Often it is necessary to multiply polynomials in s together. Although the following multiplication is quite simple: (5s + 1)(3s + 1) = 15s2 + 8s + 1, a more complex multiplication, such as (2s2 + 1.5s + 1)(3s3 + 8s2 + 4s + 1) could be quite time consuming. The following conv (convolve) command makes this straightforward. Also, the roots of the resulting polynomial are easily calculated using the roots command.
» g = conv([2 1.5 1],[3 8 4 1]) g =
6.0000 20.5000 23.0000 16.0000 5.5000 1.0000
» p = roots(g)
First, define an object, such one of the transfer functions in Example 3.6.
» num = [-10 1]; den = [45 18 1];
then plot the step response (use help step to view other options)
» step(sysc)
A number of other useful functions, such as pole and tzero are shown in Module 4.
Simulink
It is highly recommended that you interactively work through the initial sections of Module 2 to understand how SIMULINK can be used to simulate dynamic systems.
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3.14 Summary
The main objective of this chapter was to develop an understanding of dynamic process behavior.
Although Laplace transforms can be used to obtain analytical solutions to differential equations, we will not be using them for this purpose during the rest of this textbook. The concept of a transfer function is very useful for control system design and analysis. In future chapters we find that transfer functions allow the use of algebra rather than differential equations when analyzing control systems composed of a number of components (controller, valve, process, sensor, etc.).
The stability of a process is determined by the eigenvalues of the state space A matrix, or the poles of the Laplace transfer function: if all are negative, the system is stable. Complex
eigenvalues (poles) correspond to underdamped (oscillatory) behavior, characterized by damping factors less than 1. Dynamic responses are also affected by the values of the zeros (roots of the numerator polynomial in s) of a transfer function. If zeros are positive (in the right-half-plane), step responses have a characteristic inverse response.
A first-order + dead time model is the most common process representation. Time-delays are represented by an irrational term (e–qs) in the Laplace domain. First and second-order Padé approximations are sometimes used for controller design; the approximations lead to right-half-plane zeros and inverse response.
The initial and final value theorems of Laplace transforms can be used to find the limiting behavior of transfer functions without having to obtain a full solution. The process gain is a measure of the long-term change in a process output for a given input change.
State space models are easily converted to Laplace transfer function models. The use of LTI objects (see Module 4) allows easy interconversion of models in MATLAB.
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References
For more details on analytical mathematical methods (including Laplace transforms) to solve differential equations, see: Boyce, W., and R. DiPrima, Ordinary Differential Equations and Boundary Value Problems, 5th ed., Wiley, New York (1992).
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Student Exercises
1: Solve for the time-domain output of a first-order transfer function to a step input change.
2: A second-order process with one pole at the origin has the transfer function
Find the output as a function of time, for a unit step input change. Sketch the expected behavior.
3: Use the initial value theorem to find the immediate response of a lead-lag transfer function to a step input change at t = 0.
Also, use the final value theorem to find the long-term response of a lead-lag transfer function to a step input change.
4: For the following second-order process with numerator dynamics, solve for the time-domain output response to a step input change of magnitude Du at t =0.
For kp = 1°C/Lpm, t1 = 3 min, t2 = 15 min, tn = 20 min find the peak temperature and the time that it occurs.
5: Consider an input-output transfer function that mimics two first-order processes in parallel
where
If the gain of g1 is positive and the gain of g2 is negative, find the conditions (relationship between gains and time constants for the two transfer functions) that cause a right-half-plane zero (resulting in inverse response to a step input change) in gp(s).
6: Consider the state space model
Find the second-order differential equation in y. Hint: first solve for x1 from the second equation, then take the derivative and substitute into the first equation.
7: Consider the following state-space model
Which has the following input-output transfer function relationship
For a unit step change in the input, u(s) = 1/s:
Find the output at t=0 and as t approaches infinity, using the initial and final value theorems.
i.
Find the time domain solution, y(t) ii.
Sketch the time domain behavior of y(t) iii.
Are your results for i, ii, and iii consistent?
iv.
8: As a process engineer with the Complex Pole Corporation, you are assigned a unit with an exothermic chemical reactor. In order to learn more about the dynamics of the process, you decide to make a step change in the input variable, the coolant temperature, from 10°C to 15°C. Assume that the reactor was initially at a steady state. You obtain the following plot for the output variable, which is reactor
temperature (notice that the reactor temperature is in °F). Use Figure 3-9 to help answer the following questions.
What is the value of the process gain? (show units) i.
What is the value of t? (show units) ii.
What is the value of z? (show units) iii.
What is the decay ratio?
iv.
What is the period of oscillation? (show units) v.
Write the second-order transfer function.
Hint: Use Figure 3-9 to assist you.
vi.
9: Match the transfer functions with the responses to a unit step input, shown in the figure.
i.
ii.
iii.
iv.
v.
v.
10: Consider the following state space model:
Find the transfer function model.
i.
Find (analytically) the time-domain output response to a unit step input change.
Sketch the expected response to a unit step input change.
ii.
11: As a process engineer, you decide to develop a first-order + time delay model of a process using a step test. The process is initially at steady state, with an input flow rate of 5 gpm and an output of 0.75 mol/L. You make a step increase of 0.5 gpm at 3:00 p.m. and do not observe any changes until 3:07 p.m. At 3:20 p.m., the value of the output is 0.8 mol/L. You become distracted and do not have a chance to look at the output variable again, until you leave for happy hour at a local watering hole at 6:30 p.m. You note that the output has ceased to change and has achieved a new steady-state value of 0.85 mol/L. What are the values of the process parameters, with units? Show your work.
12: Use the initial and final value theorems of Laplace transforms to determine the initial and final values of the process output for a unit step input change to the following transfer functions.
i.
i.
ii.
iii.
13: Consider the following state space model for a biochemical reactor. Since there are two states (the A matrix is 2 x 2) we expect that the process transfer function will be second-order. Show that pole-zero cancellation occurs, resulting in a first-order transfer function. Find the values of the gain and time constant.
14: Match the transfer functions with the responses to a unit step input, shown in the figure.
i.
ii.
iii.
iv.
v.
v.
15: Consider Example 3.4. For an impulse input of 30 kJ, find the value of the vessel temperature immediately after the impulse input is applied.
16: Consider the following second-order transfer function
For a unit impulse input, find the output response as a function of time. What is the peak change and when does it occur?
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Chapter 4. Empirical Models
The previous two chapters have focused on (i) the development of fundamental models based on material and energy balances (Chapter 2), and (ii) understanding dynamic behavior, with an emphasis on linear systems (Chapter 3). In this chapter we discuss the development of empirical models, that is, developing models based on plant tests. For continuous control-system design (most of the focus of this text), a study of Sections 4.1–4.3 will be sufficient. The discrete models developed in Sections 4.4 and 4.5 will be useful when model predictive control (MPC) techniques are presented in Chapter 16, and when digital control is studied in Module 16.
After studying this chapter the reader should be able to:
Develop continuous first-order and integrator + dead time models from step tests
Estimate parameters for discrete-time autoregressive models based on input-output data Calculate poles and zeros of discrete-time models
Develop finite step and impulse response models The major sections of this chapter are as follows:
4.1 Introduction
4.2 First-Order + Dead Time 4.3 Integrator + Dead Time
4.4 Discrete-Time Autoregressive Models 4.5 Parameter Estimation
4.6 Discrete Step and Impulse Response Models 4.7 Summary
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4.1 Introduction
For many processes there is not enough time, or it is not worth the effort, to develop fundamental process models. Particularly if your main interest is tuning a specific control loop, it is likely that you will develop a transfer function-based model by performing a plant test. The most common plant test is to make a step change in the manipulated input (controller output) and observe the measured process output response. Then a model is developed to provide the best match between the model output and the observed plant output.
There are a number of important issues in developing an input-output model. Foremost is the selection of the proper input and output variables. For many processes this is not trivial, because a particular manipulated input may affect a number of measured outputs. Similarly, a measured output may be affected by a number of manipulated inputs. In this chapter we assume that the manipulated input and measured output have already been selected. The important discussion of the selection of a particular input to be "paired" with a particular output is postponed until
Chapter 13.
It is common to base an input-output model on step responses. In this procedure we first bring the process to a consistent and desirable steady-state operating point, then make a step change in the input variable. An important decision is the magnitude of the step change to make.
If the step change in input is too small, the measured output may not change enough to develop a reliable model. This is particularly true if the measured output is "noisy." Clearly, the magnitude of step input must be enough so that the output "signal-to-noise" ratio is high enough to obtain a good model.
If the step change in the input is too large, the output variable may change too much and produce product that is "off-specification." This is not desirable because of the severe economic penalty (i.e., the plant loses too much money while the step test is being
performed). Also, if the step input change is too large, nonlinear effects may dominate. That is, the operating condition may become significantly different than the desired condition.
Clearly there is a trade-off here. The input must be changed enough to observe a change in the output variable (it must increase above the noise level), yet not so much that the output variable change is too great (causing an economic penalty).
As a process engineer conducting a plant step test, you will usually have some basic knowledge of (or experience with) the input-output pair under consideration. Observations of the measured output with time will provide an estimate of the standard deviation or variance of the
measurement noise. A rough estimate of the process gain will enable you to select an input change magnitude so that the output change is "above the noise level." An estimate of the process gain can often be obtained through steady-state material and energy balances.
In the following sections we show how to estimate parameters for some common simple models.
By far the most commonly used model, for control-system design purposes, is the first-order + time-delay model discussed in the next section.
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4.2 First-Order + Dead Time
Recall that a first-order + dead-time process, represented by the transfer function relationship
Equation 4.1
has the following output response to a step input change,
Equation 4.2
where the measured output is in deviation variable form. The three process parameters can be estimated by performing a single step test on the process input.
The gain is found as simply the long-term change in process output divided by the change in process input. Also the time delay is the amount of time, after the input change, before a significant output response is observed. There are several easy ways to estimate the time constant for this type of model.