Stability analysis method of discrete time positive linear system based on ESPPD switching signal

Abstract

The invention discloses a stability analysis method of a discrete time positive linear system based on an ESPPD switching signal. The method comprises the following steps: constructing a state space model of the discrete time positive linear system; designing an extended switching path pair dependent switching signal; designing a stability condition, a positive stabilization condition and a positive observation condition of a discrete time positive linear system based on the extended switching path pair dependent switching signal; and verifying the stability of the discrete time positive linear system. The invention provides a novel ESPPD switching signal which comprises four dwell time switching signals and some mixed switching signals, various types of switching signals can be uniformly processed, a non-conservative stability condition and a non-conservative positive stabilization condition are provided, the conditions are presented in a linear programming form, the calculation cost is reduced, and the calculation efficiency is improved. And the applicability and the accuracy of the result are improved.

Description

Technical Field

[0001] The invention relates to an automatic control technology, in particular to a stability analysis method of a discrete time positive linear system based on an ESPPD switching signal. Background Art

[0002] Switched positive systems consist of multiple positive subsystems and switching rules, and are widely used in real life, such as networks using TCP and formation flying. Due to its wide application value, the analysis and synthesis of switched positive systems have attracted more and more attention in recent years. Early research results have explored the stability of switched positive linear systems under arbitrary switching. By using the linear co-positive Lyapunov function method and linear programming (LP) method, necessary and sufficient conditions for the stability analysis of continuous-time switched positive linear systems (CSPLSs) and discrete-time switched positive linear systems (DSPLSs) under arbitrary switching have been obtained.

[0003] Subsequently, the researchers explored the stability of CSPLSs and DSPLSs under dwell-time switching signals. Dwell-time switching signals can be divided into four types: constant dwell time, minimum dwell time, maximum dwell time, and range dwell time. It was found that the stability conditions under dwell-time signals are non-convex in terms of the system matrix, and these results cannot be directly generalized to stable controller design.

[0004] Further research involves the robust stability of uncertain CSPLSs and DSPLSs with minimum dwell time signals, and the stability and L1-gain analysis of CSPLSs and DSPLSs with range dwell time switching signals. Although some necessary and sufficient stability conditions for general switching systems under dwell time switching signals have been found, these results cannot be directly applied to the stability and stabilization problems of switched positive systems. Therefore, it is necessary to further explore the necessary and sufficient conditions for the stability and stabilization of switched positive systems under dwell time signals. Summary of the invention

[0005] Purpose of the invention: In view of the above problems, the purpose of the present invention is to provide a stability analysis method for a discrete-time positive linear system based on an ESPPD switching signal, so as to provide a more extensive and non-conservative stability condition.

[0006] Technical solution: A stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal of the present invention comprises:

[0007] Construct state-space models of discrete-time positive linear systems;

[0008] The extended switching path pair dependent switching signal is designed based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time;

[0009] Based on the extended switching path pair dependent switching signal, stability conditions, positive stabilization conditions and positive observation conditions for discrete-time positive linear systems are designed;

[0010] Verify the stability of discrete-time positive linear systems.

[0011] Furthermore, the state space model of the discrete-time positive linear system is:

[0012]

[0013] Where x(k) represents the state vector, σ(k) represents the switching signal, which is a piecewise constant function with values ​​in the set {1,2,…,N}, N represents the number of subsystems, and A σ(k) (θ(k)) is the uncertain system matrix of appropriate dimensions, and where θ1(k),θ2(k),…,θ P (k) is an unknown time-varying non-negative function that satisfies A σ(k)1 ,A σ(k)2 ,…,A σ(k)P is a known constant matrix, p∈{1,2,…,P}, p is the index variable, and P represents the number of terms in the summation process.

[0014] Furthermore, based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time, the extended switching path pair dependent switching signals are designed to include:

[0015] Given a set of basic switching paths and a set of switching paths i,j∈{1,2,…,n}, n represents the number of basic switching paths, the set of all basic switching paths is denoted as G, called the basic switching path set, and the set of all basic switching path pairs is denoted as D, called the basic switching path pair set;

[0016] If the entire switching sequence of the switching signal σ(k) starting from the initial moment consists of countless paths in the set G, and any path pair consisting of two adjacent paths belongs to the set D, then the switching signal σ(k) is defined as a basic switching path pair dependent switching signal, denoted as σ(k, G, D);

[0017] If a switching path Contains r basic switching paths, and every two adjacent switching paths form a path pair, then the switching path is defined as is the extended switching path, r is the expansion index, r ≥ 1, and the set of switching paths containing all r basic switching paths is recorded as It is called the extended switching path set;

[0018] If you expand the switching path The last basic switching path and another extended switching path The first basic switching path of forms a basic switching path pair, then define the combination To expand the switching path pairs, the set of all expanded switching path pairs is denoted as It is called the extended switching path pair set;

[0019] If the switching sequence generated by the switching signal σ(k) from 0 to ∞ is always composed of the set In the infinite number of extended switching paths, and any switching path pair consisting of any two adjacent extended switching paths is a set The switching signal σ(k) is defined as the extended switching path pair dependent switching signal, denoted as

[0020] Furthermore, the stability conditions for the designed discrete-time positive linear system include:

[0021] Based on the discrete-time positive linear system, the uncertain system matrix A σ(k) (θ(k)) is expressed as:

[0022] A σ(k) (θ(k))=A i +ΔA i (t)

[0023] In the formula, A i is a known constant matrix; ΔA i (t) is the time-varying non-negative uncertainty part and satisfies 0≤ΔA i (t)≤A i , ΔA i represents a known constant matrix;

[0024] Design non-conservative stability condition 1:

[0025] There exists a positive vector v i > 0, so that for all index variables i and σ(k) A i v i ≤v i ;

[0026] Design non-conservative stability condition 2:

[0027] There is a focus matrix X i and the positive vector vi > 0, so that for all i and σ(k) Where I is the identity matrix.

[0028] Furthermore, the positive stabilization conditions for the designed discrete-time positive linear system include:

[0029] Based on the non-conservative stability criterion, the state space model of the discrete-time positive linear system with polyhedral uncertainty is expressed as:

[0030] x(k+1)=A σ(k) x(k)+B σ(k) u(k)

[0031] Where u(k) is the control input, B σ(k) is the input matrix;

[0032] The goal of positive stabilization is to design a controller so that the system is stable and all state variables remain non-negative. The following positive stabilization condition in linear programming form is proposed:

[0033] There exists a diagonal matrix X i >0 and the controller gain matrix K i , so that for all i and σ(k) Among them B i represents the matrix used in the control system to apply the control input u(k) to the system state x(k).

[0034] Furthermore, the positive observation conditions for designing discrete-time positive linear systems include:

[0035] The output vector of the discrete-time positive linear system is expressed as:

[0036] y(k)=C σ(k) x(k)

[0037] In the formula, C σ(k) represents the output matrix of the system;

[0038] Construct an observer in the following form:

[0039]

[0040] In the formula, represents the estimated value of the system state, L σ(k) represents the gain matrix, which also depends on the value of the switching signal σ(k), represents the switching matrix, whose value depends on the value of the switching signal σ(k) at time k;

[0041] Calculate the system error, the expression is:

[0042] e(k+1)=(A σ(k) -L σ(k) C σ(k) )e(k)

[0043] The goal of positive observation is to design an observer so that the estimate of the system state is always non-negative and the dynamics of the error system is asymptotically stable. The following positive observation condition in linear programming form is proposed:

[0044] There exists a diagonal matrix X i >0 and the observer gain matrix L i , so that for all i and any σ(k) Among them C i Represents the output matrix of the system.

[0045] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0046] 1. The present invention proposes a new type of ESPPD switching signal, including four dwell time switching signals and some mixed switching signals, which can uniformly process multiple types of switching signals, propose non-conservative stability conditions and non-conservative positive stabilization conditions, and present these conditions in the form of linear programming, which reduces the calculation cost and improves the applicability and accuracy of the results;

[0047] 2. The present invention obtains less conservative stability results by adjusting the expansion exponent, providing more flexible options for controller design; and designs effective observers to accurately estimate the system state with time-varying uncertain parameters, thereby enhancing the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a flow chart of the stability analysis method of discrete-time positive linear system based on ESPPD switching signal;

[0049] Figure 2 It is the state curve diagram of the system when the constant residence time d = 5 seconds;

[0050] Figure 3 It is the state curve diagram of the system when the constant residence time d = 6 seconds;

[0051] Figure 4 It is the state curve diagram of the system when the parameter a=1.14;

[0052] Figure 5 It is the state curve diagram of the system when the parameter a=1.13;

[0053] Figure 6 It is the state curve diagram of the closed-loop system. DETAILED DESCRIPTION

[0054] In order to make the objectives, technical solutions and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments.

[0055] The stability analysis method of the discrete-time positive linear system based on the ESPPD switching signal described in this embodiment is as follows: Figure 1 As shown, including:

[0056] Construct state-space models of discrete-time positive linear systems;

[0057] The extended switching path pair dependent switching signal is designed based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time;

[0058] Based on the extended switching path pair dependent switching signal, stability conditions, positive stabilization conditions and positive observation conditions for discrete-time positive linear systems are designed;

[0059] Verify the stability of discrete-time positive linear systems.

[0060] In one example, the state space model of a discrete-time positive linear system is:

[0061]

[0062] Where x(k) represents the state vector, σ(k) represents the switching signal, which is a piecewise constant function with values ​​in the set {1,2,…,N}, N represents the number of subsystems, and A σ(k) (θ(k)) is the uncertain system matrix of appropriate dimensions, and where θ1(k),θ2(k),…,θ P (k) is an unknown time-varying non-negative function that satisfies A σ(k)1 ,A σ(k)2 ,…,A σ(k)P is a known constant matrix, p is an index variable, A σ(k)p represents the p-th matrix, θ p (k) represents the pth weight, and P represents the number of terms in the summation process.

[0063] The discrete-time positive linear system is a positive system. For any switching signal σ(k) and initial condition x(0)≥0, its state satisfies x(t)≥0, that is, all elements of the state vector are always non-negative.

[0064] Furthermore, based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time, the extended switching path pair dependent switching signals are designed to include:

[0065] Given a set of basic switching paths and a set of switching paths i,j∈{1,2,…,n}, n represents the number of basic switching paths, the set of all basic switching paths is denoted as G, called the basic switching path set, and the set of all basic switching path pairs is denoted as D, called the basic switching path pair set;

[0066] If the entire switching sequence of the switching signal σ(k) starting from the initial moment consists of countless paths in the set G, and the path pair consisting of any two adjacent paths belongs to the set D, then the switching signal σ(k) is defined as a basic switching path pair dependent switching signal, abbreviated as BSPPD switching signal, denoted as σ(k,G,D).

[0067] Based on the BSPPD switching signal, the concept of ESPPD switching signal is further introduced to adapt to more complex switching scenarios. Its definition and formation rules are as follows.

[0068] If a switching path Contains r basic switching paths (for example, j1,j2,…,j r ∈{1,2,…,n}), and every two adjacent switching paths form a path pair, then the switching path is defined as is the extended switching path, r is the expansion index, r ≥ 1, and the set of switching paths containing all r basic switching paths is recorded as It is called the extended switching path set;

[0069] If you expand the switching path The last basic switching path and another extended switching path The first basic switching path of forms a basic switching path pair, then define the combination To expand the switching path pairs, the set of all expanded switching path pairs is denoted as It is called the extended switching path pair set;

[0070] If the switching sequence generated by the switching signal σ(k) from 0 to ∞ is always composed of the set In the infinite number of extended switching paths, and any switching path pair consisting of any two adjacent extended switching paths is a set The switching signal σ(k) is defined as the extended switching path pair dependent switching signal, referred to as the ESPPD switching signal, denoted as

[0071] The expansion index r represents the number of basic switching paths included in the extended switching path. By adjusting the value of r, the complexity and length of the extended switching path can be controlled.

[0072] By introducing the ESPPD switching signal, the ESPPD switching signal can include multiple common switching signal types, such as residence time signals and mixed switching signals, and can also describe more complex switching logic, which is suitable for a wider range of application scenarios.

[0073] It should be noted that the ESPPD switching signal should also be the BSPPD switching signal. When the basic switching path set and the basic switching path pair set are given, for any r≥1, the ESPPD switching signal and the BSPPD switching signal σ(k,G,D) are equivalent.

[0074] The following example uses the ESPPD switching signal for an equivalent description.

[0075] Example 1: Maximum dwell time switching signal

[0076] Consider a switching system A with two subsystems A1 and A2. σ(k) , the maximum residence time of subsystems A1 and A2 is 1 second and 2 seconds respectively, so the switching signal σ(k) is a maximum residence time switching signal. Assume that the switching paths are: Define a basic switching path set Basic switching path pair set The maximum dwell time switching signal σ(k) can be regarded as a BSPPD switching signal σ(k,G,D).

[0077] When the expansion index r=2, the process of obtaining the ESPPD switching signal based on the BSPPD switching signal is:

[0078] Define the extended switching path as Define an extended switching path set Define an extended switching path pair set Then the ESPPD switching signal relative to the maximum dwell time switching signal σ(k) is obtained

[0079] Example 2: Mixed switching signals

[0080] Consider a switching system A with three subsystems A1, A2 and A3 σ(k) , the dwell time of subsystems A1 and A2 is 2 seconds, and the dwell time of subsystem A3 should be no less than and no more than 3 seconds. Subsystem A1 can only switch to A2, subsystem A2 can only switch to A3, and subsystem A3 can only switch to A1. It can be seen that the switching signal σ(k) is a mixed switching signal. The switching paths are defined as Define a basic switching path set Define a basic switching path pair set Then the hybrid switching signal σ(k) can be regarded as a BSPPD switching signal σ(k,G,D).

[0081] When the expansion index r=2, the process of obtaining the ESPPD switching signal based on the BSPPD switching signal is:

[0082] Define the extended switching path as Define the extended switching path set as Define the extended switching path pair set as

[0083] , then the ESPPD switching signal is obtained It is completely equivalent to σ(k, G, D). The value of the extended index r can be any positive integer. The larger the value of r, the longer the extended switching path is and the more switching information it contains.

[0084] Furthermore, the stability conditions for the designed discrete-time positive linear system include:

[0085] Based on the discrete-time positive linear system, the uncertain system matrix A σ(k) (θ(k)) is expressed as:

[0086] A σ(k) (θ(k))=A i +A i (t)

[0087] In the formula, A i is a known constant matrix; ΔA i (t) is the time-varying non-negative uncertainty part and satisfies 0≤ΔA i (t)≤ΔA i , ΔA i represents the known constant matrix used to limit ΔA i (t) to ensure that the uncertainty does not exceed this range;

[0088] In order to analyze the stability of the system, the following two non-conservative stability conditions are proposed. Design non-conservative stability condition 1:

[0089] There exists a positive vector v i > 0, so that for all index variables i and σ(k) A i v i ≤v i ;

[0090] Design non-conservative stability condition 2:

[0091] There is a focus matrix X i and the positive vector v i > 0, so that for all i and σ(k) Where I is the identity matrix.

[0092] The above two non-conservative stability conditions ensure that under a given ESPPD switching signal, the state vector of the system will not grow at each step and eventually reach asymptotic stability.

[0093] By introducing the extended index r, we further analyze the behavior of the above stability conditions under different index values. The proof shows that when the extended index r is large enough, these stability conditions are equivalent, which means that no matter whether we choose condition 1 or condition 2 for analysis, we can get the same stability conclusion.

[0094] Assume that condition 1 holds true, that is, there exists a positive vector v i >0, satisfying A i v i ≤v i By constructing a diagonal matrix X i =diag(v i ), it can be proved Therefore, condition 2 also holds.

[0095] Assume that condition 2 holds true, that is, there exists a diagonal matrix X i and the positive vector v i >0, satisfied By choosing a suitable positive vector v i , which can make A i v i ≤v i So condition 1 is also true.

[0096] As the extension index r increases, the system behavior covered by the extended switching path will get closer and closer to all possible state combinations of the real system. Eventually, when r is large enough, the behavior of the system under any switching signal can be described by these conditions. Therefore, conditions 1 and 2 are equivalent when r is large enough.

[0097] Through the above derivation, it is proved that when the expansion index r is large enough, conditions 1 and 2 are equivalent in describing the asymptotic stability of the system. This equivalence provides a variety of possible tool choices for system stability analysis and enhances the applicability and flexibility of the method.

[0098] In studying the stability problem of uncertain discrete-time switched positive linear systems (DSPLSs) with ESPPD switching signals, several sufficient positive stabilization and positive observation conditions are proposed based on non-conservative stability criteria. All results are given in the form of linear programming (LP) to facilitate the design of controllers and observers in practical applications.

[0099] Furthermore, the positive stabilization conditions for the designed discrete-time positive linear system include:

[0100] Based on the non-conservative stability criterion, the state space model of the discrete-time positive linear system with polyhedral uncertainty is expressed as:

[0101] x(k+1)=A σ(k) x(k)+B σ(k) u(k)

[0102] Where u(k) is the control input, B σ(k) is the input matrix;

[0103] The goal of positive stabilization is to design a controller so that the system is stable and all state variables remain non-negative. The following positive stabilization condition in linear programming form is proposed:

[0104] There exists a diagonal matrix X i >0 and the controller gain matrix K i , so that for all i and σ(k) Among them B i represents the matrix used in the control system to apply the control input u(k) to the system state x(k).

[0105] Furthermore, the positive observation conditions for designing discrete-time positive linear systems include:

[0106] The output vector of the discrete-time positive linear system is expressed as:

[0107] y(k)=C σ(k) x(k)

[0108] In the formula, C σ(k) represents the output matrix of the system;

[0109] Construct an observer in the following form:

[0110]

[0111] In the formula, represents the estimated value of the system state, L σ(k) represents the gain matrix, which also depends on the value of the switching signal σ(k), represents the switching matrix, whose value depends on the value of the switching signal σ(k) at time k;

[0112] Calculate the system error, the expression is:

[0113] e(k+1)=(A σ(k) -L σ(k) C σ(k) )e(k)

[0114] The goal of positive observation is to design an observer so that the estimate of the system state is always non-negative and the dynamics of the error system is asymptotically stable. The following positive observation condition in linear programming form is proposed:

[0115] There exists a diagonal matrix X i >0 and the observer gain matrix L i , so that for all i and any σ(k) Among them C i Represents the output matrix of the system.

[0116] The above positive stabilization and positive observation conditions can be solved by linear programming (LP) problems. Specifically,

[0117] LP problem for controller design:

[0118] min F1

[0119]

[0120] LP problem of observer design:

[0121] min F2

[0122]

[0123] Among them, F1 and F2 are different function objectives. By solving the LP problem, we can design a controller and observer that can make the system meet the requirements of positive stabilization and positive observation.

[0124] The stability of the discrete-time positive linear system of the present invention is verified and illustrated by the following examples.

[0125] In an example, consider a discrete-time positive linear system with four subsystems, where the switching signal has a constant switching time and the dwell time of the switching signal is different in seconds. The matrices of the four subsystems are:

[0126]

[0127] The state results of the system at a constant residence time of d = 5 seconds are as follows Figure 2 As shown, the horizontal axis k represents time, the vertical axis x(k) represents the state vector, and the blue and red colors represent different state vectors. Figure 2 The system state curve shows that it diverges at an extremely slow rate, indicating that the system is unstable under this condition. The state results of the system under a constant residence time of d = 6 seconds are as follows Figure 3 As shown in Figure 2, when the residence time d = 6 seconds, the system state curve converges at an extremely slow rate, indicating that the system is in a critical stable state. Figure 2 and Figure 3It can be obtained that when d=6 seconds, the system is at the critical point where the system changes from an unstable state to a stable state, that is, d=6 seconds serves as a switching signal to stabilize the system.

[0128] In another example, a DSPLSs with three subsystems is considered, and the dwell time of each subsystem is different in seconds. A switching signal with a constant dwell time is used, and the switching rules between subsystems are more complicated. The three subsystem matrices are:

[0129]

[0130] The state results under the adjustable system parameter a = 1.14 are as follows Figure 4 As shown, the state structure under the system parameter a = 1.13 is as follows Figure 5 As shown in the figure, the state curve shows different convergence and divergence at different dwell times. According to the simulation results, when a=1.13, the state curve converges at an extremely slow rate; and when a=1.14, the state curve diverges at an extremely slow rate. It can be seen that the stable region obtained by selecting the expansion index r=2 is non-conservative, confirming the sufficiency and necessity of the results. By selecting a suitable expansion index r, the stable region of the system can be significantly expanded, thus verifying the effectiveness of the proposed method under different complexity conditions.

[0131] In another example, different subsystems have different dwell times and switching rules, aiming to verify the stability and control effect of the method of the present invention under complex switching signals. The six subsystem matrices are:

[0132]

[0133] The controller design problem is transformed into a linear programming problem. By solving the LP problem, the control gain matrix K that meets the positive stabilization requirements can be obtained. i The controller gain is solved as follows:

[0134]

[0135] Under the designed controller gain, the state of the closed-loop system not only remains positive but also gradually converges to zero, verifying the positivity and stability of the system. This result shows that by increasing the expansion index, a less conservative controller design can be obtained, making the system stable under a wider range of switching conditions. Figure 6 It can be seen that the state of the closed-loop system under the above controller gain is positive and gradually converges to zero. The positivity and stability of the system are guaranteed by the designed controller, which verifies the validity of the results.

[0136] The above examples verify the validity of the non-conservative stability conditions and positive stabilization conditions of the discrete-time positive linear system with ESPPD switching signal proposed in the present invention. The simulation results show that with the increase of the extended index, the stability of the system is significantly improved, and the controller and observer design are more flexible and widely applicable. The above examples not only demonstrate the practicality of the present invention, but also show that the method has obvious advantages in dealing with complex switching systems.

Claims

1. A stability analysis method for discrete-time positive linear systems based on ESPPD switching signals, characterized in that: include: Construct state-space models of discrete-time positive linear systems; The extended switching path pair dependent switching signal is designed based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time; Based on the extended switching path pair dependent switching signal, stability conditions, positive stabilization conditions and positive observation conditions for discrete-time positive linear systems are designed; Verify the stability of discrete-time positive linear systems.

2. The stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal according to claim 1, characterized in that: The state space model of a discrete-time positive linear system is: x(k+1)=A σ(k) (θ(k))x(k),x(k0)≥0 Where x(k) represents the state vector; σ(k) represents the switching signal, which is a piecewise constant function with values ​​in the set {1,2,…,N}, N represents the number of subsystems, and A σ(k) (θ(k)) is the uncertain system matrix of appropriate dimensions, and where θ1(k),θ2(k),…,θ P (k) is an unknown time-varying non-negative function that satisfies A σ(k)1 ,A σ(k)2 ,…,A σ(k)P is a known constant matrix, p∈{1,2,…,P}, p is the index variable, and P is the number of terms in the summation process.

3. The stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal according to claim 2, characterized in that: Based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time, the extended switching path pair dependent switching signals include: Given a set of basic switching paths a1, a2, …, a n and a set of switching path pairs (a i ,a j ), i, j are index variables, i, j ∈ {1, 2, ..., n}, n represents the number of basic switching paths, the set of all basic switching paths is denoted as G, called the basic switching path set, and the set of all basic switching path pairs is denoted as D, called the basic switching path pair set; If the entire switching sequence of the switching signal σ(k) starting from the initial moment consists of countless paths in the set G, and any path pair consisting of two adjacent paths belongs to the set D, then the switching signal σ(k) is defined as a basic switching path pair dependent switching signal, denoted as σ(k, G, D); If a switching path a i,r Contains r basic switching paths, and every two adjacent switching paths form a path pair, then the switching path a is defined as i,r is the extended switching path, r is the expansion index, r ≥ 1, and the set of switching paths containing all r basic switching paths is recorded as It is called the extended switching path set; If the switching path a is extended i,r The last basic switching path and another extended switching path a j,r The first basic switching path of forms a basic switching path pair, then define the combination (a i,r ,a j,r ) is an extended switching path pair, and the set of all extended switching path pairs is denoted as It is called the extended switching path pair set; If the switching sequence generated by the switching signal σ(k) from 0 to ∞ is always composed of the set In the infinite number of extended switching paths, and any switching path pair consisting of any two adjacent extended switching paths is a set The switching signal σ(k) is defined as the extended switching path pair dependent switching signal, denoted as 4. The stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal according to claim 3, characterized in that: The stability conditions for designing discrete-time positive linear systems include: Based on the discrete-time positive linear system, the uncertain system matrix A σ(k) (θ(k)) is expressed as: A σ(k) (θ(k))>A i +ΔA i (t) In the formula, A i is a known constant matrix; ΔA i (t) is the time-varying non-negative uncertainty part and satisfies 0≤ΔA i (t)≤ΔA i , ΔA i is a known constant matrix; Design non-conservative stability condition 1: There exists a positive vector v i > 0, so that for all index variables i and σ(k) A i v i ≤v i ; Design non-conservative stability condition 2: There is a focus matrix X i and the positive vector v i > 0, so that for all i and σ(k) Where I is the identity matrix.

5. The stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal according to claim 4, characterized in that: The positive stabilization conditions for designing discrete-time positive linear systems include: Based on the non-conservative stability criterion, the state space model of the discrete-time positive linear system with polyhedral uncertainty is expressed as: x(k+1)=A σ(k) x(k)+B σ(k) u(k) Where u(k) is the control input, B σ(k) is the input matrix; The goal of positive stabilization is to design a controller so that the system is stable and all state variables remain non-negative. The following positive stabilization condition in linear programming form is proposed: There exists a diagonal matrix X i >0 and the controller gain matrix K i , so that for all i and σ(k) Among them B i represents the matrix used in the control system to apply the control input u(k) to the system state x(k).

6. The stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal according to claim 5, characterized in that: The positive observation conditions for designing discrete-time positive linear systems include: The output vector of the discrete-time positive linear system is expressed as: y(k)=C σ(k) x(k) In the formula, C σ(k) represents the output matrix of the system; Construct an observer in the following form: In the formula, represents the estimated value of the system state, L σ(k) represents the gain matrix, which also depends on the value of the switching signal σ(k), represents the switching matrix, whose value depends on the value of the switching signal σ(k) at time k; Calculate the system error, the expression is: e(k+1)=(A σ(k) -L σ(k) C σ(k) )e(k) The goal of positive observation is to design an observer so that the estimate of the system state is always non-negative and the dynamics of the error system is asymptotically stable. The following positive observation condition in linear programming form is proposed: There exists a diagonal matrix X i >0 and the observer gain matrix L i , so that for all i and any σ(k) Among them C i Represents the output matrix of the system.

Images