Closed-loop characteristic equation

We now finally launch into the material on controllers. State space representation is more abstract and it helps to understand controllers in the classical sense first. We will come back to state space controller design later. Our introduction stays with the basics. Our primary focus is to learn how to design and tune a classical PID controller. Before that, we first need to know how to set up a problem and derive the closed-loop characteristic equation. 

The closed-loop characteristic equation of the stirred-tank heater system is hence ... 

Recall Eq. (5-11), the closed-loop characteristic equation is the denominator of the closed-loop transfer function, and the probable locations of the closed-loop pole are given by... 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

A couple of quick observations First, Gc is the reciprocal of Gp. The poles of Gp are related to the zeros of Gc and vice versa—this is the basis of the so-called pole-zero cancellation.1 Second, the choice of C/R is not entirely arbitrary it must satisfy the closed-loop characteristic equation ... 

Let s tiy to illustrate using a system with a PI controller and a first order process function, and the simplification that Gm = Ga = 1. The closed-loop characteristic equation is... 

In this problem, the closed-loop characteristic equation is... 

The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jco in the closed-loop characteristic equation to find the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this specific proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

The second order closed-loop characteristic equation in Example 7.5 can be rearranged as... 

Example 7.7 Consider installing a PI controller in a system with a first order process such that we have no offset. The process function has a steady state gain of 0.5 and a time constant of 2 min. Take Ga = Gm = 1. The system has the simple closed-loop characteristic equation ... 

Note 2 As we reduce the integral time constant from Xi = 3 min to exactly 2 min, we have the situation of pole-zero cancellation. The terms in the closed-loop characteristic equation cancel... 

This is a big question when we use, for example, a Bode plot. Let s presume that we have a closed-loop system in which we know "everything" but the proportional gain (Fig. 8.5), and we write the closed-loop characteristic equation as... 

Example 7.2C. Let s revisit Example 7.2 (p. 7-5) with the closed-loop characteristic equation ... 

Example 7.4A. This time, let s revisit Example 7.4 (p. 7-8), which is a system with dead time. We would like to know how to start designing a PI controller. The closed-loop characteristic equation with a proportional controller is (again assuming the time unit is in min)... 

We next return to our assertion that we can choose all our closed-loop poles, or in terms of eigenvalues, ib K- This desired closed-loop characteristic equation is... 

The set point tracking controller not only becomes redundant as soon as we add feedback control, but it also unnecessarily ties the feedforward controller into the closed-loop characteristic equation. 

Let s try two more examples with the following two closed-loop characteristic equations ... 

For closedloop systems, the denominator of the transfer functions in the closedloop servo and load transfer function matrices gives the closed-loop characteristic equation. This denominator was shown in Chap. 15 to be [I + which is a scalar Nth-order polynomial in s. Therefore, the... 

The two nonlinear ordinary differential equations can be linearized around the steady-state values of the reactor compositions zA and zs. Laplace transforming gives the characteristic equation of the system. It is important to remember that we are looking at the closed-loop system with control structure CS2 in place. Therefore Eq. (2.13) is the closed-loop characteristic equation of the process ... 

Consider the process of Figure 24.1a. Couple y 1 with m2 and y2 with m, to form the two loops. Draw the corresponding block diagram. Develop the resulting closed-loop input-output relationships, similar to those given by eqs. (24.9) and (24.10). Has the closed-loop characteristic equation changed or not ... 

After introducing the necessary decouplers, can you tune the controllers of two loops separately so that the stability of the overall process is guaranteed (Hint Examine closely the closed-loop characteristic equations of two decoupled loops.)... 

Next we look at the controlled output variable Y2- Figure 9. Id shows the reduced block diagram of the system in the conventional form. We can deduce the closed-loop characteristic equation of this system by inspection. 

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