Output Feedback Stabilization with Prescribed Performance for Uncertain Nonlinear Systems in Canonical Form Charalampos P. Bechlioulis, Achilles Theodorakopoulos 2 and George A. Rovithakis 2 Abstract The problem of designing an output feedback stabilizing controller for nonlinear systems in canonical form, while guaranteeing prescribed transient and steady state performance bounds, without incorporating the actual system nonlinearities and/or their approximations, is considered in this work. The proposed design follows three steps. Assuming full state measurement, a state feedback controller is firstly constructed to solve the problem and subsequently its output is constrained via the utilization of a carefully selected saturation function to preserve the stability and performance attributes established in its absence. Finally, at a third stage the actual system states are replaced by the states of a high gain observer to formulate an output feedback controller. It is proven that only the boundedness of the closed loop system is sufficient to achieve stabilization with prescribed performance. As a consequence the proposed output feedback prescribed performance control design proves significantly less complex compared to the relevant literature, while reducing the peaking phenomenon which is typically related to the operation of the high gain observer. Simulation studies clarify and verify the approach. I. INTRODUCTION During the past several years, considerable research efforts have been devoted to deal with the output feedback stabilization control problem of nonlinear systems; see e.g., [] [5] and the references therein. In this direction, the combination of high-gain observers with state feedback controllers has evolved as an important approach for designing output feedback control schemes. In [] [6] the focus is on deriving global results under a global Lipschitz condition for the system nonlinearities, restricting the class of nonlinear systems considered. The necessity of imposing growth conditions on system nonlinearities is relaxed in [7] [5]. A significant issue associated with output feedback/high-gain observer design, is related to the peaking phenomenon, a destabilization effect caused when assigning large values to observer gains to keep the observation error small. In [7] the problem is bypassed via the utilization of a priori bounded control signals. Concerning the performance of the closed loop system, it is demonstrated in [] that as the observer gains admit higher values, system trajectories under output feedback approach arbitrarily close to those obtained under state feedback. In addition, asymptotic convergence to the origin could be achieved if accurate knowledge of system nonlinearities is exploited. Another important issue associated with the output feedback stabilization problem concerns the transient and steady state performance. In case of uncertain nonlinear systems, control schemes typically guarantee convergence of the system states to a residual set of the origin, whose size depends School of Mechanical Engineering, National Technical University of Athens, Athens 578, Greece chmpechl@mail.ntua.gr 2 Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 5424, Greece konstheo@eng.auth.gr, robi@eng.auth.gr on design parameters and some bounded though unknown terms. However, no systematic procedure exists to accurately precompute the required upper bounds, thus making the a priori selection of the aforementioned design parameters to satisfy certain steady state behavior, practically impossible. Moreover, performance issues on transient behavior are difficult to be established analytically, even in the case of known nonlinearities. Until very recently, only funnel control [6] has achieved output prescribed performance specifications with output feedback. However, the results obtained apply mainly to relative degree one nonlinear systems or to classes of low triangular systems whose nonlinearities are restricted to be bounded by functions of the system output only. In this work, an output feedback stabilizing controller using high-gain observers is proposed for nonlinear systems in the canonical form, capable of guaranteeing prescribed transient and steady state performance; without incorporating the actual system nonlinearities, not even their approximations obtained via e.g., neural networks/fuzzy systems, in the control scheme. Regarding performance it is required that system states converge to predefined arbitrarily small residual sets of the origin, with convergence rate no less than a prespecified value. The design steps are as follows: a assuming full state measurement, construct a state feedback controller to achieve prescribed performance stabilization without utilizing the actual system nonlinearities or their approximations; b constrain the state feedback controller output via saturation functions, carefully selected to preserve the achieved stability and performance attributes of the closed loop system and c replace state measurement in the control design by the states of an appropriately designed stable linear filter high-gain observer driven by the measured system output, thus forming an output feedback scheme. It is proven that the boundedness of the closed loop system i.e., the controlled system, the high gain observer and the output feedback controller trajectories is sufficient to achieve output feedback stabilization with prescribed performance. As a consequence, the proposed output feedback control design proves significantly less complex compared to existing works in the relevant literature [7] [5], without residing to extreme values of the observer gains reducing thus the peaking of the observer states. Knowledge of the actual system nonlinearities is not required to recover the performance of the state feedback controller. II. PROBLEM FORMULATION AND PRELIMINARIES We consider n-th order nonlinear systems in the canonical form, described by: ẋ = Ax + B [f x+g x u], x Ω, y = C T x where x =[x,,x n ] T R n is the state vector, u R is the control input, y Ris the measured output, f, g :
R n Rare unknown, locally Lipschitz nonlinear functions and the matrices A[ R n n, ] B R n [, C ] R n In n are given by A =, B =, C = [ ]. It is assumed that the system state initializes n from inside a compact set Ω R n that includes the origin. System arises mainly from nonlinear systems of full relative degree n as well as from augmenting dynamics of a series of integrators at the input side. Furthermore, models of mechanical and electromechanical systems where displacement variables are measured while their derivatives velocities, accelerations, etc. are not, can be represented in the form. Moreover, system satisfies the following assumption: Assumption : The sign of g x is considered known and there exists an unknown positive constant g such that g x g >, x R n. Without loss of generality it is assumed to be positive. The objective of this work is to design an output feedback control scheme, without incorporating system nonlinearities f x, g x, or even their approximations obtained from e.g., neural networks/fuzzy systems, etc., capable of practically stabilizing with prescribed transient and steady state performance. At this point, we recall some definitions and preliminary results that are necessary in the subsequent analysis. A. Prescribed Performance It will be clearly demonstrated in the Main Results Section, that the control design is heavily connected to the prescribed performance notion that was proposed to design robust state feedback controllers, for various classes of nonlinear systems, namely feedback linearizable [7], strict feedback [8], [9] and general MIMO affine in the control [2], capable of guaranteeing output tracking with prescribed performance. For completeness and compactness of presentation, this subsection summarizes preliminary knowledge on the concept of prescribed performance. Thus, consider a generic scalar error e t. Prescribed performance is achieved if e t evolves strictly within a predefined region that is bounded by decaying functions of time. The mathematical expression of prescribed performance is given, t, by the following inequalities ρ t <et <ρt where ρ t is a smooth, bounded, strictly positive and decreasing function of time satisfying lim t ρ t >, called performance function [7]. For an exponentially decreasing performance function ρ t =ρ ρ e lt + ρ, the constant ρ = ρ is selected such that ρ > e, the constant ρ = lim t ρ t represents the maximum allowable size of the tracking error e t at the steady state and the decreasing rate of ρ t, which is affected by the constant l in this case, introduces a lower bound on the required speed of convergence of e t. B. Dynamical Systems Consider the Initial Value Problem IVP: ψ = H t, ψ, ψ = ψ Ω ψ 2 with H : R + Ω ψ R n where Ω ψ R n is a non-empty open set. Definition : [2] A solution of an IVP is maximal if it has no proper right extension that is also a solution. Theorem : [2] Consider the IVP 2. Assume that H t, ψ is: a locally Lipschitz on ψ for almost all t R +, b piecewise continuous on t for each fixed ψ Ω ψ and c locally integrable on t for each fixed ψ Ω ψ. Then, there exists a maximal solution ψ t of 2 on the time interval [, τ max with τ max > such that ψ t Ω ψ, t [, τ max. Proposition : [2] Assume that the hypotheses of Theorem hold. For a maximal solution ψ t on the time interval [, τ max with τ max < and for any compact set Ω ψ Ω ψ there exists a time instant t [, τ max such that ψ t / Ω ψ. III. MAIN RESULTS Our objective is to design an output feedback control scheme that practically stabilizes the origin and guarantees prescribed transient and steady state performance, without utilizing the actual system nonlinearities. The motivation of this work originates from the design procedure and performance of high-gain observers used in nonlinear control design; according to which a globally bounded state feedback control is initially designed to achieve the desired stabilization properties and consequently a high gain observer is developed following three steps []. In the first, the ultimate boundedness of the closed loop trajectories is secured for certain values of the observer gains. In the second, by increasing further the observer gains, it is proven that the output feedback closed loop trajectories converge arbitrarily close to the nominal trajectories obtained via state feedback. Finally, assuming knowledge of the system nonlinearities, asymptotic stabilization is guaranteed for some even higher observer gain values. In our case, the design procedure is twofold: Design initially a state feedback saturated control scheme to achieve practical stabilization with prescribed transient and steady state performance, without incorporating the actual system nonlinearities. The introduction of saturation in the controller is necessary to prevent the transmission of the peaking phenomenon to the controlled system states []. Moreover, a sufficient condition is provided to guarantee stabilization of while preserving the achieved performance. 2 In the proposed state feedback realization, replace the actual system states by those of an appropriately designed stable linear filter high gain observer, driven by the measured system output and prove that the boundedness of the closed loop system i.e., the system, the filter and the output feedback controller trajectories is sufficient to achieve output feedback stabilization with prescribed performance. As a result, the proposed output feedback controller meets the objective via a significantly less complex design no need to reside to extreme values of the observer gains compared to the existing in the relevant literature, while knowledge regarding the actual system nonlinearities is not required to recover the performance achieved by state feedback. A. State Feedback Control Design Consider the linear filter s x =Λ T x = n i= d dt + ri y 3 585
where Λ=[λ λ n ] T with λ i, i =,...,n the coefficients of a Hurwitz polynomial p n + λ n p n 2 + λ 2 p + λ with n negative real roots r i <, i =,...,n. The stabilization problem of is equivalent to driving the state vector x t on the surface S = x R n : sx =} since for s x, 3 represents a stable linear differential equation whose unique solution is x =. Thus, the stabilization problem of can be reduced to that of driving the scalar quantity s x to zero. Additionally, bounds on s x can be directly translated into bounds on the state vector x t. Hence, the scalar s x represents a true measure of performance. More specifically, assume that s x t <ρt, t where ρ t =ρ ρ e rt + ρ is an exponentially decaying performance function with appropriately selected parameters ρ, ρ, r >. The following proposition dictates how the parameters r i or equivalently λ i, i =,...,n and ρ, ρ, r should be selected to guarantee exponential convergence with predefined rate of the state vector x t to a prespecified arbitrarily small neighborhood of the origin, thus achieving practical stabilization of with prescribed transient and steady state performance. Proposition 2: Consider: i the state vector x t [x t x n t] T of system, ii the metric s x t as defined in 3 and iii the performance function ρ t = ρ ρ e rt + ρ with r < r i, i =,...,n. If s x t < ρt, t then all x i t, i =,...,n converge at least e rt exponentially } fast to the sets X i = x i R: x i 2i ρ n i, i =,...,n. Proof: Consider a series of i first order linear low pass filters with output y i t and poles r j, j =,...,i, i =,...,n driven by the scalar quantity s x t i.e., i s x t = d dt + r j yi, i =,...,n. It can be j= easily verified for i =that: y t y e r t + t e r t τ s x τ dτ Employing s x t <ρt =ρ ρ e rt +ρ, t and r > r, we obtain: y t ȳ e rt + ρ r, t for a positive constant ȳ = y + ρ ρ r. Applying r recursively the same reasoning, considering the fact that r i > r, i =,...,n, we get: y i t ȳ ie rt + ρ i, i =,...,n 4 for some positive constants ȳ i = y i + ȳi r, i = i r 2,...,n. Before we proceed, notice that y n t equals to the output y t =x t of system. Hence 4 with i = n stands for the achieved output performance of system. Similarly, the remaining system states x i t, i = 2,...,n can be thought of as obtained through a cascaded series of n i first order linear low pass filters with poles r j, j =,...,n i driven by the scalar quantity s x t and a series of i first order linear high pass filters with poles r j, j = n i+,...,n. Incorporating 4 as well as the fact that that: p p+r = r p+r it can be easily verified x 2 t x 2e rt + 2ρ n 2 5 for a positive constant x 2 = ȳ n 2 + r + x n r 2. Finally, it can be proven r ȳn 2 n recursively that: x i t x ie rt + 2i ρ n i 6 for some positive constants x i = ȳ n i + r ȳn i n i+ r + x n i+ r i, i = 3,...,n with ȳ = ρ ρ, which completes the proof. In the sequel, we propose a state feedback control scheme for system that guarantees s x t <ρt, t and consequently, based on Proposition 2, its stabilization with prescribed transient and steady state performance. Lemma : Given the scalar quantity s x t defined in 3, the initialization set Ω and the required transient and steady state output performance specifications, select r i, i =,...,n and the exponentially decaying performance function ρ t = ρ ρ e rt + ρ such that: i the desired performance specifications are met as described in Proposition 2, ii s x < ρ, x Ω. The following state feedback control law: + sx ρt u x, t = k ln 7 sx ρt with k> guarantees the stabilization of and the satisfaction of the desired transient and steady state performance specifications. Proof: To prove our concept, we define the normalized scalar: ξ s x, t = s x 8 ρ t and the generalized state vector ξ = [ x T ξ s ] T. Differentiating ξ with respect to time and substituting the system dynamics and the control input 7, the closed loop dynamical system of ξ may be written as: ξ = h t, ξ Ax + B = ρt [ ] f x kg xln +ξs ξ s f x kg xln +ξs ξ s + n λi xi ξs ρ t. 9 Let us also define the open set Ω ξ = R n,. In what follows, we proceed in two phases. First, the existence of a unique maximal solution ξ t of 9 over the set Ω ξ for a time interval [,τ max i.e., ξ t Ω ξ, t [,τ max is ensured. Then, we prove that the proposed control scheme 7 guarantees, for all t [,τ max : a the boundedness of all closed loop signals of 9 as well as that b ξ t remains strictly within a compact subset of Ω ξ, which leads by contradiction to τ max = and consequently to the completion of the proof. Phase A. The set Ω ξ is nonempty and open. Moreover, the performance function ρ t has been selected to satisfy ρ > s x, x Ω. As a consequence, ξ s < which results in ξ Ω ξ. Additionally, h is continuous on t and locally Lipschitz on ξ over the 586
set Ω ξ. Therefore, the hypotheses of Theorem stated in Subsection II-B hold and the existence of a maximal solution ξ t of 9 on a time interval [,τ max such that ξ t Ω ξ, t [,τ max is ensured. Phase B. We have proven in Phase A that ξ t Ω ξ, t [,τ max and more specifically that: ξ s t,, t [,τ max. Utilizing 8 and we obtain that s x t is absolutely bounded by ρ t for all t [,τ max. Hence, employing Proposition 2 we conclude that x i t x i e rt + 2i ρ n i, i =,...,n for all t [,τ max for some positive constants x i, i =,...,n independent of τ max, which implies that x t Ω x, t [,τ max with: Ω x = x R n : x i x i + 2i ρ n i, i =,...,n j= r j }, the size of which depends solely on: i r i, i =,...,n, ii the performance function ρ t and iii the initialization set Ω. Furthermore, the signal: +ξs t e s t =ln 2 ξ s t is well defined for all t [,τ max owing to. Thus, differentiating with respect to time the positive definite and radially unbounded function V s = 2 e2 s, we obtain: 2e s V s = f x kg x ξs 2 es ρ t n + λ i x i ξ s ρ t. 3 Exploiting the fact that x t Ω x, t [,τ max and the locally Lipschitz property of f x, we conclude, by the Extreme Value Theorem, the boundedness of f x for all t [,τ max. Moreover, ρ t is bounded by construction. Hence, there exists a positive constant F independent of τ max, such that: n f x+ λ i x i ξ s ρ t F, t [,τ max. Additionally, Assumption dictates: g x g, t [,τ max. Furthermore, owing to it holds that ξε 2 >, whereas ρ t > lim t ρ t =ρ > by construction. Therefore, V s < when e s t > F kg e s t ē s =max and thus: max e s, x Ω } F, 4 kg for all t [,τ max, which from 2, taking the inverse logarithmic function leads to: < e ēs = e ēs + ξs ξs t ξs = eēs < 5 eēs + for all t [,τ max. Moreover, the control input 7 remains bounded: u x, t u = kē s, t [,τ max. 6 Up to this point, what remains to be shown is that τ max can be extended to. In this direction, notice by 5 that ξ t Ω ξ, t [,τ max, where the set: [ ] Ω e ξ =Ω x ēs, eēs e ēs + eēs + is a nonempty and compact subset of Ω ξ. Hence, assuming τ max < and since Ω ξ Ω ξ, Proposition in Subsection II-B dictates the existence of a time instant t [,τ max such that ξ t Therefore, τ max =. Thus, all closed loop signals remain bounded and moreover ξ t Ω ξ Ω ξ, t. Finally, from 8 we conclude that: / Ω ξ, which is a clear contradiction. ρ t < e ēs ρ t s x t eēs ρ t <ρt 7 e ēs + eēs + for all t and consequently, owing to Proposition 2, the stabilization of system with prescribed transient and steady state performance. Remark : From the aforementioned proof, it is worth noticing that the proposed control scheme achieves its goals without residing to the need of rendering ē s see 4 arbitrarily small. In the same spirit, the unknown system nonlinearities f x, g x affect only the size of ē s but leave unaltered the achieved convergence properties as 7 dictates. In fact, the actual transient and steady state performance is determined by the selection of r i,i=,...,n as well as of the performance function ρ t. B. Input Saturation To protect the controlled system from the peaking phenomenon [] when estimated states are utilized instead of the actual ones, the saturation of the control input signal is proposed. The following lemma provides a sufficient condition concerning the saturation level, for the stabilization of system without compromising the achieved transient and steady state performance. Lemma 2: Consider the constrained input signal: u s x, t =sat k ln + sx ρt sx ρt, ū 8 where the constant ū>is the saturation level and satv, v is a continuous saturation function defined as follows: v if v v sat v, v = sign v v if v > v. If ū>u with u as defined in 6 then the results of Lemma are still valid even if 7 is replaced by 8. Proof: It can be easily verified that if ū>u then the input is not initially saturated and additionally no saturation will occur for all t. In fact, the signal ξ s t will evolve strictly within the set [ ] ξ s,ξ s, as proven in 5. Therefore, ξ s t [ ] ξ s,ξ s Ωξs where Ω ξs = ξ s, : ln +ξs ū } 9 ξ s k which leads to the stabilization of system with prescribed transient and steady state performance. C. Output Feedback Control Design To implement the control scheme when full state measurement is not available, we use a stable linear filter driven by the measured system output to generate the state estimate ˆx =[ˆx,, ˆx n ] T as follows: ˆx = Aˆx + H μ y C T ˆx, ˆx Ω 2 with: [ ] a an H μ = diag,, μ μ n 587
where μ is a positive constant to be specified and the positive constants a i, i =,...,n are chosen such that the roots of the polynomial: p n + a p n + + a n p + a n = 2 have negative real part. Let us now define the scaled estimation errors: ζ i = xi ˆxi μ n i, i =,...,n. 22 Hence, it is obtained ˆx = x D μ ζ where D μ = diag [ μ n,,μ, ] and ζ = [ζ,,ζ n ] T. Thus, the closed loop system dynamics with the saturated input 8 employing the state estimates i.e., u = u s ˆx, t can be represented in the standard singularly perturbed form: ẋ = Ax + B [f x+gx u s x D μ ζ,t] 23 ξ s = f x+gx us x D μ ζ,t ρt n + λ i x i ξ s ρ t 24 μ ζ =A HC ζ + μb [f x+g x u s x D μ ζ,t] 25 where the matrix A HC with H =[a a n ] is Hurwitz with characteristic equation 2. The singularly perturbed system 23-25 has an exponentially stable boundary layer dζ model i.e., dτ = A HC ζ, obtained by applying the change of time variable τ = t/μ and then setting μ = in 25 and its reduced model is the closed loop system under the saturated state feedback control law 8. The μ- dependent scaling 22 causes an impulsive-like behavior in ζ as μ, but since ζ enters the slow equations 23, 24 through the saturated control law 8, the slow variables x, ξ s do not exhibit a similar impulsive like behavior [9]. The following theorem summarizes the main results of this work. Theorem 2: Consider: i system satisfying Assumption, ii the initialization set Ω R n, iii the appropriately selected performance function ρ t and parameters r i, i =,...,n that impose the required transient and steady state performance specifications, iv the stable linear filter 2 and the saturated input signal u s ˆx, t presented in 8 with saturation level ū satisfying ū>kē s as defined in 4. There exists a constant μ such that for any μ<μ the proposed output feedback control scheme guarantees the stabilization of system with prescribed transient and steady state performance. Proof: The proof has similarities with [9], [] and therefore we shall present it briefly. The main concept is based on the fact that for sufficiently small μ there exists a short transient period during which the fast variables ζ t decay to O μ values, while the slow variables x t, ξ s t remain within a subset of Ω x Ω ξs where Ω ξs was defined in 9. More specifically, owing to the uniform boundedness in μ of the right hand side of 24 there exists a finite time instant T 2, independent of μ, such that ξ s t Ω ξs, [,T 2 ] and consequently that x t Ω x it should be noticed that x t cannot escape Ω x without first ξ s t leaving Ω ξs. Employing V ζ = 2 ζt Pζ, where P = P T > is the solution of the Lyapunov equation: P A HC+A HC T P = I, and the boundedness of x, ξ s, f x, g x, u s x D μ ζ,t for all x, ξ s,ζ Ω x Ω ξs R n, r =.5 r = r =2 2 sxt 2 2 5 5 2 2 5 5 2 5 5 tsec Fig.. The scalar quantity s x t along with the imposed performance bounds. it can be easily shown from 23-25 that for any T < T 2 there exists a constant μ that depends on the system nonlinearities, the saturation level and the required performance specifications such that for all μ μ : i V ζ ζ t Ω ζ = ζ R n : 2 ζt Pζ μ 2}, t [T,T 3 where T 3 T 2 is the first time ξ s t exits from Ω ξs, ii the derivative of V s = 2 e2 s see eq. 2 along the trajectory of 24 satisfies V s for all x, ξ s,ζ Ω x Ω ξs Ω ζ we have proven that Vs, ξ s Ω ξs [ ] ξ s,ξ s and iii the derivative of V ζ = 2 ζt Pζ along the trajectory of 25 satisfies V ζ for all x, ξ s,ζ Ω x Ω ξs Ω ζ. Thus, the set Ω x Ω ξs Ω ζ is positively invariant and the trajectory ξ s t evolves strictly inside Ω ξs which leads to the fact that T 3 = and consequently that s x t <ρt, t. Hence, based on the previous subsection, output feedback stabilization with prescribed transient and steady state performance is achieved which completes the proof. Remark 2: Notice that, contrary to the common treatment in the relevant literature, the selection of the time scale μ is made towards securing only the boundedness of the closed loop trajectories since the performance is directly imposed by the appropriate choice of r i, i =,...,n and the performance function ρ t, as described in Subsection III- A. Thus, there is no need residing to extremely small values of μ in the output feedback case to recover the achieved in the state feedback case, which relaxes significantly the output feedback design procedure. IV. SIMULATION RESULTS To clarify the proposed output feedback design, consider the system: ẋ = x 2 ẋ 2 = 2 x 2 x 2 x ++2+sinx x 2 u y = x and the initialization set Ω = [, ] 2. Clearly, Σ is in canonical form and Assumption is satisfied. For the states x t, x 2 t we require steady state errors of no more Σ 588
2 x t:, x 2t: - - ˆx t:, ˆx 2t: - - r =.5 2 r =.5 5 4 2 5 5 5 5 5 r = 2 r = 5 4 2 5 5 5 5 5 r =2 2 r =2 5 4 5 5 tsec 5 5 5 tsec Fig. 2. The convergence of the system states x t, x 2 t. Fig. 3. Time evolution of the high-gain observer states ˆx t, ˆx 2 t. than.2 and convergence rate up to the exponential e 2t. To introduce the aforementioned specifications, we select according to Proposition 2, r = 6 and the performance function ρ t =. e rt +.. The estimation filter parameters are selected as a =, a 2 =25and the time scale parameter μ =.5. Finally, the control gain was set to k =5. It can be verified that a saturation level ū =5is sufficient to achieve the required performance specifications for the considered initialization set Ω. To illustrate that the performance of the proposed output feedback control scheme is directly imposed by the selection of the performance parameters, without altering the time scale parameter μ, three cases are considered with desired convergence rates e.5t, e t and e 2t i.e., r =.5, r = and r = 2. The initial system and filter state conditions were x =, x 2 = and ˆx =.8, ˆx 2 =. The evolution of s x t is shown in Fig. along with the performance bounds imposed by ρ t. The states x t and x 2 t are given in Fig. 2, while the high gain observer states are shown in Fig. 3. Obviously, output feedback stabilization with prescribed transient and steady state performance as well as reduced peaking effects is achieved, as it was predicted by the theoretical analysis, despite the presence of unknown system nonlinearities. REFERENCES [] F. Deza, E. Busvelle, J. P. Gauthier, and D. 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