Routh-Hurwitz Analysis

The stationary state of the network of reactors is stable, if all roots A, of (13.8) have a negative real part. The necessary and sufficient conditions for this to hold are the Routh-Hurwitz conditions, see Theorem 1.2. The stationary state of the network, W, undergoes a stationary instability if = 0, see (1.36), and an oscillatory instability if = 0, together with 0, A 0, Z = 1. m — 2, see (1.38). The Routh-Hurwitz analysis can be used to determine, in principle, the stability properties of the steady state of any network, even inhomogeneous networks. This advantage is, however, balanced by the fact that it is a computationally expensive task to evaluate all the coefficients C of the characteristic polynomial and the Hurwitz determinants A . In our studies of instabilities in arrays of coupled reactors, we used symbolic computation software, namely Mathematica (Wolfram Research, Inc., Champaign, IL, 2002) and Maple (Waterloo Maple Inc., Waterloo, Ontario, 2002), to obtain exact, analytical expressions for the coefficients C of the characteristic polynomial (13.8) and the Hurwitz determinants A/ for arrays of up to six coupled reactors. 

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With the Routh-Hurwitz analysis in Chapter 7, we should find that to have a stable system, we must keep Kc < 7.5. (You fill in the intermediate steps in the Review Problems. Other techniques such as root locus, direct substitution or frequency response in Chapter 8 should arrive at the same result.)... 

A Routh-Hurwitz analysis can confirm that. The key point is that with cascade control, the system becomes more stable and allows us to use a larger proportional gain in the primary controller. The main reason is the much faster response (smaller time constant) of the actuator in the inner loop.2... 

A linear stability analysis of the steady states of systems with more than two variables usually employs a Routh-Hurwitz analysis, which is based on the following theorem [309, 153, 424]. 

To determine the stability of the th mode, we conduct a Routh-Hurwitz analysis. All eigenvalues k have a negative real part, if... 

The application of the Routh-Hurwitz analysis or the direct calculation of the eigenvalues and eigenvectors of the Jacobian Jg of the network of reactors is a formidable task for moderate and large arrays of coupled reactors. There is an alternative approach, a spectral analysis of networks, that works for any size array of coupled reactors, as long as the array is homogeneous, i.e., / -, = /i for all i,i = 1,..., n, and the coupling is diffusive. In this case, the system (13.4) has a uniform steady state ... 

Generally, inhomogeneities in parameters of an array of reactors lead to nonuniform steady states. This is not the case for Lengyel-Epstein networks with inhomogeneities in the parameter a, as is clear from the structure of (13.139). The network still has a unique uniform steady state given by (13.52). We use the Routh-Hurwitz criterion to determine the stability boundaries of this USS. Note that the Routh-Hurwitz analysis is general and can deal with the case where inhomogeneities in parameters lead to nonuniform steady states. Let... 

Stability analysis methods Routh-Hurwitz criterion Apply the Routh test on the closed-loop characteristic polynomial to find if there are closed-loop poles on the right-hand-plane. 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

One may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the Routh-Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember ary formulas, and we can find the ultimate frequency, which however, can be obtained with a root locus plot or frequency response analysis—techniques that we will cover later. 

The Routh-Hurwitz criterion decides when a given polynomial has roots with a negative real part. Such information proves useful in the analysis of stability of stationary solutions to systems of ordinary differential equations. The examined polynomial is a characteristic polynomial of the stability matrix atj... 

Example 15.4 Stability Analysis with the Routh-Hurwitz Criterion... 

Re X2 3 i.e. the steady state is marginally stable. A more rigorous analysis (Murray, 1974a),using the Routh-Hurwitz criteria (Gantmacher, 1959), shows that G6 = C is indeed the boundary between stable and unstable behavior close to ( o> 0 = 16, q = 10 the relation... 

Clarke (1974a, b, 1980) gives a detailed analysis of the stability of the steady state using the structure of the Vol pert graph and several derived graphs to check the Routh-Hurwitz criterion. 

One can see that since the constant term is negative, it follows immediately from the Routh-Hurwitz criterion that the origin is an unstable equilibrium state. Furthermore, it may have no zero characteristic roots when a and b are positive. The codimension-2 point (a = b = 0) requires special considerations. We postpone its analysis to the last section, where we discuss the bifurcation of double zeros in systems with symmetry. 

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

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