Repeated roots

The next question is how to find the partial fractions in Eq. (2-25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. We will illustrate three important cases with respect to the roots of the polynomial in the denominator (1) distinct real roots, (2) complex conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 

In general, the inverse transform of repeated roots takes the form... 

When the pole p is negative, the decay in time of the entire response will be slower (with respect to only one single pole) because of the terms involving time in the bracket. This is the reason why we say that the response of models with repeated roots (e g., tanks-in-series later in Section 3.4) tends to be slower or "sluggish."... 

Repeated roots (of multi-capacity processes) lead to sluggish response. Tanks-in-series is a good example in this respect. 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

These are rough sketches of what you should obtain with MATLAB. The root locus of the system in (a) is a line on the real axis extending to negative infinity (Fig. E7.6a). The root loci in (b) approach each other (arrows not shown) on the real axis and then branch off toward infinity at 90°. The repeated roots in (c) simply branch off toward infinity. 

The two roots are the same and are called repeated roots. This is clearly seen if a value of C = I is substituted into the characteristic equation [Eq. (6.67)] ... 

The damping coelllcient is unity and there is a real, repeated root The complementary solution is... 

If some of the roots are repeated (not distinct) the complementary solution will contain exponential terms that are multiplied by various powers of t. For example, if ai is a repeated root of order 2, the characteristic equation would be... 

If H] is a repeated root of order 3, the characteristic equation would be... 

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

Equation (9.17) can be generalized for a repeated root of nth order to give... 

Let us assume a unit step change in the feed concentration C o third-order pole at s = — 1/r,). 

D. EXPONENTIAL MULTIPLIED BY TIME. In the Laplace domain we found that repeated roots l/(s -I- a) occur when we have the exponential multiplied by time. We can guess that similar repeated roots should occur in the z domain. Let us consider a very general function ... 

Looking at the three shallow intersections of the horizontal axis with the graph of / in Figure 3.4, we are reminded of the problems encountered in Chapter 1 on p. 30 and 31 with both the bisection and the Newton root finder for polynomials with repeated roots. The common wisdom is that the shallower these intersections become, the worse the roots will be computed by standard root-finding methods (see the exercises below), and multiple roots will easily be missed. 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

For this problem the contribution from the repeated root s = 0 is zero. This is not always true as shown in the next example. 

We also have the repeated root special cases ... 

If D > 0, then there are two distinct real roots. If D < 0, then there are two distinct complex roots. If D = 0, then the repeated roots are given by... 

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