RouthTable - Maple Help

DynamicSystems

 RouthTable
 generate the Routh table of a polynomial

 Calling Sequence RouthTable(p, s, opts) RouthTable(p, s, ohp, opts)

Parameters

 p - algebraic; polynomial with real (or symbolic) coefficients s - name; indeterminate of the polynomial p ohp - (optional); left or right (default) opts - (optional) equation(s) of the form option = value; specify options for the RouthTable command

Options

The opts arguments are optional arguments of the form option = value, where option is one of the names listed below. These arguments correspond to keyword parameters; the left side of an equation is the keyword, the right side is the value. Each keyword parameter has a default value that is assigned if the parameter is not passed.

The following paragraphs describe each of the keyword parameters. The first line of each paragraph specifies the format of the argument: the left side is the keyword and the right side specifies the type of the value. If the type is truefalse, then passing just the keyword is equivalent to passing keyword = $\mathrm{true}$.

 • normalize = truefalse
 Specifies that the entries in the Routh table are normalized. This results in simpler entries in the table and in the computed boolean expression. The default is equal to the value of the stablecondition keyword parameter.
 • stablecondition = truefalse
 Specifies that a boolean expression, rather than a Matrix, is to be returned. The boolean expression evaluates to true if and only if all the zeros of the polynomial are in the open LHP (open RHP if ohp is left). The default is false.

Description

 • The RouthTable command returns the Routh table of the polynomial p as a Matrix. The parameter s is the indeterminate of the polynomial p. The table can be used to determine the number of roots of p in either the open right half complex plane (open RHP) or the open left half complex plane (open LHP).
 • If the option StableCondition=true is included, the RouthTable command outputs an expression giving conditions under which the polynomial is stable. In this case, no Matrix is returned. See Options for details.
 • The polynomials are scaled at each step of the computation so that no fractions appear in the table.
 • The right column of the table consists of powers of s corresponding to the highest order power of the polynomial associated with that row. A bracketed power of s (a list) indicates that the original polynomial associated with that row was degenerate, that is, equal to zero. In that case, the derivative of the polynomial associated with the previous row is used. Note that degenerate polynomials indicate the existence of zeros z and -z for the polynomial p.
 • The number of zeros in the open RHP of the polynomial is given by the number of sign changes in the first column of the table.
 • The optional parameter ohp determines whether the sign changes in the table correspond to zeros in the open RHP or open LHP. If ohp is left the transformation s -> -s is applied to p. That moves the zeros in the LHP to RHP, and vice-versa. Consequently the number of sign changes in the first column indicates the number of zeros in the open LHP.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $p≔\left({s}^{2}+1\right)\left({s}^{2}-1\right)\left(s+2\right):$
 > $\mathrm{RouthTable}\left(p,s\right)$
 $\left[\begin{array}{cccc}{1}& {0}& {-1}& {{s}}^{{5}}\\ {2}& {0}& {-2}& {{s}}^{{4}}\\ {8}& {0}& {0}& \left[{{s}}^{{3}}\right]\\ {2}& {-2}& {0}& {{s}}^{{2}}\\ {8}& {0}& {0}& {s}\\ {-2}& {0}& {0}& {1}\end{array}\right]$ (1)

There is one sign change in the first column; therefore, there is one root in the open RHP. The $\left[{s}^{3}\right]$ indicates a degenerate polynomial. Consequently, there might be roots on the imaginary axis. Check the open LHP.

 > $\mathrm{RouthTable}\left(p,s,\mathrm{left}\right)$
 $\left[\begin{array}{cccc}{1}& {0}& {-1}& {{s}}^{{5}}\\ {-2}& {0}& {2}& {{s}}^{{4}}\\ {-8}& {0}& {0}& \left[{{s}}^{{3}}\right]\\ {-2}& {2}& {0}& {{s}}^{{2}}\\ {-8}& {0}& {0}& {s}\\ {2}& {0}& {0}& {1}\end{array}\right]$ (2)

There are two sign changes in the first column; therefore, there are two roots in the open LHP. Together with the previously determined one root in the RHP, this accounts for three roots of this polynomial of degree five, leaving two roots on the imaginary axis.

 > $\mathrm{RouthTable}\left({x}^{2}+bx+a,x,'\mathrm{stablecondition}'=\mathrm{true}\right)$
 ${0}{<}{a}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{b}$ (3)

The boolean expression is true if and only if all roots of the polynomial are in the open LHP; in this case, if the coefficients are both positive.

References

 Benedir, M., and Picinbono, B. "Extended Table for Eliminating the Singularities in Routh's  Array," IEEE Transactions on Automatic Control, Vol. 35, No. 2, (February 1990): 218-220.