Patent Application: US-8181098-A

Abstract:
a method and system for attenuating the effects of unknown , unmeasurable and time - varying exogenous disturbances on multiple - input multiple - output dynamical systems are described . the disturbance rejection system is characterized in terms of an armarkov or predictive model controller . the parameters of this controller are revised in real time at discrete time steps so as to generate an input to the dynamical system that attenuates the effect of the exogenous disturbance on any chosen set of measured outputs of the dynamical system . the method for revising the controller parameters involves the steps of defining a novel retrospective cost function based on windows of past data , calculating a gradient that is based on this cost function , and using an implementable adaptive step size that brings the controller parameters closer to optimal controller parameters after each revision . the method and system are applicable to active noise and vibration control and reject single - tone , multi - tone , sine sweeping and broadband disturbances in acoustic spaces .

Description:
to begin , we describe the armarkov model of the nth - order discrete - time finite - dimensional linear time - invariant system given by where a , b , c and d are real matrices of appropriate size , u ( k ) is of size m u and y ( k ) is of size l y , and whose markov parameters h j of size l y × m u are defined as h j  = δ  d , for   j = - 1 , ( 3 )  = δ  ca j  b ,  for   j ≥ 0 . ( 4 ) this system ( 1 ), ( 2 ) may be alternatively described by the auto - regressive moving average ( arma ) representation given by y ( k )=− a 1 y ( k − 1 )− . . . − a n y ( k − n )+ b 0 u ( k )+ . . . + b n u ( k − n ), ( 5 ) or the μ - armarkov ( arma + markov ) model or μ step ahead predictor model y  ( k ) =  ∑ j = 1 n  - α j  y  ( k - μ - j + 1 ) +  ∑ j = 1 μ  h j - 1  u  ( k - j + 1 ) +  ∑ j = 1 n  b j  u  ( k - μ - j + 1 ) , ( 6 ) where α j are scalars and b j are of size l y × m u , j = 1 , . . . , n . we note that in the special case μ = 1 , the armarkov form ( 6 ) is the same as the arma form . now , let p denote the data window length and define the extended measurement vector y ( k ) of size l y p and the armarkov regressor vector φ yu ( k ) of size l y ( p + n − 1 )+ m u ( μ + p + n − 1 ) by y  ( k )  = δ  [ y  ( k ) ⋮ y  ( k - p + 1 ) ] ,  φ yu  ( k )  = δ  [ y  ( k - μ ) ⋮ y  ( k - μ - p - n + 2 ) u  ( k ) ⋮ u  ( k - μ - p - n + 2 ) ] . ( 7 ) where the block - toeplitz armarkov weight matrix w yu of size pl y ×[ l y ( p + n − 1 )+ m u ( μ + p + n − 1 )] is defined by w yu  = δ  [ - α 1  i i y … - α n  i t y 0 t y ⋯ 0 i y h - 1 ⋯ h μ - 2 b 1 ⋯ b n 0 t y × m u ⋯ 0 t y × m u 0 t y ⋰ ⋰ ⋰ ⋮ 0 t y × m u ⋰ ⋰ ⋰ ⋰ ⋰ ⋮ ⋮ ⋰ ⋰ ⋰ 0 t y ⋮ ⋰ ⋰ ⋰ ⋰ ⋰ 0 t y × m u 0 t y ⋯ 0 t y - α 1  i t y ⋯ - α n  i t y 0 t y × m u ⋯ 0 t y × m u h - 1 ⋯ h μ - 2 b 1 … b n ]  . ( 9 ) we now develop the armarkov / toeplitz model of the two vector input , two vector output plant with sensors and actuators whose inputs are the disturbance w ( k ) and the control u ( k ), and whose outputs are the feedback measurement y ( k ) and the performance measurement z ( k ) as shown in fig1 . the armarkov form of the plant with actuators and sensors is z  ( k ) =  ∑ j = 1 n  - α j  z  ( k - μ - j + 1 ) + ∑ j = 1 μ  h zw , j - 2  w  ( k - j + 1 ) +  ∑ j = 1 n  b zw , j  w  ( k - μ - j + 1 ) + ∑ j = 1 μ  h zu , j - 2  u  ( k - j + 1 ) +  ∑ j = 1 n  b zu , j  u  ( k - μ - j + 1 ) , ( 10 ) y  ( k ) =  ∑ j = 1 n  - α j  y  ( k - μ - j + 1 ) + ∑ j = 1 μ  h yw , j - 2  w  ( k - j + 1 ) +  ∑ j = 1 n  b yw , j  w  ( k - μ - j + 1 ) + ∑ j = 1 μ  h yu , j - 2  u  ( k - j + 1 ) +  ∑ j = 1 n  b yu , j  u  ( k - μ - j + 1 ) , ( 11 ) where α j are scalars , b zw , j and h zw , j are of size l z × m w , b zu , j and h zu , j are of size l z × m u , b yw , j and h yw , j are of size l y × m w , and b yu , j and h yu , j are of size l y × m u . next , define the extended performance measurement vector z ( k ), the extended feedback measurement vector y ( k ) and the extended control vector u ( k ) by z  ( k )  = δ  [ z  ( k ) ⋮ z  ( k - p + 1 ) ] , y  ( k )  = δ  [ y  ( k ) ⋮ y  ( k - p + 1 ) ] ,  u  ( k )  = δ  [ u  ( k ) ⋮ u  ( k - p + 1 ) ] , ( 12 ) where the controller window size p c is given by μ + n + p − 1 , and the armarkov regressor vectors φ zw ( k ) and φ yw ( k ) are defined by φ zw  ( k )  = δ  [ z  ( k - μ ) ⋮ z  ( k - μ - p - n + 2 ) w  ( k ) ⋮ w  ( k - μ - p - n + 2 ) ] ,  φ yw  ( k )  = δ  [ y  ( k - μ ) ⋮ y  ( k - μ - p - n + 2 ) w  ( k ) ⋮ w  ( k - μ - p - n + 2 ) ] . ( 13 ) furthermore , define the block - toeplitz armarkov weight matrices w zw of size pl z ×[( n + p − 1 ) l z +( μ + n + p − 1 ) m w ] and w yw of size pl y ×[( n + p − 1 ) l y +( μ + n + p − 1 ) m w ] by b zu  = δ  [ - α 1  i l z … - α n  i l z 0 l z ⋯ 0 l z h zw , - 1 ⋯ h zw , μ - 2 b zw , 1 ⋯ b zw , n 0 l z × m w ⋯ 0 l z × m w 0 l z ⋰ ⋰ ⋰ ⋮ 0 l z × m w ⋰ ⋰ ⋰ ⋰ ⋰ ⋮ ⋮ ⋰ ⋰ ⋰ 0 l z ⋮ ⋰ ⋰ ⋰ ⋰ ⋰ 0 l z × m w 0 l z ⋯ 0 l z - α 1  i l z ⋯ - α n  i l z 0 l z × m w ⋯ 0 l z × m w h zw , - 1 ⋯ h zw , μ - 2 b zw , 1 … b zw , n ] , ( 14 ) w yw  = δ  [ - α 1  i l y … - α n  i l y 0 l y ⋯ 0 l y h yw , - 1 ⋯ h yw , μ - 2 b yw , 1 ⋯ b yw , n 0 l y × m w ⋯ 0 l y × m w 0 l y ⋰ ⋰ ⋰ ⋮ 0 l y × m w ⋰ ⋰ ⋰ ⋰ ⋰ ⋮ ⋮ ⋰ ⋰ ⋰ 0 l y ⋮ ⋰ ⋰ ⋰ ⋰ ⋰ 0 l y × m w 0 l y ⋯ 0 l y - α 1  i l y ⋯ - α n  i l y 0 l y × m w ⋯ 0 l z × m w h yw , - 1 ⋯ h yw , μ - 2 b yw , 1 … b yw , n ] , ( 15 ) and the block - toeplitz armarkov control matrices b zu of size pl z × p c m u and b yu of size pl y × p c l u by b zu  = δ  [ h zu , - 1 ⋯ h zu , μ - 2 b zu , 1 ⋯ b zu , 1 0 l z × m u ⋯ 0 l z × m u 0 l z × m u ⋰ ⋰ ⋰ ⋰ ⋰ ⋮ ⋮ ⋰ ⋰ ⋰ ⋰ ⋰ 0 l z × m u 0 l z × m u ⋯ 0 l z × m u h zu , - 1 ⋯ h zu , μ - 2 b zu , 1 ⋯ b zu , n ] , ( 16 ) b yu  = δ  [ h yu , - 1 ⋯ h yu , μ - 2 b yu , 1 ⋯ b yu , 1 0 l y × m u ⋯ 0 l y × m u 0 l y × m u ⋰ ⋰ ⋰ ⋰ ⋰ ⋮ ⋮ ⋰ ⋰ ⋰ ⋰ ⋰ 0 l z × m u 0 l y × m u ⋯ 0 l y × m u h yu , - 1 ⋯ h yu , μ - 2 b yu , 1 ⋯ b yu , n ] . ( 17 ) z ( k )= w zw φ zw ( k )+ b zu u ( k ), ( 18 ) y ( k )= w yw φ yw ( k )+ b yu u ( k ), ( 19 ) next , we formulate an adaptive disturbance rejection feedback algorithm for the system represented by ( 18 ) and ( 19 ). we use a strictly proper controller g c in armarkov form of order n c with μ c markov parameters , so that , analogous to ( 6 ), the control input u ( k ) is given by u  ( k ) =  ∑ j = 1 n c  - α c , j  ( k )  u  ( k - μ c - j + 1 ) +  ∑ j = 1 μ c - 1  h c , j - 1  ( k )  y  ( k - j + 1 ) +  ∑ j = 1 n c  b c , j  ( k )  y  ( k - μ c - j + 1 ) , ( 20 ) where the controller markov parameter h c , j is of size m u × l y . next , define the controller parameter block vector θ ( k ) by θ  ( k )  = δ  [ - α c , 1  ( k )  i m u ⋯ - α c , n c  ( k )  i m u h c , 0  ( k ) ⋯ h c , μ c - 2  ( k ) b c , 1  ( k ) ⋯ b c , n c  ( k ) ] , ( 21 ) where θ ( k ) is of size m u ×[ n c m u +( n c + μ c − 1 ) l y ]. now from ( 12 ) and ( 20 ) it follows that u ( k ) and u ( k ) are given by u ( k )= θ ( k ) r 1 φ uy ( k ) ( 22 ) and u  ( k ) = ∑ i = 1 p c  l i  θ  ( k - i + 1 )  r i  φ uy  ( k ) , ( 23 ) where φ uy  ( k )  = δ  [ u  ( k - μ c ) ⋮ u  ( k - μ c - n c - p c + 2 ) y  ( k - 1 ) ⋮ y  ( k - μ c - n c - p c + 2 ) ] , ( 24 ) and where l i  = δ  [ 0 ( i - 1 )  m u × m u i m u 0 ( p c - i )  m u × m u ] ( 25 ) is of size p c m u × m u with i m u denoting the identity matrix of size m u , and r i  = δ  [ 0 q1 × ( i - 1 )  m u i q1 × q1 0 q1 × ( p c - i )  m u 0 q1 × ( i - 1 )  l y 0 q1 × q2 0 q1 × ( p c - i )  l y 0 q2 × ( i - 1 )  m u 0 q2 × q1 0 q2 × ( p c - i )  m u 0 q1 × ( i - 1 )  l y i q2 × q2 0 q2 × ( p c - i )  l y ] ( 26 ) is of size [ n c m u +( n c + μ c − 1 ) l y ]×[( n c + p c − 1 ) m u +( n c + μ c + p c − 2 ) l y ], with q 1 n c m u and q 2 ( n c + μ c − 1 ) l y . thus , from ( 18 ) and ( 23 ) we obtain z  ( k ) = w zw  φ zw  ( k ) + b zu  ∑ i = 1 p c  l i  θ  ( k - i + 1 )  r i  φ uy  ( k ) . ( 27 ) next , we describe the update law for the controller parameter block vector θ ( k ). to do this , we define a retrospective performance cost function that evaluates the performance of the controller obtained from the current value of θ ( k ) based upon the measurements of the system during the previous p c steps . therefore , we define the estimated performance { circumflex over ( z )}( k ) by z ^  ( k )  = δ  w zw  φ zw  ( k ) + b zu  ∑ i = 1 p c  l i  θ  ( k )  r i  φ uy  ( k ) , ( 28 ) which has the same form as ( 27 ) but with θ ( k − i + 1 ) replaced by the current controller parameter block vector θ ( k ). using ( 28 ) we define the retrospective performance cost function j ( k )= ½ { circumflex over ( z )} t ( k ) { circumflex over ( z )} ( k ), ( 29 ) with “ t ” denoting the transpose of a vector . next , the gradient of j ( k ) with respect to θ ( k ) is given by ∂ j  ( k ) ∂ θ  ( k ) = ∑ i = 1 p c  l i t  b zu t  z ^  ( k )  φ uy t  ( k )  r i t . ( 30 ) since w ( k ) is not available , which implies that φ zw ( k ) is unknown , { circumflex over ( z )}( k ) cannot be calculated from ( 28 ). however , it follows from ( 18 ) and ( 28 ) that z ^  ( k ) = z  ( k ) - b zu  ( u  ( k ) - ∑ i = 1 p c  l i  θ  ( k )  r i  φ uy  ( k ) ) , ( 31 ) the gradient ( 30 ) is used in the update law θ  ( k + 1 ) = θ  ( k ) - η  ( k )  ∂ j  ( k ) ∂ θ  ( k ) , ( 32 ) where η ( k ) is the adaptive step size . to determine the adaptive step size η ( k ), we assume that there is a controller parameter block vector θ * that minimizes j ( k ) for all k . the method does not need to know θ *. now , we define the desired performance z ^ *  ( k )  = δ  w zw  φ zw  ( k ) + b zu  ∑ i = 1 p c  l i  θ *  r i  φ uy  ( k ) , ( 33 ) and the performance error ɛ  ( k )  = δ  z ^ *  ( k ) - z ^  ( k ) . ( 34 ) our goal is to determine η ( k ) such that θ ( k ) moves closer to θ * after each update . for convenience , we define the optimal adaptive step size η opt  ( k )  = δ   ɛ  ( k )  2 2  ∂ j  ( k ) ∂ θ  ( k )  f 2 , ( 35 ) where ∥ ∥ f denotes the matrix frobenius norm and ∥ ∥ 2 denotes the vector euclidean norm . it is shown in reference 16 that a geometrical interpretation of the procedure detailed above is now presented . using fig2 for reference , the objective of the algorithm is to move the controller parameter block vector θ ( k ) closer to the optimal controller parameter block vector θ *. the direction in which to move is the negative of the gradient ∂ j  ( k ) ∂ θ  ( k ) which is obtained from the retrospective performance cost function . the distance to move at each time step is determined by the adaptive step size η ( k ). it is shown that the step size η opt ( k ) moves θ ( k ) to the point closest to θ * along the negative gradient direction , that is , to a point such that the vectors e ( k + 1 ) and - ∂ j  ( k ) ∂ θ  ( k ) in practice , η opt ( k ) is not computable since ε ( k ) is not available from sensor measurements . the crucial innovative feature of the method in this invention is the use of an implementable adaptive step size which can be calculated from available data and is guaranteed to be within the range ( 37 ) that mathematically demonstrates that θ ( k ) moves closer to θ *. three such step sizes are given below : η 1  ( k )  = δ  1 ( [ ∑ i = 1 p c   r i  φ uy  ( k )  2  σ _  ( b zu  l i ) ] ) 2 , ( 38 ) η 2  ( k )  = δ  1  φ uy  ( k )  2 2  [ ∑ i = 1 p c  σ _  ( b zu  l i ) ] 2 , ( 39 ) n 3  ( k )  = δ  1 p c   b zu  f 2   φ uy  ( k )  2 2 ( 40 ) where { overscore ( σ )}( b zu l i ) denotes the maximum singular value of the matrix b zu l i . note that if b zu is known , then η 1 ( k ), η 2 ( k ) or η 3 ( k ) can be calculated and used to implement ( 32 ). other implementable adaptive step sizes satisfying ( 37 ) may be obtained . the step sizes η 1 ( k ), η 2 ( k ) and η 3 ( k ) satisfy the steps involved in implementing the adaptive algorithm are as follows : 0 . obtaining the matrix b zu using the identification algorithm of reference 14 , 15 or by calculating from an arma or state space representation of g zu . 1 . calculating the control signal u ( k ) from the controller parameter block vector θ ( k ) and the vector φ uy ( k ) using ( 20 ). 2 . using the signals u ( k ), z ( k ) and y ( k ) updating the estimated performance vector { circumflex over ( z )}( k ) as defined in ( 31 ). 3 . calculating the retrospective gradient ∂ j  ( k ) ∂ θ  ( k ) 4 . calculating an implementable adaptive step size such as η 1 ( k ), η 2 ( k ) or η 3 ( k ) from ( 38 ), ( 39 ) or ( 40 ). steps 1 through 5 are performed at each time step k . experimental demonstration of the armarkov adaptive disturbance algorithm for active noise control is performed on an acoustic duct of circular cross section . the duct is 80 inches long and has a diameter of 4 inches . the disturbance speaker ( w ) is located at one end of the duct and the measurement sensor ( y ), a microphone , is located 4 inches in from the same end of the duct . the performance sensor ( z ), a microphone , is positioned 6 inches in from the other end . alternative sensors for vibration control are accelerometers and piezo - electric sensors . the control actuator ( u ), a speaker , is placed 16 inches in from that end of the duct . a servovalve for flow modulation of compressed air is another form of actuation for noise control while proof mass actuators can be used for vibration control . the signals from the two microphones are amplified by a dbx 760x microphone preamplifier while the control signal is amplified by an alesis ra - 100 amplifier . both speakers are radio shack 6 inch woofers . a graphical representation of the experimental set - up is shown in fig3 . the algorithm is tested on four types of disturbances , namely , a single - tone disturbance ( 139 . 65 hz ), a two - tone disturbance ( 135 . 74 hz and 160 . 4 hz ), band - limited white noise ( up to 390 hz ) and am radio noise . the algorithm uses n = 4 and μ = 12 for the secondary path matrix b zu , and n c = 2 , μ c = 10 and p = 2 for control . the controller is implemented on a dspace ds1102 real time board running a tms320c30 dsp processor at a sampling frequency of 800 hz . the microphone signals are processed through an ithaco dl 4302 low pass filter that rolls off at 315 hz . the tonal and band - limited white noise disturbances are generated by a stanford research systems 770 fft network analyzer and amplified by an optimus sta - 825 stereo receiver . fig4 shows the acoustic response with the disturbance rejection system inactive (“ open - loop ”) and with the disturbance rejection system active (“ closed - loop ”) with a single - tone disturbance . disturbance attenuation of more than 40 db is achieved with convergence in about 1 second . the system and method provide the same level of attenuation by adaptation when the frequency of the disturbance tone is changed , as in sine sweeps , while the system is active . fig5 shows the open - loop and closed - loop performance with a two - tone disturbance . in this case , disturbance attenuation of more than 35 db is observed . fig6 shows the open - loop and closed - loop magnitude plots of the transfer function from disturbance to performance with a white noise disturbance , and noise suppression of up to 15 db is observed over a frequency range from 0 to 300 hz . finally , fig7 shows the open - loop and closed - loop frequency response with an am radio disturbance . noise reduction levels of up to 40 db are observed over the frequency range 0 to 300 hz . in contrast and improvement to the prior art , the present method has three innovative features . the first is the use of armarkov / toeplitz structures for describing both the plant and controller . while these structures have been used for predictive and neural net control as described in references 3 , 4 and 13 , the present method uses them in retrospective fashion to obtain a controller update law that learns from past data . the second innovation is the definition of the retrospective cost function and calculation of the gradient with respect to this cost function . in the prior art , instantaneous or predicted cost functions are used . the third innovation is the use of an implementable adaptive step size for the controller update which guarantees that the controller parameters move closer to the unknown optimal controller parameters at each time step . having described the invention , many modifications thereto will become apparent to those skilled in the art to which it pertains without deviation from the spirit of the invention as defined in the appended claims . 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