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.
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.
is a nonnegative (resp., positive) function if 
(resp., 
), 
.
is nonnegative and 
, using linear and quadratic Lyapunov functions, respectively.
given by (2.1) where 
is nonnegative. Then 
is asymptotically stable if and only if there exist vectors 
such that 
and 
satisfy
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
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
, 
, 
, and 
are nonnegative matrices.
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.
be partitioned as 
, where 
and 
, and note 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.
, 
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 [
, where 
is an open subset of 
that contains 
. Then 
is nonnegative with respect to
, 
, if 
for all 
, and 
. 
is nonnegative if 
for all 
, and 
.
, where 
, then 
is nonnegative if and only if 
is nonnegative [
. Then 
is an invariant set with respect to (2.8) if and only if 
is nonnegative.
, 
, 
, 
, 
is continuous and satisfies 
, and 
is continuous.
and 
, 
, the solution 
, 
, to (2.9) is nonnegative.
and 
, 
.
, 
, is sufficient to guarantee the nonnegativity of the state of (2.9).
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).
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.
and 
, it follows that the solution 
, 
, to (2.10) can be equivalently characterized by the solution 
, 
, where 
, to the discrete-time nonlinear autonomous system
. Now, since 
, 
, for 
, and 
, the result is a direct consequence of Proposition 2.8.
, 
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 
.
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 
.
  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 
, 
.
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 [
and class-
functions 
and 
such that
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
, 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 
.
and class-
functions 
and 
such that (3.3) holds. Furthermore, assume that there exists a continuous function 
such that 
, 
, and
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 
.
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.
(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
, 
to (3.1) and (3.2) is ultimately bounded.
given by
, 
, 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
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.
and 
are unknown functions with 
given by
is a known nonnegative matrix and 
is an unknown nonnegative function with respect to 
and belongs to the uncertainty set 
given by
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
with 
viewed as the input, that is, there exist a class-
function 
and a class-
function 
such that
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.
, 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
, 
, 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 
, 
.
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 [
is a dense subset of the set of continuous functions on 
. Now, as is the case in the standard neuroadaptive control literature [
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.
, 
, 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
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
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
. 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 
, 
.
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
, 
, 
, and 
with 
whenever 
, 
, 
,—with update law
satisfies
, 
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
, 
, 
, 
, and 
and 
are positive constants satisfying 
and 
, respectively. Furthermore, 
and 
, 
, for all 
.
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.
denotes the 
th row of 
.
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
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 
.
replaced by 
.
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.
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 
for a discrete-time nonlinear nonnegative dynamical system
, 
, 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 (4.4) is nonnegative and asymptotically stable, and hence, without loss of generality (see [
is an asymptotically stable compartmental matrix [
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.
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
, 
, 
,—with update law
satisfies
, 
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 
,
, 
, 
, 
, and 
and 
are positive constants satisfying 
and 
. Furthermore, 
, 
, and 
, 
, for all 
.
, 
, given by (4.15), it follows from (4.1), (4.4), and (4.14) that
and 
, using (4.5)–(4.7), (4.9), and 
, it follows from (4.2) and (A.1) that
, 
, 
, 
, 
, and 
. Furthermore, since 
is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive diagonal matrix 
and a positive-definite matrix 
such that (4.18) holds.
, consider the Lyapunov-like function
. 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 closed-loop system trajectories is given by
and 
, 
, it follows that
for all 
, that is, 
for all 
and 
, where
. Thus, it follows from Theorem 3.2 that the closed-loop 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 closed-loop 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 
.
, 
, 
, 
, 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 closed-loop system trajectories is given by
and 
for 
and 
. Now, noting 
and 
, using the inequalities
for all 
, where
for all 
, 
, where (see Figure 
, 
. Since 
for all 
, it follows that, for 
, 
,
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
. Next, let 
, where 
and assume 
and 
. Now, for every 
such that 
, 
, it follows that
, 
. Now, if there exists 
such that 
, then it follows as in the earlier case shown above that 
, 
. Hence, if 
, then
, 
, replaced by 
, 
, it can be shown that 
, 
, where 
.
is the maximum value such that 
, and define
is given by (A.30). Assume that 
(see Figure 
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 input-to-state 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 [
, a class-
function 
, and a class-
function 
such that
is given by 
, it follows that the set given by
(see Remark A.1). Now, since 
and 
are positively invariant, it follows that
and since (4.2) is input-to-state stable at 
with 
viewed as the input then it follows from Proposition 3.4 that the solution 
, 
, of the closed-loop system (4.1), (4.2), (4.15), and (4.17) is ultimately bounded for all 
.
and 
, 
, for all 
note that the closed-loop system (4.1), (4.15), and (4.17), is given by
and 
are nonnegative and, since 
whenever 
, 
, 
, 
. Hence, since 
is nonnegative with respect to 
pointwise-in-time, 
is nonnegative with respect to 
, and 
, it follows from Proposition 2.9 that 
, 
, and 
, 
, for all 
.
, 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 Stone-Weierstrass 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.
, where
, 
, given by (5.2), it follows from (4.1), (4.4), and (4.14) that
and 
and using (4.6), (4.7), and 4.9), it follows from (4.2) and (B.2) that
, and 
. Furthermore, since 
is nonnegative and asymptotically stable, it follows from Theorem 2.3 that there exist a positive diagonal matrix 
and a positive-definite matrix 
such that (5.5) holds.
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 closed-loop system trajectories is given by
and for the two cases given in (B.1), the right-hand side of (B.6) gives the following:
, then 
. Now, using (A.8), (A.9), and the inequalities
, and hence, using (A.8), (A.9), (B.7), (B.9), and (B.10), it follows that
and 
, 
, 
, it follows that
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 closed-loop 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 
.
, 
, 
, 
, 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 closed-loop system trajectories is given by
and 
for 
and 
.
, of the closed-loop system (5.4), (B.3), and (B.4) is ultimately bounded for all 
given by (A.35) and 
for 
.
, is a restatement of (5.2). Now, since 
, and 
, it follows from Proposition 2.8 that 
and 
, for all 
.