Sliding mode control method based on random communication protocol for inverted pendulum system with continuous dwell time

Abstract

The present invention relates to a sliding mode control method for an inverted pendulum system based on a random communication protocol, belonging to the field of automatic control technology. The method combines sliding mode control for an inverted pendulum system based on a continuous dwell time with sliding mode control based on a random communication protocol to construct a coordinated controller. A discrete-time uncertain switching system model with a PDT switching rule is constructed, and combined with a random communication protocol scheduling mechanism, the random switching behavior of nodes accessing the network is described through a Markov chain. A token-dependent sliding mode control law is designed to dynamically adjust control parameters to accommodate SCP scheduling and PDT switching. A sliding mode function and a Lyapunov function are designed to ensure the accessibility of the sliding mode. The present invention effectively utilizes the advantages of a networked control method based on a continuous dwell time switching rule under random communication protocol scheduling, such as strong robustness, resource optimization, and reliable stability, effectively improving the operating efficiency of the inverted pendulum system.

Description

Technical Field

[0001] The present invention belongs to the technical field of automatic control, and in particular relates to a sliding mode control method based on a random communication protocol of a continuous dwell time inverted pendulum system. Background Art

[0002] As a classic nonlinear control system, the inverted pendulum system, due to its inherent nonlinearity and instability, is an ideal platform for testing robust control strategies. In the control of an inverted pendulum system, the reliability and real-time nature of network communication have a significant impact on system performance. Traditional control methods have limitations when dealing with network communication delays, packet loss, and data conflicts.

[0003] In recent years, sliding mode control has attracted widespread attention due to its robustness to uncertainty and external disturbances. However, in networked control systems, the design and implementation of sliding mode control for inverted pendulum systems face numerous challenges, such as limited network communication resources, packet loss, and quantization. To address these challenges, researchers have proposed a variety of communication protocols. Among them, the stochastic communication protocol (SCP), due to its random node selection, excels in alleviating the network communication burden and reducing the risk of data conflicts.

[0004] Meanwhile, the persistent dwell time (PDT) switching rule, a switching strategy that combines the advantages of both dwell time and average dwell time, improves the stability and flexibility of the inverted pendulum system by balancing periodic time intervals with short random switching windows. However, the application of the PDT switching rule in combination with sliding mode control in the control of inverted pendulum systems has not been fully studied.

[0005] To address the above problems, the present invention proposes a sliding mode control method based on a random communication protocol for a continuous dwell time inverted pendulum system. The design of sliding mode control under SCP scheduling is particularly challenging. The existing technologies have the following deficiencies: (1) they do not fully consider the communication protocol scheduling problem in a networked environment; (2) they lack stability analysis of the PDT inverted pendulum system combined with SCP; and (3) they are difficult to ensure the sliding mode accessibility and the mean square exponential stability of the closed-loop system under the action of the zero-order holder (ZOH). Summary of the Invention

[0006] To address the shortcomings of existing technologies, this paper proposes a sliding mode control method for an inverted pendulum system based on a random communication protocol using a PDT (Prolonged Dwell Time) switching rule, a sliding mode control ratio, and a random communication protocol. This method addresses the control issues inherent in traditional control methods when switching systems under nonlinear disturbances (such as wind resistance and random wind disturbances) and network protocol constraints, thereby improving the robustness and stability of the inverted pendulum system under complex conditions.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A sliding mode control method based on a random communication protocol for a continuous dwell time inverted pendulum system includes:

[0009] Step 1, establish a continuous dwell time inverted pendulum system;

[0010] Step 2: Design PDT switching rules. In the system, perform slow switching and fast switching according to response requirements.

[0011] Step 3: Design a random communication protocol scheduling based on Markov chain, and use the random communication protocol scheduling to ensure that only one node is selected to access the shared network at any time;

[0012] Step 4: Design a token-dependent sliding mode control law based on a random communication protocol:

[0013] Step 5: In the update rule of the actual control input, a set of ZOHs is used to store the values ​​received by the actuator end and convert the discrete signal into a continuous signal;

[0014] Step 6: Design a sliding mode function based on the continuous dwell time switching rule to make the system trajectory converge to the sliding mode surface and obtain an inverted pendulum closed-loop system;

[0015] Step 7: Verify that the state trajectory of the inverted pendulum closed-loop system can reach the neighborhood of the specified sliding surface.

[0016] In a preferred embodiment, in step 1, a model of a continuous dwell time inverted pendulum switching system is established:

[0017] ;

[0018] in, represents the state vector of the discrete-time system at time k+1, From the current moment Status , control input and nonlinear functions related to the state Joint decision, 、 Represented as a matrix of known constants; represents norm-bounded parameter uncertainty.

[0019] In a preferred embodiment, in step 2, each stage of the PDT switching sequence includes Partial and part, Partially represents an infinite number of durations not less than The non-intersecting intervals of Part, duration not exceeding ;

[0020] Indicates The total number of switching times in the time interval satisfies:

[0021] .

[0022] In a preferred embodiment, in step 3, define Indicates The selected executor that obtains the access token of the C / A network at all times; under the random communication protocol scheduling, It is represented by a Markov chain with a state transition matrix, where the transition probability for:

[0023] ; P represents probability

[0024] in, ,s, They represent the scheduling signals executed by the executor at time k and time k+1 respectively.

[0025] In a preferred embodiment, in step 4, the sliding mode control rate for:

[0026] ;

[0027] Among them, D O is a given matrix; D p diagonal matrix, is the token-dependent controller gain to be designed, A i is a known matrix, is a matrix with full column rank, sgn is the sign function, is a given matrix.

[0028] In a preferred embodiment, in step 5, the actual control input In the update rule, a set of ZOH is used to store the values ​​received by the executor:

[0029] ;

[0030] At the time of transmission When and only when the SCP scheduling signal When updating the sliding mode control rate v of the scheduling signal executed by the actuator at time k s (k); otherwise, no update is performed and the last value received by the executor side of the ZOH storage is used ;

[0031] Actual control input on the device side Expressed as:

[0032] ;

[0033] in, represents the Kronecker function, represents a diagonal matrix.

[0034] In a preferred embodiment, in step 6, the switching signal , , , then the sliding mode function for:

[0035] ;

[0036] Where, For a given matrix, B i is a matrix with full column rank;

[0037] The inverted pendulum closed-loop system is as follows:

[0038] ;

[0039] Where, is the transition matrix, .

[0040] In a preferred embodiment, in step 7, the sliding mode domain for:

[0041] ;

[0042] in, Represents the matrix W s,i The minimum eigenvalue of is an intermediate variable; M represents the scheduling signal set; N represents the switching signal set;

[0043] Selection depends on the scheduling signal and switching rules Lyapunov function for:

[0044] ;

[0045] in, is a real matrix;

[0046] calculate ,

[0047] Express expectations, is a multidimensional matrix. When the state trajectory stays in the region Other than that, that is:

[0048] ;

[0049] Verified , it proves that the closed-loop system is driven to the area around the sliding surface by the sliding mode control rate Inside.

[0050] In a preferred embodiment, the mean square exponential stability of the inverted pendulum system under the PDT switching rule is verified:

[0051] Selection depends on the scheduling signal and switching rules Lyapunov function for:

[0052] ;

[0053] Compute the expected difference of the Lyapunov function:

[0054] ;

[0055] Represents a multidimensional matrix;

[0056] Verification:

[0057] ;

[0058] Among them, β, is the coefficient, m1 is the minimum eigenvalue, and m2 is the maximum eigenvalue;

[0059] If satisfied , then the system is mean square exponentially stable.

[0060] The beneficial effects of the present invention are:

[0061] 1) By combining PDT switching rules and SCP scheduling, the present invention effectively balances the stability and flexibility of the inverted pendulum system and reduces the performance loss caused by frequent switching.

[0062] 2) The designed token-dependent sliding mode control rate can adapt to the communication constraints in the networked environment and effectively solve the data conflict problem.

[0063] 3) Through rigorous Lyapunov stability analysis, the mean square exponential stability of the closed-loop system is proved.

[0064] 4) The proposed control method has good robustness and can handle system uncertainties and nonlinear disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1This is a control system diagram of the present invention under SCP scheduling.

[0066] Figure 2 Schematic diagram of the inverted pendulum system trolley of the present invention.

[0067] Figure 3 FIG. 4 is a possible PDT switching sequence diagram of the present invention.

[0068] Figure 4 FIG. 4 is a variation curve diagram of the switching signal during the dwell time of the present invention.

[0069] Figure 5 The random communication protocol of the present invention is scheduled 's change curve.

[0070] Figure 6 The state vector of the inverted pendulum system under sliding mode control based on the random communication protocol of the present invention is The simulation curve of .

[0071] Figure 7 is the sliding variable of the present invention The simulation curve of .

[0072] Figure 8 The actual control input of the present invention The simulation curve of . DETAILED DESCRIPTION

[0073] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0074] Example 1

[0075] like Figure 1 As shown, the control system of the present invention includes: an inverted pendulum system, a sensor, a sliding mode controller, a random communication protocol, a zero-order holder and an actuator network.

[0076] The state of the inverted pendulum system is given by Indicates that the sensor measures the state of the inverted pendulum system The sliding mode controller receives the data transmitted by the sensor and transmits the control signal to the sliding mode controller through the random communication protocol. Transmitted via a shared communication channel To the actuator, SCP scheduling in the figure is used to reduce the network burden and reduce the risk of data conflict in the controller-to-actuator (C / A) network.

[0077] The sliding mode control method of an inverted pendulum system based on a random communication protocol of the present invention comprises the following steps:

[0078] Step 1: Establish a continuous dwell time inverted pendulum system.

[0079] An inverted pendulum system consists of a pendulum attached to a horizontally moving cart, e.g. Figure 2 shown.

[0080] The following inverted pendulum mathematical model of the PDT switching system is established:

[0081] (1);

[0082] in, represents the state vector of the discrete-time system at time k+1, From the current moment Status , control input and nonlinear functions related to the state Jointly determined, while also taking into account the parameter uncertainty of the system matrix ; 、 Represented as a matrix of known constants; represents norm-bounded parameter uncertainty,

[0083] Under normal working conditions, , is the acceleration due to gravity , represents the friction coefficient of the system, represents the sampling time,

[0084] Nonlinear functions , indicating air resistance With swing angle and angular velocity is proportional to the product of .

[0085] In the case of wind disturbance, , the nonlinear term is , , represents the wind interference coefficient, represents the wind speed function. represents the state vector, is the swing angle, is the angular velocity, represents the control input, represents nonlinear interference terms (such as air resistance, wind interference), and satisfies , 、 Expressed as a known constant matrix, by The form of represents norm-bounded parameter uncertainty, where and is a known constant matrix, is satisfied , represents the identity matrix, A switching signal indicating compliance with PDT switching rules.

[0086] To simplify writing, let the switching signal ,Right now , , so formula (1) can be rewritten as:

[0087] (2);

[0088] Step 2: Design PDT switching rules:

[0089] In order to reduce the performance loss caused by frequent system switching, enhance the robustness of the system, and make the system more flexible and stable, the PDT switching rule is adopted.

[0090] PDT switching sequence is as follows Figure 3 As shown, it is composed of infinite stages, each stage contains Partial and part, Partially represents an infinite number of durations not less than It ensures that the system runs stably on a certain subsystem for a period of time and avoids frequent switching. If the system state is stable during this period, slow switching may be performed. Part, duration not exceeding , the subsystem can be switched arbitrarily, and each residence time is less than ,When the system needs to respond quickly, fast switching can be performed.

[0091] Indicates The total number of switching times in the time interval satisfies:

[0092] (3);

[0093] The denominator is used to constrain the interval The inequality (3) gives the upper bound of the number of switching times within a given time interval, reflecting the limiting effect of the dwell time and the related tolerance on the switching frequency.

[0094] Step 3: Design random communication protocol scheduling:

[0095] In order to prevent data conflicts, a random communication protocol is used to ensure that only one node is selected to access the shared network at any one time. , M represents the set of scheduling signals. Under random communication protocol scheduling, the variable can be controlled by a Markov chain with a transition probability matrix, where the transition probabilities It is defined by the following formula:

[0096] (4);

[0097] in, , for all .

[0098] s, Represents the parameters of the actuator at different times.

[0099] Step 4: For the inverted pendulum system, design a token-dependent sliding mode control law based on a random communication protocol:

[0100] (5);

[0101] Therefore, the sliding mode control rate associated with the token for:

[0102] (6);

[0103] in, is the token-dependent controller gain to be designed.

[0104] Step 5: Create update rules:

[0105] In actual control input In the update rule of , a set of zero-order holders are used to store the values ​​received by the actuator:

[0106] (7);

[0107] Update rule (4) means that at the transmission time When and only when the SCP scheduling signal When the update Otherwise, the The executor remains unchanged and the last value stored by the zero-order holder will be used Therefore, the actual control input on the device side It can be expressed as:

[0108] (8);

[0109] in, , represents a diagonal matrix, , M represents the scheduling signal set, m represents the total number, represents the Kronecker function. Without loss of generality, we set .

[0110] In the PDT inverted pendulum control system of the present invention, the zero-order holder can effectively convert discrete signals into continuous signals, thereby ensuring the continuous execution of control instructions. At the same time, combined with the random scheduling strategy of SCP, it effectively avoids data conflicts caused by multiple nodes simultaneously occupying the channel, thereby improving the reliability of communication.

[0111] Step 6: Based on the inverted pendulum system model, design a sliding mode function based on the PDT switching rule:

[0112] (9);

[0113] Where, For a given matrix, is a matrix with full column rank.

[0114] In the PDT inverted pendulum system of the present invention, a sliding mode function such as Equation (9) is designed to transform the complex control problem into a stability problem that makes the system trajectory converge to the sliding mode surface. The sliding mode domain will be introduced later. .

[0115] Using (2), (6) and (8), the closed-loop system of the inverted pendulum is obtained as follows:

[0116] (10);

[0117] Where, is the transition matrix, .

[0118] Step 7: Reachability analysis of the inverted pendulum closed-loop system:

[0119] Prove that the state trajectory of the inverted pendulum closed-loop system (11) can reach the specified sliding surface In the neighborhood of . Sliding mode domain for:

[0120] (11);

[0121] Where, Represents the matrix W s,i The minimum eigenvalue of

[0122] (12);

[0123] In the formula is a real matrix. The choice depends on the scheduling signal and switching rules Lyapunov function for:

[0124] (13);

[0125] ,T represents transpose, is a multidimensional matrix. To simplify the formula, let , we can get the following formula:

[0126] (14);

[0127] In the formula is a real matrix,

[0128] calculate , Indicates expectation, when the state trajectory stays in the area Other than that, that is:

[0129] (15);

[0130] M represents the scheduling signal set; N represents the switching signal set;

[0131] It can be verified , which means that the closed-loop system (11) is driven by the sliding mode control rate to the region around the sliding surface Inside.

[0132] Step 8, inverted pendulum system stability analysis:

[0133] Analyze the stability of the closed-loop system (11) and choose the scheduling signal and switching rules Lyapunov function for:

[0134] (16);

[0135] , T represents transpose, Represents a multidimensional matrix. Calculate the expected difference of the Lyapunov function:

[0136] (17);

[0137] Verification:

[0138] (18);

[0139] Also because , is the coefficient, ≥1;

[0140] So by iterating the equation, we can get:

[0141] (19);

[0142] represents the kp moment, is the k1 moment

[0143] 、 Represents the coefficient of the period corresponding to the superscript, greater than zero and less than 1

[0144] Combining (3) and (19), we can obtain:

[0145] (20);

[0146] Where, .

[0147] m1 is P si The minimum eigenvalue, m2 is P si Maximum eigenvalue;

[0148] make , from formula (20) we can get:

[0149] (twenty one);

[0150] If satisfied , then the system (11) is mean square exponentially stable.

[0151] Example 2

[0152] The present invention will be further described below by giving the parameters of the inverted pendulum system.

[0153] Choose the pendulum mass to be ,length , the acceleration due to gravity , the friction coefficient of the system , we choose a system with two switching subsystems. According to the physical principles, we can get:

[0154] Subsystem 1 (normal operating mode):

[0155] ,

[0156] ,

[0157] ,

[0158] Subsystem 2 (in wind interference mode), the wind interference coefficient is :

[0159] ;

[0160] ,

[0161] parameter satisfy ,Pick In addition, the parameter selection in the PDT switching rule .

[0162] In the C / A channel, assuming that only one control node is allowed to send a control signal to the actuator at a time, there are two sending situations: , then the state transfer matrix is:

[0163] ;

[0164] According to the above simulation conditions, the system is simulated to verify the trajectory tracking capability of the system.

[0165] Figure 4 It represents the PDT switching signal of the present invention The timing diagram includes (Quick Switch) and (Continuous Dwell). PDT combines the advantages of DT and ADT, allowing Flexible switching within Keep stable inside.

[0166] The system can still maintain mean square exponential stability (MSES) when switching between wind disturbance mode (subsystem 2) and normal mode (subsystem 1).

[0167] Figure 5 The SCP scheduling signal described by the Markov chain of the present invention The random switching sequence of or 2 respectively corresponding to or The random communication protocol improves the utilization of network resources. The state transition matrix is ​​used to achieve fair scheduling, avoiding the channel congestion problem under traditional protocols.

[0168] Figure 6 The state variables (angle and angular velocity ) changes over time. Solid line The dotted line represents the change trajectory of the state variables of subsystem 1 over time. The time trajectory of the state variables of subsystem 2 is shown in Figure 2. Through PDT switching rules and SCP scheduling, the system state converges to the equilibrium point (0, 0) within a finite time, verifying the effectiveness of the control strategy. Even under nonlinear disturbances (such as wind resistance and random wind disturbances) and network protocol constraints, the system remains stable, demonstrating the robustness of the controller.

[0169] Figure 7 Sliding variable of the present invention The curve changing with time reflects the distance between the system state and the preset sliding surface. Indicates The graph of sliding variables changing with time, dotted line Indicates The present invention adopts the token-based SMC law (Formula 6), and the sliding variable enters and remains in the boundary layer within a finite time. The convergence of the sliding variables proves that the control law can effectively overcome the uncertainty and data conflicts induced by the network.

[0170] Figure 8 is the actual control input The curve of change over time contains two channels ( and The control input is dynamically updated through SCP scheduling (Equation 7) only when The control signal not only ensures system stability, but also significantly reduces network bandwidth usage and avoids data conflicts.

[0171] The above results show that the coordinated control method of the present invention can effectively solve the problems of multi-mode switching, limited communication resources and nonlinear interference in the networked inverted pendulum system. It has the advantages of fast response speed, strong adaptability and strong robustness, and effectively overcomes the uncertainty and data conflict induced by the network.

[0172] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. The embodiments should therefore be considered illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be encompassed therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

Claims

1. A random communication protocol sliding mode control method based on a continuous dwell time inverted pendulum system, characterized in that: include: Step 1, establish a continuous dwell time inverted pendulum system; Step 2: Design PDT switching rules. In the system, perform slow switching and fast switching according to response requirements. Step 3: Design a random communication protocol scheduling based on Markov chain, and use the random communication protocol scheduling to ensure that only one node is selected to access the shared network at any time; Step 4: Design a token-dependent sliding mode control law based on a random communication protocol: ||D(k)-D o -D p sgn(s(k))||≤2||D p ||; The sliding mode control rate v(k) is: Among them, D O is a given matrix; D p Diagonal matrix, K δ(k),i is the token-dependent controller gain to be designed, B i is a matrix with full column rank, sgn is the sign function, X i >0 is a given matrix; Step 5: In the update rule of the actual control input, a set of ZOHs is used to store the values ​​received by the actuator end and convert the discrete signal into a continuous signal; Step 6: Design a sliding mode function based on the continuous dwell time switching rule to make the system trajectory converge to the sliding mode surface and obtain an inverted pendulum closed-loop system; Step 7: Verify that the state trajectory of the inverted pendulum closed-loop system can reach the neighborhood of the specified sliding surface; the sliding mode domain Θ is: Among them, λ min (W s，i ) represents the matrix W s,i The minimum eigenvalue of s，i is an intermediate variable; M represents the scheduling signal set; N represents the switching signal set; the Lyapunov function V1(η(k), s, i) that depends on the scheduling signal δ(k) and the switching rule σ(k) is selected as: V1(η(k),s,i)=x T (k)P s,i x(k)+u T (k-1)Q s,i u(k-1)+s T (k)W s,i s(k); Among them, P s,i ,Q s,i ,W s,i is a real matrix; Calculate E{V1((η(k+1),δ(k+1),i)|η(k),δ(k),i)}-V1(η(k),δ(k),i), where E represents the expectation and η is a multidimensional matrix. When the state trajectory stays outside the region Θ, that is: Verification shows that E{V1((η(k+1),δ(k+1),i)|η(k),δ(k),i)}-V1(η(k),δ(k),i)<0, which proves that the closed-loop system is driven by the sliding mode control rate into the area Θ around the sliding surface; Step 8: Select the Lyapunov function that depends on the scheduling signal δ(k) and the switching rule σ(k) for: Compute the expected difference of the Lyapunov function: Represents a multidimensional matrix; verification yields: Among them, β and μ are coefficients, m1 is the minimum eigenvalue, and m2 is the maximum eigenvalue; if Then the system is mean square exponentially stable.

2. The random communication protocol sliding mode control method according to claim 1, characterized in that: In step 1, a model of a continuous dwell time inverted pendulum switching system is established: x(k+1)=(A σ(k) +ΔA(k))x(k)+B σ(k) (u(k)+f σ(k) (x(k),k)): Among them, x(k+1) represents the state vector of the discrete-time system at time k+1, and x(k+1) is composed of the state x(k) at the current time k, the control input u(k), and the nonlinear function f related to the state σ(k) (x(k),k) jointly decide, A σ(k) 、B σ(k) is represented as a matrix of known constants; ΔA(k) represents the parameter uncertainty with bounded norm.

3. The random communication protocol sliding mode control method according to claim 1, characterized in that: In step 2, each stage of the PDT switching sequence includes a T part and a τ part, where the τ part represents countless durations not less than τ. PDT non-intersecting intervals; in the T part, the duration does not exceed T PDT ; N v (k0,k) represents the total number of switching times in the [k0,k) time interval, satisfying:

4. The random communication protocol sliding mode control method according to claim 1, characterized in that: In step 3, δ(k) is defined to represent the selected executor that obtains the access token of the C / A network at time k; under the random communication protocol scheduling, δ(k) is represented by a Markov chain with a state transition matrix, where the transition probability π sl for: P represents probability; Where 0≤π sl ≤1, s and l represent the scheduling signals executed by the executor at time k and time k+1 respectively.

5. The random communication protocol sliding mode control method according to claim 1, characterized in that: In step 5, the actual control input u s In the update rule of (k), a set of zero-order holders are used to store the values ​​received by the actuator: At the transmission time k, if and only if the SCP scheduling signal δ(k) = s, update the sliding mode control rate v of the scheduling signal executed by the actuator at time k s (k); Otherwise, no update is performed and the last value u received by the actuator end stored in the zero-order holder is used s (k-1); The actual control input u(k) on the device side is expressed as: u(k)=Φ δ(k) v(k)+(I-Φ δ(k) )u(k-1); Where δ(·) represents the Kronecker function, Φ δ(k) represents a diagonal matrix.

6. The random communication protocol sliding mode control method according to claim 5, characterized in that: In step 6, the switching signal Then the sliding mode function s(k, i) is: Where, X i >0 is a given matrix, B i A is a matrix with full column rank, i is a known matrix; The inverted pendulum closed-loop system is as follows: Where, is the transition matrix,

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