Stabilization for a Class of Positive Bilinear Systems

In this letter, we consider stabilizing control design for a class of positive bilinear systems. Positivity allows us to employ a max-separable function for Lyapunov analysis, enabling to estimate a region of attraction as a box, which is particularly useful for heat exchangers. We take the summation of max-separable functions with respect to the state and input as a Lyapunov candidate and design a dynamic stabilizing controller. Moreover, employing a quadratic function with respect to the input and integral action of a performance output, we propose another dynamic stabilizing controller which can additionally regulate the output.


Stabilization for a Class of Positive Bilinear Systems
Yu Kawano , Member, IEEE, and Michele Cucuzzella , Member, IEEE Abstract-In this letter, we consider stabilizing control design for a class of positive bilinear systems.Positivity allows us to employ a max-separable function for Lyapunov analysis, enabling to estimate a region of attraction as a box, which is particularly useful for heat exchangers.We take the summation of max-separable functions with respect to the state and input as a Lyapunov candidate and design a dynamic stabilizing controller.Moreover, employing a quadratic function with respect to the input and integral action of a performance output, we propose another dynamic stabilizing controller which can additionally regulate the output.

I. INTRODUCTION
A SYSTEM is called positive if its trajectory stays in the positive orthant.For instance, positivity naturally appears in chemical kinetics and population dynamics.In the linear case, exponential stability of positive systems can be verified by a max-separable Lyapunov function in a necessary and sufficient manner [1], [2], enabling scalable analysis and also to estimate a region of attraction as a box rather than an ellipse.In the nonlinear case, there are two properties corresponding to positivity: positivity and cooperativity (or more generally, monotonicity) [3], [4], where cooperativity is stronger than positivity.Cooperativity is used to proceed with max-separable type analysis; see, e.g., [5], [6], [7], [8], [9].In contrast, it is known that positivity is too weak to develop such analysis for general nonlinear systems.
Motivated by temperature control of a counter-current heat exchanger [10], we focus on stabilizing control design for Yu Kawano is with the Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan (e-mail: ykawano@hiroshima-u.ac.ip).
Michele Cucuzzella is with the Department of Electrical, Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy (e-mail: michele.cucuzzella@unipv.it).
a class of positive bilinear systems.Although there are numerous results on stabilization of bilinear systems, e.g., [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], positivity has not been incorporated except for a special case without additive terms [21], [22].It is worth emphasizing that none of [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] employs a max-separable Lyapunov function for stabilizing control design.In this letter, explicitly utilizing positivity and max-separable Lyapunov functions, we develop two Lyapunov-based stabilizing controllers for bilinear systems.For the first controller, we construct the Lyapunov candidate for the closed-loop system as the summation of a max-separable function with respect to the state and a max-separable function with respect to the input.For the second controller, instead of a max-separable function with respect to the input, we use a quadratic one.Moreover, in the quadratic case, one can further include an integral action of a performance output to achieve output regulation.Because of the construction of each Lyapunov candidate, a region of attraction with respect to the state can be estimated as a box.
Notation: The sets of real numbers and non-negative real numbers are denoted by R and R ≥0 , respectively.For matrices ) for all i = 1, . . ., n and j = 1, . . ., m.The n × n identity matrix is denoted by I n .The n×m matrix whose components are all zero is denoted by 0 n×m .The sign function is denoted by sgn(•).

II. MOTIVATING EXAMPLE
Consider a counter-current heat exchanger with three cells in two layers [10] as in Fig. 1, which can be represented as with the state x(t) = [T 1 , T 2 , T 3 , T1 , T2 , T3 ] (t) and input u(t) = q(t), where T i (t) and Ti (t) denote the temperature of the ith cell in the compartments 1 and 2, respectively, and q(t) denotes the mass flow rate in the compartment 1.The positive constants T in and Tin denote the stream temperatures at the inlet of the compartments 1 and 2, respectively, and the positive constant q denotes the mass flow rate in the compartment 2. The constants λ, c p , ρ, and ν denote the heat transfer coefficient, heat capacity, mass density of the fluid, and volume of the fluid in the heat exchanger, respectively.The control objective is to stabilize a desired equilibrium point (x * , u * ) while the state and input fulfill physical constraints, e.g., x i (t) ∈ [ Tin , T in ] ⊂ R ≥0 , i = 1, . . ., 6, and u(t) ∈ [u min , u max ] ⊂ R ≥0 for all t ≥ 0. In [19], by extending [10], a stabilizing dynamic controller is proposed based on a quadratic Lyapunov function, and a set of initial states whose corresponding trajectories satisfy physical constraints is estimated as a level set of the quadratic Lyapounov function.However, this estimation can be conservative because the level set is an ellipse contained in a box [ Tin , T in ] 6 .
As detailed below, a heat exchanger ( 1) is a positive system.Motivated by this, in this letter we develop a new control scheme for positive bilinear systems.By virtue of positivity, one can use a max-separable Lyapunov function [1], [2] for control design, which is suitable for heat exchangers since its level set is a box.

III. PROBLEM FORMULATION
Consider a bilinear system, described by where x(t) ∈ R n and u(t) := [u 1 , . . ., u m ] (t) ∈ R m denote the state and input, respectively.Motivated by the structure of the heat exchanger (1), A, B i ∈ R n×n , i = 1, . . ., m are assumed to be Metzler, i.e., their off-diagonal elements are non-negative.Also, we assume b i , G ∈ R n ≥0 , i = 1, . . ., m.Then, the bilinear system (2) is positive in the following sense.
Proposition 1: For a system (2), it follows that as long as x(t) exits.Proof: The system can also be represented as For any Thus, the statement follows from theory for positive linear systems, e.g., [23].
For stabilization of a bilinear system, it is standard in the literature to assume some open-loop stability (see, e.g., [10], [11], [13], [14], [15], [17], [19], [20]).In the case of the heat exchanger (1), A + Bu * is Hurwitz for all u * > 0 as confirmed in Section V. Thus, we impose the same assumption as [10], [11], [17], [19], [20], i.e., there exists u * ≥ 0 such that A + m i=1 B i u * i is Hurwitz. 1 Then, for fixed u * , the system (3) has the following unique equilibrium point: [23,Proposition 2], for sufficiently small λ > 0, there exist w ∈ R n such that Fixed λ > 0, solving (5) with respect to w is a linear programming problem, which can be easily solved numerically.The set of inequalities (5) ensures that is a Lyapunov function for Namely, we have where D + V x (x) denotes the upper-right Dini derivative [25] of V x (x): Denoting the set of indices the Dini derivative can be computed by This is the maximum value of the directional derivative of The function ( 6) is called a max-separable Lyapunov function [1], [2].Now, we are ready to state the main problem of this letter.Problem 1: Consider a positive bilinear system (2).Given a compact X × U ⊂ R n ≥0 × R m ≥0 , assume that there exists u * ≥ 0 such that A + m i=1 B i u * i is Hurwitz and (x * , u * ) ∈ X × U for x * satisfying (4).Then, design a controller such that the closed-loop system is asymptotically stable at (x * , u * ), and estimate a region of attraction contained in X × U .

IV. MAIN RESULTS
In this section, we design dynamic state feedback controllers based on the max-separable Lyapunov function (6) with respect to the state.With respect to the input, we first select a max-separable function and then a quadratic one.In the latter case, we further consider (robust) output regulation.

A. Max-Separable Functions With Respect to Inputs
In this subsection, we design dynamic state feedback controllers based on the following max-separable Lyapunov candidate: where V x (x) is defined by (6) and v i > 0, i = 1, . . ., m.
To describe a controller equation, we define the following set similarly to I x (x) in (8): Now, we are ready to provide the following dynamic controller: with λ u > 0 and S(x, u) := max where (•) j denotes the jth element of vector (•).
In fact, (10) is a stabilizing controller, stated below. 2  Theorem 1: For a positive bilinear system (2), let u * ≥ 0 be such that A + m i=1 B i u * i is Hurwitz, and let x * be given by (4).Then, the closed-loop system with a dynamic feedback controller (10) is globally exponentially stable at (x * , u * ).
Proof: We first consider the set (R n \ {x * }) × (R m \ {u * }).The upper-right Dini derivative of V(x, u) in ( 9), denoted by D + V(x, u), satisfies 2 Throughout this letter, we consider Filippov solutions of the corresponding differential inclusions to differential equations with sign functions.
Remark 1: According to the proof of Theorem 1, the closed-loop stability can also be shown if the control law (10a) is implemented for i ∈ I u (u), and the other inputs u j , j ∈ I u (u) are arbitrary.
A region of attraction contained in X × U can be estimated from the Lyapunov function V(x, u).Since its level set Utilizing the structure of the Lyapunov function V(x, u) in ( 9), we proceed with further analysis.Let π x ( c ) ⊂ R n denote the projection of c onto the x-space.From (9), one notices that for the same c > 0, Then, for any trajectory starting from c , the corresponding state trajectory stays in Xc .Thus, one can use the largest c max > 0 satisfying Xc max ⊂ X to estimate a region of attraction Xc max with respect to the state as a box.By a similar reasoning, the corresponding input trajectory stays in U if where Ūv is also a box.Then, we can select v in (9) to satisfy Ūv ⊂ U , and a region of attraction contained in X × U can be estimated as Xc max × Ūv .The controller (10a) has tuning parameters v and λ u .The first one can be determined to satisfy the input constraints as mentioned above, while λ u can be designed to balance the importance of input regulation and regulation of S(x, u).
When implementing the dynamic feedback controller (10), we can estimate the set of initial states such that state constraints are fulfilled and tune the controller gains such that input constraints are satisfied.If one focuses on stabilizing the equilibrium (x * , u * ) without taking these constraints into account, one can implement the following static feedback controller: where γ > 0. This is stated as follows.
Corollary 1: Under the assumptions of Theorem 1, the closed-loop system with a static feedback controller ( 14) is globally exponentially stable at x * .
Proof: The proof can be shown by utilizing V x (x) in ( 6) as a Lyapunov candidate.

B. Quadratic Functions With Respect to Inputs
In this subsection, we utilize a quadratic Lyapunov candidate with respect to the input, which allows us to additionally consider regulating a performance output y(t) = Cx(t) ∈ R p to some reference y * , where C ≥ 0. This improves robustness of the regulation property, i.e., lim t→∞ y(t) = y * against parameter uncertainties [27].
To this end, we employ for each i = 1, . . ., m the following controller: where and Note that M in ( 16) is well defined, since A + m i=1 B i u * i is Hurwitz.
One notices that the first two terms of the u i -dynamics in (15) are similar to those of (10).In fact, the first two terms are only for stabilization as stated in Corollary 2 below.The additional last term and z-dynamics are for output regulation.Based on the specific application, the importance of each term can be tuned by changing λ u > 0, q i > 0, and α i > 0. To achieve output regulation, suppose that Given A, B i , b i , u * i , i = 1, . . ., m and C, we can compute x * and M by ( 4) and ( 16).Then, ( 18) can be checked.Now, we are ready to show the closed-loop stability and output regulation as follows.
Proof: As a preliminary step, one can confirm by ( 11), (15b), and ( 16).Consider the following Lyapunov candidate that is quadratic with respect to the input: where V x (x) and V z (x, z) are defined in ( 6) and ( 17), respectively.
We consider the set According to the proof of Theorem 1 and ( 19), its upper-right Dini derivative, denoted by D + V(x, u, z), satisfies Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Similarly to the previous subsection, a region of attraction contained in X × U can be estimated from the Lyapunov function V(x, u, z) in ( 20) as a box with respect to the state.
In the previous section, we design a controller without the integral action of the output.For the comparison, we consider a special case of (15): This is also a stabilizing controller, stated below.Corollary 2: Under the assumptions of Theorem 1, a positive bilinear system (2) in closed-loop with a dynamic controller (23) is globally exponentially stable at (x * , u * ). 3 More precisely, the differential inclusion defined by the left-hand side of ( 22) contains only 0.
Proof: The statement can be show by using V x (x) + Vu (u) as a Lyapunov candidate.
It is also possible to design controllers without the integral action of the input and a static state feedback controller.Due to the space limitation, we omit to show them.
Remark 2: By virtue of the output integral action (15b), we confirm in the next section by simulation that under small parameter perturbations, the system (2) in closed-loop with a dynamic controller (15) is asymptotically stabilized at an equilibrium (x * , ũ * , z * ) of the perturbed system, satisfying y = y * ; see [27] for more details on integral action in output feedback control.
First, we apply the controller (10) with λ u = 1 × 10 −3 .As discussed immediately after Remark 1, the tuning parameter v is designed based on the set U .More precisely, we have Xc := {x ∈ R n : max i=1,...,6 {|x i − x * i |} ≤ c}, and the maximum c max > 0 satisfying Xc max ⊂ X is given by c max = 7.32.Then, for v = 0.0055, it can be verified that also the input constraints are satisfied.Secondly, to robustly regulate y = x 4 , we apply the controller (15) with λ u = 30 and α = 1.We use the same c max as above and thus, for q = 9.16 × 10 3 , also the input constraints are satisfied.
We can observe from Fig. 2(a) that when no perturbation occurs (t < 2000 s), the trajectories converge to the corresponding equilibrium values (dotted lines).Differently, when a perturbation of T in occurs (t ≥ 2000 s), only the controller ( 15) is capable to perform output regulation (see the enlargements in Fig. 2(a)).In both cases, for t ≥ 2000 s the other temperature trajectories are stable but deviate from the equilibrium point.Moreover, we note that when the controller (15) is applied, such deviations are smaller than those obtained by applying i , with an enlarged view of the output x 4 .The results on the left have been obtained by using the controller (10), while those on the right by using the controller (15).(b) Comparison between the control inputs generated by (10) and (15).the controller (10).Figure 2(b) shows the comparison between the control inputs generated by (10) and (15).

VI. CONCLUSION
In this letter, we have studied stabilization of positive bilinear systems based on Lyapunov theory.By virtue of positivity, a max-separable function is employed with respect to the state, enabling to estimate a region of attraction as a box.With respect to the input, we have used two different functions: a max-separable function and a quadratic one.Each function leads to a different dynamic stabilizing controller, and the latter achieves additionally output regulation.In theory of positive linear systems, max-separable Lyapunov functions enable sclable analysis and control design for large-scale networks.Future work includes investigating scalable control design for a network of heat exchangers.On the other hand, max-separable Lyapunov functions lead to controllers having a similar structure as sliding mode controllers.Another future work is to investigate these connections.

Manuscript received 8
September 2023; revised 28 October 2023; accepted 15 November 2023.Date of publication 21 November 2023; date of current version 11 December 2023.The work of Yu Kawano was supported in part by JST FOREST Program under Grant JPMJFR222E.The work of Michele Cucuzzella was supported by Project NODES which has received funding from the MUR-M4C2 1.5 of PNRR funded by the European Union-NextGenerationEU under Grant ECS00000036.Recommended by Senior Editor K. Savla.(Corresponding author: Yu Kawano.)

Fig. 2 .
Fig. 2. (a) Heat exchanger states x i (temperatures) and the corresponding equilibrium x *i , with an enlarged view of the output x 4 .The results on the left have been obtained by using the controller(10), while those on the right by using the controller(15).(b) Comparison between the control inputs generated by(10) and(15).
The Authors.This work is licensed under a Creative Commons Attribution 4.0 License.
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