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The Identification of Single-valued, Separable Non-linear Systems Based on a Modified Volterra Series Approach
THE IDENTIFICATION OF SINGLE-VALUED, SEPARABLE NON-LINEAR SYSTEMS BASED ON A MODIFIED VOLTERRA SERIES APPROACH G. A. Parker* and E. L. Moore** *Department of Mechanical Engineering, University of SUTTTey, Guildford GU2 5XH, u.K. **Fiat·Alliss Ltd., U.S.A.
Abstract. A functional approach has been developed to represent continuous, separable, non-linear systems of a general type based on a modified Volterra Series. The importance of this is that the effects of bias or mean signal level within the non-linear system can be separated from dynamic effects. This has particular significance in the development of identification procedures based on cross-correlation functions, as these functions can now be estimated practically without any influence from the bias level. Practical identification is described using three-level pseud·o -random input signals which are cross-correlated with the sampled system response in a mini-computer to provide an automated procedure. Good accuracy is achieved even in the presence of severe noise. Keywords. Automatic testing; computer testing; correlation theory; functional analysis; identification; non-linear control systems; random processes. INTRODUCTION Classical identification techniques have been largely based on correlation and deconvolution schemes. These have been used with some success in essential linear systems as they give good measurement of the system dynamics in a reasonably short time in the presence of plant disturbances without significantly affecting the normal operation of the process. (Hazlerigg and Noton, 1965). Among the specialised system test signals, one of the most widely used is the pseudorandom binary sequence (PRBS). As well as being easy to generate in a digital computer, PRBS is attractive for both software or hardware correlations as its binary nature removes the need for time consuming multiplications (Nikiforuk and Gupta, 1969). PRBS is a member of a broader class of odd-level maximal length sequences used for identification (Dotsenko, Faradzhev and Chatartisvhili, 1971). Due to the inverse repeat non-biased autocorrelation properties of three-level sequences (TPRS), they have been used in noisy systems with some success.
linear element (NL-L) , (L-NL) or (L-NL-L). The first of these is often referred to as a Hammestein model and the second a Wiener model. The last case may be thought of as a generalised Wiener model. Using TPRS inputs for non-linear identification based on functional representations has the advantage that either all even or all odd terms may be eliminated from the series. Gardiner (1973) has shown that further simplification can be achieved by carrying out two tests with different amplitude TPRS excitation. Krempl (1973) and Tuis (1975) further extended this approach to simplify the Volterra series by multiple level testing at the expense of increased computation and testing time. These methods have been applied to the identification of large scale industrial plants by Fasol (1974). However, such approaches do have their limitations when the input signal to the system has a DC level or bias applied to it. This is because a biased TPRS will not automatically remove all odd or even terms from the Volterra series representation of the nonlinp..a rity.
Functional representations of single-v alued non-linear functions were first introduced by Volterra (1930) and provide a generalised nonparametric method of expressing the response of a non-linear system (George, 1959). Simpson and Power (1972) reviewed these methods and showed that practically all algorithms designed to identify separable non-linear systems could be classified as follows: a non-linear element followed by a
505
The work described in this paper is concerned with the application of a modified Volterra Series to the identification of separable single-valued non-linear systems without the disadvantages mentioned above. It will be shown how the linear and non-linear characteristics may be separately determined independently of the bias level of the system input, using TPRS signals of small amplitude. This
506
G. A. Parker and E. L. Moore
leads to a considerable simplification in manipulating system response signals in both closed and open loop for many types of single valued non-linear systems and is particularly useful in developing identification procedures. As a result system excitation, data collection and correlation calculations can all be carried out automatically using a micro- or mini-computer system. THE MODIFIED VOLTERRA SERIES Figure 1 shows details of the assumed term of the generalised Wiener System model including the amplitude dependent, single-valued continuous non-linear element N Lu(t)J. It is slightly different in form to the models used by Gardiner (1973) as the two linear elements are represented by their gain K and unity gain impulse response g(t). The input-output relationship for the non-linearity is assumed to be polynomial with the form: n
v(t) = N lu(t)] = . L
ai [u(t)] i
(1)
the time independent Bi(m) coefficients and the gain K2 • Inspection of the relationships for the Bi(m) coefficients yields the following relationship: 1 di Bi(m) = i! d;i Bo(m) (7) which is useful for the derivation of the series coefficients. It is especially significant when it is used in conjunction with the non-linear system output equation as the response y(t) becomes: n
1
di
j-
.
K2 L 7I~o(m) g2( A )H~(t- A ,~)d A i=o ~. dm gl (8) The output equation has been expressed in terms of convolution operations between unbiased signals and unity gain linear elements, multiplied by coefficients which are themselves weighted derivatives of the non-linear inputoutput relationship. The derivative weights decrease in magnitude rapidly, the first five being I, 0.5, 0.167, 0.042 and 0.008. y(t)
~=o
The input signal x(t), with bias. 'm' and dynamic component x(t), produces a response u(t) from the first linear element given by: u(t)
K1m + Kl
j gl (T)~(t-T)dT
Kl [m + Hgl (t,~)J
(2)
The output from the non-linearity, v(t), can be found by substitution of Eq. (2) into Eq. (1) giving:
PSEUDO-RANDOM INPUT SIGNALS Pseudo-random sequences of various types have been extensively used for many years due to their characteristic of approximating to a white noise source in certain circumstances. The sequence is periodic and the signal amplitude switched between discrete levels with the peak-to-peak amplitude of the sequence AL given by: AL
The terms containing a , K and m may be represented by a new coefficient, B(m), defined by: n
(4)
L
r=i so that Eq. (3) representing the response from the non-linearity can be simply written as: n v (t) =
L ~=o
B. (m)H i (t,x) ~
gl
-
(5)
This signal forms the excitation to the second linear element, whose response y(t) is the output of the complete non-linear system. This is given by:
The output from the non-linear system is thus expressed in terms of a modified Volterra Series containing two sets of terms, one being represented by the Bi(m) coefficients and the gain Kz, and the other by the multidimensional convolutions. Of importance is the fact that these convolutions involve only unbiased signals and unity gain elements, while all the effects due to bias and gain are contained in
=
(L - l)A
(9)
where L is the number of levels and A is the smallest amplitude step. The commonest member of this class of signals is the two-level signal (PRBS), while threelevel (TPRS) and five-level (FPRS) signals are also used as input signals. It is to be noted that odd level sequences due to their inverse-repeat properties have zero mean values over one period of the signal and therefore when superimposed on a bias level m do not produce modification to the bias level. This is not true for PRBS and other even level sequences and can give rise to serious distortions when applied to nonlinear systems. If xL(t) denotes an L-level pseudo-random signal of amplitude A, superiril}Josed .on a bias m, then:
for L
3, 5 • . .
(10)
where ~(t) is an L-level pseudo-random signal of unit amplitude. Using this definition: Hi(t,x ) = Hi(t,Az ) = AiHi(t,z ) -L -L -L
(11)
507
Identification of single-valued, separable non-linear systems It then follows that the response y(t) of the general non-linear system given by Eq. (8) becomes: n
y(t)
=
for L
L i=o
K2
J (A)H~1 (t-A'~L)dA g2
(12)
~ (T) ~3 Y
where D1
The method assumes that the system is represented by the generalised Wiener and Hammerstein models, as shown in Fig. 1, which is excited by a multi-level biased maximal length pseudo-random sequence. Crosscorrelation is then carried out between the dynamic component of the input signal and the system response which, in conjunction with the modified Volterra Series representation, allows estimates of the small signal non-linear system gain to be obtained at any bias level, m, of the input signal. Normally correlation expressions become very complex if the input signal has a mean or bias level, which has proved to be a limitation in some identification methods. Application of the modified Volterra series in conjunction with cross-correlation will be shown not to suffer from these limitations due to the separation of bias effects. The cross-correlation of the dynamic component of the input signal x(t) with the system response y(t) may be-determined from Eq. (6) as: n ~
~y
(T)
=K
L
2
i=1
8i (m)D i
(T)
~fg2 (A)~xui(T-A)dA K -
D3
K2[A2 81 (m).D 1 (T)
Kl 1
+ A: 4 8 3 (m)D 3 (T) +
.]
f g2(A)·~Z
O~3g(T)
f
';:31
g 2 (A)
u(T-A)dA
3_
·~z 3_u 3 (T-A)dA
(16)
(17) (18)
etc. The coefficient 8 (m) in the first term of Eq. (15) and (16)1is of particular significance as, using Eq. (7), we have d
-dm N(K 1m)
d = -dm 80 (m)
(19)
which shows that 8 1 (m) is the slope with respect to the system input bias of the nonlinear gain function, existing between the system input, and the output from the nonlinear element. It follows that the overall system small signal gain is K2 times this value. This overall gain is a function of input bias which can be denoted by the symbol T(m), such that: (20)
Introducing the small signal gain into the cross-correlation using TPRS
(13)
in which notation is introduced for the integral terms involving the multi-dimensional cross-correlation functions with: =
(15)
3, 5
For example, the series for a 3-level sequence (TPRS) can be written as:
when the input is an L-level pseudo-random signal. IDENTIFICATION PROCEDURE FOR THE NON-LINEAR SYSTEM GAIN FUNCTION
=
(14)
1
As in the case of the modified Volterra Series, this provides a function in which bias effects are completely separated out in the 8i(m) coefficients and. the multidimensional cross-correlat~on terms represented by Di(T) are entirely bias free. T~is will be termed the modified cross-correlat~on function. Pseudo-random input signals of odd level sequences (three-level, five-level, etc.) exhibit the property of inverse repeat, so when they are used it follows that even terms in the series are r emoved as:
(21) The first term shows that the small signal gain appears as a multiplier of the system impulse response function g(T). Gain identification is therefore approximately achieved using cross-correlation provided the corruption of the terms within the summation sign is not large. It is seen that by the use of TPRS inputs the number of these terms has been halved, thereby providing a significant truncation of the series. The convergence of the corrupting terms can best be examined by expressing the crosscorrelation Eq. (21) in integral form over the range 0 to T, in which the period of the TPRS is 2T. n di - l 2 A2T(m) + L AiG . -::-r=T T (m) (22 1)1 0 ~ z3 dm i= 3,5)7) . . T (23) where IjJ ~xy(T)dT
J 0
which simplifies the modified crosscorrelation function given in Eq. (13) to:
G·~
1 (i-I)!
T
JDi (T)dT
0
(24)
508
G. A. Parker and E. L. Moore
Hence a good estimate of the small signal gain T(m) is obtained at any bias level m by carrying out a correlation test with TPRS and integrating the resulting function provided that the terms Gi and derivatives of T(m) decrease sufficiently rapidly. It is to be noted that Gi is entirely dependent on the linear dynamics of the system and is independent of bias due to the use of the modified Volterra Series. Experiments were carried out on many differ ent linear systems to determine the relative significance of the distortion terms in the estimate of Gi. Figure 2 shows some of these results and indicates generally that convergence is rapid but with highly oscillating systems exhibiting the slowest convergence.
The estimate of T(m) which is obtained by the use of this equation is unaffected by nonlinear terms up to and including the fourth order one, and is therefore more accurate than that obtained by writing A2T,(m) '" ljJ. The procedure of eliminating the third order non-linear term can be extended to higher order ones, but computation and testing times increase accordingly. It was found that satisfactory results can be expected with the si~gle amplitude test in many cases. The repeated amplitude constant bias level test improves this but it is not considered worthwhile in practice to go beyond two tests per operating point. ERRORS IN THE ESTIMATE OF T(m) Distortion Terms
The effects of derivatives of T(m) are more difficult to quantify as they are directly dependent on the non-linearity and the bias level selected. Except in regions of very large non-linear discontinuities, the deriva tive effects do not appear to have any signif icant effect on the estimate of T(m). Hence approximate identification is achieved with: (25) ljJ '" a ~ A2T(m)
Denoting the error in the estimate of T(m) as eo from noise-free testing with a single amplitude s ignal A and el as the corresponding error with two amplitude testing using signal amplitudes A and RA, then: A2V3 + A4v s + A6 V7 + ABv g + •• (32) 6 z 4 -fA' _ Rzv s + A (R +R )v 7
3
However, it will be noted from Fig. 2 that if the first distortion term (i=3) could be eliminated, a significant reduction in the error for the estimate of T(m) would be produced. This may be achieved by conducting two separate cross-correlation experiments with two input signal amplitudes A and, say, RA at each bias level m. Denoting the two resulting integrated cross-correlation functions by ljJl and ljJz respectivel y , and putting: Tl(m) = a~ .T(m)
(26)
3
;
di - l ~ dm i - l
A·G.
T (m)
(27)
T(m) (28)
For notational convenience l e t: di - l i ~ 1 vi = Gi dm~-l T(m)
(29)
Combining the two equations in matrix form yields:
+
[R::::': R:::~, :::j (30)
Assuming that the terms Vs and higher are negligible: T(m)
~
A4R2!R2-l) [R'ljJl - ljJzJ
a~
z3
(33) It is assumed that the amplitude A is the smallest value consistent with noise and quantisation considerations so that the a~pli tude ratio R is considered to be greater than or equal to unity. If A could be made very small, both eo and e l would tend to zero, and the identification would be free from errors due to the nonlinearity. Comparing the two error equations shows that the two amplitude mean level test has the effect of eliminating the term containing V3 and of changing the polarity of all the terms containing vs' v 7 ' and so on. Further, since R > 1, the magnitude of these terms is increased. It might be expected that as R is increased these terms become so large that the beneficial effect of eliminating the third order one is compl e tely offset. However this limit is rarely reached in practice. Experiments have shown that for values of R up to 4, the two amplitude mean level test is always advantageous. Higher values of Rare imprac tical and pointless. Nevertheless this theory does show that there is a limit to which the repeated mean level tests can be taken. For convenience a function is now defined which measures the improvement which is obtained with the two amplitude mean level test as opposed to the single amplitude test. Denoting it by 6e(m), it is given by:
M (m) (31)
I
eo
AZv 3
I- I -
el
I
(34)
{A 4 (R Z -l)v s
+ A6 (R'+R z-l)v 7 + •• }
(35)
Identification of single-valued, separable non-linear systems This function is dependent of bias m, since the coefficients vi are functions of m. When 6e(m) is positive, the repeated mean level test is advantageous. As pointed out, this is nearly always the case, but Eq. (35) does show that the improvement function decreases with amplitude A. From these arguments it can be concluded that R should be as close to one as possible. System Noise If the system response yet) is a noise signal n(t), which has mu and dynamic component ~(t), term is produced in the system correlation expression. For a term is:
corrupted by a bias level then an extra crossTPRS this
It may be shown that by adding this term to the integral cross-correlation equations given by Eq. (30) then an error in the estimate for T(m) is produced by the noise. Denoting this by e 2 it may be written as: e2
where
~r~:=~]
(36)
T
fo z n(T)dT
(37)
3-
The theoretical value of the error e as the 2 amplitude ratio R varies is shown as the full line in Fig. 3. It is asymptotic to the value mp/A, this being the error due to noise with the single amplitude test. Thus it is seen that the repeated mean level test amplifies errors due to additive noise, particularly for values of R close to unity. However, there is another source of error which is produced by quantisation and rounding in the A/D and D/A converters of the practical equipment used. The experimental points in Fig. 3 were obtained from rms values of many tests on simulated linear systems using 12 bit converters and show that as the two test amplitudes get closer together (R approaching unity) the responses diverge due to these errors. Best Choice of Value for R If a repeated mean level test is necessary the theory and experiment have shown that the errors due to non-linearity increase with R, and those due to noise decrease with R. In particular, when R is very close to one, with small signal amplitudes, noise errors add to those due to quantisation. These points are illustrated in Fig. 4, showing results from a simulation of a system containing a ninth order non-linearity in which the rms value of the overall error in the identified small signal gain is shown plotted against R. To calculate the rms error, ten tests were carried out over a portion of the system's operating range. From the figure it is seen that the optimum choice of R is
509
approximately 1.5. However the figure is only useful in demonstrating a trend, and not of general use as each system will have a different optimum value of R. In many cases the authors have found that good values of R were between 1.2 and 1.7, with base amplitudes of the order of 1/20th of the system's operating range. This has included tests on severely non-linear systems, with signal-to-noise ratios down to three. Other situations, however, can be envisaged when the noise is highly correlated with the input signals giving large values of m. This might necessitate larger values of R to reduce the noise amplification effect of the repeated mean level test. It is significant that m is not proportional to the signal-to-noise ratio; it depends rather on the average degree of correlation between the dynamic components of the system input signal and the noise. GAIN IDENTIFICATION OF SINGLE-VALUED CONTINUOUS NON-LINCAR SYSTEMS The results illustrated in Fig. 3, 4 ~orm part of a series of experiments to verify the identification procedure using computer simulated systems. Under programme control, a minicomputer sends out sets of TPRS excitation signals, with different amplitudes, and samples the system response. The stored data is then used to calculate an estimate of T(m) for that particular operating point based on Eq. (25) and (31). The computer then increments the system input bias by 6m and repeats the procedure. The range of bias values, the increments between tests, sample rates and other parameters are externally set prior to the start of the experiment. Figure 5 shows the experimental arrangement for the gain identification of a noise-free system containing a fifth order non-linearity. The system was tested with a TPRS having a sequence length of 80 bits which was rather coarse resolution for the cross-correlation function but led to reasonable computation speeds. Both single amplitude and two amplitude tests (R=1.5) were conducted to compare accuracies and the results of these are shown in Fig. 6. The difference between the two sets of estimated data represent 6e(m), the error difference defined by Eq. (34). As prediced, 6e(m) is greater for the larger values of 1 m I, the magnitude of the operating point. Further, the error in the estimate of T(m) obtained with the twoamplitude mean level test is a constant over the whole of the test range, while that for the single amplitude test increases approximately quadratically with 1 m I. Also, for the single amplitude test, the predicted outputs are greater than the actual, while for the two level test they are slightly smaller. This is due to the polarity change of the fifth and higher order error terms which is a characteristic of the identification method.
510
G. A. Parker and E. L. Moore
To test the performance of the identification algorithm under controlled random noise conditions, Gaussian noise was added to the system output such that the input signal-tonoise ratio was 5. The estimated gain curves for the single and two amplitude mean level tests are shown in Fig. 7. The presence of the noise has not prevented the elimination of the effects of the cubic error term while the much smaller effects due to the fifth and seventh order terms are masked by the noise process. The noise effects are more pronounced for the two level test which is as predicted in the error analysis. The integration of the gain function T(m) produces the dynamic component of the inputoutput curve, but it is necess~ry to make some assumptions about the non-linear characteristic to obtain the complete curve. Mathematically this is equivalent to choosing the constant of integration. If the non-linear characteristic is symmetric, then with zero steady state input, the output will be zero. This means uo=O, and the integrated characteristic must pass through zero. The value of the constant of integration, which must be added to the integrated T(m) curve, is therefore fixed. In cases when it cannot be assumed that uo=O, some other assumption is necessary, dictated by the nature of the system being identified. The estimates of the input-output relationships are shown in Fig. 8. The integration process used to derive these from the T(m) curves has almost completely eliminated the effect of high frequency variations in the gain curves, but some bias remains. In particular, for the two level test a slight positive bias is present throughout the range, although, considering the severity of the noise conditions, this is considered to be very acceptable. CONCLUSIONS A functional approach has been developed to represent continuous, separable, non-linear systems of a general type based on a modified Volterra Series. The importance of this is that the effects of bias or mean signal level within the non-linear system can be separated from the dynamic effects. This has particular significance in the development of identification procedures based on crosscorrelation functions, as these functions can now be estimated practically without any influence from the bias level. It has been shown that by the use of a maximum length TPRS excitation signal of small amplitude simplification in the modified Volterra Series, and hence the crosscorrelation computation, are obtained. Errors in the method have been discussed and it has been shown that increasing accuracy, at the expense of greater computation time, can be achieved by mUltiple amplitude testing at each bias level of the input signal.
Practical verification of the method has been shown for continuous, separable, non-linearities using a mini-computer to automatically introduce the system excitation, collect the res~ ponse data and carry out the cross-correlation procedures on simulated systems. It has been demonstrated that two amplitude testing produces very acceptable accuracy for a fifth order non-linear system despite the presence of severe noise. In low noise systems single amplitude testing would be a more satisfactory compromise between accuracy and computational speed.
REFERENCES Dotsenko, V.I., R.G. Faradzhev and G.S. Chatartisvhili (1971). Properties of maximal length sequences with p-levels. Autom. & Telemekh., 8, 189-194. Fasol, K.H. (1974). On some methods for the investigation of the dynamic behaviour of turbine-governors, turbine units and large hydraulic systems. I.A.H.R. Symposium. Gardiner, A.B. (1973). Identification of processes containing single-valued nonlinearities. Int. J. Control, 18, 1029-1039. -George, D.A. (1959). Continuous non-linear systems. Report No. 355, Research Lab. of Electronics, M.I.T. Hazlerigg, A.D.G. and A.R.M. Noton (1965). The application of cross-correlating equipment to linear system identification. I.E.E., 112, 2385-2400. Krempl, R. (1973). Anwendung von dreiwertigen pseudofMlligen Signolen zur Identifikation nichtlinearen regelungssysteme. Ph.D. Thesis, Bochum Ruhr University. Nikifaruk, P.N. and M.M. Gupta (1969). A bibliography of the properties, generation and control systems applications of shift register sequences. Int. J. Control, ~, 217-234. Simpson, R.J. and H.M. Power (1972). Correlation techniques for the identification of non-linear systems. Meas. & Control, 5, 316-321. Tuis~ L. (1975). Anwendung von Mehrwertingen PseudofMlligen Signolen zur Identifikation von nichtlinearen Regelungssysteme. Ph.D. Thesis, Bochum Ruhr University. Volterra, V. (1930). Theory of functionals. Blackie.
Fig. 1.
Generalised non-linear system (L-NL-L)
Identification of single-valued, separabl e non-linear systems
sll
1
• system 9,.(t) o system 9,,(t).9.(t) ... 4th. order lovv pass filter
10 •
G(S) = A
o
"
3.10 S'l.100S.3.10'"
10 -~
.')
(1.0·8.105)(1.0'7.105) +
10
Fig. 2.
Examples of convergence of Volterra order i.
I Ci I
constant estimates as a function of the
1.5
5 4
1.0
E;.A 3
m; 2
Q5
theory
.-.-.-~----.
1
O+-----~----.-----~
1
2
3
4
O+-----~----.----.
1
RATIO,R
Fig. 3.
Theoretical and experimental variation of non-dimensional error due to random noise as a function of the amplitude ratio, R.
Fig. 4.
2
3 4 RATIO,R
Effect of amplitude ratio, R, on the rms percentage error in the estimated non-linear gain T(m) for a ninth-order non-lineari ty.
G. A. Parker and E. L. Maore
512
xJt) N=80
U(t) 9Jt>
y(t)
V(t)
N(u)=u + U3 + U5
9Jt>
~T=1ms
Fig. 5.
Experimental arrangement for the gain identification of a system containing a fifth order non-linearity.
GAIN T(m) 10 o single amp. • two amp.
0
\
8
\
0
6
e \
•
\
e,
4
0
0
o
0
I
Fig. 6,
o
0
0
· " - .....0 0 ........
-1.0
A=0.6volt R =1.5
-0.5
2 0
--.-.0
-_.
o
I
0
o
o
0
/
o •
/
/
/~ 3 5 • U+U+U
/.
/-
/-
......
I
I
1.0 0.5 BIAS,m (volts)
Estimation of gain function T(m) for system in Fig. 5 under noise-free conditions.
Identification of single-valued, separable non-linear systems
GAI~O T(m) -.- single amp. 8 -two amp. /
Fig. 7.
o
-0.5
-1.0
I
/
6 4
I
./' I
.,..;
0.5 1.0 BIAS,m (volts)
Estimation of gain function T(m) for system in Fig. 5 in the presence of additive output Gaussian noise.
OUTPUT BIAS, my(volts)
4
single amp. • two amp.
0 0
0
3
0
2 1
-1.0
-1
1.0 BIAS, m (vol ts)
-2 o
-3
o
o
Fig. 8.
-4
Estimation of the input-output relationship corresponding to the gain function T(m) of Fig. 7.
513
Abstract. A functional approach has been developed to represent continuous, separable, non-linear systems of a general type based on a modified Volterra Series. The importance of this is that the effects of bias or mean signal level within the non-linear system can be separated from dynamic effects. This has particular significance in the development of identification procedures based on cross-correlation functions, as these functions can now be estimated practically without any influence from the bias level. Practical identification is described using three-level pseud·o -random input signals which are cross-correlated with the sampled system response in a mini-computer to provide an automated procedure. Good accuracy is achieved even in the presence of severe noise. Keywords. Automatic testing; computer testing; correlation theory; functional analysis; identification; non-linear control systems; random processes. INTRODUCTION Classical identification techniques have been largely based on correlation and deconvolution schemes. These have been used with some success in essential linear systems as they give good measurement of the system dynamics in a reasonably short time in the presence of plant disturbances without significantly affecting the normal operation of the process. (Hazlerigg and Noton, 1965). Among the specialised system test signals, one of the most widely used is the pseudorandom binary sequence (PRBS). As well as being easy to generate in a digital computer, PRBS is attractive for both software or hardware correlations as its binary nature removes the need for time consuming multiplications (Nikiforuk and Gupta, 1969). PRBS is a member of a broader class of odd-level maximal length sequences used for identification (Dotsenko, Faradzhev and Chatartisvhili, 1971). Due to the inverse repeat non-biased autocorrelation properties of three-level sequences (TPRS), they have been used in noisy systems with some success.
linear element (NL-L) , (L-NL) or (L-NL-L). The first of these is often referred to as a Hammestein model and the second a Wiener model. The last case may be thought of as a generalised Wiener model. Using TPRS inputs for non-linear identification based on functional representations has the advantage that either all even or all odd terms may be eliminated from the series. Gardiner (1973) has shown that further simplification can be achieved by carrying out two tests with different amplitude TPRS excitation. Krempl (1973) and Tuis (1975) further extended this approach to simplify the Volterra series by multiple level testing at the expense of increased computation and testing time. These methods have been applied to the identification of large scale industrial plants by Fasol (1974). However, such approaches do have their limitations when the input signal to the system has a DC level or bias applied to it. This is because a biased TPRS will not automatically remove all odd or even terms from the Volterra series representation of the nonlinp..a rity.
Functional representations of single-v alued non-linear functions were first introduced by Volterra (1930) and provide a generalised nonparametric method of expressing the response of a non-linear system (George, 1959). Simpson and Power (1972) reviewed these methods and showed that practically all algorithms designed to identify separable non-linear systems could be classified as follows: a non-linear element followed by a
505
The work described in this paper is concerned with the application of a modified Volterra Series to the identification of separable single-valued non-linear systems without the disadvantages mentioned above. It will be shown how the linear and non-linear characteristics may be separately determined independently of the bias level of the system input, using TPRS signals of small amplitude. This
506
G. A. Parker and E. L. Moore
leads to a considerable simplification in manipulating system response signals in both closed and open loop for many types of single valued non-linear systems and is particularly useful in developing identification procedures. As a result system excitation, data collection and correlation calculations can all be carried out automatically using a micro- or mini-computer system. THE MODIFIED VOLTERRA SERIES Figure 1 shows details of the assumed term of the generalised Wiener System model including the amplitude dependent, single-valued continuous non-linear element N Lu(t)J. It is slightly different in form to the models used by Gardiner (1973) as the two linear elements are represented by their gain K and unity gain impulse response g(t). The input-output relationship for the non-linearity is assumed to be polynomial with the form: n
v(t) = N lu(t)] = . L
ai [u(t)] i
(1)
the time independent Bi(m) coefficients and the gain K2 • Inspection of the relationships for the Bi(m) coefficients yields the following relationship: 1 di Bi(m) = i! d;i Bo(m) (7) which is useful for the derivation of the series coefficients. It is especially significant when it is used in conjunction with the non-linear system output equation as the response y(t) becomes: n
1
di
j-
.
K2 L 7I~o(m) g2( A )H~(t- A ,~)d A i=o ~. dm gl (8) The output equation has been expressed in terms of convolution operations between unbiased signals and unity gain linear elements, multiplied by coefficients which are themselves weighted derivatives of the non-linear inputoutput relationship. The derivative weights decrease in magnitude rapidly, the first five being I, 0.5, 0.167, 0.042 and 0.008. y(t)
~=o
The input signal x(t), with bias. 'm' and dynamic component x(t), produces a response u(t) from the first linear element given by: u(t)
K1m + Kl
j gl (T)~(t-T)dT
Kl [m + Hgl (t,~)J
(2)
The output from the non-linearity, v(t), can be found by substitution of Eq. (2) into Eq. (1) giving:
PSEUDO-RANDOM INPUT SIGNALS Pseudo-random sequences of various types have been extensively used for many years due to their characteristic of approximating to a white noise source in certain circumstances. The sequence is periodic and the signal amplitude switched between discrete levels with the peak-to-peak amplitude of the sequence AL given by: AL
The terms containing a , K and m may be represented by a new coefficient, B(m), defined by: n
(4)
L
r=i so that Eq. (3) representing the response from the non-linearity can be simply written as: n v (t) =
L ~=o
B. (m)H i (t,x) ~
gl
-
(5)
This signal forms the excitation to the second linear element, whose response y(t) is the output of the complete non-linear system. This is given by:
The output from the non-linear system is thus expressed in terms of a modified Volterra Series containing two sets of terms, one being represented by the Bi(m) coefficients and the gain Kz, and the other by the multidimensional convolutions. Of importance is the fact that these convolutions involve only unbiased signals and unity gain elements, while all the effects due to bias and gain are contained in
=
(L - l)A
(9)
where L is the number of levels and A is the smallest amplitude step. The commonest member of this class of signals is the two-level signal (PRBS), while threelevel (TPRS) and five-level (FPRS) signals are also used as input signals. It is to be noted that odd level sequences due to their inverse-repeat properties have zero mean values over one period of the signal and therefore when superimposed on a bias level m do not produce modification to the bias level. This is not true for PRBS and other even level sequences and can give rise to serious distortions when applied to nonlinear systems. If xL(t) denotes an L-level pseudo-random signal of amplitude A, superiril}Josed .on a bias m, then:
for L
3, 5 • . .
(10)
where ~(t) is an L-level pseudo-random signal of unit amplitude. Using this definition: Hi(t,x ) = Hi(t,Az ) = AiHi(t,z ) -L -L -L
(11)
507
Identification of single-valued, separable non-linear systems It then follows that the response y(t) of the general non-linear system given by Eq. (8) becomes: n
y(t)
=
for L
L i=o
K2
J (A)H~1 (t-A'~L)dA g2
(12)
~ (T) ~3 Y
where D1
The method assumes that the system is represented by the generalised Wiener and Hammerstein models, as shown in Fig. 1, which is excited by a multi-level biased maximal length pseudo-random sequence. Crosscorrelation is then carried out between the dynamic component of the input signal and the system response which, in conjunction with the modified Volterra Series representation, allows estimates of the small signal non-linear system gain to be obtained at any bias level, m, of the input signal. Normally correlation expressions become very complex if the input signal has a mean or bias level, which has proved to be a limitation in some identification methods. Application of the modified Volterra series in conjunction with cross-correlation will be shown not to suffer from these limitations due to the separation of bias effects. The cross-correlation of the dynamic component of the input signal x(t) with the system response y(t) may be-determined from Eq. (6) as: n ~
~y
(T)
=K
L
2
i=1
8i (m)D i
(T)
~fg2 (A)~xui(T-A)dA K -
D3
K2[A2 81 (m).D 1 (T)
Kl 1
+ A: 4 8 3 (m)D 3 (T) +
.]
f g2(A)·~Z
O~3g(T)
f
';:31
g 2 (A)
u(T-A)dA
3_
·~z 3_u 3 (T-A)dA
(16)
(17) (18)
etc. The coefficient 8 (m) in the first term of Eq. (15) and (16)1is of particular significance as, using Eq. (7), we have d
-dm N(K 1m)
d = -dm 80 (m)
(19)
which shows that 8 1 (m) is the slope with respect to the system input bias of the nonlinear gain function, existing between the system input, and the output from the nonlinear element. It follows that the overall system small signal gain is K2 times this value. This overall gain is a function of input bias which can be denoted by the symbol T(m), such that: (20)
Introducing the small signal gain into the cross-correlation using TPRS
(13)
in which notation is introduced for the integral terms involving the multi-dimensional cross-correlation functions with: =
(15)
3, 5
For example, the series for a 3-level sequence (TPRS) can be written as:
when the input is an L-level pseudo-random signal. IDENTIFICATION PROCEDURE FOR THE NON-LINEAR SYSTEM GAIN FUNCTION
=
(14)
1
As in the case of the modified Volterra Series, this provides a function in which bias effects are completely separated out in the 8i(m) coefficients and. the multidimensional cross-correlat~on terms represented by Di(T) are entirely bias free. T~is will be termed the modified cross-correlat~on function. Pseudo-random input signals of odd level sequences (three-level, five-level, etc.) exhibit the property of inverse repeat, so when they are used it follows that even terms in the series are r emoved as:
(21) The first term shows that the small signal gain appears as a multiplier of the system impulse response function g(T). Gain identification is therefore approximately achieved using cross-correlation provided the corruption of the terms within the summation sign is not large. It is seen that by the use of TPRS inputs the number of these terms has been halved, thereby providing a significant truncation of the series. The convergence of the corrupting terms can best be examined by expressing the crosscorrelation Eq. (21) in integral form over the range 0 to T, in which the period of the TPRS is 2T. n di - l 2 A2T(m) + L AiG . -::-r=T T (m) (22 1)1 0 ~ z3 dm i= 3,5)7) . . T (23) where IjJ ~xy(T)dT
J 0
which simplifies the modified crosscorrelation function given in Eq. (13) to:
G·~
1 (i-I)!
T
JDi (T)dT
0
(24)
508
G. A. Parker and E. L. Moore
Hence a good estimate of the small signal gain T(m) is obtained at any bias level m by carrying out a correlation test with TPRS and integrating the resulting function provided that the terms Gi and derivatives of T(m) decrease sufficiently rapidly. It is to be noted that Gi is entirely dependent on the linear dynamics of the system and is independent of bias due to the use of the modified Volterra Series. Experiments were carried out on many differ ent linear systems to determine the relative significance of the distortion terms in the estimate of Gi. Figure 2 shows some of these results and indicates generally that convergence is rapid but with highly oscillating systems exhibiting the slowest convergence.
The estimate of T(m) which is obtained by the use of this equation is unaffected by nonlinear terms up to and including the fourth order one, and is therefore more accurate than that obtained by writing A2T,(m) '" ljJ. The procedure of eliminating the third order non-linear term can be extended to higher order ones, but computation and testing times increase accordingly. It was found that satisfactory results can be expected with the si~gle amplitude test in many cases. The repeated amplitude constant bias level test improves this but it is not considered worthwhile in practice to go beyond two tests per operating point. ERRORS IN THE ESTIMATE OF T(m) Distortion Terms
The effects of derivatives of T(m) are more difficult to quantify as they are directly dependent on the non-linearity and the bias level selected. Except in regions of very large non-linear discontinuities, the deriva tive effects do not appear to have any signif icant effect on the estimate of T(m). Hence approximate identification is achieved with: (25) ljJ '" a ~ A2T(m)
Denoting the error in the estimate of T(m) as eo from noise-free testing with a single amplitude s ignal A and el as the corresponding error with two amplitude testing using signal amplitudes A and RA, then: A2V3 + A4v s + A6 V7 + ABv g + •• (32) 6 z 4 -fA' _ Rzv s + A (R +R )v 7
3
However, it will be noted from Fig. 2 that if the first distortion term (i=3) could be eliminated, a significant reduction in the error for the estimate of T(m) would be produced. This may be achieved by conducting two separate cross-correlation experiments with two input signal amplitudes A and, say, RA at each bias level m. Denoting the two resulting integrated cross-correlation functions by ljJl and ljJz respectivel y , and putting: Tl(m) = a~ .T(m)
(26)
3
;
di - l ~ dm i - l
A·G.
T (m)
(27)
T(m) (28)
For notational convenience l e t: di - l i ~ 1 vi = Gi dm~-l T(m)
(29)
Combining the two equations in matrix form yields:
+
[R::::': R:::~, :::j (30)
Assuming that the terms Vs and higher are negligible: T(m)
~
A4R2!R2-l) [R'ljJl - ljJzJ
a~
z3
(33) It is assumed that the amplitude A is the smallest value consistent with noise and quantisation considerations so that the a~pli tude ratio R is considered to be greater than or equal to unity. If A could be made very small, both eo and e l would tend to zero, and the identification would be free from errors due to the nonlinearity. Comparing the two error equations shows that the two amplitude mean level test has the effect of eliminating the term containing V3 and of changing the polarity of all the terms containing vs' v 7 ' and so on. Further, since R > 1, the magnitude of these terms is increased. It might be expected that as R is increased these terms become so large that the beneficial effect of eliminating the third order one is compl e tely offset. However this limit is rarely reached in practice. Experiments have shown that for values of R up to 4, the two amplitude mean level test is always advantageous. Higher values of Rare imprac tical and pointless. Nevertheless this theory does show that there is a limit to which the repeated mean level tests can be taken. For convenience a function is now defined which measures the improvement which is obtained with the two amplitude mean level test as opposed to the single amplitude test. Denoting it by 6e(m), it is given by:
M (m) (31)
I
eo
AZv 3
I- I -
el
I
(34)
{A 4 (R Z -l)v s
+ A6 (R'+R z-l)v 7 + •• }
(35)
Identification of single-valued, separable non-linear systems This function is dependent of bias m, since the coefficients vi are functions of m. When 6e(m) is positive, the repeated mean level test is advantageous. As pointed out, this is nearly always the case, but Eq. (35) does show that the improvement function decreases with amplitude A. From these arguments it can be concluded that R should be as close to one as possible. System Noise If the system response yet) is a noise signal n(t), which has mu and dynamic component ~(t), term is produced in the system correlation expression. For a term is:
corrupted by a bias level then an extra crossTPRS this
It may be shown that by adding this term to the integral cross-correlation equations given by Eq. (30) then an error in the estimate for T(m) is produced by the noise. Denoting this by e 2 it may be written as: e2
where
~r~:=~]
(36)
T
fo z n(T)dT
(37)
3-
The theoretical value of the error e as the 2 amplitude ratio R varies is shown as the full line in Fig. 3. It is asymptotic to the value mp/A, this being the error due to noise with the single amplitude test. Thus it is seen that the repeated mean level test amplifies errors due to additive noise, particularly for values of R close to unity. However, there is another source of error which is produced by quantisation and rounding in the A/D and D/A converters of the practical equipment used. The experimental points in Fig. 3 were obtained from rms values of many tests on simulated linear systems using 12 bit converters and show that as the two test amplitudes get closer together (R approaching unity) the responses diverge due to these errors. Best Choice of Value for R If a repeated mean level test is necessary the theory and experiment have shown that the errors due to non-linearity increase with R, and those due to noise decrease with R. In particular, when R is very close to one, with small signal amplitudes, noise errors add to those due to quantisation. These points are illustrated in Fig. 4, showing results from a simulation of a system containing a ninth order non-linearity in which the rms value of the overall error in the identified small signal gain is shown plotted against R. To calculate the rms error, ten tests were carried out over a portion of the system's operating range. From the figure it is seen that the optimum choice of R is
509
approximately 1.5. However the figure is only useful in demonstrating a trend, and not of general use as each system will have a different optimum value of R. In many cases the authors have found that good values of R were between 1.2 and 1.7, with base amplitudes of the order of 1/20th of the system's operating range. This has included tests on severely non-linear systems, with signal-to-noise ratios down to three. Other situations, however, can be envisaged when the noise is highly correlated with the input signals giving large values of m. This might necessitate larger values of R to reduce the noise amplification effect of the repeated mean level test. It is significant that m is not proportional to the signal-to-noise ratio; it depends rather on the average degree of correlation between the dynamic components of the system input signal and the noise. GAIN IDENTIFICATION OF SINGLE-VALUED CONTINUOUS NON-LINCAR SYSTEMS The results illustrated in Fig. 3, 4 ~orm part of a series of experiments to verify the identification procedure using computer simulated systems. Under programme control, a minicomputer sends out sets of TPRS excitation signals, with different amplitudes, and samples the system response. The stored data is then used to calculate an estimate of T(m) for that particular operating point based on Eq. (25) and (31). The computer then increments the system input bias by 6m and repeats the procedure. The range of bias values, the increments between tests, sample rates and other parameters are externally set prior to the start of the experiment. Figure 5 shows the experimental arrangement for the gain identification of a noise-free system containing a fifth order non-linearity. The system was tested with a TPRS having a sequence length of 80 bits which was rather coarse resolution for the cross-correlation function but led to reasonable computation speeds. Both single amplitude and two amplitude tests (R=1.5) were conducted to compare accuracies and the results of these are shown in Fig. 6. The difference between the two sets of estimated data represent 6e(m), the error difference defined by Eq. (34). As prediced, 6e(m) is greater for the larger values of 1 m I, the magnitude of the operating point. Further, the error in the estimate of T(m) obtained with the twoamplitude mean level test is a constant over the whole of the test range, while that for the single amplitude test increases approximately quadratically with 1 m I. Also, for the single amplitude test, the predicted outputs are greater than the actual, while for the two level test they are slightly smaller. This is due to the polarity change of the fifth and higher order error terms which is a characteristic of the identification method.
510
G. A. Parker and E. L. Moore
To test the performance of the identification algorithm under controlled random noise conditions, Gaussian noise was added to the system output such that the input signal-tonoise ratio was 5. The estimated gain curves for the single and two amplitude mean level tests are shown in Fig. 7. The presence of the noise has not prevented the elimination of the effects of the cubic error term while the much smaller effects due to the fifth and seventh order terms are masked by the noise process. The noise effects are more pronounced for the two level test which is as predicted in the error analysis. The integration of the gain function T(m) produces the dynamic component of the inputoutput curve, but it is necess~ry to make some assumptions about the non-linear characteristic to obtain the complete curve. Mathematically this is equivalent to choosing the constant of integration. If the non-linear characteristic is symmetric, then with zero steady state input, the output will be zero. This means uo=O, and the integrated characteristic must pass through zero. The value of the constant of integration, which must be added to the integrated T(m) curve, is therefore fixed. In cases when it cannot be assumed that uo=O, some other assumption is necessary, dictated by the nature of the system being identified. The estimates of the input-output relationships are shown in Fig. 8. The integration process used to derive these from the T(m) curves has almost completely eliminated the effect of high frequency variations in the gain curves, but some bias remains. In particular, for the two level test a slight positive bias is present throughout the range, although, considering the severity of the noise conditions, this is considered to be very acceptable. CONCLUSIONS A functional approach has been developed to represent continuous, separable, non-linear systems of a general type based on a modified Volterra Series. The importance of this is that the effects of bias or mean signal level within the non-linear system can be separated from the dynamic effects. This has particular significance in the development of identification procedures based on crosscorrelation functions, as these functions can now be estimated practically without any influence from the bias level. It has been shown that by the use of a maximum length TPRS excitation signal of small amplitude simplification in the modified Volterra Series, and hence the crosscorrelation computation, are obtained. Errors in the method have been discussed and it has been shown that increasing accuracy, at the expense of greater computation time, can be achieved by mUltiple amplitude testing at each bias level of the input signal.
Practical verification of the method has been shown for continuous, separable, non-linearities using a mini-computer to automatically introduce the system excitation, collect the res~ ponse data and carry out the cross-correlation procedures on simulated systems. It has been demonstrated that two amplitude testing produces very acceptable accuracy for a fifth order non-linear system despite the presence of severe noise. In low noise systems single amplitude testing would be a more satisfactory compromise between accuracy and computational speed.
REFERENCES Dotsenko, V.I., R.G. Faradzhev and G.S. Chatartisvhili (1971). Properties of maximal length sequences with p-levels. Autom. & Telemekh., 8, 189-194. Fasol, K.H. (1974). On some methods for the investigation of the dynamic behaviour of turbine-governors, turbine units and large hydraulic systems. I.A.H.R. Symposium. Gardiner, A.B. (1973). Identification of processes containing single-valued nonlinearities. Int. J. Control, 18, 1029-1039. -George, D.A. (1959). Continuous non-linear systems. Report No. 355, Research Lab. of Electronics, M.I.T. Hazlerigg, A.D.G. and A.R.M. Noton (1965). The application of cross-correlating equipment to linear system identification. I.E.E., 112, 2385-2400. Krempl, R. (1973). Anwendung von dreiwertigen pseudofMlligen Signolen zur Identifikation nichtlinearen regelungssysteme. Ph.D. Thesis, Bochum Ruhr University. Nikifaruk, P.N. and M.M. Gupta (1969). A bibliography of the properties, generation and control systems applications of shift register sequences. Int. J. Control, ~, 217-234. Simpson, R.J. and H.M. Power (1972). Correlation techniques for the identification of non-linear systems. Meas. & Control, 5, 316-321. Tuis~ L. (1975). Anwendung von Mehrwertingen PseudofMlligen Signolen zur Identifikation von nichtlinearen Regelungssysteme. Ph.D. Thesis, Bochum Ruhr University. Volterra, V. (1930). Theory of functionals. Blackie.
Fig. 1.
Generalised non-linear system (L-NL-L)
Identification of single-valued, separabl e non-linear systems
sll
1
• system 9,.(t) o system 9,,(t).9.(t) ... 4th. order lovv pass filter
10 •
G(S) = A
o
"
3.10 S'l.100S.3.10'"
10 -~
.')
(1.0·8.105)(1.0'7.105) +
10
Fig. 2.
Examples of convergence of Volterra order i.
I Ci I
constant estimates as a function of the
1.5
5 4
1.0
E;.A 3
m; 2
Q5
theory
.-.-.-~----.
1
O+-----~----.-----~
1
2
3
4
O+-----~----.----.
1
RATIO,R
Fig. 3.
Theoretical and experimental variation of non-dimensional error due to random noise as a function of the amplitude ratio, R.
Fig. 4.
2
3 4 RATIO,R
Effect of amplitude ratio, R, on the rms percentage error in the estimated non-linear gain T(m) for a ninth-order non-lineari ty.
G. A. Parker and E. L. Maore
512
xJt) N=80
U(t) 9Jt>
y(t)
V(t)
N(u)=u + U3 + U5
9Jt>
~T=1ms
Fig. 5.
Experimental arrangement for the gain identification of a system containing a fifth order non-linearity.
GAIN T(m) 10 o single amp. • two amp.
0
\
8
\
0
6
e \
•
\
e,
4
0
0
o
0
I
Fig. 6,
o
0
0
· " - .....0 0 ........
-1.0
A=0.6volt R =1.5
-0.5
2 0
--.-.0
-_.
o
I
0
o
o
0
/
o •
/
/
/~ 3 5 • U+U+U
/.
/-
/-
......
I
I
1.0 0.5 BIAS,m (volts)
Estimation of gain function T(m) for system in Fig. 5 under noise-free conditions.
Identification of single-valued, separable non-linear systems
GAI~O T(m) -.- single amp. 8 -two amp. /
Fig. 7.
o
-0.5
-1.0
I
/
6 4
I
./' I
.,..;
0.5 1.0 BIAS,m (volts)
Estimation of gain function T(m) for system in Fig. 5 in the presence of additive output Gaussian noise.
OUTPUT BIAS, my(volts)
4
single amp. • two amp.
0 0
0
3
0
2 1
-1.0
-1
1.0 BIAS, m (vol ts)
-2 o
-3
o
o
Fig. 8.
-4
Estimation of the input-output relationship corresponding to the gain function T(m) of Fig. 7.
513