Characteristic equations

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... 

Note that the characteristic equation wiU be unchanged for the FF + FB system, hence system stability wiU be unaffected by the presence of the FF controller. In general, the tuning of the FB controller can be less conservative than wr the case of FB alone, since smaller excursions from the set point will residt. This in turn woidd make the dynamic model Gp(.s) more accurate. 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

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 characteristic equation was defined in section 3.6.2 for a second-order system as... 

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.

Imaginary axis crossover (Rule 9) From characteristic equation (5.ii2)... 

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... 

Consider the characteristic equation of a sampled-data system... 

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. 

Unit circle crossover This can be obtained by determining the value of K for marginal stability using the Jury test, and substituting it in the characteristic equation (7.76). 

The step response shown in Figure 7.15 is for K=. Inserting K = 1 into the characteristic equation gives... 

The control problem is to design a compensator D z), which, when cascaded with G z), provides a characteristic equation... 

For the system described by equation (8.92), and using equation (8.52), the characteristic equation is given by... 

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