{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

17_SEM_KL_Ch7_W12

# 17_SEM_KL_Ch7_W12 - Fitting First and Second-Order Models...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Fitting First and Second-Order Models Using Step Tests Development of Empirical Models From Process Data • Simple transfer function models can be obtained graphically from step response data. Chapter 7 • In some situations it is not feasible to develop a theoretical (physically-based model) due to: 1. Lack of information 2. Model complexity 3. Engineering effort required. • An attractive alternative: Develop an empirical dynamic model from input-output data. • Advantage: less effort is required • If the process of interest can be approximated by a first- or second-order linear model, the model parameters can be obtained by hand calculations using the response curve. • The response of a first-order model, Y(s)/U(s)=K/(s+1), to a step change of magnitude M is: / y t KM (1 e t ) (5-18) • Disadvantage: the model is only valid (at best) for the range of data used in its development. i.e., empirical models usually don’t extrapolate very well. • A plot of the output response of a process to a step change in input is sometimes referred to as a process reaction curve or bump test. 1 2 y (t ) 1/ t 1 e KM • The initial slope is given by: d y 1 dt KM t 0 τ (7-15) • The gain can be calculated from the steady-state changes in u and y: K y y u M Figure 7.3 Step response of a first-order system and graphical constructions used to estimate the time constant, τ. where Δy is the steady-state change in y. 3 4 First-Order Plus Time Delay Model G (s) K e -θs τs 1 For this FOPTD model, we note the following characteristics of its step response: 1. The response attains 63.2% of its final response at time, t = . 2. The line drawn tangent to the response at maximum slope (t = ) intersects the y/KM=1 line at (t = ). 3. The step response is essentially complete (99.3%) at t = 5 , the “settling” time. Figure 7.5 Graphical analysis of the process reaction curve to obtain parameters of a first-order plus time delay model. 5 6 Sundaresan and Krishnaswamy’s Method There are two generally accepted graphical techniques for determining model parameters , , and K. • They proposed that two times, t1 and t2 , be found from a step response curve, corresponding to the times for 35.3% and 85.3% of the final response, respectively. Method 1: Slope-intercept method First, a slope is drawn through the inflection point of the process reaction curve in Fig. 7.5. Then and are determined by inspection. • The time delay and time constant are then estimated from the values of t1 = t35.3 and t2 = t85.3 using the following equations: θ 1.3t1 0.29t2 τ 0.67 t2 t1 Alternatively, can be found from the time that the normalized response is 63.2% complete. (7-19) • These values of and approximately minimize the difference between the measured response and the model, based on a correlation for many data sets to yield the 5 “parameter numbers” of the method: 35.3, 85.3, 1.3, -0.29, 0.67. Method 2. Sundaresan and Krishnaswamy’s Method This method avoids use of the point of inflection construction entirely to estimate the time delay. 7 8 Estimating Second-order Model Parameters Using Graphical Analysis • “In general”, a better approximation to an experimental step response can be obtained by fitting a second-order model to the data (first-order models are a “subset” of 2nd-order ones). • Figure 7.6 shows the range of shapes that can occur for the 2nd-order overdamped step response model, K G s (5-39) τ1s 1 τ 2 s 1 • Figure 7.6 includes two limiting cases: τ 2 / τ1 0 , where the system becomes first order, and τ 2 / τ1 1 , the critically damped case. Note dimensionless time used. Figure 7.6 Step response for several overdamped secondorder systems. • The larger of the two time constants, τ1 , is called the dominant time constant. 9 10 11 12 Smith’s Method – not just overdamped • Assumed model: G s Ke θs τ 2 s 2 2ζτs 1 • Procedure: 1. Determine t20 and t60 from the step response and their ratio t20/t60 – after eliminating time delay. 2. Find ζ and t60/ from Fig. 7.7 (2 y-values). 3. Using t60/ from Fig. 7.7 and then calculate (since t60 is known) ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online