Active Vibration Control Using LMI-Based Mixed H2

WA08

Proceedings of the 35th Conference on Decision and Control Koba, Japan ● Daeembar 1996

10:00

Active Vibration Control Using LMI-Based Mixed State and Output FeedbackControl with Nonlinearity Hz/

H.

Kenzo

Deaprtment

Nonami

of Mechanical

Chiba University,

and

Selim Sivrioglu

Engineering,

1-33 Yayoi-cho,

Faculty of Engineering

Inage-ku,

e-mail nonami @meneth.tm.chiba-u

Chiba 263,Japan .ac.jp

separately both theoretical and experimental by many authors. If there are uncertainties in the system model, H. control maintains good robust performance. On the other hand, H. control design is mainly concerned with frequency-domain performance and does not guarantee good transient behaviors for the closed-loop system. H2 control gives more suitable performance on system transient behaviors. Combining Hz and H- control objectives in a controller is one further step in robust control theory. Mixed problem can also be solved adding the Hz control objective to known H- control design. This procedure is quite difficult and not practical. Our design approach is numerical and has some advantage on classicalmixed control design.

Abstract In this paper, the mixed H,/ H= control is studied using Linear Matrix Inequality approach (LMI). In many linear control problems, the desigri constraints can be reformulated in terms of LMI. In general, Ha control maintains good frequency response performance but transient response of control system may not be guaranteed. Here we have formulated H. and H, control separately by means of LMI and also combined both objectives as a mixed control problem. We applied mixed H2 / H= control to an active vibration control problem and obtained the reasonable results in frequency and time domain. The design approach described here is based on numerical optimization technique which requires efficient convex optimizationsoftware.

In general, convexity is an important specification and many linear control problems can be reduced to convex optimization problems which involves linear matrix inequalities (LMI). LMI has more flexibility for combining various design constraints on the closed-loop system. Recently LMI-based control system analysis has become popular since it encompass many control subjects. The theory of the LMI-based control system analysishas been given as a very strict mathematical sense in reference [l]. Robusl control analysis using LMI approach can be found in [2] for general cases. And the reserach works [3], and [4] also give us good suggestions for finding out suboptimal H. controller using LMI approach.

1. Introduction In the last decade, robust control problems have been studied effectively in many fields of control engineering. Active vibration control is one of the main topic in these research works and still remains attractive for new control design schemes. Generally, time domain specifications like LQG control or HZcontrol and frequency domain specifications like H. control are mostly considered in active vibration control problems as valuable criterions. For robust control design, these specifications are also important to assess controller performance, In active vibration control area, it is necessary to realize both specifications of time domain and frequency domain. Here this study is concerned with robust control to satisfy the both performances of active vibration problems in a practical sense.

Mixed Hz/ H- control problem using convex optimization is formulated in reference [5]. State-feedback H2 / H. design with additional pole placement constraint is studied using LMI approach in [6]. In this study, the multiobjectiveH21 H= control is described by means of LMI-based state and output feedback and applied to an active vibration control problem. We solved multiobjective problem using very efficient convex optimization software [7,8,9] for a practical control object of the flexible structural control which was studied before [10] using both H2 and H. control separately.

Let us consider the control structure shown in Fig. 1. The plantG(s) is an Linear Time Invariant(LTI) model and given. At first, we may also assume that the state vector x is measurable. Here, w denotes the exogenous input vector and 7.1, 7.Zdenote the controlled output vectors. The closed-loop transfer functions from w to z, and z* can be expressed TZ,W and T,zW,respectively. The multiobjective H2 / Ha control may be described as follows. Find a staticstate-feedback law u = Kx such that minimizes IT,,W~ subject to ITZ,W 1. K y . This approach yields a convex suboptimal control problem whose global minimum of the performance measure J (T, J is an upper bound on the optimal performance of transIer matrix lTZ,W~. Next, we design an output feedback control

2. LMI Formulation

with State Feedback

Consider the LTI system described by X= AX+ B1W+B2U (1)

Z1= Clx + Dl,w + D1# Zz= C2X+ D21W+ D22U

system for a muhiobjective Hz i H. control. where x ● R“ is the state vector,

Pure Hz and H. control problems have been studied 0-7803-3590-2/66 $5.00 @ 1996 IEEE

161

z ]>z

z=

R“’

are the

Arranging the inequality (12), the following matrix inequality can be writtenas (A+ B2K)TP+P(A + B2K) +(C, + D, ZK)T(C1+ D,2K) )

(

w

PB1 + (Cl+ D12K)TD1, o [1ST R is eauvalent to . R>O

u.= (2) where K ~ R“Ux n is the statefeedback gain. The closedloop system is given by X=( A+ B2K)X+B1W Zz= (Cz + DZZK)X +D21W 2.1.

P–SR-lST>O

(15)

P- ‘(A+B,K)T+(A+B2K)P” ‘+P”‘(C,+D,,K)T(C,+D,,K)P-’

H2 Performance

- (B,+P-‘(C,+D,2K)T)D,, )(- fl+D,,D~,)-‘(BT,+D~,(C,+D,,K)P‘) 0 such thatfor all t dv ~ (x) + Z:z, – yzw,w 0 . Then, write for this XC, the Bounded Real inequality Lemma (39) A;XC,+ XC~C,

XCIBC,

X= AX+ BIW+BZU z~= C,x

- y..

D:

cc,

DC,

- yI

Z2= Czx + D22U

- (c,++) m,

and solve this inequality for the controller parameters @ . Since the (39) is an LMI in @, it can be solved by the same A= optimization algorithms. After finding a controller K(S) satisfying the above inequalities the following specifications are achieved by LMI based controller Al) K(s) internallystabilizes AZ) IF(S) I- < y “ LMI Based Feedback

Mixed Hz /lI.Control

-(k, +k,) ml

c, ~

~k m,

o

000

0

0

0

0

000

0

0

~

k -l_ m,

0

o

0

0

1

0

0

0

0

o

0

0

0

1

= y~c,+ OTQTPXC, + PTxcl@Q0 . Here the following condition should be satisfies for mixed problem for the tractabilityof LMI problem (43) x =Xc, = X2 Finally mixed problem can be solved with the constraint (41), and (42) plus Hmconstraints(34), (35), and (36). 5. Control

x,

/n, =l.5kg ml= 1.4kg Q = 1.4kg m,= 1.5kg kt.2985Nlm ct=0.89N-mlsec i = 1,...,4

Object

The active vibration control system as a regulator problem has been studied before using H2 and H~ design techniques scpcrately [10]. The model of the design object is shown in Fig. 2. The detailed information about this test rig is written in reference [10]. The equation of motion for this system is given by

Fig. 2 Model of t}le control object

164

6. Simulation 6.1 Cases control

with

Results state

feedback

The mixed H,/ H. state-feedback controller which is found using LMI approach has achived high damping in every mode. The frequency ressponse of the system with control and without control is shown in Fig 3. These frequency-domain results are quite good for this case. One practical aim of this is to show the design example improverneht of the time response of the system due to Hz performance objective. Let’s try H~ performance objective only for the same system and compare its time history response with the mixed H,/ Hcontrol response. Figures 4 and 5 show the impulse responses of the system with H~ and mixed HZ/ H. constraint respectively. As it can be seen from Fig. 5, impulse response of the mixed Hz / H~ control is much better than H@ control because the initial transient maximum amplitude in H2 / H* is small. This is the result of the Hz performance objeetive.

.,,oo~

.l~l

12002040

with output

feedback

Figures 6 and 7 show the results of the frequency response and the time history response each other based on H~ control using LMI control toolbox of MATLA13. y“,, is the minimum value obtained by iterative computation. We tried to compute a H~ control system using Riccati equation-based analytical approach. The both results are just the same. Figures 8 and 9 show the results of H2 contrbl. The performance in Fig.6 is better than Fig.8 because the gain of Fig.8 in the lower frequency range is high. On the other hand, the performance in Fig.9 is better than Fig.7. So, we have to take a trade-off between Hw norm and H2 norm to realize a desirable frequency response and a time history response. We show the results of mixed H,/ H. control with try and error of a trade-off concerning such two norms from Fig. 10 to Fig. 17. Figures 18 and 20 show the improvements of time history response using mixed Hz/ H. control changing y constraint of H~ norm and q constraint of Hz norm.

100

Frequency [rzd/s]

120

Fig.3Frequencyrespnseof x4/w usingmixedHz/H- controller

IIIIPUISC Rmpona,.f Ih, Sya,m wllhH-id CmIlml

0.015,

-0.OU ;

2 3 mm.[tee]

1

4

IInplsc Sqwmw .1 tk SystemwithH2M.fnf CaIIlml

0.015,

I

I .0.015 I 0

5

1

2 3 ‘mm[see]

4

Fig.5 Impulseresponseof x4 usingH, / H-

Flg.4 Impulseresponseof x4 usingH- controller

controller

Hm Control .Wx.,.

—X..x 6.2 Cases control

N)s0

Freqwncy @d/sl

Results

(-y@ = 0.000356)

. . g -,..

-t$0.

~

/..

,+,.,.,.,,,,,

,00 ,ln +% ., , ,* , n.%...)‘ “ ‘ ““ s -. w Fig,6 Frequencyresponsefor Hm control Fig.7 Impulseresponsefor H~ control

H2 Control

Results

Fig.8 FrequencyresponseforH2 control Mixed

H2/Hm

Control

(~oDt= 0.016)

Flg.9 Impulseresponsefor H2control

Results

(-y = 0.01, q.ti = 0.016 ) ,...-.........1.... ~’””””” ~““””””’l /1 ~, .....j..i .... ..i......... ,.....\ . ...

,-,

..,..,.,:.,., -v.-**lu$w........ ,.

..l..{...[ .. .t.....l .....l ... .~

,fJ..f .t....q ......i g .;q!..p/,. (~;~~;j I ; ) “:fi““y j~,1\j ;+, ,::~““+{’””/+ 0-

;,;:$ :

Wf’. w “’” & ....f.~’\j...\’.l

e Figure 21 shows the summary of two :!: !~ji,;::~ 1,; ““””’” norms trade-off. From the simulation ~....f~ ;! ,, ‘ : .$..!J ! ; y;,pill w.,.r,..;: results, it is not always best area for the ~,, ::... .+,,..$.. p....,.....+ .. :...., j ...1.. ,,\, ......; case of the shortest distance for the origin ~,,, ~llil~\[j,\ in Fig.21. The best performance area . ,3, m.%..)‘ ‘“ ‘ 4s “ which means good response for Fig.11 Impulseresponse formixedHz/Hm control frequent y domak and time history W. 10~wuenw rewonsefor ~xed H2/H~ control domain is far from the origin area. 165

I* 4,M+

,.,..

,

.s!

,,,

7, Conclusions We have formulated multiobjective (~= 0.001, qd = 0.0138 ) HZ/ H. control using LMI-based statefeedback and output-feedback approach and applied to the four stories structure. For vibration control engineers, the results obtained by us are reasonable and popular. Mixed Hz/ H. formulation encompasses many practical control problem like our design example. In LMI-based approach, many design constraints can be expressed as matrix inequalities and solved numerically very Fig.12FYequencyresponsefor mixedHz/Hm control Fig. 13 Impulseresponsefor mixedH2/Hm control efficiently. In particular, it seems that the LMI based control system design is Mixed H2/HW Control (~= 0.0005, ~op= 0.0148 ) superior to the conventional H= control —-.lm. -—-lUl#.l-.l. -—-.,.--. mwa.-wa ......... ..............: ............ ..T...,...., I I I ,, I .M.r !,. ,..,.....: ,. , system design because the solvable condition does not exist for a LMI solver and it is very easy to solve the mixed controller like Hz / H- control problem with output feedback. References 1) S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan : Linear Matrix Inequalities in System and Control Theory, SIAM Publication, 1994. 2) S. Hara, LMI Based Control System Design, Tutorial Seminar Text, SICE, pp.61-90, 1995, (in Japanese). 3) P. Gahinet, P. Apkarian : A Linear Matrix Inequality Approach To H@ Control, Int. J. of Robust and Nonlinear Control, vol. 4, pp. 421-448, 1994. 4) T. Iwasaki, and R. E. Skelton : All Controllers for the General Ho Control Problem : LMI Existence Conditions and State Space Formulas, Automatic, vol. 8, no. 8, pp.1307-1317, 1994. 5) P. P. Khargonekar, and M. A. Rotea : Mixed H2/ H. Control : A convex Optimization Approach, IEEE Trans. Aut. Contr., VO1.36, no. 7, pp. 824-837, 1991. 6) M. Chilali, and P. Gahinet: H~ Design with Pole Placement Constraints : an LMI Approach, Proc. of CDC, pp. 553-558, 1994. 7) L. E. Ghaoui, F. Delebecque, R. Nikoukhah, LMITOOL: A User Friendly Interface for LMI Optimization, User’s Guide, Beta Version, 1995. 8) L. Vanderberge, S. Boyd, Semidefinite Beta User’s Guide, Programing, Version, 1994. 9) P.Gahinet,A.Nemirovski,A. J. Laub,M.C hilali, LMI Control Toolbox, For Use withMATLAB,1995 10) W. Cui, K. Nonami, H. Nishimura, Experimental Study on Active Vibration Control of Structures by means of H. and Hz Control, JSME International Journal, VO1.37, no. 3, pp.462-467, 1994.

Fig. 14 Frequencyreeponeefor mixedH-JH~ control

Fig. 15 Imputseresponsefor mixedH.z/Hm

Mixed H2/Hm Control (7.X = 0.00035, q~ = 0.0382 )

Fig, 16 Fkequencyresponsefor mixedHz/H~ control

Time Responses Control

Comparison

of Mixed

-,.O

I

i

i

&

;

,j

*

Control

with

Hm

‘ !.....1. ...!.... ..i... ,!

m%.., ,

,*

Flg.20 7 = 0.00035,qop= 0.0382

166

H2/Hm

Ftg.19 T = 0.0~5, vwt = 0.0148

Fig.18 7 = O.ml, V4 = 0.0138

I*=I !

Fig.17 Impulserespomefor mixed H2/H~

.

4..

s

Flg.21 ‘lhkoff

betweenHm and H2 norm