Dominant pole

It is important to understand and be able to identify dominant poles if they exist. This is a skill that is used later in what we call model reduction. This is a point that we first observed in Example 2.6. Consider the two terms such that 0 < a < a2 (Fig. 2.4),... 

As time progresses, the term associated with x2 (or a2) will decay away faster. We consider the term with the larger time constant X as the dominant pole. 1... 

With higher order models, we can construct approximate reduced-order models based on the identification of dominant poles. This approach is used later in empirical controller tuning relations. 

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... 

In this example, the dominant pole is at -1/3, corresponding to the largest time constant at 3 [time unit]. Accordingly, we may approximate the full order function as... 

The choice of the time constant and dead time is meant as an illustration. The fit will not be particularly good in this example because there is no one single dominant pole in the fifth order function with a pole repeated five times. A first order with dead time function will never provide a perfect fit. 

Here, we assume that the data fitting allows us to recover the time constant of the dominant pole reasonably well, and the dead time is roughly 0.9 s. We are not adding exactly 0.2 to 0.725 as a way to emphasize that in reality, we would be doing data fitting and the result will vary. How good an approximation is depends very much on the relative differences in the time constants. 

We used the term pole-zero cancellation at the beginning of this section. We should say a few more words to better appreciate the idea behind direct synthesis. Pole-zero cancellation is also referred to as cancellation compensation or dominant pole design. Of course, it is unlikely to... 

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. 

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. 

How could we guess what the time axis should be It is not that difficult if we understand how to identify the dominant pole, the significance behind doing partial fractions, and that the time to reach 99% of the final time response is about five time constants. 

The dominant pole of this temperature control system is also determined by the thermal time constant of the microhotplate, which is approximately 20 ms. The open-loop gain of the differential analog architecture (Aql daa) is given by Eq. (5.8) ... 

As the thermal capacitance and resistance of the hotplate provide a thermal low-pass transfer function (with the dominant pole corresponding to a characteristic time of 10-20 ms, depending on the fabrication process), the ZA modulator driving the hotplate constitutes a linear noise-shaping DAC with an output in the thermal domain. 

Directional walk of atoms 42 DNA 341 Dominant pole 265 Eddy-current damper 248 Elasticity theory 365—376 Electrochemical tip etching 282—285 Electrochemistry 323... 

Environmental vibration 242—244 suppression of 7 typical spectrum 244 Equilibrium distance 38, 54 Esbjerg-Nprskov approximation 109 Feedback circuit 258—266 dominant pole 265 response function 262 steady-state response 258 transient response 261 Feenstra parameter 303—306... 

The dominant pole is created by the source impedance of the MOSFET, and the output capacitor. A second pole is created by the MOSFET s Ciss and its driving impedance. Therefore, the MOSFET... 

Note that the integrator has a single-pole — at zero frequency . Therefore, we will often refer to it as the pole-at-zero stage or section of the compensation network. This pole is more commonly called the pole at the origin or the dominant pole. 

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. 

From Eq. (7.174), we see that the ideal terms obtained with infinite GB have been corrupted by additional terms caused by finite GB. To see the effects graphically, we select several values of Q and then factor the polynomial in Eq. (7.175) for many values of GB in order to draw loci. The polynomial in Eq. (7.175) is fifth order, but three of the roots are in the far left-half normalized s plane. The dominant poles are a pair of complex poles that correspond to the ideal poles in Eq. (7.172) but are shifted because of finite GB. Figure 7.117 shows the family of loci generated, one locus for each value of Q selected. Only the lod of the dominant second quadrant pole are plotted. The other dominant pole is the conjugate. 

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. 

In these cases one always finds a dominant pole strength value. Pi > 0.6. In the case of metallic polymers one can still use equations (68)-(70), but after the substitution one... 

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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