 Research Article
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Neural Network Adaptive Control for DiscreteTime Nonlinear Nonnegative Dynamical Systems
Advances in Difference Equations volumeÂ 2008, ArticleÂ number:Â 868425 (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 setpoint regulation of discretetime nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunovbased 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 [3â€“8]. A subclass of nonnegative dynamical systems are compartmental systems [8â€“18]. 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 discretetime nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunovbased 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 Lyapunovlike 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 setpoint 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 discretetime 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 nonnegativedefinite (resp., positivedefinite) 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 discretetime linear nonnegative dynamical system
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
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 positivedefinite matrix such that
Next, consider the controlled discretetime linear dynamical system
where
is nonnegative and is nonnegative such that rank . The following theorem shows that discretetime 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
where , , , and are nonnegative matrices.
Theorem 2.4.
Consider the discretetime 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
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 discretetime nonlinear dynamical system
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 discretetime nonlinear dynamical systems of the form
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 discretetime 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 discretetime 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 timevarying extension to Proposition 2.8 needed for the main theorems of this paper. Specifically, we consider the timevarying system
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 timevarying discretetime 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 timevarying discretetime system (2.10) as an autonomous discretetime 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 discretetime nonlinear autonomous system
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 Lyapunovlike theorems for partial boundedness and partial ultimate boundedness of discretetime 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 discretetime nonlinear autonomous interconnected dynamical system
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]).

(i)
The discretetime nonlinear dynamical system (3.1) and (3.2) is bounded with respect touniformly in if there exists such that, for every , there exists such that implies for all . The discretetime nonlinear dynamical system (3.1) and (3.2) is globally bounded with respect touniformly in if, for every , there exists such that implies for all .

(ii)
The discretetime nonlinear dynamical system (3.1) and (3.2) is ultimately bounded with respect toâ€‰â€‰â€‰â€‰uniformly inâ€‰â€‰â€‰â€‰with ultimate boundâ€‰â€‰â€‰â€‰if there existsâ€‰â€‰ such that, for every , there exists such that implies , . The discretetime nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect toâ€‰â€‰â€‰â€‰uniformly inâ€‰â€‰â€‰â€‰with ultimate boundâ€‰â€‰ if, for every , there exists such that implies , .
Note that if a discretetime 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 discretetime 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 Lyapunovlike theorems for boundedness and ultimate boundedness for discretetime 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 discretetime nonlinear dynamical system (3.1) and (3.2). Assume that there exist a continuous function and class functions and such that
where is such that . Furthermore, assume that exists. Then the discretetime nonlinear dynamical system (3.1) and (3.2) is bounded with respect to â€‰â€‰uniformly inâ€‰â€‰. Furthermore, for every , implies that , , where
, and . If, in addition, and is a class function, then the discretetime 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 discretetime 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
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 inputtostate stability given in [21].
Proposition 3.4.
Consider the discretetime nonlinear interconnected dynamical system (3.1) and (3.2). If (3.2) is inputtostate 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 inputtostate stable with viewed as the input, it follows that is finite, and hence, there exist a class function and a class function such that
which proves that the solution , to (3.1) and (3.2) is ultimately bounded.
4. Neuroadaptive Control for DiscreteTime Nonlinear Nonnegative Uncertain Systems
In this section, we consider the problem of characterizing neuroadaptive feedback control laws for discretetime nonlinear nonnegative and compartmental uncertain dynamical systems to achieve setpoint regulation in the nonnegative orthant. Specifically, consider the controlled discretetime nonlinear uncertain dynamical system given by
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
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
where is a known nonnegative matrix and is an unknown nonnegative function with respect to and belongs to the uncertainty set given by
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
In addition, we assume that (4.2) is inputtostate stable at with viewed as the input, that is, there exist a class function and a class function such that
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
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 StoneWeierstrass 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 closedloop system remains in the nonnegative orthant of the state space for nonnegative initial conditions.
To ensure nonnegativity of the closedloop 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
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
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 inputtostate 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
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 discretetime 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 inputtostate 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
where
, , , and with whenever , , ,â€”with update law
where satisfies
for positive definite , and are positive constants satisfying and , , and â€”guarantees that there exists a positively invariant set such that , where , and the solution , , of the closedloop system given by (4.1), (4.2), (4.15), and (4.17) is ultimately bounded for all with ultimate bound , , where
, , , , 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 closedloop plant states remain nonnegative.
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 discretetime 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 inputtostate 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 , , ,â€”with update law
where satisfies (4.18), and are positive constants satisfying and , , â€”guarantees that there exists a positively invariant set such that , where , and the solution , , of the closedloop 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
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 closedloop 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.
5. Neuroadaptive Control for DiscreteTime 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 discretetime nonnegative systems with nonnegative control inputs. In general, unlike linear nonnegative systems with asymptotically stable plant dynamics, a given set point for a discretetime nonlinear nonnegative dynamical system
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 discretetime 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 inputtostate stable at with viewed as the input. Then the neuroadaptive feedback control law
where
and , , ,â€”with update law
where satisfies
for positive definite , and are positive constants satisfying and , , â€”guarantees that there exists a positively invariant set such that and the solution , , of the closedloop system given by (4.1), (4.2), (5.2), and (5.4) is ultimately bounded for all with ultimate bound , , where ,
, , , , 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 setpoint regulation of discretetime 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 closedloop system states associated with the plant dynamics.
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Acknowledgments
This research was supported in part by the Air Force Office of Scientific Research under Grant no. FA95500610240 and the National Science Foundation under Grant no. ECS0601311.
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Appendices
In this appendix, we prove Theorem 4.1. First, note that with , , given by (4.15), it follows from (4.1), (4.4), and (4.14) thatNow, defining and , using (4.5)â€“(4.7), (4.9), and , it follows from (4.2) and (A.1) that where , , , , , and . Furthermore, since is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive matrix and a positivedefinite matrix such that (4.18) holds.Next, to show that the closedloop system given by (4.17), (A.2), and (A.3) is ultimately bounded with respect to , consider the Lyapunovlike functionwhere . Note that (A.4) satisfies (3.3) with , , , where . Furthermore, is a class function. Now, using (4.17) and (A.2), it follows that the difference of along the closedloop system trajectories is given byNext, using it follows thatFurthermore, note that since, by assumption, and , , it follows thatHence,Now, forit follows that for all , that is, for all and , whereFurthermore, it follows from (A.12) thatHence, it follows from (A.4) and (A.15) thatwhere . Thus, it follows from Theorem 3.2 that the closedloop system given by (4.17), (A.2), and (A.3) is globally bounded with respect to uniformly in , and for every , , , where, and . Furthermore, to show that , , suppose there exists such that for all . In this case, , , which implies , . Alternatively, suppose there does not exist such that for all . In this case, there exists an infinite set . Now, with (A.13) satisfied, it follows that for all , that is, for all and , where is given by (A.14). Furthermore, note that , , and (A.16) holds. Hence, it follows from Theorem 3.3 that the closedloop system given by (4.17), (A.2), and (A.3) is globally ultimately bounded with respect to uniformly in with ultimate bound given by , where .Next, to show ultimate boundedness of the error dynamics, consider the Lyapunovlike functionNote that (A.18) satisfieswith , , , , and , where . Furthermore, is a class function. Now, using (4.18), (A.10), and the definition of , it follows that the difference of along the closedloop system trajectories is given bywhere in (A.20) we used and for and . Now, noting and , using the inequalitiesand rearranging terms in (A.20) yieldsNow, forit follows that for all , whereor, equivalently, for all , , where (see Figure ) Next, we show that , . Since for all , it follows that, for , ,Now, let and assume . If , , then it follows that , . Alternatively, if there exists such that , then, since , it follows that there exists , such that and , where . Hence, it follows thatwhich implies that . Next, let , where and assume and . Now, for every such that , , it follows thatwhich implies that , . Now, if there exists such that , then it follows as in the earlier case shown above that , . Hence, if , thenFinally, repeating the above arguments with , , replaced by , , it can be shown that , , where .Next, definewhere is the maximum value such that , and definewhere is given by (A.30). Assume that (see Figure ) (this assumption is standard in the neural network literature and ensures that in the error space there exists at least one Lyapunov level set . In the case where the neural network approximation holds in , this assumption is automatically satisfied. See Remark A.1 for further details). Now, for all , . Alternatively, for all , . Hence, it follows that is positively invariant. In addition, since (A.3) is inputtostate stable with viewed as the input, it follows from Proposition 3.4 that the solution , , to (A.3) is ultimately bounded. Furthermore, it follows from [, Theorem 1] that there exist a continuous, radially unbounded, positivedefinite function , a class function , and a class function such thatSince the upper bound for is given by , it follows that the set given byis also positively invariant as long as (see Remark A.1). Now, since and are positively invariant, it follows thatis also positively invariant. In addition, since (4.1), (4.2), (4.15), and (4.17) are ultimately bounded with respect to and since (4.2) is inputtostate stable at with viewed as the input then it follows from Proposition 3.4 that the solution , , of the closedloop system (4.1), (4.2), (4.15), and (4.17) is ultimately bounded for all .Finally, to show that and , , for all note that the closedloop system (4.1), (4.15), and (4.17), is given bywhereNote that and are nonnegative and, since whenever , , , . Hence, since is nonnegative with respect to pointwiseintime, is nonnegative with respect to , and , it follows from Proposition 2.9 that , , and , , for all .Remark A.1.In the case where the neural network approximation holds in , the assumptions and invoked in the proof of Theorem 4.1 are automatically satisfied. Furthermore, in this case the control law (4.15) ensures global ultimate boundedness of the error signals. However, the existence of a global neural network approximator for an uncertain nonlinear map cannot in general be established. Hence, as is common in the neural network literature, for a given arbitrarily large compact set , we assume that there exists an approximator for the unknown nonlinear map up to a desired accuracy. Furthermore, we assume that in the error space there exists at least one Lyapunov level set such that . In the case where is continuous on , it follows from the StoneWeierstrass theorem that can be approximated over an arbitrarily large compact set . In this case, our neuroadaptive controller guarantees semiglobal ultimate boundedness. An identical assumption is made in the proof of Theorem 5.1. In this appendix, we prove Theorem 5.1. First, define , whereNext, note that with , , given by (5.2), it follows from (4.1), (4.4), and (4.14) thatNow, defining and and using (4.6), (4.7), and 4.9), it follows from (4.2) and (B.2) that where , and . Furthermore, since is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive matrix and a positivedefinite matrix such that (5.5) holds.Next, to show ultimate boundedness of the closedloop system (5.4), (B.3), and (B.4), consider the Lyapunovlike functionwhere and with . Note that (B.5) satisfies (3.3) with , , , where . Furthermore, is a class function. Now, using (5.4) and (B.3), it follows that the difference of along the closedloop system trajectories is given byNow, for each and for the two cases given in (B.1), the righthand side of (B.6) gives the following:Hence, it follows from (B.6) that in either caseFurthermore, note that since, by assumption, and , , , it follows thatHence,Now, it follows using similar arguments as in the proof of Theorem 4.1 that the closedloop system (5.4), (B.3), and (B.4) is globally bounded with respect to uniformly in . If there does not exist such that for all , it follows using similar arguments as in the proof of Theorem 4.1 that the closedloop system (5.4), (B.3), and (B.4) is globally ultimately bounded with respect to uniformly in with ultimate bound given by , where . Alternatively, if there exists such that for all , then for all .Next, to show ultimate boundedness of the error dynamics, consider the Lyapunovlike functionNote that (B.16) satisfies (A.19) with , , , , and , where Furthermore, is a class function. Now, using (5.5), (B.13), and the definition of , it follows that the forward difference of along the closedloop system trajectories is given bywhere once again in (B.17) we used and for and .Next, using (A.21) and (B.17) yieldsNow, using similar arguments as in the proof of Theorem 4.1 it follows that the solution , of the closedloop system (5.4), (B.3), and (B.4) is ultimately bounded for all given by (A.35) and for .Finally, , is a restatement of (5.2). Now, since , and , it follows from Proposition 2.8 that and , for all .
A. Proof of Theorem 4.1
In this appendix, we prove Theorem 4.1. First, note that with , , given by (4.15), it follows from (4.1), (4.4), and (4.14) that
Now, defining and , using (4.5)â€“(4.7), (4.9), and , it follows from (4.2) and (A.1) that
where , , , , , and . Furthermore, since is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive diagonal matrix and a positivedefinite matrix such that (4.18) holds.
Next, to show that the closedloop system given by (4.17), (A.2), and (A.3) is ultimately bounded with respect to , consider the Lyapunovlike function
where . Note that (A.4) satisfies (3.3) with , , , where . Furthermore, is a class function. Now, using (4.17) and (A.2), it follows that the difference of along the closedloop system trajectories is given by
Next, using
it follows that
Furthermore, note that since, by assumption, and , , it follows that
Hence,
Now, for
it follows that for all , that is, for all and , where
Furthermore, it follows from (A.12) that
Hence, it follows from (A.4) and (A.15) that
where . Thus, it follows from Theorem 3.2 that the closedloop system given by (4.17), (A.2), and (A.3) is globally bounded with respect to uniformly in , and for every , , , where
, and . Furthermore, to show that , , suppose there exists such that for all . In this case, , , which implies , . Alternatively, suppose there does not exist such that for all . In this case, there exists an infinite set . Now, with (A.13) satisfied, it follows that for all , that is, for all and , where is given by (A.14). Furthermore, note that , , and (A.16) holds. Hence, it follows from Theorem 3.3 that the closedloop system given by (4.17), (A.2), and (A.3) is globally ultimately bounded with respect to uniformly in with ultimate bound given by , where .
Next, to show ultimate boundedness of the error dynamics, consider the Lyapunovlike function
Note that (A.18) satisfies
with , , , , and , where . Furthermore, is a class function. Now, using (4.18), (A.10), and the definition of , it follows that the difference of along the closedloop system trajectories is given by
where in (A.20) we used and for and . Now, noting and , using the inequalities
and rearranging terms in (A.20) yields
Now, for
it follows that for all , where
or, equivalently, for all , , where (see Figure 3)
Next, we show that , . Since for all , it follows that, for , ,
Now, let and assume . If , , then it follows that , . Alternatively, if there exists such that , then, since , it follows that there exists , such that and , where . Hence, it follows that
which implies that . Next, let , where and assume and . Now, for every such that , , it follows that
which implies that , . Now, if there exists such that , then it follows as in the earlier case shown above that , . Hence, if , then
Figure 3
Visualization of sets used in the proof of Theorem 4. 1.
Next, define
where is the maximum value such that , and define
where is given by (A.30). Assume that (see Figure 3) (this assumption is standard in the neural network literature and ensures that in the error space there exists at least one Lyapunov level set . In the case where the neural network approximation holds in , this assumption is automatically satisfied. See Remark A.1 for further details). Now, for all , . Alternatively, for all , . Hence, it follows that is positively invariant. In addition, since (A.3) is inputtostate stable with viewed as the input, it follows from Proposition 3.4 that the solution , , to (A.3) is ultimately bounded. Furthermore, it follows from [21, Theorem 1] that there exist a continuous, radially unbounded, positivedefinite function , a class function , and a class function such that
Since the upper bound for is given by , it follows that the set given by
is also positively invariant as long as (see Remark A.1). Now, since and are positively invariant, it follows that
is also positively invariant. In addition, since (4.1), (4.2), (4.15), and (4.17) are ultimately bounded with respect to and since (4.2) is inputtostate stable at with viewed as the input then it follows from Proposition 3.4 that the solution , , of the closedloop system (4.1), (4.2), (4.15), and (4.17) is ultimately bounded for all .
Finally, to show that and , , for all note that the closedloop system (4.1), (4.15), and (4.17), is given by
where
Note that and are nonnegative and, since whenever , , , . Hence, since is nonnegative with respect to pointwiseintime, is nonnegative with respect to , and , it follows from Proposition 2.9 that , , and , , for all .
Remark A.1.
In the case where the neural network approximation holds in , the assumptions and invoked in the proof of Theorem 4.1 are automatically satisfied. Furthermore, in this case the control law (4.15) ensures global ultimate boundedness of the error signals. However, the existence of a global neural network approximator for an uncertain nonlinear map cannot in general be established. Hence, as is common in the neural network literature, for a given arbitrarily large compact set , we assume that there exists an approximator for the unknown nonlinear map up to a desired accuracy. Furthermore, we assume that in the error space there exists at least one Lyapunov level set such that . In the case where is continuous on , it follows from the StoneWeierstrass theorem that can be approximated over an arbitrarily large compact set . In this case, our neuroadaptive controller guarantees semiglobal ultimate boundedness. An identical assumption is made in the proof of Theorem 5.1.
B. Proof of Theorem 5.1
In this appendix, we prove Theorem 5.1. First, define , where
Next, note that with , , given by (5.2), it follows from (4.1), (4.4), and (4.14) that
Now, defining and and using (4.6), (4.7), and 4.9), it follows from (4.2) and (B.2) that
where , and . Furthermore, since is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive diagonal matrix and a positivedefinite matrix such that (5.5) holds.
Next, to show ultimate boundedness of the closedloop system (5.4), (B.3), and (B.4), consider the Lyapunovlike function
where and with . Note that (B.5) satisfies (3.3) with , , , where . Furthermore, is a class function. Now, using (5.4) and (B.3), it follows that the difference of along the closedloop system trajectories is given by
Now, for each and for the two cases given in (B.1), the righthand side of (B.6) gives the following:

(1)
if , then . Now, using (A.8), (A.9), and the inequalities
(B.7)(B.8)(B.9)(B.10)it follows that
(B.11) 
(2)
otherwise, , and hence, using (A.8), (A.9), (B.7), (B.9), and (B.10), it follows that
(B.12)
Hence, it follows from (B.6) that in either case
Furthermore, note that since, by assumption, and , , , it follows that
Hence,
Now, it follows using similar arguments as in the proof of Theorem 4.1 that the closedloop system (5.4), (B.3), and (B.4) is globally bounded with respect to uniformly in . If there does not exist such that for all , it follows using similar arguments as in the proof of Theorem 4.1 that the closedloop system (5.4), (B.3), and (B.4) is globally ultimately bounded with respect to uniformly in with ultimate bound given by , where . Alternatively, if there exists such that for all , then for all .
Next, to show ultimate boundedness of the error dynamics, consider the Lyapunovlike function
Note that (B.16) satisfies (A.19) with , , , , and , where Furthermore, is a class function. Now, using (5.5), (B.13), and the definition of , it follows that the forward difference of along the closedloop system trajectories is given by
where once again in (B.17) we used and for and .
Next, using (A.21) and (B.17) yields
Now, using similar arguments as in the proof of Theorem 4.1 it follows that the solution , of the closedloop system (5.4), (B.3), and (B.4) is ultimately bounded for all given by (A.35) and for .
Finally, , is a restatement of (5.2). Now, since , and , it follows from Proposition 2.8 that and , for all .
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Haddad, W.M., Chellaboina, V., Hui, Q. et al. Neural Network Adaptive Control for DiscreteTime Nonlinear Nonnegative Dynamical Systems. Adv Differ Equ 2008, 868425 (2008). https://doi.org/10.1155/2008/868425
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DOI: https://doi.org/10.1155/2008/868425
Keywords
 Linear Dynamical System
 Compartmental System
 Nonnegative Orthant
 Ultimate Boundedness
 Nonnegative Vector