Chapter 5: Frequency Domain Analysis of Feedback Systems
Transcript of Chapter 5: Frequency Domain Analysis of Feedback Systems
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Chapter 5: Frequency Domain Analysis of Feedback Systems
In this chapter we review absolute stability of a linear feedback system with a nonlinear block through circle and Popov criterion. Then by quasi-linear approximation of nonlinear feedback systems into a linear and a nonlinear block, existence, stability and frequency and amplitude of limit cycles are analyzed, by using describing function analysis.
Nonlinear Control
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
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Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
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Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Contents
In this chapter we review absolute stability of a linear feedback system with a nonlinear block. Circle and Popov criterion in single variable, andmultivariable systems are described in detail. Then by quasi-linear approximation of nonlinear feedback systems into a linear and a nonlinear block, existence, stability and frequency and amplitude of limit cycles are analyzed, by using describing function analysis.
4
Absolute StabilityIntroduction, definitions, sector nonlinearity, Lureโs problem, Multivariable and single variable circle and Popov criteria.1
Describing Function MethodIllustrating example, assumptions and definitions, computing describing functions for common nonlinearities.2
Describing Function MethodReview of Nyquist criterion, Existence and stability of limit cycles, examples.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Frequency Domain Analysis of Feedback Systems5
โข Introduction
Feedback Connection representation of many nonlinear
physical systems
โข Separation of linear and nonlinear elements.
โ Linear + one nonlinear block
โ Genuinely nonlinear but separable
โข Assume ๐ = 0, stability of unforced system
โ Linear Control theory and frequency response.
โ Extension of Nyquist stability criteria
โข Absolute Stability
โ The origin is globally uniformly asymptotically Stable.
โ The Circle and Popov criteria
Feedback system 5.1
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Frequency Domain Analysis of Feedback Systems6
โข Absolute Stability
Consider the feedback connected system by closed-loop non-autonomous system
แถ๐ฅ = ๐ด๐ฅ + ๐ต๐ข (7.1)
๐ฆ = ๐ถ๐ฅ + ๐ท๐ข (7.2)
๐ข = โ๐(๐ก, ๐ฆ) (7.3)
โข where (๐ด, ๐ต) is controllable, (๐ด, ๐ถ) is observable,
โข And ๐ is a memoryless, possibly time varying nonlinearity
โ Piecewise continuous in ๐ก and locally Lipschitz in ๐ฆ
โข The transfer matrix of the system:
๐บ ๐ = ๐ถ ๐ ๐ผ โ ๐ด โ1๐ต + ๐ท (7.4)
โข For all nonlinearities, origin is the eq. point.
โข Lureโs problem: Study the stability for sector type nonlinearity.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability7
โข Definitions
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability8
โข Definitions
Absolute Stability
โข The stability is examined using two Ly. Functions
โ Weighted norm Lyapunov function:
โ Lure type Lyapunov function:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability9
โข Multivariable Circle Criterion
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability10
โข Circle Criterion:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability11
โข Circle Criterion:
Graphical Representation
โข Disk ๐ท(๐ผ , ๐ฝ) is shown graphically
โข Extension of Nyquist criteria by replacement of โ1/๐ to the disk ๐ท(๐ผ , ๐ฝ).
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability12
โข Circle Criterion:
Example 1: Consider the system
โข Sector Nonlinearity [๐ผ, ๐ฝ]
โ Case (a): ๐ผ < 0 < ๐ฝ
โ For absolute stability the Nyquist must lie inside circle:
โ 1st choice:
๐ท(๐ผ , ๐ฝ) = [โ1/4 ,+1/4] = [โ0.25,+0.25]
โข This is not the largest sector.
โ 2nd choice: dotted circle:
โ D(ฮฑ ,ฮฒ) = [-1/4.4 , +1/1.4]
= [-0.227, +0.714]
โข Better than the 1st choice.
4-1.4
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability13
โข Circle Criterion:
Example 1: (Cont.)
โข Sector Nonlinearity [๐ผ, ๐ฝ]
โ Case (b): 0 = ๐ผ < ๐ฝ
โ For absolute stability the Nyquist must lie to the right of line Re(s) -
0.875:
โ ๐ท(๐ผ , ๐ฝ) = [0 ,+1/0.875]
= [0 , 1.117]
โ This is not the largest sector.
-0.875
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability14
โข Circle Criterion: Example 2: Consider the system
โข Linear system + saturation
โข Saturation lies in the sector [0 , 1]:
โข According to Ex 1 case b, the system is absolutely stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability15
โข Circle Criterion:
Example 3: Let
โข Not Hurwitz
โ Nyquist plot must not enter the disk, but encircle it once in c.c.w.
direction.
โ Plot Nyquist.
โ Plot circle:
Center: (0,โ3.2) Radius: 0.1688
โ The disk is ๐ท( โ3.36 ,โ3.03)
โ The stable sector is
[0.297, 0.330]
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability16
โข Circle Criterion:
Example 4: Let
โข Nonlinearity is saturation belonging to sector [0, 1]
โ Not Hurwitz, Nyquist plot must not enter the
disk, but encircle it once in c.c.w. direction.
โ Plot Nyquist, and the circle:
Center โถ (0, โ1),Radius โถ 0.45
โ The disk is ๐ท(โ0.55,โ1.45)
โ The stable sector is [0.690, 1.818]
โ The system is not globally asymptotically
stable, and it is only absolutely stable in a
finite domain.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability17
โข Popov Criteria:
Consider the feedback connected system by closed-loop non-autonomous system
แถ๐ฅ = ๐ด๐ฅ + ๐ต๐ข (7.13)
๐ฆ = ๐ถ๐ฅ (7.14)
๐ข๐ = โ๐๐ ๐ฆ๐ , 1 โค ๐ โค ๐ (7.15)
โข where (๐ด, ๐ต) is controllable, (๐ด, ๐ถ) is observable,
โข And ๐๐ is a memoryless nonlinearity belonging to sector [0, ๐๐]
โข The transfer matrix of the system is strictly proper:
๐บ ๐ = ๐ถ ๐ ๐ผ โ ๐ด โ1๐ต
โข With the Lure-type Lyapunov function absolute stability of the system is proved.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability18
โข Multivariable Popov Criteria:
โข For ๐ + ๐ผ + ๐ ฮ ๐บ ๐ be strictly positive real ๐บ ๐ must be Hurwitz.
For scalar case the following theorem may be stated.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability19
โข Popov Criteria:
โข Applicable only for SISO and stable systems.
โข Only Sufficient Condition.
โข Graphical representation of condition (7.19)
Theorem 7.4: Consider scalar nonlinearity ๐ and the feedback system (5.1) that satisfies the conditions:
โข The matrix A is Hurwitz and the pair (A,B) is controllable.โข The nonlinearity ๐ belongs to the sector [0, ๐]โข There exists a positive number ๐พ such that
Then the origin is globally asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability20
Popov Criteria:
Example: Consider the system
โข Let ๐ = โ ๐ฆ โ ๐ผ๐ฆ, ๐ผ > 0 to make the system matrix Hurwitz.
โ Assume the sector nonlinearity โ โ ๐ผ, ๐ฝ , ๐ฝ > ๐ผ > 0
โ Then ๐ โ 0, ๐ , k = ๐ฝ โ ๐ผ. Then, the Popov condition is
โ For all positive values of ๐ผ and ๐ this is satisfied by ๐พ > 1.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Absolute Stability21
Popov Criteria:
Example: (Cont.)
โ Even for ๐ = โ, the popov condition is satisfied for all ๐
โ Hence, the system is absolutely stable for all โ in the sector ๐ผ,โ ,
where ๐ผ can be arbitrary small.
โ The Popov plot is shown for ๐ผ = 1.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Contents
In this chapter we review absolute stability of a linear feedback system with a nonlinear block. Circle and Popov criterion in single variable, andmultivariable systems are described in detail. Then by quasi-linear approximation of nonlinear feedback systems into a linear and a nonlinear block, existence, stability and frequency and amplitude of limit cycles are analyzed, by using describing function analysis.
22
Absolute StabilityIntroduction, definitions, sector nonlinearity, Lureโs problem, Multivariable and single variable circle and Popov criteria.1
Describing Function MethodIllustrating example, assumptions and definitions, computing describing functions for common nonlinearities.2
Describing Function MethodReview of Nyquist criterion, Existence and stability of limit cycles, examples.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Frequency Domain Analysis of Feedback Systems23
โข Describing Function Method
Illustrating Example: Van der Pol
โข System Dynamics: แท๐ฅ + ๐ผ ๐ฅ2 โ 1 แถ๐ฅ + ๐ฅ = 0
โข Assume, a disk type limit cycle ๐ฅ ๐ก = ๐ด๐ ๐๐(๐๐ก)
โ With amplitude ๐ด and frequency ๐.
Separate the linear and nonlinear elements, and Integrate it into
feedback system (5.2)
แท๐ฅ โ ๐ผ แถ๐ฅ + ๐ฅ = ๐ผ(โ๐ฅ2 แถ๐ฅ)
โ Consider the nonlinear term ๐ค ๐ฅ = โ๐ฅ2 แถ๐ฅ in the feedback.
๐บ(๐ )๐ค(โ )๐ฅ(๐ก)๐ค(๐ก)โ๐ฅ(๐ก)๐ = 0
โ+
Nonlinearity Linear SystemFeedback System 5.2
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method24
Illustrating Example: Van der Pol (Cont.)
โข Then the linear system is
๐บ ๐ =๐ฅ
๐ข=
๐ผ
๐ 2โ๐ผ ๐ +1
โข Approximate the nonlinear element
โ For a limit cycle ๐ฅ ๐ก โ ๐ด sin(๐๐ก) โ แถ๐ฅ = ๐ด๐ cos ๐๐ก
๐ค = โ๐ฅ2 แถ๐ฅ = โ๐ด2 sin2(๐๐ก) ๐ด๐ cos ๐๐ก
=โ1
2๐ด3๐ 1 โ cos 2๐๐ก cos ๐๐ก =
โ1
4๐ด3๐ cos(๐๐ก) โ cos(3๐๐ก)
โข Approximate the output
๐ค โโ๐ด3
4๐ cos ๐๐ก =
๐ด2
4
๐
๐๐กโ๐ด sin ๐๐ก =
๐ด2
4(โ๐ ๐ฅ)
โข Input/Output Quasi-Linear approximation
๐ค
๐ฅโ๐ด2
4โ๐
โ Amplitude dependent
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method25
Quasi-linear block diagram
โข The frequency response estimate:
๐ค = ๐ ๐ด,๐ โ๐ฅ โ ๐ ๐ด,๐ =๐ด2
4๐๐
โข Existence of limit cycle: โ ๐ฅ ๐ก โ ๐ด sin ๐๐ก , โ
๐ฅ ๐ก โ ๐ด sin ๐๐ก = ๐บ ๐๐ ๐(๐ด, ๐)(โ๐ฅ) โ 1 + ๐บ ๐๐ ๐ ๐ด,๐ ๐ฅ
โ For a nontrivial limit cycle: ๐ฅ ๐ก โ 0 โ 1 + ๐บ ๐๐ ๐ ๐ด,๐ = 0
โ Compare Nyquist criteria: 1 + ๐บ ๐๐ โ ๐ = 0
๐ผ
๐ 2 โ ๐ผ๐ + 1
๐ฅ(๐ก)๐ค(๐ก)โ๐ฅ(๐ก)๐ = 0
โ+
๐ด2
4๐
Quasi-linear Approximation ๐บ(๐ )
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method26
Illustrating Example: Van der Pol (Cont.)
โข Graphical Solution
โ Intersect ๐บ(๐๐) and โ1/๐(๐ด,๐)
โข Analytical Solution (for ๐ = ๐๐)
2( )
1G j
j
2 2
1 4 4
( , )
j
N A A j A
2
4, 1 2
jj A
A
๐ = 1
๐บ(๐๐)
โ1/๐(๐ด,๐)
1,2p
2
2 2 2
4 0 2
11 1/ 64 ( 4)
A A
A
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method
Illustrating Example: Van der Pol (Cont.)
โข Graphical Solution
The approx. limit cycle
parameters does not
depend on ๐ผ
๐ = 1
โ1/๐(๐ด, ๐)
๐บ(๐๐), ๐ผ = 2
๐บ(๐๐), ๐ผ = 1
๐บ(๐๐), ๐ผ = 4
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method
Illustrating Example: Van der Pol (Cont.)
โข The approximate L.C. parameters does not depend on ๐ผ
โข The REAL L.C. parameters depends on ๐ผ
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method29
โข Assumptions and Definitions
Systems Under D.F. Analysis
โข Separation of linear and nonlinear elements.
โ Linear + one nonlinear block
โ Genuinely nonlinear but separable
โข only one time invariant nonlinearity
โข The 1st harmonics of nonlinearity pass through the linear part
(๐บ(๐ ) is low pass).
๐บ(๐ )๐ค(โ )๐ฅ(๐ก)๐ค(๐ก)โ๐ฅ(๐ก)๐ = 0
โ+
Nonlinearity Linear SystemFeedback System 5.2
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method30
โข Assumptions and Definitions
Nonlinear function
โข For a Limit cycle the output is periodic.
โข Use Fourier Series to represent the output.
โข For odd nonlinearities ๐0 = 0.
โข By 3rd assumption consider just the first harmonics.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method31
โข Assumptions and Definitions
Nonlinear function (cont.)
โ In which
โ In complex representation
โข Describing Function
โ Extension of input/output transfer function:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method32
โข Computing Describing Functions
Common nonlinearities
โข Hardening Spring
โ Fourier series for input: :
โ Nonlinearity is odd: ๐0 = ๐1 = 0, and
โ Therefore,
โ Describing Function is:
โ Note: D.F. is real and does not depend on ๐
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method33
โข Computing Describing Functions
Common nonlinearities
โข Saturation:
โ Consider the input:
โ Saturated output:
โ In which,
โ Nonlinearity is odd: ๐0 = ๐1 = 0, and
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method34
โข Computing Describing Functions
Common nonlinearities
โข Saturation:
โ Describing Function:
โ D.F. Features:
โ Graphical Representation
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method35
โข Computing Describing Functions
Common nonlinearities
โข Relay: Special case of saturation
โ Direct determination:
โ Describing Function:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method36
โข Computing Describing Functions
Common nonlinearities
โข Piecewise Linear
โ Describing Function
1( , )b
N aa
( )N a
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method37
โข Computing Describing Functions
Common nonlinearities
โข Piecewise Linear
โ Describing Function: Graphical Representation
โข Saturation
โ Special case of piecewise linear with
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method38
โข Computing Describing Functions
Common nonlinearities
โข Dead-Zone
โ Nonlinear function
โ In which,
โ Nonlinearity is odd: ๐0 = ๐1 = 0, and
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method39
โข Computing Describing Functions
Common nonlinearities
โข Dead-Zone
โ Describing Function
โ Real, A dependent
โ Not depending on ๐
โ No phase shift
โ Graphical
Representation
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method40
โข Computing Describing Functions
Common nonlinearities
โข Backlash
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method41
โข Computing Describing Functions
โข Backlash
Nonlinear Function:
In which
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method42
โข Computing Describing Functions
โข Backlash
โ Describing Function:
โ Therefore,
โ D.F. Graphical Representation
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Contents
In this chapter we review absolute stability of a linear feedback system with a nonlinear block. Circle and Popov criterion in single variable, andmultivariable systems are described in detail. Then by quasi-linear approximation of nonlinear feedback systems into a linear and a nonlinear block, existence, stability and frequency and amplitude of limit cycles are analyzed, by using describing function analysis.
43
Absolute StabilityIntroduction, definitions, sector nonlinearity, Lureโs problem, Multivariable and single variable circle and Popov criteria.1
Describing Function MethodIllustrating example, assumptions and definitions, computing describing functions for common nonlinearities.2
Describing Function AnalysisReview of Nyquist criterion, Existence and stability of limit cycles, examples.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Method44
โข Review of Nyquist Criterion
Feedback system
โข Loop gain ๐ฟ(๐ ) = ๐ถ(๐ )๐(๐ )
โ Characteristic Equation 1 + ๐ฟ(๐ ) = 0 or ๐ฟ(๐ ) = โ1
โ Draw the Nyquist plot of ๐ฟ(๐๐) for โ < ๐ < +
โ Count the number of c.w. encirclement of the Nyquist plot around point (โ1,0). ๐
โ Compute the number of unstable poles of the loop gain ๐ฟ(๐ ). ๐
โ The number of unstable roots of the characteristic equation (or the number of unstable poles of the closed-loop transfer function) is denoted by ๐ and is found from: ๐ = ๐ + ๐
๐(๐ )๐ถ(๐ )+ -
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis45
โข Review of Nyquist Criterion
Feedback system
โข Loop gain ๐ฟ(๐ ) = ๐ถ(๐ )๐(๐ )
โ Characteristic Equation 1 + ๐ ๐ฟ(๐ ) = 0 or ๐ฟ(๐ ) = โ1 / ๐
โ Draw the Nyquist plot of ๐ฟ(๐๐) for โ < ๐ < +
โ Count the number of c.w. encirclement of the Nyquist plot around
point (โ1 / ๐ , 0). ๐
โ Compute the number of unstable poles of the loop gain ๐ฟ(๐ ). ๐
โ The number of unstable roots of the characteristic equation (or the number of unstable poles of the closed-loop transfer function) is denoted by ๐ and is found from: ๐ = ๐ + ๐
๐ฟ(๐ )๐+
-
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis46
โข Existence of Limit Cycle
Feedback System๐ฅ = โ๐ฆ
๐ค = ๐ ๐ด,๐ ๐ฅ๐ฆ = ๐บ ๐๐ ๐ค
โ ๐ฆ = ๐บ ๐๐ ๐ ๐ด,๐ (โ๐ฆ)
โข Characteristics Equation: (for ๐ฆ 0 )
๐บ ๐๐ ๐ ๐ด,๐ + 1 = 0 OR ๐บ ๐๐ = โ1
๐ ๐ด,๐
โข Solve it Graphically
โข Amplitude dependent describing function
โ Intersect ๐บ(๐๐) and โ1/๐(๐ด)
โ If intersection occurs L.C. exists
โ Amplitude of L.C. ๐ด (on โ1/๐(๐ด))
โ Frequency of L.C. ๐ (on ๐บ(๐๐))
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis47
โข Existence of Limit Cycle
Graphical Solution
โข General Describing Function (๐(๐ด,๐))
โ Plot ๐บ(๐๐)
โ Plot a set of โ1/๐(๐ด,๐) for different constant frequencies.
โ Analyze the intersection points
โ If an intersection point in which
The frequencies match L.C. exists
โ Amplitude of L.C. ๐ด (on -1/๐(๐ด,๐))
โ Frequency of L.C. the matched ๐
(on ๐บ(๐๐) and on -1/๐(๐ด,๐))
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis48
โข Stability of Limit Cycle
Feedback System
โข Characteristics Equation: (for ๐ฆ 0 )
๐บ ๐๐ ๐ ๐ด,๐ + 1 = 0 OR ๐บ ๐๐ = โ1
๐ ๐ด,๐
Compare it with Nyquist criteria:
๐ฟ ๐ ๐ + 1 = 0 or ๐ฟ(๐ ) = โ1 / ๐
โข Extended Nyquist Criteria
โ Replace the point โ1/๐ by โ1/๐(๐ด,๐) and count encirclements
โ Apply the same Nyquist criteria to determine stability
โ Note: stability decay of amplitude
โ Instability increase of amplitude
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Unstable Region
Describing Function Analysis49
โข Stability of Limit Cycle
Example:
โข Consider ๐บ(๐ ) is stable ๐ = 0
โข Suppose two L.C. ๐ฟ1, ๐ฟ2โข Stability analysis of ๐ฟ1
โ Perturb ๐ฟ1by increasing its amplitude
โข ๐ฟ1โฒ is inside ๐บ(๐๐) (unstable)
โข The amplitude increases
โข ๐ฟ1โฒ moves away from ๐ฟ1 Unstable Limit Cycle
โ Perturb ๐ฟ1by decreasing its amplitude
โข ๐ฟ1" is outside ๐บ(๐๐) (stable)
โข The amplitude decreases
โข ๐ฟ1" moves away from ๐ฟ1 Unstable Limit Cycle
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis50
'
2L
"
2L
โข Stability of Limit Cycle
Example: (cont.)
โข Stability analysis of ๐ฟ2
โ Perturb ๐ฟ2 by increasing its amplitude
โข ๐ฟ2โฒ is outside G(jฯ) (stable)
โข The amplitude decreases
โข ๐ฟ2โฒ moves toward ๐ฟ2 Stable Limit Cycle
โ Perturb ๐ฟ2 by decreasing its amplitude
โข ๐ฟ2โฒโฒ is inside G(jฯ) (unstable)
โข The amplitude increases
โข ๐ฟ2โฒโฒ moves toward ๐ฟ2 Stable Limit Cycle
Unstable Region
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
โข Example 1:
Consider ๐บ ๐ =1
๐ (๐ +1)(๐ +2)with unit
โข Nyquist Plot
๐บ ๐๐ = โ3๐ โ ๐(2 โ ๐2)
9๐3 + ๐ 2 โ ๐2 2
โ @ point A
โ ๐บ ๐๐ = 0 โ 2 โ ๐2 = 0 โ ๐ = 2
๐ด = ๐บ ๐ 2 = โ3 2
18 2 + 0= โ
1
6
(a) For unit saturation (๐ = 1)
โ ๐(๐ด) < 1 No Crossing, No Limit Cycle
Describing Function Analysis51
a) Saturation
b) Relay
โ1/๐(๐ด)
๐บ(๐๐)
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
โข Example 1: (Cont.)
(b) For unit Relay (M=1)
โ โ < โ1/๐(๐ด) < 0 Crossing @ point A
โ Amplitude:โ1
๐ ๐ด= โ
1
6โ
๐ ๐ด =4
๐๐ด= 6 โ ๐ด =
2
3๐
โ Frequency: ๐ = 2 and Limit Cycle is stable.
Describing Function Analysis52
โ1/๐(๐ด)
๐บ(๐๐)
Unstable Region
๐ด
๐ฟโฒ
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis53
โข Example 1: (Cont.)
Simulation verifications:
โ Frequency :
โ 2๐ = 10 โ ๐ = 5 and ๐ = 2๐/5 = 1.26 (estimated by 1.414)
โ Amplitude ๐ด = 2/3๐ = 0.21 (red circle)
2๐ = 10 sec
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis54
โข Example 2:
Consider ๐บ ๐ =โ๐
๐ 2+0.8๐ +8with
โข Nyquist Plot
โ @ point A
(a) For unit saturation (๐ = 1)
โ โ1
๐ ๐ด= โ1.25 โ ๐ ๐ด = 0.8
โ ๐ด 1.455 @ ๐ = 2.83 Stable Limit Cycle
a) Saturation, 1, 1
b) Dead-zone, 1, 0.5
k a
k
2 2
2 2 2
0.8 (8 )( )
0.64 (8 )
jG j
2( ( )) 0 (8 ) 0 2 2G j
( 2) 1.25A G j
โ1/๐(๐ด)
๐บ(๐๐)
โ1 Unstable Region
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Describing Function Analysis55
โข Example 2:
Simulation verifications:
โข Frequency :
โ 7๐ = 16 ๐ = 2.285 and ๐ = 2๐/๐ = 2.75 (estimated by 2.83)
โ Amplitude ๐ด = 1.45 (red ellipse)
7๐ = 16sec
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
56
Describing Function Analysis
โข Example 2: (Cont.)
โข (b) For Dead-zone ๐ฟ = 1, ๐ = 0.5
โข 0 < ๐ ๐ด < 0.5 ๐ ๐ด 0.8
โข No Crossing No Limit Cycle
โ2โ1/๐(๐ด)
๐บ(๐๐)
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
57
Describing Function Analysis
โข Example 3:
Consider Raleighโs Equation แท๐ฅ + ๐ฅ = ๐( แถ๐ฅ โ1
3แถ๐ฅ3)
โข Separation แท๐ฅ โ ๐ แถ๐ฅ + ๐ฅ = โ๐(1
3แถ๐ฅ3)
โ Consider ๐ฆ = แถ๐ฅ and ๐ ๐ฆ =1
3๐ฆ3
โ ๐ 2 โ ๐๐ + 1 ๐ฅ = โ๐๐ ๐ฆ = โ๐๐ ๐ ๐ฅ = โ๐๐ ๐ ๐ฅ
โ Hence,
โข Linear element: ๐บ ๐ =๐ฅ
๐=
๐๐
๐ 2โ๐๐ +1
โข Nonlinear function: ๐ ๐ฆ =1
3๐ฆ3
๐บ(๐ )๐(โ )๐ฅ(๐ก)๐(๐ก)โ๐ฅ(๐ก)๐ = 0
โ+
Nonlinearity Linear System
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
58
Describing Function Analysis
โข Example 3:
Raleighโs Equation (Cont.)
โข Describing Function ๐ ๐ฆ =1
3๐ฆ3
โข ๐ ๐ด =๐1
๐ด=
2
3๐A0๐๐ด sin ๐ 3 sin ๐ ๐๐ =
1
4๐ด2
โข Nyquist Plot
๐บ ๐ =๐๐
๐ 2 โ ๐๐ + 1โ ๐บ ๐๐ =
๐๐๐ 1 โ ๐2 + ๐๐๐
1 โ ๐2 2 + ๐2๐2
โ @ point A
โ ๐บ ๐๐ = 0 โ 1 โ ๐2 = 0 โ ๐ = 1
โ ๐ด = ๐บ ๐1 = โ๐2
๐2= โ1
โ Regardless of ๐
๐ด
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
59
Describing Function Analysis
โข Example 3:
Raleighโs Equation (Cont.)
โข Intersect โ1 / ๐(๐ด) with โ1
โข ๐ ๐ด =1
4๐ด2 = 1 โ ๐ด = 2
โข A Limit cycle with
โ Amplitude = 2
โ Frequency = 1
โข Stability analysis
โ G(s) is unstable ๐ = 2
โ For stable region ๐ = โ2
โ direction of โ1/๐(๐)
โข The limit cycle is STABLE
-40
-30
-20
-10
0
Magnitu
de (
dB
)
10-1
100
101
-270
-225
-180
-135
-90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
โ1/๐(๐ด)
โ1
N=-2: Stable Region
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
60
Describing Function Analysis
โข Example 3: Raleighโs Equation (Cont.)
Simulation verifications:โข Frequency :
โ 2๐ = 13.34 ๐ = 2๐/6.67 = 0.95 (estimated by 1)
โ Amplitude ๐ด = 2/3๐ = 0.21 (red circle)
๐๐ป = ๐๐. ๐๐ ๐๐๐
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Chapter 5: Frequency Domain Analysis of Feedback Systems
To read more and see the course videos visit our course website:
http://aras.kntu.ac.ir/arascourses/nonlinear-control/
Thank You
Nonlinear Control
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 21, 2020
Hamid D. Taghirad has received his B.Sc. degree in mechanical engineering
from Sharif University of Technology, Tehran, Iran, in 1989, his M.Sc. in mechanical
engineering in 1993, and his Ph.D. in electrical engineering in 1997, both
from McGill University, Montreal, Canada. He is currently the University Vice-
Chancellor for Global strategies and International Affairs, Professor and the Director
of the Advanced Robotics and Automated System (ARAS), Department of Systems
and Control, Faculty of Electrical Engineering, K. N. Toosi University of Technology,
Tehran, Iran. He is a senior member of IEEE, and Editorial board of International
Journal of Robotics: Theory and Application, and International Journal of Advanced
Robotic Systems. His research interest is robust and nonlinear control applied to
robotic systems. His publications include five books, and more than 250 papers in
international Journals and conference proceedings.
About Hamid D. Taghirad
Hamid D. TaghiradProfessor