Sistemas com satura¸c˜ao no controle II. Contexto da estabilidadesala225/ia360/PDF/sat_cont.pdf · 2005. 8. 24. · 1 Sistemas com satura¸c˜ao no controle II. Contexto da estabilidade

1

Sistemas com saturacao no controle

II. Contexto da estabilidade

Sophie Tarbouriech

LAAS - CNRS, Toulouse, France

Colaboradores principais :

J.-M. Gomes da Silva Jr (UFRGS), G. Garcia (LAAS-CNRS), I. Queinnec (LAAS-CNRS)

Sistemas com saturacao no controle - IIBrasil, UNICAMP, Agosto 2005

Outline 2

Context of stability

1. Introduction

2. Global stability

3. Semiglobal stability

4. Local stability

5. Invariant sets and local stability

6. Estimates of the region of attraction

7. Analysis problems

8. Synthesis problems


1. Introduction 3

Attraction Region of the Origin

☞ Saturated system:

x(t) = Ax(t) +Bsat(Kx(t)) (1)

☞ The stability is closely related to initial condition.

Definition 1 A set E, containing the origin in its interior, is said to be a region of

asymptotic stability for system (1) if for all x(0) ∈ E, the corresponding trajectory

converges to the origin.

Definition 2 The attraction region of the origin is:

RA(0) = {x(0) ∈ �n;x(t, x(0)) → 0 as t→ ∞}

� By definition E ⊆ RA(0).

� RA(0) corresponds in fact to the exact stability region of the saturated system.


1. Introduction 4

☞ In general, the attraction region RA(0) is non-convex.

☞ In general, such a region is impossible to determine via analytical methods.

� RA(0) could be computed after important numerical efforts:

� Computation of all the feasible trajectories

� Computation of all the equilibrium points (local study of each equilibrium

point)

� Only estimations of RA(0) can be obtained.

� Objective: to obtain such an estimated domain as large as possible

� Choice of a criterium?

☞ In some cases, we have RA(0) = �n (global stability)

☞ But in general RA(0) �= �n (local stability)


2. Global Stability 5

☞ System:

x(t) = Ax(t) +Bu(t) ; − umin ≤ u(t) ≤ umax (2)

Definition 3 The system (2) is said globally asymptotically stable (GAS) if ∀ x0 ∈ �n;

x(t;x0) → 0 as t→ ∞, where x(t;x0) denotes the trajectory of the system issued from x0.

� Related to the null-controllability: the set of null-controllability is the state-space �n

� Basin of attraction: RA(0) = �n

Theorem 1 System (2) is GAS if and only if:

(a) pair (A,B) is stabilizable

(b) the eigenvalues of matrix A are not strictly unstable.

� Theorem 1 gives a necessary and sufficient condition for the global asymptotic

stability of system (2).

� However, it is not constructive in terms of effective determination of a control law.



☞ The first question studied in the literature was the following:

� Is it possible to determine a control law

u(t) = sat(Kx(t)) (3)

that globally stabilizes system (2)?

� In general, the GAS property cannot be achieved with (3).

[Sussman, Sontag, Burgat, Tarbouriech, Teel, Ma, Suarez]

☞ Necessity to consider nonlinear control laws u(t) = sat(K(x(t))).

→ nested saturations

→ solution of a τ -parameterized family of algebraic Riccati equations

→ ...



☞ However, in the case where the open-loop system is asymptotically or critically stable,

it was proven that a control law (3) can be always determined [Tarbouriech et al] .

Theorem 2 Given a symmetric positive definite matrix P ∈ �n×n, a matrix L ∈ �m×n

and a matrix E ∈ �n×n satisfying the following conditions

(i). A′P + PA− P ≤ −L′L− EP .

(ii). EP ≥ 0 and symmetric.

(iii). Pair (B′P +√

2L,A) is observable (detectable).

a suitable controller gain is defined by

K = −Γ(B′P +√

2L)

with Γ ∈ �m×m is a diagonal matrix such that 0 < Γ ≤ Im.

� The tools used to derive such a result are quadratic and Lure Lyapunov functions

with circle and Popov criteria.

� What are the closed-loop performances?


3. Semiglobal Stability 8

☞ In practice, the evolution of a system is often restricted to a specific region of the

state space around the equilibrium point

� This region can be viewed as a zone of safe behavior or a zone where the state of

the system can move by the action of disturbance.

☞ Moreover, the global stabilizing controllers do not provide really interesting

performances around the origin with respect to the open-loop system

� In some cases, the performances can be even degraded by the global stabilizing

control laws [Pittet] .

☞ Thus, an alternative approach to the global stabilization is the semi-global

stabilization.

� This concept was introduced in the literature almost at the same time by [Lin &

Sabery] and [Alvarez-Ramirez et al] .



Definition 4 A linear system with bounded controls is semi-globally stabilizable if for any

given a priori bounded set χ0 ∈ �n, containing the origin and arbitrarily large, there exists

a control law of the type u(t) = sat(Kx(t)) such that:

1. The closed-loop system is locally asymptotically stable;

2. χ0 is included in the region of attraction of the origin of the closed-loop system.

� This definition points out that the semi-global stability is half-way between the

notions of global and local stability.

• Since χ0 can be chosen arbitrarily large, if it tends towards �n, one recovers the

case of global stability.

• Since the asymptotic stability of the system is only ensured for the states

belonging to χ0, the semi-global stabilization can be considered as a particular

case of the local stabilization.

� As in the case of Theorem 1 (GAS), a necessary and sufficient condition for SGAS

can be expressed, but it is not constructive.



Theorem 3 If the open-loop system is not strictly unstable (null-controllability) then

∀χ0 ⊂ �n, ∀umin, umax (the bounds on the control input) then there exists a control law of

the type u(t) = sat(Kx(t)) such that ∀x(0) ∈ χ0 one gets limt→∞

x(t) = 0.

� χ0 ⊆ RA(0)

� Three main approaches have been developed:

• Approach by pole placement [Suarez], [Lin & Saberi] ⇒ no saturation

• Approach by Riccati equation [Suarez], [Lin & Saberi] ⇒ no saturation

• Approach with high and low gains [Lin & Saberi], [Teel] ⇒ saturation

� Closed-loop performance ?

� Robustness ?


4. Local Stability 11

☞ As previously underlined, the global and semi-global stabilization require some

assumptions on the open-loop stability.

� When relaxing this hypothesis, that is, when the open-loop matrix A is strictly

unstable, neither the global nor the semi-global stability of the system with

bounded controls may be achieved.

� In this case, only the local stability can be investigated.

� Furthermore, note that even in the case where the open-loop matrix A is not

strictly unstable, some specifications (robustness, performances, mode decoupling,

disturbance rejection...) may not be achieved through globally or semi-globally

stabilizing controllers

☞ Given a saturating controller (3) for system (2), it is of major interest for the user to

be able to determine the region of attraction, or at least larger as possible estimates

of this region.

☞ Tool: Positive invariant and contractive domains ⇒ regions of stability



☞ The general idea is to use Lyapunov functions to build domains of stability

� polyhedral

� ellipsoidal

☞ The stability can be ensured only for a set D0 of admissible initial conditions →estimates of the region (basin) of attraction.

☞ Concerning the local stability, we can pursue two main approaches:

� determine stability regions and/or control laws in order to avoid the control

saturation

→ study of the linear behavior of the closed-loop system

� consider the occurrence of the control saturation

→ study of the nonlinear behavior of the closed-loop system

☞ Trade-off: size of the regions of stability × performance and robustness.

☞ References: [Bitsoris, Burgat, Hennet, Castelan, Gilbert, Blanchini, Tarbouriech,

Gomes da Silva Jr., Bernstein, Jabbari, Lin, Johanson, Iwasaki]



Linear Approach

☞ Given a set D0 of admissible initial conditions

☞ Objective: to avoid saturation

☞ Solution: to determine a gain K and a set S such that:

1. (A+BK) asymptotically stable

2. D0 ⊆ S ⊆ S(K,umin, umax)

3. S positively invariant w.r.t. x(t) = (A+BK)x(t)

min

1

x2

D0

max

x

S(K,u ,u )

S

☞ Different approaches can be considered

− Linear programming

− Eigenstructure assignment

− Determination of the maximal invariant set included in S(K,umin, umax)

☞ References

- polyhedral S: [Bitsoris, Burgat, Hennet, Castelan, Gilbert, Blanchini]

- ellipsoidal S: [Tarbouriech, Gomes da Silva Jr., Boyd, Lin]



Non-Linear Approach

☞ Given a set D0 of admissible initial conditions

☞ Objective: to take into account the effective saturation

☞ Solution: to determine a gain K and a set S such that:

1. (A+BK) asymptotically stable

2. D0 ⊆ S3. S contractive w.r.t. x(t) = Ax(t) +Bsat(Kx(t))

� Remark: The set S is not included in S(K,umin, umax)

S(K,u ,u )maxmin

2

D0

x

S

1

x

☞ To obtain tractable conditions ⇒ ”good” modelling of the saturation term

☞ References:

- polyhedral S: [Gomes da Silva Jr., Tarbouriech, Johanson, Milani, Alamo, Limon]

- ellipsoidal S: [Tarbouriech, Gomes da Silva Jr., Pittet, Henrion, Jabbari, Berstein,

Lin, Hindi, Iwasaki]


5. Invariant sets and local stability 15

Representations of saturations

☞ Saturation regions: Exact description in subregions of the state space

☞ Polytope of matrices: sat(u(i)) = α(i)u(i)(t) with 0 < α(i) ≤ 1

☞ Dead-zone nonlinearity: ψ(u)′T (ψ(u) + w) ≤ 0 in {u,w ∈ �m;−umin ≤ u− w ≤ umax}

Stability

☞ The stability can be studied using Lyapunov stability theory and some extensions as

Lasalle invariance principle.

Tools

☞ Differential inclusions, sector conditions

☞ Quadratic, polyhedral, Lure Lyapunov functions.

☞ Riccati equations, Linear Matrix Inequalities, S-procedure, Convex and Semi Definite

Programming.



Positive invariance

☞ We must determine a locally stable domain of initial conditions ⇒ Neighborhood of the

origin such that any initial condition taken inside this domain remains inside (invariance).

Definition 5 A set D, with 0 ∈ D, is said positively invariant for the closed-loop

saturated system if ∀x(0) ∈ D, x(t, x(0)) ∈ D, ∀t ≥ 0.

Definition 6 A set D, with 0 ∈ D, is a positively invariant set of asymptotic stability for

the closed-loop saturated system relatively to x = 0 if ∀x(0) ∈ D, x(t, x(0)) ∈ D, ∀t ≥ 0

and x(t, x(0)) → 0 when t→ ∞.



☞ (Positive) invariance and stability are different notions:

D2

x1

x(0)

x

x(0)

2

x1

Dx

A invariant set D is not A region of stability D is not

necessarily a set of stability necessarily a set of positive invariance



☞ Important property. Consider V (x) a Lyapunov function for the closed-loop saturated

system. Then the compact set

D(V, γ) = {x ∈ �n;V (x) ≤ γ−1, γ > 0}

is an invariant set

� of stability if V (x) ≤ 0, ∀x ∈ D(V, γ)

� of asymptotic stability if V (x) < 0, ∀x ∈ D(V, γ), x �= 0.

☞ One of the main difficulties resides in the choice of a ”good” Lyapunov function.


6. Estimates of the region of attraction 19

☞ Consider a quadratic Lyapunov function: V (x) = x′Px

☞ The associated invariant set is an ellipsoidal set

D(P, γ) = {x ∈ �n;x′Px ≤ γ−1, γ > 0}

☞ Let f(D(P, γ)) be a convex function associated to the geometry/size of D(P, γ)

[Gomes da Silva Jr. & Tarbouriech/ECC99]

min f(D(P, γ)) ⇒ e.g.

��

- max. minor axis: λmin(P−1)

- min. the trace of P : trace(P )

- max. volume:

�

det(P−1γ−1)

- max. in some directions

☞ The associated generic optimization problems are described as follows:

min f(D(P, γ)

subject to

stability conditions of the Propositions



Maximization of the minor axis of the ellipsoid

☞ By maximizing the minor axis, the idea is to find an ellipsoidal region of stability that

includes a ball containing the origin as large as possible.

☞ The length of the minor axis is proportional to λmin(P−1).

☞ Whether W = P−1 or P being the decision variable, one defines:

f(D(P, γ)) = λmax(P ) or f(D(W−1, γ)) = −λmin(W )

☞ That is, one considers:

min λ max λ

subject to relations or subject to relations

P ≤ λIn,+ stability conditions W ≥ λIn,+ stability conditions



Trace minimization

☞ The trace of matrix P is directly related to the size of the axis of the associated

ellipsoid.

☞ The minimization of trace(P ) leads to ellipsoids that are increased homogeneously in

all directions, since all the axis have the same weights.

☞ Thus, one can choose the following function:

f(D(P, γ)) = β0γ + β1 trace(P )

where β0 and β1 are tuning parameters.

☞ When W is a decision variable, one considers:

f(D(W−1, γ)) = β0γ + β1 trace(MW )

with �� MW In

In W

�� ≥ 0



Volume maximization

☞ The volume is proportional to

�

det(P−1γ−1) =

�

det(Wγ−1)

☞ Then, it is possible to maximize the size of the ellipsoid by minimizing the function

log(det(Pγ)).

☞ In this case, one gets:

f(D(P, γ)) = log(det(Pγ)) = log(γndet(P ))

= nlog(γ) + log(det(P ))

☞ Such a function is linear in P and γ.

☞ Note that the volume maximization can lead to flat ellipsoids in some directions.



Maximization in some directions

☞ We want to maximize the ellipsoid in some specific directions.

☞ Let us define the set of vectors defining such directions:

D = {d1, · · · , ds} , di ∈ �n , i = 1, · · · , s

☞ Hence, for each di we should satisfy:

(ηid′i)P (ηidi) ≤ γ−1 , i = 1, · · · , s (4)

or still �� γ−1 ηid

′i

ηidi W

�� ≥ 0 , i = 1, · · · , s (5)

where the objective is to maximize ηi, i = 1, · · · , s.☞ One can consider the following optimization criterion (with γ = 1)

max

si=1 βiηi

subject to conditions (4) or (5) and stability conditions


7. Analysis Problems 24

K

K

K

nD = R

imposed by

dynamics around the origin

dynamics around the origin

imposed by

saturationwith

saturation

calculation of the safe operation domain

domain linear nonlinear

domain

without

D safe operation domain

globallocal

stability

saturation

given

nonlineardynamics ?


8. Synthesis Problems 25

performances ? local

EK

EEEK K K

dynamics

Dchange of and/or of the

compromise

D and

dynamics

D

specifications

domainsafe operation

the origindynamics around

simultaneousspecifications

analysis

unchosendynamics domain

safe opertion


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Sistemas com satura¸c˜ao no controle II. Contexto da estabilidadesala225/ia360/PDF/sat_cont.pdf · 2005. 8. 24. · 1 Sistemas com satura¸c˜ao no controle II. Contexto da estabilidade