Characteristic equation function

Only certain energy values ( ) will lead to solutions of this equation. The corresponding values of the wave functions are called Eigen functions or characteristic wave functions. 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

If the reed parts of the roots of the characteristic equation are negative, there exists one and only one function V(xlt cn)for any given U(xlf , xn) this function XJ = Ud satisfies the equation... 

The dynamic behaviour of the system is thus determined by the values of the exponential coefficients. A,) and A,2, which are the roots of the characteristic equation or eigenvalues of the system and which are also functions of the system parameters. 

In addition, the time dependence of the solution, meaning the exponential function, arises from the left hand side of Eq. (2-2), the linear differential operator. In fact, we may recall that the left hand side of (2-2) gives rise to the so-called characteristic equation (or characteristic polynomial). 

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."... 

We will not write out the entire closed-loop function C/R, or in this case, T/Tsp. The main reason is that our design and analysis will be based on only the characteristic equation. The closed-loop function is only handy to do time domain simulation, which can be computed easily using MATLAB. Saying that, we do need to analysis the closed-loop transfer function for several simple cases so we have abetter theoretical understanding. 

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... 

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... 

We now repeat the problem with the s = jco substitution in the characteristic equation, and rewrite the time delay function with Euler s identity ... 

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... 

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 ... 

Let say we have a simple open-loop transfer function G0 of the closed-loop characteristic equation... 

If you remember from Chap. 6, repeated roots of the characteristic equation yielded time functions that contained an exponential multiplied by time. 

The denominator of the transfer function is exactly the same as the polynomial in s that was called the characteristic equation in Chap. 6. The roots of the denominator of the transfer function are called the poles of the transfer function. These are the values of s at which goes to infinity. 

The roots of the characteristic equation are equal to the poles of the transfer function. 

Notice that the denominator of the transfer function is again the same polynomial in s as appeared in the characteristic equation of the system [Eq. (6.73)]. The poles of the transfer function are located at s = — 2 and s = — 3. So the poles of the transfer function arc the roots of the characteristic equation. 

As noted in Chap. 6, the roots of the characteristic equation, which are the poles of the transfer function, must be real or must occur as complex conjugate pairs. In addition, the real parts of all the poles must be negative for the system to be stable. 

The dynamics of this openloop system depend on the roots of the openloop characteristic equation, i.e., on the roots of the polynomials in the denominators of the openloop transfer functions. These are the poles of the openloop transfer functions. If all the roots lie in the left half of the s plane, the system is openloop stable. For the two-heated-tank example shown in Fig. 10.16, the poles of the openloop transfer function are 5 = 1 and s = — j, so the system is openloop stable. 

Closedloop Characteristic Equation and Closedloop Transfer Functions... 

Since the characteristic equation of any system (openloop or closedloop) is the denominator of the transfer function describing it, the closedloop characteristic equation for this system is... 

This equation shows that closedloop dynamics depend on the process openloop transfer functions (G, Gv, and Gj) and on the feedback controller transfer function (fl). Equation (10.10) applies for simple single-input-single-output systems. We will derive closedloop characteristic equations for other systems in later chapters. 

In Chap. 12 we will show that we can convert from the Laplace domain (Russian) into the frequency domain (Chinese) by merely substituting ia for s in the transfer function of the process. This is similar to the direct substitution method, but keep in mind that these two operations are different. In one we use the transfer function. In the other we use the characteristic equation. 

The usual steadystate performance specification is zero steadystate error. We will show below that this steadystate performance depends on both the system (process and controller) and the type of disturbance. This is different from the question of stability of the system which, as we have previously shown, is only a function of the system (roots of the characteristic equation) and does not depend on the input. 

The tools are those developed in Chaps. 9 and 10. We use transfer functions to design feedforward controllers or to develop the characteristic equation of the system and to find the location of Us roots in the 5 plane. 

Now we solve for the closedloop transfer function for the primary loop with the secondary loop on automatic. Figure 11.3c shows the simplified block diagrana. By inspection we can see that the closedloop characteristic equation is... 

Derive the closedloop transfer function between and What is the closedloop characteristic equation ... 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

The system considered in the example above has a characteristic equation that is the denominator of the transfer function set equal to zero. This true, of course, for any system. Since the system is uncontrolled, the openloop characteristic equation is [using Eq. (15.60)]... 

Exarngde 15.15. Determine the dosedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 15.14. Use a diagonal controller structure (two SI SO.controllers) that are proportional only. 

Note that this is exactly the same characteristic equation that we found using the transfer function notation [sec Eq. (15.68)]. 

For openloop systems, the denominator of the transfer functions in the matrix gives the openloop characteristic equation. In Example 15.14 the denominator of the elements in was (s + 2X + 4). Therefore the openloop characteristic equation was... 

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... 

We can use this theorem to find out if the closedloop characteristic eqga-tion has any roots or zeros in the right half of the s plane. The s variable follows a closed contour that completely surrounds the entire right half of the s plane. Since the closedloop characteristic equation is given in Eq. (16.1), the function that we are interested in is... 

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