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ChE 170 Lecture 9 Controller tuning Criteria for good control • Most operators of processes know what they want in the form of a response to a change in set point or load. • For example, a response that gives minimum overshoot and ¼ decay ratio is often considered as a satisfactory response. • In many cases, tuning is done by trial and error until such a response is obtained. Performance Criteria for Closed-Loop Systems 1. The closed-loop system must be stable 2. The effects of disturbances are minimized, providing good disturbance rejection 3. Rapid, smooth responses to set-point changes are obtained, that is, good set-point tracking 4. Steady-state error (offset) is eliminated 5. Excessive control action is avoided. 6. The control system is robust, that is, insensitive to changes in process conditions and to inaccuracies in the process model Tradeoffs between Performance and Robustness Rapid Smooth Little oscillation Balance Performance Robustness Satisfactory performance to wide range of condition. Conservative controller settings. Tradeoffs between set point tracking and disturbance rejection Balance Excellent set point tracking will provide sluggish response to disturbance Set point Tracking Disturbance Rejection Excellent disturbance rejection will produce large overshoot in set-point changes. Performance Criteria Quarter-Amplitude Decay If a loop is oscillating, each peak deviation should be only onefourth of the previous peak deviation on the same side of set point. Also called “Quarter-wave decay”, “Quarter-wave damping” Quarter-Amplitude Decay Response Quarter-Amplitude Decay Response Quarter-wave decay ratio response •A control loop tuned to have a Quarter-wave Amplitude decay response in a set point change will show a sluggish response in a load disturbance. •A control loop tuned to have a Quarter-wave Amplitude decay response in a load upset will show a too oscillatory response in a set point change. Quarter-wave decay ratio response • For many applications, the quarter-amplitude-decay criterion provides acceptable damping of the oscillations that follow a set point change. For other applications, quarter-amplitude decay may be too oscillatory. A plant’s operations and engineering staff may be willing to accept a more sluggish response to a load upset in order to minimize the overshoot that follows a set point change. • Some operators prefer to see loops that are critically damped; that is, they want the measurement to rise rapidly to the new set point value and yet avoid overshoot. Quarter-wave decay ratio response • For many control loops, the set point is rarely changed. The purpose of these loops is to minimize the effect of disturbances. Even so, because set point changes are usually made more easily than load changes, in actual practice many loops are tuned for a suitable response to a set point change. Then, the resulting response for a load upset is accepted even though it may not be the best. • A preferable tuning strategy would be to tune the controller for best response to a load change, then use one of the set point “softening” techniques to ameliorate the effect of occasional set point changes. Performance Criteria for Closed-Loop Systems Decay ratio alone does not provide a complete performance specification. Other criteria that are sometimes used to specify, or measure, control loop performance include: Rise time—the time between a set point change and the first crossing of the set point; Overshoot ratio—the ratio of the magnitude of the first overshoot above set point to the magnitude of the set point change itself; Settling time—the time, following a set point change or disturbance, that it takes for the oscillation to become so small that the deviation does not exceed some specified amount. Integral Error Criteria Integral Error Criteria based on minimizing the integral of some function of the error. Note: Graphical interpretation of IAE. The shaded area is the IAE value. Integral Error Criteria The optimum settings minimize an integral error criterion. Note: Techniques for determine controller settings 1. 2. 3. 4. 5. 6. Direct Synthesis (DS) method Internal Model Control (IMC) Controller tuning relations Frequency response techniques Computer Simulation On-line tuning after the control system is installed Note: Techniques 1 – 5 are based on process models, however, initial controller setting are typically adjusted after the control system is installed. Techniques 1 – 5 provide a good initial controller setting for the subsequent on-line tuning. On-line tuning after the control system is installed On-site tuning Field tuning Controller tuning If preliminary settings are not satisfactory, alternative settings are obtained from simple experimental tests. And if necessary, the settings are fine-tuned by a modest trial-and-error i.e. Continuous Cycling Method Continuous Cycling Method: • Refers to sustained oscillation with a constant amplitude • Closed-loop tuning method (automatic mode) • Determined empirically to provide closed-loop response that have a ¼ decay ratio • (ultimate gain) and (ultimate period) be obtained from a closed-loop test of the actual process. • Example: • Ziegler-Nichols (Z-N) controller settings • Tyreus-Luyben controller settings Experimental determination of the ultimate gain Controller setting based on continuous cycling method The ultimate period Pu is defined as the period of the sustained cycling that would occur if a proportional controller with gain Kcu were used. = , Continuous Cycling Method Steps Step 1. After the process has reached steady state (at least approximately), eliminate the integral and derivative control action by setting to zero and to the largest possible value. Step 2. Set Kc equal to a small value (e.g., 0.5) and place the controller in the automatic mode. Step 3. Introduce a small, momentary set-point change so that the controlled variable moves away from the set point. Gradually increase Kc in small increments until continuous cycling occurs. The numerical value of Kc that produces continuous cycling (for proportional-only control) is called the ultimate gain, Kcu. The period of the corresponding sustained oscillation is referred to as the ultimate period, Pu. Continuous Cycling Method Step 4. Calculate the PID controller settings using the Ziegler-Nichols (Z-N) tuning relations Step 5. Evaluate the Z-N controller settings by introducing a small set-point change and observing the closed-loop response. Fine-tune the settings, if necessary. Disadvantages of Continuous Cycling Method • Time-consuming if several trials are required and the process dynamics are slow. The long experimental tests may result in reduced production or poor product quality. • In many applications, continuous cycling is objectionable because the process is pushed to the stability limits. • Not applicable to integrating or open-loop unstable processes because their control loops typically are unstable at both high and low values of Kc, while being stable for intermediate values. • For first-order and second-order models without time delays, the ultimate gain does not exist because the closed-loop system is stable for all values of Kc, providing that its sign is correct. However, in practice, it is unusual for a control loop not to have an ultimate gain. Direct synthesis method (Controller synthesis) Consider the block diagram for a standard feedback control system. The closed-loop transfer function for set-point changes K mGcGvG p Y Ysp 1 GcGvG p Gm For simplicity, let and assume Gc G Y Ysp 1 Gc G (12-1) (12-2) Rearranging and solving for Gc 1 Y / Ysp Gc G 1 Y / Ysp (12-3a) •Let (Y/Ysp)d be the desired closed-loop transfer function What do we choose for ? The specification of (Y/Ysp)d is the key design decision Ideally we want (Y/Ysp)d = 1, a perfect control, but this can not be achieve for feedback control For process w/o time delay, 1st order model is a reasonable choice: where Desired Closed-Loop Transfer Function For processes without time delays, the first-order model is Y Ysp 1 d c s 1 (12-4) • The model has a settling time of ~ 4 • Because the steady-state gain is one, no offset occurs for set-point changes. • By substituting (12-4) into (12-3b) and solving for Gc, the controller design equation becomes: Recall PID controller: If = Desired Closed-Loop Transfer Function If the process transfer function contains a known time delay, θs Y e Y sp d τc s 1 (12-6) • The time-delay term in (12-6) is essential because it is physically impossible for the controlled variable to respond to a set-point change at t = 0, before t = . • If the time delay is unknown, must be replaced by an estimate. • Combining Eqs. 12-6 and 12-3b gives: 28 Approximating the time-delay term in the denominator of (12-7) with a truncated Taylor series expansion: e θ s 1 θs (12-8) Substituting (12-8) into the denominator of Eq. 12-7 and rearranging gives If in a FOPTD Model Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI controller, with the following controller settings: Gc K c 1 1/ τ I s Where 1 τ Kc , K θ τc τI τ (12-11) If in an SOPTD Model Substitution into Eq. 12-9 and rearrangement gives a PID controller in parallel form, 1 Gc K c 1 τD s (12-13) τI s where: 1 τ1 τ 2 Kc , K τc τ I τ1 τ 2 , τ1τ 2 τD τ1 τ 2 (12-14) Internal Model Control (IMC) • A more comprehensive model-based design method, Internal Model Control (IMC), was developed by Morari and coworkers (Garcia and Morari, 1982; Rivera et al., 1986). • The IMC method, like the DS method, is based on an assumed process model and leads to analytical expressions for the controller settings. • These two design methods are closely related and produce identical controllers if the design parameters are specified in a consistent manner. • Current modern technique in design controller •The IMC method is based on the simplified block diagram shown in Fig. 12.6b is the TF of the ideal process model is the TF of the actual process is the ideal process model response is the actual response ∗ is the IMC controller is the TF of the ideal process model is the TF of the actual process is the ideal process model response is the actual response ∗ is the IMC controller • The model response is subtracted from the actual response Y, and the difference, Y Y * is used as the input signal to the IMC controller, Gc • In general, Y Y due to modeling errors that are not accounted for in the model. G G and unknown disturbances D 0 • It can be shown that the two block diagrams are identical if controllers Gc and Gc* satisfy the relation Gc Gc* (12-16) 1 Gc*G • Thus, any IMC controller Gc* is equivalent to a standard feedback controller Gc, and vice versa. • The following closed-loop relation for IMC can be derived from Fig. 12.6b using the block diagram algebra of Chapter 11: Y Gc*G 1 Gc* G G Ysp 1 Gc*G 1 Gc* G G D (12-17) For the special case of a perfect model, G G, (12-17) reduces to Y G c* G Y sp 1 G c* G D (12-18) The IMC controller is designed in two steps: Step 1. The process model is factored as G G G (2-19) where G contains any time delays and right-half plane zeros. • In addition, G is required to have a steady-state gain equal to one in order to ensure that the two factors in Eq. 12-19 are unique. Step 2. The controller is specified as 1 * Gc f G (12-20) where f is a low-pass filter with a steady-state gain of one. It typically has the form: f 1 τc s 1 r (12-21) In analogy with the DS method, τc is the desired closed-loop time constant. Parameter r is a positive integer. The usual choice is r = 1. For the ideal situation where the process model is perfect G G , substituting Eq. 12-20 into (12-18) gives the closed-loop expression Y G fYsp 1 fG D (12-22) Thus, the closed-loop transfer function for set-point changes is Y G f Ysp (12-23) Selection of • The choice of design parameter and IMC design methods. is a key decision in both the DS • In general, increasing produces a more conservative controller because Kc decreases while increases. • Several IMC guidelines for 12-10: and have been published for the model in Eq. Rivera et al., 1986 Chien and Fruehauf, 1990 (Skogestad, 2003) Solution: a. Table 12.1 IMC-Based PID Controller Settings for Gc(s) (Chien and Fruehauf, 1990). See the text for the rest of this table. Table 12.1 IMC-Based PID Controller Settings for Gc(s) (Chien and Fruehauf, 1990). See the text for the rest of this table. Tuning Relations Based on Integral Error Criteria • Developed for FOPTD model • Parallel form of the PID Controller (noninteractive) • Disturbance and Process Transfer function are identical (Gd=G). • Optimal set-point changes and step disturbances are different. Controller Design relations Based on the ITAE Performance Index and FOPTD Model Controller Design relations Based on the ITAE Performance Index and FOPTD Model (For disturbance and set-point change) DESIGN RELATION: P only: PI: PID: For set-point change, DESIGN RELATION FOR INTEGRAL MODE: Other tuning relation Other tuning relation