Input shaping and time-optimal control of flexible structures

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Automatica 39 (2003) 893 – 900 www.elsevier.com/locate/automatica

Brief Paper

Input shaping and time-optimal control of #exible structures
Mark A. Laua , Lucy Y. Paob;∗ b Department

a School of Engineering, Turabo University, Gurabo, PR 00778-3030, USA of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309-0425, USA

Received 25 August 2000; received in revised form 28 June 2002; accepted 13 January 2003

Abstract We consider the problem of minimum time input shaping and its relation to the classical time-optimal control problem. We demonstrate that some typical minimum time input shapers are equivalent to the time-optimal control of di3erent, although related, systems. The analytical proofs are based on the optimality criteria stipulated by the Karush-Kuhn-Tucker conditions. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Time-optimal control; Input shaping

1. Introduction Fast maneuvers of lightweight #exible structures pose very challenging control problems (Book, 1993; Junkins & Kim, 1993). Various methods for controlling #exible structures have been investigated and can be roughly divided into feedback and feedforward approaches. While feedback techniques have been demonstrated to e3ectively reduce vibration, the performance of feedback methods can often be improved by additionally using feedforward controllers that alter the actuator commands in order to achieve vibration reduction. One feedforward approach, known as input shaping, has been successfully applied for controlling #exible structures (Singer & Seering, 1990). In this method, an input command is convolved with a sequence of impulses designed to produce a shaped command that causes less residual vibration than the original unshaped command. The goal of input shaping is to determine the amplitudes and timing of the impulses to suppress or reduce residual vibration. Since input shaping introduces a time lag in the input command, the shaper that results in the shortest sequence of impulses is desired.


This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Masaki Yamakita under the direction of Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +1-303-4922360; fax: +1-303-4922758. E-mail addresses: [email protected] (M. A. Lau), [email protected] (L. Y. Pao).

Typical input shapers include Zero Vibration shapers, which are the simplest ones and constrain zero residual vibration at the end of the maneuver (Singer & Seering, 1990). To increase robustness to parameter variations, the derivatives of the Zero Vibration constraints are also constrained to zero to yield shapers known as Zero Vibration and Derivative shapers (Singer & Seering, 1990). The e3ectiveness of input shaping has been demonstrated on many di3erent applications. It was used to improve the throughput of wafer steppers and wafer handling robots (Rappole, Singer, & Seering, 1994; de Roover, Sperling, & Bosgra, 1998) and the repeatability and speed of coordinate measuring machines (Jones & Ulsoy, 1999; Singhose, Seering, & Singer, 1996). Input shaping was a major component of an experiment in #exible system control that #ew on the Space Shuttle Endeavor in March 1995 (Tuttle & Seering, 1997). A large gantry crane operating in a nuclear environment was equipped with input shaping to enable swing-free operation and precise payload positioning (Singer, Singhose, & Kriikku, 1997). Input shaping has been investigated as a means of reducing residual vibrations of long reach manipulators (Jansen, 1992; Magee & Book, 1995) for handling hazardous waste, and it has also been used for reducing sloshing in an open container of liquid while being carried by a robot arm (Feddema et al., 1997). It has been conjectured that minimum time input shaping is equivalent to time-optimal control (Pao & Singhose, 1995). In Pao and Singhose (1998), robust time-optimal controllers (Liu & Wie, 1992) for #exible structures

0005-1098/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00024-4

894

M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900

were shown to be equivalent to traditional non-robust time-optimal controllers for di3erent #exible systems. This paper proves the equivalence between minimum time input shaping and time-optimal control and hence the relationships between traditional non-robust time-optimal control, robust time-optimal control, and minimum time input shaping is established. By demonstrating this equivalence, properties and algorithms for traditional time-optimal control of #exible structures can now be applied for solving minimum time input shaping designs. While #exible structures are modeled by distributed parameter systems exactly, many studies (Feddema et al., 1997; Jones & Ulsoy, 1999; Liu & Wie, 1992; Magee & Book, 1995; Pao & Singhose, 1995, 1998; Rappole et al., 1994; de Roover et al., 1998; Singer & Seering, 1990; Singer et al., 1997; Singh, Kabamba, & McClamroch, 1989; Singhose et al., 1996; Singhose & Pao, 1997; Tuttle & Seering, 1997) have employed Inite-dimensional models of #exible structures, and several experimental demonstrations (Feddema et al., 1997; Jones & Ulsoy, 1999; Rappole et al., 1994; de Roover et al., 1998; Singer et al., 1997; Singh et al., 1989; Singhose et al., 1996; Tuttle & Seering, 1997) have shown that controllers based on such Inite-dimensional models yield good performance while generally being more computationally tractable and eJcient. As such, the goal of this paper is to show the relationship between the technique of input shaping and the well studied time-optimal control method on Inite-dimensional models of #exible structures. QuantiIcation of residual modes and spillover e3ects in input shaping and time-optimal controllers have been studied by Singh et al. (1989) and Singhose and Pao (1997). In this paper, a proof based on the Karush-Kuhn-Tucker optimality conditions is presented to prove the equivalence conjecture between minimum time input shaping and classical time-optimal control. In Section 2, we prove the equivalence conjecture for Zero Vibration (ZV) shapers. In Section 3, we extend this proof to Zero Vibration and Derivative (ZVD) shapers. Finally, concluding remarks are presented in Section 4.

where

2. ZV shapers

H(tf∗ ) = 0;

It is well known that for linear time-invariant systems, the time-optimal control proIles are bang-bang (Kirk, 1970). These proIles can be obtained by convolving a step input command with a sequence of impulses (Pao & Singhose, 1995) and this operation is referred to as input shaping. This section explores the solutions to time-optimal input shapers and their relation to traditional time-optimal control. We consider the rest-to-rest motion of a #exible structure controlled by a single input and modeled as a linear time-invariant system: ˙ = Fx(t) + gu(t); x(t)

(1)

x(t) = [x1 (t) x2 (t) x31 (t) x41 (t) : : : x3m (t) x4m (t)]T ; F = blockdiag[F0 0

g = [g0

g1

F1

:::

:::

0

Fm ]; gm ]T ;

F0 and g0 represent the pure rigid body dynamics:

 0 1 0 F0 = ; g0 = ; 0 0 1 and




Fj =



0

1

−!j2

−2 j !j

;

j = 1; 2; : : : ; m;

represent the #exible dynamics where !1 ¡ · · · ¡ !m are the structural frequencies and j are the damping ratios. 2.1. Time-optimal control formulation For the #exible structure (1), the time-optimal control problem consists of Inding the scalar control function u(t) subject to actuator limits |u(t)| 6 U0

(2)

so that the system is transferred between the two rest states x(0) = [ − L

0

0

:::

0

0]T

and

x(tf ) = 0

(3)

while the transfer time tf is minimized. Since the system is normal, existence and uniqueness of the optimal solution is guaranteed. The time-optimal control is bang-bang with a Inite number of switches. Pontryagin’s Minimum Principle provides the following suJcient and necessary conditions for optimality (Kirk, 1970): p˙∗ (t) = −FT p∗ (t);

t ∈ [0; tf∗ ];

u∗ (t) = −U0 sgn(gT p∗ (t));

t ∈ [0; tf∗ ];

(4) (5) (6)

∗

where (·) denotes optimal quantities, ‘sgn’ is the signum function, p(t) = [p1 (t) p2 (t) : : : p2m+2 (t)]T is the costate vector, and H is the Hamiltonian H(x(t); u(t); p(t)) = 1 + pT (t)[Fx(t) + gu(t)]: With a bang-bang control, the system of ODEs in (1) can be easily integrated to yield (Pao & Singhose, 1995): 2

k 

(−1)i−1 ti + (−1)k tf = 0;

i=1

2

k  i=1

(−1)i ti2 + (−1)k+1 tf2 =

2L ; U0

M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900 k 

1+2

Including the boundary conditions on the rigid body

(−1)i (ti ) + (−1)k+1 (tf ) = 0;

x2 (tn ) − x2 (t0 ) = 0;

i=1

2

k 

(−1)i (ti ) + (−1)k+1 (tf ) = 0;

(7)

i=1

where (t) = e j !j t cos (!dj t)

and

(t) = e j !j t sin (!dj t); (8)

and k is the number of switches, tf is the maneuver time to be minimized,  = sgn(L) is the initial sign of the control  u(0), and !dj = !j natural frequencies.

1 − j2 , j = 1; : : : ; m, are the damped

i=0

+

j = 1; : : : ; m;

(9)

where the amplitudes Ai and time locations ti deIne the input shaper. To account for actuator limits, we additionally impose the following constraints on the amplitude of the impulses (Pao & Singhose, 1995)  p      (10) Ai  6 1; p = 0; 1; : : : ; n − 1;    i=0

Ai = 0:

(11)

The last constraint arises from the nature of rest-to-rest maneuvers. If the residual vibration in (9) is constrained to zero, the resulting input shaper is called a ZV shaper, leading to the constraints: Ai (ti ) = 0;

j = 1; : : : ; m;

(12)

Ai (ti ) = 0;

j = 1; : : : ; m:

(13)

i=0 n  i=0

Ai ti = 0; Ai ti2

i=1

(15)

2L = ; U0

where  = sgn(A0 ) is the sign of the Irst impulse. The constraints (15) along with (12), (13), (10), and (11) deIne the ZV input shaping problem of the system (1): n 

Ai ti = 0;

i=1 n 

Ai ti2 =

A0 +

n 

2L ; U0

Ai (ti ) = 0;

j = 1; : : : ; m;

i=1 n 

Ai (ti ) = 0;

(16) j = 1; : : : ; m;

i=1

  p     Ai  6 1;   

p = 0; 1; : : : ; n − 1;

i=0

n 

Ai = 0

i=0

i=0

n 

n 

i=1

i=0

n 

completes the list of constraints for the input shaping formulation. The minimum time input shaping solution yields the amplitudes Ai and timings ti that satisfy the constraints (10)–(14) with the smallest elapsed time tn − t0 . Without loss of generality, we can let t0 = 0 and then the goal is to minimize tn = tn − t0 . It can be shown that the boundary constraints (14), along with t0 =0 can be expressed in terms of the shaper parameters as

n 

For the #exible structure in (1), a step command input of size U0 can be convolved with a sequence of impulses so that the resulting command will cause no residual vibration at the end of the maneuver. The process of convolving a command input with a sequence of impulses is referred to as input shaping. The residual vibration resulting from a sequence of impulses is given by Singer and Seering (1990)  2 n  − j !j tn  Ai (ti ) V (!j ; j ) = e 2 1=2 Ai (ti )  ;

(14)

x1 (tn ) − x1 (t0 ) = L;

i=1

2.2. Input shaping formulation

 n 

895

while tn is minimized. In Pao and Singhose (1995), it is conjectured that solving for the time-optimal control problem of (7), subject to the actuator limit (2) and the boundary conditions (3), and solving for the minimum time zero-vibration input shaper (12)–(14), subject to (10) and (11), are equivalent. However, the equivalence statements alluded to in Pao and Singhose (1995) are based on the similarities of (7) with (16) and the same numerically obtained control patterns. Nevertheless, to provide a rigorous proof of the equivalence statement, one needs to show that the solution of the time-optimal control problem also solves the minimum time input shaper problem and vice versa.

M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900

2.3. Proof of equivalence conjecture

Proposition 1. Let

To establish a proof of the equivalence conjecture, we will make use of the Karush-Kuhn-Tucker (KKT) necessary conditions for optimality (Papalambros & Wilde, 1988), which apply to the following optimization problem: Min

z(d)

subject to r(d) = 0; s(d) 6 0; where z ∈ R, d ∈ Rl , r ∈ Rp , and s ∈ Rq . For the minimum time input shaping problem, we can deIne d = [A0 A1 : : : An |t1 : : : tn ]T as the vector containing the optimization variables for (16). We also deIne the vector functions r and s containing the constraint equations in (16) as   n  A i ti     i=1     n     2L    Ai ti2 − U 0     i=1     n    ; A + A (t ) r(d) =  0 i i     i=1     n      A (t ) i i     i=1     n      A i

i=0





A0 − 1

w = [w0

w0

v3 w1

w1

:::

Proof. (a) Note that from the KKT necessary condition ∇z(d) + vT ∇r(d) + wT ∇s(d) = 0 (Papalambros & Wilde, 1988), the gradient terms with respect to ti , after division by Ai (Ai = 0), yield v1 + 2v2 ti + v3 [ !(ti ) − !d (ti )] + v4 [ !(ti ) + !d (ti )] = 0; for i = 1; : : : ; n − 1. Now consider  (t) =

d dt (t)

 (t) = v1 + 2v2 t + v3 [ !(t) − !d (t)] + v4 [ !(t) + !d (t)]; and we have that  (ti )=0, i=1; : : : ; n−1. This signiIes that (t) attains a maximum or a minimum at each switching time ti . Since we have a minimization problem, i.e., minimize tn , this condition requires that successive switching times ti and ti+1 correspond to adjacent maxima and minima of (t), respectively, or vice versa (see Fig. 1). By the continuity of (t) we can conclude that i = i+1 , i = 1; : : : ; n − 2. (b) It follows directly from (a) because 0 is not an extremal for (t) at t = t0 , but 1 is. Hence 0 = 1 . (c) We observe that n is not an extremal for (t) at t =tn , as we can infer from the gradient term with respect to tn : or

ti-1

v 5 ]T ;

v4

(a) For i = 1; : : : ; n − 2, the relations i = i+1 hold. (b) 0 = 1 . (c) n−1 = 0.

 (tn ) = −

1 = 0: An

Moreover, we also observe that n = 0, as we can conclude from the gradient term with respect to An . Therefore, we have that n−1 corresponds to an extremal for (t) at t =tn−1 while n is not an extremal. By the continuity of (t), n−1 = n and hence n−1 = 0.

where for simplicity of exposition we have considered only one p#exible mode (m = 1). Observe that the inequalities | i=0 Ai | 6 1; p = 0; 1; : : : ; n − 1, have been rewritten without | · | in s and the objective function is given by z(d) = tn . Let the vectors of Lagrange multipliers be v2

and de:ne i = (ti ). The following statements about i are true:

1 + An  (tn ) = 0

    −A0 − 1         A0 + A1 − 1       −A − A − 1 0 1 s(d) =  ;     ..   .        A0 + A1 + · · · + An−1 − 1    −A0 − A1 − · · · − An−1 − 1

v = [v1

(t) = v1 t + v2 t 2 + v3 (t) + v4 (t) + v5 ;

δ (t)

896

ti

ti+1

Time, t

wn−1

 wn−1 ]T :

Fig. 1. The switching times ti ’s occur at extrema of the function (t).

M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900

897

Proposition 2. The following statements about the multipliers wi and wi are true:

cannot be simultaneously zero or the same positive number. Therefore, two possibilities remain:

(a) For i = 0; 1; : : : ; n − 1, the relations wi − wi = 0 hold. (b) Successive di
• wi ¿ 0 and wi ¿ 0, with wi = wi . • wi ¿ 0 and wi = 0, or vice versa.

Proof. (a) Due to the particular structure of ∇s(d), the only terms that involve wi and wi originate from the partial derivatives with respect to Ai . From the gradients with respect to Ai , the KKT necessary condition ∇z(d)+vT ∇r(d)+ wT ∇s(d) = 0 yields      w0 − w0 0 1 1 ::: 1 1       1   0 1 : : : 1 1   w1 − w1               ..   .. ..   = − . :  . .            n−2   0 0 : : : 1 1   wn−2 − w  n−2     

A0 + A1 + · · · + Aj−1 + Aj − 1 = 0;

0

0

:::

0

1

 wn−1 − wn−1

n−1

By back substitution  wn−1 − wn−1 = −n−1  which implies that wn−1 − wn−1 = 0 by Proposition 1(c). The next step leads to  wn−2 − wn−2 = −n−2 + n−1 = 0

by Proposition 1(a). Invoking Proposition 1(a) repeatedly, the next steps lead to wi − wi = −i + i+1 = 0: Continuing in this manner, the last step leads to w0 − w0 = −0 + 1 = 0 by Proposition 1(b). (b) Consider the successive di3erences wi − wi = −i + i+1 ;  = −i+1 + i+2 : wi+1 − wi+1

Without loss of generality, assume that i , i+1 , and i+2 correspond to a minimum, maximum, and minimum, respectively. Clearly, −i + i+1 ¿ 0 and −i+1 + i+2 ¡ 0, and the result follows. Proposition 3. At  a minimizer d∗ , all the inequality conp straints in (16),  | i=0 Ai | 6 1, p = 0; 1; : : : ; n − 1, are p active. That is, | i=0 Ai | = 1, p = 0; 1; : : : ; n − 1. Proof. From the KKT necessary conditions (Papalambros & Wilde, 1988), the multipliers associated with the inequalities, wi and wi , satisfy wi ¿ 0 and wi ¿ 0, i =0; 1; : : : ; n−1. It follows from Proposition 2(a) that, for Ixed i, wi and wi

The Irst possibility is ruled out because this would require that, for Ixed j, both constraints

−A0 − A1 − · · · − Aj−1 − Aj − 1 = 0; be satisIed simultaneously, which is clearly impossible. Therefore, wi ¿ 0 and wi = 0, or vice versa, is the only possibility. This implies that either one of the above constraints is active, but not both, i.e., |A0 + A1 + · · · + Aj−1 + Aj | = 1: Since the argument is valid for j = 0; 1; : : : ; n − 1, the result follows. Theorem 1 (Equivalence Conjecture for ZV). The timeoptimal problem (7) and the minimum time ZV shaper problem (16) are equivalent. Proof. (a) It is obvious that the coeJcients of ti , ti2 , (ti ) and (ti ) in (7) satisfy the last two constraints ((10) and (11)) in (16), and the remaining constraints of (16) become identical to those of (7) with t0 = 0, n − 1 taking the role of the number of switches k, and tf = tn . The n + 1 impulses (A0 ; A1 ; : : : ; An ) will produce n − 1 transitions in the control when convolved with a step input. n (b) Conversely, by Proposition 3 and i=0 Ai =0 of (16), the only possible choices of impulse amplitudes are 1 −2 2 −2 : : : ±2 ∓1 −1

2

−2

2

:::

∓2

±1:

Because this is always the case for a minimizer d∗ of (16), the last two constraints ((10) and (11)) can be eliminated from (16), and the amplitudes can be substituted directly in the remaining constraints of (16), leading to the same constraints in (7), with t0 =0, k =n−1, and tf =tn . If n−1 is not equal to the number of switches k in (7), then this means that there is a di3erent bang-bang controller with a di3erent number of switches that satisIes (16) and leads to a minimal tn . Since (16) is now the same as (7) with n − 1 taking the role of k by Proposition 3, this violates Pontryagin’s minimum principle and contradicts the uniqueness of the time-optimal control. Hence, n − 1 must be the same as k and the minimum time ZV shaper solution of (16) is the same as the time-optimal control of (7). (c) Hence, from (a) and (b) we know that at a minimizer, both sets of constraints (7) and (16), without the impulse amplitude constraints, become identical. Since solutions of either (7) or (16) can be local minima, we can invoke Pontryagin’s minimum principle (4)–(6) to assess the optimality of solutions to (7) or (16).

898

M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900

Extending the proofs of Propositions 1–3 and Theorem 1 to multiple modes is straightforward by appropriately augmenting the (t) function, the constraint equations in (12) and (13) and r(d), and the Lagrange multiplier vector v.

3. ZVD shapers To increase robustness to modeling errors, the partial derivatives of the ZV constraints (12) and (13) with respect to ! and are also constrained to zero to yield the additional ZVD shaper constraints (Singer & Seering, 1990): n 

Ai ti (ti ) = 0;

j = 1; : : : ; m;

(17)

and subject to the same control limit (2) and boundary conditions (3). In essence, we show that the minimum time ZVD shaping problem is equivalent to the time-optimal control problem of a system with double poles at the locations of the original #exible poles. With a bang-bang control applied to (1) and Fj of the form given by (20), the time-optimal constraints on the switching times become 2

k  i=1

2

k 

1+2 Ai ti (ti ) = 0;

j = 1; : : : ; m:

(18)

n 

2

2L Ai ti2 = ; U0

i=0 n 

2

(−1)i ti (ti ) + (−1)k+1 tf (tf ) = 0;

k 

(−1)i ti (ti ) + (−1)k+1 tf (tf ) = 0

while tf is minimized. Similar to the ZV case of Section 2, let the vectors of Lagrange multipliers be

Ai (ti ) = 0;

j = 1; : : : ; m;

v = [v1 (19)

Ai ti (ti ) = 0;

j = 1; : : : ; m;

Ai ti (ti ) = 0;

j = 1; : : : ; m;

i=0

  p     Ai  6 1;   

p = 0; 1; : : : ; n − 1;

i=0

n 

k 

j = 1; : : : ; m;

i=0 n 

(−1)i (ti ) + (−1)k+1 (tf ) = 0;

Ai (ti ) = 0;

i=0 n 

(21)

i=1

i=0 n 

k 

i=1

i=0 n 

(−1)i (ti ) + (−1)k+1 (tf ) = 0;

i=1

2

Ai ti = 0;

k 

2L ; U0

i=1

i=0

The minimum time ZVD shaping problem of system (1) is to Ind the amplitudes Ai and timings ti such that

(−1)i ti2 + (−1)k+1 tf2 =

i=1

i=0 n 

(−1)i−1 ti + (−1)k tf = 0;

Ai = 0

i=0

while tn is minimized. We want to show that solving for the optimization problem in (19) is equivalent to solving for the time-optimal control problem of (1) with Fj now being   0 1 0 1     −!j2 −2 j !j 0 0   (20) Fj =  ;   0 0 0 1   0 0 −!j2 −2 j !j

w = [w0

v2 w0

v3

v4

w1

v5 w1

v6 ···

v 7 ]T ; wn−1

 wn−1 ]T :

Because the proof for ZVD shapers is analogous to that of ZV shapers, we state the following propositions and theorem without proof. Proposition 4. Let !(t) = v1 t + v2 t 2 + v3 (t) + v4 (t) + v5 t(t) + v6 t(t) + v7 ; and de:ne !i = !(ti ). The following are true: (a) For i = 1; : : : ; n − 2, the relations !i = !i+1 hold. (b) !0 = !1 . (c) !n−1 = 0. Proposition 5. The following statements about the multipliers wi and wi are true: (a) wi − wi = 0 for i = 0; 1; : : : ; n − 1. (b) Successive di
M. A. Lau, L. Y. Pao / Automatica 39 (2003) 893 – 900

899

Fig. 2. Control proIles and time responses of position resulting from applying the minimum time ZVD shaper control for a single-mode system and traditional time optimal control for a double-mode system.

Proposition 6. At  a minimizer d∗ , all the inequality conp straints in (19),  | i=0 Ai | 6 1, p = 0; 1; : : : ; n − 1, are p active. That is, | i=0 Ai | = 1, p = 0; 1; : : : ; n − 1. Theorem 2 (Equivalence Conjecture for ZVD). The timeoptimal problem (21) and the minimum time ZVD shaper problem (19) are equivalent. The propositions and theorem in this section show that minimum time ZVD shaping is equivalent to traditional time-optimal control of a di3erent system with double #exible poles at the original locations of the single #exible poles. Since it was shown by Pao and Singhose (1998) that Irst-order robust time-optimal controllers are equivalent to traditional time-optimal controllers of these double-mode #exible systems, it is now clear that minimum time ZVD shaping, Irst-order robust time-optimal control of the same #exible system (Liu & Wie, 1992), and traditional time-optimal control of a di3erent double-mode #exible system are all equivalent. As with the extension of robust time-optimal controllers to higher-order robustness constraints, the extension to higher-order derivative shapers (ZVDD, ZVDDD, etc.) is similar, with equivalent time-optimal control systems having triple, quadruple, etc. poles at the same locations of the original #exible poles. Hence, traditional time-optimal control techniques and algorithms can be used to solve and verify global optimality for robust time-optimal controllers as well as minimum time robust input shapers. The double-pole (or triple- or quadruple-pole, etc.) system is a mathematical system that need not exist in practice but is used to facilitate the numerical solution of minimum time robust input shapers for the original system. Fig. 2 shows the response of a single-mode #exible system to the minimum time ZVD shaped control, the response of a double-mode #exible system to the time-optimal control, as well as the identical time-optimal and minimum time ZVD

shaped control proIles obtained by solving (19) and (21). The control proIles for both responses are equivalent, as proven by Theorem 2. The responses assume the following √ parameters: L=5 m, U0 =1 m=s2 , != 2 rad=s2 , and =0; and the GAMS software package (Rosenthal, 1998) was used to numerically solve the optimization problems. The time responses for (19) and (21) shown in Fig. 2 are not identical because the same numerically obtained inputs are applied to di3erent systems. More precisely, the ZVD shaped input is applied to a system with one #exible pole at (!; ) and the (same) time-optimal control input is applied to a system with double #exible poles at (!; ). Nevertheless, Eqs. (19) and (21) will always lead to the same settling time and steady (rest) state with di3erent transients. Numerical solutions comparing the time-optimal control in (7) and minimum time ZV shaping in (16) also lead to identical control proIles, thus validating the analytical proof of Theorem 1 in Section 2. In this case, the system position responses are identical because the shaped and time-optimal inputs are the same and are applied to the same systems.

4. Conclusions We have proven that minimum time Zero Vibration (ZV) and Zero Vibration and Derivative (ZVD) shapers are equivalent to traditional time-optimal control, possibly of di3erent systems. The proof based on the Karush-Kuhn-Tucker conditions provides useful insights on the timing of the impulses of these shapers and also the nature of the constraints on the impulse amplitudes. The consequences of the proof of the equivalence conjecture presented in this paper agree with the optimal controls dictated by Pontryagin’s minimum principle. The signiIcance of proving the equivalence is that time-optimal control of #exible structures is a well studied area and many of its properties and numerical

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procedures can now be applied to the relatively newer Ield of input shaping. Acknowledgements This work was supported in part by the National Science Foundation (Early Faculty CAREER Award Grant CMS-9625086 and Grant CMS-0201459), a University of Colorado Faculty Fellowship, and the Colorado Center for Information Storage. References Book, W. J. (1993). Controlled motion in an elastic world. Journal of Dynamic Systems, Measurement and Control, 115(2), 252–261. Feddema, J. T., Dohrmann, C. R., Parker, G. G., Robinett, R. D., Romero, V. J., & Schmitt, D. J. (1997). Control for slosh-free motion of an open container. IEEE Control System Magazine, 17(1), 29–36. Jansen, J. F. (1992). Control and analysis of a single-link Bexible beam with experimental veri:cation. ORNL/TM-12198, Oak Ridge National Laboratory. Jones, S., & Ulsoy, A. (1999). An approach to control input shaping with application to coordinate measuring machines. Journal of Dynamic Systems, Measurement and Control, 121(2), 242–247. Junkins, J. L., & Kim, Y. (1993). Introduction to dynamics and control of Bexible structures. Washington, DC: AIAA Education Series. Kirk, D. E. (1970). Optimal control theory: An introduction (pp. 227–291). Prentice-Hall, Inc. Liu, Q., & Wie, B. (1992). Robust time-optimal control of uncertain #exible spacecraft. Journal of Guidance, Control, and Dynamics, 15(3), 597–604. Magee, D. P., & Book, W. (1995). Filtering micro-manipulator wrist commands to prevent #exible base motion. Proceedings of the American Control Conference (pp. 924 –928). Seattle, WA. Pao, L. Y., & Singhose, W. E. (1995). On the equivalence of minimum time input shaping with traditional time-optimal control. Proceedings of the IEEE Conference on Control Applications (pp. 1120 –1125). Albany, NY. Pao, L. Y., & Singhose, W. E. (1998). Robust minimum time control of #exible structures. Automatica, 34(2), 229–236. Papalambros, P. Y., & Wilde, D. J. (1988). Principles of Optimal Design: Modeling and Computation (pp. 183–184). Cambridge: Cambridge University Press. Rappole, B. W., Singer, N. C., & Seering, W. P. (1994). Multiple-mode input shaping sequences for reducing residual vibrations. Proceedings of the ASME Mechanisms Conference, Vol. DE–71 (pp. 11–16). de Roover, D., Sperling, F. B., & Bosgra, O. H. (1998). Point-to-point control of a MIMO servomechanism. Proceedings of the American Control Conference (pp. 2648–2651). Philadelphia, PA. Rosenthal, R. E. (1998). GAMS Tutorial. GAMS Development Corporation. Singer, N., & Seering, W. (1990). Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement and Control, 112(1), 76–82.

Singer, N., Singhose, W., & Kriikku, E. (1997). An input shaping controller enabling cranes to move without sway. Proceedings of the American Nuclear Society Topical Meeting on Robotics and Remote Systems, Atlanta, GA, USA. Singh, G., Kabamba, P. T., & McClamroch, N. H. (1989). Planar, time-optimal, rest-to-rest slewing maneuvers of #exible spacecraft. Journal of Guidance, Control and Dynamics, 12(1), 71–81. Singhose, W., Seering, W., & Singer, N. (1996). Improving repeatability of coordinate measuring machines with shaped command signals. Precision Engineering, 18, 138–146. Singhose, W. E., & Pao, L. Y. (1997). A comparison of input shaping and time-optimal #exible-body control. Control Engineering Practice, 5(4), 459–467. Tuttle, T. D., & Seering, W. P. (1997). Experimental veriIcation of vibration reduction in #exible spacecraft using input shaping. Journal of Guidance, Control and Dynamics, 20(4), 658–663.

Mark A. Lau was born in Peru in 1967. He received his B.S. in Engineering Sciences and Dipl. in Industrial Engineering from the University of Piura, Peru, in 1988 and 1991, respectively. He attended the University of Colorado at Boulder and earned his M.S. and Ph.D. in Electrical Engineering in 1997 and 2000, respectively. Dr. Lau is currently an Assistant Professor in the School of Engineering at Turabo University, Puerto Rico. He spent the 2001– 2002 academic year as a Visiting Assistant Professor in the Electrical and Computer Engineering Department at the University of West Florida. Dr. Lau received the 2001 American Control Conference Best Student Paper Award. His research interests are in the control of #exible structures, optimization, process control, MEMS, and nanotechnology.

Lucy Y. Pao was born in Washington, DC. in 1968. She received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Stanford University in 1987, 1988, and 1992, respectively. She is currently an Associate Professor in the Electrical and Computer Engineering Department at the University of Colorado at Boulder. She spent the 2001–2002 academic year as a Visiting Scholar at Harvard University. Dr. Pao received the 1996 IFAC World Congress Young Author Prize, a National Science Foundation Early Faculty CAREER Development Award (1996 – 2001), and an OJce of Naval Research Young Investigator Award (1997– 2000). She has served on the IEEE Control Systems Society Conference Editorial Board (1995 –1997) and as the American Automatic Control Council newsletter editor (1995 –2001). She has also been on organizing committees and program committees for a number of conferences and workshops. Currently, she is the Vice Chair for Invited Sessions for the 2003 American Control Conference, is the Program Chair for the 2004 American Control Conference, and is on the IEEE Control System Society’s Board of Governors. Her research interests are in the control of #exible structures, multisensor data fusion, and haptic and multimodal visual/haptic/audio interfaces.