Closed-Loop Input Shaping for Flexible Structures using Time-Delay ...

Proceedings of the 3Sm Conference on Decision & Control Phoenix, Arizona USA December 1999

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Closed-Loop Input Shaping for Flexible Structures using Time-Delay Control Vikram Kapila, Anthony Tzes, and Qiguo Yan Department of Mechanical, Aerospace, and Manufacturing Engineering Polytechnic University, Brooklyn, NY 11201

Abstract Input shaping techniques reduce the residual vibration in flexible structures by convolving the command input with a sequence of impulses. The exact cancellation of the residual structural vibration via input shaping is dependent on the amplitudes and instances of impulse application. A majority of the current input shaping schemes are inherently open-loop where impulse ap li cation at inaccurate instances can lead to system perkr: rnance degradation. In this paper, we develop a closedloop control design framework for input shaped systems. This framework is based on the realization that the dynamics of input shaped systems give rise to time delays in the input. Thus, we exploit the feedback control theory of time dela systems for the closed-loop control of input shaped flexiile structures. A Riccati equation-based and a linear matrix inequality-based frameworks are developed for the stabilization of systems with uncertain, multiple input delays. Next, the aforementioned framework is applied to an input shaped flexible structure system. This framework guarantees closed-loop system stability and performance when the impulse train is applied at inaccurate instances. A simulation study demonstrates that the closed-loop system with the proposed time delay controller outperforms the open-loop, input shaped system and the standard linear quadratic regulator when the impulse is applied at an inexact instance. 1.

Introduction

Active vibration control of flexible structures, such as flexible robotic manipulator systems, has experienced rapid growth in recent years. Flexible structure dynamics consist of underdamped poles and zeros resulting in a lightly damped impulse response (Tzes and Yurkovich 1993 . In recent years, many researchers have focused on t e development of precompensation schemes to reduce the residual vibration that result in the structure when control is applied (Bodson (1998), Magee et al. (1997), Pao and Lau (1999), Singer and Seering (1990), Singhose et al. (1996, 1997)). Amongst these schemes, the input shaping technique relies on the development of an impulse train, with appropriate magnitudes and time-intervals, that convolves in real-time with the reference input (Singer and Seering (1990), Swigert (1980), Tallrnan and Smith (1958)). The input shaping filter essentially attempts to add zeros to the system at the exact locations of the system poles. Thus, the amplitudes and instances of application of the impulses in input shaping technique de end on the damping factors and natural frequencies o f t h e system poles. Several techniques have recently been proposed that enhance the performance of input shaping filter to ac-

;

count for uncertainties in the damping factors and natural frequencies of the flexible structure (Pa0 and Lau (1999), Singh and Vadali (1993), Singhose et al. (1996, 1997)). These schemes typically lengthen the duration of the impulse sequence and result in a slower system response. Despite the advantages offered by the input shaping framework (simplicity, ease of implementation, saturation avoidance, etc.) compared to other precompensators (Tzes and Yurkovich 1989), closed-loop control for input shaped system has received only scant at tention. The dynamics of a flexible structure coupled with an input shaper give rise to time delays in the control input. Since the feedback control theory for time delay systems had not sufficiently advanced until recently (Dugard and Verriest 1997) , the aforementioned closed-loop control problem for input shaped system did not have a tractable solution. However, with the recent advancements in the feedback control theory for time delay systerns (Dugard and Verriest (1997), Haddad et al. (1997), Kapila et al. (1998), Kim et al. (1996), Moheimani and Petersen (1995), Mori et al. (1983), Niculescu et al. (1994), Niculescu et al. (1997), Shen et al. (1991), Verriest and Ivanov (1994)), it is now possible to account for the delay dynamics that arise in the input shaped flexible structures. In addition, since the time delay control theory can account for uncertainty in the amount of input delay (Dugard and Verriest (1997), Kapila et al. (1998), Kim et al. (1996), Niculescu et al. (1997 ), one can design feedback controllers for input shaped exible structures that are robust to inaccuracies in the application of instances of impulses. Note that inaccuracies in the application of instances of impulses can arise in realtime control, e.g., due to the sampled-data controller implementation which restricts the command to be applied at sampling instances. This paper is organized as follows. The input shaped flexible structure dynamic modeling is reviewed in Section 2. A full-state feedback control design framework for linear systems with multiple, arbitrary, input time delays is presented in Section 3. A Riccati equationbased and a linear matrix inequality (LMI) based delayindependent stabilization techniques for time-invariant dynamical systems with multiple input delays are developed in this section. The proposed framework also accounts for degradation in a quadratic performance functional in the presence of input delays. An illustrative numerical example is given in Section 4 which demonstrates the efficacy of the proposed approach for closed-loop input shaping. Finally, concluding remarks are provided in Section 5. Nomenclature

B

B, Brxs, Br - real numbers, r x Research supported in part by the Air Force Office of Scientific Research under Grant F49620-93-0063, Air Force Research Laboratory/VAAD, WPAFB, OH, under IPA: Visiting Faculty Grant, and the NASA/New York Space Grant Consortium under Grant 32310-5891.

0-7803-5250-5/99/$10.000 1999 IEEE

1561

Y,Pr IrjOr

s real matrices, Rrxl r x r identity matrix, r x T zero matrix r x r symmetric, positive-definite

matrices 21 < 2,

- 2, - 21 E Pr; Zl,Z, E 9'

2. 2.1.

System Model Flexible Structure Dynamics

The dynamics of a flexible structure can be described by a set of partial differential equations. Under the assumed modes method and retaining a finite number, n, of modes, the dynarnics are (Meirovitch 1980)

impulses have amplitudes A1 , j , j = 1,. . . N I , and are applied at times t l j , j = 1,. . . N I . If each impulse has an appropriate magnitude relative to the first impulse, and is delayed by a proper amount of time, then the superposition of the NI impulse responses can be made equal to zero after the a plication of the last impulse. From Singer and Seering rl990) and Txes and Yurkovich (1993) the expressions for the amplitudes and the instances of the impulses application, for j = 1,.. .N I , are

A1,j = = M*

[-#I

+H*F(t),

where M , G , and K are the symmetric- mass, damping, and stiffness matrices, respectively, q E IW” refers to the modal position vector, and F is the forcing input. The eigenvalues, Xi, of the system matrix M * are typically lightly damped and are given by

A.z -

-e. + j W i J 1 rWi

~ i + = ~ .-ciwi

- jwi

(5)

Jz,

i = 1 , . . . ,n,

where ci (= 0) is the damping factor and wi is the undamped natural frequency of the ithmode. Assume that the observed output w(t) is related to the states as

The amplitudes A I , ~j, = 1,.. . N I , of the impulses are independent of the modal frequency. However, the times of their application tl,j, j = 1,.. . N I ,depend directly on the modal frequency. Therefore, the basic assumption in the shaping scheme is that the damping ratio c1, and in particular, the natural frequency w1, are known a priori. Although the robustness of the system against w1 and perturbations is improved through the use of more impulses in the formed sequence, the system responds more slowly since the length of the impulse train is longer (tl,N1 > t l . for ~ ~ NI > Nz). The transfer function of the system comprised by the convolver (impulse shaper) and the dynamics of the flexible structure is r N.

1

Let the partial fraction expansion of the system transfer function for the dynamic model ( l ) , ( 2 ) be n

G ( s )= C * ( S I- M * ) - l H * =

k l s2

+

Law? 2ciwjs

+ w?.

(3)

The impulse response of this linear vibratory system is given by

Due to the delayed nature of the impulses’ applications, an infinite number of complex zeros are introduced to the overall transfer function and the input shaping scheme essentially cancels the effect of the system poles by providing a pair of zeros at the corresponding pole locations. As an example for NI = 2, the introduced pairs of zeros are placed at

n

sh,z

-LIn t1,2

(4)

where A0 is the aInPlitude of the impulse and t o the time of the impulse application. 2.2.

Input Shaping Control

The objective of the time domain precompensators is to produce a control input sequence such that the residual vibration components are reduced. The input shaping filter consists of a real time convolver, shown in Figure 1, which convolves the reference input with a sequence of impulses having appropriate magnitudes and spacing within the time domain. The input shaping filter essentially adds zeros to the system at the exact locations of the system poles. To illustrate and simplify the analysis, consider the case where only one flexible mode exists, i.e., n = 1. Suppose a train of N I impulses is applied to the system; these

(i:,:) -

kx fj-, t1,2 k = 1 , 3 , . . . ,

where the fundamental zero pair (k = 1) is placed at the exact pole location of the structure, or sll = XI, and s12 = Xz. Adding more impulses to the sequence introduces more pairs of complex zeros at the location of the system poles. I t should be emphasized, however, that adding more impulses adds lag to the system and there is an overall trade-off between the system’s robustness and speed of response. Although the aforementioned preshaper was presented for a system with one mode, it can be extended to include additional modes n > 1). The additional modes are handled by simply esigning impulse sequences for each mode individually, then convolving the sequences together to arrive at a composite shaped impulse sequence. The resultin composite sequence will return the system to rest in a fnite time. The convolved impulse train for a structure with n modes and N, , i = 1,. . . ,n, used impulses per mode is

7 562

d

[Ai.i6(t)+ . . . + A 1,NIS(t-ti.N1)]*...* [&.16(t) + ... + & . ~ , , 6 ( t- t n , ~ , , ),] (8)

*

where corresponds t o the convolution operator. This input impulse sequence will result in an output with no residual vibration after t = x y = l ( N i - l ) t i , ~. , If the lant is to be driven with an arbitrary input F ( t ) , a n f be free of vibration, then convolution of the input with the system’s shaped impulse sequence will produce a vibrationless response after an initial time delay (Singer and Seering 1990). 2.3.

Input Shaped System Dynamics

which satisfies the following design criteria i) the closedloop system (12), (13) is asymptotically stable and ii) the quadratic performance functional

J(K)

+

t E [o, m), ~ ( t=) K x ( ~ ) ,

(10)

The combination of the input impulse convolver with the system dynamics generates a system with the following description

+ B,

k

o

1

7=1

where R

Full-State Feedback Stabilization Problem

In this section, we introduce the full-state feedback control problem for linear systems with multiple input delays. Specifically, given the nth-order, stabilizable, dynamical system, where stabilizability is defined in the sense of Bhat and Koivo (1976), with multiple input delays k

+ B U ( t ) + 1B d , U ( t -

Td,),

7=1

E

[o,m),

U(t)

= 0,

Tdk

tE

> Tdk-l > * . ’ > Tdl > 0,

[-Td,,O],

Z(0) = $0,

rt

is a Lyapunov function that guarantees that the closedloop system (12), 13 , or, equivalently, (15), 16 is asymptotically stab e or all T~~ > 0, i = 1,.. . , burthermore, the performance functional (16) satisfies the bound J(zo,K) < V ( X O ) .

1.

\l

Note that V(z0)= zTPz0 provides an upper bound on the quadratic performance (14). Now, in the spirit of Kapila et al. (1998), V ( s 0 )can be considered as an auxiliary cost. In particular, note that t r zTPzo = t r P z o z z has the same form as the H 2 cost in standard LQR theb DDT where D E RnXd ory. Hence, we re lace and proceed by &ermining t i e controllers that minimize t r P D D T . This leads t o the following optimization problem. Auxiliary Minimization Problem. Determine K that minimizes J ( P , K ) f t r PDDT where P > 0 satisfies (17) and such that K is minimal.

~OZT

(12)

where ~ ( tE) Rm, determine a full-state feedback controller (see, e.g., Figure 2)

u(t) = K z ( t ) ,

5 ETE. Then the function k

(11)

3. Full-State Feedback Controller Synthesis for Systems with Multiple Input Delays

t

= A T ~ + ~ A + t ~ + C ~ f ~ T ~ Z ~ 2=1

Note that the dynamic model (11) corresponds t o a linear system with multiple input delays.

S ( t ) = AZ(t)

+

+ C a ; Z ~ ~ d z ~ ; l(17) ~;~+~,

+(UAi,N,)4t - C t i . , N J .

3.1.

(16)

k

+ A l , 2 ( H Ai,l)U(t- h , 2 ) + . . . i=2

i=l

(15)

Theorem 3.1. Let K be given. Suppose there exists an n x n positive-definite matrix P and scalars E , cy, > 0, i = 1,. . . ,IC, such that

i=l

k l

> 0, i = 1 , . . . , k ,

+

Ai,i)u(t)

n

Td,

A BK. where A Next, we present a theorem that is essential for the statement of the main result of this section. In order t o state this theorem, we define E f El E2K, RI f ET&, R2 f EFE2 > 0, and let El and E2 be such that ET& = 0.

(n [ ”

n

B d , ~ (-t ~ d , ) , ~ ( 0 = ) $0, 7=1

where z E W2” is the new state vector and u(t) = F ( t ) . The previous equation can be rewritten in a more compact matrix notation form

k ( t ) = A,z(t)

k

k ( t ) = Az(t)

e

A s z ( t )+ B z ~ ( t ) .

(14)

+

r o i

=

zT(t)z(t)dt,

where z ( t ) f E l z ( t ) Ezu(t),z E W , is minimized. Note that for a given full-state feedback controller K the closed-loop system (12), (13) can be written as

The transfer function from equation (3) can be cast in an equivalent state space description as

k(t)

Sorn

5

(13)

3.2.

Control Synthesis: A Riccati Equation-Based Approach

Now, we present the main theorem characterking fullstate feedback controllers for (12) with multiple input

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delays. For convenience in stating the main result of this 4 R 2 , R2 (1 section, we define the notation R2,

e +

matrix X and an m x n matrix Y satisfying

A+AT+tX+R2b

ZL"3R2. Theorem 3.2. Assume E , a, > 0, i = 1,. . . ,k , and suppose there exist an n x n positive-definite matrix P satisfying

+

0 = A ~ P PA

+ CP+ R~+

P =

I

(25)

x-1,

K = YX-'.

(26)

(19)

Then P and K satisfy (21) and the feedback interconnection of the delay system (12) and the full-state feedback controller 13) is asymptotically stable for all T ~ > + 0, i = 1,.. . ,k! . Finally, for all T,,~ > 0, i = 1,.. . ,k , the auxiliary cost J ( P , K ) satisfies

(20)

J ( P ,K ) < t r 2,

and let K be given by

K = -RTIBTP.

6

Then P satisfies (17) and K is an extrema1 of J P, K ) . Furthermore the feedback interconnection of t e delay system (12) and the full-state feedback controller (13) is asymptotically stable for all T ~ ,> 0, a = 1,.. . ,k. Finally, the cost J ( P , K ) is given by J ( P , K ) = tr P D D ~ . Theorem 3.2 provides constructive sufficient conditions that yield full-state feedback gain K for the stabiliiation of systems with input delays. When solving (19) numerically, scalars a ? ,i = 1,. . . ,k, appearing in the desi n equations can be adjusted t o determine the region of solvability of (19). 3.3. Controller Synthesis: An LMI-Based Approach

In this subsection, we begin by noting that if (17) is replaced by the matrix inequality condition k

ATP+PA+tP+C~:KTR2K i=l

then the function (18) is a Lyapunov function that guarantees that the closed-loop system (12), (13 is asymp, 0, i = 1,.. . ,k. n addition, totically stable for all T ~ > the performance bound J ( z 0 , K ) < V(z0) still holds. Next, we exploit (21) to develop an alternative framework, viz., an LMI approach, to desi II a feedback controller gain K that stabilizes the close!-loop system (12), (13) in the presence of multiple delays. Before proceeding, we relax the condition ETE2 = 0, assumed earlier, and allow for ETE2 # 0. For the statement of the next result, we define

1

2 KX, E 2 ElX+E2Y,

Y

-ET Ip

0pxm.

< 0.

PB~&~B$P

-PBR; B~ P,

P-1, AX+BY,

i

In addition, let P and K be given by

k 1=1

x f A 5

[

YT -R;:

(27)

is such that the LMI variables X E P" where 2 E satisfying (25) additionally satisfies

E

DXT ] >O.

Proof. First note that using (24), the matrix inequality (21) can be rewritten as

ATP + P A -I- CP+ KTR2,K

f

PRzbP + E T E < 0. (29)

Next, by forming X(29)X, using (22), (23), arid by a repeated application of the Schur Complement (Boyd et al. 1994), it follows that (29) is equivalent to (25). Thus, it follows that, for given E, CY? > 0, i = 1,.. . , k , the existence of P E P" and K E RmX"satisfying (24) is equivalent t o the existence of X E I P and Y E Rmx" satisfying (25). This proves that the existence of X E P" and Y E R" x n satisfying (25)yields asymptotic stability of the feedback interconnection of the delay system (12) and the full-state feedback controller (13) for all T ~ >, 0, i = l ,..., k . Next, to show that J ( P , K ) satisfies the bound (27), we consider the inequality

z>

D~PD,

(30)

which yields (28). Now, using the Schur Complement (Boyd e t al. 1994) and (22), it follows that the existence and P E !F satisfying (30) is equivalent to the of 2 E existence of 2 E and X E P'satisfying (28 . Thus, , 0, i = 1,. . . , k , t o minimize the per orniance for all T ~ > bound (27), we consider the LMI rriinimixation problerri

1

minimize tr 2 subject to X E (25) and (28).

P, 2

E

p,and

Y E R m X " satisfyiI8

(22) (23)

4.

Illustrative Numerical Example

In this section, we demonstrate an application of the time delay control theory, developed in Section 3, t o deRza f E a ? R z , R2b f E a ; 2 B d , R ; 1 B x . (24) sign feedback controllers for the closed-loop control of ?=1 ?=1 (11 . Thus, consider a single mode flexible structure wit the following model parameters, w1 = 5, C l = 0.01, Theorem 3.3. Assume € , a , > 0, i = 1,. . . , k , are L1 = 0.04. The parameters of the input shaper from given. Suppose there exist an n x n positive-definite (5) and (6) are Al.l = 0.5079, A1.z = 0.4921, arid 1564 k

k

h

t 1 , 2 = 0.6283. The open-loop impulse response of the system is shown in Figure 3. In addition, the preshaped impulse response with the aforementioned parameters is given in Figure 4 which indicates that the residual vibration after t1.2 is eliminated. The input shaped dynamic system description in (11)is given by

Dorato, P., Abdallah, C., and Cerone, V., 1995, LinearQuadratic Control: A n Introduction. Englewood Cliffs, NJ: Prentice-Hall. Dugard, L. and Verriest , E. I., 1997, Stabzlzty and Control of Time-delay Systems. New York, NY: Springer-Verlag. Haddad, W. M., Kapila, V., and Abdallah, C. T., 1997, “Stabilization of Linear and Nonlinear Systems with Time Delay,” Proc. Amer. Contr. Conf., 3220-3225, Albuquerque, NM, 1997; see also Stabality and Control of Time-Delay Systems, L. Dugard and E. Verriest, Eds., 205-217.

The response of the open-loop input shaped system with ~d = %f is given in Figure 5. It is transparent from Figure 5 that the residual vibration with inexact impulse application instance is not eliminated. Next, for (31) we design a feedback controller using the standard linear quadratic regulator (LQR) technique (Dorato et al. 1995 which does not account for time delay in the input. or the LQR design we select the state and control weights as RI = diag (100,l) and Rz = 0.1, respectively. Next, we design full-state feedback controllers using Theorems 3.2 and 3.3 with the state and control weights as in the LQR design and with 01 = 3.65. Using Theorems 3.2 and 3.3, both, the control gain K is found to be K = [ -0.7093 -12.9484 Figures 6 and 7 illustrate the closed-loop res onse with the LQR controller and the Theorem 3.2 (&uivalently, Theorem 3.3) design, res ectively. It can be seen from these responses that the Tgeorem 3.2 design (Figure 7) outperforms the open-loop input shaper (Figure 5 ) and the LQR-based closed-loop system (Figure 6). Finally, note that the LQR-based controller does not guarantee the closed-loop system stability in the presence of time delays that arise in the in ut shaped systems. In contrast, the Theorem 3.2 b a s e 1 control design ensures closed-loop stability in the presence of delays.

d

1.

5.

Conclusion

In this paper, we considered a closed-loop control design for an input shaped flexible structure. It was demonstrated that the input shaped flexible structure dynamics lead to time delays in the control in ut. For this class of systems, a Lyapunov function-basefstabilizationframework was developed. The proposed closed-loop control design scheme is delay-independent and guarantees stability against arbitrary delay in the input. The efficacy of the proposed scheme was contrasted in simulation studies with that of an LQR-based controller. Finally, as mentioned earlier, we note that the proposed scheme can be extended to account for bounded input delays using a delay-dependent stabilization framework.

Kapila, V., Haddad, W. M., and Grivas, A. D., 1998, “Stabilization of Linear Systems with Simultaneous State, Actuation, and Measurement Delays,” Proc. Amer. Contr. Conf., 2381-2385, Philadelphia, PA; see also Int. J. Contr., to appear. Kim, J. H., Jeung, E. T., and Park, H. B., 1996, “Robust Control for Parameter Uncertain Delay Systems in State and Control Input,” Automatica, vol. 32, 1337-1339. Magee, D. P., Cannon, D. W., and Book, W. J., 1997 “Combined Command Shaping and Inertial Damping for Flexure Control” Proc. Amer. Contr. Conf., 1330-1334, Albuquerque, NM. Meirovitch, L., 1980, Computatzonal Methods in Structural Dynamics. Rockville, MD: Sijthoff and Noordhoff Int. Publ. Moheimani, S. 0. R. and Petersen, I. R., 1995, “Optimal Quadratic Guaranteed Cost Control of a Class of Uncertain Time-Delay Systems,” Proc. IEEE Conf. on Dec. and Contr., 1513-1518, New Orleans, LA. Mori, T., Noldus, E., and Kuwahara, M., 1983, “A Way to Stabilize Linear Systems with Delayed State,” Automataca, vol. 19, 571-573. Niculescu, %I., de Souza, C. E., Dion, J. M., and Dugard, L., 1994, “Robust Stability and Stabilization of Uncertain Linear Systems with State Delay: Single Delay Case (I),” Proc. IFAC Workshop Robust Contr. Design, 469474, Rio de Janeiro, Brazil. Niculescu, S.-I., Fu, M., and Li, H., 1997, “Stability of Linear Systems with Input Delay: An LMI Approach,” Proc. IEEE Conf. on Dec. and Contr., 1623-1628, San Diego, CA. Paa,L. Y. and Lau, M. A. 1999, “Expected Residual Vibration of Tkaditional and Hybrid Input Shaping Designs,” J. Guid., Contr., and Dyn., vol. 22, 162-165. Shen, J. C., Chen, B. S., and Kung, F. C., 1991, “Memoryless Stabilization of Uncertain Dynamic Delay Systems: Riccati Equation Approach,” IEEE Dans. Automat. Contr., vol. 36, 638-640. Singer, N. C. and Seering, W. P., 1990, “Preshaping Command Inputs to Reduce System Vibration,” %ns. ASME, J. Dyn., Meas., and Contr., vol. 112, 76-82.

Singh, T. and Vadali, S. R. “Robust Time Delay Control,” Bans. ASME, J . Dyn., Meas., and Contr., vol. 115,

References Bhat, K. P. M. and Koivo, H. N., 1976, “Modal Characterizations of Controllability and Observability in Time Delay Systems,” IEEE %ns. Automat. Contr., vol. 21, 292-293.

303-306.

Singhose, W. E, Derezinski, S., and Singer, N. C., 1996, “Extra-Insensitive Input Shaper for Controlling Flexible Spacecraft,” J. Guid., Contr., and Dyn., vol. 19, 385-391.

Bodson, M., 1998, “An Adaptive Algorithm for the Tuning of Two Input Shaping Methods,” Automatica, vol. 34, 771-776.

Boyd, S., El-Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994.

Singhose, W. E, Porter, L. J., Tuttle, T. D., and Singer, N. C., 1997, ‘‘Ybration Reduction using Multi-Hump Input Shapers, Trans. ASME, J. Dyn., Meas., and Contr., vol. 119, 320-326. Swigert, C. J., 1980, “Shaped Torque Techniques,” J. Guid. and Contr., vol. 3, 46&467.

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Tallman, G. H. and Smith, G. H., 1958, "Analog study of Dead-Beat Posicast Control," IRE h n s . on Automat. Contr., vol. 3, 14-21. Tzes, A. and Yurkovich, S., 1989, "Adaptive Precompensators for Flexible Link Manipulator Control," in Proc. IEEE Conf. on Dec. and Contr., 2083-2088, Tampa, FL. Tzes, A. and Yurkovich, S., 1993, "An Adaptive Input Shaping Control Scheme for Vibration Suppression in Slewing Flexible Structure," IEEE h n s . Contr. Sus. Tech., vol. 1, 114-121. Verriest, E. I. and Ivanov, A. F., 1994, "Robust Stability of Systems with Delayed Feedback," Circuits, Sys., and Sig. Proc., vol. 13, 213-222.

I

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.

a5

1

IS

2

,

,

.

.

.

26

3

35

1

45

T m Wsl

Figure 4: Response with Exact Cancellation

1

Shaper

Figure 1: Open-Loop Input Shaper Figure 5: Response with Inaccurate Impulse Application

Instance

Figure 2: Closed-Loop System with Input Shaper Figure 6: Closed-Loop Response with Inaccurate Im-

pulse Application Instance: LQR Design

1,

,

.

.

.

.

.

.

,

,

,

Figure 3: Open-Loop Response

Figure 7: Closed-Loop Response with Inaccurate Im-

pulse Application Instance: Theorem 3.2 Design

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