Desired closed-loop poles

Calculate desired closed-loop poles desiredpoles=rcots(chareqn)... 

Actual closed-loop poles = desired closed-loop poles... 

The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6-20) that Gc is sort of the reciprocal of Gp. The zeros of Gc are by choice the poles of Gp. The product of GcGp cancels everything out—hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as... 

Note that the system remains stable in all cases, as it should for a first or second order system. One final question Based on the design guidelines by which the system should respond faster than the process and the system should be slightly underdamped, what are the ranges of derivative and integral time constants that you would select for the PD, PI, and PID controllers And in what region are the desired closed-loop poles ... 

Ackermanns formula improves the traditional pole assignment standard algorithm of SISO system. The open loop eigenpolynomial of system is not requisite. Ackermanns formula is used for designing the control law u, and the desired closed-loop pole is obtained from designed ideal pole distribution with differential transformation method. 

The angle deficiency at the desired closed-loop pole at s -4 > J0.2 is ohtinsd as foUoiiBs... 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

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... 

A structured model of the process (typically a Laplace tramsfer function) is used directly in a design method such as pole-placement or internal model control (IMG) to yield expressions for the controfier parameters that are functions of the process model parameters and some user-specified parameter related to the desired performance, e.g. a desired closed-loop time constant. These approaches to PID design carry restrictions on the allowable model structure, although it has been shown that a wide range of types of processes can be accommodated if the PID controller is augmented with a first order filter in series. An example of this design approach may be found in Rivera et al. (1986). 

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. 

Plants with unstable poles and significant (i.e. not much larger than the desired bandwidth of the closed-loop system) RHP zeros or significant time delays are extremely problematic. Special care also is required for unstable plants with actuator rate limitations. 

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