Proportional gain constant

Using the value of Sq = 330 kg/h, use program REFRIG2 to find the value of the proportional gain constant kp required to give stability. 

Feed temperature Latent heat of vaporization Average steam mass flow Proportional gain Set temperature of tank Time constant of thermocouple Time constant of thermowell constant of integral control 1,TFIN=30,NOCI=3 RESET GOTOl... 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

Consider the liquid flow rate controller in Fig R5.3. We want to keep the flow rate q constant no matter how the upstream pressure fluctuates. Consider if the upstream flow Q drops below the steady state value. How would you choose the regulating valve when you have (a) a positive and (b) a negative proportional gain ... 

All tuning relations provide different results. Generally, the Cohen and Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The Ciancone-Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reflects a common industrial preference for PI controllers. 

The choice of the proportional gain is affected by two quantities the product KcK, and the ratio of dead time to time constant, td/x. It may not be obvious why the product KCK is important now, but we shall see how it arises from direct synthesis in the next section and appreciate how it helps determine system stability in Chapter 8. 

The proportional gain, integral time and derivative time constants are provided by the respective terms in the transfer function. If you have trouble spotting them, they are summarized in Table... 

The integral time constant is x = xb and the term multiplying the terms in the parentheses is the proportional gain Kc. In this problem, the system damping ratio Q is the only tuning parameter. 

If saturation is not a problem, the proportional gain Kc = 7.17 (point B) is preferred. The corresponding closed-loop pole has a faster time constant. (The calculation of the time period or frequency and confirmation of the damping ratio is left as homework.)... 

Finally, you may wonder if one can use the integral or the derivative time constant as the parameter. Theoretically, we can. We may even rearrange the characteristic equation in such a way that can take advantage of pre-packaged programs that use the proportional gain as the parameter. 

The state feedback gain including integral control K is [0 1.66 -4.99], Unlike the simple proportional gain, we cannot expect that Kn+1 = 4.99 would resemble the integral time constant in classical PI control. To do the time domain simulation, the task is similar to the hints that we provide for Example 7.5B in the Review Problems. The actual statements will also be provided on our Web Support. 

A Routh-Hurwitz analysis can confirm that. The key point is that with cascade control, the system becomes more stable and allows us to use a larger proportional gain in the primary controller. The main reason is the much faster response (smaller time constant) of the actuator in the inner loop.2... 

Proportional gain, integral and derivative time constants to PI and PID controllers. Cohen-Coon was designed to handle disturbances by preventing a large initial deviation from the set point. The one-quarter decay ratio response is generally too underdamped for set point changes. 

Proportional gain, integral and derivative time constants of PID controllers. Experimental analog of the s = jco substitution calculation. Not necessarily feasible with chemical systems in practice. Tuning relations allow for choices from 1/4 decay ratio to little oscillations. 

Proportional gain, integral and derivative time constants where appropriate. The design is not necessarily PID, but where the structure of a PID controller results, this method provides insight into the selection of the controller mode (PI, PD, PID) and settings. Especially useful with system that has no dead time. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

The reader can easily show that a phase margin of 45° is enough to tune the controller in case 1 and provide the necessary safety factor for absorbing a 50% error in the dead time. The value of the proportional gain Kc for a 45° phase margin is found to be Kc = 5.05. Assume that there is an error in the time constant which has a true value of 0.25 instead of the assumed 0.5. Then the crossover frequency is found from the equation... 

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