Finite time adaptive high-order sliding mode control method based on variable condensation method

Abstract

The invention discloses a finite time adaptive high-order sliding mode control method based on a variable condensation method. Aiming at unknown time-varying parameters existing in the system, designing an adaptive law by using a variable agglutination method; in addition, in order to realize finite time stability, a finite time adaptive second-order sliding mode controller is designed by using a power integral technology. And expanding the result of the finite time adaptive second-order sliding mode controller to the design of a finite time adaptive n-order sliding mode controller (n is greater than or equal to 3). According to the control method disclosed by the invention, the limitation that system uncertainty upper bound information is needed in the design of a traditional high-order sliding mode control method is broken through; secondly, the dynamic adjustment of the control gain can be realized, the big problem of excessive gain is avoided, and the buffeting is further weakened; in addition, the controller can enable the controlled system to realize finite time stability and obtain a higher convergence speed.

Description

Technical Field The invention relates to high-order sliding mode control, and in particular to a finite-time adaptive high-order sliding mode control method based on a variable condensation method. Background Art Uncertainty and disturbance are widely present in nonlinear systems. Sliding mode control has attracted much attention due to its strong robustness. In early studies, a series of first-order sliding mode control algorithms were proposed, but their main limitations were the chattering phenomenon and the relative degree of the sliding variable was 1. Therefore, the second-order sliding mode algorithm was proposed, and the super-helical control algorithm and suboptimal control algorithm were subsequently proposed [3]. However, high-order sliding mode (n ≥ 3) can break through the limitation of the relative order of the sliding variable and can further weaken the chattering. Therefore, high-order sliding mode control has gradually become a research hotspot. However, many studies on high-order sliding mode control require information about known uncertainty boundaries, which has certain limitations in practical applications and usually leads to excessive control gain [4]. To solve this problem, various adaptive high-order sliding mode controllers have been proposed, which dynamically adjust the control gain by designing adaptive laws. However, these methods can only ensure that the sliding variable converges to a certain neighborhood of zero. A method based on a low-pass filter to approximate the minimum feasible value of the control gain was proposed [5], but it still does not completely get rid of the dependence on uncertainty boundary information. Therefore, how to break through the limitations of system uncertainty boundary information has become an important research topic. In addition, finite-time high-order sliding mode has attracted much attention due to its faster convergence speed and strong anti-interference ability. By introducing nonlinear functions in the sliding surface, the terminal sliding mode control algorithm ensures that the system state converges to the origin within a finite time. However, this method has a singularity problem, so the non-singular terminal sliding mode was proposed [6]. Subsequently, a finite-time high-order sliding mode control method based on power integral technology was proposed, which can achieve faster convergence speed and avoid the singularity problem [7]. Considering that time-varying parameters are widely present in practical control problems, such as the time-varying armature resistance and flux linkage of motors, and the time-varying inertia parameters of spacecraft, it is particularly necessary to study the design problem of finite-time high-order sliding mode control algorithm for unknown time-varying parameters. Therefore, this patent explores the design problem of finite-time adaptive high-order sliding mode controller considering the existence of unknown time-varying parameters.

[0001] Y.Feng, F.Han, and X.Yu, "Chattering free full-order sliding-modecontrol," Automatica, vol.50, no.4, pp.1310-1314, 2014.

[0002] F.Zhang,Y.Chen,and Y.Zhang,“Neural Network Boundary Approximation forUncertain Nonlinear Spatiotemporal Systems and Its Application ofTrackingControl,”IEEE Transactions on Neural Networks andLearning Systems,vol.35,no.5,pp.7238-7243,2024.

[0003] G.Bartolini,A.Ferrara,and E.Usai,“Outputtracking control ofuncertainnonlinear second-order systems,”Automatica,vol.33,no.12,pp.2203-2212,1997.

[0004] V.I.Utkin,and A.S.Poznyak,“Adaptive sliding mode control withapplication to super-twist algorithm:Equivalentcontrolmethod,”Automatica,vol.49,no.1,pp.39-47,2013.

[0005] C.Qian and W.Lin,“A continuous feedback approach to global strongstabilization ofnonlinear systems,”IEEE Transactions onAutomatic Control,vol.46,no.7,pp.1061-1079,2001.

[0006] A.Levant,“Homogeneity approachto high-order sliding mode design,”Automatica,vol.41,no.5,pp.823-830,2005.

[0007] J.Song, Y.Zou, and Y.Niu, "Finite-time stabilization via sliding modecontrol, IEEE Transactions on Automatic Control, vol.62, no.3, pp.1478-1483, 2017. Summary of the invention Purpose of the invention: The present invention is aimed at nonlinear systems and considers the design problem of finite-time adaptive high-order sliding mode controllers in the presence of unknown time-varying parameters. The controlled nonlinear system can achieve the following features: 1) break through the limitation of requiring system uncertainty information when designing the controller; 2) dynamically adjust the control gain to effectively weaken chattering; 3) achieve finite-time stability and obtain faster convergence speed. Technical solution: The finite-time adaptive high-order sliding mode control method based on the variable condensation method described in the present invention firstly uses the variable condensation method to design the adaptive law for the unknown time-varying parameters of the nonlinear system, and uses the power integral technique to design the finite-time adaptive second-order sliding mode controller; then, the result of the finite-time adaptive second-order sliding mode controller is extended to the design of the finite-time adaptive n-order sliding mode controller. The specific design process includes the following steps: 1) Design process of finite-time adaptive second-order sliding mode control method. Design the Lyapunov function as follows: Derivative of the above formula: Designing a virtual controller As follows: where β1>0 and Bringing Virtual Controllers into have to: Choose the Lyapunov function as follows: in Derivative of the above formula: The finite-time adaptive second-order sliding mode controller is designed as follows: in The adaptive law is designed as: in as well as Bring the controller in have to: The proof of finite-time stability is given below: make Where Y1(s1,s2)=V 1,1 (s1)+W 1,1 (s1,s2) and So we can get: in The finite-time stability is proved. 2) Design process of finite-time adaptive n(n≥3)-order sliding mode control method. Select the Lyapunov function V 2,1 , which has the same form as V in Problem 1 1,1 , The calculation process is the same as V 1,1 The derivation process, in which the virtual controller is designed as follows: Where β1≥n. Then, we can get: Select the Lyapunov function Same as in 1) The derivation calculation process is similar to 1), and the final result is as follows: Therefore, we can get: Assume that a suitable positive constant β can be found in step i-1 j ,j=1,…,i-1,and a positive definite function V 2，i-1 So that: The form of the Lyapunov function for the i-th step is selected as follows: in Defined as: The calculation process similar to 1) is as follows: In summary: Select the Lyapunov function as follows: right The derivative is: Design the controller u as follows: in The adaptive law is designed as: So we get: The proof process of finite-time stability is similar to the proof process of finite-time adaptive n-th order sliding mode controller. make as well as Calculated: in The proof of finite-time stability is complete. The beneficial effects of the present invention are: first, it breaks through the limitation of the traditional high-order sliding mode control method design that requires the upper limit information of the system uncertainty; second, it can realize the dynamic adjustment of the control gain to avoid the problem of excessive gain and thus weaken the chattering; in addition, the controller can make the controlled system achieve finite time stability and obtain faster convergence speed. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 Sliding variable response curve for simulation 1; Figure 2 This is the control input response curve of simulation 1; Figure 3 It is the control parameter response curve of simulation 1; Figure 4 Compare the simulated control input response curve for simulation 1; Figure 5 Sliding variable response curve for simulation 2; Figure 6 is the control parameter response curve of simulation 2; Figure 7 Control input response curve for simulation 2. Figure 8 The present invention is a flow chart of the method. DETAILED DESCRIPTION The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments, but the content of the present invention is not limited to the scope of the embodiments. like Figure 8As shown, the present application discloses a finite time adaptive high-order sliding mode control method for a type of nonlinear system with unknown time-varying parameters. First, for the second-order sliding mode, the unknown time-varying parameters are processed using the variable condensation method. In this method, the time-varying parameters are transformed into the following form: The controller design based on this transformation does not require the upper bound information of the time-varying parameters, but only the range of the time-varying parameters. Then, the finite-time adaptive high-order sliding mode controller is designed based on the power integral technique and the quasi-backstepping method, and the finite-time stability proof process based on the Lyapunov method is given. Finally, two simulation examples are given to verify the effectiveness of the proposed method. The specific process of controller design is given below. Consider the following nonlinear system s=s(t,x), in express control input; Indicates the system status. s: represents a sliding variable and the relative order is n, that is: s (n) =a(t,x)+b(t,x)u, where a(x,t): and b(x,t): are unknown functions respectively. According to the literature [2], the above sliding mode can be parameterized as: s (n) =φ1(s i )θ1(t)+θ2(t)u Where φ1(s i ): For a smooth mapping, and is an unknown time-varying parameter and satisfies the following assumptions: Assumption 1: The parameter θ1(t) is piecewise continuous and for t ≥ 0, θ1(t)∈Θ1, where Θ1 is a compact set and the “radius” of Θ1, ie Assume that θ is known and θ1 is unknown. Assumption 2: The parameter θ2(t) is bounded and non-zero, that is, there exists a positive constant Make For t ≥ 0, sign(θ2(t)) is known and unchanged. Order i =s i-1 ,i=1,...,n, the high-order sliding mode dynamics can be described as: Question 1: Considering the sliding mode with a relative order of 2, based on assumptions 1 and 2, an adaptive second-order sliding mode controller is designed so that the sliding mode can stabilize to the origin within a finite time, that is, in is the stabilization time. Design the Lyapunov function as follows: Where ρ, r1 and r2 = r1-τ are real numbers and satisfy ρ≥a≥r1>0 and a positive real number τ. Taking the derivative of V1, we get: Designing a virtual controller The following form: where β1>0 and Bringing Virtual Controllers into have to: Choose the Lyapunov function as follows: The second term of the above formula is designed based on the power integration technique, and the form is designed as follows: where r3=r2-τ≥0; the other two terms are used to estimate θ1(t) and θ2(t). γ1 and γ2 are adaptive gains, the center of variation (ie, as well as ) does not need to be known, which is the key idea of ​​variable condensation. V 1,2 The derivative is: because as well as: According to Theorem 1: According to Theorem 2: in: According to Theorem 3: consider have: Combining the above two equations, we get: According to Theorem 2: in: Applying Theorem 1 and Theorem 2, we can get: in: In summary: The finite-time adaptive second-order sliding mode controller is designed as follows: in The adaptive law is designed as: in as well as Bring the controller in have to: The proof of finite-time stability is given below: make Where Y1(s1,s2)=V 1,1 (s1)+W 1,1 (s1,s2) and According to the definition of Y1: in According to Theorem 3: Therefore, we can get: in We can find c0∈(0,1) such that 0≤Y2≤c0V2 holds, so we can get: in Therefore, the system can achieve finite-time stability. Question 2: Considering the sliding mode with a relative order of n, based on assumptions 1 and 2, an adaptive n-order sliding mode controller is designed so that the sliding mode can stabilize to the origin within a finite time, that is, Step 1: Select the Lyapunov function V 2,1 , which has the same form as V in Problem 1 1,1 , The calculation process is the same as V 1,1 The derivation process, in which the virtual controller is designed as follows: where β1≥n, Then, we can get: Step 2: Select the Lyapunov function Same as question 1 The derivation calculation process is similar to problem 1, and the final result is as follows: Designing virtual controls as follows: in Therefore, we can get: Step 3: Assume that a suitable positive constant β can be found in step i-1 j ,j=1,…,i-1, and a positive definite function V 2,i-1 get: in: The Lyapunov function for the i-th step is selected as follows: in Defined as: Carry out the calculation process similar to the second step, we have: Designing a virtual controller The following form in In summary: Step 4: Select the Lyapunov function as follows: right The derivative is: Design the controller u as follows: in The adaptive law is designed as: So we get: The proof of finite-time stability is similar to the proof of finite-time adaptive second-order sliding mode controller. make as well as Calculated: So we can get: in The finite-time stability is proved. Step 5: Simulation experiment (a) Simulation experiment 1: Consider the following system Where x1 and x2 are position and velocity respectively. S f (x2) = arctan(0.1x2); K f (t) = 1 + 0.1 sin (t); Ψ1(t) = 1 + sin(t) and Ψ2(t) = 2 + cos(0.5πt) represent unknown time-varying parameters; u is the control voltage. Let s1 = x1 - x 1d , the following sliding mode is obtained The controller u uses the finite-time adaptive second-order sliding mode controller designed in this paper, and the parameters are selected as follows: ρ=a=r1=1.2,τ=5 / 12,β1=0.55,γ1=0.1,γ2=0.1,x 1d =0.1, Figure 1 — Figure 4 is the simulation result diagram, Figure 1 It is the response curve of sliding variable under different initial values; Figure 2 It is the response curve diagram of the control input; Figure 3 To control the gain as well as Dynamic response curve of Figure 4This is a comparative experimental diagram of the control designed in this paper and the controller in the literature [1]. Under the action of the algorithm disclosed in this patent, the sliding variable can be stabilized to the origin within a finite time. Under the action of the adaptive law, the dynamic adjustment of the control gain can be achieved, thereby effectively weakening the vibration. (b) Simulation experiment 2: Consider the following nonlinear system Where x1, x2 and x3 are system states, and u is the control input. The sliding variable is selected as s1=x1, and we get Where φ1(s1,s2)=x2(0.02(x1)+0.1sin(0.1x1)); θ1(t)=1+0.1(sin(0.1t)) 2 And θ2(t)=1+0.01cos(t) is the time-varying parameter, and the control coefficient is selected as ρ=a=r1=1,τ=8 / 35,β1=3.1,β2=8, γ1=0.4,γ2=0.1. Figure 5 — Figure 7 is the simulation result diagram, Figure 5 It is the response curve of sliding variable under different initial values; Figure 6 To control the gain as well as Dynamic response curve of Figure 7 It is a control input response curve diagram. Under the action of the controller disclosed in this patent, the sliding variable can converge to 0, and under the action of the adaptive law, the control gain can be dynamically adjusted, thereby effectively weakening the vibration. theorem: Theorem 1: For q1>0,0<q2≤1 and have: Theorem 2: For any γ(y1,y2): have: Where n1>0 and n2>0. Theorem 3: For any d∈(0,1) and have: (|y1|+…|y n |) d ≤|y1| d +…+|y n | d .

Claims

1. A finite-time adaptive high-order sliding mode control method based on variable condensation method, characterized in that: Firstly, the variable aggregation method is used to design the adaptive law for the unknown time-varying parameters of the nonlinear system, and the power integral technique is used to design a finite-time adaptive second-order sliding mode controller. Then, the results of the finite-time adaptive second-order sliding mode controller are extended to the design of a finite-time adaptive n-order sliding mode controller.

2. According to the finite time adaptive high order sliding mode control method based on variable condensation method in claim 1, it is characterized in that: Consider the following nonlinear system s=s(t,x), in express control input; Indicates the system status; represents a sliding variable and the relative order is n, that is: s (n) =a(t,x)+b(t,x)u, in and are unknown smooth functions respectively; The above sliding mode is parameterized as: s (n) =φ1(s i )θ1(t)+θ2(t)u, in For a smooth mapping, and is an unknown time-varying parameter and satisfies the following assumptions: Assumption 1: The parameter θ1(t) is piecewise continuous and for t ≥ 0, θ1(t)∈θ1, where θ1 is a compact set and the "radius" of θ1 is Assume that θ is known but θ1 is unknown; Assumption 2: The parameter θ2(t) is bounded and non-zero, and there exists a positive constant Make For t ≥ 0; sign(θ2(t)) is known and unchanged; Order i =s i-1 ,i=1,...,n, the high-order sliding mode dynamic description is:

3. The finite-time adaptive high-order sliding mode control method based on variable condensation method according to claim 2 is characterized in that: The specific steps are as follows: S1. For the unknown time-varying parameters of nonlinear systems, the variable aggregation method is used to design the adaptive law, and the power integral technique is used to design the finite-time adaptive second-order sliding mode controller; Design the Lyapunov function V 1,1 The following form: Where ρ, r1 and r2 = r1-τ are real numbers and satisfy ρ≥a≥r1>0,τ>0. Designing a virtual controller The following form: where β1>0, r2=r1-τ and The above Lyapunov function V 1,1 The derivative of is: The Lyapunov function form is selected as follows: in: r3=r2-τ; The finite-time adaptive second-order sliding mode controller is designed as follows: in The adaptive law is designed as: in as well as Bring the controller in have to: S2, extend the results of finite-time adaptive second-order sliding mode controller to the design of finite-time adaptive n-order sliding mode controller, where n ≥ 3; Select the Lyapunov function V 2,1 , its form is the same as V 1,1 , The calculation process is the same as V 1,1 The derivation process yields: Select the Lyapunov function same The final result of the derivative calculation is as follows: Assume that a suitable positive constant β can be found in step i-1 j ,j=1,…,i-1, and a positive definite function V 2，i-1 Make where r i =r i-1 -τ The Lyapunov function for the i-th step is selected as follows: in Defined as: In summary: Select the Lyapunov function as follows: Design the controller u as follows: in The adaptive law is designed as follows: So we get:

4. The finite-time adaptive high-order sliding mode control method based on variable condensation method according to claim 3 is characterized in that: The finite-time stability analysis of the finite-time adaptive second-order sliding mode controller is: make Where Y1(s1,s2)=V 1,1 (s1)+W 1,1 (s1,s2) and get: in The finite-time stability is proved.

5. The finite-time adaptive high-order sliding mode control method based on variable condensation method according to claim 3 is characterized in that: The finite-time stability analysis of the finite-time adaptive n-order sliding mode controller is: make in as well as get: in The proof of finite-time stability is complete.

Images