Ideal PID controller

Finally, we can put all the components together to make a PID (or 3-mode) controller. The time-domain equation and the transfer function of an ideal PID controller are ... 

Like direct synthesis, xc is the closed-loop time constant and our only tuning parameter. The first order function raised to an integer power of r is used to ensure that the controller is physically realizable. 2 Again, we would violate this intention in our simple example just so that we can obtain results that resemble an ideal PID controller. 

Substitution of (E6-5) and (E6-6) into Eq. (6-31), and after some algebraic work, will lead to the tuning parameters of an ideal PID controller x... 

For the present problem, and based on all the settings provided by the different methods, we may select Xj = 3 s and xD = 0.5 s. We next tune the proportional gain to give us the desired response. The closed-loop equation with an ideal PID controller is now ... 

Finally, let s take a look at the probable root loci of a system with an ideal PID controller, which introduces one open-loop pole at the origin and two open-loop zeros. For illustration, we will not use the integral and derivative time constants explicitly, but only refer to the two zeros that the controller may introduce. We will also use zpk () to generate the transfer functions. 

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 PID control law considered here contains the P, PI, PD, and PID control laws as special cases. The velocity form of the discrete approximation of an ideal PID controller is given by [2]... 

It is well known that there are closed loop locations which can not be reached by constant proportional control using less than full state feedback. The common approach in the case where proportional output feedback cannot yield a satisfactory design is to add an observer to the system. A similar but somewhat different approach is to use a dynamic controller. As an example, consider the control of a second order SISO plant by an ideal PID controller cascaded with a first order filter, which is... 

In many process control applications, the control algorithm consists of three modes proportional (P), integral (I), and derivative (D). The ideal PID controller equation is... 

The PID controller combines the advantages of the three basic types very fast action and fast adjustment to the setpoint without a permanent control deviation. It is especially suitable for temperature control. An ideal PID controller can be described mathematically by Equation (2.8-2). 

We may illustrate the nature of digital PID algorithms by starting with the ideal PID controller [7, 22, 23] according to Eq. (91), where qv t),( vs are the new output and the steady-state bias values of the flow rate respectively, and e(t) = hsp — h(t) is the current error in the level height set point and measured value. 

In Example 8.12, we used the interacting form of a PID controller. Derive the magnitude and phase angle equations for the ideal non-interacting PID controller. (It is called non-interacting because the three controller modes are simply added together.) See that this function will have the same frequency asymptotes. 

We can build our own controllers, but two simple ones are available an ideal PID and a PID with approximate derivative action. 

The transfer function of a real PID controller, as opposed to an ideal one, is the PI transfer function with a lead-lag element placed in series. 

It is a long and frustrating process to adjust a controller to an evaporation source, requiring several minutes for stabilization and hours to obtain satisfactory results. Often the parameters selected for a certain rate are not suitable for an altered rate. Thus, a controller should ideally adjust itself, as the new controllers in INFICON coating measuring units do. At the beginning of installation and connection the user has the unit measure the characteristics of the evaporation source. Either a PID controller is used as the basis for slow sources or another type of controller for fast sources without significant dead time. 

The best results in evaluating coating control units are achieved with ITAE. There are overshoots, but the reaction is fast and the ripple time short. Controller setting conditions have been worked up for all integral evaluation criteria just mentioned so as to minimize the related deviations. With a manual input as well as vi/ith experimental determination of the process response coefficients, the ideal PID coefficients for the ITAE evaluation can easily be calculated from equations 6.13,6.14 and 6.15 ... 

However, the ideal control algorithm would have no overshoot, no offset, and a quick response characteristic. For this purpose, a proportional action (P), an integral action (I), and a differential action (D) were combined as a PID controller as follows. 

Proportional-plus-Integral-plus-Derivative (PID) Control The derivative mode moves the controller output as a function of the rate of change of the controlled variable, which adds phase lead to the controller, increasing its speed of response. It is normally combined with proportional and integral modes. The noninteracting or ideal form of the PID controller appears functionally as... 

In a well-engineered system, measurement noise is unlikely to affect the performance of a PI controller significantly (Section V.A.2), but it may limit the use of derivative control action and will certainly limit the performance of advanced control schemes that attempt to approach ideal control (Section V.A.5). Our approach to PID control has been to optimize PI controllers and leave the possible benefit of derivative action to the commissioning engineers. 

Tuning of a PID controller should ideally lead to values of the P, I, and D terms of the controller that result in the most favorable actual control performance. The P-term is described by the proportional gain Kp or the proportional bandwidth Xp. The effect of changing the proportional band is shown in Fig. 4.41. 

The inclusion of this algorithm has added four tuning constants. When feedforward is added to an existing feedback scheme we will show later that it may be necessary to retune the PID controller. Tuning seven constants by trial and error would be extremely time-consuming. While a little fine tuning may be necessary we should use the process dynamics to obtain the best possible estimate. Ideally we should be able to obtain an estimate which works first time. 

Table 8.1 shows the most important forms of PID controllers, controller equations, and transfer functions. The derivation of several controller equation forms is left as an exercise for the reader. The table is organized by the descriptive names used in this book, but common synonyms are also included. However, all these terms should be used with caution as a result of the inconsistent terminology that occurs in the literature. For example, referring to the parallel form (the first line of Table 8.1) as an ideal controller is misleading, because its derivative... 

Ideal Proportional-Derivative Controller. The ideal proportional-derivative (PD) controller (cf. Eq. 8-11) is rarely implemented in actual control systems but is a component of PID control and influences PID control at high frequency. Its transfer function is... 

Figure 14,6. Bode plots of ideal parallel PID controller and series PID controller with derivative filter (a = 1)...

An ideal three-mode PID (proportional, integral, and derivative) feedback controller is described by the equation ... 

---