Adaptive Neural Control Design for Nonlinear ... - Semantic Scholar
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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Adaptive Neural Control Design for Nonlinear Distributed Parameter Systems With Persistent Bounded Disturbances Huai-Ning Wu and Han-Xiong Li, Senior Member, IEEE
Abstract—In this paper, an adaptive neural network (NN) -gain performance is proposed for a control with a guaranteed class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and persistent bounded disturbances. Initially, Galerkin method is applied to the PDE system to derive a low-order ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the low-order slow model and the Lyapunov technique, an adaptive modal feedback controller is developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) with an -gain performance. In the proposed control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative of the Lyapunov function due to the unknown system nonlinearities. The outcome of the adaptive -gain control problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal controller is obtained in the sense of minimizing an upper bound of the -gain, while control constraints are respected. Furthermore, it is shown that the proposed controller can ensure the semiglobal input-to-state practical stability and -gain performance of the closed-loop PDE system. Finally, by applying the developed design method to the temperature profile control of a catalytic rod, the achieved simulation results show the effectiveness of the proposed controller. Index Terms—adaptive control, distributed parameter systems, input-to-state stability (ISS), linear matrix inequality (LMI), -gain control, neural network (NN).
I. INTRODUCTION
O
VER the past two decades, extensive research has been carried out for achieving disturbance attenuation of nonlinear lumped parameter systems (LPSs) [1]–[8]. The -norm) setting in solving the problem of classical -gain ( Manuscript received December 23, 2007; revised June 29, 2009; accepted July 20, 2009. First published September 09, 2009; current version published October 07, 2009. The work was supported in part by the City University of Hong Kong under SRG project 7002450, by the RGC of Hong Kong under GRF project CityU: 117208, by the National Natural Science Foundation of China under Grants 50775224 and 60674058, and by the Program for New Century Excellent Talents in Universities, China. H.-N. Wu is with the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, 100191, China (e-mail: [email protected]; [email protected]). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China (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/TNN.2009.2028887
disturbance attenuation is to satisfy the so-called Hamilton–Jacobi–Isaccs (HJI) inequality [2], where the external disturspace. bances should be with finite energy or belong to However, this setting cannot be applied to nonlinear systems with persistent bounded disturbances, which do not belong to space. A solution to this difficulty is to address disturbance -gain ( -norm) attenuation of nonlinear systems in the disturbance rejection setting [6]–[8]. In [6], the optimal problem for nonlinear systems was formulated as a constrained optimization problem and a generalized recursive nonlinear programming was proposed for solving this problem. Lu [7] employed the concept of invariance to present an approach -performance analysis and synthesis for nonlinear to the systems. More recently, a suboptimal -gain control method has been provided in [8] for nonlinear systems based on the Takagi–Sugeno (T–S) fuzzy model and the linear matrix inequality (LMI) optimization technique [9], [10]. Despite these -gain perefforts, it remains an open issue how to achieve an formance for nonlinear distributed parameter systems (DPSs) with persistent bounded disturbances. In practice, a significant number of industrial processes are inherently distributed in space so that their behavior depends on spatial position as well as time, such as many thermal, fluid flow, and chemical reactor processes. These systems are usually described by a set of nonlinear partial differential equations (PDEs) with mixed or homogeneous boundary conditions. Moreover, the external disturbances impinging on the processes are in general -gain disturbance attenuation persistent. Thus, achieving for nonlinear PDE systems is of theoretical, challenging, and practical importance. Due to infinite-dimensional nature of PDE systems, it is very difficult directly using the design methods of LPSs to design a controller that can be readily implemented in real time with reasonable computing power. The existing approaches to the control design of PDE systems can be classified into two types: design-then-reduce one and reduce-then-design one. The former takes full advantage of infinite-dimensional operator theory [11] for the controller design, and the resulting infinite-dimensional control solution is then lumped for implementation purpose. The latter discretizes the DPS into an approximate model consisting of a set of ordinary differential equations (ODEs) in time. Then, the design methods of LPSs are applied directly to accomplish controller design. Although there are many advantages, the design-then-reduce methods are mainly limited to linear systems [11]. The reduce-then-design approaches require that the resulting design be robust to tolerate finite-dimensional approximation errors. Since it is
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WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
relatively easy to design controllers using various concepts of finite-dimensional control theory, the reduce-then-design approaches have recently attracted a lot of attention [12]–[19]. For example, Balas [12] and Ray [13] presented the finite-dimensional linear quadratic regulator (LQR) design for linear parabolic PDE systems; Christofides et al. [14], [15] provided the low-order controller synthesis for nonlinear parabolic PDE systems using Galerkin method [20] and feedback linearizafuzzy tion [21]; Wu and Li [16] proposed a constrained observer-based control design for nonlinear parabolic PDE systems based on Galerkin method and the T–S fuzzy model. However, most of the existing results on nonlinear parabolic PDE systems require that system nonlinearities be completely known (see, for example, [14]–[16]), which cannot be applied to systems with unknown nonlinearities. More recently, Wu and Li [17] employed Galerkin method and multilayer neural networks (NNs) to develop a simple guaranteed cost control design for a class of parabolic PDE systems with unknown nonlinearities. However, this work did not consider the effect of external disturbances on the systems. Motivated by its practical -gain control design problem for a class importance, the of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances is studied in this paper along the way of reduce-then-design. However, how to deal with this problem using directly the original PDEs, i.e., following the way of design-then-reduce, is a very difficult problem to be solved that we leave for future investigation. For systems with unknown nonlinearities, recent years have witnessed a rapid growing interest in control system design by using NNs together with the adaptive control technique, and there have been many successful applications; see, e.g., [22], [23], and the references therein. By exploiting the universal approximation property of NNs over a compact set, the unknown nonlinearities can be straightforwardly substituted by an NN model, which is of known structure but contains a number of unknown parameters (synaptic weights), plus a modeling error term. Depending on the NN model used, its weights may appear linearly or nonlinearly with respect to the network nonlinearities. As a consequence, the original problem can be transformed into a nonlinear adaptive control one. This adaptive NN-based control technique has been widely accepted as a mature methodology for the control design of systems with unknown nonlinearities [5], [22]–[30]. In this work, an -gain control design is developed for a adaptive NN-based class of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances. The main feature of parabolic PDE systems is that the eigenspectrum of their spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this fact, Galerkin method is initially applied to the PDE system to derive a slow system of low-order ODEs. Based on the slow model and the Lyapunov technique, an adaptive NN-based modal feedback control design is subsequently developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) and an -gain performance of disturbance attenuation is satisfied. In our control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative
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of the Lyapunov function due to the unknown system nonlin-gain control problem earities. The outcome of the adaptive is formulated as an LMI problem, which can be efficiently solved using the existing LMI optimization technique [9], -gain [10]. Moreover, a suboptimal constrained adaptive control problem is addressed for the slow system in the sense -gain, while control of minimizing an upper bound of the constraints are respected. Furthermore, it is also proven that the proposed controller can ensure that the closed-loop PDE system -gain performance of is semiglobally ISpS and satisfies an disturbance attenuation. Finally, the simulation study on the temperature profile control of a catalytic rod is given to show the effectiveness of the proposed design method. The main contribution of this paper is that the adaptive -gain control problem of a class of parabolic PDE systems with unknown nonlinearities and persistent external disturbances is handled using Galerkin method and RBF NNs. A simple LMI approach to the design of a suboptimal adaptive -gain controller for the low-order slow system is developed -gain, while control to minimize an upper bound of the constraints are respected. Finally, it is also proven that the proposed controller can ensure the semiglobal input-to-state -gain performance of the closed-loop practical stability and PDE system. The rest of this paper is organized as follows. In Section II, we review some basic definitions and preliminary results. Section III gives the problem formulation. Section IV presents an adaptive neural control design for achieving the semiglobal -gain performance of input-to-state practical stability and disturbance attenuation. In Section V, numerical simulations on a catalytic rod are provided to show the effectiveness of the proposed controller. Finally, some conclusions are drawn in Section VI. denote the set of real and nonnegative Notations: and and denote the -dimenreal numbers, respectively. matrices, sional Euclidean space and the set of all real respectively. and stand for the absolute value for scalars , and Euclidean norm for vectors, respectively. For . if . For a symmetric matrix , means that it is positive definite (positive semidefinite, negative definite, and negaand denote the maxtive semidefinite, respectively). imum singular value and the minimum singular value of a matrix, respectively. The superscript is used for the transpose. The symbol is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g.,
II. PRELIMINARIES In this section, we recall some definitions and preliminary results, which are necessary in the design of the proposed adaptive neural controller. We begin with definitions of class and functions. is said to be Definition 1 [21]: A function of class if it is continuous, strictly increasing, and is zero at zero. A function is of class if, for each
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fixed
, the function is of class and, for each fixed , the function is decreasing and tends to zero at infinity. We introduce the following notion of the input-to-state practical stability for our application. Consider a nonlinear system given by (1) is the state, is the input, and is a function such that a well-defined solution to the differential equation exists. Then, the following definition can be made. Definition 2 [31]: The system (1) is said to be semiglobally ISpS if for each compact set and each measurable essentially defined for all , there exist a compact bounded input of class , a function of class , and a set , a function such that, for any initial condition , constant the associated solution exists for all , does not exceed , and satisfies where
be the estimate of and the weight estimation error Let . Then, the following result is given in [32] be and [33]. , and let evolve according to the Lemma 1: Let following dynamics:
Then, the following is true: , ; 1) 2) , . The following lemma will be useful in the sequel. and be real scalar functions. Lemma 2 [34]: Let Then
implies that
for any positive constant . When and are whole state space, the property is then called input-to-state practical stability. and any compact set , from the Given any universal approximation property of RBF NNs [24], it follows that there exists an integer such that for any continuous func, the following representation is true: tion
III. PROBLEM FORMULATION We consider nonlinear parabolic PDE systems in one spatial dimension with a state–space description of the form [14]
(5)
(2) subject to the boundary conditions is the ideal weight vector, is the node where number of the NN, is the vector of RBFs, and is the network approximation error satisfying
(6) (3)
and the initial condition (7)
In the following, we will use Gaussian basis functions for aphas the form proximation, that is, the th element of
is the vector of state is the domain of definition of the process, is the spatial coordinate, is the time, is the manipulated input vector, and denotes the bounded peak exogenous disturbance vector, i.e., . and are the first- and second-order spatial derivatives of , respectively. and are known smooth vector describes how functions of , where is distributed in the interval and the control action specifies the position of action of the exogenous disturbance in . is an unknown . , , , nonlinear vector function satisfying , , and are constant matrices, , , , and are is the initial condition. constant vectors, and Remark 1: It is worth noting that this study focuses only on the parabolic PDEs with interior or distributed controls, i.e., the where variables,
(4) is the center of the receptive field and is the where width of the Gaussian function. and Define the sets where and are , define the projection design parameters. For any given by operator
and and
.
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
manipulated input enters directly into the differential equation and does not appear in the boundary condition. However, in many applications, it may be desirable to adopt boundary control for parabolic PDEs, where the control actuation is applied only through the boundary conditions. Therefore, it remains a challenge to extend the proposed result in this paper to the parabolic PDEs with boundary controls, which we leave for future study. Due to the existence of unknown nonlinearities and the spatially distributed nature in processes (5)–(7), it does not allow applying the PDE (5)–(7) directly for the control design. For model-based synthesis and real-time implementation, a low-order ODE system is preferred. be a Hilbert space of 1-D functions deLet that satisfy the boundary conditions given in fined on (6), with inner product and norm
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is large. Based on this assumption, we can eigenvalues of apply Galerkin method to the system (5)–(7) to obtain an approximate low-order nonlinear ODE system. Without loss of and denote generality, we assume , , , and . First, by the separation of time and spatial variables [18], [19], the following infinite-dimensional nonlinear ODE system can be obtained:
(9) where
and where
and are two elements of and denotes the standard inner product in . The folis necessary for the lowing assumption on the function control design in this paper. is locally Lipschitz continuous, i.e., Assumption 1: and such that for any there exist positive real numbers satisfying , . as Define the spatial operator in
and the boundary conditions in (6) hold (8) For the operator
, the eigenvalue problem is defined as , , where denotes the th eigenvalue and denotes the corresponding eigen, function. Here, we assume that the eigenfunctions , have been orthonormalized. Define the eigenspectrum of as . . The following assumption is made for Assumption 2 [14]: , where 1) denotes the real part of . can be partitioned as , 2) consists of the first finite eigenvalues, i.e., where , and . and , 3) where is a small positive number. is the order of magnitude notation [21] (i.e., Here, if there exist positive real numbers and such that , ). can be partitioned into a finiteAssumption 2 states that dimensional part consisting of slow eigenvalues and a stable infinite-dimensional complement containing the remaining fast eigenvalues, and that the separation between the slow and fast
Obviously, the functions and in (9) are unknown since is unknown. By neglecting the fast modes, the following finite-dimensional slow system is derived from (9):
(10) In this study, it is assumed that the state vector of the system (5) is completely available as an output. In this situation, . For using the modal analyzer [13] can immediately yield , the th element of example, for can be calculated as follows: (11) Remark 2: Note that in practice, it is impossible to obtain . However, one can provide this the complete state at spatial information as follows [13]. First, measure positions; then . Defining and , then we have . If the sampling locations are well will not be singular, and thus chosen, (12a) This relation can be used in place of (11) in the modal analyzer. If measurements at more than spatial positions are available, using then one can use a least squares fit for (12b) in place of (11). For more detail regarding this issue, one can refer to [13].
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IV. ADAPTIVE
Fig. 1. Adaptive NN-based modal feedback controller for DPSs.
Based on the slow model (10), we consider the following modal feedback control law for the PDE system (5)–(7):
(13) is the gain matrix of linear controller while is the adaptive NN-based controller in which is the vector of adjustable weights of the network. The controller structure is shown in Fig. 1, where the modal feedback controller (13) is applied to the process, . The actual state is fed to a yielding the state modal analyzer consisting of (11). The resulting coefficients of the state variable are fed to the controller (13). be some vector as the design disturbance, Let which is introduced in the NN-based control design later. Deand consider the following -gain perforfine mance associated with the system (10) under the initial condi: tion where
-GAIN NEURAL CONTROL DESIGN
In this section, we first present an LMI-based method for the design of adaptive neural controller (13) such that the closedloop of the slow system (10) is semiglobally ISpS with respect -gain performance in (14) is satisfied for all to and the . Then, a suboptimal adaptive -gain control problem is addressed for the slow system (10) in the sense of minimizing -gain, while control constraints are an upper bound of the respected. Finally, we will show that the resulting controller can ensure that the closed-loop PDE system is semiglobally -gain performance in (15) ISpS with respect to and the is achieved for all . Substituting (13) into (10) leads to the following partially closed-loop system: (16) . where Let us consider the following Lyapunov function candidate for the system (16): (17) where . The time derivative of trajectories of the system (16) is given by
along the
(18) where
is some scalar,
, and
(14) is some given level of disturbance attenuation. where for all , then the -gain Remark 3: If performance in (14) can be written as
which means that the peak of the state response of slow system (10) can be attenuated by a level under the effect of the peak . of the persistently bounded input signal The objective of this paper is to seek an adaptive neural controller of the form (13) and determine an attenuation level as small as possible such that the closed-loop slow system is and the -gain persemiglobally ISpS with respect to formance in (14) is satisfied for all . Moreover, the resulting controller is desired to ensure that the closed-loop PDE and the folsystem is semiglobally ISpS with respect to -gain performance under the initial condition lowing is satisfied:
Note that in (18) is unknown due to unknown . To and provide a valid solution overcome the uncertainty of to the control problem considered in this paper, we employ the to approximate RBF NN of the form on the compact set . From (2), we have
(19) , the ideal network Assumption 3: On the compact set . weight vector belongs to , that is, Consider the following Lyapunov function candidate: (20) where is defined by (17), and in which is a design constant. Differentiating with respect to time and considering and (18), we obtain
(15) for all
, where
is some scalar.
(21)
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
Obviously, if
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Proof: Pre- and postmultiplying both sides of (25) by and considering (26) yield (22)
then we have
(29) (23)
Letting (30)
Define the set
where and the design disturbance be
is a small design constant. Let
(24)
then we obtain (22). This means that for some , if there and satisfying (25), then the condition exist matrices and . (22) holds for Consider the Lyapunov function candidate (20). Since (22) satisfies (23). For , holds, the time derivative of by considering (19) and (30), (23) can be rewritten as
where
(31) , then by selecting with the update law
If
(32) Then, we have the following result. Theorem 1: Consider the slow system (10). Given some small , if there exist matrices design constant and satisfying the following LMI for some scalar : (25) then there exists a modal feedback controller of the form (13) where (26)
and using Lemma 1, we can obtain from (31) that (33) , by setting with the update law (32), If it is clear that (33) holds. Moreover, from Lemma 1, it follows that (34) When
, if
, then by selecting with the update law , we have from (19), (23), and (30) that
and
and
which means that (35)
and otherwise (27) with the update law
, by setting with the update law , it If can ensure follows from (23) that (35) holds. Obviously, . that (34) holds for any in (24). From (34), Let us consider the design disturbance , and Assumption 3, we have that (36)
(28) such that the closed-loop system is semiglobally ISpS and the -gain performance in (14) with is satisfied . for all
(37)
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for all , i.e., . Since is locally Lipschitz is bounded for all . Moreover, due continuous, [see (3)] and , we have that to the boundedess of for all . Thus, . From the above analysis, we can conclude that under the control law (13) with (26)–(28), the time derivative of satisfies
where
(38) for all
It is easily observed that in (40), is a function of class and is a function of class . Denote and in Definition 2 and let
. Applying Lemma 2 to (38) yields
(41) is given by (27) for some . Thus, by Definition where 2, we conclude from (40) that the closed-loop slow system [i.e., ] is semiglobally ISpS with respect to . , it is easy to In addition, under the initial condition for all that see from (39) and
which implies that where
or
, which implies that (42) (39)
It follows from (39) and (37) that
Since inequality implies that
and
, the above
or (40)
for all , i.e., . The proof is complete. Theorem 1 provides an LMI approach to the design of adap-gain perfortive neural controller (13) with a guaranteed mance for the slow system (10). The proposed controller can be obtained by solving the LMI (25) [9], [10]. Moreover, it is noticed that solving LMI (25) for different values of can lead to . This fact implies that depends different values of if minimizing on . Thus, it is necessary to iterate over is desirable. the attenuation level Remark 4: Notice that convergence of the NN weight estimates in adaptive law (28) is not guaranteed due to the lack of persistent excitation (PE) for the system signals. Since our control design for achieving the objective of this paper does not require to compute the optimal weight vector , the convergence is not necessary. Moreover, the RBF network used in this paper can be replaced by any other linear approximator introduced in [23], while the semiglobal input-to-state practical -gain performance properties of the adaptive stability and system still hold. Remark 5: It should be pointed out that as approaches zero, in (27) may be discontinuous at the points the control law satisfying . Such discontinuous control law may not only raise questions of existence and uniqueness of solutions, but also cause chattering where, due to imperfections in switching devices or computational delays, the control has fast switching fluctuations across the switching surface [21]. -gain performance in (14), we can conTo obtain better sider the following minimization problem so that the attenuation
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
level sible for some
in Theorem 1 is reduced as small as pos:
subject to
and (25)
(43)
It is easy to observe that if
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Proof: By Schur complement, it follows that the LMIs in (49) are, respectively, equivalent to and (51) By (30), the first inequality in (51) implies that . Thus, we conclude that (48) holds. By (26) and (30), the second inequality in (51) can be written as
(44) (52) where is a scalar, then we obtain fore, the problem (43) can be reformulated as
subject to
. ThereLet implies
(25), and (44)
. From (48), we have which . Define . Then, we have
(45)
Consequently, we have
(46) However, in practical control applications, very small attenuation level in general leads to high-gain control . Thus, it is necessary to consider the tradeoff between the level and the control input . To this end, we impose the following control constraints in the design for linear controller :
(53) Obviously, from (52) and (53), we get (54)
(47) where , are some given positive scalars. where is Suppose that some given constant. Then, from (39), it is easily observed that
Thus, from (50) and (54), we have (47). Therefore, for some given constants , , and , a suboptimal constrained adaptive -gain neural control design for the system (10) can be formulated as the following LMI optimization problem: subject to
which implies that there exists a constant satisfying such that the state trajectory of the system (10) satisfies for all
(48)
The following theorem shows how the control constraints of (47) can be enforced at all times for the adaptive NN-based controller (13). , there exist matrices Theorem 2: Assume that for some and satisfying the LMI (25). The control constraints of for the adaptive NN-based (47) are enforced at all times controller (13) if the following LMIs hold for some constants and : (49) (50) where matrix.
and
denotes the
element of a
(25), (44), (49), and (50) (55)
, which can be solved efficiently using the exwhere isting LMI optimization technique [9], [10]. Moreover, it is observed that different values of can give rise to different optimized values of . Hence, we can still minimize the level by . iterating over From the above analysis, the suboptimal control design can be obtained as follows. Design procedure: Step 1: Choose a large positive constant
.
, let where Step 2: For some given is a given constant. Thus, we have . Find an interval of such that the problem (55) has a solution. , find the optimal value Step 3: By iterating over of such that in (55) is minimized. Let and be the solution to the problem (55) with . Then, substituting them into (26) and (27) yields a suboptimal -gain performance adaptive neural controller such that the
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in (46) with (47) are respected.
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is satisfied, while the control constraints
Step 4: If the interval of repeat Steps 2 and 3.
cannot be found, decrease
and
-gain Notice that Theorem 1 only shows the stability and performance of the closed-loop slow system; nothing about the closed-loop PDE system is discussed. Next, we will show that under the proposed controller in Theorem 1, the closed-loop -gain PDE system is also semiglobally ISpS and satisfies an performance. Theorem 3: Consider the PDE system (5)–(7). Assume that , there exist matrices and satisfor some given fying the LMI (25). Then, there exist positive real numbers and such that if , then the adaptive modal feedback controller (13) with (26)–(28) can ensure that the closed-loop -gain performance PDE system is semiglobally ISpS and the . in (15) is satisfied for all Proof: See the Appendix. Remark 6: It should be mentioned that the validity of the proposed approach here relies on the large separation of slow and fast modes of the spatial differential operator of the parabolic PDE system. This approach is not applicable to hyperbolic PDE systems (i.e., convection–reaction processes) whose spatial differential operator possesses different features than the one associated with parabolic PDEs. Specifically, all the eigenmodes of the spatial differential operator of hyperbolic PDEs contain the same, or nearly the same amount of energy, and thus, high-order finite-dimensional approximation is necessary to accurately describe the dynamic behavior of hyperbolic PDEs. This feature unfortunately prevents the application of the aforementioned model reduction technique, to derive a low-order ODE system that accurately describes the dynamics of the dominant modes of the PDE system. Thus, the controller design problem of the hyperbolic PDE system has to be addressed directly on the basis of the infinite-dimensional model itself (see [14]), which will be one of the subjects for further investigation. V. SIMULATION STUDY To illustrate the proposed adaptive neural control design procedure, we consider the control problem of the temperature profile of a catalytic rod [14]. The mathematical model which describes the spatio–temporal evolution of the rod temperature consists of the following parabolic PDE:
(56) subject to the Dirichlet boundary conditions (57) where denotes the temperature in the reactor, denotes denotes a heat of reaction, denotes a activation energy, is the manipulated input, a heat transfer coefficient, and is the bounded exogenous disturbance. The actuator and disturbance distribution functions are taken to be
and , respectively. The following typical values are given to the process parameters:
For the above values, it was verified that the operating steady is an unstable one in the absence of state (the open-loop state starting from initial conditions close to the moves to another stable steady state) steady state [14]. The eigenvalue problem for the spatial differential operator of the process
can be solved analytically and its solution is of the form
For this system, we consider the first two eigenvalues as the dominant ones (and thus, ). By Galerkin method, the following 2-D ODE system is derived:
(58)
where
and
Assume that the initial condition of PDE system given in (56) and (57) is (59) Then, we have . Fig. 2 shows the open-loop profile of evolution of the rod temperature starting from this . Fig. 3 shows the actual initial condition in the absence of and of the corresponding opentrajectories of states loop slow system. If the exact model of catalytic rod is known, then the existing results in [14] can be applicable for a modal feedback control synthesis in the absence of persistent bounded disturbances. However, these results cannot be applied to parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances. Since the exact model is difficult to be known in practice and the external disturbances are in general persistent, in the -gain following simulation study, we design an adaptive NN controller without the prior nonlinear model of the catalytic rod. Moreover, to show the effectiveness of the proposed method,
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
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TABLE I OPTIMIZED VERSUS THE PARAMETER
Fig. 2. Open-loop profile of evolution of rod temperature in the absence of w(t).
Fig. 4. Closed-loop profile of evolution of rod temperature when using PI controller.
w(t)
0
w(t)
0
Fig. 3. Actual trajectories of states x (t) and x (t) of open-loop slow system.
two controllers are studied for comparison. A fixed-gain proportional-plus-integral (PI) control law is first used as follows: Fig. 5. Closed-loop profile of evolution of rod temperature when using adaptive NN controller.
(60)
and integral time constant . with gain matrix The parameters of the PI controller are selected to stabilize the . PDE systems (56) and (57) in the absence of . Following the design procedure with Let and in the previous section, it is found that the . The opproblem (55) has a solution for timized in (55) versus the parameter is tabulated in Table I. By performing a simple one-parameter-search minimization of , it can be found that the optimal is given by . The solution of the problem (55) with is obtained as
Thus, we obtain . is designed as follows. The The RBF network in ) with centers network contains 25 nodes (i.e., evenly spaced in and widths . The design parameters of are , , , and . The initial weight is selected to . be Now, we apply the PI controller (60) and the proposed adaptive NN controller (13) with the above parameters to the system of (56) and (57) under the initial condition (59), respectively. and Figs. 4–7 illustrate the control results when . The closedFigs. 8–11 show the results when loop profiles of evolution of rod temperature when using two controllers are shown in Figs. 4 and 5 (Figs. 8 and 9), respectively. The actual state trajectories of the corresponding slow system and control action
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Fig. 6. Actual state trajectories of closed-loop slow system when w (t)
Fig. 7. Control actions u(t) and u (t) when w (t)
0.
Fig. 9. Closed-loop profile of evolution of rod temperature when
:
t
0 5 sin( ) using adaptive NN controller.
w(t)
=
w(t)
=
0.
Fig. 10. Actual state trajectories of closed-loop slow system when 0:5 sin(t).
Fig. 8. Closed-loop profile of evolution of rod temperature when 0:5 sin(t) using PI controller.
w(t)
=
using the PI controller when are indicated by the dotted lines in Figs. 6 and 7 (Figs. 10 and 11), respectively. The solid lines in Figs. 6 and 7 (Figs. 10 and 11) show the actual state trajectories of the slow system and conusing the proposed adaptive NN controller when trol action , respectively. From Figs. 4–6, it can be seen that although no exact model of the catalytic rod is available, the proposed adaptive NN controller stabilizes the as the PI controller. Figs. 8 PDE system in the absence of and 9 indicate that although the external disturbance is persistent, the proposed adaptive NN controller can make the rod temperature converge to a small region after a short time, and presents better performance of disturbance attenuation than that of the PI controller. is not considIf the control constraint ered, by solving the LMIs (25) and (44) with the same and
Fig. 11. Control actions u(t) and u (t) when w (t) = 0:5 sin(t).
as in the above design (i.e., and ), then we can obtain an unconstrained adaptive NN controller (13) by replacing and in the above constrained controller with
respectively. Applying this unconstrained controller to the PDE system with , the control action is indicated by the solid lines in Fig. 12, where Fig. 12(a) and (b) gives for and , respectively, and the using the proposed constrained one. It dotted lines show is found from Fig. 12 that using the unconstrained controller, while
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
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that the closed-loop slow system is semiglobally ISpS with an -gain performance is given in terms of LMIs. Moreover, by the existing LMI optimization technique, a suboptimal controller can be obtained in the sense of minimizing the upper -gain, while control constraints are respected. bound of the Furthermore, it is shown that the proposed controller can ensure -gain the semiglobal input-to-state practical stability and performance of the closed-loop PDE system. Finally, the simulation results on a catalytic rod indicate that the proposed design method is effective. APPENDIX PROOF OF THEOREM 3 From the proof of Theorem 1, it has been shown that for some , if there exist matrices and satisfying the LMI (25), then (22) holds for and . This such that implies that there exists a scalar
Fig. 12. (Un)constrained control actions u (t) when w (t) = 0:5 sin(t).
(A1) Substituting the control law (13) into the system (9) and multiplying the -subsystem by , we can obtain the following singular perturbation model: (A2) which is in fact equivalent to the partially closed loop of the original PDE system (5) and can be rewritten as
Fig. 13. Control actions 0:5 sin(t).
u (t)
and
u (t)
when
#
= 0:001 and
w(t)
(A3) =
using the constrained one. This result shows that it may lead to if we do not consider to restrain a larger control action . To study the control actions for different design parameter , we reduce from 0.1 to 0.001 and keep other simulation parameters unchanged. The control actions of applying the proposed constrained controller are given in Fig. 13, which shows that chattering phenomenon and a larger control action occur when is chosen to be very small. Thus, to avoid the occurrence of chattering and a larger control action, it is necessary to choice a suitable parameter in (27). VI. CONCLUSION -gain control problem of a In this paper, the adaptive class of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances has been studied using Galerkin method and NNs. Galerkin method is initially applied to a parabolic PDE system to derive a slow system of low-order ODEs. Based on the slow model and the Lyapunov technique, an adaptive NN-based modal feedback control design is subsequently developed. The proposed control scheme employs an RBF NN to approximate the unknown term in the derivative of the Lyapunov function along the trajectories of the slow system. The existence condition of adaptive neural controllers such
where and . It is observed that setting in (A2) yields and then into (A2) gives the slow model (10). substituting and are Lipschitz continuous, it folConsidering that lows that there exist positive real numbers , , , , and such that
(A4) and . for any Since Gaussian basis functions are bounded, every element is bounded, i.e., , where is of a constant. Using this fact and (34), it follows from (27) that (A5) shown at the bottom of the next page holds, where . It is clear from (A5) that (A6) for all
. Therefore, from (13) and (A6), we have (A7)
where
.
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According to singular perturbation theory [21], by introand setting , we can ducing the fast time scale obtain the following infinite-dimensional fast subsystem from the model (A2): (A12) (A8) From the fact that there exists a matrix Lyapunov equation [21]:
where
, , , and . Thus, from (A11) and (A12), we obtain
where
and
and the definition of , it follows that satisfies the following (A9)
is some given matrix. where Let us consider the following Lyapunov function candidate: (A10) where
is defined by (20) and
ferentiating with respect to time and considering (A3), (A4), and (A7), we obtain
. Dif, (A1),
Defining
we have that if
, then
, and thus
(A13) By applying a similar analysis for (23) in the proof of Theorem 1 to (A13), it is easy to conclude that under the control law (13) with (26)–(28), the time derivative of satisfies (A11) where and By using (A3), (A4), (A7), and (A9), we have
.
(A14) for all from (A14)
. Similar to the derivation of (39), we can obtain
which implies
(A15)
and (A5) and otherwise
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
where
,
, and
. From (A15) and (37), it is easy to obtain that
or
(A16) From (A16), we can conclude that the closed-loop PDE system is semiglobally ISpS with respect to . For example, for , in view of [12], we have from (A16) that
which means that the closed-loop PDE system is semiglobally ISpS with respect to . (i.e., and ), by considering When for all , we have from (A15)
for all , i.e., , which implies that (15) is satisfied with for all since . This completes the proof. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions on the improvement of the text. REFERENCES
[1] A. J. van der Schaft, “ L -gain analysis of nonlinear systems and nonlinear state-feedback H control,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 770–784, Jun. 1992. [2] A. Isidori and A. Astolfi, “Disturbance attenuation and H -control via measurement feedback in nonlinear systems,” IEEE Trans. Autom. Control, vol. 37, no. 9, pp. 1283–1293, Sep. 1992. [3] Z.-P. Jiang and D. J. Hill, “Passivity and disturbance attenuation via output feedback for uncertain nonlinear systems,” IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 992–997, Jul. 1998. [4] D. Liberzon, E. D. Sontag, and Y. Wang, “Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation,” Syst. Control Lett., vol. 46, pp. 111–127, 2002.
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[5] S. S. Ge and T.-T. Han, “Semiglobal ISpS disturbance attenuation with output tracking via direct adaptive design,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1129–1148, Jul. 2007. [6] R. J. P. de Figueiredo and G. Chen, “Optimal disturbance rejection for nonlinear control systems,” IEEE Trans. Autom. Control, vol. 34, no. 12, pp. 1242–1248, Dec. 1989. [7] W.-M. Lu, “Rejection of persistent L -bounded disturbances for nonlinear systems,” IEEE Trans. Autom. Control, vol. 43, no. 12, pp. 1692–1702, Dec. 1998. [8] C.-S. Tseng and C.-K. Hwang, “Fuzzy observer-based fuzzy control design for nonlinear systems with persistent bounded disturbances,” Fuzzy Sets Syst., vol. 158, pp. 164–179, 2007. [9] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [10] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox for Use With Matlab. Natick, MA: The MathWorks, Inc., 1995. [11] R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. New York: Springer-Verlag, 1995. [12] M. J. Balas, “Feedback control of linear diffusion processes,” Int. J. Control, vol. 29, pp. 523–533, 1979. [13] W. H. Ray, Advanced Process Control. New York: McGraw-Hill, 1981. [14] P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkhäuser, 2001. [15] N. H. El-Farra, A. Armaou, and P. D. Christofides, “Analysis and control of parabolic PDE systems with input constraints,” Automatica, vol. 39, pp. 715–725, 2003. [16] H.-N. Wu and H.-X. Li, “ H fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 502–516, Apr. 2008. [17] H.-N. Wu and H.-X. Li, “A Galerkin/neural-network based design of guaranteed cost control for nonlinear distributed parameter systems,” IEEE Trans. Neural Netw., vol. 19, no. 5, pp. 795–807, May 2008. [18] K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chem. Eng. Sci., vol. 56, pp. 6683–6710, 2001. [19] H. Deng, H.-X. Li, and G. Chen, “Spectral approximation based intelligent modeling for a distributed thermal process,” IEEE Trans. Control Syst. Technol., vol. 13, no. 5, pp. 686–700, Sep. 2005. [20] C. A. J. Fletcher, Computational Galerkin Methods, 1st ed. New York: Springer-Verlag, 1984. [21] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [22] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2001. [23] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches. Hoboken, NJ: Wiley-Interscience, 2006. [24] R. M. Sanner and J.-J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [25] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [26] M. M. Polycarpou and M. J. Mears, “Stable adaptive tracking of uncertain systems using nonlinearly parametrized on-line approximators,” Int. J. Control, vol. 70, no. 3, pp. 363–384, 1998. [27] G. A. Rovithakis, “Stable adaptive neuron-control design via Lyapunov function derivative estimation,” Automatica, vol. 37, no. 8, pp. 1213–1221, 2001. [28] H. Xu and P. A. Ioannou, “Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds,” IEEE Trans. Autom. Control, vol. 48, no. 5, pp. 728–742, May 2003. [29] S. S. Ge and C. Wang, “Adaptive neural control of uncertain MIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 674–692, May 2004. [30] N. Hovakimyan, B.-J. Yang, and A. J. Calise, “Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems,” Automatica, vol. 42, no. 4, pp. 513–522, 2006.
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[31] Z.-P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,” Automatica, vol. 34, pp. 825–840, 1998. [32] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 729–740, Jun. 1992. [33] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 177–188, Feb. 1996. [34] P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996. Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received the B.E. degree in automation from Shandong Institute of Building Materials Industry, Jinan, China, in 1992 and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1997. From August 1997 to July 1999, he was a Postdoctoral Researcher at the Department of Electronic Engineering, Beijing Institute of Technology, Beijing, China. In August 1999, he joined the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics). From December 2005 to May 2006, he was a Senior Research Associate at the Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Hong Kong. From October to December of each year between 2006 and 2008, he was a Research Fellow in the Department of MEEM, City University of Hong Kong. He is currently a Professor at Beihang University. His current research interests include robust and reliable control and filtering, time-delay systems, Markovian jump systems, distributed parameter systems, and fuzzy/neural modeling and control. Dr. Wu is a member of the Committee of Technical Process Failure Diagnosis and Safety, Chinese Association of Automation.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree in aerospace engineering from the National University of Defence Technology, Changsha, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997. Currently, he is a Full Professor at the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. Over the last 20 years, he has had opportunities to work in different fields, including military service, industry, and academia. His research experience and accomplishment include design and control for electronics packaging process, modeling and control of spatio–temporal dynamic system, a pioneering threedomain fuzzy logic system for modeling and control. His current research interests include intelligent modeling and control, process design and control, and distributed parameter systems. He has authored and coauthored more than 90 papers in international journals with half of them in IEEE TRANSACTIONS. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS. He was awarded the Distinguished Young Scholar by the China National Science Foundation in 2004, and a “Chang Jiang Scholar” by the Ministry of Education, China, in 2006.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Adaptive Neural Control Design for Nonlinear Distributed Parameter Systems With Persistent Bounded Disturbances Huai-Ning Wu and Han-Xiong Li, Senior Member, IEEE
Abstract—In this paper, an adaptive neural network (NN) -gain performance is proposed for a control with a guaranteed class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and persistent bounded disturbances. Initially, Galerkin method is applied to the PDE system to derive a low-order ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the low-order slow model and the Lyapunov technique, an adaptive modal feedback controller is developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) with an -gain performance. In the proposed control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative of the Lyapunov function due to the unknown system nonlinearities. The outcome of the adaptive -gain control problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal controller is obtained in the sense of minimizing an upper bound of the -gain, while control constraints are respected. Furthermore, it is shown that the proposed controller can ensure the semiglobal input-to-state practical stability and -gain performance of the closed-loop PDE system. Finally, by applying the developed design method to the temperature profile control of a catalytic rod, the achieved simulation results show the effectiveness of the proposed controller. Index Terms—adaptive control, distributed parameter systems, input-to-state stability (ISS), linear matrix inequality (LMI), -gain control, neural network (NN).
I. INTRODUCTION
O
VER the past two decades, extensive research has been carried out for achieving disturbance attenuation of nonlinear lumped parameter systems (LPSs) [1]–[8]. The -norm) setting in solving the problem of classical -gain ( Manuscript received December 23, 2007; revised June 29, 2009; accepted July 20, 2009. First published September 09, 2009; current version published October 07, 2009. The work was supported in part by the City University of Hong Kong under SRG project 7002450, by the RGC of Hong Kong under GRF project CityU: 117208, by the National Natural Science Foundation of China under Grants 50775224 and 60674058, and by the Program for New Century Excellent Talents in Universities, China. H.-N. Wu is with the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, 100191, China (e-mail: [email protected]; [email protected]). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China (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/TNN.2009.2028887
disturbance attenuation is to satisfy the so-called Hamilton–Jacobi–Isaccs (HJI) inequality [2], where the external disturspace. bances should be with finite energy or belong to However, this setting cannot be applied to nonlinear systems with persistent bounded disturbances, which do not belong to space. A solution to this difficulty is to address disturbance -gain ( -norm) attenuation of nonlinear systems in the disturbance rejection setting [6]–[8]. In [6], the optimal problem for nonlinear systems was formulated as a constrained optimization problem and a generalized recursive nonlinear programming was proposed for solving this problem. Lu [7] employed the concept of invariance to present an approach -performance analysis and synthesis for nonlinear to the systems. More recently, a suboptimal -gain control method has been provided in [8] for nonlinear systems based on the Takagi–Sugeno (T–S) fuzzy model and the linear matrix inequality (LMI) optimization technique [9], [10]. Despite these -gain perefforts, it remains an open issue how to achieve an formance for nonlinear distributed parameter systems (DPSs) with persistent bounded disturbances. In practice, a significant number of industrial processes are inherently distributed in space so that their behavior depends on spatial position as well as time, such as many thermal, fluid flow, and chemical reactor processes. These systems are usually described by a set of nonlinear partial differential equations (PDEs) with mixed or homogeneous boundary conditions. Moreover, the external disturbances impinging on the processes are in general -gain disturbance attenuation persistent. Thus, achieving for nonlinear PDE systems is of theoretical, challenging, and practical importance. Due to infinite-dimensional nature of PDE systems, it is very difficult directly using the design methods of LPSs to design a controller that can be readily implemented in real time with reasonable computing power. The existing approaches to the control design of PDE systems can be classified into two types: design-then-reduce one and reduce-then-design one. The former takes full advantage of infinite-dimensional operator theory [11] for the controller design, and the resulting infinite-dimensional control solution is then lumped for implementation purpose. The latter discretizes the DPS into an approximate model consisting of a set of ordinary differential equations (ODEs) in time. Then, the design methods of LPSs are applied directly to accomplish controller design. Although there are many advantages, the design-then-reduce methods are mainly limited to linear systems [11]. The reduce-then-design approaches require that the resulting design be robust to tolerate finite-dimensional approximation errors. Since it is
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WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
relatively easy to design controllers using various concepts of finite-dimensional control theory, the reduce-then-design approaches have recently attracted a lot of attention [12]–[19]. For example, Balas [12] and Ray [13] presented the finite-dimensional linear quadratic regulator (LQR) design for linear parabolic PDE systems; Christofides et al. [14], [15] provided the low-order controller synthesis for nonlinear parabolic PDE systems using Galerkin method [20] and feedback linearizafuzzy tion [21]; Wu and Li [16] proposed a constrained observer-based control design for nonlinear parabolic PDE systems based on Galerkin method and the T–S fuzzy model. However, most of the existing results on nonlinear parabolic PDE systems require that system nonlinearities be completely known (see, for example, [14]–[16]), which cannot be applied to systems with unknown nonlinearities. More recently, Wu and Li [17] employed Galerkin method and multilayer neural networks (NNs) to develop a simple guaranteed cost control design for a class of parabolic PDE systems with unknown nonlinearities. However, this work did not consider the effect of external disturbances on the systems. Motivated by its practical -gain control design problem for a class importance, the of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances is studied in this paper along the way of reduce-then-design. However, how to deal with this problem using directly the original PDEs, i.e., following the way of design-then-reduce, is a very difficult problem to be solved that we leave for future investigation. For systems with unknown nonlinearities, recent years have witnessed a rapid growing interest in control system design by using NNs together with the adaptive control technique, and there have been many successful applications; see, e.g., [22], [23], and the references therein. By exploiting the universal approximation property of NNs over a compact set, the unknown nonlinearities can be straightforwardly substituted by an NN model, which is of known structure but contains a number of unknown parameters (synaptic weights), plus a modeling error term. Depending on the NN model used, its weights may appear linearly or nonlinearly with respect to the network nonlinearities. As a consequence, the original problem can be transformed into a nonlinear adaptive control one. This adaptive NN-based control technique has been widely accepted as a mature methodology for the control design of systems with unknown nonlinearities [5], [22]–[30]. In this work, an -gain control design is developed for a adaptive NN-based class of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances. The main feature of parabolic PDE systems is that the eigenspectrum of their spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this fact, Galerkin method is initially applied to the PDE system to derive a slow system of low-order ODEs. Based on the slow model and the Lyapunov technique, an adaptive NN-based modal feedback control design is subsequently developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) and an -gain performance of disturbance attenuation is satisfied. In our control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative
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of the Lyapunov function due to the unknown system nonlin-gain control problem earities. The outcome of the adaptive is formulated as an LMI problem, which can be efficiently solved using the existing LMI optimization technique [9], -gain [10]. Moreover, a suboptimal constrained adaptive control problem is addressed for the slow system in the sense -gain, while control of minimizing an upper bound of the constraints are respected. Furthermore, it is also proven that the proposed controller can ensure that the closed-loop PDE system -gain performance of is semiglobally ISpS and satisfies an disturbance attenuation. Finally, the simulation study on the temperature profile control of a catalytic rod is given to show the effectiveness of the proposed design method. The main contribution of this paper is that the adaptive -gain control problem of a class of parabolic PDE systems with unknown nonlinearities and persistent external disturbances is handled using Galerkin method and RBF NNs. A simple LMI approach to the design of a suboptimal adaptive -gain controller for the low-order slow system is developed -gain, while control to minimize an upper bound of the constraints are respected. Finally, it is also proven that the proposed controller can ensure the semiglobal input-to-state -gain performance of the closed-loop practical stability and PDE system. The rest of this paper is organized as follows. In Section II, we review some basic definitions and preliminary results. Section III gives the problem formulation. Section IV presents an adaptive neural control design for achieving the semiglobal -gain performance of input-to-state practical stability and disturbance attenuation. In Section V, numerical simulations on a catalytic rod are provided to show the effectiveness of the proposed controller. Finally, some conclusions are drawn in Section VI. denote the set of real and nonnegative Notations: and and denote the -dimenreal numbers, respectively. matrices, sional Euclidean space and the set of all real respectively. and stand for the absolute value for scalars , and Euclidean norm for vectors, respectively. For . if . For a symmetric matrix , means that it is positive definite (positive semidefinite, negative definite, and negaand denote the maxtive semidefinite, respectively). imum singular value and the minimum singular value of a matrix, respectively. The superscript is used for the transpose. The symbol is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g.,
II. PRELIMINARIES In this section, we recall some definitions and preliminary results, which are necessary in the design of the proposed adaptive neural controller. We begin with definitions of class and functions. is said to be Definition 1 [21]: A function of class if it is continuous, strictly increasing, and is zero at zero. A function is of class if, for each
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fixed
, the function is of class and, for each fixed , the function is decreasing and tends to zero at infinity. We introduce the following notion of the input-to-state practical stability for our application. Consider a nonlinear system given by (1) is the state, is the input, and is a function such that a well-defined solution to the differential equation exists. Then, the following definition can be made. Definition 2 [31]: The system (1) is said to be semiglobally ISpS if for each compact set and each measurable essentially defined for all , there exist a compact bounded input of class , a function of class , and a set , a function such that, for any initial condition , constant the associated solution exists for all , does not exceed , and satisfies where
be the estimate of and the weight estimation error Let . Then, the following result is given in [32] be and [33]. , and let evolve according to the Lemma 1: Let following dynamics:
Then, the following is true: , ; 1) 2) , . The following lemma will be useful in the sequel. and be real scalar functions. Lemma 2 [34]: Let Then
implies that
for any positive constant . When and are whole state space, the property is then called input-to-state practical stability. and any compact set , from the Given any universal approximation property of RBF NNs [24], it follows that there exists an integer such that for any continuous func, the following representation is true: tion
III. PROBLEM FORMULATION We consider nonlinear parabolic PDE systems in one spatial dimension with a state–space description of the form [14]
(5)
(2) subject to the boundary conditions is the ideal weight vector, is the node where number of the NN, is the vector of RBFs, and is the network approximation error satisfying
(6) (3)
and the initial condition (7)
In the following, we will use Gaussian basis functions for aphas the form proximation, that is, the th element of
is the vector of state is the domain of definition of the process, is the spatial coordinate, is the time, is the manipulated input vector, and denotes the bounded peak exogenous disturbance vector, i.e., . and are the first- and second-order spatial derivatives of , respectively. and are known smooth vector describes how functions of , where is distributed in the interval and the control action specifies the position of action of the exogenous disturbance in . is an unknown . , , , nonlinear vector function satisfying , , and are constant matrices, , , , and are is the initial condition. constant vectors, and Remark 1: It is worth noting that this study focuses only on the parabolic PDEs with interior or distributed controls, i.e., the where variables,
(4) is the center of the receptive field and is the where width of the Gaussian function. and Define the sets where and are , define the projection design parameters. For any given by operator
and and
.
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
manipulated input enters directly into the differential equation and does not appear in the boundary condition. However, in many applications, it may be desirable to adopt boundary control for parabolic PDEs, where the control actuation is applied only through the boundary conditions. Therefore, it remains a challenge to extend the proposed result in this paper to the parabolic PDEs with boundary controls, which we leave for future study. Due to the existence of unknown nonlinearities and the spatially distributed nature in processes (5)–(7), it does not allow applying the PDE (5)–(7) directly for the control design. For model-based synthesis and real-time implementation, a low-order ODE system is preferred. be a Hilbert space of 1-D functions deLet that satisfy the boundary conditions given in fined on (6), with inner product and norm
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is large. Based on this assumption, we can eigenvalues of apply Galerkin method to the system (5)–(7) to obtain an approximate low-order nonlinear ODE system. Without loss of and denote generality, we assume , , , and . First, by the separation of time and spatial variables [18], [19], the following infinite-dimensional nonlinear ODE system can be obtained:
(9) where
and where
and are two elements of and denotes the standard inner product in . The folis necessary for the lowing assumption on the function control design in this paper. is locally Lipschitz continuous, i.e., Assumption 1: and such that for any there exist positive real numbers satisfying , . as Define the spatial operator in
and the boundary conditions in (6) hold (8) For the operator
, the eigenvalue problem is defined as , , where denotes the th eigenvalue and denotes the corresponding eigen, function. Here, we assume that the eigenfunctions , have been orthonormalized. Define the eigenspectrum of as . . The following assumption is made for Assumption 2 [14]: , where 1) denotes the real part of . can be partitioned as , 2) consists of the first finite eigenvalues, i.e., where , and . and , 3) where is a small positive number. is the order of magnitude notation [21] (i.e., Here, if there exist positive real numbers and such that , ). can be partitioned into a finiteAssumption 2 states that dimensional part consisting of slow eigenvalues and a stable infinite-dimensional complement containing the remaining fast eigenvalues, and that the separation between the slow and fast
Obviously, the functions and in (9) are unknown since is unknown. By neglecting the fast modes, the following finite-dimensional slow system is derived from (9):
(10) In this study, it is assumed that the state vector of the system (5) is completely available as an output. In this situation, . For using the modal analyzer [13] can immediately yield , the th element of example, for can be calculated as follows: (11) Remark 2: Note that in practice, it is impossible to obtain . However, one can provide this the complete state at spatial information as follows [13]. First, measure positions; then . Defining and , then we have . If the sampling locations are well will not be singular, and thus chosen, (12a) This relation can be used in place of (11) in the modal analyzer. If measurements at more than spatial positions are available, using then one can use a least squares fit for (12b) in place of (11). For more detail regarding this issue, one can refer to [13].
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IV. ADAPTIVE
Fig. 1. Adaptive NN-based modal feedback controller for DPSs.
Based on the slow model (10), we consider the following modal feedback control law for the PDE system (5)–(7):
(13) is the gain matrix of linear controller while is the adaptive NN-based controller in which is the vector of adjustable weights of the network. The controller structure is shown in Fig. 1, where the modal feedback controller (13) is applied to the process, . The actual state is fed to a yielding the state modal analyzer consisting of (11). The resulting coefficients of the state variable are fed to the controller (13). be some vector as the design disturbance, Let which is introduced in the NN-based control design later. Deand consider the following -gain perforfine mance associated with the system (10) under the initial condi: tion where
-GAIN NEURAL CONTROL DESIGN
In this section, we first present an LMI-based method for the design of adaptive neural controller (13) such that the closedloop of the slow system (10) is semiglobally ISpS with respect -gain performance in (14) is satisfied for all to and the . Then, a suboptimal adaptive -gain control problem is addressed for the slow system (10) in the sense of minimizing -gain, while control constraints are an upper bound of the respected. Finally, we will show that the resulting controller can ensure that the closed-loop PDE system is semiglobally -gain performance in (15) ISpS with respect to and the is achieved for all . Substituting (13) into (10) leads to the following partially closed-loop system: (16) . where Let us consider the following Lyapunov function candidate for the system (16): (17) where . The time derivative of trajectories of the system (16) is given by
along the
(18) where
is some scalar,
, and
(14) is some given level of disturbance attenuation. where for all , then the -gain Remark 3: If performance in (14) can be written as
which means that the peak of the state response of slow system (10) can be attenuated by a level under the effect of the peak . of the persistently bounded input signal The objective of this paper is to seek an adaptive neural controller of the form (13) and determine an attenuation level as small as possible such that the closed-loop slow system is and the -gain persemiglobally ISpS with respect to formance in (14) is satisfied for all . Moreover, the resulting controller is desired to ensure that the closed-loop PDE and the folsystem is semiglobally ISpS with respect to -gain performance under the initial condition lowing is satisfied:
Note that in (18) is unknown due to unknown . To and provide a valid solution overcome the uncertainty of to the control problem considered in this paper, we employ the to approximate RBF NN of the form on the compact set . From (2), we have
(19) , the ideal network Assumption 3: On the compact set . weight vector belongs to , that is, Consider the following Lyapunov function candidate: (20) where is defined by (17), and in which is a design constant. Differentiating with respect to time and considering and (18), we obtain
(15) for all
, where
is some scalar.
(21)
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
Obviously, if
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Proof: Pre- and postmultiplying both sides of (25) by and considering (26) yield (22)
then we have
(29) (23)
Letting (30)
Define the set
where and the design disturbance be
is a small design constant. Let
(24)
then we obtain (22). This means that for some , if there and satisfying (25), then the condition exist matrices and . (22) holds for Consider the Lyapunov function candidate (20). Since (22) satisfies (23). For , holds, the time derivative of by considering (19) and (30), (23) can be rewritten as
where
(31) , then by selecting with the update law
If
(32) Then, we have the following result. Theorem 1: Consider the slow system (10). Given some small , if there exist matrices design constant and satisfying the following LMI for some scalar : (25) then there exists a modal feedback controller of the form (13) where (26)
and using Lemma 1, we can obtain from (31) that (33) , by setting with the update law (32), If it is clear that (33) holds. Moreover, from Lemma 1, it follows that (34) When
, if
, then by selecting with the update law , we have from (19), (23), and (30) that
and
and
which means that (35)
and otherwise (27) with the update law
, by setting with the update law , it If can ensure follows from (23) that (35) holds. Obviously, . that (34) holds for any in (24). From (34), Let us consider the design disturbance , and Assumption 3, we have that (36)
(28) such that the closed-loop system is semiglobally ISpS and the -gain performance in (14) with is satisfied . for all
(37)
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for all , i.e., . Since is locally Lipschitz is bounded for all . Moreover, due continuous, [see (3)] and , we have that to the boundedess of for all . Thus, . From the above analysis, we can conclude that under the control law (13) with (26)–(28), the time derivative of satisfies
where
(38) for all
It is easily observed that in (40), is a function of class and is a function of class . Denote and in Definition 2 and let
. Applying Lemma 2 to (38) yields
(41) is given by (27) for some . Thus, by Definition where 2, we conclude from (40) that the closed-loop slow system [i.e., ] is semiglobally ISpS with respect to . , it is easy to In addition, under the initial condition for all that see from (39) and
which implies that where
or
, which implies that (42) (39)
It follows from (39) and (37) that
Since inequality implies that
and
, the above
or (40)
for all , i.e., . The proof is complete. Theorem 1 provides an LMI approach to the design of adap-gain perfortive neural controller (13) with a guaranteed mance for the slow system (10). The proposed controller can be obtained by solving the LMI (25) [9], [10]. Moreover, it is noticed that solving LMI (25) for different values of can lead to . This fact implies that depends different values of if minimizing on . Thus, it is necessary to iterate over is desirable. the attenuation level Remark 4: Notice that convergence of the NN weight estimates in adaptive law (28) is not guaranteed due to the lack of persistent excitation (PE) for the system signals. Since our control design for achieving the objective of this paper does not require to compute the optimal weight vector , the convergence is not necessary. Moreover, the RBF network used in this paper can be replaced by any other linear approximator introduced in [23], while the semiglobal input-to-state practical -gain performance properties of the adaptive stability and system still hold. Remark 5: It should be pointed out that as approaches zero, in (27) may be discontinuous at the points the control law satisfying . Such discontinuous control law may not only raise questions of existence and uniqueness of solutions, but also cause chattering where, due to imperfections in switching devices or computational delays, the control has fast switching fluctuations across the switching surface [21]. -gain performance in (14), we can conTo obtain better sider the following minimization problem so that the attenuation
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
level sible for some
in Theorem 1 is reduced as small as pos:
subject to
and (25)
(43)
It is easy to observe that if
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Proof: By Schur complement, it follows that the LMIs in (49) are, respectively, equivalent to and (51) By (30), the first inequality in (51) implies that . Thus, we conclude that (48) holds. By (26) and (30), the second inequality in (51) can be written as
(44) (52) where is a scalar, then we obtain fore, the problem (43) can be reformulated as
subject to
. ThereLet implies
(25), and (44)
. From (48), we have which . Define . Then, we have
(45)
Consequently, we have
(46) However, in practical control applications, very small attenuation level in general leads to high-gain control . Thus, it is necessary to consider the tradeoff between the level and the control input . To this end, we impose the following control constraints in the design for linear controller :
(53) Obviously, from (52) and (53), we get (54)
(47) where , are some given positive scalars. where is Suppose that some given constant. Then, from (39), it is easily observed that
Thus, from (50) and (54), we have (47). Therefore, for some given constants , , and , a suboptimal constrained adaptive -gain neural control design for the system (10) can be formulated as the following LMI optimization problem: subject to
which implies that there exists a constant satisfying such that the state trajectory of the system (10) satisfies for all
(48)
The following theorem shows how the control constraints of (47) can be enforced at all times for the adaptive NN-based controller (13). , there exist matrices Theorem 2: Assume that for some and satisfying the LMI (25). The control constraints of for the adaptive NN-based (47) are enforced at all times controller (13) if the following LMIs hold for some constants and : (49) (50) where matrix.
and
denotes the
element of a
(25), (44), (49), and (50) (55)
, which can be solved efficiently using the exwhere isting LMI optimization technique [9], [10]. Moreover, it is observed that different values of can give rise to different optimized values of . Hence, we can still minimize the level by . iterating over From the above analysis, the suboptimal control design can be obtained as follows. Design procedure: Step 1: Choose a large positive constant
.
, let where Step 2: For some given is a given constant. Thus, we have . Find an interval of such that the problem (55) has a solution. , find the optimal value Step 3: By iterating over of such that in (55) is minimized. Let and be the solution to the problem (55) with . Then, substituting them into (26) and (27) yields a suboptimal -gain performance adaptive neural controller such that the
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in (46) with (47) are respected.
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is satisfied, while the control constraints
Step 4: If the interval of repeat Steps 2 and 3.
cannot be found, decrease
and
-gain Notice that Theorem 1 only shows the stability and performance of the closed-loop slow system; nothing about the closed-loop PDE system is discussed. Next, we will show that under the proposed controller in Theorem 1, the closed-loop -gain PDE system is also semiglobally ISpS and satisfies an performance. Theorem 3: Consider the PDE system (5)–(7). Assume that , there exist matrices and satisfor some given fying the LMI (25). Then, there exist positive real numbers and such that if , then the adaptive modal feedback controller (13) with (26)–(28) can ensure that the closed-loop -gain performance PDE system is semiglobally ISpS and the . in (15) is satisfied for all Proof: See the Appendix. Remark 6: It should be mentioned that the validity of the proposed approach here relies on the large separation of slow and fast modes of the spatial differential operator of the parabolic PDE system. This approach is not applicable to hyperbolic PDE systems (i.e., convection–reaction processes) whose spatial differential operator possesses different features than the one associated with parabolic PDEs. Specifically, all the eigenmodes of the spatial differential operator of hyperbolic PDEs contain the same, or nearly the same amount of energy, and thus, high-order finite-dimensional approximation is necessary to accurately describe the dynamic behavior of hyperbolic PDEs. This feature unfortunately prevents the application of the aforementioned model reduction technique, to derive a low-order ODE system that accurately describes the dynamics of the dominant modes of the PDE system. Thus, the controller design problem of the hyperbolic PDE system has to be addressed directly on the basis of the infinite-dimensional model itself (see [14]), which will be one of the subjects for further investigation. V. SIMULATION STUDY To illustrate the proposed adaptive neural control design procedure, we consider the control problem of the temperature profile of a catalytic rod [14]. The mathematical model which describes the spatio–temporal evolution of the rod temperature consists of the following parabolic PDE:
(56) subject to the Dirichlet boundary conditions (57) where denotes the temperature in the reactor, denotes denotes a heat of reaction, denotes a activation energy, is the manipulated input, a heat transfer coefficient, and is the bounded exogenous disturbance. The actuator and disturbance distribution functions are taken to be
and , respectively. The following typical values are given to the process parameters:
For the above values, it was verified that the operating steady is an unstable one in the absence of state (the open-loop state starting from initial conditions close to the moves to another stable steady state) steady state [14]. The eigenvalue problem for the spatial differential operator of the process
can be solved analytically and its solution is of the form
For this system, we consider the first two eigenvalues as the dominant ones (and thus, ). By Galerkin method, the following 2-D ODE system is derived:
(58)
where
and
Assume that the initial condition of PDE system given in (56) and (57) is (59) Then, we have . Fig. 2 shows the open-loop profile of evolution of the rod temperature starting from this . Fig. 3 shows the actual initial condition in the absence of and of the corresponding opentrajectories of states loop slow system. If the exact model of catalytic rod is known, then the existing results in [14] can be applicable for a modal feedback control synthesis in the absence of persistent bounded disturbances. However, these results cannot be applied to parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances. Since the exact model is difficult to be known in practice and the external disturbances are in general persistent, in the -gain following simulation study, we design an adaptive NN controller without the prior nonlinear model of the catalytic rod. Moreover, to show the effectiveness of the proposed method,
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
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TABLE I OPTIMIZED VERSUS THE PARAMETER
Fig. 2. Open-loop profile of evolution of rod temperature in the absence of w(t).
Fig. 4. Closed-loop profile of evolution of rod temperature when using PI controller.
w(t)
0
w(t)
0
Fig. 3. Actual trajectories of states x (t) and x (t) of open-loop slow system.
two controllers are studied for comparison. A fixed-gain proportional-plus-integral (PI) control law is first used as follows: Fig. 5. Closed-loop profile of evolution of rod temperature when using adaptive NN controller.
(60)
and integral time constant . with gain matrix The parameters of the PI controller are selected to stabilize the . PDE systems (56) and (57) in the absence of . Following the design procedure with Let and in the previous section, it is found that the . The opproblem (55) has a solution for timized in (55) versus the parameter is tabulated in Table I. By performing a simple one-parameter-search minimization of , it can be found that the optimal is given by . The solution of the problem (55) with is obtained as
Thus, we obtain . is designed as follows. The The RBF network in ) with centers network contains 25 nodes (i.e., evenly spaced in and widths . The design parameters of are , , , and . The initial weight is selected to . be Now, we apply the PI controller (60) and the proposed adaptive NN controller (13) with the above parameters to the system of (56) and (57) under the initial condition (59), respectively. and Figs. 4–7 illustrate the control results when . The closedFigs. 8–11 show the results when loop profiles of evolution of rod temperature when using two controllers are shown in Figs. 4 and 5 (Figs. 8 and 9), respectively. The actual state trajectories of the corresponding slow system and control action
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Fig. 6. Actual state trajectories of closed-loop slow system when w (t)
Fig. 7. Control actions u(t) and u (t) when w (t)
0.
Fig. 9. Closed-loop profile of evolution of rod temperature when
:
t
0 5 sin( ) using adaptive NN controller.
w(t)
=
w(t)
=
0.
Fig. 10. Actual state trajectories of closed-loop slow system when 0:5 sin(t).
Fig. 8. Closed-loop profile of evolution of rod temperature when 0:5 sin(t) using PI controller.
w(t)
=
using the PI controller when are indicated by the dotted lines in Figs. 6 and 7 (Figs. 10 and 11), respectively. The solid lines in Figs. 6 and 7 (Figs. 10 and 11) show the actual state trajectories of the slow system and conusing the proposed adaptive NN controller when trol action , respectively. From Figs. 4–6, it can be seen that although no exact model of the catalytic rod is available, the proposed adaptive NN controller stabilizes the as the PI controller. Figs. 8 PDE system in the absence of and 9 indicate that although the external disturbance is persistent, the proposed adaptive NN controller can make the rod temperature converge to a small region after a short time, and presents better performance of disturbance attenuation than that of the PI controller. is not considIf the control constraint ered, by solving the LMIs (25) and (44) with the same and
Fig. 11. Control actions u(t) and u (t) when w (t) = 0:5 sin(t).
as in the above design (i.e., and ), then we can obtain an unconstrained adaptive NN controller (13) by replacing and in the above constrained controller with
respectively. Applying this unconstrained controller to the PDE system with , the control action is indicated by the solid lines in Fig. 12, where Fig. 12(a) and (b) gives for and , respectively, and the using the proposed constrained one. It dotted lines show is found from Fig. 12 that using the unconstrained controller, while
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
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that the closed-loop slow system is semiglobally ISpS with an -gain performance is given in terms of LMIs. Moreover, by the existing LMI optimization technique, a suboptimal controller can be obtained in the sense of minimizing the upper -gain, while control constraints are respected. bound of the Furthermore, it is shown that the proposed controller can ensure -gain the semiglobal input-to-state practical stability and performance of the closed-loop PDE system. Finally, the simulation results on a catalytic rod indicate that the proposed design method is effective. APPENDIX PROOF OF THEOREM 3 From the proof of Theorem 1, it has been shown that for some , if there exist matrices and satisfying the LMI (25), then (22) holds for and . This such that implies that there exists a scalar
Fig. 12. (Un)constrained control actions u (t) when w (t) = 0:5 sin(t).
(A1) Substituting the control law (13) into the system (9) and multiplying the -subsystem by , we can obtain the following singular perturbation model: (A2) which is in fact equivalent to the partially closed loop of the original PDE system (5) and can be rewritten as
Fig. 13. Control actions 0:5 sin(t).
u (t)
and
u (t)
when
#
= 0:001 and
w(t)
(A3) =
using the constrained one. This result shows that it may lead to if we do not consider to restrain a larger control action . To study the control actions for different design parameter , we reduce from 0.1 to 0.001 and keep other simulation parameters unchanged. The control actions of applying the proposed constrained controller are given in Fig. 13, which shows that chattering phenomenon and a larger control action occur when is chosen to be very small. Thus, to avoid the occurrence of chattering and a larger control action, it is necessary to choice a suitable parameter in (27). VI. CONCLUSION -gain control problem of a In this paper, the adaptive class of parabolic PDE systems with unknown nonlinearities and persistent bounded disturbances has been studied using Galerkin method and NNs. Galerkin method is initially applied to a parabolic PDE system to derive a slow system of low-order ODEs. Based on the slow model and the Lyapunov technique, an adaptive NN-based modal feedback control design is subsequently developed. The proposed control scheme employs an RBF NN to approximate the unknown term in the derivative of the Lyapunov function along the trajectories of the slow system. The existence condition of adaptive neural controllers such
where and . It is observed that setting in (A2) yields and then into (A2) gives the slow model (10). substituting and are Lipschitz continuous, it folConsidering that lows that there exist positive real numbers , , , , and such that
(A4) and . for any Since Gaussian basis functions are bounded, every element is bounded, i.e., , where is of a constant. Using this fact and (34), it follows from (27) that (A5) shown at the bottom of the next page holds, where . It is clear from (A5) that (A6) for all
. Therefore, from (13) and (A6), we have (A7)
where
.
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According to singular perturbation theory [21], by introand setting , we can ducing the fast time scale obtain the following infinite-dimensional fast subsystem from the model (A2): (A12) (A8) From the fact that there exists a matrix Lyapunov equation [21]:
where
, , , and . Thus, from (A11) and (A12), we obtain
where
and
and the definition of , it follows that satisfies the following (A9)
is some given matrix. where Let us consider the following Lyapunov function candidate: (A10) where
is defined by (20) and
ferentiating with respect to time and considering (A3), (A4), and (A7), we obtain
. Dif, (A1),
Defining
we have that if
, then
, and thus
(A13) By applying a similar analysis for (23) in the proof of Theorem 1 to (A13), it is easy to conclude that under the control law (13) with (26)–(28), the time derivative of satisfies (A11) where and By using (A3), (A4), (A7), and (A9), we have
.
(A14) for all from (A14)
. Similar to the derivation of (39), we can obtain
which implies
(A15)
and (A5) and otherwise
WU AND LI: ADAPTIVE NEURAL CONTROL DESIGN FOR NONLINEAR DISTRIBUTED PARAMETER SYSTEMS
where
,
, and
. From (A15) and (37), it is easy to obtain that
or
(A16) From (A16), we can conclude that the closed-loop PDE system is semiglobally ISpS with respect to . For example, for , in view of [12], we have from (A16) that
which means that the closed-loop PDE system is semiglobally ISpS with respect to . (i.e., and ), by considering When for all , we have from (A15)
for all , i.e., , which implies that (15) is satisfied with for all since . This completes the proof. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions on the improvement of the text. REFERENCES
[1] A. J. van der Schaft, “ L -gain analysis of nonlinear systems and nonlinear state-feedback H control,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 770–784, Jun. 1992. [2] A. Isidori and A. Astolfi, “Disturbance attenuation and H -control via measurement feedback in nonlinear systems,” IEEE Trans. Autom. Control, vol. 37, no. 9, pp. 1283–1293, Sep. 1992. [3] Z.-P. Jiang and D. J. Hill, “Passivity and disturbance attenuation via output feedback for uncertain nonlinear systems,” IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 992–997, Jul. 1998. [4] D. Liberzon, E. D. Sontag, and Y. Wang, “Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation,” Syst. Control Lett., vol. 46, pp. 111–127, 2002.
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[5] S. S. Ge and T.-T. Han, “Semiglobal ISpS disturbance attenuation with output tracking via direct adaptive design,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1129–1148, Jul. 2007. [6] R. J. P. de Figueiredo and G. Chen, “Optimal disturbance rejection for nonlinear control systems,” IEEE Trans. Autom. Control, vol. 34, no. 12, pp. 1242–1248, Dec. 1989. [7] W.-M. Lu, “Rejection of persistent L -bounded disturbances for nonlinear systems,” IEEE Trans. Autom. Control, vol. 43, no. 12, pp. 1692–1702, Dec. 1998. [8] C.-S. Tseng and C.-K. Hwang, “Fuzzy observer-based fuzzy control design for nonlinear systems with persistent bounded disturbances,” Fuzzy Sets Syst., vol. 158, pp. 164–179, 2007. [9] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [10] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox for Use With Matlab. Natick, MA: The MathWorks, Inc., 1995. [11] R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. New York: Springer-Verlag, 1995. [12] M. J. Balas, “Feedback control of linear diffusion processes,” Int. J. Control, vol. 29, pp. 523–533, 1979. [13] W. H. Ray, Advanced Process Control. New York: McGraw-Hill, 1981. [14] P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkhäuser, 2001. [15] N. H. El-Farra, A. Armaou, and P. D. Christofides, “Analysis and control of parabolic PDE systems with input constraints,” Automatica, vol. 39, pp. 715–725, 2003. [16] H.-N. Wu and H.-X. Li, “ H fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 502–516, Apr. 2008. [17] H.-N. Wu and H.-X. Li, “A Galerkin/neural-network based design of guaranteed cost control for nonlinear distributed parameter systems,” IEEE Trans. Neural Netw., vol. 19, no. 5, pp. 795–807, May 2008. [18] K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chem. Eng. Sci., vol. 56, pp. 6683–6710, 2001. [19] H. Deng, H.-X. Li, and G. Chen, “Spectral approximation based intelligent modeling for a distributed thermal process,” IEEE Trans. Control Syst. Technol., vol. 13, no. 5, pp. 686–700, Sep. 2005. [20] C. A. J. Fletcher, Computational Galerkin Methods, 1st ed. New York: Springer-Verlag, 1984. [21] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [22] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2001. [23] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches. Hoboken, NJ: Wiley-Interscience, 2006. [24] R. M. Sanner and J.-J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [25] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [26] M. M. Polycarpou and M. J. Mears, “Stable adaptive tracking of uncertain systems using nonlinearly parametrized on-line approximators,” Int. J. Control, vol. 70, no. 3, pp. 363–384, 1998. [27] G. A. Rovithakis, “Stable adaptive neuron-control design via Lyapunov function derivative estimation,” Automatica, vol. 37, no. 8, pp. 1213–1221, 2001. [28] H. Xu and P. A. Ioannou, “Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds,” IEEE Trans. Autom. Control, vol. 48, no. 5, pp. 728–742, May 2003. [29] S. S. Ge and C. Wang, “Adaptive neural control of uncertain MIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 674–692, May 2004. [30] N. Hovakimyan, B.-J. Yang, and A. J. Calise, “Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems,” Automatica, vol. 42, no. 4, pp. 513–522, 2006.
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[31] Z.-P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,” Automatica, vol. 34, pp. 825–840, 1998. [32] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 729–740, Jun. 1992. [33] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 177–188, Feb. 1996. [34] P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996. Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received the B.E. degree in automation from Shandong Institute of Building Materials Industry, Jinan, China, in 1992 and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1997. From August 1997 to July 1999, he was a Postdoctoral Researcher at the Department of Electronic Engineering, Beijing Institute of Technology, Beijing, China. In August 1999, he joined the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics). From December 2005 to May 2006, he was a Senior Research Associate at the Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Hong Kong. From October to December of each year between 2006 and 2008, he was a Research Fellow in the Department of MEEM, City University of Hong Kong. He is currently a Professor at Beihang University. His current research interests include robust and reliable control and filtering, time-delay systems, Markovian jump systems, distributed parameter systems, and fuzzy/neural modeling and control. Dr. Wu is a member of the Committee of Technical Process Failure Diagnosis and Safety, Chinese Association of Automation.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree in aerospace engineering from the National University of Defence Technology, Changsha, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997. Currently, he is a Full Professor at the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. Over the last 20 years, he has had opportunities to work in different fields, including military service, industry, and academia. His research experience and accomplishment include design and control for electronics packaging process, modeling and control of spatio–temporal dynamic system, a pioneering threedomain fuzzy logic system for modeling and control. His current research interests include intelligent modeling and control, process design and control, and distributed parameter systems. He has authored and coauthored more than 90 papers in international journals with half of them in IEEE TRANSACTIONS. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS. He was awarded the Distinguished Young Scholar by the China National Science Foundation in 2004, and a “Chang Jiang Scholar” by the Ministry of Education, China, in 2006.
