Stability analysis method of discrete-time positive linear systems based on esppd switching signal

Abstract

The application discloses a stability analysis method of a discrete-time positive linear system based on an ESPPD switching signal, and 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 the 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 application proposes a new ESPPD switching signal, which contains four types of dwell time switching signals and some mixed switching signals, can uniformly process various types of switching signals, proposes non-conservative stability conditions and non-conservative positive stabilization conditions, presents the conditions in the form of linear programming, reduces the calculation cost, and improves the applicability and accuracy of the results.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of automation control, in particular to a stability analysis method of discrete-time positive linear systems based on ESPPD switching signals. BACKGROUND

[0002] Switched positive systems are composed of multiple positive subsystems and switching rules, and are widely used in real life, such as networks using TCP and formation flight, etc. Due to its wide application value, in recent years, the analysis and synthesis of switched positive systems have attracted more and more attention. Early research results have discussed the stability of switched positive linear systems under arbitrary switching, and through the use of 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, researchers have discussed 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 is found that the stability conditions under dwell time signals are non-convex in terms of system matrices, and these results cannot be directly extended to the design of stable controllers.

[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 switched systems under dwell time switching signals have been found, these results cannot be directly used for the stability and stabilization of switched positive systems. Therefore, it is still necessary to further explore the necessary and sufficient conditions for the stability and stabilization of switched positive systems under dwell time signals. SUMMARY

[0005] The present application aims to provide a stability analysis method of discrete-time positive linear systems based on ESPPD switching signals, in order to provide more extensive and non-conservative stability conditions.

[0006] Technical solution: The stability analysis method of discrete-time positive linear systems based on ESPPD switching signals of the present application comprises:

[0007] constructing a state space model of the discrete-time positive linear system;

[0008] Designing extended switching path pair-dependent switching signals based on four kinds of dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time;

[0009] Designing stability condition, positive stabilization condition and positive observation condition of discrete-time positive linear systems based on extended switching path pair-dependent switching signals;

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

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

[0012]

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

[0014] Further, designing extended switching path pair-dependent switching signals based on four kinds of dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time includes:

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

[0016] If the entire switching sequence of the switching signal σ(k) from the initial time consists of infinite paths in the set G, and any two adjacent path pairs belong 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 each two adjacent switching paths forms a path pair, then the switching path is defined as a switching path pair-dependent switching signal, denoted as σ(k, G, D). For the extended switching path, r is the extension index, r≥1, and all the switching path sets containing r basic switching paths are denoted as The set of extended switching paths is called the extended switching path set;

[0018] If the last basic switching path of the extended switching path and the first basic switching path of another extended switching path form a basic switching path pair, then the combination is defined as For the extended switching path pair, all the extended switching path pairs are denoted as The set of extended switching path pairs 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 infinite extended switching paths in the set and any switching path pair composed of any two adjacent extended switching paths is an element of the set , then the switching signal σ(k) is defined as the extended switching path pair dependent switching signal, denoted as

[0020] Further, the stability condition of the discrete-time positive linear system is designed as:

[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 a time-varying non-negative uncertain part, and satisfies 0≤ΔA i (t)≤A i , and ΔA i represents a known constant matrix;

[0024] Non-conservative stability condition 1 is designed as:

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

[0026] Non-conservative stability condition 2 is designed as:

[0027] There is a focal matrix X i and a positive vector vi > 0, such that for all i and σ(k) satisfy where I is the identity matrix.

[0028] Further, the positive stabilization condition for a discrete-time positive linear system includes:

[0029] Based on the non-conservative stability criterion, the state-space model of a discrete-time positive linear system with polytopic uncertainty is represented 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 objective of positive stabilization is to design a controller such that the system is stable and all state variables remain non-negative at all times. The following linear programming form of positive stabilization condition is proposed:

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

[0034] Further, the positive observation condition for a discrete-time positive linear system includes:

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

[0036] y(k) = C σ(k) x(k)

[0037] where C σ(k) represents the output matrix of the system;

[0038] An observer is constructed in the form:

[0039]

[0040] where represents the estimated value of the system state, L σ(k) represents the gain matrix, which is also dependent 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] The system error is calculated, expressed as:

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

[0043] The positive observation aims to design an observer, so that the estimation of the system state always remains non-negative, while the dynamic of the error system is asymptotically stable, and the positive observation condition in the form of linear programming is proposed as follows:

[0044] There exist a diagonal matrix X i > 0 and an observer gain matrix L i , such that for all i and arbitrary σ(k) satisfy Where C i represents the output matrix of the system.

[0045] Advantages: compared with the prior art, the present application has the following advantages:

[0046] 1. The present application proposes a new ESPPD switching signal, which contains four types of residence time switching signals and some mixed switching signals, can unify the processing of various types of switching signals, proposes non-conservative stability conditions and non-conservative positive stabilization conditions, presents these conditions in the form of linear programming, reduces the calculation cost, and improves the applicability and accuracy of the results.

[0047] 2. The present application adjusts the extended index to obtain less conservative stability results, provides more flexible selection for controller design; designs an effective observer to accurately estimate the system state of the time-varying uncertain parameter, and enhances the robustness of the system. BRIEF DESCRIPTION OF DRAWINGS

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

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

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

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

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

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

[0054] In order to make the purposes, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to 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 the embodiment has a flow chart as shown in Figure 1 The method comprises the following steps:

[0056] Constructing a state space model of the discrete-time positive linear system;

[0057] Designing an extended switching path pair dependent switching signal based on four kinds of dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time;

[0058] Designing stability conditions, positive stabilization conditions and positive observation conditions of the discrete-time positive linear system based on the extended switching path pair dependent switching signal;

[0059] Verifying the stability of the discrete-time positive linear system.

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

[0061]

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

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

[0064] Further, designing the extended switching path pair dependent switching signal based on the four kinds of dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time comprises:

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

[0066] If the entire switching sequence of the switching signal σ(k) from the initial time consists of an infinite number of paths in the set G, and any two adjacent path pairs belong 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] On the basis of the BSPPD switching signal, the concept of ESPPD switching signal is further introduced to adapt to more complex switching scenarios, and 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 each two adjacent switching paths form a path pair, then the switching path is defined as an extended switching path, r is the expansion index, r≥1, and the set of all switching paths containing r basic switching paths is denoted as , which is called the extended switching path set.

[0069] If the last basic switching path of the extended switching path and the first basic switching path of another extended switching path form a basic switching path pair, then the combination is defined as an extended switching path pair, and the set of all extended switching path pairs is denoted as , which 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 an infinite number of extended switching paths in the set , and any switching path pair composed of any two adjacent extended switching paths is an element of the set , then the switching signal σ(k) is defined as an extended switching path pair dependent switching signal, abbreviated as ESPPD switching signal, denoted as

[0071] The expansion index r represents the number of basic switching paths contained in the extended switching path, and 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 dwell 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. It can be seen that 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 follows: 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] The expansion switching path is defined as The expansion switching path set is defined as The expansion switching path pair set is defined as

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

[0084] Further, the stability condition for designing a discrete-time positive linear system includes:

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

[0086] A σ(k) (θ(k)) = A i + ΔA i (t)

[0087] In the formula, A i is a known constant matrix; ΔA i (t) is a time-varying non-negative uncertain part, and satisfies 0 ≤ ΔA i (t) ≤ ΔA i , ΔA i represents a known constant matrix used to limit the upper and lower bounds of ΔA i (t), to ensure that the uncertain part will 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, such that for all index variables i and σ(k) satisfy A i v i ≤ v i ;

[0090] Design non-conservative stability condition 2:

[0091] There exist a focal matrix X i and a positive vector v i > 0, such that for all i and σ(k) satisfy 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 does not grow at each step and eventually reaches asymptotic stability.

[0093] By introducing the extended index r, further analysis of the behavior of the above stability conditions under different index values is conducted. The proof shows that when the extended index r is large enough, these stability conditions are equivalent, which means that the same stability conclusion can be drawn regardless of choosing condition 1 or condition 2 for analysis.

[0094] Assume that condition 1 holds, i.e., there exists a positive vector v i > 0 that satisfies A i v i ≤ v i . By constructing a diagonal matrix X i = diag(v i ), it can be proven that and thus condition 2 also holds.

[0095] Assume that condition 2 holds, i.e., there exists a diagonal matrix X i and a positive vector v i > 0 that satisfies By choosing a suitable positive vector v i , it can make A i v i ≤ v i hold, and thus condition 1 also holds.

[0096] As the extended index r increases, the system behavior covered by the extended switching path will be 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, condition 1 and condition 2 are equivalent when r is large enough.

[0097] Through the above derivation, it is proven that when the extended index r is large enough, condition 1 and condition 2 are equivalent in describing the asymptotic stability of the system, which provides multiple possible tool choices for system stability analysis and enhances the applicability and flexibility of the method.

[0098] In the study of the stability of uncertain discrete-time switched positive linear systems (DSPLSs) with ESPPD switching signals, based on non-conservative stability criteria, several sufficient positive stabilization and positive observation conditions are proposed, all of which are given in the form of linear programming (LP), which is convenient for designing controllers and observers in practical applications.

[0099] Further, a positive stabilization condition for a discrete-time positive linear system comprises:

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

[0101] x(k+1) = A σ(k) x(k) + B σ(k) u(k)

[0102] where u(k) is a control input, B σ(k) is an input matrix;

[0103] The objective of positive stabilization is to design a controller such that the system is stable and all state variables remain non-negative at all times, and a positive stabilization condition in the form of a linear programming is proposed as follows:

[0104] There exist a diagonal matrix X i > 0 and a controller gain matrix K i such that for all i and σ(k) satisfies where B i represents a matrix in the control system for applying the control input u(k) to the system state x(k).

[0105] Further, a positive observation condition for a discrete-time positive linear system comprises:

[0106] An output vector of a discrete-time positive linear system is represented as:

[0107] y(k) = C σ(k) x(k)

[0108] where C σ(k) represents an output matrix of the system;

[0109] An observer is constructed in the form as follows:

[0110]

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

[0112] A system error is calculated, expressed as:

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

[0114] The objective of positive observation is to design an observer such that the estimation of the system state is always non-negative while the dynamics of the error system is asymptotically stable, and the positive observation condition in the form of linear programming is proposed as follows:

[0115] There exists a diagonal matrix X i > 0 and an observer gain matrix L i such that for all i and arbitrary σ(k) satisfy where C i represents the output matrix of the system.

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

[0117] The LP problem for controller design:

[0118] min F1

[0119]

[0120] The LP problem for observer design:

[0121] min F2

[0122]

[0123] where F1 and F2 are different function objectives. By solving the LP problems, the controller and observer that make the system meet the requirements of positive stabilization and positive observation can be designed.

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

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

[0126]

[0127] The state results of the system under constant dwell time d = 5 seconds are shown in Figure 2 , where the horizontal coordinate k represents time, the vertical coordinate x(k) represents state vector, and blue and red represent different state vectors. Figure 2 From the system state curve, it can be seen that it diverges at an extremely slow rate, indicating that the system is unstable under this condition. The state results of the system under constant dwell time d = 6 seconds are shown in Figure 3 . When the dwell time d = 6 seconds, the system state curve converges at an extremely slow rate, indicating that the system is in a critical stable state. By Figure 2 and Figure 3The critical point at which the system transitions from unstable to stable state is obtained when d = 6 seconds, i.e., d = 6 seconds as the switching signal that stabilizes the system.

[0128] In another example, a DSPLS containing three subsystems is considered, and the dwell time of each subsystem is different in seconds, a switching signal with constant dwell time is adopted, and the switching rule between the subsystems is complex. The three subsystem matrices are respectively:

[0129]

[0130] The state results under the adjustable system parameter a = 1.14 are shown in Figure 4 , and the state structure under the system parameter a = 1.13 is shown in Figure 5 . The state curve shows different convergence and divergence under different dwell times. According to the simulation results, when a = 1.13, the state curve converges at an extremely slow rate; 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, thereby 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 proposed method under complex switching signals. The six subsystem matrices are respectively:

[0132]

[0133] The controller design problem is converted into a linear programming problem, and by solving the LP problem, the control gain matrix K i that satisfies the positive stabilization requirement can be obtained. The controller gain obtained by solving is 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. The results show that by increasing the expansion index, a less conservative controller design scheme can be obtained, making the system stable under a wider range of switching conditions. According to 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, and the positivity and stability of the system are guaranteed by the designed controller, verifying the effectiveness of the results.

[0136] The effectiveness of the non-conservative stability condition and the positive stabilization condition of the discrete-time positive linear system with ESPPD switching signal proposed in the application is verified through the above examples. The simulation results show that the stability of the system is significantly improved with the increase of the extended index, and the controller and observer design is more flexible and widely applicable. Through the above examples, not only the practicability of the application is shown, but also it is shown 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; The extended switching path pair dependent switching signals are designed based on four dwell time switching signals including constant dwell time, minimum dwell time, maximum dwell time and range dwell time. Given a set of basic switching paths a1, a2, L, a n and a set of switching path pairs (a i ,a j ), i, j are index variables, i, j ∈ {1, 2, L, 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 recorded 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, 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 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: 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, L, 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),L,θ P (k) is an unknown time-varying non-negative function that satisfies A σ(k)1 ,A σ(k)2 ,L,A σ(k)P is a known constant matrix, p∈{1,2,L,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 1, characterized in that: Design stability conditions for 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) Where A i is a known constant matrix; ΔA i (t) is a time-varying non-negative uncertainty part that 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, such 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, such that for all i and σ(k) Where I is the identity matrix.

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 positive stabilization conditions for designing discrete-time positive linear systems include: Based on the non-conservative stability criterion, the state space model of a 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).

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 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) Where C σ(k) represents the output matrix of the system; Construct an observer in the following form: Where, 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.

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