Lyapunov Technique and Backstepping for Nonlinear Neutral Systems
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[4] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “Statespace solution to standard and control problems,” IEEE Trans. Autom. Control, vol. AC-34, pp. 831–847, 1989. [5] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to control,” Int. J. Robust Nonlin. Control, vol. 4, pp. 421–448, 1994. [6] C. Scherer, Theory of Robust Control. Delft, The Netherlands: Mechanical Engineering Systems and Control Group, Delft University of Technology, 2001, Lecture Note. [7] C. Scherer and S. Weiland, Linear Matrix Inequalities in Control. Delft, The Netherlands: Delft University of Technology and Eindhoven University of Technology, 2005, Lecture Notes. [8] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [9] P. Apkarian and P. Gahinet, “A convex characterization of gain-scheduled controllers,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 853–864, May 1995. [10] M. C. de Oliveira and R. E. Skelton, “Stability test for constrained linear systems,” in Perspectives in Robust Control, S. O. Reza Moheimani, Ed. London, U.K.: Springer, 2001, vol. 268, Lecture Notes in Control and Information Sciences. [11] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkhauser, 2003. [12] J. Hespanha, “Root-mean-square gains of switched linear systems,” IEEE Trans. Autom. Control, vol. 48, pp. 2040–2045, 2003. [13] J. Geromel and P. Colaneri, “ and dwell time specifications of continuous-time switched linear systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 207–212, Jan. 2010. [14] M. Margaliot and J. Hespanha, “Root-mean-square gains of switched linear systems: A variational approach,” Automatica, vol. 44, pp. 2398–2402, 2008. [15] K. Hirata and J. Hespanha, “ -induced gain analysis for a class of switched systems,” in Proc. 48th Conf. Decision and Control, 2009, pp. 2138–2143. [16] L. Wu and W. X. Zheng, “Weighted model reduction for linear switched systems with time-varying delay,” Automatica, vol. 45, pp. 186–193, 2009. [17] L. Wu, D. W. C. Ho, and C. W. Li, “Sliding mode control of switched hybrid systems with stochastic perturbation,” Syst. Control Lett., vol. 60, pp. 531–539, 2011. [18] L. Wu, Z. Feng, and W. X. Zheng, “Exponential stability analysis for delayed neural networks with switching parameters: Average dwell time approach,” IEEE Trans. Neural Networks, vol. 121, pp. 1396–1407, 2010. [19] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Logic-based switching control of a nonholonomic system with parametric modeling uncertainty,” Syst. Control Lett., vol. 38, pp. 167–177, 1999. [20] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Hysteresis-based switching algorithms for supervisory control of uncertain systems,” Automatica, vol. 39, pp. 263–272, 2003. [21] S. Boyarski and U. Shaked, “Time-convexity and time-gain-scheduling in finite-horizon robust -control,” in Proc. 48th CDC09, Shanghai, China., 2009, pp. 2765–2770. [22] L. I. Allerhand and U. Shaked, “Robust stability and stabilization of linear switched systems with dwell time,” IEEE Trans. Autom. Control, vol. 56, pp. 381–386, 2011. [23] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA.: The MathWorks Inc., 1995. [24] I. R. Petersen, “Quadratic stabilizability of uncertain linear systems: Existence of a nonlinear stabilizing control does not imply existence of a stabilizing control,” IEEE Trans. Autom. Control, vol. AC-30, no. 3, pp. 291–293, Mar. 1985. [25] J. Ackermann, “Longitudinal control of fighter aircraft F4E with additional canards,” in A Collection of Plant Models and Design Specifications for Robust Control, K. P. Sondergeld, Ed. Oberpfaffenhoffen, Germany: DFVLR, 2012. [26] U. Shaked, “A LPV approach to robust and static output-feedback design,” IEEE Trans. Autom. Control, vol. 48, pp. 866–872, 2003. [27] P. L. D. Peres, J. C. Geromel, and S. R. Souza, “ control design by static output-feedback,” in Proc. IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, 1994, pp. 243–248. [28] J. Geromel and P. Colaneri, “Stability and stabilization of continuoustime switched linear systems,” SIAM J. Control Optim., vol. 45, no. 5, pp. 1915–1930, 2006.
Lyapunov Technique and Backstepping for Nonlinear Neutral Systems Frédéric Mazenc and Hiroshi Ito
Abstract—For nonlinear systems with delay of neutral type, we propose a new technique of stability and robustness analysis. It relies on the construction of functionals which make it possible to establish estimates of the solutions different from, but very similar to, estimates of input-to-state stability (ISS) or integral ISS (iISS) type. These functionals are themselves different from, but very similar to, ISS or iISS Lyapunov-Krasovskii functionals. The approach applies to systems which do not have a globally Lipschitz vector field and are not necessarily locally exponentially stable. We apply this technique to carry out a backstepping design of stabilizing control laws for a family of neutral nonlinear systems. Index Terms—Backstepping., delay, neutral system, nonlinear, stability.
I. INTRODUCTION Research devoted to systems with delay shows a lot of interest in systems of neutral type. Indeed, these systems are encountered in a wide range of engineering problems which include heat exchangers [14], population ecology [15, Ch. 2], distributed networks [5]. These systems have the feature of incorporating retarded derivatives of state variable. Typically they are of the type (1) , where is a constant delay [27], but they where can be more complicated: in particular they may be time-varying, include several pointwise delays, distributed or time-varying delays and disturbances. Many papers have explored many problems for these systems in the particular case where the function is linear. In some contributions, the frequency domain approach has been used to provided with sufficient stability conditions: see [4], [9], [19]. In the time-domain, LMI conditions ensuring the asymptotic stability of the origin of the studied systems have been proposed in [3], [6]–[8], [22] and in other papers. Further studies have extended these conditions to cases where the function is nonlinear. In most of them, the nonlinearities are regarded as disturbances and the stability of the system is deduced from the presence of stabilizing linear terms: results of this type, for instance, are given in [13], [25], [27]. A common feature of these works is that they apply only to systems that are locally exponentially stable and globally Lipschitz. The results presented in [15, Ch. 9] are different from all those we have mentioned: the local asymptotic stability of a family of nonlinear systems, which are not necessarily locally exponentially stable, is established by using a Lyapunov functional. The present study owes a Manuscript received September 14, 2011; revised March 29, 2012, June 14, 2012, and June 20, 2012; accepted June 22, 2012. Date of publication June 29, 2012; date of current version January 19, 2013. This work was supported in part by Grant-in-Aid for Scientific Research (C) of JSPS under Grant 22560449. Recommended by Associate Editor P. Pepe. F. Mazenc is with EPI INRIA DISCO, L2S-CNRS-Supélec, 91192 Gif-surYvette, France, (e-mail: [email protected]). H. Ito is with the Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2012.2206709
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great deal to [15, Ch. 9]: the first main objective we pursue is to complement [15, Ch. 9] in two directions: 1) we aim at establishing results of global asymptotic stability and 2) we wish to construct Lyapunov-Krasovskii functionals from which robustness properties can be derived. To achieve these goals, we propose a new technique of stability and robustness analysis for nonlinear neutral systems based on the design of functionals of a new type. The family of systems to which our technique applies is general in the sense that it includes systems that are not globally Lipschitz and not locally exponentially stable. The robustness result we establish is different, but very similar to the input-to-state stability (ISS) or integral ISS (iISS) robustness. Our technique of proof relies on the construction of functionals which are not Lyapunov-Krasovskii functionals as defined in preceding studies, for instance in [10], [12], [16], [21] or iISS Lyapunov-Krasovskii functionals as defined for instance in [11], [20], but they are very similar to these functionals. Our approach to neutral systems also differs from the analysis method of Lyapunov-Krasovskii type developed in [21] in the sense that this technical note does not require the formulation into coupled delay differential and difference equations with an auxiliary variable in considering the robustness. This first robust stability result in this technical note will help us to achieve our second goal, which is the extension of the backstepping approach to nonlinear neutral systems. Thus, we will complement that way the backstepping result for systems with delay presented in [17], [18] and [2]. To the best of our knowledge, the result we will propose is the first backstepping result for neutral systems and it is not covered by any of the scarce results of construction of stabilizing feedbacks for nonlinear neutral systems available in the literature, as for instance those in [24] and [26]. For the sake of simplicity, we have considered the case of systems with only one input, but extensions to the multi-input case can be easily deduced from our design. Our technical note is organized as follows. We state and prove the main result in Section II. We present a backstepping design in Section III. Some applications of the theoretical results are given in Section IV. Concluding remarks are drawn in Section V.
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as . A continuous function is of class provided that for each fixed , the function belongs to class , and for each fixed , the function is nonincreasing and as . A class function is said to be equal to a quadratic function in a neighborhood of the origin , such that for if there exist two constants . all II. GENERAL RESULT FOR NEUTRAL SYSTEMS We consider a functional differential equation
(3) , with initial conditions in at , with locally Lipschitz, with continuous and is a constant delay. We introduce two assumptions. Generally speaking, the first one imposes a limitation of growth type on the dependency of with respect and the second one also limits the influence of to and guarantees the existence of a Lyapunov function whose derivative along the solutions of the studied system satisfies an inequality of iISS and are regarded as inputs. type when both Assumption 1: There exist a nonnegative real number , a continand a function of class such uous function , , that for all with
(4) , two functions Assumption 2: There are a function of class and of class , a continuous positive definite function and two of class so that, for all functions and (5) and for all
,
Notation and Definitions (6) We let denote the Euclidean norm of matrices and vectors of defined on an interval any dimension. Given , let denote its (essential) supremum over . A function , where is a real numbers and is a real number , is piecewise continuously differentiable if it is continuous on or and, for any real number , is continuous the interval , except at a finite number of points. We let denote over -valued functions defined on a given interval the set of all that are continuous, piecewise continuously differentiable and with an essentially bounded first derivative . For a function with or , for all , the function is defined by for all . The solution of a time-delay system described by a functional differential equation: (2) where is a continuous function, with an initial condition at , will be denoted by (instead of as rigorously done for instance in [10, Ch. 2]). The notation will be simplified whenever no confusion can arise from the context. In particular, along the trajecthe time derivative of a Lyapunov functional . A continuous tories of a system (2) will be denoted simply by belongs to class provided that it is function . It belongs to class if, in addition, strictly increasing and
Moreover, for all (7) where
is the constant provided by Assumption 1, for all
(8) where is the function provided by Assumption 1, and there exists a function of class such that, for all (9) We are ready to state and prove the main result of the technical note. Theorem 1: Assume that the system (3) satisfies Assumptions 1 and be a positive real number such that the inequality 2. Let (10) is satisfied. Then the derivative of the functional (11)
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along the trajectories of (3) satisfies (12) for almost all
In the sequel, all equalities and inequalities of and should be , almost everywhere. Elementary understood to hold for all calculations yield
, with
(19) (13)
Since
, defined in (11), satisfies, along the trajectories of (3)
and (14) 1) Discussion on Theorem 1: , one can determine a constant • It is worth noting that for any such that the inequality (10) is satisfied. Therefore the asymptotic stability of the system (3) with identically equal to zero which is implied by Theorem 1 is delay independent. Notice . that (10) is satisfied for all • The requirement (7) imposes a restriction on the size of . This assumption can possibly be relaxed, but it cannot be removed: some systems which satisfy Assumptions 1 and 2 except (7) are not locally asymptotically stable. For instance the 1-D system is exponentially unstable when is smaller than a certain threshold. • A remarkable feature of Theorem 1 is that it applies to systems that are not necessarily locally exponentially stable or globally Lipschitz. • The family of functionals in (11) and the ones of [15, Ch. 9] differ mostly because each functional admits a derivative along the solutions of the system studied that is smaller than a negative , which is not the case of the functionals definite function of provided in [15, Ch. 9]. The advantage of knowing functionals satisfying (12) is that they lead to robustness properties: from the features of and , and by adapting the proofs of [1], [23] (see also [16, Ch. 2]), one can deduce from (12) that all the solutions and there are a function of of (3) are defined over and a function of class such that, for any solution class of (3), for all times , the inequality (15)
(20) by (6) in Assumption 2 and (20), we obtain
(21) From (4) in Assumption 1 and the fact that the function decreasing, we deduce that
is non-
(22) Since the function
is nondecreasing, the inequality
(23) is satisfied. Consequently
(24) Now, (7) and (8) in Assumption 2 ensure that and
. It follows that
holds. Moreover, if the function provided by Assumption 2 is of class , there are a function of class and a function of class such that, for any solution of (3), for all times , , the inequality
(16) holds. In both cases, the functions and are delay-dependent. Proof: Theorem 3.3 in [15, Ch. 3] ensures that for any initial , the system (3) admits a unique solution , which condition is piecewise continuously differentiable and defined over the interval , where is a finite positive real number or . If the max, then is not essentially imum of such intervals satisfies . bounded over defined by Now, we introduce the operator
(25) Since
satisfies (10), the inequality (26)
with
defined in (14) is satisfied. Consequently, for all (27)
(17) is a constant such that (10) holds. It satisfies, along the where trajectories of (3) (18)
By virtue of the continuity of and (20), it follows that, if real number, for all
is a finite
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Then, from Assumption 2, we deduce that, for all , , with . Next, using Assumption 1, we deduce that, almost everywhere over
From this inequality, one can deduce by using the steps method that is . Thus cannot be a real number, i.e., essentially bounded over . Hence, all the solutions of (3) are defined over . Finally, we observe that (9) in Assumption 2 and (26) give (29) which leads to the inequality (12) with
defined in (13).
III. BACKSTEPPING FOR NEUTRAL SYSTEMS In this section, we illustrate the usefulness of Theorem 1 by using it to solve a stabilization problem for a particular family of neutral systems in feedback form. We consider a neutral system of the type (30) , is a constant, , is a locally Lipschitz where is the input. nonlinear function and To begin with, we impose an assumption, which is natural in a backstepping context: of class such that Assumption A: There exists a function the system (31) is locally satisfies Assumptions 1 and 2 and the function Lipschitz. We also introduce an assumption which is instrumental in our design of stabilizing control laws and has no connection with the classical backstepping approach: Assumption B: Let , , be the functions provided by Assumptions 1 and 2. The functions and are equal to a quadratic function in a neighborhood of the origin. There exist a class function and two positive real numbers , such that
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Remark 1a: Assumptions A and B ensure that in (33) is well defined and locally Lipschitz. Remark 1b: Many extensions of Theorem 2 can be expected. For instance, it can be immediately extended to systems of the form
where is a continuous functional since the change of feedback gives a new system of the form (30). Remark 1c: There is a lot of flexibility in the backstepping design, so that we can modify the control (33). Assumption B can be replaced by functions , , satisfying other types of local homogeneous properties. defined in Proof: To begin with, we observe that the function because the func(34) is locally Lipschitz and positive over and are locally Lipschitz and are equal to positive tions definite quadratic functions in a neighborhood of the origin. Next, arguing as we did in the proof of Theorem 1, one can prove , the system (30) in that for any initial condition closed-loop with the feedback (33) admits a unique solution , which is piecewise continuously differentiable and defined over the , where is a finite positive real number or . If interval the maximum of such intervals satisfies , then is not . essentially bounded over Next, we construct a Lyapunov functional for (30) in closed-loop with the feedback (33). In the sequel, all equalities and inequalities of , and should be understood to hold almost everywhere as long as the solutions are well-defined. The change of variable transforms the system (30) into (35) Assumption A ensures that Theorem 1 applies to the -subsystem of of the (35). It follows that there are a functional satisfying such that type (11) and a constant the time derivative of along the solutions of the -subsystem of (35) satisfies (36) where fined by
. It follows that the functional de-
(37) (32) and for all . Moreover, the functions , , , are locally Lipschitz. We are ready to state and prove the following result: Theorem 2: Assume that the system (30) satisfies Assumptions A and B. Then the control law
admits a time derivative along the solutions of (35) which satisfies: (38) We introduce now the candidate control Lyapunov-Krasovskii funcfor the system (35) defined by tional (39)
(33) where is a constant such that (10) is satisfied and such that, for all function continuous over
is the (40) (34)
globally asymptotically stabilizes the origin of the system (30).
Then its time derivative along the solutions of (35) satisfies
By using (32), we obtain
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(41) The property that be proved by considering the cases is of class ) ensures that
holds for and
(which can because
Assumption 3: The function there exists a function of class
is positive definite and such that, for all , (49)
We are ready to state and prove the following result: Corollary 1: Assume that the system (47) satisfies Assumption 3. Then, when (50)
(42) the derivative of the functional
It follows that
(51)
(43) Since, for all
satisfies (52)
, we have
the inequality
(44) is satisfied. From , for all
and (11) it follows that . Therefore (45)
for all follows that
along the solutions of (47) with initial conditions in
,
. Since the function
is nondecreasing, it
(46) Arguing as we did at the end of the proof of Theorem 1, one can prove that the solutions are defined over . Since is of class , it converges to zero when goes to the follows that infinity. This fact and the properties of allow us to conclude.
, with , where for almost all is the function in (49) and , where is any . real number such that Remark 2: The functional defined in (51) is nonnegative because is positive definite. the function Remark 3: One can prove that Assumption 3 is satisfied if and only if is positive definite and . the function is Thus, Corollary 1 does not need the requirement that radially unbounded, which is imposed in [10, Sec. 9.8]. Proof of Corollary 1: Let us check now that, under the conditions of Corollary 1, the assumptions of Theorem 1 are satisfied. is positive definite, it follows that the Since the function and function is positive definite and are positive real numbers or . Since is of class , (49) implies such that for all that there exists a constant . It follows that is radially unbounded. and of class such that, Therefore, there are two functions , . for all Next, the right hand side of (47) satisfies Assumption 1 with , and for all . We observe that, for all (53) We deduce from Young’s inequality that
IV. APPLICATIONS
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In this section, we illustrate Theorem 1 via simple examples. A. Application to a Particular Family of One-Dimensional Systems We consider the following 1-D neutral system:
, and with over, we have established that there are two functions that the inequalities (5) are satisfied. Next, for all
(47) where , where is of class , where is a real number and where is a continuous function which is studied in [15, Ch. 9] and in [10, Sec. 9.8] in the case where is not present. This system is used to modelize a shunted power transmission line. For this system, we establish a stability result via a Lyapunov construction based on Theorem 1. With a view to it, we introduce the function
and
. Moresuch (55)
From (50), it follows that, for all
, (56)
Moreover, from (50), it follows that, for all (57)
(48) and the assumption:
It follows that Assumption 2 is satisfied. Hence, Theorem 1 applies. This allows us to conclude.
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B. Examples For the important family of neutral systems Corollary 1 focuses on, this section presents an example which illustrate the robustness of the types (16) and (15), respectively. The second result illustrates Theorem 2. 1) First Example: Consider the 1-D system (58) and is not present, the origin of this system is not Even when is not locally exponentially stable. Moreover, the function globally Lipschitz. , satisfy The functions with . Since this function is of class , Assumption 3 is satisfied. Thus, Corollary 1 provides with the functional
where is any real number such that along the trajectories of (58) satisfies
. Its derivative
with , and . Since , we deduce that the solutions of (58) the function is of class satisfy an inequality of the type (16). 2) Second Example: Consider the 2-D system
(59) . Let with Then Assumption 1 is satisfied by the system
for all
.
(60) , , . Moreover, Assumption with , 2 is satisfied by the system (60) with , , , provided . Thus, the system (59) satisfies Assumptions A and B . From Theorem 2, we deduce that if with the origin of the system (59) is globally asymptotically stabilized by the control law
with lows
. The above choice of to be bounded in terms of .
(61) al-
V. CONCLUSION We have proposed a new stability analysis technique for neutral systems based on the construction of an adequate Lyapunov-Krasovskii functional. The technique applies to broad families of neutral systems which include systems which are not locally exponentially stable and not globally Lipschitz and makes it possible to construct stabilizing control laws for systems in feedback form under appropriate assumptions. Much remains to be done. In particular, extensions to
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time-varying systems, to systems with several delays are expected, as long as applications to other control problems for neutral systems than the one solved in Section III.
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[4] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “Statespace solution to standard and control problems,” IEEE Trans. Autom. Control, vol. AC-34, pp. 831–847, 1989. [5] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to control,” Int. J. Robust Nonlin. Control, vol. 4, pp. 421–448, 1994. [6] C. Scherer, Theory of Robust Control. Delft, The Netherlands: Mechanical Engineering Systems and Control Group, Delft University of Technology, 2001, Lecture Note. [7] C. Scherer and S. Weiland, Linear Matrix Inequalities in Control. Delft, The Netherlands: Delft University of Technology and Eindhoven University of Technology, 2005, Lecture Notes. [8] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [9] P. Apkarian and P. Gahinet, “A convex characterization of gain-scheduled controllers,” IEEE Trans. Autom. Control, vol. 40, no. 5, pp. 853–864, May 1995. [10] M. C. de Oliveira and R. E. Skelton, “Stability test for constrained linear systems,” in Perspectives in Robust Control, S. O. Reza Moheimani, Ed. London, U.K.: Springer, 2001, vol. 268, Lecture Notes in Control and Information Sciences. [11] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkhauser, 2003. [12] J. Hespanha, “Root-mean-square gains of switched linear systems,” IEEE Trans. Autom. Control, vol. 48, pp. 2040–2045, 2003. [13] J. Geromel and P. Colaneri, “ and dwell time specifications of continuous-time switched linear systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 207–212, Jan. 2010. [14] M. Margaliot and J. Hespanha, “Root-mean-square gains of switched linear systems: A variational approach,” Automatica, vol. 44, pp. 2398–2402, 2008. [15] K. Hirata and J. Hespanha, “ -induced gain analysis for a class of switched systems,” in Proc. 48th Conf. Decision and Control, 2009, pp. 2138–2143. [16] L. Wu and W. X. Zheng, “Weighted model reduction for linear switched systems with time-varying delay,” Automatica, vol. 45, pp. 186–193, 2009. [17] L. Wu, D. W. C. Ho, and C. W. Li, “Sliding mode control of switched hybrid systems with stochastic perturbation,” Syst. Control Lett., vol. 60, pp. 531–539, 2011. [18] L. Wu, Z. Feng, and W. X. Zheng, “Exponential stability analysis for delayed neural networks with switching parameters: Average dwell time approach,” IEEE Trans. Neural Networks, vol. 121, pp. 1396–1407, 2010. [19] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Logic-based switching control of a nonholonomic system with parametric modeling uncertainty,” Syst. Control Lett., vol. 38, pp. 167–177, 1999. [20] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Hysteresis-based switching algorithms for supervisory control of uncertain systems,” Automatica, vol. 39, pp. 263–272, 2003. [21] S. Boyarski and U. Shaked, “Time-convexity and time-gain-scheduling in finite-horizon robust -control,” in Proc. 48th CDC09, Shanghai, China., 2009, pp. 2765–2770. [22] L. I. Allerhand and U. Shaked, “Robust stability and stabilization of linear switched systems with dwell time,” IEEE Trans. Autom. Control, vol. 56, pp. 381–386, 2011. [23] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA.: The MathWorks Inc., 1995. [24] I. R. Petersen, “Quadratic stabilizability of uncertain linear systems: Existence of a nonlinear stabilizing control does not imply existence of a stabilizing control,” IEEE Trans. Autom. Control, vol. AC-30, no. 3, pp. 291–293, Mar. 1985. [25] J. Ackermann, “Longitudinal control of fighter aircraft F4E with additional canards,” in A Collection of Plant Models and Design Specifications for Robust Control, K. P. Sondergeld, Ed. Oberpfaffenhoffen, Germany: DFVLR, 2012. [26] U. Shaked, “A LPV approach to robust and static output-feedback design,” IEEE Trans. Autom. Control, vol. 48, pp. 866–872, 2003. [27] P. L. D. Peres, J. C. Geromel, and S. R. Souza, “ control design by static output-feedback,” in Proc. IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, 1994, pp. 243–248. [28] J. Geromel and P. Colaneri, “Stability and stabilization of continuoustime switched linear systems,” SIAM J. Control Optim., vol. 45, no. 5, pp. 1915–1930, 2006.
Lyapunov Technique and Backstepping for Nonlinear Neutral Systems Frédéric Mazenc and Hiroshi Ito
Abstract—For nonlinear systems with delay of neutral type, we propose a new technique of stability and robustness analysis. It relies on the construction of functionals which make it possible to establish estimates of the solutions different from, but very similar to, estimates of input-to-state stability (ISS) or integral ISS (iISS) type. These functionals are themselves different from, but very similar to, ISS or iISS Lyapunov-Krasovskii functionals. The approach applies to systems which do not have a globally Lipschitz vector field and are not necessarily locally exponentially stable. We apply this technique to carry out a backstepping design of stabilizing control laws for a family of neutral nonlinear systems. Index Terms—Backstepping., delay, neutral system, nonlinear, stability.
I. INTRODUCTION Research devoted to systems with delay shows a lot of interest in systems of neutral type. Indeed, these systems are encountered in a wide range of engineering problems which include heat exchangers [14], population ecology [15, Ch. 2], distributed networks [5]. These systems have the feature of incorporating retarded derivatives of state variable. Typically they are of the type (1) , where is a constant delay [27], but they where can be more complicated: in particular they may be time-varying, include several pointwise delays, distributed or time-varying delays and disturbances. Many papers have explored many problems for these systems in the particular case where the function is linear. In some contributions, the frequency domain approach has been used to provided with sufficient stability conditions: see [4], [9], [19]. In the time-domain, LMI conditions ensuring the asymptotic stability of the origin of the studied systems have been proposed in [3], [6]–[8], [22] and in other papers. Further studies have extended these conditions to cases where the function is nonlinear. In most of them, the nonlinearities are regarded as disturbances and the stability of the system is deduced from the presence of stabilizing linear terms: results of this type, for instance, are given in [13], [25], [27]. A common feature of these works is that they apply only to systems that are locally exponentially stable and globally Lipschitz. The results presented in [15, Ch. 9] are different from all those we have mentioned: the local asymptotic stability of a family of nonlinear systems, which are not necessarily locally exponentially stable, is established by using a Lyapunov functional. The present study owes a Manuscript received September 14, 2011; revised March 29, 2012, June 14, 2012, and June 20, 2012; accepted June 22, 2012. Date of publication June 29, 2012; date of current version January 19, 2013. This work was supported in part by Grant-in-Aid for Scientific Research (C) of JSPS under Grant 22560449. Recommended by Associate Editor P. Pepe. F. Mazenc is with EPI INRIA DISCO, L2S-CNRS-Supélec, 91192 Gif-surYvette, France, (e-mail: [email protected]). H. Ito is with the Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2012.2206709
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great deal to [15, Ch. 9]: the first main objective we pursue is to complement [15, Ch. 9] in two directions: 1) we aim at establishing results of global asymptotic stability and 2) we wish to construct Lyapunov-Krasovskii functionals from which robustness properties can be derived. To achieve these goals, we propose a new technique of stability and robustness analysis for nonlinear neutral systems based on the design of functionals of a new type. The family of systems to which our technique applies is general in the sense that it includes systems that are not globally Lipschitz and not locally exponentially stable. The robustness result we establish is different, but very similar to the input-to-state stability (ISS) or integral ISS (iISS) robustness. Our technique of proof relies on the construction of functionals which are not Lyapunov-Krasovskii functionals as defined in preceding studies, for instance in [10], [12], [16], [21] or iISS Lyapunov-Krasovskii functionals as defined for instance in [11], [20], but they are very similar to these functionals. Our approach to neutral systems also differs from the analysis method of Lyapunov-Krasovskii type developed in [21] in the sense that this technical note does not require the formulation into coupled delay differential and difference equations with an auxiliary variable in considering the robustness. This first robust stability result in this technical note will help us to achieve our second goal, which is the extension of the backstepping approach to nonlinear neutral systems. Thus, we will complement that way the backstepping result for systems with delay presented in [17], [18] and [2]. To the best of our knowledge, the result we will propose is the first backstepping result for neutral systems and it is not covered by any of the scarce results of construction of stabilizing feedbacks for nonlinear neutral systems available in the literature, as for instance those in [24] and [26]. For the sake of simplicity, we have considered the case of systems with only one input, but extensions to the multi-input case can be easily deduced from our design. Our technical note is organized as follows. We state and prove the main result in Section II. We present a backstepping design in Section III. Some applications of the theoretical results are given in Section IV. Concluding remarks are drawn in Section V.
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as . A continuous function is of class provided that for each fixed , the function belongs to class , and for each fixed , the function is nonincreasing and as . A class function is said to be equal to a quadratic function in a neighborhood of the origin , such that for if there exist two constants . all II. GENERAL RESULT FOR NEUTRAL SYSTEMS We consider a functional differential equation
(3) , with initial conditions in at , with locally Lipschitz, with continuous and is a constant delay. We introduce two assumptions. Generally speaking, the first one imposes a limitation of growth type on the dependency of with respect and the second one also limits the influence of to and guarantees the existence of a Lyapunov function whose derivative along the solutions of the studied system satisfies an inequality of iISS and are regarded as inputs. type when both Assumption 1: There exist a nonnegative real number , a continand a function of class such uous function , , that for all with
(4) , two functions Assumption 2: There are a function of class and of class , a continuous positive definite function and two of class so that, for all functions and (5) and for all
,
Notation and Definitions (6) We let denote the Euclidean norm of matrices and vectors of defined on an interval any dimension. Given , let denote its (essential) supremum over . A function , where is a real numbers and is a real number , is piecewise continuously differentiable if it is continuous on or and, for any real number , is continuous the interval , except at a finite number of points. We let denote over -valued functions defined on a given interval the set of all that are continuous, piecewise continuously differentiable and with an essentially bounded first derivative . For a function with or , for all , the function is defined by for all . The solution of a time-delay system described by a functional differential equation: (2) where is a continuous function, with an initial condition at , will be denoted by (instead of as rigorously done for instance in [10, Ch. 2]). The notation will be simplified whenever no confusion can arise from the context. In particular, along the trajecthe time derivative of a Lyapunov functional . A continuous tories of a system (2) will be denoted simply by belongs to class provided that it is function . It belongs to class if, in addition, strictly increasing and
Moreover, for all (7) where
is the constant provided by Assumption 1, for all
(8) where is the function provided by Assumption 1, and there exists a function of class such that, for all (9) We are ready to state and prove the main result of the technical note. Theorem 1: Assume that the system (3) satisfies Assumptions 1 and be a positive real number such that the inequality 2. Let (10) is satisfied. Then the derivative of the functional (11)
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along the trajectories of (3) satisfies (12) for almost all
In the sequel, all equalities and inequalities of and should be , almost everywhere. Elementary understood to hold for all calculations yield
, with
(19) (13)
Since
, defined in (11), satisfies, along the trajectories of (3)
and (14) 1) Discussion on Theorem 1: , one can determine a constant • It is worth noting that for any such that the inequality (10) is satisfied. Therefore the asymptotic stability of the system (3) with identically equal to zero which is implied by Theorem 1 is delay independent. Notice . that (10) is satisfied for all • The requirement (7) imposes a restriction on the size of . This assumption can possibly be relaxed, but it cannot be removed: some systems which satisfy Assumptions 1 and 2 except (7) are not locally asymptotically stable. For instance the 1-D system is exponentially unstable when is smaller than a certain threshold. • A remarkable feature of Theorem 1 is that it applies to systems that are not necessarily locally exponentially stable or globally Lipschitz. • The family of functionals in (11) and the ones of [15, Ch. 9] differ mostly because each functional admits a derivative along the solutions of the system studied that is smaller than a negative , which is not the case of the functionals definite function of provided in [15, Ch. 9]. The advantage of knowing functionals satisfying (12) is that they lead to robustness properties: from the features of and , and by adapting the proofs of [1], [23] (see also [16, Ch. 2]), one can deduce from (12) that all the solutions and there are a function of of (3) are defined over and a function of class such that, for any solution class of (3), for all times , the inequality (15)
(20) by (6) in Assumption 2 and (20), we obtain
(21) From (4) in Assumption 1 and the fact that the function decreasing, we deduce that
is non-
(22) Since the function
is nondecreasing, the inequality
(23) is satisfied. Consequently
(24) Now, (7) and (8) in Assumption 2 ensure that and
. It follows that
holds. Moreover, if the function provided by Assumption 2 is of class , there are a function of class and a function of class such that, for any solution of (3), for all times , , the inequality
(16) holds. In both cases, the functions and are delay-dependent. Proof: Theorem 3.3 in [15, Ch. 3] ensures that for any initial , the system (3) admits a unique solution , which condition is piecewise continuously differentiable and defined over the interval , where is a finite positive real number or . If the max, then is not essentially imum of such intervals satisfies . bounded over defined by Now, we introduce the operator
(25) Since
satisfies (10), the inequality (26)
with
defined in (14) is satisfied. Consequently, for all (27)
(17) is a constant such that (10) holds. It satisfies, along the where trajectories of (3) (18)
By virtue of the continuity of and (20), it follows that, if real number, for all
is a finite
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Then, from Assumption 2, we deduce that, for all , , with . Next, using Assumption 1, we deduce that, almost everywhere over
From this inequality, one can deduce by using the steps method that is . Thus cannot be a real number, i.e., essentially bounded over . Hence, all the solutions of (3) are defined over . Finally, we observe that (9) in Assumption 2 and (26) give (29) which leads to the inequality (12) with
defined in (13).
III. BACKSTEPPING FOR NEUTRAL SYSTEMS In this section, we illustrate the usefulness of Theorem 1 by using it to solve a stabilization problem for a particular family of neutral systems in feedback form. We consider a neutral system of the type (30) , is a constant, , is a locally Lipschitz where is the input. nonlinear function and To begin with, we impose an assumption, which is natural in a backstepping context: of class such that Assumption A: There exists a function the system (31) is locally satisfies Assumptions 1 and 2 and the function Lipschitz. We also introduce an assumption which is instrumental in our design of stabilizing control laws and has no connection with the classical backstepping approach: Assumption B: Let , , be the functions provided by Assumptions 1 and 2. The functions and are equal to a quadratic function in a neighborhood of the origin. There exist a class function and two positive real numbers , such that
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Remark 1a: Assumptions A and B ensure that in (33) is well defined and locally Lipschitz. Remark 1b: Many extensions of Theorem 2 can be expected. For instance, it can be immediately extended to systems of the form
where is a continuous functional since the change of feedback gives a new system of the form (30). Remark 1c: There is a lot of flexibility in the backstepping design, so that we can modify the control (33). Assumption B can be replaced by functions , , satisfying other types of local homogeneous properties. defined in Proof: To begin with, we observe that the function because the func(34) is locally Lipschitz and positive over and are locally Lipschitz and are equal to positive tions definite quadratic functions in a neighborhood of the origin. Next, arguing as we did in the proof of Theorem 1, one can prove , the system (30) in that for any initial condition closed-loop with the feedback (33) admits a unique solution , which is piecewise continuously differentiable and defined over the , where is a finite positive real number or . If interval the maximum of such intervals satisfies , then is not . essentially bounded over Next, we construct a Lyapunov functional for (30) in closed-loop with the feedback (33). In the sequel, all equalities and inequalities of , and should be understood to hold almost everywhere as long as the solutions are well-defined. The change of variable transforms the system (30) into (35) Assumption A ensures that Theorem 1 applies to the -subsystem of of the (35). It follows that there are a functional satisfying such that type (11) and a constant the time derivative of along the solutions of the -subsystem of (35) satisfies (36) where fined by
. It follows that the functional de-
(37) (32) and for all . Moreover, the functions , , , are locally Lipschitz. We are ready to state and prove the following result: Theorem 2: Assume that the system (30) satisfies Assumptions A and B. Then the control law
admits a time derivative along the solutions of (35) which satisfies: (38) We introduce now the candidate control Lyapunov-Krasovskii funcfor the system (35) defined by tional (39)
(33) where is a constant such that (10) is satisfied and such that, for all function continuous over
is the (40) (34)
globally asymptotically stabilizes the origin of the system (30).
Then its time derivative along the solutions of (35) satisfies
By using (32), we obtain
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(41) The property that be proved by considering the cases is of class ) ensures that
holds for and
(which can because
Assumption 3: The function there exists a function of class
is positive definite and such that, for all , (49)
We are ready to state and prove the following result: Corollary 1: Assume that the system (47) satisfies Assumption 3. Then, when (50)
(42) the derivative of the functional
It follows that
(51)
(43) Since, for all
satisfies (52)
, we have
the inequality
(44) is satisfied. From , for all
and (11) it follows that . Therefore (45)
for all follows that
along the solutions of (47) with initial conditions in
,
. Since the function
is nondecreasing, it
(46) Arguing as we did at the end of the proof of Theorem 1, one can prove that the solutions are defined over . Since is of class , it converges to zero when goes to the follows that infinity. This fact and the properties of allow us to conclude.
, with , where for almost all is the function in (49) and , where is any . real number such that Remark 2: The functional defined in (51) is nonnegative because is positive definite. the function Remark 3: One can prove that Assumption 3 is satisfied if and only if is positive definite and . the function is Thus, Corollary 1 does not need the requirement that radially unbounded, which is imposed in [10, Sec. 9.8]. Proof of Corollary 1: Let us check now that, under the conditions of Corollary 1, the assumptions of Theorem 1 are satisfied. is positive definite, it follows that the Since the function and function is positive definite and are positive real numbers or . Since is of class , (49) implies such that for all that there exists a constant . It follows that is radially unbounded. and of class such that, Therefore, there are two functions , . for all Next, the right hand side of (47) satisfies Assumption 1 with , and for all . We observe that, for all (53) We deduce from Young’s inequality that
IV. APPLICATIONS
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In this section, we illustrate Theorem 1 via simple examples. A. Application to a Particular Family of One-Dimensional Systems We consider the following 1-D neutral system:
, and with over, we have established that there are two functions that the inequalities (5) are satisfied. Next, for all
(47) where , where is of class , where is a real number and where is a continuous function which is studied in [15, Ch. 9] and in [10, Sec. 9.8] in the case where is not present. This system is used to modelize a shunted power transmission line. For this system, we establish a stability result via a Lyapunov construction based on Theorem 1. With a view to it, we introduce the function
and
. Moresuch (55)
From (50), it follows that, for all
, (56)
Moreover, from (50), it follows that, for all (57)
(48) and the assumption:
It follows that Assumption 2 is satisfied. Hence, Theorem 1 applies. This allows us to conclude.
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B. Examples For the important family of neutral systems Corollary 1 focuses on, this section presents an example which illustrate the robustness of the types (16) and (15), respectively. The second result illustrates Theorem 2. 1) First Example: Consider the 1-D system (58) and is not present, the origin of this system is not Even when is not locally exponentially stable. Moreover, the function globally Lipschitz. , satisfy The functions with . Since this function is of class , Assumption 3 is satisfied. Thus, Corollary 1 provides with the functional
where is any real number such that along the trajectories of (58) satisfies
. Its derivative
with , and . Since , we deduce that the solutions of (58) the function is of class satisfy an inequality of the type (16). 2) Second Example: Consider the 2-D system
(59) . Let with Then Assumption 1 is satisfied by the system
for all
.
(60) , , . Moreover, Assumption with , 2 is satisfied by the system (60) with , , , provided . Thus, the system (59) satisfies Assumptions A and B . From Theorem 2, we deduce that if with the origin of the system (59) is globally asymptotically stabilized by the control law
with lows
. The above choice of to be bounded in terms of .
(61) al-
V. CONCLUSION We have proposed a new stability analysis technique for neutral systems based on the construction of an adequate Lyapunov-Krasovskii functional. The technique applies to broad families of neutral systems which include systems which are not locally exponentially stable and not globally Lipschitz and makes it possible to construct stabilizing control laws for systems in feedback form under appropriate assumptions. Much remains to be done. In particular, extensions to
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time-varying systems, to systems with several delays are expected, as long as applications to other control problems for neutral systems than the one solved in Section III.
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