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1 ISS0080 Automation and Process Control Lecture 5 1 Process models the desire to describe reality Model of the process, model simplication, identication. model reality is complex process Replaces the original; Maintains the essential characteristics, describes the process; One process can be described by the several models. Model is used on purpose: - Similarity of appearance - material model; - Understanding of the process, predicting of the behavior to change the process (properties) in desired direction eciency, safety, control, etc. How to develop a model? Experimental data / empirical model / so called "Black box" Computationally/ functional model / where based on the components and the laws of nature: dierential equations, state-space representation, and transfer function Combined method, so called "gray box". Type of the model will depend on assumptions initially made to dene the system. In general, the more assumptions we make, the simpler the structure of the model will be[3]. How to check a model? The model is always approximate, the model must be "Accurate enough" to describe the ; K.Vassiljeva

2 Lecture 5 ISS0080 Automation and Process Control "Simple" to calculate. Model should be specied or simplied if needed. The model is specied if its features can be changed in the area that interests us (closed system behavior). The model is simplied if its parameters are insignicant. Our goal: We are looking for the models that describe what is important for the process in a closed system. These models allow you to choose the controller, the descriptive reality as similar to the behavior of a closed system, will provide answers to key questions: When the system is stable? What are the characteristics of the system? Example 1 Heater Model of the system models of the components Actuator Heater Sensor W (s) = K T 1 s K 2 (1 + T 2 s)(1 + T 3 s) K T 4 s More oscillatory Im Faster response Unstable poles Inverese response zeros Re K.Vassiljeva

3 ISS0080 Automation and Process Control Lecture 5 jω p 4 p 3 p 2 p 1 σ Time constants: T 1, T 2, T 3, T 4 (poles p = 1/T ) Do we need so accurate (complex) model for the control? Simplication: the model reduction State-space representation A, B, C, D non-critical states? Transfer function W (s) not important poles? 1. Considering the dominant pole p 1 (min p max T ), Do not consider others, First Order model. Too simple to describe the system with required accuracy, can be used to describe the components. 2. Take into account two important poles p 1, p 2 Do not consider others, Second Order model. more accurate model, not exact, there are better models. In real word where are not purely rst or second order systems, there are additional precesses which impacts should be considered. 3. Several distant poles p 1, p 2,... with time constants T 1, T 2,... Time constants are summarized and provided as delay W (s) = e s T i Important dynamical feature of the object provided as time constant, insignicant - as delays (models FOPDT, SOPDT) 1.1 FOPDT model First Order Plus Dead Time W (s) = K 1 + T s e τ s (1) where K p - process gain: the ultimate value of the response (new steady-state) for a unit step change in the input. T - Time constant: measure of time needed for the process to adjust to a change in the input. τ - Delay: the time at which output of the system begins to change minus the time at which the input step change was made [3]. Three parameters: K, T, τ simple but moderately complex. K.Vassiljeva

4 Lecture 5 ISS0080 Automation and Process Control Describes the dynamics of the system with sucient accuracy controllers work on this basis. Easily obtained with simplication of the complex models; Easy to identify. The problem: if the highest insignicant time constant (T 2 ) is close to the important time constant (T 1 ) Example 2 Approximation with FOPDT T p s e Tps ; (2) 1 T z s e Tzs (3) If important T 1 is the largest time constant T 1 > T 2 > T 3, the third order system can be approximated as follows: K p (1 + T 1 s)(1 + T 2 s)(1 + T 3 s) K p 1 + T 1 s T 2 s T 3 s K p (1 + T 1 s) e T2s e T3s Example 3 Skogestad method [6] = K p (1 + T 1 s) e (T2+T3)s The largest neglected time constant should be divided between the smallest retained time constant and the time delay: for the rst order model: T 10 = T 1 + T 2 /2, τ = τ 0 + T 2 /2 + i 3 T pi + j T nj (4) for the second order model: T 10 = T 1, T 20 = T 2 + T 3 /2, τ = τ 0 + T 3 /2 + i 4 if we have positive numerator against neighboring denominator T z /T p for T z T p τ 1 + T z s 1 + T p s T z /τ for T z τ T p 1 for τ T z T p T z /T p T p/t p T p T z for T p T z 5τ T pi + j where T p = min(t p, 5τ) T z T nj (5) (6) K.Vassiljeva

5 ISS0080 Automation and Process Control Lecture SOPDT model K p (1 + T 1 s)(1 + T 2 s)(1 + T 3 s)(1 + T 4 s) K p e (T3/2+T4)s (1 + T 1 s)[1 + (T 2 + T 3 /2)s] Second Order Plus Dead Time W (s) = Ke τ s (1 + T 1 s)(1 + T 2 s) (7) gain, two time constants, delay used if T 2 > τ setting of the parameters is not an easy task! 1.3 Empirical model: identication Theoretical models, based on chemistry and physics of the process, may not be practical for complex processes if model requires large number of equations with a lot of process parameters and unknown parameters. An alternative approach is to develop an empirical model directly from experimental data. In general, empirical dynamic models are simpler than theoretical ones [5]. If we would like to tune the control loop, it is more likely you will design a model by performing a plant test. Process model is based on experimental data: Test, observation of the I/O signals - "black box" Planning Test Model objectives: that is, how will be used, who will be the user? A priori information: stable, static; Operating point, the input (step response, value), what to measure; How much time it takes, safety requirements. Presence of the non-linearity. Are other inputs stable? Structure of the model: what is known? Aim is to obtain model with acceptable accuracy. K.Vassiljeva

6 Lecture 5 ISS0080 Automation and Process Control What is needed to control the process. CRI - Control Relevant Identication. Parameters estimation Graphical, statistical (regression analysis). Evaluation of the model How accurately the model describes the data? Verication Additional check with other data. Models Types model structure test type step impulse 2 impulses PRBS model parameters 1. order X X K p, T 2. order X K p, T/ω, β FOPDT X K p, T, τ SOPDT - K p, T 1, T 2, τ ARX - X Step response or process reaction curve Often used, easy to understand; Does not contain information about the high-frequency behavior; Does not have theoretical advantages; Works well with a noise; if signal/noise ratio <5 it is hard to found the derivative. Not perfect step change in the input causes the error rise time t s = 0.1 τ of the input signal causes the measurement error of a delay τ up to 20%. The input must be changed enough to observe a change in the output variable (it must increase above the noise level), but not so much that the output change is too great (economical reasoning) [1]. Simple identication tests provide simple process models, which can be used to design the control system with limited features. Sometimes that is enough K.Vassiljeva

7 ISS0080 Automation and Process Control Lecture 5 Other methods Two impulses Pseudo-Random Binary Sequence (PRBS) + u + u T i T i t t - u - u (a) Two impulses (b) PRBS Figure 1: Test signals 1.4 Experimental estimation of the FOPDT model parameters u = 1(t) Δu y(t) step response of a static object input step change u at time instance 0 Register a reaction curve in response to a step change in the input from one steady-state value to another y 1 y 2 (see Fig. 2) The process gain K p K p = y y(t ) = u u(t ) (8) Maximum Slope Method Locate the inection point of the process reaction curve and draw a tangent (puutuja / êàñàòåëüíàÿ) along it. The intersection of the tangent line and the time axis y 1 corresponds to the estimate of time delay τ = [0 A]. The intersection with the nal steady-state line helps to calculate the time constant T = [A B]. K.Vassiljeva

8 Lecture 5 ISS0080 Automation and Process Control y(t) T y Δy Δy y 1 0 τ A C B t Figure 2: Process reaction curve Numerical Application of 63% Method Time delay can be found like in previous method: τ = [0 A]. Time constant T can be obtained by looking at the 63% response time. Calculate the 0.63 y of the output signal. Mark the time instance then output value is equal to it, so T = [A C]. More accurate technique. Two-Point Method Observes the output relative change y ( %) y = (y y 1 )/(y2 y 1 ). Here the time required for the process output to make 28.3% and 63.2% of the long-term change is denoted by t 28.3% and t 63.2%, respectively. The time constant and time delay can be estimated the following way In case of the 20% and 80% levels T = 1.5(t 63.2% t 28.3% ) (9) τ = t 63.2% T (10) T = 0.721(t 80% t 20% ) (11) τ = 1.161t 20% 0.161t 80% (12) The primary limitation to using step responses to identify FOPDT transfer functions is the amount of time required to assure that the process is approaching a new steady state. That is, K.Vassiljeva

9 ISS0080 Automation and Process Control Lecture 5 y(t) y 2 1 Δy y 1 0 t 1 t 2 t Figure 3: Two-Point Method the major limit is the time required to determine the gain of the process. For large time constant processes it is often desirable to use a simpler model that does not require a long step test time [1]. K.Vassiljeva

10 Lecture 5 ISS0080 Automation and Process Control K.Vassiljeva

11 Bibliography [1] B. Wayne Bequette Process Control: Modeling, Design, and Simulation. Prentice Hall, [2] William L. Luyben, Process Modeling, Simulation, and Control for Chemical Engineers. McGraw-Hill, [3] Jose A. Romagnoli and Ahmet Palazoglu Introduction to Process Control. Taylor and Francis, [4] F. G. Shinskey, Process Control Systems: application, design, and tuning. McGraw-Hill, [5] Edgar Seborg and Doyle Mellichamp, Process Dynamics and Control. John Wiley & Sons, Inc, [6] Sigurd Skogestad, "Simple analytic rules for model reduction and PID controller tuning", Journal of Process Control, Volume 13, Issue 4, June 2003, Pages

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