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# routh_t

Routh's table

### Syntax

[r [,num] ]=routh_t(p) [r [,num] ]=routh_t(h ,kp) r=routh_t(h ,k [,normalized])

### Arguments

- p
a real polynomial

- h
a real SISO transfer system

- k
a real polynomial or a scalar

- kp
a scalar: proportional controller constant

- normalized
a boolean (%t (default value) or %f)

- r
a matrix or a list: Routh array elements

- num
a scalar: the number of sign changes

### Description

`r=routh_t(p)`

computes Routh's table of the
polynomial `p`

.

`r=routh_t(h,k)`

computes Routh's table of
denominator of the system described by transfer matrix SISO
`h`

with the feedback by the gain
`k`

.

If `k=poly(0,'k')`

we will have a polynomial or
a rational matrix with dummy variable `k`

,
formal expression of the Routh table.

If `normalized=%f`

we will have a polynomial matrix
with non normalized elements. In the other case, we will have a rational
and normalized matrix.

The second output argument `num`

returns the number of sign changes
in the first column of Routh's table. Note that, this argument value will only have sense
when the table is normalized.

Hint: If `h=1/p` , then `routh_t(h, kp)` is equivalent to
`routh_t(p+kp)` . |

### Examples

s=%s; P=5*s^3-10*s^2+7*s+20; routh_t(P) // Transfer function with formal feedback, normalized case routh_t((1+s)/P,poly(0,'k')) // Transfer function with formal feedback, non normalized case routh_t((1+s)/P,poly(0,'k'),%f) // One of the coefficients in the polynomial equals zero P1=2*s^3-24*s+32; routh_t(P1) // A row full of zeros P2=s^4-6*s^3+10*s^2-6*s+9; routh_t(P2) // The number of roots in the rhp could be retrieved as the second output argument P3=5*s^3-10*s^2+7*s; [r,num]=routh_t(1/P3,20) if num==0 disp("System is stable") else mprintf("There is %g sign changes in entries of first column.\nTherefore, system is unstable.", num) end //

### See also

### Bibliography

http://controls.engin.umich.edu/wiki/index.php/RouthStability

http://www.jdotec.net/s3i/TD_Info/Routh/Routh.pdf

Comments on the Routh-Hurwitz criterion, Shamash, Y.,Automatic Control, IEEE T.A.C Volume 25, Issue 1, Feb 1980 Page(s): 132 - 133

### History

Version | Description |

5.5.0 | New output argument added: num (SEP #104). |

5.4.0 | New input argument added: normalized (SEP #89). |

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