Method for adaptively cancelling in real time elastic modes in discrete-time signals

Abstract

A method of aeroservoelastic coupling suppression, and particularly, the field of real time adaptive cancellation of elastic modes in discrete-time signals which measure the dynamics of a flexible structure. The flexible structure comprises a structure with elastic variable characteristics, and more particularly, a structure with non-linear aerodynamics. A method is disclosed for adaptively cancelling, in real time, N elastic modes in discrete-time signals which measure the dynamics of the flexible structure. Also disclosed is a computer program implemented on a computing device, a system and an aircraft implementing the mentioned method.

Claims

1. A method for adaptively cancelling, in real time, N elastic modes in discrete-time signals which measure the dynamics of a flexible structure, the flexible structure having elastic variable characteristics, and the method comprising two main blocks: a) a first block of the method comprising the following steps: providing a total number N.sup.+ of elastic modes to be cancelled, being i a generic elastic mode to be cancelled, i.sup.+: i[1, N], and two active measurement sources (A, B) which location is respectively defined by location vectors l.sub.A and l.sub.B, where l.sub.A, l.sub.B.sup.3, b) sampling two output measurement discrete-time scalar signals y.sup.A and y.sup.B of the dynamics of the flexible structure, being y.sub.n.sup.A and y.sub.n.sup.B the n.sup.th sample of the respective signal measured by the two active measurement sources (A, B), said sampling being performed with a sample time t, and the n.sup.th sample of a signal being n.sup.+: n[1, ), c) providing natural frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i, and an integer parameter M.sub.n1.sup.i, being said frequency estimate and the parameter respectively: c1) if n=1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.0.sup.i; M.sub.n1.sup.i=M.sub.0.sup.i, being M.sub.0.sup.i an integer multiple of $.Math.\frac{2}{{\hat{}}_0^i,t}.Math.,$ being the method applied for the first time, c2) if n1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.n1.sup.i; M.sub.n1.sup.i=M.sub.n1.sup.i, being M.sub.n1.sup.i an integer multiple of $.Math.\frac{2}{{\hat{}}_{n-1}^{i}t}.Math.,$ d) filtering the output measurement discrete-time scalar signals y.sup.A and y.sup.B, by means of a conditioning filter H.sub.c.sup.i, obtaining filtered measurement outputs for the elastic mode i, Y.sup.A,i and Y.sup.B,i, calculated by the following expressions in the Z-domain:
Y.sup.A,i(z)=H.sub.c.sup.i(z)y.sup.A(z)
Y.sup.B,i(z)=H.sub.c.sup.i(z)y.sup.B(z) wherein H.sub.c.sup.i(z) corresponds to the transfer function in the Z-domain of the conditioning filter H.sub.c.sup.i, said conditioning filter being a parametric band-pass digital filter centered at the frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i provided in step c), e) obtaining an n.sup.th sample of an estimated cancellation parameter per elastic mode i to be cancelled, {circumflex over (K)}.sub.n.sup.i, according to the following expression: ${\hat{K}}_n^i=\frac{\underset{j=n-M_{n-1}^i+1}{\overset{n}{.Math.}}\left({\left(Y_j^{B,i}\right)}^2-Y_j^{A,i}{Y}_j^{B,i}\right)}{\underset{j=n-M_{n-1}^i+1}{\overset{n}{.Math.}}\left({\left(Y_j^{A,i}\right)}^2+{\left(Y_j^{B,i}\right)}^2-2Y_j^{A,i}{Y}_j^{B,i}\right)}$ wherein the value of the parameter M.sub.n1.sup.i corresponds to the value provided in step c), f) generating an n.sup.th sample of a discrete-time scalar signal y.sup.X, namely y.sub.n.sup.X, according to the following expression:
y.sub.n.sup.X={circumflex over (K)}.sub.n.sup.iy.sub.n.sup.A+(1{circumflex over (K)}.sub.n.sup.i)y.sub.n.sup.B g) obtaining a filtered discrete-time scalar signal Y.sup.F by means of the following bank filtering expression in the Z-domain:
Y.sup.F(z)=y.sup.X(z)H.sub.1(z)+y.sup.A(Z)(1H.sub.1(z)) wherein H.sub.1 is a second order band-pass filter centered in the natural frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i provided in step c), and obtaining directly from the filtered discrete-time scalar signal Y.sup.F an n.sup.th sample of the filtered discrete-time scalar signal Y.sup.F, namely Y.sub.n.sup.F, h) obtaining an n.sup.th sample of an estimate of the mode displacements for each elastic mode i, {circumflex over ()}.sub.n.sup.i, by means of the following expression:
{circumflex over ()}.sub.n.sup.i=y.sub.n.sup.AY.sub.n.sup.F i) calculating an adjusted value of {circumflex over ()}.sub.n1.sup.i, namely {circumflex over ()}.sub.n1.sup.i, by introducing the value of {circumflex over ()}.sub.n.sup.i from step h) in a frequency tracking module, obtaining the adjusted value {circumflex over ()}.sub.n1.sup.i, j) saving the adjusted value {circumflex over ()}.sub.n.sup.i calculated in step i) k) performing, from i=1 to i=N, the steps c) to j), the second block of the method comprising the following steps: l) inheriting from step h) the values of the n.sup.th sample of the estimate of the mode displacements, {circumflex over ()}.sub.n.sup.i for each elastic mode i, m) obtaining an n.sup.th sample of an elastic mode cancellation output function Y.sub.n.sup.output according to the following expression: $Y_n^{\mathrm{output}}={}_n^1{y}_n^A+{}_n^2{y}_n^B+\underset{j=1}{\overset{N-1}{.Math.}}{}_n^{j+2}{\hat{}}_n^j==\left[\begin{array}{cccccc}{y}_n^A& y_n^B& {\hat{}}_n^1& {\hat{}}_n^2& \mathrm{.Math.}& {\hat{}}_n^{N-1}\end{array}\right]\left[\begin{array}{c}{}_n^1\\ \mathrm{.Math.}\\ {}_n^N\\ {}_n^{N+1}\end{array}\right]$ said n.sup.th sample of the elastic mode cancellation output function Y.sub.n.sup.output being obtained by the following steps: m1) calculating the inverse of an estimated spatial filter matrix {circumflex over ()} by the following expression: ${\hat{}}^{-1}={\left({\left[\begin{array}{cccc}1& 1& 1& 1\\ \frac{{\hat{K}}_n^1}{{\hat{K}}_n^1-1}& \mathrm{.Math.}& \frac{{\hat{K}}_n^N}{{\hat{K}}_n^N-1}& 1\\ 1-{}_n^{1,1}& \mathrm{.Math.}& 1-{}_n^{N,1}& 1\\ \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}\\ 1-{}_n^{1,N-1}& \mathrm{.Math.}& 1-{}_n^{N,N-1}& 1\end{array}\right]}^T\right)}^{-1}$ wherein: {circumflex over (K)}.sub.n.sup.i are values of the n.sup.th sample of the estimated cancellation parameter for each elastic mode i to be cancelled, inherited from step e) .sub.n.sup.i,p, p.sup.+: p[1, N1]; i+: i[1, N], is an adjustable discrete-time parameter for each elastic mode i, selected to achieve a non-singular spatial filter matrix {circumflex over ()}, m2) calculating a spatial filter parameter vector .sub.n.sup.N+1, wherein: ${}_n=\left[\begin{array}{c}{}_n^1\\ \mathrm{.Math.}\\ {}_n^N\\ {}_n^{N+1}\end{array}\right]={\hat{}}^{-1}\left[\begin{array}{c}0\\ \mathrm{.Math.}\\ 0\\ 1\end{array}\right]$ m3) generating a set of N1 parametric discrete-time virtual signals {circumflex over ()}.sup.p, being {circumflex over ()}.sub.n.sup.p the n.sup.th sample of the virtual signals {circumflex over ()}.sup.p, being p.sup.+: p[1, N1], said n.sup.th sample of the virtual signals {circumflex over ()}.sub.n.sup.p configured for complementing y.sub.n.sup.A and y.sub.n.sup.B according to the following expression: ${\hat{}}_n^p=y_n^A-\underset{i=1}{\overset{N}{.Math.}}{}_n^{i,p}{\hat{}}_n^i$ m4) substituting in the elastic mode cancellation output function y.sub.n.sup.output expression the values obtained in the previous steps k1)-k3), obtaining the value of the elastic mode cancellation output function Y.sub.n.sup.output $Y_n^{\mathrm{output}}={}_n^1{y}_n^A+{}_n^2{y}_n^B+\underset{j=1}{\overset{N-1}{.Math.}}{}_n^{j+2}{\hat{}}_n^j==\left[\begin{array}{cccccc}{y}_n^A& y_n^B& {\hat{}}_n^1& {\hat{}}_n^2& \mathrm{.Math.}& {\hat{}}_n^{N-1}\end{array}\right]\left[\begin{array}{c}{}_n^1\\ \mathrm{.Math.}\\ {}_n^N\\ {}_n^{N+1}\end{array}\right]$ n) cancelling the elastic modes i by means of the elastic mode cancellation output function Y.sub.n.sup.output, o) performing, from n=1 to n=, the steps b) to n), introducing the saved adjusted {circumflex over ()}.sub.n.sup.i values of step j) in step c).

2. The method for adaptively cancelling in real time N elastic modes in discrete-time signals according to claim 1, wherein the conditioning filter H.sub.c.sup.i of step d) is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.i and one or two narrow stop-band filters (SB) centered at {circumflex over ()}.sub.n1.sup.i1 and {circumflex over ()}.sub.n1.sup.i+1, and wherein H.sub.c.sup.i(z) is the transfer function in the Z-domain of the conditioning filter H.sub.c.sup.i corresponding to the following expression:
if i=1
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i+1)
else, if i=2, . . . ,N1
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i1)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i+1)
else, if i=N
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i1) and wherein step c) of the method further comprises providing natural frequency estimates {circumflex over ()}.sub.n1.sup.i1 and {circumflex over ()}.sub.n1.sup.i+1 of the elastic mode i1 and i+1, being: c1) if n=1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.0.sup.i; {circumflex over ()}.sub.n1.sup.i1={circumflex over ()}.sub.0.sup.i1; {circumflex over ()}.sub.n1.sup.i+1={circumflex over ()}.sub.0.sup.i+1; M.sub.n1.sup.i=M.sub.0.sup.i, being M.sub.0.sup.i an integer multiple of $.Math.\frac{2}{{\hat{}}_0^i,t}.Math.,$ being the method applied for the first time, c2) if n1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.n1.sup.i; {circumflex over ()}.sub.n1.sup.i1={circumflex over ()}.sub.n1.sup.i1; {circumflex over ()}.sub.n1.sup.i+1={circumflex over ()}.sub.n1.sup.i+1; M.sub.n1.sup.i=M.sub.n1.sup.i, being M.sub.n1.sup.i an integer multiple of $.Math.\frac{2}{{\hat{}}_{n-1}^{i}t}.Math..$

3. The method for adaptively cancelling in real time N elastic modes in discrete-time signals according to claim 2 wherein the narrow band-pass filter is defined by a bandwidth .sub.BP and the one or two narrow stop-band filters are defined by a bandwidth .sub.SB respectively, following the expression:
if i=1
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i,.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i+1,.sup.i+1)
else, if i=2, . . . ,N1
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i,.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i1,.sup.i1)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i+1,.sup.i+1)
else, if i=N
H.sub.c.sup.i(z)=H.sub.BP(z,{circumflex over ()}.sub.n1.sup.i,.sup.i)H.sub.SB(z,{circumflex over ()}.sub.n1.sup.i1) wherein is the filter width of the conditioning filter H.sub.c.sup.i, fulfilling:
.sup.i=.sub.BP
.sup.i1=.sup.i+1=.sub.SB.

4. The method for adaptively cancelling in real time N elastic modes in discrete-time signals according to claim 1, wherein the conditioning filter H.sub.c.sup.i of step d) is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.i and N1 narrow stop-band filters (SB) centered at {circumflex over ()}.sub.n1.sup.q being q.sup.+: (q[1, N]|qi), being H.sub.c.sup.i(z) the transfer function in the Z-domain of the conditioning filter H.sub.c.sup.i corresponding to the following expression: $H_c^i\left(z\right)=H_{\mathrm{BP}}\left(z,{\hat{}}_{n-1}^i\right).Math.\underset{\begin{array}{c}q=1\\ qi\end{array}}{\overset{N}{.Math.}}{H}_{\mathrm{SB}}\left(z,{\hat{}}_n^q\right)$ and wherein step c) of the method further comprises providing natural frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i, and natural frequencies estimates of the rest of the elastic modes {circumflex over ()}.sub.n1.sup.q, with q.sup.+: {q[1, N]|qi}, being: c1) if n=1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.0.sup.i; {circumflex over ()}.sub.n1.sup.q={circumflex over ()}.sub.0.sup.q with q.sup.+: {q[1, N]|qi}; M.sub.n1.sup.i=M.sub.0.sup.i, being M.sub.0.sup.i an integer multiple of $.Math.\frac{2}{{\hat{}}_0^i,t}.Math.,$ being the method applied for the first time, c2) if n1; then {circumflex over ()}.sub.n1.sup.i={circumflex over ()}.sub.n1.sup.i; {circumflex over ()}.sub.n1.sup.q={circumflex over ()}.sub.n1.sup.q with q.sup.+: {q[1, N]|qi}; M.sub.n1.sup.i=M.sub.n1.sup.i, being M.sub.n1.sup.i an integer multiple of 2/{circumflex over ()}.sub.n1.sup.i t, $.Math.\frac{2}{{\hat{}}_{n-1}^{i}t}.Math..$

5. The method for adaptively cancelling in real time N elastic modes in discrete-time signals according to claim 4 wherein the narrow band-pass filter is defined by a bandwidth .sub.BP and the N1 narrow stop-band filters are defined by a bandwidth .sub.SB respectively, following the expression: $H_c^i\left(z\right)=H_{\mathrm{BP}}\left(z,{\hat{}}_{n-1}^i{}^i\right).Math.\underset{\begin{array}{c}q=1\\ qi\end{array}}{\overset{N}{.Math.}}{H}_{\mathrm{SB}}\left(z,{\hat{}}_n^q,{}^q\right)$ wherein is the filter width of a band-pass or a stop-band filter, fulfilling:
.sup.i=.sub.BP
.sup.q=.sub.SB qi.

6. A non-transitory computer-readable medium having computer-executable program instructions stored thereon, comprising instructions for the implementation, by a processor of a computing device, of the method according to claim 1, when said program is executed by said processor.

7. A system comprising: a flexible structure and N elastic modes to be cancelled in real time in discrete-time signals which measure the dynamics of the flexible structure, and a computer device, the computer device being configured to apply a method for adaptively cancelling in real time N elastic modes in said discrete-time signals according to claim 1.

8. An aircraft comprising a system according to claim 7.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) These and other characteristics and advantages of the invention will become clearly understood in view of the detailed description of the invention which becomes apparent from a preferred embodiment of the invention, given just as an example and not being limited thereto, with reference to the drawings.

(2) FIG. 1 shows a diagram with the steps of the first block of the method for cancelling elastic modes according to the present invention.

(3) FIG. 2 shows a diagram with the complete method according to a first embodiment of the present invention.

(4) FIG. 3 shows a diagram with the complete method according to a second embodiment of the present invention.

(5) FIG. 4 shows a diagram with the complete method according to a third embodiment of the present invention.

(6) FIG. 5 shows a particular embodiment of a flying boom of a tanker aircraft on which the elastic mode cancellation method of the present invention is applied to the feedback signals of the Control Laws.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(7) The present description discloses a method for adaptively cancelling in real time N elastic modes in discrete-time signals which measure the dynamics of a flexible structure, the flexible structure having elastic variable characteristics.

(8) Elastic mode cancellation in the discrete-time signals which measure the dynamics of a structure can be addressed either theoretically or experimentally.

(9) When considering theoretical studies, there is the need of performing an analytic solving in order to obtain results for the cancellation of the selected elastic modes.

(10) An example of a theoretical study is disclosed below.

(11) Analytic solving for cancelling N elastic modes in discrete-time signals containing the measured dynamics of a flexible structure.

(12) The theoretical solving is mainly composed by two different steps which are:

(13) identifying the elastic modes characteristics by a collection of estimation kernels, and

(14) providing an extended spatial filter algebraic solver.

(15) Additionally, the theoretical solving estimates, in parallel with the steps, the exogenous boundary condition acting on the system and accordingly configures the constraints applicable to the distributed constrained estimation kernels.

(16) Elastic Modes Estimation Kernels

(17) For cancelling N elastic modes in discrete-time signals which measure the dynamics of a flexible structure with elastic variable characteristics, N elastic modes estimation kernels are required, wherein each kernel is intended to provide an estimate of the elastic modes displacement, and an estimate of K.sub.n.sup.i parameter required to cancel a specific elastic mode i, with i.sup.+: i[1, N], given two output measurement discrete-time scalar signals y.sup.A and y.sup.B, measured by two measurement sources (A, B) respectively, according to the location of the sources and the characteristics of the structure.

(18) The n.sup.th sample y.sub.n.sup.A and y.sub.n.sup.B of the output measurement discrete-time scalar signals y.sup.A and y.sup.B can be expressed in terms of the sensed rigid and elastic system dynamics as
y.sub.n.sup.A=r.sub.n+.sub.i=1.sup.N.sub.n.sup.i(l.sub.A).sub.n.sup.i+v.sub.n.sup.A
y.sub.n.sup.B=r.sub.n+.sub.i=1.sup.N.sub.n.sup.i(l.sub.B).sub.n.sup.i+v.sub.n.sup.B

(19) where

(20) l.sub.A, l.sub.B .sup.3 denote the location vectors of the two measurement sources (A, B) respectively,

(21) n a subscript denoting the n.sup.th sample of a discrete-time signal, being n.sup.+: n[1, )

(22) i a generic index referring the elastic mode to be cancelled, i.sup.+: i[1, N],

(23) r.sub.n is a rigid dynamics output measurement of each active measurement sources (A, B),

(24) .sub.n.sup.i is an elastic displacement of each of the elastic modes i,

(25) v.sub.n.sup.A and v.sub.n.sup.B are measurement noises of each active measurement source (A, B) respectively, and

(26) .sub.n.sup.i(l) is an elastic mode shape of each of the elastic modes i, according to the location vector l of the active measurement sources (A, B), l.sup.3.

(27) Therefore, the samples y.sub.n.sup.A and y.sub.n.sup.B comprise three different components, which are a rigid dynamics component, an elastic component and a noise component directly related with the measurement sources.

(28) The perfect cancellation parameter, as it is theoretically determined, K.sub.n.sup.i of each elastic mode can be computed imposing that the linear combination of the samples y.sub.n.sup.A and y.sub.n.sup.B results in the suppression of the sensed elastic system dynamics .sub.n.sup.i(l.sub.A).sub.n.sup.i of the elastic mode i while maintaining unaltered the sensed rigid dynamics denoted by r.sub.n according to the following expressions

(29) $K_n^i{}_n^i\left(l_A\right){}_n^i+\left(1-K_n^i\right){}_n^i\left(l_B\right){}_n^i=0$$K_n^i{y}_n^A+\left(1-K_n^i\right)y_n^B==r_n+\underset{j=1ji}{\overset{N}{.Math.}}\left[K_n^i{}_n^j\left(l_A\right){}_n^j+\left(1-K_n^i\right){}_n^j\left(l_B\right){}_n^j\right]+K_n^i{v}_n^A+\left(1-K_n^i\right)v_n^B$

(30) Additionally, it is necessary that the elastic mode shape of each of the elastic modes fulfills the following condition:
.sub.n.sup.i(l.sub.A).sub.n.sup.i(l.sub.B),

(31) According to the previous expressions, the theoretical elastic mode shape cancellation parameter K.sub.n.sup.i per elastic mode i to be cancelled is obtained by means of the following equation

(32) $K_n^i=\frac{{}_n^i\left(l_B\right)}{{}_n^i\left(l_B\right)-{}_n^i\left(l_A\right)}$

(33) These parameters provide the solution of the theoretical function which allows the cancellation of the selected N elastic modes in discrete-time signals which measure the dynamics of the flexible structure.

(34) Considering the parameters, an analytical solution can be related with an experimental solution of the parameters, namely the estimated parameters {circumflex over (K)}.sub.n.sup.i.

(35) The relation between the theoretical solution and the experimental solution can be expressed as follows:
{circumflex over (K)}.sub.n.sup.i=K.sub.n.sup.i+{tilde over ()}.sub.n.sup.i(.sub.a.sup.i,.sub.B.sup.i,v.sub.n.sup.A,v.sub.n.sup.B)

(36) wherein {tilde over ()}.sub.n.sup.i is the n.sup.th sample of a discrete-time error bias function for elastic mode i which depends on the bandwidth power ratios .sub.A.sup.i and .sub.B.sup.i between the rigid dynamic output measurement r.sub.n and the measured elastic displacement of each of the elastic modes i for each active measurement source (A, B) respectively, and also depends on the measurement noises v.sub.n.sup.A and v.sub.n.sup.B of the active measurement source (A, B).

(37) The bandwidth power ratios .sub.A.sup.i and .sub.B.sup.i must fulfill the following requirement in order to minimize the estimation error {tilde over ()}.sub.n.sup.i:

(38) ${}_A^i=\frac{{}_{{}_i-/2}^{{}_i+/2}\underset{n=-}{\overset{}{.Math.}}{r}_n{e}^{-jn}d}{{}_{{}_i-/2}^{{}_i+/2}\underset{n=-}{\overset{}{.Math.}}{}_n^i\left(l_A\right){}_n^i{e}^{-jn}d}1$${}_B^i=\frac{{}_{{}_i-/2}^{{}_i+/2}\underset{n=-}{\overset{}{.Math.}}{r}_n{e}^{-jn}d}{{}_{{}_i-/2}^{{}_i+/2}\underset{n=-}{\overset{}{.Math.}}{}_n^i\left(l_B\right){}_n^i{e}^{-jn}d}1$

(39) being:

(40) a generic frequency

(41) .sub.i a frequency of the elastic mode i

(42) j an imaginary number equal to {square root over (1)}

(43) a generic frequency bandwidth

(44) Conditions required in the previous equations can be always satisfied if is small enough, considering a small enough value that of <0.1.sub.i.

(45) With these results, a second step is performed considering the actuation of an extended spatial filter algebraic solver, which will finally provide with the output function solution for cancelling the elastic modes using the input discrete-time signals y.sup.A and y.sup.B.

(46) However, such a solving for the cancellation of the elastic modes in discrete-time signals is complicated to fulfill, as it is very difficult to distinguish which part of the signals y.sup.A and y.sup.B comes from the measured rigid dynamics of the system, and which part comes from the measured elastic dynamics of the system. This is why there is not immediate obtainment of the different components of the signals (r.sub.n, .sub.i=1.sup.N.sub.n.sup.i (l.sub.A).sub.n.sup.i and v.sub.n.sup.A for measurement source A) in order to select the elastic component for cancelling it, obtaining as a result the measured rigid dynamics component in signals y.sup.A and y.sup.B with zero phase-loss and attenuation in addition with the measurement noises.

(47) Therefore, a theoretical solution for the cancelling of elastic modes is complicated to perform. On the contrary, the present method provides for a real-time solver which allows the cancellation of the elastic modes according to different changing conditions.

(48) Estimate solving for cancelling in real time N elastic modes in discrete-time signals containing the measured dynamics of a flexible structure.

(49) The present method for cancelling elastic modes in discrete-time signals is divided in two main blocks, wherein different steps are fulfilled in order to obtain a final equation which allows the cancellation of the elastic modes.

(50) FIG. 1 shows a diagram with the steps of the first block for the method of canceling N elastic modes.

(51) As it can be observed, the n.sup.th sample y.sub.n.sup.A and y.sub.n.sup.B of two output measurement discrete-time scalar signals y.sup.A and y.sup.B are supplied to the conditioning filter H.sub.c.sup.i in order to obtain filtered measurement outputs for the elastic mode i, Y.sup.A,i and Y.sup.B,i.

(52) These solutions allow the performance of a least mean square error optimization in order to obtain an n.sup.th sample of an estimated cancellation parameter per elastic mode i to be cancelled, {circumflex over (K)}.sub.n.sup.i.

(53) The obtainment of these parameters is shown in FIG. 1 by means of a label STEP e).

(54) The estimated cancellation parameter along with the n.sup.th sample y.sub.n.sup.A will provide the n.sup.th sample y.sub.n.sup.X of a discrete-time scalar signal y.sup.X.

(55) The method uses a complementary filter bank which performs two different operations. The first operation is the filtering of the n.sup.th sample of a discrete-time scalar signal, namely y.sub.n.sup.X, by means of a second order band-pass filter centered in the natural frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i. The second operation is the filtering by means of the former complementary filter of the n.sup.th sample y.sub.n.sup.A.

(56) In this preferred embodiment, the signal supplied in the mentioned steps is the n.sup.th sample y.sub.n.sup.A as shown in FIG. 1.

(57) The solution of the two filtering steps is used for obtaining a filtered discrete-time scalar signal Y.sup.F, being its n.sup.th sample, namely Y.sub.n.sup.F.

(58) This n.sup.th sample of the filtered discrete-time scalar signal, Y.sub.n.sup.F along with the n.sup.th sample y.sub.n.sup.A, provides an n.sup.th sample with of an estimate of the mode displacements for each elastic mode i, {circumflex over ()}.sub.n.sup.i. This is shown in FIG. 1 by means of label STEP h).

(59) In parallel, the results of the n.sup.th sample of an estimate of the mode displacements for each elastic mode i, {circumflex over ()}.sub.n.sup.i, are used as input to a frequency tracking module, the tracking module consisting of any algorithm that computes the frequency at which a signal has its maximum of power spectral density, for obtaining an adjusted value of {circumflex over ()}.sub.n1.sup.i, namely {circumflex over ()}.sub.n.sup.i which will be saved to be used in later steps of the method. This is shown in FIG. 1 by means of a label STEP i) and label STEP j) respectively.

(60) The results of these adjusted natural frequency estimates affect the filtering transfer functions, therefore the filters used in the different step of the methods.

(61) FIG. 2 shows a diagram with the complete method according to a first embodiment of the present invention.

(62) As it can be observed, the totality of the steps of the method are shown in the diagram along with the location of the looping steps which allow the method to readjust considering the changes of the conditions of the elastic system structure.

(63) The diagram shows how, starting with the n.sup.th sample y.sub.n.sup.A and y.sub.n.sup.B of two output measurement discrete-time scalar signals, steps a) and b) of the method, the rest of the steps are fulfilled obtaining the needed results {circumflex over (K)}.sub.n.sup.i and {circumflex over ()}.sub.n.sup.i.

(64) The conditioning filter H.sub.c.sup.i used in this particular embodiment is a parametric band-pass digital filter centered at the frequency estimate {circumflex over ()}.sub.n1.sup.i of the elastic mode i provided in each of the different steps c) performed for each of the elastic modes i, which can be seen in FIG. 2.

(65) Additionally, the diagram shows the second block of the method, which ends in the obtainment of Y.sub.n.sup.output, function which allows the cancellation of the elastic modes components in the discrete-time signals y.sup.A and y.sup.B.

(66) FIG. 3 shows the diagram shown in FIG. 2 considering a particular composition of the conditioning filter H.sub.c.sup.i.

(67) The conditioning filter H.sub.c.sup.i used in this particular embodiment is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.i and two narrow stop-band filters (SB) centered at {circumflex over ()}.sub.n1.sup.i1 and {circumflex over ()}.sub.n1.sup.i+1 for i=2 . . . N1, and for the particular case where i=1 it is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.i and one narrow stop-band filter (SB) centered at {circumflex over ()}.sub.n1.sup.2, and for the particular case where i=N it is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.N and one narrow stop-band filter (SB) centered at {circumflex over ()}.sub.n1.sup.N1.

(68) FIG. 3 shows in each of the different steps c) performed for each of the elastic modes i how the countered-loop from i=1 to i=N affects the present filter.

(69) As it can be observed, step c) is affected every time by the corresponding natural frequencies estimate of the mode i and the modes i1, i+1.

(70) FIG. 4 shows the diagram shown in FIGS. 2 and 3 considering a particular composition of the conditioning filter H.sub.c.sup.i.

(71) The conditioning filter H.sub.c.sup.i used in this particular embodiment is a combination of a narrow band-pass filter (BP) centered at {circumflex over ()}.sub.n1.sup.i and N1 narrow stop-band filters (SB) centered at {circumflex over ()}.sub.n1.sup.q being q.sup.+: {q[1, N]|qi}.

(72) FIG. 4 shows in each of the different steps c) performed for each of the elastic modes i how the countered-loop from i=1 to i=N affects the present filter H.sub.c.sup.i.

(73) As it can be observed, step c) is affected every time by the corresponding natural frequencies estimate of the mode i and the mode q being q.sup.+: {q[1, N]|qi}.

Particular Example

(74) FIG. 5 shows a flying boom of a tanker aircraft (A330 MRTT) comprising two active measurement sources, which are sensor A and sensor B. The flying boom comprises elastic mode shapes and undamped elastic mode frequencies which experience great variations during transitions from free air to coupled condition and vice versa due to the discrete changes in the exogenous boundary conditions acting on the structure.

(75) For testing the method of the present invention, the method has been applied on the flying boom Control Laws, obtaining as results the adaptive cancellation in real time of the first two elastic modes in the discrete-time signals which measure the dynamics of the flying boom structure, therefore being the parameter N=2.

(76) The present method can be implemented as part of the Control Laws of any flexible system with more than one sensor located along the structure. Additionally, the present method can be used not only to filter the elastic component in the Control Laws feedback signals, but also to control the elastic dynamics by using parallel control laws with a feedback composed by the elastic modes displacements identified by the estimation kernels, or simply by using the discrete-time signal y.sup.AY.sup.output and its derivatives as control variables.

(77) The present invention includes a non-transitory computer-readable medium having computer-executable program instructions stored thereon, comprising instructions for the implementation, by a processor of a computing device, of the method described above, when the program is executed by the processor.

(78) While at least one exemplary embodiment of the present invention(s) is disclosed herein, it should be understood that modifications, substitutions and alternatives may be apparent to one of ordinary skill in the art and can be made without departing from the scope of this disclosure. This disclosure is intended to cover any adaptations or variations of the exemplary embodiment(s). In addition, in this disclosure, the terms comprise or comprising do not exclude other elements or steps, the terms a or one do not exclude a plural number, and the term or means either or both. Furthermore, characteristics or steps which have been described may also be used in combination with other characteristics or steps and in any order unless the disclosure or context suggests otherwise. This disclosure hereby incorporates by reference the complete disclosure of any patent or application from which it claims benefit or priority.