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Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems

Carlos E. de Souza Department of Systems and Control National Laboratory for Scientific Computing (LNCC/MCTI) Petr´

• polis, RJ, Brazil

Seminar presented at Universit´ e Catholique de Louvain (UCL) September 25, 2014

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 1

Focus of this Seminar

Deals with the problem of regional stabilization for, possibly

• pen-loop unstable, discrete-time nonlinear quadratic systems

subject to persistent magnitude bounded disturbances. The class of quadratic systems is an extension of bilinear systems, where quadratic terms in the state variables are also considered. Nonlinear quadratic systems can precisely represent a large class

• f systems, such as, distillation columns, induction motors, heating

and air conditioning systems, and Lokta-Volterra type predator-prey systems.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 2

Quadratic systems also find application as a quadratic approxi- mation around an operation point (via a Taylor series expansion)

• f a general nonlinear system.

We address the design of a static nonlinear state feedback controller achieving local input-to-state stability with a guaranteed bounding set for the system state trajectories under nonzero initial conditions and persistent magnitude bounded disturbances. The control gain can be quadratic in the state variables (which are potentially less conservative than linear controllers), and the design is in terms of numerically tractable linear matrix inequality (LMI) problems (state-dependent LMIs).

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 3

Motivation for Input-to-State Stability (ISS)

ISS (Sontag, TAC 1989) is a stability notion that characterizes the input/output (i/o) and internal behavior of nonlinear systems subject to a disturbance input. The notion of ISS encompasses the paradigms of i/o and state-space stability. ISS describes the dependence of the size of the system state trajectory on the magnitude of the initial state and disturbance input.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 4

Linear Matrix Inequalities (LMIs)

• Standard form:

F(x) = F0 + x1F1 + · · · + xpFp > 0 – x = [ x1 · · · xp ]′ ∈ Rp is the variable to be determined; – Fi’s are given symmetric matrices; – The inequality means that the matrix F(x) is positive definite.

• Matrices as variables: e.g., consider the matrix inequality:

A′P + PA < 0 . A is a given matrix; P = P ′ is the variable. The previous inequality can be written as an LMI in the entries of P.

• LMIs can be numerically efficiently solved via polynomial-time

algorithms.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 5

OUTLINE

Problem Formulation Features of the Control Method Controller Design An Example Concluding Remarks

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 6

PROBLEM FORMULATION

Consider the discrete-time nonlinear quadratic system S: x+ = A(x)x(k) + B(x)u(k) + E(x)d(k), k ∈ Z+, x+ := x(k+1), x(0) = x0, x ∈ X ⊂ Rn

• u(k) ∈ Rnu : control input;

d(k) ∈ D⊂ Rnw : disturbance input;

• X : polytopic region of the state-space containing the origin x=0;
• D : set of admissible disturbances

D = {s ∈ Rnw : ∥s∥ ≤ δ}

δ : a given positive scalar defining the “size" of D.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 7

• A(x), B(x), E(x) : affine matrix functions of x = [ x1 · · · xn ]′ :

A(x)=A0+

n

• i=1

xiAi, B(x)=B0+

n

• i=1

xiBi, E(x)=E0+

n

• i=1

xiEi Ai, Bi, Ei, i = 0, 1, . . . , n : given constant real matrices.

Admissible Stabilizing Controllers

We consider static nonlinear state feedback controllers K : Rn → Rnu u(x) = K(x)x, K(x) = K1(x) + K2(x) K1(x) = K0 + n

i=1 xiKi,

K2(x): quadratic in x

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 8

Regional Stabilization Problem

For a given state-space region X, and sets D and R0 of admissible initial states satisfying R0 ⊂ X (given, or to be found), determine a controller K such that: the closed-loop system is locally input-to-state stable; and the state trajectory x(k) driven by any x0 ∈ R0 and d(k) ∈ D stays confined to some state-space region R containing the equilibrium point x = 0 and satisfying R0 ⊆ R ⊂ X. – The set R is referred to as trajectories bounding set. – In the absence of d, R is a stability region for the closed-loop system, i.e., a set of initial states such that x(k)→0 as k → ∞.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 9

Input-to-State Stability

Consider the nonlinear system S0

x+ = f(x, ν), x ∈ Rn, ν ∈ Rnν, x(0) = x0, f(0, 0) = 0

• Definition. System S0 is input-to-state stable if there exist positive

scalars ρ1 and ρ2, a KL-function α1(·, ·), and a K-function α2(·) s.t.

∥x(k, x0, ν)∥ ≤ α1(∥x0∥, k) + α2(∥ν∥∞), ∀ k ∈ Z+, ∀ x0 : ∥x0∥ ≤ ρ1, ν : ∥ν∥∞ ≤ ρ2

Input-to-State Stability

Consider the nonlinear system S0

x+ = f(x, ν), x ∈ Rn, ν ∈ Rnν, x(0) = x0, f(0, 0) = 0

• Definition. System S0 is input-to-state stable if there exist positive

scalars ρ1 and ρ2, a KL-function α1(·, ·), and a K-function α2(·) s.t.

∥x(k, x0, ν)∥ ≤ α1(∥x0∥, k) + α2(∥ν∥∞), ∀ k ∈ Z+, ∀ x0 : ∥x0∥ ≤ ρ1, ν : ∥ν∥∞ ≤ ρ2

∥ν∥∞ := supk∈Z+ ∥ν(k)∥; K-function: positive definite and strictly increasing; KL-function β(s, t): (i) β(·, t) is a K-function for each t ≥ 0; (ii) decreases to 0 as t → ∞ for each s ≥ 0.

Input-to-State Stability

Consider the nonlinear system S0

x+ = f(x, ν), x ∈ Rn, ν ∈ Rnν, x(0) = x0, f(0, 0) = 0

• Definition. System S0 is input-to-state stable if there exist positive

scalars ρ1 and ρ2, a KL-function α1(·, ·), and a K-function α2(·) s.t.

∥x(k, x0, ν)∥ ≤ α1(∥x0∥, k) + α2(∥ν∥∞), ∀ k ∈ Z+, ∀ x0 : ∥x0∥ ≤ ρ1, ν : ∥ν∥∞ ≤ ρ2

• The equilibrium x = 0 is locally asymptotically stable.
• x(t) is bounded and x(k) → 0 if ν(k) → 0 as k → ∞.
• ISS generalizes the bound ∥x(k)∥ ≤ c1∥x0∥αk + c2∥ν∥∞, which

holds for asymptotically stable linear systems x+ =Ax+Bν.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 10

Key Lemma (Jiang and Wang, Automatica, 2005)

Consider the nonlinear system

x+ = f(x, ν), f(0, 0) = 0, x ∈ Rn, ν ∈ U ⊂ Rnν

Let V : Λ → R+ be continuous and b1, . . . , b4 positive scalars s.t.

b1∥x∥2 ≤ V (x) ≤ b2∥x∥2, ∀ x∈ Λ ⊂ Rn; ∆V (x):=V (x+)−V (x) ≤ −b3∥x∥2 + b4∥ν∥2, ∀ x∈ Λ, ν ∈ U

Then, the system is locally input-to-state stable. ( V (x): ISS-Lyapunov function )

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 11

FEATURES OF THE CONTROL METHOD

Scaled System Representation

x+ = A(x)x(k) + B(x)u(k) + δE(x)w(k), w(k) ∈ W, ∀ k ∈ Z+, W =

• s ∈ Rnw : ∥s∥ ≤ 1
• Decomposition of B(x)

B(x) = Ψ(x)′

• B0

B

• ,

Ψ(x)=

• In

Π(x)

• ,

Π(x) := x ⊗ In =

• x1In · · · xnIn

′, B=

• B′

1 · · · B′ n

′

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 12

Polytopic Region X

X is assumed to be a symmetric polytope with respect to the state-space origin.

• Intersection of half-spaces representation

X = {x ∈ Rn : | c′

ix | ≤ 1, i = 1, . . . , r },

ci ∈ Rn, i = 1, . . . , r : define the 2r faces of X

• Convex hull representation

X = Co{ v1, v2, . . . , vκ } vi ∈ Rn, i = 1, . . . , κ : vertices of X

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 13

ISS-Lyapunov Function Candidate

V (x) = x′Px, P = P ′ > 0

Input-to-State Stability Condition

In order to have the condition ∆V (x) ≤ −b3∥x∥2 + b4∥w∥2 of the Key Lemma satisfied, we impose the following condition:

∆V (x) ≤ β

• ∥w∥2 − V (x)
• , ∀ x∈ X, w∈ W

β: positive scalar to be found.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 14

Set R0 of Admissible Initial States

R0 :=

• x ∈ Rn : x′P0 x ≤ 1
• P0 = P ′

0 > 0 such that R0 ⊂ X.

Trajectories Bounding Set R

Defined by a normalized level set of the Lyapunov function V (x): R := { x ∈ Rn : x′P x ≤ 1 } and such that R0 ⊆ R ⊂ X.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 15

Towards an LMI-based Design

• Variable transformations

ξ1 = Q−1x, ξ2 = Z−1(x)x+, Q = P −1

Z(x): nonsingular quadratic matrix in x to be found. Without loss of generality, we let

Z(x) = Ψ(x)′ZΨ(x), ( Ψ(x) = [ In (x ⊗ In)′ ]′ ) Z : constant matrix to be found s.t. Z(x) is nonsingular over X.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 16

• Controller parameterization

F(x) = K(x)Q ⇔ K(x) = F(x)Q−1

As F(x) is quadratic in x, w.l.o.g. we can let

F(x) = F1(x) + F2(x) F1(x) = F0 + n

i=1 xiFi,

F2(x) = Πu(x)′F Π(x) Πu(x) = x ⊗ Inu, Π(x) = x ⊗ In F0, F1, . . . , Fn : constant matrices to be found.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 17

• The condition ∆V (x) ≤ β
• ∥w∥2−V (x)
• is cast as the inequality

η′ Φ(x, δ)η < 0, ∀ η ∈ R2(n+n2)+nw : Ω(x)η = 0, η ̸= 0,

• Φ(x, δ) = Φ(x, δ) + Γ(x)′ Q−1Γ(x)

Φ(x, δ): a matrix affine in x, δ and in the matrices Z, F and Fi, i=0, . . . , n, and bilinear in Q and β Γ(x): affine matrix in x and Z

• Ω(x): affine matrix in x

η:= η(x) =

• ξ′

1

(Π(x)ξ1)′ ξ′

2

(Π(x)ξ2)′ w′ ′ ,

• ξ1 = Q−1x, ξ2 = Z−1(x)x+, Π(x) = x ⊗ In
• Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 18

Finsler’s Lemma. The previous constrained inequality η′ Φ(x, δ)η < 0, ∀ η ∈ R2(n+n2)+nw : Ω(x)η = 0, η ̸= 0 holds if there exists a matrix L such that

• Φ(x, δ) + He
• L

Ωa(x)

• < 0
• He(X) := X + X′
• Ωa(x) =

⎡ ⎢ ⎢ ⎣

• Ω(x)
• N(x)

N(x)

• ⎤

⎥ ⎥ ⎦ Ωa(x)η = 0

• N(x): linear in x and s.t. N(x)Π(x)=0 ( linear annihilator of Π(x) )

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 19

• The linear annihilator N(x) of Π(x) is introduced to reduce

the conservatism of using state-dependent LMIs to ensure the feasibility of nonlinear state-dependent inequalities (the matrix

• Φ(x) and the vector η(x) are coupled via x).
• Taking into account that

Π(x) := x ⊗ In =

• x1In · · · xnIn

′ the following linear annihilator of Π(x) is considered: N(x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x2In −x1In 0n · · · 0n 0n x3In −x2In · · · 0n . . . ... ... ... . . . 0n · · · 0n xnIn −xn−1In ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 20

• By Schur’s complements and convexity arguments
• Φ(x, δ) + He
• L

Ωa(x)

• < 0, ∀ x ∈ X,
• Φ(x, δ) = Φ(x, δ) + Γ(x)′ Q−1Γ(x)

⇐ ⇒ ⎡ ⎣ Φ(vi, δ) + He

• L

Ωa(vi)

• Γ(vi)′

Γ(vi) −Q ⎤ ⎦< 0, i = 1, . . . , κ (C.1) (vi are the vertices of X )

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 21

Example of the Potential of Linear Annihilator

(Trofino, ACC’00) Let the inequality x′M(x, P)x < 0, ∀ x ∈ X ⊆ R2, x̸=0 M(x, P) = A(x)′P + PA(x), P =P ′ >0 x=

• x1

x2

• ,

A(x) =

• −2−2x2

x1+x2 x1+x2 −2−x1

• ,

X(κ)=

• x ∈ R2 : |xi|≤κ, i=1, 2
• .

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 22

Example of the Potential of Linear Annihilator

(Trofino, ACC’00) Let the inequality x′M(x, P)x < 0, ∀ x ∈ X ⊆ R2, x̸=0 M(x, P) = A(x)′P + PA(x), P =P ′ >0 x=

• x1

x2

• ,

A(x) =

• −2−2x2

x1+x2 x1+x2 −2−x1

• ,

X(κ)=

• x ∈ R2 : |xi|≤κ, i=1, 2
• .

The above inequality arises in the Lyapunov stability analysis of the system ˙ x = A(x)x.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 22

• Applying the condition

M(x, P) < 0, ∀ x∈ V(X) ⇒ x′M(x, P)x< 0 is guaranteed to hold for κ ≤ 0.5.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 23

• Applying the condition

M(x, P) < 0, ∀ x∈ V(X) ⇒ x′M(x, P)x< 0 is guaranteed to hold for κ ≤ 0.5.

• By the condition

M(x, P) + He

• L Nx(x)
• < 0, ∀ x∈ V(X),

Nx(x) = [x2 −x1] (Nx(x)x = 0) ⇒ x′M(x, P)x< 0 is guaranteed to hold for κ → ∞.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 23

Inclusion R0 ⊆ R

P0 − P ≥ 0 ⇐ ⇒

• P0

In In Q

• ≥ 0

(C.2) ( P = Q−1 )

Invariance of R (R⊂ X)

1 − c′

i Q ci ≥ 0, i = 1, . . . , r

(C.3) ( ci’s define the faces of X )

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 24

CONTROLLER DESIGN

Given a polytopic region X and sets R0 and D, suppose that for some β ∈ (0, 1) there exist matrices F, F0, F1, . . . , Fn, L, Q and Z satisfying the LMIs (C.1)-(C.3). Then, the controller u(x)=F(x)Q−1x ( F(x)=F0+ x1F1 + · · · + xnFn+ Πu(x)′F Π(x) ) ensures that: (a) the closed-loop system is locally input-to-state stable and V (x) = x′Q−1x is an ISS-Lyapunov function in X; (b) the state trajectory x(k) driven by any x0 ∈ R0 and d(k) ∈ D remains in R=

• x∈ Rn : x′Q−1x ≤ 1
• for all k∈ Z+, and x(k)→ 0

if d(k)→ 0 as k → ∞. Moreover, whenever d(k)≡ 0, R is a stability region, i.e. for any x0 ∈ R and d(k) ≡ 0, then lim

k→∞ x(k) = 0.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 25

Applications of the Stabilization Result

Maximizing the set R0 for a given D

R0 can be maximized by setting R0 =R and maximizing

• det(Q)

( which is proportional to the volume of R0). This can be achieved by solving the following LMI optimization problem for a fixed β ∈ (0, 1): max

F,F0,F1,...,Fn,L,Q,Z log(det(Q))

subject to (C.1) and (C.3).

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 26

Applications of the Stabilization Result

Maximizing the set R0 for a given D

R0 can be maximized by setting R0 =R and maximizing

• det(Q)

( which is proportional to the volume of R0). This can be achieved by solving the following LMI optimization problem for a fixed β ∈ (0, 1): max

F,F0,F1,...,Fn,L,Q,Z log(det(Q))

subject to (C.1) and (C.3).

Maximizing the set D for a given R0

Solve the following LMI optimization problem for a fixed β ∈ (0, 1): max

F,F0,F1,...,Fn,L,Q,Z,δ δ

subject to (C.1) - (C.3) and δ >0.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 26

Applications of the Stabilization Result

Maximizing the set R0 for a given D

R0 can be maximized by setting R0 =R and maximizing

• det(Q)

( which is proportional to the volume of R0). This can be achieved by solving the following LMI optimization problem for a fixed β ∈ (0, 1): max

F,F0,F1,...,Fn,L,Q,Z log(det(Q))

subject to (C.1) and (C.3).

Maximizing the set D for a given R0

Solve the following LMI optimization problem for a fixed β ∈ (0, 1): max

F,F0,F1,...,Fn,L,Q,Z,δ δ

subject to (C.1) - (C.3) and δ >0.

The maximal values of det(Q) and δ are obtained by performing

a line search on β over (0, 1).

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 26

AN EXAMPLE

Consider the system of the example in Bitsoris and Athanasopoulos (IFAC’ 08) with the addition of quadratic terms in the state variables and a disturbance input d(k) ∈ D : x(k+1) = A(x)x(k) + B(x)u(k) + E(x)d(k) where x=

• x1

x2

• , A(x)=
• 0.8 + 0.2x1

0.5 0.4 1.2 − 0.2x2

• ,

B(x)=

• 1 + 0.45(x1+ x2)

2 + 0.3(x1− x2)

• , E(x)=
• 0.5(1+ x1)

0.3(1+ x2)

• .

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 27

Problem 1

We aim to design a state-feedback control law u = K(x)x, with K(x) quadratic in x, to ensure local ISS of the controlled system while maximizing the set R0 of admissible initial states with X and D given a priori. For that, we set R=R0. We have considered X =

• x ∈ R2 : | xi | ≤ α, i = 1, 2
• , α = 1.2

D = { s ∈ R : | s | ≤ δ }, δ = 0.1 Control law obtained: u = −

• 0.0056x3

1 + 0.0027x2 1x2 + 0.0018x2 1 − 0.0258x1x2 2

− 0.0516x1x2 + 0.2800x1 − 0.0059x3

2 − 0.0427x2 2 + 0.5040x2

• Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 28

x1 x2

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

R x(0)

state trajectory

• Fig. 1: The achieved set R0 =R and a closed-loop

state trajectory for d(k) ≡ 0.1.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 29

The set R can be further enhanced by iteratively increasing the size of X ( by increasing α ). The largest X is obtained for α = 2.2. Control law obtained: u = −

• 0.0069x3

1 + 0.0026x2 1x2 − 0.00276x2 1 + 0.0125x1x2 2

− 0.1438x1x2 + 0.3495x1 + 1.23x10−4x3

2 − 0.0801x2 2

+ 0.5663x2

• Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 30

x1 x2

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3

stable trajectory

R

unstable trajectory

X

• Fig. 2: The maximized polytope X, the set R, and two

closed-loop state trajectories for d(k) ≡ 0.1.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 31

Problem 2

First, a control law is designed to maximize the size of the set D = { s ∈ R : | s | ≤ δ } for given sets X and R0 ⊂ X. For the maximum achieved δ, a control law is then designed to minimize the trajectories bounding set R, with R0 ⊆ R ⊂ X. We have considered X =

• x ∈ R2 : | xi | ≤ 0.7, i = 1, 2
• ,

R0 =

• x ∈ R2 : 4x′x ≤ 1
• .

The maximum achieved value of δ was δ = 0.3166. Control law obtained: u = −

• 0.0183x3

1 + 0.0068x2 1x2 + 0.0014x2 1 − 0.0048x1x2 2

− 0.1394x1x2 + 0.3797x1 + 0.0003x3

2 − 0.0844x2 2 + 1.1269x2

• Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 32

x1 x2

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

R R0 x(0)

state trajectory

• Fig. 3: The sets R0 and R, and a closed-loop state trajectory

driven by x0 =[ −0.5 0 ]′ and d(k) ≡ 0.3166.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 33

CONCLUDING REMARKS

An LMI method has been proposed for designing a static nonlinear state feedback controller for locally stabilizing, possibly open-loop unstable, discrete-time nonlinear quadratic systems subject to magnitude bounded disturbances. The controller achieves local input-to-state stability with a guaranteed bounding set for the system state trajectories under nonzero initial conditions and magnitude bounded disturbances.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 34

CONCLUDING REMARKS

An LMI method has been proposed for designing a static nonlinear state feedback controller for locally stabilizing, possibly open-loop unstable, discrete-time nonlinear quadratic systems subject to magnitude bounded disturbances. The controller achieves local input-to-state stability with a guaranteed bounding set for the system state trajectories under nonzero initial conditions and magnitude bounded disturbances. The stabilization method has been extended to: – locally ensure an optimized upper bound on the ℓ∞-induced gain (‘peak-to-peak gain’) of the closed-loop system; – cope with control saturation.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 34

This work has been recently accepted for publication in the International Journal of Robust and Nonlinear Control: C.E. de Souza, D. Coutinho, J. M. Gomes da Silva Jr., “Local Input-to-State Stabilization and ℓ∞-Induced Norm Control of Discrete-Time Quadratic Systems", published online, 2014.

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 35

THANK YOU !

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 36