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In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a
necessary and sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a material conditional, conditional or implicational relationship between two Statement (logic), statements. For example, in the Conditional sentence, conditional stat ...
condition for the
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Exponential stability ** Linear stability **Lyapunov stability ** Marginal s ...
of a linear time-invariant (LTI)
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
or
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician
Edward John Routh Edward John Routh (; 20 January 18317 June 1907) was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
proposed in 1876 to determine whether all the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
of the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
have negative real parts. German mathematician
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...
independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants (\Delta_i) than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial. The importance of the criterion is that the roots ''p'' of the characteristic equation of a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
with negative real parts represent solutions ''ept'' of the system that are stable ( bounded). Thus the criterion provides a way to determine if the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
of a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
have only stable solutions, without solving the system directly. For discrete systems, the corresponding stability test can be handled by the Schur–Cohn criterion, the Jury test and the Bistritz test. With the advent of computers, the criterion has become less widely used, as an alternative is to solve the polynomial numerically, obtaining approximations to the roots directly. The Routh test can be derived through the use of the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
and
Sturm's theorem In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real number, real R ...
in evaluating Cauchy indices. Hurwitz derived his conditions differently.


Using Euclid's algorithm

The criterion is related to
Routh–Hurwitz theorem In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem ...
. From the statement of that theorem, we have p-q=w(+\infty)-w(-\infty) where: * p is the number of roots of the polynomial f(z) with negative real part; * q is the number of roots of the polynomial f(z) with positive real part (according to the theorem, f is supposed to have no roots lying on the imaginary line); * ''w''(''x'') is the number of variations of the generalized Sturm chain obtained from P_0(y) and P_1(y) (by successive Euclidean divisions) where f(iy)=P_0(y)+iP_1(y) for a real ''y''. By the
fundamental theorem of algebra The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
, each polynomial of degree ''n'' must have ''n'' roots in the complex plane (i.e., for an ''ƒ'' with no roots on the imaginary line, ''p'' + ''q'' = ''n''). Thus, we have the condition that ''ƒ'' is a (Hurwitz) stable polynomial if and only if ''p'' − ''q'' = ''n'' (the
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
is given below). Using the Routh–Hurwitz theorem, we can replace the condition on ''p'' and ''q'' by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of ''ƒ''.


Using matrices

Let ''f''(''z'') be a complex polynomial. The process is as follows: # Compute the polynomials P_0(y) and P_1(y) such that f(iy)=P_0(y)+iP_1(y) where ''y'' is a real number. # Compute the Sylvester matrix associated to P_0(y) and P_1(y). # Rearrange each row in such a way that an odd row and the following one have the same number of leading zeros. # Compute each
principal minor In linear algebra, a minor of a matrix (mathematics), matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square ma ...
of that matrix. # If at least one of the minors is negative (or zero), then the polynomial ''f'' is not stable.


Example

Let f(z) = az^2 + bz + c (for the sake of simplicity we take real coefficients) where c\neq 0 (to avoid a root in zero so that we can use the Routh–Hurwitz theorem). First, we have to calculate the real polynomials P_0(y) and P_1(y): f(iy) = -ay^2 + iby + c = P_0(y) + iP_1(y) = -ay^2 + c + i(by). Next, we divide those polynomials to obtain the generalized Sturm chain: \begin P_0(y) &= \left( \tfrac y \right)P_1(y)+c, &&\implies P_2(y) = -c, \\ pt P_1(y) &= \left( \tfrac y \right)P_2(y), &&\implies P_3(y) = 0, \end and the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
stops. Notice that we had to suppose different from zero in the first division. The generalized Sturm chain is in this case \Bigl( P_0(y), P_1(y), P_2(y) \Bigr) = (c-ay^2, by ,-c). Putting y=+\infty, the sign of (c-ay^2) is the opposite sign of and the sign of is the sign of . When we put (y=-\infty), the sign of the first element of the chain is again the opposite sign of and the sign of is the opposite sign of . Finally, has always the opposite sign of . Suppose now that is Hurwitz-stable. This means that w(+\infty)-w(-\infty)=2 (the degree of ). By the properties of the function , this is the same as w(+\infty)=2 and w(-\infty)=0. Thus, and must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.


Routh–Hurwitz criterion for second, third and fourth-order polynomials

For a second-order polynomial P(s) = a_2s^2 + a_1s + a_0 = 0, all coefficients must be positive, where a_i > 0 for (i = 0, 1, 2) . For a third-order polynomial P(s) = a_3s^3 + a_2s^2 + a_1s + a_0 = 0, all coefficients must be positive, where \begin 0 &< a_i, \quad \text i = 0, 1, 2, 3; \\ 0 &< a_2a_1 - a_3a_0. \end For a fourth-order polynomial P(s) = a_4s^4 + a_3s^3 + a_2s^2 + a_1s + a_0 = 0, all coefficients must be positive, where \begin 0 &< a_i, \quad \text i = 0, 1, 2, 3, 4; \\ 0 &< a_2a_1 - a_3a_0; \\ 0 &< a_3a_2a_1 - a_4a_1^2 - a_3^2a_0. \end (When this is derived you do not know all coefficients should be positive, and you add a_3a_2 > a_1.) In general the Routh stability criterion states a polynomial has all roots in the open left half-plane if and only if all first-column elements of the Routh array have the same sign. All coefficients being positive (or all negative) is necessary for all roots to be located in the open left half-plane. That is why here a_n is fixed to 1, which is positive. When this is assumed, we can remove a_3a_2 > a_1 from fourth-order polynomial, and conditions for fifth- and sixth-order can be simplified. For fifth-order we only need to check that \Delta_2>0, \Delta_4>0 and for sixth-order we only need to check \Delta_3>0, \Delta_5>0 and this is further optimised in Liénard–Chipart criterion. Indeed, some coefficients being positive is not independent with principal minors being positive, like a_2 > 0 check can be removed for third-order polynomial.


Higher-order example

A tabular method can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain. For an th-degree polynomial whose all coefficients are the same signs D(s) = a_ns^n + a_s^ + \cdots + a_1s + a_0 the table has rows and the following structure: \begin a_n & a_ & a_ & \dots \\ a_ & a_ & a_ & \dots \\ b_1 & b_2 & b_3 & \dots \\ c_1 & c_2 & c_3 & \dots \\ \vdots & \vdots & \vdots & \ddots \end where the elements b_i and c_i can be computed as follows: \begin b_i &= \frac \\ pt c_i &= \frac \end When completed, the number of sign changes in the first column will be the number of roots whose real part are non-negative. \begin 0.75 & 1.5 & \ 0 \ & \ 0 \ \\ -3 & 6 & 0 & 0 \\ 3 & 0 & 0 & 0 \\ 6 & 0 & 0 & 0 \end In the first column, there are two sign changes (, and ), thus there are two roots whose real part are non-negative and the system is unstable. The characteristic equation of an example servo system is given by: b_0s^4 + b_1s^3 + b_2s^2 + b_3s + b_4 = 0 For which we have the following table: \begin b_ 0 & b_2 & \quad b_4 \quad & \quad 0 \quad \\ pt b_1 & b_3 & 0 & 0 \\ pt \frac & \frac = b_4 & 0 & 0 \\ \frac & 0 & 0 & 0 \\ pt b_4 & 0 & 0 & 0 \end for stability, all the elements in the first column of the Routh array must be positive when b_0 > 0. And the conditions that must be satisfied for stability of the given system as follows: \begin 0 &< b_1, \\ pt 0 &< b_1b_2 - b_0b_3, \\ pt 0 &< (b_1b_2 - b_0b_3)b_3-b_1^2b_4, \\ pt 0 &< b_4. \end We see that if (b_1b_2 - b_0b_3)b_3 - b_1^2b_4 \geq 0 then b_1b_2 - b_0b_3 > 0 is satisfied. Another example is: s^4+6s^3+11s^2+6s+200=0 We have the following table : \begin 1 & 11 & 200 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 20 & 0 & 0 \\ -19 & 0 & 0 & 0 \\ 20 & 0 & 0 & 0 \end there are two sign changes. The system is unstable, since it has two right-half-plane poles and two left-half-plane poles. The system cannot have jω poles since a row of zeros did not appear in the Routh table. For the case s^4 + s^3 + 3s^2 + 3s + 3 = 0 We have the following table with zero appeared in the first column which prevents further calculation steps: \begin 1 & 3 & 3 \\ 1 & 3 & 0 \\ 0 & 3 & 0 \end we replace 0 by \varepsilon > 0 and we have the table \begin 1 & 3 & \quad 3 \quad \\ 1 & 3 & 0 \\ \varepsilon & 3 & 0 \\ 3 - \frac & 0 & 0 \\ 3 & 0 & 0 \end When we make \varepsilon \rightarrow +0, there are two sign changes. The system is unstable, since it has two right-half-plane poles and two left-half-plane poles. Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability. In that case the coefficients of the "Routh array" in a whole row become zero and thus further solution of the polynomial for finding changes in sign is not possible. Then another approach comes into play. The row of polynomial which is just above the row containing the zeroes is called the "auxiliary polynomial". s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0 We have the following table: \begin 1 & 8 & 20 & 16 \\ 2 & 12 & 16 & 0 \\ 2 & 12 & 16 & 0 \\ 0 & 0 & 0 & 0 \end In such a case the auxiliary polynomial is A(s) = 2s^4 + 12s^2 + 16\, which is again equal to zero. The next step is to differentiate the above equation which yields the polynomial B(s) = 8s^3 + 24s^1. The coefficients of the row containing zero now become "8" and "24". The process of Routh array is proceeded using these values which yield two points on the imaginary axis. These two points on the imaginary axis are the prime cause of marginal stability.


See also

*
Control engineering Control engineering, also known as control systems engineering and, in some European countries, automation engineering, is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with d ...
*
Derivation of the Routh array The Routh array is a Routh–Hurwitz_stability_criterion#Higher-order_example, tabular method permitting one to establish the stable polynomial, stability of a system using only the coefficients of the characteristic polynomial. Central to the f ...
*
Nyquist stability criterion In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
*
Routh–Hurwitz theorem In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem ...
* Root locus *
Transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
* Liénard–Chipart criterion (variant requiring fewer computations) * Kharitonov's theorem (variant for unknown coefficients bounded within intervals) * Jury stability criterion (analog for discrete-time LTI systems) * Bistritz stability criterion (analog for discrete-time LTI systems)


References

* Felix Gantmacher (J.L. Brenner translator) (1959). ''Applications of the Theory of Matrices'', pp 177–80, New York: Interscience. * * * * * Stephen Barnett (1983). ''Polynomials and Linear Control Systems'', New York: Marcel Dekker, Inc.


External links


A MATLAB script implementing the Routh-Hurwitz test


{{DEFAULTSORT:Routh-Hurwitz stability criterion Stability theory Electronic feedback Electronic amplifiers Signal processing Polynomials