Roots of characteristic equation

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 ... 

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

For a critically-damped system, the roots of characteristic equation are equal, or ... 

The above equation is called the characteristic equation of the system. It is the system s most important dynamic feature. The values of s that satisfy Eq. (6.67) are called the roots of the characteristic equation (they are also called the eigenvalues of the system). Their values, as we will shortly show, will dictate if the system is fast or slow, stable or unstable, overdamped or underdamped. Dynamic analysis and controller design consists of finding out the values of the roots of the characteristic equation of the system and changing their values to give the desired response. The rest of this book is devoted to looking at roots of characteristic equations. They are an extremely important concept that you should fully understand. ... 

This relationship between the poles of the transfer function and the roots of characteristic equation is an extremely important and useful one. 

Figure 15. The dependence of roots of characteristic equation of system (37) on toxicant concentration.

We should note that all equilibrium states listed above are stable nodes. It is obvious from Figure 15 in which substantial parts of roots of characteristic equation calculated on the... 

Nn Positive roots of characteristic equation Nu Nusselt number for heat or mass convection P Pressure, atm... 

Let us consider different particular forms of relation (5.272). If the surfactant in the solution exists mainly in form of micelles, and using the approximation (5.263) for the roots of characteristic equation (5.262), we obtain... 

Radial coordinate in a (solid) catalyst particle or in a reactor tube Generation rate Vector of generation rates Generation rate of key component k Vector of generation rates of key component k Dimensionless generation rate. Equation 5.101 Roots of characteristic equation (second-order linear differential equations)... 

Damping Coefficient Characterization of Response Roots of Characteristic Equation ... 

Thus Xj is a root of the equation, polynomial in A, det(M — XI) = 0 and computing the eigenvalues is equivalent to finding the roots of that polynomial, which is called the characteristic polynomial. 

Roots ofthe Characteristic Equation for the System of Example 15.7... 

Suppose that the two feedback controllers Gci and Gc2 are tuned separately (i.e., keeping the loop under tuning closed, and the other open). Then we cannot guarantee stability for the overall control system, where both loops are closed. The reason is simple Tuning each loop separately, we force the roots of the characteristic equations (24.12) for the individual loops to acquire negative real parts. But the roots of these equations are different from the roots of the characteristic equation (24.13), which determines the stability of the overall system with both loops closed. 

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

In certain problems it may be necessary to locate all the roots of the equation, including the complex roots. This is the case in finding the zeros and poles of transfer functions in process control applications and in formulating the analytical solution of linear nth-order differential equations. On the other hand, different problems may require the location of only one of the roots. For example, in the solution of the equation of state, the positive real root is the one of interest. In any case, the physical constraints of the problem may dictate the feasible region of search where only a subset of the total number of roots may be indicated. In addition, the physical characteristics of ihe problem may provide an approximate value of the desired root. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equiUbrium, or the three roots (vapor, Hquid, sohd) characteristic of the triple point. 

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... 

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. 

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... 

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