Multiobjective Robust Control of Flexible Structure with Observer

MUL TIOBJECTlVE ROBUST CONTROL OF FLEXIBLE STRUCTUR...

14th World Congress oflFAC

P-8a-04-6

Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China

MUL TIOBJECTlVE ROBUST CONTROL OF FLEXIBLE STRUCTURE WITH OBSERVER Yao-wuDu

Hai-pingDu

Xi-zhi Shi

State Key Laboratory o/Vibration, Shock & Noise Shanghai Jiao Tong University, Shanghai, 200030, P. R. China E-mail. [email protected]

Abstract: In this paper the multiobjective robust control method for the vibration control of flexible structure is studied. The parameter uncertainty of the system is formulated as a H 00 optimal problem and the state feedback H2 / Hco mixed norm control with pole placement constraint is solved by LMI (linear matrix inequalities) . Robustness of observer to the parameters uncertainty of system matrix is also formulated as a H", optimal problem with pole placement constraints and solved by LMI. Simulation results show that this controller design approach is robustness under large parameters perturbation. Copyright (!:) 1999lFAC Keywords: Large space structure, Active Control, Multiobjective optimization, Robust Control, Observer

1. INTRODUCTION Active vibration control of flexible strucrure has received considerable attention in the recently literature. Control of flexible structure differs from many conventional linear system, because there are two major difficulties in the controller design: unstructured uncertainty and structure uncertainty ( BaJas, 1978; Hyland, et al. 1993 ) . Unstructured uncertainty is bring out by high frequency unmodeled system dynamics and is fully studied by many researchers. Various control methods such as frequency-shaped LQG (H 2 ) control (Gupta, 1992) ,LQGILTR method (Prasad, et aL 1991 ) , H~ control ( Chase, 1996; Safanov, 1989 ) method are applied to solve the spillover problem caused by unstructured uncertainty. Structured uncertainty which is caused by the parameter pertw'bation due to the inherent modeling error or nonlinearity in system is another obstacle to effectively design controller for active vibration control of flexible structure. The robust stability of close-loop system under structured uncertainty is fonnulated as an Hoo optimal control problem by

Coeffrey et aL ( 1996 ). But his approach is based on state feedback and not take account for the uncertainty effect on the observer. In many case for vibration control, the state variables is hard to obtain due to the physical restriction. So observer is needed. The parameter uncertainty's effect on observer is critical in design a observed based controller and it must be considered in active vibration control of space flexible structure. In this paper, in order to not only obtain good frequency domain specification of close-loop system such as robust stability but also obtain time domain performance such as LQR specification, multiobjective robust controller is applied to the control of parameter uncertainty system. The state feedback H2 ! H"" mixed norm control with close-loop pole placement constraint is formulated in LMI (linear matrix inequalities ) and solved by convex optimization method. The robustness of state observer to the uncertainty of system matrix is formulated as a Hoo optimal control problem with pole placement constraint and is also solved by LMI. Computer simulation of vibration control of a simple supported beam is conducted and the results show that our method is effective to tackle the parameter

8054

Copyright 1999 IFAC

ISBN: 0 08 043248 4

MUL TIOBJECTlVE ROBUST CONTROL OF FLEXIBLE STRUCTUR...

14th World Congress oflFAC

uncertainty.

A
F(t)F T (t) < I

2.1 Mathematical model offlexible system The equation of motion for most flexible structure can be transformed into the following state space equation by modal synthesis and modal truncation.

(4 )

where D, E is known constant matrix which represent the structure of uncertainty, F( t) is the normbounded with the uncertainty; which is defined as having maximum singular value less than or equal to 1 for all time. The system with parameter uncertainty and state feedback control is as follows

( 1)

X=(A+M.)X+BU Y=CX

.) 2 = ( q 1 , ••. , q n, q.l ' ... q n ERn

is the

nominal system model, M is the uncertainty of A. The definition of the uncertainty M is the normbounded as follows M = DF(t)E (3 )

This section is divided into two sub-sections. Subsection I discussed the mathematical model of flexible structure system and sub-section II discussed the state feedback multiobjective robust control of structure system.

where X

( 2 )

+ M(t)

where A
2. MODELING AND MULTIOBJECTIVE ROBUST CONTROL

X=AX+BU y=cx

=A

( 5 )

U=KX

is the state In practical sense, the state in modal control is hard to

vector, q { and q I is the modal amplitude and modal velocity respectively, U E R m is the control vector, and Y ER P is the output vector. A, B, C are nominal system matrix as follows 0

A

In

= [ -An _2~AJl2 o

B=

0

]

0 0

'1 (XIft)

,,,(Xm)

'n(Xm)

c{'·~Z')

'n (Zl)

0

'l(ZP)

'n(Zp)

0

A 112

According to the theorem developed by Khargonekar ( 1990) that close-loop system under parameter uncertainty is quadric stable under the following condition:

. . !1

IjF(SI -A- BKrI~L < 1

= diag( ()} 1 , ••• , ()} n) , ()}

j

X=AX+BU+Ew

is

the ith modal

frequency, ~ is the damping coefficient, m is the number of actuators, Xj is the placement of ith actuator, p is the number of sensors, Zj is the i

( 6 )

By defining the auxiliary signal w and Zjnf transforming the system matrix into following form

2 = dia,.{ &\()} 2. ' • • • , ()} " ) ,

placement of ith sensor. ,

2.2 State feedback robust controller design The control objective of system described in equation 5 is that the close loop is robust stability under parameter tDlcertainty and LQR specification is minimized with pole placement constraint.

'l(XI)

where A

obtain, a state observer is needed and the design method is discussed in section 3.

(x) is the ith modal shape.

( 7 )

Zinf =FX Y=X,U=KX

So the robust stability condition for system is converted to the

IITzUofWII""

<1

In engineering practice, the accurate model of flexible

In order to satisfy the LQR specification in time

structure system is hard to obtain especially for the modal damping parameters and modal frequency. So it is assumed that the parameter tDlcertainty only occurred at the system matrix A has form as followings

domain as follows: J =

r

(X T QX + UT RU)dt

( 8 )

8055

Copyright 1999 IFAC

ISBN: 0 08 043248 4

MUL TIOBJECTIVE ROBUST CONTROL OF FLEXIBLE STRUCTUR ...

X=ATX+CTU+W U=LTX

( 18 )

Z=DTX

It is obviously that the H ~ nonn of the transfer function from W to Z is equal to

liTx-wlL ' So we

can solve standard H"" problem to obtain LT that minimize the [A,LC] and

fiT 110>' Because that the system [AT ,C T LT] has the same pole, so in X-w

14th World Congress ofTFAC

is demonstrated. The dotted line denoted open loop frequency response and the solid line denoted the close-loop frequency response with state feedback control. The open loop frequency response has three peak in the bode figure; while in the close-loop case, the amplitude peaks disappeared. The impulse response of system is shown in fig. 3; dotted line denoted open loop system response and the solid line denoted the state feedback control system response. It will be seen that the state feedback controller can effectively suppress the vibration of the beam. Bode Diagrams

addition to the minimize the H OCJ norm, we can specified the pole location constraint to ensure the observer has fast dynamics than that of controller. proof end Now the robust observer problem is converted into H"", optimal problem with pole pJacement constraint and can be solved by LMI. 4. SIMULATION STUDY In this section, the control system simulation of feedback control of transverse vibration of a simple supported beam is conducted. For convenience, the beam parameters m (mass per unit ), I (moment of inertial ), E (modules of elasticity) and L (length of the beam) is set to 1lllity. According to the boundary condition of the beam, the modal shape

r/li (x) = sin(im:), modal frequency

tiJ;

=

(illy.

Frequency (rad/sec)

Fig. 2 Frequency response of system impulse Response OCJ4~

____________

~

It is 003 ,

assumed that the damping coefficient .; is 0.01. Consider two-input two-output feedback control system as fig. 1. The two actuators are placed at 1112 and 1/6; two sensors are placed at 5/6 and 11/12. The modal frequency and damping coefficient are assumed to have 20% variation.

~

noz Q)

-g

.1::!

(i.

E

11

'lQ"

o.

<:(

-.0.02

.008

v.C4

L -_ _ _ _ _ _ _ _ _ _ _ _----1

Time (sec)

Fig. 1 Simple supported beam model

Fig. 3 Impulse response of system

The controller and observer is designed according to the method in section 2 and section 3. Matlab LMI toolbox is used to solve the controller and observer designed problem and as a tool for computer simulation (Gahinet, 1995 ). The simulation result is shown from fig. 2 to fig. 6. In fig. 2 the frequency response of nominal system from input 1 to output 1

When consider the system parameter uncertainty and the observer based control, the simulation results is shown in figure 4 to figure 6. In figure 4 the comparison of impulse response for nominal system ( dotted line ) and uncertainty system (solid line ) with robust controller and robust observer is illustrated. The parameter perturbation is relatively

8056

Copyright 1999 IF AC

ISBN: 008 0432484

MUL TIOBJECTlVE ROBUST CONTROL OF FLEXIBLE STRUCTUR...

X=ATX+CTU + W U=LTX

( 18 )

Z=DTX It is obviously that the H w

nonn of the transfer

function from W to Z is equal to

liT

X-w 11""

' So we

can solve standard H", problem to obtain LT that minimize the

liT IL.

[A, LC] and

[AT ,C LT]

X-w

T

Because that the system

14th World Congress oflFAC

is demonstrated. The dotted line denoted open loop frequency response and the solid line denoted the close-loop frequency response with state feedback control. The ,open loop frequency response has three peak in the bode figure; while in the close-loop case, the amplitude peaks disappeared. The impulse response of system is shown in fig. 3; dotted line denoted open loop system response and the solid line denoted the state feedback control system response. It will be seen that the state feedback controller can effectively suppress the vibration of the beam.

has the same pole, so in

BOde Diagrams

addition to the minimize the Her> nonn, we can specified the pole location constraint to ensure the observer has fast dynamics than that of controller.

proof end

[i]

~ 0"100

Now the robust observer problem is converted into Her> optimal problem with pole placement constraint and can be solved by LMI.

~

.~ ·,so '----'-..:..:.:.:.;;;.----'-'-_ _ _----'--'-'--'''--~'____''_'

~

:::i:

0r-~~~~~~~~~~---~~

4. SIMULATION STUDY In this section, the control system snnulation of

feedback control of transverse vibration of a simple supported beam is conducted. For convenience, the beam parameters m (mass per unit ), I (moment of inertial ), E (modules of elasticity) and L (length of the beam) is set to unity. According to the bOlmdary condition of the beam, the modal shape

tP;{x) = Sin(i1lX).

modal frequency

O)j

=

(i.nY·

·200 '-----'-'-'-''''''----'--'-'-'-'-=-----'---'-~"'--_'_'_____'_'.::..J

' o"

Frequency (rad/sec) Fig. 2 Frequency response of system impulse Response 0 0 4 . - -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _- .

Tt is

assumed that the damping coefficient ,; is 0.01. Consider two-input two-output feedback control system as fig . 1. The two actuators are placed at 1112 and 1/6; two sensors are placed at 5/6 and 11112. The modal frequency and damping coefficient are assumed to have 20% variation.

·0 .04 ' - -_ _. -_

_

_ _ _ _ _ _ _ _ _---'

w

Time (sec)

Fig. 1 Simple supported beam model

The controller and observer is designed according to the method in section 2 and section 3. Matlab LMI toolbox is used to solve the controller and observer designed problem and as a tool for computer simulation (Gahlnet, 1995 ). The simulation result is shoml from fig. 2 to fig. 6. In fig. 2 the frequency response of nominal system from input 1 to output I

Fig. 3 Impulse response of system When consider the system parameter uncertainty and the observer based control, the simulation results is shown in figure 4 to figure 6. In figure 4 the comparison of impulse response for nominal system ( dotted line) and uncertainty system (solid line ) with robust controller and robust observer is illustrated. The parameter perturbation is relatively

8057

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ISBN: 0 08 043248 4

14th World Congress oflFAC

MULTIOBJECTIVE ROBUST CONTROL OF FLEXIBLE STRUCTUR.. .

very big. From the figure 4, the control effect of proposed controller for the uncertainty system is deteriorated, but the stability is remained and the control effect is fairly good compared with the open loop response.

system with robust observer is stable under system uncertainty while response is unstable with conventional observer. Impulse Response

Impulse Response om5,-_ _ _ _ _ _ _ _ _ _ _ _ _-----. D.015

o, ~

I \

I \

I I ,

d.lI'J \1

J

..o.oos

-0.D1

=--------------1 -aOI5'-_ _ _ _ _ _ _ _ _ _ _ _ _---'

o

r

~O \

~

time (sec.)

_ _ _ _ _ _ _ _~----~

o

10

Time (sec.)

Fig. 6 Comparison of observer design

Fig. 4 Impulse of perturbation system

5. CONCLUSION Bode Diagrams

iD

' \'::0

:E~

.~

-l.t-D "----'----'-'--"--_

-'--'-'-'-'---'---'-'-'--"'--'------'--=

~ :2

5il .=

:sw

~ .c

-400 .

a..

~L,-'-~~~~~~-_-'-~~~~~

10

IQ

10

Frequency (rad/sec)

Fig. 5 Bode plot of system with observer In order to illustrate the robusmess of our proposed observer designed method, we compared the impulse response of uncertainty system with robust observer

( solid line) and conventional observer designed by pole placement (dotted line ) . The pole of robust observer is restricted in a disk whose center is located in -25 + Oi and radius is 3. The conventional observer's pole are also placed in this disk. The bode figure of system from input 1 to output 1 is show in figure 5. From figure 5, it can be seen that it has little difference between robust observer and conventional observer. Figure 6 shows impulse response of system with two types of observer. The impulse response of

This paper consider the robust control of vibration flexible structure with parameter uncertainty based on observer. A novel controller and observer scheme for vibration control is proposed. The robust stability of state feedback control under parameters is converted to a H of.> optimal control problem. In order to achieve good performance in time domain, LQR or equivalent H2 specification is include in the robust controller design and a mixed H2! H", optimal control with pole placement is formulated. This problem can be solved with convex optimization method-LMI . Robustness designed method of observer under parameter uncertainty is converted into H of.> optimal control problem with pole placement constraint and can also be solved by LMI. Computer simulation of vibration control of simple supported beam illustrated that the controller design method is effective and can be used to tackle the parameter uncertainty in the vibration control of flexible structure.

REFERENCE Balas M. J., (1978) Feedback Control of Flexible System, IEEE Transaction on Automatic Control, VoJ. AC-23, No. 4, pp. 673-679 Chase J. Geoffrey, H. Allison Smith, (1996) Robust H", Control Considering Actuator Saturation I: Theory, Journal ofEngineering Mechanics,Oct., pp: 976-983

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MUL TIOBJECTIVE ROBUST CONTROL OF FLEXIBLE STRUCTUR ...

14th World Congress ofTFAC

Hyland D. H., Jilllkins J. L., Longman R. W., Active control teclmology for large space structure, Journal of Guidance, Control, and Dynamics, Vol. 16,1993, pp:801-821 Gupta N. K., ( 1992) Frequency-shaped Cost Functionals: Extemation of LinearQuadratic-Gaussian Methods, Journal of Guidance, Contro~ and Dynamics, vol. 15, No.4, pp801-816 Gahinet P., LMl toolbox for matlab, (1995) Mathworks Inc Khargonekar P., Petersen I and Zhou K., ( 1990 ) Robust stabilization of uncertain linear systems: quadratic stability and H", control theory. IEEE Trans. Aut. Contr., Vol. 35, No. 7, pp. 356-361 Khargonekar P. P., RoteaM. A., (1991) Mixed H2 / H", : A Convex Optimization Approach, IEEE TrallS. Aut. Contr., Val. 36, No. 7, pp. 824 Mahmound Chilali, Pascal Gahinet, (1994) H Design with Pole Placement Constraints: An LMI Approach, Procdeeings ofthe 33rd eDc., pp: 553-558 Prasad J.V.R., Anthony J. Calise, Edward V. Byrns, ( 1991 ) Active Vibration Control Using Fixed Order Dynamic Compensation with Frquency Shaped Cost Functionals, IEEE Control System Maginze, pp:71-78 00

Safonov M.G., R.Y. Chiang and H. FJashner,

(1988) He>:> robust control synthesis for a space structure, Proc. ACC, Atlanta,

Selim Sivrioglu, Kenzo Nonami, (1997) Active Vibration Control by Means ofLMI-based Mixed H ... / R", State Feedback Control, JSME Int. J., Ser. C, Vol. 40, No. 2, pp. 239-244

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