On Multivariable Cell Structures and Leonov Functions for

On Multivariable Cell Structures and LeonovFunctions for Global Synchronization Analysis in

Power Systems

J. Schiffer1

Joint work with D. Efimov2, R. Ortega3 and N. Barabanov4

1 Brandenburg University of Technology Cottbus-Senftenberg2 INRIA and University ITMO3 Laboratoire des Signaux et Systemes, Supelec4 North Dakota State University

Motivation (1) - Increasingly stressed power systems

Transmission

Distribution

Electricalgrid

Synchronous generatorinterfaced power plants

Operation &monitoring system

Loads

Frequency

2 / 40

Motivation (1) - Increasingly stressed power systems

Transmission

Distribution

Electricalgrid

Synchronous generatorinterfaced power plants

Operation &monitoring system

Solar homes& loads

More renewableenergy production

Frequency → Power systems oftenoperate closer to theirstability limits

→ Need for quicklyverifiable analyticconditions to assesstransient stability

2 / 40

Motivation (2) - Power system model

Second-order multi-machine powersystem with N ≥ 1 nodes and m ≥ 1power lines (in error coordinates)

˙η = B>ω,M ˙ω = −Dω − B

(∇U(η + η∗)−∇U(η∗)

),

with potential function U : Rm → R,

U(η) = −m∑`=1

a` cos(η`),

a` = ViVk |Bik | > 0

Synchronized motion: (η∗, 0N )

Dynamics are 2π-periodic in variables η

η ∈ Rm phaseangles

ω ∈ RN relativeangularfrequencies

M = diag(Mi ) inertiaconstants

D = diag(Di ) dampingconstants

B ∈ RN×m node-edgeincidencematrix

3 / 40

Motivation (2) - Classical transient stability analysis

Main analytic approach for transient stability studies:Lyapunov-based methods

”Classical” energy function:

V (η, ω) =12ω>Mω + U(η + η∗)− η>∇U(η∗)

with

V = −ω>Dω ≤ 0

Usual main technical restrictions:To ensure V (η, ω) is continuously differentiable, need to defineη ∈ Rm

But then V (η, ω) not bounded from below and only locally positivedefinite

→ Lyapunov theory + LaSalle (+ some further technicalities) yieldonly local stability result

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Motivation (3)

→ There is a strong need for new advanced formal methods andtechniques for power system analysis and design

Main objectives of this talk

Break with prevalent conventional purely local treatment of powersystem dynamics

Discuss methods for global power system stability analysis

By doing so, provide different perspectives and insights intostructural power system properties

5 / 40

Motivation (3)

→ There is a strong need for new advanced formal methods andtechniques for power system analysis and design

Main objectives of this talk

Break with prevalent conventional purely local treatment of powersystem dynamics

Discuss methods for global power system stability analysis

By doing so, provide different perspectives and insights intostructural power system properties

5 / 40

Outline

1 State periodic systems

2 (Classical) Cell structure approach

3 Application example: Global synchronization analysis ofSingle-Machine-Infinite-Bus System

4 Cell structures and Leonov functions - Multivariable case

5 Application example: Global synchronization analysis ofmulti-machine power system

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State periodic systems

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Let f : Rn → Rn with f (0) = 0

Consider the system

x(t) = f (x(t)), x(0) = x0, t ≥ 0with state x(t)

Assumption

Let x = (z, θ), where z ∈ Rk and θ ∈ Rq , n = k + q, k > 0, q > 0

The vectorfield f is 2π-periodic with respect to θ

Periodicity of f with respect to θ implies existence of multipleequilibria (in addition to origin)

Examples: AC power systems and microgrids, phase-lockedloops, complex networks of oscillators,. . .

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Let f : Rn → Rn with f (0) = 0

Consider the system

x(t) = f (x(t)), x(0) = x0, t ≥ 0with state x(t)

Assumption

Let x = (z, θ), where z ∈ Rk and θ ∈ Rq , n = k + q, k > 0, q > 0

The vectorfield f is 2π-periodic with respect to θ

Periodicity of f with respect to θ implies existence of multipleequilibria (in addition to origin)

Examples: AC power systems and microgrids, phase-lockedloops, complex networks of oscillators,. . .

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(Classical) Cell structure approach

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Cell structure approach - Scalar case

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Application example: Global synchronizationanalysis of Single-Machine-Infinite-Bus System

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Single generator infinite bus (SMIB) scenario

SG

edqGenerator

Power line∼

vdqInfinite bus

SMIB model with constant field currentderived from first principles

θ = ω − ωs,

Jω = −Dω + Tm − b (iq cos(θ) + id sin(θ)) ,

Lid = −Rid − Lωs iq + bω sin(θ)− vd ,

Liq = −Riq + Lωs id + bω cos(θ)− vq

Model shown in dq-coordinates withdq-transformation angle ϕ = ωst

θ ∈ R rel. rotor angleω ∈ R elec. freq. of

rotorωs ∈ R grid freq.idq ∈ R2 stator currentvdq ∈ R2 infinite bus

voltageedq ∈ R2 SG voltageJ ∈ R>0 moment

of inertiaD ∈ R>0 damping

coefficientTm ∈ R mechanical

torqueL ∈ R>0 stator + line

inductanceR ∈ R>0 stator + line

resistanceb ∈ R constant

12 / 40

Almost global stability analysis

Stability analysis consists of two main steps1) Establish convergence of bounded trajectories (to asymptotically

stable equilibria)

2) Derive sufficient conditions for boundedness of trajectories

Need a continuous Lyapunov-like function to obtain convergenceresult → view system evolving in R4 → θ is not a–priori bounded!

Analysis conducted under following assumption

The parameters of the SMIB system are such that there existtwo equilibria and b > 0.

Error coordinates

θ = θ − θs, ω = ω − ωs, id = id − isd , iq = iq − isq

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Stability analysis consists of two main steps1) Establish convergence of bounded trajectories (to asymptotically

stable equilibria)

2) Derive sufficient conditions for boundedness of trajectories

Need a continuous Lyapunov-like function to obtain convergenceresult → view system evolving in R4 → θ is not a–priori bounded!

Analysis conducted under following assumption

The parameters of the SMIB system are such that there existtwo equilibria and b > 0.

Error coordinates

θ = θ − θs, ω = ω − ωs, id = id − isd , iq = iq − isq

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Convergence of bounded solutions

Proposition (Convergence of bounded solutions)

Consider the SMIB system verifying the inequality

4RD[(Lωs)2 + R2

]> (Lbωs)2.

Every bounded solution tends to an equilibrium point

Claim established by constructing Lyapunov–like function andinvoking LaSalle’s invariance principle

Physical interpretation

4DG(R2 + X2)2 > X2b2 ⇔ 4DRX> |B|b2

→ High damping factor D and high R/X ratio are beneficial toensure convergence

→ High value of |b|, i.e., high excitation and consequently largeEMF amplitude, deteriorate the likelihood of convergence

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Convergence of bounded solutions

Proposition (Convergence of bounded solutions)

Consider the SMIB system verifying the inequality

4RD[(Lωs)2 + R2

]> (Lbωs)2.

Every bounded solution tends to an equilibrium point

Claim established by constructing Lyapunov–like function andinvoking LaSalle’s invariance principle

Physical interpretation

4DG(R2 + X2)2 > X2b2 ⇔ 4DRX> |B|b2

→ High damping factor D and high R/X ratio are beneficial toensure convergence

→ High value of |b|, i.e., high excitation and consequently largeEMF amplitude, deteriorate the likelihood of convergence

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Boundedness of solutions via cell structure principle

Result below is direct corollary of Theorem 16 in G. Leonov,”Phase synchronisation. Theory and applications”, 2006

Proposition (Boundedness of solutions)

Let χ = (z, θ) and z = (ω, id , iq)

Suppose there exists a function V : R× R3 → R such that

V (03, 0) = 0, V (z, 0) > 0 ∀z ∈ R3 \ 03

Assume there exist positive real numbers ε and λ such that alongthe solutions of the SMIB system the function

V (χ) = V (χ)− ε

2θ2

verifies˙V (χ) ≤ −λV (χ).

Then, all solutions χ of the SMIB system are bounded

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Main result (2): Almost global stability

εmin = infε∈R>0

c∫ θ

0

[sin(θs − φ+ s)− sin(θs − φ)

]ds ≤ ε

2θ2, ∀θ ∈ R

,

g(λ) = 4(

R − Lλ2

)[((Lωs)2 + R2

)(D − Jλ

2

)− 2εmin

λ

]

Assumption

There exists λmax > 0—a point of local maximum of the functiong(λ)—such that

2R > λmax L and g(λmax ) > (Lbωs)2

Theorem (Almost global stability)

Consider SMIB system verifying above assumption

The equilibrium point (θs, ωs, isd , isq ) satisfying |θs − φ| < π

2 (modulo2π) is locally asymptotically stable and almost globally attractive,i.e., for all initial conditions, except a set of measure zero, thesolutions of the SMIB system tend to that equilibrium point

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εmin = infε∈R>0

c∫ θ

0


]ds ≤ ε

2θ2, ∀θ ∈ R

,

g(λ) = 4(

R − Lλ2

)[((Lωs)2 + R2

)(D − Jλ

2

)− 2εmin

λ

]

Assumption







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εmin = infε∈R>0

c∫ θ

0


]ds ≤ ε

2θ2, ∀θ ∈ R

,

g(λ) = 4(

R − Lλ2

)[((Lωs)2 + R2

)(D − Jλ

2

)− 2εmin

λ

]

Assumption







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References and extensions

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Cell structures and Leonov functions -Multivariable case

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Multivariable cell structure (MCS) approachMain technical difficulty in case of several periodic states:cells can not only be connected via equilibria→ their intersection is not compact (see also Noldus’ paper)

Solution provided via MCS and Leonov function concept in1,2

Main properties of a Leonov function (loosely speaking)It is sign-indefinite with respect to periodic states and radiallyunbounded with respect to non-periodic states

It is negative definite with respect to distance to a set, whichseparates equilibria of system

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′′0

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′1

Ω′2

Ω′4

1D. Efimov, J. Schiffer A new criterion for boundedness of solutions for a class of periodic systems, ECC’20182D. Efimov, J. Schiffer On boundedness of solutions of state periodic systems: a multivariable cell structure approach,

IEEE TAC’1919 / 40

Multivariable cell structure (MCS) approachMain technical difficulty in case of several periodic states:cells can not only be connected via equilibria→ their intersection is not compact (see also Noldus’ paper)

Solution provided via MCS and Leonov function concept in1,2

Main properties of a Leonov function (loosely speaking)It is sign-indefinite with respect to periodic states and radiallyunbounded with respect to non-periodic states

It is negative definite with respect to distance to a set, whichseparates equilibria of system

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′′0

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′1

Ω′2

Ω′4

1D. Efimov, J. Schiffer A new criterion for boundedness of solutions for a class of periodic systems, ECC’20182D. Efimov, J. Schiffer On boundedness of solutions of state periodic systems: a multivariable cell structure approach,

IEEE TAC’1919 / 40

Multivariable cell structure (MCS) approach

Properties of a Leonov function V : R2N−1 → R (for power system):

1) α(|ω|)−ψ(|η|)− g≤ V (η, ω) for all col(η, ω) ∈ R2N−1

2) infcol(η,ω)∈W V (η, ω) > 0 and supcol(η,ω)∈U V (η, ω) ≤ 0

3) V + λ(V ) ≤ 0 for all col(η, ω) ∈ R2N−1

α ∈ K∞, ψ ∈ K, g ≥ 0 is a constant and λ : R→ R is a continuousfunction satisfying λ(0) = 0 and λ(s)s > 0 for all s 6= 0

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′′0

θ1

θ2

|z|

ππ

2π2π

3π

3π

Ω′0

W

Ω′1

Ω′2

Ω′4

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MCS approach - Relaxed requirements

Introduce the following sets with π ≤ c <2π, ε ∈ R>0 and ξ ∈ R>0

Ω = col(η, ω) ∈ R2N−1 : V ≤ 0,

Ω′ε,c = col(η, ω) ∈ R2N−1 : V ≤ ε, |η|∞ < c,

Z = col(η, ω) ∈ R2N−1 : |ω| > ξ

Corollary

Suppose that there exists a Leonov function V : R2N−1 → R for thepower system, such that:

1) supη∈RN−1 ψ(|η|) < +∞,

2) the inequality V + λ(V ) ≤ 0 is verified only for

col(η, ω) ∈ (R2N−1 \ Ω) ∩ (Z ∪ Ω′ε,c).

Then for all initial conditions col(η(0), ω(0)) ∈ R2N−1 the correspondingtrajectories col(η, ω) are bounded ∀t ≥ 0.

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For further details see . . .

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Application example: Global synchronizationanalysis of multi-machine power system

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Power system model - Standard coordinates

Second-order multi-machine powersystem with N ≥ 1 nodes and m ≥ 1power lines

θ = ω,

Mω = −Dω −∇U(θ) + Pnet,

with potential function U : RN → R,

U(θ) = −∑

i,k∈N×Naik cos(θi − θk ),

aik = ViVk |Bik | > 0

Gradient: ∇U(θ) = BAsin(B>θ)

Dynamics are 2π-periodic in θ

θ ∈ Rm phaseangles

ω ∈ RN angularfrequencies



A = diag(aik ) matrixof edgeweights

B ∈ RN×m networkincidencematrix

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Main contributions

Sufficient conditions for almost global synchronization of amulti-machine power system model

Model assumptions:Radial network topology

→ Incidence matrix B has full column rank (m = N − 1)

Synchronous generators are represented by standard swingequation, i.e., model on previous slide

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Power system model - Change of coordinates

Synchronized motion

θs(t) = θs0 + ωst , ωs = 1Nω

∗

Change of coordinates

η = B>θ ∈ R(N−1)

(projection of θ on subspace orthogonal to 1N )

Then,

∇U(θ) = BAsin(B>θ) = BAsin(η) = B∇U(η)

Also, for any α ∈ R,

η∗ = B>(θs0 + ωst + α1N ) = B>θs

0

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Nec. and suff. conditions for existence of equilibria

Proposition

The radial power system possesses equilibria if and only if∥∥∥∥∥A−1B+

(Pnet −

1>N Pnet

1>N K−11NK−11N

)∥∥∥∥∥∞≤ 1 (1)

Then, also all equilibria are isolated

Furthermore, if and only if (1) is satisfied with strict inequality,then the equilibria are given by col(ηs,i

0 , ω∗1N ) (modulo 2π), whereηs,i

0 , i = 1, . . . , 2(N−1), are permutations of the vectors ηs0 and

ηs0 = π1(N−1) − η

s0,

with

ηs0 = arcsin

(A−1B+

(Pnet −

1>N Pnet

1>N K−11NK−11N

))

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Local stability properties of equilibria

Proposition

The equilibrium ηs0 is locally asymptotically stable

All other equilibria are unstable

The Jacobian of the power system dynamics evaluated at anyunstable equilibrium point has at least one eigenvalue withpositive real part

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Power system model - Error coordinates

Second-order multi-machine powersystem with N ≥ 1 nodes and m ≥ 1power lines (in error coordinates)

˙η = B>ω,M ˙ω = −Dω − B

(∇U(η + η∗)−∇U(η∗)

),

with potential function U : Rm → R,

U(η) = −m∑`=1

a` cos(η`),

a` = ViVk |Bik | > 0

Nominal equilibrium point: (η∗, 0N )

Dynamics are 2π-periodic in variables η

η ∈ Rm phaseangles

ω ∈ RN relativeangularfrequencies



B ∈ RN×m node-edgeincidencematrix

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Stability analysis consists of three main steps

1) Derive necessary and sufficient conditions for existence ofequilibria and qualify their stability properties

2) Derive sufficient conditions for boundedness of trajectories usingmultivariable cell structure (MCS) approach(view system evolving in R2N−1 → η is not a–priori bounded!)

3) Establish global convergence of bounded trajectories with”classical” energy function

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Leonov function candidate for power system

Leonov function candidate

V (η, ω) =

[Bζ(η)

ω

]> [Φ ΦD

DΦ M + DΦD

]︸︷︷︸

:=Ψ

[Bζ(η)

ω

]− κ

+ 2[U(η + η∗)− U(η∗)−∇U>(η∗)tanh(η)],

ζ(η) =∇U(η + η∗)−∇U(η∗)

Design matrix Φ ∈ RN×N , Φ = Φ> > 0

Design parameter κ ∈ R≥0

tanh(·) ∈ [−1, 1]n and cos(·) ∈ [−1, 1]n denote element-wise tanh-and cos-function, respectively

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Conditions for almost global stability

Assumption (1)

There exist c ∈ [π, 2π), Φ > 0, ν > 0, β > 0 and µ > 0, such that

Q(η) >0,

infcol(η,ω)∈W

V (η) >0,

supcol(η,0N )∈Ω

−β|Bζ(η+η∗)|2 +1ν|Bdiag(tanh2(η))∇U(η∗)|2 < 0

Lower bound for V for all (η, ω) ∈ R2N−1:

V (η) =2[U(η + η∗)− U(η∗)−∇U>(η∗)tanh(η)] + λmin(Ψ)|Bζ|2 − κ ≤ V ,

Ω =col(η, ω) ∈ R2N−1 : V ≤ 0, 0 < |η|∞ ≤ c

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Main result: Almost global stability of power systems

Proposition (Global boundedness of trajectories)

Consider the power system with Assumption 1

Select

κ = 2N−1∑i=1

|ai sin(η∗i )|

Then, the function V is a Leonov function for the power system

Furthermore, all solutions of the power system are bounded



Denote by X the set of asymptotically stable equilibria

The set X is almost globally asymptotically stable, i.e., for allinitial conditions, except a set of measure zero, the solutions ofthe power system asymptotically converge to a point in X

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Main result: Almost global stability of power systems

Proposition (Global boundedness of trajectories)


Select

κ = 2N−1∑i=1

|ai sin(η∗i )|

Then, the function V is a Leonov function for the power system

Furthermore, all solutions of the power system are bounded



Denote by X the set of asymptotically stable equilibria

The set X is almost globally asymptotically stable, i.e., for allinitial conditions, except a set of measure zero, the solutions ofthe power system asymptotically converge to a point in X

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Discussion on main synchronization condition

Lemma (Q(η) > 0 for uniform damping and inertia constants)

Suppose that D = dIN , M = mIN , d > 0, m > 0

Set β = 1d and select d and ν, such that

1− ν

d− 2

mρd2 −

m2ρ2

d4 > 0

Then Q(η) > 0 for all η∗ ∈ RN−1 and all η ∈ RN−1 as well as someµ > 0

Constant

ρ = supη∈RN−1

‖BAcos(η + η∗)B>‖2 = λmax(BAB>)

ρ can be interpreted as an upper bound on networkinterconnection strength

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Numerical example 1: IEEE 9 bus test system

G1

G2 G3

Kron-reduced model of IEEE 9 bus test system with threegenerators

All system data taken from P. Anderson and A. Fouad, PowerSystem Control and Stability, J.Wiley & Sons, 2002

Admittance between nodes 2 and 3 omitted

Conditions feasible for values up to |η∗1 | = 28

and |η∗2 | = 25

(with c = π)

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Numerical example 1: η∗ = col(22,−25

)

−6.28 −3.14 0 3.14 6.28−6.28

−3.14

0

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η1

η 2

Contour plot of−β|Bζ(η + η∗)|2

+ 1ν |Bdiag(tanh2(η))∇U(η∗)|2

for |η|∞ ≤ 2π

Dashed curvesrepresent level set V = 0

Plots show that withc = π conditions on Ω

and W in Assumption 1are satisfied

36 / 40

Numerical example 2: Microgrid

DG3

L3

3

Y13

1

L1

DG1

Y12

2

DG2

L2

Based on Subnetwork 1 of the CIGRE MV benchmark model(Rudion et al.’06)

37 / 40

Numerical example 2: η∗ = col(20,−21

)

−6.28 −3.14 0 3.14 6.28−6.28

−3.14

0

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η1

η 2

Contour plot of−β|Bζ(η + η∗)|2

+ 1ν |Bdiag(tanh2(η))∇U(η∗)|2

for |η|∞ ≤ 2π

Dashed curvesrepresent level set V = 0

Plots show that withc = π conditions on Ω

and W in Assumption 1are satisfied

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Further details and numerical verification procedure

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Summary & future workMain contribution

Recently developed multivariable cell structure (MCS) approachand concept of Leonov functions

Cell structures provide useful tool for global synchronizationanalysis in AC power systems

Almost global synchronization results for SMIB and multi-machinepower system established by combining LaSalle’s invarianceprinciple with (M)CS approach

Proposed conditions can be verified in a reasonably broad range ofoperating scenarios

Future workProvision of additional robustness measures to account for modeluncertainties

Extension of analysis to more detailed models with meshedtopologies

Derive Control Leonov Functions (CLeFs) and use these to designglobally stabilizing controllers

40 / 40

Documents

On Multivariable Cell Structures and Leonov Functions for