Adaptive and Learning Control for SI Engine Model With ... - CiteSeerX

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 14, NO. 1, FEBRUARY 2009

93

Adaptive and Learning Control for SI Engine Model With Uncertainties Huajin Tang, Member, IEEE, Larry Weng, Zhao Yang Dong, Senior Member, IEEE, and Rui Yan, Member, IEEE

Abstract—Air–fuel ratio control is a challenging control problem for port-fuel-injected and throttle-body-fuel-injected spark ignition (SI) engines, since the dynamics of air manifold and fuel injection of the SI engines are highly nonlinear and often with unmodeled uncertainties and disturbance. This paper presents nonlinear control approaches for multi-input multi-output engine models, by developing adaptive control and learning control design methods. Theoretical proofs are established that ensure that proposed controllers are able to give asymptotical tracking performance. As a comparison, the method applying global linearizing controller can give accurate tracking for the engine model without uncertainty and disturbance, but it fails to keep tracking performance when uncertainty is incorporated into the system. Adaptive control and learning control approaches are capable of dealing with both constant uncertainty and time-varying periodic uncertainty. Simulation results illustrate the efficacy of the proposed controllers. Index Terms—Adaptive control, control engineering, uncertainty.

m ˙ ap m ˙ at pa Ta ki n m ˙f X τf m ˙ ff Hu m ˙ fv ηi m ˙ fi Pb pi θm bt

NOMENCLATURE Air mass flow rate into cylinder (in kilogram per second). Air mass flow rate past throttle plate (in kilogram per second). Atmosphere pressure (1.013 bar). Atmosphere temperature (in Kelvin). Constants. Crank shaft speed (in kilo revolution per minute). Cyclinder port fuel flow (in kilogram per second). Fraction of m ˙ f i which is deposited on manifold as fuel film. Fuel evaporation time constant (0.25 s). Fuel film mass flow (in kilogram per second). Fuel heating value (4.3 × 104 kJ/kg). Fuel vapor mass flow (in kilogram per second). Indicated efficiency. Injected fuel mass flow (in kilogram per second). Load power (in kilowatt). Manifold air pressure (in bar). Max break torque spark timing.

Manuscript received May 5, 2008; revised August 2, 2008. Current version published February 13, 2009. Recommended by Technical Editor J. R. Wagner. H. Tang is with the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore 138632 (e-mail: [email protected]). L. Weng is with the School of Information Technology and Electrical Engineering, University of Queensland, Brisbane, Qld. 4072, Australia (e-mail: [email protected]). Z. Y. Dong is with the Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). R. Yan is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117570, Singapore. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2008.2004806

λ Pl θ α I

Normalized air–fuel ratio. Pumping and friction power (in kilowatt). Spark advance angle (in degrees). Throttle opening angle (in degrees). Total moment of inertia (0.5(2π/60)2 kgm2 ). I. INTRODUCTION

UTOMOTIVE engine control is one of the most complex control problems for control engineers and researchers. There are increasing requirements on the engine control to reduce exhaust emissions and fuel consumption while maintaining the best engine performance. To satisfy these requirements, a variety of variables need to be controlled such as engine speed, engine torque, fuel injection timing, air–fuel ratio (AFR), etc. These variables of an engine are coupled and the engine possesses several different operation modes including acceleration, deceleration, idle, running, and aging process. Thus, the engine dynamics is naturally multivariable and highly nonlinear, and the controller design is a difficult problem. The wide operating range, the inherent nonlinearities of the induction process, and the large modeling uncertainties make the design of the fuel injection controller very difficult. A lot of research work has been done for the design and analysis of engine controls. The objective of a typical control system for an internal combustion (IC) engine is not only to maintain the desired engine speed, but also to satisfy regulations relating to exhaust emissions. Moreover, the feedback-control strategy should be robust so that the designed performance could be maintained when the engine is operating at different operating points, with different aging history of the components, or under vastly different environmental conditions. A key development was the introduction of closed-loop control in the 1970s [1], [2]. As more capable microprocessors become available through 1990s, it is practical to entertain the use of various nonlinear control and estimation algorithms for automotive engine control practice [3]–[6], where linear quadratic control, Kalman filtering, observer-based, and model predictive control methods were developed. Most of the control approaches are based on a single variable output system, i.e., only AFR is controlled. Observer-based fuel injection control algorithm [7] was developed to achieve better AFR control performance by fast response and small amplitude chattering of the AFR as compared to using sliding mode control method. However, the success of the suggested observation control methods depends largely on the accuracy of the plant model. In [8], a sliding mode control method is applied to the engine model derived from the fuel wall wetting dynamics of engine intake manifold [9] and Hendricks’ model [10]. The engine

A

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model is a nonlinear, multi-input multi-output (MIMO), coupled system having three states x = (m ˙ f , n, pi ) and three inputs u = (m ˙ f i , θ, α). By using the sliding mode techniques to an MIMO IC engine model, Khan and Spurgeon [8] demonstrated the undesirable chattering effects at the actuators. The particular choice of sliding surface is input dependent that produces a dynamic controller. Such dynamic policies are desirable in sliding mode control as they effectively filter out the chattering of the control signal. The designed controller is valid over the entire operating range. The AFR is stabilized to its ideal value and the speed and manifold pressure track satisfactorily given trajectories. The researchers in Ford Motor Company [11], [12] studied a diesel engine model and considered an MIMO control system. They proposed a control design method by employing a recently developed control-Lyapunov-function-based nonlinear control because it possesses a guaranteed robustness property equivalent to gain and phase margins. By applying the input– output linearization method and the control Lyapunov function, the controller can be designed. However, when the system contains uncertainties, the proposed controller is not able to keep tracking performance. In recent years, a number of theoretical advances in nonlinear control systems have been achieved to a level where they can be applied to solve practical control design problems. Although the nonlinear control methods applying Lyapunov theory and adaptive control have been studied in various mechatronics areas, from actuator control to power steering and large-scale system [13]–[16], there is a lack of complete and systematic treatments of applying adaptive control [17] and learning control [18], [19] methodologies to engine control. In this paper, we present the controller design methods by using adaptive control and learning control strategies. After a typical swingle-input single-output (SISO) model of spark ignition (SI) engine is studied, where the control objective is to regulate the AFR alone, then an MIMO model is also analyzed. Uncertainties and disturbance are important characteristics of the engine; therefore, the controller should have a good robust property to uncertainties and disturbance. We design the controllers for the engine based on this consideration and present the approach to deal with the uncertainties by devising suitable control algorithm. Rigorous proofs are established and simulation results are also presented to validate our theoretical results. The rest of this paper is organized as follows. The engine control model is described in Section II, where both SISO and MIMO systems are defined. Global linearizing controller is discussed in Section III. Adaptive control and learning control approaches for MIMO-based engine control are introduced in Sections IV and V, respectively. In Section VI, simulation results are given to demonstrate the developed controller advantages. Finally, a conclusion is drawn. II. ENGINE MODEL DESCRIPTION There are a number of IC engine dynamic models in the literature. Among them the mean value engine model (MVEM) developed by Hendricks et al.. [10], [20]–[22] is mathematically compact and can be parameterized for different engines easily. Thus, the MVEM is adopted widely for engine control systems [23].

A. Mean Value Engine Model The engine model comprises three subsystems: fueling, air intake, and engine speed dynamic systems. These subsystems are described briefly in the following. 1) Fueling System: The fluid film flow model describes the dynamics of the fluid flow through the manifold. The fluid flow has two components: fuel vapor flow and fuel film flow, denoted ˙ f f , respectively. The total flow m ˙ f is not measurable. by m ˙ f i, m The dynamics of the submodel is described as m ¨ ff =

1 (−m ˙ f f + Xm ˙ f i) τf

(1a)

m ˙ f v = (1 − X)m ˙ fi

(1b)

m ˙f =m ˙ ff + m ˙ fv

(1c)

X = kx1 + kx2 λ=

m ˙ ap m ˙ ap (max)

m ˙ ap = h(·) 14.67m ˙f

(1d) (1e)

where X m ˙ f i is the rate at which injected fuel is deposited on manifold as fuel film. 2) Engine Speed Dynamics: The submodel is derived from energy conservation laws, and is described as n˙ =

1 (Hu ηi m ˙ f (t − τd ) − Pl − Pb ) nI

Pl = n(k1 + k2 n + k3 n2 ) + n(−k4 + k5 n)pi Pb = kb n3

(2)

where ηi = ηin ηip ηi λ ηiθ ηin (n) = k6 (1 − k7 n−0.36 ) ηip (n) = k8 + k9 pi − k10 p2i ηi λ = −k11 + k12 λ − k13 λ2 ηiθ = e−θ

2

2 /2θ m bt

ηm bt = min(min(θ1 , θ2 ), 45) θ1 = k14 pi + k15 + n47 θ2 = k16 pi + k17 + n47
4.7n, n < 4.8 n47 = 4.7 × 4.8, otherwise. 3) Air Flow System: The air mass flowing through the manifold is depicted by the following equations: p˙i =

RTi (−m ˙ ap + m ˙ at ) Vi

Vd (k18 pi + k19 )n 120RTi pa β2 (pr )β1 (α) = k20 β2 (pr )β1 (α) = mat1  (Ta )

m ˙ ap = m ˙ at

β1 (α) = 1 − cos(α − α0 )

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TANG et al.: ADAPTIVE AND LEARNING CONTROL FOR SI ENGINE MODEL WITH UNCERTAINTIES

   1 ppr 1 − ppr 2 , if p ≥ p r c β2 (pr ) = pn  1, otherwise.

(3)

Regulation AFR has received overwhelming attention in engine control since maintaining AFR at the stoichiometric value can obtain the best balance between power output and fuel consumption, and also, effectively reduce the pollution emission, because a stoichiometric value produces relatively low CO and NOx emissions while enabling good after treatment conversion efficiency. First, the SI engine model considered is an SISO sys˙ f , n, pi ). In tem, which has three states x = (x1 , x2 , x3 ) = (m the work of Muller et al. [24], the cylinder pressure information was used to achieve good AFR control. The SISO plant has a single input u = m ˙ f i and the AFR a single output y = λ ¨f = x˙ 1 = m

1 (−m ˙ f +m ˙ f i ) + (1 − X)m ¨ fi τf

1 (Hu ηi m ˙ f − Pl − Pb ) In 1 1 =− (Pl + Pb ) + Hu ηi x1 Ix2 Ix2

x˙ 2 = n˙ =

RTi (−m ˙ ap + m ˙ at ) Vi

x˙ 3 = p˙i = =−

RTi RTi m ˙ ap + k20 β2 (pi )(1 − cos(α − α0 )) Vi Vi

=−

RTi RTi m ˙ ap + k20 β2 (x3 )(1 − cos(α − α0 )). (4) Vi Vi

In the previous equation of x˙ 3 , the first term represents the impact of air flow into the cylinder and the second term represents that of the air flow past the throttle plate (namely, the ˙ f is assumed to be orifice equation). The time delay τd for m zero. Remarks: The transport delay is neglected here. Typical transport delays are on the order of 250 ms, and cannot be neglected in practice. It is a most challenging problem for the AFR control of SI engines and we will consider nonzero time delay cases in our future work. The SISO system can be written in a controller canonical form, assuming X to be constant
SISO:

x˙ = f (x) + G˜ u

(5)

y = h(x)

where  −x

1

  1 G = 0 0 u ˜=

B. SISO and MIMO Engine Plants



  τf     1 1   f = − (pl + pb ) + Hu ηi x1  Ix2   x2 I   RT RTi i k20 β2 (x3 )(1 − cos(α − α0 )) − m ˙ ap Vi Vi

95

1 u + (1 − X)u. ˙ τf

(6)

Note that h(x) = λ = =

m ˙ ap Vd 1 = (k18 pi + k19 )n 14.67m ˙f 14.67 120RTi

Vd x2 (k18 x3 + k19 ). 14.7 × 120RTi x1

(7)

Remarks: In our assumption, the AFR is not directly measurable and it is calculated by (7). That is to say, AFR is modeled as a continuous function of x1 , x2 , x3 , and thus, its value is determined by the three state variables. However, with the advance of the sensors such as universal exhaust gas oxygen (UEGO), the measurement of AFR becomes available, and the dynamic state equations and output equations of the MIMO control system can be simplified; thus, a simplified control algorithm can be developed and applied. For a more practical and accurate control, the MIMO control system should be considered. The system’s states are x = (m ˙ f , n, pi ) defined as before. The system has three inputs u = (m ˙ f i , θ, α) and three outputs y = (λ, n, pi ). The block diagram of the control system is illustrated in Fig. 1. The system equation is described by x˙ = f (x) + G˜ u MIMO: (8) ˆ y = h(x) where

 −x

1



  τf     1  f = − (pl + pb )     x2 I   RT i m ˙ ap − Vi G = diag([g1 , g2 , g3 ])   1 0 0   1 0  Hu ηi x1 0  = Ix 2     RTi k20 β2 (x3 ) 0 0 Vi   1 u + (1 − X)u˙  τf   . u ˜= 2 2  e−θ /2θ m b t 1 − cos(α − α0 )

(9)

III. GLOBALLY LINEARIZING CONTROLLER DESIGN The goal of the controller is to act on the system inputs (e.g., fuel injection, spark advance angle, throttle plate characterized by m ˙ f i , θ, α, respectively) to the plant, so that the engine behaves according to the specifications.

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Fig. 1.

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Block diagram of the control system.

The current production electric control unit (ECU) uses lookup tables with compensation of a proportional–integral (PI) feedback loop controller to control AFR. This method cannot produce desirable precise control performance due to the high nonlinearity of SI engines [7], [25].

Now, if Lg h(x) = 0 for some state variables x, then the relative order of the system (i.e., the number of times that the output has to be differentiated to obtain an explicit relation between y and u) is unity. Let the linear input–output closed-loop dynamics induced by the controller be of the form y˙ = −β0 y + β1 v

A. Review of GLC In this section, a new method of nonlinear control system, called globally linearizing controller (GLC), is applied to the engine control. The control canonical form is written as x˙ = f (x) + g(x)u

(10a)

y = h(x)

(10b)

where x is the state vector, y is the output vector, and u is the input vector. The basic concept of input–output linearization is to find a control law such that the output y is related linearly to the input u, i.e., to find a suitable equation eliminating the nonlinearity between y and u. The proposed control law is of the form u = φ(x, v)

x˙ = f (x) + g(x)φ(x, v)

(12)

y = h(x)

(13)

has linear input–output behavior. Moreover, one would select the control law based on a prespecified transfer function for the input–output linear closed-loop system. To find the control law (11), differentiating the output y in (10) yields ∂h(x) ∂h dy = x˙ = (f (x) + g(x)u) dt ∂x ∂x = Lf h(x) + Lg h(x)u

where β0 and β1 are constant coefficients. Through differential geometry theory, a static state feedback control given later can be used to globally linearize the nonlinear system u = φ(x, v) =

(14)

−β0 h(x) + β1 v − Lf h(x) . Lg h(x)

Recall the SISO control plant (5)
x˙ = f (x) + G˜ u y = h(x). From previous results, let K = Vd /14.7 × 120RTi , the Lie derivatives are calculated as Lh f =

3

Lhi fi

i=1

  (k18 x3 + k19 )x2 x1 − x21 τf   k18 x3 + k19 1 1 +K (pl + pb ) + Hu ηi x1 − x1 x2 I Ix2   RTi RTi k20 β2 (x3 )(1 − cos(α − α0 )) − m ˙ ap +K Vi Vi

= −K

×

k18 x2 x1

Lh G =

3

(18) Lhi gi = −K

i=1

where ∂h ∂h f (x) Lg h(x) = g(x) (15) ∂x ∂x are the Lie derivatives of h(x) with respect to f (x) and g(x), respectively. Lf h(x) =

(17)

B. Controller Design for SISO System

(11)

where φ(x, v) is an algebraic function of the state variables x and external reference input v, such that the closed-loop system

(16)

(k18 x3 + k19 )x2 . x21

(19)

Define e = y − r, and thus ∂h(x) ∂h x˙ = (f (x) + g(x)u) ∂x ∂x = Lf h(x) + Lg h(x)u.

e˙ = y˙ =

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(20)

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97

Let the induced linearization dynamics be written as e˙ = −β0 e.

IV. ADAPTIVE CONTROL (21)

Therefore, the controller is derived as u ˜ = φ(x, v) =

−β0 e − Lf h(x) . Lg h(x)

(22)

C. Controller Design for MIMO System Here, we extend the GLC method to the MIMO system controller design. Recalling in the MIMO system (8), the output is ˜ y = (λ, x2 , x3 ) = h(x). The system error is written as   λ−r e =  x2 − x2d  . (23) x3 − x3d Differentiating e1 , e2 , and e3 yields ∂h x˙ = Lh (f + G˜ u) e˙ 1 = λ˙ = ∂x = Lh f +

3

Lhi G˜ ui + Lh1 G˜ u1

(24a) (24b)

e˙ 3 = x˙ 3 − x˙ 2d = f3 + g3 u ˜3 − x˙ 3d .

(24c)

Similarly, it is also derived that Lhi fi

i=1

(k18 x3 + k19 )x2 1 =K × x1 τf   k18 x3 + k19 pl + pb +K − x1 x2 I   k18 x2 RTi +K m ˙ ap − x1 Vi Lh G =

3

(25)

(k18 x3 + k19 )x2 x21

i = 2, 3.

i=2

e˙ 3 = g3 (g3−1 f3 + u ˜3 − g3−1 x˙ 3d + g3−1 d3 ). (29) The control objective is to look for a suitable controller u ˜i such that the tracking error ei (t) converges to zero as t → ∞ even if the system has some constant uncertainties. In order to realize the aims, the adaptive controller u ˜i is designed as follows: u ˜1 = −C1 (Lh1 g1 )−1 e1 − (Lh1 g1 )−1 Lh f − (Lh1 g1 )−1

3

Lhi gi u ˜i − θˆ1

i=2

u ˜2 = (26)

By writing the induced dynamics as: e˙ i = −βi ei , i = 1, 2, 3, the controllers can be obtained as    ˜i −β1 e1 − Lh f − 3i=2 Lhi gi u (27a) u ˜1 = Lh1 (−βi ei − fi + x˙ id ) , gi

e˙ 3 = x˙ 3 − x˙ 2d = f3 + g3 u ˜3 − x˙ 3d + d3 . It is noted that the value of each Lhi gi is dependent on the state variables x. It is reasonably assumed that Lhi gi and gi are nonzero for some range of state variables; therefore, the inverse functions for Lhi gi and gi exist in such range of state variables. The previous equations can be rewritten as  3

e˙ 1 = Lh1 g1 (Lh1 g1 )−1 · Lh f + (Lh1 g1 )−1 Lhi gi u ˜i

e˙ 2 = g2 (g2−1 f2 + u ˜2 − g2−1 x˙ 2d + g2−1 d2 )

Hu ηi + K(k18 x3 + k19 ) Ix2

u ˜i =

e˙ 2 = x˙ 2 − x˙ 2d = f2 + g2 u ˜2 − x˙ 2d + d2

+u ˜1 + g1−1 d1

Lhi gi

k18 x2 RTi k20 β2 (x3 ) +K × . x1 Vi

Lhi gi u ˜i + Lh1 g1 u ˜1 + Lh1 d1



i=1

= −K

3

i=2

e˙ 2 = x˙ 2 − x˙ 2d = f2 + g2 u ˜2 − x˙ 2d

3

ˆ Outputs of the system are given by y = (λ, x2 , x3 ) = h(x), where λ = h(x). Define e1 = λ − r, e2 = x2 − x2d , e3 = x3 − x3d , then the error dynamics are derived as ∂h x˙ e˙ 1 = ∂x = Lh (f + G˜ u) + Lh1 d = Lh f +

i=2

Lh f =

In this section, we design the controller based on adaptive control method to deal with uncertainties in the engine dynamic system. We assume that there exists a constant uncertainty (disturbance) in the air-fuel dynamics  ˜ 1 + d1   x˙ 1 = f1 + g1 u (28) x˙ 2 = f2 + g2 u ˜ 2 + d2   x˙ 3 = f3 + g3 u ˜ 3 + d3 .

(27b)

−C2 g2−1 e2

− g2−1 f2 + g2−1 x˙ 2d − θˆ2

u ˜3 = −C3 g3−1 e3 − g3−1 f3 + g3−1 x˙ 3d − θˆ3

(30)

and the parametric updating law is ˙ θˆ1 = q1 · Lh1 g1 e1 ˙ θˆ2 = q2 g2 e2 ˙ θˆ3 = q3 g3 e3 with the feedback gains qi > 0, i = 1, 2, 3.

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(31)

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The globally asymptotical tracking property is ensured by the following theorem. Theorem 1: For the air-fuel dynamics with constant uncertainties, the adaptive controller (30) and parametric updating law (31) guarantee that lim ei (t) = 0,

t→∞

i = 1, 2, 3.

Proof: Let θ1 = d1 and θi = gi−1 di , i = 2, 3. To prove the convergence of the tracking error, we define the following Lyapunov function: V = V1 + V 2 + V 3

(32)

where V1 =

1 2 1 e1 + (θ1 − θˆ1 )2 2 2q1

V2 =

1 2 1 e2 + (θ2 − θˆ2 )2 2 2q2

1 1 V3 = e23 + (θ3 − θˆ3 )2 . 2 2q3

(33)

e˙ 1 = −C1 e1 + e1 (θ1 − θˆ1 )Lh1 g1 e˙ 2 = −C2 e2 + g2 (θ2 − θˆ2 ) (34)

Differentiating V with respect to time t gives V˙ = V˙ 1 + V˙ 2 + V˙ 3 .

(35)

Furthermore, according to (34), we obtain 1 ˙ V˙ 1 = e1 e˙ 1 + (θ1 − θˆ1 )(−θˆ1 ) q = −C1 e21 +

1 ˙ (θ1 − θˆ1 )(q1 Lh1 g1 e1 − θˆ1 ) q1

1 ˙ (θ2 − θˆ2 )(q2 e2 g2 − θˆ2 ) q2

=

1 ˙ + (θ3 − θˆ3 )(q3 e3 g3 − θˆ3 ). q3

Since V is a continuous function of ei , θi , i = 1, 2, 3, this guarantees that V (0) is bounded for any initial values e(0) and θ(0), i.e., V (0) < ∞. It is also noted that 0 ≤ V (t) ≤ V (0). Integrat∞ 2 ing the previously inequality yields 0 Ci ei (t)dt ≤ V (0) <  2 ∞. This implies that ei (t)dt is bounded and ei is bounded, by noting that Ci > 0. Second, e˙ i is bounded from (34). Therefore, we conclude that limt→∞ ei (t) = 0 according to Barbalat’s Lemma [26]. It shows that the tracking error converges to zero asymptotically, i.e., the adaptive controller guarantees the globally asymptotical tracking.


In this section, we design the controller based on learning control method to deal with time-varying periodic uncertainties in the engine dynamic system. As described in the previous section, we first consider that the state dynamics with uncertainties are written as  ˜1 + d1 (t)   x˙ 1 = f1 + g1 u ˜2 (39) x˙ 2 = f2 + g2 u   x˙ 3 = f3 + g3 u ˜3 where the uncertainty d1 (t) is time varying and assumed to be periodic with period T , i.e., d1 (t) = d1 (t − T ). ˆ Outputs of the system are given by y = [λ, x2 , x3 ]T = h(x), where λ = h(x). Define e1 = λ − r, e2 = x2 − x2d , e3 = x3 − x3d , then e˙ 1 is derived as

= Lh f +

3

Lhi gi u ˜i + Lh1 g1 u ˜1 + Lh1 d1 .

(40)

i=2

Now, there is a disturbance term Lh1 d in e˙ 1 . It is rewritten as (36) e˙ 1 = Lh1 g1 ((Lh1 g1 )−1 · Lh f + (Lh1 g1 )−1

Therefore, we have

3

Lhi gi u ˜i

i=2

V˙ = −C1 e21 − C2 e22 − C3 e23

+u ˜1 + g1−1 d1 ).

1 ˙ + (θ1 − θˆ1 )(q1 Lh1 g1 e1 − θˆ1 ) q1 +

(38)

∂h x˙ ∂x = Lh (f + G˜ u) + Lh1 d1

1 ˙ V˙ 3 = e3 e˙ 3 + (θ3 − θˆ3 )(−θˆ3 ) q −C3 e23

≤ 0.

e˙ 1 =

1 ˙ V˙ 2 = e2 e˙ 2 + (θ2 − θˆ2 )(−θˆ2 ) q = −C2 e22 +

V˙ = −C1 e21 − C2 e22 − C3 e23

V. LEARNING CONTROL

Substituting the adaptive controller (30) into (29), we obtain

e˙ 3 = −C3 e3 + g3 (θ3 − θˆ3 ).

Considering the parametric updating law in (31), we further have

The learning controller is

1 ˙ (θ2 − θˆ2 )(q2 e2 g2 − θˆ2 ) q2

1 ˙ + (θ3 − θˆ3 )(q3 e3 g3 − θˆ3 ). q3

(41)

u ˜1 = −C1 (Lh1 g1 )−1 e1 − (Lh1 g1 )−1 · Lh f (37)

− (Lh1 g1 )−1

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3

i=2

Lhi gi u ˜i − g1−1 dˆ1 (t)

(42)

TANG et al.: ADAPTIVE AND LEARNING CONTROL FOR SI ENGINE MODEL WITH UNCERTAINTIES

and the parametric updating law is given as follows: dˆ1 (t) =

t ∈ [0, T )

qLh1 e1 , dˆ1 (t − T ) + qLh1 e1 ,

where q > 0 is a constant. To facilitate the convergence analysis, the following positivedefinite function will be adopted:  t  1 2 1   e1 + d˜2 (τ )dτ,  2 2q 0 1 V (t) =  t  1 1 2  e1 + d˜2 (τ )dτ, 2 2q t−T 1

t−T

˜ is −2d(t)f (t) − f 2 (t). Proof: From Property 1, the upper right-hand derivative of  t d˜2 (τ )dτ t−T

is

for t ∈ [0, T )

d˜2 (t) − d˜2 (t − T ).

(44) for t ∈ [T, ∞)

Using the relation (49)

where d˜1 (t) = d1 (t) − dˆ1 (t). In the following, we summarize two important properties associated with functionals, which will be used in subsequent derivations with the Lyapunov–Krasovskii functional. Property 1: Let φ ∈ R and T > 0 be a finite constant. The upper right-hand derivative of 

Then, the upper right-hand derivative of  t d˜2 (τ )dτ

(43)

t ∈ [T, ∞)

99

ˆ − T )][d(t − T ) − d(t ˆ − T )] d˜2 (t − T ) = [d(t − T ) − d(t ˆ + f (t)][d(t) − d(t) ˆ + f (t)] = [d(t) − d(t) ˜ + f 2 (t). = d˜2 (t) + 2f (t)d(t) Substituting the previous relation into (50) yields ˜ −2d(t)f (t) − f 2 (t).

t

φ2 (τ )dτ




t−T

is φ2 (t) − φ2 (t − T ). Proof: The upper right-hand derivative of the integral is  t+∆ t lim + sup

φ2 (τ )dτ − ∆t

t+∆ t−T

∆ t→0

t t−T

φ2 (τ )dτ

.

(45)



t+∆ t



t−T

φ2 (τ )dτ =

t

φ2 (τ )dτ +

t+∆ t−T

t−T +∆ t



φ2 (τ )dτ t−T

t+∆ t

φ2 (τ )dτ

+

and substitute into (45), then we have lim + sup

t+∆ t−T

∆ t→0

φ (τ )dτ − ∆t

 t+∆ t

= lim + sup ∆ t→0

t

2

t→∞

t−T

1 V˙ (t) = e1 e˙ 1 + d˜2 2q

t

 t+∆ t

The convergence property of the proposed adaptive control method is summarized in the following theorem. Theorem 2: The control law (42) with the parametric updating law parameter law (43) warrants the asymptotical convergence  t e21 (τ )dτ = 0. lim Proof: The proof consists of three parts. Part I proves the finiteness of V in [0, T ). Part II proves the negativeness of V in [T, ∞). Part III derives the asymptotical convergence of the tracking error e1 (t). Part I. Finiteness of V in [0, T ): Let us first derive the upper right-hand derivative of V for t ∈ [0, T ), which is

Note the fact 

(50)

t t−T

φ (τ )dτ

 t−T +∆ t

φ2 (τ )dτ − t−T ∆t

= φ2 (t) − φ2 (t − T ).

1 = −C1 e21 + eLh1 d˜ + d˜2 2q

2

1 1 = −C1 e21 + dˆd˜ + d˜2 q 2q

φ2 (τ )dτ (46)


ˆ d(t), ˜ f (t) ∈ R, and assume that Property 2: Let d(t), d(t), the following relations hold: d(t) = d(t − T )

(47)

˜ = d(t) − d(t) ˆ d(t)

(48)

ˆ = d(t) ˆ + f (t). d(t)

(49)

1 1 1 = −C1 e21 + (dˆ − d)d˜ + dd˜ + d˜2 q q 2q = −C1 e21 −

1 ˜2 1 ˜ d + dd. 2q q

(51)

Since for any c > 0, 1 ˜ c ˜2 1 2 dd ≤ d + d . q q 4cq Let 0 < c/q < 1/2q V˙ ≤ −C1 e21 −



1 c − 2q q

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1 2 d . d˜2 + 4cq

(52)

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 14, NO. 1, FEBRUARY 2009

Since d(t) is periodic, there exists a finite bound dM ≥ d(t), ∀t ∈ [0, T ]. Thus, V˙ < 0 outside the region  
 1 c ˜2 2 2 2 ˜ − (e1 , d) ∈ R | C1 e1 + d ≤ dM . 2q q Therefore, V (t) is finite in the interval [0, T ). Part II. Negativeness of V in [T, ∞): The upper right-hand derivative of V, according to Property 1 for t ∈ [T, ∞), should be 1 V˙ (t) = e1 e˙ 1 + (d˜2 (t) − d˜2 (t − T )). (53) 2q Note that e1 e˙ 1 = e1 Lh1 g1 (−C1 (Lh1 g1 )−1 e1 + g1−1 (d1 − dˆ1 )) ˜ = −C1 e21 + e1 Lh1 d.

(54)

According to Property 2 and the parameter updating law (43), we can further derive the following relationship: 1 ˜2 (d (t) − d˜2 (t − T )) 2q =

1 ˆ ˆ ˆ ˆ − d(t ˆ − T )) (d(t − T ) − d(t))(2(d(t) − d(t)) + d(t) 2q

=

1 ˆ ˆ d(t) ˜ − 1 (d(t ˆ − T ) − d(t)) ˆ 2 (d(t − T ) − d(t)) q 2q

1 ˆ ˆ 2. − T ) − d(t)) = e1 Lh1 d˜ − (d(t 2q

(55)

Substituting the previous equations to V˙ yields 1 V˙ (t) = −C1 e21 − L2h1 e21 2q ≤ −C1 e21 ≤ 0.

(56)

Part III. Asymptotical convergence: Now let us derive the convergence property  t e21 (τ )dτ = 0 lim t→∞

t−T

using the fact (56) that V˙ for t ∈ [T, ∞) is negative semidefiniteness. Suppose that  t e21 (τ )dτ = 0. lim t→∞

t−T

Then, there exists an > 0, a t0 ≥ T , and a sequence ti → ∞ t with i = 1, 2, . . . and ti+1 ≥ ti + T such that t ii−T e21 (τ )dτ > when ti > t0 . Hence, from (56), we obtain for t > T i  tj

˜ ˜ ))− lim lim V (t, e1 , d)≤V (T, e1 (T ), d(T e21 (τ )dτ. i→∞

i→∞

j =1

t j −T

˜ )) is finite, the previous relation implies Since V (T, e1 (T ), d(T ˜ → −∞ lim V (t, e1 , d)

i→∞

Fig. 2. System outputs y when facing some constant disturbance. It is shown that the control system fails to give correct tracking. (a) AFR λ. (b) Crank shaft speed n (in kilo revolution per minute). (c) Intake manifold air pressure p i (in bar).

a contradiction to the positiveness property of limt→∞ ˜ limi→∞ V (t, e1 , d). This completes the proof.
Remark 1: The previous analysis can be extended straightforward to the following uncertainties systems: ˜1 + d1 (t) x˙ 1 = f1 + g1 u x˙ 2 = f2 + g2 u ˜2 + d2 (t) x˙ 3 = f3 + g3 u ˜3 + d3 (t)

(57)

where the uncertainty di (t) is time varying and assumed to be periodic with period Ti , i.e., di (t) = di (t − Ti ).

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TANG et al.: ADAPTIVE AND LEARNING CONTROL FOR SI ENGINE MODEL WITH UNCERTAINTIES

Fig. 3. Adaptive control law ensures asymptotic tracking performance when there is a constant disturbance in the control plant. The outputs y, i.e., the AFR, crank shaft speed, and intake manifold air pressure, are shown. (a) AFR λ. (b) Crank shaft speed n (in kilo revolution per minute). (c) Intake manifold air pressure p i (in bar).

101

Fig. 4. Corresponding adaptive control signals u, i.e., the injected fuel mass flow, spark advance angle, and throttle opening angle. (a) Injected fuel mass flow (in kilogram per second). (b) Spark advance angle (in degrees). (c) Throttle opening angle (in degrees).

and the parametric updating law

The learning controller

−1

−1

u ˜1 = −C1 (Lh1 g1 ) e1 − (Lh1 g1 ) − (Lh1 g1 )−1

3

dˆ1 (t) =

· Lh f



Lhi gi u ˜i − g1−1 dˆ1 (t)

dˆ2 (t) =

i=2



u ˜2 = −C2 e2 − g2−1 f2 + g2−1 x˙ 2d − dˆ2 (t) u ˜3 = −C3 e3 − g3−1 f3 + g3−1 x˙ 3d − dˆ3 (t)

(58)

dˆ3 (t) =

q1 Lh1 e1 , dˆ1 (t − T1 ) + qLh1 e1 ,

t ∈ [0, T ) t ∈ [T, ∞)

q2 g2 e2 , dˆ2 (t − T2 ) + q2 g2 e2 ,

t ∈ [0, T2 )

q3 g3 e3 , dˆ3 (t − T3 ) + q3 g3 e3 ,

t ∈ [0, T3 )

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t ∈ [T2 , ∞) t ∈ [T3 , ∞)

(59)

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 14, NO. 1, FEBRUARY 2009

Fig. 5. Learning control law is applied to the system with periodic uncertainty and ensures desired tracking performance. The outputs y, i.e., the AFR, crank shaft speed, and intake manifold air pressure, are shown. (a) AFR λ. (b) Crank shaft speed n (in kilo revolution per minute). (c) Intake manifold air pressure p i (in bar).

with qi > 0 will guarantee that  t lim e2i (τ )dτ = 0, t→∞

i = 1, 2, 3.

t−T

Fig. 6. Corresponding learning control signals u, i.e., the injected fuel mass flow, spark advance angle, and throttle opening angle. (a) Injected fuel mass flow (in kilogram per second). (b) Spark advance angle (in degrees). (c) Throttle opening angle (in degrees).

0.272, k3 = 0.0135, k4 = 0.969, k5 = 0.206, k6 =0.558, k7 = 0.392, k8 = 0.9301, k9 = 0.2154, k10 = 0.1657, k14 = 47.31, k17 = 57.34,K18 =0.952, k19 = k15 = 2.6214, k16 = −56.55, √ 0.075, k20 = 7.32Pa / Ta .

VI. SIMULATION RESULTS The simulations are carried out for a 1275cc British Leyland engine for which data have been reported by Hendricks et al. [10] and by Khan and Spurgeon [8]. The values of the parameters used in the simulations are adopted from their work, as given in the following: kg·m2 , τf = 0.25, X = 0.22, Vi = I = (2π/60)2 × 0.5 −6 646 × 10 , Hu = 4.3 × 104 , Vd = 1.275, R = 287 × 10−5 , Ta = 297, Pa = 1.013, k1 = 1.673, k2 = Ti = 308,

A. For Constant Uncertainties Without using the adaptive control law, the control system using the GLC method is not able to give the desired AFR and track the speed and air pressure correctly when facing disturbances. In the following, the simulation results are provided when the disturbance d = (1, 2, 0.1)T . As a comparison, the adaptive control law derived in the previous section is applied. The system outputs are still be able

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TANG et al.: ADAPTIVE AND LEARNING CONTROL FOR SI ENGINE MODEL WITH UNCERTAINTIES

Fig. 7. Block diagrams for adaptive control (a) and learning control (b). In ˆ both approaches, the adaptation equations for the parameters d(t) are first derived, and then, the control signal u(t) is derived based on the adaption ˆ and tracking errors e(t). As illustrated, adaptive control applies signals d(t) differential equations to estimate the adaption signals, while learning control applies difference equations to estimate them.

to achieve the desired objective. According to Theorem 1, the adaptive controller is able to ensure an asymptotic tracking performance. It is verified by the simulation results (see Fig. 3). As can be seen from Fig. 3, though the response is underdamped undesirably in the beginning [see the subplot of Fig. 3(a)], the normalized stoichiometric AFR is rapidly maintained as 1 after 0.5 s. The control signals, e.g., the applied fuel mass flow, spark advance angle, and throttle opening angle, are depicted in Fig. 4. B. For Time-Varying Periodic Uncertainties In this example, we will apply the designed learning controller to give the desired AFR and track the speed and air pressure correctly when facing time-varying periodic disturbances. In the following, the simulations are performed with the disturbance d = (0.1 sin(2πt), 0, 0)T . According to Theorem 2, the learning controller can guarantee asymptotic tracking when there is time periodic disturbance in the control plant. It can be seen from Fig. 5(a) that the AFR is maintained at a good level, only with a miniature deviation less than 0.1% of the magnitude. On the other hand, the crank shaft speed and intake manifold air pressure are exactly tracked to the desired signals [see Fig. 5(b) and (c)]. VII. CONCLUSION This paper studied the SI engine control problems. First, we introduced the GLC control design method for both SISO and MIMO systems and analyzed the control performance. To deal with uncertainties in the engine modeling, two controller design methods based on adaptive control and learning control were proposed by considering two types of uncertainties: constant and time-varying periodic uncertainty. With rigorous analysis, we proved the convergence properties of the adaptive and learning control laws. The simulation results show that the controller

103

applying GLC can give accurate tracking for the engine model without uncertainty and disturbance; however, it fails to keep good tracking performance in the cases of uncertainty and disturbance. As a comparison, the controller using adaptive control and learning control is able to achieve the control objectives in the event of uncertainty and disturbance. There are some rooms for improvements on the current research. For example, AFR was assumed not directly measurable and modeled by (7). With the advance of the sensors, such as UEGO, the measurement of AFR becomes available, and a simplified control algorithm can be developed after the state and output equations are simplified. On the other hand, the transport delay was neglected here. Typical transport delays are on the order of 250 ms and cannot be neglected in practice. To deal with the transport delay in theory is still a most challenging problem in AFR control of SI engines. We will consider such nonzero time delay problem in our future work. APPENDIX For the illustrative purpose, the block diagrams for the adaptive control and learning control strategies are depicted in Fig. 7. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped to improve this paper significantly. REFERENCES [1] J. G. Rivard, “Closed-loop electronics fuel injection control of IC engine,” SAE Tech. Paper, 1973, paper 730005. [2] M. Athans, “The role of modern control theory for automotive engine control,” SAE Tech. Paper, 1978, paper 780852. [3] M. Athans, J. F. Cassidy, and W. H. Lee, “On the design of electronic automotive engine controls using linear quadratic control theory,” IEEE Trans. Contr. Syst. Technol., vol. 25, no. 5, pp. 901–912, Oct. 1980. [4] J. Poulsen, E. Hendricks, M. Fons, P. B. Jensen, M. B. Olsen, and C. Jepsen, “A new family of nonlinear observers for Si engine AFR control,” SAE Tech. Paper, 1997, paper 970615. [5] N. P. Fekete, J. D. Powell, and C. F. Chang, “Observer-based air-fuel ratio control,” IEEE Control Syst., vol. 18, no. 5, pp. 72–83, Oct. 1998. [6] N. Giorgetti, G. Ripaccioli, A. Bemporad, I. V. Kolmanovsky, and D. Hrovat, “Hybrid model predictive control of direct injection stratified charge engines,” IEEE/ASME Trans. Mechatron., vol. 11, no. 5, pp. 499–506, Aug. 2006. [7] S. B. Choi and J. K. Hendrick, “An observer-based controller design method for improving air/fuel characteristics of spark ignition engines,” IEEE Trans. Control Syst. Technol., vol. 6, no. 3, pp. 325–334, May 1998. [8] M. K. Khan and S. K. Spurgeon, “MIMO control of an IC engine using dynamic sliding modes,” in Proc. Intell. Syst. Control, 2003, vol. 388, pp. 132–137. [9] C. F. Aquino, “Transient A/F control characteristics of the 5 liter central fuel injected engine,” SAE Tech. Paper, 1981, paper 810494. [10] E. Hendricks, A. Chevalier, M. Jensen, S. C. Sorenson, D. Trumpy, and J. Asik, “Modelling of the intake manifold filling dynamics,” SAE Tech. Paper, pp. 122–146, 1996, paper 960037. [11] M. JankoviC, M. JankoviC, and I. Kolmanovsky, “Constructive Lyapunov control design for turbocharged diesel engines,” IEEE Trans. Control Syst. Technol., vol. 8, no. 2, pp. 288–299, Mar. 2000. [12] M. Larsen, M. Jankovic, and P. V. Kokotovic, “Indirect passivation design for a diesel engine model,” in Proc. IEEE Int. Conf. Control Appl., Alaska, 2000, pp. 309–314. [13] H. Son and K.-M. Lee, “Distributed multipole models for design and control of PM actuators and sensors,” IEEE/ASME Trans. Mechatron., vol. 13, no. 2, pp. 228–238, Apr. 2008.

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[14] A. S. Putra, S. Huang, K. K. Tan, S. K. Panda, and T. H. Lee, “Self-sensing actuation with adaptive control in applications with switching trajectory,” IEEE/ASME Trans. Mechatron., vol. 13, no. 1, pp. 104–111, Feb. 2008. [15] E. Yang and D. Gu, “Nonlinear formation-keeping and mooring control of multiple autonomous underwater vehicles,” IEEE/ASME Trans. Mechatron., vol. 12, no. 2, pp. 164–178, Apr. 2007. [16] P. R. Pagilla, R. V. Dwivedula, and N. B. Siraskar, “A decentralized model reference adaptive controller for large-scale systems,” IEEE/ASME Trans. Mechatron., vol. 12, no. 2, pp. 154–163, Apr. 2007. [17] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [18] J.-X. Xu, “A new periodic adaptive control approach for time-varying parameters with known periodicity,” IEEE Trans. Automat. Control, vol. 49, no. 4, pp. 579–583, Apr. 2004. [19] J.-X. Xu and R. Yan, “On repetitive learning control for periodic tracking tasks,” IEEE Trans. Automat. Control, vol. 51, no. 11, pp. 1842–1848, Nov. 2006. [20] E. Hendricks and S. C. Sorenson, “Mean value modelling of spark ignition engines,” SAE Tech. Paper, 1990, paper 900616. [21] E. Hendricks and S. C. Sorenson, “SI engine controls and mean value modelling,” SAE Tech. Paper, 1991, paper 910258. [22] E. Hendricks, “A generic mean value engine model for spark ignition engines,” in Proc. 41st SIMS Simul. Conf.: SIMS 2000, Lyngby, Denmark, Sep. 18–19. [23] J. Asgari, J. Deur, J. Petric, and D. Hrovat, “Recent advances in controloriented modeling of automotive power train dynamics,” IEEE/ASME Trans. Mechatron., vol. 11, no. 5, pp. 513–523, Oct. 2006. [24] N. Muller, S. Leonhardt, and R. Isermann, “Methods for engine supervision and control based on cylinder pressure information,” IEEE/ASME Trans. Mechatron., vol. 4, no. 3, pp. 235–245, Sep. 1999. [25] C. Manzie, M. Palaniswami, D. Ralph, H. Watson, and X. Yi, “Model predictive control of a fuel injection system with a radial basis function network observer,” J. Dyn. Syst. Meas. Control Trans. ASME, vol. 124, no. 4, pp. 648–658, 2002. [26] J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.

Huajin Tang (M’02) received the B.Eng. degree in mechanical engineering from Zhe Jiang University, Hangzhou, China, the M.Eng. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore, Singapore, in 1998, 2001, and 2005, respectively. From 2004 to 2006, he was an R&D Engineer at STMicroelectronics, Singapore. From 2006 to 2008, he was a Postdoctoral Fellow at Queensland Brain Institute, University of Queensland, where he was engaged in research on computational neuroscience and vision. He is currently a Research Fellow in the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore. His current research interests include machine learning, computational and biological intelligence, dynamic system, and control. He is the author or coauthor of several papers published in journals and conferences, and one monograph in computational intelligence. Dr. Tang has served as a Program Committee Member for many international conferences and a reviewer for many international journals.

Larry Weng received the Bachelor’s degree in mechanical engineering and the Ph.D. degree in engine control systems from the University of Queensland (UQ), Brisbane, Australia, in 2001 and 2008, respectively. He is the founder of ActiveTorque Technology. He is a former member of the UQ Sustainable Energy Research Group. He was also a member of the UQ Formula SAE Racing Team for three years. It was during this experience, while he was looking to improve engine efficiency, that he got the inspiration for ActiveTorque.

Zhao Yang Dong (M’99–SM’06) received the Ph.D. degree in electrical engineering from the University of Sydney, Sydney, Australia, in 1999. He is currently an Associate Professor at The Hong Kong Polytechnic University, Kowloon, Hong Kong. He previously held academic positions at the University of Queensland, Australia. He also held industrial positions with Powerlink Queensland and Transend Networks, Tasmania, Australia. His research interests include power system dynamics and stability, power system planning, computational intelligence for power system analysis, electricity market modeling and planning, and data mining and its application in power engineering.

Rui Yan (M’02) received the B.S. and M.S. degrees from the Department of Mathematics, Si Chuan University, Chengdu, China, in 1998 and 2001, respectively, and the Ph.D. degree from the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, in 2006. She is currently a Research Fellow at the Department of Electrical and Computer Engineering, National University of Singapore. Her research interests include advanced nonlinear control, linear and nonlinear dynamic systems, and power system stability analysis and control.

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