Routh-Hurwitz stability criterion
Encyclopedia
The Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability
of a single-input, single-output (SISO), linear
time invariant (LTI) control system
. More generally, given a polynomial
, some calculations using only the coefficients of that polynomial can lead to the conclusion that it is not stable. For the discrete case, see the Jury test
equivalent.
The criterion establishes a systematic way to show that the linearized equations of motion of a system have only stable solutions exp(pt), that is where all p have negative real parts. It can be performed using either polynomial divisions or determinant
calculus.
The criterion is derived
through the use of the Euclidean algorithm and Sturm's theorem
in evaluating Cauchy indices
.
. Indeed, from the statement of that theorem, we have
where:
By the fundamental theorem of algebra
, 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 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.
Notice that we had to suppose b different from zero in the first division. The generalized Sturm chain is in this case
. Putting 
, the sign of 
is the opposite sign of a and the sign of by is the sign of b. When we put 
, the sign of the first element of the chain is again the opposite sign of a and the sign of by is the opposite sign of b. Finally, -c has always the opposite sign of c.
Suppose now that f is Hurwitz-stable. This means that
(the degree of f). By the properties of the function w, this is the same as 
and 
. Thus, a, b and c must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.
the table has n + 1 rows and the following structure:
| 
>
>-
|
>-
|
>-
|
>-
|
where the elements
and 
can be computed as follows:
When completed, the number of sign changes in the first column will be the number of non-negative poles.
Consider a system with a characteristic polynomial
We have the following table:
| width="40px"| 1 >
2
3
0
|-
| 4
5
6
| 0.75 1.5
0
| −3 6
0
| 3 0
| 6 0
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two non-negative roots where the system is unstable.
" Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability. In that case the coefficients of the "Routh Array" 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 "Auxiliary Polynomial".
We have the following table:
| width="40px"| 1 >
8
20
16
|-
| 2
12
16
| 2 12
16
| 0 0
0
In such a case the Auxiliary polynomial is
which is again equal to zero. The next step is to differentiate the above equation which yields the following polynomial. 
. 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.
Stable polynomial
A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...
of a single-input, single-output (SISO), linear
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
time invariant (LTI) control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
. More generally, given a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
, some calculations using only the coefficients of that polynomial can lead to the conclusion that it is not stable. For the discrete case, see the Jury test
Jury stability criterion
The Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh-Hurwitz stability criterion...
equivalent.
The criterion establishes a systematic way to show that the linearized equations of motion of a system have only stable solutions exp(pt), that is where all p have negative real parts. It can be performed using either polynomial divisions or determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
calculus.
The criterion is derived
Derivation of the Routh array
The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial...
through the use of the Euclidean algorithm and Sturm's theorem
Sturm's theorem
In mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
in evaluating Cauchy indices
Cauchy index
In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh-Hurwitz theorem, we have the following interpretation: the Cauchy index of...
.
Using Euclid's algorithm
The criterion is related to Routh–Hurwitz theoremRouth–Hurwitz theorem
In mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz-stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.-Notations:...
. Indeed, from the statement of that theorem, we have
where:
- p is the number of roots of the polynomial ƒ(z) located in the left half-plane;
- q is the number of roots of the polynomial ƒ(z) located in the right half-plane (let us remind ourselves that ƒ 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
and 
(by successive Euclidean divisions) where 
for a real y.
By the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, 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
Stable polynomial
A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...
if and only if p − q = n (the proof 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
and 
such that 
where y is a real number. - Compute the Sylvester matrixSylvester matrixIn mathematics, a Sylvester matrix is a matrix associated to two polynomials that provides information about those polynomials. It is named for James Joseph Sylvester.-Definition:...
associated to
and 
. - 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 minorMinor (linear algebra)In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
of that matrix. - If at least one of the minors is negative (or zero), then the polynomial f is not stable.
Example
- Let
(for the sake of simplicity we take real coefficients) where 
(to avoid a root in zero so that we can use the Routh–Hurwitz theorem). First, we have to calculate the real polynomials 
and 
:
-
-
- Next, we divide those polynomials to obtain the generalized Sturm chain:
-
-
yields 
-
yields 
and the Euclidean division stops.
-
-
Notice that we had to suppose b different from zero in the first division. The generalized Sturm chain is in this case
. Putting , the sign of is the opposite sign of a and the sign of by is the sign of b. When we put , the sign of the first element of the chain is again the opposite sign of a and the sign of by is the opposite sign of b. Finally, -c has always the opposite sign of c.Suppose now that f is Hurwitz-stable. This means that
(the degree of f). By the properties of the function w, this is the same as and . Thus, a, b and c 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,
, all the roots are in the left half plane (and the system with characteristic equation 
is stable) if all the coefficients satisfy 
. - For a third-order polynomial
, all the coefficients must satisfy 
, and 
- For a fourth-order polynomial
, all the coefficients must satisfy 
, and 
and 
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 nth-degree polynomial
the table has n + 1 rows and the following structure:
>
>-
|
>-
|
>-
|
>-
|
where the elements
and can be computed as follows:
When completed, the number of sign changes in the first column will be the number of non-negative poles.
Consider a system with a characteristic polynomial
We have the following table:
|-
| 4
| 0.75
| −3
| 3
| 6
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two non-negative roots where the system is unstable.
" Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability. In that case the coefficients of the "Routh Array" 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 "Auxiliary Polynomial".
We have the following table:
|-
| 2
| 2
| 0
In such a case the Auxiliary polynomial is
which is again equal to zero. The next step is to differentiate the above equation which yields the following polynomial. . 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 engineeringControl engineeringControl engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...
- Derivation of the Routh arrayDerivation of the Routh arrayThe Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial...
- Nyquist stability criterionNyquist stability criterionWhen designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...
- Routh–Hurwitz theoremRouth–Hurwitz theoremIn mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz-stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.-Notations:...
- Root locusRoot locusRoot locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback system. This is a technique used in the field of control systems developed by Walter R...
- Transfer functionTransfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
- Jury stability criterionJury stability criterionThe Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh-Hurwitz stability criterion...