Random nonlinear system nonsingular finite time tracking control method

Abstract

The invention provides a nonsingular finite time tracking control method for a random nonlinear system, and belongs to the technical field of random nonlinear system control. Establishing a random nonlinear system model with unknown sensor sensitivity; defining coordinate transformation to obtain a tracking error, and constructing a preset performance function and a preset performance constraint; obtaining a converted tracking error through a tracking error conversion mechanism; designing a barrier Lyapunov function by using the converted tracking error; designing a finite time controller based on a backstepping method in combination with a preset performance function, an obstacle Lyapunov function and a kernel function method; and controlling the random nonlinear system with unknown sensor sensitivity by using the finite time controller, so that an actual trajectory tracks a target trajectory. The finite time controller is designed in combination with the preset performance function and the obstacle Lyapunov function, the problems of unknown measurement sensitivity and unknown parameters are solved, the problem of singularity is eliminated, and the design limitation of the fractional order feedback controller is relaxed.

Description

Technical Field

[0001] The present invention relates to the technical field of random nonlinear system control, in particular to a non-singular finite-time tracking control method for a random nonlinear system. Background Art

[0002] In practical engineering, most systems are often affected by random factors such as environmental noise, modeling errors, and component aging. Furthermore, sensor sensitivity errors can occur due to limitations in operating conditions and sensor aging. Consequently, control technologies such as adaptive control have emerged for random nonlinear systems. To better design controllers, these control technologies are often combined with backstepping techniques.

[0003] However, when backstepping is introduced into controller design, a "differential explosion" phenomenon occurs, increasing the computational complexity. Furthermore, in practical engineering, rapid system convergence and response are required. Finite-time controllers based on backstepping suffer from control "singularities," which can affect system performance.

[0004] Therefore, a non-singular finite-time tracking control method for stochastic nonlinear systems is needed. Summary of the Invention

[0005] In view of this, the present invention provides a non-singular finite-time tracking control method for a random nonlinear system, which achieves precise control of system performance by combining preset performance constraints with backstepping method, thereby improving the stability and response speed of the system.

[0006] To this end, the present invention provides the following technical solutions:

[0007] A non-singular finite-time tracking control method for a stochastic nonlinear system, comprising:

[0008] Establish a random nonlinear system model with unknown sensor sensitivity;

[0009] Define coordinate transformation to obtain tracking error, and construct preset performance function and preset performance constraints;

[0010] Obtaining the converted tracking error through a tracking error conversion mechanism;

[0011] The barrier Lyapunov function is designed using the transformed tracking error;

[0012] Design of finite-time controller based on backstepping method combined with preset performance function, barrier Lyapunov function and kernel function method;

[0013] The finite time controller is used to control the random nonlinear system with unknown sensor sensitivity so that the actual trajectory tracks the target trajectory.

[0014] Furthermore, the random nonlinear system model with unknown sensor sensitivity is:

[0015]

[0016] s j (t)=σ j x j (t),j=1,2,...,n

[0017] in, represents an unknown bounded time-varying parameter, represents an unknown smooth nonlinear function, and y(t)=s1(t) represent the system state, input, and output respectively; ω represents the standard Wiener process, satisfying E{dω(t)}=0; s i (t) represents the non-real state value of the system measured by the sensor and is used to design the feedback controller; σ i represents the measurement sensitivity, which is assumed to be positive.

[0018] Furthermore, obtaining the tracking error through coordinate transformation includes:

[0019] z1=s1-y d

[0020] z i =s i -α i-1d ,i=2,…,n

[0021] Among them, s1 represents the system output; s i Indicates the non-real state value of the system measured by the sensor; y d represents the reference trajectory.

[0022] Furthermore, the preset performance function:

[0023] The smooth strictly decreasing function is used as the preset performance function PPFμ i (t);

[0024] And the preset performance function satisfies:

[0025]

[0026]

[0027] in, μ i0 and μ i∞ Represents a positive constant.

[0028] Furthermore, the preset constraints include:

[0029]

[0030] Among them, H Li , H hi The (0,1] interval represents the design parameters; the maximum overshoot and undershoot are affected by -H Li μ i (0), H hi μ i (0) and μ i (0) restrictions.

[0031] Furthermore, the error conversion mechanism includes:

[0032]

[0033] η i =δη Li +(1-δ)η hi

[0034] Among them, ρ i represents a positive design parameter, when z i When (t)≥0, δ=1, otherwise δ=0.

[0035] Furthermore, the design of the finite-time controller based on the backstepping method combined with the preset performance function, the barrier Lyapunov function and the core function method also includes:

[0036] Constructing a first-order filter with nonlinear terms to reduce the computational burden of step-by-step derivation;

[0037] The first-order filter with nonlinear terms:

[0038]

[0039] ζ i =α i-1d -α i-1

[0040] Among them, α i-1d , α i-1 (i=2,3,...,n) represent the output and input of the nonlinear filter respectively; ζ i Represents the filtering error.

[0041] Advantages and positive effects of the present invention:

[0042] This paper combines a preset performance function with a barrier Lyapunov function to design a finite-time controller, addressing unknown measurement sensitivity and unknown parameters, eliminating the "singularity" problem, and relaxing the design constraints of fractional-order feedback controllers. A lemma based on the kernel function method is proposed to solve the nonparametric decomposition problem of nonlinear functions. Furthermore, adaptive techniques are employed to replace the Nussbuam function to process unknown control parameters. The proposed control algorithm also exhibits strong robustness to random disturbances.

[0043] The present invention introduces a symmetric barrier Lyapunov function to directly handle the full-state tracking error constraint. At the same time, without changing the barrier Lyapunov function structure, the conclusions of the present invention can be further extended to tracking control problems without performance constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0045] Figure 1 Flowchart of the analytical method for nonsingular finite-time tracking control of random nonlinear systems with unknown measurement sensitivity;

[0046] Figure 2 The output signal y1 and the reference trajectory y in the simulation of this embodiment d ;

[0047] Figure 3 is the tracking error z1 and the preset bound in the simulation of this embodiment;

[0048] Figure 4 is the x2 trajectory of the system state in the simulation of this embodiment;

[0049] Figure 5 1 is a diagram illustrating the control input u in the simulation of this embodiment;

[0050] Figure 6 The adaptive law in the simulation of this embodiment trajectory;

[0051] Figure 7 The adaptive law in the simulation of this embodiment trajectory. DETAILED DESCRIPTION

[0052] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0053] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the numbers used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0054] The present invention provides a non-singular finite-time tracking control method for a random nonlinear system. The method avoids the "singularity" problem in the backstepping design process by adopting an improved error conversion technology based on a preset performance function; avoids the "differential explosion" problem by designing a first-order filter; addresses the problem of unavailability of non-real measurement values ​​of the system state, and enables the system to achieve a control target within a finite time through a feedback control algorithm, including: adopting a symmetric barrier Lyapunov function to meet the requirements of the full-state tracking error constraint; and using a finite-time stability criterion to prove that the proposed control strategy can ensure the boundedness of the system variables and that the state tracking error remains within a pre-designed range within a finite time.

[0055] Combine Figure 1 The method of the present invention comprises: S1, collecting information: obtaining a reference signal and a controller state at an initial moment, designing a measurement sensitivity, and obtaining a nonlinear system control input at an initial moment, establishing a random nonlinear system model with unknown sensor sensitivity, and determining a non-strict feedback system;

[0056] S2. Design a coordinate transformation to obtain a tracking error model, establish a preset performance function and preset performance constraints; combine the error conversion mechanism to obtain the converted tracking error; use the converted error barrier Lyapunov function to extend the backstepping method to the design of controllers for non-strict feedback systems;

[0057] S3. Design a method based on the core function method to handle unknown smooth nonlinear functions in the system. Convert the unknown smooth nonlinear function into a known smooth function and unknown constants, and then handle it in the subsequent backstepping controller design.

[0058] S4. Construct a first-order filter with nonlinear terms to solve the "differential explosion" problem;

[0059] S5. The converted tracking error obtained by the transformation formula based on the tracking error model and the preset performance function is used to construct a suitable barrier Lyapunov function and analyze whether it can be achieved to determine whether finite-time semi-global actual finite-time stability is achieved;

[0060] S6, based on steps S1, S2, S3, S4 and S5, combining backstepping, preset performance function and core function decomposition method to design a controller, using the designed actual control signal to make the actual trajectory track the target trajectory, and constrain the tracking error within the preset performance function;

[0061] S7, perform stability analysis to prove that the system can remain stable within a limited time.

[0062] The steps of the present invention are specifically described as follows:

[0063] S1. Information Collection: Obtain the reference signal and the controller state at the initial moment, design the measurement sensitivity, and obtain the nonlinear system control input at the initial moment. Establish a random nonlinear system with unknown sensor sensitivity and determine the non-strict feedback system. In practice, collect information about production process errors and differences in the operating environment to design unknown measurement sensitivity. To avoid system unavailability caused by unknown measurement sensitivity, establish and handle random nonlinear systems with unknown measurement sensitivity as follows:

[0064]

[0065] s j (t)=σ j x j (t),j=1,2,…,n (1)

[0066] in, represents an unknown bounded time-varying parameter, represents an unknown smooth nonlinear function, and y(t)=s1(t) represent the system state, input, and output respectively. ω refers to the standard Wiener process, satisfying E{dω(t)}=0. s i (t) represents the non-real state value of the system measured by the sensor, and is used to design the feedback controller. i represents the measurement sensitivity, which is assumed to be positive.

[0067] Considering that the system is feasible, assumption 1 needs to be met.

[0068] Assumption 1: Ideal tracking trajectory y d and Everything has boundaries.

[0069] S2: Design coordinate transformation to obtain tracking error, establish preset performance function and preset performance constraint, and combine with error conversion mechanism to obtain converted tracking error, and use the converted error to design obstacle Lyapunov function.

[0070] S21. Introduce a smooth strictly decreasing function as the preset performance function PPFμ i (t);

[0071] satisfy and

[0072] in, μ i0 and μ i∞ Represents a positive constant.

[0073] S22. The preset constraints are:

[0074]

[0075] Among them, H Li , H hi The (0,1] interval represents the design parameters, and the maximum overshoot and undershoot are affected by -H Li μ i (0), H hi μ i (0) and μ i (0) restrictions.

[0076] S23. Design error conversion mechanism:

[0077]

[0078] η i =δη Li +(1-δ)η hi (3)

[0079] Among them, ρ i represents a positive design parameter, when z i When (t)≥0, δ=1, otherwise δ=0.

[0080]

[0081] according to And the above formula, inequality This is helpful for the subsequent design of the Lyapunov barrier function.

[0082] S24. Define the coordinate transformation as:

[0083] z1=s1-y d

[0084] z i =s i -α i-1d ,i=2,…,n(4)

[0085] By combining the symmetric barrier Lyapunov function with the error transformation, the singularity problem is avoided.

[0086] S3: Under the condition that Assumption 2 is satisfied, a method based on the core function method is designed to handle the unknown smooth nonlinear function in the system. The unknown smooth nonlinear function is converted into a known smooth function and unknown constants, and then processed in the subsequent backstepping controller design.

[0087] Assumption 2: Given a known smooth function unknown non-negative constant a i1 ≥0, a i2 ≥0 such that:

[0088] and

[0089] satisfy

[0090] If x i It is bounded. It is also bounded, i = 1, 2,…, n.

[0091] For system (1), the function and satisfy:

[0092]

[0093]

[0094] in, is a smooth function; a i1 ≥0,a i2 ≥0 indicates an unknown constant, ψ i1 ≥0 and ψ i2 ≥0 indicates a parametric function.

[0095] Proof: By applying Lemma 4 and Assumption 1, it is easy to prove that:

[0096]

[0097]

[0098] in,

[0099] S4: A first-order filter with nonlinear terms is constructed to address the "differential explosion" problem, which occurs when the virtual controller continuously derivates during backstepping controller design, leading to a dramatic increase in computational complexity. Therefore, the nonlinear terms are used to offset the derivative terms and reduce the computational burden.

[0100]

[0101] ζ i =α i-1d -α i-1 (8)

[0102] Among them, α i-1d , α i-1 (i=2,3,...,n) represent the output and input of the nonlinear filter respectively. i Represents the filtering error.

[0103] S5: Based on the tracking error model and the preset performance function, the transformation formula is defined to obtain the converted tracking error, which is used to construct a suitable barrier Lyapunov function. By taking the differential operator, the simplified analysis is performed and whether it satisfies the requirements. This is how to determine whether finite-time semi-global actual finite-time stability is achieved.

[0104] Considering that the system uses the backstepping method for controller design, it is necessary to set the barrier Lyapunov function at each step:

[0105]

[0106] Among them, r1>0, r2>0 are design parameters. is an unknown parameter, for estimated value. Represents the estimation error. and is a dummy parameter.

[0107]

[0108] Among them, the parameter r i+1 >0,r i1 >0,r i2 >0. Indicates M i ,N i The estimation error of . Definition is an unknown parameter.

[0109] Obstacle Lyapunov function V n for:

[0110]

[0111] Among them, the parameter r n+1 >0,r n1 >0,r n2 >0. Indicates M n ,N n The estimated error of is as follows. Indicates unknown parameters.

[0112] For the system dx(t)=f(x(t))dt+g(x(t))dw, if there exists a function V(x) and a positive constant c>0,0<l<1,D>0 satisfying:

[0113]

[0114] The convergence time is:

[0115]

[0116] Where V(x(0)) represents the initial value and 0<κ<1 represents a constant. Then, when The system can maintain semi-globally practical finite-time stability (SGPFS).

[0117] S6: Based on steps S1, S2, S3, S4 and S5, the controller design is performed by combining the backstepping method, the preset performance function and the core function decomposition method as follows:

[0118] Using the kernel function method and Young's inequality to deal with nonlinear terms, we can obtain:

[0119]

[0120] Among them, ∈ ij >0,j=1,2…,4 represents the design parameters;

[0121]

[0122] Its design adaptive law and the virtual control signal α i as follows:

[0123]

[0124] in,

[0125]

[0126] in:

[0127]

[0128] is a scalar function.

[0129] We can get:

[0130]

[0131] in:

[0132]

[0133] Design of actual control law and adaptive law:

[0134] Using a similar design procedure as in S6, we can obtain:

[0135] Its design adaptive law And the virtual control signal u is as follows:

[0136]

[0137] in,

[0138] in,

[0139]

[0140] is a scalar function,

[0141]

[0142] in:

[0143]

[0144]

[0145] S7: Perform stability analysis, simplify the final result in S6, and then prove that the system can remain stable within a limited time. Specifically include the following:

[0146] Adding the n-1 term of formula (16) to formula (18) yields:

[0147]

[0148] in:

[0149]

[0150]

[0151] definition:

[0152]

[0153] Among them: ζ(0)=[ζ2(0),ζ3(0),...,ζ n (0)] T ,

[0154] e(0)=[e1(0),e2(0),...,e n (0)] T ,

[0155]

[0156]

[0157] This means that the closed-loop system is semiglobally finite-time stable.

[0158] We can further obtain that when t≥T * ,

[0159] Then we can get:

[0160] |z1|=|s1-y d |<|η1|,|z i |=|s i -α i-1d |<|η i |,i=2,...,n. Further, we can get s i Bounded. Considering s i (t)=σ i x i (t), σ i is a constant, then we can get x i Boundedness of . Prove that for the random system (1), under the premise of satisfying the assumptions 1-2, when Design controller α i ,i=1,2…,n,u and adaptive law In the presence of unknown measurement sensitivity, the internal signal of the closed-loop system remains bounded, and the tracking error reaches the preset limit within a finite time.

[0161] The effectiveness of the control method of the present invention is verified by the following simulation experiments:

[0162] Consider the following system:

[0163]

[0164] in:

[0165]

[0166] θ2=2sin(t), σ1=0.9, σ2=1.3;

[0167] The design parameters are selected as follows:

[0168]

[0169] c1=188, c2=188, ε1=1.1, ε2=1.1, r1=0.02, r2=0.02, r3=0.02, r 21 =1.1,r 22 =1.1

[0170]

[0171] The initial conditions and the desired tracking signal are chosen as:

[0172]

[0173] and y d =0.2sin(0.5t).

[0174] The simulation results are as follows Figures 2 to 7 shown. Figure 2 The tracking performance is verified. The tracking error waveform within the preset range is as follows Figure 3 shown. Figure 4 The performance of x2 is given. The input signal u and the adaptive law respectively Figure 5-7 Obviously, the above simulation results confirm the effectiveness of the proposed control scheme.

[0175] This invention not only solves the finite-time "singularity" problem and relaxes the limitations of fractional-order feedback controllers, but also constructs a new finite-time feedback control algorithm to handle unknown measurement sensitivity and unknown parameters. Furthermore, a new lemma based on the kernel function method is proposed to solve the nonparametric decomposition problem of nonlinear functions. This makes the control algorithm proposed in this invention highly robust to random disturbances.

[0176] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A non-singular finite-time tracking control method for a random nonlinear system, characterized in that: include: Establish a random nonlinear system model with unknown sensor sensitivity; Define coordinate transformation to obtain tracking error, and construct preset performance function and preset performance constraints; Obtaining the converted tracking error through a tracking error conversion mechanism; The barrier Lyapunov function is designed using the transformed tracking error; Design of finite-time controller based on backstepping method combined with preset performance function, barrier Lyapunov function and kernel function method; The finite time controller is used to control the random nonlinear system with unknown sensor sensitivity so that the actual trajectory tracks the target trajectory.

2. The non-singular finite-time tracking control method for a random nonlinear system according to claim 1, characterized in that: The random nonlinear system model with unknown sensor sensitivity is: s j (t)=σ j x j (t),j=1,2,…,n in, represents an unknown bounded time-varying parameter, represents an unknown smooth nonlinear function, and y(t)=s1(t) represent the system state, input, and output respectively; ω represents the standard Wiener process, satisfying E{dω(t)}=0; s i (t) represents the non-real state value of the system measured by the sensor and is used to design the feedback controller; σ i represents the measurement sensitivity, which is assumed to be positive.

3. The non-singular finite-time tracking control method for a random nonlinear system according to claim 1, characterized in that: The obtaining of the tracking error by coordinate transformation includes: z1=s1-y d z i =s i -α i-1d ,i=2,…,n Among them, s1 represents the system output; s i Indicates the non-real state value of the system measured by the sensor; y d represents the reference trajectory.

4. The non-singular finite-time tracking control method for a random nonlinear system according to claim 1, characterized in that: The preset performance function: The smooth strictly decreasing function is used as the preset performance function PPFμ i (t); And the preset performance function satisfies: in, μ i0 and μ i∞ Represents a positive constant.

5. The non-singular finite-time tracking control method for a random nonlinear system according to claim 4, characterized in that: The preset constraints include: Among them, H Li , H hi The (0,1] interval represents the design parameters; the maximum overshoot and undershoot are affected by -H Li μ i (0), H hi μ i (0) and μ i (0) restrictions.

6. The non-singular finite-time tracking control method for a random nonlinear system according to claim 1, characterized in that: The error conversion mechanism includes: or i =also Li +(1-d)n hi Among them, ρ i represents a positive design parameter, when z i When (t)≥0, δ=1, otherwise δ=0.

7. The non-singular finite-time tracking control method for a random nonlinear system according to claim 1, characterized in that: The design of the finite-time controller based on the backstepping method combined with the preset performance function, the barrier Lyapunov function and the core function method also includes: Constructing a first-order filter with nonlinear terms to reduce the computational burden of step-by-step derivation; The first-order filter with nonlinear terms: g i =a i-1d -a i-1 Among them, α i-1d , α i-1 (i=2,3,...,n) represent the output and input of the nonlinear filter respectively; ζ i Represents the filtering error.

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