Open Access

Neural Network Adaptive Control for Discrete-Time Nonlinear Nonnegative Dynamical Systems

  • Wassim M. Haddad1Email author,
  • VijaySekhar Chellaboina2,
  • Qing Hui1 and
  • Tomohisa Hayakawa3
Advances in Difference Equations20082008:868425

https://doi.org/10.1155/2008/868425

Received: 27 January 2008

Accepted: 8 April 2008

Published: 9 April 2008

Abstract

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences, and they typically involve the exchange of nonnegative quantities between subsystems or compartments, wherein each compartment is assumed to be kinetically homogeneous. In this paper, we develop a neuroadaptive control framework for adaptive set-point regulation of discrete-time nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. In addition, the neuroadaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space for nonnegative initial conditions.

1. Introduction

Neural networks have provided an ideal framework for online identification and control of many complex uncertain engineering systems because of their great flexibility in approximating a large class of continuous maps and their adaptability due to their inherently parallel architecture. Even though neuroadaptive control has been applied to numerous engineering problems, neuroadaptive methods have not been widely considered for problems involving systems with nonnegative state and control constraints [1, 2]. Such systems are commonly referred to as nonnegative dynamical systems in the literature [38]. A subclass of nonnegative dynamical systems are compartmental systems [818]. Compartmental systems involve dynamical models that are characterized by conservation laws (e.g., mass and energy) capturing the exchange of material between coupled macroscopic subsystems known as compartments. The range of applications of nonnegative systems and compartmental systems includes pharmacological systems, queuing systems, stochastic systems (whose state variables represent probabilities), ecological systems, economic systems, demographic systems, telecommunications systems, and transportation systems, to cite but a few examples. Due to the severe complexities, nonlinearities, and uncertainties inherent in these systems, neural networks provide an ideal framework for online adaptive control because of their parallel processing flexibility and adaptability.

In this paper, we extend the results of [2] to develop a neuroadaptive control framework for discrete-time nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states as well as the neural network weighting gains. The neuroadaptive controllers are constructed without requiring knowledge of the system dynamics while guaranteeing that the physical system states remain in the nonnegative orthant of the state space. The proposed neuro control architecture is modular in the sense that if a nominal linear design model is available, the neuroadaptive controller can be augmented to the nominal design to account for system nonlinearities and system uncertainty. Furthermore, since in certain applications of nonnegative and compartmental systems (e.g., pharmacological systems for active drug administration) control (source) inputs as well as the system states need to be nonnegative, we also develop neuroadaptive controllers that guarantee the control signal as well as the physical system states remain nonnegative for nonnegative initial conditions.

The contents of the paper are as follows. In Section 2, we provide mathematical preliminaries on nonnegative dynamical systems that are necessary for developing the main results of this paper. In Section 3, we develop new Lyapunov-like theorems for partial boundedness and partial ultimate boundedness for nonlinear dynamical systems necessary for obtaining less conservative ultimate bounds for neuroadaptive controllers as compared to ultimate bounds derived using classical boundedness and ultimate boundedness notions. In Section 4, we present our main neuroadaptive control framework for adaptive set-point regulation of nonlinear uncertain nonnegative and compartmental systems. In Section 5, we extend the results of Section 4 to the case where control inputs are constrained to be nonnegative. Finally, in Section 6 we draw some conclusions.

2. Mathematical Preliminaries

In this section we introduce notation, several definitions, and some key results concerning linear and nonlinear discrete-time nonnegative dynamical systems [19] that are necessary for developing the main results of this paper. Specifically, for we write (resp., ) to indicate that every component of is nonnegative (resp., positive). In this case, we say that is nonnegative or positive, respectively. Likewise, is nonnegative or positive if every entry of is nonnegative or positive, respectively, which is written as or , respectively. In this paper it is important to distinguish between a square nonnegative (resp., positive) matrix and a nonnegative-definite (resp., positive-definite) matrix. Let and denote the nonnegative and positive orthants of , that is, if , then and are equivalent, respectively, to and . Finally, we write to denote transpose, for the trace operator, (resp., ) to denote the minimum (resp., maximum) eigenvalue of a Hermitian matrix, for a vector norm, and for the set of all nonnegative integers. The following definition introduces the notion of a nonnegative (resp., positive) function.

Definition 2.1.

A real function is a nonnegative (resp., positive) function if (resp., ), .

The following theorems give necessary and sufficient conditions for asymptotic stability of the discrete-time linear nonnegative dynamical system
(2.1)

where is nonnegative and , using linear and quadratic Lyapunov functions, respectively.

Theorem 2.2 (see [19]).

Consider the linear dynamical system given by (2.1) where is nonnegative. Then is asymptotically stable if and only if there exist vectors such that and satisfy
(2.2)

Theorem 2.3 (see [6, 19]).

Consider the linear dynamical system given by (2.1) where is nonnegative. Then is asymptotically stable if and only if there exist a positive diagonal matrix and an positive-definite matrix such that
(2.3)
Next, consider the controlled discrete-time linear dynamical system
(2.4)
where
(2.5)
is nonnegative and is nonnegative such that rank . The following theorem shows that discrete-time linear stabilizable nonnegative systems possess asymptotically stable zero dynamics with viewed as the output. For the statement of this result, let denote the spectrum of , let , and let in (2.4) be partitioned as
(2.6)

where , , , and are nonnegative matrices.

Theorem 2.4.

Consider the discrete-time linear dynamical system given by (2.4), where is nonnegative and partitioned as in (2.6), and is nonnegative and is partitioned as in (2.5) with rank . Then there exists a gain matrix such that is nonnegative and asymptotically stable if and only if is asymptotically stable.

Proof.

First, let be partitioned as , where and , and note that
(2.7)

Assume that is nonnegative and asymptotically stable, and suppose that, ad absurdum, is not asymptotically stable. Then, it follows from Theorem 2.2 that there does not exist a positive vector such that . Next, since is nonnegative it follows that for any positive vector . Thus, there does not exist a positive vector such that , and hence, it follows from Theorem 2.2 that is not asymptotically stable leading to a contradiction. Hence, is asymptotically stable. Conversely, suppose that is asymptotically stable. Then taking and , where is nonnegative and asymptotically stable, it follows that , and hence, is nonnegative and asymptotically stable.

Next, consider the discrete-time nonlinear dynamical system
(2.8)

where , is an open subset of with , and is continuous on . Recall that the point is an equilibrium point of (2.8) if . Furthermore, a subset is an invariant set with respect to (2.8) if contains the orbits of all its points. The following definition introduces the notion of nonnegative vector fields [19].

Definition 2.5.

Let , where is an open subset of that contains . Then is nonnegative with respect to , , if for all , and . is nonnegative if for all , and .

Note that if , where , then is nonnegative if and only if is nonnegative [19].

Proposition 2.6 (see [19]).

Suppose . Then is an invariant set with respect to (2.8) if and only if is nonnegative.

In this paper, we consider controlled discrete-time nonlinear dynamical systems of the form
(2.9)

where , , , , is continuous and satisfies , and is continuous.

The following definition and proposition are needed for the main results of the paper.

Definition 2.7.

The discrete-time nonlinear dynamical system given by (2.9) is nonnegative if for every and , , the solution , , to (2.9) is nonnegative.

Proposition 2.8 (see [19]).

The discrete-time nonlinear dynamical system given by (2.9) is nonnegative if and , .

It follows from Proposition 2.8 that a nonnegative input signal , , is sufficient to guarantee the nonnegativity of the state of (2.9).

Next, we present a time-varying extension to Proposition 2.8 needed for the main theorems of this paper. Specifically, we consider the time-varying system
(2.10)

where is continuous in and on and , , and is continuous. For the following result, the definition of nonnegativity holds with (2.9) replaced by (2.10).

Proposition 2.9.

Consider the time-varying discrete-time dynamical system (2.10) where is continuous on for all and is continuous on for all . If for every , is nonnegative and is nonnegative, then the solution , , to (2.10) is nonnegative.

Proof.

The result is a direct consequence of Proposition 2.8 by equivalently representing the time-varying discrete-time system (2.10) as an autonomous discrete-time nonlinear system by appending another state to represent time. Specifically, defining and , it follows that the solution , , to (2.10) can be equivalently characterized by the solution , , where , to the discrete-time nonlinear autonomous system
(2.11)
(2.12)

where . Now, since , , for , and , the result is a direct consequence of Proposition 2.8.

3. Partial Boundedness and Partial Ultimate Boundedness

In this section, we present Lyapunov-like theorems for partial boundedness and partial ultimate boundedness of discrete-time nonlinear dynamical systems. These notions allow us to develop less conservative ultimate bounds for neuroadaptive controllers as compared to ultimate bounds derived using classical boundedness and ultimate boundedness notions. Specifically, consider the discrete-time nonlinear autonomous interconnected dynamical system
(3.1)
(3.2)

where , is an open set such that , , is such that, for every , and is continuous in , and is continuous. Note that under the above assumptions the solution to (3.1) and (3.2) exists and is unique over .

Definition 3.1 (see [20]).
  1. (i)

    The discrete-time nonlinear dynamical system (3.1) and (3.2) is bounded with respect to uniformly in if there exists such that, for every , there exists such that implies for all . The discrete-time nonlinear dynamical system (3.1) and (3.2) is globally bounded with respect to uniformly in if, for every , there exists such that implies for all .

     
  2. (ii)

    The discrete-time nonlinear dynamical system (3.1) and (3.2) is ultimately bounded with respect to     uniformly in     with ultimate bound     if there exists   such that, for every , there exists such that implies , . The discrete-time nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to     uniformly in     with ultimate bound   if, for every , there exists such that implies , .

     

Note that if a discrete-time nonlinear dynamical system is (globally) bounded with respect to uniformly in , then there exists such that it is (globally) ultimately bounded with respect to uniformly in with an ultimate bound . Conversely, if a discrete-time nonlinear dynamical system is (globally) ultimately bounded with respect to uniformly in with an ultimate bound , then it is (globally) bounded with respect to uniformly in . The following results present Lyapunov-like theorems for boundedness and ultimate boundedness for discrete-time nonlinear systems. For these results define , where and is a given continuous function. Furthermore, let , , , denote the open ball centered at with radius and let denote the closure of , and recall the definitions of class- , class- , and class- functions [20].

Theorem 3.2.

Consider the discrete-time nonlinear dynamical system (3.1) and (3.2). Assume that there exist a continuous function and class- functions and such that
(3.3)
(3.4)
where is such that . Furthermore, assume that exists. Then the discrete-time nonlinear dynamical system (3.1) and (3.2) is bounded with respect to   uniformly in   . Furthermore, for every , implies that , , where
(3.5)

, and . If, in addition, and is a class- function, then the discrete-time nonlinear dynamical system (3.1) and (3.2) is globally bounded with respect to uniformly in and for every , , , where is given by (3.5) with .

Proof.

See [20, page 786].

Theorem 3.3.

Consider the discrete-time nonlinear dynamical system (3.1) and (3.2). Assume there exist a continuous function and class- functions and such that (3.3) holds. Furthermore, assume that there exists a continuous function such that , , and
(3.6)

where is such that . Finally, assume exists. Then the nonlinear dynamical system (3.1), (3.2) is ultimately bounded with respect to uniformly in with ultimate bound where . Furthermore, . If, in addition, and is a class- function, then the nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to uniformly in with ultimate bound .

Proof.

See [20, page 787].

The following result on ultimate boundedness of interconnected systems is needed for the main theorems in this paper. For this result, recall the definition of input-to-state stability given in [21].

Proposition 3.4.

Consider the discrete-time nonlinear interconnected dynamical system (3.1) and (3.2). If (3.2) is input-to-state stable with viewed as the input and (3.1) and (3.2) are ultimately bounded with respect to uniformly in , then the solution , , of the interconnected dynamical system (3.1)-(3.2), is ultimately bounded.

Proof.

Since system (3.1)-(3.2) is ultimately bounded with respect to (uniformly in ), there exist positive constants and such that , . Furthermore, since (3.2) is input-to-state stable with viewed as the input, it follows that is finite, and hence, there exist a class- function and a class- function such that
(3.7)

which proves that the solution , to (3.1) and (3.2) is ultimately bounded.

4. Neuroadaptive Control for Discrete-Time Nonlinear Nonnegative Uncertain Systems

In this section, we consider the problem of characterizing neuroadaptive feedback control laws for discrete-time nonlinear nonnegative and compartmental uncertain dynamical systems to achieve set-point regulation in the nonnegative orthant. Specifically, consider the controlled discrete-time nonlinear uncertain dynamical system given by
(4.1)
(4.2)
where , , and , , are the state vectors, , , is the control input, is nonnegative with respect to but otherwise unknown and satisfies , , is nonnegative with respect to but otherwise unknown and satisfies , , and is a known nonnegative input matrix function. Here, we assume that we have control inputs so that the input matrix function is given by
(4.3)

where is a positive diagonal matrix and is a nonnegative matrix function such that , . The control input in (4.1) is restricted to the class of admissible controls consisting of measurable functions such that , . In this section, we do not place any restriction on the sign of the control signal and design a neuroadaptive controller that guarantees that the system states remain in the nonnegative orthant of the state space for nonnegative initial conditions and are ultimately bounded in the neighborhood of a desired equilibrium point.

In this paper, we assume that and are unknown functions with given by
(4.4)
where is a known nonnegative matrix and is an unknown nonnegative function with respect to and belongs to the uncertainty set given by
(4.5)
where and is an uncertain continuous function such that is nonnegative with respect to . Furthermore, we assume that for a given there exist and such that
(4.6)
(4.7)
In addition, we assume that (4.2) is input-to-state stable at with viewed as the input, that is, there exist a class- function and a class- function such that
(4.8)

where denotes the Euclidean vector norm. Unless otherwise stated, henceforth we use to denote the Euclidean vector norm. Note that is an equilibrium point of (4.1) and (4.2) if and only if there exists such that (4.6) and (4.7) hold.

Furthermore, we assume that, for a given , the th component of the vector function can be approximated over a compact set by a linear in the parameters neural network up to a desired accuracy so that for , there exists such that , , and
(4.9)

where , , are optimal unknown (constant) weights that minimize the approximation error over , , , are a set of basis functions such that each component of takes values between 0 and 1, , , are the modeling errors, and , where , , are bounds for the optimal weights , .

Since is continuous, we can choose , , from a linear space of continuous functions that forms an algebra and separates points in . In this case, it follows from the Stone-Weierstrass theorem [22, page 212] that is a dense subset of the set of continuous functions on . Now, as is the case in the standard neuroadaptive control literature [23], we can construct the signal involving the estimates of the optimal weights as our adaptive control signal. However, even though , , provides adaptive cancellation of the system uncertainty, it does not necessarily guarantee that the state trajectory of the closed-loop system remains in the nonnegative orthant of the state space for nonnegative initial conditions.

To ensure nonnegativity of the closed-loop plant states, the adaptive control signal is assumed to be of the form , , where is such that each component of takes values between 0 and 1 and , whenever for all , , where and are the th element of and , respectively. This set of functions do not generate an algebra in , and hence, if used as an approximator for , , will generate additional conservatism in the ultimate bound guarantees provided by the neural network controller. In particular, since each component of and takes values between 0 and 1, it follows that
(4.10)

This upper bound is used in the proof of Theorem 4.1 below.

For the remainder of the paper we assume that there exists a gain matrix such that is nonnegative and asymptotically stable, where and have the forms of (2.6) and (2.5), respectively. Now, partitioning the state in (4.1) as , where and , and using (4.3), it follows that (4.1) and (4.2) can be written as
(4.11)
(4.12)
(4.13)
Thus, since is nonnegative and asymptotically stable, it follows from Theorem 2.4 that the solution of (4.12) with , where and satisfy , is globally exponentially stable, and hence, (4.12) is input-to-state stable at with viewed as the input. Thus, in this paper we assume that the dynamics (4.12) can be included in (4.2) so that . In this case, the input matrix (4.3) is given by
(4.14)

so that . Now, for a given desired set point and for some , our aim is to design a control input , , such that and for all , where , and and , , for all . However, since in many applications of nonnegative systems and, in particular, compartmental systems, it is often necessary to regulate a subset of the nonnegative state variables which usually include a central compartment, here we only require that , .

Theorem 4.1.

Consider the discrete-time nonlinear uncertain dynamical system given by (4.1) and (4.2) where and are given by (4.4) and (4.14), respectively, is nonnegative with respect to , is nonnegative with respect to , and is nonnegative with respect to and belongs to . For a given assume there exist nonnegative vectors and such that (4.6) and (4.7) hold. Furthermore, assume that (4.2) is input-to-state stable at with viewed as the input. Finally, let be such that is nonnegative and is nonnegative and asymptotically stable. Then the neuroadaptive feedback control law
(4.15)
where
(4.16)
, , , and with whenever , , ,—with update law
(4.17)
where satisfies
(4.18)
for positive definite , and are positive constants satisfying and , , and —guarantees that there exists a positively invariant set such that , where , and the solution , , of the closed-loop system given by (4.1), (4.2), (4.15), and (4.17) is ultimately bounded for all with ultimate bound , , where
(4.19)
(4.20)

, , , , and and are positive constants satisfying and , respectively. Furthermore, and , , for all .

Proof.

See Appendix A.

A block diagram showing the neuroadaptive control architecture given in Theorem 4.1 is shown in Figure 1. It is important to note that the adaptive control law (4.15) and (4.17) does not require the explicit knowledge of the optimal weighting matrix and constants and . All that is required is the existence of the nonnegative vectors and such that the equilibrium conditions (4.6), and (4.7) hold. Furthermore, in the case where is an unknown positive diagonal matrix but , , where is known, we can take the gain matrix to be diagonal so that , where is such that , . In this case, taking in (4.4) to be the identity matrix, is given by which is clearly nonnegative and asymptotically stable, and hence, any positive diagonal matrix satisfies (4.18). Finally, it is important to note that the control input signal , , in Theorem 4.1 can be negative depending on the values of , . However, as is required for nonnegative and compartmental dynamical systems the closed-loop plant states remain nonnegative.
Figure 1

Block diagram of the closed-loop system.

Next, we generalize Theorem 4.1 to the case where the input matrix is not necessarily nonnegative. For this result denotes the th row of .

Theorem 4.2.

Consider the discrete-time nonlinear uncertain dynamical system given by (4.1) and (4.2), where and are given by (4.4) and (4.14), respectively, is nonnegative with respect to , is nonnegative with respect to , and is nonnegative with respect to and belongs to . For a given , assume there exist a nonnegative vector and a vector such that (4.6) and (4.7) hold with . Furthermore, assume that (4.2) is input-to-state stable at with viewed as the input. Finally, let be such that , , and is nonnegative and asymptotically stable. Then the neuroadaptive feedback control law (4.15), where is given by (4.16) with , , , and with whenever , , ,—with update law
(4.21)

where satisfies (4.18), and are positive constants satisfying and , , —guarantees that there exists a positively invariant set such that , where , and the solution , , of the closed-loop system given by (4.1), (4.2), (4.15), and (4.21) is ultimately bounded for all with ultimate bound , , where is given by (4.19) with replaced by in and , . Furthermore, and , , for all .

Proof.

The proof is identical to the proof of Theorem 4.1 with replaced by .

Finally, in the case where is an unknown diagonal matrix but the sign of each diagonal element is known and , , where is known, we can take the gain matrix to be diagonal so that , where is such that , . In this case, taking in (4.4) to be the identity matrix, is given by which is nonnegative and asymptotically stable.

Example 4.3.

Consider the nonlinear uncertain system (4.1) with
(4.22)
where are unknown. For simplicity of exposition, here we assume that there is no internal dynamics. Note that and in (4.22) can be written in the form of (4.4) and (4.3) with , , , and . Furthermore, note that is unknown and belongs to . Since for there exists such that (4.6) is satisfied, it follows from Theorem 4.2 that the neuroadaptive feedback control law (4.15) with and update law (4.21) guarantees that the closed-loop systems trajectory is ultimately bounded and remains in the nonnegative orthant of the state space for nonnegative initial conditions. With , , , , , , and initial conditions and , Figure 2 shows the state trajectories versus time and the control signal versus time.
Figure 2

State trajectories and control signal versus time.

5. Neuroadaptive Control for Discrete-Time Nonlinear Nonnegative Uncertain Systems with Nonnegative Control

As discussed in the introduction, control (source) inputs of drug delivery systems for physiological and pharmacological processes are usually constrained to be nonnegative as are the system states. Hence, in this section we develop neuroadaptive control laws for discrete-time nonnegative systems with nonnegative control inputs. In general, unlike linear nonnegative systems with asymptotically stable plant dynamics, a given set point for a discrete-time nonlinear nonnegative dynamical system
(5.1)

where , , and , may not be asymptotically stabilizable with a constant control . Hence, we assume that the set point satisfying is a unique equilibrium point in the nonnegative orthant with and is also asymptotically stable for all . This implies that the equilibrium solution to (5.1) with is asymptotically stable for all .

In this section, we assume that in (4.4) is nonnegative and asymptotically stable, and hence, without loss of generality (see [19, Proposition 3.1]), we can assume that is an asymptotically stable compartmental matrix [19]. Furthermore, we assume that the control inputs are injected directly into separate compartments so that and in (4.14) are such that is a positive diagonal matrix and , where , , is a known positive diagonal matrix function. For compartmental systems, this assumption is not restrictive since control inputs correspond to control inflows to each individual compartment. For the statement of the next theorem, recall the definitions of and , , given in Theorem 4.1.

Theorem 5.1.

Consider the discrete-time nonlinear uncertain dynamical system given by (4.1) and (4.2), where and are given by (4.4) and (4.14), respectively, is nonnegative and asymptotically stable, is nonnegative with respect to , is nonnegative with respect to , and is nonnegative with respect to and belongs to . For a given assume there exist positive vectors and such that (4.6) and (4.7) hold and the set point is asymptotically stable with constant control for all . In addition, assume that (4.2) is input-to-state stable at with viewed as the input. Then the neuroadaptive feedback control law
(5.2)
where
(5.3)
and , , ,—with update law
(5.4)
where satisfies
(5.5)
for positive definite , and are positive constants satisfying and , , —guarantees that there exists a positively invariant set such that and the solution , , of the closed-loop system given by (4.1), (4.2), (5.2), and (5.4) is ultimately bounded for all with ultimate bound , , where ,
(5.6)

, , , , and and are positive constants satisfying and . Furthermore, , , and , , for all .

Proof.

See Appendix B.

6. Conclusion

In this paper, we developed a neuroadaptive control framework for adaptive set-point regulation of discrete-time nonlinear uncertain nonnegative and compartmental systems. Using Lyapunov methods, the proposed framework was shown to guarantee ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains while additionally guaranteeing the nonnegativity of the closed-loop system states associated with the plant dynamics.

Declarations

Acknowledgments

This research was supported in part by the Air Force Office of Scientific Research under Grant no. FA9550-06-1-0240 and the National Science Foundation under Grant no. ECS-0601311.

Authors’ Affiliations

(1)
School of Aerospace Engineering, Georgia Institute of Technology
(2)
Department of Mechanical and Aerospace Engineering, University of Tennessee
(3)
Department of Mechanical and Environmental Informatics (MEI), Tokyo Institute of Technology, O'okayama

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© Wassim M. Haddad et al. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.