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Q(s) bn sn bn1sn1 bn2 sn2 b1s b0 0
(6.2)
If the b0 term is zero, divide by s to obtain the equation in the form of Eq. (6.2). The b's are real coefficients, and all powers of s from sn to s0 must be present in the characteristic equation.
2008.3
Fundamentals of Control Theory
Chap 6 Control-system characteristics
Chap 6 Control system characteristics
6.1 6.2 6.3 6.4 6.5 6.6 Introduction Routh’s stability criterion Mathematical and physical forms Feedback System Types Analysis of System Types Summary
6.2 Routh’s stability criterion
The rest of the rows are formed in this way down to the s0 row. The complete array is triangular, ending with the s0 row. Notice that the s1 and s0 rows contain only one term each. Once the array has been found, Routh’s criterion states that the number of roots of the characteristic equation with positive real parts is equal to the number of changes of sign of the coefficients in the first column. Therefore, the system is stable if all terms in the first column have the same sign.
This process is continued until no more d terms are present.
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
We cቤተ መጻሕፍቲ ባይዱn solve the characteristic equation to find the exact roots, such as s1,s2, ... ,sn, then if all the characteristic roots have minus real parts, the system is stable, otherwise if there is one root with positive real parts, the system will be unstable, if there is one or more roots with zero real parts, the system will be critical stable.
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion
The Routh’s criterion is an algebra criterion, which utilize the relations between the roots of the characteristic equation and its coefficients to determine the locations of the roots in s plane, without solving for the actual value of the roots. The characteristic equation of the system is:
This pattern is continued until the rest of the c’s are all equal to zero. Then the d row is formed by using the sn-1 and sn-2 row. The constants are
c1bn3 bn1c2 c1bn5 bn1c3 c1bn7 bn1c4 d1 , d2 , d3 c1 c1 c1
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction The characteristic equation of the system is:
Q(s) bn sn bn1sn1 bn2 sn2 b1s b0 0
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion
A necessary but not sufficient condition for stable roots is that all the coefficients in Eq. (6.2) be positive. If any coefficients other than b0 are zero, or if all the coefficients do not have the same sign, then there are pure imaginary roots or roots with positive real parts and the system is unstable. In that case it is unnecessary to continue if only stability or instability is to be determined. When all the coefficients are present and positive, the system may or may not be stable because there still may be roots on the imaginary axis or in the right-half s plane.
Fundamentals of Control Theory
Chap 6 Control-system characteristics
Fundamentals of Mechanical Control Theory
School of Mechanical Engineering
湖南工业大学机械工程学院
sn
s n 1
bn bn1 c1
bn2 bn3 c2
bn4 bn5 c3
bn6 bn7
s n2 s n 3
d1
d2
s1 s0
School of Mechanical Engineering
j1 k1
湖南工业大学机械工程学院
Fundamentals of Control Theory
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction In some cases, it is difficult to find the exact solution of the characteristic equation. For example, there is a characteristic equation as follow:
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion The coefficients of the characteristic equation are arranged in the pattern shown in the first two rows of the following Routhian array. These coefficients are then used to evaluate the rest of the constants to complete the array.
D(s) s5 s 4 10s3 72s 2 152s 240 0
It will be difficult to some extent to find the exact solution. In most cases, if we only want to judge whether a system is stable or not, it is usually unnecessary to find the exact roots of the characteristic equation. But what methods can we use to solve this problem? Routh, Nyquist and Bode have done large number of work to do this.
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion The constants c1, c2, c3, etc., in the third row are evaluated as follows:
bn1bn2 bnbn3 bn1bn4 bnbn 5 bn 1bn 6 bnbn 7 c1 , c2 , c3 bn1 bn1 bn1
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction To a controlled system, there are three basic requirements: stability, sensibility and accuracy. The stability is the most important. If the system is unstable, there will be no necessary to discuss the other two requirements. But how to judge whether a controlled system is stable or unstable? According to we have learned before, if there is a system which has the transfer function as follow:
P(s) am s m am1s m1 am2 s m2 a1s a0 G( s) (m n) n n 1 n2 Q(s) bn s bn1s bn2 s b1s b0
School of Mechanical Engineering
(6.2)
If the b0 term is zero, divide by s to obtain the equation in the form of Eq. (6.2). The b's are real coefficients, and all powers of s from sn to s0 must be present in the characteristic equation.
2008.3
Fundamentals of Control Theory
Chap 6 Control-system characteristics
Chap 6 Control system characteristics
6.1 6.2 6.3 6.4 6.5 6.6 Introduction Routh’s stability criterion Mathematical and physical forms Feedback System Types Analysis of System Types Summary
6.2 Routh’s stability criterion
The rest of the rows are formed in this way down to the s0 row. The complete array is triangular, ending with the s0 row. Notice that the s1 and s0 rows contain only one term each. Once the array has been found, Routh’s criterion states that the number of roots of the characteristic equation with positive real parts is equal to the number of changes of sign of the coefficients in the first column. Therefore, the system is stable if all terms in the first column have the same sign.
This process is continued until no more d terms are present.
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
We cቤተ መጻሕፍቲ ባይዱn solve the characteristic equation to find the exact roots, such as s1,s2, ... ,sn, then if all the characteristic roots have minus real parts, the system is stable, otherwise if there is one root with positive real parts, the system will be unstable, if there is one or more roots with zero real parts, the system will be critical stable.
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion
The Routh’s criterion is an algebra criterion, which utilize the relations between the roots of the characteristic equation and its coefficients to determine the locations of the roots in s plane, without solving for the actual value of the roots. The characteristic equation of the system is:
This pattern is continued until the rest of the c’s are all equal to zero. Then the d row is formed by using the sn-1 and sn-2 row. The constants are
c1bn3 bn1c2 c1bn5 bn1c3 c1bn7 bn1c4 d1 , d2 , d3 c1 c1 c1
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction The characteristic equation of the system is:
Q(s) bn sn bn1sn1 bn2 sn2 b1s b0 0
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion
A necessary but not sufficient condition for stable roots is that all the coefficients in Eq. (6.2) be positive. If any coefficients other than b0 are zero, or if all the coefficients do not have the same sign, then there are pure imaginary roots or roots with positive real parts and the system is unstable. In that case it is unnecessary to continue if only stability or instability is to be determined. When all the coefficients are present and positive, the system may or may not be stable because there still may be roots on the imaginary axis or in the right-half s plane.
Fundamentals of Control Theory
Chap 6 Control-system characteristics
Fundamentals of Mechanical Control Theory
School of Mechanical Engineering
湖南工业大学机械工程学院
sn
s n 1
bn bn1 c1
bn2 bn3 c2
bn4 bn5 c3
bn6 bn7
s n2 s n 3
d1
d2
s1 s0
School of Mechanical Engineering
j1 k1
湖南工业大学机械工程学院
Fundamentals of Control Theory
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction In some cases, it is difficult to find the exact solution of the characteristic equation. For example, there is a characteristic equation as follow:
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion The coefficients of the characteristic equation are arranged in the pattern shown in the first two rows of the following Routhian array. These coefficients are then used to evaluate the rest of the constants to complete the array.
D(s) s5 s 4 10s3 72s 2 152s 240 0
It will be difficult to some extent to find the exact solution. In most cases, if we only want to judge whether a system is stable or not, it is usually unnecessary to find the exact roots of the characteristic equation. But what methods can we use to solve this problem? Routh, Nyquist and Bode have done large number of work to do this.
Chap 6 Control-system characteristics
6.2 Routh’s stability criterion The constants c1, c2, c3, etc., in the third row are evaluated as follows:
bn1bn2 bnbn3 bn1bn4 bnbn 5 bn 1bn 6 bnbn 7 c1 , c2 , c3 bn1 bn1 bn1
School of Mechanical Engineering
湖南工业大学机械工程学院
Fundamentals of Control Theory
Chap 6 Control-system characteristics
6.1 Introduction To a controlled system, there are three basic requirements: stability, sensibility and accuracy. The stability is the most important. If the system is unstable, there will be no necessary to discuss the other two requirements. But how to judge whether a controlled system is stable or unstable? According to we have learned before, if there is a system which has the transfer function as follow:
P(s) am s m am1s m1 am2 s m2 a1s a0 G( s) (m n) n n 1 n2 Q(s) bn s bn1s bn2 s b1s b0
School of Mechanical Engineering