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sliding mode control implementation for robust tracking of uncertain flexible structures
MECIMTRONICS PERGAMON
Mechatronics 8 (1998) 121 142
Time-optimal/sliding mode control implementation for robust tracking of uncertain flexible structures Nader Jalili, Nejat Olgac* Department o/Mechanical Engineering. University O/('onnecticut. Storrs. ('T06269-3139. US.A. Received 28 February 1997; revised 6 October 1997; accepted 13 October 1997
Abstract
An improvement strategy in robust control is proposed for uncertain linear or nonlinear systems. This two-stage control scheme first modifies the original desired trajectory, and then imposes robustness against uncertainties in tracking the new trajectory. For the trajectory modification stage, two different schemes are considered: Time Optimal-Rigid Body Motion (TO) and Shaped Input (SI). The robustness stage is performed using Sliding Mode Control with Perturbation Estimation (SMCPE), an advanced form of SMC. This routine brings some strong features as demonstrated by examples. A rotating hub with flexible beam attachment is taken as the first example, and an undercontrolled two-mass system with a linear spring as the second. The comparative studies show superior results for the combination of TO-SMCPE over SI-SMCPE. Moreover, this two-stage control exhibits stable and highly advantageous performance even for cases where H~ type robust control structure is declared unstable. i~,: 1998 Elsevier Science Ltd. All rights reserved.
Nomenclature
Ai A
6, fl B AB
C,.k
t a p p e d - d e l a y filter weighting u n i f o r m b e a m cross-sectional area i-th c o l u m n o f original m a t r i x B, ~ " rigid b o d y m o d e l c o n t r o l gain matrix, ~ ..... original system c o n t r o l gain matrix, ~ . . . . p e r t u r b a t i o n on B, ~ . . . . sliding h y p e r p l a n e coefficients
* Corresponding author. E-mail: olgac~a eng2.uconn.edu. 0957-4158.,.'98 $19.00 (": 1998 Elsevier Science Ltd. All rights reserved. PII:S0957 4158(97)00049 4
122
d
E f
Af H 1 11, l, k; k L DI I
Itl,
m,, M, P
p*(t) ro sI I
It
AT T(t) U
x,,(t) x,~,,(t) ~,,,,,(t) x*(t) x~t) x,,,(t) xza,,,(t) ;. [tJq~ U)
P ~csL T (1
O,t( t )
. \ J a l i l i . . \ . OIqac ."~'l('('hatruntc.s ,'~' .' 199A' ; 121 142
disturbance vector. 9¢'" trajectory tracking e r r o r beam Young's modulus o f elasticity rigid body model nonlinear driving term. '.R' original system nonlinear driving term vector. '.R'" perturbation on f. W" Hamiltonian beam cross-sectional moment of inertia hub mass moment of inertia hub-beam total mass moment of inertia, eqn (A9) positive constant (defined in s-dynamic) spring stiffness (in two-mass example) beam length mass of the driver body mass of the driven body upper bound lk)r i-th control actuator lower bound tk)r i-th control actuator positive constant (detined in s-dynamic) Lagrange multiplier vector. :R" hub radius /-th sliding hyperplane time optimal settling time of rigidized model time delay in tapped-delay filter control torque acting on hub control gain vector. '.R"' longitudinal deformation of the beam transverse deformation of the beam unit-step function target trajectory vector ',.v,,At)l. / = I . . . . . n time-optimal modified desired trajectory vector '¢x,,~,,,(t)]. j = 1. . . . . , input-shaping modified desired trajectory vector li,,~,,,(t)~,, j = I . . . . . n rigidized time optimal trajectory vector virtual rigid body time optimal trajectory vector../= I . . . . . n target trajectory for two-mass example modified trajectory for two-mass example scaling parameter for rigid body model natural frequency of the i-th mode of the beam dominant natural frequency of the two-mass system beam mass per unit volume perturbation vector. 9~"' perturbation estimation vector. '.R"' sampling period positive real constant target trajectory for hub-beam example
N. Jalili. N. Olyac/ Mechatronics 8 ( 1998s 121-142
123
1. Introduction
The primary motivation for this study is to present an improvement strategy for robust nonlinear control. For systems which are hard to model numerous control schemes have been developed especially in the last few decades [1-4]. A common difficulty appears in almost all of these strategies, which is the trade-off between the robustness and the trajectory tracking accuracy. That is, for a given hardware setting if the control algorithm were to be robust against sizable uncertainties it would inevitably be inferior in tracking accuracy. The point of approach in this study is to soften this trade-off by combining the compromising steps both from the robustness requirements as well as the definition of the desired trajectory. So the new control strategy brings a mixture of an optimal trajectory description and a robust control law driving the system to track this trajectory. In summary, the objective of this study is to introduce a robust nonlinear control method for accurate and fast trajectory tracking. These requirements, robustness and tracking, constitute a conflict and invite a compromise. For instance, fast trajectory tracking will not be desirably accurate especially when there is substantial uncertainty in the plant. A trade-off among these objectives arises. The nature of the trade-off is the basis of the study presented here. We also consider the robustness in this context against modeling and parametric uncertainties in the plant. The problem is addressed in two stages. The first stage is for trajectory modification. A compromise on the original desired trajectory is negotiated considering the control constraints. The resultant modified trajectory is then used in the second stage: robust tracking against plant uncertainties. A simple depiction of this procedure is shown in Fig. ! where xu(t) = { x j d ( t ) } , j = 1, 2 . . . . . n is the general position vector. Without loss of generality xa(t) is taken as a point-to-point motion, throughout this study. For the trajectory modification stage we follow two different perspectives. (a) In the first, we consider the actuator limitations (such as saturation and bandwidth) and renegotiate the target trajectory (this name is used for the "original desired trajectory", xd). This process is handled assuming no flexibilities in the system, i.e., rigid body dynamics. We try to follow the target trajectory, x~(t), within the minimum time possible, given the actuator constraints. That is, a time optimal trajectory control problem (TO) is solved under the given actuator limitations. This effort results in the modified trajectory, xu,, (Fig. 1). (b) In the second, residual oscillation of the controlled motion is taken into account.
Originaldeared
uajec~ ~t)
Trajectorynxxtification ~ a) T ~ r i g i d body mode t r a ~ L ...........................
............... Rolx~ control ]
O1"
Stage I ~,,,(t) Stage 2 Fig. 1. Proposed two-stagecontrol scheme.
x,,~(0
124
. \ .lalih, .\
(~,htm ,lh'chatr,mc~ p,
,'¢ .' 199,~.~ 121 142
,41
;
.-~.A2 e- AIs
~.
t, A~ e 2a/,,
.~:~
p, A,,~ e - (,n - 1)~7~ .
Fig. 2. "l'appcd-delay lihcr scheme [.r.;,a.~land {.~.,:(.~)arc the [.aplacc transform of the target and modilicd ira cctories, rcspectively~.
The input shaping strategy of Singcr and Seering [5] is followed with no actuator limitations. In this strategy thc tlexibilities in the system arc taken at their nominally known values, and tapped-delay lilter structure is implemented in order to suppress the residual oscillations of the structure at the end o f the control task (scc Fig. 2). This control crcatcs the modificd trajectory of Fig. 1. ~,l,,,(t). In the second stage of the control the modified trajectory is taken as the basis for a robust control activity against uncertainties in the system model. An advanced version of Variable Structure Systcm (VSS) control, Sliding M o d e Control with Perturbation Estimation ( S M C P E ) is used for this purpose [3]. This strategy has an interesting feature. It requires only the upper bounds o f estimation errors on system uncertainties not the bounds o f the uncertainties themselves. With this inlormation the routine can follow the desired trajectory despite the modeling uncertainties. Depending on the composition o f the control law (Fig. I), some favorable fcaturcs appear such as compliance with the actuator limitations, suppressing residual oscillations at the end o f motion, and effectively competing against the plant uncertainties. The c o m p o n e n t s o f the new routine are explained in the following sections, along with some rcpresentative example cases demonstrating the advantages.
2. S t a g e I: trajectory m o d i f i c a t i o n
This section outlines the effect of altering the desired trajectory on the system performance. As explained in Section I, two paths are taken for this purpose. The2,' are presented next. 2. I.
l'ime-optimal
comrol.lor
r i g i d i z e d sy.~lem
The objective here is to find a constrained control strategy in order to guide the system as close to target trajectory, %, as possible in the shortest time. A further
N. Jalili, N. Olyac/Mechatronics 8 {1998) 121-142
125
simplification in the system representation is the elimination of flexibilities, i.e., converting the plant into a rigid body system. This is done to obtain a closed form solution for the control as will be detailed below. Consider the "rigidized" system representation ignoring the uncertainties as ~(t) -: ['(x(t), t) + [~(x(t), t)u(t)
(1)
where x = [ x j , x z . . . . . . r,]r is the state vector. It should be noted that, this transition to rigidized mode reduces the number of independent coordinates for the original system. Actuator constraints are taken as M , _ <~ u,(t) <~ M,+
i=
1. . . . . m
(2)
The objective here is to find the optimal control in order to bring the state from xi(0) to Xid(tr) in minimum time t/subject to the constraints in eqn (2). Utilizing the Pontryagin Minimum Principle, the optimal control is stated as [6]
u,*(t) =
/
M,.,
for
p*r(t)6,(t) < 0
M,
for
p*r(t)6~(t) > 0
undetermined
for
p*7(t)6i(t) = 0
(3)
Accompanying co-state equations are given as p*(t) -
~
~ (x*(tl,u'lt).p'lt)J)
i = 1 , 2 . . . . ,D1
p*. l(t) = 0 where p*(t), i = 1 . . . . . defined
(4) m is the i-th Lagrange multipliers, and the Hamiltonian H is
H(x(t), u(t), p(t), t) = I + pr(t)[~(x(t), t) + B(x(t), t)u(t)]
(5)
We consider that the solution (3) always yields determinate cases and at the instants of p*r6i = 0 we choose u*(t) = 0, momentarily. The control in the form of eqn (3) is known as b a n g - b a n g control. The system response to the control is determined by solving the differential equation ~*(t) = ~'(x*(t), t) + I~(x*(t), t)u*(t)
(6)
x*(t) forms the modified trajectory Xd,,, = [Xia,,},j = 1, 2 . . . . . r. Further improvements can be done on this trajectory. For instance, a scaling factor )' can be used on some system parameters involved, such as inertia elements. We start with 7 = I yielding x*(t) = x,~m(t) which has an optimal settling time t~. For 7 > 1, in the case of large inertia, the motion becomes smoother, yielding tr~. > tj, and 7 < 1 causes t n < t~, ultimately, 7 = 0 (no inertia) giving x;*(t) = Xd(t) which is the target trajectory. By maintaining ), > 1 the optimal trajectories are further smoothed. That is, the bigger the inertia the larger the settling time, consequently the smoother the modified trajectory Xdm. This dictates that, 7 > 1 gives a plausible parametric study
126
.V..lalili. .\. OI.qac :llechatr~mlc.~
,'~' ' 199,~',
121 142
which offers smoother control actions without violating the control constraints. This aspect will be further clarified in the simulations section. 2.2. Shaped input control This method employs shaped control inputs to reduce or eliminate endpoint vibration. To achieve this, an impulse sequence is convolved with the c o m m a n d input producing an aperiodic response. The routine requires reasonably good information about natural frequencies o f the lincarized system [5.7]. As shown in Fig. 2, this technique simply utilizes a tapped-delay tilter with proper weightings A, and time delays AT. The convolution of the target trajectory .v,At) and sequence o f m impulses results in a new multiswitch, multilevel function •f,a,,,(/) = 5~ A,.v,,,(t---(i • l ) A T ) u , ( t - - ( i .- I )AT) ,-
(7)
I
where .4, is the magnitude of the i-th impulse tit t = ( i - - I ) A T and uAt) represents a unit-step function. A proper sequence o f impulses, whose power spectrum has a notch at a structural resonant frequency, can be found such that the residual vibration o f the response of an u n d a m p e d second-order system to a sequence o f m impulses will not occur after t = (m 1)A Twhere A T = ~ xo and ~,~is the d o m i n a n t natural frequency of the llexible behavior. This effort creates the second class o f modified trajectories, •'~-,a,,,(t). to be considered in the following robustizing control scheme. It should be noted that this derivation is aimed for linear systems alone. N o general statement can be made regarding the applications to nonlinear systems.
3. S t a g e 2: S M C P E ,
an o v e r v i e ~
The modified desired trajectories generated in Section 2 are taken as the control objectives here. However. the control insensitivity against modeling uncertainties is considered as the primary exercise. The nonlinear control routine S M C P E is adopted as the robust control methodology. We present, next, a brief s u m m a r y o f it, following Elmali and OIgac [3]. A general class o f nonlinear systems is considered x .... = f ( X , . X . . . . . .
X.,)--
Af(X,.
X ......
X.,) + [B(X,. X: .....
X,,,)
+AB(XI.X: ..... X,,)]u+d(n
(8)
where X: = [.v,..L . . . . . . vl" i,]~ 6,.R,. i - I . . . . . m is the state sub-vector and .v, i = I. . . . . m arc m independent coordinates. The intluence of the u n k n o w n terms, which tire assumed to be b o u n d e d by known functions, is gathered into a single quantity named perturbation vector • (t) = A f + A B u ( t ) + d ( t )
= x"" - B u ( t ) - f
(9)
N. Jalili. N. OI.qac.,Mechatronics 8 (1998) 121-142
127
The control objective is to track the modified desired trajectory xj~(t), j = 1. . . . . m in the presence of unknown perturbations, modeling uncertainties, and unmodeled dynamics. To this purpose, first, the sliding hyperplanes are selected as Hurwitz polynomials of the tracking errors of the associate sub-vector as nt -
1
sj = ~ %.k+ "" le,"
j=
1. . . . . . m
(10)
k~O
where Cj.,, = 1, and ej = xj-X,a,,. In order to enforce the conventional attractivity condition [8], the following s-dynamics is proposed .¢j= - P s j - k
isgn(s,)
j=
1. . . . . m
(11)
where sgn (~)) = s/l~)l. This proposition is in line with Lyapunov convergence concept. The guarantee of convergence in finite time despite the uncertainties is detailed in [3]. The control arising from eqn ( I I) is u = B
~[- Ps-k
sat (s) - f -
~ +"~"~Ad.,-- ~P~,]
(12)
where k=diag[k,
k2
...
k,,],
sat (s) = [sat (s ~)
sat(s2)
...
sat(sin)] r, n-
s=[s,
s2
...
s,,,] r,
~=[~,
~2
..-
~,,]r
and
I
~ , = ~ C,.kel k) (13) k
I
It is important to note that the sgn (sl) was smoothed to sat
(si)
~" .s)/le.,l IsA ~< c, / (sgn (si) Is, I > ~:,
(14)
in order to avoid control chatter, e.j is a time-varying boundary layer for s-dynamic. Also, in eqn (12) W~, is an estimate of perturbation W. Using eqn (9), this estimate can be computed, which requires calculation of x ~"~and control feedback. In practice, such an algorithm is implemented digitally and the sampling speed is selected high enough to ensure that u(t) _~ u ( t - t). Additionally, in the absence of measurements of x ~") (e.g. acceleration measurement for second order dynamics) an approximation is utilized as x~")(1) ~ [x (° ' ~ ( t ) - x ~" ' ~ ( t - r ) ] / t
(15)
These are reasonable assumptions considering the fact that the control sampling speed is at least an order of magnitude higher than both the hardware bandwidth and the frequency contents of the desired trajectory. Typical example in robot controls, 200 Hz sampling is very common, and the trajectories hardly go beyond 2-3 Hz. It is shown both numerically and experimentally [9] that, SMCPE can maintain a desired level of accuracy in trajectory tracking despite the uncertainties. The combination of the trajectory modification and robustizing control enables us to form a
N. ,lalih. N. O l y a c :14~,~hatronic~ ,~' ;
128
199A' )
121. 142
mixed treatment tbr speed and accuracy in nonlinear systems as examples below demonstrate.
4. Simulations and numerical examples To show the effectiveness of the proposed control strategy, two examples are presented. I-:irst. we consider a rotating hub-beam structure with uncertainties on flexible modes of the beam. In the second example, a simple two-mass system with one force actuator is considered For both examples a reference point-to-point motion is takcn as target trajcctory.
4.1. Rolatht.q hub-beam system This example includes a uniform llexible bcam attached to a rigid hub, moving in the horizontal plane, as shown in Fig. 3 . 0 , , is an inertial coordinate of referencc with its origin at the center of hub, and 0,, is a rotating frame of reference attached to the beam. The .v'-axis is the spatial coordinate along thc length of the beam and tangent to the beana center line at C. while v' is the transverse coordinate. A control torque, T(t). acting on the hub (normal to the plane of motion) is the only external forcing. Thc hub-beam equations of motion (sec Appendix A) are given as
(',,
[2O'+h'(O'-"'~')l'q' t
I
i +/,A('' ,
l=];,+,t,i,,T /~
(]' :: ('pl { ~_'j [2t~'(C' " + q'(O'- -- ")~'' )]q~' " /JAb('1, ] = ]I +,(I T
i = 1,2 . . . . . n
(16)
where variables )71 and .0, arc sell" explanatory and the remaining terms arc delined
y'
'~ Y
ttx
X'
x
FO Fig 3. I-he hub-beam s3stem contiguration.
N. Jalili. N. OlgaciMechatronics 8 (1998) 121 142
129
in Appendix A. Ultimately, the hub-beam system is represented by the following equation.
(17)
= ~(x, ~) + g(x, ~)u wherc
f=[.L
.T, .L
... L]".
~:to,,
fr,
fr_...,
fr,]T
(18)
which is a nonlinear second-order system where state x = [0 g~ g2 . . . g,]~, and u is the input torque. The control objective is to track a desired trajectory Oa(t) in the presence of unknown perturbations, modeling uncertainties, and unmodeled dynamics. In this work, the controller considers the flexibility around rigid body mode only. For numerical results, Oa(t) = n/2, t > 0, and zero initial conditions are taken. The hub-beam system properties are given in Appendix B. The nonlinear model truncated with n = 3, described in eqn (16), was used in the simulations.
4.1.1. S M C P E
control.
We reverse the order here and present the robustizing control SMCPE first as the common element for all modified trajectories. It is appended to the modified desired trajectory creation stages, see Fig. 1, which are described in Sections 4.1.2 and 4.1.3. The sliding-mode controller considers the beam flexibility around rigid body mode only. This does not impose a serious practical restriction, because the higher modal frequencies are usually distinctly remote from the operational frequency interval. Therefore they should not come into play. According to eqn (16), neglecting the beam flexibility (gj = .4j = 0 or U,. = u, = 0), the rigid-body model with the presence of perturbations can be written as O( t) = .)7olu, = 0, = ,, + A]~,(x, ~ ) + [ffolu, = ~, . o + Afro(x, ~)]u + d( t)
(19)
wheref~ and fro are only functions of O = [0 0], while A]~ and Afro contain all the remaining terms. On the other hand, Af0 and Afrorepresent the uncertainties on f) and fro, respectively, d(t) is the external disturbance on the system. s-dynamics presented in eqn (10) for n = 2, can be written as s = d+ae
(20)
where e(t) = O ( t ) - O a ( t ) . In practice, a is selected for maximum tracking accuracy taking into account unmodeled dynamics and actuator hardware limitations [10]. For the simulations no such restrictions are considered, i.e., ideal actuator, high sampling frequency, and perfect measurements are assumed. The control law follows eqn (12) as
130
.\..lalih. ),.. O(qacMechatrmut.~ ?~", 1998~ 121 142 u(t)
4.1.2.
= lh
[ -- k sat (s) -./;, I. . . . . . .
Time-optimal
SM('PE
- erg, + 0,~ -- ~ . , , ]
(21)
control.
The target trajectory is moditied first, using time optimal control on the rigid body model of the system 0 -- u.l,
(22)
The control objective is to bring 0 ( 0 ) = 0 to 0 ( t , ) = 0,~ = n..2 within the shortest possible time given the constraint of lul ~< I. The resulting time optimal response of this system can be obtained from eqn (6) as I
2/, t" O*(t) = O,t,,,(t) =
- I t, 21. t'-+ I , t + ( I ' 1 0,~
0 ~< t <~ t~,,2 t/ 2L
(23) t,..2<~t<~t~ t > l,
where t, = 2\. 1,(t,i. This system response is used as the objective trajectory in S M C P E , the robustizing phase of the control. In order to further smooth the modified trajectory we perform, next, an inertia scaling study. The rigid body model for the virtual system with scaled inertia is given by ~; = !' 7/,
(24)
where the system inertia is conceived as ,'1,. The corresponding settling time to 0,/is 1,. = 2\./1,0,1
(25)
Clearly. the optimal settling time, t,:, is m o n o t o n o u s l y varying with 7. Since it has such a direct influence. ,, may also be used as a fictitious p a r a m e t e r to improve the modified trajectory. The ideal trajectory is represented by ,' = 0. That is, it can only happen if there is no inertia. Realizable control is performed for this example with 7 = I, 10 and 100. The effects of these selections on the modified desired trajectories are shown in Fig. 4(a). The robustizing stage of the routine, S M C P E , is performed next, according to Section 4.1. I. The system behavior is given in Fig. 4(b) for various y values. Due to the s m o o t h e r modified trajectory ;' = 100 case exhibits the best trajectory overshoot, control effort (Fig. 4(c)), and undesired transverse beam oscillations (Fig. 4(d)) characteristics a m o n g the four. ]'he effects of the modified trajectory on the transverse beam oscillations is especially obvious. The additional parameterization such as 7 can form a valuable desired utility for the controller.
N. Jalili, N. OIgac/Mechatronics 8 (1998) 121 142
131
(a)
(b)
2.0
/ .........
1.6
'-~i ~ i: ~ ' -
a
I
-
12
]; / /
0.8
i
!i
;io
04
,"i / o,]/ i///
~y= I00
2
3
4
5
0
-y=l
,:loo
o o
OO
I
~ .
I
2
3
4
Tire~(~c)
Tm'~ (s~.)
(d)
(c)
1.2 . . . .
12 - - .
1
O8
0.8 7=0
•
~ 04. .~
"¢=I00 ~/
.\' //
i0,4
A
I I ~'
= M-°
.~0o
I t/l-,
i
_
J
"\y~10
t~
y=10
-0g •
-os t . i (
"i'=1
-12 ]~-. -12 4 ~ - 0
"y=l
.. . . . .
I
2 Time ( ~ )
3
4
0
I
2
3
4
--4 $
Ttr~ ( ~ . )
Fig. 4. TO-SMCPE for hub-beam system for y = 0 (thin solid), 7 = I (dotted), y = 10 (dashed), and ), = 100 (thick solid).
4.1.3. Shaped-input S M C P E control. As stated before, this m e t h o d c o n s i d e r s the linearized m o d e l o f the system, such as: = A~+Bu
y = C~
(26)
w h e r e ~ = [0 .qt .q2 . . . 0 9~ .q2 . . . ] . A i s t h e J a c o b i a n m a t r i x o f i ' w i t h r e s p e c t to ~ at ~ = 0, a n d u = 0, a n d B is the J a c o b i a n m a t r i x o f ~ with respect to u at the s a m e p o i n t . F o r n = 3, the rigid b o d y m o d e a n d three c o n s e c u t i v e m o d e s are t a k e n i n t o a c c o u n t . T h e n , the s y s t e m r e d u c e s to a n e i g h t h - o r d e r l i n e a r m o d e l .
N. , l a h l i . . \ . ()lgav Mcchatronic,~ A' i 1993; 121 142
132
Since the governing equation of motion is truncated to three modes of the flexible beam, a sequencc of impulses corresponding to these three modes are used in input shaping. These three impulsc sequcnces can bc convolved to form a single sequence to reduce vibration in thrce separate modes [5]. Ensuing modified desired trajectory and the system response to SMCPE control tire shown in Fig. 5. It is important to note that in the time history' of the control torque, Fig. 5(c) {dotted curve) there are some.jitters as a consequence of the shaped-input controller. These discontinuities are the filtered effects or the discontinuities of the modified
(h)
(a)
20
20,
16 I
i ~"
}
12
0
8
0
9 0 *~/
~' 04
O0 0
.~
2
I
.
II
4
.
.
.
.
1
.
.
.
.
.
3
~
-1
5
I .no (~',~)
hnw: (we)
(d)
(~)
I.
(J 04
I :,
,02
o..
0 02
,
A
-~
;__.
•
i£
~ .
il O0
I
-O02
I
-0 02
4) 04
[i U4
0
I
2
~
I
5
rlrn~ f~¢¢ i
Fig. 5. Comparison between T O - S M ( ' P E bean1
s~stem
.
.
.
.
1
.
2
.
. I
. 4
I'm~ (~ec)
l(~l ;, -
101i I l h i c k solid), a n d S I - S M C P E
(thin solid) for lhe h u b -
N. Jalili, N. OIgac/Mechatronics 8 (1998) 121 142 X1
133
X2
I U
/ Fig. 6. Two-mass system.
desired trajectory (Fig. 5(a)/dotted curve) through S M C P E routine. They do not appear in the T O - S M C P E since the time-optimal pre-filter and its first derivative are both continuous functions. This is a very strong advantage of the T O - S M C P E combination. 4.2. T w o - m a s s s y s t e m
A simple linear system of Fig. 6 is considered as the second example. We present this to compare the performance of T O - S M C P E and SI-SMCPE algorithms in particular from the robustness point of view. The system uncertainty is taken only in the stiffness of the spring connecting the two masses. Both bodies have unit mass. The only control u acts on body 1 and the position of body 2 is the output to be controlled. The equations of motion are ml,~l + k ( x l - - x 2 )
= u
m2.f2 -- k ( x l - x D
= 0
(27)
where x~ and x2 are the positions of the body 1 and 2, respectively. The control design objective here is to achieve a fast settling time for a step-type desired trajectory (x2j(O) = 0, x2u(tj) = 1) and robust performance against the stiffness uncertainty. 4.2.1. S M C P E
control.
With proper mathematical manipulations, eqn (27) can be transformed into x~,.~ =
ml__-k-m2k,q, + - -k mlm2
u
(28)
mlm2
s-dynamics presented in eqn (10) for n = 4, can be written as s(t) = "~2(t) + ClO'2(t) + C2O:(t) + C~e2(t)
(29)
where e2(t) = x2(t) - x2j(t), and Ct, (72, Cs are all constant and selected in such a way that s forms a Hurwitz polynomial. Since we have a single input, the subscript j in eqn (10) from s-dynamics is dropped. By selecting ,~ as defined in eqn (I i) the control law is written as
134
:\'..lalili..%".
u(t)
= I 2
I
u(t--~)+
tl! I DI 2
k
OlgacMechutr.nic~ ,~' ~ 199~' J 121 142
[
P.s-.ksat(s:)
+ ("2-#2,/+ ("~-~_'J]+ m~
Ill 2
~-
171 I
-- ('~.~, + 2,,,~ + ('~Tx'-,/ -
-- ("2
"
-
(-V I -- -g2 ) -]" ( '1 (-'('2 -- .i- I )
(3(I)
In this expression the targel trajectory is considered, therefore x'~i/ = ~2,; = .#2,; = 5::,/= 0 and .x2,~ = I. 4.2.2.
Time-optimal SM('I'E
control.
The two-stage control T O - S M C P E is employed f'or this system. First, the modified desired trajectory is created for optimal time with the constraint [u[ ~< I. The rigidized model of the nominal system equation of motion is (31)
.~, = .~'~ = u (m~ -4-m:)
The time optimal response of this system can be determined similar to the hubbeam system, i.e.. eqn (23) by simply substituting 1~ by (m~ + m 2 ) . 0,M) by x2,~(t), and #,~,,,(t) by x2,~,,,(t). The time optimal trajectory is then used in the second stage as the desired trajectory for SMCPE. Equation (30) is used for the control law with x_,,~,,,. .{it -~2,~,,,, .~-:,a,,,substituted from eqn (23) and 22a,,, = -\2,~,,, = O. Figure 7 shows the effectiveness of p a r a m e t e r 7 for S M C P E method in two different categories, one when there is an input saturation and another when the input is unconstrained. As clearly seen, different scaling t:actors of ,', similar to the first example, yield responses which arc superior to those in the case of ,' = 0, i.e.. target trajectory being used directly in S M C P E . 4.2.3.
Shaped-input SM('PE
control.
Fhe natural frequency of the flexible mode of the linear system in Fig. 6 is /k(mt +m~)
~t) = Xj
....
Ill t D12
(32)
For this example, four impulse sequences are convolved to form a modified desired trajectory. Ensuing tr~uectory and the nominal system response to S M C P E control are shown in Fig. 8. We restate that. the preshaping c o m m a n d input requires good infi)rmation about natural frequencies of the system. It is needed to properly suppress the residual vibration of the system. In this example, the system frequency is dependent on the spring stiffness which is considered uncertain. The effect of this uncertainty on the performance of the control is studied next. 4.2.4.
P e r t u r b a t i o n on sprinq st([]hess.
To show the robustness feature of the two-stage routines against modeling uncertainties, we consider a range of spring stiffness uncertainty around the nominal k = 1. When similar dynamics was treated by Wie and Liu [11], the stable zone of
N. Jalili, N. OIgaciMechatronics 8 (1998) 121-142
135
(a)
1.2
/y=0
._.~/ ......
ID
~" 0.8 "1
/ " / "1'= I
/ "y=lO
I 0.6 "I~ 04 '
"
02J "
0.0 0
4
8
16
12
20
"l'in~(t~¢.)
:b)
1.2
/ ¢~____ y
(c)
=
~
1.2
/~--y=l ,0
/r:lo
-/
-5 , : ~ - - - - - - ~
°'/i,J2 /L °61 / .:o
!
i
~ 04
.i/.....-7-~
IO
o,
7~=o ; /=
°'i ! /' 02 ~
0.2 ~ 0.0 g
12
Tm~(sec.)
16
20
,
"
O0 0
4
12
S Time
16
20
(~.)
Fig. 7. TO-SMCPE for two-mass system for 7 = 0 (thin solid), 7 = 1 ( d o t t e d ) , ~, = 10 (dashed), and ), = 2 0 (thick solid). (b) and (d) when the control input is constrained and P = 0, (c) a n d (e) when the control is unconstrained and P = 10.
0.44 < k < 3.27 was found for H~ controller. Here we consider the perturbation on k both inside and outside this interval. For a better comparison, TO-SMCPE corresponding to ": = 20 is taken, which performs the best a m o n g others. Figure 9 shows the system response for three different values of k with our T O - S M C P E and SI-SMCPE routines and H~ robust algorithm as employed by Wie and Liu [11]. For brevity we only give the respective results of the latter without any discussion o f the routine. As claimed in the referenced paper, the H~ controller is not stable outside the region 0.44 < k < 3.27. On the other hand, both TO-SMCPE and SI-SMCPE exhibit stable behavior for wider range of spring stiffness uncertainty. This is assured by the
.V. Jalili, ..%:.O l g a c Mechatroni('.~ <~ r 199,~'; 121 142
136
(d)
Ca) 12: .
.
.
.
.
I
06
:
E
.
10
r=l
il o.o ':
.
,I
lU
i:.' .
g=O
,
o,
t
E Z
.L
"/=20
.~
oo
y=20
i
LJ
,, -12
el(i -05
. . . . . . . . . . . . . . . . 4
i 0
2(1
16
12
g
4
2
6
g
Io
Time (see)
Time (sec)
(e) 9
• 300 "~-
y-I 15
.v=-0
.[ i 200
y=20
3
~
0
I
E
E
C\i , / ~x : !
100
.',=20
! -
o
.i i ~
y=lO
70
(10
-I(lO
0
12
4
I{l
20
I
2
3
4
"rin~ (sec.)
Time (~e~)
I:ig 7. c:mtim.,d.
basic feature of SMCPE: the routine is not sensitive to the system uncertainties but the error in their estimates. A point to note is that the settling time for both TO-SMCPE and SI-SMCPE are approximately the same. However, in the time history o f control, it is shown that there are sudden changes in magnitude at each instant of application of the impulse in the shaped input controller and at switching times of the TO-SMCPE. For the SISMCPE, they originate from the discontinuities in the pre-shaped desired trajectory. The discontinuities in TO-SMCPE appear because of the s-dynamic defined in eqn (29). In this special case, sliding function uses the second derivative of the timeoptimized trajectory. According to eqn (23), this derivative has two discontinuities at
N. Jalili, N. 01,qaciMechatronics 8 (1998) 121 142
137
(a) Modified desired trajectory, X2,t="
10 08 08 04
02
4
8
12
16
20
Time (see.)
(19) Output of SI-SMCPE, x2.
10
f
08
/
06 04
02
/
/
./
f
J /
//
/
4
8
12
16
20
16
20
lime (see.)
(c) Control input,
0
4
8
12
it.
Fig. 8. Shaped-input S M C P E for t w o - m a s s system for P = 0 (solid) and P = 10 (dotted).
138
•\.
Jalih,
.V.
(Hqa~. Mechatronw.~ ,~ ! 1 9 9 8 : 1 2 1
142
(a)
12 08 ]
•
0 8
/ /
(]4 1
i'
=o
I
00 02 4) 4 4
g 12 rime (.,.~: )
I(,
4
2o
8 12 "l'im¢(s~:)
16
20
8
16
20
(hi 12
................. 08
" 10
.
08 O6
~
~//
i
o 4
O0 ~
.
_
_
'~ o2 O0
,
0
4
•
8 lime
,
12 (sex)
. . . . 16
-04 . 21)
o
4
12
Titt~ (s,e~)
Fig. 9. ( ' o m p a r i s o n b e t w e e n I O - S M C P I . I (thick solid), SI-SMCPE (thin solid), and I1, (dotted). (a) nominal:/,- - 1. (b) perturbed inside the stable region: k - 0.7, and perturbed outside (2 cases), (c) k = 0.4 and ( d ) / , = 3.5.
switching times at t,..2 and t,. Further improvements on the modilied trajectory can be made by smoothing the second derivative at these instants.
5. Conclusions This study demonstrates the feasibility of using rigidized system model and limeoptimal response as a way to alter the target trajectory. This method provides a set of modified and smoother desired trajectories with longer settling times. This compromised trajectory can also be obtained by introducing a virtual model for the rigidized system (for instance, by using an inertia scaling, 7).
139
N. Jalili, N. Olqac/Mechatronic's 8 (1998) 121 142 (c) 1.0 m
r
~. 2.5] 03
i
5I i 0.0 0.3
-05
-05 .~--
0
,
8
16 T ~
24
40
32
-I.0 • ---~ 0
24
16
8
32
40
(~.) T~(~.)
12
10 ]
t
i
'0,1
~ 06 I~
I I
04
02
,- ...... / I 40
0.0
8
16 Time
32
i 00
-0.4
,"
~--
0
(~,)
,
,
g
,
,
.~
--~
16 Tin~:
. . . . .
24
~ - - - - - ~
32
40
(~c.)
Fig. 9. continued
Two different examples have been studied and two separate two-stage controllers were applied. The TO-SMCPE combination presents superior results when compared with other robust controllers such as SI-SMCPE and H~. The first example, hubbeam setting, demonstrates the feasibility of the two-stage control routine for a general class of dynamic systems with non-linear governing equations. The second example shows that, while the H~ controller is not stable for some values of the uncertainties two versions of SMCPE exhibit stable behavior. For both examples it is also shown that, SI-SMCPE provides the same maneuverability and robustness characteristics compared with TO-SMCPE. However, it is limited to the linear systems and it requires good information about system natural frequencies, while the TOSMCPE does not. Also, the time history of control input in SI-SMCPE shows some
140
.~ll. .lalili.
N. O l # a c ,~,l(,chalrontc.~ ,~ , 1998 j 121 142
jumps at each instant of application of the shaping impulses, while the one in TOSMCPE has bccn significantly reduced due to continuity of thc modified desired trajectory. These are very strong advantages of the rigid mode time optimal,SMCPE combination.
Appendix A: Equations of motion for rotating hub-beam system A modeling effort following Chang and Jayasuriya [12]. is presented hcrc for the hub-bcam shown in Fig. 3. The transverse and longitudinal deformations of a beam element arc denoted by u, (.v. t) and u,(.v. 1), respectivcly, where .v is the distance to the beam element measured from point C in Fig. 3 and t is time. The hub angular rotation, i.c.. 0{t) is measured counterclockwise from thc incrtial x-axis. The Euler Bernoulli assumptions are used. The kinetic and potential energies of this system are stated first in order to apply Lagrange's formulation. 11),0-'+ p4 KE = 2 2 " .,
(
\ ~'t ~
+u~O-"--2¢)t,, ,'u,
;~1
+(r,,+.\+u,/-'0-'+
t ' E = 2EA
,) \ ~ . v /
d.v+ 2 E l
\ {! / + 2 0 ( r . - t - . x 4 - u , )
,,?v' " d . \ L[J
.
('~l jd.v
(AI)
(A2)
/
It is assumcd that u, and u, can be approximated as the finite sums u,(.v, t) = ~'L(t)¢p,(.\) :
(A3)
I
u, (.v, t) -- ~ .q,(t)~0,(.v) i.
(A4)
I
where q~,(.v) and tki(.v) are i-th longitudinal and j-th transverse modal shapes (eigenfunctions) of a clamped-tYee and fixed-free beam. respectively./i(t) and g~(t) are the corresponding generalized coordinates of the beam. In the case of a beam slewing through a finite angle, such as in robot motion, the longitudinal detbrmation and its coupling to the transverse modes and rigid body may be neglected, i.e.. JM) = 0 is assumed. The work done by the external torque T on the hub is given by W,,~.= TO. Using the orthogonality relationship between the eigenfunctions and further using the Lagrange's equations of" motion, the following coupled equations are obtained:
.4,+(~,~,.-O'-)y,= - b , 0
i = 1.2 . . . . . n
(A5)
N. Jalili, N. Olgac/Mechatronics 8 (1998) 121..142
141
[ (lt/pA) +j~, g~ ] O+ ( 2 j ~ gl.0j) 0 + j ~ b,ff) = T/pA
(A6)
=
where
9,
= E , r( pA Jo
j = i,2 ..... .
bj = I~ (ro +x)~j(x)dx
(A7)
j = 1,2 . . . . . n
(A8)
It = & + pA {[(ro + L) 3-r]]/3}
(A9)
The hub-beam dynamics of eqns (A3)--(A9), can be transformed into a more suitable form for controller synthesis
1
O=
~=, [20j+b,(O -09;)]9, + ~
T=]~,+goT
pA Cp T=.f+9~T
g, = ,-P U=, [2b,g) + qj(O" - to.q2,)]gj
i= 1,2 . . . . . n
(AI0)
w h e r e f and .~7~are self explanatory terms and
Cp = It/pA + ~ (q~ - b f ) j=l =
tl'
~Cp+b~ ( b~bi
for./=/
(All)
f o r j ¢: i
Appendix B: Modal parameters of the flexible hub-beam system
Properties
Symbol
Value
Unit
Aluminum density Aluminum Young's modulus Beam width Beam height Beam length Beam cross-section area Beam moment of inertia Beam mass moment of inertia Hub mass moment of inertia Hub radius
P E b h L A I
2710 71 x 109 7 x 10 -~ 2 x 10 -2 0.575 i.56 x 10 -5 8.533 x 10 - t 3 0.1988 x 10 ..2 0.2087 x 10- 2 0.065
Kg/m -~ N/m 2 m m m m2 m4 Kg m ~ Kg m: m
I,, ro
142
N. Jalili. N. Olqar..lleN~atron,',~ ~ ~ 1998J 121 142
References [1] Slotine JJ. Sliding controller design for ntmlincar ~,ystem. International Journal of ('ontrol 1984:40:421 34. [21 Utkin VI. Yang KI). Methods for constructing discontinuity planes in muhidimensional variable structure systems. Translated from Avlomatika 1 Telemekhanika 1978:10:72 7. [3] Elmali It. Olgac N. Sliding Mode Control with Perturbation Estimation (SMCPE): a new approach. International Journal of Control 1992:56( I ):923 4 I. 14} Moura J1". Roy RG, Olgac N. Optimum trajectory prc-shaping llor sliding mode control. Japan U.S.A. Symposturn on Flexible Automation. Boston, MA. July 1996. [5] Singer N('. Seering WP. Preshaping command inputs to reduce system vibration. Transactions of the ASM E. Journal of Dyn.. Sys., Measur. and Control. 1990:112:76- 82. [61 Kirk DE. ()pt#~tal Control Th~'orr. Prentice-Hall Inc.. 1970. [7] Singh T. Golnaraghi MI:, Dubcy RN. Sliding-modeshaped-input control of flexible/rigid link robots. Journal of Sound and Vibration 1994171 (21:185 200. [8] Slotine J J, Li W. Applied Nonlinear Control. Prentice-Hall Inc., 1991. [9] Roy RG. Moura JT, Olgac N. }:requcncy shaped sliding modes--theory and experiments. IEEE Transactions on ('ontrol Systems Technology. July 1997. pp. 394--401. [10] Moura J']'. Roy RG. Olgac N. Sliding mode control with perturbation estimation and frequency shaped sliding sur|~.ces. Transactions of the ASME. Journal of Dyn.. Sys., Measur. and Control 1997:119(31:584 8. [11] Wie B, Liu Q. Comparison between robusti|icd feedforward and feedback for achieving parameter robustness. Journal of Guidance, Control, and Dynamics 1992:15(4):935 43. [ 12] Chang PM, Jayasuriya S. An evaluation of several controller synthesis methodologies using a rotating llexible beana as a test bed. Transactions o[" the ASME. Journal of Dyn., Sys.. Measur. and Control 1995:117:36(I 73.
Mechatronics 8 (1998) 121 142
Time-optimal/sliding mode control implementation for robust tracking of uncertain flexible structures Nader Jalili, Nejat Olgac* Department o/Mechanical Engineering. University O/('onnecticut. Storrs. ('T06269-3139. US.A. Received 28 February 1997; revised 6 October 1997; accepted 13 October 1997
Abstract
An improvement strategy in robust control is proposed for uncertain linear or nonlinear systems. This two-stage control scheme first modifies the original desired trajectory, and then imposes robustness against uncertainties in tracking the new trajectory. For the trajectory modification stage, two different schemes are considered: Time Optimal-Rigid Body Motion (TO) and Shaped Input (SI). The robustness stage is performed using Sliding Mode Control with Perturbation Estimation (SMCPE), an advanced form of SMC. This routine brings some strong features as demonstrated by examples. A rotating hub with flexible beam attachment is taken as the first example, and an undercontrolled two-mass system with a linear spring as the second. The comparative studies show superior results for the combination of TO-SMCPE over SI-SMCPE. Moreover, this two-stage control exhibits stable and highly advantageous performance even for cases where H~ type robust control structure is declared unstable. i~,: 1998 Elsevier Science Ltd. All rights reserved.
Nomenclature
Ai A
6, fl B AB
C,.k
t a p p e d - d e l a y filter weighting u n i f o r m b e a m cross-sectional area i-th c o l u m n o f original m a t r i x B, ~ " rigid b o d y m o d e l c o n t r o l gain matrix, ~ ..... original system c o n t r o l gain matrix, ~ . . . . p e r t u r b a t i o n on B, ~ . . . . sliding h y p e r p l a n e coefficients
* Corresponding author. E-mail: olgac~a eng2.uconn.edu. 0957-4158.,.'98 $19.00 (": 1998 Elsevier Science Ltd. All rights reserved. PII:S0957 4158(97)00049 4
122
d
E f
Af H 1 11, l, k; k L DI I
Itl,
m,, M, P
p*(t) ro sI I
It
AT T(t) U
x,,(t) x,~,,(t) ~,,,,,(t) x*(t) x~t) x,,,(t) xza,,,(t) ;. [tJq~ U)
P ~csL T (1
O,t( t )
. \ J a l i l i . . \ . OIqac ."~'l('('hatruntc.s ,'~' .' 199A' ; 121 142
disturbance vector. 9¢'" trajectory tracking e r r o r beam Young's modulus o f elasticity rigid body model nonlinear driving term. '.R' original system nonlinear driving term vector. '.R'" perturbation on f. W" Hamiltonian beam cross-sectional moment of inertia hub mass moment of inertia hub-beam total mass moment of inertia, eqn (A9) positive constant (defined in s-dynamic) spring stiffness (in two-mass example) beam length mass of the driver body mass of the driven body upper bound lk)r i-th control actuator lower bound tk)r i-th control actuator positive constant (detined in s-dynamic) Lagrange multiplier vector. :R" hub radius /-th sliding hyperplane time optimal settling time of rigidized model time delay in tapped-delay filter control torque acting on hub control gain vector. '.R"' longitudinal deformation of the beam transverse deformation of the beam unit-step function target trajectory vector ',.v,,At)l. / = I . . . . . n time-optimal modified desired trajectory vector '¢x,,~,,,(t)]. j = 1. . . . . , input-shaping modified desired trajectory vector li,,~,,,(t)~,, j = I . . . . . n rigidized time optimal trajectory vector virtual rigid body time optimal trajectory vector../= I . . . . . n target trajectory for two-mass example modified trajectory for two-mass example scaling parameter for rigid body model natural frequency of the i-th mode of the beam dominant natural frequency of the two-mass system beam mass per unit volume perturbation vector. 9~"' perturbation estimation vector. '.R"' sampling period positive real constant target trajectory for hub-beam example
N. Jalili. N. Olyac/ Mechatronics 8 ( 1998s 121-142
123
1. Introduction
The primary motivation for this study is to present an improvement strategy for robust nonlinear control. For systems which are hard to model numerous control schemes have been developed especially in the last few decades [1-4]. A common difficulty appears in almost all of these strategies, which is the trade-off between the robustness and the trajectory tracking accuracy. That is, for a given hardware setting if the control algorithm were to be robust against sizable uncertainties it would inevitably be inferior in tracking accuracy. The point of approach in this study is to soften this trade-off by combining the compromising steps both from the robustness requirements as well as the definition of the desired trajectory. So the new control strategy brings a mixture of an optimal trajectory description and a robust control law driving the system to track this trajectory. In summary, the objective of this study is to introduce a robust nonlinear control method for accurate and fast trajectory tracking. These requirements, robustness and tracking, constitute a conflict and invite a compromise. For instance, fast trajectory tracking will not be desirably accurate especially when there is substantial uncertainty in the plant. A trade-off among these objectives arises. The nature of the trade-off is the basis of the study presented here. We also consider the robustness in this context against modeling and parametric uncertainties in the plant. The problem is addressed in two stages. The first stage is for trajectory modification. A compromise on the original desired trajectory is negotiated considering the control constraints. The resultant modified trajectory is then used in the second stage: robust tracking against plant uncertainties. A simple depiction of this procedure is shown in Fig. ! where xu(t) = { x j d ( t ) } , j = 1, 2 . . . . . n is the general position vector. Without loss of generality xa(t) is taken as a point-to-point motion, throughout this study. For the trajectory modification stage we follow two different perspectives. (a) In the first, we consider the actuator limitations (such as saturation and bandwidth) and renegotiate the target trajectory (this name is used for the "original desired trajectory", xd). This process is handled assuming no flexibilities in the system, i.e., rigid body dynamics. We try to follow the target trajectory, x~(t), within the minimum time possible, given the actuator constraints. That is, a time optimal trajectory control problem (TO) is solved under the given actuator limitations. This effort results in the modified trajectory, xu,, (Fig. 1). (b) In the second, residual oscillation of the controlled motion is taken into account.
Originaldeared
uajec~ ~t)
Trajectorynxxtification ~ a) T ~ r i g i d body mode t r a ~ L ...........................
............... Rolx~ control ]
O1"
Stage I ~,,,(t) Stage 2 Fig. 1. Proposed two-stagecontrol scheme.
x,,~(0
124
. \ .lalih, .\
(~,htm ,lh'chatr,mc~ p,
,'¢ .' 199,~.~ 121 142
,41
;
.-~.A2 e- AIs
~.
t, A~ e 2a/,,
.~:~
p, A,,~ e - (,n - 1)~7~ .
Fig. 2. "l'appcd-delay lihcr scheme [.r.;,a.~land {.~.,:(.~)arc the [.aplacc transform of the target and modilicd ira cctories, rcspectively~.
The input shaping strategy of Singcr and Seering [5] is followed with no actuator limitations. In this strategy thc tlexibilities in the system arc taken at their nominally known values, and tapped-delay lilter structure is implemented in order to suppress the residual oscillations of the structure at the end o f the control task (scc Fig. 2). This control crcatcs the modificd trajectory of Fig. 1. ~,l,,,(t). In the second stage of the control the modified trajectory is taken as the basis for a robust control activity against uncertainties in the system model. An advanced version of Variable Structure Systcm (VSS) control, Sliding M o d e Control with Perturbation Estimation ( S M C P E ) is used for this purpose [3]. This strategy has an interesting feature. It requires only the upper bounds o f estimation errors on system uncertainties not the bounds o f the uncertainties themselves. With this inlormation the routine can follow the desired trajectory despite the modeling uncertainties. Depending on the composition o f the control law (Fig. I), some favorable fcaturcs appear such as compliance with the actuator limitations, suppressing residual oscillations at the end o f motion, and effectively competing against the plant uncertainties. The c o m p o n e n t s o f the new routine are explained in the following sections, along with some rcpresentative example cases demonstrating the advantages.
2. S t a g e I: trajectory m o d i f i c a t i o n
This section outlines the effect of altering the desired trajectory on the system performance. As explained in Section I, two paths are taken for this purpose. The2,' are presented next. 2. I.
l'ime-optimal
comrol.lor
r i g i d i z e d sy.~lem
The objective here is to find a constrained control strategy in order to guide the system as close to target trajectory, %, as possible in the shortest time. A further
N. Jalili, N. Olyac/Mechatronics 8 {1998) 121-142
125
simplification in the system representation is the elimination of flexibilities, i.e., converting the plant into a rigid body system. This is done to obtain a closed form solution for the control as will be detailed below. Consider the "rigidized" system representation ignoring the uncertainties as ~(t) -: ['(x(t), t) + [~(x(t), t)u(t)
(1)
where x = [ x j , x z . . . . . . r,]r is the state vector. It should be noted that, this transition to rigidized mode reduces the number of independent coordinates for the original system. Actuator constraints are taken as M , _ <~ u,(t) <~ M,+
i=
1. . . . . m
(2)
The objective here is to find the optimal control in order to bring the state from xi(0) to Xid(tr) in minimum time t/subject to the constraints in eqn (2). Utilizing the Pontryagin Minimum Principle, the optimal control is stated as [6]
u,*(t) =
/
M,.,
for
p*r(t)6,(t) < 0
M,
for
p*r(t)6~(t) > 0
undetermined
for
p*7(t)6i(t) = 0
(3)
Accompanying co-state equations are given as p*(t) -
~
~ (x*(tl,u'lt).p'lt)J)
i = 1 , 2 . . . . ,D1
p*. l(t) = 0 where p*(t), i = 1 . . . . . defined
(4) m is the i-th Lagrange multipliers, and the Hamiltonian H is
H(x(t), u(t), p(t), t) = I + pr(t)[~(x(t), t) + B(x(t), t)u(t)]
(5)
We consider that the solution (3) always yields determinate cases and at the instants of p*r6i = 0 we choose u*(t) = 0, momentarily. The control in the form of eqn (3) is known as b a n g - b a n g control. The system response to the control is determined by solving the differential equation ~*(t) = ~'(x*(t), t) + I~(x*(t), t)u*(t)
(6)
x*(t) forms the modified trajectory Xd,,, = [Xia,,},j = 1, 2 . . . . . r. Further improvements can be done on this trajectory. For instance, a scaling factor )' can be used on some system parameters involved, such as inertia elements. We start with 7 = I yielding x*(t) = x,~m(t) which has an optimal settling time t~. For 7 > 1, in the case of large inertia, the motion becomes smoother, yielding tr~. > tj, and 7 < 1 causes t n < t~, ultimately, 7 = 0 (no inertia) giving x;*(t) = Xd(t) which is the target trajectory. By maintaining ), > 1 the optimal trajectories are further smoothed. That is, the bigger the inertia the larger the settling time, consequently the smoother the modified trajectory Xdm. This dictates that, 7 > 1 gives a plausible parametric study
126
.V..lalili. .\. OI.qac :llechatr~mlc.~
,'~' ' 199,~',
121 142
which offers smoother control actions without violating the control constraints. This aspect will be further clarified in the simulations section. 2.2. Shaped input control This method employs shaped control inputs to reduce or eliminate endpoint vibration. To achieve this, an impulse sequence is convolved with the c o m m a n d input producing an aperiodic response. The routine requires reasonably good information about natural frequencies o f the lincarized system [5.7]. As shown in Fig. 2, this technique simply utilizes a tapped-delay tilter with proper weightings A, and time delays AT. The convolution of the target trajectory .v,At) and sequence o f m impulses results in a new multiswitch, multilevel function •f,a,,,(/) = 5~ A,.v,,,(t---(i • l ) A T ) u , ( t - - ( i .- I )AT) ,-
(7)
I
where .4, is the magnitude of the i-th impulse tit t = ( i - - I ) A T and uAt) represents a unit-step function. A proper sequence o f impulses, whose power spectrum has a notch at a structural resonant frequency, can be found such that the residual vibration o f the response of an u n d a m p e d second-order system to a sequence o f m impulses will not occur after t = (m 1)A Twhere A T = ~ xo and ~,~is the d o m i n a n t natural frequency of the llexible behavior. This effort creates the second class o f modified trajectories, •'~-,a,,,(t). to be considered in the following robustizing control scheme. It should be noted that this derivation is aimed for linear systems alone. N o general statement can be made regarding the applications to nonlinear systems.
3. S t a g e 2: S M C P E ,
an o v e r v i e ~
The modified desired trajectories generated in Section 2 are taken as the control objectives here. However. the control insensitivity against modeling uncertainties is considered as the primary exercise. The nonlinear control routine S M C P E is adopted as the robust control methodology. We present, next, a brief s u m m a r y o f it, following Elmali and OIgac [3]. A general class o f nonlinear systems is considered x .... = f ( X , . X . . . . . .
X.,)--
Af(X,.
X ......
X.,) + [B(X,. X: .....
X,,,)
+AB(XI.X: ..... X,,)]u+d(n
(8)
where X: = [.v,..L . . . . . . vl" i,]~ 6,.R,. i - I . . . . . m is the state sub-vector and .v, i = I. . . . . m arc m independent coordinates. The intluence of the u n k n o w n terms, which tire assumed to be b o u n d e d by known functions, is gathered into a single quantity named perturbation vector • (t) = A f + A B u ( t ) + d ( t )
= x"" - B u ( t ) - f
(9)
N. Jalili. N. OI.qac.,Mechatronics 8 (1998) 121-142
127
The control objective is to track the modified desired trajectory xj~(t), j = 1. . . . . m in the presence of unknown perturbations, modeling uncertainties, and unmodeled dynamics. To this purpose, first, the sliding hyperplanes are selected as Hurwitz polynomials of the tracking errors of the associate sub-vector as nt -
1
sj = ~ %.k+ "" le,"
j=
1. . . . . . m
(10)
k~O
where Cj.,, = 1, and ej = xj-X,a,,. In order to enforce the conventional attractivity condition [8], the following s-dynamics is proposed .¢j= - P s j - k
isgn(s,)
j=
1. . . . . m
(11)
where sgn (~)) = s/l~)l. This proposition is in line with Lyapunov convergence concept. The guarantee of convergence in finite time despite the uncertainties is detailed in [3]. The control arising from eqn ( I I) is u = B
~[- Ps-k
sat (s) - f -
~ +"~"~Ad.,-- ~P~,]
(12)
where k=diag[k,
k2
...
k,,],
sat (s) = [sat (s ~)
sat(s2)
...
sat(sin)] r, n-
s=[s,
s2
...
s,,,] r,
~=[~,
~2
..-
~,,]r
and
I
~ , = ~ C,.kel k) (13) k
I
It is important to note that the sgn (sl) was smoothed to sat
(si)
~" .s)/le.,l IsA ~< c, / (sgn (si) Is, I > ~:,
(14)
in order to avoid control chatter, e.j is a time-varying boundary layer for s-dynamic. Also, in eqn (12) W~, is an estimate of perturbation W. Using eqn (9), this estimate can be computed, which requires calculation of x ~"~and control feedback. In practice, such an algorithm is implemented digitally and the sampling speed is selected high enough to ensure that u(t) _~ u ( t - t). Additionally, in the absence of measurements of x ~") (e.g. acceleration measurement for second order dynamics) an approximation is utilized as x~")(1) ~ [x (° ' ~ ( t ) - x ~" ' ~ ( t - r ) ] / t
(15)
These are reasonable assumptions considering the fact that the control sampling speed is at least an order of magnitude higher than both the hardware bandwidth and the frequency contents of the desired trajectory. Typical example in robot controls, 200 Hz sampling is very common, and the trajectories hardly go beyond 2-3 Hz. It is shown both numerically and experimentally [9] that, SMCPE can maintain a desired level of accuracy in trajectory tracking despite the uncertainties. The combination of the trajectory modification and robustizing control enables us to form a
N. ,lalih. N. O l y a c :14~,~hatronic~ ,~' ;
128
199A' )
121. 142
mixed treatment tbr speed and accuracy in nonlinear systems as examples below demonstrate.
4. Simulations and numerical examples To show the effectiveness of the proposed control strategy, two examples are presented. I-:irst. we consider a rotating hub-beam structure with uncertainties on flexible modes of the beam. In the second example, a simple two-mass system with one force actuator is considered For both examples a reference point-to-point motion is takcn as target trajcctory.
4.1. Rolatht.q hub-beam system This example includes a uniform llexible bcam attached to a rigid hub, moving in the horizontal plane, as shown in Fig. 3 . 0 , , is an inertial coordinate of referencc with its origin at the center of hub, and 0,, is a rotating frame of reference attached to the beam. The .v'-axis is the spatial coordinate along thc length of the beam and tangent to the beana center line at C. while v' is the transverse coordinate. A control torque, T(t). acting on the hub (normal to the plane of motion) is the only external forcing. Thc hub-beam equations of motion (sec Appendix A) are given as
(',,
[2O'+h'(O'-"'~')l'q' t
I
i +/,A('' ,
l=];,+,t,i,,T /~
(]' :: ('pl { ~_'j [2t~'(C' " + q'(O'- -- ")~'' )]q~' " /JAb('1, ] = ]I +,(I T
i = 1,2 . . . . . n
(16)
where variables )71 and .0, arc sell" explanatory and the remaining terms arc delined
y'
'~ Y
ttx
X'
x
FO Fig 3. I-he hub-beam s3stem contiguration.
N. Jalili. N. OlgaciMechatronics 8 (1998) 121 142
129
in Appendix A. Ultimately, the hub-beam system is represented by the following equation.
(17)
= ~(x, ~) + g(x, ~)u wherc
f=[.L
.T, .L
... L]".
~:to,,
fr,
fr_...,
fr,]T
(18)
which is a nonlinear second-order system where state x = [0 g~ g2 . . . g,]~, and u is the input torque. The control objective is to track a desired trajectory Oa(t) in the presence of unknown perturbations, modeling uncertainties, and unmodeled dynamics. In this work, the controller considers the flexibility around rigid body mode only. For numerical results, Oa(t) = n/2, t > 0, and zero initial conditions are taken. The hub-beam system properties are given in Appendix B. The nonlinear model truncated with n = 3, described in eqn (16), was used in the simulations.
4.1.1. S M C P E
control.
We reverse the order here and present the robustizing control SMCPE first as the common element for all modified trajectories. It is appended to the modified desired trajectory creation stages, see Fig. 1, which are described in Sections 4.1.2 and 4.1.3. The sliding-mode controller considers the beam flexibility around rigid body mode only. This does not impose a serious practical restriction, because the higher modal frequencies are usually distinctly remote from the operational frequency interval. Therefore they should not come into play. According to eqn (16), neglecting the beam flexibility (gj = .4j = 0 or U,. = u, = 0), the rigid-body model with the presence of perturbations can be written as O( t) = .)7olu, = 0, = ,, + A]~,(x, ~ ) + [ffolu, = ~, . o + Afro(x, ~)]u + d( t)
(19)
wheref~ and fro are only functions of O = [0 0], while A]~ and Afro contain all the remaining terms. On the other hand, Af0 and Afrorepresent the uncertainties on f) and fro, respectively, d(t) is the external disturbance on the system. s-dynamics presented in eqn (10) for n = 2, can be written as s = d+ae
(20)
where e(t) = O ( t ) - O a ( t ) . In practice, a is selected for maximum tracking accuracy taking into account unmodeled dynamics and actuator hardware limitations [10]. For the simulations no such restrictions are considered, i.e., ideal actuator, high sampling frequency, and perfect measurements are assumed. The control law follows eqn (12) as
130
.\..lalih. ),.. O(qacMechatrmut.~ ?~", 1998~ 121 142 u(t)
4.1.2.
= lh
[ -- k sat (s) -./;, I. . . . . . .
Time-optimal
SM('PE
- erg, + 0,~ -- ~ . , , ]
(21)
control.
The target trajectory is moditied first, using time optimal control on the rigid body model of the system 0 -- u.l,
(22)
The control objective is to bring 0 ( 0 ) = 0 to 0 ( t , ) = 0,~ = n..2 within the shortest possible time given the constraint of lul ~< I. The resulting time optimal response of this system can be obtained from eqn (6) as I
2/, t" O*(t) = O,t,,,(t) =
- I t, 21. t'-+ I , t + ( I ' 1 0,~
0 ~< t <~ t~,,2 t/ 2L
(23) t,..2<~t<~t~ t > l,
where t, = 2\. 1,(t,i. This system response is used as the objective trajectory in S M C P E , the robustizing phase of the control. In order to further smooth the modified trajectory we perform, next, an inertia scaling study. The rigid body model for the virtual system with scaled inertia is given by ~; = !' 7/,
(24)
where the system inertia is conceived as ,'1,. The corresponding settling time to 0,/is 1,. = 2\./1,0,1
(25)
Clearly. the optimal settling time, t,:, is m o n o t o n o u s l y varying with 7. Since it has such a direct influence. ,, may also be used as a fictitious p a r a m e t e r to improve the modified trajectory. The ideal trajectory is represented by ,' = 0. That is, it can only happen if there is no inertia. Realizable control is performed for this example with 7 = I, 10 and 100. The effects of these selections on the modified desired trajectories are shown in Fig. 4(a). The robustizing stage of the routine, S M C P E , is performed next, according to Section 4.1. I. The system behavior is given in Fig. 4(b) for various y values. Due to the s m o o t h e r modified trajectory ;' = 100 case exhibits the best trajectory overshoot, control effort (Fig. 4(c)), and undesired transverse beam oscillations (Fig. 4(d)) characteristics a m o n g the four. ]'he effects of the modified trajectory on the transverse beam oscillations is especially obvious. The additional parameterization such as 7 can form a valuable desired utility for the controller.
N. Jalili, N. OIgac/Mechatronics 8 (1998) 121 142
131
(a)
(b)
2.0
/ .........
1.6
'-~i ~ i: ~ ' -
a
I
-
12
]; / /
0.8
i
!i
;io
04
,"i / o,]/ i///
~y= I00
2
3
4
5
0
-y=l
,:loo
o o
OO
I
~ .
I
2
3
4
Tire~(~c)
Tm'~ (s~.)
(d)
(c)
1.2 . . . .
12 - - .
1
O8
0.8 7=0
•
~ 04. .~
"¢=I00 ~/
.\' //
i0,4
A
I I ~'
= M-°
.~0o
I t/l-,
i
_
J
"\y~10
t~
y=10
-0g •
-os t . i (
"i'=1
-12 ]~-. -12 4 ~ - 0
"y=l
.. . . . .
I
2 Time ( ~ )
3
4
0
I
2
3
4
--4 $
Ttr~ ( ~ . )
Fig. 4. TO-SMCPE for hub-beam system for y = 0 (thin solid), 7 = I (dotted), y = 10 (dashed), and ), = 100 (thick solid).
4.1.3. Shaped-input S M C P E control. As stated before, this m e t h o d c o n s i d e r s the linearized m o d e l o f the system, such as: = A~+Bu
y = C~
(26)
w h e r e ~ = [0 .qt .q2 . . . 0 9~ .q2 . . . ] . A i s t h e J a c o b i a n m a t r i x o f i ' w i t h r e s p e c t to ~ at ~ = 0, a n d u = 0, a n d B is the J a c o b i a n m a t r i x o f ~ with respect to u at the s a m e p o i n t . F o r n = 3, the rigid b o d y m o d e a n d three c o n s e c u t i v e m o d e s are t a k e n i n t o a c c o u n t . T h e n , the s y s t e m r e d u c e s to a n e i g h t h - o r d e r l i n e a r m o d e l .
N. , l a h l i . . \ . ()lgav Mcchatronic,~ A' i 1993; 121 142
132
Since the governing equation of motion is truncated to three modes of the flexible beam, a sequencc of impulses corresponding to these three modes are used in input shaping. These three impulsc sequcnces can bc convolved to form a single sequence to reduce vibration in thrce separate modes [5]. Ensuing modified desired trajectory and the system response to SMCPE control tire shown in Fig. 5. It is important to note that in the time history' of the control torque, Fig. 5(c) {dotted curve) there are some.jitters as a consequence of the shaped-input controller. These discontinuities are the filtered effects or the discontinuities of the modified
(h)
(a)
20
20,
16 I
i ~"
}
12
0
8
0
9 0 *~/
~' 04
O0 0
.~
2
I
.
II
4
.
.
.
.
1
.
.
.
.
.
3
~
-1
5
I .no (~',~)
hnw: (we)
(d)
(~)
I.
(J 04
I :,
,02
o..
0 02
,
A
-~
;__.
•
i£
~ .
il O0
I
-O02
I
-0 02
4) 04
[i U4
0
I
2
~
I
5
rlrn~ f~¢¢ i
Fig. 5. Comparison between T O - S M ( ' P E bean1
s~stem
.
.
.
.
1
.
2
.
. I
. 4
I'm~ (~ec)
l(~l ;, -
101i I l h i c k solid), a n d S I - S M C P E
(thin solid) for lhe h u b -
N. Jalili, N. OIgac/Mechatronics 8 (1998) 121 142 X1
133
X2
I U
/ Fig. 6. Two-mass system.
desired trajectory (Fig. 5(a)/dotted curve) through S M C P E routine. They do not appear in the T O - S M C P E since the time-optimal pre-filter and its first derivative are both continuous functions. This is a very strong advantage of the T O - S M C P E combination. 4.2. T w o - m a s s s y s t e m
A simple linear system of Fig. 6 is considered as the second example. We present this to compare the performance of T O - S M C P E and SI-SMCPE algorithms in particular from the robustness point of view. The system uncertainty is taken only in the stiffness of the spring connecting the two masses. Both bodies have unit mass. The only control u acts on body 1 and the position of body 2 is the output to be controlled. The equations of motion are ml,~l + k ( x l - - x 2 )
= u
m2.f2 -- k ( x l - x D
= 0
(27)
where x~ and x2 are the positions of the body 1 and 2, respectively. The control design objective here is to achieve a fast settling time for a step-type desired trajectory (x2j(O) = 0, x2u(tj) = 1) and robust performance against the stiffness uncertainty. 4.2.1. S M C P E
control.
With proper mathematical manipulations, eqn (27) can be transformed into x~,.~ =
ml__-k-m2k,q, + - -k mlm2
u
(28)
mlm2
s-dynamics presented in eqn (10) for n = 4, can be written as s(t) = "~2(t) + ClO'2(t) + C2O:(t) + C~e2(t)
(29)
where e2(t) = x2(t) - x2j(t), and Ct, (72, Cs are all constant and selected in such a way that s forms a Hurwitz polynomial. Since we have a single input, the subscript j in eqn (10) from s-dynamics is dropped. By selecting ,~ as defined in eqn (I i) the control law is written as
134
:\'..lalili..%".
u(t)
= I 2
I
u(t--~)+
tl! I DI 2
k
OlgacMechutr.nic~ ,~' ~ 199~' J 121 142
[
P.s-.ksat(s:)
+ ("2-#2,/+ ("~-~_'J]+ m~
Ill 2
~-
171 I
-- ('~.~, + 2,,,~ + ('~Tx'-,/ -
-- ("2
"
-
(-V I -- -g2 ) -]" ( '1 (-'('2 -- .i- I )
(3(I)
In this expression the targel trajectory is considered, therefore x'~i/ = ~2,; = .#2,; = 5::,/= 0 and .x2,~ = I. 4.2.2.
Time-optimal SM('I'E
control.
The two-stage control T O - S M C P E is employed f'or this system. First, the modified desired trajectory is created for optimal time with the constraint [u[ ~< I. The rigidized model of the nominal system equation of motion is (31)
.~, = .~'~ = u (m~ -4-m:)
The time optimal response of this system can be determined similar to the hubbeam system, i.e.. eqn (23) by simply substituting 1~ by (m~ + m 2 ) . 0,M) by x2,~(t), and #,~,,,(t) by x2,~,,,(t). The time optimal trajectory is then used in the second stage as the desired trajectory for SMCPE. Equation (30) is used for the control law with x_,,~,,,. .{it -~2,~,,,, .~-:,a,,,substituted from eqn (23) and 22a,,, = -\2,~,,, = O. Figure 7 shows the effectiveness of p a r a m e t e r 7 for S M C P E method in two different categories, one when there is an input saturation and another when the input is unconstrained. As clearly seen, different scaling t:actors of ,', similar to the first example, yield responses which arc superior to those in the case of ,' = 0, i.e.. target trajectory being used directly in S M C P E . 4.2.3.
Shaped-input SM('PE
control.
Fhe natural frequency of the flexible mode of the linear system in Fig. 6 is /k(mt +m~)
~t) = Xj
....
Ill t D12
(32)
For this example, four impulse sequences are convolved to form a modified desired trajectory. Ensuing tr~uectory and the nominal system response to S M C P E control are shown in Fig. 8. We restate that. the preshaping c o m m a n d input requires good infi)rmation about natural frequencies of the system. It is needed to properly suppress the residual vibration of the system. In this example, the system frequency is dependent on the spring stiffness which is considered uncertain. The effect of this uncertainty on the performance of the control is studied next. 4.2.4.
P e r t u r b a t i o n on sprinq st([]hess.
To show the robustness feature of the two-stage routines against modeling uncertainties, we consider a range of spring stiffness uncertainty around the nominal k = 1. When similar dynamics was treated by Wie and Liu [11], the stable zone of
N. Jalili, N. OIgaciMechatronics 8 (1998) 121-142
135
(a)
1.2
/y=0
._.~/ ......
ID
~" 0.8 "1
/ " / "1'= I
/ "y=lO
I 0.6 "I~ 04 '
"
02J "
0.0 0
4
8
16
12
20
"l'in~(t~¢.)
:b)
1.2
/ ¢~____ y
(c)
=
~
1.2
/~--y=l ,0
/r:lo
-/
-5 , : ~ - - - - - - ~
°'/i,J2 /L °61 / .:o
!
i
~ 04
.i/.....-7-~
IO
o,
7~=o ; /=
°'i ! /' 02 ~
0.2 ~ 0.0 g
12
Tm~(sec.)
16
20
,
"
O0 0
4
12
S Time
16
20
(~.)
Fig. 7. TO-SMCPE for two-mass system for 7 = 0 (thin solid), 7 = 1 ( d o t t e d ) , ~, = 10 (dashed), and ), = 2 0 (thick solid). (b) and (d) when the control input is constrained and P = 0, (c) a n d (e) when the control is unconstrained and P = 10.
0.44 < k < 3.27 was found for H~ controller. Here we consider the perturbation on k both inside and outside this interval. For a better comparison, TO-SMCPE corresponding to ": = 20 is taken, which performs the best a m o n g others. Figure 9 shows the system response for three different values of k with our T O - S M C P E and SI-SMCPE routines and H~ robust algorithm as employed by Wie and Liu [11]. For brevity we only give the respective results of the latter without any discussion o f the routine. As claimed in the referenced paper, the H~ controller is not stable outside the region 0.44 < k < 3.27. On the other hand, both TO-SMCPE and SI-SMCPE exhibit stable behavior for wider range of spring stiffness uncertainty. This is assured by the
.V. Jalili, ..%:.O l g a c Mechatroni('.~ <~ r 199,~'; 121 142
136
(d)
Ca) 12: .
.
.
.
.
I
06
:
E
.
10
r=l
il o.o ':
.
,I
lU
i:.' .
g=O
,
o,
t
E Z
.L
"/=20
.~
oo
y=20
i
LJ
,, -12
el(i -05
. . . . . . . . . . . . . . . . 4
i 0
2(1
16
12
g
4
2
6
g
Io
Time (see)
Time (sec)
(e) 9
• 300 "~-
y-I 15
.v=-0
.[ i 200
y=20
3
~
0
I
E
E
C\i , / ~x : !
100
.',=20
! -
o
.i i ~
y=lO
70
(10
-I(lO
0
12
4
I{l
20
I
2
3
4
"rin~ (sec.)
Time (~e~)
I:ig 7. c:mtim.,d.
basic feature of SMCPE: the routine is not sensitive to the system uncertainties but the error in their estimates. A point to note is that the settling time for both TO-SMCPE and SI-SMCPE are approximately the same. However, in the time history o f control, it is shown that there are sudden changes in magnitude at each instant of application of the impulse in the shaped input controller and at switching times of the TO-SMCPE. For the SISMCPE, they originate from the discontinuities in the pre-shaped desired trajectory. The discontinuities in TO-SMCPE appear because of the s-dynamic defined in eqn (29). In this special case, sliding function uses the second derivative of the timeoptimized trajectory. According to eqn (23), this derivative has two discontinuities at
N. Jalili, N. 01,qaciMechatronics 8 (1998) 121 142
137
(a) Modified desired trajectory, X2,t="
10 08 08 04
02
4
8
12
16
20
Time (see.)
(19) Output of SI-SMCPE, x2.
10
f
08
/
06 04
02
/
/
./
f
J /
//
/
4
8
12
16
20
16
20
lime (see.)
(c) Control input,
0
4
8
12
it.
Fig. 8. Shaped-input S M C P E for t w o - m a s s system for P = 0 (solid) and P = 10 (dotted).
138
•\.
Jalih,
.V.
(Hqa~. Mechatronw.~ ,~ ! 1 9 9 8 : 1 2 1
142
(a)
12 08 ]
•
0 8
/ /
(]4 1
i'
=o
I
00 02 4) 4 4
g 12 rime (.,.~: )
I(,
4
2o
8 12 "l'im¢(s~:)
16
20
8
16
20
(hi 12
................. 08
" 10
.
08 O6
~
~//
i
o 4
O0 ~
.
_
_
'~ o2 O0
,
0
4
•
8 lime
,
12 (sex)
. . . . 16
-04 . 21)
o
4
12
Titt~ (s,e~)
Fig. 9. ( ' o m p a r i s o n b e t w e e n I O - S M C P I . I (thick solid), SI-SMCPE (thin solid), and I1, (dotted). (a) nominal:/,- - 1. (b) perturbed inside the stable region: k - 0.7, and perturbed outside (2 cases), (c) k = 0.4 and ( d ) / , = 3.5.
switching times at t,..2 and t,. Further improvements on the modilied trajectory can be made by smoothing the second derivative at these instants.
5. Conclusions This study demonstrates the feasibility of using rigidized system model and limeoptimal response as a way to alter the target trajectory. This method provides a set of modified and smoother desired trajectories with longer settling times. This compromised trajectory can also be obtained by introducing a virtual model for the rigidized system (for instance, by using an inertia scaling, 7).
139
N. Jalili, N. Olqac/Mechatronic's 8 (1998) 121 142 (c) 1.0 m
r
~. 2.5] 03
i
5I i 0.0 0.3
-05
-05 .~--
0
,
8
16 T ~
24
40
32
-I.0 • ---~ 0
24
16
8
32
40
(~.) T~(~.)
12
10 ]
t
i
'0,1
~ 06 I~
I I
04
02
,- ...... / I 40
0.0
8
16 Time
32
i 00
-0.4
,"
~--
0
(~,)
,
,
g
,
,
.~
--~
16 Tin~:
. . . . .
24
~ - - - - - ~
32
40
(~c.)
Fig. 9. continued
Two different examples have been studied and two separate two-stage controllers were applied. The TO-SMCPE combination presents superior results when compared with other robust controllers such as SI-SMCPE and H~. The first example, hubbeam setting, demonstrates the feasibility of the two-stage control routine for a general class of dynamic systems with non-linear governing equations. The second example shows that, while the H~ controller is not stable for some values of the uncertainties two versions of SMCPE exhibit stable behavior. For both examples it is also shown that, SI-SMCPE provides the same maneuverability and robustness characteristics compared with TO-SMCPE. However, it is limited to the linear systems and it requires good information about system natural frequencies, while the TOSMCPE does not. Also, the time history of control input in SI-SMCPE shows some
140
.~ll. .lalili.
N. O l # a c ,~,l(,chalrontc.~ ,~ , 1998 j 121 142
jumps at each instant of application of the shaping impulses, while the one in TOSMCPE has bccn significantly reduced due to continuity of thc modified desired trajectory. These are very strong advantages of the rigid mode time optimal,SMCPE combination.
Appendix A: Equations of motion for rotating hub-beam system A modeling effort following Chang and Jayasuriya [12]. is presented hcrc for the hub-bcam shown in Fig. 3. The transverse and longitudinal deformations of a beam element arc denoted by u, (.v. t) and u,(.v. 1), respectivcly, where .v is the distance to the beam element measured from point C in Fig. 3 and t is time. The hub angular rotation, i.c.. 0{t) is measured counterclockwise from thc incrtial x-axis. The Euler Bernoulli assumptions are used. The kinetic and potential energies of this system are stated first in order to apply Lagrange's formulation. 11),0-'+ p4 KE = 2 2 " .,
(
\ ~'t ~
+u~O-"--2¢)t,, ,'u,
;~1
+(r,,+.\+u,/-'0-'+
t ' E = 2EA
,) \ ~ . v /
d.v+ 2 E l
\ {! / + 2 0 ( r . - t - . x 4 - u , )
,,?v' " d . \ L[J
.
('~l jd.v
(AI)
(A2)
/
It is assumcd that u, and u, can be approximated as the finite sums u,(.v, t) = ~'L(t)¢p,(.\) :
(A3)
I
u, (.v, t) -- ~ .q,(t)~0,(.v) i.
(A4)
I
where q~,(.v) and tki(.v) are i-th longitudinal and j-th transverse modal shapes (eigenfunctions) of a clamped-tYee and fixed-free beam. respectively./i(t) and g~(t) are the corresponding generalized coordinates of the beam. In the case of a beam slewing through a finite angle, such as in robot motion, the longitudinal detbrmation and its coupling to the transverse modes and rigid body may be neglected, i.e.. JM) = 0 is assumed. The work done by the external torque T on the hub is given by W,,~.= TO. Using the orthogonality relationship between the eigenfunctions and further using the Lagrange's equations of" motion, the following coupled equations are obtained:
.4,+(~,~,.-O'-)y,= - b , 0
i = 1.2 . . . . . n
(A5)
N. Jalili, N. Olgac/Mechatronics 8 (1998) 121..142
141
[ (lt/pA) +j~, g~ ] O+ ( 2 j ~ gl.0j) 0 + j ~ b,ff) = T/pA
(A6)
=
where
9,
= E , r( pA Jo
j = i,2 ..... .
bj = I~ (ro +x)~j(x)dx
(A7)
j = 1,2 . . . . . n
(A8)
It = & + pA {[(ro + L) 3-r]]/3}
(A9)
The hub-beam dynamics of eqns (A3)--(A9), can be transformed into a more suitable form for controller synthesis
1
O=
~=, [20j+b,(O -09;)]9, + ~
T=]~,+goT
pA Cp T=.f+9~T
g, = ,-P U=, [2b,g) + qj(O" - to.q2,)]gj
i= 1,2 . . . . . n
(AI0)
w h e r e f and .~7~are self explanatory terms and
Cp = It/pA + ~ (q~ - b f ) j=l =
tl'
~Cp+b~ ( b~bi
for./=/
(All)
f o r j ¢: i
Appendix B: Modal parameters of the flexible hub-beam system
Properties
Symbol
Value
Unit
Aluminum density Aluminum Young's modulus Beam width Beam height Beam length Beam cross-section area Beam moment of inertia Beam mass moment of inertia Hub mass moment of inertia Hub radius
P E b h L A I
2710 71 x 109 7 x 10 -~ 2 x 10 -2 0.575 i.56 x 10 -5 8.533 x 10 - t 3 0.1988 x 10 ..2 0.2087 x 10- 2 0.065
Kg/m -~ N/m 2 m m m m2 m4 Kg m ~ Kg m: m
I,, ro
142
N. Jalili. N. Olqar..lleN~atron,',~ ~ ~ 1998J 121 142
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