The characteristic equation

Consider the transfer function given by equation 7.109. For a unit step change in the set point R, the Laplace transform of the controlled variable C is given by  

Equation 7.119 is the characteristic equation of the system shown in Fig. 7.34 and is dependent only upon the open-loop transfer function G (j)H(j) and is therefore the same for both set point and load changes (equations 7.109 and 7.110). 

The determination of the nature of the roots of the characteristic equation (or the poles of the corresponding system transfer function) forms the basis of many techniques used to establish the nature of the stability of the system. In order to calculate the step response, equation 7.118 must be split into partial fractions for inversion, thus  

The roots of the characteristic equation may be real and/or complex, depending on the form of the open-loop transfer function. Suppose at to be complex, such that  

A qualitative assessment of the stability of a given system can be made conveniently by considering the positions of the system poles (i.e. the roots of the characteristic equation) on the complex plane. This is illustrated in the following example. 

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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. 

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

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... 

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

Equation (3-111) is the secular equation for this problem. [Pauling and Wilson discuss the meaning of the term secular in this context.] Expansion of the determinant gives a polynomial in X that is called the characteristic equation ... 

For Scheme XVIII, find the secular equation, the characteristic equation, and the eigenvalues. 

Adiabatic compression (termed adiabatic isentropic or constant entropy) of a gas in a centrifugal machine has the same characteristics as in any other compressor. That is, no heat is transferred to or from the gas during the compression operation. The characteristic equation... 

Isothermal compression takes place when the heat of compression is removed during compression and when the temperature of the gas stays constant. The characteristic equation is... 

Thus, if the characteristic equation of the invariance matrix of the p order composition of (j) 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... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

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