Delay-dependent stability of reset control systems with anticipative reset conditions

Proceedings of the 7th IFAC Symposium on Robust Control Design The International Federation of Automatic Control Aalborg, Denmark, June 20-22, 2012

Delay-dependent stability of reset control systems with anticipative reset conditions Jos´ e Antonio Gonz´ alez Prieto ∗ Antonio Barreiro ∗ Sebasti´ an Dormido ∗∗ Sophie Tarbouriech ∗∗∗ ∗

Departamento de Ingenier´ıa de Sistemas y Autom´ atica, E.T.S.I.Industriales Vigo ([email protected], [email protected]). ∗∗ Dept. Inform´ atica y Autom´ atica, UNED Madrid ([email protected]). ∗∗∗ CNRS, LAAS, 7 Avenue Colonel Roche, F-31400 Toulouse, France and Univ de Toulouse, LAAS, F-31400 Toulouse, France. ([email protected]). Abstract: Typical reset conditions can be the zero crossing of the tracking error signal e(t) = 0 or when input and output of the reset controller have opposite signs eT (t)u(t) ≤ 0. This paper studies the delay-dependent stability properties for delayed systems with state-dependent reset impulses when anticipation reset actions are used on the reset event conditions. The analysis includes the application of a Lyapunov-Krasovskii functional technique in the time domain for dealing with stability conditions and a frequency domain approach. The effectiveness and the drawback of such an approach are illustrated on an example. Keywords: Reset control systems, delay-dependent stability, Lyapunov-Krasovskii functional. 1. INTRODUCTION The idea of reset control was originated with the Clegg integrator in Clegg [1958] and was formalized in Horowitz [1975] and Horowitz [1974] in the seventies. Since those early works, it was clear that the benefit was to outperform linear solutions and to overcome fundamental limitations by means of an extremely simple action: resetting to zero the states of the controller when the tracking error is zero. Somewhat later, new formalized results were presented in Beker [2001], Beker et al. [2004], Beker et al. [2001] and Chait et al. [2002] for general reset systems based on the Horowitz’s reset condition. New strategies for reset control were proposed and studied in Aangenent et al. [2009], Loquen et al. [2008], Neˇsi´c et al. [2008] and Zaccarian et al. [2005] using another formulation of the reset condition, based on the sign of the input and output controller signals. Recently, new ideas are being proposed in the application of reset controllers on systems with delays, as in Ba˜ nos and Barreiro [2009], Ba˜ nos et al. [2007], Ba˜ nos and Barreiro [2007], Barreiro et al. [2011] and Tarbouriech et al. [2004], where some stability results are provided based on Lyapunov-Krasovskii analysis. The introduction of some slight modifications on the reset conditions based on a kind of reset anticipation as in Ba˜ nos et al. [2011], Ba˜ nos et al. [2009] and Prieto et al. [2011a] have been presented to show some performance improvement compared to conventional reset. As well as stability of delay differential equations with fixed time impulses have been deeply study as in Liu et al. [2007a] and Liu et al. [2007b], the literature regarding delay differential equations with state-dependent impulses 978-3-902823-03-8/12/$20.00 © 2012 IFAC

219

is less developed (see, for example, Zhang et al. [2005] and Liu et al. [2006]). Hence the design of rules for guaranteeing the stability of such impulsive systems is still in progress. The current paper takes place in this context. This paper studies the stability properties for reset systems with delays considering the anticipative effects on the reset conditions. For this purpose, the paper is organized as follows. In section 2 we introduce the problem statement and set the structure of the closed loop. Next, in section 3 the Lyapunov-Krasovskii theory is applied for two different reset signals (without and with anticipation). Section 4 shows an illustrative example where we consider the anticipative design by a frequency-domain methodology. Notation: In is the n × n identity matrix and 0 may either denote the scalar zero or a matrix of zeros with appropriate dimensions. For a real matrix H, H T denotes its transpose and H > 0 means that H is symmetric and positive definite. For a block matrix, the symbol ? represents symmetric blocks outside the main diagonal block. C(h,n) = C([−h, 0], Rn ) denotes the Banach space of continuous vector functions mapping the interval [−h, 0] into Rn with the topology of uniform convergence. k · k refers to either the Euclidean vector norm or the induced matrix 2-norm. k φ kc = sup k φ(t) k stands for the −h≤t≤0

norm of a function φ(t) ∈ C(h,n) . When the delay is finite v then “sup” can be replaced by “max”. C(h,n) is the set v defined by C(h,n) = {φ(t) ∈ C(h,n) ; || φ ||c < v, v > 0}. For a state x(t) ∈ Rn we denote the distributed state v by some piecewise continuous function xt ∈ C(h,n) , where xt (θ) = x(t + θ) for θ ∈ [−h, 0].

10.3182/20120620-3-DK-2025.00078

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

The application of S-Procedure and Finsler’s lemma will be a common procedure in this paper. References to these theorems can be found in Boyd et al. [1994]. 2. PROBLEM STATEMENT Let us consider a system formed by a plant (P) given by  x˙ p (t) = Ap xp (t) + Bp u(t − h) P : (1) yp (t) = Cp xp (t) with xp (t) ∈ Rp , u(t − h) ∈ R , yp (t) ∈ R, Ap ∈ Rp×p , Bp ∈ Rp , Cp ∈ R1×p , and a reset controller (RC) given by   x˙ r (t) = Ar xr (t) + Br e(t), xr (t) ∈ F r RC : xr (t+ ) = Aµ xr (t), xr (t) ∈ J r (2)  u(t) = Cr xr (t) with F r the flow surface and J r the reset surface, so that the hits of xr (t) on J r causes the the reset actions. We will consider a partition of the reset states, separating the states that may be reset (r2 ) and the states that remain unchanged (r1 ), with r = r1 + r2 and
T xr (t) ∈ Rr , xTr1 (t) xTr2 (t) ! Ir1 ×r1 0r1 ×r2 Aµ = ∈ Rr×r 0r2 ×r1 0r2 ×r2 with xr1 (t) ∈ Rr1 , xr2 (t) ∈ Rr2 , e(t) ∈ R , u(t) ∈ R, Ar ∈ Rr×r , Br ∈ Rr , Cr ∈ R1×r . Closed-loop dynamics, obtained using (1), (2), the relation e(t) = −yp (t) and a delay connection (e−ht ) between the controller and the plant, are  x(t) ˙ = Ax(t) + Ah x(t − h), x(t) ∈ F cl   + x(t ) = AR x(t), x(t) ∈ J cl (3) CL :   y(t) = Cx(t) x(t) = φ(t), t ∈ [−h, 0]
T
T x(t) = xTp (t) xTr (t) = xTp (t) xTr1 (t) xTr2 (t) ∈ Rp+r ! Ap 0(p,r) A= ∈ R(p+r)×(p+r) −Br Cp Ar ! 0p×p Bp Cr Ah = ∈ R(p+r)×(p+r) 0r×p 0r×r   Ip×p 0p×r1 0p×r2     AR = 0r1 ×p Ir1 ×r1 0r1 ×r2  ∈ R(p+r)×(p+r)   0r2 ×p 0r2 ×r1 0r2 ×r2 1×(p+r)

C = (Cp 01×r1 01×r2 ) ∈ R v with the initial conditions given by φ(t) ∈ C(h,p+r) , and cl cl F (J ) the closed-loop flow (reset) surface. Finally, we also define the augmented state vector as
T xh (t) = xT (t) xT (t − h) ∈ R2(p+r) (4) 3. STABILITY ANALYSIS To addres stability of system (3) we consider a LyapunovKrasovskii functional V (xt ) and use the following Theorem to check stability conditions. 220

v Theorem 1. Let V (xt ) : C(h,p+r) −→ R be a continuously differentiable Lyapunov-Krasovskii functional, with x(t) a solution of (3) at time t and initial condition at t0 given v by φ(t) ∈ C(h,p+r) , where xt0 (θ) = φ(t0 + θ) for θ ∈ [−h, 0]. The zero equilibrium x(t) = 0 is globally asymptotically stable if for some  > 0, V (xt ) satisfies

V (xt ) ≥ kxt (0)k

2

x(t) ∈ F cl

dV (xt ) 2 ≤ −kxt (0)k dt

x(t) ∈ F cl

− ∆V (xt ) = V (x+ t ) − V (xt ) ≤ 0

x(t) ∈ J cl

(5) (6) (7)

Proof. The proof is omitted for brevity, but it can be found in Ba˜ nos and Barreiro [2007] (with references to some details in Beker et al. [2004] and Ba˜ nos and Barreiro [2009]) with a slight variation. In this paper the equations (5) and (6) are restricted to the flow surface F cl and equation (7) is restricted to the reset surface J cl . We will use the following Lyapunov-Krasovskii functional V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) V1 (xt ) = xT (t)P x(t) Z 0 V2 (xt ) = xT (t + θ)Sx(t + θ)dθ −h Z 0 Z 0 V3 (xt ) = f T (x(t + θ + ξ))Zf (x(t + θ + ξ))dξdθ −h

θ

with Z = Z T and f (x(t+θ+ξ)) = Ax(t+θ+ξ)+Ah x(t+θ+ξ−h). Proposition 2. The system (3) is stable if there exists P = P T > 0, X = X T , Z = Z T , Y, S such that xTh (t)QF 1 xh (t) > 0, x(t) ∈ F cl

(8)

xTh (t)QF 2 xh (t)

< 0, x(t) ∈ F

cl

(9)

∆V (xt ) = V (xt+ ) − V (xt ) ≤ 0, x(t) ∈ J

cl

(10)

with 

QF 2

X Y



 QF 1 =  YT Z   N P Ah + hAT ZAh − Y  = T F −S + hAh ZAh

N = P A + AT P + hAT ZA + S + hX + Y + Y T

(11)

(12) (13)

and where we will apply the partition of the matrix P as  p×p p×r1 p×r2  P11 ∈ R

 P = 

T P12 T P13

P12 ∈ R

P13 ∈ R

 

P22 ∈ Rr1 ×r1 P23 ∈ Rr1 ×r2  ∈ R(p+r)×(p+r) T P23

P33 ∈ Rr2 ×r2

Proof. The proof for (8) and (9) can be found in Gu et al. [2003] page 173-175. Condition (10) is the same as in (7) in Theorem 1. Our objective in this paper is to study the stability conditions for system (3) setting different J cl definitions, where, in the anticipative reset cases, we will use an anticipative parameter τ that means the time before a

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

reset signal e(t) crosses the zero value. This implies that the anticipative reset condition becomes e(t + τ ) ≈ e(t) + τ e(t) ˙ =0





    QAV = 0r1 ×p 0r1 ×r1 −P23  ∈ R(p+r)×(p+r)  

(14)

This reset surface definition is quite different from the reset band definition employed in Ba˜ nos et al. [2011], Ba˜ nos et al. [2009] and Prieto et al. [2011a]. Both anticipative reset definitions are compared, in the (e, e) ˙ plane, in Fig. 1. We will also study for system (3) the stability problem

0p×p 0p×r1 −P13

T T −P13 −P23 −P33

obtained from the application of the reset jump conditions defined in (3). Applying the Finsler’s lemma the condition (10) holds if ∃η ∈ R such that QAV − ηC T C ≤ 0

(19)

Some properties related to the Hβ -condition can be obtained from (19), and they are detailed in Prieto et al. [2011b].

de(t) dt

Mrb+ tan ( ) =

(18)

1

3.2 Reset with anticipation τ based on e(t) +

-

e(t)

+

-

de(t) =0 dt

e(t) +

M Mrb+ − Fig. 1: Anticipative reset surfaces : Mrb ∪Mrb → Reset band, Mτ → Derivative approximation

The reset surface, if we consider an anticipation parameter τ , is defined as  J cl = x(t) ∈ Rp+r : e(t + τ ) = 0  ' x(t) ∈ Rp+r : e(t) + τ e(t) ˙ =0 n o ⊂ xh (t) ∈ R2(p+r) : Chτ xh (t) = 0 = ker(Chτ ) (20) with Chτ = (Hp 01×r 01×p Gr ) ∈ R1×2(p+r)

based on the signal condition

and

eT (t)u(t) ≤ 0

Hp = Cp + τ Cp Ap ∈ R1×p

such that the anticipative reset condition for this case is T eT (t + τ )u(t + τ ) ≈ (e(t) + τ e(t)) ˙ (u(t) + τ u(t)) ˙ ≤0

(15)

3.1 Reset without anticipation based on e(t) The reset surface is defined as n o J cl = x(t) ∈ R(p+r) : e(t) = 0 n o = x(t) ∈ R(p+r) : Cx(t) = 0 n o ⊂ xh (t) ∈ R2(p+r) : Ch xh (t) = 0 = ker(Ch ) (16)
with Ch , C 01×(p+r) . Stability conditions (8) and (9) are restricted to x(t) ∈ F cl ⇒ Ch xh (t) 6= 0 ⇒ xTh (t)QMh xh (t) > 0, with QMh = Ch T Ch given by   C T C 0(p+r)×(p+r) ≥0 QMh =  (17) F 0(p+r)×(p+r) As QMh ≥ 0, the restriction defines an hyperplane with no volume on the xh (t) space so, from continuity, it doesn’t affect the conditions (8) and (9), that have to be verified in the whole space. This implies that we cannot prove stability for delays that make the base linear system unstable. Stability condition (10) takes the form xT (t)QAV x(t) ≤ 0 restricted to Cx(t) = 0, with QAV = by

(21)

Gr = τ Cp Bp Cr ∈ R1×r

Stability conditions (8) and (9) are restricted to x(t) ∈ F cl ⇒ Chτ xh (t) 6= 0 ⇒ xTh (t)QτMh xh (t) > 0, with QτMh = Chτ T Chτ given by  T  Hp Hp 0p×r 0p×p HpT Gr      0r×p 0r×r 0r×p 0r×r    QτMh =  (22)   F F 0p×p 0p×r      F

F

F GTr Gr

As previously, the restriction doesn’t change stability conditions (8) and (9), so we only can prove stability conditions up to the limit of the maximun delay for the base linear system. We define   QAV 0(p+r)×(p+r)  ∈ R2(p+r)×2(p+r) (23) QAVh =  F 0(p+r)×(p+r) with QAV given in (18), so stability condition (10) takes the form xTh (t)QAVh xh (t) ≤ 0 restricted to Chτ xh (t) = 0. Applying the Finsler’s lemma, the condition (10) holds if ∃η ∈ R such that QAVh − ηQτMh ≤ 0

See Prieto et al. [2011b] for details on the properties related to (24). 3.3 Reset without anticipation based on eT (t)u(t)

ATR P AR

− P given The reset surface is defined as 221

(24)

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

 J cl = x(t) ∈ Rp+r : eT (t)u(t) ≤ 0  = x(t) ∈ Rp+r : −xTp (t)CpT Cr xr (t) ≤ 0  = x(t) ∈ Rp+r : xT (t)QR x(t) ≤ 0 n o ⊂ xh (t) ∈ R2(p+r) : xTh (t)QRh xh (t) ≤ 0

(25)

with   CpT Cr 0p×p − 2  (p+r)×(p+r) QR =  ∈R

(26)

0r×r

F and QRh

  QR 0(p+r)×(p+r)  ∈ R2(p+r)×2(p+r) = F 0(p+r)×(p+r)

(27)

By the application of the S-Procedure the stability conditions (8) and (9) holds if ∃δ1 ≥ 0, δ2 ≥ 0 ∈ R such that QF 1 − δ1 QτRρh > 0 (34) τρ QF 2 + δ2 QRh < 0 (35) Stability condition (9) takes the form xTh (t)QAVh xh (t) ≤ 0 restricted to xTh (t)QτRh xh (t) ≤ 0. Again by the SProcedure application, the condition (10) holds if ∃η ≥ 0 such that QAVh − ηQτRh ≤ 0 (36) See Prieto et al. [2011b] for details on the properties related to (36). 4. ILLUSTRATIVE EXAMPLE

Stability conditions (8) and (9) are restricted to x(t) ∈ F cl ⇒ xTh (t)QRh xh (t) > 0. We can use a very small real ρ > 0 such that the restriction can be written as x(t) ∈ F cl ⇒ xTh (t)QRh xh (t) ≥ xTh (t)ρIxh (t) ⇒ xTh (t)QρRh xh (t) ≥ 0 with QρRh = QRh − ρI (28) Now, applying the S-Procedure with the new restriction expression, the stability conditions (8) and (9) hold if ∃δ1 ≥ 0, δ2 ≥ 0 ∈ R such that QF 1 − δ1 QρRh > 0 (29) QF 2 + δ2 QρRh < 0 (30) Stability condition (9) takes the form xTh (t)QAVh xh (t) ≤ 0 restricted to x(t) ∈ J cl ⇒ xTh (t)QRh xh (t) ≤ 0 with QAVh defined in (23) and QRh in (27). Applying again the SProcedure, the condition (10) holds if ∃η ≥ 0 such that QAVh − ηQRh ≤ 0 (31) See Prieto et al. [2011b] for details on the properties related to (31).

Consider a feedback system according to the set up of Section 2. Here the plant (P) has a transfer function P (s) given by s+1 P (s) = s(s + 0.2) and the feedback compensator RC is a FORE compensator with base LTI system given by 1 RCLT I (s) = s+1 As the reset controller is a FORE, it can be shown (see Prieto et al. [2011b]) that the dynamic behaviour induced by the reset surface based on e(t) = 0 and e(t)u(t) ≤ 0 are equivalent. The same equivalence is not true for anticipative reset surfaces. It can be shown that the base 0.5

0.4

0.3

0.2

0.1

Imag (jω)

3.4 Reset with anticipation τ based on eT (t)u(t) The reset surface, if we consider an anticipation parameter τ , is given by  J cl = x(t) ∈ Rp+r : eT (t + τ )u(t + τ ) ≤ 0  T ' x(t) ∈ Rp+r : (e(t) + τ e(t)) ˙ (u(t) + τ u(t)) ˙ ≤0 n o ⊂ xh (t) ∈ R2(p+r) : xTh (t)QτRh xh (t) ≤ 0

−0.1

−0.2

τ=0.00 τ=0.36 τ=0.72 τ=1.44

−0.3

−0.4

−0.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Real

with

QτRh

0

2H T G −H T G 0  T p r p×p Gp Hr p p     0r×r 0r×p −GT r Hr  1 F =   ∈ R2(p+r)×2(p+r) 2  F F 0 0 p×p p×r   F

F

F

Fig. 2: Open-loop Nyquist diagram with h = 0.60, τ ∈ [0.00, 1.44]

0r×r (32)

and Hp = Cp + τ Cp Ap ∈ R1×p Hr = τ Cp Bp Cr ∈ R1×r Gp = τ Cr Br Cp ∈ R1×p Gr = Cr + τ Cr Ar ∈ R1×r Employing the same technique as in the previous case we use a very small real ρ > 0 and define QτRρh = QτRh − ρI (33) 222

LTI control system is stable for h ≤ 0.20 = hlimit . Using the realizations of P (s) and RCLT I (s) given by ! ! ! −0.2 0 1 1 Ap = ; Bp = ; CpT = 1 0 0 1 Ar = −1; Br = 1; Cr = 1 the matrices A and Ah , from Eq. (3), are     −0.2 0 0 0 01         0 0 ; A= 1 Ah = 0 0 0     −1 −1 −1 0 00

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

The objective of this example is to check the stability requirements given in Proposition 2 with the application of the conditions for the proposed reset surfaces, to compare the results beyond the base LTI delay stability limit (hlimit ), so we choose a delay value h = 0.60. First, we

0.8 0.6 0.4

Feasibility

1

0.5

Error

FEASIBLE

Base linear system Standard reset e(t) = 0 Anticipative reset e(t+τ) = 0 Standard reset e(t) u(t) <= 0 Anticipative reset e(t+τ) u(t+τ) <= 0

1

0.2 MARGINALLY FEASIBLE

0 −0.2

0 −0.4

−0.5 −0.6

−1

0

5

10

−0.8

15

time

UNFEASIBLE

−1

0.8

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

τ 0.6

Control

0.4

Fig. 5: LMI Feasibility for h = 0, 60, τ ∈ [0.50, 1.00]

0.2

τ=0.00 τ=0.36 τ=0.72 τ=1.44

0 −0.2 −0.4 −0.6

0

5

10

15

time

Fig. 3: Error and control with h = 0.60, xp1 (0) = 0.5, xp2 (0) = −0.9, τ ∈ [0.00, 1.44]

have to determine the anticipation parameter τ , so we will use the Describing Function frequency methodology to set up the FORE controller frequency response for an anticipative reset surface based on e(t + τ ) = 0, as can be found in Prieto et al. [2011a]  −π  DF (F ORE) =

1 2ωejψ (ω cos ψ + sin ψ)(1 + e 1+j jω + 1 π(ω 2 + 1)

ω

)

where we introduce the anticipation parameter τ by applying the relation ωτ = ψ. Employing this DF (F ORE)

the same initial conditions and anticipative reset surface, is displayed on Fig. 4 and shows how the anticipation works to partially compensate the effect of the delay, by making the reset points fits near the Cp xp (t) = 0 surface for the selected τ values. Using this τ estimation range we can check the stability requirements given in Proposition 2 and the conditions obtained for different reset surfaces, by looking at the feasibility of the LMI problem for the selected τ range. Fig. 5 shows the feasibility of the properties for the presented reset cases, where, as we have expected, the feasibility of the e(t + τ ) = 0 reset surface can’t be proved for this time delay, but the eT (t + τ )u(t + τ ) ≤ 0 solution becomes feasible for a range 1.02 ≤ τ ≤ 1.20, so we can set the stability conditions even for an unstable (caused by the delay) base LTI closed-loop system. Numerical issues related to the solver and code can be found in Prieto et al. [2011b]. Fig. 6 shows the error and control values when we set 0.6

0.6

0.4

← Cp xp = 0 ⇒ xp1= −xp2

0.2

Error

0.4

0.2

0 −0.2 −0.4 −0.6

0

−0.8

0

5

10

15

xp

2

time −0.2 0.4 0.2

Control

−0.4

−0.6

τ=0.00 τ=0.36 τ=0.72 τ=1.44

−0.8

−1

−0.5

−0.4

0 −0.2

e(t+τ1)=0

−0.4

t0 −0.3

−0.2

−0.1

0 xp1

0.1

0.2

0.3

0.4

−0.6

e(t+τ2)u(t+τ2)<=0 0

5

10

15

time

0.5

Fig. 4: Phase plane with h = 0.60, xp1 (0) = 0.5, xp2 (0) = −0.9, τ ∈ [0.00, 1.44]

expression we can see on Fig. 2 the Nyquist Diagram for the open loop. Fig. 3 shows the time simulation results when the system is started with non zero initial conditions and with an anticipative reset surface based on e(t+τ ) = 0. In both cases we have selected τ ∈ [0.00, 1.44], and we can observe that a stabilized effect is presented for an estimated range 0.40 ≤ τ ≤ 1.20, with an approximated optimal value of τ = 0.72. The phase plane evolution, for 223

Fig. 6: Error and control with h = 0.60, xp1 (0) = 0.5, xp2 (0) = −0.9, τ1 = 0.72, τ2 = 1.05

e(t + τ1 ) = 0 ⇒τ1 = 0.72 T

e (t + τ2 )u(t + τ2 ) ≤ 0 ⇒τ2 = 1.05 Clearly the reset surface based on e(t + τ )T u(t + τ ) ≤ 0 presents better performance, acting in a kind of multiple anticipative reset actions (some ideas of this multiple anticipation are presented in Prieto et al. [2011a]) and avoiding, in an anticipative form, that the error and the

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

control signals have opposite signs. Finally Fig. 7 shows the phase plane evolution for both anticipative reset surfaces.

0.6

← Cp xp = 0 ⇒ xp1= −xp2 0.4

xp2

0.2

0

−0.2

−0.4

e(t+τ1)=0, τ1 = 0.72 −0.6 −0.5

e(t+τ2)u(t+τ2)<=0, τ2 = 1.05 −0.4

−0.3

−0.2

−0.1

0 xp1

0.1

0.2

0.3

0.4

0.5

t

Fig. 7: Phase plane with h = 0.60, xp1 (0) = 0.5, xp2 (0) = −0.9, τ1 = 0.72, τ2 = 1.05

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