Last lecture Overview of the elements of measurement systems. Sensing elements. Signal conditioning elements. Signal processing elements. Data presentation elements. Static characteristics of measurement systems. Range, span, resolution. Straight line vs. non-linearities. Sensitivity. Environmental effects, wear and aging. Hysteresis. Error bands. Today s menu Statistical characteristics: Accuracy. Repeatability. Tolerance. Uncertainty analysis (in the steady state). 1 2 Some definitions Accuracy How close are we, on average, to the true value of the measured quantity? Repeatability How much does the measured value vary around its average over time. Let s start by looking at repeatability of single sensing elements. Repeatability of sensing elements Recall the voltage and current measurements from last lecture: ik () R uk () 3 4
Measuring the voltage 10000 times, we obtained the following distribution 0.04 0.03 Sample distribution Estimated Gaussian PDF Some comments on the sensing element (voltage meter) in this example: The mean value of the measured value is equal to the true voltage. Conclusion: The sensing element is accurate. The variance around the mean value is very high. This means the voltage meter suffers from a large lack of repeatability. Probability 0.02 0.01 0 2 0 2 4 u 5 6 The most common cause of the variation in output O is random fluctuations with time in the environmental inputs I M and I I. If the coupling constants, K M and K I are non-zero, this causes variations in the output O. By making reasonable assumptions about the probability density functions of the inputs, I,I M,I I, we can find (or at least approximate) the probability density function of the output O. Very often, the probability density function of the inputs can be assumed to be the Normal probability distribution or the Gaussian distribution, i.e. p(x) = 1 ] [ σ 2π exp (x x)2, 2σ 2 where x is the mean or expected value (center of the distribution) and σ is the standard deviation (spread of the distribution). 7 8
General Gaussian distribution ( x =0,σ =1). Recall the general equation for the output of a measurement system O = KI + a + N(I)+K M I M I + K I I I. P σ,σ = P 2σ,2σ = P 3σ,3σ = σ σ 2σ 2σ 3σ 3σ p(x)dx 0.683 p(x)dx 0.955 p(x)dx 0.997 P(x) 3σ 2σ σ 0 σ 2σ 3σ x A small deviation in the output O can be approximated as ΔO = ( ) ( ) O O ΔI + ΔI M + I I M ( O I I ) ΔI I which means ΔO is approximated by a linear combination of the deviations of the inputs, I,I M,I I. 9 10 It can be shown that, if y is a linear combination of the independent variables x 1,x 2,x 3, i.e. y = a 1 x 1 + a 2 x 2 + a 3 x 3, and the if x 1,x 2,x 3 have normal distributions with standard deviations σ 1,σ 2,σ 3, respectively, then the output will also have a normal distribution, with standard deviation σ = a 2 1 σ2 1 + a2 2 σ2 2 + a2 3 σ2 3. That is, a linear combination of Gaussian variables is also Gaussian, with standard deviation as the equation above. We can now express the standard deviation of the output of a system element in terms of the standard deviations of the input, as σ O = ( O I ) 2 ( ) 2 O σi 2 + σi 2 I M + M ( O I I The mean (or expected value) of the output is given by: Ō = KĪ + a + N(Ī)+K M Ī M Ī + KĪ I, and the corresponding probability density function is: p(o) = ] 1 (O Ō)2 exp [ σ O 2π 2σO 2. ) 2 σ 2 I I. 11 12
Some comments In some cases the standard deviation of the inputs are known or can be estimated. In other cases, the standard deviation of the output can be estimated by repeated experiments. The decomposition of the output s variance into contributions from the different inputs, gives a detailed understanding of where to put the effort in order to minimize the deviation. Tolerance A manufacturer of some sensing elements, for example resistors, can state that the resistor: Has a resistance R 0 = 100 Ω with a tolerance of ±0.15 Ω. This means he has to reject all components with resistance outside the interval 99.85 Ω <R 0 < 100.15 Ω. OR he could state that all components lie in that interval, with some probability (typically about 99 %). The user can now choose to either calibrate each component or, more realistically, use the manufacturer s tolerance to estimate the resulting deviation of his/her system. 13 14 Accuracy Accuracy is quantified using the measurement error Err = measured value true value. A system is said to be perfectly accurate if Err =0, or, in the presence of deviation of its elements, it is said to be unbiased if Err = E{Err} = Err p(err)derr =0. Accuracy (cont d...) Comments on the equations in the book In the book, the word mean (as in average) is used as equal to the expected value. This is not true! The expected value of a variable x is defined as μ x = E{x} = xp(x)dx, while the average (or as it s often named, the sample mean) is x = 1 N N x n, n=1 where N is the number of observations. 15 16
Accuracy (cont d...) Comments on the equations in the book (cont d...) If the errors are Gaussian, then the sample mean is in fact an unbiased consistent estimate of the true mean, that is lim N 1 N N x n = μ x. n=1 Another, more important, error in the book concerns the standard deviation... Accuracy (cont d...) Comments on the equations in the book (cont d...) Strictly speaking, the variance σx 2 of a random variable x is defined as σx 2 = E{(x μ x ) 2 } = while the book defines it as σ 2 x = 1 N N (x n x) 2. n=1 (x μ x ) 2 p(x)dx, 17 18 Accuracy (cont d...) Comments on the equations in the book (cont d...) Example A temperature measurement system If the true mean μ x is unknown and replaced with the sample mean, then the sample variance is an estimate of the true variance, given by s 2 = 1 N 1 N (x n x) 2. n=1 For small N, the equation in the book gives biased estimates of the variance. For very large N and Gaussian noise, the results are the same. T o C R T o T i ma M C True temperature Platinum resistance temperature detector Current transmitter Recorder Measured temperature 19 20
Example (cont d...) Determining the overall uncertainty Work flow Find the model equations for each of the system elements (usually specified by the manufacturer). Find the nominal values (mean values) of the model quantities. Find the individual standard deviations (or tolerance measures) for the model quantities. Calculate the overall mean and standard deviation of each system element. Combine all system elements to obtain an estimate of the total accuracy and standard deviation. This is sometimes referred to as the total uncertainty of the system. Example (cont d...) Model equations for the temperature measurement system The resistance temperature detector: ( R T = R 0 1+αT + βt 2 ). The current transmitter: i = KR T + K M R T ΔT a + K I ΔT a + a, where ΔT a is a deviation in ambient temperature from 20 C. The recorder: T M = Ki + a. 21 22 Example (cont d...) Going through the calculations we see that estimated temperature has a bias of 0.05 C (at T = 117 C) and a standard deviation σ TM =0.49 C. A 2σ interval contains approximately 95 % of all measurements. Modeling using error bands In situations where element non-linearity, hysteresis, an environmental effects are small, they can simply be represented by a uniform error distribution, or error band. Measured temperature, T M 130 125 120 115 110 105 100 100 105 110 115 120 125 130 True temperature, T Estimated temperature 2σ interval 23 In a system of N such elements, the overall error distribution will approach the Normal distribution (by the central limit theorem). In these cases, the calculations of the error propagation simplifies. See pages 39 40 in the book for details. 24
Error reduction techniques For different types of elements and effects, different error reduction techniques can be applied. These techniques will be covered later in the course, when we go through specific sensing elements. Read section 3.3 in the text book on your own. Summary Repeatability, accuracy, and tolerance of sensing elements. Variance and mean of system quantities. Uncertainty analysis of measurement systems. Error bands and confidence intervals. All this for systems in the steady state. 25 26 Next lecture Dynamic characteristics of measurement systems. First and second order elements. Identification of dynamics: Step responses. Sinusoidal responses. Dynamic errors. Dynamic compensation. Recommended exercises 2.5. 3.1, 3.2, 3.4 3.9. Also, implement 3.8 in MATLAB and plot as a function of the force, with a 2σ interval. 27 28
Questions? 29