Lecture 11 ME 176 5 Stability
21
ME 176 Control Systems Engineering Department of Mechanical Engineering Stability
-
Upload
leonidesdeocampo -
Category
Documents
-
view
4.145 -
download
4
Transcript of Lecture 11 ME 176 5 Stability
ME 176Control Systems Engineering
Department of
Mechanical Engineering
Stability
Background: Design Process
Department of
Mechanical Engineering
Background: Analysis & Design Objectives
"Analysis is the process by which a system's performance is determined."
"Design is the process by which a systems performance is created or changed."
Transient ResponseSteady State ResponseStability
Department of
Mechanical Engineering
Background: Stability
Characteristic : most important system requirement.Scope :
Linear - the relationship between the input and the output of the system satisfies the superposition property. If the input to the system is the sum of two component signals:
In general:
If, then,
Department of
Mechanical Engineering
Background: Stability
Characteristic : most important system requirement.Scope :
Time invariant systems - are systems that can be modeled with a transfer function that is not a function of time except expressed by the input and output.
"Meaning, that whether we apply an input to the system now or T seconds from now, the output will be identical, except for a time delay of the T seconds. If the output due to input x (t ) is y (t ), then the output due to input x (t − T ) is y (t − T ). More specifically, an input affected by a time delay should effect a corresponding time delay in the output, hence time-invariant."
Department of
Mechanical Engineering
Background: Definitions In terms of natural response:
Stable : natural response approaches 0 as time approaches infinity.
Unstable : natural response grows within bounds as time approaches infinity.
Marginally Stable : natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.
In terms of total response:
Stable : if every bounded input yields a bounded output.
Unstable : if any bounded input yields an unbounded output.
Department of
Mechanical Engineering
Background: Definitions In terms of poles:
Stable : for closed-loop transfer functions with poles only in the left hand plane.
Unstable : for closed-loop transfer functions with at least one pole in the right half-plane and/or poles of multiplicity greater than 1on the imaginary axis.
Marginally Stable : for closed-loop transfer functions with only imaginary axis poles of multiplicity 1 and poles in the left half-plane.
Department of
Mechanical Engineering
Routh-Hurwitz Criteria: A method which allows one to tell how many closed-loops system poles are in the left half-plane, in the right half-plane, and on the imaginary axis. Steps:
1. Generate the data table, called a Routh table. 2. Interpret the Routh table, to tell how where poles are located.
Power of method is not in the analysis, but in the design. For unknown parameters, it allows for a closed form expression for the range of the unknown parameters.
Department of
Mechanical Engineering
Routh-Hurwitz Criteria: Generating Table
Department of
Mechanical EngineeringPowers of s in the denominator in decreasing order
Coefficientsof s.
determinant entries of previous rows.
Routh-Hurwitz Criteria: Generating Table
Department of
Mechanical Engineering
1 31
0
10 1 1030 103--- ------ 0
Routh-Hurwitz Criteria: Interpreting Table
Number of roots of the polynomial that are in the right half-plane is equal to the number of sign changes in the first column.
Department of
Mechanical Engineering
1 31
0
10 1 1030 103--- ------ 0
Negative
Routh-Hurwitz Criteria: Special Cases Having zero only in the first column of the row (epsilon method)
Department of
Mechanical Engineering
1 3
2
5
6 3
7/2 0
3 0
3 0
0 0
0
+
+
+
-
+
+
Positive
+
+
-
+
+
+
Routh-Hurwitz Criteria: Special Cases Having zero only in the first column of the row (reciprocal method)
Department of
Mechanical Engineering
3 6
5
2
3 1
1.4 0
1 0
0
0 0
0
4.2
1.33
-1.75
1
+
+
+
-
+
+
Positive
+
+
-
+
+
+
Negative
Routh-Hurwitz Criteria: Special Cases Entire row consists of zeros
Department of
Mechanical Engineering
1 6
7 1
8
42 6 56 8
0 12 3 0 0 0
8 0
0
0 0
0
0 4 1
3
1/3
8
---
---
---
--- -- --- -- --- --
Routh-Hurwitz Criteria: Special Cases Entire row consists of zeros : Analysis
"A row of zero is caused when there is a factor of an even polynomial" Characteristics of even polynomials (only even powers of s):
Department of
Mechanical Engineering
1. Roots on jw axis mean there are zero roots, and it is only with such roots can there be roots on jw axis.
2. Row previous to the zeros contains the even polynomial which a factor of original polynomial.
3. Everything from zero row down is a test only of the even polynomial.
Routh-Hurwitz Criteria: Special Cases Entire row consists of zeros
Department of
Mechanical Engineering
Routh-Hurwitz Criteria: Example Find number of poles on the right hand, left hand, and jw axis of the s-plane.
Department of
Mechanical Engineering
Characteristics: 2 sign changes.No zero rows.
Pole locations:
two right hand polestwo left hand poles
System Analysis:
Unstable
Routh-Hurwitz Criteria: Example Find number of poles on the right hand, left hand, and jw axis of the s-plane.
Department of
Mechanical Engineering
Characteristics for e as positive: 2 sign changes No zero rows.
Pole locations:
two right hand polestwo left hand poles
System Analysis:
Unstable
Routh-Hurwitz Criteria: Example Find number of poles on the right hand, left hand, and jw axis of the s-plane.
Department of
Mechanical Engineering
Characteristics for e as positive: 2 sign changes No zero rows.
Pole locations:
two right hand polestwo left hand poles
System Analysis:
Unstable
Routh-Hurwitz Criteria: Example Find number of poles on the right hand, left hand, and jw axis of the s-plane.
Department of
Mechanical Engineering
Characteristics: Zero rows at power 5.Sign change after power 5.
Pole locations:
two right hand polesfour left hand poles 2 at jw axis
System Analysis:
Unstable
Routh-Hurwitz Criteria: Design Find K, that will cause the system to be stable, unstable, marginally stable.
Department of
Mechanical Engineering
Characteristics: K<1386 : StableK>1386 : UnstableK = 1386 : Marginally Stable
case K=1386
