On the Use of A Priori Knowledge in Adaptive Inverse Control

Transcription

1 54 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 47, NO 1, JANUARY 2000 On the Use of A Priori Knowledge in Adaptive Inverse Control August Kaelin, Member, IEEE, and Daniel von Grünigen Abstract In adaptive inverse control the coefficients of the controller are adaptively adjusted by an algorithm which is controlled by the input and the error signal Using a priori knowledge about the plant, the controller can be splitted into a long fixed and into a short adaptive filter The controller can be made more efficient by feeding back the error signal only in a desired frequency range A modified objective function allows the minimization of the filtered squared error plus the norm of the controller The technique is demontrated an active noise control (ANC) system where it suppresses the sound pressure level by more than 20 db(a) Index Terms Active noise control, adaptive filters, adaptive inverse control, filtered error LMS algorithm, root locus method Fig 1 Basic concept of adaptive inverse control I INTRODUCTION ADAPTIVE inverse control techniques have been succesfully applied to a variety of control problems [1], [2] The system of Fig 1 illustrates the basic concept of adaptive inverse control The aim of such a system is to cause the plant output to track the output of the reference model If the plant is minimum phase and the reference model has a transfer function equal to one, then the transfer function of the ideal controller is the inverse of the plant In order to find a realizable controller even if the inverse does not exist, an objective function has to be considered In considering, eg, the mean square error, such a controller can be found adaptively as indicated in Fig 1 The controller, which is usually designed as an adaptive transversal filter, may become very long, requiring hundreds of taps This turns the adaptation algorithm slow and increases the computational costs To overcome this problem, we propose to split the controller into a long fixed part and into a short adaptive part, as depicted in Fig 2 The coefficients of the fixed controller are computed in an offline procedure before the adaptive system goes into operation The adaptive controller has to equalize only variations of the plant that occur during operation There are many practical situations where only a part of the error spectrum is relevant An example of such a situation is an active noise control (ANC) system, where only the audible part of the spectrum is perceived as error signal The additional a-filter passes only that part of the error signal to the adaptation Manuscript received January 26, 1999 This paper was recommened by Associate Editor L Fortuna A Kaelin was the Signal and Information Processing Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland He is now the Siemens AG, Zurich, Switzerland ( kaelin@isieeethzch) D von Grünigen is the University of Applied Sciences, Burgdorf, Switzerland ( danielvongruenigen@hta-bubfhch) Publisher Item Identifier S (00)00725-X Fig 2 Block diagram of an adaptive inverse control system using a priori knowledge incorporated in the blocks ^c and a algorithm, which is meaningful for a certain application The optimal designed controller then minimizes the squared error only in a desired frequency range in cost of an irrelevant error power spectrum The conventional objective is to minimize the mean squared error (MSE) In doing so, the norm of the controller can grow considerably, especially when the the plant is ill conditioned In the next section, a controller is derived based on an appropriately modified objective function We call this solution the modified Wiener solution In Section III, we split the controller into a fixed and into an adaptive part We will show how the fixed controller and the initial adaptive controller can be computed and derive an adaptation algorithm for the adaptive part We will prove its convergence in the mean, if the step size is chosen properly In Section V we finally present two examples: a low-order example to verify the theory and a real ANC system to show its usefulness in practice A Notation, Preliminaries All signals are real discrete-time signals denoted, eg, by the time index A signal vector of context-dependent length is defined by (1) /00$ IEEE

2 KAELIN AND VON GRÜNIGEN: ON THE USE OF A PRIORI KNOWLEGE IN ADAPTIVE INVERSE CONTROL 55 Here, a system is linear and defined by its impulse response, eg, Such an impulse response is assumed to be of finite length and described by the corresponding coefficient vector The convolution of two coefficient vectors, eg, and,isa vector of length, denoted by (2) range as mentioned in the introduction The second term prevents a prohibitive growth of the controller coefficients if the overall system is ill conditioned, as explained below The positive-valued paramter weights the squared controller norm Its choice is a compromise: a small value yields a small MSE of at the cost of a possibly high controller norm The optimum controller is denoted as modified Wiener solution, In deriving this solution, we express as The convolution of an impulse response is defined by (3) the signal (4) as depicted in Fig 3 Due to linearity and and time invariance, the convolution operators are commutative, which allows the formulation (9) (10) With, and relations (5) and (6), (10) can now be rewritten as (11) With product, ie,, this convolution sum can be written as a scalar (5) Assuming to be shorter than, it has to be appropriately augmented zeros With and uncorrelated, the objective function (8) yields Convolving the impulse response the signal results in the convolution matrix of a cascaded system (6) (7) II MODIFIED WIENER SOLUTION Referring to the inverse control system in Fig 2, we assume the independent noise processes and to be stationary and zero mean The available signal respresents the output of the plant disturbed by measurement noise The desired signal is generated by a model driven by the input signal The plant itself is driven by the controller output The overall controller is itself driven by the input signal The available signal is subtracted from, yielding the error signal An error filter yields which finally adapts the controller In deriving the optimal controller for the inverse control system in Fig 2, we start by discussing the optimum solution for the overall controller Refering to Fig 3, the here considered objective function is (8) (12) where controller coefficient vector; convolution matrix containing the vector analog to (7); coefficient vector of the error filter; length noise data vector at the output of the plant In proceeding, is derivated respect to Setting the derivative to zero yields (13) For a stationary input process we define the autocorrelation matrix as Hence (14) We note here that the normal equation (14) can be solved by means of the so-called Levinson algorithm [4] due to the Toepliz structure of the matrix to be inverted Formally, the optimum overall controller, here denoted as modified Wiener solution, is given by (15) The minimum value of the objective function can be otained by inserting into (12): the mean square of the filtered error signal,, modified the square of the controller norm [3] Filtering the error signal allows an appropriate weigthing of the relevant frequency

3 56 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 47, NO 1, JANUARY 2000 iterative manner by alternately minimizing the objective function (18) respect to and In this way the nonlinear optimization problem is reduced to the solution of linear matrix equations In doing so, we first minimize (18) respect to Analog to (14), this results in the following normal equation in the th step: Fig 3 Inverse control system one controller It finally yields (16) The minimum value of is given by the average power of the -filtered output noise, This value is achieved only when, ie, when the controlled plant,, is equal to the model, In general, by solving (14), this minimum value cannot be achieved III DESIGN OF THE SPLITTED CONTROLLER In many applications, such as ANC, a controller hundreds of taps is required The optimal controller (15) is based on the knowledge of the systems and of the statistics of the input signal For an unknown plant this controller might be replaced by an adaptive filter Making all taps adaptive results in a controller which converges very slowly and requires considerable computational power However, in many situations, the plant is relatively constant and varies only slowly, eg, due to temperature variations For such a situation, we propose to split the controller into a cascade of a short adaptive filter and a long fixed filter,as depicted in Fig 2 The fixed filter is designed offline based on the available a priori knowledge of the plant, the model, the weighting filter, and the signal statistics During operation the short adative filter tracks small changes of the plant and variations in the signal statistics For the splitted controller, we propose to modify the objective function (8) as follows: (17) Minimizing in (17) optimizes the fixed filter and the initial adaptive filter In order to make these filters unique, we normalize the initial adaptive controller The optimization problem can now be stated as follows: (18) subject to (19) an appropriately chosen constant Equations (18) and (19) constitute a constrained minimization problem which could be solved the method of Lagrange multipliers [5] As a computationally simpler alternative we propose to solve it in an (20) the convolution matrix according to (7) containing the vector Due to the Toeplitz structure of the normal equation, the numerically robust and efficient Levinson algorithm [4] can be used for the solution Substituting the solution in (18), a similar normal equation can be obtained for the (21) convolution matrix containing the vector In order to fullfil constraint (19), we finally normalize the th iteration as follows: and To initialize the procedure, we set We stop after, eg, ten iterations, or after the measure is close to its minimum IV ADAPTATION ALGORITHM (22) (23) (24) Once and are determined, we are seeking for an algorithm which adapts to the slowly changing plant and/or to the signal statistics A Derivation of the Adaptation Algorithm The task of the adaptation algorithm is to minimize the objective function (8), here rewritten as (25) (26) According to Fig 2, the controller is splitted as discussed in Section III Substituted in (10) results for the filtered error signal in (27)

4 KAELIN AND VON GRÜNIGEN: ON THE USE OF A PRIORI KNOWLEGE IN ADAPTIVE INVERSE CONTROL 57 Assuming a slowly changing adaptive controller, the convolution operators are still considered to be commutative, ie, In the following analysis, we will search for the condition under which (36) is fullfilled In doing so, we first rewrite (27) as or Using (5), the convolution operators are replaced by scalar products, yielding (37) With and as illustrated in Fig 4, we obtain in vector notation (29) In order to derive an algorithm which is based on the gradient of the objective function (25), we derivate it respect to With the help of (29), this yields (30) The method of steepest descent [6] recursively updates the coefficient vector the help of the gradient (30) as follows: Introducing its transpose in (35) results (34) in (38) (39) (31) denoting the step size An LMS-like algorithm can be obtained by replacing the gradient (30) its instantaneous estimate, ie, (32) We note that the update of is performed the help of the -filtered error signal (see Fig 2) Hence, we call it the filtered-error LMS algorithm, in contrast to the known filteredor filtered- LMS algorithm [2] B Convergence of the Mean of the Adaptive Controller Coefficient Vector Minimizing the objective function in (25) respect to yields, in a procedure similar to Section II, the following modified Wiener solution: (33) The vector is the optimal controller vector for the system consisting of the vectors,,, and for the correlation matrix of the input signal The system is described by the convolution matrix based on and the coefficient vector Now we introduce the weight error vector (34) It is noted that the vector of length can be written as (40) the two vectors and of appropriate length Similarly (41) With these relations and the independence assumption [6] it follows for the mean of (39) (42) Here denotes the input correlation matrix for an input vector of length In proceeding, we introduce the Wiener solution (33) in the above equation yielding and subtract from both sides of (32), yielding (35) (43) A necessary condition for a stable adaptive inverse system is the mean of the weight error vector converging to the zero vector (36) Defining the constants for else (44)

5 58 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 47, NO 1, JANUARY 2000 Fig 4 Adaptive inverse control system, simplified for analysis purpose simplifies the weight error differerence equation to can be de- According to [6], the autocorrelation matrix composed into (45) Fig 5 Example of a root locus A(z) := z (z 0 1), B(z) := (z 0 z )(z 0 z ), and > 0 the characteristic equations are given as (54) (46) an orthonormal matrix, ie,, and a diagonal matrix containing the positive real-valued eigenvalues of (36) With this orthogonal transformation the recursive equation (45) can be rewritten as follows: (47) where Here, the decouple from the off-diagonal terms and so we have Now, the convergence condition can be guaranteed if the zeros of the characteristic equations are in the unit circle The dependency of these zeros on the parameter is described the root locus An example is depicted in Fig 5 for For the zeros of (54) are the zeros of, ie, for else (55) If the branch starting from goes the the left, ie, its argument is equal to, then there is a stability range According to root locus theory [7] we find for (56) (48) In proceeding, (48) are transformed into the characteristic equations -domain yielding (49) or (57) By introducing the parameter and the polynomials else (50) (51) (52) where are the zeros of The coefficients of are all real and nonnegative (see (44) and (53) With this condition, it can easily be shown, that the real zeros are all nonpositive Conjugate complex and nonpositive zeros turn the sum of arguments in the last step of (57) to zero, ie, The boundary in (56) is the smallest postive value corresponding to roots on the unit circle These values can be computed the Theorem given in the Appendix It allows to be written as (58) (53)

6 KAELIN AND VON GRÜNIGEN: ON THE USE OF A PRIORI KNOWLEGE IN ADAPTIVE INVERSE CONTROL 59 (59) The elements are given by being real zeros of the polynomial in the range Itis defined by (60) (61) and a second kind Chebyshev polynomial of order as described in the Appendix The stability range for the step size is now given (51) as Fig 6 Behavior of the filtered-error LMS algorithm a precise estimation of the plant (above) and an erroneous estimation of the plant (below) (62) where is the maximum eigenvalue of the input correlation matrix V EXAMPLES A A Simple Adaptive Inverse Control System In order to verify the effectiveness of the filtered-error LMS algorithm we simulated the proposed splitted inverse controller (see Fig 2) for the plant 1 The error weighting filter and the model have coefficient vectors, and, respectively The input signal and the plant noise are zero mean white signals variances and, respectively The remaining parameters were chosen as follows Adaptation step, weight parameter of the controller, desired norm of the adaptive controller, length of the fixed controller, and length of the adaptive controller The fixed controller has been found by iteratively applying (20) (22) We stopped after ten iterations Fig 6 shows the behavior of the adaptive algorithm (32) initialized The objective function (see (25)) has been averaged over 100 runs In Fig 6 above, the true plant is assumed to be known for the computation of the fixed controller and the adaptation process of the controller In Fig 6 below, we show the robustness of the adaptation algorithm by introducing a 50% error in the true plant coefficients by adding white noise of variance As shown in Fig 6 the introduced plant-error delays the convergence and the adapted controller has now an error in the range of 005% Even the observed robustness of the filterederror LMS algorithm due to plant estimation errors has not been proven, it has been observed in many practical examples Fig 7 ANC system Such an example is given in the following section B An Active Noise Control System Suppressing acoustic noise on the principle of destructive interference is a classic discipline in noise control The standard solution for this problem is an adaptive feedforward controller which drives an antisource [8] A practical example of such an active noise control (ANC) system is depicted in Fig 7 It consists of an experimental duct having a diameter of 30 cm and a length of 5 m driven by loudspeakers For experimental purpose, an adaptive inverse controller as discussed above has been realized on a floating-point DSP [9] The obtained results are based on recorded roots blower noise used as sound source signal Considering Fig 7, we can distinguish three electroacustic paths: coefficient vector characterizes the impulse response of the primary path from the input microphone to the error microphone, the secondary path from the controller output to the error microphone, and the feedback path from the controller output to the input microphone Using these three paths, the block diagram of the considered duct system is depicted in the dashed box in Fig 8 The path is an estimation of the feedback path and is assumed here to be equal to resulting in If we define the plant as 2 (63) 1 This is an approximation of a first-order lowpass filter the transfer function P(z) =1=(1 0 0:5z ): 2 The vector g is assumed to be zero-padded in order to have the length of f 3 m

7 60 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 47, NO 1, JANUARY 2000 Fig 8 Block diagram of the duct system plant noise and compensation feedback path incorporated Fig 10 Power spectrum of the measured error signal for roots blower noise (a) Without controller (b) With optimum fixed controller (c) With added adaptive controller Fig 9 Block diagram of the ANC system then our adaptive-inverse-control concept can be applied In doing so, we added a large fixed controller of length and a small adaptive controller of length, as depicted in Fig 9 Before the ANC system goes into operation, the fixed and the initial adaptive controller have to be computed off line This is done by first identifying the three acoustical paths using white noise as input signal Then the optimal controllers have been computed analog to the example in Section V-A Thereby, the filter has been used Such an A -filter is well known in acoustics [10] because it only weighs the error in the sensitive frequency range of a human listener During online operation, the adaptive controller is updated by the filtered-error LMS algorithm (32) the adaptation constant and the weighting coefficient Fig 10 shows the power spectrum of the error signal for the mentioned recorded roots blower noise and out controllers Using only the fixed controller, which has been optimized for white noise input, a suppression level of 21 db(a) was obtained Adding our small adapitve filter 32 taps, the controller adapts to the harmonic input signal by increasing the suppression level to 31 db(a) This practical example illustrates how available a priori knowledge can be incorporated into a large-order fixed controller Online derivations, such as changes in the input spectrum, can easily be adapted to by using a small-order FIR filter The proposed objective function consists of the expectation of the squared filtered error plus the weighted norms of the controllers A filtered error signal is used for the adaption It allows to efficiently minimize the error power in the frequency range of interest The splitting of the controller decreases the computational complexity and accelerates the rate of convergence The long fixed controller and the initial short adaptive controller can be computed the fast and stable Levinson algorithm, using available a priori knowledge of the system and the input signal Implemented for a practical ANC system, the new controller has demonstrated its excellent performance APPENDIX INTERSECTIONS OF THE ROOT LOCUS WITH THE UNIT CIRCLE Theorem: Assume the characteristic equation the real polynomials (64) (65) and root locus gain Let have at least one zero inside and at least one zero outside the unit circle, ie,, Then, the gains at which the root locus crosses the unit circle are given by (66) VI CONCLUSIONS A controller, consisting of an adaptive and a fixed part, has been developed for an adaptive inverse control system (67)

8 KAELIN AND VON GRÜNIGEN: ON THE USE OF A PRIORI KNOWLEGE IN ADAPTIVE INVERSE CONTROL 61 The polynomial is defined by (68) Considering, for, we rewrite the above equation as follows: (76) (69) Next, we introduce the definition (77) and a second kind Chebyshev polynomial of order Proof: We start by investigating the roots of the characteristic (64) on the unit circle, ie, This leads to where is a Chebyshev polynomial of the second kind [11, p 1054], given by or, equivalently (70) Replaced in (76) yields (78) Equating the real and the imaginary part yields (71) (72) (79) With, the square of the above expression results in the polynomial as given in (68) Now, in order to obtain the angles at which the root locus crosses the unit circle, we have to determine its real zeros, in the range Introduced in the first equation of (72), the gains (66) are finally obtained Solving the first equation for results in and replacing it in the second ACKNOWLEDGMENT The authors would like to thank C Steinebrunner and M Schwab for realizing a real time ANC system and M Schwab and I Oesch for drawing the figures (73) In proceeding, we use the well-known trigonometric relation and obtain which simplifies to (74) (75) REFERENCES [1] B Widrow and SD Stearns, Adaptive Signal Processing Englewood Cliffs, NJ: Prentice-Hall, 1985 [2] B Widrow and E Walach, Adaptive Inverse Control Englewood Cliffs, NJ: Prentice-Hall, 1996 [3] G C Goodwin and K S Sin, Adaptive Filtering, Prediction and Control Englewood Cliffs, NJ: Prentice-Hall, 1984 [4] G H Golub and C F van Loan, Matrix Computations Baltimore, MD: John Hopkins Univ Press, 1996 [5] G Strang, Introduction to Applied Mathematics Cambridge, MA: Wellesley-Cambridge, 1986 [6] S Haykin, Adaptive Filter Theory Englewood Cliffs, NJ: Prentice- Hall, 1996 [7] G S Moschytz, Linear Integrated Networks, Fundamentals New York, NY: Van Nostrand Reinholt, 1974 [8] S J Elliot and P A Nelson, Active noise control, IEEE Signal Processing Mag, vol 10, no 4, pp 12 35, Oct 1993 [9] A Kaelin, D Cachin, and F Egger, An adaptive feedforward a-weightoptimized controller for the active reduction of stochastic sound, in Proc Fourth Int Congr Sound and Vibration, vol 1, J M Crocker and N I Ivanov, Eds, St Petersburg, Russia, 1996, pp

9 62 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 47, NO 1, JANUARY 2000 [10] S J Yang and S J Elliot, Machinery Noise Measurement London, UK: Oxford Univ Press, 1989 [11] I S Gradshteyn and I M Ryzhik, Table of Integrals, Series and Products New York, NY: Academic, 1994 August N Kaelin received the diploma and the Dr ScTechn degrees in electrical engineering from the Swiss Federal Institute of Technology (ETH) in 1983 and 1990, respectively He has been at the Institute for Signal and Information Processing at ETH from 1983 to 1998, from 1992 to 1998 as an Assistant Professor for Signal Processing Since 1998 he has lead a research group in new technologies at Siemens AG, Zurich, Switzerland His general interests are in analog and digital filters, estimation and detection theory, and system identification His specific reseach ares include various aspects of adaptive filters and its application in communication systems, hearing aids, and ANC Daniel C von Grünigen was born in Saanen, Switzerland, in 1947 He received the diploma and the Dr Sc Techn degrees in electrical engineering from the Swiss Federal Institute of Technology (ETH) in 1976 and 1983, respectively Since 1983, he has been a Professor at the University of Applied Sciences in Burgdorf, Switzerland He has been an academic guest at the Institute for Signal and Information Processing at ETH in 1994 and 1995 His interests are in the field of digital signal processing, especially in adaptive systems