Mario INNOCENTI, email: mario.innocenti@unipi.it

Dipartimento di Ingegneria dell’InformazioneTel: 0502217319,

RICEVIMENTO: Mercoledì 15:30-18:30

Controllo dei Sistemi Incerti Modulo II del corso Identificazione e Controllo dei Sistemi Incerti (263 II, 12 CFU)

Laurea Magistrale in Ingegneria Robotica e dell’AutomazioneUniversità di Pisa

Chapter 1: Introduction, Motivation, Background

Chapter 1: Introduction, Motivation, Background

• Mammarella (PhD, WVU)‐ GMV Space Systems, Madrid Spain

• Bracci (PhD, UNIPI) ‐ FlyBy Livorno Italy

• Munafò (PhD, UNIPI)‐ CMRE‐NATO, La Spezia, Italy

• Tucci (PhD, UNIPI) – Prof. Associato, DESTEC UNIPI

• Crisostomi (PhD, UNIPI) – Prof. Associato, DESTEC UNIPI

• Grioli (PhD, UNIPI) – Qrobotics, Pisa, Italy

• Petrone – Fabio Perini SpA, Lucca Italy

• Raciti – GNC, Alenia – Aermacchi, Varese, Italy

• Mazzi – PurePowerControl –Navacchio, Italy• Tripicchio – CGS Advanced Robotics 

Research Center, SSSUP La Spezia, Italy

• Biondi – Giorgio Gori srl, Livorno Italy

• Schiavi (PhD, UNIPI) – Evidence srl, Pisa Italy

• Cartocci  – Calzaturificio Fucecchio, Italy

• Agostini  – McLaren Corse, Guilford UK

• Medaglia  – Senior BI Finance Analyst• Cambridge U. Press, UK

• Santerini  – Institute BioRobotics, SSSUP Pontedera Italy

• Soleri  – Oto Melara Defence, La Spezia Italy

• Morellato  – Morellato Energia SAS, Pisa Italy

• DeCaria – Globant, Montevideo, Uruguay, Los Angeles USA

• Joalè  – Algorab, Trento Italy

• Greco  – Ingegnere, Lari Italy

Anno Accademico 1999 - 2000

Chapter 1: Introduction, Motivation, Background

Topics• Introduction, review of basic material• MIMO tools• Frequency shaping techniques• Robustness characterization and analysis• Synthesis of robust controllers

References• Robust Control Design with MATLAB, 2nd edition, Gu, Petkov, 

Kostantinov, Springer, 2013 (including downloadable Matlab software).• Multivariable Feedback Control analysis and design, 2nd edition, 

Skogestad, Postlethwaite, Wiley 2005.• Access to Matlab/Simulink• Lecture Notes, additional material online

Oral Exam• Material covered in class• Final Project (with Matlab)

Chapter 1: Introduction

Acknowledgments:Some of the material presented in these lecture notes has been made available by the individual authors, and official open source organizations. The use is permitted for teaching purposes  with no copyright infringement

• MIT OpenCourseWare program (Drs. Frazzoli, Metrevski, How)• Stanford Engineering Everywhere (Drs. Boyd, Lall, Tomlin, Bryson)• Princeton University (Dr. Stengel)• National University of Singapore (Dr. B. Chen)• Norwegian Institute of Technology (Dr. Skogestad)• CalTech (Drs. Doyle, Murray, Astrom, Packard)• Delft University of Technology (Drs. Kwakernaak, Scherer)• The Boeing Company Phantom Works (Dr. Lavretski)• University of Washington in St. Louis (Dr. Wise)• Utah State University (Dr. C. Q. Chen)• Washington State University (Dr. Saberi)• University of Minnesota (Dr. Balas)• Air Force Research Laboratory (Drs. Banda, Ridgley)• Honeywell Research Laboratory (Dr. Stein)

Chapter 1: Introduction

The course has some Math …..

Chapter 1: Introduction

Graphical History of Control Systems

Chapter 1: Introduction

W.R. Evans

M. Khomyakov

E.A. Sperry

Chapter 1: Introduction

Advanced Fighter Technology Integration AFTI – F16

The Advanced Fighter Technology Integration (AFTI) F-16 phase I tests began following its arrival at Dryden on July 15, 1982. The initial flights checked out the airplane's stability and control systems. These included a triplex digital flight control computer system, and the two triangular "chin" canards mounted under the aircraft's intake, which form an inverted "V"-shape. These canards allow the AFTI F-16 to make flat turns. By late December 1982, tests began of the Voice Command System. This allowed the pilot to change switch positions, display formats, and modes simply by saying the correct word. The initial tests were of the system's ability to recognize words. Later tests were made under increasing levels of noise, vibrations, and G-forces. These showed a 90 percent success rate. Later tests were also made of a helmet-mounted sight. The AFTI F-16's 100th flight was made on July 15, 1983. The phase I tests concluded soon after, and on July 30, 1983, the aircraft left Dryden and was flown back to the General Dynamics facility at Fort Worth, Texas. In all, 118 flights had been made, totaling 177.3 hours of flight time.

Chapter 1: Introduction

Standard Maneuver ()1. Command or ~ )2. Change 3. Change 4. Change Altitude

Chapter 1: Introduction

. . .

1

2

5

4

3

6

7

To ActuatorsCONTROLLER

From Reference and Commands

From Sensors

1, 2

3, 4, 56, 7

Chapter 1: Introduction

Chapter 1: Introduction

R. Bellmann

Angelo Miele

R. Kalman

M. Lyapunov

Chapter 1: Introduction

MIMO Frequency Domain solution strategies

Chapter 1: Introduction

L.A. Zadeh A. Isidori K. Astrom

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

A further step ahead: Computation Intelligence / Self Consciousness ?

In computer science, artificial intelligence (AI), sometimes called machine intelligence, is intelligence demonstrated by machines, in contrast to the natural intelligence displayed by humans and animals. Computer science defines AI research as the study of "intelligent agents": any device that perceives its environment and takes actions that maximize its chance of successfully achieving its goals. Colloquially, the term "artificial intelligence" is used to describe machines that mimic "cognitive" functions that humans associate with other human minds, such as "learning" and "problem solving"

Chapter 1: Introduction

Reinforcement learning (RL) is an area of machine learning concerned with how software agents ought to take actions in an environment so as to maximize some notion of cumulative reward. Reinforcement learning is considered as one of three machine learning paradigms, alongside supervised learning and unsupervised learning.

Chapter 1: Introduction

Data mining is the process of discovering patterns in large data sets involving methods at the intersection of machine learning, statistics, and database systems. Data mining is an interdisciplinary subfield of computer science and statistics with an overall goal to extract information (with intelligent methods) from a data set and transform the information into a comprehensible structure for further use.

Chapter 1: Motivation & Limits

Model – based Control System Design

Chapter 1: Motivation & Limits

Chapter 1: Motivation & Limits

Geology

Automation

Aerospace

Robotics

Mechanical

Fun

Science

Civil

Height: 509 m

Chapter 1: Motivation & Limits

Chapter 1: Motivation & Limits

Multivariable System

1. How many inputs?2. How many measurements?3. How many state variables?4. How many states (regulated outputs) we 

want to control5. How many states we need to identify?6. How many applications?

Replace Human work force with automated transport1. Workers too expensive2. Workers want to be carried ( we are all 

generals, no soldiers..)3. Workers can’t be found (too much effort)4. People in Robotics need a job too..5. ….

Chapter 1: Motivation & Limits

Summary

We will consider dynamic systems represented by models that can be used for control synthesis

We will consider dynamic systems represented by models simple enough that can be used for control synthesis

The system(s) is (are) described by linear ordinary differential equations, yielding the so-called linear time invariant representation

The system(s) is (are) a continuous function of time, yielding a continuous controller, which may be discretized at a later stage

The models are affected by errors producing uncertainties, which will be accommodated by the controller synthesis

The system(s) is (are) multi input, multi output so that not all classical tools can be readily applied (especially the graphical ones)

Chapter 1: Motivation & Limits

G(s)dje t-

Loop Delay G(s)

Nonlinearities

Model‐based Control Analysis and Synthesis for stability and performance in the presence of a ‘set’ of Uncertainties and Errors

Uncertaintyas Frequency DependentFunction

G(s)F(s)?

MODEL

REAL SYSTEM

s+2(s+1+a)(s+2+b)

UnknownParameters

Chapter 1: Motivation & Limits

Robust Control Techniques

Model Predictive Control

MPC is based on iterative, finite‐horizon optimization of a plant model. At time t the current plant state is sampled and a cost minimizing control strategy is computed for a relatively short time horizon in the future: [t,t+T]. The process is repeated with new data.

Variable Structure Control

Variable structure control is a form of discontinuous nonlinear control. The method alters the dynamics of a nonlinear system by application of a high‐frequency switching control. The state‐feedback control law is not a continuous function of time; it switches from one smooth condition to another. It has low sensitivity to matched uncertainties

Adaptive Control

Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. Adaptive control is different from robust control in that it does not need a priori information about the bounds on these uncertain or time‐varying. Adaptive control is concerned with control law changing itself.

Norm based Optimal Control

Robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some set. Robust methods aim to achieve robust performance and/or stability in the presence of bounded modelling errors. Robust control systems may be represented by high order transfer functions required to simultaneously accomplish desired disturbance rejection performance with robust closed loop operation.

Chapter 1: Mathematical Background

These are some background topics to be reviewed

2. Time domain representation of linear systems• State space representation, solution, • Relationship with frequency response

3. Elements of System theory• Matrix theory• Structural properties

4. Linear Optimal Control• Linear Quadratic Regulator (LQR)• State Estimation• Frequency Response properties of LQR• Optimal State Estimation (KBF)

1. SISO Design• Frequency Shaping • Performance limitations

Chapter 1: Background

General Feedback Loop

Plant, Model, System, Process,…Controller, Regulator, Compensator,…

Feedback Loop

Chapter 1: Background

• Controller: provides the decision strategy (algorithmic, analytical)• Sensors: provide the information on the “state” of the system, convert units to be 

understood by the controller, reduce power required. • Actuators: implement the controller decision strategy, convert units from 

controller to the plant, provides power required.

Chapter 1: Background

Basic Single Input Single Output (SISO) Control

The majority of control problems and tools refer to linear time invariant systems (LTI) described by linear ordinary differential equations with constant coefficients (LODE)

The most common domain is frequency (for time varying signals)

( )( )

( )Y s

G sU s

= ( ) ( ) ( )Y s G s U s= ⋅

1

0

( ) ( ) ( ) ( ) [ ( ) ( )]t

y t g t u d y t G s U st t t -= - =ò L

Chapter 1: Background

Standard Performance Characterization (Nominal)

( )Ks ( )Gs+

-

++

+

+

+

+

r(s) (s)

u(s)

y(s)

n(s)

di(s) do(s)

m(s)

• Typical single loop unity feedback configuration

• Typical single loop disturbance accomodation (feedforward and feedback)

Chapter 1: Background

• Control objective: output must follow tthe input in the presence of disturbance• System tranfer functions: Go1(s) and Go2(s)• Controller transfer functions: Gf(s) and C(s)

{ }2 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

o g o f gY s G s D s G s G s D s C s R s Y sé ù= + + -ê úë û

2 1 2 1

2 1 2 1

(1 )( ) ( ) ( )

1 1o o f o o

go o o o

G G G G G CY s D s R s

G G C G G C

+= +

+ +

What happens if we choose

1

1( )

( )fo

G sG s

= -

Chapter 1: Background

• Examples of multiloop feedback configurations 

• Inner loop to improve stability (higher bandwidth), outer loop for performance (lower bandwidth)

Example: design a controller to have zero steady state error to a unit step for the unstable system:

1( )

( 1)G s

s s=

-

Chapter 1: Background

• Type 1 loop transfer function• Closed loop system is always unstable for any proportional feedback.

2( )

kT s

s s k=

- +

• The controller must have a dominant zero (PD)

1 2 1 22

( ) ( ) ( 1)s

K s k s k k kk

= + = +

Chapter 1: Background

• The gain k2 selects the zero position (k2 > 0) for example k2 = 2.• The gain  k1 > kcr > 1 defines the loop gain in order to maintain closed loop 

stability

k1 = 9 2

9( 2)( )

8 18

sT s

s s

+=

+ +

Chapter 1: Background

• The controller is made causal with a pole outside the bandwisth (one order of magnitude) 

1 2( ) ( ) 9( 2)K s k s k s= + = +

Z=N+P=-1+1=0

1 2( ) 9( 2)

( ) 10( )

(1 )50

k s k sK s k

s P s

+ += =

++

Z=N+P=-1+1=0

• Several choices for the pole location and loop gain: for example, maintain rise time and overshoot the same as those of pure lead (PD).  

Chapter 1: Background

Chapter 1: Background

1 2

1 1 1( ) ( ) ( )

( 1) ( 1)G s G s G s

s s s s= = = ⋅

- -

1( 1)s -

1s

( )u s ( )y s ( )y s

Available for feedbackAvailable for feedback

1. Design a controller to stabilize subsystem G1(s)

1( 1)s -

2k

( )y s±

2

1( )

( 1)T s

s k=

+ -

2 2 21; 2 1

CRk k k P> = = =-

Chapter 1: Background

Design a unity feedback outer loop controller to stabilize the complete system ( ) ( )2T s G s

1s2

1( 1)s k+ -1

k±2

2k =

1

2

( ) ( )[ ( 1)]

kG s K s

s s k=

+ -1

22 1

( )( 1)

kT s

s k s k=

+ - +

1 2

1,2

1; 2

0.5 0.866

k k

p j

= ==-

Chapter 1: Background

Trade off

Single Loop

1. Simpler Feedback structure2. Controller failure critical for 

stability3. 1 Sensor4. 1 Actuator5. Controller Structure 

complex (dynamic system)

Dual Loop

1. Feedback structure more complex

2. Only outer loop failure critical for stability

3. 2 Sensors4. 1 Actuator5. Controller Structure 

simpler (2 static gains)

Chapter 1: Background

Loop Shaping Approach (Frequency Response Design)

In the following, we refer (unless otherwise stated) to linear systems whose looptransfer function satisfies Bode Criteria for minimum phase systems. This allows the use of the magnitude frequency response (the phase is automatically known) rather than using the full application of Nyquist criterion. 

i) Nominal closed loop stabilitypole location

ii) Transient and steady state responseSteady state error to canonical signals, tracking of a time varying signal within a frequency range (rise time, settling time, overshoot, bandwidth, crossover frequency, dominant poles, high gain feedback,...)

iii) Disturbance and noise rejection/attenuationAmplitude attenuation at given frequencies, low high frequency loop gain

iv) Control energy limitations Amplitude attenuation due to saturation limits  

v) Points i – iv in the presence of ‘’specific’’ errors in the loopstability margins,...

Standard Performance Parameters and Frequency Shaping Design

Chapter 1: Background

• Transfer functions of Interest

( ) ( ) ( )

( )( )

1 ( )

1( )

1 ( )

( ) ( ) ( )u

L s G s K s

L sT s

L s

S sL s

S s K s S s

=

=+

=+

=

• Loop Transfer Function

• Complementary Sensitivity

• Input Sensitivity

• Output sensitivity

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

o i

o i

o i

y s T s r s n s S s d s G s S s d s

s S s r s n s S s d s G s S s d s

u s K s S s r s n s d s T s d s

e

é ù= - + +ê úë ûé ù= - - -ê úë û

é ù= - - -ê úë û

• Note:

( ) ( ) 1S j T jw w+ =

Chapter 1: Background

• Use the superposition principle

( ) 1, 0,r r

S jw e w wé ù£ < Î ê úë û( ) 1

( )( ) 1 ( ) ( )s

S sr s G s K se

= =+

• Tracking a command r(t) within a frequency range

11 ( ) ( ) rG j K j

ew w

£+

11 ( ) ( ) 1

r

G j K jw we

+ ³

1 ( ) ( ) ( ) ( )G j K j G j K jw w w w+ @

1( ) ( ) 1

rr

G j K jw w be

³ =»

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

o i

o i

o i

y s T s r s n s S s d s G s S s d s

s S s r s n s S s d s G s S s d s

u s K s S s r s n s d s T s d s

e

é ù= - + +ê úë ûé ù= - - -ê úë û

é ù= - - -ê úë û

Chapter 1: Background

• Use the same idea for the other requirements

( ) 1, 0,d d

S j w e w wé ù£ < Î ê úë û1

( ) ( ) 1d

d

G j K jw w be

³ =

• Disturbance rejection 

1 2( ) 1, ,n n nT jw e w w wé ù£ < Î ë û ( ) ( ) ( ) 1n

T j G j K jw w w e@ £

• Noise attenuation and stability at high frequency

• Actuator Constraints

( ) ( ) 1G j K jw w ,p qw w wé ùÎ ë û

( ) 1 1 1( ) 1 1

( ) ( ) ( )p

u jT j

r j G j G jw

ww w w e

= » ⋅ =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

o i

o i

o i

y s T s r s n s S s d s G s S s d s

s S s r s n s S s d s G s S s d s

u s K s S s r s n s d s T s d s

e

é ù= - + +ê úë ûé ù= - - -ê úë û

é ù= - - -ê úë û

Chapter 1: Background

𝑮 𝒋𝝎

Chapter 1: Background

Chapter 1: Background

Chapter 1: Background

1( )

1G s

s=

+ Example

Design a controller K(s) that maintains closed loop stability  G(s) and satisfies the following requirements

i. Zero Static position error

ii. ‐40 dB disturbance rejection in the frequency range: 10, 0,0.01d

radsw -é ù é ù=ê ú ê úë û ë û

iii. ‐100 dB error to a command in the frequency range:

3 10, 0, 10r

radsw - -é ùé ù =ê ú ê úë û ë û

iv. ‐40 dB noise attenuation in the frequency range:

v. Stability Margins:  045 , 15PM GM dB³ ³

1 2 3 12

, 10 , 10n n

radsw w -é ù é ù=ê ú ê úë û ë û

Chapter 1: Background

• The constraint on the steady state error requires integral control

1( ) ( ) ; 0

1k

G s K s ks s

= >+

( ) ( ) ; 100G j K j kw w =

• The Phase Margin constraint is not satisfied

Chapter 1: Background

• Select a lag compensator ( ) , 0p s z

K s z pz s p

+= > >

+ 3 110 ; 10p z- -= =

Chapter 1: Background

Chapter 1: Background

Performance Limitations

Physical properties of a system impose performance limitations on the controller structure (loop delay, pole – zero location,…)

Chapter 1: Background

Consider a point mass system (double integrator)

v = x2a = u r = x1

-y

• Input: Force/Acceleration• Output: Linear combination of 

variables. Position can be computed from available data

2

1( )

sG s

s

-=

• In the first case, a standard lead compensator can be designed easily with arbitrary high loop gain

• In the second case, a stabilizing controller may be more difficult to find with standard classical control tools, due to the loss of phase introduced by the unstable zero. In any case, there are intrinsic limitations on the loop gain.

• Input: Force/Acceleration• Output: Position 2

1( )G s

s=

= y

Chapter 1: Background

• Check how large the loop gain can be made in both cases 

Chapter 1: Background

s=tf('s');g1=1/(s^2);g2=(s‐1)/(s^2);controlSystemDesigner(g1)controlSystemDesigner(g2) k=‐0.043503*(1+10*s)/(1+0.05*s)

Chapter 1: Background

• Algebraic Limitations: Performance requirements do not satisfy loop shaping (for instance tracking of a high frequency signal and simultaneous sensitivity to a low frequency noise)

( )( ) 0.01

( )j

S jr je w

ww

= £ ( )( ) 0.05

( )y j

T jn j

ww

w= £

1 ( ) ( )( ) ( ) 1

1 ( ) ( ) 1 ( ) ( )G s K s

S s T sG s K s G s K s

+ = + =+ +

( ) 1 ( )S s T s= - ( ) 1 ( )S j T jw w³ -

( ) 0.05T jw £ ( ) 1 0.05 0.95 0.01S jw ³ - =

Chapter 1: Background

• Analytical Limitations: Stability related to the pole – zero location and Bode’s criteria (waterbed effects)

Chapter 1: Background

• Consider a stable function with stable inverse (j), selected by design, such that it constitues a sensitivity upper bound at all frequencies

• The control design is to find a stabilizing controller K(s), such that:

( ) ( )S j jw w£ L 1( ) ( )j W jw w-L =1( ) ( ) 1S j jw w-L £

( ) ( ) 1;S s W s s C+

£ Î

Note 1: by construction of , W(s) is analytic on the RHP and so is S(s) if K(s) is a stabilizing controllerNote 2: this implies that S(s) has no singularities in the RHP and it is limited in magnitude

Chapter 1: Background

( )z se • Fictitious Output( )

( ) ( ) ( ) ( )( )

( )1 ( )

W sz s S

K ss W s s s

G se e e= =+

Control design requirements from loop shaping:

• The closed loop system must be asymptotically stable • The fictitious output must follow the loop shaping 

requirements• Remember that S(s)W(s) = W(s)S(s) for SISO systems

Chapter 1: Background

Difficulty in satisfying the requirements comes from unstable/non minimum phase loop transfer function due to waterbed effects and sensitivity limitations. 

• Theorem #1: Given a system with minimum phase loop transfer function, then:

0 0

0

0

1ln ( ) ln lim ( )

1 ( ) 21

ln Re( ) lim ( )1 ( ) 2

s

ks

S j d d sL sL j

d p sL sL j

pw w w

wp

w pw

¥ ¥

¥

¥

¥

ìïïïïïïïïï= = -íï +ïïïïï = -ï +ïïî

ò ò

åò

Stable, relative degree 2 or higher 

Stable

Unstable (pk unstable poles)

• This means that if the sensitivity is to be reduced in a certain frequency range, then it must be increased in another frequency range. This is referred to as the waterbed effect.

Chapter 1: Background

• Theorem #2: Given a system with non minimum phase loop transfer function, then (Poisson integral theorem):

2 2 2 2

*

01

ln ( ) , 1,..( ) ( )

.k k

k k k

nupk i

zik

uk i

z pS j d Ln for k n

z p

g g

g d w g dw w p

w

¥

=

é ùê +

⋅ = =-

ú+ê ú+ - + -ê úë ûò

k k k

up

uz

z j

n

n

g dìï = +ïïï =íïï =ïïî

Unstable zero (obviously with its conjugate)Number of unstable poles

Number of unstable zeros

Notes:• For stable systems the RHS is equal to zero (balance in the sensitivity)• The part in red is positive and the RHS is greater than 1, so the weighted integral of the sensitivity 

will be positive if there are both NMP zeros and RHS poles• The excess positive area may appear in either a large sensitivity peak (decreased stability margin) or 

sensitivity that is greater than unity over a broader frequency range (decreased closed‐loop performance) or both.

• The right‐hand side of the equation becomes very large when there is an unstable pole close to the unstable zero. This condition makes the system very hard to control.

• The closed‐loop bandwidth should not exceed the ' magnitude of the smallest NMP open‐loop zero. Otherwise a very large sensitivity peak will occur, leading to fragile loops (no‐robust) and large undershoots and overshoots.

• The maximum sensitivity peak should be limited in order to reduce disturbance amplification and maintain a satisfactory stability margin (between 1.3 and 2).

Chapter 1: Background

• Example

( ) ( )( 1)( 2)

kG s K s

s s s=

+ +

1 ( 1)( 2)( )

1 ( ) ( ) ( 1)( 2)s s s

S sG s K s s s s k

+ += =

+ + + +

Chapter 1: Background

• Example (Student independent study)

𝐿 𝑗𝜔𝑠 1

𝑠 2 𝑠 10

𝐿 𝑗𝜔𝑠 1

𝑠 2 𝑠 10

𝐿 𝑗𝜔22 𝑠 1

𝑠 2 𝑠 10

Chapter 1: Background

Relative Stability (Stability Margins)

• Note to the reader: traditionally, in classical control stability, definitions are based on  the input – output representation and therefore refer to “external” stability properties (more on that later).

• For now let us assume external stability  internal stability

Asymptotic Stability ‘Marginal Stability’

Instability

Chapter 1: Background

• Relative Stability: it refers to the closed loop stability properties in the presence unmodeled dynamics (dominant poles approximations), or other errors quantifiable using the open loop frequency response.

• Tools:• Nyquist Criterion• Stability margins• Sensitivity peak

No

Chapter 1: Background

• One of the most common consequence of Nyquist Criterion is the definition of Stability Margins

Nyquist Criterion

'( ) ( ) 1 ( ) ( ) ( )OL

F j F j G j K j G jw w w w w= - = =

Given a complex function F’(j) defined as:

When s follows the closed curve N (Nyquist contour) in a clockwise direction for  = [-∞, +∞], F’(j) is mapped into a closed curve N that encircles the critical point(‐1, 0) a number of times N with N = Z – P,  in a clockwise direction. Where:

Z is the number of closed loop poles within N (unstable)P is the number of closed loop poles within N (unstable).

Chapter 1: Background

1. Stability margins provide a necessary and sufficient condition for closed loop stability, if the loop transfer function satisfies the first Bode theorem (stable open loop system and a single crossing of 0 dB)

( )OL

G sr(s) y(s)-

(s)

2. Stability margins relate stability and the influence on it by a bounded uncertainty complex function described by its magnitude and phase, and entering the loop in a multiplicative form

( )OL

G s( )E sr(s) y(s)-

(s)

Chapter 1: Background

( )( ) ( ) ( ); ( ) : ( ) j L jOL OL

G j E j G j E j E j e ww w w w w = =

( ) ( )( ) ( ) ( ) ( ) OLj E j G j

OL OLE j G j E j G j e w ww w w w +=

0

0

1( ) ( ) -180

( )

( ) 180 ( ) at ( ) 1 0

H OL H

OL H

C OL C OL C

E j G jG j

E j G j G j dB

w ww

w w w

ìïï £ =ïïíïï £- - = =ïïî

• Degree of Stability or Robustness Margin m: for a stable open loop system GOL(s), the degree of stability is the minimum vector distance of GOL(s) from the critical point (‐1, 0).

• Special Cases

Chapter 1: Background

1 ( ) 1 ( )m OL

G j L jw wD = + = +

1( )m

S jw-D =

• The higher the sensitivity peak and the smaller is m. This yields a less robust design !

Chapter 1: Background

• Systems with conditional stability

0.1( ) ( )

( 0.5)( 2)( 10)s

K s G s ks s s s

+=

- + +

1 2

10.9 156.88

10.9 156.88CR CR

k

k k k

< <

= < < =

Chapter 1: Background

• The closed loop system is unstable fork < k1cr = 10.9

1 2

1

PZ N P

N

ìï =ï = + =íï =ïî

Chapter 1: Background

• The closed loop system is asymptotically stable for 10.9 = k1cr < k < k2cr = 156.88

10

1

PZ N P

N

ìï =ï = + =íï = -ïî

Chapter 1: Background

• The closed loop is unstable for 156.88 = k2cr < k

1 2

1

PZ N P

N

ìï =ï = + =íï =ïî

Chapter 1: Background

• References:

1. Bode sensitivity handout2. Skogestad – “MULTIVARIABLE FEEDBACK CONTROL Analysis and design”, 

chapters 1, 2, 3, 5 (relevant parts)3. Any text on classical control

Chapter 1: Extras

Analytic functions

( , ) ( , ) ( , )f x y u x y jv x y

z x jy

ìï = +ïíï = +ïî• Given a complex function of a complex variable f(z):

• A necessary condition for f = u + iv to be analytic is that f depends only on z. In terms of the of the real and imaginary parts u, v of f, this implies: