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A sliding manifold approach for vibration reduction of flexible systems
Automatica 35 (1999) 1689}1696
Brief Paper
A sliding manifold approach for vibration reduction of #exible systems夽 A. Cavallo*, G. De Maria, R. Setola Dipartimento di Ingegneria dell'Informazione, Seconda Universita% degli Studi di Napoli, via Roma 29 81031, Aversa, Italy Received 19 December 1996; revised 16 March 1999
Abstract In this paper an active vibration controller for #exible systems is developed using a sliding manifold approach. The controller is designed to `eliminatea the system modes whose frequencies are close to the disturbance frequencies. The sliding surface consists of a time-invarying part which `eliminatesa the selected modes without a!ecting the remaining dynamics, and a time-varying part which bounds the control amplitude and its rate in the transient. Moreover, the features of the controller allows one to reduce the excitation of the high-frequency modes to reduce the spillover phenomenon. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Flexible structures; Robust control; Singular perturbations; Spillover; Sliding manifold
1. Introduction In the past decades there has been an increasing interest around active vibration control of #exible systems. Active vibration methods have been developed to overcome some of the limits of the classical passive methods (Fuller & von Flotow, 1995), in particular, the need to reduce the vibration at `lowa frequencies, i.e., in the correspondence of the "rst system natural frequencies where, generally, the main part of the elastic energy is stored. However, to develop an active vibration controller it is useful to point out some peculiarities of #exible systems (Balas, 1982) (i) these structures are in"nite-dimensional systems, e.g., they are described by means of linear partial di!erential operators; (ii) they are passive systems;
夽 This work was supported by POP97 grant, MISURA 5.4.2., Regione Compania and PRW97-MURST grant. This paper was not presented at any IFAC meeting. This paper was recommened for publication by Associate Editor C.V. Hollot under the direction of Editor R. Tempo. * Corresponding author. Tel: 0039-81-5010-252; fax: 0039-81-5037042. E-mail address: [email protected] (A. Cavallo)
(iii) they present an ideally in"nite number of resonance peaks with very small damping factors; (iv) some parameters, like damping factors, moment of inertia, etc., are uncertain; (v) their resonance peaks emphasize the disturbances whose frequencies are close to the system natural frequencies; (vi) to operate with "nite-dimensional systems the partial di!erential model is replaced by a "nite set of ordinary di!erential equations, neglecting, generally, the high-frequency dynamics. A direct consequence of item (vi) is that, since sensors and actuators act both on modelled dynamics and neglected ones, the latter ones may be driven to instability (spillover phenomenon, Balas, 1978). More speci"cally, the action exerted by actuators on the unmodelled dynamics is called control spillover, whereas the component of the unmodelled dynamics measured by the sensor is called observation spillover (Meirovitch, 1990). An approach that is una!ected by the spillover phenomenon is the co-located feedback (Preumont, Dufor & MaleH kian, 1992). This approach yields a moderate increase of the damping factor of some modes. However, considering that practical devices have "nite bandwidth, and due to the impossibility of a perfect co-location of the devices, their passivity property may no longer be guaranteed (Goh & Caughey, 1985).
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 8 - 0
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Before describing our approach, we brie#y discuss the problem speci"cations: E in many situations (e.g. pressure wave at fuselage in turbofan aircraft) the disturbances are narrow band signals whose spectrum can be easily determined. In this case, the worst situation occurs when the disturbance frequencies are close to some natural frequencies of the structure; E the controller should be robust with respect to parametric uncertainty and disturbances; E control spillover is avoided if the unmodelled dynamics are not a!ected by the control action; in particular, the control signals should be smooth and with a limited rate of variation (Junkins & Kim, 1993). In this paper we present a control strategy addressing each of the above points. In particular, E the closed-loop system does not possess resonances that can be excited by narrow-band disturbances; E the controller is based on singular perturbation concepts and thus exhibits high robustness properties; E to avoid excitation of high-frequency unmodelled dynamics, special attention is devoted to the initial control transients (the `reaching phasea of the control), and the limitation of controller rate variation. It is known (Utkin, 1990) that high-gain feedback guarantees robust performances and insensitivity with respect to disturbances and parametric variations. However, this control strategy requires some state dynamics to become faster as the gain increases. During the initial transient (reaching phase) these `fasta states quickly approach the fast manifold, resulting in control signal peaks which can excite the high-frequency dynamics thus generating considerable control spillover. Using a singular perturbation approach in Cavallo, De Maria and Nistri (1993a) it is shown that the robustness property of high gain control may be obtained, without the above drawbacks, by de"ning the control signal as the solution of a di!erential equation depending on a small parameter e. Speci"cally, the whole state is `slowa, while the control signal is a `fasta and continuously di!erentiable function. An alternative line of attack to the problem of eliminating the reaching phase can be found in Utkin (1996), where a strategy based on a pre-computation of a control law stabilizing the nominal system is presented. However, even in this case, the control exhibits a fast transient (control reaching phase) which generates control spillover. In Junkins and Kim (1993) a multiplier function is proposed to smooth the control resulting from a bang}bang strategy, and in Cavallo, De Maria and Nistri (1998a) a multiplicative action on the derivative of the control, again in the singular perturbation framework, has been proposed.
In this paper, by using the above strategy, a controller which is able to increase the damping factor of r selected modes of the structure (where r is the number of available inputs) leaving the remaining dynamics una!ected has been proposed. Moreover, it is robust to uncertainties and generates low control spillover. This approach has been experimentally tested on a fuselage frame of a DC9 aircraft (Cavallo, De Maria, Leccia & Setola, 1998b).
2. Flexible structures In this paper we deal with the transversal vibration of a pinned}pinned beam in the con"guration shown in Fig. 1. The system dynamics are described by the partial di!erential equation (Meirovitch, 1967)
* *v(z, t) *v(z, t) P>PB EI(z) #m(z) " f (z, t) H *z *z *t H with boundary conditions v(0, t)"0,
(1)
v(¸, t)"0,
*v "0, "0, (2) *z X X* where v(z, t) is the de#ection at abscissa z, f (z, t), H j"1,2,r#r external inputs: r control input and B r disturbances, ¸ the beam length, m(z) the mass per unit B length, I(z) the moment of inertia and E the Young's modulus. It is well known (Meirovitch, 1967) that the solution of (1) can be approximated using the modal approach and limiting ourselves to the "rst n modes. *v *z
P>PB gK (t)#2m u g (t)#ug " N (t), G G G G G G GH H
i"1, 2,2,n, (3)
where g (t) are the modal coordinates, m the ith modal G G damping ratio and N (t) is the projection of f (z, t) on the GH H ith mode. In this paper, we assume pointly force disturbances located at abscissae z , j"1,2,r and piezoelectric plate H B devices as actuators at abscissae z , j"r #1,2,r #r. H B B
Fig. 1. Flexible Structure.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
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In particular, these devices exert on the structure on which they are bonded a couple of moments applied in the correspondence of the piezoelectric plates' extreme (say the abscissae z\ and z>) (Crawley & de Luis, 1987) H H For the disturbance then we have
where u G, i"1,2,r, are the fundamental frequencies of B the disturbance, b (t) and (t) are bounded functions HG HG with a rate of variation `smalla with respect to the corresponding u G. B
N (t)"< (z )d (t), GH G H H
3. Control strategy
j"1,2,r , B
(4)
where < (z) is the normalized shape of the ith mode and G d (t) is the time behavior of the disturbance (Meirovitch, H 1967). For the control action, the force exerted by the piezoelectric controls can be modelled as
N (t)" GH
d < (z>) d < (z\) G H ! G H u (t); H dz dz
Disturbances (10) emphasize the r modes of the beam whose natural frequencies are close to u G, hence the B control action should `eliminatea these peaks, i.e. to increase their damping factor. Moreover, the control must be smooth enough not to excite high-frequency unmodelled dynamics. To this end we de"ne the sliding manifold.
(5)
S"+(x, t) : s(x, t)"0, where s(x, t)"H(!x#e5Rx ), (11)
where u (t) is the time behavior of the jth control input. H These equations may be collected into the state-space model
H3RP"L, =3RL"L, x "x(0) and de"ne the control law as the solution of the di!erential equation (Cavallo et al., 1998a)
x "AM x #BM u#BM d, S B
(6)
eu "g(t, e)
(7)
where e is a positive small parameter,
j"r #1,2,r #r,
x (0)"x ,
y"CM x
where u is the r-vector of the control inputs and d the r -vector of disturbances. Moreover, the state vector B x 3RL is (g ,2, g , g ,2, g )2 and L L
0
AM "
L"L K
I L , Q
0 BM " L"P , S BI S
BM " B
0 B L"P , BI B
(8)
where K3RL"L and Q3RL"L are diagonal matrices and BI 3RL"P, BI 3RL"PB are the matrices obtained from (5) S B and (4) considering, respectively, the control inputs and the disturbances. The output vector y3RN is given by
L < (z )g (t) H H H y(t)" $ " 0 CI x N"L L < (z )g (t) H H N H
(9)
and we assume that the de#ection velocity sensors are located in the middle of the corresponding actuator piezoelectric plates. Finally, we assume r disturbances of the form B P d (t)" b (t) sin(u Gt# (t)), H HG B HG G
j"1,2,r , B
(10)
*s(x, t) *s(x, t) # x , *x *t
I !e\C\>R if e'e , g(t, e)" P e \ I ! e\C >R if 04e4e , P e
(12)
(13)
'3RP"P is a nonsingular matrix, (3RP"P and e is a positive scalar. On the basis of the following theorem, we may guarantee that the control strategy (12) stabilizes the system and ensures robustness against parameter uncertainties. Theorem 1. Consider the system of diwerential equations x "Ax#B u, x(0)"x , S *s(x, t) *s(x, t) eu "g(t, e) # x , u(0)"u . *t *x
(14) (15)
Let c , c , l, i be given positive numbers. Assume that (i) Re j (HB )'c ;
S (ii) Re j (B (HB )\HA!A)'c #l;
S S (iii) Re j (!=)'c ;
(iv) Re j (()'c ;
where Re j (X) denotes the minimum of the real part of
the eigenvalues of the matrix X. Then there exist e 'e '0 such that for any e3(e , e ], the solution (x(t, e), u(t, e)) to (14)}(15) is such that ""x(t, e)""4i#ae\AR
(16)
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A. Cavallo et al. / Automatica 35 (1999) 1689}1696
with t3[0,R), a is a positive constant depending on the data and
1 u(t, e)" (I!e\C\>R)s(x(t, e), t) e
R !( e\C\>Os(x(q, e), q) dq #u .
(17)
Proof. See Cavallo et al. (1998a). Actually, it is possible to prove, (Cavallo et al., 1993a), by Singular perturbation theory, that in the proposed strategy the control is the `fasta variable, while the whole state is `slowa. In the control law (17), the term e5Rx imposes that s(x(0),0)"0, hence u(0,e)"0 for any e'0, then no peaking phenomena is possible. Possible suggestions for the choice of = may be found in Cavallo, De Maria and Nistri (1993b). Next, as stated in the introduction, we must bound the derivative of the control. The function g(t,e) (13) imposes a bound on the rate of convergence of the actual control to the equivalent one. In particular, ( and ' impose prescribed values of the "rst and second control derivative at t"0, indeed the following approximations can be shown to hold:
where K 3RP"P, Q 3RP"P, K 3RL\P"L\P, Q 3RL\P"L\P, B 3RP"P, B 3RL\P"P are de"ned by (8) in accordance with the sorting of the state vector. Let us consider the matrix D3RL"L which transforms system (20) into the following modi"ed real diagonal form:
) !)
)
0 P"L\P D\AD")" ) 0 , P"L\P 0 0 ) L\P"P L\P"P
(21)
where the ) , i"1, 2, are diagonal matrices whose entries G are, respectively, the real and imaginary parts of the modes. On the basis of the structure of the matrix A, it is easy to verify that the matrix D\ has the following structure:
d d
d
0 P"L\P D\" d 0 , (22) P"L\P 0 0 * L\P"P L\P"P where the d are diagonal matrices whose entries are d , G GH i"1,2,4, j"1,2,r, and * indicates no speci"ed entry.
Theorem 2. Assume that B is nonsingular and select the matrix H in (11) as % 0 )D\, P"L\P where M3RP"P is a nonsingular matrix,
u (0, e)+'\H(=!A)x ,
(18)
H"M(%
1 uK (0, e)+ (H(=!A)x . e
(19)
% "diag+k ,2,k ,, i"1,2 G G GP with k 3+!1, 1,. GH Then it is possible to choose the parameter M, % and % in order to satisfy hypothesis (i) and (ii) of Theorem 1.
So we can select ' to bound the initial rate of the control and ( to shape the second derivative of the control. Next, we use the matrix H to de"ne the sliding surface. Exploiting the peculiarities of the #exible systems, this parameter may be selected in order to `eliminatea the r-modes whose frequencies are close to the disturbance frequencies without a!ecting the other dynamics of the structure. First of all we rewrite system (6) into an equivalent form obtained by means of a suitable sorting of the state variables, namely the "rst 2r variables are those associated with the modes which should be `eliminateda.
I P Q
0 P"P K
0 0 P"L\P P"L\P 0 0 P"L\P P"L\P x x " 0 0 0 I L\P"P L\P"P L\P"P L\P 0 0 K Q L\P"P L\P"P
#
0 P"P B
0 L\P"P B
u"Ax#B u, S
(20)
(23)
Proof. See the appendix. Remark. Choosing the matrix H as in Theorem 2 guarantees that the `reduced-order systema (KokotovitcH , Khalil & O'Reilly, 1986), obtained from (14)}(15) letting e"0, has the same poles as the openloop one, but the "rst r modes have been `eliminateda. Moreover, the matrix M may be used to de"ne the eigenvalues of the `boundary layer systema. The control law (12) with the sliding manifold de"ned by (11) and the matrix H selected as in Theorem 2 reduces the vibration of the structure `eliminatinga the modes which emphasize the disturbance. In conclusion, the proposed control law designs a sliding surface which guarantees that only some modes of the structure are modi"ed, leaving una!ected the remaining ones. Moreover, during the reaching phase, the control is guaranteed to be smooth, thus avoiding excitation of high-order neglected dynamics.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
4. Observer The control law (17) is a state feedback. In the present case the state elements are the values of the modal coordinates g (t) and they are not directly measurable. In G Cavallo, De Maria and Nistri (1994), the following observer is proposed: x( "(A!G(q)C)x( #B u S #G(q)y!G(q)e*Ry , x( (0)"0, (24) where x( is the estimated state, y"Cx and y "y(0) is the initial plant output, and ¸3RN"N is a Hurwitz matrix, p is the number of available measurements, q a positive scalar, and G(q) is chosen as G(q)"P(q)C2R\,
(25)
P(q) is the solution of the Algebraic Riccati Equation AP(q)#P(q)A2#Q(q)!P(q)C2R\CP(q)"0,
(26)
where R is a positive-de"nite matrix and Q(q)"Q #qB
(28)
\
5. Simulation results In this section we show the properties of the proposed controllers on a uniform cross section beam with the following numerical values: ¸"3 (m); m"19.61 (kg/m); I"6.28;10\ (m), m "0.01; E"2.07;10 (N/m). G The model used for the control design has been obtained as shown in Section 2 considering the "rst 4 modes. As shown in Fig. 1, there are three disturbances acting at abscissae z"+¸/6, ¸/2, 6¸/7, and two piezoelectric plate actuators of length * "30 (mm) located at abscisX sae z"+¸/4, 3¸/5,; moreover, for the design of the observer we assume the presence of two vertical de#ection velocity sensors located in the correspondence of the middle of the piezoelectric plates. The disturbances, expressed in (N), acting on the beam are d (t)"30(1#0.1sin 0.2t)sin 85t#sin(320t#sin 0.1t), (32)
#sin(320t#sin 0.1t),
!( e\C >OHx( (q, q) dq . It is possible to show (Cavallo et al., 1994) that
(29)
lim (x( !x)"!(B !C)>B !e*Ry , S S O
(30)
(33)
p d (t)"30(1#0.1sin 0.2t)sin 85t# 4
p #sin 320t# #sin 0.1t . 3
where !"< R\ , ( ) )> is the pseudo-inverse operator, and lim lim (x , u )"(x, u). C O
Note that the last term in (24) plays the same role as the e5Rx term in (11).
p d (t)"30(1#0.1sin 0.2t)sin 85t# 2
1 u (t, e, q)"! (I!e\C\>R)Hx( (t, q) e R
1693
(34)
The fundamental frequencies is u "85 (rad/s), B u "320 (rad/s), they are close to the system's "rst two B resonances, respectively (see Table 1). Then the control action will be focused on `eliminatinga these dynamics.
(31)
The subscript `obsa refers to the presence of the observer in the loop, while the variables on the right-hand side of (31) are de"ned in (16)}(17). In order to apply this result two hypotheses should be satis"ed: the system must be observable and of minimum phase. We assume that the locations of sensors guarantee the observability, moreover it is possible to show that, in the present sensors}actuators con"guration, the system is of minimum-phase if in the model the modes i"1,2,n are taken into account with n42¸/* where * is the X X length of the piezoelectric plate.
Table 1 Eigenvalues of the 4-modes beam model (nominal), of the 8-modes model and that of the 8-modes model in the presence of perturbations nominal (4-modes)
8-modes
perturbed
!0.9$89.3j !3.6$357.1j !8$803.6j !14.3$1428j
!0.9$89.3j !3.6$357.1j !8$803.6j !14.3$1428j !22.3$2232j !32.1$3214j !43.8$4375j !57.1$5714j
!0.6$82.3j !2.3$329.3 !5.2$740.9j !9.2$1317j !14.4$2963j !20.7$2963j !28.2$4033j !36.9$5268j
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A. Cavallo et al. / Automatica 35 (1999) 1689}1696
Fig. 2. De#ection rate at abscissa z"¸/2. The control action starts at time instant t"5.
Fig. 4. The de#ection rate at abscissa z"¸/2 of 8-modes perturbed plant.
when the system has substantially reached its steady state. This has been done in order to emphasize the in#uence of the controller on the high frequencies, i.e. control spillover. In Fig. 3 open- and closed-loop maximum singular values are compared. It is evident that the controller increases the damping factor of the "rst two modes leaving una!ected the other dynamics. If the state is not available an observer is needed. In particular, the observer is designed as in Section 4 with
!0.89 !89.3
¸"
<"I , Fig. 3. Maximum singular values of the 8-modes model open loop (solid line) and closed loop (dashed line).
In particular, the following parameters have been used: e"7;10\; '"10\ ) I ; ("6;10\ I , !1 0 % " , % "I , 0 d1
89.3
!0.89
, q"1000, Q "I ,
R"0.1I .
(35)
In order to show the robustness property of the control scheme with respect to uncertainties and spillover we have applied the previously described observercontroller on the 8-modes model and assuming, also, the following multiplicative errors on corresponding parameters *I"0.85, *m"0.7.
(36)
The results are depicted in Fig. 4.
!58.7 !136.9
M"
!58.7
136.9
.
The matrix = has been selected as the solution of an LQ problem (Cavallo et al., 1993b) for plant (20) with weighting matrices Q "I , R "10\I . */ */ The controller designed on the 4-modes model has been tested on a more accurate 8-modes model (see Fig. 2). Note that the control action starts at time t"5 s
6. Conclusion Vibration control of #exible structures has gained more and more importance in many engineering application "elds. In the design of these controllers the poor knowledge of the system and the need to operate with a "nite-dimensional model should be taken into account.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
In particular, the presence of neglected dynamics may drive the controlled plant to instability (spillover phenomenon). In this paper a vibration controller is designed using singular perturbation theory. In particular, the controller imposes that the system lays on a time-varying manifold in spite of possible uncertainties and disturbances. A suitable de"nition of the manifold allows one to dramatically increase the damping factor of prescribed modes leaving una!ected the other dynamics. Moreover, the control signal is smooth in order to reduce the control spillover.
1695
It is immediate to recognize that the last 2(n!r) eigenvalues are una!ected by the control action. Then we may focus our attention on the eigenvalues associated with the "rst r-modes. In particular, we known that in the sliding mode the system must lie on the sliding surface, i.e. Hx"0N(% % 0)D\ x"0N% x #% x "0, K K
(A.4)
where x "[x x x ] is the state vector in the K K K K modi"ed real diagonal form partitioned in accordance with (20). From (A.4) we have the following constraint:
Appendix Proof of Theorem 2. Hypotheses (i) and (ii) of Theorem 1 do not depend on function g and on the exponential function e5Rx ; therefore in the proof we may ignore these components of the control law. From (23) one has
B HB "M(% % 0)D\ S 0
(A.1)
Hypothesis (i) of Theorem 1 requires that the matrix HB S is nonsingular. This implies that % d #% d is nonsin gular. From (22) it is evident that the rows of d and d cannot be simultaneously zero. If one neglects the case "d """d ", a suitable selection G G of the matrix M ensures hypothesis (i), independently of the matrices % . Indeed, hypothesis (ii) imposes that the G following matrix is Hurwitz (A.2)
It is easy to see, using (21) and expliciting the matrices structure, that the eigenvalues of (A.2) are those of the matrix
!K "
!K
0 , ) *
(A.3)
where
) !d %K (% ) !% ) ) ) !d %K (% ) #% ) ) !) !d %K (% ) !% ) ) ) !d %K (% ) #% ) )
and %K "(% d #% d )\.
(A.6)
d k u G G G , j "!a !k k u #2 G G G G G k d #k d G G G G
i"1,2, r. (A.7)
It is easy to check that, whatever the values of d , d , a , u are, by a suitable selection of k and G G G G G k the eigenvalues j , i"1,2,r may be forced to belong G G to the left half-plane. Finally, we consider the case "d """d ". In this situG G ation the nonsingularity of matrix (A.7) depends on the k values. In particular, one must choose GH
k k " G G !k
if d "d , G G if d "!d G G G
(A.8)
in both cases (A.7) degenerates in j "!a , hence hyG G pothesis (ii) of Theorem 1 is satis"ed. 䊐
References
!K "
where we have used the property of the % matrices: G %\"% . G G After some algebra, one may see that the dynamics of x is de"ned by the matrix K
while the behavior of x is de"ned by the algebraic K relation (A.5). Eq. (A.6) is a diagonal matrix, since the addends are diagonal and its eigenvalues are
B
A!B (HB )\HA. S S
(A.5)
) !% % ) !2d (% d #% d )\% )
0
"M(% d #% d )B .
x "!% % x , K K
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Cavallo, A., De Maria, G., & Nistri, P. (1993b). Linear tracking problems by a sliding manifold approach. Proceedings of the 2nd European Control Conference, Groningen, The Netherland (pp. 854}858). Cavallo, A., De Maria, G., & Nistri, P. (1994). Sliding mode techniques with robust observer. Proceedings of the Robust Control via Variable Structure & Lyapunov Techniques, Benevento, Italy (pp. 261}265). Cavallo, A., De Maria, G., & Nistri, P. (1998a). Robust Control Design with Integral Action and Limited Rate Control. IEEE Transactions on Automatic Control, to appear. Cavallo, A., De Maria, G., Leccia, E., & Setola, R. (1998b). Robust vibration control of a DC9 aircraft frame. Proceedings of 37th Conference on Decision and Control, Tampa, FL, USA. Crawley, E. F., & de Luis, J. (1987). Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal, 25(10), 1373}1385. Fuller, C., & von Flotow, A. H. (1995). Active control of sound and vibration. IEEE Control Systems, 15, 9}19. Goh, C., & Caughey, T. (1985). On the stability problem caused by "nite actuator dynamics in the collocated control of large space structures. International Journal of Control, 41, 787}802. Junkins, J. L., Kim, Y. (1993). Introduction to dynamics and control of yexible structures. AIAA education series. Washinton, DC. KokotovitcH , P., Khalil, H. K., & O'Reilly, J. (1986). Singular Perturbation Methods in Control: Analysis and Design. London: Academic Press. Meirovitch, L. (1967). Analytical methods in vibrations. New York: Macmillan Publishing. Meirovitch, L. (1990). Dynamic and control of structures. New York: Wiley-Interscience Publication. Preumont, A., Dufor, J., & MaleH kian, C. (1992). Active damping by a local force feedback with piezoelectric actuators. Journal of Guidance, Control and Dynamics, 15, 390}395. Utkin, V. (1990). Sliding modes in control and optimization. Berlin: Springer. Utkin, V. (1996). Integral sliding mode in systems operating under uncertainty conditions. Proceedings of 35th Conference on Decision and Control, Kobe, Japan (pp. 4591}4596).
Roberto Setola was born in Naples, Italy, in 1969. He received his M.S. and Ph.D. degree in Electronic Engineering from the University of Naples Federico II, in 1992 and 1996, respectively. Since 1997 he has held a post-doc position at the University of Sannio. From 1995 to 1997 he was a contract professor at the University of Naples Federico II and from 1998 to date he has been a contract professor of Process Control at the Second University of Naples. His research interest is in the areas of active vibration control design and of non-linear observation techniques. Giuseppe De Maria was born in Napoli, Italy, in 1948, he received the Laurea in Electronic Engineering in 1973. He joined the Department of Informatica e Sistemistica of University of Napoli `Federico IIa as an associate professor of Automatic Control. In 1992 he joined the Department of Ingegneria dell'Informazione of the second University of Napoli, as a full professor of Automatic Control and System Theory. Now he is the head of the department. His research interests include robust control, attitude control of aerospace systems and active vibration control of #exible systems. He is member of IEEE Control system society. Alberto Cavallo was born in Napoli, Italy, in 1964. He received his Laurea degree and Ph.D. in Electronic Engineering from the University of Napoli `Federico IIa, in 1989 and 1992, respectively. He joined the `Dipartimento di Ingegneria dell'Informazionea of the Second University of Napoli as a Researcher Associate. Currently he is an Associate Professor of Automatic Control at the University of Sannio. His research interests include robust control, aerospace control systems and active vibration control techniques.
Brief Paper
A sliding manifold approach for vibration reduction of #exible systems夽 A. Cavallo*, G. De Maria, R. Setola Dipartimento di Ingegneria dell'Informazione, Seconda Universita% degli Studi di Napoli, via Roma 29 81031, Aversa, Italy Received 19 December 1996; revised 16 March 1999
Abstract In this paper an active vibration controller for #exible systems is developed using a sliding manifold approach. The controller is designed to `eliminatea the system modes whose frequencies are close to the disturbance frequencies. The sliding surface consists of a time-invarying part which `eliminatesa the selected modes without a!ecting the remaining dynamics, and a time-varying part which bounds the control amplitude and its rate in the transient. Moreover, the features of the controller allows one to reduce the excitation of the high-frequency modes to reduce the spillover phenomenon. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Flexible structures; Robust control; Singular perturbations; Spillover; Sliding manifold
1. Introduction In the past decades there has been an increasing interest around active vibration control of #exible systems. Active vibration methods have been developed to overcome some of the limits of the classical passive methods (Fuller & von Flotow, 1995), in particular, the need to reduce the vibration at `lowa frequencies, i.e., in the correspondence of the "rst system natural frequencies where, generally, the main part of the elastic energy is stored. However, to develop an active vibration controller it is useful to point out some peculiarities of #exible systems (Balas, 1982) (i) these structures are in"nite-dimensional systems, e.g., they are described by means of linear partial di!erential operators; (ii) they are passive systems;
夽 This work was supported by POP97 grant, MISURA 5.4.2., Regione Compania and PRW97-MURST grant. This paper was not presented at any IFAC meeting. This paper was recommened for publication by Associate Editor C.V. Hollot under the direction of Editor R. Tempo. * Corresponding author. Tel: 0039-81-5010-252; fax: 0039-81-5037042. E-mail address: [email protected] (A. Cavallo)
(iii) they present an ideally in"nite number of resonance peaks with very small damping factors; (iv) some parameters, like damping factors, moment of inertia, etc., are uncertain; (v) their resonance peaks emphasize the disturbances whose frequencies are close to the system natural frequencies; (vi) to operate with "nite-dimensional systems the partial di!erential model is replaced by a "nite set of ordinary di!erential equations, neglecting, generally, the high-frequency dynamics. A direct consequence of item (vi) is that, since sensors and actuators act both on modelled dynamics and neglected ones, the latter ones may be driven to instability (spillover phenomenon, Balas, 1978). More speci"cally, the action exerted by actuators on the unmodelled dynamics is called control spillover, whereas the component of the unmodelled dynamics measured by the sensor is called observation spillover (Meirovitch, 1990). An approach that is una!ected by the spillover phenomenon is the co-located feedback (Preumont, Dufor & MaleH kian, 1992). This approach yields a moderate increase of the damping factor of some modes. However, considering that practical devices have "nite bandwidth, and due to the impossibility of a perfect co-location of the devices, their passivity property may no longer be guaranteed (Goh & Caughey, 1985).
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 8 - 0
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Before describing our approach, we brie#y discuss the problem speci"cations: E in many situations (e.g. pressure wave at fuselage in turbofan aircraft) the disturbances are narrow band signals whose spectrum can be easily determined. In this case, the worst situation occurs when the disturbance frequencies are close to some natural frequencies of the structure; E the controller should be robust with respect to parametric uncertainty and disturbances; E control spillover is avoided if the unmodelled dynamics are not a!ected by the control action; in particular, the control signals should be smooth and with a limited rate of variation (Junkins & Kim, 1993). In this paper we present a control strategy addressing each of the above points. In particular, E the closed-loop system does not possess resonances that can be excited by narrow-band disturbances; E the controller is based on singular perturbation concepts and thus exhibits high robustness properties; E to avoid excitation of high-frequency unmodelled dynamics, special attention is devoted to the initial control transients (the `reaching phasea of the control), and the limitation of controller rate variation. It is known (Utkin, 1990) that high-gain feedback guarantees robust performances and insensitivity with respect to disturbances and parametric variations. However, this control strategy requires some state dynamics to become faster as the gain increases. During the initial transient (reaching phase) these `fasta states quickly approach the fast manifold, resulting in control signal peaks which can excite the high-frequency dynamics thus generating considerable control spillover. Using a singular perturbation approach in Cavallo, De Maria and Nistri (1993a) it is shown that the robustness property of high gain control may be obtained, without the above drawbacks, by de"ning the control signal as the solution of a di!erential equation depending on a small parameter e. Speci"cally, the whole state is `slowa, while the control signal is a `fasta and continuously di!erentiable function. An alternative line of attack to the problem of eliminating the reaching phase can be found in Utkin (1996), where a strategy based on a pre-computation of a control law stabilizing the nominal system is presented. However, even in this case, the control exhibits a fast transient (control reaching phase) which generates control spillover. In Junkins and Kim (1993) a multiplier function is proposed to smooth the control resulting from a bang}bang strategy, and in Cavallo, De Maria and Nistri (1998a) a multiplicative action on the derivative of the control, again in the singular perturbation framework, has been proposed.
In this paper, by using the above strategy, a controller which is able to increase the damping factor of r selected modes of the structure (where r is the number of available inputs) leaving the remaining dynamics una!ected has been proposed. Moreover, it is robust to uncertainties and generates low control spillover. This approach has been experimentally tested on a fuselage frame of a DC9 aircraft (Cavallo, De Maria, Leccia & Setola, 1998b).
2. Flexible structures In this paper we deal with the transversal vibration of a pinned}pinned beam in the con"guration shown in Fig. 1. The system dynamics are described by the partial di!erential equation (Meirovitch, 1967)
* *v(z, t) *v(z, t) P>PB EI(z) #m(z) " f (z, t) H *z *z *t H with boundary conditions v(0, t)"0,
(1)
v(¸, t)"0,
*v "0, "0, (2) *z X X* where v(z, t) is the de#ection at abscissa z, f (z, t), H j"1,2,r#r external inputs: r control input and B r disturbances, ¸ the beam length, m(z) the mass per unit B length, I(z) the moment of inertia and E the Young's modulus. It is well known (Meirovitch, 1967) that the solution of (1) can be approximated using the modal approach and limiting ourselves to the "rst n modes. *v *z
P>PB gK (t)#2m u g (t)#ug " N (t), G G G G G G GH H
i"1, 2,2,n, (3)
where g (t) are the modal coordinates, m the ith modal G G damping ratio and N (t) is the projection of f (z, t) on the GH H ith mode. In this paper, we assume pointly force disturbances located at abscissae z , j"1,2,r and piezoelectric plate H B devices as actuators at abscissae z , j"r #1,2,r #r. H B B
Fig. 1. Flexible Structure.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
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In particular, these devices exert on the structure on which they are bonded a couple of moments applied in the correspondence of the piezoelectric plates' extreme (say the abscissae z\ and z>) (Crawley & de Luis, 1987) H H For the disturbance then we have
where u G, i"1,2,r, are the fundamental frequencies of B the disturbance, b (t) and (t) are bounded functions HG HG with a rate of variation `smalla with respect to the corresponding u G. B
N (t)"< (z )d (t), GH G H H
3. Control strategy
j"1,2,r , B
(4)
where < (z) is the normalized shape of the ith mode and G d (t) is the time behavior of the disturbance (Meirovitch, H 1967). For the control action, the force exerted by the piezoelectric controls can be modelled as
N (t)" GH
d < (z>) d < (z\) G H ! G H u (t); H dz dz
Disturbances (10) emphasize the r modes of the beam whose natural frequencies are close to u G, hence the B control action should `eliminatea these peaks, i.e. to increase their damping factor. Moreover, the control must be smooth enough not to excite high-frequency unmodelled dynamics. To this end we de"ne the sliding manifold.
(5)
S"+(x, t) : s(x, t)"0, where s(x, t)"H(!x#e5Rx ), (11)
where u (t) is the time behavior of the jth control input. H These equations may be collected into the state-space model
H3RP"L, =3RL"L, x "x(0) and de"ne the control law as the solution of the di!erential equation (Cavallo et al., 1998a)
x "AM x #BM u#BM d, S B
(6)
eu "g(t, e)
(7)
where e is a positive small parameter,
j"r #1,2,r #r,
x (0)"x ,
y"CM x
where u is the r-vector of the control inputs and d the r -vector of disturbances. Moreover, the state vector B x 3RL is (g ,2, g , g ,2, g )2 and L L
0
AM "
L"L K
I L , Q
0 BM " L"P , S BI S
BM " B
0 B L"P , BI B
(8)
where K3RL"L and Q3RL"L are diagonal matrices and BI 3RL"P, BI 3RL"PB are the matrices obtained from (5) S B and (4) considering, respectively, the control inputs and the disturbances. The output vector y3RN is given by
L < (z )g (t) H H H y(t)" $ " 0 CI x N"L L < (z )g (t) H H N H
(9)
and we assume that the de#ection velocity sensors are located in the middle of the corresponding actuator piezoelectric plates. Finally, we assume r disturbances of the form B P d (t)" b (t) sin(u Gt# (t)), H HG B HG G
j"1,2,r , B
(10)
*s(x, t) *s(x, t) # x , *x *t
I !e\C\>R if e'e , g(t, e)" P e \ I ! e\C >R if 04e4e , P e
(12)
(13)
'3RP"P is a nonsingular matrix, (3RP"P and e is a positive scalar. On the basis of the following theorem, we may guarantee that the control strategy (12) stabilizes the system and ensures robustness against parameter uncertainties. Theorem 1. Consider the system of diwerential equations x "Ax#B u, x(0)"x , S *s(x, t) *s(x, t) eu "g(t, e) # x , u(0)"u . *t *x
(14) (15)
Let c , c , l, i be given positive numbers. Assume that (i) Re j (HB )'c ;
S (ii) Re j (B (HB )\HA!A)'c #l;
S S (iii) Re j (!=)'c ;
(iv) Re j (()'c ;
where Re j (X) denotes the minimum of the real part of
the eigenvalues of the matrix X. Then there exist e 'e '0 such that for any e3(e , e ], the solution (x(t, e), u(t, e)) to (14)}(15) is such that ""x(t, e)""4i#ae\AR
(16)
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A. Cavallo et al. / Automatica 35 (1999) 1689}1696
with t3[0,R), a is a positive constant depending on the data and
1 u(t, e)" (I!e\C\>R)s(x(t, e), t) e
R !( e\C\>Os(x(q, e), q) dq #u .
(17)
Proof. See Cavallo et al. (1998a). Actually, it is possible to prove, (Cavallo et al., 1993a), by Singular perturbation theory, that in the proposed strategy the control is the `fasta variable, while the whole state is `slowa. In the control law (17), the term e5Rx imposes that s(x(0),0)"0, hence u(0,e)"0 for any e'0, then no peaking phenomena is possible. Possible suggestions for the choice of = may be found in Cavallo, De Maria and Nistri (1993b). Next, as stated in the introduction, we must bound the derivative of the control. The function g(t,e) (13) imposes a bound on the rate of convergence of the actual control to the equivalent one. In particular, ( and ' impose prescribed values of the "rst and second control derivative at t"0, indeed the following approximations can be shown to hold:
where K 3RP"P, Q 3RP"P, K 3RL\P"L\P, Q 3RL\P"L\P, B 3RP"P, B 3RL\P"P are de"ned by (8) in accordance with the sorting of the state vector. Let us consider the matrix D3RL"L which transforms system (20) into the following modi"ed real diagonal form:
) !)
)
0 P"L\P D\AD")" ) 0 , P"L\P 0 0 ) L\P"P L\P"P
(21)
where the ) , i"1, 2, are diagonal matrices whose entries G are, respectively, the real and imaginary parts of the modes. On the basis of the structure of the matrix A, it is easy to verify that the matrix D\ has the following structure:
d d
d
0 P"L\P D\" d 0 , (22) P"L\P 0 0 * L\P"P L\P"P where the d are diagonal matrices whose entries are d , G GH i"1,2,4, j"1,2,r, and * indicates no speci"ed entry.
Theorem 2. Assume that B is nonsingular and select the matrix H in (11) as % 0 )D\, P"L\P where M3RP"P is a nonsingular matrix,
u (0, e)+'\H(=!A)x ,
(18)
H"M(%
1 uK (0, e)+ (H(=!A)x . e
(19)
% "diag+k ,2,k ,, i"1,2 G G GP with k 3+!1, 1,. GH Then it is possible to choose the parameter M, % and % in order to satisfy hypothesis (i) and (ii) of Theorem 1.
So we can select ' to bound the initial rate of the control and ( to shape the second derivative of the control. Next, we use the matrix H to de"ne the sliding surface. Exploiting the peculiarities of the #exible systems, this parameter may be selected in order to `eliminatea the r-modes whose frequencies are close to the disturbance frequencies without a!ecting the other dynamics of the structure. First of all we rewrite system (6) into an equivalent form obtained by means of a suitable sorting of the state variables, namely the "rst 2r variables are those associated with the modes which should be `eliminateda.
I P Q
0 P"P K
0 0 P"L\P P"L\P 0 0 P"L\P P"L\P x x " 0 0 0 I L\P"P L\P"P L\P"P L\P 0 0 K Q L\P"P L\P"P
#
0 P"P B
0 L\P"P B
u"Ax#B u, S
(20)
(23)
Proof. See the appendix. Remark. Choosing the matrix H as in Theorem 2 guarantees that the `reduced-order systema (KokotovitcH , Khalil & O'Reilly, 1986), obtained from (14)}(15) letting e"0, has the same poles as the openloop one, but the "rst r modes have been `eliminateda. Moreover, the matrix M may be used to de"ne the eigenvalues of the `boundary layer systema. The control law (12) with the sliding manifold de"ned by (11) and the matrix H selected as in Theorem 2 reduces the vibration of the structure `eliminatinga the modes which emphasize the disturbance. In conclusion, the proposed control law designs a sliding surface which guarantees that only some modes of the structure are modi"ed, leaving una!ected the remaining ones. Moreover, during the reaching phase, the control is guaranteed to be smooth, thus avoiding excitation of high-order neglected dynamics.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
4. Observer The control law (17) is a state feedback. In the present case the state elements are the values of the modal coordinates g (t) and they are not directly measurable. In G Cavallo, De Maria and Nistri (1994), the following observer is proposed: x( "(A!G(q)C)x( #B u S #G(q)y!G(q)e*Ry , x( (0)"0, (24) where x( is the estimated state, y"Cx and y "y(0) is the initial plant output, and ¸3RN"N is a Hurwitz matrix, p is the number of available measurements, q a positive scalar, and G(q) is chosen as G(q)"P(q)C2R\,
(25)
P(q) is the solution of the Algebraic Riccati Equation AP(q)#P(q)A2#Q(q)!P(q)C2R\CP(q)"0,
(26)
where R is a positive-de"nite matrix and Q(q)"Q #qB
(28)
\
5. Simulation results In this section we show the properties of the proposed controllers on a uniform cross section beam with the following numerical values: ¸"3 (m); m"19.61 (kg/m); I"6.28;10\ (m), m "0.01; E"2.07;10 (N/m). G The model used for the control design has been obtained as shown in Section 2 considering the "rst 4 modes. As shown in Fig. 1, there are three disturbances acting at abscissae z"+¸/6, ¸/2, 6¸/7, and two piezoelectric plate actuators of length * "30 (mm) located at abscisX sae z"+¸/4, 3¸/5,; moreover, for the design of the observer we assume the presence of two vertical de#ection velocity sensors located in the correspondence of the middle of the piezoelectric plates. The disturbances, expressed in (N), acting on the beam are d (t)"30(1#0.1sin 0.2t)sin 85t#sin(320t#sin 0.1t), (32)
#sin(320t#sin 0.1t),
!( e\C >OHx( (q, q) dq . It is possible to show (Cavallo et al., 1994) that
(29)
lim (x( !x)"!(B !C)>B !e*Ry , S S O
(30)
(33)
p d (t)"30(1#0.1sin 0.2t)sin 85t# 4
p #sin 320t# #sin 0.1t . 3
where !"< R\ , ( ) )> is the pseudo-inverse operator, and lim lim (x , u )"(x, u). C O
Note that the last term in (24) plays the same role as the e5Rx term in (11).
p d (t)"30(1#0.1sin 0.2t)sin 85t# 2
1 u (t, e, q)"! (I!e\C\>R)Hx( (t, q) e R
1693
(34)
The fundamental frequencies is u "85 (rad/s), B u "320 (rad/s), they are close to the system's "rst two B resonances, respectively (see Table 1). Then the control action will be focused on `eliminatinga these dynamics.
(31)
The subscript `obsa refers to the presence of the observer in the loop, while the variables on the right-hand side of (31) are de"ned in (16)}(17). In order to apply this result two hypotheses should be satis"ed: the system must be observable and of minimum phase. We assume that the locations of sensors guarantee the observability, moreover it is possible to show that, in the present sensors}actuators con"guration, the system is of minimum-phase if in the model the modes i"1,2,n are taken into account with n42¸/* where * is the X X length of the piezoelectric plate.
Table 1 Eigenvalues of the 4-modes beam model (nominal), of the 8-modes model and that of the 8-modes model in the presence of perturbations nominal (4-modes)
8-modes
perturbed
!0.9$89.3j !3.6$357.1j !8$803.6j !14.3$1428j
!0.9$89.3j !3.6$357.1j !8$803.6j !14.3$1428j !22.3$2232j !32.1$3214j !43.8$4375j !57.1$5714j
!0.6$82.3j !2.3$329.3 !5.2$740.9j !9.2$1317j !14.4$2963j !20.7$2963j !28.2$4033j !36.9$5268j
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A. Cavallo et al. / Automatica 35 (1999) 1689}1696
Fig. 2. De#ection rate at abscissa z"¸/2. The control action starts at time instant t"5.
Fig. 4. The de#ection rate at abscissa z"¸/2 of 8-modes perturbed plant.
when the system has substantially reached its steady state. This has been done in order to emphasize the in#uence of the controller on the high frequencies, i.e. control spillover. In Fig. 3 open- and closed-loop maximum singular values are compared. It is evident that the controller increases the damping factor of the "rst two modes leaving una!ected the other dynamics. If the state is not available an observer is needed. In particular, the observer is designed as in Section 4 with
!0.89 !89.3
¸"
<"I , Fig. 3. Maximum singular values of the 8-modes model open loop (solid line) and closed loop (dashed line).
In particular, the following parameters have been used: e"7;10\; '"10\ ) I ; ("6;10\ I , !1 0 % " , % "I , 0 d1
89.3
!0.89
, q"1000, Q "I ,
R"0.1I .
(35)
In order to show the robustness property of the control scheme with respect to uncertainties and spillover we have applied the previously described observercontroller on the 8-modes model and assuming, also, the following multiplicative errors on corresponding parameters *I"0.85, *m"0.7.
(36)
The results are depicted in Fig. 4.
!58.7 !136.9
M"
!58.7
136.9
.
The matrix = has been selected as the solution of an LQ problem (Cavallo et al., 1993b) for plant (20) with weighting matrices Q "I , R "10\I . */ */ The controller designed on the 4-modes model has been tested on a more accurate 8-modes model (see Fig. 2). Note that the control action starts at time t"5 s
6. Conclusion Vibration control of #exible structures has gained more and more importance in many engineering application "elds. In the design of these controllers the poor knowledge of the system and the need to operate with a "nite-dimensional model should be taken into account.
A. Cavallo et al. / Automatica 35 (1999) 1689}1696
In particular, the presence of neglected dynamics may drive the controlled plant to instability (spillover phenomenon). In this paper a vibration controller is designed using singular perturbation theory. In particular, the controller imposes that the system lays on a time-varying manifold in spite of possible uncertainties and disturbances. A suitable de"nition of the manifold allows one to dramatically increase the damping factor of prescribed modes leaving una!ected the other dynamics. Moreover, the control signal is smooth in order to reduce the control spillover.
1695
It is immediate to recognize that the last 2(n!r) eigenvalues are una!ected by the control action. Then we may focus our attention on the eigenvalues associated with the "rst r-modes. In particular, we known that in the sliding mode the system must lie on the sliding surface, i.e. Hx"0N(% % 0)D\ x"0N% x #% x "0, K K
(A.4)
where x "[x x x ] is the state vector in the K K K K modi"ed real diagonal form partitioned in accordance with (20). From (A.4) we have the following constraint:
Appendix Proof of Theorem 2. Hypotheses (i) and (ii) of Theorem 1 do not depend on function g and on the exponential function e5Rx ; therefore in the proof we may ignore these components of the control law. From (23) one has
B HB "M(% % 0)D\ S 0
(A.1)
Hypothesis (i) of Theorem 1 requires that the matrix HB S is nonsingular. This implies that % d #% d is nonsin gular. From (22) it is evident that the rows of d and d cannot be simultaneously zero. If one neglects the case "d """d ", a suitable selection G G of the matrix M ensures hypothesis (i), independently of the matrices % . Indeed, hypothesis (ii) imposes that the G following matrix is Hurwitz (A.2)
It is easy to see, using (21) and expliciting the matrices structure, that the eigenvalues of (A.2) are those of the matrix
!K "
!K
0 , ) *
(A.3)
where
) !d %K (% ) !% ) ) ) !d %K (% ) #% ) ) !) !d %K (% ) !% ) ) ) !d %K (% ) #% ) )
and %K "(% d #% d )\.
(A.6)
d k u G G G , j "!a !k k u #2 G G G G G k d #k d G G G G
i"1,2, r. (A.7)
It is easy to check that, whatever the values of d , d , a , u are, by a suitable selection of k and G G G G G k the eigenvalues j , i"1,2,r may be forced to belong G G to the left half-plane. Finally, we consider the case "d """d ". In this situG G ation the nonsingularity of matrix (A.7) depends on the k values. In particular, one must choose GH
k k " G G !k
if d "d , G G if d "!d G G G
(A.8)
in both cases (A.7) degenerates in j "!a , hence hyG G pothesis (ii) of Theorem 1 is satis"ed. 䊐
References
!K "
where we have used the property of the % matrices: G %\"% . G G After some algebra, one may see that the dynamics of x is de"ned by the matrix K
while the behavior of x is de"ned by the algebraic K relation (A.5). Eq. (A.6) is a diagonal matrix, since the addends are diagonal and its eigenvalues are
B
A!B (HB )\HA. S S
(A.5)
) !% % ) !2d (% d #% d )\% )
0
"M(% d #% d )B .
x "!% % x , K K
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Roberto Setola was born in Naples, Italy, in 1969. He received his M.S. and Ph.D. degree in Electronic Engineering from the University of Naples Federico II, in 1992 and 1996, respectively. Since 1997 he has held a post-doc position at the University of Sannio. From 1995 to 1997 he was a contract professor at the University of Naples Federico II and from 1998 to date he has been a contract professor of Process Control at the Second University of Naples. His research interest is in the areas of active vibration control design and of non-linear observation techniques. Giuseppe De Maria was born in Napoli, Italy, in 1948, he received the Laurea in Electronic Engineering in 1973. He joined the Department of Informatica e Sistemistica of University of Napoli `Federico IIa as an associate professor of Automatic Control. In 1992 he joined the Department of Ingegneria dell'Informazione of the second University of Napoli, as a full professor of Automatic Control and System Theory. Now he is the head of the department. His research interests include robust control, attitude control of aerospace systems and active vibration control of #exible systems. He is member of IEEE Control system society. Alberto Cavallo was born in Napoli, Italy, in 1964. He received his Laurea degree and Ph.D. in Electronic Engineering from the University of Napoli `Federico IIa, in 1989 and 1992, respectively. He joined the `Dipartimento di Ingegneria dell'Informazionea of the Second University of Napoli as a Researcher Associate. Currently he is an Associate Professor of Automatic Control at the University of Sannio. His research interests include robust control, aerospace control systems and active vibration control techniques.