Ebook Process dynamics and control (4th edition): Part 2

Continued part 1, part 2 of ebook Process dynamics and control (4th edition) provide readers with content about: frequency response analysis and control system design; feed forward and ratio control; advanced process control; enhanced single-loop control strategies; digital sampling, filtering, and control; multi loop and multi variable control; real-time optimization; model predictive control;... Please refer to the ebook for details!

Chapter 14 Frequency Response Analysis and Control System Design CHAPTER CONTENTS 14.1 Sinusoidal Forcing of a First-Order Process 14.2 Sinusoidal Forcing of an nth-Order Process 14.3 Bode Diagrams 14.3.1 14.3.2 14.3.3 First-Order Process Integrating Process Second-Order Process 14.3.4 Process Zero 14.3.5 Time Delay 14.4 Frequency Response Characteristics of Feedback Controllers 14.5 Nyquist Diagrams 14.6 Bode Stability Criterion 14.7 Gain and Phase Margins Summary In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness assessment Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979) We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function model of any linear process Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability 244 analysis Next we introduce two useful metrics for relative stability, namely gain and phase margins These metrics indicate how close a control system is to instability A related issue is robustness, that is, the sensitivity of control system performance to process variations and to uncertainty in the process model The design of robust feedback control systems is considered in Appendix J 14.1 SINUSOIDAL FORCING OF A FIRST-ORDER PROCESS We start with the response properties of a first-order process when forced by a sinusoidal input and show how the output response characteristics depend on the frequency of the input signal This is the origin of 14.1 the term frequency response The responses for firstand second-order processes forced by a sinusoidal input were presented in Chapter Recall that these responses consisted of sine, cosine, and exponential terms Specifically, for a first-order transfer function with gain K and time constant τ, the response to a general sinusoidal input, x(t) = A sin ωt, is KA (ωτe−t∕τ − ωτ cos ωt + sin ωt) (5-23) ω2 τ2 + where y is in deviation form If the sinusoidal input is continued for a long time, the exponential term (ωτe−t/τ ) becomes negligible The remaining sine and cosine terms can be combined via a trigonometric identity to yield y(t) = y𝓁 (t) = √ KA ω2 τ2 + sin (ωt + ϕ) (14-1) where ϕ = −tan−1 (ωτ) The long-time response y𝓁 (t) is called the frequency response of the first-order system and has two distinctive features (see Fig 14.1) The output signal is a sine wave that has the same frequency, but its phase is shifted relative to the input sine wave by the angle ϕ (referred to as the phase shift or the phase angle); the amount of phase shift depends on the forcing frequency ω The sine wave has an amplitude  that is a function of the forcing frequency: KA  = √ ω2 τ2 + (14-2) Dividing both sides of Eq 14-2 by the input signal amplitude A yields the amplitude ratio (AR) AR =  K =√ A ω2 τ2 + (14-3a) which can, in turn, be divided by the process gain to yield the normalized amplitude ratio (ARN ): ARN = AR =√ K ω τ2 + (14-3b) Next we examine the physical significance of the preceding equations, with specific reference to the blending P A A Output, y Time shift, Δt Input, x Time, t Figure 14.1 Attenuation and time shift between input and output sine waves The phase angle ϕ of the output signal is given by ϕ = Δt/P × 360∘, where Δt is the time shift and P is the period of oscillation Sinusoidal Forcing of a First-Order Process 245 process example discussed earlier In Chapter 4, the transfer function model for the stirred-tank blending system was derived as X ′ (s) = K3 K1 K2 X ′ (s) + W ′ (s) + W ′ (s) (4-84) τs + 1 τs + τs + 1 Suppose flow rate w2 is varied sinusoidally about a constant value, while the other inlet conditions are kept constant at their nominal values; that is, w′1 (t) = x′1 (t) = Because w2 (t) is sinusoidal, the output composition deviation x′ (t) eventually becomes sinusoidal according to Eq 5-24 However, there is a phase shift in the output relative to the input, as shown in Fig 14.1, owing to the material holdup of the tank If the flow rate w2 oscillates very slowly relative to the residence time τ(ω ≪ 1/τ), the phase shift is very small, approaching 0∘ , whereas the normalized amplitude ̂ ratio (A/KA) is very nearly unity For the case of a low-frequency input, the output is in phase with the input, tracking the sinusoidal input as if the process model were G(s) = K On the other hand, suppose that the flow rate is varied rapidly by increasing the input signal frequency For ω ≫ 1/τ, Eq 14-1 indicates that the phase shift approaches a value of −π/2 radians (−90∘ ) The presence of the negative sign indicates that the output lags behind the input by 90∘ ; in other words, the phase lag is 90∘ The amplitude ratio approaches zero as the frequency becomes large, indicating that the input signal is almost completely attenuated; namely, the sinusoidal deviation in the output signal is very small These results indicate that positive and negative deviations in w2 are essentially canceled by the capacitance of the liquid in the blending tank if the frequency is high enough High frequency implies ω ≫ 1/τ Most processes behave qualitatively similar to the stirred-tank blending system, when subjected to a sinusoidal input For high-frequency input changes, the process output deviations are so completely attenuated that the corresponding periodic variation in the output is difficult (perhaps impossible) to detect or measure Input–output phase shift and attenuation (or amplification) occur for any stable transfer function, regardless of its complexity In all cases, the phase shift and amplitude ratio are related to the frequency ω of the sinusoidal input signal In developments up to this point, the expressions for the amplitude ratio and phase shift were derived using the process transfer function However, the frequency response of a process can also be obtained experimentally By performing a series of tests in which a sinusoidal input is applied to the process, the resulting amplitude ratio and phase shift can be measured for different frequencies In this case, the frequency response is expressed as a table of measured amplitude ratios and phase shifts for selected values of ω However, the method is very time-consuming 246 Chapter 14 Frequency Response Analysis and Control System Design because of the repeated experiments for different values of ω Thus other methods, such as pulse testing (Ogunnaike and Ray, 1994), are utilized, because only a single test is required In this chapter, the focus is on developing a powerful analytical method to calculate the frequency response for any stable process transfer function Later in this chapter, we show how this information can be used to design controllers and analyze the properties of the closed loop system responses 14.2 The shortcut method can be summarized as follows: Step Substitute s = jω in G(s) to obtain G(jω) Step Rationalize G(jω), i.e., express G(jω) as the sum of real (R) and imaginary (I) parts R + jI, where R and I are functions of ω, using complex conjugate multiplication Step The √ output sine wave has amplitude −1 ̂ A = A R2 + I and phase angle √ϕ = tan (I/R) 2 The amplitude ratio is AR = R + I and is independent of the value of A SINUSOIDAL FORCING OF AN nTH-ORDER PROCESS This section presents a general approach for deriving the frequency response of any stable transfer function The physical interpretation of frequency response is not valid for unstable systems, because a sinusoidal input produces an unbounded output response instead of a sinusoidal response A rather simple procedure can be employed to find the sinusoidal response After setting s = jω in G(s), by algebraic manipulation we can separate the expression into real (R) and imaginary (I) terms (j indicates an imaginary component): G(jω) = R(ω) + jI(ω) ϕ = tan (I∕R) G(s) = (14-6b) Both  and ϕ are functions of frequency ω A simple but elegant relation for the frequency response can be derived, where the amplitude ratio is given by √  AR = (14-7) = |G| = R2 + I A The absolute value denotes the magnitude of G, and the phase shift (also called the phase angle or argument of G, ∠G) between the sinusoidal output and input is given by (14-8) ϕ = ∠G = tan−1 (I∕R) Because R(ω) and I(ω) (and hence AR and ϕ) can be obtained without calculating the complete transient response y(t), these characteristics provide a convenient shortcut method to determine the frequency response of transfer functions Equations 14-7 and 14-8 can calculate the frequency response characteristics of any stable G(s), including those with time-delay terms τs + (14-9) SOLUTION First substitute s = jω in the transfer function G(jω) = 1 = τjω + jωτ + (14-10) Then multiply both numerator and denominator by the complex conjugate of the denominator, that is, −jωτ + −jωτ + −jωτ + = 2 (jωτ + 1)(−jωτ + 1) ω τ +1 G(jω) = (14-5)  and ϕ are related to I(ω) and R(ω) by the following relations (Seborg et al., 2004): √ (14-6a)  = A R2 + I −1 Find the frequency response of a first-order system, with (14-4) Similar to Eq 14-1, we can express the long time response for a linear system (cf Eq 14-1) as y𝓁 (t) =  sin(ωt + ϕ) EXAMPLE 14.1 (−ωτ) +j 2 = R + jI ω2 τ2 + ω τ +1 = where (14-11) R= ω2 τ2 + (14-12a) I= −ωτ ω2 τ2 + (14-12b) and From Eq 14-7, √ AR = |G(jω|) = ( )2 ω2 τ2 +1 ( + −ωτ +1 )2 ω2 τ2 Simplifying, √ AR = (1 + ω2 τ2 ) = √ (ω2 τ2 + 1)2 ω2 τ2 + ϕ = ∠G(jω) = tan−1 (−ωτ) = −tan−1 (ωτ) (14-13a) (14-13b) If the process gain had been a positive value K instead of 1, AR = √ K ω2 τ2 + (14-14) and the phase angle would be unchanged (Eq 14-13b) Both the amplitude ratio and phase angle are identical to those values calculated in Section 14.1 using the time-domain derivation 14.3 From this example, we conclude that direct analysis of the complex transfer function G(jω) is computationally easier than solving for the actual long-time output response j𝓁 (t) The computational advantages are even greater when dealing with more complicated processes, as shown in the following Start with a general transfer function in factored form G (s)Gb (s)Gc (s) · · · (14-15) G(s) = a G1 (s)G2 (s)G3 (s) · · · G(s) is converted to the complex form G(jω) by the substitution s = jω: G (jω)Gb (jω)Gc (jω) · · · G(jω) = a (14-16) G1 (jω)G2 (jω)G3 (jω) · · · The magnitude and phase angle of G(jω) are as follows: |Ga (jω)‖Gb (jω)‖Gc (jω)| · · · |G(jω)| = (14-17a) |G1 (jω)‖G2 (jω)‖G3 (jω)| · · · ∠G(jω) = ∠Ga (jω) + ∠Gb (jω) + ∠Gc (jω) + · · · − [∠G1 (jω) + ∠G2 (jω) + ∠G3 (jω) + · · ·] (14-17b) Equations 14-17a and 14-17b greatly simplify the computation of |G(jω)| and ∠G(jω) and, consequently, AR and ϕ, for factored transfer functions These expressions eliminate much of the complex algebra associated with the rationalization of complicated transfer functions Hence, the factored form (Eq 14-15) may be preferred for frequency response analysis On the other hand, if the frequency response curves are generated using software such as MATLAB, there is no need to factor the numerator or denominator, as discussed in Section 14.3 Calculate the amplitude ratio and phase angle for the overdamped second-order transfer function G(s) = K (τ1 s + 1)(τ2 s + 1) SOLUTION Using Eq 14-15, let Ga = K G1 = τ1 s + G2 = τ2 s + Substituting s = jω Ga (jω) = K G1 (jω) = jωτ1 + G2 (jω) = jωτ2 + The magnitudes and angles of each component of the complex transfer function are |Ga | = K √ |G1 | = √ω2 τ21 + |G2 | = ω2 τ22 + ∠Ga = ∠G1 = tan−1 (ωτ1 ) ∠G2 = tan−1 (ωτ2 ) 247 Combining these expressions via Eqs 14-17a and 14-17b yields |Ga (jω)| |G1 (jω)‖G2 (jω)| K √ = √ ω2 τ21 + ω2 τ22 + AR = |G(jω)| = (14-18a) ϕ = ∠G(jω) = ∠Ga (jω) − (∠G1 (jω) + ∠G2 (jω)) = −tan−1 (ωτ1 ) − tan−1 (ωτ2 ) (14-18b) 14.3 BODE DIAGRAMS The Bode diagram (or Bode plot) provides a convenient display of the frequency response characteristics in which AR and ϕ are each plotted as a function of ω Ordinarily, ω is expressed in units of radians/time to simplify inverse tangent calculations (e.g., Eq 14-18b) where the arguments must be dimensionless, that is, in radians Occasionally, a cyclic frequency, ω/2π, with units of cycles/time, is used Phase angle ϕ is normally expressed in degrees rather than radians For reasons that will become apparent in the following development, the Bode diagram consists of: (1) a log–log plot of AR versus ω and (2) a semilog plot of ϕ versus ω These plots are particularly useful for rapid analysis of the response characteristics and stability of closed-loop systems 14.3.1 EXAMPLE 14.2 Bode Diagrams First-Order Process In the past, when frequency response plots had to be generated by hand, they were of limited utility A much more practical approach now utilizes spreadsheets or control-oriented software such as MATLAB to simplify calculations and generate Bode plots Although spreadsheet software can be used to generate Bode plots, it is much more convenient to use software designed specifically for control system analysis Thus, after describing the qualitative features of Bode plots of simple transfer functions, we illustrate how the AR and ϕ components of such a plot are generated by a MATLAB program in Example 14.3 For a first-order model, K/(τs + 1), Fig 14.2 shows a general log–log plot of the normalized amplitude ratio versus ωτ, for positive K For a negative valve of K, the phase angle is decreased by −180∘ A semilog plot of ϕ versus ωτ is also shown In Fig 14.2, the abscissa ωτ has units of radians If K and τ are known, ARN (or AR) and ϕ can be plotted as a function of ω Note that, at high frequencies, the amplitude ratio drops to an infinitesimal level, and the phase lag (the phase angle expressed as a positive value) approaches a maximum value of 90∘ Some books and software define AR differently, in terms of decibels The amplitude ratio in decibels 248 Chapter 14 Frequency Response Analysis and Control System Design ωb = 1/τ ϕ = ∠G(jω) = ∠K − tan−1 (j∞) = −90∘ (14-21) 14.3.3 Normalized amplitude 0.1 ratio, ARN Second-Order Process A general transfer function for a second-order system without numerator dynamics is G(s) = 0.01 0.01 0.1 ωτ 10 100 –30 ωb = 1/τ Phase angle –60 ϕ (deg) –90 –120 0.01 0.1 ωτ 10 100 Figure 14.2 Bode diagram for a first-order process ARdb is defined as ARdb = 20 log AR (14-19) The use of decibels merely results in a rescaling of the Bode plot AR axis The decibel unit is employed in electrical communication and acoustic theory and is seldom used today in the process control field Note that the MATLAB bode routine uses decibels as the default option; however, it can be modified to plot AR results, as shown in Fig 14.2 In the rest of this chapter, we only derive frequency responses for simple transfer functions (integrator, first-order, second-order, zeros, time delay) Software should be used for calculating frequency responses of more complicated transfer functions 14.3.2 Integrating Process The transfer function for an integrating process was given in Chapter Y(s) K G(s) = = (5-32) U(s) s Because of the single pole located at the origin, this transfer function represents a marginally stable process The shortcut method of determining frequency response outlined in the preceding section was developed for stable processes, that is, those that converge to a bounded oscillatory response for a sinusoidal input Because the output of an integrating process is bounded when forced by a sinusoidal input, the shortcut method does apply for this marginally stable process: |K| K (14-20) AR = |G(jω)| = || || = | jω | ω K τ2 s2 + 2ζτs + (14-22) Substituting s = jω and rearranging into real and imaginary parts (see Example 14.1) yields K AR = √ (14-23a) 2 (1 − ω τ )2 + (2ζωτ)2 [ ] −2ζωτ (14-23b) ϕ = tan−1 − ω2 τ2 Note that, in evaluating ϕ, multiple results are obtained because Eq 14-23b has infinitely many solutions, each differing by n180∘ , where n is a positive integer The appropriate solution of Eq 14-23b for the second-order system yields −180∘ < ϕ < Figure 14.3 shows the Bode plots for overdamped (ξ > 1), critically damped (ξ = 1), and underdamped (0 < ξ < 1) processes as a function of ωτ The lowfrequency limits of the second-order system are identical to those of the first-order system However, the limits are different at high frequencies, ωτ ≫ ARN ≈ 1∕(ωτ)2 ϕ ≈ −180o (14-24a) (14-24b) For overdamped systems, the normalized amplitude ̂ ratio is attenuated (A/KA < 1) for all ω For underdamped systems, the amplitude√ratio plot exhibits a maximum (for values of < ζ < 2∕2) at the resonant frequency √ − 2ζ2 (14-25) ωr = τ (ARN )max = √ 2ζ − ζ2 (14-26) These expressions can be derived by the interested reader The resonant frequency ωr is that frequency for which the sinusoidal output response has the maximum amplitude for a given sinusoidal input Equations 14-25 and 14-26 indicate how ωr and (ARN )max depend on ξ This behavior is used in designing organ pipes to create sounds at specific frequencies However, excessive resonance is undesirable, for example, in automobiles, where a particular vibration is noticeable only at a certain speed For industrial processes operated without feedback control, resonance is seldom encountered, although some measurement devices are designed to exhibit a limited amount of resonant behavior On the other hand, feedback controllers can be tuned to give the controlled process a slight amount of oscillatory 14.3 249 10 ζ=1 ζ = 0.2 0.1 ARN 0.01 0.001 0.0001 0.01 Bode Diagrams ARN 0.01 Slope = –2 0.1 ωτ 10 0.4 0.8 0.1 Slope = –2 0.001 0.01 100 0.1 ωτ 10 100 –45 ϕ (deg) –90 –45 –135 –180 0.01 ϕ (deg) ωτ 0.4 –135 ζ=1 0.1 –90 ζ = 0.2 10 100 –180 0.01 0.1 ωτ 0.8 10 100 Figure 14.3 Bode diagrams for second-order processes Right: underdamped Left: overdamped and critically damped or underdamped behavior in order to speed up the controlled system response (see Chapter 12) 14.3.4 Process Zero A term of the form τs + in the denominator of a transfer function is sometimes referred to as a process lag, because it causes the process output to lag the input (the phase angle is negative) Similarly, a process zero of the form τs + (τ > 0) in the numerator (see Section 6.1) causes the sinusoidal output of the process to lead the input (ϕ > 0); hence, a left-half plane (LHP) zero often is referred to as a process lead Next we consider the amplitude ratio and phase angle for this term Substituting s = jω into G(s) = τs + gives G(jω) = jωτ + from which AR = |G(jω)| = √ ω2 τ2 + ϕ = ∠G(jω) = tan−1 (ωτ) (14-27) (14-28a) (14-28b) Therefore, a process zero contributes a positive phase angle that varies between and +90∘ The output signal amplitude becomes very large at high frequencies (i.e., AR → ∞ as ω → ∞), which is a physical impossibility Consequently, in practice a process zero is always found in combination with one or more poles The order of the numerator of the process transfer function must be less than or equal to the order of the denominator, as noted in Section 6.1 Suppose that the numerator of a transfer function contains the term − τs, with τ > As shown in Section 6.1, a right-half plane (RHP) zero is associated with an inverse step response The frequency response characteristics of G(s) = − τs are √ (14-29a) AR = ω2 τ2 + ϕ = −tan−1 (ωτ) (14-29b) Hence, the amplitude ratios of LHP and RHP zeros are identical However, an RHP zero contributes phase lag to the overall frequency response because of the negative sign Processes that contain an RHP zero or time delay are sometimes referred to as nonminimum phase systems because they exhibit more phase lag than another transfer function that has the same AR characteristics (Franklin et al., 2014) Exercise 14.11 illustrates the importance of zero location on the phase angle 14.3.5 Time Delay The time delay e−θs is the remaining important process element to be analyzed Its frequency response characteristics can be obtained by substituting s = jω: G(jω) = e−jωθ (14-30) which can be written in rational form by substitution of the Euler identity G(jω) = cos ωθ − j sin ωθ (14-31) 250 Chapter 14 Frequency Response Analysis and Control System Design 10 AR AR 0.1 0.01 0.1 ωθ 10 ω (rad/min) –180 ϕ (deg) ϕ (deg) –360 –540 0.01 0.1 10 ωθ −θs Figure 14.4 Bode diagram for a time delay, e Figure 14.5 Bode plot of the transfer function in Example 14.3 From Eq 14.6, √ AR = |G(jω)| = cos2 ωθ + sin2 ωθ = ( ) sin ωθ ϕ = ∠G(jω) = tan−1 − cos ωθ (14-32) or ϕ = −ωθ (14-33) Because ω is expressed in radians/time, the phase angle in degrees is −180ωθ/π Figure 14.4 illustrates the Bode plot for a time delay The phase angle is unbounded, that is, it approaches −∞ as ω becomes large By contrast, the phase angles of all other process elements are smaller in magnitude than some multiples of 90∘ This unbounded phase lag is an important attribute of a time delay and is detrimental to closed-loop system stability, as is discussed in Section 14.6 EXAMPLE 14.3 Generate the Bode plot for the transfer function G(s) = ω (rad/min) 5(0.5s + 1)e−0.5s (20s + 1)(4s + 1) where the time constants and time delay have units of minutes SOLUTION The Bode plot is shown in Fig 14.5 The steady-state gain (K = 5) is the value of AR when ω → The phase angle at high frequencies is dominated by the time delay The MATLAB code for generating a Bode plot of the transfer function is shown in Table 14.1 In this code the normalized AR is used (ARN ) Table 14.1 MATLAB Program to Calculate and Plot the Frequency Response in Example 14.3 %Make a Bode plot for G = (0.5s + 1)e^–0.5s/(20s + 1) %(4s + 1) close all gain = 5; tdead = 0.5; num = [0.5 1]; den = [80 24 1]; G = tf (gain∗ num, den) %Define the system as a transfer %function points = 500; %Define the number of points ww = logspace (−2, 2, points); %Frequencies to be evaluated [mag, phase, ww] = bode (G,ww); % Generate numerical %values for Bode plot AR = zeros (points, 1); % Preallocate vectors for Amplitude %Ratio and Phase Angle PA = zeros (points, 1); for i = : points AR(i) = mag (1,1,i)/gain; %Normalized AR PA(i) = phase (1,1,i) – ((180/pi) ∗ tdead∗ ww(i)); end figure subplot (2,1,1) loglog(ww, AR) axis ([0.01 100 0.001 1]) title (‘Frequency Response of a SOPTD with Zero’) ylabel(‘AR/K’) subplot (2,1,2) semilogx(ww,PA) axis ([0.01 100 −270 0]) ylabel(‘Phase Angle (degrees)’) xlabel(‘Frequency (rad/time)’) 14.4 14.4 Frequency Response Characteristics of Feedback Controllers FREQUENCY RESPONSE CHARACTERISTICS OF FEEDBACK CONTROLLERS In order to use frequency response analysis to design control systems, the frequency-related characteristics of feedback controllers must be known for the most widely used forms of the PID controller discussed in Chapter In the following derivations, we generally assume that the controller is reverse-acting (Kc > 0) If a controller is direct-acting (Kc < 0), the AR plot does not change, because |Kc | is used in calculating the magnitude However, the phase angle is shifted by −180∘ when Kc is negative For example, a direct-acting proportional controller (Kc < 0) has a constant phase angle of −180∘ As a practical matter, it is possible to use the absolute value of Kc to calculate ϕ when designing closed-loop control systems, because stability considerations (see Chapter 11) require that Kc < only when Kv Kp Km < This choice guarantees that the open-loop gain (KOL = Kc Kv Kp Km ) will always be positive Use of this convention conveniently yields ϕ = 0∘ for any proportional controller and, in general, eliminates the need to consider the −180∘ phase shift contribution of the negative controller gain Proportional Controller Consider a proportional controller with positive gain Gc (s) = Kc (14-34) In this case, |Gc (jω)| = Kc , which is independent of ω Therefore, (14-35) AR = Kc and ϕ = 0∘ (14-36) Proportional-Integral Controller A proportionalintegral (PI) controller has the transfer function, ( ( ) ) τI s + 1 = Kc (14-37) Gc (s) = Kc + τI s τI s Substituting s = jω gives ( ( ) ) j = Kc − Gc (jω) = Kc + τI jω ωτI (14-38) Thus, the amplitude ratio and phase angle are √ √ (ωτI )2 + 1 AR = |Gc (jω)| = Kc + = K c (ωτI )2 ωτI (14-39) ϕ = ∠Gc (jω) = tan−1 (−1∕ωτI ) = tan−1 (ωτI ) − 90∘ (14-40) Based on Eqs 14-39 and 14-40, at low frequencies, the integral action dominates As ω → 0, AR → ∞, and ϕ → −90∘ At high frequencies, AR = Kc and ϕ = 0∘ ; neither is a function of ω in this region (cf the proportional controller) 251 Ideal Proportional-Derivative Controller The ideal proportional-derivative (PD) controller (cf Eq 8-11) is rarely implemented in actual control systems but is a component of PID control and influences PID control at high frequency Its transfer function is Gc (s) = Kc (1 + τD s) (14-41) The frequency response characteristics are similar to those of an LHP zero: √ AR = Kc (ωτD )2 + (14-42) ϕ = tan−1 (ωτD ) (14-43) Proportional-Derivative Controller with Filter As indicated in Chapter 8, the PD controller is most often realized by the transfer function ( ) τD s + Gc (s) = Kc (14-44) ατD s + where α has a value in the range 0.05–0.2 The frequency response for this controller is given by √ (ωτD )2 + AR = Kc (14-45) (αωτD )2 + ϕ = tan−1 (ωτD ) − tan−1 (αωτD ) (14-46) The pole in Eq 14-44 bounds the high-frequency asymptote of the AR lim AR = lim |Gc (jω)| = Kc ∕α = 2∕0.1 = 20 (14-47) ω→∞ ω→∞ Note that this form actually is an advantage, because the ideal derivative action in Eq 14-41 would amplify high-frequency input noise, due to its large value of AR in that region In contrast, the PD controller with derivative filter exhibits a bounded AR in the high-frequency region Because its numerator and denominator orders are both one, the high-frequency phase angle returns to zero Parallel PID Controller The PID controller can be developed in both parallel and series forms, as discussed in Chapter Either version exhibits features of both the PI and PD controllers The simpler version is the following parallel form (cf Eq 8-14): ) ( ( ) + τI s + τI τD s2 Gc (s) = Kc + + τD s = Kc τI s τI s (14-48) Substituting s = jω and rearranging gives ( ) [ ( )] 1 Gc (jω) = Kc + + jωτD = Kc + j ωτD − jωτI ωτI (14-49) 252 Chapter 14 Frequency Response Analysis and Control System Design 102 moves the amplitude ratio curve up or down, without affecting the width of the notch Generally, the integral time τI is larger than τD , typically τI ≈ 4τD Ideal With derivative filter AR 101 100 10–3 10–2 10–1 100 101 ω (rad/min) 100 50 ϕ (deg) –50 –100 10–3 10–2 10–1 100 101 ω (rad/min) Parallel PID Controller with a Derivative Filter The parallel controller with a derivative filter was described in Chapter and Table 8.1 (14-50) Figure 14.6 shows a Bode plot for an ideal PID controller, with and without a derivative filter (see Table 8.1) The controller settings are Kc = 2, τI = 10 min, τD = min, and α = 0.1 The phase angle varies from −90∘ (ω → 0) to +90∘ (ω → ∞) A comparison of the amplitude ratios in Fig 14.6 indicates that the AR for the controller without the derivative filter in Eq 14-48 is unbounded at high frequencies, in contrast to the controller with the derivative filter (Eq 14-50), which has a bounded AR at all frequencies Consequently, the addition of the derivative filter makes the series PID controller less sensitive to high-frequency noise For the typical value of α = 0.10, Eq 14-50 yields at high frequencies: ARω→∞ = lim |Gc (jω)| = Kc ∕α = 20Kc ω→∞ This controller transfer function can be interpreted as the product of the transfer functions for PI and PD controllers Because the transfer function in Eq 14-52 is physically unrealizable and amplifies high-frequency noise, a more practical version includes a derivative filter 14.5 Figure 14.6 Bode plots of ideal parallel PID controller and ideal parallel PID controller filter (α = 0.1) ( with derivative ) Ideal parallel: Gc (s) = + + 4s 10s ( ) 4s Parallel with derivative filter: Gc (s) = + + 10s 0.4s + ) ( τD s Gc (s) = Kc + + τI s ατD s + Series PID Controller The simplest version of the series PID controller is ( ) τ1 s + (14-52) Gc (s) = Kc (τD s + 1) τ1 s (14-51) When τD = 0, the parallel PID controller with filter is the same as the PI controller of Eq 14-37 By adjusting the values of τI and τD , one can prescribe the shape and location of the notch in the AR curve Decreasing τI and increasing τD narrows the notch, whereas the opposite changes broaden it Figure 14.6 indicates that the center of the notch is located at √ ω = 1∕ τI τD where ϕ = 0∘ and AR = Kc Varying Kc NYQUIST DIAGRAMS The Nyquist diagram is an alternative representation of frequency response information, a polar plot of G(jω) in which frequency ω appears as an implicit parameter The Nyquist diagram for a transfer function G(s) can be constructed directly from |G(jω)| and ∠G(jω) for different values of ω Alternatively, the Nyquist diagram can be constructed from the Bode diagram, because AR = |G(jω)| and ϕ = ∠G(jω) The advantages of Bode plots are that frequency is plotted explicitly as the abscissa, and the log–log and semilog coordinate systems facilitate block multiplication The Nyquist diagram, on the other hand, is more compact and is sufficient for many important analyses, for example, determining system stability (see Appendix J) Most of the recent interest in Nyquist diagrams has been in connection with designing multiloop controllers and for robustness (sensitivity) studies (Maciejowski, 1989; Skogestad and Postlethwaite, 2005) For single-loop controllers, Bode plots are used more often 14.6 BODE STABILITY CRITERION The Bode stability criterion has an important advantage in comparison with the alternative of calculating the roots of the characteristic equation in Chapter 11 It provides a measure of the relative stability rather than merely a yes or no answer to the question “Is the closed-loop system stable?” Before considering the basis for the Bode stability criterion, it is useful to review the General Stability Criterion of Section 11.1: A feedback control system is stable if and only if all roots of the characteristic equation lie to the left of the imaginary axis in the complex plane Thus, the imaginary axis divides the complex plane into stable and unstable regions Recall that the characteristic equation was defined in Chapter 11 as + GOL (s) = (14-53) where the open-loop transfer function in Eq 14-53 is GOL (s) = Gc (s)Gv (s)Gp (s)Gm (s) 14.6 Before stating the Bode stability criterion, we introduce two important definitions: A critical frequency ωc is a value of ω for which ϕOL (ω) = −180∘ This frequency is also referred to as a phase crossover frequency A gain crossover frequency ωg is a value of ω for which AROL (ω) = The Bode stability criterion allows the stability of a closed-loop system to be determined from the open-loop transfer function Bode Stability Criterion Consider an open-loop transfer function GOL = Gc Gv Gp Gm that is strictly proper (more poles than zeros) and has no poles located on or to the right of the imaginary axis, with the possible exception of a single pole at the origin Assume that the open-loop frequency response has only a single critical frequency ωc and a single gain crossover frequency ωg Then the closed-loop system is stable if the open-loop amplitude ratio AROL (ωc ) < Otherwise, it is unstable The root locus diagrams of Section 11.5 (e.g., Fig 11.27) show how the roots of the characteristic equation change as controller gain Kc changes By definition, the roots of the characteristic equation are the numerical values of the complex variable, s, that satisfy Eq 14-53 Thus, each point on the root locus also satisfies Eq 14-54, which is a rearrangement of Eq 14-53: (14-54) GOL (s) = −1 The corresponding magnitude and argument are (14-55) |G (jω)| = and ∠G (jω) = −180∘ OL OL For a marginally stable system, ωc = ωg and the frequency of the sustained oscillation, ωc , is caused by a pair of roots on the imaginary axis at s = ±ωc j Substituting this expression for s into Eq 14-55 gives the following expressions for a conditionally stable system: AROL (ωc ) = |GOL (jωc )| = ϕOL (ωc ) = ∠GOL (jωc ) = −180∘ (14-56) (14-57) for some specific value of ωc > Equations 14-56 and 14-57 provide the basis for the Bode stability criterion Some of the important properties of the Bode stability criterion are It provides a necessary and sufficient condition for closed-loop stability, based on the properties of the open-loop transfer function The Bode stability criterion is applicable to systems that contain time delays The Bode stability criterion is very useful for a wide variety of process control problems However, for any GOL (s) that does not satisfy the required conditions, the Nyquist stability criterion discussed in Appendix J can be applied Bode Stability Criterion 253 10000 100 AROL 0.01 –90 ϕOL –180 (deg) –270 –360 0.001 0.01 0.1 ω (radians/time) 10 100 Figure 14.7 Bode plot exhibiting multiple critical frequencies For many control problems, there is only a single ωc and a single ωg But multiple values for ωc can occur, as shown in Fig 14.7 In this somewhat unusual situation, the closed-loop system is stable for two different ranges of the controller gain (Luyben and Luyben, 1997) Consequently, increasing the absolute value of Kc can actually improve the stability of the closed-loop system for certain ranges of Kc For systems with multiple ωc or ωg , the Bode stability criterion has been modified by Hahn et al (2001) to provide a sufficient condition for stability As indicated in Chapter 11, when the closed-loop system is marginally stable, the closed-loop response exhibits a sustained oscillation after a set-point change or a disturbance Thus, the amplitude neither increases nor decreases In order to gain physical insight into why a sustained oscillation occurs at the stability limit, consider the analogy of an adult pushing a child on a swing The child swings in the same arc as long as the adult pushes at the right time and with the right amount of force Thus the desired sustained oscillation places requirements on both timing (i.e., phase) and applied force (i.e., amplitude) By contrast, if either the force or the timing is not correct, the desired swinging motion ceases, as the child will quickly protest A similar requirement occurs when a person bounces a ball To further illustrate why feedback control can produce sustained oscillations, consider the following thought experiment for the feedback control system shown in Fig 14.8 Assume that the open-loop system is stable and that no disturbances occur (D = 0) Suppose that the set-point is varied sinusoidally at the critical frequency, ysp (t) = A sin (ωc t), for a long period of time Assume that during this period, the measured output, ym , is disconnected, so that the feedback loop is broken before the comparator After the initial transient dies out, ym will oscillate at the excitation frequency ωc , because the response of a linear system to a sinusoidal input is a sinusoidal output at the same Appendix E Process Control Modules APPENDIX CONTENTS E.1 Introduction E.2 Module Organization E.3 Hardware and Software Requirements E.4 Installation E.5 Running the Software E.1 INTRODUCTION The Process Control Modules (PCM), originally developed at the University of Delaware, have been designed to address the key engineering educational challenge of realistic problem solving within the constraints of a typical lecture course in process dynamics and control (Doyle III et al., 1998; Doyle III, 2001) A listing of the modules appears in Table E.1 These modules have been updated and adapted by Dr Eyal Dassau at the University of California, Santa Barbara, to be used in conjunction with the third edition of Process Dynamics and Control The primary objectives in creating these MATLAB® modules were to develop the following: • Realistic computer simulation case studies, based on physical properties that exhibited nonlinear, high-order dynamic behavior in a rapid simulation environment • A convenient graphical interface for students that allowed real-time interaction with the evolving virtual experiment • A set of challenging exercises that reinforce the conventional lecture material through active learning and problem-based methods E.2 MODULE ORGANIZATION Eight distinct chemical and biological process applications, which range from simple single input-single output (SISO) processes to more complex × control loops, are formulated with a modular approach The progression of the modules follows a typical undergraduate process dynamics and control course, starting with low-order dynamic system analysis and continuing through multivariable controller synthesis Table E.1 Organization of Process Control Modules (PCM) Module Furnace Distillation Column Bioreactor Four Tanks Fermentor Diabetes First and Second Order Systems Discrete Modes Operator Interface Operator Interface PID PID Feedforward Feedforward Multivariable Multivariable MPC Decoupling Operator Interface Operator Interface Operator Interface Operator Interface First Order System PID PID PID PID Second Order System Feedforward Feedforward Multivariable Multivariable Decoupling MPC Aliasing Model ID MPC System Identification #1 PID-Furnace System Identification #2 PID-Column PID-Four Tanks MPC IMC-Furnace IMC-Column 489 490 Appendix E E.3 HARDWARE AND SOFTWARE REQUIREMENTS Process Control Modules The Process Control Modules are a set of MATLAB/ Simulink routines that require either a full license or the Student Version of MATLAB and Simulink The current version of the modules has been tested with version 2011b of MATLAB and Simulink The minimum recommended system configuration is a Windows PC with GB RAM E.4 INSTALLATION The Process Control Modules (PCM) software can be downloaded from www.wiley.com/college/seborg onto the user’s computer Then double-click on the PCM file, and follow the instructions on the installer to install the software Note that MATLAB should be installed in order to use these modules During the installation, users can create a shortcut icon to the software on their desktop (recommended) Figure E.1 PCM main interface E.5 RUNNING THE SOFTWARE There are two ways to execute the software: the first is to double-click the PCM button on the desktop, which launches MATLAB and the PCM interface (Fig E.1), and the other way is to open MATLAB manually and to call the PCM software by pointing to the PCM installation folder and typing “PCM,” followed by the Enter key The website for this textbook contains a more detailed tutorial on PCM, including case studies for the furnace and distillation column modules REFERENCES Doyle III, F J., E P Gatzke, and R S Parker, Practical Case Studies for Undergraduate Process Dynamics and Control Using Process Control Modules, Comp Appls Eng Educ., 6, 181 (1998) Doyle III, F J Process Control Modules: A Software Laboratory for Control Design, Prentice Hall PTR, Upper Saddle River, NJ, 2000 Appendix F Review of Basic Concepts From Probability and Statistics APPENDIX CONTENTS F.1 Probability Concepts F.2 Means and Variances F.2.1 F.2.2 Means and Variances for Probability Distributions Means and Variances for Experimental Data F.3 Standard Normal Distribution F.4 Error Analysis In this appendix, basic probability and statistics concepts that are necessary for the safety analysis of Chapter 10 and the quality control charts of Chapter 21 are reviewed Analogous expressions are available for the union of more than two events (Montgomery and Runger, 2013) If A and B are independent, then the probability that both occur is P(A ∩ B) = P(A) P(B) F.1 PROBABILITY CONCEPTS (for independent events) The term probability is used to quantify the likely outcome of a random event For example, if a fair coin is flipped, the probability of a head is 0.5 and the probability of a tail is 0.5 Let P(A) denote that probability that a random event A occurs Then P(A) is a number in the interval ≤ P(A) ≤ 1, such that the larger P(A) is, the more likely it is that A occurs Let A′ denote the complement of A, that is, the event that A does not occur Then, P(A′ ) = − P(A) (F-1) Now consider two events, A and B, with probabilities P(A) and P(B), respectively The probability that one or both events occurs (A ∪ B) can be expressed as P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (F-2) If A and B are mutually exclusive, this means that if one event occurs, the other cannot; consequently, their intersection is the null set A ∩ B = ϕ Then P(A ∩ B) = and Eq F-2 becomes P(A ∪ B) = P(A) + P(B) (for mutually exclusive events) (F-3) (F-4) Similarly, the probability that n independent events, E1 , E2 , … , En , occur is P(E1 ∩ E2 ∩ · · · ∩ En ) = P(E1 ) P(E2 ) · · · P(En ) (for independent events) (F-5) These probability concepts are illustrated in two examples EXAMPLE F.1 A semiconductor processing operation consists of five independent batch steps where the probability of each step having its desired outcome is 0.95 What is the probability that the desired end product is actually produced? SOLUTION In order to make the product, each individual step must be successful Because the steps are independent, the probability of a success, P(S), can be calculated from Eq F-5: P(S) = (0.95)5 = 0.77 491 492 Appendix F Review of Basic Concepts From Probability and Statistics EXAMPLE F.2 In order to increase the reliability of a process, a critical process variable is measured on-line using two sensors Sensor A is available 95% of the time, while Sensor B is available 90% of the time Suppose that the two sensors operate independently and that their periods of unavailability occur randomly What is the probability that neither sensor is available at any arbitrarily selected time? SOLUTION Let A denote the event that Sensor A is not available and B denote the event that Sensor B is not available The event that neither Sensor is available can be expressed as (A ∪ B)′ Then from Eqs F-1 and F-2, P(A ∪ B)′ = − P(A ∪ B) P(A ∪ B)′ = − [P(A) + P(B) − P(A ∩ B)] σ2X = E[(X − uX )2 ] = ∞ ∫−∞ (x − μX )2 f (x)dx (F-8) The positive square root of the variance is the population standard deviation, σX These calculations are illustrated in Example F.3 EXAMPLE F.3 A mass fraction of an impurity X varies randomly between 0.3 and 0.5 with a uniform probability distribution: 0.2 Determine its population mean and population standard deviation f (x) = P(A ∪ B)′ = − [0.95 + 0.90 − (0.95)(0.90)] = 0.005 F.2 The expected value is also called the population mean or average It is an average over the expected range of values, weighted according to how likely each value is The population variance of X, σ2X , indicates the variability of X around its population mean It is defined as SOLUTION MEANS AND VARIANCES Next, we consider two important statistical concepts, means and variances, and how they can be used to characterize both probability distributions and experimental data Substituting f(x) into Eq F-7 gives F.2.1 Thus μX is the midpoint of the [0.3, 0.5] interval for X To determine σx , substitute f(x) into Eq F-8: ) ( ∞ 0.5 (x − uX )2 f (x)dx = (x − 0.4)2 σ2X = dx ∫−∞ ∫0.3 0.2 ( )( )| 1 |0.5 σ2X = (x − 0.4)3 | = 0.00333 |0.3 0.2 | σX = 0.0577 Means and Variances for Probability Distributions In Section F.1, we considered the probability of one or more events occurring The same probability concepts are also applicable for random variables such as temperatures or chemical compositions For example, the product composition of a process could exhibit random fluctuations for several reasons, including feed disturbances and measurement errors A temperature measurement could exhibit random variations due to turbulence near the sensor Probability analysis can provide useful characterizations of such random phenomena Consider a continuous random variable, X, with an assumed probability distribution, f(x), such as a Gaussian distribution The probability that X has a numerical value in an interval [a, b] is given by (Montgomery and Runger, 2013), 0.5 ∞ ( 0.2 xf (x)dx = x ∫−∞ ∫0.3 ( )( )| 1 |0.5 μX = x | = 0.4 |0.3 0.2 | μX = F.2.2 ) dx Means and Variances for Experimental Data A set of experimental data can be characterized by its sample mean and sample variance (or simply, its mean and variance) Consider a set of N measurements, {x1 , x2 , … , xN } Its mean, x, and variance, s2 , are defined as (Montgomery and Runger, 2013) 1∑ x≜ x N i=1 i N P(a ≤ X ≤ b) = b ∫a f (x)dx (F-6) where x denotes a numerical value of random variable, X By definition, the expected value of X, μX , is defined as ∞ μX = E(X) = ∫−∞ xf (x)dx (F-7) s2 ≜ N ∑ (x − x)2 N − i=1 i (F-9) (F-10) The standard deviation s is the positive square root of the variance References The mean is the average of the data set while the variance and standard deviation characterize the variability in the data As an important example of error analysis, consider a linear combination of p variables, Y= F.3 X − μX Z= σX (F-11) Statistics book contains tables of the cumulative standard normal distribution, Φ(z) By definition, Φ(z) is the probability that Z is less than a specified numerical value, z (Ogunnaike, 2010; Montgomery and Runger, 2013): Φ(z) = P(Z ≤ z) (F-12) Example 10.3 illustrates an application of Φ(z) F.4 ERROR ANALYSIS In engineering calculations, it can be important to determine how uncertainties in independent variables (or inputs) lead to even larger uncertainties in dependent variables (or outputs) This analysis is referred to as error analysis Due to the uncertainties associated with input variables, they are considered to be random variables The uncertainties can be attributed to imperfect measurements or uncertainties in unmeasured input variables Error analysis is based on the statistical concepts of means and variances, considered in the previous section P ∑ ci Xi (F-13) i=1 STANDARD NORMAL DISTRIBUTION The normal (or Gaussian) probability distribution plays a central role in both the theory and application of statistics It was introduced in Section 21.2.1 For probability calculations, it is convenient to use the standard normal distribution, N(0, 1), which has a mean of zero and a variance of one Suppose that a random variable X is normally distributed with a mean μX and variance σ2X Then, the corresponding standard normal variable Z is 493 where Xi is an independent random variable with expected value μi and variance σ2i Then, Y has the following mean and variance (Montgomery and Runger, 2013): P ∑ ci μi (F-14) μY = i=1 σ2Y = P ∑ c2i σ2i (F-15) i=1 Equations F-14 and F-15 show how the variability of the individual Xi variables determines the variability of their linear combination, Y EXAMPLE F.4 Experimental tests are to be performed to determine whether a new catalyst A is superior to the current catalyst B, based on their yields for a chemical reaction Denote the yields by XA and XB and their standard deviations by σA = 3% and σB = 2% What is the standard deviation for the difference in yields, XA − XB ? SOLUTION Let Y = XA − XB , an expression in the form of Eq F-13 with cA = and cB = −1 Thus Eq F-15 becomes σ2Y = σ2A + σ2B and σY = √ √ σ2A + σ2B = (3%)2 + (2%)2 = 3.6% Thus, the standard deviation of the difference is larger than the individual standard deviations REFERENCES Montgomery, D C., and G C Runger, Applied Statistics and Probability for Engineers, 6th ed., John Wiley and Sons, Hoboken, NJ, 2013 Ogunnaike, B A., Random Phenomena: Fundamentals of Probability and Statistics for Engineers, CRC Press, Boca Raton, FL, 2010 Index A absorption column, 28 actuator, 150, 151, 156 adaptation, on-line, 292 adaptive control, 279, 289, 292 applications, 289, 290, 291, 293 commercial systems, 293 programmed, 290, 292 self-tuning, 292 adaptive tuning, 292 ADC, 300, 320, 466 advanced control techniques, 284 alarm flood, 165, 168 alarms classification, 163 limits, 163 management, 165 switch, 163 aliasing, 302, 321 ammonia synthesis, 265 amplitude ratio, 121, 245 analog controller, 123, 125, 129, 131, 133 analog instrumentation, 124, 151 analog to digital converter, 133, 316, 466 analog signal, 466 analytical predictor (AP), 295 analyzers, 134 annunciator, 163, 168 anti-aliasing filter, 302 anticipatory control, 128 anti-reset windup, 127, 128, 424 approximation finite difference, 133, 327 higher-order systems, of, 88, 92, 100 least squares, 107 linearization, 61 Padé, 91 Taylor series, 61, 92, 93 argument, 246, 253 artificial neural net (ANN), 113 ARX model, 116 assignable cause, 396, 398, 399 auctioneering control, 287 automatic mode, 132, 143 autoregressive model, 116, 320 auto-tuning, 284 average run length, 401, 403 averaging level control, 221 B back calculation method, 127 backlash (valve), 158 backward difference, 125, 311, 318 bandwidth, J11 bang-bang control, 136 batch control system, 415, 430 alarms, 416, 417 ANSI-IS, 95, 415 bioprocesses, 414, 428 cyclical batch control, 428 batch production management, 415, 416, 427 binary logic diagram, 417, 418 campaign, 428 control during the batch, 415, 421 flexible manufacturing, 430 Gantt chart, 429 information flow diagram, 417 ladder logic diagram, 417, 418 levels, 415 rapid thermal processing, 425, 427 reactive scheduling, 430 reactor control, 422, 425 recipe, 428, 430 run-to-run control, 415, 416, 426, 427 scheduling and planning, 428, 429 semiconductor processing, 422, 425, 427, 430 sequential function chart, 417, 418 sequential logic, 416, 417, 421 SP-88 terminology, 427 batch distillation, 413 batch reactor control, 422, 425 batch sequence, 416 batch-to-batch control, 416 beermaking, 436 Bernoulli equation, 25 beta-gamma controller, 141 bias, 424, 426, 427 biggest log-modulus (BLT) tuning, 341 binary logic diagram, 417, 418 biological switch, 462 bioreactor, 41, 50, 436 bioreactor sensor, 154 black box modeling, 114 blending process, 15, 18, 56, 176, 278, 326, 476 block diagram algebra, 58, 59, 269 analysis, 482 feedback control, reduction, 179, 281 representation, 176 blood glucose, 442, 443 blood pressure control, 444 Bode diagrams breakpoint, 260, 261 of controllers, 252, 262 Bode sensitivity integral, 252, J14 Bode stability criterion, 263 boilers adaptive control, 279, 292 feedforward control, 274 inverse response of reboiler, 89 RTO, 363 selective control, 279, 287, 293 split-range control, 287, 288 bracket (on optimum), 357 4break frequency, 343 Bristol’s relative gain array, 332, 334 bumpless transfer, 132 C calibration, instrument, 163 campaign, 428 cancer treatment, 445 cardiac-assist device, 446 capability index, 404 capacitance probe, 153 cascade control design, 279, 282 frequency response, 337 loop configuration, 288, 358, G3, H3 primary controller, 280, 283, 284 secondary controller, 280, 283, 284 catalytic converters, 246 Central Dogma, 452, 453, 461 chemotaxis, 457, 462 Center for Chemical Process Safety (CCPS), 161 Central Limit Theorem, 402 characteristic equation, 188, 330 digital control, 300, 307, 313 characteristic polynomial, 87 characteristic roots, 481 chemical reactors ammonia synthesis, 265 batch, 413 catalytic, 287 continuous stirred-tank reactor (CSTR), 34 fluidized catalytic cracker, 352, 355 trickle-bed, 91 tubular, 89, 297 chemometrics, 114 chromatographic analysis, 154, 481 circadian clock, 452, 455 closed loop block diagrams, 176 frequency response, J10 gain, 188 performance criteria, J11 poles, 188 prediction, 369 response, 181–186 stability, 186 transfer function, 63, 178, 179, 180, 188, J2 495 496 Index coincidence point, 373, 385 combustion process adaptive control, 279, 292 ratio control, 264 comparator, 177 compensation, dynamic, 268 complementary sensitivity, J10 composition control, 176 composition sensor, 154, 176 computer hardware, 465 interface, 466 representation of information, 466 software, 470 computer control, 329, 465 conditional stability, 253 condition number, 338, 339, 340 connection weight (neural net), 114 conservation laws, 17 constrained optimization, 356, 359, 362 constraint control, 299 constraints feasible region, 359 hard, 382, 386 in optimization, 352, 355 soft, 382, 385 continuous cycling method, 215 continuous stirred-tank reactor (CSTR), 26 cascade control, 279 dynamics, G3 feedback control, G1 linearization, 68 modeling, 26, 67, G3 recycle, G1 transfer function, 67, 69, G3 contour mapping, J7 control algorithm, 123, 128, 303, 311, 315 cascade, 279, 282, 293 chart, 396 configuration, 327, 328, 330 constraint, 341 degrees of freedom, 19, 230 during the batch, 417, 421 feedback, 6, 123, 127, 132, 133, 135, 176, 180 feedforward, 263 hardware, 124, 131, 466 horizon, 370, 374, 378, 380, 381, 384, 389 hierarchy, 8, 350, H1 law, 123, 131 multiloop, 327, 328 multivariable, 327, 342 model predictive (MPC), 368 plantwide See plantwide control regulatory, 180 run-to-run, 415, 416, 426 control loop interactions, 327, 343 control loop troubleshooting, 222 control objectives, 232, 235 control performance monitoring, 408 control-relevant model, 107 control requirements, 232 control strategies, 199 control structure, G3, H2 control systems adaptive, 279, 292 advanced, 279, 284, 293 cascade, 291, G3, H3 design, 10, 201, H1, H4 economic justification, 358, 362, 364 effect of process design, 235 feedback, 123 feedforward, 263 feedforward-feedback, 272 inferential, 279, 286, 289, 293 model-based, 201 multiple-loop, 327, 328 multivariable, 327, 342 nonlinear, 300 plantwide control See plantwide control ratio, 264 robustness, 380 selective, 279, 287, 293 split-range, 287, 288 troubleshooting, 230 variable selection, 239, H3 control valve, 156 air-to-close, 157 air-to-open, 157 dynamic model, 164, 185 fail-closed, 157 fail-open, 157 flow characteristics, 158 globe, 157 plug, 157 pneumatic, 157 quarter-turn, 157 rangeability, 159 rotary, 157 sizing, 158 trim, 152 controlled cycling, 223, 226 controlled variable(s) selection of, 232, 355, G1, G2, H2, H3 controller analog, 125 automatic, 124, 128, 132 beta-gamma, 131 bias, 375 digital, 129, 131, 133, 134, 303, 307 direct-acting, 131, 132 error gap, 290 frequency response, 258 gain, 125, 126, 130–132, 135, 136 gain scheduling, 289, 291, 292 historical perspective, 124 manual, 132 on-off, 136 parameter scheduling, 289, 290 performance, 200 predictive, 368 proportional-integral-derivative (PID), 123, 124, 127, 129 relay, 218 reverse-acting, 131 robustness, J13 saturation, 126, 134 transfer function, 137, 177 tuning See controller tuning two degrees of freedom, 222 ultimate gain, 215 controller design direct synthesis (DS), 201 frequency response, 251 integral error criteria, 210 internal model control (IMC), 205 controller pairing, 334, 339, H2, H3 controller parameters/settings, 126, 128, 130 controller tuning, 199 AMIGO method, 211, 213, 224 feedforward controller, 273 Hägglund-Åström, 211 IMC, 206 integral error criteria, 210 multiloop control system, 327, 335, 341 on-line, 214 predictive control, 384, 385 relay auto-tuning, 218 Skogestad, 206, 209 Skogestad IMC method (SIMC), 209, 210, 224 Tyreus-Luyben, 224 Ziegler-Nichols, 211, 215 conversion of signals continuous to discrete-time, 308 converters analog to digital, 466 digital to analog, 466 instrument, 151 convolution model, 370 core reactor/flash unit model, G4 coriolis meter, 153 critical controller gain See ultimate gain critical frequency, 253 critical point, J7 critically damped, 75 cross controllers, 342 crossover frequency, 253, 256 crystallizer, 438, 439, 449 CSTR, see continuous stirred-tank reactor current-to-pressure transducer, 135, 151, 177 CUSUM control chart, 403, 404 cycle time, 425 cycling, continuous, 215, 466 D Dahlin’s algorithm, 315 modified version, 316, 317 damping coefficient, 75, 78 data fitting, 109, 110, 111 data highway, 142 data network, 141 data reconciliation, 351, 353, 354 data validation, 114, 370 DCS (distributed control system), 469 deadband, 136 dead time See also time delay, 90 decay ratio, 77, 211, 215, 224 one-quarter, 215, 224, 232 decentralized integral controllability, 351 decibel, 343 decoupling control, 342, 343 partial, 342 static, 342 defuzzification, 291 degree of fulfillment, 287 degrees of freedom control, 230, H2 effect of feedback control, 232, H3 modeling, 20 delta function (unit impulse), 40, 41 Index derivative approximation of, 129, 311 control action, 128–130 kick, 130, 134 Laplace transform, 41 mode filter, 129 time, 129, 136 design of control systems, 199, 266, H1, H4 design, plant, G4 detuning control loops, 341 deviation variable, 59 dialysis, kidney, diabetes mellitus, 442, 443, 449 difference equations, 125, 313 differential equations discretization, 125, 312 numerical solution, 482 solution by Laplace transforms, 42–45 differential pressure transducer, 153 digester, batch, digital communication, 471 digital control block diagram, 302, 312, 318 control hardware, 303 data acquisition, 303, 314 distributed control, 465 interface, 466 programmable logic controller, 468 stability analysis, 312 digital control algorithms, 146 analytical predictor, 319 conversion of continuous controller settings, 313 Dahlin, 315 direct synthesis, 313, 315, 319 disturbance estimation, 319 integral error criteria, 311 internal model control, 303, 318 minimal prototype, 315 modified Dahlin, 316 PID, 145, 303, 311, 314 pole placement, 317 reset windup, 145, 146 ringing, 317, 319 time-delay compensation, 316 tuning, 304, 313, 317 Vogel-Edgar, 317 digital controllers, PID, 145 approximation of analog controllers, 145, 302 derivative kick, 146 PID, 145 digital filters, 303, 446 digital signal binary representation, 466 converter, 466 multiplexer, 467 pulse train, 467 transmission, 471 digital-to-analog converter, 145, 301 digital versions of PID controllers, 145 Dirac delta function (unit impulse), 40, 41 direct-acting controller, 142 direct substitution method, 200 direct synthesis method, 201, 224, 331 Dahlin’s algorithm, 315 Vogel-Edgar, 317 discrete event analysis, 415 discrete-time signal, 307, 312, 315 discrete-time system closed-loop system, 329, 335 effect of hold element, 317 exact, 115 identification, 116 stability analysis, 312 z-transform, 307, 309, 312 discrete transfer function, 307, 312, 315, 320 discretization of ordinary differential equation, 115 of partial differential equation, 30 distance-velocity lag See time delay distillation control, alternative configurations, 348, G2 decoupling, 337, 342, 501 feedback, G2 feedforward, 266 heat integration, G3, G4 inferential, 279, 286, 293 inverse response, 89 override, 287, 289 selection of manipulated variables, 339 distributed control system (DCS), 469 distributed-parameter systems, 16 disturbance changes, closed-loop, 180 disturbance rejection, 200, G3 disturbance variable, 68, 368, 376, 378, 384 autoregressive, 320 moving average, 304, 320 non-stationary, 320 predictor, 316, 319 stationary, 320 transfer function, 280 DMC, 369 dominant time constant, 93, 206, 286, 290 double-exponential filter, 304 drift, 304, 319 Drosophila melanogaster, 455 drug delivery, 442 drug target, 453 DS, method, 201 duty cycle, 467 dynamic behavior of various processes first order, 68 higher order, 92 instruments, 152, 163 integrating process, 73 inverse response system, 88, 89 second order, 75 time delay, 89 dynamic compensation, 268 dynamic error, 163 dynamic matrix, 375 Dynamic Matrix Control (DMC), 369 dynamic model, 14 E E coli, 454, 457 entrainment, 456 error analysis, 503 eukaryote, 460 event tree analysis, 172 economic optimization, 369 497 EEPROM, 474 eigenvalue, 96, 97, 338 emergency shutdown system (ESD), 161 empirical model, 15, 105–108, 113, 114, 117 end point, 416 enterprise resource planning, 351, 447 environmental regulations, equal concern factor, 401 equal-percentage valve, 158–160 error control, 136 instrument, 162 error criteria See integral-error criteria error gap controller, 290 error signal, 136 etcher, plasma, Ethernet, 469, 471, 472 Euler identity, 249 Euler integration, 115 evaporator, 234 event-based control, 315 evolutionary operation (EVOP), 359 EWMA control chart, 402, 403, 404 exact discretization, 115, 312 Excel, 112, 116, 360, 362 Excel Solver, 108, 361, 363, 366 exponential filter, 303 exponential function approximations, 90 Laplace transform, 39 exponentially-weighted moving average (EWMA) filter, 447 F failure, computer, 465 failure probability, 171 failure rate, 163 fault detection, 169, 398 fault tree analysis, 172 FDA, 453 feasible region, 359 fed-batch, 29 feedback control adaptive, 279, 292 block diagram, design, 199, 251 disturbance changes, 180, 201 historical perspective, 135 multiple input-multiple output (MIMO) system, 326, 341 performance criteria, 200 regulator problem, 180 servo problem, 179 set-point changes, 179 transfer functions, 176–181 feedback loop, 267 dynamics, 143 hidden, 329 feedback path, 179 feedback trim, 283 feedforward control, 263 configuration, 272 design, 266 disturbance rejection, 270 lead-lag unit, 270 physically unrealizable, 280, 282 498 Index feedforward control (continued) stability considerations, 280 tuning, 284 feedforward-feedback control, 272 feedforward variable, 368 fermentor, 436, 437, 448 fiber optics, 153 fieldbus, 162, 471 field tuning, 214 filters analog, 319 derivative mode, 139 digital, 319 anti-aliasing, 302 double exponential, 304 effect on PID controller, 314 EWMA, 304 exponential, 303 moving-average, 304, 320 moving-window, 305 noise-spike, 305 rate-of-change, 305 Savitzky-Golay, 305 second order, 304 final control element, 136, 137, 141, 143, 145, 156 final value theorem Laplace domain, 45 z-domain, 323 finite-difference, 125, 311 finite impulse response (FIR) model, 117 finite step response model, 112 first-order hold, 301 first-order-plus-time-delay (FOPTD) model, 110 first-order process responses, 70 first-order system, 70, 109–111 fitting data, 109–111, 116 flash drum, 258 flexible manufacturing, 430 flooded condenser, 249 flooding, 287 flow characteristic curve, valve, 159–161 flow control, 135, 139, 228, 280, 288, 296 flow-head relation, 159, 160 flow/inventory control G1 flow rate sensors, 153 fluidized catalytic cracker, 352, 355 food industry, 438 FOPTD model, 110 forcing function, 69 forward path, 178 fraction incomplete response method, 111 freedom, degrees of, 19 frequency response analysis Bode diagrams, 257 closed-loop, J11 feedback controller, 251 gain and phase margins, 256 Nichols chart, J12 Nyquist diagram, J7 open loop, 251, 252 shortcut method, 246 fuel-air ratio control, 264 furnace cascade control, 279 thermal cracking, 354 fuzzification, 291 fuzzy logic, 290, 291 fuzzy logic controller (FLC), 291 G gain closed-loop, 182 controller, 136 critical See ultimate gain crossover frequency, 253 discrete-time system, 307, 310, 312 margin, 256 matrix, 360 open-loop, 57, 187 process, 58, 63 transfer function, 57, 281, 284, 291, 438 transmitter, 261 ultimate, 254 variable, 289 z-transform, 307, 313 gain margin, 256 gain scheduling, 289, 292 gain/time constant form, 58, 88 Gantt chart, 429 gap action, 290 gas absorption, 28 gas chromatograph, 154, 467 gas-liquid separator, 327 gas pressure control loop, 221 Gaussian distribution, 414 gene regulation, 452, 453, 454 generalized predictive control (GPC), 369 generalized reduced gradient (GRG), 362, 363 general stability criterion, 252 grade change, 292 granulator, 439, 440, 441, 448 graphical user interface, 469, 474 H half-rule, 92 hard constraint, 382, 386 hardware computer system, 467 control loop, 468 instrumentation, 209 real-time optimization, 352, 358 HART protocol, 472 HAZOP, 169, 416 heat exchanger, cascade control, 280, 281 double-pipe, 31 evaporator, 234 modeling, 31 heat shock response, 451, 452 Heaviside expansion, 43 HIV/AIDS treatment, 445 hidden feedback loop, 329 hidden oscillation, 316 hierarchy, control, 8, 350, H1 higher-order process (system), 92, 318 homeostasis, horizons, 382 Hotelling’s T2 statistic, 407, 408 hysteresis, 158 I IAE, 210 ideal controller, 137, 139–141 ideal decoupler, 342 idealized sampling, 301 identification, process, 117 If-then statement, 291 IID assumption, 401, 402 ill-conditioned, 338, 396 IMC See Internal Model Control impulse inputs, 41 modulation, 301 response, 41 response model, 117 sampler, ideal, 308 impulse function Laplace transform, 41 z-transform, 307, 310, 312, 451 incomplete response method, 109 individuals chart, 399 inferential control, 279, 286, 293 information flow diagram, 417, 420 initial value theorem Laplace domain, 45 in phase, 245, 254 input blocking, 381, 384 dynamics, 87 variables, 68, 69 input-output interface, 466, 471 input-output model continuous-time transfer function, 54 discrete-time, 307, 310, 315 installed valve characteristics, 159 instrument accuracy, 162 signal level, 152 smart, 163 instrumentation symbols, 499 insulin, 442, 443 intracranial pressure, 448 integral of the absolute error (IAE), 210 integral control, 139, 141, 142 reset windup, 138 integral error criteria, 210 integrals approximation of, 145 Laplace transform, 46 integral of the squared error (ISE), 210 integral of the time-weighted absolute error (ITAE), 210 integral time, 137 integral windup, 138 integrating process, 122 control characteristics, 193, 311, 320 response, 122 integration analytical methods, 38 numerical techniques, 481 interacting tanks, 102 Index interacting control loops, 341 decoupling of, 337 interacting processes, 94, 341 interaction index, 332, 333 interface, 151 computer-process, 466 interlock, 164, 415 Internal Model Control, 207 digital, 318, 319, 451 PID settings, 207 relationship to Direct Synthesis, 205 internal set point, 177 internal stability, J3 Internet Protocol (IP), 470, 472 internodal communication, 471 intersample ripple, 316 inverse Laplace transform, 39 partial fraction expansion, 43 inverse response, 88, 89 inverse z-transform, 308, 309 IP (Internet Protocol), 470, 472 ISA instrumentation standards, 208 ISE, 210 ISO (International Standards Organization) certification, 428, 485 ITAE, 210 linear model, 54 linear programming (LP), 376 constraints, 359 Excel solution, 361 feasible region, 359 objective function, 360 simplex method, 360 linear regression, 106–109 linear system, 58 linguistic variable, 304 liquid level dynamic model, 25, 61, 68 sensors, 152–153 load See disturbance variable local area network (LAN), 471 logic controllers, 415, 418 long-time (large-time) response, 71, 245, 247 loop failure tolerance, 341 loop gain, 62, 182 loop integrity, 335 loop shaping, J10 lot, 416, 429 low-pass filter, 205 low selector switch, 287 LP See linear programming lumped parameter system, 29 K Kappa number, kidney dialysis, kinase, 474 L lab-on-a-chip, 154 lac Z gene, 454 ligands, 457 ladder diagram, 418 ladder logic diagram, 417, 418, 419 lag, distance-velocity See time delay lambda tuning, 202, 211, 315 LAN (local area network), 471 Laplace transforms, 38–51 definition, 39 inverse of, 39 of derivatives, 39 of integrals, 46 partial fraction expansion, 43 properties, 39, 45 sampled signal, 300, 312 table, 40 layers (neural nets), 114 layers of protection, 161 lead, 249 lead-lag unit, 87, 269 least-squares estimation, 107 level control, 183, 236, G1 averaging control, 235 surge control, 235 levels of process control, 8, 351 limit checking, 9, 396, 397 limits, control three sigma, 416 six sigma, 396, 405 line driving, 208 linearization, 61 M makespan, 429 magnetic resonance analysis, 143 magnitude, 246 management-of-change-process, 161 manipulated variable, 3, 68, 232 manual mode, 132 manufacturing automation protocol (MAP), 471 Maple, 49, 51 marginal stability, 248, 254, J4 Markov parameter, 372 mass flow controller (MFC), 144 mass flow meter, 144 mass spectroscopy, 146 master controller, 281 material recycle, G3 MATHEMATICA, 44, 49 MATLAB Bode 44, 250, 257 equation solving, 480 matrix operations, 480 MPC Toolbox, 386 Parameter estimation, 112 scripts, 481 Simulink, 482 solving ODEs, 482 Symbolic Math Toolbox, 44, 49, 50, 331 toolboxes, 482 measured variables, 95, 232 measurement dynamics, 147 error, 147 location, 142 instrumentation, 140 messenger RNA, 452 microprocessor, 468 MIMO system, 98, 117, 326, 454 minimal prototype control, 315 minimum variance control, 319 mixing process, 425 mobile worker, 476 models and modeling control-relevant, 107 convolution, 370 degrees of freedom, 16 development, 16, 105 discrete-time, 115, 320, 370 dynamic, 14, 17, 107 empirical, 15, 109 error, 107 general principles, 16 input-output, 54, 105, 326 lumped parameter, 106 parsimony, 129, 372 procedure, 62, 107 semi-empirical, 15 steady-state, theoretical, 15 model-based control, 8, 20 model predictive control, 368 calculations, 379 constraints, 381 design, 374 Dynamic Matrix Control, 369 horizons, 373 implementation, 389 MIMO system, 377 move suppression, 385 set-point calculation, 382 Toolbox, 386 tuning, 384 model validation, 117 monitoring, 395 motif, 454, 462 move suppression, 395 moving-average filter, 304 moving range, 400 MPC See model predictive control multiloop control strategies, 327, 342 multiple-input, multiple-output system control system, 321 block diagram analysis, 328 decoupling control, 342 hidden feedback loop, 328 input-output model, 98, 377 linearization, 63 non-square system, 327 process interaction, 327 reducing loop interactions, 342 relative gain array, 332 square system, 327, 339 stability analysis, 92, 330 transfer-function matrix, 95, 328 variable pairing, 331 multiplexer (MUX), 467 multirate sampling, 301 multivariable control system, 326 decoupling of loops, 342 interaction of loops, 327 variable pairing, 331 multivariable transmitter, 144 multivariate control chart, 407 MUX (multiplexer), 467 499 500 Index N negative feedback, neural net, 113 Newton-Raphson method, 64 Nichols chart, J11 noise, 110, 129 noise filter, 212 noise-spike filter, 305 noninteracting processes, 94 two tank system , 94 nonlinear control system, 279 models, 61 optimization, 360 programming, 362, 363 regression, 112 transformation, 289 nonminimum-phase system, 240, 312 non-self-regulating process, 74 normal distribution, 398 normalized amplitude ratio, 245 numerator dynamics, 58, 88 numerical methods approximation of derivatives, 133, 311 approximation of integrals, 133, 311 parameter estimation in transfer function models, 118, 119 solution of equations, 481 Nyquist contour, K1 diagram, 252 stability criterion, J7, K1 O objective function, 107, 369 object linking and embedding, 470 observer, 286 offset, 129, 182, 287 one-dimensional search, 357 one-way interaction, 335 on-off controller, 136 open loop frequency response, 253 gain, 182 transfer function, 180, 253 open standards, 31 operating costs, 352 limits, 355 objectives, 9, 355 range, 152, 351 window, 360 operator interface, 31, 131 operator training, 31, 404 optimization constrained, 359 EVOP, 359 formulation, 334 multivariable, 359 real-time (RTO), 9, 350 Simplex, 360 single-variable unconstrained, 356 software, 363 outlier, 400 outrigger canoe, 124 output variable, 17 overdamped process, 75 overdamped response, 75 override control, 287 overshoot, 77 overspecified model, 20 P P&IDs (piping and instrumentation diagrams), 487 Padé approximation, 91 pairing of variables, 236, 347 parameter estimation, 106 partial decoupling, 342 partial differential equations, 16 partial fraction expansion, 38 partial least squares (PLS), 408 particle size distribution, 440 pattern tests, 402 PCA, 408 PCM (Process Control Modules), 501 perfect control, 12, 202 performance criteria, 210 performance index, 210, 397 period of oscillation, 77, 245 pharmaceutical industry, 453 phase angle, 245 crossover frequency, 253 lag, 245 lead, 249 margin, 256 shift, 245 pH control, 291 phosphorylation, 453, 473 photolithography, 400 physical realizability, 58, 129, 310 physically unrealizable controller, 129, 252, 270 PI controller, 127 PID controller, 129 digital version of, 34, 133 expanded form, 130 parallel form, 129 series form, 129 PIDPlus controller, 303, 311, 321 piping and instrumentation diagrams, 487 planning and scheduling, 8, 350 plant-model mismatch (model error), 308 plantwide control design, G1, H1, H4 case study, G4, H5 energy management, H4 hierarchical procedure, H1 inventory control, G2 production rate control, H2, H10, H12 quality control, H10 recycle loops, G7, H10 specification of objectives, H4 structural analysis, H4, H9 plasma etcher, 240 plug and play, 469 PLC (programmable logic controller), 468 PLS, 408 pneumatic controller, 124 control valve, 148 instrument, 124 signal transmission, 140 poles, 86, 326 pole-zero cancellation, 188 position form, digital controller, 133 positive feedback, 5, 452, G15 PRBS (pseudo-random binary sequence), 113 pre-act, 128 predictive control See model predictive control prediction horizon, 378 predictive emission monitoring system (PEMS), 146 pre-filter, 303 preload (batch control), 424 pressure sensor, 143 primary controller, 280 primary loop, cascade control, 280 principal component analysis (PCA), 402 Principle of the Argument, K1 Principle of Superposition, 39, 57 probability concepts, review, 491 process control, dynamics, economics, 354 gain matrix, 332, 333 identification, 105 interactions, 327 measure, 332 interface, 141 monitoring, 325 reaction curve, 109 safety, 160 thermal mixing, 98, 100 variables, 1, 229 process capability index, 404 process control language (PCL), 474 Process Control Modules (PCM), 501 distillation, 122, 227, 257, 278, 501 furnace, 122, 227, 237, 278, 501 processes batch, 2, 415 continuous, fed-batch, 2, 29 semibatch, 2, 18 stirred-tank blending, 4, 15, 18, 123, 176, 267 process reaction curve method, 218 Profibus, 142, 466 programmable logic controller (PLC), 468 prokaryote, 460 promoter, 460 proportional band, 126 proportional (P) control, 4, 125 proportional derivative (PD) control, 129 proportional-integral (PI) control, 127 proportional-integral-derivative (PID) control, 129 proportional kick, 130 protection See safety pseudo-random binary sequence (PRBS), 113 pulse duration output (PDO), 467 pulse function See rectangular pulse pulse testing, 246 Index Q quadratic interpolation, 357 quadratic programming, 362 quality control charts, 396 individuals chart, 399 s chart, 406 x̄ chart, 398 quantization, 466 quasi-steady-state operation, G14 quick-opening valve, 152 R radar and radiation level sensors, 143, 145 ramp input, 69 responses, 72 random input, 70 range, 124, 399 range control, 352 rapid thermal processing, 425 rate control action See derivative control rate-of-change filter, 223 ratio control, 264, G10 ratio station, 265 reactive scheduling, 430 reactor See also chemical reactors batch, 3, 22, 422, 425 continuous, 26 semibatch, trickle-bed, 91 real-time clock, 470 real-time optimization (RTO), 350 applications, 354 basic requirements, 352 constrained optimization, 359 Excel Solver, 361 linear programming, 359 models, 355 nonlinear programming, 362 operating profit, 332 operating window, 364 quadratic programming, 362 Real Translation Theorem, 46 receding horizon approach, 360, 370 reconstruction of continuous signals, 300 rectangular pulse, 41, 75 reference trajectory, 380 regression techniques, 106 regulator problem, 180 relative disturbance gain, 337 gain array, 332 stability, 252, 256 relay auto-tuning, 218, 341 relay ladder logic, 418 reliability analysis, 171 repeatability, instrument, 155 reset time, 126 reset windup, 127 residual, 107 resistance temperature detector (RTD), 143 resonant frequency, 248 resonant peak, H15 response mode, 87 response time, 75 reverse-acting controller, 131 RGA See relative gain array rework, 422, 430 right-half plane (RHP) pole, 87, K2 right-half plane (RHP) zero, 88 ringing, controller, 316 rise time, 77 risk assessment, 171 robustness, 215 robust performance, J14 robust stability, J14 root locus diagram, 191 Routh array, 190 Routh stability criterion, 190 RTO (real-time optimization), 367 rung (ladder logic), 433 Runge-Kutta integration, 33 run-to-run control, 443 S safety, 160 safety instrumented system (SIS), 161 safety interlock system, 161 sampled-data system stability, 312 sample mean, 307 sample variance, 169, 397 sampling, 300 aliasing, 302 multirate, 301 period, selection, 301 time-delay approximation, 313 saturation of controller, 126, 424 SCADA (supervisory control and data acquisition), 468 S cerevisiae, 454 scheduling and planning, 9, 350 s control chart, 400 search multivariable, 359 nonlinear programming, 358 one-dimensional, 356 SCM (supply chain management), 351 secondary controller, 280 secondary control loop, 280 secondary measurement, 280 second-order-plus-time-delay (SOPTD) model, 203 selection controlled variables, 232, 359 manipulated variables, 232, 359 measurement device, 142 measured variables, 233 sampling period, 301 selective control, 287 selectors, 287 self-adaptive control(ler), 292 self-regulating process, 178 self-tuning control, 292 semiconductor processing, 4, 240, 400 sensitivity 6, 129, 244, 252, J1 sensitivity function, J1 sensors, 141 composition, 145 flow-rate, 143 level, 144 pressure, 143 temperature, 143 separation concentration ratio, G4 sequential function chart, 417 sequential logic, 416 serially correlated, 402 service factors, 408, 409 servo problem, 179 set point, 124 changes, closed-loop, 179 ramping, 134, 416 trajectory, 380, 421 weighting, 209, 213, 224 settling time, 77 Shannon’s sampling theorem, 302 Shewhart control chart, 396 signals conditioning, 303 discrete-time, 141 processing of, 303 reconstruction of, 300 signal transduction, 457 signal transmission, 141 Simplex See linear programming simulation, 105 dynamic, 51 equation-oriented, 31 modular, 31 Simulink closed-loop simulation, 204 discrete-time system, 313 single-input, single-output (SISO) system, 98, 246 singular value analysis, 338 sinusoidal response of processes, 72, 246 six sigma approach, 396 sizing control valves, 150 Skogestad’s “half rule,” 92 slack parameter, 382 slack variable, 382 slave controller, 280 slope-intercept method, 218 slowdown effect, G13 slurry flow control, 288 smart devices, 155, 469, 470, 474, 476 Smith predictor technique, 284 Smith’s second-order method, 181 snowball effect, G7 soft sensor, 114, 146 span, transmitter, 142 SPC See statistical process control special cause, 396 specification limits, 404 split-range control, 287 spreadsheet software, 360, 470 SQC See statistical quality control stability analysis, 96, 188, 190, 330 closed-loop, 330 conditional, J4 criteria Bode, 252 general, 188 Nyquist, J7 Routh, 190 sampled-data, 190, 312 definitions, 96, 187 501 502 Index stability (continued) direct substitution method, 190 feedforward control, 270 marginal, 253 multivariable, 96, 330 pole (root) location, 87, 338 relative, 256 root-locus, 191 Routh method, 190 standard normal distribution, 398 standard transfer function forms gain/time constant form, 88 pole/zero form, 94 start-up, 1, 417 state-space model, 95, 372 state variables, 95 statistics review, 395 statistical process control, 396 statistical quality control, 396 steady-state control See real-time optimization steady-state gain, 70 steady-state gain matrix, 322 step function, 39 Laplace transform, 39 z-transform, 308 step input, 69 response, 71, 109 response coefficient, 371, 372, 376, 378 response model, 370–373, 377, 378 step test method, 109, 218 stick-slip, 149 stirred-tank heating system, 21 electrical heating, 23 steam heating, 24 transfer function, 59 stirred-tank reactor See continuous stirred-tank reactor stochastic process, 70 successive quadratic programming, 363 superposition principle, see Principle of Superposition supervisory control and data acquisition (SCADA), 468 supply chain management (SCM), 351 surge tank, 26, 73 sustained oscillation, 187 surface acoustic wave (SAW), 143 SVA, 338 switch, alarm, 163 Symbolic Math Toolbox, 49, 331 system identification, 116 systems biology, 451 temperature sensor, 143 theoretical models, 15 thermocouple dynamic response, 147 thermowell, 147 three-mode controller See PID controller threshold parameter, 397 time constant, 58, 70 time delay, 50, 89 Laplace transform, 47 Padé approximation, 91 polynomial approximation, 91 time-delay compensation, 284 time to first peak, 83 totalizer, 422 transcription, 452 transducers, 141 transfer function, 54 additive property, 58 approximation of higher-order, 92 closed-loop, 178 controller, 125–130 control valve, 150 definition of, 54 disturbance, 178 empirical determination of, 104, 218 final control element, 148 gain, 57 matrix, 95, 327 multiplicative property, 59 open-loop, 180 poles and zeros of, 86 process, 180 properties, 57 transient behavior, 1, 51 translation in time (time delay), 46 translation theorem, 46 transmission line, 124 transmitter, 141 transportation lag, 90 transport delay, 90 trim heat exchanger, G16 triply redundant, 164 troubleshooting control loops, 222 truth table, 417 tryptophan synthesis, 461 tubular reactor, 89, 287 auctioneering control, 287 hot spot control, 287 inverse response, 89 tuning, controller See controller tuning turbine flow meter, 153 two-point composition control, two-position (on-off) control, 136 Tyreus-Luyben tuning method, 257 T U target, 351, 369 Taylor series approximation, 61 TCP/IP, 471 temperature control, 221 ultimate gain, 215 ultimate period, 215 ultrasound level sensor, 143 undamped natural frequency, 75 underdamped process, 76 underdamped response, 75 underspecified model, 20 unit step, 39 unrealizable controller decoupling, 342 digital, 310 feedback, 129, 252 feedforward, 269 unstable closed-loop system, 186 open-loop process, 37 unsteady-state operation See dynamic behavior V validation of models, 117 valve, control See control valve valve coefficient, 151 valve positioner, 150 velocity form of digital controller, 133 virtual sensor, 114 Vogel-Edgar control algorithm, 317 W warning limits, 402 Western Electric rules, 402 windup integral, 127 reset, 127 wireless battery, 473 event-based control, 315 feedback control, 314, 473 final control elements, 473 HART, 470, 472 level sensor, 143 network, 473 PIDPlus controller, 473 Wood-Berry column, 402 Z zero, transmitter, 147 zero-order hold, 301 zeros, 87 Ziegler-Nichols method, 215 Ziegler-Nichols settings, 215 zone control, 382 zone rules, 402 z-transform, 307 approximate conversion method, 311 definition, 308 long division, 317 physical realizability, 310 properties, 308 table, 328 variable, 308 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... +j 2 = R + jI ? ?2 ? ?2 + ω τ +1 = where (14-11) R= ? ?2 ? ?2 + (14-12a) I= −ωτ ? ?2 ? ?2 + (14-12b) and From Eq 14-7, √ AR = |G(jω|) = ( )2 ? ?2 ? ?2 +1 ( + −ωτ +1 )2 ? ?2 ? ?2 Simplifying, √ AR = (1 + ? ?2 ? ?2 )... D2 + Gp2 Gv Gc2 E2 (16 -2) ̃ sp2 − Ym2 = Gc1 E1 − Gm2 Y2 E2 = Y (16-3) E1 = −Gm1 Y1 (16-4) Eliminating all variables except Y1 and D2 gives Gp1 Gd2 Y1 = D2 + Gc2 Gv Gp2 Gm2 + Gc1 Gc2 Gv Gp2 Gp1... functions for the outer and inner loops are Gc1 Gc2 Gv Gp1 Gp2 Km1 Y1 = Ysp1 + Gc2 Gv Gp2 Gm2 + Gc1 Gc2 Gv Gp2 Gp1 Gm1 (16-6) Gc2 Gv Gp2 Y2 = (16-7) ̃ + Gc2 Gv Gp2 Gm2 Ysp2 For disturbances in