Daniel Liberzon

FoRCE online seminar, Mar 23, 2018 1 of 23

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

Nonlinear Observers Robust to Measurement Errors and their Applications in Control and Synchronization

INFORMATION FLOW in CONTROL SYSTEMS

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INFORMATION FLOW in CONTROL SYSTEMS

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INFORMATION FLOW in CONTROL SYSTEMS

• Coarse sensing

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity• Coarse sensing

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• Security considerations

• Coarse sensing

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• Security considerations • Event-driven actuators

• Coarse sensing

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• Security considerations • Event-driven actuators

• Coarse sensing

• Theoretical interest

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• Security considerations • Event-driven actuators

• Coarse sensing

• Theoretical interest

Limited information errors

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INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity

• Security considerations • Event-driven actuators

• Coarse sensing

• Theoretical interest

Limited information errors

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need robust algorithms

OBSERVER–BASED OUTPUT FEEDBACK CONTROL

Plant

Controller

Sensors

Observer

x y

x

u

++ errors (e.g.,

quantization)

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OBSERVER–BASED OUTPUT FEEDBACK CONTROL

Plant

Controller

Sensors

Observer

x y

x

u

++ errors (e.g.,

quantization)

error propagation

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not much is known about this problem

OBSERVER–BASED OUTPUT FEEDBACK CONTROL

Plant

Controller

Sensors

Observer

x y

x

u

++ errors (e.g.,

quantization)

error propagation

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not much is known about this problem

OBSERVER–BASED OUTPUT FEEDBACK CONTROL

Plant

Controller

Sensors

Observer

x y

x

u

++ errors (e.g.,

quantization)

error propagation

Input-to-state stability (ISS) provides a framework for quantifying robustness (graceful error propagation)

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TALK OUTLINE

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TALK OUTLINE

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• Fresh look at input-to-state stability (ISS): asymptotic ratio

TALK OUTLINE

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• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

TALK OUTLINE

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• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

TALK OUTLINE

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• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

HyungboShim

TALK OUTLINE

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• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

HyungboShim

TALK OUTLINE

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[L–Shim, An asymptotic ratio characterization of ISS, TAC 2015]• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

HyungboShim

TALK OUTLINE

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[L–Shim, An asymptotic ratio characterization of ISS, TAC 2015]• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

[Shim–L, Nonlinear observers robust to measurement disturbances in an ISS sense, TAC 2016], see also [Shim–L–Kim, CDC 2009]

(

HyungboShim

TALK OUTLINE

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[L–Shim, An asymptotic ratio characterization of ISS, TAC 2015]• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

• electric power generators [Ajala–Domínguez-Garcia–L, 2018]

• Lorenz chaotic system [Andrievsky–Fradkov–L, CDC 2017, SCL 2018]

[Shim–L, Nonlinear observers robust to measurement disturbances in an ISS sense, TAC 2016], see also [Shim–L–Kim, CDC 2009]

(

TALK OUTLINE

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• Application to output feedback control design

• Applications to robust synchronization

• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

ISS existence of ISS Lyapunov function:⇔

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

ISS existence of ISS Lyapunov function:⇔V

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)C1pos. def., rad. unbdd, function satisfying

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

ISS existence of ISS Lyapunov function:⇔

or equivalently

V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

V

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)C1pos. def., rad. unbdd, function satisfying

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

ISS existence of ISS Lyapunov function:⇔

or equivalently

V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)GASx = f(x,0) ISS, e.g.:x = f(x, d)

V

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)C1pos. def., rad. unbdd, function satisfying

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

x = −x+ xd ( unbdd for )x d ≡ 2

ISS existence of ISS Lyapunov function:⇔

or equivalently

V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)GASx = f(x,0) ISS, e.g.:x = f(x, d)

V

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)C1pos. def., rad. unbdd, function satisfying

[Sontag ’89]INPUT–to–STATE STABILITY (ISS)

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x = f(x, d)System is ISS if its solutions satisfy

class- fcnK∞γ ∈ K∞,where β(·, t) ∈ K∞, β(r, ·)& 0

|x(t)| ≤ β(|x(0)|, t) + γ³kdk[0,t]

´

x = −x+ xd ( unbdd for )x d ≡ 2(may have even if )x↑∞ d→ 0x = −x+ x2d

ISS existence of ISS Lyapunov function:⇔

or equivalently

V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)GASx = f(x,0) ISS, e.g.:x = f(x, d)

V

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)C1pos. def., rad. unbdd, function satisfying

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

Theorem: ISS asymptotic-ratio ISS Lyapunov function⇔∃

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Proof of follows from characterization of ISS via (2)⇒

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

Theorem: ISS asymptotic-ratio ISS Lyapunov function⇔∃

ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Proof of follows from characterization of ISS via (2)⇒Proof of proceeds by constructing as in (1)ρ⇐

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Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)

Theorem: ISS asymptotic-ratio ISS Lyapunov function⇔∃

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d

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(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

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(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

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V = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

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= −x2 + x2 d2

1+d2+ xdV = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

α3(|x|)6 of 23

= −x2 + x2 d2

1+d2+ xdV = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

α3(|x|)g(|x|, |d|)

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= −x2 + x2 d2

1+d2+ xdV = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

α3(|x|)g(|x|, |d|)

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= −x2 + x2 d2

1+d2+ xdV = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

lim supr→∞

g(r,s)α3(r)

< 1 ∀ s ≥ 0

V ≤ −α3(|x|) + g(|x|, |d|)where , is continuous non-negative, α3 ∈ K∞ g

g(r, ·) is non-decreasing for each , with , andr g(r,0)=0

Example (scalar): , x = − 11+d2

x+ d V (x) := 12x2

α3(|x|)g(|x|, |d|)No info about ISS gain

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= −x2 + x2 d2

1+d2+ xdV = − x2

1+d2+ xd

(1)(2)V ≤ −α(|x|) + χ(|d|) (α,χ ∈ K∞)

|x| ≥ ρ(|d|) ⇒ V < 0 (ρ ∈ K∞)ASYMPTOTIC–RATIO ISS LYAPUNOV FUNCTIONS

Definition: pos. def., rad. unbdd, function is anasymptotic-ratio ISS Lyapunov function if

C1 V

TALK OUTLINE

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• Application to output feedback control design

• Applications to robust synchronization

• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

ROBUST OBSERVER DESIGN PROBLEM

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ROBUST OBSERVER DESIGN PROBLEM

Plantxu

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ROBUST OBSERVER DESIGN PROBLEM

Sensorsy

++d

Plantxu

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ROBUST OBSERVER DESIGN PROBLEM

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)

Sensorsy

++d

Plantxu

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ROBUST OBSERVER DESIGN PROBLEM

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)

Sensorsy

++d

Plantxu

Observer x

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ROBUST OBSERVER DESIGN PROBLEM

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)Observer: z = F (z, y, u), x = H(z, y) (z ∈ Rm)

Sensorsy

++d

Plantxu

Observer x

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ROBUST OBSERVER DESIGN PROBLEM

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)Observer: z = F (z, y, u), x = H(z, y) (z ∈ Rm)Full-order observer: x = z, m = n ; reduced-order: m < n

Sensorsy

++d

Plantxu

Observer x

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ROBUST OBSERVER DESIGN PROBLEM

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)Observer: z = F (z, y, u), x = H(z, y) (z ∈ Rm)Full-order observer: x = z, m = n ; reduced-order: m < n

Sensorsy

++d

+–ePlant

xuObserver x

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ROBUST OBSERVER DESIGN PROBLEM

State estimation error: e := x− x = H(z, h(x, d))− x

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)Observer: z = F (z, y, u), x = H(z, y) (z ∈ Rm)Full-order observer: x = z, m = n ; reduced-order: m < n

Sensorsy

++d

+–ePlant

xuObserver x

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ROBUST OBSERVER DESIGN PROBLEM

State estimation error: e := x− x = H(z, h(x, d))− x

Plant: x = f(x, u), y = h(x, d) (x ∈ Rn)Observer: z = F (z, y, u), x = H(z, y) (z ∈ Rm)Full-order observer: x = z, m = n ; reduced-order: m < n

Sensorsy

++d

+–ePlant

xuObserver x

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Robustness issue: can have whenyet for arbitrarily small

e→ 0e%∞ d 6= 0

d ≡ 0

DISTURBANCE–to–ERROR STABILITY (DES)

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DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

|e(t)| ≤ β(|e(0)|, t) + γ³kdk[0,t]

´β ∈ KL,γ ∈ K∞

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

|e(t)| ≤ β(|e(0)|, t) + γ³kdk[0,t]

´β ∈ KL,γ ∈ K∞

Known conditions for this [Sontag-Wang ’97, Angeli ’02] are very strong

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

|e(t)| ≤ β(|e(0)|, t) + γ³kdk[0,t]

´β ∈ KL,γ ∈ K∞

Known conditions for this [Sontag-Wang ’97, Angeli ’02] are very strong

Also, DES is coordinate dependent as global error convergenceis coordinate dependent:

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

|e(t)| ≤ β(|e(0)|, t) + γ³kdk[0,t]

´β ∈ KL,γ ∈ K∞

Known conditions for this [Sontag-Wang ’97, Angeli ’02] are very strong

Also, DES is coordinate dependent as global error convergenceis coordinate dependent: z → x 6⇒ Φ(z)→ Φ(x)

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ISS-like robustness notion: call observer DES if

DISTURBANCE–to–ERROR STABILITY (DES)Plant: x = f(x, u), y = h(x, d)

Observer: z = F (z, y, u), x = H(z, y)

Estimation error: e := x− x

|e(t)| ≤ β(|e(0)|, t) + γ³kdk[0,t]

´β ∈ KL,γ ∈ K∞

Known conditions for this [Sontag-Wang ’97, Angeli ’02] are very strong

Also, DES is coordinate dependent as global error convergenceis coordinate dependent: z → x 6⇒ Φ(z)→ Φ(x)Path toward less restrictive, coordinate-invariant robustnessproperty: impose DES only as long as are boundedx, u

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QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

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QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if ∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u,

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u,

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

y = x+ d,

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u, z = −z+ y2u

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

y = x+ d,

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u, z = −z+ y2u

e = −e+2xud+ ud2

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

y = x+ d,

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u, z = −z+ y2u

e = −e+2xud+ ud2 qDES but not DES

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

y = x+ d,

QUASI–DISTURBANCE–to–ERROR STABILITY (qDES)

Example: x = −x+ x2u, z = −z+ y2u

e = −e+2xud+ ud2 qDES but not DES

x = f(x, u), y = h(x, d)

z = F (z, y, u), x = H(z, y)e = x− x

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Definition: observer is quasi-Disturbance-to-Error Stable (qDES) if

whenever kuk[0,t], kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

∀K > 0 ∃βK ∈ KL, γK ∈ K∞ such that

y = x+ d,

The qDES property is invariant to coordinate changes

REDUCED–ORDER qDES OBSERVERS

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y = x1

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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+ d

y = x1

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

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y = x1

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(x1, x2 + e, u)− f2(x1, x2, u)

i

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y = x1

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(x1, x2 + e, u)− f2(x1, x2, u)

i

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Assume this is , then we have an asymptotic observer: when

≤ −α3(|e|)e→ 0 d ≡ 0

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

i

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

iassumed to be ≤ −α3(|e|)

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

iassumed to be ≤ −α3(|e|)

assume this has norm ≤ α4(|e|)

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

iassumed to be ≤ −α3(|e|)

assume this has norm ≤ α4(|e|)upper-bounded by φK(|d|)

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y = x1+ d

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

iassumed to be ≤ −α3(|e|)

assume this has norm ≤ α4(|e|)upper-bounded by φK(|d|)

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y = x1+ d

Then V ≤ −α3(|e|) + α4(|e|)φK(|d|)

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

e := z − x2, V = V (e)

V = ∂V∂e

hf2(y, x2 + e, u)− f2(x1, x2, u)

i= ∂V

∂e

hf2(y,x2+e,u)−f2(y,x2,u)

i+∂V

∂e

hf2(y,x2,u)−f2(x1,x2,u)

iassumed to be ≤ −α3(|e|)

assume this has norm ≤ α4(|e|)upper-bounded by φK(|d|)

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y = x1+ d

Then V ≤ −α3(|e|) + α4(|e|)φK(|d|)whenever kuk[0,t], kxk[0,t] ≤ K

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s ⇐ limr→∞

α4(r)α3(r)

= 0

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s ⇐ limr→∞

α4(r)α3(r)

= 0

If we have such that α3(r) ≥ α(r)α4(r)α∈K∞

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s ⇐ limr→∞

α4(r)α3(r)

= 0

V ≤−[α(|e|)−φK(|d|)]·α4(|e|)If we have such that α3(r) ≥ α(r)α4(r)α∈K∞then

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s ⇐ limr→∞

α4(r)α3(r)

= 0

V ≤−[α(|e|)−φK(|d|)]·α4(|e|)If we have such that α3(r) ≥ α(r)α4(r)α∈K∞

< 0 when |e| > α−1 ◦ φK(|d|)then

REDUCED–ORDER qDES OBSERVERSPlant (after a coordinate change):x1 = f1(x1, x2, u)

x2 = f2(x1, x2, u)

Observer:z = f2(y, z, u)

x1 = y

x2 = z

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y = x1+ d

V ≤ −α3(|e|) + α4(|e|)φK(|d|)Asymptotic ratio condition:

lim supr→∞

α4(r)α3(r)

φK(s) < 1 ∀s ⇐ limr→∞

α4(r)α3(r)

= 0

V ≤−[α(|e|)−φK(|d|)]·α4(|e|)If we have such that α3(r) ≥ α(r)α4(r)α∈K∞

< 0 when |e| > α−1 ◦ φK(|d|)then

Can estimate ISS gain but only if is knownα

TALK OUTLINE

• Application to output feedback control design

• Applications to robust synchronization

14 of 23

• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

Plant

Controller Observer

Sensorsx y

x

u

OBSERVER–BASED OUTPUT FEEDBACK REVISITED

15 of 23

Controller Observer

Sensorsx y

x

u

OBSERVER–BASED OUTPUT FEEDBACK REVISITED

x=f(x, u)

15 of 23

Controller Observer

x y

x

u

OBSERVER–BASED OUTPUT FEEDBACK REVISITED

x=f(x, u) y=h(x, d)

15 of 23

x y

x

u

OBSERVER–BASED OUTPUT FEEDBACK REVISITED

x=f(x, u) y=h(x, d)

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z=F (z, y, u)x = zController

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])• Assume controller is ISS w.r.t. : e

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])• Assume controller is ISS w.r.t. : e

|x(t)| ≤ β(|x(0)|, t) + γ³kek[0,t]

´

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])• Assume controller is ISS w.r.t. : e

|x(t)| ≤ β(|x(0)|, t) + γ³kek[0,t]

´[Freeman, Fah, Jiang et al., Sanfelice–Teel, Ebenbauer et al.]

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])• Assume controller is ISS w.r.t. : e

|x(t)| ≤ β(|x(0)|, t) + γ³kek[0,t]

´[Freeman, Fah, Jiang et al., Sanfelice–Teel, Ebenbauer et al.]

Cascade argument: closed-loop system is quasi-ISS

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

OBSERVER–BASED OUTPUT FEEDBACK REVISITEDx y

x

u

z=F (z, y, u)x = z

• Assume observer is qDES w.r.t. :d

kuk, kxk≤K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])• Assume controller is ISS w.r.t. : e

|x(t)| ≤ β(|x(0)|, t) + γ³kek[0,t]

´[Freeman, Fah, Jiang et al., Sanfelice–Teel, Ebenbauer et al.]

Cascade argument: closed-loop system is quasi-ISS¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

x=f(x, u) y=h(x, d)

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u=k(z)=k(x+e)

APPLICATION to QUANTIZED OUTPUT FEEDBACK

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APPLICATION to QUANTIZED OUTPUT FEEDBACK¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

16 of 23

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

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quantizer

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

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quantizer

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

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quantizer

∃K = K(M) ∆, |x(0)|, |z(0)|• & upper bounds on s.t.

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

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quantizer

∃K = K(M) ∆, |x(0)|, |z(0)|• & upper bounds on s.t.

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

• remain|x(t)|, |u(t)| ≤ K

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

16 of 23

quantizer

∃K = K(M) ∆, |x(0)|, |z(0)|• & upper bounds on s.t.

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

• remain|x(t)|, |u(t)| ≤ K• lim sup

t→∞

¯³x(t)z(t)

´¯≤ γK(∆)

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

16 of 23

quantizer

∃K = K(M) ∆, |x(0)|, |z(0)|• & upper bounds on s.t.

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

• remain|x(t)|, |u(t)| ≤ K• lim sup

t→∞

¯³x(t)z(t)

´¯≤ γK(∆)

• Contraction is guaranteed if quantization is fine enough

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

16 of 23

quantizer

∃K = K(M) ∆, |x(0)|, |z(0)|• & upper bounds on s.t.

APPLICATION to QUANTIZED OUTPUT FEEDBACK

y = q(h(x))

|h(x)| ≤M ⇒ |d| ≤∆

• remain|x(t)|, |u(t)| ≤ K• lim sup

t→∞

¯³x(t)z(t)

´¯≤ γK(∆)

• Contraction is guaranteed if quantization is fine enough

• Can achieve asymptotic stability by dynamic “zooming”

¯³x(t)z(t)

´¯≤ βK

³¯³x(0)z(0)

´¯, t´+ γK

³kdk[0,t]

´kuk, kxk≤K

d – quantization error

= h(x) + d

16 of 23

quantizer

TALK OUTLINE

• Application to output feedback control design

• Applications to robust synchronization

17 of 23

• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

ROBUST SYNCHRONIZATION and qDES OBSERVERS

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ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2)

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ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2)

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x1

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2)

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x1

d

++

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2)

18 of 23

x1

d

++y

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2)

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

Equivalently: follower is a reduced-order qDES observer for leader

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

Equivalently: follower is a reduced-order qDES observer for leader

Sufficient condition from before: s.t.∃V =V (e)

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

Equivalently: follower is a reduced-order qDES observer for leader

Sufficient condition from before: s.t.∃V =V (e)¯∂V∂e

¯≤α4(|e|),

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

Equivalently: follower is a reduced-order qDES observer for leader

Sufficient condition from before: s.t.∃V =V (e)¯∂V∂e

¯≤α4(|e|),

∂V∂e (e)

³f2(x1, z)−f2(x1, x2)

´≤−α3(|e|),

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

ROBUST SYNCHRONIZATION and qDES OBSERVERS

Leaderx2 = f2(x1, x2)x1 = f1(x1, x2) e := z − x2

Robust synchronization: s.t.∀K > 0 ∃βK∈KL, γK∈K∞

whenever kxk[0,t] ≤ K|e(t)| ≤ βK(|e(0)|, t) + γK(kdk[0,t])

Equivalently: follower is a reduced-order qDES observer for leader

Sufficient condition from before: s.t.∃V =V (e)¯∂V∂e

¯≤α4(|e|),

∂V∂e (e)

³f2(x1, z)−f2(x1, x2)

´≤−α3(|e|), and

lim supr→∞

α4(r)α3(r)

= 0 (asymptotic ratio condition)

18 of 23

x1

d

++y

Follower

z = f2(y, z). . .

APPLICATION EXAMPLE #1

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APPLICATION EXAMPLE #1

19 of 23

θ2

loadGenerator 1 Generator 2

APPLICATION EXAMPLE #1

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θ1 = ω1ω1 = u1−`(t)−D1ω1 θ2

Generator 1 Generator 2

APPLICATION EXAMPLE #1

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θ1 = ω1ω1 = u1−`(t)−D1ω1 θ2

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

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θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

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θ1 θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Objective: connect 2nd generator when θ1 ≈ θ2, ω1 ≈ ω2

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Objective: connect 2nd generator when θ1 ≈ θ2, ω1 ≈ ω2

e := ω2 − ω1V = e2 gives DES (ISS) from to d

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Objective: connect 2nd generator when θ1 ≈ θ2, ω1 ≈ ω2

e := ω2 − ω1V = e2 gives DES (ISS) from to d

ω1 ω2• and will synchronize with error + load variations kD1kdk

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Objective: connect 2nd generator when θ1 ≈ θ2, ω1 ≈ ω2

e := ω2 − ω1V = e2 gives DES (ISS) from to d

ω1 ω2• and will synchronize with error + load variations kD1kdk

θ1 ≈ θ2• due to phase drift, will have at some time

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Objective: connect 2nd generator when θ1 ≈ θ2, ω1 ≈ ω2

e := ω2 − ω1V = e2 gives DES (ISS) from to d

ω1 ω2• and will synchronize with error + load variations kD1kdk

⇒ can connect 2nd generatorθ1 ≈ θ2• due to phase drift, will have at some time

Measurements:PMU corruptedby disturbance

19 of 23

θ1

d

++

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

ω1→u1(t, θ1)=

`(t) =

control input (mechanical power) With integral control: desired frequency

electrical load (slowly varying)

APPLICATION EXAMPLE #1

Extensions:

• phase-dependent damping D1 = D1(θ1)

analysis more challenging, but can still showstate boundedness and qDES from to ed

20 of 23

θ2 = ω2ω2 = u2 −D2ω2

θ1 = ω1ω1 = u1−`(t)−D1ω1

Generator 1 Generator 2

• network case (microgrids)

θ1

d

++

APPLICATION EXAMPLE #2

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APPLICATION EXAMPLE #2

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Lorenz system

APPLICATION EXAMPLE #2

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Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

APPLICATION EXAMPLE #2

21 of 23

Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

APPLICATION EXAMPLE #2

21 of 23

Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

d

++y

APPLICATION EXAMPLE #2

21 of 23

Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

Observer

z2 = θy − z2 − yz3z3 = −βz3 + yz2

d

++y

APPLICATION EXAMPLE #2

21 of 23

Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

Observer

z2 = θy − z2 − yz3z3 = −βz3 + yz2

d

++y

Can show is bounded using V (x) = x21 + x22 + (x3−σ−θ)2x

APPLICATION EXAMPLE #2

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Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

Observer

z2 = θy − z2 − yz3z3 = −βz3 + yz2

d

++y

Can show is bounded using V (x) = x21 + x22 + (x3−σ−θ)2x

Can show qDES from to using d e :=³z2−x2z3−x3

´V (e) = e22 + e23

APPLICATION EXAMPLE #2

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Lorenz system

x1 = σx2 − σx1x2 = θx1 − x2 − x1x3x3 = −βx3 + x1x2

x1

Observer

z2 = θy − z2 − yz3z3 = −βz3 + yz2

d

++y

Can show is bounded using V (x) = x21 + x22 + (x3−σ−θ)2x

Can show qDES from to using d e :=³z2−x2z3−x3

´V (e) = e22 + e23

For arising from time sampling and quantization, we can derive an explicit bound on synchronization error which is inversely proportional to data rate (see paper for details)

d

TALK OUTLINE

22 of 23

• Fresh look at input-to-state stability (ISS): asymptotic ratio

• Observers robust to measurement disturbances:formulation and Lyapunov condition

• Application to output feedback control design

• Applications to robust synchronization

• electric power generators

• Lorenz chaotic system

FUTURE WORK

Nonlinear qDES observer design:• Identify system classes to which Lyapunov conditions apply

• Develop more constructive procedures for observer design

Quantized output feedback control:• Relax ISS controller assumption

• Study other coupled oscillator network models • Look for examples in other areas (e.g., vehicle formations)

Robust synchronization:

Papers and preprints available at liberzon.csl.illinois.edu23 of 23