Generalized Internal Model Control for a Single Flexible Beam*

Copyright ® IFAC Robust Control Design, Prague, Czech Republic, 2000

Generalized Internal Model Control for a Single Flexible Beam * Daniel U. Campos-Delgado, Yun-Ping Huang and Kemin Zhou Department of Electrical and Computer Engineering Louisiana State University Baton Rouge, LA 70803, USA

Abstract: This paper presents an experimental study of a flexible beam using a new controller architecture. This controller architecture separates the control design into two steps: (a) a performance controller and (b) a robustness controller. The distinguished feature of this new controller architecture is that the control system will behave like a system with only the performance controller when there is no uncertainty and the robust controller will only act where there is a discrepancy between the true system and the mathematical model used in design. Our experiments show the advantage of this new controller architecture. Copyright ©2000 IFAC Keywords: Robustness, Generalized Internal Model, Controller Parameterization, Flexible Structure, Hoo Control, LQG Control

case scenario which may never happen. Hence it is not surprising to see that such control system does not performance very well even though it is robust to model uncertainties. Motivated from these design limitations, one of the author has recently introduces a new controller architecture in Zhou (1999), called Generalized Internal Model Control (GIMC), which can overcome the classical conflict between performance and robustness in the traditional feedback framework. This controller architecture uses the well-known Youla controller parameterization in a completely different way. First of all, a high performance controller, say Ko, can be designed using any method, and then a robustification controller, say Q, can be designed to guarantee robust stability and performance using any standard robust control techniques. The feedback control system will be solely controlled by the high performance controller Ko when there is no model uncertainties and external disturbances, and the robustification controller Q will only be active when there is model uncertainties or external disturbances. In this paper, we shall apply this GIMC control strategy to a simple flexible beam experiment at Louisiana State University. This experiment con-

1. Introduction

The control of flexible structure vibration is very important in many practical applications. There are basically two ways to control vibration: passive or active. We shall only discuss the active approach using feedback control. Many active control strategies have been studied over the years. In particular, Hoc based robust control techniques have been widely studied in the last decade or so. One of the major problems in controlling a flexible structure is to obtain a reasonably accurate model. However, due to the distributed nature of the problem, the mathematical model is described by some partial differential equations which make the model infinite dimensional. Thus to do a control design, one must make some approximations. Hence one must take the modeling uncertainty into consideration in control design. This is probably the reason why robust control design techniques are essential in designing a high performance flexible structure control system. Unfortunately, the standard Hoc based robust control design is based on the worst • This research was supported in part by grants from ARO (DAAH04-96-1-0193), AFOSR (F49620-94-1-0415) , and LEQSF (DOD/LEQSF(1996-99)-04)

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sists of a flexible beam attached to a motor shaft and a digital camera that measures the deflection of the tip of the beam; this is the same setup developed and studied at University of Waterloo, see Krishnan and Vidyasagar (1988), Wang and Vidyasagar (1991), Wang and Vidyasagar (1992), Ravichandran, Pang and Wang (1993) and also Jung (1998). The paper is organized as follows. Section 2 gives a description of the setup of the Flexible Link Experiment. In section 3, the mathematical model is derived assuming mode shapes. Section 4 describes the new GIMC architecture in detail. In section 5, the GIMC controller is designed and implemented for the flexible beam. Both numerical simulations and experimental results are reported.

(1)

where d(x, t) is the time-dependent deflection along the beam for 0 ::; x ::; I, E is the modulus of elasticity, I is the cross-sectional area-moment of inertia, p is the density, and A is the crosssectional area of the beam. The solution of equation (1) can be found using the method of separation of variables: d(x, t) = (x) . q(t) which leads to

4

d (x) _ /34(X)

dx 4 2

d q(t)

dt 2

2. Experiment Description and Mathematical Model-

+W

=0

(2)

0

(3)

2 ( ) _

qt -

Ww

2 ing . The associated boundary where /3 4 = ~ = The single flexible link experiment consists of conditions are a direct drive motor, 12000 line encoder, EG&G (4) "(1)=0 1024 element linear CCD digital camera, Nikon 111 (I) = 0 (5) Lens and a 20 ampere power amplifier. A highpowered LED is mounted at the free tip of the (6) (0) = 0 aluminum link, the other end of the link is fixed 2 (7) El· " (0) + w J m ' (0) = 0 to the motor hub. The CCD camera mounted on top of the motor detects the tip deflection. A senwhere J m is the hub inertia. Equations (4) and (5) sor in the motor shaft encodes the motor angle represent the boundary conditions for the free end, relative to the initial angle. These two measure(6) is the condition for the clamped end, and (7) ments are processed by a personal computer which is due to the fact that the bending moment at the implements the controllers. The controller output joint is balanced by the inertial forces . Note that is the voltage applied to the motor. This results there are infinite many solutions to (2) depending in a single-input, multiple-output (SIMO) system. on the solutions for /3i that are solved from: However, the two measurements can be combined to describe the net tip position, which leads to a single-input, single-output (SISO) system. (1+cos/31 cosh/3l) + '::'~3 (cosh/3l sin/3l-sinh{3l cos/3l) = 0

,.

(8) which means that there are also infinite many natural frequencies of vibration that are related to /3i

.,

by Wi = /3ij'fI. Now , the solution to (2) can be written as

(

-ki (COS/3iX - cosh/3ix

+ ~ Sinh/3i x ) ]

(9)

where

=

k

y

,

FIG . 1.

Schematic of the setup

sin/3i l + sinh/3i l K 2 A· , cos/3i l + cosh/3il smh/3i l

/!:/33.

= 2pA J

m

and Di is a constant such that the modes shapes are orthonormal. Thus the deflection can be expressed as

To derive a mathematical model of the system, consider the schematic diagram shown in Figure 1 with motor angle {}, angular tip deflection Q ~ dll, tip deflection d, and link length l. It is noted that the net tip position is y = (}·l + d. The equation of motion for the free vibration of the beam is

00

d(x, t) =

L i(X) . qi(t) i=1

646

y

where qi(t) are time-dependent generalized coordinates. To derive a state-space model, an energy method is used. Define qo(t) = B(t) for the rigid body mode. The total kinetic energy T, the elastic potential energy V due to the bending beam, and the dissipative energy F due to the damping moment and the damping force can be expressed as:

•

FIG . 2. Standard Feedback Configuration

Lemma 1: Suppose that Ko stabilizes internally the feedback system shown in Figure 2. Let Ko and P have the following coprime factorizations Ko = UV- 1 =

i=l

K

~(Jm + JL)2~iwiq;

8L uqi

8F uqi

= N M-I

= M-I N.

= (V -

QN) - l(U

+ QM)

for some Q E Hoc such that det(V(oo) Q(oo)N(oo)) =I O.

where ~i = (ao + alw?)/2wi are damping ratios in which ao and al are proportionality factors. Defining the Lagrangian as L = T - V and using the Lagrange equation 8 8L ut uqi

P

Then every controller K that internally stabilizes the feedback system shown in Figure 2 can be written in the following form :

v = ~(Jm + JL) fw;q; F =

V-1U,

y

r

.

!:l~-~+~=Qi' l=0,1,2, . . .

where Qi are the generalized external forces, which gives the dynamic equation of the system that lead to the infinite dimensional state-space equations

FIG. 3. Youla Controller Parameterization

Note that the feedback system with a controller

q = Aq + BTinput, Y = Cq

(10) K

where q = [qO,qO,ql,ql, . .. ,V·

A = diag ([

~ ~], [

B = J 1 .1 ([0,1]' [0, 4>~ (0)], [0, 4>~(0)], ... ,) C=( l

I

0

I/ll(l)

0

...

I/ln(l)

0

QN)-l(U

+ QM)

(11)

can be implemented either as shown in Figure 2 after obtaining a total transfer function K or as shown in Figure 3 with five blocks. For a fixed Q, it is clear that there is no advantage in using the implementation in Figure 3 and, in fact, this implementation is usually not desirable since it needs much higher order controller implementation. A new implementation of the controller (11) is proposed in Zhou (1999), it is called Generalized Internal Mode Control (GIMC) , see Figure 4. The distinguished feature of this feedback implementation is that the control system will be solely controlled by the high performance controller Ko = V- I U when there is no model uncertainties and external disturbances while the robustification controller Q will only be active when there are model uncertainties or external disturbances. Suppose that Po has the following state space realization

o

[ -~~ m+

= (V -

. .. )

Remark 1: In this study, the damping ratio is assumed to be ~ = 0.001. In practice, the damping ratio depends on the considered mode and is higher than 0.001. Remark 2: The driving motor of the flexible link can not move instantaneously fast and do not respond to quite small driving signals, these nonlinearities are characterized by a voltage saturation of ±5 V and a deadzone of ±0.1 V.

Po Controller Structure

= [ ~ I~

]

and assume that (A, B) is stabilizable and (C, A) is detectable. Let F and L be such that A + BF is stable and A + LC is stable. Then the left cop rime factorization Po = M-I N can be chosen as

Consider a standard feedback configuration shown in Figure 2 where P is a linear time invariant plant and K is a linear time invariant controll er. The following controller parameterization is a variation of the well-known Youla controller parameterization Zhou (1999).

[N M] = [ 647

LC A +C

I B ~LD

7].

(12)

r

y

e

where Q is restricted to a lower order transfer function. This problem can be approximately solved using Hankel norm model reduction method or balance truncation method, see glover (1994), Zhou, Doyle, and Glover (1996). In summary, using this new structure a high performance robust controller can be designed in three main steps: Step 1:

FIG . 4. Generalized Internal Model Control Structure

Denote the state vector of [N note that f=Nu-My.

M]

by

x and Step 2:

(13)

Then we have

+ LC)x + (B + LD)u (Cx + Du) - y

x

(A

f

Ly (14)

(15)

Step 3:

i.e., f is the estimated output error. On the other hand, the GIMC controllers do not necessarily need to be implemented using five blocks of transfer functions. If Ko has the following stabilizable and detectable realization

Ko = [

~: I ~: ]

then there exists an Lk such that Ak stable and Ko = V-IU with

Design Ko = V-IU to satisfy the system performance specifications with a nominal plant model Po . Design Q to satisfy the system robustness requirements. Note that the controller Q will not affect the system nominal performance. Apply model reduction techniques to Q [N M], for instance using Hankel norm approximation or balance truncation method.

Controller Design

+ LkCk

For design purpose, a plant Po is obtained from the state-space representation of the flexible beam (10) by taking just 1 mode. So, the resulting nominal plant Po is a fourth order system. The nominal controller Ko is selected to be observer-based type

is

Then the GIMC structure can be redrawn as shown in Figure 5 where [ I -

Bk

V - ] _ [ A

M

-

+ LC C

I B +DLD

+ LkDk Dk (16)

;]

.

y

r

where the state-feedback gain F and observer gain L are computed using LQ R techniques to achieve good transient behavior. One restriction to the design is the saturation in the control voltage, so the nominal controller had to address it. Consequently, the weights for the synthesis of F and L were chosen Q = diag(l x 104 ,1,2,2)

in both cases, but R = 7 and 10 for F and L respectively. Doing so, the step response for the nominal controller Ko with plant Po is plotted in Figure 6. Next, to obtain [N !VI] and [ I - VU] we used again Po and Ko, and applied formulas (12) and (16) respectively. Land Lk were chosen to get normalized coprime representations. On the other hand, the true plant P(s) is assumed to be a left cop rime factor perturbed systern

FIG. 5. Alternative Implementation of GIMC

It is noted that the GIMC structure may result in a high order controller. Thus it might be convenient to do a controller order reduction for the inner loop. The following controller reduction scheme is used:

with the model uncertainty satisfying

(17) 648

,.sr----.-----.-----.-----.-----,

coprime perturbations. Furthermore, the same weighting function W(s) is used in both cases. Finally, the complete GIMC architecture conVU] sists mainly of two transfer functions

[I -

based on a observer-based controller and Q that comes from solving (17) , computed using balanced truncation method. The simulation taking as a true plant P(s) from (10) with 6-modes is presented in Figure 9. In addition, to check robustness of our design the damping ratio in P(s) was reduced by a factor 1/100, see Figure 10. Similarly, the eigenvalues for P(s) were reduced in a factor 0.5 and 0.75 , the responses are plotted in Figures 11 and 12 respectively.

_o.s'-_ _---':_ _ _--'--_ _ _L -_ _---'-_ _- - - '

o

0.5

' .5

2.5

time (sec)

-2 _3'-------':----'-----L------'----J o 0.5 , .5 2.5 time (sec)

FIG.

6. Step Response for Po with Nominal Controller Ko

1.5

/

E c

In this way, the control strategy is to design Q for robust stability of the uncertain feedback system, see Figure 7. Keeping in mind, that our nominal controller Ko = V-1U guarantees good performance if the uncertainty is small enough. As a consequence, in order to apply standard B oo techniques we need to obtain the Generalized Linear Fractional Transformation Form for the system, see Figure 8. The weighting function W(s) E B oo is chosen as a high-pass filter to address the uncertainty in the high frequency due to neglected modes.

....---.. ......:.~ '

I

/

.

.. ......

....

t:/ ·.···r"'/~/f---------"' ===~-----i ,I oo "

.

, time (sec)

;;

i

2

:1

,

~R

0

~

--,V ... - ,

~--'--=,.--'-~----------j ',----~-----

_2' - - - - - ' - - - - - - ' : - - L - - - - ' - - - - - ' - - - ' - - - - ' - - - - - '

o

4

'ime (sec)

9. Step Response for P(s) , GIMC (solid) and Hoc (dashed)

FIG.

.. ... .: I j

f

FIG .

7. GIMC with Left Coprime Perturbed System

4 lime (sec)

Meanwhile, to be able to compare the performance of the GIMC structure, a Boo controller for robust stability is designed based also on left ~

w

V" --

----------1

--........ -. :-::---'-:::=--'--.......

0

3-,

_2 ' - - - - - ' - - - - - - ' : - - L - - - ' - - - - - ' - - - ' - - - - ' - - - - - '

o

•

,,," (sec)

10. Step Response for P(s) with ( = 0.00001, GIMC (solid) , Hoc (dashed) and Ko (dash-dot)

FIG.

f

FIG.

q

From the simulations, it is clear that the GIMC controller performed better than the Boo design under even large uncertainty. This was the expected result, since for small uncertainty, the

8. Generalized Linear Fractional Transformation 649

GIMC controller performs exactly the same as Ko.

! ..

t• ...

..........--~----:---".•-.;:-._-_. - --;----;';;-----i

FIG. 13. Experimental Step Response, GIMC (solid), Hoo (dashed) and Ko (dash-dot) time (sec)

! ".

I ", .........--~----:----:;----:;----7,,----! 'w ... c_,

FIG. 14. Experimental Step Responses for two different beams

lime (sec)

FIG. 11 . Step Response for P(s) with perturbation in the eigenvalues, GIMC (solid), Hoo (dashed) and Ko (dash-dot)

Mode!. IEEE International Conference on Robotics and Automation. pp. 9-14 Wang, D . and M. Vidyasagar (1991). Transfer Function for a Single Flexible Link . The International Journal of Robotics Research. Vo!. 10, No. 5. Wang, D . and M . Vidyasagar (1992). Passive Control of a Stiff Flexible Link. The International Journal of Robotics Research. Vo!. 11 , No . 6. Ravichandran, T., G.K.H. Pang and D. Wang (1993) . Robust Hoo Optimal Control of a Single Flexible Link. Control-Theory And Advanced Technology. Vo!. 9, No . 4, pp .887-908. Jung, M. (1998). Control of a Single Flexible Link. Thesis for Master of Science in Electrical Engineering . Louisiana State University. Zhou, K. (1999). A New Controller Architecture for High Performance, Robust, Adaptive, and Fault Tolerant Contro!' submitted to IEEE Transactions on Automatic Control. Glover, K. (1984) . All Optimal Hankel-Norm Approximation of Linear Multivariable Systems and Their £00 - Error Bounds. International Journal of Control. Vo!. 39, pp. 1115-1193. Zhou, K. , J .C . Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall , Upper Saddle River, New Jersey.

lime (sec)

,.

, .,

i,

i' i',

\

"

I

:I I, I

, ',. , , , , I i , , , , ,. , ,, (I " " I

• J

\

I

I

I

,

" -50L---'---.l.--~--...L4---:---':----7---'-' tine (sec)

FIG . 12. Step Response for P(s) with perturbation in the eigenvalues, GIMC (solid), Hoo (dashed) and Ko (dash-dot)

Now, the experimental results are showed in Figure 13. In reality the motor does not act as a pure integrator, so the responses are normalized to its steady-state value. On the other hand, in order to compare the robustness in the experiment, the flexible beam (beam #1) was removed and substituted with a thinner one (beam #2) but with the same length. Therefore the parameters as Jm and JI, and consequently the frequency of the vibration modes change. The responses in Figure 14 express the good robustness of the controllers. REFERENCES Krishnan, H. and M. Vidyasagar (1988). Control of a SingleLink Beam Using a Hankel-Norm Based Reduced Order

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