Routh

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). 

Another important approaeh to eontrol system design was developed by Evans (1948). Based on the work of Maxwell and Routh, Evans, in his Root Loeus method, designed rules and teehniques that allowed the roots of the eharaeteristie equation to be displayed in a graphieal manner. 

Routh s array can be written in the form shown in Figure 5.3. In Routh s array Figure 5.3... 

Routh s method is easy to apply and is usually used in preference to the Hurwitz technique. Note that the array can also be expressed in the reverse order, commencing with row. v". 

Te.st 1 All eoeffieients are present and have the same sign. Proeeed to Test 2, i.e. Routh s array... 

From Routh s array, marginal stability oeeurs at A" = 70. 

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

Hence, to give a GM of 2 and a PM of 50°, the controller gain must be set at 1.0. If it is doubled, i.e. multiplied by the GM, then the system just becomes unstable. Check using the Routh stability criterion ... 

In the same way that the Routh-Hurwitz criterion offers a simple method of determining the stability of continuous systems, the Jury (1958) stability test is employed in a similar manner to assess the stability of discrete systems. 

Routh, E.J. (1905) Dynamics of a System of Rigid Bodies, Macmillan Co., London. 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

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. 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

We first introduce the time honored (/.< ., ancient ) Routh-Hurwitz criterion for stability testing. 

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. 

If the characteristic polynomial passes the coefficient test, we then construct the Routh array to find the necessary and sufficient conditions for stability. This is one of the few classical techniques that we do not emphasize and the general formula is omitted. The array construction up to a fourth order polynomial is used to illustrate the concept. 

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. 

MATLAB does not even bother wit a Routh function. Such an M-file is provided on our Web... 

In the case of a second order system, the first column of the Routh array reduces to simply the coefficients of the polynomial. The coefficient test is sufficient in this case. Or we can say that both the coefficient test and the Routh array provide the same result. 

In this case, we have added one column of zeros they are needed to show how b2 is computed. Since b2 = 0 and c, = a0, the Routh criterion adds one additional constraint in the case of a third order polynomial ... 

The two additional constraints from the Routh array are hence... 

Example 7.2 If we have only a proportional controller (i.e., one design parameter) and real negative open-loop poles, the Routh-Hurwitz criterion can be applied to a fairly high order system with ease. For example, for the following closed-loop system characteristic equation ... 

With the Routh-Hurwitz criterion, we need immediately xr > 0 and Kc > 0. (The, v term requires Kc > -1, which is overridden by the last constant coefficient.) The Routh array for this third order polynomial is... 

If the given value of Xj is larger than 0.5, the inequality simply infers that Kc must be larger than some negative number. To be more specific, if we pick X = l,1 the Routh criterion becomes... 

From the imaginary part equation, the ultimate frequency is u = Vl 1. Substituting this value in the real part equation leads to the ultimate gain Kc u = 60, which is consistent with the result of the Routh criterion. 

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