# Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria

*Complex Eng Syst*2023;3:11.

## Abstract

This paper studies the issues of region stability of switched two-dimensional linear dissipative Hamiltonian systems. Such switched systems are composed of two stable subsystems with two different equilibrium points. Since the equilibrium points of two subsystems are different, and the state matrices of subsystems may not commute, it is difficult to address such switched systems. This paper considers the case that the switching path corresponding to the switched systems is a switching line passing through the equilibrium points of two different subsystems. A suitable region containing all the equilibrium points of subsystems is first determined. Based on the concept of region stability of switched systems with multiple equilibrium points, this paper proposes some sufficient conditions of region stability and asymptotically region stability for such kind of switched linear dissipative Hamiltonian systems via the maximum energy function method. The above main results obtained can be applied to some classes of electronic circuits, such as switching DC/DC converters and AC/DC converters. As an application and illustration, a switching DC circuit and two numerical examples are carried out to show the effectiveness of the region stability results obtained in this paper.

## Keywords

*,*dissipative Hamiltonian systems

*,*switched line

*,*regional stability

*,*asymptotic regional stability

*,*multiple equilibrium points

*,*maximum energy function method

## 1. INTRODUCTION

Hybrid systems are a kind of complex systems owning both continuous-time states and discrete time states. As an important class of hybrid systems, a switched system is composed of finite/infinite subsystems and a switching path/signal, which is used to select one subsystem to activate at any instant. Since 1990, it has been found that switched systems have been widely used in many physical and practical fields, such as power systems ^{[1]}, multi-agent technology ^{[2–5]}, hybrid electric vehicles ^{[6]}, traffic control systems ^{[7]}, and so on. Owing to the structural complexity and extensive application of switched systems, it is of great significance to study switched systems. In 1999, Liberzon and Morse published a review on the issues of stability analysis of switched systems ^{[8]}. The review summarized that there are three basic problems with the stability of the switched system as follows. One is the stability of switched systems under any switching paths; Another is the stability of the system under given switching paths; The third is to determine suitable switching paths such that switched systems are stable under the determined switching paths. After that, a large number of researchers have paid increasing attention to the domain of switched systems and proposed plenty of results for switched systems. Meanwhile, some important methods are developed to analyze the stability of switched systems, such as the common Lyapunov function method (CLF) ^{[9]}, the multi-Lyapunov functions method (MLF) ^{[10]}, the multi-storage functions method (MSFs) ^{[11]}, and so on.

Almost all of the above system stability analysis and comprehensive results obtained for switched systems are based on the same assumption that all subsystems have a common single equilibrium point, i.e., the origin, in their common state domain ^{[12–16]}. However, due to the complexity of the environment and system structures, for each subsystem of switched systems, there will be one or more equilibria in its state domain. Moreover, the equilibrium points of each subsystem in many switched systems may not be the same one. Since there is short of necessary stability analysis methods/techniques, it is indeed a challenge and has a practical significance to investigate switched systems with multiple equilibria (ME). Although it is more difficult to study switched systems with ME than switched systems with a common single equilibrium point, many researchers have presented some stability results in the literature. For example, in the reference ^{[17]}, several sufficient conditions of region stability, global asymptotic region stability, and region instability are proposed for switched linear time-invariant systems under arbitrary periodical/quasi-periodical switching paths with respect to a region. The corresponding region contains all the multiple equilibrium points of such kind of switched linear systems with ME. The results of the boundedness and practical stability are obtained for switched systems with ME in the literature ^{[18]} by studying the robustness to external disturbances of such switched systems.

As an effective method, the Hamiltonian function method has been widely used to study nonlinear systems. Such a method should be based on system energy control and system stability analysis. In practice, a general affine nonlinear system can be converted into a port-controlled Hamiltonian (PCH) system based on the Hamiltonian realization method ^{[19]}. The PCH systems can be accepted as a form of unified mathematical structures for various physical systems, such as electric power systems and mechanical systems. The Hamiltonian function of such a PCH system has an explicitly physical significance and has often been used as the total energy of physical systems. It is then often selected as an appropriate Lyapunov function. Based on the above advantages of Hamiltonian systems, researchers obtain lots of results of Hamiltonian systems, such as the results of trajectory tracking control ^{[20, 21]}, ^{[22]}, etc.

Recently, as an important kind of switched systems, switched Hamiltonian systems (SHSs) have also been studied. This may be because such kind of switched systems can be used to model many practical systems composed of finite difference modes. In fact, studying the SHS may induce to provide an effective method for analyzing such a class of switched systems. However, compared to the existing vast results of switched systems with subsystems not being the formation of Hamiltonian systems, there are only a few results presented for SHSs in the open literature. Except for the literature ^{[23]} studying the stability issues of switched linear Hamiltonian systems, some other studies ^{[24, 25]} analyze the issues of system stability, system stabilization, and

It should be pointed out that all the aforementioned results obtained for switched PCH systems are also based on the assumption that all subsystems must have a common single equilibrium point–the origin of the system state space. To the best knowledge of the authors, there are fewer results reported for SHSs with ME in the open literature, including the following one. The literature ^{[26]} tries to model a power system with a series of faults in the form of switched impulsive Hamiltonian systems (SIHSs) with ME and proposes some necessary and sufficient conditions of RS and ARS for the power system with respect to the region via the maximum energy function method, which was first introduced in the literature ^{[15]}. It is especially witnessed that there do not exist any results of switched linear Hamiltonian systems with ME in the literature. Therefore, studying switched linear Hamiltonian systems with multiple equilibrium points not only enriches the theoretical results of system analysis and system control of switched systems but also has important practical significance.

This paper studies the region stability issues of switched linear PCH systems with multiple equilibrium points. Due to the complexity involved in studying such a switched system composed of subsystems having multiple equilibrium points and non-commutative subsystems' state matrices, we exclusively consider three specific cases of the switched linear Hamiltonian system. One is the case that there are only two subsystems. Another is the case that each subsystem has only a unique equilibrium point, and the equilibrium points of the two subsystems are different. The third is the case that the involved switching path is the straight line passing through the equilibrium points of the two subsystems. To address these cases, we utilize the concepts of region stability and asymptotic region stability of switched systems with multiple equilibrium points. Furthermore, by means of the maximum energy method, we propose the main contributions of this paper as follows: (1) several sufficient conditions of the region stability and asymptotic region stability are given for switched linear PCH systems with respect to a region containing all multiple equilibrium points under the specific switching path; (2) an application of switching DC electric circuits and two numerical examples are carried out to illustrate the effectiveness and practicality of the theoretical results obtained in this paper. Figure 1 shows the framework of the content of this paper. Compared to the existing region stability results proposed in the literature ^{[17]}, the region stability criteria obtained in this paper have the following significant advantages: (1) they are suitable to the case that any pairs of all the state matrices of subsystems do not commute between each other. However, for the sufficient conditions of the region stability results given in the literature ^{[17]}, all the state matrices of subsystems of switched linear systems with ME are assumed to be commutative matrices; (2) dissimilar to that of region stability proposed in the literature ^{[17]}, the sufficient conditions of region stability obtained in this paper do not require any information on the dwell-times of any subsystems; (3) the sufficient conditions of region stability presented in this paper are very easy to check whether the given switched linear Hamiltonian systems with switching lines are (asymptotically) region stable or not.

The rest of this paper is organized as follows. Section 2 gives the system expression of switched linear Hamiltonian systems with multiple equilibrium subsystems, definitions, and other preliminaries, including notations. Section 3 proposes the main contributions of this paper, i.e., some region stability criteria of switched linear Hamiltonian systems with multiple equilibrium points. Section 4 illustrates numerical examples and an application for switching DC circuits to show the validity of the obtained new results, which is followed by the conclusion in Section 5.

Notation:

## 2. PRELIMINARIES

This section gives the preliminaries needed necessarily for studying switched linear PCH systems with multiple equilibrium points in the next sections. Section 2.1 introduces the switched systems model considered in this paper and some preparatory knowledge and notation. Section 2.2 introduces some relevant definitions and a proposition that will be used in the sequel.

### 2.1. System description and preliminaries

Consider a switched linear Hamiltonian system with multiple equilibrium subsystems as follows.

where

**Remark 1***Note that due to the special dissipative Hamiltonian structures of subsystems of system* (1)*, i.e., the matrices * (1)

*are all revolved around the corresponding equilibrium points of subsystems anticlockwise or clockwise.*

#### 2.1.1. Switching line

To facilitate the subsequent analysis, the switching path

Throughout this paper, the switching path

A1. Switching phenomena of system (1) only appear on a straight line that passes through the two equilibrium points of the first subsystem and the second subsystem of system (1). Such a straight line is called a *switching line* in this paper and is denoted by

A2. Any one of all the subsystems cannot be excluded from those activated subsystems as time goes to infinity.

A3. The switching times of the switching path

A4. The whole state trajectory

**Remark 2***It should be pointed out from Remark 1 that the trajectories of every subsystem of system* (1) *must pass through the switching line intercepting the equilibrium points *

**Remark 3***The explanations and motivations of the above four assumptions are addressed as follows. As is well-known, there are many event-driven practical/physical dynamical systems that can be modeled by switched systems with special switching paths/strategies. Among these switching paths/ strategies caused by event-driven, there is a kind of switching path–switching lines. This is the motivation for Assumption A1. Since we just consider the final tendency of the trajectory of system* (1) *in this paper, it is not necessary to consider those subsystems that are not activated anymore after a certain finite time. Therefore, all the subsystems of* (1) *must be often activated as time goes to infinity, i.e., Assumption A2. The aim of Assumption A3 is to exclude the Zeno phenomena–the chatting, i.e., infinite switching occurs in any finite time interval. Assumption A4 is used to avoid the jump phenomenon of the state of system* (1).

Moreover, we let

(1) If the equilibrium points

where the two parameters of

(2) If the equilibrium points

**Remark 4***Since the switching line * (2)

*passes through the equilibrium points*$$ x^{e1} $$ and $$ x^{e2} $$ of the two different subsystems and the two different points are both determined in advance, both the slope $$ k $$ and the intercept $$ b $$ in(3)

*of the straight line*$$ l_1 $$ can be easily obtained via the general method of analytic geometry.

Based on the above, we know that the trajectory

#### 2.1.2. The Hamiltonian functions

In this paper, the Hamiltonian functions of the two subsystems of system (1) are assumed to be the following quadratic forms:

where

Since the switching states

where

where

For the case that

where

### 2.2. Some definitions and propositions

This subsection refers to some definitions and gives a proposition that is needed to analyze the region stability of system (1) in the next section below.

**Definition 1** (*The maximum energy function) ^{[15]}. The following function * (1)

**Definition 2***(The maximum switching energy sequence) ^{[15]}. The following sequence is known as the maximum switching energy sequence of the switching path *

**Proposition 1***Consider system (1). The Hamiltonian functions *

*where *

**Proof:** Letting

We obtain from the Hamiltonian function

Since every

where

such that the following two hold.

(A.) By means of the orthogonal matrix

where

(B.) The Hamiltonian function

where

It can be obtained from (13), (19), (16), and (15) that the square norm of the vector

We know from (17)-(20) that the following two hold.

and

where

It follows from (14), (18), (21), and (22) that

Therefore, we know from (17), (20), and (23) that for any

which is exact (12). The proof of Proposition 1 is thus completed.

Since every subsystem of system (1) has a unique equilibrium point

or

Similar to that of region stability defined in the reference ^{[17]}, based on the region

**Definition 3*** ^{[17]} Consider system* (1)

*with the region*$$ {\Omega} $$ defined in(24)

*or*

*defined in*(25)

*under a special kind of switching path*$$ \sigma(t) $$ , i.e., the switching line $$ l_1 $$ in(2).

*System*(1)

*under the switching line*$$ l_1 $$ in(2)

*is said to be*

● *Region stable with respect to the region * (24)

*or the region*$$ \Psi $$ in(25)

*, if for*$$ \forall\varepsilon>0, \;\exists\delta:=\delta(\varepsilon)>0 $$ such that the following formula holds for any $$ x_{0} $$ ,

*or*

● *Asymptotically region stable with respect to the region * (24)

*or the region*$$ \Psi $$ in(25)

*, if both*(26)

*or*(27)

*and the following limit hold*

## 3. REGION STABILITY ANALYSIS

This section will study the stability issue of switched two-dimensional linear Hamiltonian systems with ME. Based on the concept of region stability defined in Section 2, we propose several sufficient conditions of region stability and asymptotic region stability for system (1), respectively.

### 3.1. Some lemmas

This subsection introduces some lemmas that will be used in the next subsection. Firstly, it can be obtained from Proposition 1 that the following result holds.

**Lemma 1***Consider system* (1) *under the switching line* (2). *The Hamiltonian functions *

*where * (24)

*and*(10),

*respectively.*

**Proof:** It is easy to see from the maximum energy function defined in (10) of Definition 1 and the equation (12) in Proposition 1 and Condition (24) that Lemma 1 holds true.

**Lemma 2***Consider system* (1) *under the switching line* (2). *If *

*where *.

**Proof:** Since for

where

It follows from the fact that the trajectory

One obtains from (32) that

Since the first element

It is then obtained from (31) and (34) that for any

One obtains from (31) and (35) that for all

From (35) and (36), it follows that

which is (30). Thus, Lemma 2 holds true.

**Lemma 3**^{[27]} Let

*and*

*If the two infinite sequences *

**Lemma 4*** ^{[26]} System* (1)

*under the switching line*$$ l_1 $$ in(2)

*is region stable with respect to the region*$$ \Omega $$ in (24) if and only if for any $$ x_{0}\in\mathbb{D} $$ , the following holds:

*where the *

**Remark 5***Note that the proofs of Lemmas 1-4 are just related to the compact property of the regions of * (24)

*and*$$ \Psi $$ in(25)

*containing all the equilibrium points of the subsystems. Therefore, if the region*$$ \Omega $$ in(24)

*is replaced by the region*$$ \Psi $$ in(25)

*, then all Lemmas 1-4 hold too.*

### 3.2. Regional stability results

Based on Definition 3, we obtain from Proposition 1, Lemmas 1-4 that the two main results of this paper are proposed in series as follows.

For the horizontal and vertical ordinates of the equilibrium points

**Theorem 1***Consider system* (1) *with the compact region * (24)

*and the switching line*$$ l_1 $$ in(2).

*For the case that*$$ x_{1}^{e1}\neq x_{1}^{e2} $$ , system(1)

*under the switching line*$$ l_1 $$ in(2)

*is*

*(i) region stable with respect to the region * (24)

*, if*$$ J_1\not= 0 $$ , $$ J_2\not=0 $$ , $$ R_1{\geqslant} 0 $$ , $$ R_2{\geqslant} 0 $$ , and

*where * (6)

*over the interval*$$ (x_{1}^{e1}, \, x_{1}^{e2}) $$ in the horizontal axis, i.e.,

*where * (7).

*(ii) asymptotic region stable with respect to the region * (41)

*are all satisfied.*

**Proof:** Without loss of generality, we just show that the two Statements

For any trajectory

(1) We show the conclusion of Theorem 1 holds for the case that

Firstly, we will show that the Statement

It follows from (43) that

which can also be expressed as

Since

It can be obtained from (30) in Lemma 2, (45), and (46) that the two sequences

Based on the above analysis, one obtains from (10) in Definition 1 and the fact that the trajectory

It is easy to see from (47) that

From which and (41), we know that

It follows from (49) that (40) in Lemma 4 is satisfied for system (1). Then, by Lemma 4, we know that system (1) under the switching line

Secondly, we will show that Statement

From which, it can be known that the discriminant of the roots of the equation (50) is as follows.

which means that two curves

It follows from (51) that the equation (50) has the following two solutions:

and

where

Next, we will find the intersection points of the switching line

It can be obtained from (54) that the discriminant of the roots of the equation (54) is as follows.

It then follows from (55) that the solutions of the equation (54) are as follows.

The two intersection points can be denoted by

Similarly, solving

The two intersections are then denoted by

One obtains from (56) and (57) that the switching line

In the following, we consider the minimum value of the maximum energy function

(C1) As

From (59), one obtains that the

Similarly, we know from (59) that the

It is obvious from (50) and (52) that

From (59) and (60), it shows that the minimum value of

where

(C2) As

It can be obtained from (62) that the

Similarly, we know that the

Therefore, it can be obtained from (59) and (60) that the minimum value of the maximum energy function

Then, one obtains from (61) and (63) that the minimum value of the maximum energy function

It follows from (41) and the condition that

and

where

From (59), (60), (62), (65), and (66), one obtains that

where

It is obvious from (64) and (67) that for any switching states

which implies that (40) in Lemma 4 is satisfied. By Lemma 4, we know that for Case (b), system (1) under the switching line

Thirdly, we show that Statement

Based on the above statements, we know that there is a time interval

It is known from (69) that Statement

Finally, Statement

(S1) As

The subsystem

We know from (71) that

From Lemma 2, the condition of

On the other hand, one obtains from Lemma 2 and (72) that

Then, as

where

By the squeeze theorem, one obtains from (73) and (74) that

where

As

It is obvious from (72) and (76) and the squeeze theorem that

where

Since

(S2) As

where the two functions

Similar to that proof of Situation (S1), one obtains from Lemma 2 and the condition of

As for Case (b), it is easy to see from the conditions of Statement

Finally, we show that Statement

Based on the above, by Definition 3, we obtain from the condition of

(2) We show the conclusion of Theorem 1 also holds for the case that

In this case, letting

Solving the above equation (80), we obtain its solution denoted by

It can be obtained from (81) that

Thus, the proof of Theorem 1 is finished.

For the case that horizontal ordinates of the equilibrium points

**Theorem 2***Consider system* (1) *with the compact region * (25)

*and the switching line*$$ l_1 $$ in(2).

*For the case that*$$ x_{2}^{e1}\neq x_{2}^{e2} $$ , system(1)

*under the switching line*$$ l_1 $$ in(2)

*is*

(*i*) *region stable with respect to the region * (25)

*, if*$$ J_1\not= 0 $$ , $$ J_2\not=0 $$ , $$ R_1{\geqslant} 0 $$ , $$ R_2{\geqslant} 0 $$ , and

*where * (8)

*over the interval*$$ (x_{2}^{e1}, \, x_{2}^{e2}) $$ in the vertical axis, i.e.,

*where * (9).

(*ii*) *asymptotic region stable with respect to the region * (82)

*are all satisfied*.

**Proof:** Similar to that proof of Theorem 1, we show from

## 4. AN APPLICATION AND TWO NUMERICAL EXAMPLES

In this section, the main results obtained in Section 3 are applied to a switching electric circuit in Subsection 4.1. Two numerical examples are carried out to verify the effectiveness and practicability of Theorems 1 and 2, respectively, proposed in Section 3.

### 4.1. An application to a switching DC electric circuit

Consider an ideal switching DC electric circuit, as shown in Figure 2. In which

To do that, we let

where the corresponding matrices of the first and second subsystems are as follows.

The Hamiltonian functions of the two subsystems are, respectively, listed as follows.

in which the matrices

The equilibrium points of the first subsystem and the second subsystem are

which is a special switching path

One obtains from (85) and (87) that

and

From (89) and (90), it can be seen that although the two subsystems of system (84) are two essential linear systems, their state matrices ^{[17]}. Moreover, there are not any other stability criteria reported in the open literature that can be used to check the stability of system (84). However, by the main results of this paper, we can check the stability of system (84) as follows.

It can be obtained from (87), (7), (42), and the equilibrium points

and

It is obvious from (91) and (92) that (41) of Theorem 1 is satisfied. One knows from (85) that the following hold:

To show the above conclusion, we will simulate system (84) as follows. An initial state is chosen as

Figure 3. The response of the switching path

Figure 4. The trajectory of system (84) under the switching path

### 4.2. Two numerical examples

**Example 1***Consider a switched linear Hamiltonian system with two subsystems as follows.*

*governed by a switching path *

*In system (94), the Hamiltonian functions of the two subsystems are listed as*

*where the matrices *

*The corresponding matrices of the first and second subsystems are as follows.*

We obtain from (97) and (98) that

and

From (99) and (100), it is obvious that although the two subsystems of system (94) are two essential linear systems, their state matrices ^{[17]}. Moreover, there is not any stability criteria reported in the open literature. However, we can check the stability of system (94) as follows.

According to (7), (97), and (41), we obtain that

and

It is obvious from (101) and (102) that (41) of Theorem 1 is satisfied.

It is obvious from (98), (101), and (102) that

To show the above conclusion by simulations, we choose the following two initial states:

Figure 6. The trajectory of system (94) under the switching path

Figure 7. The trajectory of system (94) under the switching path

Figure 8. The trajectory of system (94) under the switching path

Figure 9. The trajectory of system (94) under the switching path

Figure 10. The response of the switching path

Figure 11. The response of the switching path

**Example 2***Consider a switched linear Hamiltonian system with two subsystems as follows.*

*governed by a switching path *

*In system (104), the Hamiltonian functions of the two subsystems are listed as*

*where the matrices *

*The corresponding matrices of the first and second subsystems are as follows.*

One obtains from (104), (107), and (108) that the following two:

and

From (109) and (110), it is obvious that although the two subsystems of system (104) are also two linear systems, their state matrices ^{[17]}. Moreover, there is not any stability criteria reported in the open literature. However, we can check the stability of system (104) as follows.

According to (7), (107), and (41), we obtain that

and

It is obvious from (108), (111), and (112) that

To show the above conclusion by simulations, we choose the following two initial states:

Figure 12. The trajectory of system (104) under the switching path

Figure 13. The trajectory of system (104) under the switching path

Figure 14. The trajectory of system (104) under the switching path

Figure 15. The trajectory of system (104) under the switching path

Figure 16. The response of the switching path

## 5. CONCLUSIONS

We have studied the region stability of two-dimensional switched linear Hamiltonian systems with multiple equilibrium points. For the case that there are two subsystems and the switching path is a switching line, by the maximum energy function method, we have proposed some sufficient conditions of region stability and asymptotic region stability of such kind of switched systems. The stability criteria given are easily-test. An application of switching DC electric circuits and two numerical examples have illustrated the effectiveness and practicality of the two theorems obtained in this paper. The limitations of the stability results obtained in this paper are the following two: (1) Switched linear Hamiltonian systems with multiple equilibrium points are two-dimensional. (2) The special switching paths of switching lines. To remove the above limitations, the investigation of region stability of high-dimensional SHSs with ME under arbitrary switching paths will be our next work.

## DECLARATIONS

### Authors' contributions

Made substantial contributions to supervision, writing, review, editing, and methodology: Zhu L

Performed writing-original draft, software, validation, and visualization: Liu T

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work was supported by Shandong Natural Science Foundation of China under Grant (No. ZR2021MF012) and Cultivating Foundation of Qilu University of Technology under Grant (No. 2022PYI010).

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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Liu T, Zhu L. Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria. *Complex Eng Syst* 2023;3:11. http://dx.doi.org/10.20517/ces.2023.13

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Liu T, Zhu L. Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria. *Complex Engineering Systems*. 2023; 3(3): 11. http://dx.doi.org/10.20517/ces.2023.13

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Tongsu Liu, Liying Zhu. 2023. "Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria" *Complex Engineering Systems*. 3, no.3: 11. http://dx.doi.org/10.20517/ces.2023.13

**ACS Style**

Liu, T.; Zhu L. Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria. *Complex. Eng. Syst.* **2023**, *3*, 11. http://dx.doi.org/10.20517/ces.2023.13

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