SIGNAL PROCESSING MEANS FOR IDENTIFYING SYSTEMS SUBJECT TO PERIODIC DISTURBANCES

Abstract: In a signal processing system for a plant such as a motor or machine, subject to disturbances which are approximately periodic and wherein a signal (T(t)) time related to the disturbances is available, the plant response is sampled in an ADC triggered by the signal (T(t)) in order to synchronise the sampling process with the cyclic disturbances, and a processing means derives from the sampled signals an estimate of the disturbances and thereby of that part of the plant output due to unknown inputs (Fig. 2), the estimate of disturbances possibly being used to drive an active vibration control system employed in combination with the sampling system.

Description: Title; Signal Processing Means for Identifying Systems Subject to Periodic Disturbances. Field of invention This invention relates to a signal processing system for characterising a plant subject to periodic or almost periodic forcing. The expression plant is employed herein to describe any- closed system such as an industrial plant, a motor vehicle, or aeroplane, or machine etc. Background to the invention In many situations a plant is required to produce some desired output, d_(t). The plant is controlled by inputs u(t) but the measured plant output £(t) is subject to disturbance y_(t) and noise n_(t), (see Figure 1). To achieve good control it is necessary to estimate a model or inverse-model of the plant as described in B. Widrow '& S.D. Stearns, "Adaptive Signal Processing", Prentice Hall Inc., 1985 for example. If the plant characteristics are time varying the plant model must be updated. There exist several methods for doing this as described in G. C. Goodwin & R.L. Payne, "Dynamic System Identification: Experiment Design and Data Analysis", Academic Press, 1977 and A.P. Sage & J.L. Melsa, "System Identification", Academic Press, 1971 for example, but they assume that the disturbances are stochastic ie random, and are not suitable for application where the disturbance is substantially periodic. Summary of the invention In a plant in which the disturbances (y_(t)) are substantially periodic and a signal (T(t)), which is time- related to the disturbance (y_(t)), is available, a signal processing system is provided to estimate the disturbances and to characterise the plant, wherein a sampling of the plant response is performed in an analogue to digital converter (ADC) and the signal T(t) is used as a trigger to synchronise the sampling process of the Analogue to Digital Converter. In particular the signal processing system estimates that part of the measured plant output which is due to unknown inputs and estimates the relationship between the plant output and the known plant inputs. The sampled output Y due to an input (u(t) is approximated by equation (3.1). In (3.1) (a,, a_, a,, ) is the sampled plant response. The first simplification occurs when the input to the plant u(t) is periodic. If u is samp•^led M times p_rer c_yrcle then u n+ . M , =u n so that Y n can be written as in equation (3.3). Hence, instead of needing to estimate the whole impulse response, a , we only need the M terms of α in equation (3.4). Also because the disturbance y(t) is periodic, it has only M independent terms after sampling. We introduce the vectors given by equations (3.5) and (3.6) and the matrix U with elements given by equation (3.7). U is a cyclic matrix, and the plant output £ can be written as given in equation (3.8). The noise, n(t), can be reduced by averaging over many cycles so that it can be ignored in the following description,. If the plant, the disturbance and the inputs are slowly varying compared with the cyclic variations, the output value of equation (3.8) at time t can be well approximated by equation (3.9). Alternatively, if the plant is constant but the period of the disturbance and the input is changing, the impulse response a(t) will be sampled at slightly different times so that a_ will change as the period changes. Thus the plant can again be approximated by (3.9). Therefore a constant plant subject to inputs of slowly changing periodicity can be modelled as a time-varying plant subject to periodic inputs, provided that the sampling of the ADC is synchronised by the external trigger signal T(t). This has considerable advantages over methods which take no account of periodicity. For example, i) Averaging can be used to reject the noise ιι(t) ii) The vector .a describes the plant response and contains far fewer parameters than would be required to characterise the whole impulse response. iii) The cyclic nature and low order of the equation (3.9) reduces the computational effort in the estimation of a_ and y_(t), particularly since transforms, such as the finite Fourier transform for example, can be used to diagonalise the matrix U. In order to estimate a_ and y_ we require that small changes are made to U on a time-scale shorter than the plant or periodicity variations, so that- (3.9) becomes the equation given at (3.10) where X(t) is given by expression (3.11) and b_ by (3.12) and I is the MxM identity matrix. The small changes to U can either be as the result of changing control signals or, to avoid ill-conditioning, the result of psuedo-random changes made by the processor. An orthogonal transform of (3.10) gives the expression shown in (3.13). In (3.13), n is the transform variable, which corresponds to the harmonic number if a finite Fourier transform is used, and x T a nd b are as given in (3.14) and (3.15). In (3.14) and (3.15) the superposed bar denotes the transform, and the superposed T denotes the transpose. The extension to a multichannel system with S inputs is given by expression (3.16) for each plant output i. The vectors on the right hand side are defined in (3.17) and (3.18). Expression (3.16) and the single channel version, (3.13), are in a form which allows the use of standard algorithms for parameter estimation. One solution is as given by (3.19) as amplified by (3.20) and (3.21). The solutions to (3.20) and (3.21) can be obtained recursively as in G.C. Goodwin 's- R.L. Payne, "Dynamic System Identification: Experiment Design and Data Analysis", Academic Press, 1977 and A.P. Sage ! & J.L. Melsa, "System Identification", Academic Press, 1971. One embodiment of the invention is shown in Figure 2. The system shown differs from previous systems in that the sampling in the ADCs is controlled by the external trigger signal T(t) which is time related to the disturbance y_(t). A preferred configuration employs transform and inverse- transform modules. The transform has the property that it diagonalises circular matrices such as that occuring in equation (3.8). The solution can be obtained without the use of orthogonal transforms but this would generally require more computation. A signal processing system embodying the invention can be in equation (3.8). The solution can be obtained without the use of orthogonal transforms but this would generally require more computation. A signal processing system embodying the invention can be used in co juction with an active control system. The desired signal is then zero, the disturbance y_(t) is the output from sensors due to the sound or vibration . to be cancelled, the signals _u(t) are the drive signals sent to the actuators and the plant represents the acoustic or vibration path from the actuators to the sensors characteristics. The required actuator drives are as given by equation (3.22). In (3.22) the matrix A has components given by (3.23) and the vector y has components given by (3.24). A is a generalised inverse of A . If a complex finite Fourier transform is used then A + n is as given by equation (3.25). In (3.25) the star denotes conjugate transpose and it has been assumed that the number of sensors is not less than the number of actuators. The drives can also be obtained iteratively for example using the LMS algorithm. At the k-th step the value of u is given by equation ( 3.26 ) where JJL is a convergence factor. The signal processing system can be rearranged to give A directly. Equation (3.16) can be rewritten using (3.17) and (3.18) as (3.27), which leads to (3.28). This can be solved in a similar fashion to give estimates ooff AA + nn iand and this g ■*■ ives an inverse model of the plant. The incorporation of the signal processing system in an active control system is shown in Figure 3. The input to the control system is also sampled by an ADC. This is triggered by the signal T(t), as is the Digital to Analogue Converter (DAC) which converts the digital output from the control system into the analogue drive signals to ■the actuators. The ADC can be common to both the control system and the signal processor which produces the estimates of the plant and the disturbance. The transform and inverse transform modules could also be common to both systems . Any sensor failure will result in one row of the matrix A becoming close to zero, while an actuator failure will result in a column of A becoming small. The system can monitor changes in the matrix A and identify and indicate failures in sensors or actuators. This is an important aid to system maintenance and repair. If more than one periodic source is present an extra trigger signal for each additional source can be incorporated to allow the system to discriminate against these sources. -<?- FORMULA SHEET Y - ' § n in —-co τι-π_ m m-—- > ii-m m (3.1) Y - U a n k—-« m—1 n-m m+kM (3.3) a m - I k--o a m+ ,kM (3. * 4) •Y. - ( i. y 2 m (3.5) FORMULA SHEET -2' a ) ra (3.6) U - u run n-m (3.7) Ua + y + n(t) (3.8) e.(t) - U(t)a(t) + y.(t) (3.9) χ(t) - U(t) a + x (3.10) - X(t)b -,0 FORMULA SHEET (3.12) e n (t) x (t) b ,n - 1 , ,M (3.13) x T — [ύ(n,t) l] (3.14) b - r n > i (3.15) "- y(n) J - π - FORMULA SHEET e > - < (3.16) "Ξ - [~l (n,t),ϋ 2 (n,t), ,0 s (n,t),l] (3.17) bj - yi (n) ] τ 3 - 18) b — n 1 - φ x ' x 1 Φ e L x < 3 ' 19 > ^x - 1 -i w ^ ( <3*20) - IZ- FORMULA SHEET 6 X I 2 i ψe x " N k-1 W e n V (3.21) u — n . n*n (3.22) hh 1 (n) (3.23) - 1 (3.24) FORMULA SHEET , * . - 1 * (A A ) A n n n' n (3.25) k k- * u — u n -μ (3.26) n T e^t) - A n u n t) + Xn (3.27) A π ϋ " A n * ΓL ' S^C t) (3.28) £-, ( t ) - t < -*_*_ ) [ ]-v < >

Claims: - I if - Claims 1. A signal processing system for a plant subject to disturbances (^(G) ) which are substantially periodic and wherein a signal (T(t)) time related to the disturbances is available, characterised in that, in order to estimate the disturbances and thereby characterise the plant, the plant response is sampled in an analogue to digital converter (ADC) and the signal T(t) is employed as a trigger and to synchronise the sampling pror.pss i.f r.he ADC. 2. A system according to claim 1, wherein, by means of the sampling process, the system estimates that part of a measured plant output due to unknown inputs and thereafter estimates the relationship between the total plant output and known plant inputs. 3. A system according to claim 1 or claim 2, wherein the sampled plant output (Yn) due to an input (u(t)) is approximated by equation (3.1) hereinbefore given and defined, is simplified for a periodic input (u(t)) which is periodic, whereby (Yn) is estimated in accordance witn equation (3.3) hereinbefore given, which reduces to equation (3. ) hereinbefore given and defined. 4. A system according to claim 3, wherein vectors given by equations (3.5) and (3.6) together with a cyclic matrix ( U ) with elements given by equation (3.7) are introduced into equation { 3. ) so that the plant output is rneasureα in accordance with equation (3.8) , all said pqudti on*-- being hereinbefore given. 5" * . A system accoding to claim A, wherein the plant output (a(t)) at a time (t) is estimated in accordance with equation (3.9) hereinbefore given, both in a plant where the disturbances and/or inputs vary slowly compared with the cyclic variations, the sampling periodicity then being constant, and in a constant plant where the disturbances and/or inputs are changing, the sampling periodicity then being slowly varied. 6. A system according to claim 5, wherein noise (n(t)) is reduced to a negligible factor in the estimated output by averaging over many cycles. 7. A system according to claim 5 or claim 6, wherein transforms, such as finite Fourier transforms, are employed to diagonalise the matrix U. 8. A system according to any of claims 1 to 7, employed in conjunction with an active vibration control system having sensors for measuring unwanted sounds and vibrations to be cancelled, and wherein the signals (U(t)) of equation (3.9) are used as drive signals defined in accordance with equation (3.22) hereinbefore given and defined for actuators for actively cancelling said sounds and vibrations. 9. A system according to claim 8, wherein an inverse transform of the matrix (An) of equation (3.22) is developed to provide an input to the control system, in accordance with equation (3.25) hereinbefore given. 10. A system in accordance with claim 8, wherein the input to the control system is also sampled by an ADC triggered by the signal T(t) and a digital to analogue converter (DAC) is provided to convert the digital output of the control system into analogue drive signals for the actuators.

Inventors: EATWELL GRAHAM PAUL, ROSS COLIN FRASER, EATWELL, GRAHAM, PAUL, ROSS, COLIN, FRASER
Applicants: TOPEXPRESS LTD, TOPEXPRESS LIMITED
Application year: 1989
Main classification: G05B21
IPC codes: G05B13, G05B21