Lyapunov-Based Boundary Control for A Class of ... - Semantic Scholar
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
701
Lyapunov-Based Boundary Control for A Class of Hyperbolic Lotka–Volterra Systems Lacra Pavel, Senior Member, IEEE, and Liz Chang, Student Member, IEEE
Abstract—This paper considers a boundary feedback control problem for two first-order, nonlinearly coupled, hyperbolic partial differential equations with Lotka–Volterra type coupling. Boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value. Static and dynamic boundary controllers are designed based on a special Lyapunov functional that is related to an entropy function. The time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions. A unique classical solution is shown to exist globally in time and (asymptotic) exponential convergence to the desired steady-state solution is shown in the ( 0 ) 2 -norm. The boundary control design is illustrated with simulations. Index Terms—Boundary control, Lotka–Volterra, Lyapunov functional, nonlinearly coupled, partial differential equations.
I. INTRODUCTION
F
INITE-DIMENSIONAL models are often inadequate for control design for distributed systems [24], [34]. In particular, hyperbolic systems have features such as finite speed of propagation that require approximation by high-order finite dimensional models. While hyperbolic systems of second-order describe oscillatory systems such as strings or beams, first-order hyperbolic systems describe physical problems of transport-reaction type such as traffic flows, chemical reactors, and heat exchangers [46]. The distinct feature of hyperbolic partial differential equations is that all the eigenmodes of the spatial differential operator typically contain nearly the same amount of energy; as a result a very large number of modes is required to accurately approximate their dynamic behavior. This feature distinguishes hyperbolic partial differential equations and suggests addressing the control problem on the basis of the infinite-dimensional model itself [7], [30]. In this paper, a boundary feedback control problem for two coupled first-order hyperbolic partial differential equations with a nonlinear coupling of the Lotka–Volterra type is considered. Manuscript received April 04, 2009; revised March 18, 2010, May 09, 2011, and July 29, 2011; accepted September 09, 2011. Date of publication September 19, 2011; date of current version February 29, 2012. This work was supported in part by the National Science and Engineering Research Council. Recommended by Associate Editor J. J. Winkin. L. Pavel is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]). L. Chang is with Microsoft, Inc, Seattle, WA USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2168909
Lotka–Volterra type nonlinearity is commonly used to model biological systems and networks [18], predator-prey systems [17] evolutionary dynamics of species [19], or stimulated Raman induced nonlinearity in optics [4], [14]. For example, optical Raman amplifiers there have been many recent efforts to control transients using simple PID feedback, feedforward control, or all-optical gain clamping, but these designs rely on empirical selection of parameters [4], [8], [31], [47]. A systematic control design is performed in [14], based on a finite-dimensional approximation of the linearized partial differential equation model around the desired steady-state. From a 588th-order finite-difference model a tenth-order reduced model is obtained by open-loop balanced truncation, and -controller is designed. However, there is no a tenth-order guarantee that the stabilization and regulation properties will hold for the closed-loop hyperbolic system. Our work focuses on boundary control design techniques that can be performed directly on semilinear hyperbolic systems with Lotka–Volterra type nonlinear coupling. Boundary control action is used on one of the channels only, and simple static and dynamic boundary controllers are designed to drive the state at the end of the spatial domain to a desired constant reference value on the other channel. Boundary control has been an area of much interest recently, one reason being the fact that it addresses the problem on the infinite-dimensional model itself. Results for set point stabilization and/or disturbance rejection for the linear wave equation are developed in [30], [33]. The back-stepping method is an elegant approach that requires numerical computation of the kernel [23]. The zero dynamics-based approach provides a systematic methodology, and typically results in infinite-dimensional controllers [5], [6]. We use a Lyapunov-based approach which is related to the recently employed methods in [2], [10], [11], [13], [39], and [46]. In [2] concepts from thermodynamics are used to construct a Lyapunov functional. A semilinear hyperbolic system of partial differential equations with no source terms is considered in [46]. A quasi-linear hyperbolic homogeneous system of two conservation laws is treated in [11]. A quadratic Lyapunov functional with weighting factor as in [46] is used for analysis with a proportional boundary control in [11] and extended to integral action control in [13]. In [10], this approach is extended to multiple conservation laws based on a generalized quadratic Lyapunov functional. In [39], the stability properties of a binary Lipschitz nonlinearity are anreaction-diffusion system with alyzed via quadratic Lyapunov functionals.
0018-9286/$26.00 © 2011 IEEE
702
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
The hyperbolic system considered here has nonlinear growth (coupling) terms of the Lotka–Volterra type and does not satisfy Lipschitz condition. This precludes the application of the quadratic type Lyapunov functionals as in these works. In contrast, inspired by a Lyapunov function used for Lotka–Volterra ordinary differential equations [19], [35], a special Lyapunov functional that can be used directly on the nonlinear hyperbolic system is introduced. This Lyapunov functional is related to entropy and is matched to the Lotka–Volterra system. Moreover, it incorporates spatial weighting as well as a spatial factor as in [46]. This is beneficial for the design of two simple boundary controllers—proportional and dynamic with integral action. It is shown that the parameters of these controllers can be selected so that the time-derivative of the entropy function calculated along the trajectories of the closed-loop system is negative definite. Preliminary results appeared in [37]. The paper is organized as follows. Section II presents the model and boundary control problem. Section III gives conditions for the closed-loop system to be well-posed, i.e., for the mixed initial-boundary value problem to admit a unique classical solution globally in time. Main results are given in Section IV for the two boundary controllers. The boundary controllers’ parameters are selected based on a Lyapunov functional studied in Section V. Proofs of the main results are given in Section VI and Section VII, showing (asymptotic) exponential stability of the equilibrium of the closed-loop systems in the -norm. The control design is verified through simulations in Section VIII and conclusions are given in Section IX. , , The following notation is used. Let . Let denote the Lebesgue space of -valued square-integrable functions on , and use as a compact notation for this space of vector-valued is a Hilbert space with the inner product defunctions. and the -norm denoted by noted by
the completion of the space of continuously differen-norm denoted tiable functions on , equipped with the usual . Similarly, let by
where denotes the Euclidean norm in . denote the Lebesgue space of -valued measurLet able functions on that are essentially bounded with the norm
the completion of in the -norm. let denote Sobolev spaces for higher . Similarly, The following embedding relations hold among Sobolev spaces
II. MODEL AND PROBLEM FORMULATION Consider the 2 2 first-order semilinear hyperbolic system with nonlinear reaction:
(1) ,
, with initial conditions (2)
The nonlinear reaction on the right side of (1) is of the Lotka–Volterra type and is encountered in biological systems and networks, predator-prey, and competing species interac, , respectively, denoting tions [17], [18], [44], with the predator and prey population densities. System (1) can also represent coupled wave equations for modeling the dynamics of stimulated Raman scattering [4], [14]. In a normalized co-propagating Raman amplifier, with unit attenuation and and denote the coupling coefficients, signal and pump powers propagating at wavelengths , and , with characteristic speeds along the length of the amplifier denoted by . This setting is the representative case taken for the boundary control design problem considered here. Typically the signal and a certain desired input is maintained constant level (set point) is required for the signal output by ma. Thus, system nipulating only the boundary condition on (1) has the boundary conditions (3)
and the space of with the norm
where
-valued continuous functions on
in . For continuous functions the -norm is equal to the -norm. Let denote the usual Sobolev space [1], [40]
where is the control action to be designed based on the boundary measurement . For a constant control action , let denote the steady-state solution and boundary conditions which satisfies the set point (4) The following assumption will be used. Assumption 1: There exists a unique steady-state solution , , , that satisfies and the boundary conditions (4). the set point
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
703
with boundary conditions from (3), (4) (10) where to be designed, and with initial conditions from (2)
,
is
(11) Fig. 1. Boundary controlled system.
Such a steady-state solution involves solving a set of ODEs and can be found for example by using extremum seeking to , given feasible compute the optimal input pump power , , [15]. Depending on the form of these boundary conditions, the steady-state solution may be stable or unstable. Therefore, we focus on the following problem: Problem 1: Find a boundary controller (see Fig. 1) such that with boundary conditions (3) set point stabilization is achieved, i.e., for any , feasible
and for any sufficiently small and smooth initial condition the closed-loop system has a unique classical solution converging to the desired steady-state solution in some appropriate norm. In order for the initial conditions in (2) to be compatible with the boundary conditions (3) and (4) they must , . satisfy satisfies From (1) the steady-state solution
In the coordinates the closed-loop boundary controlled system (8)–(11), is a semilinear hyperbolic system with a nonlinear reaction of the Lotka–Volterra type. The control objectives stated earlier motivate two simple boundary controllers—a proportional boundary control (12) and a dynamic boundary control with integral action
(13) small, ensuring tracking to the desired set points. where With the boundary controllers specified in (12) and (13), the boundary control problem is restated to finding conditions on the parameters of (12), (13) such that, for any small smooth enough initial condition, the corresponding closed-loop system converging to the (8)–(11) has a unique classical solution origin in some appropriate norm. In the next section conditions are given for the closed-loop system (8)–(11) to be well-posed, i.e., for the mixed initial-boundary value problem to admit a unique classical solution globally in time. III. EXISTENCE AND UNIQUENESS GLOBALLY IN TIME
(5) with boundary conditions (4), and hence . This steady-state is denoted in vector form by (6) The following change of variables is made
In this section, conditions are given for classical solutions of the mixed initial-boundary value problem (8)–(11) to exist globally in time. First, the closed-loop system (8)–(11) is formulated in an abstract setting as in [12], [38]. It is noted that the abstract theory of boundary control systems started with [16] and was significantly developed in the last few decades in work such as [25], [41], and [45]. The formulation here resembles, for the case , that for linear passive boundary control systems in [30, Ch. 5, p. 259], and that for closed-loop dissipative systems in [26]. The following operators are introduced. Let us denote by “ ” the operator defined by
(7) Using (1), (5), and (7) the closed-loop system is written in the coordinates as
(8)
(14) i.e., point-wise multiplication. The same notation is used for matrix and vector-valued functions of compatible dimensions. Recall (8) with the matrix-valued coefficients , . Using the point-wise multiplication notation, let denote the linear differential operator defined by
where
(15) so that for all (9)
704
with domain Similarly, let
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
. denote the nonlinear operator defined by
where that for any that
(16) . It can be shown , there exists such
conditions. The boundary conditions are dealt with as in [29]. , for Specifically, with a change of variable , (21) is converted into a modified inhomogeneous linear hyperbolic Cauchy problem (22) with homogeneous zero boundary conditions
(17) where
and is locally Lipschitz on , [36]. Let the operator be defined by (18) with domain . Thus, is obtained from via the perturbation that is locally Lipschitz . on Then for (8)–(11), can be regarded as a function of time . For each let so that the closed-loop system (8)–(11) is written as
(19) where for classical solutions we take where
and
(20) with the boundary condition embedded. Note that (19) can be written in the form
where and , (20). is used to prove that for any The Lipschitz property of a classical solution to (19) exists, i.e., a solution that is with respect to time in and with respect to , [36]. This local existence result is given in the time in next proposition. , Proposition 1: For every initial condition there exists some such that (19) has a unique classical defined for every , such that solution
A detailed proof is given in [36], using an approach similar to the one used in [10] and [29]. The idea of the proof is to start from (19) and consider the linearized equation
(21) for any given , where is some subspace of funcwith values in , . tions of time over , (21) is an inhomogeneous linear hyperFor any given bolic boundary value problem with inhomogeneous boundary
, , is an appropriately defined function of and . The operator is a closed linear operator on and it generates a -semigroup on (by Lemma 3.2, [21]). To this equation standard existence results for classical solutions can be applied, (Corollary 4.2.5, [38] and Lemma 3.2 in [21]). Then reverting the change of variable it for any can be shown that (21) has a unique solution ; hence, it defines a mapping , with , given into itself. The proof follows by using Banach that maps contraction mapping theorem, perturbation of linear operators and Sobolev embedding, together with Gronwall inequality. develKey ingredients are a priori energy estimates on oped using Moser-type estimates [44] for the nonlinearity on and Lispchitz continuity on . We note that a similar result is sketched in [10] (Proposition 2.1) for quasilinear hyperbolic systems with no source terms . Existence and uniqueness of classical solutions globally in time depend on properties of (16). For the Cauchy problem in (19) corresponding to (uncontrolled case), i.e., for the open loop system, existence of a global classical solution depends on the global Lipschitz properties of (see Theorem 6.1.7, [38], or Proposition 4.2, Ch. 15 [44]). Here (16) has -norm bound and is loLipschitz properties dependent on a . Global in time existence depends on cally Lipschitz on the existence of a uniform bound on the -norm of the local solution. , and —the Proposition 2: Assume that for every with unique local classical solution of (19) on
for some
, the following a priori uniform bound holds for independent of : (23)
Then, (19) admits a unique global classical solution, i.e.,
The detailed proof of Proposition 2 is given in [36]. The proof follows an approach as in Theorem 6.1.7, [38] and Proposition 4.2, Ch. 15 [44]. The Sobolev embedding theorem, [40], which , then and shows that if (24)
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
for some is instrumental in the proof. This fact is used -norm dependent Moser-type estimates together with the to show that if the uniform bound [44] that satisfies on (23) holds, then the following a priori estimate on the -norm holds of
where is independent of . Hence, cannot blow-up in finite time and is a classical solution on . The result in Proposition 2 is similar to the one for parabolic reaction-diffusion equations (see Proposition 4.2 and 4.3, Ch. -norm. 15 [44]) and the search for a subset invariant in the Specifically, solutions exist globally in time in (classical solution) for those initial conditions belonging to a subset -norm. This subset s shown to be invariant bounded in the -norm. under the solution operator in the We will show that for (19) such an invariant subset can be found by using a Lyapunov function, for sufficiently small . Note that for weak solutions we could take and use density arguments, and for the overall space use the -norm and the -norm. , and work with classical We only treat the case which are with respect to time in , solutions with respect to time in . For sufficiently small and , by working with classical solution (differentiable in time) and the Lyapunov functional (and ), we show exponential (and asymptotic) convergence to in the -norm or -norm), respectively. (and
705
where is a real number. Such a function has been introduced in [22]. Function is continuously differentiable on , positive , , and radially unbounded, i.e., definite on as . The term , is instrumental in obtaining a strict Lyapunov function, as in [11] and [46]. The first boundary controller considered is (12). In the original coordinates the steady-state is denoted by , (6). In the -coordinates defined in (7), the closed-loop system is given by (8)–(10) and (28) The main result for the proportional boundary controller (12) is given next. Theorem 1: Consider the closed-loop system (8)–(11), (28) under Assumption 1 and let . Assume . Let any such that that , , and select any such that . If is chosen such that
and (29) where , then such that 1) For any initial condition , where is the constant in (24), there exists some such that , and (25) satisfies the entropy function
IV. ASYMPTOTIC AND EXPONENTIAL STABILITY In this section, the main results giving conditions on the two boundary controllers (12), (13) are presented. The proofs of the main results are given in Section VI and Section VII and are based on a special Lyapunov functional related to entropy. (7) can be regarded as deviation from the Recall that . Consider the entropy-like function desired steady-state (25) and the Lyapunov functional
(30) i.e., the Lyapunov functional
satisfies
as long as the solution exists and . such that, for any initial Moreover, there exists a condition with , , (30) the classical solution exists globally over and the origin is asymptotically stable holds for -norm. in the 2) There exists a number such that, for any initial condition with , the -norm of the classical solution satisfies
where (26) and
Let
be defined for any given
by (27)
for some constant , hence the origin is exponentially stable in the -norm. Remark 1: For and a weak solution, convergence in the -norm could be shown based on density arguments and continuity of solutions with respect to initial conditions as in [29], or [30, p. 270]. We only work with the classical solution ( with respect to time) herein and use Lyapunov , arguments to show that for sufficiently small -norm [part (i)] and exponential asymptotic stability in the
706
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
stability in the -norm [part 2)] hold. We do not yet have ex-norm, which would imply expoponential stability in the -norm; we can only show asymptotic nential stability in the stability in the -norm [part 1)]. depends on Remark 2: Selection of the proportional gain parameters , , and such that the sufficient bounds in (29) are satisfied. The first of these correspond to boundary dissipative conditions, as given in [3] for the linearized Saint-Venant equations, while the second one shows explicit dependence on the desired steady-state values at the ends of the domain. Note that given , , and selected based on the desired steadystate values and , the bounds in (29) describe the non-empty intersection of two intervals around 0, so that there exists always a feasible in this interval to satisfy (29). Moreand the over, there is a tradeoff between the size of the gain domain over which stability in ensured as seen from the reciprocal relationship between the bound on and . The proof of Theorem 1 is given in Section VI. Unlike the quadratic-type Lyapunov functions considered in [10], [11], and (25), (27) [46], the special entropy-like Lyapunov employed here is matched to the structure of Lotka–Volterra nonlinearity. Together with the spatially dependent weighting factors in (25) this is beneficial to develop sharp decay estimates. The proof follows by showing that, for sufficiently smooth, small initial conditions , an a priori uniform bound -norm of the solution of (8)–(11), (28) and (23) on the boundary term holds. Properties of , and (25)–(27) are instrumental to show this. Then by Proposition 2 a unique classical solution exists globally in time, and its asymptotic properties can be studied. This asymptotic analysis relies on invariant or -norm) and of Lyapunov sub-level sets of (for the functional (for the -norm). Note that asymptotical stability -norm (part 1) of Theorem 1) implies that the in the holds as . For the set point condition functional it is shown that decays exponentially along the unique classical solution of the , closed-loop system. It is related to the -norm of
then for any initial condition with , for some , the classical solution exists globally over and the origin is exponentially stable in the -norm. The proof of Theorem 2 is given in Section VII. Remark 3: Selection of parameters can proceed as follows: as in Theorem 1. Then we can first choose , , , and select based on the choice of , based on the choice of , , , and . The bounds , and are simply the roots of a quadratic expression (see the proof of Theorem 2). is based on the value of Then the range of the integral gain these previously chosen parameters. As in Theorem 1, there is a tradeoff between the size of these gains and the domain over which stability in ensured. V. LYAPUNOV FUNCTIONAL AND ITS PROPERTIES In this section, we study properties of Lyapunov functional related to the entropy function , (25). Proofs of the main results Theorem 1 and 2 rely on properties of and . We show or . However, that is well-defined on subsets in we use as a Lyapunov functional for classical solutions, so will be used for subsets within . First, recall that for any given real number , , deis well-defined over fined in (27), , continuously differentiable in its argument, pos, , and radially unbounded, i.e., itive definite on as . , consider the function (27), for Lemma 1: Given any . any real such that finite, there exists finite such 1) For every that
and
and this is exploited to show exponential decay of the -norm (part 2) of Theorem 1). For the dynamic boundary controller (13) proceed similarly and shift the coordinates with respect to the desired steady-state . In the shifted coordinates , where stands for , the closed-loop system is given by (8), (9), and
(31) The following is the main result for the dynamic controller with integral action (13). Theorem 2: Consider the system (8), (9), (31) and assume and are selected as in Theorem 1. all parameters , set and select For any . Then there exist some predetermined such that if is chosen such that
when
. 2) For any
such that
where . The proof of Lemma 1 is given in the Appendix. The domain can be arbitrarily enlarged by increasing (decreasing ); hence, 2) Lemma 1 is semi-global. , (27), for a Remark 4: Lemma 1 gives properties of , given any . This result real number such that can be extended for functions of as follows. Given any such that for , let . Consider any function such that for . Thus, for , so is well-defined that for each and properties in Lemma 1 hold point-wise for . Moreover, for any such that for , function (where denotes composition) is well-defined and continuous with respect to on .
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
Similar extensions hold for (26) and vector functions of . Given , , with , let , and . Consider a function of such that for , i.e., , . Thus and (26), (27) are well-defined. Moreover, properties in Lemma 1 hold point-wise for for each and is positive definite. If then continuous with respect to on . such that For any for , operator is well-defined and continuous in -norm ( -norm). This follows from properties the in Lemma 1 and classical results on nonlinear substitution operators and calculus of variations, (Proposition 2.1, p. 160, [32]). Moreover, since is compact, is uniformly continuous into , and bounded from each bounded subset of (by Proposition 2.3, p. 161, [32]). Recall (25) and the functional defined by
Evaluated at
,
707
(33) is written as
(34) and is well-defined and continuous for any for , . that The following result gives estimates for (32) and , such that Lemma 2: Given , , and , let , and consider , (32), and , (33). 1) Let any such that , and for any let such that For every , , the following holds:
such (33). , and , . , and
(32) (35) or, evaluated for such that , 2) Let any let , . Then for every for any such that , where , the following holds: where is a function of , , such that for , , where and are defined as in (26). Note that ignoring the term, , i.e., evaluates to the -norm of the element . Thus, based on the foregoing remarks for , is continuous in the -norm ( -norm). This follows from classical results in calculus of variations (Lemma 4.3.1, p. 188, [20]). Remark 5: Note that (and ) are well-defined and contin(and ), such that uous for for , . However, we will only use (and ) to study classical solutions, so we consider such that for , . Thus, by embedding we satisfy these bounds; specifically we work with invariant sublevel sets of and in the proof of Theorem 1. , (20) and consider , (32), defined for any Recall such that classical solution of (19) for , . Based on (19) or (8), we define another functional
, and
(36) Proof of Lemma 2: For simplicity of notation throughout the proof let and . For any we have (37) that 1) We show that (35) holds for any satisfies (37) and the associated bounds, i.e., , , and . Since is dense in and is continuous in the -norm, by density arguments it will follow that satisfies (37) that (35) holds for any and the associated bounds, i.e., for any that satisfies the associated bounds. Consider thus any that satisfies (37) and such , . Hence, , that , and is well-defined. We show first that satisfies the following inequality
(33) (38) where
where
is defined as a weighted , i.e.,
708
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
In order to show (38) evaluate (9). Note that
and, since
using (34) and
as in
Using (9) the second term in
(40) can be written as
Thus, using the previous two expressions
,
(42) Using the foregoing to substitute for (34) yields
into
,
, and by assumption (since ), it follows . Then claim (38) follows from
, that indeed (41). Using the compatibility condition (37) into (38) yields
Hence, (39) where
and (39), using definition of (32) yields
Therefore, since
(40) . Integrating by parts in , together with and
since , , . Lemma 1, 1) and estimates on can be used in the second and third term on the right-hand side of the foregoing to show that, for any and given as in the statement, (35) holds. This and density arguments conclude the proof of part 1). Part 2) is proved by using the definition of , (32) and Lemma 1, 2) for (see the Appendix). Remark 6: Note that , in 1) Lemma 2 depend only on and , respectively. (35) indicates how a boundary control map should be designed. In 2), Lemma 2 , depend on . The result can be made semi-global by increasing , at the expense of reducing and increasing . VI. PROOF OF THEOREM 1
(41) (40),
Claim (38) is proved if it can be shown that . is first simpliIn order to show this, expression (40) for fied. With notation , as in (6), (9), the steady-state (5) are rewritten as
where the first term in
and (40) is written as
From the definition of
. Then
In this section, the proof of Theorem 1 is developed, based on properties of the Lyapunov functional , (32), and the entropy , (25). Let function (43) with as given in the theorem’s statement. Proof of 1) (Part A): First (30) is proved. From Proposition there exists such that (19), 1, for any or (8)–(11), (28), admits a unique classical solution for . (see also (32)). First, we Recall (25) and , evaluated show that the time derivative of , (25), at time along the solution of (19) passing at time through is equal to the functional , (33), evaluated , i.e., at the point
and (26), for all (44) For a classical solution of (19) of class based on (25), the following holds:
Using this together with (26) into foregoing yields after some manipulation
on
,
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
or
709
. Specifically, such solution exists globally in time over inside a a bound is shown to hold for any initial condition sublevel set of (26), (43). Consider defined for each by
Hence, for a classical solution
of (19) of class
on
(45) holds for all . By a density argument, (45) holds also of (19). Thus, based on for a classical solution (19) or (8)
which, by using the definition (33) of , gives (44). Recall the estimates in 1), 2) in Lemma 2 that hold for , as long as the associated bounds are satisfied, i.e., on , (43). Next we show that the following holds for (46) From (20), (28), 1) in Lemma 2, for any
, and
Then using (19), the definition of the operator (9) yields
(15) and
in
Note that the right-hand side is similar to the integrand term on the right-hand side of (34). Then after some manipulations as in the proof of 1), Lemma 2, i.e., by using and the two relations after (34) to re-express the right-hand side in the above yields
which is similar to the integrand term on the right-hand side of (39). Using in the first term of the above leads to
. From as given (35) holds, i.e., (49) where
holds for every
such that
,
(47) ,
and . These bounds are satisfied for any , as given in the theorem’s statement, if is picked as with in the second condition in (29). Hence,
for any and, since also by (29), (46) follows. with Pick any initial condition . Then , by Sobolev embedding theorem , (43). Then by (46), and (24), so that . Moreover (46) holds for , i.e., (48) remains in , (43). as long as Then using (44), (48) and
yields
as long as remains in . Finally, (30) is obtained by Gronwall’s inequality and using again , and it holds stays in . as long as Proof of 1) (Part B): The foregoing results hold locally in . Next it is shown that for sufficiently small initime over tial conditions , there exists finite, independent of such that the bound (23) holds, so that by Proposition 2 the
(50) Recalling (40), note that evaluated for , yields
, . Thus, by , and, from (49), differential inequality
, (50) is the same as , (40) . Then using (42) evaluated for
, , satisfies the scalar linear
(51) . Moreover, from with initial conditions (28), it follows that . and Using the estimates for in 1), Lemma 1, with , it follows that . Thus, on , Recall (51) and consider the solution linear equation
. to the scalar
(52) and boundary conditions with initial conditions . Note that (52) has a traveling wave . Consider , solution,
710
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
, hence the constant solution of (52) , . Now, from (51), (52),
,
(53)
, , with boundary condition . for Note that (55) is a scalar linear hyperbolic equation with dissipation term and boundary condition [28, p. 136]. Its classical solution exists globally and , . Then, by the same and this above, if scalar comparison lemma [27] for , , then , for every , . , ; hence, This shows that as . is continuous in the -norm it follows that Since as , and asymptotic stability in the -norm is proved. This completes part B and the proof of 1). Proof of 2): Recall that 1) shows exponential decay for , (30), and asymptotic stability in the -norm. Based on the a priori estimates on with respect to the -norm of solutions (especially the lower bound in 2), Lemma 2 which holds on a decays smaller domain), it is shown that the -norm of exponentially on a smaller domain . where From 2) in Lemma 2 the bounds in (36) hold on . Let and , . For given , pick correspondingly the set any such that and then pick as in the proof of Part B in 1) above. Thus, is such that . Let an arbitrary and pick any (54) holds for . Note that , where is the one in part 1). Again by Sobolev embedding theorem with , it follows that [40], for any and
(54)
so that
satisfies
and . By a scalar comparison lemma for differential inequalities (Theorem 1.2-4, p. 19, [27]), , , it follows that, if then , , , on ; hence, using the definition of , that , , , . By applying 1) in Lemma 1, it follows , , , or that , , , as long as the solution remains in . Hence, (23) holds and it remains to show how to pick and such that the solution remains in initial condition . This can be shown by continuity of . Since is continuous in the -norm and , for every there such that for any exists , holds. By applying , it can be shown that there again 1) in Lemma 1 for every exists finite such that . Pick such , so that . This gives an upper bound on that and the corresponding . . Then for any Pick with , by Sobolev embedding theorem and (24), , hence and . For picked as above it follows that , , and
In other words, for as long as the solution exists
stays inside . From (53) it is seen that i.e., the solution independent of . By Proposition 2 the (23) holds with . In unique classical solution exists globally in time over addition (54) holds for , for any other and a slightly smaller . so its The classical solution exists globally in time on , since asymptotic properties can be studied. For stays inside by (54), it follows that (30) holds for , . Thus, decays exponentially as , hence exponentially, as . Based on or as properties of and it can be shown that in the -norm. This can be shown as follows. Recall defined in (50). Since , , for , there exists such that , ; hence,
Proceeding similarly as in (49), (51), (52) consider solution to
the
(55)
, as needed for (54). For any such , by using the right-hand side of (36)
, since
for
. Using (54) on it follows that . Thus, by (30), , . Since , by using the left-hand side , yields of (36) together with 2) in Lemma 2 for ,
For any such that
Since
for all
, since
from (30) it follows
and remains within , (36) can be used on both sides so that
. Thus, , i.e.,
,
,
, for
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
and exponential asymptotic stability in the -norm is proved, concluding the proof of 2). Remark 7: is radially unbounded so that in the proof can be arbitrarily large by modifying . Thus, the domain can be arbitrarily large and the result gives semi-global exponential and stability in -norm. For simplicity the symmetric set is used in the proof. Similar arguments (but with more tedious notations) could be used for the larger rectangular , , with , for set . For and arbitrarily small, (30) is semi-global . The stronger inequality is used over instead of simply showing that is negative definite, enabling exponential decay of the strict Lyapunov function. In addition, under the condition that satisfies (29), altering the controller gain determines the size of the asymptotic stability domain. VII. PROOF OF THEOREM 2 For the closed-loop system (8), (9), (31) in the coordinates, consider an augmented Lyapunov functional (56) is a design where is the original Lyapunov function and parameter that scales the contributions of the augmented state . Define the function . Then as in the can be comproof of Theorem 1, the time derivative of of the augmented closedputed along the solution loop system (8), (9) and (31) and shown to satisfy
and
where
is defined in (33). Consider
so that
711
where
Matrix is positive definite as shown next. For any chosen . Also as in the statement it follows that
With for the unknown, the equation has two dis, , for any , tinct real roots . Moreover, it can be shown and and . that Thus, for any , and , so that indeed is positive definite. Then from (57) it follows that
which has exactly the same form as (46). Then, the proof follows similar steps as the proof of Theorem 1. Remark 8: Selection of parameters can proceed as follows: first choose , , and as in Theorem 1. Then select based on the choice of , based on the choice of , , , and . As seen in the proof above the bounds , and are simply the roots of a quadratic expression . is based on the value of Then the range of the integral gain these previously chosen parameters. As in Theorem 1 there is a tradeoff between the size of these gains and the domain over which stability in ensured ( or ). Remark 9: The results developed in this paper can be immesystems, , diately extended to fully actuated i.e., such that there are control channels and signal chan, nels and decoupled boundary controller , or, , or , , . For under actusystems, , a similar ated methodology can be used but controller conditions are not immediately generalized and need to be separately analyzed. VIII. SIMULATION RESULTS
With
given as from (31), by using Lemma 2 for it can be shown as in the proof of Theorem 1 that the following holds [see (47)] on
where
,
. This can be written as
(57)
In this section, MATLAB simulation results are presented based on the hyperbolic partial differential equations solver, [42], [43]. The associated initial-boundary value problems were solved using central finite difference method with a time step of and a spatial discretization step of . The open-loop system (1) was first simulated until it reached steady state, then , the boundary conditions for the controllers (12) with at , and (13) with , , and were applied. The first scenario studied is a 40% drop/add in reference . Fig. 2, 3 show output signal and pump evolution for the dynamic controller. Fig. 4 shows the signal output for the two controllers in the first scenario. The proportional controller produced a large offset, while the dynamic controller tracked the reference accurately with a steady-state error of less than 0.5%.
712
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
Fig. 2. Signal output evolution for reference change (dynamic control).
tial differential equations with Lotka–Volterra nonlinear coupling. These equations serve as model for such physical phenomena as biological systems, population dynamics or stimulated Raman scattering. A special Lyapunov functional related to an entropy function for Lotka–Volterra systems was introduced. The time derivative of the Lyapunov-based function was made strictly negative by an appropriate choice of boundary conditions. Based on this Lyapunov functional static and dynamic boundary controllers were designed to drive the state at the end of the spatial domain to desired constant reference, based on only one control at one end. Based on global in time existence and uniqueness of classical solutions of the closed-loop system, exponential decay of the entropy function -norm was shown. On and asymptotic stability in the -norm a smaller domain exponential stability in the was shown. The design was illustrated with simulations. Future work will address generalization to systems, counter-propagating and un-normalized cases as well as rejection of disturbances. APPENDIX Proof of Lemma 1: 1) The first part follows immediately, based on the properties and graph of the Lambert W (or product log) function, [9]. . The left side of the 2) Consider inequality follows immediately by showing that has a local , i.e., that is positive definite in locally. minimum at is continuous and twice differentiable on Notice that . Differentiate with respect to to obtain
Fig. 3. Pump output (control) evolution for reference change (dynamic control).
Then For any
when or , for . such that , it follows that hence and . Thus, attains its local minimum and its local maximum at , beyond which at decreases. Therefore, for any . The right side of the inequality follows similarly. Proof of Lemma 2: , with and as in Part 2): For any the statement the following hold:
Similarly, Fig. 4. Signal output evolution for reference change.
IX. CONCLUSION In this paper, a boundary feedback control problem has been considered for two coupled first-order hyperbolic par-
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
Thus,
where , is well-defined and continuous
and . By Lemma 1,
Then, summing up after , yields for
Integrating over , and any
, using the definition of , yields (36).
, that
and
,
ACKNOWLEDGMENT The authors would like to thank the anonymous referees and the associate editor for their constructive comments that have significantly improved the manuscript. REFERENCES [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces. New York: Academic, 1975. [2] A. A. Alonso and B. E. Ydstie, “Stabilization of distributed systems using irreversible thermodynamics,” Automatica, vol. 37, no. 11, pp. 1739–1755, 2001. [3] G. Bastin, J.-M. Coron, and B. d’Andréa Novel, “Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks,” in Lecture Notes for Pre-Congress Workshop on Complex Embedded and Networked Control Syst., 17th IFAC World Congr., Seoul, Korea, Jul. 2008. [4] J. Bromage, “Raman amplification for fibre communication systems,” J. Lightw. Technol., vol. 22, no. 1, pp. 79–93, 2004. [5] C. Byrnes, D. Gilliam, and C. Hu, “Set-point boundary control for a Kuramoto-Sivashinsky equation,” in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp. 75–80. [6] C. Byrnes, D. Gilliam, A. Isidori, and V. Shubov, “Set point boundary control for a nonlinear distributed parameter system,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, Dec. 2003, pp. 312–317. [7] P. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkhauser, 2000. [8] C. J. Chen et al., “Control of transient effects in distributed and lumped Raman amplifiers,” Electron. Lett., vol. 37, no. 21, pp. 1304–1305, Oct. 2001. [9] R. M. Corless et al., “On the Lambert W function,” Adv. Comput. Math., vol. 5, pp. 329–359, 1996. [10] J.-M. Coron, G. Bastin, and B. d’Andréa Novel, “Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,” SIAM J. Control Optim., vol. 47, no. 3, pp. 1460–1498, May 2008. [11] J.-M. Coron, B. d’Andréa Novel, and G. Bastin, “A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,” IEEE Trans. Autom. Control, vol. 52, no. 1, pp. 2–11, Jan. 2007. [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Texts in Applied Mathematics. New York: Springer-Verlag, 1995, vol. 21. [13] V. Dos Santos, G. Bastin, J.-M. Coron, and B. d’Andréa Novel, “Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments,” Automatica, vol. 44, no. 5, pp. 1310–1318, May 2008.
713
[14] P. Dower and P. Farrell, “On linear control of backward pumped Raman amplifiers,” in Proc. IFAC Symp. Syst. Ident., Newcastle, Australia, Mar. 2006, pp. 547–552. [15] P. Dower, P. Farrell, and D. Nesic, “An application of extremum seeking in cascaded optical amplifier control,” in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp. 4472–4477. [16] H. O. Fattorini, “Boundary control systems,” SIAM J. Control, vol. 6, pp. 349–388, 1968. [17] K. R. Fister, “Optimal control of harvesting coupled with boundary control in a predator-prey system,” Applicable Analysis, vol. 77, no. 1, pp. 11–28, 2001. [18] B. S. Goh, G. Leitmann, and T. L. Vincent, “Optimal control of a preypredator system,” Math. Biosci., vol. 19, no. 3–4, pp. 263–386, 1974, American Elsevier Publishing Company. [19] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge, U.K.: Cambridge Univ. Press, 1998. [20] J. Jost and X. Li-Jost, Calculus of Variations. Cambridge, U.K.: Cambridge Univ. Press, 1998. [21] T. Kato, “The Cauchy problem for quasi-linear symmetric hyperbolic systems,” Arch. Rational Mech. Anal., vol. 58, no. 3, pp. 181–205, 1975. [22] M. Krstic, D. Fontaine, P. V. Kokotovic, and J. Paduano, “Useful nonlinearities and global bifurcation control of jet engine surge and stall,” IEEE Trans. Autom. Control, vol. 43, no. 12, pp. 1739–1745, Dec. 1998. [23] M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays,” in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, Dec. 2007, pp. 225–230. [24] I. Lasiecka, Mathematical Control Theory of Coupled PDEs. Philadelphia, PA: SIAM, 2002. [25] L. Lasiecka and R. Triggiani, “A cosine operator approach to modeling L (0; T ; L (G))—Boundary input hyperbolic equations,” Appl. Math. Optim., vol. 7, pp. 35–93, 1981. [26] L. Lasiecka and R. Triggiani, “ L (6)-regularity of the boundary to boundary operator B L for hyperbolic and Petrowski PDEs,” Abstract Appl. Anal., vol. 19, pp. 1061–1139, 2003. [27] A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering. Norwell, MA: Kluwer, 1989. [28] T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems. New York: Wiley, 1994, Res. Appl. Math. 32, Mason. [29] W.-J. Liu and M. Krstic, “Global boundary stabilization of the Korteweg-de Vries-Burgers equation,” Comput. Appl. Math., vol. 21, no. 1, pp. 315–354, 2002. [30] Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems With Applications. New York: SpringerVerlag, 1999. [31] M. Karasek et al., “Protection of surviving channels in all-optical gainclamped Raman fibre amplifier: Modelling and experimentation,” Opt. Commun., vol. 231, no. 1–6, pp. 309–317, Feb. 2004. [32] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces. New York: Wiley, 1976. [33] O. Morgul, “Stabilization and disturbance rejection for the wave equation,” IEEE Trans. Auto. Control, vol. 43, no. 1, pp. 89–95, 1998. [34] K. A. Morris, “Control of systems governed by partial differential equations,” in The Control Theory Handbook, W. S. Levine, Ed. Boca Raton, FL: CRC, 2010. [35] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez, “Stabilization of food-chain systems using a port-controlled Hamiltonian description,” in Proc. Amer. Control Conf., Chicago, IL, Jun. 2000, pp. 2245–2249. [36] L. Pavel, “Global classical solvability of initial-boundary problems for hyperbolic Lotka–Volterra systems in Sobolev spaces,” in Proc. 48th IEEE Conf. Decision Control, Shanghai, China, Dec. 2009, pp. 5514–5519. [37] L. Pavel and L. Chang, “Lyapunov-based boundary control for 2 2 hyperbolic Lotka–Volterra systems,” in Proc. 48th IEEE Conf. Decision Control, Shanghai, China, Dec. 2009, pp. 3406–3411. [38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983, vol. 44. [39] S. Rionero, “A rigorous reduction of the L -stability of the solutions to a nonlinear binary reaction-diffusion system of PDE’s to the stability of the solutions to a linear binary system of ODE’s,” J. Math. Anal. Appl., vol. 319, no. 2, pp. 377–397, Jul. 2006. [40] J. C. Robinson, Infinite-Dimensional Dynamical Systems. Cambridge, U.K.: Cambridge Univ. Press, 2001.
2
714
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
[41] D. Salamon, “Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach,” Trans. Amer. Math. Soc., vol. 300, pp. 383–431, 1987. [42] L. F. Shampine, “Solving hyperbolic PDEs in MATLAB,” Appl. Numer. Anal. Comput. Math., vol. 2, no. 3, pp. 346–358, 2005. [43] SimulationCode. [Online]. Available: http://www.scg.utoronto.ca/ pavel/simulationscodetac10-130.zip [44] M. Taylor, Partial Differential Equations, Tome III, Nonlinear Equations. : Springer-Verlag, 1996, vol. 117. [45] G. Weiss, “Admissibility of unbounded control operators,” SIAM J. Control Optim., no. 27, pp. 527–545, 1989. [46] C.-Z. Xu and G. Sallet, “Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,” ESAIM: Control, Optimis. Calculus of Var., vol. 7, pp. 421–442, 2002. [47] X. Zhou, M. Feuer, and M. Birk, “Submicrosecond transient control for a forward-pumped Raman fiber amplifier,” IEEE Phot. Techn. Lett., vol. 17, no. 10, pp. 2059–2061, Oct. 2005.
Lacra Pavel (SM’04) received the Ph.D. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1996. She spent a year at the Institute for Aerospace Research as a NSERC PostDoctoral Fellow and four years in the communications industry. She joined the University of Toronto, Toronto, ON, Canada, in August 2002, where she is now an Associate Professor in the Department of Electrical and Computer Engineering. Her research interests include game theory and optimization, robust and decentralized control and their applications to complex systems such as communications networks and multi-agent networks.
Liz Chang (S’10) received the B.A.Sc. and the M.A.Sc. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 2007 and 2010, respectively. She is currently with Microsoft, Inc., Seattle, WA.
701
Lyapunov-Based Boundary Control for A Class of Hyperbolic Lotka–Volterra Systems Lacra Pavel, Senior Member, IEEE, and Liz Chang, Student Member, IEEE
Abstract—This paper considers a boundary feedback control problem for two first-order, nonlinearly coupled, hyperbolic partial differential equations with Lotka–Volterra type coupling. Boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value. Static and dynamic boundary controllers are designed based on a special Lyapunov functional that is related to an entropy function. The time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions. A unique classical solution is shown to exist globally in time and (asymptotic) exponential convergence to the desired steady-state solution is shown in the ( 0 ) 2 -norm. The boundary control design is illustrated with simulations. Index Terms—Boundary control, Lotka–Volterra, Lyapunov functional, nonlinearly coupled, partial differential equations.
I. INTRODUCTION
F
INITE-DIMENSIONAL models are often inadequate for control design for distributed systems [24], [34]. In particular, hyperbolic systems have features such as finite speed of propagation that require approximation by high-order finite dimensional models. While hyperbolic systems of second-order describe oscillatory systems such as strings or beams, first-order hyperbolic systems describe physical problems of transport-reaction type such as traffic flows, chemical reactors, and heat exchangers [46]. The distinct feature of hyperbolic partial differential equations is that all the eigenmodes of the spatial differential operator typically contain nearly the same amount of energy; as a result a very large number of modes is required to accurately approximate their dynamic behavior. This feature distinguishes hyperbolic partial differential equations and suggests addressing the control problem on the basis of the infinite-dimensional model itself [7], [30]. In this paper, a boundary feedback control problem for two coupled first-order hyperbolic partial differential equations with a nonlinear coupling of the Lotka–Volterra type is considered. Manuscript received April 04, 2009; revised March 18, 2010, May 09, 2011, and July 29, 2011; accepted September 09, 2011. Date of publication September 19, 2011; date of current version February 29, 2012. This work was supported in part by the National Science and Engineering Research Council. Recommended by Associate Editor J. J. Winkin. L. Pavel is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]). L. Chang is with Microsoft, Inc, Seattle, WA USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2168909
Lotka–Volterra type nonlinearity is commonly used to model biological systems and networks [18], predator-prey systems [17] evolutionary dynamics of species [19], or stimulated Raman induced nonlinearity in optics [4], [14]. For example, optical Raman amplifiers there have been many recent efforts to control transients using simple PID feedback, feedforward control, or all-optical gain clamping, but these designs rely on empirical selection of parameters [4], [8], [31], [47]. A systematic control design is performed in [14], based on a finite-dimensional approximation of the linearized partial differential equation model around the desired steady-state. From a 588th-order finite-difference model a tenth-order reduced model is obtained by open-loop balanced truncation, and -controller is designed. However, there is no a tenth-order guarantee that the stabilization and regulation properties will hold for the closed-loop hyperbolic system. Our work focuses on boundary control design techniques that can be performed directly on semilinear hyperbolic systems with Lotka–Volterra type nonlinear coupling. Boundary control action is used on one of the channels only, and simple static and dynamic boundary controllers are designed to drive the state at the end of the spatial domain to a desired constant reference value on the other channel. Boundary control has been an area of much interest recently, one reason being the fact that it addresses the problem on the infinite-dimensional model itself. Results for set point stabilization and/or disturbance rejection for the linear wave equation are developed in [30], [33]. The back-stepping method is an elegant approach that requires numerical computation of the kernel [23]. The zero dynamics-based approach provides a systematic methodology, and typically results in infinite-dimensional controllers [5], [6]. We use a Lyapunov-based approach which is related to the recently employed methods in [2], [10], [11], [13], [39], and [46]. In [2] concepts from thermodynamics are used to construct a Lyapunov functional. A semilinear hyperbolic system of partial differential equations with no source terms is considered in [46]. A quasi-linear hyperbolic homogeneous system of two conservation laws is treated in [11]. A quadratic Lyapunov functional with weighting factor as in [46] is used for analysis with a proportional boundary control in [11] and extended to integral action control in [13]. In [10], this approach is extended to multiple conservation laws based on a generalized quadratic Lyapunov functional. In [39], the stability properties of a binary Lipschitz nonlinearity are anreaction-diffusion system with alyzed via quadratic Lyapunov functionals.
0018-9286/$26.00 © 2011 IEEE
702
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
The hyperbolic system considered here has nonlinear growth (coupling) terms of the Lotka–Volterra type and does not satisfy Lipschitz condition. This precludes the application of the quadratic type Lyapunov functionals as in these works. In contrast, inspired by a Lyapunov function used for Lotka–Volterra ordinary differential equations [19], [35], a special Lyapunov functional that can be used directly on the nonlinear hyperbolic system is introduced. This Lyapunov functional is related to entropy and is matched to the Lotka–Volterra system. Moreover, it incorporates spatial weighting as well as a spatial factor as in [46]. This is beneficial for the design of two simple boundary controllers—proportional and dynamic with integral action. It is shown that the parameters of these controllers can be selected so that the time-derivative of the entropy function calculated along the trajectories of the closed-loop system is negative definite. Preliminary results appeared in [37]. The paper is organized as follows. Section II presents the model and boundary control problem. Section III gives conditions for the closed-loop system to be well-posed, i.e., for the mixed initial-boundary value problem to admit a unique classical solution globally in time. Main results are given in Section IV for the two boundary controllers. The boundary controllers’ parameters are selected based on a Lyapunov functional studied in Section V. Proofs of the main results are given in Section VI and Section VII, showing (asymptotic) exponential stability of the equilibrium of the closed-loop systems in the -norm. The control design is verified through simulations in Section VIII and conclusions are given in Section IX. , , The following notation is used. Let . Let denote the Lebesgue space of -valued square-integrable functions on , and use as a compact notation for this space of vector-valued is a Hilbert space with the inner product defunctions. and the -norm denoted by noted by
the completion of the space of continuously differen-norm denoted tiable functions on , equipped with the usual . Similarly, let by
where denotes the Euclidean norm in . denote the Lebesgue space of -valued measurLet able functions on that are essentially bounded with the norm
the completion of in the -norm. let denote Sobolev spaces for higher . Similarly, The following embedding relations hold among Sobolev spaces
II. MODEL AND PROBLEM FORMULATION Consider the 2 2 first-order semilinear hyperbolic system with nonlinear reaction:
(1) ,
, with initial conditions (2)
The nonlinear reaction on the right side of (1) is of the Lotka–Volterra type and is encountered in biological systems and networks, predator-prey, and competing species interac, , respectively, denoting tions [17], [18], [44], with the predator and prey population densities. System (1) can also represent coupled wave equations for modeling the dynamics of stimulated Raman scattering [4], [14]. In a normalized co-propagating Raman amplifier, with unit attenuation and and denote the coupling coefficients, signal and pump powers propagating at wavelengths , and , with characteristic speeds along the length of the amplifier denoted by . This setting is the representative case taken for the boundary control design problem considered here. Typically the signal and a certain desired input is maintained constant level (set point) is required for the signal output by ma. Thus, system nipulating only the boundary condition on (1) has the boundary conditions (3)
and the space of with the norm
where
-valued continuous functions on
in . For continuous functions the -norm is equal to the -norm. Let denote the usual Sobolev space [1], [40]
where is the control action to be designed based on the boundary measurement . For a constant control action , let denote the steady-state solution and boundary conditions which satisfies the set point (4) The following assumption will be used. Assumption 1: There exists a unique steady-state solution , , , that satisfies and the boundary conditions (4). the set point
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
703
with boundary conditions from (3), (4) (10) where to be designed, and with initial conditions from (2)
,
is
(11) Fig. 1. Boundary controlled system.
Such a steady-state solution involves solving a set of ODEs and can be found for example by using extremum seeking to , given feasible compute the optimal input pump power , , [15]. Depending on the form of these boundary conditions, the steady-state solution may be stable or unstable. Therefore, we focus on the following problem: Problem 1: Find a boundary controller (see Fig. 1) such that with boundary conditions (3) set point stabilization is achieved, i.e., for any , feasible
and for any sufficiently small and smooth initial condition the closed-loop system has a unique classical solution converging to the desired steady-state solution in some appropriate norm. In order for the initial conditions in (2) to be compatible with the boundary conditions (3) and (4) they must , . satisfy satisfies From (1) the steady-state solution
In the coordinates the closed-loop boundary controlled system (8)–(11), is a semilinear hyperbolic system with a nonlinear reaction of the Lotka–Volterra type. The control objectives stated earlier motivate two simple boundary controllers—a proportional boundary control (12) and a dynamic boundary control with integral action
(13) small, ensuring tracking to the desired set points. where With the boundary controllers specified in (12) and (13), the boundary control problem is restated to finding conditions on the parameters of (12), (13) such that, for any small smooth enough initial condition, the corresponding closed-loop system converging to the (8)–(11) has a unique classical solution origin in some appropriate norm. In the next section conditions are given for the closed-loop system (8)–(11) to be well-posed, i.e., for the mixed initial-boundary value problem to admit a unique classical solution globally in time. III. EXISTENCE AND UNIQUENESS GLOBALLY IN TIME
(5) with boundary conditions (4), and hence . This steady-state is denoted in vector form by (6) The following change of variables is made
In this section, conditions are given for classical solutions of the mixed initial-boundary value problem (8)–(11) to exist globally in time. First, the closed-loop system (8)–(11) is formulated in an abstract setting as in [12], [38]. It is noted that the abstract theory of boundary control systems started with [16] and was significantly developed in the last few decades in work such as [25], [41], and [45]. The formulation here resembles, for the case , that for linear passive boundary control systems in [30, Ch. 5, p. 259], and that for closed-loop dissipative systems in [26]. The following operators are introduced. Let us denote by “ ” the operator defined by
(7) Using (1), (5), and (7) the closed-loop system is written in the coordinates as
(8)
(14) i.e., point-wise multiplication. The same notation is used for matrix and vector-valued functions of compatible dimensions. Recall (8) with the matrix-valued coefficients , . Using the point-wise multiplication notation, let denote the linear differential operator defined by
where
(15) so that for all (9)
704
with domain Similarly, let
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
. denote the nonlinear operator defined by
where that for any that
(16) . It can be shown , there exists such
conditions. The boundary conditions are dealt with as in [29]. , for Specifically, with a change of variable , (21) is converted into a modified inhomogeneous linear hyperbolic Cauchy problem (22) with homogeneous zero boundary conditions
(17) where
and is locally Lipschitz on , [36]. Let the operator be defined by (18) with domain . Thus, is obtained from via the perturbation that is locally Lipschitz . on Then for (8)–(11), can be regarded as a function of time . For each let so that the closed-loop system (8)–(11) is written as
(19) where for classical solutions we take where
and
(20) with the boundary condition embedded. Note that (19) can be written in the form
where and , (20). is used to prove that for any The Lipschitz property of a classical solution to (19) exists, i.e., a solution that is with respect to time in and with respect to , [36]. This local existence result is given in the time in next proposition. , Proposition 1: For every initial condition there exists some such that (19) has a unique classical defined for every , such that solution
A detailed proof is given in [36], using an approach similar to the one used in [10] and [29]. The idea of the proof is to start from (19) and consider the linearized equation
(21) for any given , where is some subspace of funcwith values in , . tions of time over , (21) is an inhomogeneous linear hyperFor any given bolic boundary value problem with inhomogeneous boundary
, , is an appropriately defined function of and . The operator is a closed linear operator on and it generates a -semigroup on (by Lemma 3.2, [21]). To this equation standard existence results for classical solutions can be applied, (Corollary 4.2.5, [38] and Lemma 3.2 in [21]). Then reverting the change of variable it for any can be shown that (21) has a unique solution ; hence, it defines a mapping , with , given into itself. The proof follows by using Banach that maps contraction mapping theorem, perturbation of linear operators and Sobolev embedding, together with Gronwall inequality. develKey ingredients are a priori energy estimates on oped using Moser-type estimates [44] for the nonlinearity on and Lispchitz continuity on . We note that a similar result is sketched in [10] (Proposition 2.1) for quasilinear hyperbolic systems with no source terms . Existence and uniqueness of classical solutions globally in time depend on properties of (16). For the Cauchy problem in (19) corresponding to (uncontrolled case), i.e., for the open loop system, existence of a global classical solution depends on the global Lipschitz properties of (see Theorem 6.1.7, [38], or Proposition 4.2, Ch. 15 [44]). Here (16) has -norm bound and is loLipschitz properties dependent on a . Global in time existence depends on cally Lipschitz on the existence of a uniform bound on the -norm of the local solution. , and —the Proposition 2: Assume that for every with unique local classical solution of (19) on
for some
, the following a priori uniform bound holds for independent of : (23)
Then, (19) admits a unique global classical solution, i.e.,
The detailed proof of Proposition 2 is given in [36]. The proof follows an approach as in Theorem 6.1.7, [38] and Proposition 4.2, Ch. 15 [44]. The Sobolev embedding theorem, [40], which , then and shows that if (24)
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
for some is instrumental in the proof. This fact is used -norm dependent Moser-type estimates together with the to show that if the uniform bound [44] that satisfies on (23) holds, then the following a priori estimate on the -norm holds of
where is independent of . Hence, cannot blow-up in finite time and is a classical solution on . The result in Proposition 2 is similar to the one for parabolic reaction-diffusion equations (see Proposition 4.2 and 4.3, Ch. -norm. 15 [44]) and the search for a subset invariant in the Specifically, solutions exist globally in time in (classical solution) for those initial conditions belonging to a subset -norm. This subset s shown to be invariant bounded in the -norm. under the solution operator in the We will show that for (19) such an invariant subset can be found by using a Lyapunov function, for sufficiently small . Note that for weak solutions we could take and use density arguments, and for the overall space use the -norm and the -norm. , and work with classical We only treat the case which are with respect to time in , solutions with respect to time in . For sufficiently small and , by working with classical solution (differentiable in time) and the Lyapunov functional (and ), we show exponential (and asymptotic) convergence to in the -norm or -norm), respectively. (and
705
where is a real number. Such a function has been introduced in [22]. Function is continuously differentiable on , positive , , and radially unbounded, i.e., definite on as . The term , is instrumental in obtaining a strict Lyapunov function, as in [11] and [46]. The first boundary controller considered is (12). In the original coordinates the steady-state is denoted by , (6). In the -coordinates defined in (7), the closed-loop system is given by (8)–(10) and (28) The main result for the proportional boundary controller (12) is given next. Theorem 1: Consider the closed-loop system (8)–(11), (28) under Assumption 1 and let . Assume . Let any such that that , , and select any such that . If is chosen such that
and (29) where , then such that 1) For any initial condition , where is the constant in (24), there exists some such that , and (25) satisfies the entropy function
IV. ASYMPTOTIC AND EXPONENTIAL STABILITY In this section, the main results giving conditions on the two boundary controllers (12), (13) are presented. The proofs of the main results are given in Section VI and Section VII and are based on a special Lyapunov functional related to entropy. (7) can be regarded as deviation from the Recall that . Consider the entropy-like function desired steady-state (25) and the Lyapunov functional
(30) i.e., the Lyapunov functional
satisfies
as long as the solution exists and . such that, for any initial Moreover, there exists a condition with , , (30) the classical solution exists globally over and the origin is asymptotically stable holds for -norm. in the 2) There exists a number such that, for any initial condition with , the -norm of the classical solution satisfies
where (26) and
Let
be defined for any given
by (27)
for some constant , hence the origin is exponentially stable in the -norm. Remark 1: For and a weak solution, convergence in the -norm could be shown based on density arguments and continuity of solutions with respect to initial conditions as in [29], or [30, p. 270]. We only work with the classical solution ( with respect to time) herein and use Lyapunov , arguments to show that for sufficiently small -norm [part (i)] and exponential asymptotic stability in the
706
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
stability in the -norm [part 2)] hold. We do not yet have ex-norm, which would imply expoponential stability in the -norm; we can only show asymptotic nential stability in the stability in the -norm [part 1)]. depends on Remark 2: Selection of the proportional gain parameters , , and such that the sufficient bounds in (29) are satisfied. The first of these correspond to boundary dissipative conditions, as given in [3] for the linearized Saint-Venant equations, while the second one shows explicit dependence on the desired steady-state values at the ends of the domain. Note that given , , and selected based on the desired steadystate values and , the bounds in (29) describe the non-empty intersection of two intervals around 0, so that there exists always a feasible in this interval to satisfy (29). Moreand the over, there is a tradeoff between the size of the gain domain over which stability in ensured as seen from the reciprocal relationship between the bound on and . The proof of Theorem 1 is given in Section VI. Unlike the quadratic-type Lyapunov functions considered in [10], [11], and (25), (27) [46], the special entropy-like Lyapunov employed here is matched to the structure of Lotka–Volterra nonlinearity. Together with the spatially dependent weighting factors in (25) this is beneficial to develop sharp decay estimates. The proof follows by showing that, for sufficiently smooth, small initial conditions , an a priori uniform bound -norm of the solution of (8)–(11), (28) and (23) on the boundary term holds. Properties of , and (25)–(27) are instrumental to show this. Then by Proposition 2 a unique classical solution exists globally in time, and its asymptotic properties can be studied. This asymptotic analysis relies on invariant or -norm) and of Lyapunov sub-level sets of (for the functional (for the -norm). Note that asymptotical stability -norm (part 1) of Theorem 1) implies that the in the holds as . For the set point condition functional it is shown that decays exponentially along the unique classical solution of the , closed-loop system. It is related to the -norm of
then for any initial condition with , for some , the classical solution exists globally over and the origin is exponentially stable in the -norm. The proof of Theorem 2 is given in Section VII. Remark 3: Selection of parameters can proceed as follows: as in Theorem 1. Then we can first choose , , , and select based on the choice of , based on the choice of , , , and . The bounds , and are simply the roots of a quadratic expression (see the proof of Theorem 2). is based on the value of Then the range of the integral gain these previously chosen parameters. As in Theorem 1, there is a tradeoff between the size of these gains and the domain over which stability in ensured. V. LYAPUNOV FUNCTIONAL AND ITS PROPERTIES In this section, we study properties of Lyapunov functional related to the entropy function , (25). Proofs of the main results Theorem 1 and 2 rely on properties of and . We show or . However, that is well-defined on subsets in we use as a Lyapunov functional for classical solutions, so will be used for subsets within . First, recall that for any given real number , , deis well-defined over fined in (27), , continuously differentiable in its argument, pos, , and radially unbounded, i.e., itive definite on as . , consider the function (27), for Lemma 1: Given any . any real such that finite, there exists finite such 1) For every that
and
and this is exploited to show exponential decay of the -norm (part 2) of Theorem 1). For the dynamic boundary controller (13) proceed similarly and shift the coordinates with respect to the desired steady-state . In the shifted coordinates , where stands for , the closed-loop system is given by (8), (9), and
(31) The following is the main result for the dynamic controller with integral action (13). Theorem 2: Consider the system (8), (9), (31) and assume and are selected as in Theorem 1. all parameters , set and select For any . Then there exist some predetermined such that if is chosen such that
when
. 2) For any
such that
where . The proof of Lemma 1 is given in the Appendix. The domain can be arbitrarily enlarged by increasing (decreasing ); hence, 2) Lemma 1 is semi-global. , (27), for a Remark 4: Lemma 1 gives properties of , given any . This result real number such that can be extended for functions of as follows. Given any such that for , let . Consider any function such that for . Thus, for , so is well-defined that for each and properties in Lemma 1 hold point-wise for . Moreover, for any such that for , function (where denotes composition) is well-defined and continuous with respect to on .
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
Similar extensions hold for (26) and vector functions of . Given , , with , let , and . Consider a function of such that for , i.e., , . Thus and (26), (27) are well-defined. Moreover, properties in Lemma 1 hold point-wise for for each and is positive definite. If then continuous with respect to on . such that For any for , operator is well-defined and continuous in -norm ( -norm). This follows from properties the in Lemma 1 and classical results on nonlinear substitution operators and calculus of variations, (Proposition 2.1, p. 160, [32]). Moreover, since is compact, is uniformly continuous into , and bounded from each bounded subset of (by Proposition 2.3, p. 161, [32]). Recall (25) and the functional defined by
Evaluated at
,
707
(33) is written as
(34) and is well-defined and continuous for any for , . that The following result gives estimates for (32) and , such that Lemma 2: Given , , and , let , and consider , (32), and , (33). 1) Let any such that , and for any let such that For every , , the following holds:
such (33). , and , . , and
(32) (35) or, evaluated for such that , 2) Let any let , . Then for every for any such that , where , the following holds: where is a function of , , such that for , , where and are defined as in (26). Note that ignoring the term, , i.e., evaluates to the -norm of the element . Thus, based on the foregoing remarks for , is continuous in the -norm ( -norm). This follows from classical results in calculus of variations (Lemma 4.3.1, p. 188, [20]). Remark 5: Note that (and ) are well-defined and contin(and ), such that uous for for , . However, we will only use (and ) to study classical solutions, so we consider such that for , . Thus, by embedding we satisfy these bounds; specifically we work with invariant sublevel sets of and in the proof of Theorem 1. , (20) and consider , (32), defined for any Recall such that classical solution of (19) for , . Based on (19) or (8), we define another functional
, and
(36) Proof of Lemma 2: For simplicity of notation throughout the proof let and . For any we have (37) that 1) We show that (35) holds for any satisfies (37) and the associated bounds, i.e., , , and . Since is dense in and is continuous in the -norm, by density arguments it will follow that satisfies (37) that (35) holds for any and the associated bounds, i.e., for any that satisfies the associated bounds. Consider thus any that satisfies (37) and such , . Hence, , that , and is well-defined. We show first that satisfies the following inequality
(33) (38) where
where
is defined as a weighted , i.e.,
708
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
In order to show (38) evaluate (9). Note that
and, since
using (34) and
as in
Using (9) the second term in
(40) can be written as
Thus, using the previous two expressions
,
(42) Using the foregoing to substitute for (34) yields
into
,
, and by assumption (since ), it follows . Then claim (38) follows from
, that indeed (41). Using the compatibility condition (37) into (38) yields
Hence, (39) where
and (39), using definition of (32) yields
Therefore, since
(40) . Integrating by parts in , together with and
since , , . Lemma 1, 1) and estimates on can be used in the second and third term on the right-hand side of the foregoing to show that, for any and given as in the statement, (35) holds. This and density arguments conclude the proof of part 1). Part 2) is proved by using the definition of , (32) and Lemma 1, 2) for (see the Appendix). Remark 6: Note that , in 1) Lemma 2 depend only on and , respectively. (35) indicates how a boundary control map should be designed. In 2), Lemma 2 , depend on . The result can be made semi-global by increasing , at the expense of reducing and increasing . VI. PROOF OF THEOREM 1
(41) (40),
Claim (38) is proved if it can be shown that . is first simpliIn order to show this, expression (40) for fied. With notation , as in (6), (9), the steady-state (5) are rewritten as
where the first term in
and (40) is written as
From the definition of
. Then
In this section, the proof of Theorem 1 is developed, based on properties of the Lyapunov functional , (32), and the entropy , (25). Let function (43) with as given in the theorem’s statement. Proof of 1) (Part A): First (30) is proved. From Proposition there exists such that (19), 1, for any or (8)–(11), (28), admits a unique classical solution for . (see also (32)). First, we Recall (25) and , evaluated show that the time derivative of , (25), at time along the solution of (19) passing at time through is equal to the functional , (33), evaluated , i.e., at the point
and (26), for all (44) For a classical solution of (19) of class based on (25), the following holds:
Using this together with (26) into foregoing yields after some manipulation
on
,
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
or
709
. Specifically, such solution exists globally in time over inside a a bound is shown to hold for any initial condition sublevel set of (26), (43). Consider defined for each by
Hence, for a classical solution
of (19) of class
on
(45) holds for all . By a density argument, (45) holds also of (19). Thus, based on for a classical solution (19) or (8)
which, by using the definition (33) of , gives (44). Recall the estimates in 1), 2) in Lemma 2 that hold for , as long as the associated bounds are satisfied, i.e., on , (43). Next we show that the following holds for (46) From (20), (28), 1) in Lemma 2, for any
, and
Then using (19), the definition of the operator (9) yields
(15) and
in
Note that the right-hand side is similar to the integrand term on the right-hand side of (34). Then after some manipulations as in the proof of 1), Lemma 2, i.e., by using and the two relations after (34) to re-express the right-hand side in the above yields
which is similar to the integrand term on the right-hand side of (39). Using in the first term of the above leads to
. From as given (35) holds, i.e., (49) where
holds for every
such that
,
(47) ,
and . These bounds are satisfied for any , as given in the theorem’s statement, if is picked as with in the second condition in (29). Hence,
for any and, since also by (29), (46) follows. with Pick any initial condition . Then , by Sobolev embedding theorem , (43). Then by (46), and (24), so that . Moreover (46) holds for , i.e., (48) remains in , (43). as long as Then using (44), (48) and
yields
as long as remains in . Finally, (30) is obtained by Gronwall’s inequality and using again , and it holds stays in . as long as Proof of 1) (Part B): The foregoing results hold locally in . Next it is shown that for sufficiently small initime over tial conditions , there exists finite, independent of such that the bound (23) holds, so that by Proposition 2 the
(50) Recalling (40), note that evaluated for , yields
, . Thus, by , and, from (49), differential inequality
, (50) is the same as , (40) . Then using (42) evaluated for
, , satisfies the scalar linear
(51) . Moreover, from with initial conditions (28), it follows that . and Using the estimates for in 1), Lemma 1, with , it follows that . Thus, on , Recall (51) and consider the solution linear equation
. to the scalar
(52) and boundary conditions with initial conditions . Note that (52) has a traveling wave . Consider , solution,
710
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
, hence the constant solution of (52) , . Now, from (51), (52),
,
(53)
, , with boundary condition . for Note that (55) is a scalar linear hyperbolic equation with dissipation term and boundary condition [28, p. 136]. Its classical solution exists globally and , . Then, by the same and this above, if scalar comparison lemma [27] for , , then , for every , . , ; hence, This shows that as . is continuous in the -norm it follows that Since as , and asymptotic stability in the -norm is proved. This completes part B and the proof of 1). Proof of 2): Recall that 1) shows exponential decay for , (30), and asymptotic stability in the -norm. Based on the a priori estimates on with respect to the -norm of solutions (especially the lower bound in 2), Lemma 2 which holds on a decays smaller domain), it is shown that the -norm of exponentially on a smaller domain . where From 2) in Lemma 2 the bounds in (36) hold on . Let and , . For given , pick correspondingly the set any such that and then pick as in the proof of Part B in 1) above. Thus, is such that . Let an arbitrary and pick any (54) holds for . Note that , where is the one in part 1). Again by Sobolev embedding theorem with , it follows that [40], for any and
(54)
so that
satisfies
and . By a scalar comparison lemma for differential inequalities (Theorem 1.2-4, p. 19, [27]), , , it follows that, if then , , , on ; hence, using the definition of , that , , , . By applying 1) in Lemma 1, it follows , , , or that , , , as long as the solution remains in . Hence, (23) holds and it remains to show how to pick and such that the solution remains in initial condition . This can be shown by continuity of . Since is continuous in the -norm and , for every there such that for any exists , holds. By applying , it can be shown that there again 1) in Lemma 1 for every exists finite such that . Pick such , so that . This gives an upper bound on that and the corresponding . . Then for any Pick with , by Sobolev embedding theorem and (24), , hence and . For picked as above it follows that , , and
In other words, for as long as the solution exists
stays inside . From (53) it is seen that i.e., the solution independent of . By Proposition 2 the (23) holds with . In unique classical solution exists globally in time over addition (54) holds for , for any other and a slightly smaller . so its The classical solution exists globally in time on , since asymptotic properties can be studied. For stays inside by (54), it follows that (30) holds for , . Thus, decays exponentially as , hence exponentially, as . Based on or as properties of and it can be shown that in the -norm. This can be shown as follows. Recall defined in (50). Since , , for , there exists such that , ; hence,
Proceeding similarly as in (49), (51), (52) consider solution to
the
(55)
, as needed for (54). For any such , by using the right-hand side of (36)
, since
for
. Using (54) on it follows that . Thus, by (30), , . Since , by using the left-hand side , yields of (36) together with 2) in Lemma 2 for ,
For any such that
Since
for all
, since
from (30) it follows
and remains within , (36) can be used on both sides so that
. Thus, , i.e.,
,
,
, for
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
and exponential asymptotic stability in the -norm is proved, concluding the proof of 2). Remark 7: is radially unbounded so that in the proof can be arbitrarily large by modifying . Thus, the domain can be arbitrarily large and the result gives semi-global exponential and stability in -norm. For simplicity the symmetric set is used in the proof. Similar arguments (but with more tedious notations) could be used for the larger rectangular , , with , for set . For and arbitrarily small, (30) is semi-global . The stronger inequality is used over instead of simply showing that is negative definite, enabling exponential decay of the strict Lyapunov function. In addition, under the condition that satisfies (29), altering the controller gain determines the size of the asymptotic stability domain. VII. PROOF OF THEOREM 2 For the closed-loop system (8), (9), (31) in the coordinates, consider an augmented Lyapunov functional (56) is a design where is the original Lyapunov function and parameter that scales the contributions of the augmented state . Define the function . Then as in the can be comproof of Theorem 1, the time derivative of of the augmented closedputed along the solution loop system (8), (9) and (31) and shown to satisfy
and
where
is defined in (33). Consider
so that
711
where
Matrix is positive definite as shown next. For any chosen . Also as in the statement it follows that
With for the unknown, the equation has two dis, , for any , tinct real roots . Moreover, it can be shown and and . that Thus, for any , and , so that indeed is positive definite. Then from (57) it follows that
which has exactly the same form as (46). Then, the proof follows similar steps as the proof of Theorem 1. Remark 8: Selection of parameters can proceed as follows: first choose , , and as in Theorem 1. Then select based on the choice of , based on the choice of , , , and . As seen in the proof above the bounds , and are simply the roots of a quadratic expression . is based on the value of Then the range of the integral gain these previously chosen parameters. As in Theorem 1 there is a tradeoff between the size of these gains and the domain over which stability in ensured ( or ). Remark 9: The results developed in this paper can be immesystems, , diately extended to fully actuated i.e., such that there are control channels and signal chan, nels and decoupled boundary controller , or, , or , , . For under actusystems, , a similar ated methodology can be used but controller conditions are not immediately generalized and need to be separately analyzed. VIII. SIMULATION RESULTS
With
given as from (31), by using Lemma 2 for it can be shown as in the proof of Theorem 1 that the following holds [see (47)] on
where
,
. This can be written as
(57)
In this section, MATLAB simulation results are presented based on the hyperbolic partial differential equations solver, [42], [43]. The associated initial-boundary value problems were solved using central finite difference method with a time step of and a spatial discretization step of . The open-loop system (1) was first simulated until it reached steady state, then , the boundary conditions for the controllers (12) with at , and (13) with , , and were applied. The first scenario studied is a 40% drop/add in reference . Fig. 2, 3 show output signal and pump evolution for the dynamic controller. Fig. 4 shows the signal output for the two controllers in the first scenario. The proportional controller produced a large offset, while the dynamic controller tracked the reference accurately with a steady-state error of less than 0.5%.
712
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
Fig. 2. Signal output evolution for reference change (dynamic control).
tial differential equations with Lotka–Volterra nonlinear coupling. These equations serve as model for such physical phenomena as biological systems, population dynamics or stimulated Raman scattering. A special Lyapunov functional related to an entropy function for Lotka–Volterra systems was introduced. The time derivative of the Lyapunov-based function was made strictly negative by an appropriate choice of boundary conditions. Based on this Lyapunov functional static and dynamic boundary controllers were designed to drive the state at the end of the spatial domain to desired constant reference, based on only one control at one end. Based on global in time existence and uniqueness of classical solutions of the closed-loop system, exponential decay of the entropy function -norm was shown. On and asymptotic stability in the -norm a smaller domain exponential stability in the was shown. The design was illustrated with simulations. Future work will address generalization to systems, counter-propagating and un-normalized cases as well as rejection of disturbances. APPENDIX Proof of Lemma 1: 1) The first part follows immediately, based on the properties and graph of the Lambert W (or product log) function, [9]. . The left side of the 2) Consider inequality follows immediately by showing that has a local , i.e., that is positive definite in locally. minimum at is continuous and twice differentiable on Notice that . Differentiate with respect to to obtain
Fig. 3. Pump output (control) evolution for reference change (dynamic control).
Then For any
when or , for . such that , it follows that hence and . Thus, attains its local minimum and its local maximum at , beyond which at decreases. Therefore, for any . The right side of the inequality follows similarly. Proof of Lemma 2: , with and as in Part 2): For any the statement the following hold:
Similarly, Fig. 4. Signal output evolution for reference change.
IX. CONCLUSION In this paper, a boundary feedback control problem has been considered for two coupled first-order hyperbolic par-
PAVEL AND CHANG: LYAPUNOV-BASED BOUNDARY CONTROL FOR A CLASS OF HYPERBOLIC LOTKA–VOLTERRA SYSTEMS
Thus,
where , is well-defined and continuous
and . By Lemma 1,
Then, summing up after , yields for
Integrating over , and any
, using the definition of , yields (36).
, that
and
,
ACKNOWLEDGMENT The authors would like to thank the anonymous referees and the associate editor for their constructive comments that have significantly improved the manuscript. REFERENCES [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces. New York: Academic, 1975. [2] A. A. Alonso and B. E. Ydstie, “Stabilization of distributed systems using irreversible thermodynamics,” Automatica, vol. 37, no. 11, pp. 1739–1755, 2001. [3] G. Bastin, J.-M. Coron, and B. d’Andréa Novel, “Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks,” in Lecture Notes for Pre-Congress Workshop on Complex Embedded and Networked Control Syst., 17th IFAC World Congr., Seoul, Korea, Jul. 2008. [4] J. Bromage, “Raman amplification for fibre communication systems,” J. Lightw. Technol., vol. 22, no. 1, pp. 79–93, 2004. [5] C. Byrnes, D. Gilliam, and C. Hu, “Set-point boundary control for a Kuramoto-Sivashinsky equation,” in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp. 75–80. [6] C. Byrnes, D. Gilliam, A. Isidori, and V. Shubov, “Set point boundary control for a nonlinear distributed parameter system,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, Dec. 2003, pp. 312–317. [7] P. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkhauser, 2000. [8] C. J. Chen et al., “Control of transient effects in distributed and lumped Raman amplifiers,” Electron. Lett., vol. 37, no. 21, pp. 1304–1305, Oct. 2001. [9] R. M. Corless et al., “On the Lambert W function,” Adv. Comput. Math., vol. 5, pp. 329–359, 1996. [10] J.-M. Coron, G. Bastin, and B. d’Andréa Novel, “Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,” SIAM J. Control Optim., vol. 47, no. 3, pp. 1460–1498, May 2008. [11] J.-M. Coron, B. d’Andréa Novel, and G. Bastin, “A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,” IEEE Trans. Autom. Control, vol. 52, no. 1, pp. 2–11, Jan. 2007. [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Texts in Applied Mathematics. New York: Springer-Verlag, 1995, vol. 21. [13] V. Dos Santos, G. Bastin, J.-M. Coron, and B. d’Andréa Novel, “Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments,” Automatica, vol. 44, no. 5, pp. 1310–1318, May 2008.
713
[14] P. Dower and P. Farrell, “On linear control of backward pumped Raman amplifiers,” in Proc. IFAC Symp. Syst. Ident., Newcastle, Australia, Mar. 2006, pp. 547–552. [15] P. Dower, P. Farrell, and D. Nesic, “An application of extremum seeking in cascaded optical amplifier control,” in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp. 4472–4477. [16] H. O. Fattorini, “Boundary control systems,” SIAM J. Control, vol. 6, pp. 349–388, 1968. [17] K. R. Fister, “Optimal control of harvesting coupled with boundary control in a predator-prey system,” Applicable Analysis, vol. 77, no. 1, pp. 11–28, 2001. [18] B. S. Goh, G. Leitmann, and T. L. Vincent, “Optimal control of a preypredator system,” Math. Biosci., vol. 19, no. 3–4, pp. 263–386, 1974, American Elsevier Publishing Company. [19] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge, U.K.: Cambridge Univ. Press, 1998. [20] J. Jost and X. Li-Jost, Calculus of Variations. Cambridge, U.K.: Cambridge Univ. Press, 1998. [21] T. Kato, “The Cauchy problem for quasi-linear symmetric hyperbolic systems,” Arch. Rational Mech. Anal., vol. 58, no. 3, pp. 181–205, 1975. [22] M. Krstic, D. Fontaine, P. V. Kokotovic, and J. Paduano, “Useful nonlinearities and global bifurcation control of jet engine surge and stall,” IEEE Trans. Autom. Control, vol. 43, no. 12, pp. 1739–1745, Dec. 1998. [23] M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays,” in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, Dec. 2007, pp. 225–230. [24] I. Lasiecka, Mathematical Control Theory of Coupled PDEs. Philadelphia, PA: SIAM, 2002. [25] L. Lasiecka and R. Triggiani, “A cosine operator approach to modeling L (0; T ; L (G))—Boundary input hyperbolic equations,” Appl. Math. Optim., vol. 7, pp. 35–93, 1981. [26] L. Lasiecka and R. Triggiani, “ L (6)-regularity of the boundary to boundary operator B L for hyperbolic and Petrowski PDEs,” Abstract Appl. Anal., vol. 19, pp. 1061–1139, 2003. [27] A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering. Norwell, MA: Kluwer, 1989. [28] T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems. New York: Wiley, 1994, Res. Appl. Math. 32, Mason. [29] W.-J. Liu and M. Krstic, “Global boundary stabilization of the Korteweg-de Vries-Burgers equation,” Comput. Appl. Math., vol. 21, no. 1, pp. 315–354, 2002. [30] Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems With Applications. New York: SpringerVerlag, 1999. [31] M. Karasek et al., “Protection of surviving channels in all-optical gainclamped Raman fibre amplifier: Modelling and experimentation,” Opt. Commun., vol. 231, no. 1–6, pp. 309–317, Feb. 2004. [32] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces. New York: Wiley, 1976. [33] O. Morgul, “Stabilization and disturbance rejection for the wave equation,” IEEE Trans. Auto. Control, vol. 43, no. 1, pp. 89–95, 1998. [34] K. A. Morris, “Control of systems governed by partial differential equations,” in The Control Theory Handbook, W. S. Levine, Ed. Boca Raton, FL: CRC, 2010. [35] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez, “Stabilization of food-chain systems using a port-controlled Hamiltonian description,” in Proc. Amer. Control Conf., Chicago, IL, Jun. 2000, pp. 2245–2249. [36] L. Pavel, “Global classical solvability of initial-boundary problems for hyperbolic Lotka–Volterra systems in Sobolev spaces,” in Proc. 48th IEEE Conf. Decision Control, Shanghai, China, Dec. 2009, pp. 5514–5519. [37] L. Pavel and L. Chang, “Lyapunov-based boundary control for 2 2 hyperbolic Lotka–Volterra systems,” in Proc. 48th IEEE Conf. Decision Control, Shanghai, China, Dec. 2009, pp. 3406–3411. [38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983, vol. 44. [39] S. Rionero, “A rigorous reduction of the L -stability of the solutions to a nonlinear binary reaction-diffusion system of PDE’s to the stability of the solutions to a linear binary system of ODE’s,” J. Math. Anal. Appl., vol. 319, no. 2, pp. 377–397, Jul. 2006. [40] J. C. Robinson, Infinite-Dimensional Dynamical Systems. Cambridge, U.K.: Cambridge Univ. Press, 2001.
2
714
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 3, MARCH 2012
[41] D. Salamon, “Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach,” Trans. Amer. Math. Soc., vol. 300, pp. 383–431, 1987. [42] L. F. Shampine, “Solving hyperbolic PDEs in MATLAB,” Appl. Numer. Anal. Comput. Math., vol. 2, no. 3, pp. 346–358, 2005. [43] SimulationCode. [Online]. Available: http://www.scg.utoronto.ca/ pavel/simulationscodetac10-130.zip [44] M. Taylor, Partial Differential Equations, Tome III, Nonlinear Equations. : Springer-Verlag, 1996, vol. 117. [45] G. Weiss, “Admissibility of unbounded control operators,” SIAM J. Control Optim., no. 27, pp. 527–545, 1989. [46] C.-Z. Xu and G. Sallet, “Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,” ESAIM: Control, Optimis. Calculus of Var., vol. 7, pp. 421–442, 2002. [47] X. Zhou, M. Feuer, and M. Birk, “Submicrosecond transient control for a forward-pumped Raman fiber amplifier,” IEEE Phot. Techn. Lett., vol. 17, no. 10, pp. 2059–2061, Oct. 2005.
Lacra Pavel (SM’04) received the Ph.D. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1996. She spent a year at the Institute for Aerospace Research as a NSERC PostDoctoral Fellow and four years in the communications industry. She joined the University of Toronto, Toronto, ON, Canada, in August 2002, where she is now an Associate Professor in the Department of Electrical and Computer Engineering. Her research interests include game theory and optimization, robust and decentralized control and their applications to complex systems such as communications networks and multi-agent networks.
Liz Chang (S’10) received the B.A.Sc. and the M.A.Sc. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 2007 and 2010, respectively. She is currently with Microsoft, Inc., Seattle, WA.
