Control Engineering Lecture# 10 & 11 30th April’2008

Stability Analysis BIBO: Bounded Input Bounded Output systems.BIBO: Bounded Input Bounded Output systems. For LTI systems this requires that all poles of the closed-For LTI systems this requires that all poles of the closed-

looploop transfer function lie in the left half of the complex plane.transfer function lie in the left half of the complex plane. Determine if the transfer function has any poles either on the Determine if the transfer function has any poles either on the

imaginary axis or in the right half of the s-plane.imaginary axis or in the right half of the s-plane.

Routh-Hurwitz criterion: determine if any roots of a Routh-Hurwitz criterion: determine if any roots of a polynomial lie outside the lelf half of the complex plane. It polynomial lie outside the lelf half of the complex plane. It does not find the exact locations of the roots.does not find the exact locations of the roots.

Other methods find the exact locations of the roots. For first Other methods find the exact locations of the roots. For first and second order systems, analytical method can be used. and second order systems, analytical method can be used. For higher order systems, computer programs or simulation For higher order systems, computer programs or simulation are required.are required.

Some basic resultsSome basic results:: Second order system:Second order system:

3213231212

213

321012

23

2

21212

21012

2

)()(

))()(()(

:systemorder For third

)(

))(()(

pppsppppppsppps

pspspsasasassP

ppspps

pspsasassP

We see that the coefficients of the polynomial are given by:We see that the coefficients of the polynomial are given by:

Suppose that all the roots are real and on the left half plane, Suppose that all the roots are real and on the left half plane, then all coefficients of the polynomial are positive.then all coefficients of the polynomial are positive.

If all the roots are real and in the left half plane then no If all the roots are real and in the left half plane then no coefficient can be zero.coefficient can be zero.

The only case for which a coefficient can be negative is The only case for which a coefficient can be negative is when there is at least one root in the right half plane.when there is at least one root in the right half plane.

time.aat 3 taken roots of nscombinatio possible

all of products theof sum theof negative

time.aat 2 taken roots of

nscombinatio possible all of products theof sum

roots. all of sum theof negative

3

2

1

n

n

n

a

a

a

The above is also true for complex roots.The above is also true for complex roots.

1) If any coefficient is equal to zero, then not all roots are in1) If any coefficient is equal to zero, then not all roots are in

the left half plane.the left half plane.

2) If any coefficient is negative, then at least one root is in2) If any coefficient is negative, then at least one root is in

the right half plane.the right half plane.

3) The converse of rule 2) is not always true.3) The converse of rule 2) is not always true.

Example:Example:

plane. halfright in the are (complex)

roots But two positive. are tscoefficien all

)4)(2(82)( 223 sssssssP

Routh-Hurwitz Stability Criterion

All the coefficients must be positive if all the roots are in the All the coefficients must be positive if all the roots are in the left half plane. Also it is necessary that all the coefficients for left half plane. Also it is necessary that all the coefficients for a stable system be nonzero.a stable system be nonzero.

These requirements are necessary but not sufficient. That is These requirements are necessary but not sufficient. That is we know the system is unstable if they are not satisfied; yet if we know the system is unstable if they are not satisfied; yet if they are satisfied, we must proceed further to ascertain the they are satisfied, we must proceed further to ascertain the stability of the system.stability of the system.

For example,For example,

The Routh-Hurwitz is a necessary and sufficient criterion for The Routh-Hurwitz is a necessary and sufficient criterion for the stability of linear systems.the stability of linear systems.

positive. are tscoefficien allyet unstable is system the

)4)(2(82)( 223 sssssssq

The Routh-Hurwitz criterion applies to a polynomial of the The Routh-Hurwitz criterion applies to a polynomial of the form:form:

The Routh-Hurwitz array:The Routh-Hurwitz array:

0 assume

.......)(

0

011

1

a

asasasasP nn

nn

10

11

212

43213

43212

75311

642

. . .

. . .

....

....

....

....

ms

ls

kks

ccccs

bbbbs

aaaas

aaaas

n

n

nnnnn

nnnnn

Columns of s are only for accounting.Columns of s are only for accounting. The b row is calculated from the two rows above it.The b row is calculated from the two rows above it. The c row is calculated from the two rows directly above it.The c row is calculated from the two rows directly above it. Etc…Etc… The equations for the coefficients of the array are:The equations for the coefficients of the array are:

Note: the determinant in the expression for the ith coefficient Note: the determinant in the expression for the ith coefficient in a row is formed from the first column and the (i+1)th in a row is formed from the first column and the (i+1)th column of the two preceding rows.column of the two preceding rows.

...... , 1

1

....... , 1

1

31

51

12

21

31

11

51

4

12

31

2

11

bb

aa

bc

bb

aa

bc

aa

aa

ab

aa

aa

ab

nnnn

nn

nn

nnn

nn

n

The number of polynomial roots in the right half plane is equal to the The number of polynomial roots in the right half plane is equal to the number of sign changes in the first column of the arraynumber of sign changes in the first column of the array ..

Example:Example:

Since there are two sign changes on the first column, there are two roots Since there are two sign changes on the first column, there are two roots of the polynomial in the right half plane: system is unstable.of the polynomial in the right half plane: system is unstable.

Note:Note: The Routh-Hurwitz criterion shows only the stability of the The Routh-Hurwitz criterion shows only the stability of the system, it does not give the locations of the roots, therefore no system, it does not give the locations of the roots, therefore no information about the transient response of a stable system is derived information about the transient response of a stable system is derived from the R-H criterion. Also it gives no information about the steady from the R-H criterion. Also it gives no information about the steady state response. Obviously other analysis techniques in addition to the R-state response. Obviously other analysis techniques in addition to the R-H criterion are neededH criterion are needed..

8 s

6- s

8 1 s

2 1 s

:isarray Routh The

)4)(2(82)(

0

1

2

3

223 sssssssP

From the equations, the array cannot be completed if the first From the equations, the array cannot be completed if the first element in a row is zero. Because the calculations require element in a row is zero. Because the calculations require divisions by zero. We have 3 cases:divisions by zero. We have 3 cases:

Case 1: none of the elements in the first column of the array Case 1: none of the elements in the first column of the array is zero. This is the simplest case. Follow the algorithm as is zero. This is the simplest case. Follow the algorithm as shown in the previous slides.shown in the previous slides.

Case 2: The first element in a row is zero, with at least one Case 2: The first element in a row is zero, with at least one nonzero element in the same row. In this case, replace the nonzero element in the same row. In this case, replace the first element which is zero by a small number . All the first element which is zero by a small number . All the elements that follow will be functions of . After all the elements that follow will be functions of . After all the elements are calculated, the signs of the elements in the first elements are calculated, the signs of the elements in the first column are determined by letting approach zero. column are determined by letting approach zero.

Example:Example:

1011422)( 2345 ssssssP

There are 2 sign changes regardless of is positive or There are 2 sign changes regardless of is positive or negative. Therefore the system is unstable.negative. Therefore the system is unstable.

10 s

6 s

10 12

- s

6 s

10 4 2 s

11 2 1 s

0

1

2

3

4

5

results. verify the

shouldYou ts.coefficienother thecalculate and

bput wee therefor,6b ,0b

:elements thecalculate When we

121

Case 3: All elements in a row are zero.Case 3: All elements in a row are zero. Example:Example:

Here the array cannot be completed because of the zero Here the array cannot be completed because of the zero element in the first column.element in the first column.

Another example:Another example:

0

1

2

2

0

1 1

1)(

s

s

s

ssP

0

1

2

3

23

0

2 1

2 1

:isarray The

22)(

s

s

s

s

ssssP

Case 3 polynomial contains an even polynomial as a Case 3 polynomial contains an even polynomial as a factor. It is called the factor. It is called the auxiliary polynomial. auxiliary polynomial. In the first In the first example, the example, the auxiliary polynomial auxiliary polynomial isis

and in the second exampleand in the second example, auxiliary polynomial , auxiliary polynomial isis

Case 3 polynomial may be analyzed as follows:Case 3 polynomial may be analyzed as follows:

Suppose that the row of zeros is the row, then theSuppose that the row of zeros is the row, then the

auxiliary polynomial is differentiated with respect to s,auxiliary polynomial is differentiated with respect to s,

and the coefficients of the resulting polynomial used to and the coefficients of the resulting polynomial used to replace the zeros in the row. The calculation of the replace the zeros in the row. The calculation of the array then continues as in the case 1.array then continues as in the case 1.

Example: Example:

12 s

22 s

is

is

completed. then isarray Routh theand row,

in the 0 replaces 2 therefore,2 is derivative The

2)(

:row thefrom obtained is polynomial

auxiliary thezeros, contains row theSince

0

2 1

2 1

2 3 1

:isarray Routh The

223)(

1

2

2

1

0

1

2

3

4

234

s

s

ssP

s

s

s

s

s

s

s

sssssP

aux

plane. halfright

on the roots hasit or example), in this (as axisimaginary

on theeither roots have it will isThat nonstable. is system

thearray,Routh in the zeros of row a is re When the:Note

below. Note

see However, plane. halfright in the roots no are thereHence

2

2 1

2 1

2 3 1

:becomes nowarray Routh The

223)(

0

21

2

3

4

234

s

s

s

s

s

sssssP