DE4216811A1 - Method for the numerical differentiation of a digital sensor signal - Google Patents

Abstract

According to a digital sensor signal numerical differentiation process, the differentiation interval is variable and the contradictory conditions dependent on differentiation interval length, namely small relative errors resulting from isolation of the digital signal and a small phase shift between signal and differentiated signal, are continuously checked in order to ascertain whether they lie within a determined range.

Description

State of the art

The invention is based on a method for numerical differences tiation of a digital sensor signal.

With digital regulations, the actual sensor is often used in addition signals also require the differentiated sensor signals. This dif referenced sensor signals can be identified, for example, by nume win differentiation. At the first nume the differentiated size through difference formation two samples with a fixed time interval and division of the differences limit determined by this time interval. The first procedure arises when the rates of change are too small a large discretization error relative to the absolute worth the differentiated size. With discretization error is that Error meant when digital sensor signals are sampled and either one or an adjacent discrete place of the signal can be used.  

A second numerical method involves measuring time a fixed change in the size to be differentiated Lich, for example, the time taken for a fixed path interval is required for position sensors. By forming the The differentiated size can be obtained from the reciprocal value.

A relatively large discretization arises with this method errors with large rates of change to differentiate the size. The relative discretization can be used in both methods who reduces errors by increasing the specified intervals However, this leads to the fact that the differentiated signal ge an enlarged phase shift compared to the original signal exercise that is intolerable in a regulatory process.

A combination of the two methods mentioned would solve these problems possibly avoid, but there would be complex switching conditions and, if necessary, also to consider an unexplained transition area.

Thus the quantities obtained through numerical differentiation at a digital control can be used without any problems two requirements are met:

1. The phase ver lust must be minimized.

2. The ent through the discretization of the digital sensor signal standing relative errors should be especially small changes speeds of the differentiated size must be as low as possible.

Is the differentiation through the first method namely through Forming the difference between two samples is the first demand by a small time interval between the two samples  ten met, while the second requirement a large time interval required. These two competing demands can be met the previously known numerical differentiations.

Advantages of the invention

The inventive method for numerically differentiating a digital sensor signals has the advantage that it is the two requirements after a small phase loss and a small relative Discretization errors met at the same time because the Differentia tion interval should be variable and thus to the prevailing Conditions can be adjusted.

It is particularly advantageous that the differentiation interval as there is a sliding differentiation interval, the current one Differentiation interval depending on at least one before outgoing is formed.

By doubling the last differentiation interval, the current particularly easy with the help of sliding operations that run in a microprocessor.

The measures specified in the subclaims result in an advantage liable further developments and improvements of the bean in claim 1 spoke procedure.

drawing

The invention is illustrated in the drawing and is in the after following description explained.

In Fig. 1 the sample values are shown on an incremental scale, in Fig. 2 a flow chart is given which shows the steps of the method according to the invention, in Fig. 3 the relationships are illustrated using an example.

Description of the embodiment

In the case of a digital sensor, for example an incremental dimension rod or absolute encoder, measured values are obtained as shown in FIG. 1. In this example, which has a resolution of a = 10 µm, a series of discrete measurement values 10 µm apart are obtained. If such systems are scanned, apart from the error caused by the discretization (quantization error) which leads to the so-called discretization noise, no further errors and no further measurement noise occur.

If the two successive values X1 = X (t) and X2 = X (t + T) are measured with the distance X _ex , a differentiated signal can be obtained by subtracting the two successive values X1 and X2 and the Difference is divided by the sampling time T. To distinguish them, the individual values X1, X2 are designated as X1, X2 and X1 and X2.

Since the measured values are available discretely, a maximum quantization error of ± 1 increment per sampling time can occur, so that the exact differentiated variable V _ex applies:

(X2-X1-1) / T <V _ex <X2-X1 + 1) / T.

It was considered that approximation error applies to the maximum quantization: .DELTA.X _max- = (X ^· * .DELTA.T) -1 increment
or .DELTA.X _{max +} = (X ^* * .DELTA.T) +1 increment.

The following then applies to the error: V _Fmax- = -1 increment / T
or: V _{Fmax +} = + 1 increment / T.

The differentiated variable V _ex thus lies within a range which is referred to below as the confidence range.

FIG. 2 shows a flow chart which indicates how the differentiation process works. To explain this flowchart, the following definitions should first be given:
T: sampling time
X (N): Nth sample value of the variable X to be differentiated
XD: result of differentiation
Xd: intermediate result of differentiation
Ugr: lower limit of the confidence interval
Ogr: upper limit of the confidence interval
ΔN: current length of the time interval
ΔNalt: Length of the time interval with previous differentiation
DiffN: Measure for cutout limitation
ΔEnd: maximum length of the time interval when the section is limited
ΔMax: maximum length of the time interval.

In a first step S1, the differentiated size becomes minimum the differentiation interval and the quantization error speaking confidence range determined. To do this, first in Step S11, which is a sub-step of step S1, the upper one Bound of the trust area Ogr read, as the largest represent real positive number. The lower limit of trust continues range and the current length of the time interval ΔN  set as the maximum value from (ΔNalt / DiffN, T) and the maximum Length of the time interval with cutout limitation ΔEnd. ΔEnd will as a minimum from the length of the time interval for previous differences rentiation ΔNalt and the measure for the cut-out limitation DiffN un ter taking into account the maximum lengths of the time interval ΔMax educated.

With these values, the intermediate result is in the next step S12 the differentiation is determined by taking the Nth sample to dif referencing variable X is the N-ΔN-th sample of the difference size X is subtracted and the difference by the current Length of the time interval is divided.

In the next step S13 it is checked whether this is determined in step S12 intermediate result of differentiation Xd within the trust range.

If this is the case, a new trust becomes in the next step S14 ens range VB determined, the upper and lower limits Ugr and Ogr of this area of trust according to the formula

[Ugr, Ogr] = [Xd-1 / ΔN, Xd + 1 / ΔN] [Ugr, Ogr]

is determined.

In the next step S15 it is checked whether the current length of the Time interval ΔN less than the maximum length of the time interval with cutout limitation. If this is the case, then in step S16 doubles the current length of the time interval ΔN and that The method begins again with step S2.  

It is recognized in step S13 that the intermediate result of the differences tiation Xd is outside the confidence interval VB, is in Step S17 halves the current length of the time interval ΔN and the intermediate result of the differentiation Xd from the difference of the Nth sample of the variable X to be differentiated and the N-ΔNth Sample by the new current length of the time interval is divided. I.e. the result of the sliding differences rentiation becomes the one determined in the previous iteration section Value determined at which the boundaries of the confidence limits are not yet were exceeded.

In step S18, the variable .DELTA.N _old , which corresponds to the length of the time interval in the case of previous differentiation and thus the old differentiation interval, is updated for the next sampling step, ie .DELTA.Nalt is set equal to .DELTA.N as the new differentiation interval. The result of the differentiation XD is equal to the last valid intermediate result of the differentiation Xd.

The completion step S18 is also performed if in step S15 it is determined that the length of the time interval ΔN precedes the has reached the maximum length of the time interval ΔEnd.

In the method shown in FIG. 2, which runs as a program in a computer, the boundary conditions are first defined or read in in a step S11, which is referred to as step S11.

These boundary conditions then become the diffe in the first step S2 limited size Xd initially with minimal differentiation interval, that corresponds to the sampling time T is determined, at the same time the confidence interval corresponding to the quantization error. In the next steps, the differentiation interval will be each magnified and the quantization error reduced accordingly.  

If the differentiation interval is doubled, the quantisie halved.

After each step it is checked whether the newly calculated value is internal is within the confidence interval determined in the previous step. Is if this is the case, a new trust range is determined. This will the intersection of the old confidence interval and the quantisie error that corresponds to the new differentiation interval, educated.

This procedure is terminated as soon as the calculated value is exceeded is within the previously determined range of trust or one predetermined maximum interval was reached. The at that time The point of provisional differentiated size Xd is then determined as final differentiated size XD recognized.

Because the process stops immediately as soon as the one in the previous step calculated boundaries of the confidence interval, the Phase errors are kept as low as possible without the risk of that the signal quality suffers at constant speed.

The method shown in Fig. 2 can be realized particularly easily if the time intervals are each doubled. It is then possible to carry out the specified algorithm with the aid of shifting operations that are easy to carry out in a microprocessor.

Should find the optimal time interval at each sampling time with each differentiation, all time intervals begin with the smallest time interval T up to the termination criterion calculated. If there is not enough computing time available, then there is  also the possibility of only a section of the admissible Be to look rich, taking into account the surroundings of the previous one Sampling time determined time interval can be used.

The procedure described was carried out on a simulated system a sampling time of T = 1.5 msec. and doubling each Differentiation interval checked.

In Fig. 3, an example is given in which in a) the curve plotted for differentiating variable X to the sampling time T. In b) the differentials calculated for N = 8 are entered together with tolerance data.

In 3c) the confidence range for the exact value V _{ex is given} for four steps and in Fig. 3d) the calculated speed value X ^· , the limit of the confidence interval for the current differentiation interval and the limits of the actual confidence range for four steps are given. The confidence interval for the current differentiation interval is limited by the upper limit Ogr, act and the lower limit Ugr, act, for the limits of the actual confidence interval apply Ogr, tat and Ugr, tat accordingly.

The method described can be applied to any digital signals expand, which should be differentiated. The process is ongoing usually in a computing device, e.g. B. a microprocessor, which has the required registers or memory.

Claims (6)

1. A method for the numerical differentiation of a digital sensor signal, in which samples of the output signal are determined and the differentiation follows by forming two samples from one another which are removed in a differentiation interval, characterized in that the differentiation intervals (ΔN) are variable. 2. The method according to claim 1, characterized in that the differences Rent intervals (ΔN) are set so that the Discretization error and the possible phase error within tolerable limits remain. 3. The method according to claim 1 or 2, characterized in that the Differentiation intervals (ΔN) can be determined so that a Confidence area (VB), which extends over an area of by the differentiated size and the maximum Quantization error is limited, is not exceeded.   4. The method according to any one of the preceding claims, characterized in that each differentiation interval (ΔN) is formed as a function of at least one previous differentiation interval (ΔN _old ). 5. The method according to claim 4, characterized in that the new differentiation interval (ΔN) is formed by doubling the previous differentiation interval (ΔN _old ). 6. The method according to claim 5, characterized in that the Ver doubling through shift operations in registers of a microprocess sors.