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L2 -induced gain for discrete-time switched Lur’e
systems via a suitable Lyapunov function
Julien Louis, Marc Jungers, Jamal Daafouz, Eugênio Castelan
To cite this version:
Julien Louis, Marc Jungers, Jamal Daafouz, Eugênio Castelan. L2 -induced gain for discrete-time
switched Lur’e systems via a suitable Lyapunov function. 16th IFAC workshop on control applications
of optimization, CAO’2015, Oct 2015, Garmisch-Partenkirchen, Germany. pp.277-282. �hal-01185652�
L
2-induced gain for discrete-time switched
Lur’e systems via a suitable Lyapunov function
Julien Louis,∗,∗∗Marc Jungers,∗,∗∗ Jamal Daafouz,∗,∗∗,∗∗∗ Eugênio B. Castelan∗∗∗∗
∗Centre de Recherche en Automatique de Nancy, UMR 7039 –
Université de Lorraine, 2 avenue de la Forêt de Haye 54516 Vandœuvre-les-Nancy, France
(emails: {julien.louis;marc.jungers;jamal.daafouz}@univ-lorraine.fr)
∗∗CNRS, CRAN, UMR 7039, France ∗∗∗Institut Universitaire de France
∗∗∗∗DAS-CTC-UFSC
P.O. Box 476, 88040-900 Florianópolis, SC, Brazil (email: [email protected])
Abstract: This paper deals with discrete-time switched Lur’e systems including an additional saturated input and a bounded energy disturbance. Two control design issues are investigated related to the L2-induced gain and the rejection of disturbance. An common optimization
problem under LMI conditions and differing from the cost function to optimize, is provided to offer a solution to the both problems. The main approach is to upper bound a criterion with a suitable Lur’e type Lyapunov function. Numerical examples are given to underline the efficiency of our approach and to allow a discussion with the literature.
Keywords: Switched systems; Lur’e type systems; L2-induced gain; Lur’e type Lyapunov
functions.
1. INTRODUCTION
Since its introduction in the literature by Lur’e (Lur’e and Postnikov, 1944; Kalman, 1963), the Lur’e problem, that is the stability analysis of the interconnection between a linear system and a nonlinearity verifying a cone bounded sector condition has attracted considerable attention. This should be probably explained by the large class of practical systems, which can be modeled by such a nonlinear system. One of the widely encountered nonlinearity in practice is the saturation, which generates a specific field of re-search (Khalil, 2002; Hu and Lin, 2001; Kapila and Grigo-riadis, 2002; Tarbouriech et al., 2011). Taking into account the presence of the saturation implies the issues of local stability associated with estimating the basin of attraction of the origin. Recently a new Lur’e type Lyapunov function has been provided. It is particularly suitable for studying the global or local stability or control synthesis of discrete-time Lur’e systems and it exhibits possibly non-convex and disconnected level sets (Gonzaga et al., 2012b). The property of disconnection of the level sets generates several interesting issues for discrete-time Lur’e systems obtained by the discretization of continuous-time ones. One can cite among them the questions treated in (Louis et al., 2013, 2015b,a).
A switched version of the discrete-time Lur’e systems has been proposed in (Jungers et al., 2011). It consists of a set of finite number of Lur’e systems and a switching law that indicates at each time which Lur’e system is active. This type of systems is of interest because the considered non-linearity depends on the active mode and is characterized
by a switched cone bounded sector condition. The Lya-punov Lur’e function provided in (Gonzaga et al., 2012b, 2011b) has been extended to the case of switched systems: when the switching law is a perturbation (Gonzaga et al., 2012a, 2011a), or when it is a control input (Jungers et al., 2013). See (Gonzaga et al., 2013) in (Daafouz et al., 2013) for an overview about this topic.
To go beyond the stability issues, the performance aspect of nonlinear systems has been considered, with L2-induced
gain and disturbance rejection (Hindi and Boyd, 1998), or quadratic performance index explicitly depending on the saturation (Schutter, 2000). This last type of quadratic performance index including a saturated control input has been used to formalize the issue of global stabilization of a linear system with a saturated input (Goebel, 2005). Itera-tive procedure using sum-of-square problems to obtain the optimal control for input-saturated system with saturation dependent cost function could be also provided (Baldi et al., 2015).
In this paper, a switched discrete-time Lur’e system with additional saturated input and L2-bounded disturbance is
considered. The contribution is twofold. More precisely, two switched control design problems are investigated aiming for the first one at obtaining the largest set of initial conditions ensuring the stability and for the second one at minimizing the L2-induced gain. The main originality is
here to upper bound a quadratic cost function depending explicitly on the nonlinearity and the saturation by using a suitable Lur’e type Lyapunov function in order to improve the results with respect to the literature.
The paper is organized as follows. Section 2 defines the studied system and settles the two problems which are coped with. Furthermore some technical lemmas are pre-sented in order to build the main result given in Sec-tion 3. SecSec-tion 4 is devoted to the numerical illustraSec-tion of the contribution and allowing a large discussion with respect to the aforementioned literature, before concluding remarks in Section 5.
Notation. For a matrix A ∈ Rn×m, A0 denotes its transpose. If A = A0 ∈ Rn×n, then A < 0 (A ≤ 0)
means that A is negative- (semi-)definite. The components of any vector x ∈ Rn are denoted by x
(i), ∀i = 1, . . . , n.
Inequalities between vectors are component-wise: x ≤ 0 means that x(i)≤ 0 and x ≤ y means that x(i)− y(i)≤ 0.
In(resp. 0n) denotes the n × n identity (resp. null) matrix.
The symbol ? stands for symmetric blocks in matrices. The Hermitian operator is denoted He(·).
2. PROBLEM STATEMENT AND PRELIMINARIES Consider the discrete-time switched Lur’e system, with N nonlinear modes, including a saturated input and a disturbance: xk+1= Aσkxk+ Gσkϕσk(yk) + Bσksat(uk) + Eσkwk, (1) yk= Cσykxk, (2) zk= Cσzkxk+ Lσkϕσk(yk) + Dσksat(uk) + E z σkwk, (3) uk= Kσkxk+ Γσkϕσk(yk), (4)
where xk∈ Rnis the state, yk ∈ Rpis the output, zk ∈ Rpz
is the performance output, wk ∈ Rr is the disturbance of
bounded energy and uk ∈ Rm is the control input. The
switching law σ(·) is defined to take arbitrarily its value in a finite set, σ : N → IN = {1, · · · , N }.
The nonlinearities ϕσk(·) : R
p→ Rp are decentralized and
satisfy their own cone bounded sector condition defined by N diagonal positive definite matrices Ωσk∈ R
p×pby
ϕ0σ
k(yk)Sσk(ϕσk(yk) − Ωσkyk) ≤ 0, ∀σk ∈ IN, (5)
where Sσk and Ωσk ∈ R
p×p are diagonal positive definite
matrices (see (Khalil, 2002)).
The saturation operator is defined by
(sat(u))(`)= sign(u(`)) min(ρ(`), |u(`)|), ∀` ∈ {1, · · · , m},
(6) with ρ > 0 is the symmetric saturation level.
The performance index related to the system (1)-(4) is defined to be quadratic. This paper will investigate more particularly the L2-induced gain associated with the
criterion:
J =X
k∈N
(zk0zk− γwk0wk) . (7)
Thanks to the definition of the performance output (3), the criterion J given by (7) is an explicit function of the nonlinearity and of the saturation.
The disturbance {wk}k∈N is assumed to have a bounded
energy less than or equal to δ−1 ∈ R+, that is it belongs
to the set Wδ = ( w : N → Rr; X k∈N wk0wk ≤ δ−1 ) . (8)
The heart of the contribution is to consider a class of Lur’e type Lyapunov functions (Gonzaga et al., 2012b), which initial value is an upper bound of the criterion J and finally to find the minimal upper bound into this class. This function is defined by ∀i ∈ IN
Vi: Rn× Rp → R, (xk; ϕi(Ciyxk)) 7→ x0kPixk+ 2ϕ0i(C y ix)∆iΩσkC y ixk, (9) where Pσk ∈ R
n×nis a symmetric positive definite matrix
and ∆σk∈ R
p×pis a diagonal positive semidefinite matrix.
It is noteworthy that the structure of the modal function Vi(·, ·) is decomposed into two terms: a quadratic term
with respect to the state and a cross term between the output and the nonlinearity. As it will be discussed in the sequel, the switched quadratic Lyapunov function is a particular case of such a function.
The switching level set (SLS) associated with V and δ−1 is given by
LV(δ−1) =x ∈ Rn; Vi(xk; ϕi(C y
ixk)) ≤ δ−1, ∀i ∈ IN .
(10) The set LV(δ−1) is thus the intersection of the level set of
each modal Lyapunov function related to the level δ−1. The criterion defined by (7) is related to the issue of L2-induced gain. The L2-induced gain of the closed-loop
system (1)–(4) is defined by (see (van der Schaft, 2000) for more details) max x0=0; wk∈Wδ kzkk2 kwkk2 . (11)
It is also obtained by the smallest √γ such that the criterion given by (7) is negative.
In order to clarify the notations and uniform the issues, the following notations and tools are presented. The dual nonlinearity of the saturation is the dead-zone
Ψ(u) = u − sat(u). (12) The saturation will be formulated in the following by the dead-zone in order to emphasize a local cone bounded sector condition. Furthermore, the Lyapunov Lur’e type function depending on the nonlinearity, we introduce for more convenience the extended state:
ηk= x0k ϕ0σk(yk) Ψ
0(u
k) w0k ϕ0σk+1(yk+1)
0
. (13) We use the shorten notation i = σk and j = σk+1.
In this way, we also introduced ϕi = ϕσk(C
y σkxk), ϕj = ϕσk+1(C y σk+1xk+1), Vi(xk) = Vσk(xk; ϕ(C y σkxk)) and Vj(xk+1) = Vσk+1(xk+1; ϕ(C y
σk+1xk+1)) and the matrices
Mi=Acli Gicl −Bi Eix 0n×py (14) Yi=C y i 0py×py 0py×m 0py×r 0py×py , (15) Zi=Cz,cli Lcli −Di Eiz 0pz×py , (16) Ki=Ki Γi −Im 0m×r 0m×py , (17)
with the closed-loop matrices
Acli = Ai+ BiKi, Gcli = Gi+ BiΓi, (18)
Cz,cli = Ciz+ DiKi, Lcli = Li+ DiΓi. (19)
xk+1= Miηk, (20) yk= Yiηk, (21) zk= Ziηk. (22) In addition, we have yk+1= C y jMiηk, (23) sat(uk) = Kiηk. (24)
The issues treated in this paper are the following Prob-lem 1 and ProbProb-lem 2.
Problem 1. (Disturbance). Consider the closed-loop sys-tem (1)–(4) and a fixed scalar δ > 0. The purpose is to design a control law (4) in order to ensure that the trajectories starting from x0 in a bounded set, as large
as possible, are bounded for any disturbance belonging to Wδ.
Problem 2. (Disturbance rejection). Consider the closed-loop system (1)–(4) and a fixed scalar δ > 0. The purpose is to design a control law (4) in order to ensure that the trajectories starting from x0 = 0 are bounded for any
disturbance belonging to Wδ and, in addition, to design
the lowest estimation of the L2-induced gain from the
disturbance wk to the performance output zk.
Before presenting the main result in Section 3, technical lemmas are given.
Lemma 1. Consider the switching matrices Ki, J1,i ∈
Rm×n, Γi, J2,i ∈ Rm×p and the following notations
ˆ
Ki = [Ki0 Γ0i] 0
and ˆJi = J1,i0 J2,i0
0 , with i ∈ IN. If ˆ xk = (x0k ϕ 0 i) 0 is an element of LV(δ−1) ⊂ S( ˆKi(·) − ˆ
Ji(·), ρ) then uk defined by (4), the nonlinearity Ψ(uk)
satisfies the following inequality
2Ψ(uk)0Ti−1(Ψ(uk) − J1,ixk− J2,iϕi) ≤ 0, (25)
for any diagonal positive definite switching matrix Ti ∈
Rm×m.
Proof 1. See (Gonzaga et al., 2012a).
In order to clarify the rest of the paper and more precisely Lemma 2 , the following notations are defined, Λj = ΩjCyj,
Υi= (Si− ∆i)ΩiCyi, Ma i,j= Pj ? ? ? ? ? ? 0 −γIpz ? ? ? ? ? 0 Cz,cli 0 −Pi ? ? ? ? 0 Lcl i 0 Υi −2Si ? ? ? 0 −TiDi0 J1,i J2,i −2Ti ? ? 0 Ez i 0 0 0 0 −I r ? 0 0 0 0 0 0 2∆i , (26) Mb i = In 0pz×n −Acl i 0 −Gcl i 0 TiB0i −Ex i 0 0py×n , Mc i,j= 0n×py 0pz×py −(ΛjAcli )0 −(ΛjGcli)0 (ΛjBiTi)0 −(ΛjExi)0 Ipy . (27)
Lemma 2. For a given scalar γ > 0, i, j ∈ IN, consider
the existence of matrices Ki and J1,i ∈ Rm×n, Γi and
J2,i∈ Rm×py, FUj and XU ∈ Rn×n, diagonal positive
def-inite matrices FΘ
j and XΘ ∈ Rpy×py, symmetric positive
definite matrices Pi ∈ Rn×n, positive diagonal matrices
∆i ∈ Rpy×py, Si ∈ Rpy×py, Ti ∈ Rm×m and a scalar γ such that Ma i,j− He (D) ? ? Mb i 0 − XU 0+ FU j 0(q+py)×n 0 −He FU j ? Mc i,j 0− 0(q+n)×py XΘ0+ FΘ j 0 0 −He FΘ j < 0 (28) with D = diag XU, 0 q, Xθ and q = n + pz+ py+ m + r, implies that Vj(xk+1) − Vi(xk) + zk0zk γ − w 0 kwk
− 2Ψ0(uk)(Ti)−1(Ψ(uk) − J1,ixk− J2,iϕi)
− 2ϕ0iSi(ϕi− Ωiyk) − 2ϕ0jWj(ϕj− Ωjyk+1) < 0. (29)
Proof 2. Under the condition for the existence of the Equation (28), there exist multipliers
Fj = In 0(q+py)×n In 0py×n F U j 0(q+n)×py Ipy 0n×py Ipy F Θ j , (30) such that Ma i,j ? ? Mb i 0 0n ? Mc i,j 0 0 0 py + He NU j 0 NΘ j 0 −In 0 0 −Ipy Fj0 < 0, (31) is satisfied, with NU j = −Uj 0n×(q+py) and NΘ j = 0py×(q+n) −Θj.
Moreover, the kernel of NU j 0 NΘ j 0 −In 0 0 −Ipy 0 =N U j −In 0 NΘ j 0 −Ipy is defined by NU j −In 0 NΘ j 0 −Ipy ⊥ = In+py+q NU j NΘ j . (32)
Then the Finsler’s lemma (de Oliveira and Skelton, 2001) is used to rewrite the Inequality (31) such that
In+py+q NU j NΘ,j 0 Ma i,j ? ? Mb i 0 0n ? Mc i,j0 0 0py In+py+q NU j NΘ,j . (33)
Thus with the matrices defined by Equations (26) and (27), Inequality (33) leads to Inequality (34).
Noticing that Pj− Uj− Uj0 < 0 and Pj> 0, such that Ujis
of full rank, imply that −Uj0Pj−1Uj ≤ Pj− Uj− Uj0.
Com-bining the change of basis diagUj; In+py; Ti; Ipz+r+py,
using a Schur complement twice and the change of variable Wj= Θj− ∆j:
Pj− Uj0− Uj ? ? ? ? ? ? 0 −γIpz ? ? ? ? ? Acli0Uj Cz,cli 0 −Pi+ Qi ? ? ? ? Gcl i 0 Uj Lcli 0 Υi −2Si ? ? ? −TiBi0Uj −TiD0i J1,i J2,i −2Ti ? ? Exi0Uj Ezi 0 0 0 −Ir ? 0 0 ΠjAcli ΠjGcli −ΠjBiTi ΠjEix −2(Θj− ∆j) < 0, where Πj = ΘjΩjCiy. (34) −Pi ? ? ? ? Υi −2Si ? ? ? Ti−1J1,i Ti−1J2,i −2Ti−1 ? ? 0 0 0 −Ir ? ΠjAcli ΠjGcli −ΠjBi ΠjEix −2Wj +M0iPjMi+Z 0 iZi γ < 0, (35) where Πj= (Wj+ ∆j) ΩjCiy.
By multiplying this last inequality on the left by ηk and on
the right by its transpose, the Inequality (29) is ensured. Lemma 3. Consider for i ∈ IN, that there exist symmetric
positive definite matrices Pi ∈ Rn×n, matrices Ki, J1,i ∈
Rm×n and Γi, J2,i∈ Rm×p, a positive scalar α and a fixed
positive scalar δ such that ∀i ∈ IN and ∀l ∈ {1; · · · ; m},
Pi ? ? ? Υa i 2Sai ? ? (Ki− J1,i)(l) (Γi− J2,i)(l) δρ2(l) ? (Ki− J1,i)(l) (Γi− J2,i)(l) 0 αρ2(l) > 0, (36) with Υa i = (∆i− Sai)ΩiCiy. Then LV(µ−1) ⊂ S( ˆKi(·) − ˆJi(·), ρ), (37)
with µ−1= δ−1+ α−1, which is defined in Lemma 1. Proof 3. The Schur complement, applied on LMI (36), leads to the following inequality
Pi ? Υai 2Sai > 1 α+ 1 δ 1 ρ2 (l) (K i− J1,i)(l) (Γi− J2,i)(l) 0(K i− J1,i)(l) (Γi− J2,i)(l) . (38)
In the sequel, by multiplying the previous Inequality on the right by ˆxk = x0k ϕ0i(C
y ixk)
0
and on the left by its transpose, we get the following inequality
x0kPixk+ 2ϕ0i∆iΩiCiyxk+ 2ϕ0iS a i (ϕi− ΩiCiyxk) > 1 α+ 1 δ 1 ρ2 (l) ˆK i− ˆJi (l) ˆ xk 2 , (39) where ˆKi= [Ki0 Γ0i] 0
and ˆJi=J1,i0 J2,i0
0 . Thus Vj(xk+1) − Vi(xk) > 1 µρ2 (l) ˆKi− ˆJi (l)xˆk 2 . (40)
Then ∀xk ∈ LV(µ−1), Vi(xk) < µ−1, which implies that
xk∈ S( ˆKi− ˆJi, ρ). Moreover the inclusion (37) is obtained.
3. MAIN RESULTS
We gather in the following theorem the solutions of the two problems 1 and 2.
Theorem 1. For a given scalar δ > 0, i, j ∈ IN, by
consid-ering matrices Ki and J1,i∈ Rm×n, Γi and J2,i∈ Rm×py,
FU
j and XU ∈ Rn×n, diagonal positive definite matrices
FΘ
j and XΘ∈ Rpy×py, symmetric positive definite
matri-ces Pi ∈ Rn×n, positive diagonal matrices ∆i ∈ Rpy×py,
Si∈ Rpy×py, Ti∈ Rm×mand positive scalars γ and α, the
convex optimization problem min
Ki,Γi,J1,i,J2,i,FUj,XU,FΘj,XΘ,Pi,∆i,Si,Ti
ξ1γ + ξ2α (41)
subject to LMIs (28) and (36)
leads to a switching control law, solution of Problem 2, related to the maximization of the disturbance rejection with the weighting factors ξ1= 1 and ξ2= 0. The solution
of Problem 1 with ξ1= 0 and ξ2= 1.
Proof 4. The LMIs (28) being satisfied, Lemma 2 allows to write Inequality (29). Since LMIs (36) are verified, Lemma 3 implies Inequalities (25). After simplification, using the global cone bounded sector condition (5) and the local bounded sector condition (25), we have
Vj(xk+1) − Vi(xk) + zk0zk γ − w 0 kwk< 0. (42) By recurrence, Vσk0+1(xk0+1) − Vσ0(x0) + k0 X k=0 z0 kzk γ − w 0 kwk < 0. (43)
On assuming that x0∈ LV(α−1) and with assumption of
the Inequality (8) the last inequality leads to Vσk0+1(xk0+1) ≤ Vσ0(x0) + k0 X k=0 w0kwk≤ 1 α+ 1 δ. (44) Thus the limit limk0→+∞Vσk0(xk0) exists and is positive
and bounded by µ−1. Then the system (1)–(4) is stable. Moreover ∀k ∈ N wk = 0, then Inequality (42) leads to
prove the local asymptotic stability of the system (1)–(4). In a second time, on assuming that x0 = 0 (which
implies that Vσ0(x0) = 0) and with assumption of the
Inequality (8) the last inequality leads to Vσk0+1(xk0+1) ≤ k0 X k=0 w0kwk≤ 1 δ. (45) Thus the limit limk0→+∞Vσk0(xk0) exists and is positive
and bounded by 1
δ. Then, on summing Inequality (42), we
prove that 1 γ X k∈N z0kzk < X k∈N w0kwk. (46)
In other words, the L2-induced gain of the closed-loop
provides the minimal γ which gives us an estimate of the L2-induced gains.
4. ILLUSTRATION
Let us consider the following numerical example to illus-trate the solution to Problem 1 and 2.
Example 1: A1=0.4 0.40.2 1 , A2=0.5 −0.30.5 −1.4 , B1=0.50.5 , B2= 1.2 1.3 , G1= 1 1.2 , G2= 1.2 1.3 , E1x= 0.4 0.48 , E2x= 0.4 0.44 , C1y= [1 −0.9] , C2y= [−0.5 0.9] , C1z= [1 1] , C2z= [1 1] , ρ = 1.5, Lz1= 1, Lz2= 2, Dz1= 1, D2z= 1, Ez1= 0.3, E2z= 0.22, ϕ1(yk) = Ω1yk 2 1 + sin(10y 2 k) , Ω1= 0.8, ϕ2(yk) = Ω2yk 2 (1 + sin(3yk)) , Ω2= 0.9. The disturbance considered is defined by
wk=
0.6 sin(17k), 0 ≤ k ≤ 10
0, k > 10. (47) The energy of this disturbance is equal to 1.7616. Then Equation (8) implies that δ = 0.5. In this case, the method investigated in (Jungers et al., 2011) is not able to give an estimate of the L2-induced gain. We can note that
the disturbance is rejected. In order to compare, with the previous work of (Jungers et al., 2011), Fig. 1 presents the evolution of the L2-induced gain obtained by the both
method for different values of δ. As we can see, on the example, the method presented in this paper improves the estimate of the L2-gain around 55% for δ ∈ [2; 100]. For
δ < 2, the advantage is more important. We can also notice that for δ < 0.6, the quadratic method is not feasible, when the new method gives result for δ > 0.3. This can be graphically pointed out by the fact that the both curves admit a vertical asymptote respectively around δ = 0.3 and δ = 0.6.
However, the method introduced on this paper does not always give a better estimate of the L2-induced gain. Even
if the advanced Lyapunov function (9) is an extension of the quadratic one, the tools and manipulations which are used to obtain the LMI conditions to solve the problem are different. More precisely, the change of variables, the Schur complement and the Finsler’s lemma are not handled in the same way due to the presence of a sum of two terms in the Lur’e type Lyapunov function. Hence, the different LMIs are associated with sufficient conditions that are different and not comparable. As a consequence, the method provided here is not always more efficient than the quadratic version. In order to emphasize this remark, we consider the following example, based on a single modification of Example 1.
Example 2: same as Example 1 but with E1x= 0.4 0.08 . 0 0.5 1 1.5 2 2.5 3 3.5 4 0 50 100 150 200 250 δ γAV , γQ
Fig. 1. Evolution of the estimations of the L2-gain given
by the new method γAV and the quadratic one γQ
(blue stars line) in function of δ. (Example 1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4 5 6 7 8 9 10 11 δ γAV , γQ
Fig. 2. Evolution of the estimations of the L2-gain given
by the new method γAV and the quadratic one γQ
(blue stars line) in function of δ. (Example 2) In Fig. 2, we can see that for some δ, the quadratic method leads to a better result. The different sufficient conditions are not straightforwardly comparable.
The second illustration concerns the solution of Problem 1 in Fig. 3. We can see that the level set LV(δ−1) is the
intersection of the two modal level sets and is disconnected. Here the trajectory starting from x0 = 0 with a bounded
energy disturbance remains in the main part of the level set LV(δ−1) containing the origin. Nevertheless the trajectory
may jump in the other parts (which do not contain the origin), even if it is not the case here in Fig. 3 by considering Example 2.
5. CONCLUSION
A suitable Lur’e type Lyapunov function has been used to consider the performance aspect of switched discrete-time Lur’e system affected by a saturated input control and a bounded energy disturbance. With several techniques among the Finsler’s lemma, LMI constraints are obtained for an optimization problem leading to find state and nonlinearity feedbacks law minimizing the L2-induced gain
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 x1 x2
Fig. 3. Disconnected estimate LV(δ−1) obtained by
Theo-rem 1 (black solid line). And an example of trajectory (blue cross line) under the disturbance wk.
stability under a bounded energy disturbance. Numerical illustrations are also given to underline the applicability of our results. Finally a discussion and a comparison with the results of the literature are performed.
ACKNOWLEDGEMENTS
This work was initiated during a visit of Julien Louis at UFSC in the framework of the project CAPES/COFECUB n◦ 701/11. This work was partially supported by the project CAPES/COFECUB n◦ 701/11, by ANR Project COMPACS -"Computation Aware Control Systems", ANR-13-BS03-004, by the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n◦ 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems" and by Région Lorraine. E. B. Castelan has a financial support by CNPq Brazil.
REFERENCES
Baldi, S., Valmorbida, G., Papachristodoulou, A., and Kosmatopoulos., E. (2015). Piecewise polynomial policy iterations for synthesis of optimal control laws in input-saturated systems. In Proceedings of the 2015 American Control Conference. Chicago.
Daafouz, J., Tarbouriech, S., and Sigalotti, M. (eds.) (2013). Hybrid Systems with Constraints. Wiley-ISTE.
de Oliveira, M.C. and Skelton, R. (2001). Stability tests for constrained linear systems. In S.O.R. Mo-heimani (ed.), Perspectives in Robust Control, Lec-tures Notes in Control an Information Sciences, 241– 257. Springer-Verlag.
Goebel, R. (2005). Stabilizing a linear system with sat-uration through optimal control. IEEE Transactions on Automatic Control, 50(5), 650–655.
Gonzaga, C.A.C., Jungers, M., and Daafouz, J. (2012a). Stability analysis and stabilisation of switched nonlin-ear systems. International Journal of Control, 85(7), 822–829.
Gonzaga, C.A.C., Jungers, M., and Daafouz, J. (2012b). Stability analysis of discrete-time Lur’e systems. Au-tomatica, 48(9), 2277–2283.
Gonzaga, C.A.C., Jungers, M., and Daafouz, J. (2013). Hy-brid Systems with Constraints, chapter Advanced Lya-punov functions for Lur’e systems and their switched extension. In Daafouz et al. (2013).
Gonzaga, C.A.C., Jungers, M., Daafouz, J., and Castelan, E.B. (2011a). A new class of Lyapunov functions for nonstandard switching systems: the stability analysis problem. In 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’11), 411–416. Orlando, Florida, USA.
Gonzaga, C.A.C., Jungers, M., Daafouz, J., and Castelan, E.B. (2011b). Stabilization of discrete-time nonlinear systems subject to input saturations: a new Lyapunov function class. In IFAC World Congress. Milan, Italy. Hindi, H. and Boyd, S. (1998). Analysis of linear sys-tems with saturation using convex optimization. In Proceedings of the 37th IEEE Conference on Decision and Control, 903–908.
Hu, T. and Lin, Z. (2001). Control systems with actuator saturation. Analysis and design. Birkh¨auser.
Jungers, M., Castelan, E.B., Tarbouriech, S., and Daafouz, J. (2011). Finite L2-induced gain and λ-contractivity
of discrete-time switching systems including modal nonlinearities and saturating actuator. Nonlinear Analysis: Hybrid Systems, 5(2), 289–300.
Jungers, M., Gonzaga, C.A.C., and Daafouz, J. (2013). Min-switching local stabilization for discrete-time switching systems with nonlinear modes. Nonlinear Analysis: Hybrid Systems, 9, 18–26.
Kalman, R. (1963). Lyapunov functions for the problem of Lur’e in automatic control. Proceedings of National Academy of Sciences, (49), 201–205.
Kapila, V. and Grigoriadis, K. (2002). Actuator saturation control. Marcel Dekker, Inc.
Khalil, H.K. (2002). Nonlinear Systems - Third Edition. Prentice Hall.
Louis, J., Jungers, M., and Daafouz, J. (2013). On using disconnected level sets Lyapunov functions in the context of sampled-data systems. In Proceedings of the 52nd IEEE Conference on Decision and Control, 630–635. Firenze, Italy.
Louis, J., Jungers, M., and Daafouz, J. (2015a). Stabiliza-tion of Lur’e type systems via their Euler approximate model. International Journal of Control, Online. Louis, J., Jungers, M., and Daafouz, J. (2015b). Switching
control consistency of switched Lur’e systems with application to digital control design with non uniform sampling. In European Control Conference 2015, 1748–1753. Linz, Austria.
Lur’e, A.I. and Postnikov, V.N. (1944). On the theory of stability of control systems. Applied Mathematics and Mechanics, 8(3), 3–13.
Schutter, B.D. (2000). Optimal control of a class of linear hybrid systems with saturations. SIAM J. Control Optim., 39(3), 835–851.
Tarbouriech, S., Garcia, G., Gomes da Silva Jr., J.M., and Queinnec, I. (2011). Stability and Stabilization of Linear Systems with Saturating Actuators. Springer. van der Schaft, A. (2000). L2-Gain and Passivity
