Dominant closed-loop poles

The technique of using the damp ratio hne 0 = cos in Eq. (2-34) is apphed to higher order systems. When we do so, we are implicitly making the assumption that we have chosen the dominant closed-loop pole of a system and that this system can be approximated as a second order underdamped function at sufficiently large times. For this reason, root locus is also referred to as dominant pole design. 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

Since the dominant closed-loop poles are located very close to the jCO axis the response speed is very slow conpared with that of the closed-loop sjrstem shown in Figure 8-73[Pg.160]

When we used root locus for controller design in Chapter 7, we chose a dominant pole (or a conjugate pair if complex). With state space representation, we have the mathematical tool to choose all the closed-loop poles. To begin, we restate the state space model in Eqs. (4-1) and (4-2) ... 

From the root locus plots, it is clear that the system may become unstable when x = 0.05 s. The system is always stable when = 5 s, but the speed of the system response is limited by the dominant pole between the origin and -0.2. The proper choice is xt = 0.5 s in which case the system is always stable but the closed-loop poles can move farther, loosely speaking, away from the origin. 

Considering asymptotically stability, design the overshoot a < 5% and peak time < 10s. There are 3 closed-loop poles in 3-order system. Firstly, dominant poles and A on the left side half open complex plane are selected. Then, the other dominant pole A should be on the left side half open complex plane far away from A and Aj for weakening the impact of system. Thus, the system can be simplified as 2-order system with 2 dominant poles. 

The performances specification in tracking and further control to clarify the dominant auxiliary poles of the closed loop are successively represented in table (2). 

With respect to the problem of choosing a jmd r for the control signed performance specification, we recommend that their values be related to the process dynamics in order to achieve a desired closed-loop response. Three examples are presented here to show how to choose a and t in such a way as to cancel the dominant process pole using the zero of the lead-lag element in Equation (6.11). 

In Chapter 2, it was stated that if the process is greater than first order but without time delay, a reasonable choice for the scaling factor p can be based on the dominant time constant of the process. In this case, we can let ar = i to cancel this dominant pole in G(s), which gives t = allowing us to dioose a to bring about the desired closed-loop response speed. 

The poles of the closed loop system transfer function may be real and/or complex conjugate pairs. For systems with more than one pole, the pole which has the slowest response is dominant over other poles after some time. For stable systems, the dominant pole is the pole nearest to the imaginary axis (the pole with largest value of cr/lcol), and it is used to determine the stability of the system. The stability of the system depends on the value of cr. For the system to be stable, all the poles of the closed loop transfer function must have negative real parts (cr < 0). The system becomes unstable if a pole crosses the imaginary axis and enters into the... 

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