Event-triggered finite-time second-order sliding mode control method for output constraint problem

Abstract

The application discloses an event-triggered finite-time second-order sliding mode control method for an output limited problem, adopts a new barrier Lyapunov function to process output constraints, proposes a new second-order sliding mode control algorithm, so that the output satisfies the constraint condition, and is successfully applied to a nonlinear system considering output constraints; the output variable is taken as a sliding variable, so that the output limited is constrained in the form of the sliding variable limited; in the case of ensuring that the output satisfies the constraint, it is proved that the sliding variable will not deviate from the constraint region and can be stabilized to the origin in a finite time; the barrier Lyapunov function is introduced into the event-triggered condition design, is combined with the second-order sliding mode control algorithm, ensures the stability of the output constraint nonlinear system, retains the advantage of fast convergence speed of the traditional continuous-time control, realizes the non-continuous-time control based on the event triggering, greatly reduces the controller update frequency, and completes the exclusion proof of the Zeno behavior.

Description

Technical Field

[0001] This invention belongs to the field of control technology, specifically relating to a finite-time second-order sliding mode control method. Background Technology

[0002] Sliding mode control is widely used to handle nonlinear systems due to its robustness and simple physical implementation. However, the chattering problem of traditional first-order sliding mode control and its requirement for the relative order of the system limit its practical application. To overcome the shortcomings of traditional sliding mode control, second-order sliding mode control, as the simplest and most special high-order sliding mode control, has received considerable attention in the past few decades. Second-order sliding mode control possesses the ideal characteristic of achieving finite-time convergence of closed-loop systems. Furthermore, second-order sliding mode control may also alleviate the chattering problem present in first-order sliding mode control. Compared with traditional sliding mode control, second-order sliding mode control provides higher accuracy for closed-loop systems.

[0003] Due to the inherent physical limitations of real-world systems, system output constraints are ubiquitous [ ]. The design of controllers for nonlinear systems with output constraints has attracted widespread attention from scholars both domestically and internationally. If the system does not meet the physical constraints, it will lead to a decline in system performance, or even system instability. We can see from existing research on the output constraint problem of nonlinear systems that very few results relate to second-order sliding mode control. This is because most second-order sliding mode control algorithms are based on some homogeneous properties of the closed-loop system, and output constraints will violate these properties. In fact, output constraints also significantly increase the difficulty of SOSM stability analysis.

[0004] In addition, the control methods in the above studies all consider the traditional time-triggered control strategy, that is, the time-driven controller uses clock cycle sampling to obtain the state information of the system, and the information update relies on high-frequency periodic sensor sampling. This control mode requires real-time interconnection of data between various modules inside the system. Even when the control command does not need to be updated, the system still needs to occupy a large amount of communication resources. The traditional time-triggered control strategy obviously brings a huge network communication burden, which can easily lead to channel congestion in severe cases and cause the controller to fail. To avoid the above problems, an event-triggered control strategy is proposed. According to the designed event-triggered mechanism, the controller is only triggered when a certain trigger condition is met (e.g., the control error is large), and then the control law is updated. On the premise of not affecting the performance of the control system, the event-triggered control mode makes the system no longer rely on real-time communication of signals, but executes control tasks on demand, thereby significantly reducing the amount of internal communication of the system and reducing the computational load. At present, there are few results that combine event triggering with the control problem of nonlinear systems with output constraints. This is because the difficulties of event-triggered control lie in the design of event-triggered conditions and the exclusion proof of Zeno behavior. The event-triggered condition determines when the controller updates, and the output constraint problem of nonlinear systems increases the requirement for the controller update frequency, thus greatly increasing the difficulty of event-triggered condition design and Zeno behavior exclusion proof. Summary of the Invention

[0005] In order to overcome the deficiencies of the prior art, the present invention provides an event-triggered finite-time second-order sliding mode control method for output constraint problems. A novel barrier Lyapunov function is used to handle output constraints, and a new second-order sliding mode control algorithm is proposed, which enables the output to meet the constraint conditions and is successfully applied to nonlinear systems considering output constraints; the output variable is used as the sliding variable, so that the output constraint is implemented in the form of sliding variable constraint; under the condition that the output meets the constraint, it is proved that the sliding variable will not leave the constraint region and can stabilize to the origin in finite time; the barrier Lyapunov function is introduced into the design of event-triggered conditions and combined with the second-order sliding mode control algorithm, which ensures the stability of the output-constrained nonlinear system while retaining the advantage of fast convergence rate of traditional continuous-time control, realizes event-triggered based discontinuous-time control, greatly reduces the update frequency of the controller, and completes the exclusion proof of Zeno behavior.

[0006] The technical solution adopted by the present invention to solve its technical problem comprises the following steps:

[0007] Step 1: Premise assumption;

[0008] define the following three lemmas are introduced:

[0009] Lemma 1: Assume a>0 and 0<b≤1, for The following formula holds true:

[0010]

[0011] Lemma 2: Let θ(x,y)>0 be a function of m and n. For any positive constants a1 and a2, we have:

[0012]

[0013] Lemma 3: For The following inequalities apply to Hengchengli:

[0014] (|m1|+…|m n |) ι ≤|m1| ι +…+|m n | ι

[0015] Step 2: Establish a nonlinear system model under output constraints;

[0016] The nonlinear system model under output constraints is described as follows:

[0017]

[0018]

[0019] in, These are the system state and control input, respectively. f(t,x) and g(t,x) are smooth functions, and y is the system output with a correlation r = 2 relative to the control input u(t). It is any positive real number;

[0020] By taking the system's output variable as the sliding variable, the output constraint becomes the sliding variable constraint, and the system models (1) and (2) are rewritten as follows:

[0021]

[0022]

[0023] Taking the second derivative with respect to the sliding variable s, we get:

[0024]

[0025] in and a(t,x) and b(t,x) are uncertain, but they satisfy the following assumptions:

[0026] Assumption 1: For the sliding mode dynamics equation (5), there exists a positive number and a positive function M(x) such that:

[0027]

[0028] Step 3: Design of event-triggered condition;

[0029] The event triggering error is constructed as follows:

[0030] e=w(t)-u(t) (7)

[0031] The definition of w(t) is shown in formula (14);

[0032] The event triggering condition is as follows:

[0033] g=||e||-ηΘ (8)

[0034]

[0035]

[0036] wherein η∈(0,1), μ≥a≥λ1, λ3=λ1-2ε≥0, ε>0, σ1>0, κ1 and κ2 are positive numbers;

[0037] The triggering instant sequence, 0<t0<t1<t2<t3<…, is defined as follows:

[0038] t k+1 =min{t>t k |g>0},k=0,1,2,… (11)

[0039] If g>0, the controller and the actuator will be executed. Therefore, the triggering condition (8) satisfies the following relation:

[0040] ||e||≤ηΘ (12)

[0041] Step 4: Establishing an event-triggered second-order sliding mode finite-time controller for the output constraint problem of nonlinear systems;

[0042] Considering that the initial value of system (3) enables the sliding variable s to be within the constraint (4) at the initial time, there exist two positive constants σ1 and σ2 such that the event-triggered second-order sliding mode finite-time controller ensures that in system (3), the establishment in finite time is achieved, and the sliding variable s satisfies the constraint condition (4) for all t≥0;

[0043]

[0044]

[0045]

[0046]

[0047] in λ2=λ1-ε>0, μ≥a≥λ1, λ3=λ1-2ε≥0, σ1>0,

[0048] Furthermore, γ1(σ1) and γ4(σ1) are defined as follows:

[0049]

[0050]

[0051] in

[0052] The beneficial effects of this invention are as follows:

[0053] 1) The method of this invention adopts the backstepping controller design concept, making the controller design more flexible and convenient;

[0054] 2) This method adopts an event-triggered mechanism, which enables the controller to be triggered only when a certain triggering condition is met (e.g., the control error is large), thereby updating the control law. This avoids the network communication burden caused by the traditional time-triggered control strategy, greatly saves resource consumption, and improves system stability.

[0055] 3) This method innovates the second-order sliding mode method, relaxes the assumptions in the conventional second-order sliding mode control algorithm, and the proposed control strategy is more universal and general.

[0056] 4) This method uses a novel obstacle Lyapunov function to handle output constraints and innovatively proposes a new second-order sliding mode control algorithm, which enables the output to meet the constraint conditions and is successfully applied to nonlinear systems considering output constraints.

[0057] 5) In the design process, this method cleverly uses the output variable as the sliding variable, so that the output is constrained in the form of the sliding mode variable, which greatly facilitates the output constrained control problem.

[0058] 6) This method, while ensuring that the output meets the constraints, can be mathematically proven to ensure that the sliding mode variable will not leave the constrained region and will stabilize to the origin in a finite time.

[0059] 7) This method innovatively introduces the obstacle Lyapunov function into the design of event-triggered conditions and combines it with the second-order sliding mode control algorithm. While ensuring the stability of the output-constrained nonlinear system, it retains the advantage of fast convergence speed of traditional continuous-time control, realizes event-triggered non-continuous-time control, greatly reduces the controller update frequency, saves computing and communication resources, and completes the proof of the exclusion of Zeno behavior. Attached Figure Description

[0060] Figure 1 This is a flowchart of the method of the present invention.

[0061] Figure 2 This is a diagram showing the changes in restricted state variables according to an embodiment of the present invention.

[0062] Figure 3 This is a diagram showing the changes in control variables according to an embodiment of the present invention.

[0063] Figure 4 This is a diagram showing the event triggering times in an embodiment of the present invention.

[0064] Figure 5 This is a diagram showing the number of triggers in an embodiment of the present invention.

[0065] Figure 6 This is a diagram showing the changes in constrained state variables in the control of the output constraint problem of a non-event-triggered nonlinear system according to an embodiment of the present invention.

[0066] Figure 7 This is a graph showing the changes in control variables for controlling the output limitation problem of a non-event-triggered nonlinear system according to an embodiment of the present invention.

[0067] Figure 8 This is a diagram showing the changes in constrained state variables in a non-event-triggered, non-output-constrained nonlinear system control according to an embodiment of the present invention.

[0068] Figure 9 This is a diagram showing the changes in control variables for a non-event-triggered, non-output-constrained nonlinear system control according to an embodiment of the present invention. Detailed Implementation

[0069] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0070] The present invention provides an event-triggered finite-time second-order sliding mode control method for output constraint problems. Based on the characteristics of a novel barrier Lyapunov function, research is conducted on the problems of high control difficulty, low control accuracy, low convergence speed, violation of output constraint conditions, chattering in the control process, large calculation load, and high design difficulty of event-triggered conditions existing in the output constraint problem of nonlinear systems. The method of the present invention successfully innovates the second-order sliding mode method, relaxes the assumption conditions in conventional traditional second-order sliding mode control algorithms, and the proposed control strategy is more universal and general; a novel barrier Lyapunov function is used to handle output constraints, and a new second-order sliding mode control algorithm is innovatively proposed, so that the output satisfies the constraint conditions and is successfully applied to nonlinear systems considering output constraints; in the design process of the new second-order sliding mode control algorithm, the output variable is skillfully used as the sliding variable, so that the output constraint is implemented in the form of sliding variable constraint; under the condition that the output is guaranteed to satisfy the constraint, it is proved that the sliding variable will not leave the constraint region and can stabilize to the origin in finite time, and the stability analysis and proof are completed; the barrier Lyapunov function is innovatively introduced into the design of event-triggered conditions and combined with the second-order sliding mode control algorithm, which ensures the stability of output-constrained nonlinear systems while retaining the advantage of fast convergence of traditional continuous-time control, realizes event-triggered discontinuous-time control, greatly reduces the update frequency of the controller, saves computing and communication resources, and completes the exclusion proof of Zeno behavior at the same time.

[0071] The technical solution adopted by the present invention to solve its technical problem is characterized by comprising the following four design steps: (1) premise assumptions (2) establishing a nonlinear system model under output constraints (3) design of event-triggered conditions (4) establishing an event-triggered second-order sliding mode finite-time controller for the output constraint problem of nonlinear systems.

[0072] Step 1: Premise assumptions;

[0073] Define The following three lemmas are introduced.

[0074] Lemma 1 Assume that a>0 and 0<b≤1, for The following expressions hold:

[0075]

[0076] Lemma 2 Let θ(x,y)>0 be a function of m and n. For any positive constants a1 and a2, the following can be obtained;

[0077]

[0078] Lemma 3 For The following inequality holds for always holds:

[0079] (|m1|+…|m n |) ι ≤|m1| ι +…+|m n | ι

[0080] Step 2: Establish a nonlinear system model under output constraints;

[0081] The nonlinear system model under output constraints is described as follows:

[0082]

[0083]

[0084] in, Here, f(t,x) and g(t,x) are the system state and control input, respectively. f(t,x) and g(t,x) are smooth functions, and y is the system output with a correlation r = 2 relative to the control input u(t). It is any positive real number.

[0085] Since the sliding variable typically consists of the system output and the reference signal, both of which are measurable, the system output variable can be taken as the sliding variable. Thus, the output constraint becomes the sliding variable constraint, and system models (1) and (2) are rewritten as follows:

[0086]

[0087]

[0088] Taking the second derivative with respect to the sliding variable s, we get:

[0089]

[0090] in and a(t,x) and b(t,x) are uncertain, but they satisfy the following assumptions.

[0091] Assumption 1: For the sliding mode dynamics equation (5), there exists a positive number. And a positive function M(x) such that:

[0092]

[0093] Step 3: Design event triggering conditions;

[0094] To reduce resource consumption and communication burden between the internal controller and actuators, the event-triggered error is constructed as follows:

[0095] e = w(t)-u(t) (23)

[0096] The definition of w(t) will be given later.

[0097] The event-triggered condition is as follows:

[0098] g = ||e||-ηΘ (24)

[0099]

[0100]

[0101] where η ∈ (0, 1), μ ≥ a ≥ λ1, λ3 = λ1-2ε ≥ 0, ε > 0, σ1 > 0, κ1 and κ2 are positive numbers, which are defined in the subsequent proof process.

[0102] The trigger moment sequence 0 < t0 < t1 < t2 < t3 < … can be defined as follows:

[0103] t k+1 = min{t>t k |g>0}, k = 0, 1, 2, … (27)

[0104] If g > 0, the controller and actuator will be executed, therefore, trigger condition (8) will satisfy the following relationship.

[0105] ||e|| ≤ ηΘ (28)

[0106] Step 4: Establish an event-triggered second-order sliding mode finite-time controller for output constraint problems of nonlinear systems;

[0107] Considering that the initial value of system (3) makes the sliding variable s within the constraint (4) at the beginning, there are two positive constants σ1 and σ2, so that the event-triggered second-order sliding mode finite-time controller ensures that in system (3) it is established in finite time, and the sliding variable s satisfies the constraint condition (4) for all t ≥ 0.

[0108]

[0109]

[0110]

[0111]

[0112] wherein λ2 = λ1-ε > 0, μ ≥ a ≥ λ1, λ3 = λ1-2ε ≥ 0, σ1 > 0, γ1(σ1) and γ4(σ1) are defined in the subsequent proof.

[0113] Considering the output-constrained nonlinear system (3)(4) and triggering condition (8) under assumption 1, the second-order sliding mode finite-time control law based on event triggering is given by (13)-(16), which can guarantee that the system does not violate the constraint condition and tends to be stable within a finite time, while reducing resource and energy consumption. Zeno's phenomenon can be successfully avoided, and the control problem of the output-constrained nonlinear system can be successfully solved.

[0114] Example:

[0115] like Figure 1 As shown, an event-triggered finite-time second-order sliding mode control method for output-constrained problems is proposed, and its mathematical derivation is as follows:

[0116] The systematic expression (5) can be rewritten in the following form:

[0117]

[0118] It only needs to be proven that the initial state of s1 lies within the constraint. Under the premise of the above output-constrained nonlinear system (17), the above output-constrained nonlinear system will converge and stabilize to the origin in a finite time, and Zeno's phenomenon will not exist.

[0119] Mathematical derivation step 1: Select a Lyapunov function:

[0120]

[0121] Where μ, λ1, and ε are positive numbers satisfying μ ≥ λ1 ≥ 0 and ε > 0, and the Lyapunov function V1(s1) is defined on the region Above. Take the derivative of V1(s1) along system (17) as follows:

[0122]

[0123] make Equation (19) becomes:

[0124]

[0125] Where λ2=λ1-ε>0, It is the virtual control law that was subsequently determined.

[0126] Based on the backstepping method, a virtual controller is constructed as follows: Where σ1>0, μ≥a≥λ1, From (20), we can directly deduce that:

[0127]

[0128] Mathematical derivation step 2: Consider the following Lyapunov function:

[0129]

[0130] Where λ3 = λ2 - ε ≥ 0, r3 is defined on the interval [0, 1]. Therefore, the function V2(s1,s2) can be considered as a region. The Lyapunov function on the system. Differentiating the Lyapunov function V2(s1,s2) along the system (17) yields:

[0131]

[0132] in

[0133] Estimate the right-hand side of equation (23) above:

[0134] Given the fact that 0 < λ1 / a ≤ 1 and

[0135] Applying Lemma 1

[0136]

[0137] Using Lemma 2 and Equation (24), we can obtain:

[0138]

[0139] Where γ1(σ1) is defined as:

[0140]

[0141] On the other hand, from Lemma 1 we can obtain:

[0142]

[0143] Will Substituting, we get:

[0144]

[0145] By using Lemma 3, it is not difficult to conclude that:

[0146]

[0147] With the help of equation (17), and combining (27) and (28), we get:

[0148]

[0149] By Lemma 2, we can obtain from (29) that there exist two positive constants γ2(σ1) and γ3(σ1) such that:

[0150]

[0151] in

[0152] Substituting (30) into (26) gives:

[0153]

[0154] Note that ψ(s1) ≥ 1. Using Lemma 2, we get again:

[0155]

[0156] Where γ4(σ1) is defined as

[0157]

[0158] Substituting formulas (25) and (33) into (23), we get:

[0159]

[0160] By using assumption 1, formula (13) and facts We can obtain:

[0161]

[0162] Substituting (7), (9), (10), (12), (13), and (14) into (35), and combining this with Young's inequality, we obtain:

[0163]

[0164] Organize (36), for all

[0165]

[0166] Where η∈(0,1). Next, we will prove that the closed-loop system (17) is stable in finite time;

[0167] Note that for any For all x ≥ tan(x), this property implies the existence of a small region around the origin. For all All

[0168]

[0169] Through simple calculation, we can obtain:

[0170]

[0171] in It is very clear that:

[0172]

[0173] choose Based on the fact Ψ(s1)≥1, for The following is true:

[0174]

[0175] According to the finite-time Lyapunov theory, in the region The system (17) defined above can be stabilized for a finite time by the controller (36).

[0176] Finally, the boundedness of the sliding variable s1 will be verified. By substituting V2(s1,s2)=V1(s1)+W2(s1,s2) and (37) into the equation, the boundedness of the sliding variable s1 will be verified. The following is true:

[0177] V1(s1(t))≤V2(s1(t),s2(t))≤V2(s1(0),s2(0))

[0178] Given the definition of V1(s1), we can obtain:

[0179]

[0180] It can be rewritten as

[0181]

[0182] Right now

[0183]

[0184] Obviously, according to formula (39), we can obtain

[0185]

[0186] This directly led to

[0187] Note that controller (36) can be rewritten as (13). This means that under constraint (4), controllers (13)-(16) can guarantee that system (3) is established in finite time.

[0188] The theoretical proof for establishing Zeno's free actions is then given as follows.

[0189] Based on the above analysis, we can conclude that s, It is bounded. Therefore, we can obtain It is also bounded. Let's define a function χ(t) as follows.

[0190] χ(t)=||e||

[0191] Then, using the facts, we can obtain:

[0192]

[0193] Where δ is The upper boundary.

[0194] Considering triggering condition (12), the following relationship is always satisfied:

[0195]

[0196] Then, we can get

[0197] T min =t k+1 -t k ≥ηΘ>0 (58)

[0198] Therefore, Zeno-free behavior is guaranteed by formula (42).

[0199] Q.E.D.

[0200] Example

[0201] Consider the following specific nonlinear system:

[0202]

[0203] in g(t,x)≥1. The state constraints are as follows:

[0204]

[0205] Table 1 Controller and Triggering Condition Parameters

[0206]

[0207] The controller and trigger condition parameters are shown in Table 1.

[0208] Design a controller such that state x1 converges to zero while satisfying constraint (44).

[0209] We can let the sliding variable s = x1. From formula (43), we have the following dynamic equation for the sliding variable:

[0210]

[0211] in b(t,x) = g(t,x). This means that we can obtain:

[0212]

[0213] Considering the state constraints (44), the constraints imposed on the sliding variable s

[0214] In the embodiment, the initial state was chosen as (x1(0), x2(0)) = (1.95, 1.5), and the uncertainty was assumed to be... g(t,x)=1+0.5(sin(t)) 2 The sampling time is set to 0.0001s.

[0215] Results analysis:

[0216] The simulation results of the novel event-triggered finite-time second-order sliding mode control method for the output-constrained problem of nonlinear systems are shown in the figure. Figure 2 — Figure 6 .

[0217] from Figure 2 It can be clearly observed that the restricted state variable x1 converges to 0, and does not exceed the boundary value during the entire convergence process. Furthermore, when the constrained state variable x1 approaches its boundary value, it will be constrained because the gain function also increases as x1 approaches its boundary value, providing stronger control. This can be demonstrated through... Figure 3 This has been verified.

[0218] Figure 3 The time history of the controller is displayed. It is easy to see that as the state variable x1 approaches the boundary, the control input signal becomes very large, which verifies the previous theoretical analysis.

[0219] Figure 4 It is the event trigger interval. Figure 4 , Figure 5 , Figure 6 , Figure 7 In summary, it is easy to see that the event-triggered controller greatly reduces the number of controller triggers, saves energy consumption for the system, and ensures that the system remains stable under conditions where output variables are limited.

[0220] By comparison Figure 8 and Figure 2 It is not difficult to see that the controller proposed in this invention has a good control effect on nonlinear systems considering output constraints, and the output constraint boundary value is not exceeded throughout the entire process. Through comparison... Figure 9 and Figure 3It can be clearly observed that the control strategy proposed in this invention greatly reduces the update frequency of the controller and saves internal energy consumption of the system. At the same time, by utilizing the characteristics of the tangent function, it avoids the disadvantage of long convergence time of event-triggered discontinuous control and can achieve system stability in a short and finite time.

Claims

1. An event-triggered finite-time second-order sliding mode control method for output-constrained problems, characterized in that, Includes the following steps: Step 1: Premise Assumptions; definition We introduce the following three lemmas: Lemma 1: Assumption as well as ,for The following formula holds true: Lemma 2: Let Become and The function, for any positive constant , ,get: Lemma 3: For The following inequalities apply to Hengchengli: Step 2: Establish a nonlinear system model under output constraints; The nonlinear system model under output constraints is described as follows: (1) (2) in, , These are the system status and control inputs, respectively. , It is a smooth function. It is the system output and is relative to the control input. There is a relevance ; , is any positive real number; By taking the system's output variable as the sliding variable, the output constraint becomes the sliding variable constraint, and the system models (1) and (2) are rewritten as follows: (3) (4) For sliding variables Taking the second derivative, we get: (5) in and , and It is uncertain, but the following assumptions are met: Assumption 1: For the sliding mode dynamics equation (5), there exists a positive number and positive function Make: (6) Step 3: Design event triggering conditions; The event triggering error is constructed as follows: (7) The definition is shown in equation (14); The event is triggered under the following conditions: (8) (9) (10) in , , , , , , , It is a positive number; Trigger time sequence, It is defined as follows: (11) if The controller and actuator will be executed, therefore, the trigger condition formula (8) satisfies the following relationship: (12) Step 4: Establish an event-triggered second-order sliding mode finite-time controller for the output-constrained problem of nonlinear systems; Considering the initial values ​​of the system model (3), the sliding variable Initially, within the constraint formula (4), there exist two positive constants. and This ensures that the event-triggered second-order sliding mode finite-time controller guarantees the system model (3) The finite time establishment, while the sliding variable Satisfy all The constraint condition formula (4); (13) (14) (15) (16) in , , , , , ; The and Defined as: in , .

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