ADAPTIVE FEEDBACK LINEARIZED VARIABLE DISPLACEMENT ENGINE INTAKE MANIFOLD PRESSURE CONTROL

ADAPTIVE FEEDBACK LINEARIZED VARIABLE DISPLACEMENT ENGINE INTAKE MANIFOLD PRESSURE CONTROL Alex Gibson, Ilya Kolmanovsky and John Michelini Ford Research and Advanced Engineering Ford Motor Company Dearborn MI, 48121 Email:{agibson8, ikolmano, jmichel1}@ford.com Abstract: This paper develops a parameterization of the intake manifold filling dynamics which are then used in the development of adaptive feedback linearization intake manifold pressure control strategies. Nonlinear simulation results are used to describe the pressure control and parameter estimation response of the designs. These results demonstrate that adaptive feedback linearization can be used to develop manifold pressure control designs that are capable of providing good pressure tracking response over a range of engine speeds and displacements. It is also shown that in the presence of large transients, the adaptive control response can be improved by using manifold pressure time constant values that are based upon steady state calibration values, during transients. I Introduction With the recent introduction of variable displacement engines, there has been considerable interest in the design, vibration, [1] and [2], and fuel economy, [3], aspects of these engines This paper addresses the impact of displacement changes on the intake manifold pressure, see Fig. 1, and its control. In a variable displacement engine, the engine displacement can change by a factor of two, e.g. an eight to four cylinder transition. During such a transition, the intake manifold pressure is increased or decreased to match the engine torque in the preceding mode. During the initial transient the manifold filling dynamics prevent the intake manifold pressure from instantly matching the desired pressure. This manifold pressure error produces a corresponding error between the desired and actual engine torque. The error in the desired engine torque is then compensated for by applying spark advance or retard, [4]. The use of spark advance or retard during the transitions can lead to increased fuel consumption and therefore should be minimized. It can be shown that the dependence on spark during these transitions can be minimized by using manifold pressure control to reduce the filling response time, [4] and [5]. In this paper adaptive feedback linearization is used to develop manifold pressure control and filling time constant estimation designs using an isothermal manifold pressure model. These designs are then tested with a nonlinear engine model that includes the

manifold pressure and temperature dynamics. Related work on adaptive control of engine air path includes [6] and adaptive air charge estimation has been widely studied in the literature, see [7].

.

mth

.

mcyl

Pm

Fig. 1 Engine Intake and Exhaust Manifolds Variable displacement engine simulation results are used to study the response of the manifold pressure adaptive feedback linearized control and manifold filling time constant estimation designs. In Section II a parameterization of the intake manifold filling dynamics is developed. The manifold filling dynamics model, developed in Sections II and III, is then used in the development of feedback linearized intake manifold pressure compensation and filling time constant estimation designs. Linear control

design is then used to motivate the closed loop control input selection. And nonlinear simulation results are then presented to describe the pressure tracking, variable engine displacement and parameter estimation response characteristics of the feedback and feedforward designs. II Intake Manifold Pressure Model The intake manifold pressure and temperature dynamics, [8] and [9], without heat release or transfer are given by: Tm =

{

RTm min C p Tin − mout C p Tout − (m in − mout )C v Tm } PmVm C v

(1) R {min γ in Tin − mout γ out Tout } Pm = Vm

(2)

Where Pm, Tm and Vm are the manifold pressure, temperature and volume, min , mout , Tin , Tout and γin , γout are the mass flow rate, temperature and polytropic constants associated with the air flowing into and out of the manifold. And Cp and Cv are the heat capacity at constant pressure and volume respectively and for an ideal gas, γ = Cp/Cv and R = Cp – Cv, [10]. For a normally aspirated, i.e. non-boosted, engine it has been shown that the intake manifold dynamics can be approximated as an isothermal system, with γin = γout = 1 and Tin = Tout = Tm, [11]. Then assigning the throttle flow to min and the cylinder flow to mout , the intake manifold dynamics can be approximated by, [5]:

Pm =

RTm (mth − mcyl ) Vm

(3)

Using P = ρRT to eliminate the manifold density, the following equation can be derived:

Pm +

RTm ηvVd N Pm = mth 120 ⋅ Vm Vm

The throttle flow rate, mth , is largely a function of the throttle cross sectional area and the pressure ratio across the throttle, Pm/P0. If Pm/P0 is greater than the critical pressure ratio where:

Pm P0

γ

2 (γ −1) = γ +1

crit

mcyl =

ρ mη vVd N 2 ⋅ 60

(4)

Where N is the engine speed in RPM, Vd is the displaced volume of the engine and ηvVd is the effective charge volume for a given engine speed, load and valve timing, where ηv is the volumetric efficiency. For a variable displacement engine the displaced volume, Vd, varies with the number of firing cylinders. For example, the displaced volume in eight cylinder mode is double the displaced volume in four cylinder mode.

,

(6)

e.g. 0.528 for γ = 1.4 and 0.546 for γ = 1.3, [9], then 1

mth =

C D AT P0 Pm P0 RT0

γ

2γ P 1− m γ −1 P0

(γ −1)

γ

1 2

(7)

For choked flow, i.e. Pm/P0 is less than the critical pressure ratio [9]: mth =

C D AT P0 RT0

γ

1/ 2

2 γ +1

( γ +1) / 2 ( γ −1)

(8)

Where AT is the throttle area, CD is the discharge coefficient, R is the ideal gas constant, γ is the polytropic constant, and P0 and T0 are the up-stream pressure and temperature. If the throttle area is equated to a normalized area multiplied by the maximum area, DT2 , then Eq. 4 (5) can be reorganized in the following manner: i.e. AT = ATn ATmax , where AT max = π

Where mth is the throttle mass air flow rate and mcyl is the mean value mass air flow rate into the cylinders. The mean value cylinder flow, for a four stroke engine, is given by, [9]:

(5)

Pm = −

1

τm

Pm + K m ATn

(9)

where τm and Km are given by:

τm =

120 ⋅ Vm ηvVd N

Km =

RTm mth Vm ATn

(10) (11)

It is interesting to note that mth ATn is the wide open throttle flow rate which is mainly a function of the pressure ratio across the throttle.

III τm and Km vs. Engine Speed and Displacement The intake manifold dynamics parameter τm , given by Eqs. (9) and (10), is plotted as a function of engine speed and engine displacement, i.e. eight vs. four cylinder operation, in Fig. 2, for an effective volumetric efficiency of one. In Fig. 2 it is shown that the gradient of τm with respect to engine speed is greatest at low engine speeds, e.g. 600 to 1500 RPM. And that a factor of two decrease in engine displacement increases τm by the same factor. In addition to engine speed and displacement τm is a function of volumetric efficiency, ηv, which varies with engine speed, load, intake and exhaust manifold pressures and residual level, [9], see Eq. (10). In Fig. 3 the Km vs. intake manifold pressure ratio relationship, given by Eqs. (7), (8) and (11), is shown. 0.9

0.8

0.7

0.5

Four Cylinder Operation Eight Clinder Operation

m

τ (sec)

0.6

0.4

Assuming τm and Km are known within well defined bounds then exact feedback linearization, [12] and [13], can be applied to Eq. (9). Applying feedback linearization: Pm = −

1

Pm + K m ATn = −

τm

1

τd

Pm +

1

τd

u

(12)

Therefore:

ATn =

1 Km

1

−

τm

1

Pm +

τd

1

τd

u

(13)

Where τd is a constant. If τm and/or Km are not known then either or both of these parameters can be estimated, [13]. If both τm and Km are estimated then persistent excitation may be required, which may not be possible to always generate in this application. The parameter Km can be readily calculated from Eq. (11) and available measured signals. The time response parameter τm can vary as a function of engine operating point, as noted in Section III. Therefore τm is chosen as the estimation parameter. Then for adaptive feedback linearization:

0.3

0.2

ATn =

0.1

0

1000

1500

2000

2500

3000 3500 4000 Engine Speed(RPM)

4500

5000

5500

6000

Fig. 2 Manifold Filling Time Constant vs. Engine Speed for Eight and Four Cylinder Operation

1 Km

50

1

τd

Pm +

1

τd

u

(14)

Where θˆ is an estimate of 1/τm, with the desired, or modeled, intake manifold dynamics: Pd = −

55

θˆ −

1

Pd +

τd

1

τd

u

(15)

45 40

Then substituting Eq. (14) into Eq. (9):

Choked Flow

Pm = −

30

m

K (Bar/sec)

35

25 20

)

1 Pm + θˆ − θ Pm + u

τd

(16)

Where θ is 1/τm actual. By setting the state estimate error to e = Pm – Pd and the parameter estimate error to ~ θ = θˆ − θ the state error dynamics are given by:

15

(

10 5

(

1

τd

0.4

0.5

0.6

P /P m

0.7

0.8

0.9

)

1

0

Fig. 3 Km vs. Intake Manifold Pressure Ratio IV Adaptive Feedback Linearization In the intake manifold dynamics given by Eq. (9), τm and Km may be viewed as functions of either time or time and the state, Pm. Therefore the manifold filling dynamics conform to the definition of a feedback linearizable system, [12]. Then given the feedback linearized dynamics, a constant gain design can be developed to control the intake manifold pressure over the engine operating range.

e = Pm − Pd = −

~ 1 e + θ Pm τd

(17)

Selecting the Lyapunov function:

V =

1 2 1 ~2 e + θ 2 2γ

(18)

then ~ 1 ~~ 1 1~ V = ee + θ θ = − e 2 + θ + Pm e θ

γ

τd

γ

(19)

~

θ = θˆ − θ = −γPm e

(20)

Using Eq. (15) to calculate Pd and a measurement of Pm, equation (20) can be integrated to calculate θˆ .

And then with a calculated value of Km, Eq. (14) can be used to convert the time varying nonlinear dynamics given by Eq. (9) into the time invariant linear dynamics given by Eq. (15). The control input u is then selected to provide the desired pressure tracking and disturbance rejection characteristics for the linearized system. When using Eq. (20) to calculate θˆ it is assumed that either θ is small relative to θˆ or that θ is zero, e.g. the

parameter θ is a constant. The assumption that θ is close to zero is only satisfied at steady state operating conditions. It is shown in Section VII that by using pre-calculated values of τm, referred to as τm calibration, during the period just after the transient, and then restarting the τm adaptation, the excursions in the τm estimate can be minimized. And if continuous adaptation of τm is used during transients, the excursions in the τm estimate and the errors in the commanded pressure response can be significant. V ~

Control Input Selection 1 Assuming θ << , where τd is a positive constant,

τd

the feedback linearized dynamics given by Eq. (16) can be approximated by: Pm = −

1

τd

Pm +

1

τd

u

(21)

The control input u can be derived by combining Eq. (21) with a linear model that effectively represents the throttle dynamics and by applying linear control design, see Fig 7. KFF Pdes

+

-

K(s)

+ +

1

1

τThs + 1

τ ds + 1

Throttle Dynamics

Manifold Dynamics

Pm

characteristics. Thus, the input u is set to u = u = K FF Pdes + K ( s)(Pm − Pdes ) , where K(s) is the controller to be designed, KFF is the feedforward gain and Pdes is the manifold pressure set-point. As shown in Fig. 8 the command following response can be improved by either using feedforward, i.e. KFF = 1, or by adding integral control, i.e. K ( s ) = K P + K I s . 1.4

K(s) = 0 w/K

FF

Using linear control design it can be shown that with proportional feedback alone, i.e. K(s) = KP, it is not possible to select a gain that provides both good command following and well damped response

K(s) = Kp + Ki/s K(s) = Kp + Ki/s w/K FF P input

1

des

0.8

0.6

0.4

0.2

0

−0.2

−0.4

0.5

1

1.5

2

2.5

Time(sec)

Fig. 5 Commanded Pressure Response w/Proportional and Proportional plus Integral Control with and without Feedforward VI Nonlinear Simulation To test the effectiveness of the manifold pressure control and parameter estimation, developed in Sections IV and V, the designs were integrated into a nonlinear variable displacement engine simulation. The simulation includes the intake manifold pressure and temperature dynamics, Equ. (1) and (2), and a cylinder-by-cylinder variable displacement engine model, [8]. The manifold pressure compensation is run as a separate digital control function with 20 msec sampling, [4]. The comparison of modeled and measured vehicle data is given in Fig. 6.

0.9

0.85

0.8

0.75

0.7

0.65

Fig. 4 Intake Manifold Pressure Feedback Model

K(s) = Kp K(s) = Kp w/KFF

1.2

Linear Commanded Pressure Response

0 if:

Intake Manifold Pressure(Bar)

For τd > 0, V

Measured Vehicle Data Simulation

2

4

6

8 Time(sec)

10

12

Fig. 6 Measured vs. Simulated Vehicle Tip-Out/Tip-In Manifold Pressure Response

14

0.8

Intake Manifold Pressure(Bar)

0.75

0.7

0.65

Fixed Gain AFL w/τm Cal AFL wo/τm Cal Pdes

0.6

7

8

9

10

11

12

13

Time(sec)

Fig. 7 Adaptive Feedback Linearized, with and without τm calibration vs. Fixed Gain at 1,000 RPM 0.28

0.275

0.27

m

Estimated τ (sec)

0.75

0.7

0.65

Fixed Gain AFL w/τm Cal AFL wo/τm Cal P des

0.6

7

8

9

10

11

12

13

Time(sec)

Fig. 9 Adaptive Feedback Linearized, with and without τm calibration, vs. Fixed Gain at 3,000 RPM VII.2 Engine Displacement Response During an eight to four cylinder transition of a variable displacement engine, the effective engine displacement is reduced by a factor of two which causes the manifold pressure to increase due to the drop in the cylinder flow rate. Typically it is desirable to match the eight and four cylinder operating torques before and after the transition. This torque matching can be partially accomplished through manifold pressure control. Note that, for a variable displacement eight cylinder engine, after accounting for the differences in friction, pumping losses and indicated torque, the four cylinder torque output may be roughly ten to fifteen percent more efficient. This greater efficiency, during four cylinder operation, translates into a desired pressure of 85 to 90 percent of roughly double the eight cylinder manifold pressure.

wo/τ Calibration m w/τm Calibration

0.265

0.26

0.255

0.25

0.245

0.24

0.235

0.8

Intake Manifold Pressure(Bar)

VII Nonlinear Simulation Results VII.1 Feedback Pressure Tracking The constant engine displacement closed loop pressure tracking response, at 1,000 RPM, is shown in Fig. 7 for adaptive feedback linearized, with and without τm calibration, and a fixed gain proportional plus integral control. As shown all three feedback designs provide good pressure tracking response. In Fig. 8 the continuously calculated τm estimate oscillates during the period just after the pressure transients, as predicted in Section IV. Yet this variation in τm has little impact on the pressure tracking response, because the amplitude of the τm variation is small relative to the mean value.

7

8

9

10

11

12

13

14

Time(sec)

Fig. 8 Estimates of τm with and without τm Calibration The feedback pressure tracking responses shown in Fig. 9, are for an adaptive feedback linearized, with and without τm calibration, and fixed gain proportional plus integral control at 3,000 RPM. As shown in Fig. 9 the fixed gain control shows a substantial loss in responsiveness vs. adaptive feedback linearized control solutions and the 1,000 RPM result, see Fig. 7. The result shown in figures 7 and 9 demonstrates that feedback linearization provides a structured τm and Km compensation method. While the fixed gain controller could be tuned for 3,000 RPM, it can be shown that this would lead to degraded performance at 1,000 RPM.

VII.2.1 Engine Displacement Feedforward During a change in engine displacement, the desired manifold pressure control tracks a pressure command and rejects the displacement disturbance. As the displacement change is known in advance, feedforward can be used. The feedforward can be implemented by adding a feedforward term to the throttle command or by adding a feedforward term to the input of the feedback linearized throttle command, i.e. adding uff to u in Eq. (14). If the feedback linearized throttle command input u is set to uff, then the feedback linearized control becomes a dynamic feedforward design, where Eq. (14) is used to map the manifold dynamics into the desired dynamics. As the desired dynamics used in the feedback linearization process has an input gain of one, Eq. (15), and the throttle model has unity gain, the feedforward term uff is equal to the commanded pressure.

Figure 10 shows the feedforward pressure response during an eight to four and then a four to eight cylinder displacement transition at an engine speed of 1,000 RPM. During the eight to four cylinder transition the intake manifold pressure command is

35

30

25

Cyl 1 Cyl 2 Cyl 3 Cyl 4 Cyl 5 Cyl 6 Cyl 7 Cyl 8

20

15

10

5

0

−5

7

8

9

10

11

12

13

Time(sec)

Fig. 12 Net Cylinder Torque During an Eight to Four and Four to Eight Cylinder Transition Figure 12 shows the simulated cylinder torque response for the dynamic feedforward control response shown in figure 10. As shown in Fig. 12, cylinders 2, 4, 6 and 8 in the firing order are deactivated during the eight to four cylinder transition and then re-activated during the four to eight cylinder transition. In an actual engine both manifold pressure and spark control would be used to better match the engine output torque after each displacement change, in a manner similar to that described in [5].

0.9

0.8 Intake Manifold Pressure(Bar)

40

Net Cylinder Torque(Nm)

increased from 50 to 85 Kpa, based upon a 15 percent increase in four cylinder torque production efficiency. Figure 10 shows the pressure response with static feedforward, i.e. no dynamic compensation, and dynamic feedforward, u = Pdes in eq. (14), with and without τm calibration. As shown in Fig. 10 the dynamic feedforward response without τm calibration, exhibits more over and under shoot. Further the static feedforward response, feedforward alone, has a significantly longer pressure rise time. The increased over and under shoot of the dynamic feedforward response without τm calibration can be explained by the lag and overshoot in the τm estimate during the transients, see Fig. 11. As shown in Fig. 11 the τm estimate nearly doubles when the engine displacement is reduced by a factor of two, as predicted by Eq. (11) and shown in Fig. 2.

0.7

0.6

0.5

FF Alone AFL w/u = P & τ Cal des m AFL w/u = Pdes & wo/τm Cal P

0.4

des

7

8

9

10

11

12

13

Time(sec)

Fig. 10 Dynamic Feedforward with and without τm Calibration vs. Feedforward at 1,000 RPM

Figure 13 shows the simulated throttle command and actual throttle angular displacement, for the dynamic feedforward, with τm calibration, response shown in Fig. 10. As shown in Fig. 13, the steady state change in the throttle angle is small, as the displacement change drives the manifold pressure in the desired direction.

1.1

1

30

0.9

w/τm Calibration wo/τm Calibration

Commanded Actual

25 Throttle Angle(degrees)

m

Estimated τ (sec)

0.8

0.7

0.6

0.5

20

15

0.4

10 0.3

0.2

7

8

9

10

11

12

13

Time(sec)

Fig. 11 Estimates of τm with and without τm Calibration, During Displacement Transient

14

5 7

8

9

10 Time(sec)

11

12

Fig. 13 Commanded and Actual Throttle Angle Response During Displacement Transition

13

0.9

0.8 Intake Manifold Pressure(Bar)

VII.2.2 Engine Displacement Closed Loop Figure 14 shows the adaptive feedback linearized proportional plus integral pressure control response, with and without τm calibration, for the engine displacement transition. As shown in Fig. 14 the adaptive feedback linearized closed loop response, without τm calibration exhibits more over and under shoot. Again this can be explained by the oscillation in the τm estimate.

0.7

0.6

0.5

0.9

0.85

AFL w/τ Cal, u = PI + P m des AFL wo/τ Cal, u = PI + P m des P des

0.4

0.8

8

9

10

11

12

13

Time(sec)

Fig. 15 Integrated Feed-forward/back, with and without τm calibration, at 1,000 RPM

0.7

0.65 0.85

0.6

0.75

des

0.5

0.45

0.8

AFL w/τm Cal AFL wo/τ Cal m P

0.55

7

8

9

10

11

12

13

Time(sec)

Fig. 14 Adaptive Feedback Linearized Closed Loop Control, with and without τm calibration

Intake Manifold Pressure(Bar)

Intake Manifold Pressure(Bar)

7 0.75

0.7 0.65 0.6 0.55 0.5

AFL w/τm Cal, N = 1,000 RPM AFL w/τm Cal, N = 3,000 RPM P

0.45

VII.2.3 Integrated Feedforward/Feedback Figure 15 shows the engine displacement pressure responses of the adaptive feedback linearized controllers in which feedforward control has been added to the proportional plus integral feedback in an integrated manner. These responses were generated by setting the control input u, in Eq. (14), to the commanded pressure just after the displacement transition and then setting u to the proportional plus integral control input after the transition. As expected the adaptive feedback linearized response, without τm calibration, exhibits more over and under shoot. Figure 16 shows the integrated, with τm calibration, engine displacement pressure response for engine speeds of 1,000 and 3,000 RPM. As shown, the integrated control performs well at both 1,000 and 3,000 RPM.

des

0.4 0.35

7

8

9

10

11

12

13

Time(sec)

Fig. 16 Integrated Feed-forward/Feedback, with τm calibration, at 1,000 and 3,000 RPM Summary and Conclusions In Sections II and III a parameterization of the manifold filling dynamics time constant, τm, and gain, Km, is developed. In Sections IV adaptive feedback linearization is applied to manifold pressure control and filling time constant, τm, estimation. In Section V linear control is used to motivate the feedback linearized control input selection. Section VI provides a description of the nonlinear simulation used to test the control and estimation designs. Section VII presents simulation results which are used to describe the pressure tracking, disturbance rejection and τm estimation response characteristics.

Based upon the fixed displacement feedback pressure tracking results shown in Figs. 7 and 9 one may conclude that the adaptive feedback linearized closed loop control provides good pressure tracking response. As shown in Fig. 8 the change in τm is relatively small, for the range of pressure changes considered, which allows the continuously adapting solution to perform well. For applications in which the change in τm is small, i.e. fixed displacement pressure tracking, continuously adapting feedback linearization may offer the advantage of minimal calibration and

automatic online adaptation, assuming a sufficiently robust measure or estimate of intake manifold pressure is available. In Figs. 10 through 16 an eight to four cylinder engine displacement transition is used to study the response characteristics of feedback linearized pressure control using both feedback and feedforward designs. It is shown in these responses that the large change in τm, during a factor of two change in the engine displacement, can lead to a degraded adaptive feedback linearized pressure response if continuous adaptation of the manifold filling time constant is used. And that the pressure response can be significantly improved by using τm calibration, described in Section IV. Based upon the responses shown in figures 7 through 16 one could conclude that the dynamic feedforward design, feedback linearization with feedforward alone, provides sufficient pressure control response and potentially eliminates the need for an intake manifold pressure sensor. For applications in which all of the pressure control events are based upon pressure commands or known or measured disturbances, this may be correct. Otherwise the design with integrated feedforward/feedback and τm calibration, provides equivalent response characteristics while maintaining feedback control for the rejection of unknown or unmeasured disturbances.

References [1] Falkowski, A., McElwee, M. and Bonne, M., 2004, "Design and Development of the DaimlerChrysler 5.7L Hrmi® Engine MultiDisplacement Cylinder Deactivation System," SAE Paper 2004-01-2106. [2] Lee, P. and Rahbar, A., 2005, "Active Tuned Absorber for Displacement-On-demand Vehicles," SAE paper 2005-01-2545. [3] Leone, T. and Pozar, M., 2001, "Fuel Economy Benefit of Cylinder Deactivation" SAE Paper 2001-01-3591. [4] Gibson, A.O., Kolmanovsky, I.V., 2006," Modeling and Analysis of Engine Torque Modulation for Shift Quality Improvement," SAE Paper 2006-01-1073. [5] Kolmanovsky, I., Sun, J. and Druzhinina, M., 2002, "Nonlinear Charge Control in Direct injection Gasoline Engines," IFAC Proceedings, Barcelona, Spain [6] C. Nesbit, J.K. Hedrick, "Adaptive Engine Control," Proc. of the 1991 American Controls Conference, pp. 2072-2076, 1991. [7] Stotsky, A., Kolmanovsky, I., and Eriksson, S., “Composite adaptive and input observer-based approaches to the cylinder flow estimation in spark ignition automotive engines,” Int J of Adapt Con and Sig Proc, vol. 18, no. 2, pp. 125-144, March, 2004. [8] Gibson, A.O., Kolmanovsky, I.V., 2003, "Modeling Positive Intake Valve Overlap Air Charge Response in Engines," ACC Proceedings, Denver, pp. 755-760. [9] Heywood, J., 1988, Internal Combustion Engine Fundamentals, Second Edition, McGraw-Hill, New York, NY. [10] John, J., 1969, Gas Dynamics, Allyn and Bacon, Boston, MA. [11] E. Hendricks, Isothermal versus Adiabatic Mean Value SI Engine Models, Pro of 3rd IFAC Workshop on Adv in Automotive Con, Karksruhe, Germany, 2001. [12] Khalil, H., 1992, Nonlinear Systems, Macmillan, New York, NY. [13] Astrom, K. and Wittenmark, B., 1995, Adaptive Control, Second Edition, Addison-Wesley, Reading, MA.