TAP CHi KHOA IIQC & C O N G NGHf CAC TRUING DAI HQC KV THU^T * S 6 91 - 2012
ROBUST AND ADAPTIVE TRACKING CONTROLLER DESIGN FOR GEARING TRANSMISSION SYSTEMS BY USING ITS THIRD ORDER MODEL
THIET KE B O DIEU KHIEN T H I C H NGHI BEN vONG CHO HE TRUYEN DONG
QUA B A N H R A N G NHd M O HINH BAC BA Nguyen Doan Phuoc Le Thi Thu Ha
Hanoi University of Science and Technology Thai Nguyen University of Technology
Received August 01, 2012; accepted September 28, 2012
ABSTRACTTo eliminate the effect of backlash, shaft elasticity and fdction in control of transmission system are conventionally nonlmear estimators for their compensations often used. This paper proposes a new design procedure of an adaptive and robust tracking controller for geanng mechanical transmission systems without using of such estimators and compensators The proposed controller is designed by using the third order model of gearing transmission system. The asymptotic tracking behavior of the system in the presence of all uncertainties caused by backlash, faction or cogwheel elasticity is proved The simulation results are provided lo illustrate the satisfactory pedormance of the closed loop system.
TOM TAT
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1 INTRODUCTION frictions and then compensate these by states The effect of immeasurable frictions, feedback control [I],[3],[4],[5].
unpredictable elasticity of shaits and imprecise Such proposed control strategy, however, description of backlash is inevitable in can only be used good either for systems with mechanical transmission systems, which limits shaft elasticity or with backlash separately [6].
the performance of control system in practical Moreover, a good tracking performance of applications. Those inevitable uncertainties can systems, in which all uncertainties like reduce the lifetime of the whole system or even immeasurable friction, unpredictable elasticity
of shafts and backlash are simultaneous present, cannot be achieved with such compensative disturb the system behavior. Therefore,
damping the torsional vibration due to the shaft
or cogwheel elasticity and suppressing the techniques.
effect of friction or backlash are the most
important control problems of mechanical ^ ° overcome this problem is an systems in general and of gearing transmission avoidance of estimators and compensators systems in particular. included in controller necessary. Therefore will
According lo improve the control performance of gearing transmission systems have to be suppressed the frictions, shaft elasticity and backlash between cogwheel. The
most promising approach for suppression of estimators and compensators for elimination cf those uncertainties can be obtained with
estimators of backlash, shaft elasticity and
be now applied here an adaptive robust control based on the sliding mode technique and the certainty equivalence principle, which allows to improve the overall tracking performance of the closed loop system without using of additional estimators and compensat
uncertainties in systems.
TAP CHi KHOA HQC & C 6 N G NGH? CAC T R U 6 N G DAI HOC KV THUAT • S6 91 - 2012 The sliding mode control is one of the
robust control theories to suppress the effect of bounded noises or disturbances in systems. In addition, the certainty equivalence is also the most successfully used principle in adaptive controller designs for uncertain nonlinear systems in the presence of unknown constants in the systems' model. In this connection, the paper combines both the sliding mode technique and the certainty equivalence principle for designing an adaptive robust tracking controller for gearing transmission systems, in which the unpredictable elasticity of cogwheels and the imprecise description of backlash between cogwheels are considered as unknown constant parameters, whereas immeasurable shaft friction and the load capacity are regarded as bounded time dependent noises and disturbances in the system.
This paper is organized as follows. In section 2 is included the third order model of gearing transmission systems. The section 3 describes design procedure of robust adaptive controller for system based on this third order model. In section 4 the experimental results and simuladons are includes. Finally are included in section 5 some conclutions and commentaries about future research.
2. M O D E L L I N G O F G E A R I N G T R A N S M I S S I O N S Y S T E M S
Consider a gearing transmission system with a controller depicted in Figure 1. The driving motor provides a control torque Mj which is transmitted to the load M^ through two wheel gears and two elastic shafts.
Let M|-^ and M^2 denote the friction moment on each shaft. Both shafts have the same elasticity factor denoted by c . Let ^ | and (P2 be the rotational angles of corresponding shaft and a the backlash between cogwheels.
The Euler-Lagrange model of this gearing transmission system is given as follows (see, for example, [2]).
J2<P2 -cr2C0S^a{^2 +'2\'P\)-~^c ~^/2
where r, and r^ are the outer radii of corresponding wheels i and 2, I'lj-'Ti' is the transmission rate of the two wheels and
./|, J j , Jj are the inertia moments of wheel 1, wheel 2 and the driving motor respecdvely and l\=Jj+J\ denotes the sum of inertia moments of wheel I and the driving motor.
While 7 | , J j , j|2, '21' 'i a"<l '2 ' " Euler- Lagrange model (1) can be considered as known parameters, the other parameters such as shaft elasticityc, friction moments M y , , M ^ , , load moment M^, backlash a are all uncertainties or disturbances of the system.
Therefore in the following, all unknown constant parameters of the model will be denoted by O^., whereas disturbances by d^ .
By using
= cr{cos'a -cr^cos^a Mj^ - 6 i ^ i +1^1(97,,0 M^ -Mj2 - - ^ 2 ^ 2 -d2{9id) where
6j, bj - known constants, 9., $2 - unknown constants.
y " ' - i"'derivative of X, p, q - finite positive integers,
(2)
(1)
Figure 1. Configuration of gearing transmission system d^{^^,l) , (^2(^2>') - unknown disturbances, the Euler-Lagrange model (1) becomes
J^p^ +0i{^i ^'\i<Pi) = ^d ~b\^\ -d^
•^2'Pi - ^2 iVi + 'I'lVi) = -b2<p2 ~ ^2 (3)
TAI- Clli KIIOA l i p c & C.6NG NGIIf CAC T R U W N G DAI HQC K» T H I I ^ T * s 6 91 -2012
(5) From the second equation of (3), it is easy to see thai
PI = ' l 2 [ ' ' l ( - ' ! « ' ! + * ! f ' ! + < ' ; ) - « ' 2 ]
= 9 , j i , + f l > j - ; „ f ) j + r f , (4) with
</, =l|jffjrfj, a, =(9;./2- ^4='l2^j''2 and
Vl =Sj(02+Sl«'2-'l2ft+<')
where
d,=j, ,J,=j,.
From (3), (4) and (5), it follows that A/,, = 7,fl,f>™ + {j,B, +b,S,)(ii,+
-1- (7|rf5 + tf.rfj + b,d., + d,) Next, let states vector x, truncated states vector X, input control signal u, vector of unknown constants dj, unknown constant 0 and unknown disturbance d(x,t) be defined as follows
ll = Mj
• " • I
1 - ^ 4 ^ (f^
Vl V,
\Vij (x,-^
• I , A'l
f'"'l
'qy.
J,0^
d(x,t) = 0^0^-h^
b^9^+0^9^-J^i^,
-(dA+9^d^-*-b,d,+d^)
The Euler-Lagrange model (3) of the gearing transmission system, can now be rewritten in the form of uncertain states fourth order model:
Xi^i if 1 < * < 3
djx + d{x,t) + 0^u (6)
with d(x,t) being bounded by a number S>0, that is
\d(x,i)\^S for all x,t. (!) Because reference signal w{i) is often
the desired speed (o^ = JCJ, not the the rotational angle ^^ = J : [ , the third order model:
\x,=x, (8)
\x^ =0'jX + d(x,t) + O^u
can be employed in controller design instead of fourth order model (6). Furthermore the using of third order model (8) will definitely guarantee for obtaining of a more simple controller than by using of original fourth order model (6).
3. C O N T R O L L E R D E S I G N
Let be first w(t) the reference signal, so the reference trajectory for third order system (8) will be:
and the vector of reference error is
e = {e,e,ef (IQ) Where e^w-x^ =^-'P2 ( H )
To control the states vector x(i) of system (8) to asymptotically track the reference trajectory w(l) given in (9) based on sliding mode control, first the following sliding surface is used:
s{e) = a^e + a , e + e = afe (12) where all elements a,, a^ of vector
n ' - { a , , u , , 1 ) ' '
are chosen such that the following polynomial
/ ' ( a ) = a | + a 2 « + « ^ (13) will be Hurwitz. Because polynomial (13) is
from second order, necessary and sufficient for H u r w i t z i a n o f p{a) is a, > 0 and a^ > 0 .
TAP CHf KHOA HQC & CONG NGHE CAC TRUING D^l HQC KV THU^T • Sd 91 - 2012 Moreover, by using sliding surface given
in (12), in order to ensure the asymptotic tracking performance
e->-0 and |e|<co
the nesecessary and sufficient condhion is 5(e) - 0
Thus, the initial tracking control aim can now be replaced with
s{e)->0 and |5(e)|<co for / > 0 Now consider the following candidate control Lyapunov function (CLF)
with its derivative being given by V - SS = s[uye + a2'e + w -x^^
= s(a,e + a2e + w-0jx-d{x,t)-d^u]
Therefore, if the following controller is used u = 0;'(a,e + a2e + w-9jx + Asgnis)) (14) with any A.>d then
V = s («,£ + Qje + w- 0jx - d{x,l) - 9^u\
- -sd(x, t) - As sgn (5(e))
<\s\\d{x,l}\-Assgn{s{e))
<{S-A)\s\<0,
which sufficiently ensures the boundedness of
\s(e)\ as well as the asymptotic decay to zero of
r r*
(18)
Controller (15)
3
U Plant
(8) X
In practice, the controller (14), however, cannot be used because of the unknown constant vector 9^ and unknown parameter 9g . To overcome this limitation, the certainty equivalence principle will be employed.
First, the unknown constants 9^ and 9^
in (14) are replaced by time functions 0.(1) and 0^(f), respectively, yielding
e +1^2^ + ii' - 9jx + Asgn(s) (15) with any chosen parameter X>S .
With this replacement, the derivative of the sliding surface (12) is now given by
s ^a.e + a^e + w-x.
= a^e + a^e + vi^ - T^J^a; + d(x, I) + 9^u\
= a,e + a2e + w -
-[0Jx+d(x,t) + (0^-0^,)u + 9^u']
= (9j-dfJ x + {6^-9^)u-d{x,l)-Xsgn(s)
= djx + d^u- d(x, t)-X sgn(_s) where
S^ =9j -9f and 5^=9^ -0^ . It can be noted further that
S^^S^ and S^=0^. (16) Second, by using an adaptive CLF
candidate
j? = -s^ +-S':p-'S, + — Sl (17) 2 2 ^ - ^ 2 ^ ^ ' where F G R ^ " ^ is any symmetric poshive
definite matrix and ^ is an arbitrary positive constant.
By using the derivative of the sliding surface i and the fact (16), one subsequently obtains directly from (17)
Figure 2. Configuration of the closed loop system
TAP CHi KHOA HQC & CONG NGHf CAC T R U O N G DAI HQC K^ T H l l ^ T * s 6 91-2012
= sl S'^x + d^u - d{x,t)- Asgn{s)^
+s',F-^e, +-&A
I / ^ . . - -sd{xj) -5,^sgn(iO + S'f i.sx + P''0, j +
+ (5, S H + - ^ , .
Now, by using the following adaptive adjustments for the lime functions 9^(1) and
0 (t) of controller (15)
(18) 9j ^ -¥sic)x
0^=-is{e)u
the derivative V becomes negative definite V - -sd{x,t)-sXsgn{s) < \s\\d(x,l)\ -A\s\
<{S-A)\s\<0
which is sufficient for ensuring that |i'(e)|<Qo and j(e)—>0
Figure 2, shows the main configuration of the closed loop system, in which the designed controller, including sliding mode controller (15) and adaptive parameters laws (18), always drives tlie output J ' = . V T = ^ asymptotically convergent to any three times differentiable desired trajectory w{t).
-.. N U M E R I C A L E X A M P L E
Consider a gearing transmission system with third order states model (8), where d{x,t) is a white noise with | | c / | [ ^ - 0 , l . Let design parameters be chosen as follows:
' I 0 0^
F - 0 2 0 , ^ - 0 . 1 , A - 0 . 9 , a , = a 3 = 1 . 8
^0 0 3 ,
The tracking error and the system output are shown in Figure 3 . , the vector 9y and 0^ from
the adaptive adjustors, are also given in Figure 4. and in Figure 7. -Figure 7. respectively.
Figure 3. Desired trajectory and system output
Figure 4. Adjusted parameter ' compared
•With '
--^v^
Figure 5 Adjusted
9,m
compared with9 121
Figured. Adjusted ^^ * compared with
e,m
TAP CHi KHOA HOC & C 6 N G NGH? CAC TRUING DAI HOC KY THUAT • SO 91 - 2012