Estimation and enlargement of domains of attraction Alexandre - - PowerPoint PPT Presentation

Estimation and enlargement of domains of attraction

Alexandre Sanfelice Bazanella

On leave from Universidade Federal do Rio Grande do Sul Electrical Engineering Department Porto Alegre - RS, BRAZIL

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My research topics

• Design of controllers with restricted structure

– for multivariable processes – automation of design procedures – optimization of the experimental procedures

• Modeling and identification for control

– applications for electric power systems - load models

• Domains of attraction

– enlargement of domains of attraction by control ∗ includes estimation – passivity-based controllers – dynamic extension controllers: robust regulation and enhanced properties – applications of nonlinear control and identification, particularly to electric power systems

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This seminar - domains of attraction

• The problem setting: objectives, definitions and Lyapunov’s methods
• Estimation of domains of attraction with Lyapunov functions:

– how-to and why – some history – the issues

• Numerical determination of Lyapunov functions with LMI’s
• Passivity-based control for enlargement of the domain of attraction: LgV and

IDA Examples/applications:

• pendulum
• electric power systems

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Problem setting

• Consider autonomous nonlinear systems described by

˙ x = f(x) x ∈ ℜn (1)

• The (unique) solutions of (1) with initial condition x(0) = x0 are x(x0,t)
• The equilibria of (1) are the points xe : f(xe) = 0
• System stability concerns:

– existence of (desired) equilibrium - operating point – asymptotic stability (a.s.) of the operating point – sufficiently large domain of attraction (DOA) of the operating point

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Related control problem

• "Autonomous" control-affine nonlinear systems

˙ x = f(x)+g(x)u x ∈ ℜn u ∈ ℜm (2)

• Design u = φ(x) such that:

– the operating point is kept: g(xe)φ(xe) = 0 – the operating point is a.s. – the DOA of the operating point is as large as possible

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Lyapunov’s indirect method

• Define the Jacobian

J(x) = ∂f(x) ∂x – the equilibrium is a.s. if σ(J(xe)) ⊂ C − – the equilibrium is unstable if σ(J(xe))

C + = /

– if σ(J(xe)) ⊂ C − & σ(J(xe))

• ∂C − = /

then we don´t know

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Lyapunov’s direct method

Lyapunov’s direct method: the equilibrium is a.s. if there exists a (Lyapunov) function V(·) such that

∂V(x) ∂x |x=xe= 0 & ∂2V(x) ∂x2

|x=xe> 0 ∀x ∈ R+ ⊃ xe (xe is a minimum of V(x)) ˙ V(x) = ∂V(x)

∂x f(x) < 0

∀x ∈ R− ⊃ xe The (LaSalle’s) invariance principle: allows to prove a.s. when ˙ V(x) ≤ 0 Converse Lyapunov theorems: if the equilibrium is a.s. then there exists a Lyapunov function

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Two assumptions for this presentation

– A1 - Whenever we prove a.s. with a Lyapunov function, ˙ V(x) < 0 ∗ for simplicity only, all results that follow apply mutatis mutandis when the invariance principle is used – A2 - The Jacobians at all equilibria have no eigenvalues with zero real part; not all results that follow apply otherwise

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The domain of attraction (DOA)

The domain of attraction of an a.s. equilibrium is the set of all initial conditions which converge to it:

D(xe) = {x0 : lim

t→∞x(x0,t) = xe}

• The DOA is an open, connected, invariant set
• The boundary of the DOA is an invariant set
• The boundary of the DOA contains critical elements (other equilibria, limit-

cycles, etc)

• Lyapunov’s direct method allows to obtain analytic estimates of the DOA

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Estimation of DOA

• By topological characterization
• By invariant set theory
• By Lyapunov’s direct method, which we prefer because it provides:

– analytical expressions for the DOA – control design tools

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Lyapunov estimation of DOA

Let :

• xe = 0 and V(xe) = 0, for simplicity
• La = {x : V(x) ≤ a} be the level sets of the Lyapunov function
• R− = {x : ˙

V(x) < 0}

• La and R− be connected, so we don’t get overwhelmed by the notation

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Lyapunov estimation of DOA

• Note that ˙

V(x) = 0 ∀x ∈ ∂R−

• The level sets contained in R− are invariant sets
• Trajectories starting at a bounded level set converge to the equilibrium
• Then ˆ

D = supLa⊆R−La, with La bounded, is the best estimate that can be

• btained with a given Lyapunov function
• The perfect Lyapunov function: ˆ

D = D

• A good Lyapunov function: ˆ

D ≈ D

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Lyapunov estimation of DOA

• How to obtain Lyapunov functions?

– Quadratic: V(x) = xTPx ∗ can always be obtained solving Lyapunov equation: JT(xe)P+JT(xe)P = −Q P,Q > 0 – Krasovskii: V(x) = f(x)TPf(x) – Lur’e-Postnikov – etc

• The methods are not always successful in finding a Lyapunov function
• Even when they succeed in finding a Lyapunov function, it can be a bad one

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The issues

• There is no general systematic way to derive a good Lyapunov function
• Estimates of DOA’s can be very poor
• "Bad" Lyapunov functions have little value for control design: improving the

estimate does not imply improving the real DOA

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Example - the pendulum

M

• L

T

˙ x1 = x2 ˙ x2 = −a1sin(x1)−a2x2 +b where x1 = δ is the angle, x2 is the speed, b is the torque, ai > 0. There are infinite equilibria given by xe = [x1e x2e]T = [arcsin b

a1 0]T

Let a1 = 2, a2 = 0.1 and b = 0 and study the stability of xo

e = [0 0]T

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Example - the pendulum

Consider a quadratic Lyapunov function VQ(x) = xTPx By solving the Lyapunov equation for Q = I we get P =

• 15.025

0.25 0.25 7.5

• The Lyapunov derivative is

˙ VQ(x) = 2xTPf(x) = −x1sinx1 −x2

2 +15(x1x2 −sinx1x2)

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Example - the pendulum

−4 −3 −2 −1 1 2 3 4 −4 −3 −2 −1 1 2 3 4 x1 x2

Figure 1: Level curves of the quadratic Lyapunov function and the boundary of

R−

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−4 −3 −2 −1 1 2 3 4 −4 −3 −2 −1 1 2 3 4 x1 x2

Figure 2: Real DOA and its estimate with the quadratic Lyapunov function

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Example - the pendulum

Now consider the Lure-Postnikov (energy-like) Lyapunov function V(x) = 1 2x2

2 +

x1e

x1

(b−a1sin(ς))dς = 1 2x2

2 +b(x1e −x1)+a1(cosx1e −cosx1)

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6 x1 x2

Figure 3: Level curves of the Lur’e-Postnikov Lyapunov function

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Its derivative is ˙ V(x) = −a2x2

2 ≤ 0 and a.s. is established by the invariance

principle. Then R− = ℜ2 and ˆ

D is the largest bounded level curve

−4 −3 −2 −1 1 2 3 4 −4 −3 −2 −1 1 2 3 4 x1 x2

Figure 4: Real and estimated DOA’s for the pendulum

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Parenthesis - how did I plot the real DOA?

• The stable manifold of an equilibrium Ms(xe) = {x0 : limt→+∞x(x0,t) = xe}
• The unstable manifold of an equilibrium Mu(xe) = {x0 : limt→−∞x(x0,t) = xe}
• The stable and unstable manifolds of an equilibrium are, by definition,

invariant sets. Theorem 1. Let the jacobian J(xe) = ∂f(x)

x

|x=xe have rs eigenvalues in the LHP and ru in the RHP . Then dimMs(xe) = rs and dimMu(xe) = ru. Moreover, the subspace generated by the "stable" (unstable) eigenvectors of J(xe) is tangent to Ms(xe) (Mu(xe)).

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Parenthesis - how did I plot the real DOA?

Theorem 2. The boundary of the domain of attraction ∂D is the union of the stable manifolds M i

s of all the critical elements that lie at its boundary:

∂D =

• i=1,2,...

M i

s

For the pendulum there are two equilibria at the boundary of the DOA: xu

e =

[π 0]T and −xu

• e. So

∂D = {x0 : lim

t→−∞x(x0,t) = xu e}

• {x0 : lim

t→−∞x(x0,t) = −xu e}

Simulate ˙ x = −f(x) with x(0) = xu

e +εvs where vs is the eigenvector associated

to the stable eigenvalue of J(xu

e) and repeat for −xu e

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Short history of DOA estimation

• 1957 (1964) V.I. Zubov - series expansion of the exact boundary of DOA -

numerically unstable

• 1971 E.J. Davidson and E.M. Kurak - Automatica:

the best quadratic Lyapunov functions

• 1985 A. Vanelli and M. Vidyasagar - Automatica: similar to Zubov´s
• 1988 H.D. Chiang et all - IEEE TAC: characterization of DOA’s
• 1989 H.D. Chiang and J.S. Thorp - IEEE TAC: iterative improvement of the

Lyapunov function

• 2001 N.G. Bretas and L.F. Alberto - IEEE TC&S: generalization of the

invariance principle

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LMI based Lyapunov functions

Transform the nonlinear system into a singular system with linear differential part and rational algebraic part: ˙ x = A1x+A2ξ(x), Ω1(x)x+Ω2(x)ξ(x) = 0 (3) where Ω1(x) and Ω2(x) are affine matrix functions of x. For this class of systems, there exists a Lyapunov function in the form V(x) = xTP(x)x where P(x) is a quadratic function of x; this Lyapunov function can be found by solving a set of LMI’s.

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LMI based Lyapunov functions

Example: the pendulum again        ˙ x =   1 a1  x+   a2 β1 1  ξ = Ω1x+Ω2ξ (4) where auxiliary vectors and matrices have been defined as Ω1 =       τ2 −a4 x3 I3       , Ω2 =       −1 1 −a5 τ2 −1 −x 1 −τ2       , x =   x1 x2 x3  , ξ =     τ2x2 τ1 τ2 1    . (5)

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LMI based Lyapunov functions

Theorem 3. Let Bx be a given polytope. Suppose that the matrices P, Lk (for k = 1,...,ne), R and S are a solution to the following LMIs at V (Bx): P+RC +C′R′ > 0   1

• a′

k

• ak
• (P+LkC +C′L′

k)

  ≥ 0 , for k = 1,...,ne

• (A′

a1P+PAa1)

PAa2 A′

a2P

• +SE +E′S′ < 0

Then, V(x) = x

′P(x)x is a Lyapunov function for Bx and ϒ {x : V(x) ≤ 1} ⊂ D. INMA - FSA - UCL Louvain-la-Neuve c

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LMI based Lyapunov functions

• Maximization of the DOA can be included
• The procedure is systematic, the problem is convex and very good Lyapunov

functions have been obtained in several examples

• However the result is conservative and the computations are heavy

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Control design with Lyapunov functions

A function V(·) is a control Lyapunov function (CLF) if: ∂V(x) ∂x f(x) < 0 ∀x = 0 : ∂V(x) ∂x g(x) = 0 A stabilizing control law is then given by Sontag’s formula Other stabilizing control laws exist

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Even shorter history of control Lyapunov functions

Theory has been concerned almost exclusively with global a.s.

• 1966 - V.I. Zubov: introduction of LgV control
• 1978 - Jurdjevic ans Quinn - J. of Differential Equations: characterization of

LgV control, passivity

• 1983 - E.D. Sontag - SIAM JC&O: introduction of the concept of control

Lyapunov functions

• 1996 - R. Freeman and P

.V. Kokotovic: disturbance rejection

• 1997 - R. Sepulchre and P

.V. Kokotovic: constructive nonlinear control

• 1999 - A.S. Bazanella, A.S. e Silva and P

.V. Kokotovic - IEEE TAC: passivity- based dynamic controller to enlarge the DOA

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LgV control

Let V(x) be a Lyapunov function for (1):

• V(0) = 0,

V(x) > 0 ∀x ∈ R+

• ˙

Vol(x) = ∂V(x)

∂x f(x) = LfV(x) < 0 ∀x ∈ R−

Consider the control u = −k∂V(x)

∂x g(x) = −kLgV(x), k > 0 in (2). Then V(x) is a

Lyapunov function for the closed-loop system: ˙ Vcl(x) = ∂V(x)

∂x [f(x)+g(x)u] = ˙

Vol(x)−k[∂V(x)

∂x g(x)]2 < ˙

Vol(x) < 0 ∀x ∈ R− It follows that

• the closed-loop system is more damped
• the ESTIMATE of the DOA is larger in closed-loop

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IDA control design

The interconnection and damping assignment (IDA) design applies to systems in the port–controlled Hamiltonian form: ˙ x = (J−R)∂V ∂x +gu where R = R⊤ ≥ 0 is the damping matrix and J = −J⊤ is the interconnection matrix With the system described in this form, the Lyapunov derivative can be written as ˙ V(x) = −(∂V ∂x (x))TR(x)∂V ∂x (x) ≤ 0

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IDA control design

Stabilization in IDA control is achieved assigning the closed–loop dynamics ˙ x = (Jd −Rd)∂Vd ∂x (x) where Vd(x) is the desired closed-loop Lyapunov function, Jd = −J⊤

d and Rd =

R⊤

d ≥ 0 are desired interconnection and damping matrices.

Requires solution of a PDE LgV control is a special case in which Jd = J

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Example - the SMIB

Application to power systems - assess and enhance the system’s security The "synchronous machine infinite bus system" (SMIB) ˙ x1 = x2 ˙ x2 = P−b1x3sin(x1)−Dx2 ˙ x3 = b7cos(x1)−b9x3 +E

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Example - the SMIB

The best general Lyapunov function to date is the Lur’e-Postnikov V(x) = 1 2x2

2 +b1x3(cosx1∗ −cosx1)−P(x1 −x1∗)+ b1b9

2b7 (x3 −x3∗)2 Figure 5: Estimated DOA for the SMIB

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Example - the SMIB

The IDA control law: u = −kvb1(cosx1∗ −cosx1)−α1α2(b3 b1 +kv)˜ x1 −α1x2 −(b3 b1 α2 −b4 +kvα2)˜ x3

−1 −0.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 x1 x3

Figure 6: Open-loop and closed-loop estimate of the DOA’s for the SMIB (projections)

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Example - the SMIB

The Lyapunov function was not that good, so ...

−2 −1 1 2 3 4 0.5 1 1.5 2 2.5 3 −20 −15 −10 −5 5 10 15 20 25 x1 x3 x2 X com R Xe com R Xi com R X sem R Xe sem R Xi sem R

Figure 7: The estimate is much smaller than the real DOA for the SMIB

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Passivity-based control - conclusions

• IDA enlarges R− and allows shaping of La, so it tends to enlarge ˆ

D

• In all the many examples tried IDA enlarged ˆ

D and D

• LgV enlarges ˆ

D; in some examples it does not enlarge D!

• LgV and IDA are only for open-loop stable systems
• Other (similar) passivation procedures exist for other classes of systems

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Dynamic extensions

Under parameter disturbances, we don’t know the equilibrium Then the control law u = φ(x,xe) can not be implemented Solution : consider the equilibrium as an unknown parameter and adapt: u = φ(x,θ) ˙ θ = ψ(x,θ) The adaptation law ψ(x,θ) can be designed for local stability This solution generalizes the usual solution for this problem (wash-out filters), which easily fails

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Properties of the dynamic extension

For passivity-based controllers the dynamic extension inherits the properties

• f the controller:
• the closed-loop system is more damped
• the ESTIMATE of the DOA is larger in closed-loop (for LgV)

Examples: REAL DOA is larger with the dynamic controller than with the

• riginal static (both for LgV and IDA)

Further generalization: Hassouneh et all, ACC2004

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Concluding remarks

• Nonlinear control theory has only recently turned into nonglobal a.s. - a more

realistic objective

• Systematic ways to obtain good (control) Lyapunov functions for open-loop

a.s. systems - numerical methods

• Design tools that provide closed-loop Lyapunov functions

– very encouraging results for some classes of systems – much more to be done

• Dynamic extension controllers provide smoother control, robust regulation

and larger DOA’s - an open field

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