Transfer function poles

As with any filter, we can find the poles by finding the roots of the transfer-function polynomial. 

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The most important dynamic aspect of any system is its stability. We learned in Chapter 2 that stability is dictated by the location of the roots of the characteristic equation of the system. In Chapter 7 we learned that the roots of the denominator of the system transfer function (poles) are exactly the same as the roots of the characteristic equation. Thus, for the system to be stable, the poles of the transfer function must lie in the left half of the s plane (LHP). 

Dynamic Character. The extent of complexity associated with the dynamic responses (for linear systems step response, frequency response, or transfer function poles and zeros) From a classification of Simple for systems exhibiting first-order, and other relatively low-order behavior, through a classification of Moderate for systems exhibiting higher order, but still relatively benign, behavior, to a classification of Difficult for systems exhibiting problematic dynamics such as inverse response, time delays, etc. that impose severe limitations on the best possible control system performance. 

Then a second computer analysis has been used in this chapter to determine a very accurate analytical model for the frequency response of the circuit in terms of the general frequency parameter f Such an analysis is frequently performed by hand where one must solve a set of coupled equations, keeping all important frequency dependent terms. This becomes very difficult and beyond our ability to correctly perform the math for more than a simple circuit. However, with the approach used here, considerably more complicated problems can be addressed. By performing similar analyses with different circuit parameters, one could map out the dominant dependences of the transfer function poles and zeros on the values of all the circuit parameters. 

The position of the closed-loop poles in the. v-plane determine the nature of the transient behaviour of the system as can be seen in Figure 5.5. Also, the open-loop transfer function may be expressed as... 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

An example of the use of an ISIM sub-model representing a complex pole transfer function is also given in the ISIM manual (obtainable separately). 

The poles of the transfer function (roots of the denominator) are at -1, -l j, -3, -4, -5, -8 and -10. Let us assume that we seek a third order system that follows as closely as possible the behavior of the high order system. Namely, consider... 

The poles reveal qualitatively the dynamic behavior of the model differential equation. The "roots of the characteristic equation" is used interchangeably with "poles of the transfer function."... 

For the time domain function to be made up only of exponential terms that decay in time, all the poles of a transfer function must have negative real parts. (This point is related to the concept of stability, which we will address formally in Chapter 7.)... 

The most important time dependence of e-t/Tl arises only from the pole of the transfer function in Eq. (2-38). Again, we can "spell out" the function if we want to ... 

Of course, Gd(s) and Gp(s) are the transfer functions, and they are in pole-zero form. Once again( ), we are working with deviation variables. The interpretation is that changes in the inlet temperature and the steam temperature lead to changes in the tank temperature. The effects of the inputs are additive and mediated by the two transfer functions. 

After this exercise, let s hope that we have a better appreciation of the different forms of a transfer function. With one, it is easier to identify the pole positions. With the other, it is easier to extract the steady state gain and time constants. It is veiy important for us to leam how to interpret qualitatively the dynamic response from the pole positions, and to make physical interpretation with the help of quantities like steady state gains, and time constants. 

Take note (again ) that the characteristic polynomials in the denominators of both transfer functions are identical. The roots of the characteristic polynomial (the poles) are independent of the inputs. It is obvious since they come from the same differential equation (same process or system). The poles tell us what the time-domain solution, y(t), generally would "look" like. A final reminder no matter how high the order of n may be in Eq. (3-4), we can always use partial fractions to break up the transfer functions into first and second order terms. 

Here, the pole of the transfer function G(s) is at the origin, s = 0. The solution of (3-12), which we could have written down immediately without any transform, is... 

The transfer function has the distinct feature that a pole is at the origin. Since a step input in either q in or q would lead to a ramp response in h, there is no steady state gain at all. 

The real part of a complex pole in (3-19) is -Zjx, meaning that the exponential function forcing the oscillation to decay to zero is e- x as in Eq. (3-23). If we draw an analogy to a first order transfer function, the time constant of an underdamped second order function is x/t,. Thus to settle within 5% of the final value, we can choose the settling time as 1... 

Let say we have a high order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a first order with dead time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple... 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

One important observation that we should make immediately the characteristic polynomial of the matrix A (E4-7) is identical to that of the transfer function (E4-2). Needless to say that the eigenvalues of A are the poles of the transfer function. It is a reassuring thought that different mathematical techniques provide the same information. It should come as no surprise if we remember our linear algebra. 

This transfer function has closed-loop poles at -0.29, -0.69, and -10.02. (Of course, we computed them using MATLAB.)... 

In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root locus and frequency response analysis, we can make more rational explanations, including why the function is called a lead-lag element. We ll see that this is a nice strategy which is preferable to using the ideal PD controller. 

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... 

The closed-loop system is stable if all the roots of the characteristic polynomial have negative real parts. Or we can say that all the poles of the closed-loop transfer function he in the left-hand plane (LHP). When we make this statement, the stability of the system is defined entirely on the inherent dynamics of the system, and not on the input functions, fn other words, the results apply to both servo and regulating problems. 

The important point is that the phase lag of the dead time function increases without bound with respect to frequency. This is what is called a nonminimum phase system, as opposed to the first and second transfer functions which are minimum phase systems. Formally, a minimum phase system is one which has no dead time and has neither poles nor zeros in the RHP. (See Review Problems.)... 

Construct a polynomial from its roots Partial fraction expansion Find the roots to a polynomial Transfer function to zero-pole form conversion Zero-pole form to transfer function conversion... 

Transfer function to zero-pole form, tf2zp o... 

Zero-pole form to transfer function, zp2tf o... 

A transfer function can be written in terms of its poles and zeros. For example,... 

MATLAB is object-oriented. Linear time-invariant (LTI) models are handled as objects. Functions use these objects as arguments. In classical control, LTI objects include transfer functions in polynomial form or in pole-zero form. The LTI-oriented syntax allows us to better organize our problem solving we no longer have to work with individual polynomials that we can only identify as numerators and denominators. 

The function pole () finds the poles of a transfer function. For example, try ... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

The functions also handle multiple transfer functions. Let s make a second transfer function in pole-zero form,... 

Note In the text, we emphasize the importance of relating pole positions of a transfer function to the actual time-domain response. We should get into the habit of finding what the poles are. The time response plots are teaching tools that reaffirm our confidence in doing analysis in the Laplace-domain. So, we should find the roots of the denominator. We can also use the damp () function to find the damping ratio and natural frequency. 

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