Robotics process control book Part 9 doc

4.3 Frequency Analysis 137 The graph of G(jω) G(jω) = |G(jω)|e j arg G(jω) = [G(jω)] + j[G(jω)] (4.3.36) in the complex plane is called the Nyquist diagram. The magnitude and phase angle can be expressed as follows: |G(jω)| =  {[G(jω)]} 2 + {[G(jω)]} 2 (4.3.37) |G(jω)| =  G(jω)G(−jω) (4.3.38) tan ϕ = [G(jω)] [G(jω)] (4.3.39) ϕ = arctan [G(jω)] [G(jω)] (4.3.40) Essentially, the Nyquist diagram is a polar plot of G(jω) in which frequency ω appears as an implicit parameter. The function A = A(ω) is called magnitude frequency response and the function ϕ = ϕ(ω) phase angle frequency response. Their plots are usually given with logarithmic axes for frequency and magnitude and are referred to as Bode plots. Let us investigate the logarithm of A(ω) exp[jϕ(ω)] ln G(jω) = ln A(ω) + jϕ(ω) (4.3.41) The function ln A(ω) = f 1 (log ω) (4.3.42) defines the magnitude logarithmic amplitude frequency response and is shown in the graphs as L(ω) = 20 log A(ω) = 20 0.434 ln A(ω). (4.3.43) L is given in decibels (dB) which is the unit that comes from the acoustic theory and merely rescales the amplitude ratio portion of a Bode diagram. Logarithmic phase angle frequency response is defined as ϕ(ω) = f 2 (log ω) (4.3.44) Example 4.3.1: Nyquist and Bode diagrams for the heat exchanger as the first order system The process transfer function of the heat exchanger was given in ( 4.3.2). G(jω) is given as G(jω) = Z 1 T 1 jω + 1 = Z 1 (T 1 jω −1) (T 1 jω) 2 + 1 = Z 1 (T 1 ω) 2 + 1 − j Z 1 T 1 ω (T 1 ω) 2 + 1 = Z 1  (T 1 ω) 2 + 1 e −j arctan T 1 ω The magnitude and phase angle are of the form A(ω) = Z 1  (T 1 ω) 2 + 1 ϕ(ω) = −arctan T 1 ω Nyquist and Bode diagrams of the heat exchanger for Z 1 = 0.4, T 1 = 5.2 min are shown in Figs. 4.3.2, 4.3.3, respectively. 138 Dynamical Behaviour of Processes 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 Re Im Figure 4.3.2: The Nyquist diagram for the heat exchanger. 10 −2 10 −1 10 0 −25 −20 −15 −10 −5 Frequency [rad/min] Gain [dB] 10 −2 10 −1 10 0 −80 −60 −40 −20 0 Frequency [rad/min] Phase [deg] Figure 4.3.3: The Bode diagram for the heat exchanger. 4.3 Frequency Analysis 139 0 -20 1 10 100 L[dB] ω [rad/min] 20log Z 1 ω 1 Figure 4.3.4: Asymptotes of the magnitude plot for a first order system. 4.3.3 Frequency Characteristics of a First Order System In general, the dependency [G(jω)] on [G(jω)] for a first order system described by (4.3.2) can easily be found from the equations u = [G(jω)] = Z 1 (T 1 ω) 2 + 1 (4.3.45) v = [G(jω)] = − Z 1 T 1 ω (T 1 ω) 2 + 1 (4.3.46) Equating the terms T 1 ω in both equations yields (v −0) 2 −  u − Z 1 2  =  Z 1 2  2 (4.3.47) which is the equation of a circle. Let us denote ω 1 = 1/T 1 . The transfer function ( 4.3.2) can be written as G(s) = ω 1 Z 1 s + ω 1 . (4.3.48) The magnitude is given as A(ω) = Z 1 ω 1  ω 2 1 + ω 2 (4.3.49) and its logarithm as L = 20 log Z 1 + 20 log ω 1 − 20 log  ω 2 1 + ω 2 . (4.3.50) This curve can easily be sketched by finding its asymptotes. If ω approaches zero, then L → 20 log Z 1 (4.3.51) and if it approaches infinity, then  ω 2 1 + ω 2 → √ ω 2 (4.3.52) L → 20 log Z 1 + 20 log ω 1 − 20 logω. (4.3.53) This is the equation of an asymptote that for ω = ω 1 is equal to 20 log Z 1 . The slope of this asymptote is -20 dB/decade (Fig 4.3.4). 140 Dynamical Behaviour of Processes Table 4.3.1: The errors of the magnitude plot resulting from the use of asymptotes. ω 1 5 ω 1 1 4 ω 1 1 2 ω 1 ω 1 2ω 1 4ω 1 5ω 1 δ(dB) 0.17 -0.3 -1 -3 -1 -0.3 -0.17 0 - π /2 1 10 100 ϕ ω 1 ω - π /4 [rad/min] Figure 4.3.5: Asymptotes of phase angle plot for a first order system. The asymptotes ( 4.3.51) and (4.3.53) introduce an error δ for ω < ω 1 : δ = 20 log ω 1 − 20 log  ω 2 1 + ω 2 (4.3.54) and for ω > ω 1 : δ = 20 log ω 1 − 20 log  ω 2 1 + ω 2 − [20 log ω 1 − 20 log ω] (4.3.55) The tabulation of errors for various ω is given in Table 4.3.1. A phase angle plot can also be sketched using asymptotes and tangent in its inflex point (Fig 4.3.5). We can easily verify the following characteristics of the phase angle plot: If ω = 0, then ϕ = 0, If ω = ∞, then ϕ = −π/2, If ω = 1/T 1 , then ϕ = −π/4, and it can be shown that the curve has an inflex point at ω = ω 1 = 1/T 1 . This frequency is called the corner frequency. The slope of the tangent can be calculated if we substitute for ω = 10 z (log ω = z) into ϕ = arctan(−T 1 ω): ˙ϕ = 1 1 + x 2 , x = −T 1 ω dϕ dz = −2.3 1 + (−T 1 10 z ) 2 T 1 10 z dϕ d log ω = −2.3 1 + (−T 1 ω) 2 T 1 ω for ω = 1/T 1 dϕ d log ω = −1.15 rad/decade -1.15 rad corresponds to −66 o . The tangent crosses the asymptotes ϕ = 0 and ϕ = −π/2 with error of 11 o 40  . 4.3 Frequency Analysis 141 4.3.4 Frequency Characteristics of a Second Order System Consider an underdamped system of the second order with the transfer function G(s) = 1 T 2 k s 2 + 2ζT k s + 1 , ζ < 1. (4.3.56) Its frequency transfer function is given as G(jω) = 1 T 2 k   1 T 2 k − ω 2  2 +  2ζ T k  2 ω 2 exp  j arctan − 2ζ T k ω 1 T 2 k − ω 2  (4.3.57) A(ω) = 1 T 2 k   1 T 2 k − ω 2  2 +  2ζ T k  2 ω 2 (4.3.58) ϕ(ω) = arctan − 2ζ T k ω 1 T 2 k − ω 2 . (4.3.59) The magnitude plot has a maximum for ω = ω k where T k = 1/ω k (resonant frequency). If ω = ∞, A = 0. The expression M = A(ω k ) A(0) = A max A(0) (4.3.60) is called the resonance coefficient. If the system gain is Z 1 , then L(ω) = 20 log |G(jω)| = 20 log       Z 1 (jω) 2 ω 2 k + 2 ζ ω k jω + 1       (4.3.61) L(ω) = 20 log     Z 1 T 2 k (jω) 2 + 2ζT k jω + 1     (4.3.62) L(ω) = 20 log Z 1 − 20 log  (1 −T 2 k ω 2 ) 2 + (2ζT k ω) 2 (4.3.63) It follows from (4.3.63) that the curve L(ω) for Z 1 = 1 is given by summation of 20 log Z 1 to normalised L for Z 1 = 1. Let us therefore investigate only the case Z 1 = 1. From (4.3.63) follows L(ω) = −20 log  (1 −T 2 k ω 2 ) 2 + (2ζT k ω) 2 . (4.3.64) In the range of low frequencies (ω  1/T k ) holds approximately L(ω) ≈ −20 log √ 1 = 0. (4.3.65) For high frequencies (ω  1/T k ) and T 2 k ω 2  1 and (2ζT k ω) 2  (T 2 k ω 2 ) 2 holds L(ω) ≈ −20 log(T k ω) 2 = −2 20 log T k ω = −40 log T k ω. (4.3.66) Thus, the magnitude frequency response can be approximated by the curve shown in Fig 4.3.6. Exact curves deviate with an error δ from this approximation. For 0.38 ≤ ζ ≤ 0.7 the values of δ are maximally ±3dB. The phase angle plot is described by the equation ϕ(ω) = −arctan 2ζT k 1 −T 2 k ω 2 . (4.3.67) At the corner frequency ω k = 1/T k this gives ϕ(ω) = −90 o . Bode diagrams of the second order systems with Z 1 = 1 and T k = 1 min are shown in Fig. 4.3.7. 142 Dynamical Behaviour of Processes 0 -40 1 10 100 L[dB] ω ω = 1/Τ k k [rad/min] Figure 4.3.6: Asymptotes of magnitude plot for a second order system. 10 −1 10 0 10 1 −60 −40 −20 0 20 40 Frequency (rad/s) Gain (dB) 10 −1 10 0 10 1 −200 −150 −100 −50 0 Frequency (rad/s) Phase (deg) ζ=0.05 ζ=0.2 ζ=0.5 ζ=1.0 Figure 4.3.7: Bode diagrams of an underdamped second order system (Z 1 = 1, T k = 1). 4.3 Frequency Analysis 143 Re Im - π /2 ω = ω→ 0 0 Figure 4.3.8: The Nyquist diagram of an integrator. 4.3.5 Frequency Characteristics of an Integrator The transfer function of an integrator is G(s) = Z 1 s (4.3.68) where Z 1 = 1/T I and T I is the time constant of the integrator. We note, that integrator is an astatic system. Substitution for s = jω yields G(jω) = Z 1 jω = −j Z 1 ω = Z 1 ω e −j π 2 . (4.3.69) From this follows A(ω) = Z 1 ω , (4.3.70) ϕ(ω) = − π 2 . (4.3.71) The corresponding Nyquist diagram is shown in Fig. 4.3.8. The curve coincides with the negative imaginary axis. The magnitude is for increasing ω decreasing. The phase angle is independent of frequency. Thus, output variable is always delayed to input variable for 90 o . Magnitude curve is given by the expression L(ω) = 20 log A(ω) = 20 log Z 1 ω (4.3.72) L(ω) = 20 log Z 1 − 20 log ω. (4.3.73) The phase angle is constant and given by ( 4.3.71). If ω 2 = 10ω 1 , then 20 log ω 2 = 20 log 10ω 1 = 20 + 20 log ω 1 , (4.3.74) thus the slope of magnitude plot is -20dB/decade. Fig. 4.3.9 shows Bode diagram of the integrator with T I = 5 min. The values of L(ω) are given by the summation of two terms as given by (4.3.73). 4.3.6 Frequency Characteristics of Systems in a Series Consider a system with the transfer function G(s) = G 1 (s)G 2 (s) . . . G n (s). (4.3.75) Its frequency response is given as G(jω) = n  i=1 A i (ω)e jϕ i (ω) (4.3.76) G(jω) = exp  j n  i=1 ϕ i (ω)  n  i=1 A i (ω), (4.3.77) 144 Dynamical Behaviour of Processes 10 −1 10 0 10 1 10 2 −60 −50 −40 −30 −20 −10 0 10 Frequency [rad/min] Gain [dB] 10 −1 10 0 10 1 10 2 −91 −90.5 −90 −89.5 −89 Frequency [rad/min] Phase [deg] Figure 4.3.9: Bode diagram of an integrator. and 20 log A(ω) = n  i=1 20 log A i (ω), (4.3.78) ϕ(ω) = n  i=1 ϕ i (ω). (4.3.79) It is clear from the last equations that magnitude and phase angle plots are obtained as the sum of individual functions of systems in series. Example 4.3.2: Nyquist and Bode diagrams for a third order system Consider a system with the transfer function G(s) = Z 3 s(T 1 s + 1)(T 2 s + 1) . The function G(jω) is then given as G(jω) = Z 3 jω(T 1 jω + 1)(T 2 jω + 1) . Consequently, for magnitude follows L(ω) = 20 log Z 3 ω  (T 1 ω) 2 + 1  (T 2 ω) 2 + 1 L(ω) = 20 log Z 3 − 20 log ω −20 log  (T 1 ω) 2 + 1 − 20 log  (T 2 ω) 2 + 1 and for phase angle: ϕ(ω) = − π 2 − arctan(T 1 ω) −arctan(T 2 ω) Nyquist and Bode diagrams for the system with Z 3 = 0.5, T 1 = 2 min, and T 2 = 3 min are given in Figs. 4.3.10 and 4.3.11. 4.3 Frequency Analysis 145 −2.5 −2 −1.5 −1 −0.5 0 −50 −40 −30 −20 −10 0 10 Re Im Figure 4.3.10: The Nyquist diagram for the third order system. 10 −3 10 −2 10 −1 10 0 10 1 −100 −50 0 50 100 Frequency [rad/min] Gain [dB] 10 −3 10 −2 10 −1 10 0 10 1 −300 −250 −200 −150 −100 −50 Frequency [rad/min] Phase [deg] Figure 4.3.11: Bode diagram for the third order system. 146 Dynamical Behaviour of Processes 4.4 Statistical Characteristics of Dynamic Systems Dynamic systems are quite often subject to input variables that are not functions exactly spec- ified by time as opposed to step, impulse, harmonic or other standard functions. A concrete (deterministic) time function has at any time a completely determined value. Input variables may take different random values through time. In these cases, the only characteristics that can be determined is probability of its influence at certain time. This does not imply from the fact that the input influence cannot be foreseen, but from the fact that a large number of variables and their changes influence the process simultaneously. The variables that at any time are assigned to a real number by some statement from a space of possible values are called random. Before investigating the behaviour of dynamic systems with random inputs let us now recall some facts about random variables, stochastic processes, and their probability characteristics. 4.4.1 Fundamentals of Probability Theory Let us investigate an event that is characterised by some conditions of existence and it is known that this event may or may not be realised within these conditions. This random event is char- acterised by its probability. Let us assume that we make N experiments and that in m cases, the event A has been realised. The fraction m/N is called the relative occurrence. It is the experimen- tal characteristics of the event. Performing different number of experiments, it may be observed, that different values are obtained. However, with the increasing number of experiments, the ratio approaches some constant value. This value is called probability of the random event A and is denoted by P (A). There may exist events with probability equal to one (sure events) and to zero (impossible events). For all other, the following inequality holds 0 < P (A) < 1 (4.4.1) Events A and B are called disjoint if they are mutually exclusive within the same conditions. Their probability is given as P (A ∪ B) = P(A) + P(B) (4.4.2) An event A is independent from an event B if P (A) is not influenced when B has or has not occurred. When this does not hold and A is dependent on B then P (A) changes if B occurred or not. Such a probability is called conditional probability and is denoted by P (A|B). When two events A, B are independent, then for the probability holds P (A|B) = P (A) (4.4.3) Let us consider two events A and B where P (B) > 0. Then for the conditional probability P (A|B) of the event A when B has occurred holds P (A|B) = P (A ∩ B) P (B) (4.4.4) For independent events we may also write P (A ∩ B) = P(A)P (B) (4.4.5) 4.4.2 Random Variables Let us consider discrete random variables. Any random variable can be assigned to any real value from a given set of possible outcomes. A discrete random variable ξ is assigned a real value from [...]... between random variables where Cov(ξi , ξj ) = Cov(ξj , ξi ) (4.4. 49) The main diagonal of a covariance matrix contains variances of random variables ξ i : 2 Cov(ξi , ξi ) = E [(ξi − E[ξi ])(ξi − E[ξi ])] = σi (4.4.50) 152 Dynamical Behaviour of Processes ξ 1 2 3 t1 t2 tn tm t Figure 4.4.3: Realisations of a stochastic process 4.4.3 Stochastic Processes When dealing with dynamic systems, some phenomenon... infinity, we speak about a stochastic (random) process A stochastic process is given as a set of time-dependent random variables ξ(t) Thus, the concept of a random variable ξ is broadened to a random function ξ(t) It might be said that a stochastic process is such a function of time whose values are at any time instant random variables A random variable in a stochastic process yields random values not only... statistically completely determined by the density functions f 1 , , fn and the relationships among them The simplest stochastic process is a totally independent stochastic process (white noise) For this process, any random variables at any time instants are mutually independent For this process holds f2 (x1 , t1 ; x2 , t2 ) = f (x1 , t1 )f (x2 , t2 ) (4.4.53) as well as fn (x1 , t1 ; x2 , t2 ; ; xn , tn... values of a stochastic process ξ(t) in the time instants t1 and t2 4.4 Statistical Characteristics of Dynamic Systems 153 Sometimes, also n-dimensional density function f2 (x1 , t1 ; x2 , t2 ; ; xn , tn ) is introduced and is analogously defined as a probability that a process ξ(t) passes through n points with deviation not greater than dx1 , dx2 , , dxn A stochastic process is statistically... belongs to the set of random variables ξ(t) is called the realisation of a stochastic process A stochastic process in some fixed time instants t1 , t2 , , tn depends only on the outcome of the experiment and changes to a corresponding random variable with a given density function From this follows that a stochastic process can be determined by a set of density functions that corresponds to random variables... function, the expected value of a stochastic process is given by ∞ µ(t) = E[ξ(t)] = xf1 (x, t)dx (4.4.55) −∞ In (4.4.55) the index of variables of f1 is not given as it can be arbitrary Variance of a stochastic process can be written as ∞ D[ξ(t)] = −∞ 2 [x − µ(t)]2 f1 (x, t)dx (4.4.56) D[ξ(t)] = E[ξ (t)] − (E[ξ(t)])2 (4.4.57) Expected value of a stochastic process µ(t) is a function of time and it is... t1 ; x2 , t2 )dx1 dx2 (4.4. 59) For the auto-correlation function follows Rξ (t1 , t2 ) = Covξ (t1 , t2 ) − µ(t1 )µ(t2 ) (4.4.60) Similarly, for two stochastic processes ξ(t) and η(t), we can define the correlation function Rξη (t1 , t2 ) = E[ξ(t1 )η(t2 )] (4.4.61) and the covariance function Covξη (t1 , t2 ) = E[(ξ(t1 ) − µ(t1 ))(η(t2 ) − µη (t2 ))] (4.4.62) If a stochastic process with normal distribution... difference between the expected value of the squared random variable and squared expected value of random variable Because the following holds always E[ξ 2 ] ≥ (E[ξ])2 , (4.4. 19) 4.4 Statistical Characteristics of Dynamic Systems 1 49 F(x) 1 F(b) F(a) a) 0 a b x f(x) b) 0 a b x Figure 4.4.2: Distribution function and corresponding probability density function of a continuous random variable variance is... where the variable f (x) = dF (x) dx (4.4.27) 150 Dynamical Behaviour of Processes is called probability density Figure 4.4.2b shows an example of f (x) Thus, the distribution function F (x) may be written as x F (x) = f (x)dx (4.4.28) −∞ Because F (x) is non-decreasing, the probability density function must be positive f (x) ≥ 0 (4.4. 29) The probability that a random variable is within an interval (a,... function of time and it is the mean value of all realisations of a stochastic process The variance D[ξ(t)] gives information about dispersion of realisations with respect to the mean value µ(t) Based on the information given by the two-dimensional density function, it is possible to find an influence between the values of a stochastic process at times t1 and t2 This is given by the auto-correlation function . Behaviour of Processes 10 −1 10 0 10 1 10 2 −60 −50 −40 −30 −20 −10 0 10 Frequency [rad/min] Gain [dB] 10 −1 10 0 10 1 10 2 91 90 .5 90 − 89. 5 − 89 Frequency [rad/min] Phase [deg] Figure 4.3 .9: Bode. simplest stochastic process is a totally independent stochastic process (white noise). For this process, any random variables at any time instants are mutually independent. For this process holds f 2 (x 1 ,. variable. Because the following holds always E[ξ 2 ] ≥ (E[ξ]) 2 , (4.4. 19) 4.4 Statistical Characteristics of Dynamic Systems 1 49 0 a b x f(x) F(a) F(b) F(x) a) b) 0 x a b 1 Figure 4.4.2: Distribution