Process Control & Instrumentation (CH 3040)

First-order systems Process Control & Instrumentation (CH 3040) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January - April 010 Lectures: Mon, Tue, Wed, Fri Extra class: Thu A first-order system a transfer function of the form: where Kp and τp are the gain and timeconstants of the system respectively The impulse, step and sinusoidal response are given by: where g(t) = K p τ p e t τp K p τ p s +1 = K p/τ p s +1/τ p y s (t) = K p (1 e t τp ) y f (t) = B sin(ω 0 t + φ) at steady-state B A = K p ; φ = tan τ 1 (τ p ω 0 ) p ω0 +1 Example s +1 u(t) = sin(π(0.)t) Sinusoidal (Frequency) Response Since sinusoids are also characterized by their frequencies, the steady-state response to such signals are also known as frequency response The LTI system, due to its linearity, will produce the same shape as the input, but at steadystate. The input will be, however, scaled and shifted The transient portion will be due to its inertia, which will die down if the system is stable. Consider a first-order system G(s) excited with a sine input u(t) = Asin(ω0t) Y (s) = Kp Aω 0 τ ps+1 s +ω0 = y(t) =c 1 e t τp + c e jω0t + c 3 e jω0t The transient portion of the response dies off leaving behind the complex exponentials! Clearly c and c3 have to be complex conjugates of each other, i.e., c = c3* c = lim G(s) A A ; c = lim G(s) s jω0 jω 0 s jω0 jω 0 Thus, the steady-state response is, using polar representation: G(jω 0 )= G(jω 0 ) e j G(jω0) y s.s. (t) =(c e jω0t )=( A G(jω 0) e j G(jω0) e j π e jω 0t )=A G(jω 0 ) sin(ω 0 t + G(jω 0 ) Frequency response of LTI systems A very interesting result emerges: The steady-state response of an LTI system to a sinusoidal input is a scaled and shifted sine wave of the same frequency (as the input) u(t) = Asin(ω0t) LTI yss(t) = Bsin(ω0t + Φ) For any stable system, the steady-state output can be calculated by noting B A (ω 0) = G(jω 0 ) (Amplitude Ratio) ω0 refers to the input frequency φ(ω 0 ) = G(jω 0 ) (Phase shift) Observe that both amplification/attenuation and phase shift vary with the input frequency For causal systems, the phase shift Φ 0 (output can lag or be in phase with the input) Example (first-order system): B A = K p ; φ = tan τ 1 (τ p ω 0 ) p ω0 +1 3 4

First-order system The amplitude ratio is usually plotted as decibels (db) (after Graham Bell) db = 0log10(AR). Thus a zero decibel corresponds to no attenuation/amplification ωc: Corner frequency Bode plots Plots of db vs. ω and phase shift vs. ω were first proposed by Bodé Hence these plots are popularly known as Bodé plots The ω on the x-axis refers to the input frequency and is on a log-scale For a first-order system, K p db = 0 log 10 τp ω0 +1 = 0 log 10 K p 10 log 10 (τ p ω 0 + 1) 0 log 10 K p when τ p ω 0 1 0 log 10 K p 0 log 10 τ p ω 0 when τ p ω 0 1 φ = tan 1 (τ p ω 0 ) 0 when τ p ω 0 1 π when τ pω 0 1 Example s +1 The db value falls off at -0dB/octave at high freqs. The asymptotes of the db plot at low and high freqs. meet at a frequency - corner frequency The log scale for the x-axis helps in a linear curve for the db in the roll-off regime Bode plots of higher-order systems can be easily constructed from the knowledge of plots for lower-order systems Any higher-order system can be expressed as first- and/or second-order systems in series The AR ratio for the overall system is a product of the ARs of the individual systems Consequently, dboverall = Sum of dbindividuals (due to the property of logarithms) All first-order systems produce lagged outputs (lag systems) The corner frequency is, in fact, ωc = 1/τp A single pole will always shift high freq. inputs by -90º Phaseoverall = Sum of (phase shifts)individuals (angles add up in polar representations) 5 6 Frequency response function (FRF) The quantity G(jω) carries vital information on how the system treats inputs of different frequencies For this reason, it is known as the frequency response function In general, the response of an LTI system an input depends on its frequency! From this viewpoint, every LTI system is a (linear) filter (only certain inputs are passed ). In the first-order case, the high-frequency inputs are relatively filtered out => first-order systems are low-pass filters The FRF is of great value in communications, control, identification, etc. A good understanding of the frequency-domain behaviour of LTI systems is critical to design of filters, receivers, etc. In control and linear systems theory, FRFs provide powerful ways of handling delay systems, stability criteria, controller design, handling disturbances and modelling. In identification, the bandwidth of the system is useful in designing optimal input signals. FRF and Impulse Response The FRF is related to the impulse response through the Fourier Transform where y(t) = y(t)e jωt dt = g(τ)u(t τ) dτ = Y (ω) = G(ω)U(ω) G(ω) = Thus, knowing the FRF is equivalent to knowing the IR of the system Observe that Fourier Transform is a special case of Laplace transform => The quantity G(jω) is also known as the A.C. gain g(τ)u(t τ)e jωt dτ dt g(τ)e jωτ dτ = FRF = Fourier Transform of IR FRF = lim s jω G(s) The D.C. Gain is easily obtained from the A.C. gain by evaluating at ω = 0 (constant signal!) 7 8

Use of FRFs in control A quote Stability criteria are most generally provided in frequency-domain Bode s stability criterion (easily handles delays unlike the traditional root-locus techniques) Nyquist s stability criterion (the most general criterion for linear systems) Frequency response techniques have been increasingly proven to be powerful tools in control system design and analysis A quote due to A.C. Hall Robustness analysis Using Nyquist s or Bode s stability criteria, one can determine the margins of stability once a controller has been designed Uncertainty/disturbance/modelling error descriptions are easily characterized in frequency domain (low-frequency, high-frequency disturbances, mismatch in the FRFs, etc.) Ability to decouple the effects of delays, time-constants and gains is a very powerful feature of the frequency response functions. Gain does not affect phase of the system, but only affects the magnitude plot Delay does not affect the magnitude plot, while only shapes the phase plot Source: Frequency Response (R. Oldenburger, ed.), p. 4, MacMillan, New York, 1956. Time-constants affect both the magnitude and phase plots 9 10 Second-order systems Second-order systems (two poles/states) are classified according to their pole locations: Responses of second-order systems The step responses of second-order systems are shown below Poles (location) Type Impulse response Purely imaginary (IA) Complex, in the LHP Clearly all can be seen as variants of a purely oscillatory second-order system The general transfer function is: Oscillatory Underdamped Sinusoidal with frequency = natural frequency of the system ωn Damped sinusoid (damping not enough to suppress the oscillatory nature) Repeated real, in the LHP Critically damped Exponential (damping just about sufficient) Real, in the LHP Overdamped Sum of exponentials (damping more than sufficient) Overdamped 100s + 30s +1 Underdamped 100s + 10s +1 ζ =0 (no damping); ζ < 1 (underdamped); ζ =1 (critical damping); ζ > 1 (overdamping); 11 1

Frequency response of underdamped systems Frequency response of overdamped systems Underdamped systems exhibit a peak in the magnitude plot For zero damped systems, resonance occurs when the input frequency coincides with the natural frequency of the system The db roll-off is -40 db/octave at high frequencies, while the overall phase shift at high frequencies is -180º. The peak in magnitude plot is only observed when damping ratio is Zero damping AR max = K p ζ 1 ζ, ω r = ω n 1 ζ 0 < ζ < 1 The overdamped system behaves like two first-order systems in series Hence the values of db and phase are merely the sum of db and phase values of the two respective first-order systems The overall roll-off rate is 40 db/ octave at high frequencies τ p1 = 6.18, ω c1 =0.038, 100s + 30s +1 τ p =3.8 sec ω c =0.618 rad/sec One observes two corner frequencies, one at ωc = 1/τ1 and the other at ωc = 1/τ 13 14 A pure delay system is described by y(t) = u(t - D), D! 0 The transfer function is e -Ds How is delay defined? Delay systems Delay is the time taken for the system to respond for the first time after the input is changed. Delay is NOT settling time or time-constant It is a characteristic of the system that is different from its inertia Why do we encounter delays in systems? Transportation lags: Change in input is introduced at a distance away from the system Measurement lags: Output is measured at a distance away from the point of change Higher-order dynamics: Higher-order systems exhibit such sluggishness that appears as delay The first two are true delays present in a system, while the third one is an apparent delay u(t) 0 y(t) 0 D t t Bode plots of delay systems Delays in systems only influence the phase plot of the systems Delays cause a linear roll-off in phase Delays present serious challenges and limitations in controller design Shower example: The shower head is usually about 1- m from the point of mixing => measurement delay. Assume the process is initially at steady-state. A hotter stream is desired so that more hot water is let in, but no immediate change is observed. Unaware of the measurement delay, the hot water valve is opened further, by when the effect of the earlier mixing arrives. After the delay, the hotter stream arrives, forcing the hot water valve to be closed partially, but due to delay, the valve is closed more than by the necessary amount! - this continues forever producing oscillatory response G(jω) = e jdω = G(jω) = 1 and G(jω) = Dω A delay system is an all-pass filter 15 16

Apparent Delays Approximations of Delays Higher-order systems produce such sluggishness that can appear as delays Traffic flow example: At the lights, the signal changes from red to green (step change). Assuming N (large) vehicles in front of your vehicle, to an observer (usually behind you) observing only the signal change and your movements, he/she feels that you have responded only after a finite time. None among the N drivers exhibit a delay, but this is due to collective inertia. N tanks connected in series: Imagine N tanks connected in series and an observer stands at the last tank to observe the level change for a change in the inlet flow to the first tank. To the observer, the response of the N th tank is delayed. From a mathematical viewpoint, delays are infinite-order systems e Ds = 1 e Ds = 1 State-space viewpoint: = 1+Ds + D! s + 1 lim 1+ Ds N N In the standard state-space form, a delay system is described by infinite states! N A model containing delays may have to be approximated as a lower-order model for purposes of controller design For this purposes, very often different approximation of delays are used Single-zero approximation: e Ds 1 Ds Single-pole approximation: e Ds 1 1+Ds 1 D Pade s first-order approximation: e Ds s 1+ D s Pade s second-order approximation: e Ds 1 D s + D 1 s 1+ D s + D 1 s (For small delays) (Error of O(s 3 )) (Error of O(s 5 )) Each of these approximations has its use in controller design The above approximations are used to approximate parts of higher-order systems as delays 17 18 Revisiting phase plots It is useful to examine the Bode plots of certain basic elements so that for a given system, one can construct an approximate overall Bode plot Constructing a Bode plot: Example (s + 1) (10s + 1)(4s + 1)(s + 1) e 3s Consider, the Bode plot can be easily constructed from its parts - a lead, a constant (gain element), a delay and three lags. Constant system First-order (Lag element) First-order neg. gain FRF of unstable systems have little meaning - because they do not exist theoretically! First-order unstable Lead element Integrator 19 0

Putting them together Effects of Zeros The Bode diagrams of the individual elements can now be added up to arrive at the overall Bode plot for the third-order (plus delay) system Until now we have examined the effects of pole locations on LTI systems Poles are a characteristic of the systems, arising due to inertia (independent of the inputs) However, zeros can have a significant influence on the response depending on the locations What are zeros anyway? They tell us which inputs are blocked by the system (also known as transmission zeros) Why do they arise? Due to the way (i) the inputs interact with the system (the input derivatives affect the output), (ii) two or more subsystems affect each other Due to approximations of delays They can drastically limit the performance and stability margins of a controller A study of effect of zeros is useful in controller design In filter and controller design, zeros are used to nullify or weaken the effects of poles 1 First-order systems with a zero Consider a first-order system with a zero The system can be viewed as a sum of two subsystems (1/5) + 4/5 Thus, both feedthrough (static) and inertial effects The net effect is a discontinuity in the step response Such systems are known as jump systems The zero affects both db and phase plots (s + 1) Example: Zero located closer to the origin (10s + 1) The zero is located closer to the origin than the pole is. The system exhibits high-pass filter characteristics Since a LHP zero produces lead, while an LHP pole induces lag, first-order systems are known as lead-lag systems An LHP zero introduces a 90º lead at high frequencies An RHP zero introduces a 90º lag at high frequencies The location of the zero strongly determines the filtering characteristics u(t) = sin(0.t) u(t) = sin(10t) 3 4

Minimum-phase and Non-minimum phase systems Two systems can have the same amplitude ratio but a different phase For a given AR, the system with the least phase is said to be minimum phase Delays and RHP zeros cause the systems to be non-minimum phase Example: (s + 1) What problems arise due to non-minimum phase? vs. (s 1) Second-order systems with zero The responses of second-order systems with zero depends on the location of the zero (i.e., in LHP or RHP) (5s + 1) (10s + 1)(4s + 1) = 5/3 10s +1 + 1/3 4s +1 = y s(t) = (5/3)(1 e 0.1t )+(1/3)(1 e 0.5t ) ( 5s + 1) (10s + 1)(4s + 1) = 5 10s +1 3 4s +1 = y s (t) = 5(1 e 0.1t ) 3(1 e 0.5t ) Non-minimum phase is due to RHP zero(s) and/or delays The ideal controller is the inverse of process Non-minimum phase systems are only partially invertible (RHP zeros / delays will cause unstable poles / non-causal parts in process inverse) Thus, non-invertible portions cannot be controlled and result in performance limitations 5 6 Inverse Response FRFs of systems with and w/o inverse response u(t) K 1 τ 1 s +1 K τ s +1 Second-order systems with a zero in RHP exhibit a phenomenon known as the inverse response when a step input is applied. + - y(t) K 1 K > τ 1 τ > 1 The net phase shift is larger for systems with inverse response (70º) than that of those without the inverse response Once again this is due to the non-minimum phase behaviour Observe how the phase plot begins from π and ends up at π/ radians A system is said to exhibit inverse response when the first direction of step response is in a direction opposite to that of the final response (slope of the step response at t = 0) The slope of the response at t = 0 can be calculated from the initial value theorem. The inverse response is due to the fact that a second-order system with RHP zero is equivalent to two competing subsystems operating at different time scales Inverse response leads to non-minimum phase behaviour Any LTI system will exhibit inverse response if it has odd number of RHP zeros 7 8

Higher-order systems Higher-order systems produce more sluggish step responses (larger no. of inertial systems) The initial sluggishness appears as delay to an observer of such systems The larger lags at high frequencies for higher-order systems explains the initial sluggishness Comparison of responses with increase in order Comparison of Bode plots with increase in order 9