Characteristic equation complex roots

The roots of the characteristic equation can be real or complex. But if they are complex they must appear in complex conjugate pairs. The reason for this is illustrated for a second-order system with the characteristic equation... 

As noted in Chap. 6, the roots of the characteristic equation, which are the poles of the transfer function, must be real or must occur as complex conjugate pairs. In addition, the real parts of all the poles must be negative for the system to be stable. 

The stability of any system is determined by the location of the roots of its characteristic equation (or the poles of its transfer function). The characteristic equation of a continuous system is a polynomial in the complex variable s. If all the roots of this polynomial are in the left half of the s plane, the system is stable. For a continuous closedloop system, all the roots of 1 + must lie in the left... 

The stability of a sampled-data system is determined by the location of the roots of a characteristic equation that is a polynomial in the complex variable z. This characteristic equation is the denominator of the system transfer function set equal to zero. The roots of this polynomial (the poles of the system transfer function) are plotted in the z plane. The ordinate is the imaginary part of z, and the abscissa is the real part of z. 

Eq.(31) can be used to adjust the parameter of the PI controller LC at Figure 12. For example, complex roots of the characteristic equation of (30) give the following dimensionless volume of the reactor ... 

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. 

It is shown in Section 7.10.1 that a continuous system is unstable if any root of the associated characteristic equation (i.e. any pole of the system transfer function) lies in the right half of the complex s-plane (Fig. 7.93a). If this root is s, then i, can be expressed in terms of its real and imaginary parts, i.e. ... 

In order to determine the number of roots of the z-transformed characteristic equation that lie outside the unit circle, a procedure analogous to the Routh-Hurwitz approach for continuous systems (Section 7.10.2) can be used. The Routh-Hurwitz criterion cannot be applied directly to the characteristic equation f(z) = 0. However, by mapping the interior of the unit circle in the z-piane on to the left half of a new complex variable -plane, the Routh-Hurwitz criterion can be applied as for continuous systems to the corresponding characteristic equation in terms of the new variable<4,). This mapping can be achieved using the bilinear transformation07 ... 

This helps indicate why groups of multiplicative type are important. But it should be said that solvability is definitely a necessary hypothesis. Let S for example be the group of all rotations of real 3-space. For g in S we have gtf — 1, so all complex eigenvalues of g have absolute value 1. The characteristic equation of g has odd degree and hence has at least one real root. Since det( ) = 1, it is easy to see that 1 is an eigenvalue. In other words, each rotation leaves a line fixed, and thus it is simply a rotation in the plane perpendicular to that axis (Euler s theorem). Each such rotation is clearly separable. But obviously the group is not commutative (and not solvable). Finally, since U is nilpotent, we have the following result. 

Evaluation of the rate coefficients from the relaxation times of a complex system generally involves a matrix formulation of this type. Solution of eqn. (22) gives two relaxation times, T and rn, which are the roots of the characteristic equation... 

In some cases, the roots can be complex. When the roots of the characteristic equation are complex, the general solution takes the form... 

Manometers and pressure springs may be described dynamically to a first approximation by second-order differential equations for which the roots of the characteristic equation are conjugate complex. As shown in Section III, 8, lc, since the roots are complex, these systems have an oscillatory mode, and the response of the system to step forcing, for example, is a damped sinusoid. 

The root loci are merely the plots, in the complex plane, of the roots of the characteristic equation as the gain Kc is varied from zero to infinity. As such they are very useful in determining the stability... 

A discrete system is stable if all its poles (i.e., the roots of its characteristic equation) lie inside or on the unit circle in the complex plane (i.e., they have a magnitude less than or equal to unity). 

Representing the roots of the characteristic equation in the complex domain offers a simple way to perform a stability analysis. The system is stable if and only if all the poles are located in the open left-half-plane (LHP). If there is at least one pole in the right-half-plane (RHP), the system is unstable. The representation is similar wiA the well-known root-locus plot used to evaluate the stability of a closed-loop system. 

The Nyquist stability criterion developed in Chapter 11 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem that says that the difference between the number of zeros and poles of a function inside a closed contour can be found by plotting the function and looking at the number of times it encircles the origin. We can use this theorem to find out if the closedloop characteristic equation has any roots or zeros in the right half of the s plane. The s variable follows a closed contour that completely surrounds the entire right half of the 5 plane. Since the closedloop characteristic equation is given in Eq. (I 2.36), the function of interest is... 

Our objective is to And a complex symmetric formulation that contains the seed of the relativistic frame invariants. The trick is to entrench an apposite matrix of operators whose characteristic equation mimics the Klein-Gordon equation (or in general the Dirac equation). Intuitively, one might infer that we have realised the feat of obtaining the negative square root of the aforementioned operator matrix. Thus, the entities of the formulation are operators and furthermore since they permit... 

The solutions, Eq. (1.10), correspond each to a root of the characteristic equation Eqs. (1.6-1.8). Although the general setting of the complex symmetric forms ensures biorthogonality, the eigenvectors for % (and similarly for T) obtain simply as... 

Upon expansion, an 8th-order polynomial equation in A arises. Eight roots of the polynomial exist. Even if all the values in the characteristic matrix are real, some roots may be complex. When complex roots occur, they appear in pairs. The roots of the polynomial are called eigenvalues of the characteristic matrix. The polynomial equation is called the eigenvalue equation. 

To answer this question , some mathematical theorems can be useful. In particular, the theorem of localization of the characteristic numbers (or roots) of matrices proved by Gershgorin is most appropriate. This theorem, applied to Hermitian matrices, states that the region of localization of the roots of the secular equation (37) of the order n satisfies the following inequality ("Gershgorin circles" on the complex plane) ... 

The stochastic aspect of a complex bifurcation arising in a two variables chemical system is studied. The dynamics reduces, in a suitable region of the phase space, to a normal form for which both roots of the characteristic equation vanish simultaneously. In conditions close to this degenerate situation, the normal form can be viewed as a perturbation of an exactly soluble hamiltonian system, of hamiltonian h, which exhibits a homoclinic trajectory, h = 0. BAESENS and IMICOLIS [l ] have shown that the phase portrait of the dissipative sytem displays two steady states that coalesce, a focus F and a saddle S. [ Moreover, as one moves in the parameter space, a limit cycle surrounding F, bifurcates from a homoclinic trajectory and then disappears by Hopf bifurcation. ... 

Figure 11.25 provides a graphical interpretation of this stability criterion. Note that all of the roots of the characteristic equation must he to the left of the imaginary axis in the complex plane for a stable system to exist. The qualitative effects of these roots on the transient response of the closed-loop system are shown in Fig. 11.26. The left portion of each part of Fig. 11.26 shows representative root locations in the complex plane. The corresponding figure on the right shows the contributions these poles make to the closed-loop response for a step change in set point. Similar responses would occur for a step change in a disturbance. A system that has all negative real roots will have a stable.

Figure 11.25 Stability regions in the complex plane for roots of the characteristic equation.

The imaginary axis divides the complex plane into stable and unstable regions for the roots of the characteristic equation, as indicated in Fig. 11.26. On the imaginary axis, the real part of s is zero, and thus we can write 5- = yo). Substituting s = yco into the characteristic equation allows us to find a stability limit such as the maximum value of Kc (Luyben and Luyben, 1997). As the gain Kc is increased, the roots of the characteristic equation cross the imaginary axis when Kc = Kcm-... 

Before considering the basis for the Bode stability criterion, it is useful to review the General Stability Criterion of Section 11.1 A feedback control system is stable if and only if all roots of the characteristic equation lie to the left of the imaginary axis in the complex plane. 

The root locus diagrams of Section 11.5 (e.g.. Fig. 11.27) show how the roots of the characteristic equation change as controller gain Kc changes. By definition, the roots of the characteristic equation are the numerical values of the complex variables that satisfy Eq. 14-53. Thus, each point on the root locus also satisfies (14-54), which is a rearrangement of (14-53) ... 

In general, the ith root of the characteristic equation can be expressed as a complex number, = a( bj. Note that complex roots occur as complex conjugate... 

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. 

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