Observer-Based Modal Control of Flexible Systems using Distributed

Proceedings of the 40th IEEE Conference on Decision and Control Orlando, Florida USA, December 2001

FrM04-4 Observer-Based Modal Control of Flexible Systems Using Distributed Sensing S. P. Nagarkatti, C. D. Rahn† , D. M. Dawson‡ and E. Zergeroglu Lucent Technologies Optical Fiber Solutions Sturbridge, MA 01550 snagarkatti, [email protected]

†

Abstract In this paper, we design a modal controller for conservative ßexible systems that uses distributed sensing. A spatial Þlter design based on the eigenfunctions coupled with a velocity observer prevents spillover instabilities in the closed-loop system. The proposed control is proven to stabilize a discrete set of controlled modes without destabilizing the remaining, residual modes. We then apply the theory to a single ßexible link robot arm and experimentally demonstrate the feasibility of the proposed control strategy. The experiments use high speed video feedback with image processing to determine the spatial beam curve. The controller quickly damps the Þrst modal response without causing instability in the remaining modes. 1.

Introduction

Vibration and noise reduce the perceived quality, productivity, and efficiency of many mechanical systems. Vibration can cause defects and limit production speeds during manufacturing and produce premature failure of the Þnished product due to fatigue. Potential contact with a vibrating system or hearing damage from a noisy machine can produce a dangerous, unhealthy, and uncomfortable operating environment. At the scale of most mechanical applications, the material composing the system components acts as a continuum. Components that do not deform appreciably under the applied loading may be approximated as rigid bodies. The remaining, distributed components can be modeled by partial differential equations (PDEs) or approximated by ordinary differential equations (ODEs) using numerical discretization techniques such as the Finite Element Method (FEM). Discretization may be the only option for distributed components with complex geometry or built-up assemblies. Many manufacturing, aerospace, acoustic, robotic, transportation, and power transmission applications, however, have the geometric simplicity that make PDE models the most accurate and concise representation of the system dynamics. Many control design tools exist for discretized ordinary differential equation models of ßexible systems (e.g. observer-based state feedback [5]). A substantial difficulty in the design of these controllers is the choice of the discretization order. Reduction of the inÞnite dimensional continuum model to a Þnite dimensional (N th order) discrete model means that certain motions (∞ − N ) are neglected. Typically, modal analysis motivates the model reduction. With sufficient system damping, higher order modes can be neglected if the controller rolls off (i.e. the controller gain drops sharply) at high frequency. Choice of N too small results in spillover instability that occurs when the controller, designed for the Þnite dimensional model, senses and actuates higher order modes, driving them unstable [1]. Reduction of the control gain to eliminate spillover often results in poor performance. Choice of N too large results in a high order compensator that can be difficult and costly to implement. Control design based on distributed parameter models eliminates control spillover instabilities. The physical displacement, slope, and curvature of the continuum constitute the state variables rather than numerically generated node displacements or modes. Thus, the system model closely links to the underlying mechanics. The resulting controllers are often simple, physically motivated, intuitive, and easier to implement and tune. Unfortunately, the relatively few control techniques for distributed parameter models (e.g. Lyapunov techniques [3] and semigroup theory

0-7803-7061-9/01/$10.00 © 2001 IEEE

‡

Pennsylvannia State University Mechanical and Nuclear Eng., University Park, PA 16802 [email protected]

Clemson University Electrical and Computer Eng., Clemson, SC 29634-0915 [email protected]

[9]) have not been developed into simple design tools. To avoid the spillover instabilities and complexity associated with discretized and distributed controllers, respectively, recent researchers have proposed controllers that use distributed sensing and actuation. Burke and Hubbard [2] use spatially varying piezoelectric Þlm to control all modes of a ßexible beam. Tzou and Tseng [12] use the direct and converse effects in two piezoelectric layers to measure and actuate a distributed shell model. Lee and Moon [7] design the spatial distribution of piezoelectric laminates to create distributed sensors and actuators that measure and excite speciÞc modes of plates and beams. Liu and Tzou [8] develop distributed photostrictive actuators. In this paper, we develop a modal controller for conservative ßexible systems that avoids spillover instabilities by employing distributed sensing. The control approach can be used with the modal sensors described in Lee and Moon [7] or with distributed sensing from high speed video. The controller includes a general spatial Þlter design based on the system eigenfunctions. Observerbased feedback exponentially stabilizes N controlled modes. A Lyapunov-like stability analysis proves that the residual modes remain bounded in closed-loop operation. We experimentally implement the proposed control for the Þrst mode of a ßexible single link robot using high-speed video feedback and demonstrate the closed-loop performance. 2.

Mathematical Model

The partial differential equations describing the vibration of linear, ßexible systems with negligible damping and gyroscopic effects can be written in the following form: w ¨ + A0 w = B0 f ,

(1)

where w (x, t) ∈ H denotes the composite displacement vector that contains all dynamic distributed and lumped displacements, A0 is the stiffness matrix operator, B0 is a bounded control input operator, f ∈ Rm is the control input vector, t is time, and x is the spatial coordinates vector. The Hilbert space H imposes integrability and static boundary condition constraints on the solutions of Eq. (1) and includes the inner product h·, ·i. We assume that operator A0 is symmetric and positive deÞnite as follows: Property 1: The matrix operator A0 is symmetric in the sense that hA0 v, wi = hv, A0 wi for all v, w ∈ H. Property 2: The matrix operator A0 is positive-deÞnite in the sense that hw, A0 wi > γ kwk2 for all w ∈ H, w 6= 0, where kwk2 = hw, wi and γ is a positive scalar constant.

Property 3: The operator B0 is bounded in the sense that kB0 f k ≤ α kf k2 ,

where k·k2 is the Euclidean norm and α is a positive scalar constant. To obtain the modal equations of motion, we assume the existence of a separable solution for Eq. (1) of the form

4268

w (x, t) = W (x) exp (jωt) ,

(2)

where W (x) denotes the eigenfunction vector, ω denotes the nat√ ural frequencies and j = −1. Substitution of Eq. (2) into Eq. (1) with f = 0 yields ¢ ¡ (3) A0 − ω2 I W = 0,

where I is the identity matrix. Under the assumption of distinct ωi , Lemma 1 in Appendix A shows that the eigensolutions (Wi , ωi ) of Eq. (3) are orthogonal as follows: hWk , Wl i = 1 hA0 Wk , Wl i =

ω2k

.

(4)

∀k = l

k=1

(6)

hWk , vi = hA0 Wk , vi = 0 ∀k ≤ N.

(20)

where Ks is the feedback gain matrix. With the control law of Eq. (20), the state equation becomes z. zú = (A − BKs ) z + BKs˜

(5)

Wk η k (t) + v (x, t) ,

(19)

Observed state feedback is designed as follows: z, f = −Ksˆ

where the subscript k denotes the k th controlled mode, η k (t) denotes the kth modal coordinate, and v (x, t) is orthogonal to the Þrst N controlled modes

(21)

From Eqs. (19) and (21), we formulate the dynamics of the observed-state feedback control system      zú A − BKs z BKs =   , (22) . 0 A − LC ˜ z ˜ z

with the characteristic equation

This implies that and

(17)

produces the state estimate ˆ z (t). The gain matrix L is used to tune the observer time response. The observation error is deÞned as ˜ z (t) = z (t) −ˆ z (t) (18) and the observation error dynamics are ˜ z= (A − LC) ˜ z.

The displacement is transformed into the Þrst N modes and a residual displacement v (x, t) as follows: N X

.

ˆ z= Aˆ z + Bf + L (y − Cˆ z)

∀k = l

hWk , Wl i = hA0 Wk , Wl i = 0 ∀k 6= l.

w=

A state observer

(23)

hv, A0 wi = hv, A0 vi

(7)

|sI − A + BKs | |sI − A + LC| = 0.

hv, wi = hv, vi .

(8)

The matrices L and Ks are designed to ensure negative closedloop roots of the characteristic Eq. (23) such that z and ˜ z in Eq. (22) exponentially decay to zero. Hence, from Eq. (18), ˆ z is exponentially stable in the sense that

hv, ú A0 wi = hv, ú A0 vi

(9)

Separability means

and hv, ú wi ¨ = hv, ú v ¨i . (10) To obtain the modal equations of motion, we evaluate the inner product ¨ + A0 w = B0 f i ∀k = 1, .., N. (11) hWk , w Substitution of Eq. (5) into Eq. (11) and use of the orthogonality relations in Eqs. (4) and (6) produce the N linearly independent equations η ¨k (t) + ω 2k η k (t) = bT kf

∀k = 1, 2, ..., N,

(12)

where bT k f = hWk , B0 f i and kbk k2 6= 0 ∀k ≤ N so that the modes are controllable. These modal equations of motion form the open-loop system η ¨ + Ω2N η = B0 f , (13) ª T 2 2 N ×N , B0 = [b1 , b2 , ..., bN ] ∈ where ΩN = diag ωk ∈ R RN ×m , and η = [η 1 , η 2 , ..., η N ]T ∈ RN . ©

3.

Control Design

The control objective is to exponentially regulate the controlled modes in Eq. (13) without destabilizing the residual Eq. (27). We assume that the distributed measurements w (x, t) are available for all x and t. A full-state observer is constructed to estimate w ú (x, t) from the w (x, t) data. The modal coordinates are calculated from the mode shapes and distributed measurements as follows: η k = hWk , wi . Az + Bf

y=

Cz,

zk2 ≤ βξ 1 exp (−ξ 2 t) , kf k2 = k[Ks ] ˆ 4.

(25)

Residual Mode Stability

The weak form of the residual equations of motion results from the inner product hv, ú w ¨ + A0 w = B0 f i (26) producing hv, ú v ¨i = − hv, ú A0 vi + hv, ú B0 f i , (27) where the orthogonality property in Eq. (6) has been used. To establish the residual mode stability result, we formulate the energy of the system as Er =

1 1 kvk ú 2 + hv, A0 vi , 2 2

(28)

where the Þrst and second terms in Eq. (28) represent the kinetic and potential energies, respectively. The time derivative of Eq. (28) yields Eú r = hv, ú v ¨i + hv, ú A0 vi = hv, ú Bf i , (29) where Eq. (27) and the distributive property of the inner product is used. Eq. (29) is upper bounded as Eú r ≤ |hv, ú B0 f i| ≤ kvk ú kB0 f k ,

(30)

where the Cauchy-Schwarz inequality is used. Use of Property 3 and substitution of the bound in Eq. (25) into Eq. (27) produces kB0 f k ≤ αβξ1 exp (−ξ 2 t) .

(15) where y=η is the output, the state ¤T £ ∈ R2N and the matrices z = ηT ηú T     0 0 I £    , and C = I  , B= A= 2 B0 −ΩN 0

(24)

where β = kKs k2 .

(14)

The modal equations (13) are written in state-space form zú =

kˆ zk2 ≤ ξ 1 exp (−ξ 2 t) ,

where ξ1 and ξ2 are positive scalar constants. Substitution of the bound of Eq. (24) into Eq. (20) yields

vector

0

¤

.

(16)

(31)

After substituting Ineq. (31) into Ineq. (30), we obtain q q Eú r ≤ kvk ú 2 αβξ 1 exp (−ξ 2 t) ≤ kvk ú 2 + kvk2 αβξ 1 exp (−ξ2 t) . (32) Application of Property 2 to Ineq. (32) produces Ãs ! 1 Eú r ≤ hv, ú vi ú + hv, A0 vi αβξ1 exp (−ξ2 t) . (33) γ

4269

After substituting Eq. (28) into Ineq. (33), we obtain the upper bound s ) ( √ 2 p Eú r ≤ max 2, Er αβξ1 exp (−ξ 2 t) , (34) γ which is rewritten as d ³ p ´ 2 Er ≤ max dt

(

s ) √ 2 2, αβξ 1 exp (−ξ2 t) . γ

The integration of Ineq. (35) yields p Er ≤ ξ3 [1 − exp (−ξ2 t)] + c2 ,

(35)

(38)

where the positive scalar constant c1 = ξ3 + c2 . Upon squaring both sides of Ineq. (38), we obtain Er ≤ γ 3 exp (−γ 4 t) + c0 , where positive scalar constants © ª γ 3 = max ξ 23 , 2ξ 3 c1 , γ 4 = min {ξ 2 , 2ξ2 } , c0 = c21 .

(39)

1 (γ exp (−γ 4 t) + c0 ) 2γ 3

(41)

Flexible Link Robot Arm Example

The ßexible link robot arm shown in Figure 1 is modeled as an Euler-Bernoulli beam clamped to a rotating hub at one end and carrying a payload at the other end. A control torque input is applied to the hub with the dual control objectives of providing a desired angular displacement while simultaneously regulating the link vibration. Based on standard Euler-Bernoulli beam modeling assumptions and for small displacement, the Þeld equation for the beam is given by EI wxxxx = 0 ∀x ∈ (0, L) (42) wtt + ρ with the following boundary conditions w (0, t) = 0, EI wxx (0, t) = τ h (t) , Jh EI wxxx (L, t) = 0, wtt (L, t) − m

wxtt (0, t) −

and wxtt (L, t) +

EI wxx (L, t) = 0, Jm

where the Hilbert space H is deÞned as n H := [a, b, c, d]T | a ∈ H 2 (0, L), b, c, d ∈ R b = ax (0) , c = a (L) , d = ax (L) , a (0) = 0} , where and

(43) (44) (45)

(46)

where w (x, t) = q (x, t) − xθd (47) is the transformed link displacement, θd is the constant desired angular setpoint displacement of the hub, q (x, t) denotes the actual link displacement, ρ and EI denote the link mass/length and bending stiffness, respectively, m and Jm denote the payload mass and inertia, respectively, Jh is the hub inertia, τ h (t) is the applied hub torque, and subscripts x and t indicate partial derivatives with respect to x and t, respectively. The open-loop system for the ßexible link robot system given by Eqs. (42) — (46) can be written in the form of Eq. (1) where the composite displacement vector ¤T £ w (L, t) wx (L, t) wx (0, t) , (48) w = w (x, t)

© ª H 2 (0, L) = g : [0, L] → R | g, g 0 , g 00 ∈ L2 .

n o R 2 L2 (0, L) = g : [0, L] → R | L 0 g dx < ∞ .

(49)

(50)

(51)

(52)

(53) (54)

From Lemmas 2 and 3 in AppendixnB, we see that A0 satisÞes o EI EI EI EI Properties 1 and 2 with γ = min 4ρL 4 , 4mL3 , 4Jm L , 4J L . h Clearly, B0 satisÞes Property 3 with α = 1. 6.

6.1.

and the residual modes remain bounded in closed-loop operation. 5.

An inner product is deÞned as follows R hw, ui = ρ 0L wudx + mw (L, t) u (L, t) +Jm wx (L, t) ux (L, t) + Jh wx (0, t) ux (0, t) ,

(40)

Thus, from Eq. (28), Ineq. (39), and Property 2, we obtain kvk2 ≤

the control input f = τ h , and the input operator £ ¤T 1 0 0 . B0 = 0

(36)

where c2 is the constant of integration and the positive scalar constant s ) ( √ 1 2 αβξ1 ξ 3 = max 2, . (37) 2 γ ξ2 Ineq. (36) is rewritten as p Er ≤ ξ 3 exp (−ξ2 t) + c1 ,

the stiffness matrix operator is deÞned by h 1 w (x, t) − J1 wxx (0, t) A0 w = EI ρ xxxx h iT 1 1 −m wxxx (L, t) w (L, t) , Jm xx

Experimental Validation

Description of Experimental Setup

The experimental testbed shown in Figure 2 consists of thin ßexible beam 1 actuated at one end by a switched reluctance motor (SRM) and carrying a payload mass of 0.1 [kg]. The following control hardware is used: (i) a Dalsa CAD-6 camera that captures 955 frames per second with 8-bit gray scale at a 256×256 pixel resolution; (ii) a Road Runner Model 24 video capture board; and (iii) two Pentium II-based personal computers (PCs) operating under QNX (micro kernel-based, real-time operating system). A 102,400count resolver is mounted on the SRM to measure hub angular displacement wx (0, t). Data acquisition and control implementation are performed at 1 [kHz] via the Quanser MultiQ I/O Board and interfacing circuitry. The Dalsa camera, with lens of 0.08 [m] focal length, is mounted 1.2 [m] above the robot workspace. One PC hosts the video capture board and acquires and processes the visual data from the high-speed camera. This raw data is processed to obtain distributed displacement measurements. Figure 3 (a) shows a photograph of the experimental setup as seen by the camera. The camera viewing area is square and, to simplify the image processing, it does not include the entire link. A snapshot, the beam centerline pixel data, and a best Þt cubic polynomial are shown in Figures 3 (b) — (d). The four time-varying polynomial coefficients are then transmitted via a fast, dedicated TCP/IP connection to the control PC, where the control algorithm and other I/O operations associated with the ßexible link robot are implemented. The various parameters associated with the ßexible link robot system are calculated using standard test procedures and engineering handbook tables to be, m = 0.31 [kg], L = 1 [m], EI = 2.933 [N—m 2 ], Jh = 0.034 [kg—m 2 ], Jm = 0.1562 [kg—m 2 ], ρ = 0.239 [kg/m]. The Þrst modal coordinate η 1 is numerically computed online according to Eq. (14). The time derivative calculations are implemented using a standard, backwards difference/Þltering algorithm for the full-state feedback implementation only. The control algorithms are written in C++ and implemented using the QMotor [4] real-time control environment.

6.2.

Eigenvalues and Eigenfunctions

For the ßexible link robot experiments, we choose N = 1 to focus control on the Þrst ßexible mode. Implementation of the proposed controller in Eq. (20) requires measurement of η1 (t) as shown in 1 The ßexible beam is 0.05 [m] in width and 0.005 [m] in thickness.

4270

Eq. (14). Consequently, determination of the mode shape W1 is essential. After substituting Eq. (2) into Eq. (42), we obtain s EI 2 ω= (55) β , ρ where the eigenfunctions that satisfy Eqs. (42) — (43) are of the form W (x) = C1 sin (βx) + C2 cos (βx) (56) +C3 sinh (βx) + C4 cosh (βx) ,

are comparable. However, with regards to the vibration regulation, the observer-based modal controllers outperform the PD controller. SpeciÞcally from subplots (a) and (b) in Figure 4, we observe that at x = 0.45L and 0.85L, the beam exhibits large displacements with a signiÞcant Þrst ßexible mode component with the PD controller (see Figure 5 (b)); however, these displacements are very quickly reduced to less than ±1 pixels within 4 [sec] and almost completely eliminated by 8 [sec]. Figure 5 compares the results of experiments 1—4. The observer-based feedback greatly outperforms the PD control and is robust to large variations in payload mass (see Figure 5 (b)).

where are C1 , C2 , C3 , and C4 are constant coefficients. The eigenvalues are obtained by the substitution of Eq. (56) into the boundary conditions (43) — (46). The resulting characteristic equation ¢ 2β 4 (EI)3 £¡ −mJh Jm β 7 − ρ2 Jh β 3 3 ¡ ρ2 ¢ + ¡−2ρ mβ sin (βL) sinh (βL) ¢ + ¡ρJh Jm β 6 + ρmJm β 4 + ρmJh β 4 − ρ3 ¢ sin (βL) cosh (βL) 6 4 4 3 + ¡ρJh Jm β − ρmJm β − ρmJh β + ¢ ρ cos (βL) sinh (βL) ¤ + mJh Jm β 7 − 2ρ2 Jm β 3 − Jh ρ2 β 3 cos (βL) cosh (βL) = 0 (57) has a solution for the experimental parameters of −

β 1 = 1.483.

(58)

The constant coefficients in Eq. (56) are numerically computed to yield the eigenfunction W1 (x) =

−0.9546 sin (β 1 x) − 0.1959 cos (β 1 x) +0.1098 sinh (β 1 x) + 0.1959 cosh (β 1 x) .

7.

For positive deÞnite, self-adjoint ßexible systems in the form of Eq. (1) with distinct natural frequencies, spatial Þlters given by Eq. (14) produce the modal displacements. These spatial Þlters can be implemented using spatially varying piezoelectric laminates as in [7] and boundary measurements or high speed video feedback. Modal displacement and estimated velocity feedback exponentially stabilizes the Þrst N modes without destabilizing the residual modes assuming the input operator is bounded. The theory is applied to Þrst mode control of a single ßexible link robot. The feedback includes hub rotation, payload displacement, and payload rotation terms measured using high speed video with polynomial curve-Þtting and a hub encoder. The experimental results demonstrate hub position regulation and fast transient decay without spillover instability. References

(59)

[1] Balas, M., 1978, “Active Control of Flexible Systems,” Journal of Optimization Theory and Applications, Vol. 25, No. 3, pp. 415-436.

Remark 1 The eigenvalue in Eq. (58) corresponds to a natural frequency of 1.23 [Hz] for the Þrst ßexible mode. This analytical result is experimentally validated to be 1.22 [Hz] by energizing the robot actuator with a small sinusoidal torque trajectory with a time-varying frequency from 0.1 [Hz] to 4 [Hz] over an interval of 10 [sec] and extracting the frequency of peak link angular displacement.

6.3.

[2] Burke, S., and Hubbard, J., “Distributed Actuator Control Design for Flexible Beams,” Automatica, Vol. 24, No. 5, pp. 619-627, September 1988. [3] Canbolat, H., Dawson, D., Rahn, C., and Vedagarbha, P., “Boundary Control of a Cantilevered Flexible Beam with Point-Mass Dynamics at the Free End,” Mechatronics, Vol. 8, No. 3, pp. 163-186, 1998.

Experimental Results

[4] Costescu, N., Dawson, D., and Loffler, M., 1999, “QMotor 2.0 - A Real-Time PC Based Control Environment”, IEEE Control Systems Magazine, pp. 68—76.

The objective of the experiment is to regulate the angular displacement of the ßexible link robot to a setpoint of 20 [deg] starting from an initial angle of 0 [deg]. Experiment 1: For comparison, a PD control law

[5] Gould, L. A., and Murray-Lasso, M. A., “On the Modal Control of Distributed Systems with Distributed Feedback,” IEEE Transactions on Automatic Control, Vol AC-11, No. 4, October 1966.

τ h (t) = −ka wx (0, t) − kb wxt (0, t) , is implemented with ka = 2.0 and kb = 0.01 using the hub encoder feedback. The control gains are tuned to achieve the best performance. Experiment 2: Next, the proposed controller deÞned by Eq. . (20) is implemented, where η ˆ1 is obtained by integrating the equation .. ¡ ¡ ¢ ¢ 2 2 (60) η ˆ1 = − 1 + ω21 + kobs η ˆ1 + τ h + 1 + kobs η1 ,

[6] Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities, Cambridge University Press, 1959. [7] Lee, C.-K., and Moon, F. C., “Modal Sensors/Actuators,” Journal of Applied Mechanics, Vol. 57, June 1990, pp. 434441. [8] Liu, B., and Tzou, H.-S., “Distributed Photostrictive Actuation and OptoPiezothermoelasticity Applied to Vibration Control of Plates,” Journal of Vibration and Acoustics, Vol. 120, October 1998, pp. 937-943.

.

η ˆ1 is obtained by integrating η ˆ1 , and the differential Eq. (60) is constructed from Eq. (17). The set of control gains that render the best closed-loop performance are © ª 3.0, 0.007 Ks = and kobs = 25.

Experiment 3: To establish robustness, the controller deÞned in Eq. (20) is tested with a higher payload of m = 0.55 [kg]. The same set of control gains in Experiment 2 are used to allow for a fair comparison. Experiment 4: Experiment 2 is repeated again with no payload and the same set of control gains as in Experiment 3. Subplots (a) and (b) in Figure 4 show the camera measured beam displacements at x = 0.45L and x = 0.85L, respectively, measured in pixels to quantify and compare the vibration regulation performance of the controllers. Subplots (c) and (d) in Figure 4 show the hub angle and hub torque, respectively. Subplot (e) in Figure 4 shows the time response of the Þrst modal coordinate, while Figure 4 (f) shows the time response of the Þrst mode estimate deÞned in Eq. (60). From the experimental results, we observe that the hub angular displacement regulation performances of all the experiments

Conclusions

[9] Luo, Z. H., and Guo, B. Z., 1997, “Shear Force Feedback Control of a Single-Link Flexible Robot with a Revolute Joint”, IEEE Trans. Automatic Control, Vol. 42, No. 1, pp. 53—65. [10] Meirovitch, L., 1967, Analytical Methods in Vibrations, New York, MacMillan. [11] Ogata, K., 1990, Modern Control Engineering, New Jersey, Prentice Hall. [12] Tzou, H. S., and Tseng, C. I., “Distributed Piezoelectric Sensor/Actuator Design for Dynamic Measurement/Control of Distributed Parameter Systems: A Piezoelectric Finite Element Approach,” Journal of Sound and Vibration, Vol 138, No. 1, pp. 17-34, 1990. Appendix - Lemmas and Operator Properties Lemma 1 Given that the matrix operator A0 in Eq. (49) is symmetric and positive-deÞnite with distinct eigenvalues, the solutions of Eq. (3) are orthogonal.

4271

Proof: The eigenvalue problem of Eq. (3) can be written as A0 Wk = ω2k Wk .

(61)

After taking the inner product with Wj , we get hWj , A0 Wk i =

ω 2k

hWj , Wk i .

0

(62)

The symmetry of matrix operator A0 allows us to rewrite Eq. (62) as (63) hA0 Wj , Wk i = ω 2k hWj , Wk i . Similarly, we can rewrite Eq. (61) as A0 Wj =

ω2j Wj .

Cancelling common terms and applying the boundary condition of Eq. (43), we obtain Z L 2 wxx (x) dx, (74) hw, A0 wi = EI

(64)

After taking the inner product with Wk , we get hA0 Wj , Wk i = ω2j hWj , Wk i .

(65)

After subtracting Eq. (65) from Eq. (63), we get ¢ ¡ 2 ωk − ω2j hWj , Wk i = 0

(66)

with distinct eigenvalues (i.e., no repeated natural frequencies), then  hWj , Wk i = 0  ∀k 6= j, (67)  hA0 Wj , Wk i = 0  hWj , Wk i = 1  ∀k = j, (68)  hA0 Wj , Wk i = ω2i where Eq. (63) has been used and the symmetric nature of A0 has been exploited. ¤ Lemma 2 The operator A0 given in Eq. (49) is symmetric. Proof: Based on the deÞnition of Eq. (51), we evaluate the inner product hR hw, A0 ui = EI 0L w (x) uxxxx (x) dx − w (L) uxxx (L) +wx (L) uxx (L) − wx (0) uxx (0)] , (69) where w, u ∈ H are deÞned in Eq. (52). After integrating by parts the Þrst term on the right-hand side of Eq. (69) four times, we obtain hR hw, A0 ui = EI 0L wxxxx (x) u (x) dx + wuxxx |L 0

which is rewritten as hw, A0 wi =

R R EI L 2 EI L 2 0R wxx (x) dx + 4 0R wxx (x) dx 4 L 2 EI L 2 w (x) dx + + EI xx 0 0 wxx (x) dx. 4 4

(75)

Remark 2 We use the following standard inequalities to obtain a lower bound for Eq. (75)  R 1 RL 2 2 (x, t) dx  w (x, t) dx ≤ 0L wxx   L4 0    R L 2 ∀x ∈ [0, L] . (76) 2 3 w (x) ≤ L 0 wxx (x, t) dx       R 2 (x, t) dx wx2 (x, t) ≤ L 0L wxx Note that the boundary conditions are utilized to formulate the inequalities (76) as illustrated in [6].

After applying the above standard inequalities to Eq. (75), we obtain RL 2 EI EI 2 hw, A0 wi ≥ 4L 4 0 w (x) + 4L3 w (x, t) (77) EI 2 EI 2 + 4L wx (x, t) + 4L wx (x, t) dx. The last three terms on the right hand side of Ineq. (77) are pointwise quantities (i.e., ∀x ∈ [0, L]). Hence, it follows that w 2 (x, t) ≥ w2 (L, t) , wx2 (x, t) ≥ wx2 (0, t)

wx2 (x, t) ≥ wx2 (L, t) , ∀x ∈ [0, L] .

(78)

After substituting Ineqs. (78) into Ineq. (77), we obtain n oh R EI EI EI EI hw, A0 wi ≥ min 4ρL ρ 0L w 2 (x) dx 4 , 4mL3 , 4Jm L , 4J L h ¤ 2 2 +mw (L, t) + Jm wx (L, t) + Jh wx2 (0, t) , (79) which is rewritten by using Eq. (51) as ½ ¾ EI EI EI EI 4 hw, A0 wi ≥ min , , , kwk2 = γ kwk2 4 3 4ρL 4mL 4Jm L 4Jh L (80) and is of the form given in Property 2. ¤

L L − wx uxx |L 0 + wxx ux |0 − wxxx u|0 −w (L) uxxx (L) + wx (L) uxx (L) − wx (0) uxx (0)] . (70) After cancelling common terms and applying the boundary condition of Eq. (43), we obtain hR hw, A0 ui = EI 0L wxxxx (x) u (x) dx + wxx (L) ux (L) −wxx (0) ux (0) − wxxx (L) u (L)] = hA0 w, ui . (71) ¤

Lemma 3 The operator A0 given in Eq. (49) is positive-deÞnite.

Figure 1 — Schematic diagram of the ßexible link robot arm.

Proof: Based on the deÞnition of Eq. (51), we evaluate the inner product hR hw, A0 wi = EI 0L w (x) wxxxx (x) dx − w (L) wxxx (L) +wx (L) wxx (L) − wx (0) wxx (0)] . (72) Integrating by parts the Þrst term on the right-hand side of Eq. (72) twice, we obtain hR hw, A0 wi = EI 0L wxx (x) wxx (x) dx + wwxxx |L 0 (73) − wx wxx |L − w (L) wxxx (L) 0

+wx (L) wxx (L) − wx (0) uxx (0)] .

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Figure 4 — Experimental modal control with an observer for the modal coordinate time derivative: (a) Beam displacement at x = 0.45L, (b) Beam displacement at x = 0.85L, (c) Hub angle, (d) Applied hub torque, (e) Estimated modal coordinate ηˆ1 , and (f) Estimated . modal velocity ηˆ1 .

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Figure 5 — Comparison of beam displacement at x = 0.45L: PD control (dashed line) and Full order observerbased modal control (thick solid line), with a heavier payload mass (dash—dot line), and with no payload mass (thin solid line): (a) Full time response and (b) MagniÞed response.

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