k0 . a

11th IFAC ALCOSP July 3-5, 2013. Caen, France

u

G ( s ) e − Ts

ya

y

ω

Adaptive NN

yf

H (s )

Exsosystem

r +

ea

v

ue +

−k

u

+

−

Plant

y +

ya

+

G a (s )

PFC

yf

Fig. 6. Overall augmented controlled system Now, let’s consider an equivalent block diagram of the control system given in Fig. 4 as shown in Fig. 5, where, the resulting PFC: H(s) is given by H(s) = F (s)e−T s − P (s) (12) and ya (t) is the output of the overall augmented system (see Fig. 6.): Ga (s) = G(s)e−T s + H(s)

(13)

In this case, since the PFC can be obtained by H(s) = F (s)e−T s − P (s) ˜ a (s)e−T s − G ˜ a (s)) = F (s)e−T s − (G

˜ a (s) − G(s)e−T s , =G the resulting augmented system can be given as −T s

Ga (s) = G(s)e ˜ a (s) =G

Fig. 7. Block diagram of the designed control system vnn (t) = W T S(ω) T

where W = [w1 , · · · , wl ] ∈ R is the weight vector, l is the number of NN nodes (weight number) and S(ω) = [s1 (ω), · · · , sl (ω)]T is the radial basis function vector. This basis function vector S(ω) is generally designed by the Gaussian functions such as   −(ω − µi )T (ω − µi ) , i = 1, 2, · · · , l si (ω) = exp ηi2 (19)

(14)

where µi = [µi1 , · · · , µiq ]T is the center of the receptive field and ηi is the width of the Gaussian function.

(15)

It has been clarified (Ge et al., 2002) that, for a sufficiently large l and a compact set Ωω ⊂ Rq , there exists an ideal constant weight vector W ∗ such that  W ∗ , arg min sup |v ∗ − W T S(ω)| (20) l W ∈R ω ∈Ωω and thus the ideal input v ∗ (t) can be approximated by

+ H(s)

˜ a (s) given in (7) to be ASPR, Thus if we design the G the obtained augmented controlled system Ga (s) with the PFC H(s) should be ASPR.

T

v ∗ (t) = W ∗ S(ω) + ǫ(ω) |ǫ(ω)| ≤ ǫ∗

4. ADAPTIVE CONTROL SYSTEM DESIGN

For the lock-up clutch system given in (4), let’s consider an augmented system with a PFC H(s): (16)

Suppose that this augmented system is ASPR. Further suppose that a reference signal yr (t), which the system’s output is required to follow, can be generated by the following reference model: ˙ ω(t) = s(ω) yr (t) = p(ω) s(0) = 0 , p(0) = 0

(21)

where ǫ(ω) is an approximation error.

4.1 Basic Design of Adaptive Output Feedback System

ya (t) = Ga (s)[u(t)] Ga (s) = G(s)e−T s + H(s)

(18)

l

(17)

where ω(t) ∈ Rq is a state vector of the reference model. We assume that the model (17) satisfies the neutral stability property (Isidori, 1995). That is, the linear approximation S = [ ∂∂s ω ]ω =0 of the vector field s(ω) at ω = 0 has all its eigenvalues on the imaginary axis. Under this assumption it has been clarified that there exists an ideal control input v ∗ (t) such that v ∗ (t) can be given by a function of ω as v ∗ (t) = c(ω) (Isidori, 1995). In this case, the ideal input v ∗ (t) can be approximated by a radial basis function (RBF) neural networks (NN) as follows: 474

Under these conditions, an adaptive output feedback control input is designed as follows with an adaptive NN feedforward input in order to attain a good tracking performance for the actual output y(t) (Mizumoto et al., 2009): u(t) ue (t) v(t) ˙ k(t) ˆ˙ (t) W

= = = =

ue (t) + v(t) −k(t)¯ ea (t) T ˆ W (t) S(ω) γ¯ e2a − σk(t),

(22) γ>0, σ>0

ˆ (t), Γ > 0 , σw > 0 = −ΓS(ω)¯ e a − σw W

Fig. 7 shows the overall block diagram of the designed adaptive output feedback control system. 4.2 Practical PFC design As given in (1) and (2), G(s) of the lock-up clutch system is a third order system with relative degree of three. Therefore we first design a virtual PFC F (s) by using the ladder network type method given in Iwai et al. (1994) and Iwai and Mizumoto (1994) as follows: F (s) = F1 (s) + F2 (s) (23) β2 β1 , F2 (s) = α2 , α 2 > 0 F1 (s) = (s + α1 )2 s + α2

11th IFAC ALCOSP July 3-5, 2013. Caen, France

˜ a (s) = Using this PFC F (s), the augmented system G G(s) + F (s) can be ASPR by setting sufficiently small β1 ≫ β2 > 0. Thus, the desired PFC can be designed as follows if the controlled system is known. H(s) = (G(s) + F (s)) − G(s)e−T s (24) In the practical case, the accurate system model would not be obtained, and the PFC is designed using a nominal model of the considered system. That is, denoting the nominal model by G0 (s)eT0 s , the PFC can be designed by H(s) = (G0 (s) + F (s)) − G0 (s)e−T0 s ˜ a0 (s) − G0 (s)e−T0 s =G

˜ a0 (s) = G0 (s) + F (s) G

(25) (26)

In this case, the practical augmented system with the designed PFC H(s) can be represented by Ga (s) = G(s)e−T s + H(s) ˜ a0 (s) − G0 (s)e−T0 s = G(s)e−T s + G ˜ a0 (s) + (G(s) − G0 (s))e =G

˜ a0 (s)(1 + ∆(s)) =G

(27)

where ∆(s) = ∆1 (s) + ∆2 (s) and ˜ a0 (s)−1 (G(s) − G0 (s))e−T s ∆1 (s) = G ˜ a0 (s)−1 G0 (s)(e−T s − e−T0 s ) ∆2 (s) = G

(28) (29)

Concerning the ASPR-ness of the resulting augmented system with unmodelled dynamics, the following Theorem has been provided (Mizumoto and Iwai, 1996). Theorem 1. If the following conditions are satisfied, then the resulting augmented system Ga (s) is ASPR. (a) (b) (c)

We design the PFC using the nominal system model G0 (s). At first, F(s) is designed using ladder network method so as to render G0 (s) + F (s) ASPR as follows: 0.6 5 + (30) F (s) = (s + 10)2 s + 10 Then the PFC H(s) is finally designed by H(s) = (G0 (s) + F (s)) − G0 (s)e−T s

−T s

+G0 (s)(e−T s − e−T0 s )

we know a nominal system model at an nominal engine revolution speed as follows: 944.7 , T = 200[ms] G0 (s) = 3 s + 31.05s2 + 212.7s + 1608 In the simulation, if the engine revolution speed is changed to 1.5 times of nominal revolution speed, the system will be varied to 1005 G1 (s) = 3 , T = 200[ms] s + 61.2s2 + 391.2s + 3538 and if the engine revolution speed increase twice, then the system will be changed to 1106 , T = 200[ms] G2 (s) = 3 s + 111.4s2 + 688.7s + 6754

˜ a0 (s) is ASPR. G ∆(s) ∈ RH∞ . k∆(s)k∞ < 1.

Remark: Unfortunately, since the obtained ∆(s) has a time-delay factor e−T s , the condition (b) in Theorem 1 does not satisfied, and then one can not check ASPRness of the augmented system properly from Theorem 1. However, in the cases where the time-delay factor can be approximated by an appropriate higher order lag system with a rational transfer function, for the approximated system, one can check the ASPR-ness of the system. 5. ILLUSTRATIVE EXAMPLE In order to examine the effectiveness of the proposed adaptive output feedback control system design method for slip control of a lock-up clutch system with timedelay, we confirm effectiveness of the proposed method by numerical simulations. In this simulation, we assume that the system parameter is varied according to the engine revolution speed and 475

(31)

The design parameters in the adaptive controller are set as γ = 50, σ = 0.1, Γ = 340, σw = 0.001 We compared the performance of the proposed method with the one of a PID control method. The PID controller was given by 1 CP ID = KP + KI + KD s, s KP = 0.54, KI = 3.7, KD = 0.9 The PID parameters were adjusted so as to obtain a better performance for the nominal system with nominal engine revolution speed. Figs 8 to 13 show simulation results. Fig. 8 shows the result with the proposed method for a nominal system which is referred to design the PFC, Fig. 9 is the result for system with 1.5 times of engine speed from the engine speed of nominal system and Fig. 10 is the result with twice of engine speed. We can confirm that the slip revolution speed of the system quickly follows the reference revolution speed for each case with the different engine speeds. On the other hand, as shown in Figs 11 to 13, in the cases with a fixed PID controller which is adjusted for the nominal system, the control performances are deteriorated according to the change of the engine speed. 6. CONCLUSION In this paper, we considered designing an adaptive output feedback control system with a simple structure for the slip control of a lock-up clutch system with a time delay. A concrete PFC design method based on the idea of Smith predictor was shown for the slip control of lock-up clutch system with time-delay and an adaptive output feedback control of the slip system of lock-up clutch was designed. The effectiveness of the proposed method is confirmed through numerical simulations for a slip control of lockup clutch system.

11th IFAC ALCOSP July 3-5, 2013. Caen, France

350

350 control signal reference

300

250

output

output

250 200 150 100

200 150 100

50 0

control signal reference

300

50

0

5

10

15

20

25

30

35

0

40

0

5

10

15

Time [s]

Fig. 8. Simulation results with adaptive control for a system with a nominal engine speed

40

250

output

output

35

control signal reference

300

250 200 150 100

200 150 100

50

50

0

5

10

15

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30

35

0

40

0

5

10

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Time [s]

20

25

30

35

40

Time [s]

Fig. 9. Simulation results with adaptive control for a system with 1.5 times of engine speed from the nominal engine speed

Fig. 12. Simulation results with PID controller for a system with 1.5 times of engine speed from the nominal engine speed

350

350 control signal reference

300

control signal reference

300 250

output

250

output

30

350 control signal reference

300

200 150 100

200 150 100

50 0

25

Fig. 11. Simulation results with PID controller for a basic rotation number of engine

350

0

20

Time [s]

50

0

5

10

15

20

25

30

35

0

40

Time [s]

0

5

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25

30

35

40

Time [s]

Fig. 10. Simulation results with adaptive control for a system with twice of engine speed from the nominal engine speed

Fig. 13. Simulation results with PID controller for a system with twice of engine speed from the nominal engine speed

REFERENCES

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Barkana, I. (1987). Parallel feedforward and simplified adaptive control. International Journal of Adaptive Control and Signal Processing, 1(2), 95–109. Fradkov, A.L. (1994). Adaptive stabilization for minimumphase multi-input plants without output derivatives measurment. Physics-Doklady, 39(8), 550–552. Fradkov, A.L. (1996). Shunt output feedback adaptive controller for nonlinear plants. Proc. of 13th IFAC World Congress, San-Francisco, July, K, 367–372. Ge, S.S., Wang, C.C., Lee, T.H., and Zhang, T. (2002). Stable AdaptiveNeural Network Control. Kluwer Academic Pub. Isidori, A. (1995). Nonlinear Control Systems. Springer, 3rd edition. Iwai, Z. and Mizumoto, I. (1994). Realization of simple adaptive control by using parallel feedforward compen-

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