An adaptive finite-time command filter control and simulation method for uncertain nonlinear systems

Abstract

The present invention discloses an adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, which relates to the field of nonlinear system stability and finite-time control, and includes the following steps: First, a novel actual finite-time stability criterion is established as the theoretical basis for finite-time control design. Second, the original constrained nonlinear system is converted into an unconstrained system through nonlinear mapping. Third, using the established actual finite-time stability criterion and command filtering technology, a finite-time adaptive control scheme based on a command filter is proposed. While strictly complying with the extended full-state constraints, the present invention ensures excellent tracking performance, and all system signals remain actual finite-time stable. Finally, stability analysis and simulation further verify the effectiveness of the proposed control strategy.

Description

Technical Field

[0001] The present invention relates to the field of nonlinear system stability and finite-time control, and in particular to an adaptive finite-time command filtering control of a nonlinear system with novel state constraints and unknown disturbances. Background Art

[0002] Finite-time control is an important research direction in modern control theory. Its core goal is to converge the system state to a desired value or equilibrium point within a finite time. By designing an appropriate control law, finite-time control can ensure that the system state reaches a stable state within a finite time and maintains robustness after reaching stability. To achieve finite-time stability, early researchers used sliding mode control techniques to design controllers. However, this approach can lead to chattering. To overcome this problem, a finite-time stability strategy based on Lyapunov theory was proposed. Based on this mature theory, researchers have conducted a series of in-depth studies on the finite-time control of nonlinear systems. However, the existing results have imposed specific requirements on the system form, such as low-order systems, known control gain functions, or linear growth characteristics. This situation changed only with the development of a practical finite-time control strategy. This strategy uses a simple and easy-to-implement controller structure. This strategy makes it easier to apply backstepping techniques to design finite-time controllers for nonlinear systems. Subsequently, scholars have conducted extensive research based on practical finite-time stability theory, achieving a series of successful results. It is worth noting that the above finite-time control studies have a common feature, which is to prove that the Lyapunov function V(x) satisfies:

[0003]

[0004] And the finite time T s is determined by the formula:

[0005]

[0006] It is easy to see that time T s It is related to the initial value of the system. At the same time, the controller designed with this control strategy will have a singularity problem. In order to avoid this phenomenon, it is of great significance to propose a new finite-time control method. Therefore, the present invention designs an adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems. Summary of the Invention

[0007] The purpose of the present invention is to solve the problems in the prior art and to propose an adaptive finite-time command filtering control and simulation method for an uncertain nonlinear system.

[0008] An adaptive finite-time command filter control and simulation method for an uncertain nonlinear system is disclosed, wherein a finite-time controller is designed for a class of state-constrained nonlinear systems to achieve effective tracking control, and the method comprises the following steps:

[0009] S1: Establishing a practical finite-time stability criterion;

[0010] Among them, for a limited time:

[0011]

[0012] Among them, the design parameters υ>0, 0<β<1;

[0013] S2: The original constrained nonlinear system is transformed into an unconstrained system by introducing a nonlinear mapping, wherein the original constrained nonlinear system can be described as

[0014]

[0015] in is a known control gain function, η j (t)∈R ι is an unknown time-varying bounded parameter, is a known smooth function, x j is the system state, u is the system input, y is the system output, δ j (t) represents the bounded external perturbation;

[0016] S3, using the established practical finite-time stability criterion and command filtering technology, a finite-time adaptive control scheme based on command filter is proposed.

[0017] In the above-mentioned adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, the method further includes:

[0018] S4: Perform stability analysis on the designed finite-time adaptive control scheme.

[0019] In the above-mentioned adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, the method further includes:

[0020] S5: Verify the effectiveness of the control strategy through simulation.

[0021] In the above-mentioned adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, in step S2, a nonlinear mapping is introduced:

[0022]

[0023] where Γ j (x j ,Kj1 (y d ,t),K j2 (y d ,t)) is a smooth and strictly increasing function for ξ j Taking the derivative we get:

[0024]

[0025] in:

[0026]

[0027] The original system is converted into the following form:

[0028]

[0029] in:

[0030]

[0031] Δ j (x j ,y d ,t)=λ j1 (x j ,y d ,t)δ j (t)

[0032] definition:

[0033]

[0034] The coordinate transformation is introduced as follows:

[0035] z1=ξ1-ω1,z j =ξ j -ω j ,j=1,…,n,

[0036]

[0037] Among them ω i It is given by the following equation:

[0038] ω j (0) = α j-1 (0),j=2,…,n,

[0039] At the same time, in order to eliminate errors The impact on the system, construct the following error compensation signal:

[0040]

[0041] Among them C jis the design constant,

[0042]

[0043] In the above-mentioned adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, in the finite-time adaptive control scheme proposed in step S3, the command filtering method is used to design the control law and the adaptive rate as follows:

[0044]

[0045] In the above-mentioned adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems, in step S4, stability analysis includes, for a parameterized nonlinear system with extended constraints, under virtual control, actual control law, and adaptive control rate, ensuring that all signals of the closed-loop system are practically finite-time stable and all states do not violate pre-defined constraints. The specific steps include:

[0046] Proof: Define a compact set

[0047]

[0048] in is a positive constant and is only used for stability analysis;

[0049]

[0050] Choose the following Lyapunov function:

[0051]

[0052] Taking the derivative of L with respect to time t, we get

[0053]

[0054] in:

[0055] ∈ j =ω j -α j-1 ,j=2,…,n; Obviously, is a positive number, so:

[0056]

[0057] thereby:

[0058]

[0059] Select the design parameters as follows:

[0060]

[0061] in:

[0062]

[0063] Then we have:

[0064]

[0065] make:

[0066]

[0067] Applying Lemma 1:

[0068] When Ξ, is any real number, q1,q2,ι are any given positive constants, then the following inequality holds:

[0069]

[0070] but:

[0071]

[0072] Applying Lemma 2 again:

[0073] For a given The following inequality holds:

[0074]

[0075] We can get:

[0076]

[0077] if Then e j , Bounded, through analysis we can see that there are positive constants M1, M2, M3 such that:

[0078]

[0079] So we get

[0080]

[0081] in,

[0082]

[0083] if For a given set of γ1,…,γ n , we can choose sufficiently large λ1,…,λ n To obtain a sufficiently large υ, and because ρ and λ j , j=1,…,n are irrelevant, so we can get:

[0084]

[0085] This leads to This means that if Then for all t≥0, we have Complete stability analysis.

[0086] In the above-mentioned adaptive finite-time command filter control and simulation method for uncertain nonlinear systems, in step S5, a two-stage chemical reactor system is used for simulation, and its dynamic control model is described as follows:

[0087]

[0088] Where u represents the control input signal, P A and P A It is a synthetic compound, R A and R B represents the circulation flow rate; K A and K B is the reaction rate constant; T A and T B is the residence time of the reactor; V A and V B is the volume of the reactor; δ1 and δ2 represent bounded external disturbances;

[0089] make:

[0090] y=x1=P A

[0091] x2=P B

[0092] So the above system can be transformed into:

[0093]

[0094] in,

[0095]

[0096] φ1(x1)=[-x1,x2] T

[0097]

[0098] The system parameters are:

[0099] K A =K B =R A =R B =V A =V B =0.5

[0100] T A =T B =2

[0101] The expected tracking trajectory is: y d =0.5sin(t)

[0102] The extended time-varying constraint function is defined as:

[0103] K 11 (y d ,t)=2sin(-0.3y d )+e -2t +0.5K 12 (y d ,t)=2sin(-0.1y d )+e -3t +0.8

[0104]

[0105] The design parameters are:

[0106] C1=1

[0107] C2=2

[0108] σ2=σ2=0.1

[0109] λ1=λ2=0.02

[0110] γ1=γ2=50

[0111] τ2=0.02

[0112]

[0113] The initial values ​​are:

[0114] x1(0)=0.6

[0115] x2(0)=0.4

[0116] ω2(0)=0.2

[0117]

[0118] The system is simulated using the designed virtual controller α1 and the actual controller u.

[0119] Compared with existing technologies, this invention offers the following advantages: This embodiment designs a finite-time controller for a class of constrained nonlinear systems to achieve effective tracking control. Deployed in three systematic steps, this innovative control algorithm ensures excellent tracking performance while strictly adhering to the extended full-state constraints. All system signals remain practically finite-time stable. Finally, stability analysis and simulations further validate the effectiveness of the proposed control strategy. BRIEF DESCRIPTION OF THE DRAWINGS

[0120] Figure 1 The output y and expected trajectory y of the system in the simulation experiment of the adaptive finite-time command filter control and simulation method of uncertain nonlinear system proposed by the present invention d picture.

[0121] Figure 2 This is a control u curve diagram in the adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems proposed by the present invention.

[0122] Figure 3 These are the curves of the adaptive laws θ1 and θ2 in the adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems proposed by the present invention.

[0123] Figure 4 This is a state and constraint diagram for an adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems proposed by the present invention.

[0124] Figure 5 This is the state phase plane diagram of the adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems proposed by the present invention.

[0125] Figure 6 This is a compensation signal curve diagram in the adaptive finite-time command filtering control and simulation method for uncertain nonlinear systems proposed by the present invention. DETAILED DESCRIPTION

[0126] Reference Figure 1-6 , an adaptive finite-time command filter control and simulation method for uncertain nonlinear systems, including the following parts:

[0127] 1. Finite-time stability theory

[0128] For the sake of illustration, consider the following nonlinear system

[0129]

[0130] Where x∈R n is the system state, δ∈R mis uncertainty, which is a time-varying bounded disturbance or an unknown constant, f(t,x,c):[0,∞)×R n ×R m →R n is a continuous function, and the origin is assumed to be the equilibrium point of system (1).

[0131] Lemma 1: For a nonlinear system (1), if there exists a radially unbounded positive definite function L(x):R n →R makes the solution x(x0,t) of equation (1) satisfy:

[0132]

[0133] in:

[0134] υ>0,0<β<1

[0135] π represents the ratio of circumference to circumference, and ρ represents a non-negative constant.

[0136] Then the system (1) is practically finite-time stable when:

[0137]

[0138] Solution of the system: x(x0,t) is confined to the bounded set Ω={x:L(x)≤ρ / υπ}.

[0139] Lemma 2: Ξ, is any real number, q1,q2,ι are any given positive constants, then the following inequality holds:

[0140]

[0141] Lemma 3: For a given:

[0142]

[0143] The following inequality holds:

[0144]

[0145] At this time, consider the nonlinear expansion-constrained nonlinear system

[0146]

[0147] in:

[0148] in is a known control gain function,

[0149] η j (t)∈R ι is an unknown time-varying bounded parameter,

[0150] is a known smooth function,

[0151] x j is the system status,

[0152] u is the system input,

[0153] y is the system output,

[0154] δ j (t) represents a bounded external perturbation.

[0155] Control objective: Design a finite-time controller u based on command filtering technology so that the system output can track the expected trajectory well and all states do not violate predefined constraints, and all signals are practical finite-time stable.

[0156] Assumption 1: Expected trajectory vector is continuous and available, where is a known compact set.

[0157] Assumption 2: K i1 (y d ,t)>0 and K i2 (y d ,t)>0 is a known bounded smooth function, i=1,…,n, and it is related to y d The partial derivatives of and t are also bounded.

[0158] When designing a controller based on the improved dynamic surface method

[0159] Introducing nonlinear mapping:

[0160]

[0161] in:

[0162] Γ j (x j ,K j1 (y d ,t),K j2 (y d ,t)) is a smooth and strictly increasing function, so (6) is a one-to-one mapping.

[0163] Yes j Taking the derivative we get

[0164]

[0165] in:

[0166]

[0167]

[0168] The original system is converted into the following form

[0169]

[0170] in,

[0171]

[0172] Δ j (x j ,y d ,t)=λ j1 (x j ,y d ,t)δ j (t) (15)

[0173] definition:

[0174]

[0175] The coordinate transformation is introduced as follows:

[0176] z1=ξ1-ω1,z j =ξ j -ω j ,j=1,…,n, (16)

[0177]

[0178] Among them ω i It is given by the following equation:

[0179] ω j (0) = α j-1 (0),j=2,…,n, (18)

[0180] At the same time, in order to eliminate errors The impact on the system, construct the following error compensation signal:

[0181]

[0182] Among them C j is the design constant,

[0183] The control law and adaptive rate are designed using the command filtering method as follows:

[0184]

[0185] Stability analysis

[0186] Theorem: For the parameterized nonlinear system with extended constraints described by Equation (5), under the virtual control (19) and (20), the actual control law (21), and the adaptive control rates (22)-(24), all signals of the closed-loop system are practically finite-time stable, and all states do not violate the pre-defined constraints.

[0187] Proof: Define a compact set

[0188]

[0189] in is a positive constant and is only used for stability analysis.

[0190] Choose the following Lyapunov function:

[0191]

[0192] Taking the derivative of L with respect to time t, we get

[0193]

[0194] where ∈ j =ω j -α j-1 ,j=2,…,n。

[0195] Obviously, is a positive number, so we have

[0196]

[0197] thereby

[0198]

[0199] Select the design parameters as follows:

[0200]

[0201] and Then we have:

[0202]

[0203] make:

[0204]

[0205] Applying Lemma 2, we can obtain:

[0206]

[0207] Applying Lemma 3 again, we get:

[0208]

[0209] if Then e j , Bounded, through analysis we can see that there are positive constants M1, M2, M3 such that:

[0210]

[0211] So we get:

[0212]

[0213] in,

[0214]

[0215] if For a given set of γ1,…,γ n , we can choose sufficiently large λ1,…,λ n To obtain a sufficiently large υ, and because ρ and λ j , j=1,…,n are irrelevant, so we can get:

[0216]

[0217] This leads to

[0218] This means that if Then for all t≥0, we have

[0219] In summary, the variable e of the closed-loop system j , is practically finite-time stable. And, α j and ω j+1 It is also practical and stable for a limited time. Then ξ j Bounded. Formula (2) shows that all states are bounded and do not violate the constraints.

[0220] Simulation research

[0221] To illustrate the effectiveness of the finite time control method proposed in this example, a two-stage chemical reactor system was simulated. Its dynamic control model is described as follows:

[0222]

[0223] Where u represents the control input signal, P A and P A It is a synthetic compound, R Aand R B represents the circulation flow rate; K A and K B is the reaction rate constant; T A and T B is the residence time of the reactor; V A and V B is the volume of the reactor; δ1 and δ2 represent bounded external perturbations.

[0224] Let y = x1 = P A ,x2=P B . Then the above system can be transformed into:

[0225]

[0226] in,

[0227]

[0228] φ1(x1)=[-x1,x2] T

[0229]

[0230] The system parameters are

[0231] K A =K B =R A =R B =V A =V B =0.5

[0232] T A =T B =2

[0233] The expected tracking trajectory is:

[0234] y d =0.5sin(t)

[0235] The extended time-varying constraint function is defined as:

[0236] K 11 (y d ,t)=2sin(-0.3y d )+e -2t +0.5

[0237] K 12 (y d ,t)=2sin(-0.1y d )+e -3t +0.8

[0238]

[0239] The design parameters are C1=1, C2=2, σ2=σ2=0.1, λ1=λ2=0.02, γ1=γ2=50, τ2=0.02, The initial values ​​are x1(0)=0.6, x2(0)=0.4, ω2(0)=0.2, The system is simulated using the designed virtual controller α1 and the actual controller u. The system output y and the expected trajectory y d like Figure 1 As shown, the control u curve is as follows Figure 2 As shown, the curves of adaptive laws θ1 and θ2 are as follows Figure 3 As shown, Figure 4 shows the states and their constraints, Figure 5 The phase plane diagram of states x1 and x2 is shown, and the compensation signal curve is as follows Figure 6 From the simulation results, it can be seen that this design method has achieved a relatively ideal tracking control effect. All signals are bounded and the state does not violate the constraint conditions.

[0240] In summary, this embodiment designs a finite-time controller for a class of constrained nonlinear systems to achieve effective tracking control. It is carried out in three systematic steps: First, a novel practical finite-time stability criterion is established as the theoretical basis for finite-time control design. Second, the original constrained nonlinear system is converted into an unconstrained system through nonlinear mapping. Third, using the established practical finite-time stability criterion and command filtering technology, a finite-time adaptive control scheme based on command filter is proposed. This innovative control algorithm ensures excellent tracking performance while strictly complying with the extended full-state constraints, and all system signals remain practical finite-time stable. Finally, stability analysis and simulation further verify the effectiveness of the proposed control strategy.

[0241] It is understood from common technical knowledge that the present invention may be implemented by other embodiments that do not depart from its spirit or essential features. Therefore, the embodiments disclosed above are, in all respects, merely illustrative and not exclusive. All modifications within the scope of the present invention or equivalent to the scope of the present invention are intended to be encompassed by the present invention.

Claims

1. An adaptive finite-time command filter control and simulation method for uncertain nonlinear systems, characterized in that: Designing a finite-time controller for a nonlinear system with state constraints to achieve effective tracking control involves the following steps: S1: Establishing a practical finite-time stability criterion; Among them, for a limited time: Among them, the design parameters υ>0, 0<β<1; S2: The original constrained nonlinear system is transformed into an unconstrained system by introducing a nonlinear mapping, wherein the original constrained nonlinear system can be described as in is a known control gain function, η j (t)∈R ι is an unknown time-varying bounded parameter, is a known smooth function, x j is the system state, u is the system input, y is the system output, δ j (t) represents the bounded external perturbation; S3, using the established practical finite-time stability criterion and command filtering technology, a finite-time adaptive control scheme based on command filter is proposed; In step S2, a nonlinear mapping is introduced: where Γ j (x j ,K j1 (y d ,t),K j2 (y d ,t)) is a smooth and strictly increasing function for ξ j Taking the derivative we get: in: The original system is converted into the following form: in: Δ j (x j ,y d ,t)=λ j1 (x j ,y d ,t)δ j (t) definition: The coordinate transformation is introduced as follows: z1=ξ1-ω1,z j =ξ j -oh j ,j=1,…,n, Among them ω i It is given by the following equation: At the same time, in order to eliminate errors The impact on the system, construct the following error compensation signal: Among them C j is the design constant, 2. The method for adaptive finite-time command filtering control and simulation of an uncertain nonlinear system according to claim 1, characterized in that: The method also includes: S4: Perform stability analysis on the designed finite-time adaptive control scheme.

3. The method for adaptive finite-time command filtering control and simulation of an uncertain nonlinear system according to claim 1, characterized in that: The method also includes: S5: Verify the effectiveness of the control strategy through simulation.

4. The method for adaptive finite-time command filtering control and simulation of an uncertain nonlinear system according to claim 2, characterized in that: In step S4, the stability analysis includes, for a parameterized nonlinear system with extended constraints, verifying that all signals of the closed-loop system are stable in a practical finite time under virtual control, actual control law, and adaptive control rate, and that all states do not violate pre-defined constraints. The specific steps include: Proof: Define a compact set in is a positive constant and is only used for stability analysis; Choose the following Lyapunov function: Taking the derivative of L with respect to time t, we get in: ∈ j =ω j -α j-1 ,j=2,…,n; Obviously, is a positive number, so: thereby: Select the design parameters as follows: in: Then we have: make: Applying Lemma 1: When Ξ, is any real number, q1,q2,ι are any given positive constants, then the following inequality holds: but: Applying Lemma 2 again: For a given The following inequality holds: We can get: if Then e j , Bounded, there exist positive constants M1, M2, M3 such that: So we get in, if For a given set of γ1,…,γ n , we can choose sufficiently large λ1,…,λ n To obtain a sufficiently large υ, and because ρ and γ j , j=1,…,n are irrelevant, so we can get: This leads to This means that if Then for all t≥0, we have Complete stability analysis.

5. The method for adaptive finite-time command filtering control and simulation of uncertain nonlinear systems according to claim 3, characterized in that: In step S5, a two-stage chemical reactor system is used for simulation, and its dynamic control model is described as follows: Where u represents the control input signal, P A and P A It is a synthetic compound, R A and R B represents the circulation flow rate; K A and K B is the reaction rate constant; T A and T B is the residence time of the reactor; V A and V B is the volume of the reactor; δ1 and δ2 represent bounded external disturbances; make: y=x1=P A x2=P B So the above system can be transformed into: in, φ1(x1)=[-x1,x2] T The system parameters are: K A =K B =R A =R B =V A =V B =0.5 T A =T B =2 The expected tracking trajectory is: and d =0.5sin(t) The extended time-varying constraint function is defined as: K 11 (the d ,t)=2sin(-0.3y d )+e -2t +0.5 K 12 (the d ,t)=2sin(-0.1y d )+e -3t +0.8 The design parameters are: C1＝1 C2＝2 σ2=σ2=0.1 λ1=λ2=0.02 γ1=γ2=50 τ2=0.02 The initial values ​​are: x1(0)＝0.6 x2(0)＝0.4 ω2(0)=0.2 The system is simulated using the designed virtual controller α1 and the actual controller u.

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