Characteristic equation roots

Figure 11.26 Contributions of characteristic equation roots to closed-loop response.

In case if amplitude-frequency characteristics of TF can be expressed through polynomial characteristic equation, stability of the system is determined by value of roots of characteristic equation. There are two rules that can be used in this case ... 

Linear Differential Equations with Constant Coeffieients and Ri ht-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation nr + am + b = 0 obtained by substituting the trial solution y = in the equation. 

Distinct Real Roots If the roots of the characteristic equation... 

Example The differential equation My" + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2 VkM. the roots of the characteristic equation... 

This polynomial in. v is called the Characteristic Equation and its roots will determine the system transient response. Their values are... 

Fig. 3.16 Effect that roots of the characteristic equation have on the clamping of a second-order system.

The roots of the characteristic equation given in equation (5.5) were shown in section 3.6.2. to be... 

The oniy difference between the roots given in equation (5.9) and those in equation (5.i0) is the sign of the reai part. If the real part cr is negative then the system is stabie, but if it is positive, the system wiii be unstabie. This iioids true for systems of any order, so in generai it can be stated If any of the roots of the characteristic equation have positive reai parts, then the system wiii be unstabie . 

The work of Routii (i905) and Hurwitz (i875) gives a method of indicating the presence and number of unstabie roots, but not their vaiue. Consider the characteristic equation... 

This is a controi system design technique deveioped by W.R. Evans (i948) that determines the roots of the characteristic equation (ciosed-ioop poies) when the open-ioop gain-constant K is increased from zero to infinity. 

Fig. 5.7 Root-locus diagram for a first-order system. Roots of characteristic equation...

Table 5.1 Roots of second-order characteristic equation for different values of K...

Fig. 5.10 Roots of the characteristic equation fora second-order system shown in the s-plane.

As with the continuous systems described in Chapter 5, the root locus of a discrete system is a plot of the locus of the roots of the characteristic equation... 

Number of distinct root loci This is equal to the order of the characteristic equation. 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

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... 

The nature of equilibrium in such cases is characterized by the nature of the roots of the following characteristic equation... 

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... 

If all roots of the characteristic equation have negative real parts, the point of equilibrium, = 0 is asymptotically stable whatever are the terms Xt. 

If among roots of the characteristic equation there is at least one with positive reed part, the point of equilibrium is unstable whatever the... 

If the characteristic equation does not have any roots with positive real parts, but has some roots with zero real parts, the terms in Xt may influence stability. This case belongs to the so-called critical case that requires special investigation. 

If the roots of the characteristic equation for the abridged system (Xt = 0) have moduli less than one, the zero solution of the system,... 

If among the roots of the characteristic equation of the abridged system there is at least one root with a modulus greater than one, the unperturbed motion is unstable for any Xt satisfying the stated conditions if A is sufficiently smcdl. 

An eigenvalue or characteristic root of a symmetric matrix A of dimension p is a root of the characteristic equation ... 

From the form of this equation we deduce that the characteristic equation has two positive roots ... 

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