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Adaptive vibration isolation for axially moving strings: theory and experiment
Automatica 38 (2002) 379–390
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Adaptive vibration isolation for axially moving strings: theory and experiment Yugang Lia , Dan Arona , Christopher D. Rahnb; ∗ a Department
b Department
of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA of Mechanical and Nuclear Engineering, The Pennsylvania State University, 150A Hammond Building, University Park, PA 16802, USA Received 14 April 1999; revised 24 March 2000; received in .nal form 24 August 2001
Abstract High-speed transport of continuous materials such as belts, webs, .laments, or bands can cause unwanted vibration. Vibration control for these systems often focuses on restricting the response resulting from external disturbances (e.g. support roller eccentricity or aerodynamic excitation) to areas not requiring high precision positioning. This paper introduces vibration controllers for an axially moving string system consisting of a controlled span coupled to a disturbed span via an actuator. The system model includes a partial di4erential equation for the two spans and an ordinary di4erential equation for the actuator. Exact model knowledge and adaptive isolation controllers, based on Lyapunov theory, regulate the controlled span from bounded disturbances in the adjacent, uncontrolled span. Assuming distributed damping in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively, while ensuring bounded uncontrolled span displacement and control force. Experiments demonstrate the e4ectiveness of the proposed controller in isolating the controlled span from disturbances and damping the controlled span displacement. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Distributed parameter systems; Axially moving string; Vibration isolation; Lyapunov theory
1. Introduction Continuous materials such as belts, webs, .laments, or bands, can vibrate excessively during high-speed axial transport, reducing the productivity of the associated manufacturing process and quality of the .nished product. Support roller eccentricity, material non-uniformity, aerodynamic disturbances, or manufacturing processes (e.g. slitting, calendering, or printing) can cause the unwanted vibration. Active vibration control has the potential to reduce vibration and improve the axial transport process. Ulsoy (1984) develops the .rst controllers for an axially moving system. An observer-based state feedback algorithm based on a reduced-order, discretized model of the in.nite dimensional, axially moving string system is shown to be a viable method for active vibration control. This
paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +1-814-865-6237; fax: +1-814-863-7222. E-mail address: [email protected] (C.D. Rahn).
The research demonstrates, however, the possible instabilities resulting from control and observation spillover when point sensors and actuators drive the closed-loop system unstable by measuring and forcing high frequency modes not included in the reduced-order system model. To prevent these instabilities, Yang and Mote (1991) develop a class of asymptotically stabilizing controllers for distributed parameter models of axially moving strings and beams. The use of a quasi-passive controller along with a carefully positioned sensor and actuator ensure asymptotic decay of all vibration modes. A number of researchers have developed boundary controllers based on distributed parameter models of strings. MorgAul (1994) introduces dynamic boundary feedback controllers for the wave equation that include proportional and strictly positive real derivative feedback. Joshi and Rahn (1995) develop and experimentally implement boundary controllers for a linear gantry crane model with a Cexible cable. Baicu, Rahn, and Nibali (1996) experimentally demonstrate how a three-term boundary control law can be used to stabilize the out-of-plane vibration of a Cexible cable. Renshaw,
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 1 9 - 9
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Y. Li et al. / Automatica 38 (2002) 379–390
Rahn, Wickert, and Mote (1998) illustrate that Lyapunov theory can be applied to axially moving strings without using the expected material derivative in the time differentiation of the Lyapunov functional. Lee and Mote (1996), Shahruz (1997), and Fung, Wu, and Wu (1999) develop boundary control laws for axially moving strings. The use of boundary control to damp vibration in an axially moving system can be diIcult to implement in practice. The size and weight of the support=guide rollers and their inherent dynamic coupling with adjacent spans may limit the applicability of boundary control in axially moving systems. Queiroz, Dawson, Rahn, and Zhang (1999) overcome these practical limitations by developing exact model knowledge and adaptive control algorithms that damp the vibration of a distributed axially moving string system using a two-degree-of-freedom actuator located between the support rollers. It is often desirable to isolate disturbances from quiet parts of a system that require precise positioning without vibration. The machine or process is carefully designed to ensure minimal disturbances apply to the quiet part, often at great expense. Other parts of the structure can vibrate appreciably without adversely a4ecting system performance. Coupling in the system, however, causes this vibration to propagate to the quiet part. In web handling machines, for example, great care is taken to minimize disturbances in printing regions where precise positioning is required. The rollers are carefully balanced and aerodynamic excitation sources are minimized. Other parts of the web handling machines are less sensitive to vibration and do not require such expensive measures. Many researchers have studied vibration isolation for systems other than axially moving strings. Karnopp (1995) shows how passive methods can be used for vibration isolation through optimization of system parameters (e.g. spring and damping constants) and addition of vibration absorbers. Using an adaptive truss within
a Cexible steel structure, Clark and Robertshaw (1997) actively controlled the structure vibration without the use of a system model. Ertur, Li, and Rahn (1999) use adaptive vibration isolation control strategies based on Lyapunov theory to regulate quiet parts of a discrete system from non-colocated disturbances. The objective of this paper is to develop a vibration isolation system for web handling machines based on a distributed axially moving string model. Using a distributed control approach based on Lyapunov’s method eliminates the possibility of spillover instability. The system model is a pinned–pinned axially moving string divided into two spans by a transverse force actuator. Exact model knowledge and adaptive isolation controllers, based on Lyapunov theory, regulate a (controlled) span from bounded disturbances in the adjacent (uncontrolled) span. Assuming distributed damping in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively, while ensuring bounded uncontrolled span displacement and control force. Experiments demonstrate the e4ectiveness of the proposed controllers in isolating the controlled span from disturbances and in damping the controlled span displacement. 2. System model Fig. 1 shows a string moving at constant speed v between pinned supports. An actuator consisting of force fc (t) and mass m controls the string displacement u(s; t) for s ∈ (0; sc ) (controlled span) where s is the material position, t is time, and sc is the actuator location. A disturbance force fu (t) and boundary displacement uL (t) act on the uncontrolled span s ∈ (sc ; L), where L is the total span length. We assume that fu (t) and uL (t); up to second time derivatives, are bounded (fu ; uL ; u˙ L ; uA L ∈ L∞ ).
Fig. 1. Schematic diagram of the axially moving string system.
Y. Li et al. / Automatica 38 (2002) 379–390
The kinetic energy of the system is sc 1 (vus + ut )2 ds T= 2 0 L 1 1 (vus + ut )2 ds + mut2 (sc ; t); + 2 sc 2
(1)
where and v are the constant mass=length and transport speed of the string, respectively, and subscripts indicate partial di4erentiation. The kinetic energy of the axial moving string divides into the .rst two terms in Eq. (1) because of the discontinuity in us (s; t) at s = sc . The third term in Eq. (1) comes from the actuator mass. Under assumptions of small string displacement and perfect Cexibility, the system potential energy is sc L 1 1 us2 ds + P us2 ds; (2) V= P 2 0 2 sc where P is the constant string tension. The virtual work done by the external forces is L L (3) W = fc u(sc ; t) + fu u ds − cut u ds; sc
0
where c is a viscous damping coeIcient. We assume distributed relative velocity damping throughout the span. Other damping models (e.g. absolute velocity, material, or boundary damping) could also be assumed. Substitution of Eqs. (1) – (3) into Hamilton’s Principle, t1 (T − V + W ) dt = 0; (4)
knowledge. In Theorem 2, we illustrate how the control law can be redesigned as an asymptotically regulating adaptive controller that compensates for uncertainty in the system parameters P0 and m. First, Lemma 1 shows that the uncontrolled span displacement is bounded if u(sc ; t) and its derivatives are bounded. 3.1. Boundedness of the uncontrolled span Lemma 1. If fu (t) ∈ L∞ and u(sc ; t) and uL (t) ∈ L∞ up to second time derivatives; then u(s; t) ∈ L∞ for s ∈ [sc ; L]. Proof. Substitution of s − sc s−L w(s; t) = u(s; t) − u(sc ; t) uL (t) − L − sc sc − L for s ∈ [sc ; L] wtt + 2 vwst − P0 wss + cwt = y(s; t)
(5)
and the boundary conditions: u(0; t) = 0;
u(L; t) = uL (t);
u(sc− ; t) = u(sc+ ; t); mutt (sc ; t) + P0 (us− − us+ ) = fc ;
(6) (7) (8)
where us− = us (sc− ; t); us+ = us (sc+ ; t), H (s) is the Heaviside step function, and we assume P0 = P − v2 ¿ 0. 3. Control formulation The control objective is to drive the string displacement u(s; t) for s ∈ (0; sc ) to zero with a bounded control force fc . In Theorem 1, we design an exponentially regulating control law for the system based on exact model
(10)
with boundary conditions w(sc ; t) = w(L; t) = 0;
(11)
where y(s; t) = fu −
(s − sc ) (s − L) uA L (t) − u(s A c ; t) (L − sc ) (sc − L)
2 v[u˙ L (t) − u(s ˙ c ; t)] L − sc s − sc s−L u(s ˙ c ; t) −c u˙ L (t) − L − sc sc − L −
yields the .eld equation: for s ∈ (0; sc ) ∪ (sc ; L)
(9)
into Eq. (5) yields
t0
utt + 2 vust + cut − P0 uss = fu H (s − sc )
381
(12)
is bounded by the lemma assumptions. A Lyapunov functional for the uncontrolled span is chosen to be Vu (t) = Es+ (t) + Ec+ (t); where 1 Es+ = 2
and Ec+
= 1
L
sc
L
sc
1 wt2 ds + P0 2
(13)
L
sc
ws2 ds
wwt ds:
with 1 ¿ 0 a small positive constant. From Queiroz et al. (1999), since u(0) = 0, L 4L2 L 2 u2 d x 6 2 u d x for x ∈ (0; L): 0 x 0
(14)
(15)
(16)
Substitution of inequality (16) and 2uv 6 u2 + v2
(17)
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Y. Li et al. / Automatica 38 (2002) 379–390
The terms of the second integral in Eq. (27) simplify via integration by parts, the boundary conditions, and inequalities (2) and (4) as follows: L L wP0 wss ds = − P0 ws2 ds; (28)
into Eq. (15) yields L + |Ec (t)| 6 1 (w2 + wt2 ) ds sc
6 1
L
sc
wt2 +
4L2 2 w 2 s
ds
max(1; 4L2 =2 ) + Es (t): 6 21 min( ; P0 )
sc
(18)
L
sc
min( ; P0 ) ; 2 max(1; 4L2 =2 )
L
sc
wwt ds 6
wy(sc ; t) ds 6
−2 v
0 6 1 Es+ (t) 6 Vu (t) 6 2 Es+ (t);
(20)
where 1 = 1 − 21 2 = 1 + 2
max(1; 4L2 =2 ) ¿ 0; min( ; P0 )
max(1; 4L2 =2 ) ¿ 1: min( ; P0 )
(21) (22)
The time derivative of Vu (t) + + V˙ u (t) = E˙ s + E˙ c
(23)
depends on L L + 2 ˙ Es = − cwt ds + wt y(s; t) ds; sc
sc
1 2uv 6 u + v2 produces
for any ¿ 0
6 − (c − )
L
sc
1 wt2 ds +
(24)
L
sc
= 1
sc
y2 (s; t) ds:
(26)
+ 1
L
sc
sc
1 2
wt2 ds + c1 L
4L2 2
y2 (sc ; t) ds +
sc
L
sc
ws2 ds;
4L2 2 2
L
sc
wwst ds 6
v 3
L
sc
wt2 ds + v3
L
sc
(29)
ws2 ds; (30)
ws2 ds; (31)
where 1 ; 2 , and 3 are arbitrary positive constants. Substitution of Eqs. (26) – (31) into Eq. (23) yields c v ˙ + V u (t) 6 − c − − 1 + 1 3 L L 1 1 + wt2 ds + y2 (sc ; t) ds 2 sc sc 2 2 4L 4L −1 P0 − c 2 1 − 2 2 − v3 L ws2 ds 6 − 3 Es+ (t) + ; (32) where
3 = min
c − − 1 +
c 1
+
v 3
2 1 P0 − c 4L 2 1 −
P0
4L2 2 2
; − v3
¿0
if 1 ; 1 ; 2 , and 3 are suIciently small, and L 1 1 2 + max y (s; t) ds ¡ ∞ for all t = 2 sc (34) from Eq. (12). Use of Eq. (20) produces
wt2 ds
L
(33)
sc
L
L
sc
(25)
The remaining term in V˙ u (t) is L + ˙ E c = 1 (wt2 + wwtt ) ds
sc
where the .eld equation (5), boundary conditions (11), and integration by parts have been applied. Application of inequality 2
c 1
(19)
then
+ E˙ s
−c
If we choose 1 ¡
sc
V˙ u (t) 6 − Vu (t) + ;
w(P0 wss − cwt + y(sc ; t) − 2 vwst ) ds: (27)
(35)
where =
3 : 2
(36)
Y. Li et al. / Automatica 38 (2002) 379–390
Solution of Eq. (35) produces Vu (t) 6 Vu (0)e−t + ∈ L∞ :
|Ec− | 6 2 sc
(37)
Using the inequality l 2 u (x) 6 l us2 (s) ds for x ∈ (0; l) 0
if u(0) = 0
∈ L∞ :
sc
0
6 2 sc
sc
0
|us ut + vus2 | ds
(ut2 + (1 + v)us2 ) ds
1 sc 2 (ut + us2 ) ds 2 0 sc 4 sc (1 + v) 1 2 2 6 ( ut + P0 us ) ds ; min( ; P0 ) 2 0
6 4 sc (1 + v)
(38)
in Eq. (20) and using Eq. (37) then yields L 1 P0 w 2 6 1 P0 ws2 ds 6 1 Es+ (L − sc ) sc 6 Vu (t) 6 Vu (0)e−t +
383
so (39)
Thus, w ∈ L∞ , so u(s; t) ∈ L∞ for s ∈ [sc ; L]. 3.2. Exact model knowledge control law
−!Es− 6 Ec− 6 !Es− ;
(48)
where 4 sc (1 + v) ¡ 1; != min( ; P0 )
(49)
if is suIciently small. Thus,
To facilitate the subsequent control design and analysis, we de.ne an auxiliary variable,
1e (Es− + 2 ) 6 Vc (t) 6 2e (Es− + 2 );
(t) = ut (sc ; t) + us− :
m ¿ 0; 1e = min 1 − !; 2 m ¿ 1: 2e = max 1 + !; 2 The time derivative of Eq. (44) is
(40)
After di4erentiation of (t) with respect to time, multiplication by m, and substitution of the boundary condition (8), Eq. (40) becomes m (t) ˙ = fc + P0 (us+ − us− ) + must− :
(41)
Theorem 2. The bounded control force fc = − P0 (us+ − us− ) − must− − ks
(42)
Proof. For simplicity, we neglect the stabilizing e4ect of damping in the controlled span. Substitution of the exact model knowledge (EMK) controller (42) into Eq. (41) yields (43)
where Es− = and
1 2
+
sc
0
Ec− = 2
sc
Ec− (t)
+
2 1 2 m ;
1 ut2 ds + P0 2
sc
0
us2 ds
(45)
= − vut2 (sc ) −
− P0
P0 2 P0 (u (sc ) + us2 (sc− )) + 2 2 t 2
(54)
0
0
sc
us2 ds
(55)
so
P0 ˙ V c (t) = − ks − 2 2 sc 2 − ut ds + P0 0
−
with and small positive constants. Inequality (17) and integration by parts bound Eq. (46) as follows:
−
0
(53)
and sc sc 1 ˙− E c = 2 sust ut ds + 2P0 sus uss ds 0 0 sc = sc ut2 (sc ) − ut2 ds + P0 sc us2 (sc− )
(46)
sus (ut + vus ) ds
(52)
Thus, 1 ˙− E s = − vut2 (sc ) + P0 ut (sc )us (sc− )
(44)
(51)
− − V˙ c (t) = E˙ s + E˙ c − ks 2 :
A Lyapunov functional is de.ned as Vc (t) = Es− (t)
(50)
where
exponentially stabilizes the string displacement for s ∈ [0; sc ].
m (t) ˙ = − ks :
(47)
0
sc
us2
ds
P0 − sc ut2 (sc ) 2 − P0 sc us2 (sc− ):
v +
P0 2
(56)
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Y. Li et al. / Automatica 38 (2002) 379–390
3.3. Adaptive control law
If we select ¿ 2sc , sc ( ut2 + P0 us2 ) ds V˙ c (t) 6 − ks − P0 2 − 2 0 6 −3e (Es + 2 ) 6 −
where 3e = min
2
ks − P0 ; 2
3e Vc (t); 2e
(57)
The EMK control law (42) depends on the exact knowledge of the system parameters m and P0 . To compensate for unknown or slowly time-varying parameters, an adaptive controller is developed. The open loop dynamics for the variable (t) are rewritten as
¿0
(58)
m (t) ˙ = fc + W ;
if ¡ 2ks =P0 . Thus 3e Vc (t) 6 Vc (0) exp − t 2e and using inequality (38) sc P0 1e 2 1e P0 u 6 us2 ds 6 1e Es− 6 Vc (t) 2sc 2 0 3e 6 Vc (0) exp − t 2e and u(s; t) → 0 for s ∈ [0; sc ] as t → ∞. We assume that u(s; t) belongs to a space s of functions where u ∈ L∞ implies us ; uss ∈ L∞ and 0 c ut2 ds ∈ L∞ implies ut ; ust ∈ L∞ . Thus, u(s; t); ut (s; t); us (s; t); ust (s; t), uss (s; t) ∈ L∞ for x ∈ [0; sc ]. The .eld equation (5) then implies utt ∈ L∞ for x ∈ [0; sc ]. Finally, from Lemma 1, us+ ∈ L∞ . Thus, all terms in the exact model knowledge control law are bounded, so fc ∈ L∞ .
(59)
where = [m; P0 ]T ;
W = [ust− ; us+ − us− ]:
The adaptive controller asymptotically stabilizes the controlled span displacement and estimates the unknown parameter vector . Theorem 3. The bounded control law fc = − W ˆ − ks
(61)
and the adaptation law ˆ˙ = !WT ;
(62)
where # = #T ¿ 0; ensure u(s; t) → 0 for s ∈ [0; sc ] as t → ∞. Proof. Substitution of Eq. (61) into Eq. (59) produces m (t) ˙ = − ks + W ˜ ;
(63)
where ˆ = [m; ˆ Pˆ 0 ]T
Remark 1. Rede.nition of
and the parameter estimation error
= ut (sc ; t) + us−
˜ =
does not alter the closed loop stability. The additional gain allows .ne tuning of the controller. Remark 2. Friction at the actuator could raise the tension in the uncontrolled span relative to the controlled span. Substitution of the upstream P0− and downstream P0+ tension in the .eld equation (5), boundary condition (8), functionals (13) and (44), and control law (42) are the only changes needed to incorporate unequal constant tension in the two spans. Remark 3. The .rst term in the control force fc is a feedforward term that cancels the transverse component of the tension P0 due to the small string angles us+ and us− . The feedforward term must− cancels a term in m . ˙ The last term of fc is a feedback term that ensures the closed-loop system boundary dynamics become m ˙ + ks = 0. Thus, goes to zero exponentially and ut → −us− at s = sc , producing a damped boundary condition for the controlled span.
(60)
− ˆ:
(64)
We de.ne the Lyapunov functional Vca (t) = Es− + Ec− + 12 m 2 +
1 2
˜ T !−1 ˜ ¿ 0
(65)
and, as in the previous proof, 0 6 1a (Es− + 2 + ˜ 2 ) 6 Vca (t) 6 2a (Es− + 2 + ˜ 2 );
(66)
where · indicates the Euclidean norm and m 1a = min 1 − !; ; !−1 ¿ 0; 2 (67) m 2a = max 1 + !; ; !−1 ¿ 1: 2 Repetition of the simplifying steps in the previous proof and substitution of the adaptive law (61) yields V˙ ca (t) 6 − 3e (Es− + 2 ) = − g 6 0:
(68)
Eqs. (65) and (68) mean that Vca (t) 6 Vca (0), or Vca (t) ∈ L∞ . Therefore, ; ˜ ; Es− and ˆ ∈ L∞ , res sulting in u(s) ∈ L∞ for s ∈ [0; sc ] and 0 c ut2 ds ∈ L∞ .
Y. Li et al. / Automatica 38 (2002) 379–390
Under the assumptions of the previous proof, ut , utt , us , ust and uss ∈ L∞ for s ∈ [0; sc ], so sc sc 1 ˙− Es = ut utt ds + P0 us ust ds ∈ L∞ : (69) 0
0
us+
From Lemma 1, ∈ L∞ . Using Eqs. (63) and (64) yields ˙ ∈ L∞ . This results in −
˙ ∈ L∞ : g(t) ˙ = − 3 (E˙ s + 2 )
(70)
Thus g → 0 as t → ∞ (Slotine, Weiping, & Li, 1991) so u(s; t) → 0 as t → ∞ for all s ∈ [0; sc ]. Eq. (61) shows fc ∈ L ∞ .
4. Experimental veri'cation While the theory applies for many actuators, we use a dancer arm actuator in the experiments. The torque & applied to the dancer arm and dancer arm inertia relate to the actuator model force and mass as follows: f = &l; m=
J ; l2
where l is the length of the dancer arm and J is the dancer arm rotary inertia with respect to the motor shaft. Fig. 2 shows the belt, drive motor, pulleys, dancer arms, and laser sensors used to experimentally verify the proposed control algorithms and compare with standard PID control. The string model is appropriate for the heavy, low bending sti4ness belt operating under moderate tension (8 N). The drive motor and pulleys
385
move the belt at 1:0 m=s during all tests. The excitation frequency ! equals the .rst natural frequency of the open loop system (10 Hz). A variety of excitation frequencies and amplitudes were tested with similar results. The motor driven disturbance arm provides repeatable, periodic forcing fu = A sin(!t) to the uncontrolled span. The other dancer arm is actuated by a brushed DC motor that provides the control torque. The controllers given by Eqs. (42) and (61) require measurement of the angles on either side of the actuator, the angular rate on the controlled side of the actuator, and the angular rate of the actuator. An encoder on the actuator motor measures the actuator angle. Backwards di4erencing of the encoder signal provides the actuator angular rate. Laser sensors measure the belt displacement on either side of the actuator. From the actuator angle and dancer arm length, we can calculate actuator displacement. We assume the angle is constant between the actuator and the laser sensor. Backwards di4erencing of the controlled-side angle provide ust− . A mechatronic workstation consisting of a Pentium 166 MHz personal computer (PC) running QNX operating system and the software Qmotor implements the control algorithms. Filtered backwards differencing for velocity measurements and trapezoidal integration are used. Fig. 3 shows the open loop response to an impulse disturbance near the controlled span pulley provided by an impact hammer (Fig. 3(a) – (c)) and the periodic dancer arm disturbance input on the uncontrolled span (Fig. 3(d) – (f)). The angle response on the controlled side has a 1:1◦ root mean squared (RMS) amplitude. Table 1 summarizes the control gains and performance of the open loop and controlled experiments.
Fig. 2. Experimental setup.
386
Y. Li et al. / Automatica 38 (2002) 379–390
Fig. 3. Open loop response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of us (sc ; t); (d) periodic disturbance response of u(sc ; t); (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span). Table 1 Controller gains and performance Controller
Gains
Performance (RMS)
Open loop
NA
us (sc− ; t) = 0:49◦ , u(sc ; t) = 0:40 mm
PID
Kp = 10; Kd = 9:7, Ki = 2:6
us (sc− ; t) = 0:22◦ , u(sc ; t) = 0:09 mm, Voltage = 0:65 V
Exact model knowledge
ks = 5; a = 1 P0 = 8 N; J = 0:0002 kg m2
us (sc− ; t) = 0:20◦ , u(sc ; t) = 0:12 mm, Voltage = 0:21 V
Adaptive
ks = 5; a = 0:8 .p = 2; .J = 1:1
us (sc− ; t) = 0:22◦ , u(sc ; t) = 0:11 mm, Voltage = 0:27 V
4.1. PID control For comparison purposes, Fig. 4 shows the system response using simple PID control as follows: t u(sc ; &) d&; (71) fc = Kp u(sc ; t) + Kd ut (sc ; t) + KI 0
where the control gains are designed to provide the best disturbance rejection to the periodic input from the uncontrolled span. The PID control signi.cantly reduces the
quiet part response (e) from 0:49◦ (RMS) to approximately 0:22◦ (RMS), and reduces the actuator response (g) to approximately 0:09 mm (RMS) using the 0:65 V (RMS) control voltage (h) in response to the periodic disturbance. However, the PID controller reduces the damping associated with the impulse disturbance (a) using the control voltage (d) with a longer transient response time (2 s) than open loop (0:7 s). 4.2. Exact model knowledge control Fig. 5 shows the exact model knowledge control response to the same impulsive and periodic disturbance as the PID control experiment. The quiet part transient resulting from impulse inputs (a) decays more than 3 times faster (0:6 s) than PID using the smaller control voltage (d) and also slightly faster than open loop. The controlled span (Fig. 5(e)) and dancer arm (Fig. 5(g)) responses are approximately 0:2◦ (RMS) and 0:12 mm (RMS), respectively, to the periodic input using a signi.cantly reduced 0:21 V (RMS) control voltage (d). 4.3. Adaptive control Fig. 6 shows the adaptive control response to the same disturbances as the PID control experiment. After a
Y. Li et al. / Automatica 38 (2002) 379–390
387
Fig. 4. PID control response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of u(sc ; t); (d) impulse response of control voltage; (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span); (g) periodic disturbance response of u(sc ; t); (h) periodic disturbance response of control voltage.
parameter convergence transient, the controlled angle (a) is less than 0:23◦ (RMS) and the dancer arm displacement is less than 0:11 mm in response to the periodic inputs, using the 0:27 V (RMS) control input (d). Fig. 6(e) and (f) show the convergence of the system parameters from zero to near the actual values of 8 N and 0:0002 kg m2 for P0 and J , respectively. After parameter convergence, the impulse response is almost identical to the EMK result. Fig. 7 shows the adaptive control response to the impulse disturbance. The quiet part transient resulting from impulse inputs (a) decays 2 times faster (1 s) than PID control. 4.4. Results and discussion The objectives of the proposed control approach are to isolate the controlled span from bounded disturbances in the uncontrolled span and damp vibration in the controlled span. The controller acts as a one-way valve, con.ning vibration energy in the uncontrolled span and eliminating vibration energy from the controlled span. The actuator used in this research applies a force to control the displacement of the axially moving string at the actuating point. If the actuator is locked in place, it works as a pinned boundary, and, for a perfectly Cexible
string, the controlled span is then isolated from disturbances in the uncontrolled span, completely decoupling the controlled and uncontrolled spans. Fig. 4(c) and (g) show that PID control e4ectively regulates the actuator displacement to zero, eliminating the coupling between the two spans and isolating the controlled span from disturbances in the uncontrolled span. The vibration in the uncontrolled span (Fig. 4(f)) is clearly larger than the controlled span but some residual vibration remains in the controlled span. Vibration propagation around the backside of the belt loop, bending sti4ness coupling, and once per cycle excitation from the belt splice cause the residual vibration. The PID controller can be thought of as the best-case vibration isolator. It is, therefore, expected that the EMK and adaptive controllers should have similar vibration isolation performance as PID control. Note, however, that the EMK and adaptive controllers require signi.cantly less control e4ort than PID control. PID control reduces the damping in the controlled span when compared to open loop (see Figs. 3(a) and 4(a)), violating a major control objective. In order to obtain best-case vibration isolation, the PID controller pins the string, eliminating the energy dissipation associated with actuator friction. Fig. 4(c) shows little
388
Y. Li et al. / Automatica 38 (2002) 379–390
Fig. 5. Exact model knowledge control response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of u(sc ; t); (d) impulse response of control voltage; (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span); (g) periodic disturbance response of u(sc ; t); (h) periodic disturbance response of control voltage.
Fig. 6.
Y. Li et al. / Automatica 38 (2002) 379–390
389
Fig. 7. Adaptive control response to impulse disturbance: (a) us (sc− ; t) (controlled span); (b) u(sc ; t); (c) control voltage.
actuator motion in response to the impulse input on the controlled span. Reduction of the control gains increases vibration damping but reduces vibration isolation. The EMK and adaptive controllers, however, provide much improved damping (Figs. 5(a) and 7(a)) without sacri.cing isolation performance. The actuator moves in response to the impulse disturbance on the controlled side (Fig. 5(c)) but not the periodic disturbance on the uncontrolled side (Fig. 5(g)). The proposed control approach is limited by the modeling assumptions. First, the distributed aerodynamic damping assumption guarantees bounded control e4ort, but for suIciently high disturbance and=or low damping, the actuator may saturate, reducing the isolation e4ectiveness. Of course, we can add additional damping (passive or active) to the uncontrolled span or substitute a more powerful actuator to prevent actuator saturation due to high disturbances. Second, the control system cannot operate above the critical speeds of either span. Finally, the transport speed, tension, and density of the material are assumed constant. The adaptive controller, however, can compensate for slowly varying parameters. 5. Conclusions For bounded disturbances in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively. With damping or non-resonant excitation, the uncontrolled span displacement and control force are bounded. The adaptive controller also estimates the string tension and actuator mass, allowing on-line monitoring of the process parameters. Experimental results demonstrate that the exact model knowledge and adaptive controllers outperform PID in impulse response damping and disturbance rejection control eIciency. ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Fig. 6. Adaptive control response to periodic disturbance: (a) us (sc− ; t) (controlled span); (b) us (sc+ ; t) (uncontrolled span); (c) u(sc ; t); (d) control voltage; (e) actuator inertia estimate Jˆ; (f) tension estimate Pˆ 0 .
Acknowledgements This research was supported by the National Science Foundation, Union Camp Corporation, and Square-D Corporation.
References Baicu, C. F., Rahn, C. D., & Nibali, B. D. (1996). Active boundary control of elastic cables: Theory and experiment. Journal of Sound and Vibration, 198(1), 17–26. Clark, W. W., & Robertshaw, H. H. (1997). Force feedback in adaptive trusses for vibration isolation in Cexible structures. Journal of Dynamic Systems, Measurement, and Control, 119, 365–371. Ertur, D., Li, Y., & Rahn, C. (1999). Adaptive vibration isolation for Cexible structures. ASME Journal of Vibration and Acoustics, 121(4), 440–445. Fung, R.-F., Wu, J.-W., & Wu, S.-L. (1999). Exponential stabilization of an axially moving string by linear boundary feedback. Automatica, 35, 177–181. Joshi, S., & Rahn, C. D. (1995). Position control of a Cexible gantry crane: Theory and experiment. Proceedings of the American Control Conference (pp. 2820 –2824). Seattle, WA. Karnopp, D. (1995). Active and semi-active vibration isolation. Transactions of the ASME, 117, 177–185. Lee, S. Y., & Mote, Jr., C. D. (1996). Vibration control of an axially moving string by boundary control. ASME Journal of Dynamic Systems, Measurement, and Control, 118, 66–74. MorgAul, O. (1994). A dynamic control law for the wave equation. Automatica, 30(11), 1785–1792. Queiroz, M. de, Dawson, D., Rahn, C., & Zhang, F. (1999). Adaptive vibration control of an axially moving string. ASME Journal of Vibration and Acoustics, 121, 41–49. Renshaw, A., Rahn, C., Wickert, J., & Mote Jr., C. D. (1998). Energy and conserved functionals for axially moving materials. ASME Journal of Vibration and Acoustics, 120(2), 634–636. Shahruz, M. S. (1997). Suppression of vibration in a nonlinear axially moving string by boundary control. ASME Design Engineering Technical Conferences, 106(1), 6–14. Slotine, J. J. E., Weiping, & Li, (1991). Applied Nonlinear Control (p. 407). Englewood Cli4s, NJ: Prentice-Hall. Ulsoy, A. G. (1984). Vibration control in rotating or translating elastic systems. ASME Journal of Dynamic Systems, Measurement, and Control, 106(1), 6–14. Yang, B., & Mote, Jr., C. D. (1991). Active vibration control of the axially moving string in the S domain. ASME Journal of Applied Mechanics, 58, 189–196.
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Y. Li et al. / Automatica 38 (2002) 379–390 Christopher D. Rahn was born in Bellefonte, Pennsylvania in 1963. He graduated from the University of Michigan with a Bachelors degree in mechanical engineering in 1985. He then obtained a Masters degree from the University of California, Berkeley in 1986. After three years as a Research and Development Engineer at Space Systems=LORAL, he returned to Berkeley to pursue a
Ph.D. After graduating from Berkeley in 1992, Dr. Rahn joined the Department of Mechanical Engineering at Clemson University. In 2000, Dr. Rahn became an Associate Professor of Mechanical Engineering at the Pennsylvania State University. His research interests include dynamic modeling, analysis, and control for distributed nonlinear systems with application to robotics, mechatronics, and manufacturing.
www.elsevier.com/locate/automatica
Adaptive vibration isolation for axially moving strings: theory and experiment Yugang Lia , Dan Arona , Christopher D. Rahnb; ∗ a Department
b Department
of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA of Mechanical and Nuclear Engineering, The Pennsylvania State University, 150A Hammond Building, University Park, PA 16802, USA Received 14 April 1999; revised 24 March 2000; received in .nal form 24 August 2001
Abstract High-speed transport of continuous materials such as belts, webs, .laments, or bands can cause unwanted vibration. Vibration control for these systems often focuses on restricting the response resulting from external disturbances (e.g. support roller eccentricity or aerodynamic excitation) to areas not requiring high precision positioning. This paper introduces vibration controllers for an axially moving string system consisting of a controlled span coupled to a disturbed span via an actuator. The system model includes a partial di4erential equation for the two spans and an ordinary di4erential equation for the actuator. Exact model knowledge and adaptive isolation controllers, based on Lyapunov theory, regulate the controlled span from bounded disturbances in the adjacent, uncontrolled span. Assuming distributed damping in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively, while ensuring bounded uncontrolled span displacement and control force. Experiments demonstrate the e4ectiveness of the proposed controller in isolating the controlled span from disturbances and damping the controlled span displacement. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Distributed parameter systems; Axially moving string; Vibration isolation; Lyapunov theory
1. Introduction Continuous materials such as belts, webs, .laments, or bands, can vibrate excessively during high-speed axial transport, reducing the productivity of the associated manufacturing process and quality of the .nished product. Support roller eccentricity, material non-uniformity, aerodynamic disturbances, or manufacturing processes (e.g. slitting, calendering, or printing) can cause the unwanted vibration. Active vibration control has the potential to reduce vibration and improve the axial transport process. Ulsoy (1984) develops the .rst controllers for an axially moving system. An observer-based state feedback algorithm based on a reduced-order, discretized model of the in.nite dimensional, axially moving string system is shown to be a viable method for active vibration control. This
paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +1-814-865-6237; fax: +1-814-863-7222. E-mail address: [email protected] (C.D. Rahn).
The research demonstrates, however, the possible instabilities resulting from control and observation spillover when point sensors and actuators drive the closed-loop system unstable by measuring and forcing high frequency modes not included in the reduced-order system model. To prevent these instabilities, Yang and Mote (1991) develop a class of asymptotically stabilizing controllers for distributed parameter models of axially moving strings and beams. The use of a quasi-passive controller along with a carefully positioned sensor and actuator ensure asymptotic decay of all vibration modes. A number of researchers have developed boundary controllers based on distributed parameter models of strings. MorgAul (1994) introduces dynamic boundary feedback controllers for the wave equation that include proportional and strictly positive real derivative feedback. Joshi and Rahn (1995) develop and experimentally implement boundary controllers for a linear gantry crane model with a Cexible cable. Baicu, Rahn, and Nibali (1996) experimentally demonstrate how a three-term boundary control law can be used to stabilize the out-of-plane vibration of a Cexible cable. Renshaw,
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 1 9 - 9
380
Y. Li et al. / Automatica 38 (2002) 379–390
Rahn, Wickert, and Mote (1998) illustrate that Lyapunov theory can be applied to axially moving strings without using the expected material derivative in the time differentiation of the Lyapunov functional. Lee and Mote (1996), Shahruz (1997), and Fung, Wu, and Wu (1999) develop boundary control laws for axially moving strings. The use of boundary control to damp vibration in an axially moving system can be diIcult to implement in practice. The size and weight of the support=guide rollers and their inherent dynamic coupling with adjacent spans may limit the applicability of boundary control in axially moving systems. Queiroz, Dawson, Rahn, and Zhang (1999) overcome these practical limitations by developing exact model knowledge and adaptive control algorithms that damp the vibration of a distributed axially moving string system using a two-degree-of-freedom actuator located between the support rollers. It is often desirable to isolate disturbances from quiet parts of a system that require precise positioning without vibration. The machine or process is carefully designed to ensure minimal disturbances apply to the quiet part, often at great expense. Other parts of the structure can vibrate appreciably without adversely a4ecting system performance. Coupling in the system, however, causes this vibration to propagate to the quiet part. In web handling machines, for example, great care is taken to minimize disturbances in printing regions where precise positioning is required. The rollers are carefully balanced and aerodynamic excitation sources are minimized. Other parts of the web handling machines are less sensitive to vibration and do not require such expensive measures. Many researchers have studied vibration isolation for systems other than axially moving strings. Karnopp (1995) shows how passive methods can be used for vibration isolation through optimization of system parameters (e.g. spring and damping constants) and addition of vibration absorbers. Using an adaptive truss within
a Cexible steel structure, Clark and Robertshaw (1997) actively controlled the structure vibration without the use of a system model. Ertur, Li, and Rahn (1999) use adaptive vibration isolation control strategies based on Lyapunov theory to regulate quiet parts of a discrete system from non-colocated disturbances. The objective of this paper is to develop a vibration isolation system for web handling machines based on a distributed axially moving string model. Using a distributed control approach based on Lyapunov’s method eliminates the possibility of spillover instability. The system model is a pinned–pinned axially moving string divided into two spans by a transverse force actuator. Exact model knowledge and adaptive isolation controllers, based on Lyapunov theory, regulate a (controlled) span from bounded disturbances in the adjacent (uncontrolled) span. Assuming distributed damping in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively, while ensuring bounded uncontrolled span displacement and control force. Experiments demonstrate the e4ectiveness of the proposed controllers in isolating the controlled span from disturbances and in damping the controlled span displacement. 2. System model Fig. 1 shows a string moving at constant speed v between pinned supports. An actuator consisting of force fc (t) and mass m controls the string displacement u(s; t) for s ∈ (0; sc ) (controlled span) where s is the material position, t is time, and sc is the actuator location. A disturbance force fu (t) and boundary displacement uL (t) act on the uncontrolled span s ∈ (sc ; L), where L is the total span length. We assume that fu (t) and uL (t); up to second time derivatives, are bounded (fu ; uL ; u˙ L ; uA L ∈ L∞ ).
Fig. 1. Schematic diagram of the axially moving string system.
Y. Li et al. / Automatica 38 (2002) 379–390
The kinetic energy of the system is sc 1 (vus + ut )2 ds T= 2 0 L 1 1 (vus + ut )2 ds + mut2 (sc ; t); + 2 sc 2
(1)
where and v are the constant mass=length and transport speed of the string, respectively, and subscripts indicate partial di4erentiation. The kinetic energy of the axial moving string divides into the .rst two terms in Eq. (1) because of the discontinuity in us (s; t) at s = sc . The third term in Eq. (1) comes from the actuator mass. Under assumptions of small string displacement and perfect Cexibility, the system potential energy is sc L 1 1 us2 ds + P us2 ds; (2) V= P 2 0 2 sc where P is the constant string tension. The virtual work done by the external forces is L L (3) W = fc u(sc ; t) + fu u ds − cut u ds; sc
0
where c is a viscous damping coeIcient. We assume distributed relative velocity damping throughout the span. Other damping models (e.g. absolute velocity, material, or boundary damping) could also be assumed. Substitution of Eqs. (1) – (3) into Hamilton’s Principle, t1 (T − V + W ) dt = 0; (4)
knowledge. In Theorem 2, we illustrate how the control law can be redesigned as an asymptotically regulating adaptive controller that compensates for uncertainty in the system parameters P0 and m. First, Lemma 1 shows that the uncontrolled span displacement is bounded if u(sc ; t) and its derivatives are bounded. 3.1. Boundedness of the uncontrolled span Lemma 1. If fu (t) ∈ L∞ and u(sc ; t) and uL (t) ∈ L∞ up to second time derivatives; then u(s; t) ∈ L∞ for s ∈ [sc ; L]. Proof. Substitution of s − sc s−L w(s; t) = u(s; t) − u(sc ; t) uL (t) − L − sc sc − L for s ∈ [sc ; L] wtt + 2 vwst − P0 wss + cwt = y(s; t)
(5)
and the boundary conditions: u(0; t) = 0;
u(L; t) = uL (t);
u(sc− ; t) = u(sc+ ; t); mutt (sc ; t) + P0 (us− − us+ ) = fc ;
(6) (7) (8)
where us− = us (sc− ; t); us+ = us (sc+ ; t), H (s) is the Heaviside step function, and we assume P0 = P − v2 ¿ 0. 3. Control formulation The control objective is to drive the string displacement u(s; t) for s ∈ (0; sc ) to zero with a bounded control force fc . In Theorem 1, we design an exponentially regulating control law for the system based on exact model
(10)
with boundary conditions w(sc ; t) = w(L; t) = 0;
(11)
where y(s; t) = fu −
(s − sc ) (s − L) uA L (t) − u(s A c ; t) (L − sc ) (sc − L)
2 v[u˙ L (t) − u(s ˙ c ; t)] L − sc s − sc s−L u(s ˙ c ; t) −c u˙ L (t) − L − sc sc − L −
yields the .eld equation: for s ∈ (0; sc ) ∪ (sc ; L)
(9)
into Eq. (5) yields
t0
utt + 2 vust + cut − P0 uss = fu H (s − sc )
381
(12)
is bounded by the lemma assumptions. A Lyapunov functional for the uncontrolled span is chosen to be Vu (t) = Es+ (t) + Ec+ (t); where 1 Es+ = 2
and Ec+
= 1
L
sc
L
sc
1 wt2 ds + P0 2
(13)
L
sc
ws2 ds
wwt ds:
with 1 ¿ 0 a small positive constant. From Queiroz et al. (1999), since u(0) = 0, L 4L2 L 2 u2 d x 6 2 u d x for x ∈ (0; L): 0 x 0
(14)
(15)
(16)
Substitution of inequality (16) and 2uv 6 u2 + v2
(17)
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Y. Li et al. / Automatica 38 (2002) 379–390
The terms of the second integral in Eq. (27) simplify via integration by parts, the boundary conditions, and inequalities (2) and (4) as follows: L L wP0 wss ds = − P0 ws2 ds; (28)
into Eq. (15) yields L + |Ec (t)| 6 1 (w2 + wt2 ) ds sc
6 1
L
sc
wt2 +
4L2 2 w 2 s
ds
max(1; 4L2 =2 ) + Es (t): 6 21 min( ; P0 )
sc
(18)
L
sc
min( ; P0 ) ; 2 max(1; 4L2 =2 )
L
sc
wwt ds 6
wy(sc ; t) ds 6
−2 v
0 6 1 Es+ (t) 6 Vu (t) 6 2 Es+ (t);
(20)
where 1 = 1 − 21 2 = 1 + 2
max(1; 4L2 =2 ) ¿ 0; min( ; P0 )
max(1; 4L2 =2 ) ¿ 1: min( ; P0 )
(21) (22)
The time derivative of Vu (t) + + V˙ u (t) = E˙ s + E˙ c
(23)
depends on L L + 2 ˙ Es = − cwt ds + wt y(s; t) ds; sc
sc
1 2uv 6 u + v2 produces
for any ¿ 0
6 − (c − )
L
sc
1 wt2 ds +
(24)
L
sc
= 1
sc
y2 (s; t) ds:
(26)
+ 1
L
sc
sc
1 2
wt2 ds + c1 L
4L2 2
y2 (sc ; t) ds +
sc
L
sc
ws2 ds;
4L2 2 2
L
sc
wwst ds 6
v 3
L
sc
wt2 ds + v3
L
sc
(29)
ws2 ds; (30)
ws2 ds; (31)
where 1 ; 2 , and 3 are arbitrary positive constants. Substitution of Eqs. (26) – (31) into Eq. (23) yields c v ˙ + V u (t) 6 − c − − 1 + 1 3 L L 1 1 + wt2 ds + y2 (sc ; t) ds 2 sc sc 2 2 4L 4L −1 P0 − c 2 1 − 2 2 − v3 L ws2 ds 6 − 3 Es+ (t) + ; (32) where
3 = min
c − − 1 +
c 1
+
v 3
2 1 P0 − c 4L 2 1 −
P0
4L2 2 2
; − v3
¿0
if 1 ; 1 ; 2 , and 3 are suIciently small, and L 1 1 2 + max y (s; t) ds ¡ ∞ for all t = 2 sc (34) from Eq. (12). Use of Eq. (20) produces
wt2 ds
L
(33)
sc
L
L
sc
(25)
The remaining term in V˙ u (t) is L + ˙ E c = 1 (wt2 + wwtt ) ds
sc
where the .eld equation (5), boundary conditions (11), and integration by parts have been applied. Application of inequality 2
c 1
(19)
then
+ E˙ s
−c
If we choose 1 ¡
sc
V˙ u (t) 6 − Vu (t) + ;
w(P0 wss − cwt + y(sc ; t) − 2 vwst ) ds: (27)
(35)
where =
3 : 2
(36)
Y. Li et al. / Automatica 38 (2002) 379–390
Solution of Eq. (35) produces Vu (t) 6 Vu (0)e−t + ∈ L∞ :
|Ec− | 6 2 sc
(37)
Using the inequality l 2 u (x) 6 l us2 (s) ds for x ∈ (0; l) 0
if u(0) = 0
∈ L∞ :
sc
0
6 2 sc
sc
0
|us ut + vus2 | ds
(ut2 + (1 + v)us2 ) ds
1 sc 2 (ut + us2 ) ds 2 0 sc 4 sc (1 + v) 1 2 2 6 ( ut + P0 us ) ds ; min( ; P0 ) 2 0
6 4 sc (1 + v)
(38)
in Eq. (20) and using Eq. (37) then yields L 1 P0 w 2 6 1 P0 ws2 ds 6 1 Es+ (L − sc ) sc 6 Vu (t) 6 Vu (0)e−t +
383
so (39)
Thus, w ∈ L∞ , so u(s; t) ∈ L∞ for s ∈ [sc ; L]. 3.2. Exact model knowledge control law
−!Es− 6 Ec− 6 !Es− ;
(48)
where 4 sc (1 + v) ¡ 1; != min( ; P0 )
(49)
if is suIciently small. Thus,
To facilitate the subsequent control design and analysis, we de.ne an auxiliary variable,
1e (Es− + 2 ) 6 Vc (t) 6 2e (Es− + 2 );
(t) = ut (sc ; t) + us− :
m ¿ 0; 1e = min 1 − !; 2 m ¿ 1: 2e = max 1 + !; 2 The time derivative of Eq. (44) is
(40)
After di4erentiation of (t) with respect to time, multiplication by m, and substitution of the boundary condition (8), Eq. (40) becomes m (t) ˙ = fc + P0 (us+ − us− ) + must− :
(41)
Theorem 2. The bounded control force fc = − P0 (us+ − us− ) − must− − ks
(42)
Proof. For simplicity, we neglect the stabilizing e4ect of damping in the controlled span. Substitution of the exact model knowledge (EMK) controller (42) into Eq. (41) yields (43)
where Es− = and
1 2
+
sc
0
Ec− = 2
sc
Ec− (t)
+
2 1 2 m ;
1 ut2 ds + P0 2
sc
0
us2 ds
(45)
= − vut2 (sc ) −
− P0
P0 2 P0 (u (sc ) + us2 (sc− )) + 2 2 t 2
(54)
0
0
sc
us2 ds
(55)
so
P0 ˙ V c (t) = − ks − 2 2 sc 2 − ut ds + P0 0
−
with and small positive constants. Inequality (17) and integration by parts bound Eq. (46) as follows:
−
0
(53)
and sc sc 1 ˙− E c = 2 sust ut ds + 2P0 sus uss ds 0 0 sc = sc ut2 (sc ) − ut2 ds + P0 sc us2 (sc− )
(46)
sus (ut + vus ) ds
(52)
Thus, 1 ˙− E s = − vut2 (sc ) + P0 ut (sc )us (sc− )
(44)
(51)
− − V˙ c (t) = E˙ s + E˙ c − ks 2 :
A Lyapunov functional is de.ned as Vc (t) = Es− (t)
(50)
where
exponentially stabilizes the string displacement for s ∈ [0; sc ].
m (t) ˙ = − ks :
(47)
0
sc
us2
ds
P0 − sc ut2 (sc ) 2 − P0 sc us2 (sc− ):
v +
P0 2
(56)
384
Y. Li et al. / Automatica 38 (2002) 379–390
3.3. Adaptive control law
If we select ¿ 2sc , sc ( ut2 + P0 us2 ) ds V˙ c (t) 6 − ks − P0 2 − 2 0 6 −3e (Es + 2 ) 6 −
where 3e = min
2
ks − P0 ; 2
3e Vc (t); 2e
(57)
The EMK control law (42) depends on the exact knowledge of the system parameters m and P0 . To compensate for unknown or slowly time-varying parameters, an adaptive controller is developed. The open loop dynamics for the variable (t) are rewritten as
¿0
(58)
m (t) ˙ = fc + W ;
if ¡ 2ks =P0 . Thus 3e Vc (t) 6 Vc (0) exp − t 2e and using inequality (38) sc P0 1e 2 1e P0 u 6 us2 ds 6 1e Es− 6 Vc (t) 2sc 2 0 3e 6 Vc (0) exp − t 2e and u(s; t) → 0 for s ∈ [0; sc ] as t → ∞. We assume that u(s; t) belongs to a space s of functions where u ∈ L∞ implies us ; uss ∈ L∞ and 0 c ut2 ds ∈ L∞ implies ut ; ust ∈ L∞ . Thus, u(s; t); ut (s; t); us (s; t); ust (s; t), uss (s; t) ∈ L∞ for x ∈ [0; sc ]. The .eld equation (5) then implies utt ∈ L∞ for x ∈ [0; sc ]. Finally, from Lemma 1, us+ ∈ L∞ . Thus, all terms in the exact model knowledge control law are bounded, so fc ∈ L∞ .
(59)
where = [m; P0 ]T ;
W = [ust− ; us+ − us− ]:
The adaptive controller asymptotically stabilizes the controlled span displacement and estimates the unknown parameter vector . Theorem 3. The bounded control law fc = − W ˆ − ks
(61)
and the adaptation law ˆ˙ = !WT ;
(62)
where # = #T ¿ 0; ensure u(s; t) → 0 for s ∈ [0; sc ] as t → ∞. Proof. Substitution of Eq. (61) into Eq. (59) produces m (t) ˙ = − ks + W ˜ ;
(63)
where ˆ = [m; ˆ Pˆ 0 ]T
Remark 1. Rede.nition of
and the parameter estimation error
= ut (sc ; t) + us−
˜ =
does not alter the closed loop stability. The additional gain allows .ne tuning of the controller. Remark 2. Friction at the actuator could raise the tension in the uncontrolled span relative to the controlled span. Substitution of the upstream P0− and downstream P0+ tension in the .eld equation (5), boundary condition (8), functionals (13) and (44), and control law (42) are the only changes needed to incorporate unequal constant tension in the two spans. Remark 3. The .rst term in the control force fc is a feedforward term that cancels the transverse component of the tension P0 due to the small string angles us+ and us− . The feedforward term must− cancels a term in m . ˙ The last term of fc is a feedback term that ensures the closed-loop system boundary dynamics become m ˙ + ks = 0. Thus, goes to zero exponentially and ut → −us− at s = sc , producing a damped boundary condition for the controlled span.
(60)
− ˆ:
(64)
We de.ne the Lyapunov functional Vca (t) = Es− + Ec− + 12 m 2 +
1 2
˜ T !−1 ˜ ¿ 0
(65)
and, as in the previous proof, 0 6 1a (Es− + 2 + ˜ 2 ) 6 Vca (t) 6 2a (Es− + 2 + ˜ 2 );
(66)
where · indicates the Euclidean norm and m 1a = min 1 − !; ; !−1 ¿ 0; 2 (67) m 2a = max 1 + !; ; !−1 ¿ 1: 2 Repetition of the simplifying steps in the previous proof and substitution of the adaptive law (61) yields V˙ ca (t) 6 − 3e (Es− + 2 ) = − g 6 0:
(68)
Eqs. (65) and (68) mean that Vca (t) 6 Vca (0), or Vca (t) ∈ L∞ . Therefore, ; ˜ ; Es− and ˆ ∈ L∞ , res sulting in u(s) ∈ L∞ for s ∈ [0; sc ] and 0 c ut2 ds ∈ L∞ .
Y. Li et al. / Automatica 38 (2002) 379–390
Under the assumptions of the previous proof, ut , utt , us , ust and uss ∈ L∞ for s ∈ [0; sc ], so sc sc 1 ˙− Es = ut utt ds + P0 us ust ds ∈ L∞ : (69) 0
0
us+
From Lemma 1, ∈ L∞ . Using Eqs. (63) and (64) yields ˙ ∈ L∞ . This results in −
˙ ∈ L∞ : g(t) ˙ = − 3 (E˙ s + 2 )
(70)
Thus g → 0 as t → ∞ (Slotine, Weiping, & Li, 1991) so u(s; t) → 0 as t → ∞ for all s ∈ [0; sc ]. Eq. (61) shows fc ∈ L ∞ .
4. Experimental veri'cation While the theory applies for many actuators, we use a dancer arm actuator in the experiments. The torque & applied to the dancer arm and dancer arm inertia relate to the actuator model force and mass as follows: f = &l; m=
J ; l2
where l is the length of the dancer arm and J is the dancer arm rotary inertia with respect to the motor shaft. Fig. 2 shows the belt, drive motor, pulleys, dancer arms, and laser sensors used to experimentally verify the proposed control algorithms and compare with standard PID control. The string model is appropriate for the heavy, low bending sti4ness belt operating under moderate tension (8 N). The drive motor and pulleys
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move the belt at 1:0 m=s during all tests. The excitation frequency ! equals the .rst natural frequency of the open loop system (10 Hz). A variety of excitation frequencies and amplitudes were tested with similar results. The motor driven disturbance arm provides repeatable, periodic forcing fu = A sin(!t) to the uncontrolled span. The other dancer arm is actuated by a brushed DC motor that provides the control torque. The controllers given by Eqs. (42) and (61) require measurement of the angles on either side of the actuator, the angular rate on the controlled side of the actuator, and the angular rate of the actuator. An encoder on the actuator motor measures the actuator angle. Backwards di4erencing of the encoder signal provides the actuator angular rate. Laser sensors measure the belt displacement on either side of the actuator. From the actuator angle and dancer arm length, we can calculate actuator displacement. We assume the angle is constant between the actuator and the laser sensor. Backwards di4erencing of the controlled-side angle provide ust− . A mechatronic workstation consisting of a Pentium 166 MHz personal computer (PC) running QNX operating system and the software Qmotor implements the control algorithms. Filtered backwards differencing for velocity measurements and trapezoidal integration are used. Fig. 3 shows the open loop response to an impulse disturbance near the controlled span pulley provided by an impact hammer (Fig. 3(a) – (c)) and the periodic dancer arm disturbance input on the uncontrolled span (Fig. 3(d) – (f)). The angle response on the controlled side has a 1:1◦ root mean squared (RMS) amplitude. Table 1 summarizes the control gains and performance of the open loop and controlled experiments.
Fig. 2. Experimental setup.
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Fig. 3. Open loop response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of us (sc ; t); (d) periodic disturbance response of u(sc ; t); (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span). Table 1 Controller gains and performance Controller
Gains
Performance (RMS)
Open loop
NA
us (sc− ; t) = 0:49◦ , u(sc ; t) = 0:40 mm
PID
Kp = 10; Kd = 9:7, Ki = 2:6
us (sc− ; t) = 0:22◦ , u(sc ; t) = 0:09 mm, Voltage = 0:65 V
Exact model knowledge
ks = 5; a = 1 P0 = 8 N; J = 0:0002 kg m2
us (sc− ; t) = 0:20◦ , u(sc ; t) = 0:12 mm, Voltage = 0:21 V
Adaptive
ks = 5; a = 0:8 .p = 2; .J = 1:1
us (sc− ; t) = 0:22◦ , u(sc ; t) = 0:11 mm, Voltage = 0:27 V
4.1. PID control For comparison purposes, Fig. 4 shows the system response using simple PID control as follows: t u(sc ; &) d&; (71) fc = Kp u(sc ; t) + Kd ut (sc ; t) + KI 0
where the control gains are designed to provide the best disturbance rejection to the periodic input from the uncontrolled span. The PID control signi.cantly reduces the
quiet part response (e) from 0:49◦ (RMS) to approximately 0:22◦ (RMS), and reduces the actuator response (g) to approximately 0:09 mm (RMS) using the 0:65 V (RMS) control voltage (h) in response to the periodic disturbance. However, the PID controller reduces the damping associated with the impulse disturbance (a) using the control voltage (d) with a longer transient response time (2 s) than open loop (0:7 s). 4.2. Exact model knowledge control Fig. 5 shows the exact model knowledge control response to the same impulsive and periodic disturbance as the PID control experiment. The quiet part transient resulting from impulse inputs (a) decays more than 3 times faster (0:6 s) than PID using the smaller control voltage (d) and also slightly faster than open loop. The controlled span (Fig. 5(e)) and dancer arm (Fig. 5(g)) responses are approximately 0:2◦ (RMS) and 0:12 mm (RMS), respectively, to the periodic input using a signi.cantly reduced 0:21 V (RMS) control voltage (d). 4.3. Adaptive control Fig. 6 shows the adaptive control response to the same disturbances as the PID control experiment. After a
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Fig. 4. PID control response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of u(sc ; t); (d) impulse response of control voltage; (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span); (g) periodic disturbance response of u(sc ; t); (h) periodic disturbance response of control voltage.
parameter convergence transient, the controlled angle (a) is less than 0:23◦ (RMS) and the dancer arm displacement is less than 0:11 mm in response to the periodic inputs, using the 0:27 V (RMS) control input (d). Fig. 6(e) and (f) show the convergence of the system parameters from zero to near the actual values of 8 N and 0:0002 kg m2 for P0 and J , respectively. After parameter convergence, the impulse response is almost identical to the EMK result. Fig. 7 shows the adaptive control response to the impulse disturbance. The quiet part transient resulting from impulse inputs (a) decays 2 times faster (1 s) than PID control. 4.4. Results and discussion The objectives of the proposed control approach are to isolate the controlled span from bounded disturbances in the uncontrolled span and damp vibration in the controlled span. The controller acts as a one-way valve, con.ning vibration energy in the uncontrolled span and eliminating vibration energy from the controlled span. The actuator used in this research applies a force to control the displacement of the axially moving string at the actuating point. If the actuator is locked in place, it works as a pinned boundary, and, for a perfectly Cexible
string, the controlled span is then isolated from disturbances in the uncontrolled span, completely decoupling the controlled and uncontrolled spans. Fig. 4(c) and (g) show that PID control e4ectively regulates the actuator displacement to zero, eliminating the coupling between the two spans and isolating the controlled span from disturbances in the uncontrolled span. The vibration in the uncontrolled span (Fig. 4(f)) is clearly larger than the controlled span but some residual vibration remains in the controlled span. Vibration propagation around the backside of the belt loop, bending sti4ness coupling, and once per cycle excitation from the belt splice cause the residual vibration. The PID controller can be thought of as the best-case vibration isolator. It is, therefore, expected that the EMK and adaptive controllers should have similar vibration isolation performance as PID control. Note, however, that the EMK and adaptive controllers require signi.cantly less control e4ort than PID control. PID control reduces the damping in the controlled span when compared to open loop (see Figs. 3(a) and 4(a)), violating a major control objective. In order to obtain best-case vibration isolation, the PID controller pins the string, eliminating the energy dissipation associated with actuator friction. Fig. 4(c) shows little
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Fig. 5. Exact model knowledge control response: (a) impulse response of us (sc− ; t) (controlled span); (b) impulse response of us (sc+ ; t) (uncontrolled span); (c) impulse response of u(sc ; t); (d) impulse response of control voltage; (e) periodic disturbance response of us (sc− ; t) (controlled span); (f) periodic disturbance response of us (sc+ ; t) (uncontrolled span); (g) periodic disturbance response of u(sc ; t); (h) periodic disturbance response of control voltage.
Fig. 6.
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Fig. 7. Adaptive control response to impulse disturbance: (a) us (sc− ; t) (controlled span); (b) u(sc ; t); (c) control voltage.
actuator motion in response to the impulse input on the controlled span. Reduction of the control gains increases vibration damping but reduces vibration isolation. The EMK and adaptive controllers, however, provide much improved damping (Figs. 5(a) and 7(a)) without sacri.cing isolation performance. The actuator moves in response to the impulse disturbance on the controlled side (Fig. 5(c)) but not the periodic disturbance on the uncontrolled side (Fig. 5(g)). The proposed control approach is limited by the modeling assumptions. First, the distributed aerodynamic damping assumption guarantees bounded control e4ort, but for suIciently high disturbance and=or low damping, the actuator may saturate, reducing the isolation e4ectiveness. Of course, we can add additional damping (passive or active) to the uncontrolled span or substitute a more powerful actuator to prevent actuator saturation due to high disturbances. Second, the control system cannot operate above the critical speeds of either span. Finally, the transport speed, tension, and density of the material are assumed constant. The adaptive controller, however, can compensate for slowly varying parameters. 5. Conclusions For bounded disturbances in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively. With damping or non-resonant excitation, the uncontrolled span displacement and control force are bounded. The adaptive controller also estimates the string tension and actuator mass, allowing on-line monitoring of the process parameters. Experimental results demonstrate that the exact model knowledge and adaptive controllers outperform PID in impulse response damping and disturbance rejection control eIciency. ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Fig. 6. Adaptive control response to periodic disturbance: (a) us (sc− ; t) (controlled span); (b) us (sc+ ; t) (uncontrolled span); (c) u(sc ; t); (d) control voltage; (e) actuator inertia estimate Jˆ; (f) tension estimate Pˆ 0 .
Acknowledgements This research was supported by the National Science Foundation, Union Camp Corporation, and Square-D Corporation.
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Y. Li et al. / Automatica 38 (2002) 379–390 Christopher D. Rahn was born in Bellefonte, Pennsylvania in 1963. He graduated from the University of Michigan with a Bachelors degree in mechanical engineering in 1985. He then obtained a Masters degree from the University of California, Berkeley in 1986. After three years as a Research and Development Engineer at Space Systems=LORAL, he returned to Berkeley to pursue a
Ph.D. After graduating from Berkeley in 1992, Dr. Rahn joined the Department of Mechanical Engineering at Clemson University. In 2000, Dr. Rahn became an Associate Professor of Mechanical Engineering at the Pennsylvania State University. His research interests include dynamic modeling, analysis, and control for distributed nonlinear systems with application to robotics, mechatronics, and manufacturing.