A Hyperbolic PDE ODE System with Delay Robust Stabilization

1. International Research Research and Development (IJTSRD) International Open Access Journal ODE System with Delay-Robust Stabilization R. Priyanka,S. Ramadevi Department of Mathematics, Vivekanandha College of Arts and Sciences for Women Elayampalayam, Thiruchengode, Namakkal, Tamil Nadu International Journal of Trend in Scientific Scientific (IJTSRD) International Open Access Journal ISSN No: 2456 ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume www.ijtsrd.com | Volume - 2 | Issue – 5 A Hyperbolic PDE-ODE System obust Stabilization Department of Mathematics, (Autonomous), Elayampalayam of Arts and Sciences for Women Tamil Nadu, India ABSTRACT This paper is concerned with a new development of a delay-robust stabilizing feedback control law for linear ordinary differential equation coupled with two linear first order hyperbolic equations in the actuation path. A second change of variables that reduce stabilization problem of the PDE-ODE system to that of a time-delay system for which a forecaster can be constructed. Hence, by choosing the pole placement on the ODE when constructing the forecaster, enabling a trade-off between convergence rate and delay-robustness. The proposed feedback law is finally proved to be robust to small delays in the actuation. Problem Formulation In this section we detail the notations used through this paper. For any integer classical euclidean norm on ??([0,1],ℝ), or ??([0,1]) if no confusion arises, the space of real-valued functions defined on absolute value is integrable. This space is equipped with the standard ?? norm, that is, for any ??([0,1]) ? ? ?(?)|??. paper is concerned with a new development of a robust stabilizing feedback control law for linear ordinary differential equation coupled with two linear first order hyperbolic equations in the actuation path. A second change of variables that reduces the In this section we detail the notations used through this paper. For any integer ? > 0,∥.∥ℝ? is the classical euclidean norm on ℝ?. We denote by if no confusion arises, the valued functions defined on [0,1] whose absolute value is integrable. This space is equipped norm, that is, for any ? ∈ ODE system to that delay system for which a forecaster can be constructed. Hence, by choosing the pole placement on the ODE when constructing the forecaster, off between convergence rate and robustness. The proposed feedback law is finally proved to be robust to small delays in the ∥ ? ∥??= ? |? ? We denote ??([0,1],ℝ) the space of real square-integrable functions defined on standard ?? norm, i.e., for any the space of real-valued integrable functions defined on [0,1] with the norm, i.e., for any ? ∈ ??([0,1]),ℝ) Keyword: Hyperbolic partial differential equation, Time-delay systems, Stabilization. Hyperbolic partial differential equation, ? ?= ? ? ?(?)??. ∥ ? ∥?? ? INTRODUCTION In this paper we develop a linear feedback that achieves delay-robust stabilization of a system of two hetero directional first-order hyperbolic Partial Differential Equations (PDEs) coupled through the boundary to an Ordinary Differential Equation (ODE). It has been observed, that for many feedback systems, the introduction of arbitrarily small time delays in the loop may cause instability under linear state feedback. In particular, for coupled linear hyperbolic systems, recent contributions have highlighted the necessity of a change of order to achieve delay-robust stabilization. The main contribution of this paper is to provide a new design for a state-feedback law for a PDE-ODE system that ensures the delay-robust stabilization. The original system can then be rewritten as a distributed delay equation for which it is possible to derive a stabilizing control law. ? In this paper we develop a linear feedback control law robust stabilization of a system of The set ?∞([0,1],ℝ) denotes the space of bounded real-valued functions defined on standard ?∞ norm, i.e., for any ∥ ? ∥??= ????∈[ In the following, for (?,?,?) define the norm ∥ (?,?, ∥ ? ∥ℝ?. The set ??([0,1]) stands for the space of real valued functions defined on [0,1 differentiable and whose ??ℎ The set ? is defined as ? = {(?,?) ∈ [0,1] denotes the space of bounded order hyperbolic Partial valued functions defined on [0,1]with the norm, i.e., for any ? ∈ ?∞([0,1],ℝ) Differential Equations (PDEs) coupled through the boundary to an Ordinary Differential Equation many feedback [?,?]|?(?)|. systems, the introduction of arbitrarily small time- delays in the loop may cause instability under linear state feedback. In particular, for coupled linear hyperbolic systems, recent contributions have highlighted the necessity of a change of paradigm in robust stabilization. The main contribution of this paper is to provide a new design ??ℝ?, we ) ∈ ???([0,1])? ?) ∥=∥ ? ∥??+∥ ? ∥??+ stands for the space of real valued 1] that are ? times derivative is continuous. ODE system that robust stabilization. The original a distributed delay equation for which it is possible to derive a stabilizing ]? ?.?.? ≤ ?}. @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Aug 2018 Page: 1988

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 and we consider only weak ?? solutions to the system. The initial condition of the ODE is denoted ??. Remark that this system naturally features several couplings that can be extended to the case ? = 0 with a slight modification transformation. Control Problem The goal of this paper is to design a feedback control law ? = ?[(?,?,?)] where ?:???([0,1])? ℝ is a linear operator, such that: The state (?,?,?) of the resulting feedback system exponentially converges to its zero equilibrium (stabilization problem), i.e. there exist ??≥ 0 and ? > 0 such that for any initial condition (??,??,??) ∈ (??[0,1])??ℝ? ∥ (?,?,?) ∥≤ ??????∥ (??,??,??) ∥ , ? ≥ 0. The resulting feedback system is robustly stable with respect to small delays in the loop (delay- robustness), i.e. there exists ?∗> 0 such that for any ? ∈ [0,?∗], the control law ?(? − ?) still stabilizes. A control law that satisfies these two constraints is said to delay-robustly stabilize system In this paper, we make the two following assumptions: Assumption 1: The pair (?,?) is stabilizable, i.e. there exists a matrix ? such that ? + ?? is Hurwitz. Assumption 2: The proximal reflection ? and the distal reflection ? satisfy |??| < 1. The first assumption stabilizability of the whole system, while the second assumption is required for the existence of a delay- robust linear feedback assumption is not restrictive since, if is not fulfilled, one could prove using arguments similar to those in Auriol that the open-loop transfer function has an infinite number of poles in the complex closed right half-plane. Consequently, one cannot find any linear state feedback law ?(.) that delay robustly stabilizes. ?(?) stands for the space of real-valued continuous functions on ?. For a positive real ? and two reals ? < ?, a function ?defined on [?,?] is said to ?- Lipschitz if for all (?,?) ∈ [?,?]?, it satisfies |?(?) − ?(?)| ≤ ?|? − ?|. The symbol ?? (or I if no confusion arises) represents the ??? identity matrix. We use the notation ??(?) for the Laplace transform of a function ?(?), provided it is well defined. The set ? stands for the convolution Banach algebra of BIBO – stable generalized functions in the sense of Vidyasagar. A function g(.) belongs to ? if it can be expressed as of the backstepping ??ℝ?→ ∞ ?(? − ??), ?(?) = ??(?) + ??? ??? Where ??∈ ??(ℝ?,ℝ),∑ ⋯??? ?(.) is the direct distribution? The associated norm is < ∞,0 = ??< ??< |??| ??? . ∥ ? ∥?=∥ ??∥??+ ?|??| ??? The set ?? of Laplace transforms of elements in ? is also Banach algebra with associated norm ∥ ? ? ∥??=∥ ? ∥?. System Under Consideration We consider a class of systems consisting of an ODE coupled to two hetero directional first-order linear hyperbolic systems in the actuation path. We consider systems of the form: ??(?,?) + ???(?,?) = ???(?)?(?,?) ??(?,?) − ???(?,?) = ???(?)?(?,?) ?̇(?) = ??(?) + ??(?,0), Evolving in {(?,?)?.? ? > 0,? ∈ [0,1]}, with the boundary conditions ?(?,0) = ??(?,0) + ??(?) ?(?,1) = ??(?,1) + ?(?), Where ? ∈ ℝ? is the ODE state, ?(?,?) ∈ ℝ ??? ?(?,?) ∈ ℝ are the PDE states and ?(?) is the control input. The in-domain coupling terms ??? and ??? belong to ??([0,1]), the boundary coupling terms ? ≠ 0 (distal reflexion) and ? (proximal reflexion), and the velocities ? and ? are constants. Furthermore, the velocities verify −? < 0 < ?. The initial conditions of the state (?,?) are denoted ?? and ?? and are assumed to belong to ??([0,1],ℝ) is necessary for the control. This second @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 1989

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Lemma Take A,B and K verifying Assumption 1 and any k such that holds. Then, the control law ? ??(?) + ∫ ??(???) ??? ? Exponentially stabilizes?(?). Furthermore, the state feedback ????(?) = ?????(?) − (? − ?)?? Exponentially stabilizes ?(? Proof For the state-forecaster feedback ?????( loop system in ?̇(?) = ??(?) + ???????? − CONCLUSION In this paper, a delay-robust stabilizing feedback control law was developed for a coupled hyperbolic PDE-ODE system. The proposed method combines using the back stepping approach with a second forecaster-type feedback. The second feedback control is obtained after a suitable change of variables that reduces the stabilization problem of the PDE ODE system. REFERENCES 1.Auriol, J., Aarsnes, U. J. Meglio. F. (2018). Delay— for two hetero directional linear coupled hyperbolic PDEs. IEEE Automatic Control. Take A,B and K verifying Assumption 1 and any k such that holds. Then, the control law ?????(?) = robust stabilizing feedback control law was developed for a coupled hyperbolic ODE system. The proposed method combines ? ? ?????(?)?? )???, ? ?? approach with a second feedback. The second feedback control is obtained after a suitable change of variables that reduces the stabilization problem of the PDE- . Furthermore, the state )??(? −? (?). ?) J. F., Martin, P., & Di —robust control design directional linear coupled IEEE ? (.), the closed- hyperbolic PDEs. Transactions Transactions on on ? ??, ? ?. ? ?satisfies ? ? ≥ 2.Datko, R., Lagnese, J., & polis, M. example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM Journal on Control and Optimization. Control and Optimization. & polis, M. P. (1986). An ?̇(?) = (? + ??)?(?), ? ≥ Exponential stability is guaranteed by the fact that (? + ??) is Hurwitz. By construction of using ??(?) = 1 − (? − ?)????, we have that solution of ?̇(?) − (? − ?)??̇(? − ?) = (? − ?)???(? − ?) + ?????−?+ ??????−1?, satisfies for any example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM Exponential stability is guaranteed by the fact that is Hurwitz. By construction of ?(?) and 3.Niculescu, S-I. (2001a). Delay effects on stability: a robust control approach, Vol.269. Springer Science & Business Media. Science & Business Media. I. (2001a). Delay effects on stability: a robust control approach, Vol.269. Springer we have that ?(?), ??(?) − (? − 4.Sipahi, R., Niculescu. S. Michiels, W., & Gu, K. (2011). Stability and stabilization of systems with time delay. IEEE Control Systems. . S.-I., Abdallah, C. T., Michiels, W., & Gu, K. (2011). Stability and of systems with time delay. IEEE satisfies for any ?≥?, 5.Zhong, Q.-C. (2006). Robust Control of time delay systems. Springer-Verlag. C. (2006). Robust Control of time- Verlag. ?(?). 1,?(?) is also ?(?) = (? − ?)??(? − ?) + ? Since |(? − ?)?| < 1?? |??| + |??| < exponentially stable.