Flexible Structure Control by Eigenstructure Assignment

Copyright © IFAC Control of Industrial Systems, Belfort. France, 1997

FLEXIBLE STRUCTURE CONTROL BY EIGENSTRUCTURE ASSIGNMENT

J.F. Magni* , Y. Le Gorrec t * , D. Alazard* and C. Chiappat * • C.E.R. T - D.E.R.A, B.? 4025, F31055 TOULOUSE CEDEX, FRANCE

t ENSAE, Av. E. Be/in, F31055 TOULOUSE CEDEX, FRANCE

Abstract. This paper presents a generalization to multi input of a technique used for designing control laws of single input flexible structures. The objective consists of assigning the eigenvalue motions due to (scalar) loop gain variations. In the single input case, phase is changed at some frequencies. In the proposed generalization, phase lead and lag is replaced by eigenvector "assignment" which consists of designing a dynamic feedforward having ad hoc eigenstructure. It remains to tune the scalar loop gain. The design problem considered is shown to reduce to solving a set of linear equalities and/or inequalities. A quadratic criterion which enables to minimize energy involved in assignment is also proposed. Key Words. Eigenstructure assignment, phase control, flexible structure, linear multivariable systems.

into account various physical considerations which are very vague but also very rich for the robustness of the final control law w.r.t. flexible modes. For instance, this approach inhibits poles zeros cancelations and respects the low frequency gain level required by the rigid control law. In order to generalize this approach to the MIMO case, we could think that H 2 and H oo synthesis or more generally the use of the standard form and optimization algorithms could provide powerful tools. Lots of papers deal with this problem (Wie et al., 1992), (Stoughton, 1992), (Tahk and Speyer, 1987), (Coville et al., 1991), (Goh and Griibel, 1992). The difficulty is that these physical considerations cannot be simply and clearly specified in the standard set-up. It turns out that these synthesis methods lead to too conservative solutions or to solutions including local dynamic inversion (Sefton and Glover, 1990). Furthermore the order of the final controller is too high and may cause troubles to design engineers with regard to implementation problems. From the frequency domain approach (Nichols plots) point of view, it is not easy to imagine a generalization to MIMO systems. In the root locus domain, usually, phase shift is viewed as a rotation of the direction of motion of poles from

1. INTRODUCTION

In recent years the level of flexibility in aeronautic and spatial systems increased notably. As a consequence, the frequencies of the lowest structural (flexible) modes are closer and closer to the bandwidth required to control the rigid body motion. Therefore, gain control which consists of introducing in the loop a structural low pass filter in order to prevent control/structure interaction, is not practicable because the frequency gap between rigid and flexible modes becomes too narrow. In many cases, the synthesis model must take into account the first flexible modes. In the SISO case, phase control, based on Nichols plots, root loci or others graphic analysis tools, seems to be an interesting alternative approach because phase variations are more easily performed than gain variations. Phase lag or phase lead are then introduced in the loop in order to place the main low frequency resonances between two critical points on the Nichols plot or to create stabilizing branches from the first flexible poles on the roots locus. Usually, the trade-off between phase control for the first modes (low frequency) and gain control for the others modes (high frequency) has to be tuned. By such an approach, engineers take implicitly 653

their nominal locations. In order to obtain a straightforward generalization to MIMO systems, we suggest to forget this interpretation and to consider these rotations as an eigenstructure property. A simple result which states that, when a gain is tuned, the motion of eigenvalues is characterized by a function of left and right eigenvectors (see (2)), will be recalled. It will be the key result for the proposed design method. Several approaches based on eigenstructure assignment for dealing with flexibility are already proposed in literature, see for example (Garrard and Liebst, 1985), (Liebst et al., 1986), (Livet et al., 1994), (Song and Jayasuriya, 1993) but the philosophy used therein is very different from ours, here, we do not assign the system eigenvectors but add ad hoc components to the open loop eigenvectors. To provide a design tool as efficient as possible, we address the following points:

values as inequality constraints can be considered. In the second strategy, the motion of eigenvalues is constrained by equalities, but in turn, the feedforward poles become free.

2. NOTATION AND PROBLEM STATEMENT We shall consider the following linear system with n states, m inputs, p outputs:

x = Ax+Bu

(1)

y=Cx

where Y is the vector of measurements, A E 1R. nxn, B E 1R. nxm, C E 1R. pxn. This system may already be in closed loop. For example, if we consider a flexible structure, a controller of the rigid body may be included in (1). It is well known that variations of the matrix A, say ~A, induces variation of the eigenvalue >. given at first order by

• The problem is defined in terms of linear constraints. Moreover, inequalities instead of equalities are considered insofar as possible. For example, it is often sufficient to give a minimum amount of the expected motion to the left of low damped poles. In that way "freedom is offered to the poles in order to combine their relative motions" . • Robustness being one of the most important issue for flexible control design, the possibility of dealing simultaneously with several models (robustness from multimodel approach) is considered. • Criteria which correspond more or less to energy requirement are considered. This is useful to choose the best solution when the problem is underspecified. It is also necessary in order to decide of the solvability (too much energy required for a small amount of eigenvalue motion) of a given problem.

= u6Av

6A

(2)

where v and u are right and left eigenvectors of A relative to the eigenvalue >., u and v are normalized such that uv

=1

(3)

(The variations ~A will be considered later as being induced by gain variation.) To check this result it suffices to consider (A +~A)(v + ~v) =

which links ~>. to this equation is A~v

(>. + ~>.)(v + ~v)

~A.

The first order form of

+ ~Av = >.~v + ~>.v

Multiplying by u on the left u>.~v

+ u~Av = u>.~v + u~>.v

from which, using (3), we obtain the result stated above in (2). This result will be used in this paper. We shall consider System (1) in closed loop as follows. Let define

Note that the proposed approach is not only devoted to generalize "phase control". Gain control (i. e. rool off at high frequencies) can be combined by assigning zero motion to some eigenvalues or by minimizing a relevant criterion. Similar ideas help to prevent change at low frequencies, for example when the proposed technique is used as a second loop, the first one (rigid control loop) being already implemented. This paper is organized as follows. First notations are given and a well known first order eigenvalue perturbation result is recalled. Then the equations to be solved for finding the feedforward controller are derived. Finally two strategies in which the linearity of the problem is preserved are proposed. In the first strategy, the open loop poles of the feedforward network are chosen a priory, more freedom is available for assigning motion of eigen-

= =

Xc Ye

Acx c Ccxc

(4)

A c E 1R. ncxn c , Bc E 1R. ncx p , Cc E 1R. mxn c and Dc E 1R. mxp (ne will result from a minimal realization of the transfer matrix in (9)). Systems (1) and (4) are connected as follows U

= Ye = py

Uc

in which p is a scalar gain to be tuned between zero to some positive Value. The overall closed loop system is

~ ] = ([ A0 [ Xc

BCc] Ac

+p

[BDcC BcC

0]) [ Xcx 0

In this closed loop state space form, A c, Bc, Cc 654

]

(5)

and Dc are the unknowns that must be chosen in such a way that, when p is tuned, the motion of some eigenvalues of A is as desired (see Figure 1).

Proof. This result proceeds easily from the first order perturbation result recalled in §2 and from Lemma 3.1:

~A = p[u +

so,

(A,B,e)

+

- uBCc(A c - AI)-I] [ B:cgv ]

~A

Fig. 1. Generalized phase control scheme,

= p UBGc(A)CV

Note that "eigenvector assignment" consists of the assignment ofthe entries (U2) of the left eigenvectors which correspond to the dynamic state space extension.

p is tuned

from zero to some positive value.

3.2. Choice of the unknowns

Two possibilities are offered to compute the matrices A c, Bc, Cc and Dc. The unknowns in (7,8) can be considered as being the coefficients of the matrices A c, Bc, Cc and Dc. In this case, two kinds of non linearities are introduced. First, A c is inverted, second, parameters of A c, Bc and Cc are multiplied. Of course we can choose a standard form for the pair (A c, Bc). If the spectrum of A c is fixed a priori, the considered standard form may be the multi-companion form, unknowns remain only in Cc and Dc, the problem becomes linear. But, note that we have to choose arbitrarily the controllability indices and the spectrum of A c being fixed a priori, some degrees of freedom are lost. Also, the definition of a meaningful quadratic criterion is more difficult than in the transfer approach considered now. The unknowns may also be the coefficient of the transfer matrix Gc(s). A realization additional step is needed to find the state space representation. In (9), the unknowns are the coefficients ai and bijk :

3. DERIVATION OF THE EQUATIONS FOR EIGENVALUE MOTION ASSIGNMENT 3.1. Main result

Let consider an eigenvalue A of A and let denote u and v the corresponding left and right eigenvectors. For p = 0, A is also an eigenvalue of the state matrix in Equation (5). Let us first give the corresponding left and right eigenvectors. Lemma 3.1 Let consider A, u and v: eigenvalue, left and right eigenvectors of the matrix A, U and v satisfy (3). Then the left and right eigenvectors of the matrix

[

A

BCc]

°

Ac

corresponding to the eigenvalue A are respectively given by ii

= [u

- uBCc(A c - AI)-I] and v

=[~ ]

= 1. Consider ii = [UI U2]

and are such that iiv

Proof.

[UI U2] so u1A

= AUt

[~ BA~c] = A [UI (which means that

u1BCc + U2Ac therefore U2

Ut

U2] (9)

= u) and

= AU2

= -uBCc(A.: -

>"1)-1

(6)

3.3. Problems that can be considered in a linear setting

The result concerning ii is obvious. Theorem 3.2 The motion of the eigenvalue A of A when p is tuned in (5) is given at the first order by ~>..

= p uBGc(>")Cv

(7)

From now on the unknowns will be the coefficients and bijk as defined in (9). Let define simplified notations in order to manipulate together Equations (7) and (9). Let introduce the real and imaginary parts of the numerator and denominator of

(8)

UBGc(A)CV:

ai

where Gc(s) is defined by Gc(s)

= Cc(sI -

Ac)-I Bc

+ Dc

655

4. DESIGN PROCEDURES

uBGc(>\)Cv

=

NR(>..,b ijk ) + ~N/(>..,bijk) (10) DR(>..,ai) + JD/(>..,ai)

To define efficient design procedures it is necessary to minimize a criterion.

When>" is fixed (therefore u and v are also fixed), N R, N/ are linear in the coefficients bijk and DR, D / are linear in the coefficients ai. The most interesting problem to be considered would be

4.1. Definition of quadratic criteria It is important to choose some best solution when

• to give freedom to the coefficients of the denominator. • to consider linear inequalities preferably to equalities, for example: ~(UBGc(A)CV)

t:.11 $

$ t:.R

~(UBGc(A)CV)

t:.l1~(UBGc(A)CV)

$

the problem consisting of several systems of equalities or inequalities of the forms (12) or (13) is underspecified. Moreover, it is important to have a measure of the applicability of a solution, in other words we need to know if for a small motion of the eigenvalues we need to use a "small or a large feedforward effect". The most natural criterion concerns the static gain of the transfer matrix Gc(s)

$

$ t:.12 ~(UBGc(A)CV)

(11)

t:.12~(uBGc(A)CV)

Unfortunately this problem is not realistic because as shown in Comments 3.1 and 3.2, we must choose between both above objectives.

(14) Gc(O) reduces to

Comment 3.1 Consider an inequality, for example the first one in (ll):1R(uBG c(>")Cv) $ AR with notations as in (10),

1R (NR(>", bijk) + ~N/(>", bi j k )) DR(>..,ai) + JD/(>..,ai) we obtain

Gc (0) =

:s AR

NR(A, bijk)DR(A, ai) + NI (A, bijk)DI(A, ai) $ (DR(A, ai)2 + DI(A, ai)2)t:.R

b01p

ball [ ~_bO_~_l

]

bo_m-,:p-q-=-

ao

This criterion is quadratic in the unknowns. Combining this criterion to systems either of the form (12) or of the form (13), the problem to solve turns out to be a problem of quadratic optimization under linear constraints. Similar criteria can be considered, for example to impose roll off at a frequency WH. We may consider

(12)

This inequality becomes linear in the coefficients bijk if the coefficients ai are fixed a priori. Clearly, under the same hypothesis, all kinds of inequalities, as for example those in (11), correspond to linear constraints. The conclusion is that inequalities can be considered provided that the coefficients ai are fixed.

J

= IIG c (wH)11 2

or a combination of similar criteria (when the coefficients ai are fixed, it is easy to check that these criteria are quadratic in the coefficients bij k.

Comment 3.2 Consider a system of equalities, for example uBGc(>")Cv = AR + jA/ with notations as in (10), this system becomes

4.2. Constraints on denominator coefficients

NR(>", bijk) = ARDR(>", ai) - A/D/(>.., ai) (13) N/(>",bijk) = ARD/(>..,ai) + A/DR(>",ai) it is linear in all the unknowns so that we can also optimize the coefficients ai. To preserve stability, constraints on these coefficients must be considered. A possible set of linear constraints that insure a given degree of stability is proposed in §4.2. The conclusion is that if we want to optimize coefficients ai and bijk together, we must consider equalities for pole motion.

656

It is worth noting that it is really easy to make a good initial choice for the coefficients ai. For that purpose it suffices to choose the roots, the coefficients proceed from the roots. The roots must be far enough on the left in the complex plane so that, when the loop gain p is tuned, the feedforward poles do not interfere with the system root locus. As for the imaginary parts, they must be in an interval around the frequencies at which an effect is expected. Also the roots must not be too far from the imaginary axis in order to insure filtering (because the numerator introduces derivations). Let aOi denote the initial choice obtained from an initial choice of roots. Next step consists of defining a box in which the coefficient ai may vary, say ai E [aOi - cbi , aOi + cb;], in such a

3Or----,--,.....---..--,.....---..--,.....---..--,.....---..--,.....--,

way that the real part of all roots remains less than -Q (for some positive number Q). For that purpose, it suffices to consider the stability of (ao

x.

20

.'

+ coo) + ... + (aq-l + cOq-I)(s + a)q-l + (s + a)q 10

The dependency of the resulting polynomial coefficients is affine with respect to the variations c:c5i . It follows that the maximal value of c: denoted i can be found by using the Edge Theorem, see for example (Ferreres and Magni, 1996) (i can also be chosen easily by trials and errors). It remains to define the values of c5 i . From applications, it appears that values of c5 i proportional to aOi is a good choice. Finally, the linear constraints to be considered as mentioned in Comment 3.2 are ai < aOi + ic5 i and aOi - ic5 i < ai These constraints are conservative but the optimal value of the coefficient obtained in that way help to improve the initial choice.

.

..

-10

x

•

-20 x·

~':-o---c_9'--~-ll'----:_7,...-~_o'----:-5,...--'_4---c-'_3--'-2--'-1--'----l Real Axis

Fig. 2. Poles map in open loop and after closing the first loop.

20

0.;

10

01;:;:

0- •• : •••••••• ; •••••••• : •••••••••• ······-• •

······'!o·····

4.3. Proposed design procedure -10

.......... -

'0'

-20

We would like to release the constraints relative to unknown parameters (including ai) and to preserve the linear-quadratic formulation. So, we propose, when it is necessary, to alternate between the solution of two linear-quadratic optimization problems, i. e. to proceed as follows. Step 1. Choose a priori the coefficients ai, i. e. choose the roots of the denominator of Gc (s). Then optimize a criterion as in §4.1 under inequality (and equality if relevant) constraints as in (12). Step 2. From the solution obtained at step 1, improve the initial choice of coefficients ai. For that purpose, consider the eigenvalue motion obtained at the first step as equality constraints and optimize again but with freedom on the parameters ai (see §4.2). Go to step one if necessary and so on. An alternative approach consists of using non linear parameters optimization, keeping in mind that the feedforward network obtained at the first step above is a good initialization. The exact constraints on coefficients ai are obtained by using Routh criterion after a translation of the imaginary axis. We do not have to mind any more about the linearity of constraints, in this way both objectives stated at the beginning of §3.3 can be met together (provided that local minima, if any, are not too troublesome). From a practical point of view, the choice of ai made a priori has always shown to be sufficient. So, in most cases the design procedure reduces to step 1 above.

-

-

Fig. 3. Root locus when p is varied.

5. APPLICATION: LATERAL MOTION OF A FLEXIBLE AIRCRAFT

5.1. System and problem statement In this example, we consider the lateral dynamics of a large flexible aircraft. The model includes the four standard rigid-body states (angle of side slip, roll rate, yaw rate, roll angle), two actuators states, and six states for modelization of aeroelastic modes. The inputs are the aileron and rudder deflections. The four rigid-body states are measured. First the eigenvalues of the rigid-body subsystem are shifted to the left. The gain used for this purpose has a destabilizing effect on some flexible modes. The design of an additional controller which aims to move back these destabilized flexible modes to their open loop values, without effect on the rigid-body dynamics, is considered now. From now on, the model considered will include the "rigid" control loop. This model is described by its pole map in Figure 2. In this figure, "*" denotes open loop poles, "+" denotes the poles after closing the first loop. So, we can point out which flexible modes have to be shifted back by 657

the second loop, and where.

Table 1 Evolution of the system eigenvalues.

5.2. Active flexible control design We must assign the motions of the poles which moved in Figure 2 (denoted A;) in order to push them back to their open loop location. For that we define several inequalities of the form

Re(u;BGc(A;)CV;) < -.6.; -.6. j < Im(u;BGc(A;)CV;) < .6. j

Re(u;BGc(A~)CV;)

Cio.cd loop without flexible contol

-10 -10

-9.16 + 0.941 -9.16 - 0.94i

-0.13 + 1.07, -0.13 - 1.07, -0.92 0.0057

-1.28 + 0.661 -1.28 - 0.661 -1.30 -0.98

Clo.ed loop with flexible contol s:ain p = 10

+ 0.74i

-11.14

-11.14 -

-1.28

0.741

+ 0.68,

-1.28 -

0.68i

-13.94

+ 19.72i 19.721

-20.80 + 18.27i -20.80 - 18.27i

-0.31 - 1l.Sh -0.31

+

ll.Sh

-0.35 - 17.74i -0.35

+

-1.66 -

-1.66

17.741

28.141

+ 28.14i

-0.30 - 11.15i -0.30 + 1l.15i 0.29 - 16.89i

0.29

+ 16.89.

-1.31 - 27.31i

-1.31

+ 27.31.

actuat. elge-DV. rigid. eigenv.

-1.31 -0.98 -13.94 -

These constraints are considered for the three flexible modes. These poles are very sensitive so we must guide them by fixing imaginary part variation to zero (.6. j = 0) . With .6.; = 0.1 for the first pole, .6.; = 0.05 for the second pole and .6.; = 0.05 for the third pole the loop gain p should be about 10 to recover open loop flexible dynamics. Furthermore, we do not want to move the rigidbody poles. Depending on the choice of the denominator of the feedforward transfer, the second loop has or not a negligible effect on the low frequency design. (In other words, the minimization of the static gain as in (14) is not sufficient in some cases, contrary to most other applications we treated.) To prevent this effect in any cases, it suffices to fix to zero real and imaginary parts of low frequency poles (denoted AD.

Im(u;BGc(ADCv;)

Open loop

-1.02 - 11.27i -1.02 + 11.27i -1.69 - 16.32i

feedfwd. eigenv.

flexible eigcnv.

+ 16.32. -1.90 - 27.10i -1.90 + 27.10. -1.69

ear quadratic design techniques. AIAA Journal of Guidance, Control and Dynamics Vol 8(3),304-311. Goh, C.H. and G. Griibel (1992). Synthesis of robust multivariable controllers for large flexible structures. IFAC Symposium on Automatic Control in Aerospace. Liebst, B.S., W.L. Garrard and W.M. Adams (1986). Design of an active flutter supression system. AIAA Journal of Guidance, Control and Dynamics Vol 9(1), 64-71. Livet, T., F. Kubica, P. Fabre and J.F. Magni (1994). Robust flight control design for highly flexible aircraft by pole migration. In Proc. IFAC Symposium on Automatic Control in Aerospace, Palo Alto, California. Magni, J.F. and Y. Le Gorrec (n.d.). Robust dynamic feedback control. Submitted for publication. Sefton, J. and K. Glover (1990). Pole/zero cancellations in the general Hoo problem with reference to a two block design. In: Systems and Control Letters. Vol. 14. pp. 295-306. Song, B.K. and S. Jayasuriya (1993). Active vibration control using eigenvector assignment for mode localization. In Proc. American Control Conference, San Francisco pp. 1020-1024. Stoughton, R.M. (1992). A loop-shaping approach to control system design using an Hoo design engine, applied to Titan IV stage 2. In Proc. AIAA Conf. on Guidance Navigation and Control. Tahk, M. and J. L. Speyer (1987). Modeling of parameter variations and asymptotic LQG synthesis. IEEE Transactions on Automatic Control. Wie, B., Q. Liu and F. Bauer (1992). Classical and robust Hoo control redesign for the hubble space telescope. In Proc. AIAA Conf. on Guidance Navigation and Control.

=0

=0

It remains to choose the denominator of Gc(s). In fact for all the roots of the denominator in the rectangle defined by: real part between -30 and -10, imaginary part between + 10 and +30, the results are almost similar (so it is not necessary to optimize the choice of the coefficients of the denominator (a;)). The results are analyzed by mean of the root locus given Figure 3 and by the table of values given in Table 1. In Figure 3, it appears that rigid modes are unchanged (as expected). The eigenvalues about 11 rd/s, 16 rd/s and 27 rd/s have been shifted to the left without undesirable effects on other poles.

REFERENCES Coville, A., H. Abou-Kandil and P. Coustal (1991). Commande robuste basee sur le theoreme des petits gains et la positivite. In Proc. European Control Conference. Ferreres, G. and J.F. Magni (1996). Robustness analysis using the mapping theorem without frequency gridding. In Proc. IFAC World Congress, San Francisco Vol H, 7-12. Garrard, W.L. and B.S. Liebst (1985). Active flutter supression using eigenspace and lin658