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.

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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, Lure’s 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



Frequency Domain Analysis of Feedback Systems5

• Introduction

Feedback Connection representation of many nonlinear

physical systems

• Separation of linear and nonlinear elements.

– Linear + one nonlinear block

– Genuinely nonlinear but separable

• Assume 𝑟 = 0, stability of unforced system

– Linear Control theory and frequency response.

– Extension of Nyquist stability criteria

• Absolute Stability

– The origin is globally uniformly asymptotically Stable.

– The Circle and Popov criteria

Feedback system 5.1




• Absolute Stability

Consider the feedback connected system by closed-loop non-autonomous system

ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢 (7.1)

𝑦 = 𝐶𝑥 + 𝐷𝑢 (7.2)

𝑢 = −𝜓(𝑡, 𝑦) (7.3)

• where (𝐴, 𝐵) is controllable, (𝐴, 𝐶) is observable,

• And 𝜓 is a memoryless, possibly time varying nonlinearity

– Piecewise continuous in 𝑡 and locally Lipschitz in 𝑦

• The transfer matrix of the system:

𝐺 𝑠 = 𝐶 𝑠𝐼 − 𝐴 −1𝐵 + 𝐷 (7.4)

• For all nonlinearities, origin is the eq. point.

• Lure’s problem: Study the stability for sector type nonlinearity.



Absolute Stability7

• Definitions



Absolute Stability8

• Definitions

Absolute Stability

• The stability is examined using two Ly. Functions

– Weighted norm Lyapunov function:

– Lure type Lyapunov function:



Absolute Stability9

• Multivariable Circle Criterion



Absolute Stability10

• Circle Criterion:





Graphical Representation

• Disk 𝐷(𝛼 , 𝛽) is shown graphically

• Extension of Nyquist criteria by replacement of −1/𝑘 to the disk 𝐷(𝛼 , 𝛽).





Example 1: Consider the system

• Sector Nonlinearity [𝛼, 𝛽]

– Case (a): 𝛼 < 0 < 𝛽

– For absolute stability the Nyquist must lie inside circle:

– 1st choice:

𝐷(𝛼 , 𝛽) = [−1/4 ,+1/4] = [−0.25,+0.25]

• This is not the largest sector.

– 2nd choice: dotted circle:

– D(α ,β) = [-1/4.4 , +1/1.4]

= [-0.227, +0.714]

• Better than the 1st choice.

4-1.4





Example 1: (Cont.)

• Sector Nonlinearity [𝛼, 𝛽]

– Case (b): 0 = 𝛼 < 𝛽

– For absolute stability the Nyquist must lie to the right of line Re(s) -

0.875:

– 𝐷(𝛼 , 𝛽) = [0 ,+1/0.875]

= [0 , 1.117]

– This is not the largest sector.

-0.875




• Circle Criterion: Example 2: Consider the system

• Linear system + saturation

• Saturation lies in the sector [0 , 1]:

• According to Ex 1 case b, the system is absolutely stable.





Example 3: Let

• Not Hurwitz

– Nyquist plot must not enter the disk, but encircle it once in c.c.w.

direction.

– Plot Nyquist.

– Plot circle:

Center: (0,−3.2) Radius: 0.1688

– The disk is 𝐷( −3.36 ,−3.03)

– The stable sector is

[0.297, 0.330]





Example 4: Let

• Nonlinearity is saturation belonging to sector [0, 1]

– Not Hurwitz, Nyquist plot must not enter the

disk, but encircle it once in c.c.w. direction.

– Plot Nyquist, and the circle:

Center ∶ (0, −1),Radius ∶ 0.45

– The disk is 𝐷(−0.55,−1.45)

– The stable sector is [0.690, 1.818]

– The system is not globally asymptotically

stable, and it is only absolutely stable in a

finite domain.




• Popov Criteria:

Consider the feedback connected system by closed-loop non-autonomous system

ሶ𝑥 = 𝐴𝑥 + 𝐵𝑢 (7.13)

𝑦 = 𝐶𝑥 (7.14)

𝑢𝑖 = −𝜓𝑖 𝑦𝑖 , 1 ≤ 𝑖 ≤ 𝑝 (7.15)

• where (𝐴, 𝐵) is controllable, (𝐴, 𝐶) is observable,

• And 𝜓𝑖 is a memoryless nonlinearity belonging to sector [0, 𝑘𝑖]

• The transfer matrix of the system is strictly proper:

𝐺 𝑠 = 𝐶 𝑠𝐼 − 𝐴 −1𝐵

• With the Lure-type Lyapunov function absolute stability of the system is proved.




• Multivariable Popov Criteria:

• For 𝑀 + 𝐼 + 𝑠Γ 𝐺 𝑠 be strictly positive real 𝐺 𝑠 must be Hurwitz.

For scalar case the following theorem may be stated.




• Popov Criteria:

• Applicable only for SISO and stable systems.

• Only Sufficient Condition.

• Graphical representation of condition (7.19)

Theorem 7.4: Consider scalar nonlinearity 𝜙 and the feedback system (5.1) that satisfies the conditions:

• The matrix A is Hurwitz and the pair (A,B) is controllable.• The nonlinearity 𝜓 belongs to the sector [0, 𝑘]• There exists a positive number 𝛾 such that

Then the origin is globally asymptotically stable.




Popov Criteria:

Example: Consider the system

• Let 𝜓 = ℎ 𝑦 − 𝛼𝑦, 𝛼 > 0 to make the system matrix Hurwitz.

– Assume the sector nonlinearity ℎ ∈ 𝛼, 𝛽 , 𝛽 > 𝛼 > 0

– Then 𝜓 ∈ 0, 𝑘 , k = 𝛽 − 𝛼. Then, the Popov condition is

– For all positive values of 𝛼 and 𝑘 this is satisfied by 𝛾 > 1.




Popov Criteria:

Example: (Cont.)

– Even for 𝑘 = ∞, the popov condition is satisfied for all 𝜔

– Hence, the system is absolutely stable for all ℎ in the sector 𝛼,∞ ,

where 𝛼 can be arbitrary small.

– The Popov plot is shown for 𝛼 = 1.



Contents


22



Describing Function MethodReview of Nyquist criterion, Existence and stability of limit cycles, examples.3




• Describing Function Method

Illustrating Example: Van der Pol

• System Dynamics: ሷ𝑥 + 𝛼 𝑥2 − 1 ሶ𝑥 + 𝑥 = 0

• Assume, a disk type limit cycle 𝑥 𝑡 = 𝐴𝑠𝑖𝑛(𝜔𝑡)

– With amplitude 𝐴 and frequency 𝜔.

Separate the linear and nonlinear elements, and Integrate it into

feedback system (5.2)

ሷ𝑥 − 𝛼 ሶ𝑥 + 𝑥 = 𝛼(−𝑥2 ሶ𝑥)

– Consider the nonlinear term 𝑤 𝑥 = −𝑥2 ሶ𝑥 in the feedback.

𝐺(𝑠)𝑤(⋅)𝑥(𝑡)𝑤(𝑡)−𝑥(𝑡)𝑟 = 0

−+

Nonlinearity Linear SystemFeedback System 5.2



Describing Function Method24

Illustrating Example: Van der Pol (Cont.)

• Then the linear system is

𝐺 𝑠 =𝑥

𝑢=

𝛼

𝑠2−𝛼 𝑠+1

• Approximate the nonlinear element

– For a limit cycle 𝑥 𝑡 ≈ 𝐴 sin(𝜔𝑡) ⇒ ሶ𝑥 = 𝐴𝜔 cos 𝜔𝑡

𝑤 = −𝑥2 ሶ𝑥 = −𝐴2 sin2(𝜔𝑡) 𝐴𝜔 cos 𝜔𝑡

=−1

2𝐴3𝜔 1 − cos 2𝜔𝑡 cos 𝜔𝑡 =

−1

4𝐴3𝜔 cos(𝜔𝑡) − cos(3𝜔𝑡)

• Approximate the output

𝑤 ≈−𝐴3

4𝜔 cos 𝜔𝑡 =

𝐴2

4

𝑑

𝑑𝑡−𝐴 sin 𝜔𝑡 =

𝐴2

4(−𝑠𝑥)

• Input/Output Quasi-Linear approximation

𝑤

𝑥≈𝐴2

4−𝑠

– Amplitude dependent




Quasi-linear block diagram

• The frequency response estimate:

𝑤 = 𝑁 𝐴,𝜔 −𝑥 ⇒ 𝑁 𝐴,𝜔 =𝐴2

4𝑗𝜔

• Existence of limit cycle: ∃ 𝑥 𝑡 ≈ 𝐴 sin 𝜔𝑡 , ∋

𝑥 𝑡 ≈ 𝐴 sin 𝜔𝑡 = 𝐺 𝑗𝜔 𝑁(𝐴, 𝜔)(−𝑥) ⇒ 1 + 𝐺 𝑗𝜔 𝑁 𝐴,𝜔 𝑥

– For a nontrivial limit cycle: 𝑥 𝑡 ≠ 0 → 1 + 𝐺 𝑗𝜔 𝑁 𝐴,𝜔 = 0

– Compare Nyquist criteria: 1 + 𝐺 𝑗𝜔 ⋅ 𝑘 = 0

𝛼

𝑠2 − 𝛼𝑠 + 1

𝑥(𝑡)𝑤(𝑡)−𝑥(𝑡)𝑟 = 0

−+

𝐴2

4𝑝

Quasi-linear Approximation 𝐺(𝑠)





• Graphical Solution

– Intersect 𝐺(𝑗𝜔) and −1/𝑁(𝐴,𝜔)

• Analytical Solution (for 𝑝 = 𝑗𝜔)

2( )

1G j

j

2 2

1 4 4

( , )

j

N A A j A

2

4, 1 2

jj A

A

𝜔 = 1

𝐺(𝑗𝜔)

−1/𝑁(𝐴,𝜔)

1,2p

2

2 2 2

4 0 2

11 1/ 64 ( 4)

A A

A



Describing Function Method


• Graphical Solution

The approx. limit cycle

parameters does not

depend on 𝛼

𝜔 = 1

−1/𝑁(𝐴, 𝜔)

𝐺(𝑗𝜔), 𝛼 = 2

𝐺(𝑗𝜔), 𝛼 = 1

𝐺(𝑗𝜔), 𝛼 = 4



Describing Function Method


• The approximate L.C. parameters does not depend on 𝛼

• The REAL L.C. parameters depends on 𝛼




• Assumptions and Definitions

Systems Under D.F. Analysis

• Separation of linear and nonlinear elements.

– Linear + one nonlinear block

– Genuinely nonlinear but separable

• only one time invariant nonlinearity

• The 1st harmonics of nonlinearity pass through the linear part

(𝐺(𝑠) is low pass).

𝐺(𝑠)𝑤(⋅)𝑥(𝑡)𝑤(𝑡)−𝑥(𝑡)𝑟 = 0

−+

Nonlinearity Linear SystemFeedback System 5.2





Nonlinear function

• For a Limit cycle the output is periodic.

• Use Fourier Series to represent the output.

• For odd nonlinearities 𝑎0 = 0.

• By 3rd assumption consider just the first harmonics.





Nonlinear function (cont.)

– In which

– In complex representation

• Describing Function

– Extension of input/output transfer function:




• Computing Describing Functions

Common nonlinearities

• Hardening Spring

– Fourier series for input: :

– Nonlinearity is odd: 𝑎0 = 𝑎1 = 0, and

– Therefore,

– Describing Function is:

– Note: D.F. is real and does not depend on 𝜔






• Saturation:

– Consider the input:

– Saturated output:

– In which,







• Saturation:

– Describing Function:

– D.F. Features:

– Graphical Representation






• Relay: Special case of saturation

– Direct determination:







• Piecewise Linear

– Describing Function

1( , )b

N aa

( )N a






• Piecewise Linear

– Describing Function: Graphical Representation

• Saturation

– Special case of piecewise linear with






• Dead-Zone

– Nonlinear function

– In which,







• Dead-Zone

– Describing Function

– Real, A dependent

– Not depending on 𝜔

– No phase shift

– Graphical

Representation






• Backlash





• Backlash

Nonlinear Function:

In which





• Backlash


– Therefore,

– D.F. Graphical Representation



Contents


43



Describing Function AnalysisReview of Nyquist criterion, Existence and stability of limit cycles, examples.3




• Review of Nyquist Criterion

Feedback system

• Loop gain 𝐿(𝑠) = 𝐶(𝑠)𝑃(𝑠)

– Characteristic Equation 1 + 𝐿(𝑠) = 0 or 𝐿(𝑠) = −1

– Draw the Nyquist plot of 𝐿(𝑗𝜔) for − < 𝜔 < +

– Count the number of c.w. encirclement of the Nyquist plot around point (−1,0). 𝑁

– Compute the number of unstable poles of the loop gain 𝐿(𝑠). 𝑃

– The number of unstable roots of the characteristic equation (or the number of unstable poles of the closed-loop transfer function) is denoted by 𝑍 and is found from: 𝑍 = 𝑁 + 𝑃

𝑃(𝑠)𝐶(𝑠)+ -



Describing Function Analysis45

• Review of Nyquist Criterion

Feedback system

• Loop gain 𝐿(𝑠) = 𝐶(𝑠)𝑃(𝑠)

– Characteristic Equation 1 + 𝑘 𝐿(𝑠) = 0 or 𝐿(𝑠) = −1 / 𝑘

– Draw the Nyquist plot of 𝐿(𝑗𝜔) for − < 𝜔 < +

– Count the number of c.w. encirclement of the Nyquist plot around

point (−1 / 𝑘 , 0). 𝑁

– Compute the number of unstable poles of the loop gain 𝐿(𝑠). 𝑃

– The number of unstable roots of the characteristic equation (or the number of unstable poles of the closed-loop transfer function) is denoted by 𝑍 and is found from: 𝑍 = 𝑁 + 𝑃

𝐿(𝑠)𝑘+

-




• Existence of Limit Cycle

Feedback System𝑥 = −𝑦

𝑤 = 𝑁 𝐴,𝜔 𝑥𝑦 = 𝐺 𝑗𝜔 𝑤

⇒ 𝑦 = 𝐺 𝑗𝜔 𝑁 𝐴,𝜔 (−𝑦)

• Characteristics Equation: (for 𝑦 0 )

𝐺 𝑗𝜔 𝑁 𝐴,𝜔 + 1 = 0 OR 𝐺 𝑗𝜔 = −1

𝑁 𝐴,𝜔

• Solve it Graphically

• Amplitude dependent describing function

– Intersect 𝐺(𝑗𝜔) and −1/𝑁(𝐴)

– If intersection occurs L.C. exists

– Amplitude of L.C. 𝐴 (on −1/𝑁(𝐴))

– Frequency of L.C. 𝜔 (on 𝐺(𝑗𝜔))




• Existence of Limit Cycle

Graphical Solution

• General Describing Function (𝑁(𝐴,𝜔))

– Plot 𝐺(𝑗𝜔)

– Plot a set of −1/𝑁(𝐴,𝜔) for different constant frequencies.

– Analyze the intersection points

– If an intersection point in which

The frequencies match L.C. exists

– Amplitude of L.C. 𝐴 (on -1/𝑁(𝐴,𝜔))

– Frequency of L.C. the matched 𝜔

(on 𝐺(𝑗𝜔) and on -1/𝑁(𝐴,𝜔))




• Stability of Limit Cycle

Feedback System

• Characteristics Equation: (for 𝑦 0 )

𝐺 𝑗𝜔 𝑁 𝐴,𝜔 + 1 = 0 OR 𝐺 𝑗𝜔 = −1

𝑁 𝐴,𝜔

Compare it with Nyquist criteria:

𝐿 𝑠 𝑘 + 1 = 0 or 𝐿(𝑠) = −1 / 𝑘

• Extended Nyquist Criteria

– Replace the point −1/𝑘 by −1/𝑁(𝐴,𝜔) and count encirclements

– Apply the same Nyquist criteria to determine stability

– Note: stability decay of amplitude

– Instability increase of amplitude



Unstable Region



Example:

• Consider 𝐺(𝑠) is stable 𝑃 = 0

• Suppose two L.C. 𝐿1, 𝐿2• Stability analysis of 𝐿1

– Perturb 𝐿1by increasing its amplitude

• 𝐿1′ is inside 𝐺(𝑗𝜔) (unstable)

• The amplitude increases

• 𝐿1′ moves away from 𝐿1 Unstable Limit Cycle

– Perturb 𝐿1by decreasing its amplitude

• 𝐿1" is outside 𝐺(𝑗𝜔) (stable)

• The amplitude decreases

• 𝐿1" moves away from 𝐿1 Unstable Limit Cycle




'

2L

"

2L


Example: (cont.)

• Stability analysis of 𝐿2

– Perturb 𝐿2 by increasing its amplitude

• 𝐿2′ is outside G(jω) (stable)

• The amplitude decreases

• 𝐿2′ moves toward 𝐿2 Stable Limit Cycle

– Perturb 𝐿2 by decreasing its amplitude

• 𝐿2′′ is inside G(jω) (unstable)

• The amplitude increases

• 𝐿2′′ moves toward 𝐿2 Stable Limit Cycle

Unstable Region



• Example 1:

Consider 𝐺 𝑠 =1

𝑠(𝑠+1)(𝑠+2)with unit

• Nyquist Plot

𝐺 𝑗𝜔 = −3𝜔 − 𝑗(2 − 𝜔2)

9𝜔3 + 𝜔 2 − 𝜔2 2

– @ point A

ℑ 𝐺 𝑗𝜔 = 0 → 2 − 𝜔2 = 0 → 𝜔 = 2

𝐴 = 𝐺 𝑗 2 = −3 2

18 2 + 0= −

1

6

(a) For unit saturation (𝑘 = 1)

– 𝑁(𝐴) < 1 No Crossing, No Limit Cycle


a) Saturation

b) Relay

−1/𝑁(𝐴)

𝐺(𝑗𝜔)



• Example 1: (Cont.)

(b) For unit Relay (M=1)

– − < −1/𝑁(𝐴) < 0 Crossing @ point A

– Amplitude:−1

𝑁 𝐴= −

1

6→

𝑁 𝐴 =4

𝜋𝐴= 6 → 𝐴 =

2

3𝜋

– Frequency: 𝜔 = 2 and Limit Cycle is stable.


−1/𝑁(𝐴)

𝐺(𝑗𝜔)

Unstable Region

𝐴

𝐿′





Simulation verifications:

– Frequency :

– 2𝑇 = 10 → 𝑇 = 5 and 𝜔 = 2𝜋/5 = 1.26 (estimated by 1.414)

– Amplitude 𝐴 = 2/3𝜋 = 0.21 (red circle)

2𝑇 = 10 sec




• Example 2:

Consider 𝐺 𝑠 =−𝑠

𝑠2+0.8𝑠+8with

• Nyquist Plot

– @ point A

(a) For unit saturation (𝑘 = 1)

– −1

𝑁 𝐴= −1.25 → 𝑁 𝐴 = 0.8

– 𝐴 1.455 @ 𝜔 = 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

−1/𝑁(𝐴)

𝐺(𝑗𝜔)

−1 Unstable Region




• Example 2:

Simulation verifications:

• Frequency :

– 7𝑇 = 16 𝑇 = 2.285 and 𝜔 = 2𝜋/𝑇 = 2.75 (estimated by 2.83)

– Amplitude 𝐴 = 1.45 (red ellipse)

7𝑇 = 16sec



56

Describing Function Analysis


• (b) For Dead-zone 𝛿 = 1, 𝑘 = 0.5

• 0 < 𝑁 𝐴 < 0.5 𝑁 𝐴 0.8

• No Crossing No Limit Cycle

−2−1/𝑁(𝐴)

𝐺(𝑗𝜔)



57


• Example 3:

Consider Raleigh’s Equation ሷ𝑥 + 𝑥 = 𝜖( ሶ𝑥 −1

3ሶ𝑥3)

• Separation ሷ𝑥 − 𝜖 ሶ𝑥 + 𝑥 = −𝜖(1

3ሶ𝑥3)

– Consider 𝑦 = ሶ𝑥 and 𝜙 𝑦 =1

3𝑦3

– 𝑠2 − 𝜖𝑠 + 1 𝑥 = −𝜖𝜙 𝑦 = −𝜖𝜙 𝑠𝑥 = −𝜖𝑠𝜙 𝑥

– Hence,

• Linear element: 𝐺 𝑠 =𝑥

𝜙=

𝜖𝑠

𝑠2−𝜖𝑠+1

• Nonlinear function: 𝜙 𝑦 =1

3𝑦3

𝐺(𝑠)𝜙(⋅)𝑥(𝑡)𝜙(𝑡)−𝑥(𝑡)𝑟 = 0

−+

Nonlinearity Linear System



58


• Example 3:

Raleigh’s Equation (Cont.)

• Describing Function 𝜙 𝑦 =1

3𝑦3

• 𝑁 𝐴 =𝑏1

𝐴=

2

3𝜋A0𝜋𝐴 sin 𝜃 3 sin 𝜃 𝑑𝜃 =

1

4𝐴2

• Nyquist Plot

𝐺 𝑠 =𝜖𝑠

𝑠2 − 𝜖𝑠 + 1→ 𝐺 𝑗𝜔 =

𝜖𝑗𝜔 1 − 𝜔2 + 𝜖𝑗𝜔

1 − 𝜔2 2 + 𝜖2𝜔2

– @ point A

ℑ 𝐺 𝑗𝜔 = 0 → 1 − 𝜔2 = 0 → 𝜔 = 1

– 𝐴 = 𝐺 𝑗1 = −𝜖2

𝜖2= −1

– Regardless of 𝜖

𝐴



59


• Example 3:

Raleigh’s Equation (Cont.)

• Intersect −1 / 𝑁(𝐴) with −1

• 𝑁 𝐴 =1

4𝐴2 = 1 → 𝐴 = 2

• A Limit cycle with

– Amplitude = 2

– Frequency = 1

• Stability analysis

– G(s) is unstable 𝑃 = 2

– For stable region 𝑁 = −2

– direction of −1/𝑁(𝑎)

• 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)

−1/𝑁(𝐴)

−1

N=-2: Stable Region



60


• Example 3: Raleigh’s Equation (Cont.)

Simulation verifications:• Frequency :

– 2𝑇 = 13.34 𝜔 = 2𝜋/6.67 = 0.95 (estimated by 1)

– Amplitude 𝐴 = 2/3𝜋 = 0.21 (red circle)

𝟐𝑻 = 𝟏𝟑. 𝟑𝟒 𝒔𝒆𝒄



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

http://aras.kntu.ac.ir/courses/nonlinear-control/



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

http://www.sharif.ir/

http://www.mcgill.ca/

http://en.kntu.ac.ir/international

http://aras.kntu.ac.ir/

https://en.kntu.ac.ir/faculties/electrical-engineering/

http://en.kntu.ac.ir/

http://ijr.kntu.ac.ir/

https://us.sagepub.com/en-us/nam/international-journal-of-advanced-robotic-systems/journal202567

https://aras.kntu.ac.ir/publications/

chapter 5: frequency domain analysis of feedback systems

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