Roots with positive real parts

In the array given in (5.16) there are two sign ehanges in the eolumn therefore there are two roots with positive real parts. Henee the system is unstable. 

Use the Routh-Hurwitz eriterion to determine the number of roots with positive real parts in the following eharaeteristie equations... 

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. 

It is often difficult to determine quickly the roots of the characteristic equation. Hurwitz(I,) and Routh( 0) developed an algebraic procedure for finding the number of roots with positive real parts and consequently whether the system is unstable or not. 

Routh-Hurwitz criterion The number of roots with positive real parts of a real polynomial equation is the number of sign changes in the following sequence ... 

According to the first test of the Routh-Hurwitz criterion for stability (see Section 15.3), eq. (24.14) has at least one root with positive real part if any of its coefficients are negative. Thus the closed-loop behavior of the process is unstable if Kc i and Kc2 take on such values that make the last term of eq. (24.14) negative [all other terms in eq. (24.14) are always positive] ... 

If Kc = 100 and t, = 0.1, the third element becomes -398 < 0, which means that the system is unstable. We have two sign changes in the elements of the first column. Therefore, we have two roots with positive real parts (see Example 15.2). 

Should any pole of the system lie on the imaginary axis (i.e. when its real part is zero) then the system is conditionally stable. Poles lying to the right of the imaginary axis (corresponding to roots of the characteristic equation with positive real parts) indicate unbounded behaviour, i.e. that the system is unstable. 

For Kc larger than the critical value (where the system becomes unstable) the roots that cause the instability are complex conjugates with positive real parts. Consequently, the unstable response of the closed-loop system to an input change will be oscillatory with growing amplitude. 

Then, shift the (m-f- l)-th row to the left over k positions, so that the element rm+i,k+i becomes the first one in the line, and multiply all other entries of this row through by (-1). Since the first entry is now non-zero, one proceeds as in the regular case. Eventually, the number of roots of S(A) with positive real parts will be equal to the number of sign changes in the first column added to the sum of deficiency numbers over all irregular rows. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

The Routh criterion states that in order to have a stable system, all the coefficients in the first column of the array must be positive definite. If any of the coefficients in the first column is negative, there is at least one root with a positive real part. The number of sign changes is the number of positive poles. 

It is important to choose the branch of the square root with a positive real part in the integrations above. Summation over quantum numbers k proceeds as before with the result for the energy density that... 

To utilize this equation, we specify 81,82, and Ca, and then calculate the roots a that satisfy the equation for each value of the wave number a. If any of the roots has a positive real part, the system is unstable. If the real parts of all roots are negative, on the other hand, the system is stable. The wave number with the largest real part for a is the fastest-growing infinitesimal disturbance. 

The product of the roots of equation (6.102), (1 + 2f)/(sp), is positive. Consequently, there exists at least one root with a positive real part. Accordingly, the stationary state (xt, yt, zj = (0, 0, 0) is unstable. We will not examine the possibility of appearance of a limit cycle from this... 

The stationary state (x2, y2, z2) will be stable when all the roots of equation (6.106) have negative real parts. We will investigate the conditions under which this stationary state loses stability, that is under which at least one solution with a positive real part appears. Next, in the region of control parameters corresponding to instability of the state (x2, y2, z2) we shall examine possible catastrophes of codimension 2. It follows from the classification given in Section 5.5 that the bifurcations of codimension one and two of a sensitive state corresponding to the requirement = 0 are theoretically possible the Hopf bifurcation for which a sensitive state is of... 

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are ... 

In order to determine whether the system is stable or unstable, the two polynomials are combined, as shown in the Method of Solution, using as the multiplier of the polynomial from the numerator of the transfer function. Function NRsdivision (which uses the Newton-Raphson method with synthetic division algorithm) or function roots (which uses the eigenvalue algorithm) is called to calculate the roots of the overall polynomial function and the sign of all roots is checked for positive real parts. A flag named stbl indicates that the system is stable (all negative roots stbl = 1) or unstable (positive root stbl = 0). 

Quantity C is always positive. In the absence of inhibition by the substrate (0 = 1 or c < 1 with 0 < 1), quantity A is also positive. The roots real part becomes positive. The steady state therefore becomes unstable as a focus (Hopf bifurcation) the passage through the critical point of instability corresponds to the occurrence of sustained oscillations in the course of time. 

The linear model has a unique positive solution. It is possible to show (see, eg, Yablonskii et al., 1991, p. 126), that the roots have negative real values or complex values with a negative real part A < 0). Complex roots are related to damped oscillations. Therefore, the relaxation process may lead to a single steady state with possibly damped oscillations. However, in reality the influence of the imaginary part of the roots will be insignificant and the damped oscillations will not be observable. 

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