Nonlinear Control Lecture 7: Passivityele.aut.ac.ir/~abdollahi/Lec_6_N11.pdfOutline Passivity De...

Outline Passivity Definition L2 and Lyapunov Stability Feedback and Passivity Theorems

Nonlinear ControlLecture 7: Passivity

Farzaneh Abdollahi

Department of Electrical Engineering

Amirkabir University of Technology

Fall 2011

Farzaneh Abdollahi Nonlinear Control Lecture 7 1/26


Passivity DefinitionState Model

L2 and Lyapunov Stability

Feedback and Passivity TheoremsFeedback and L2 StabilityFeedback and A.S



Passivity DefinitionI Consider a memoryless function

y = h(t, u) (1)

whereI h : [0,∞)× Rp → Rp

I u: input; y : output

I Exp. Resistive element: u is voltage; y iscurrent

I It is passive if the inflow of power isalways nonneg.

I ∴ uy ≥ 0 for all (u, y)I Geometrically it means the u − y curves

lie in first and third quadrantI The simplest option is linear resistor

(u = Ry)



I If u and y are vectors, the power flow onto the network will be

uT y =

p∑i=1

uiyi =

p∑i=1

uihi (u)

I For time-varying system, as long as the passivity condition is satisfied forall time, it is called passive.

I Extreme case of passivity : uT y = 0 , this system is lossless.I Input strictly passivity: if a fcn. h satisfies uT y ≥ uTφ(u), and

uTφ(u) > 0, ∀u 6= 0I since uT y = 0 only if u = 0



Input Feedforward Passive

I Let us define a new output: y = y −φ(u):

uT y = uT [y − φ(u)] ≥ uTφ(u)− uTφ(u) = 0

I ∴ any fcs. satisfying uT y ≥ uTφ(u) canbe transformed into a passive fcn. viainput feedforward.

I This fcs is input feedforward passive



Output Feedback Passive

I Suppose uT y ≥ yTρ(y).

I Let us define a new input: u = u − ρ(y):

uT y = [u − ρ(y)]T y ≥ yTρ(y)− yTρ(y) = 0

I ∴ any fcs. satisfying uT y ≥ yTρ(y) canbe transformed into a passive fcn. viaoutput feedback.

I This fcs is output feedback passiveI If yTρ(y) > 0,∀y 6= 0 the fcn. is called

output strictly passiveI since uT y = 0 only if y = 0



Passivity of Memoryless Fcn.

I The system y = h(t, u) isI passive if uT y ≥ 0I lossless if uT y = 0I input-feedforward passive if uT y ≥ uTφ(u) for some fcn φI input strictly passive if uT y ≥ uTφ(u) and uTφ(u) > 0,∀u 6= 0I output-feedback passive if uT y ≥ yTρ(y) for some fcn ρI output strictly passive if uT y ≥ yTρ(y) and yTρ(y) > 0,∀y 6= 0



State Model

I Consider a dynamical system with state model

x = f (x , u) (2)

y = h(x , u)

I f : Rn × Rp → Rn is local lip.I h : Rn × Rp → Rp is cont.I f (0, 0) = 0 and h(0, 0) = 0I # inputs = # outputs



Motivated Example: RLC Circuit

I Consider the RLC circuitwith linear C and L andnonlinear R

I The nonlinear resistors arerepresented by:i1 = h1(v1); v2 = h2(i2); i3 =h3(v3)

I Input u: voltage; output y :current

I power flow into the network:uy

I Define x1: current throughL; x2: voltage across C



I ∴ state modelLx1 = u − h2(x1)− x2

C x2 = x1 − h3(x2)

y = x1 + h1(u)

I The system is passive if the absorbed energy by the network is greaterthan the stored energy in the network over the same period:∫ t

0u(s)y(s)ds ≥ V (x(t))− V (x(0)) (3)

where V (x) = 1/2Lx21 + 1/2Cx2

2 : stored energy

I Strict inequality of (3) yields difference between the absorbed energy andincreased stored energy equals to dissipative energy in the resistors

I (3) hold for every t ≥ 0 for all t u(t)y(t) ≥ V (x(t), u(t))

I i.e. the power flow must be greater than or equal to the rate of change ofthe stored energy



Motivated Exp. Cont’dI Take the derivative of V along the system traj:

V = uy − uh1(u)− x1h2(x1)− x2h3(x2)

I ∴uy = V + uh1(u) + x1h2(x1) + x2h3(x2)

I If h1, h2 and h3 are passive ⇒uy ≥ V ; system is passiveI Otherwise

I Case 1: If h1 = h2 = h3 = 0, uy = V no energy dissipation, system islossless

I Case 2: If h2, h3 ∈ sector [0,∞] ( passive fcns.) uy ≥ V + uh1(u)I If uh1(u) > 0 for all u 6= 0 it is input strick passive (absorbed energy is

greater than increased stored energy unless u(t) ≡ 0)I If uh1(u) < 0 for some u it can be made passive by an input ff

I Case 3: If h1 = 0 and h3: passive fcn ( ∈ sector [0,∞]) uy ≥ V + yh2(y)I If yh2(y) > 0 for all y 6= 0 it is output strick passive (absorbed energy is

greater than increased stored energy unless y(t) ≡ 0)I If yh2(y) < 0 for some u it can be made passive by an output fb



Motivated Exp. Cont’d

I Case 4: If h1 ∈ [0,∞] and h2, h3 ∈ (0,∞)

uy ≥ V + x1h2(x1) + x2h3(x2)I where x1h2(x1) + x2h3(x2) is pos. def.I It is state strict passive or simply strick passive (absorbed energy is greater

than increased stored energy unless x(t) ≡ 0)



Passivity Based on State Model

I The system (2) is passive if there exist a cont. diff. p.s.d fcn V (x) (called storage fcn) s.t.

uT y ≥ V =∂V

∂xf (x , u), ∀(x , u) ∈ Rn × Rn

Moreover, it isI Lossless if uT y = VI Input-feedforward passive if uT y ≥ V + uTφ(u) for some fcn φI Input strictly passive if uT y ≥ V + uTφ(u) and uTφ(u) > 0,∀u 6= 0I Output-feedback passive if uT y ≥ V + yTρ(y) for some fcn ρI Output strictly passive if uT y ≥ V + yTρ(y) and yTρ(y) > 0,∀y 6= 0I Strictly passive if uT y ≥ V + ψ(x) for some p.d. ψ

I In all cases, the inequality should hold for all (x , u)



ExampleI Consider a cascade connection of an

integrator and a passive memoryless fcn.x = u, y = h(x)

I h is passive ∫ x

0 h(σ)dσ ≥ 0, ∀x

I Storage fcn: V (x) =∫ x

0 h(σ)dσ

I ∴V = h(x)x = yu it is loss less

I Now replace the integrator with1/(as + 1), a > 0

I The state model is:ax = −x + u, y = h(x)

I V = a∫ x

0 h(σ)dσ V = h(x)(−x + u) =yu − xh(x) ≤ yu

I ∴ It is passive.

I When xh(x) > 0 it is strictly passiveFarzaneh Abdollahi Nonlinear Control Lecture 7 14/26



I Lemma: If the system (2) is output strictly passive withuT y ≥ V + δyT y for some δ > 0 then it is finite-gain L2 stable with L2

gain less than or equal to 1/δ

I Definition: The system (2) is zero-state observable if no solution ofx = f (x , 0) can stay identically in S = {x ∈ Rn|h(x , 0) = 0} other thanthe trivial solution x(t) ≡ 0

I Example: For linear system x = Ax , y = Cx

I Observability is equivalent to y(t) = CeAtx(0) ≡ 0⇔{

x(0) = 0x(t) ≡ 0




I Lemma: If system (2), is passive with a p.d. storage fcn. V (x), then theorigin of x = f (x , 0) is stable.

I Lemma: For system (2), the origin of x = f (x , 0) is a.s if the system isI strictly passive orI output strictly passive and zero-state observable

I Furthermore, if the storage fcn. is radially unbounded, then the origin willbe g.a.s



Example

I Consider SISO system

x1 = x2

x2 = −ax31 − kx2 + u

y = x2

I a > 0, k > 0

I Consider p.d. radially unbounded V (x) = (1/4)ax41 + (1/2)x2

2

I V = −ky 2 + yu

I By ρ(y) = ky , it is output strictly passive L2 f.g.s. with gain less thanor equal to 1/k

I When u = 0, y(t) ≡ 0⇒ x2 ≡ 0⇒ x1 ≡ 0 zero-state observable

I ∴ it is g.a.s



Feedback and Passivity Theorems

I Consider a fb connection of H1 and H2

I in time-invariant dynamical systemrepresented by state model

xi = fi (xi , ei )

yi = hi (xi , ei ) (4)

I or (possibly time-varying) memorylessfcn

yi = hi (t, ei ) (5)

I Objective: Analyze stability of fbconnection, using passivity properties offb components (H1 and H2)




1. If both components H1 and H2 are dynamical systems, the closed-loopstate model is

x = f (x , u)

y = h(x , u) (6)

I where x = [x1 x2]T , u = [u1 u2]T , y = [y1 y2]T

I Assuming fi (0, 0) = 0 and hi (0, 0) = 0 f (0, 0) = 0 and h(0, 0) = 0I Assuming fi and hi are locally lip. f and h are locally lip.




2. If H1 is a dynamical system and H2 is memoryless fcn., the closed loopstate model is

x = f (t, x , u)

y = h(t, x , u) (7)

I where x = x1, u = [u1 u2]T , y = [y1 y2]T

I Assume f (t, 0, 0) = 0 and h(t, 0, 0) = 0I Assume f is p.c. in t and locally lip. in (x , u), and h is p.c. in t and cont. in

(x , u)

3. If both components are memoryless fcnsI It can be considered as a special case when x does not exit



Feedback and L2 Stability

I Theorem: The feedback connection of two passive system is passiveI Proof: Let V1(x1) and V2(x2) are storage fcns of H1 and H2 respectively.I If either components are memoryless fcn, take Vi = 0I Then eT

i yi ≥ Vi

I Considering fb. connectioneT

1 y1 + eT2 y2 = (u1 − y2)T y1 + (u2 + y1)T y2 = uT

1 y1 + uT2 y2

I ∴uT y = uT1 y1 + uT

2 y2 ≥ V1 + V2 = V

I Lemma: The fb connection of two output strictly passive systems with

eTi yi ≥ Vi + δiy

Ti yi + εie

Ti ei , i = 1, 2

is finite-gain L2 stable with gain if ε1 + δ2 > 0, ε2 + δ1 > 0



Example

I Consider H1 :

{x = f (x) + G (x)e1

y1 = h(x)and H2 : y2 = ke2 where k > 0,

ei , yi ∈ Rp

I Suppose there is a p.d. fcn V1(x) s.t.∂V1∂x f (x) ≤ 0, ∂V1

∂x G (x) = hT (x), ∀x ∈ Rn

I Both components are passive

I and eT2 y2 = keT

2 e2 = γkeT2 e2 + (1−γ)

k yT2 y2, 0 < γ < 1

I ∴ ε1 = δ1 = 0, ε2 = γk, δ2 = (1−γ)k

I It is finite gain L2 stable



Feedback and A.S.

I Stability of origin is trivial if both components are passive. (Tell me why?.

I Let us focus on a.s.I Theorem: Consider fb connection of two T.I. dynamical systems of the

form (4). The origin of the closed-loop system (6) ( when u = 0) is a.s.if

I both components are strictly passiveI or both components are output strictly passive and zero-state observableI or one component is strictly passive and the other one is output strictly

passive and zero-state observable

I Furthermore, if storage fcn of each component is radially unbounded, theorigin is g.a.s



ExampleI Consider fb connection with:

H1 :

x1 = x2

x2 = −ax31 − kx2 + e1

y1 = x2

, H2 :

x3 = x4

x4 = −bx3 − x34 + e2

y2 = x4

a, b, k > 0

I Use V1 = (a/4)x41 + (1/2)x2

2 V1 = −ky 21 + y1e1

I ∴ H1 is output strictly passive.

I when e1 = 0, y1 ≡ 0⇔ x2 ≡ 0⇔ x1 ≡ 0

I ∴ H1 is zero-state observable

I Use V2 = (b/2)x23 + (1/2)x2

4 V2 = −y 42 + y2e2

I ∴ H2 is output strictly passive.

I when e2 = 0, y2 ≡ 0⇔ x4 ≡ 0⇔ x3 ≡ 0

I ∴ H2 is zero-state observable

I V1,V2 are radially unbounded the closed-loop system is g.a.sFarzaneh Abdollahi Nonlinear Control Lecture 7 24/26


Example

I Reconsider the pervious system but change the output of H1 toy1 = x2 + e1

I Hence V1 = −k(y1 − e1)2 − e21 + y1e1 H1 is passive Not strictly

passive or output strictly passive

I Consider Lyap fcn of closed-loop sys.V = V1 + V2 = 1

4 ax41 + 1

2 x22 + 1

2 bx23 + 1

2 x24

I V = −kx22 − x4

4 − x24 ≤ 0

I Also, V = 0⇒ x2 = x4 = 0x2 ≡ 0⇒ x1 ≡ 0x4 ≡ 0⇒ x3 ≡ 0

I V is radially unbounded the closed-loop system is g.a.s



Feedback and A.S.

I Theorem: Consider fb connection of a strictly passive, T.I. dynamicalsystems of the form (4) and a passive (possibly time-varying) memorylessfcn of the form (5). The origin of the closed-loop system (7) ( whenu = 0) is u.a.s. Furthermore, if storage fcn of the dynamical system isradially unbounded, the origin is g.u.a.s


Nonlinear Control Lecture 7: Passivityele.aut.ac.ir/~abdollahi/Lec_6_N11.pdfOutline Passivity De...

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