Robust PI controller design for a constant turning force system

Int. J. Mach. Tools Manulact.

Vol. 31, No. 3, pp.257-272, 1991.

0890--6955/9153.00 + .00

Printed in Great Britain

Pergamon Press plc

ROBUST PI CONTROLLER DESIGN FOR A CONSTANT TURNING FORCE SYSTEM BOR-SEN CHEN* and YIH-FANG CHANG? (Received 25 May 1989;in finalform 16 August 1990) Al~traet~While a cutting tool is cutting a workpiece at various cutting depths, it goes through a nonlinear time-varying process. The stability of the turning force control system in this process can not be assured by the classical control theory. In this paper, the force control system is considered to be a linear one with a nonlinear time-varying perturbation. Using the functional analysis and Bellman-Gronwall inequality, a simple robust stability criterion is derived from the state space approach. For the simplicity of implementation, the PI controller is employed to treat this stabilization problems, whose design procedure for specifying the control parameters iis also proposed according to the robust stability criterion. The illustrative results of the computer simulation and practical experiments show that the control parameters, obtained by the design procedure, show better output responses than other control parameters in gain space.

NOMENCLATURE

a(t) A b d e(t) l~t) F¢(t)

Fd(t) F,(t) g h

depth of cut (mm) matrix of the dynamic equations in equation (23) maximum of the gain a(t)f(t~-t minimum of the gain a(t)f(t~-i error between Fr(t) and Fd(t), i.e. Fr(t)-Fd(t) feed of the cutting tool (mm/rev) real cutting force (N) feedback signal reference cutting force integral parameter of the PI controller proportional parameter of the PI controller - k expresses the largest real part of the eigenvalues of the matrix A, expressed as - k A m/ax[Re hi(A)]

gdf K, K~ kl n

P Re(.)

T,,

u,(t)

Vf(t)

xl(O X2(t) x,(t) x.(t) y(t) y,(t) a,(a(t),f(t)) h

constant of KaKt specific cutting force of the cutting process (N/mm2) constant of the servosystem (rad2mm/s 3) magnitude of the kle -k', which can cover the norm of the response of the transition matrix e -A' rol:ational speed of the spindle (rev/min) order of the feed of cutting process the real part of a complex number s period of revolution of the spindle T. = 60/n (s/rev) input command of the servosystem feedrate of the cutting tool (mm/s) state variable re(v) dv of the dynamic equation state variable 2Vr(t) of the dynamic equation state variable k2(t) of the dynamic equation state variable y(t) of the dynamic equation state Fc(t)/Kr nc,tation of a(t)f(t) p nc,nlinear and time-varying perturbation of a(t)f(t~' variable of the eigenvalue

*Department of Electrical Engineering, National Tsing Hua University, Kuang Fu Road, Hsinchu, Taiwan, Republic of China. ?Department of Precision Machinery Control, Division of Automatic Control, Mechanical Industry Research Laboratories, Indu~trial Technology Research Institute, Chung Hsing Road, Chutung, Hsinchu, Taiwan, Republic of China. 257

258

m. [~7~1

~11 II

BoR-SEN CHEN and YIH-FANGCHANG damping ratio of the servosystem natural frequency of the servosystem norm of a real vector, in this paper, we define a real vector x = [x,,x~... x,] T, and I~1 a. i m~l Ix,I ("T" denotes the transpose of a matrix) "sup" denotes the supremum, the largest possible number of II II induced matrix norm corresponding to the vector norm and is defined as: Ilal[ a

sup I[axll ~ a sup

I[AXIIwhere A is a (n × n) matrix, in this paper, [JAIlA max

~la,jl

(column sum) 1.

INTRODUCTION

SELECaaN6 large operation parameters such as the cutting depth and the feed will significantly increase the production rate of the CNC machine tools. However, with the increase of the cutting force due to the increase of the cutting depth and the feed, the thermal expansion on the tool tip [1], tool and workpiece deflection [2], and machining chatter [3] will rise such that the workpiece precision is reduced. Consequently, maintaining the cutting force on the tool tip at the appropriate value despite variations in depth of cut is one of the methods to guarantee that the dimension error is permissible and the production rate is maximum. For turning process, the force control system which is equipped on a CNC lathe can regulate the feed to maintain the cutting force within a preset value: however, the cutting depth is changed as a result. The main merit of the control system is that the system, without skillful operation by a senior part-programer, can bring the machine tool to its maximum productivity during the cutting process [4--6]. Over the past few years, some investigations have indicated that in an adaptive control system with contraints (ACC) if the operation parameters are changed, the open-loop gain will vary and the stability cannot be assured during the cutting process [4-11]. The time-varying cutting depth and spindle speed, as well as the nonlinear feed change make it difficult to analyse the system stability by a conventional stability criterion such as the one proposed by Routh [12,13]. Consequently, the adaptive sense defined in the control literature is widely used in this nonlinear time-varying cutting process. 1.1.

Literature

review

Stute and Goetz pointed out that the control parameters of the PI controller can be simultaneously multiplied by a factor to maintain the open-loop gain from the control signal to the feedback signal [7]. However, the method of selecting the initial control parameters was not presented. Koren and Masory [5] presented a brief adaptive identification method of tracking the process gain for an integral controller. The open-loop gain, spindle speed, and sampling time are also analysed by experiments [6] or by digital control theories [8] to ensure the system stability. This control algorithm can maintain the open-loop gain under variations in the course of cutting. Nevertheless, they did not indicate that how the presented gain space influence the transient performance. Tomizuka, Oh and Dornfeld [9] first applied a discrete time model reference adaptive control (MRAC) algorithm based on independent tracking and regulation [14] to the synthesis of both the adaptation algorithm and the controller structure. They concentrated on tracking algorithm of the MRAC system to maintain the stability of the milling system with large gain variation. Daneshmend and Pak [10] analysed the nonadaptive force control loop and briefly presented an improved MRAC design technique to regulate the feed force in turning operations. The MRAC scheme based on the technique of Landau and Lozano [14] was implemented on a mutli-microprocessor based CNC system. The research focused on the comparison of the various algorithms derived from the MRAC design technique. They still not present a research to improve the transient performance which should be studied by further researches.

Robust PI Controller Design

259

As mentioned above, these researches [4-11] concentrated their efforts on the simplification and realization of the parameter adaptive control algorithm for the force control system. When [he bound of the variable gain in the system are not known in advance, or when they are wide, adaptive controllers [4-11] could perform more consistent tracking algorithm to obtain optimal control parameters according to a mean value of cutting depths and spindle speeds with respect to any optimality criteria such as the minimal integral square error [12]. However, before control parameters reach the optimal values, the transient stability is not considered in these researches [4-11]. Some of the stabilities based on the Aizerman's conjectures are analysed to allow some kind of gain change in the system from the viewpoint of maximum gain [15,16], but these methods cannot explicitly express the tolerance of the gain perturbation in the nonlinear time-varying system. That is why there are some different conservative stability criteria which are derived from some control theorems and which are presented for high-degree (more difficult to stabilize) nonlinear time-varying systems [15,16]. Even the PI comrollers are utilized in the force control system, it is very important to find a simple robust stability-analysis method to obtain the control parameters with good reliability to tolerate this kind of perturbation. 1.2. Objective of the paper In this paper, when the spindle rotates at constant speed, the nonlinear and timevarying gain of the cutting process can be divided into an average value and a nonlinear time-varying perturbation. Based on functional analysis [15,16] and the Bellman-Gronwall inequality [15,17], a lucid design algorithm is proposed to synthesize a robust PI controller to tolerate the nonlinear feed change and time-varying cutting depths. The design procedure of the controller is presented step-by-step. The robust stability will be verified by the computer stimulation. The PI controller is installed on a digital computer to implement the force control on a converted conventional lathe. Some practical results show that the robust PI controllers are capable of better performance than those in other region of the gain space. 2.

MODEL OF THE SYSTEM

In this paper, as pointed out in the previous investigations [4-8], the turning force control system shown in Fig. 1 is composed of the PI controller, the servosystem, the feed relation, tlhe cutting process dynamics and the sensing element. The PI controller

Fr(t~ ~ e(t) ÷

,

CUTTING PI CONTROLLER PROCESS Us(t)=fge(v)dv*he(t)SERVOSYSTEM Vf(t)---l~nI f(t}l"~":- I~c(t)*Fclt)=a(t)Kf f(t)PI Fc(O ~f(t)÷2~nVf(t)*Wn2Vf(t)=KmUs(t) DYNAMOMETER

Fd(t) FIG. 1. Block diagram of the constant turning force system.

BoR-SEN CHEN and Ym-FA~G C~,No

260

consists of an integral part and a proportional part, with the dynamic relation to be expressed as:

U~(t) = ge(v) dv + he(t)

(1)

where Us(t) is the control signal to the servosystem and e(t) is the deviation between reference input Fdt ) and feedback signal Fd(t), i.e.

e(t) = Fr(t)-Fd(t).

(2)

Fr(t) is-preset by an off-line theoretical or experimental analysis subject to the machinilig ability of the tool and the lathe. In this paper, we will propose a design procedure to find out the suitable control parameters, g and h, to robustly stabilize the force control system. Consisting simply of the motor and the inertial load of the feed screw, the servosystem is presented as [4--8]: 12f(t) + 2~e.l~'f(t) + o~ Vf(t) = KmU~(t)

(3)

where ~ is the damping ratio, t o . (rad/s) is the natural frequency of the servosystem, and Km (rad2mm/s3) is a constant. Vf(t) (mm/s) is the feedrate of the cutting tool driven by the servosystem. The main cutting force is related to the feed corresponding to the moving distance of a tool per revolution of the spindle, and the relation between feedrate vf(t) and feed f(t)[mm/rev] is written as [6,8]: fit)

=

T,,Vf(t)

(4)

where T, (s/rev) is the period of revolution of the spindle, i.e. 60 r. = --, n

(5)

n (rpm) denotes the rotation speed of the spindle. Because the spindle usually rotates at a constant value during the cutting process, the time-varying nature of the value n is not considered in this paper. The dynamic equation of the cutting force F¢(t) (N) with respect to the feed fit) can be described by the first order lag[6,8]:

2

/re(t) + F¢(t) = a(t)Kff(t)P

(6)

where a(t) (mm) is the cutting depth which is a time-varying variable during the cutting process. The specific cutting force Kf (N/mm 2) is assumed to be a constant because the variation percentage is much smaller than that of the variable a(t). p denotes a constant. Both depend on the workpiece material and the shape of the tool tip. A dynamometer is equipped to detect the cutting force which is converted to a digital value Fa(t) as:

Fa(t) = KaFc(t)

(7)

where Ko (l/N) is a conversion factor. The statistical assessment results of the static cutting force vs the feed for several values of cutting depths from 0.5 to 3 mm are illustrated in Fig. 2, where the symbols

Robust PI Controller Design Fc[NI 3000

261

+ : a=3.0mm |

2700

:

i

1112.5111!

;~

3.0

~

25 "

¢ : r2.0mm •

2400

: e=l.Smm .-1.0ram

"/

/

x :

$ : a=O.Smm

2100-

/ 2.0

1000" 1500•

1.5

1200 •

k

•

'

500 GO0

'-

1.0

0.5

300 0.1

0.2

P[ mm/rev ]

0,3

0,4

0.5

Fro. 2. Statistical assessment and experimental cutting force vs feed for some different cutting depths.

@, x , . , *, # and + express the experimental cutting force at the cutting depths 0.5, 1, 1.5, 2, 2.5 and 3 mm, respectively, when the spindle speed is 60 rpm and the workpiece diameter is 50 mm. In the force control system, the cutting depth is subject to variation in the range from 0 to 3 mm, and the feed is manipulated to maintain the cutting force (e.g. along a horizontal dotted line in Fig. 2). Let y(t) (mm 2) be defined as:

y(t) = : Fc(t)/Kf

(8)

where the symbol "=:" is expressed as "is defined as". Then, equation (6) can be expressed as:

~ (t) + y(t) = a(t)f(t) p = [a(t)f(t)p-1]f(t) = :Y] (t).

(9)

From equation (9), the variable cutting depth a(t) is the time-varying term of this cutting process;, and f(t)p is the nonlinear term. Thus, the cutting proces is nonlinear and time-varying. The perturbation in equation (9) can be divided into the average value and the pure perturbation, i.e. yl(t) can be divided as follows: yl(t) =

[a(t)f(t)p-1]f(t) b~id f(t ) + Aa(a(t),f(t))

(10)

where b is the maximum of the gain a(t)f(t) p-1 and d is the minimum of the gain a(t)f(t)a-L (b+d)/2 is the average gain under control. Aa(a(t),f(0) is the nonlinear and time-varying perturbation which is a function of a(t) and f(t) and is generated when the value of the state a(t)f(t) p is not equal to (b+d)f(t)/2. In Fig. 3(a), the I/O relation between f(t) and yl(t) is described. Figure 3(b) shows the I/O relation between f(t) and pure perturbation Aa(a(t)d(t)), where (b-d)~2 denotes the maximum difference of the perturbed gain and the average gain (b+d)/2.

262

BoR-SEN CnEN and YIH-FANG CHANG

y(t) b

d f (t)

/x o(o(t),f(t))

+b2d

f(t)

_b-d

2

ng.Z(b) FIG. 3. Input/output diagram of the cutting process inside the sector[d,b] centered at

(b-d)/2.

(b+d)/2 with radius

Substituting equation (10) into equation (9), we can obtain the following equation.

~-~p(t) + y(t)= ~-~-f(t) + A~(a(t),f(t)). In terms of new variable Fd(/) =

Kdey(t)

where Kd~ =

KdKe.

(11)

y(t), Fd(t) in equation (7) can be rewritten as: (12)

3.

CONTROLLER DESIGN

Before writing the dynamic equations in compact form, we need some notations as follows:

xl(t) = e(v) dv

(13)

x2(t) = 2Vf(t)

(14)

x3(t) = .~2(t) x,(t) = y(t).

(15) (16)

Robust PI Controller Design

263

Then, the dynamic equation of the PI controller can be expressed as: Xl(t)=

fi(,;v )d v = fo[Fr(v)-

)]

v dv.

(17)

The control signal Us(t) can be expressed as:

Us(t) = gxl(t) + he(t) = gxl(t) + h[Fr(t)-Fd(t)l.

(18)

The dynamic equation (3) of the servosystem can be rewritten as: :~3(t) = -to~x2(t) - 2l~w,xa(t) + 2KmUs(t).

(19)

Substituting equations (4) and (16) into (11), the dynamic equation (11) of the nonlinear and time-varying cutting process can be expressed as:

b+d

2

b+d

2

JZ4(t) = ~ 2 - - X 2 ( / ) -- Tn X4(/) ÷ Aa(a(t),x2(t)) = ~)~'-X2(/) -- ~nX4(t) + Aa(.)

(20)

where Aa(.) = : Aa(a(t),x2(t)).

(21)

Equation (12) can be rewritten as: Fd(t) = gdfx4(t).

(22)

The block diagram Fig.1 can be recast into Fig. 4 by replacing the state variables. After substiluting equation (22) into (18), the dynamic equations of this system can be expressed by substituting (18) into (19) as: J((t) -- AX(t) + F(a(t),XE(t)) + BFr(t) where

A =

(23)

0

0

0

-- Kdf

0

0

1

0

2gKm

--(On2

,9

(b+d)/2

--2~tO, -2hKdfm

0

(24)

-2/Tn

xl(t)=fe(v)dv Frtt)

e(t)

x3(t)=~2(t)

x2(t)

Uslt)=gxl(t)*he(t)

b,d x4(t) ~ "~- x2(t)

---

2 Tn x4(t) +

6alatt~'x-IO;''" Z''"

x3(t)= ~nZx2(t)-2~nX3(t)+ZKmUs(t)

Fd(t)

FIG. 4. Block diagram of the constant turning force system in state space model.

264

BoR-SENCHENand YIH-FANGCHANG

X = [xl(t) x2(t) xa(t) x,(t)] T, F(a(t),x2(t)) = [0 0 0 Aa(a(t),x2(t))]T, B = [1 0 2hKm 0]T ("T" denotes the transpose operation of a matrix), and Kdfm = KdKfKm. For convenience of typing, nonlinear and time-varying column matrix of F(a(t),xz(t)) is denoted by the following:

F(. ) =: F( a( t),xz( t) ).

(25)

The total nonlinear and time-varying perturbation is combined in F(.), which is considered as an external input acting on the linearized system with average gain (b+d)/2 within sector [d,b]. The solution of equation (23) is:

Io

fo

X(t) = eAtXo + eA~t-v)F(.)dv + ea('-v)BFr(v) dr.

(26)

We cannot directly solve the second term on the right hand side of equation (26). However, by performing the norm operation on both sides, the norm of the term can be expressed in terms of the norm of state vector X(t) and combined with the norm in the left state vector of equation (26) by Bellman-Gronwall inequality. For easy calculation of the norm, the norm in this paper is defined as the maximum column sum [13, 16]. Performing the norm operation to both sides of equation (26) and using the triangle inequality of the norm properties, we get:

IL (t)ll

IleAqlIL oll +

fill

rlF(.)ll dv +

f[IIea(t-v)]] Ilnll tlFr(v)ll dr.

(27)

Suppose we can choose control parameters g and h of the PI controller such that all the eigenvalues of A in equation (24) are on the left half s-plane, then the transition matrix eat satisfies the following inequality [13]. IIcAll -< k, e-kr

(28)

where - k A mjax[Re hi(A)], i.e. - k is the real part of the eigenvalues that lies nearest to the imaginary axis, k~ is the magnitude which can cover the norm of the free response of the transition matrix eAt.

Remark. kl in equation (28) can be estimated from the maximum value of [[eAl[/e-kt<--kl for all t. As shown in Fig. 3(b), it is seen that nonlinear and time-varying bounded within two lines with slopes ±(b-d)~2, i.e. Ilaa(a(t),Xz(t)) II-<

~ I[xz(t)ll.

Aa(a(t),x2(l))

is

(29)

According to the above equation and the norm definition of a real vector, IIF(.)]] can be expressed as: Ilg(.)lI -< ~ - ~ I~x2(/)ll

b-d - - -2I ~ ( t ) l l .

(30)

Robust PI Controller Design

Because h is always positive,

IIBll can

265

be obtained as:

IIBII= (1 ~- 2hKm).

(31)

By substituting equations (28), (30) and (31) into equations (27), equation (27) can be rearranged as:

I~c(t)ll ~ ~c~e-~1~oll + f~ b-~ k:e -~'-v' I~(011 dv +(1 + 2hrm) sup II F~(u)ll k~e -k(t-O dv. u~[O,t]

(32)

Multiplying e k~ o n both sides and integrating the third term on the right-hand side, equation (32)can be expressed as: j

ft b - d

IlX(/)llek'~ klllXoll ÷ ~ (ek'--l) + Jo 2 kdlX(v)ll e*~dv

(33)

where J=k~(l+2hKm)

sup IIFr(u)ll. u~lO.tl

(34)

By Bellman--Gronwall inequality [15,17], and from equation (33), we can obtain a simple equation (see Appendix): j e Ct

IIS(t)llekt :~ kdlSoll e Ct + - -

C-k

j e kt

(35)

C-k"

Multiplying e -k' to both sides of equation (35), we get

IIg(/)ll ~ '~llISolle (c-k)t

+~

J

e (c-k)' -

J C-k"

(36)

If C - k < 0 , we obtain:

IIg(t)ll ~

k~ k (l+2hgm) sup IIF,(u)ll, when t----~oo u~[O,t]

(37)

which expresses the boundedness of the X(t) as t approaches infinity. Note that since C - k < O , the right-hand side of equation (37) is positive. From the above analysis, we arrive at the following result. In the force: control system of Fig. 1, if control parameters g and h in the system matrix A of equation (24) are chosen such that the robust stability criterion, C - k < O , is satisfied, i.e,., -k < -kl(b-d)/2

(38)

then the force control system is robustly stabilizable. From equation (38) and the definition of - k , the robust stability criterion can be obtained as: max[Re kj(A)]< - k l ( b - d ) / 2 . ! HTH 31-3-B

(39)

266

BoR-SEN CHEN and YIH-FANG CHANG

According to the above criterion, if the system is a linear time invariant system, i.e. b = d, equation (39) is reduced to max[Re hi(A)]
Remark. This stability criterion is only a sufficient condition. The system may be stable at certain g-h pairs without satisfying criterion (39) in the practical force control system. 4.

COMPUTER SIMULATION AND EXPERIMENTAL RESULTS

In this paper, we discuss a cutting tool that cuts a workpiece at cutting depths which vary like a square wave along the axial direction of the workpiece. Figure 5 shows the profile of the workpiece to be cut, where the width and the depth of the grooves are 8 mm and 2 mm, respectively. Data used to simulate and implement are given as follows: The material of the workpiece: $45C carbon steel tool: P10 (0,5,11,18,15,0.8). The characteristics of the dynamometer are: Kd = 4.0 N -1. The cutting process parameters are: Kf = 1732 N/mm 2 p = 0.85.

Robust PI Controller Design

workpiece

267

E EE _

lO.50mm

Vf

dynomometerI

FIG. 5. Profile of the workpiece to be cut.

In order to prevent the tool breakage arising from maximum overshoot of the transient variation in the cutting force while the cutting depth is being changed, the cutting force is to be maintained at Fc = 900 N. The reference input should be set at: F~ = 3600, because the real cutting force will be multiplied by the conversion factor Kd = 4 N-1. The magnitudes of the feedback signal will be four times those of the actual cutting force, i.e. Fd =: KdFc. The PI control algorithm can be easily implemented on a digital computer on which an A/D conve:rter is installed to receive the feedback cutting force signal from the dynamometer. A D/F converter (digital to frequency), installed on the computer, can generate pulses to drive the motor by which the cutting tool is moved in the feed direction. The discrete control algorithm used in the digital computer can be obtained by Tustin's approximation of the PI formula [18]. As the analysis is carried out in continuous time, with both the simulation and the implementation based on computer control, high sampling frequencies becomes obligatory. In this paper, the control cycle time is set at 0.02 s, which is also treated as the sampling rate in simulation. Because different sampling rates during the control process will change the stability of the dynamic systems, i.e the characteristic parameters such as damping ratios and the natural frequencies of them will be changed, the sampling rate would be fixed at a constant and then K m = 1296 rad 2 m m / s 3, ~ = 0.42, and to, = 36 rad/s can be obtained and fixed when the sampling rate is at 0.02 s. The control parameters of the robust PI controller can be designed according to the following procedure. Step (a): Subject to F: = 900 N, the feed should be regulated to f = 0.13 mm/rev when the cutting depth is at 3 mm. The maximum gain of afp-1 can be obtained from these values, i.e. b = 3 × 0.13 -°-15 = 4.08. The minimum gain d is considered as zero. Step (b): According to the above data, Region I in Fig. 6((b+d)/2=2.04) can be such that all - k in it are smaller than zero. It represents the stable region of the linearized system with average gain and without perturbation. Step (c): By comparing all - k in Region I, the minimum - k , denoted as x in Fig. 6, can be found at g = 0.4 × 10-2, h = 1.6 x 10 -4 ( - k 1 = -13). The control parameters, designed by using the condition of the maximum cutting depth, i.e. Region II in Fig. 6(b=d=4.08), are capable of better stability than those in the region outside the dot--dash line in Fig. 6. To highlight the advantage of the robust

268

BoR-SEN CHEN and YIH-FANG CHANG

gxtO2

1.50

1

7-,

/I 0.50 x 1.00

\ \'\,

...... 2

4

6

::

\

~,,,,,, 8 10 12 hxlO4

14

~ ~ ~, IG lS

Fro. 6. Boundary g-h gain space Routh-Hurwitz criterion and minimum(-k) robustness criterion. Region I: linear cutting system cutting at constant gain b = d = 2.04 mm.rev. Region II: linear cutting system cutting at constant gain b = d = 4.08 mm.rev. Point x: robust cutting system with variable cutting depth inside the sector [d,b] centered at (b+d)/2 and with radius (b-d)~2, (b = 4.08, d = 0).

PI c o n t r o l l e r , the responses of the system with the robust PI c o n t r o l l e r are c o m p a r e d with those of PI c o n t r o l l e r with g = 1.1 × 10 -2 a n d h = 1.6 × 10 -4 o n the b o u n d of R e g i o n II in Fig. 6.

Fc[N] 2100

g=0.4x10-2

h=1.6x10-4

1800

n=GOOrpm

1500 1200 7(a)

900, 600. 300. i

1.0

2.10

3.10

I

4.0 Llaec]

5.10

F[mm/rev] 0.50-

i G. 0

7.t0

9-0.4xl0-2 h-l .GxlO-4 n=GOOrp=

0.450.400.350.30 7(b)

0.25 0.20 0"15~L

J

0.10 t " 0'05 r 1.~0

t 2.0

3.10

4,io L(sec]

t 5.0

6.10

7.10

Fro. 7. (a) Simulated cutting force responses Fc(t). (b) Simulated feed responses f(t) of PI control with g = 0.4 x 10 -2, h = 1.6 x 10 -4 with respect to cutting depth change shown in Fig. 9.

Robust PI Controller Design

269

As shown in the computer simulation of Figs 7 and 8, the control parameters with minimum - k are capable of better output responses in Fig. 7 than those in Fig. 8. The cutting depth variation with respect to time is shown in Fig. 9. The feed changes to 0.13 (mm/rev) in order to maintain the cutting force when the tool engages the workpiece at 3 mm. The feed is regulated to 0.46 (mm/rev) when the cutting depth changes to 1 mm, at which the cutting force can be maintained. The response is slow because the open loop gain with the cutting depth at 1 mm is smaller than the one with a 3 mm cutting depth. When the cutting depth is increased to 3 mm, the first overshoots in two figures are very large. Because of the heavy load of the servosystem, the states of the robust force

Fc[NI 2100-

g=t .lxlO-2 h=t .6x10-4 n=6OOrpm

1BOO.

1500. 12008(o)

900. 600. 3001.10

I

2.tO

3.0

I

4.0 Llsecl

5JO

Ftmm/revl

t

6.tO

7.0

g=1.1xtO-2 h=l.GxlO-4

0.50. 0.'45.

n--6OOrpm

0.40. 0.35. S(b)

0.30 0.25 0.20 0.15

0.10 0.05 1.0

2.'0

3.:0

5.'0

4.:0 t[secl

6.'0

7.0

FIG. 8. (a) Simulaled cutting force responses F~(t) (b) Simulated feed responses f(t) of PI control with g = 1.1 × 11)-2, h = 1.6 x 10-4 with respect to cutting depth change shown in Fig. 9.

e(eml

3

I 1.0

I 2.0

I 3.0

I 4.0 L[see]

I 5.0

I G.O

FIG. 9. Cutting depth variation of simulation.

7 .lO

270

Box-SEN CHEN and YIH-FANO CHANG

control system are hard to reach the equilibrium ones "during" the cutting depth jumps from 1 mm to 3 mm. The advantageof the robust control system is that the controller can fast track the equilibrium state "after" the cutting depth jumps from 1 mm to 3 mm (3 times). It is obvious that the output responses after the first overshoots of the robust PI control in Fig. 7 are stabler than those in Fig. 8. Figures 10 and 11 illustrate the actual output responses of Fig. 7 and Fig. 8 respectively. Just like simulation, the results in Fig. 10 show better output responses than those in Fig. 11 do. The small peaks before the maximum overshoots in the practical force responses is brought about by cutting the ring-shaped chip which is generated at the end of the previous cutting process at a 3 mm cutting depth. If PI controller is designed by the maximum cutting depth a(t) = 3 mm (b=d=4.08), the actual responses, shown in Fig. 12 with g = 1.1 × 10 -2, h = 1.6 × 10 -4 in Region II, manifest the stabilization when the cutting depth is maintained at 3 mm. However, the practical responses in Fig. 11, with the same control parameters, indicate that the control parameters in Region II cannot guarantee the robust stability when the cutting depths vary like a square wave as shown in Fig. 9. 5.

CONCLUSIONS

According to the investigation carried out for this paper, the robust PI controller is designed to ensure the stability of this turning force control system with a nonlinear time-varying cutting process. Some conclusions are summarized as follows: (a) The norm characteristics of the transition matrix eA' and the Bellman-Gronwall inequality are employed to analyse the robust stability of the nonlinear time-varying

Fc[NI 2100.

g=O.4xlO-2 h=l.~xlo -4

1800

1500. 1200. |O(a)

900 600 30O 1.0

2.~0

3.10

I

4.0 5.10 f,isecl

flmm/revl 0.50]

I

G.0

7.10

g=O.4xlO-2 h=l .GxlO-4

0.451 0.40-

n--BOOrpm

0.35o.3oi io(b)

0.25-

0.200.15. 0.100.05. I

I

I

1.0

2.0

3.0

I

I

4.0 5.0 LIsecl

I

i

6.0

7.0

Fit}. 10. (a) Actual cutting force responses Fc(t), (b) actual feed responses fit) of practical PI control with g = 0 . 4 x 10 -2 , h = 1.6 x 10 -4 .

Robust PI Controller Design

FeIN] 2100-

271

9=1.lx10-2 h=t ,GxlO -4 n=b'OOr-pm

1800. 1500. II(a)

1200. 900. 6003001

1.0

2.0

3.0

4.0 tl=ecl

I

5.0

Flmm/revl 0.50.

i

G.O

7 .iO

g,,I.lxiO-2 h=t.5xi0-4

O,45. 0.40.

n-600rpm

/vV'%d ~

0.35. 1 I(b)

0.30. 0.25. 0.20. 0.150.10. 0.051.0

2.0

3,0

4,0 t[sec]

5,0

6,0

7.0

Fro. 11. (a) Actual cutting force responses Fc(t), (b) actual feed responses f(t) of practical PI control with g = 1.1 x 10 -2 , h = 1.6 x 10-'.

FclN] 2100

9-I .Ix10-2

18001 / 1500t

h=l .GxlO "4

n=600rP=

=°or 300÷ l

, 1,0

FIG. 12. Actual cutting force responses

i

2,0

Fc(t) when

i

3.0

i

~

4.0 5.0 t[t~-,c]

i

G.O

, 7.0

cutting depths maintain at 3 mm (g = 1.1 x 10 -2, h = 1.6 x 10-4).

force control system. The robust stability criterion for the system can be obtained by these methods. (b) The control parameters of the robust PI controllers can be obtained according to the robust criterion. (c) From the ~'omputer simulation and the experimental results of the force control

272

BoR-SEN CHEN and YIH-FANGCHANG

system for a converted conventional lathe, the controller designed by criterion (39) possesses good robustness and good reliability in order to override the effects of the nonlinear feed variation and time-varying cutting depth perturbation from 1 mm to 3 ram. In this force control system, the spindle speed is set at 600 rpm, which is not an absolutely optimum operation condition when a workpiece is cut with another cutting tool. An improved robust controller can be designed to treat this force control system with variable spindle speeds during the cutting process. Acknowledgements--The authors are pleased to acknowledge Professor H. H. Lin's help in editing this paper. REFERENCES [1] Y. TAKEUCm, M. SAgAMOTOand T. SARA, Improvement in the working accuracy of an NC lathe by compensating for thermal expansion, Prec. Engng, 4, 19-24 (January 1982). [2] Y. EL-KARAMANY,Turning long workpieces by changing the machining parameters, Int. J. Mach. Tool Des. Res. 24, 1-10 (1984). [3] T. INAMURA,T. SENDAand T. SARA,Computer control of chattering in turning operation, Ann. CIRP 26, 181-186 (1977). [4] O. MASORYand Y. KOREN, Adaptive control system for turning, Ann. CIRP 29, 281-284 (1980). [5] Y. KORENand O. MASORY,Adaptive control with process estimation, Ann. CIRP 30, 373-376 (1981). [6] A. G. ULSOY, Y. KORENand F. RASMUSSEN,Principal developments in the adaptive control of machine tools, ASME J. Dyn. Syst. Meas. Control 105, 107-112 (June 1983). ['7] G. SrUTE and F. R. GOETZ, Adaptive control system for variable gain in ACC systems, Proc. 16th Int. Machine Tool Design and Research Conf., pp. 117-121 (1975). [8] O. MASORYand Y. KOREN, Stability analysis of a constant force adaptive control system for turning, ASME J. Engng Ind. 107, 295-300 (November 1985). [9] M. TOMmUr,A, J. H. OH and D. A. DORNFELD,Model reference adaptive control of the milling process. In Control of Manufacturing Processes and Robotic Systems (Edited by D. E. HARTand W. J. BOOK), pp. 55-63. Dynamics and Control Division, ASME, New York (1983). [10] L. K. DANESHMENDand H. A. PAK, Model reference adaptive control of feed force in turning, ASME J. Dyn. Syst. Meas. Control 108, 215-222 (September 1986). [11] O. MASORY,Real-time estimation of cutting process parameters in turning, ASME J. Engng Ind. 106, 218--221 (August 1984). [12] G. F. FRANKLIN,J. D. POWELLand A. EMAMI-NAEINI,Feedback Control of Dynamic Systems. AddisonWesley, Reading, MA (1986). [13] C. T. CttEN, Linear System Theory and Design. CBS College, New York (1984). [14] I. D. LANDAUand R. LOZANO,Unification of discrete time model reference adaptive control designs, Automatica 17, 593-611 (1981). [15] J. C. Hsu and A. U. MAYER, Modern Control Principles and Applications. McGraw-Hill, New York (1968). [16] M. VIDYASAGAR,Nonlinear Systems Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, U.S.A. (1978). [17] E. HILLE, Lectures on Ordinary Differential Equations. Addison-Wesley, New York (1969). [18] K. J. ASTROMand B. WITrENMARK,Computer Controlled Systems--Theory and Design. Prentice-Hall, Englewood Cliffs, New York (1984).

APPENDIX

Theorem 1:

generalized BeUman-Gronwall inequality [15,17]

For R(t)>0, C>0 Q(t) <- R(t) +

i

CQ(v) dv

(A.1)

o

then

f

Q(t)<-R(t) + ' Ce ct'-'~ R (v) dr.

(A.2)

In the case of equation (33), Q(t), R(t) and C are expressed as: Q(t) = I~(t)l] ek' R(t) = k,llXoll+

(A.3)

~-(ek ' - l )

C = kj(b-d)/2.

(A.4) (A.5)