Time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraint

Abstract

The invention discloses a time-varying parameterization nonlinear system dynamic surface control and simulation method with extended time-varying constraint, and relates to the field of nonlinear system stabilization and control, and the method comprises the steps: introducing a tangent function as nonlinear mapping in a controller design process, a nonlinear parameter system with time-varying parameters and extended full-state time-varying constraints is converted into a new unconstrained parameterized system. In addition, an improved DSC with an error compensation signal is used, and a new adaptive control technology is provided for a converted nonlinear system. The proposed design scheme avoids assuming that a virtual control coefficient and an upper limit thereof are known, and proves that all signals in a closed-loop system are in semi-global end consistency bounded. A simulation result verifies the effectiveness of the method.

Description

Technical Field

[0001] The present invention relates to the field of nonlinear system stability and control, and in particular to a time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints. Background Art

[0002] Adaptive control of parameterized nonlinear systems has always been a hot issue in the field of control. In the prior art, the finite-time control problem of constant parameter systems with strict feedback form and the event-based adaptive control problem of parametric nonlinear systems with time-varying parameters have been studied respectively. Through BLF and back-stepping design methods, an adaptive inversion control method based on nonlinear mapping is proposed for nonlinear systems with output constraints, and based on one-to-one nonlinear mapping, a pure feedback system with full-state constraints is converted into a new type of unconstrained pure feedback system.

[0003] The constraints considered in the adaptive control field of nonlinear systems in the above-mentioned prior art are either constants or functions of time, and there are few functions about the expected trajectory. It is difficult to deal with this extended full-state constraint problem using traditional obstacle Lyapunov functions. In addition, there are relatively few studies on the stability control of nonlinear systems that simultaneously consider time-varying parameters and extended constraints. Based on this, the present invention designs a time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints. Summary of the invention

[0004] The purpose of the present invention is to solve the problems in the prior art and to propose a time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints.

[0005] A time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints is designed. The goal is to design an improved dynamic surface controller so that the system output can track the expected trajectory well and all states do not violate the predefined constraints. All signals are semi-globally consistent and terminally bounded.

[0006] These include the following steps:

[0007] S1: A parametric system with time-varying parameters and extended full-state constraints is transformed into an unconstrained nonlinear system through a constructed nonlinear mapping;

[0008] S2: Introducing a compensation signal to offset the error caused by replacing the virtual control derivative with the output of the linear filter;

[0009] S3: Design control laws and adaptive laws for the improved dynamic surface control method.

[0010] S4: Verify the improved dynamic surface control method and simulate and verify the method through examples.

[0011] In the above-mentioned time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, the nonlinear system is set as:

[0012]

[0013] in, is a known control gain function, η i (t)∈R ι is an unknown time-varying bounded parameter, is a known smooth function, x is the system state, u is the system input, and y is the system output;

[0014] And set the desired trajectory vector is continuous and available, where is a known compact set, and set 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.

[0015] In the above-mentioned time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, nonlinear mapping is introduced:

[0016]

[0017] Among them: F i (x i ,K i1 (y d ,t),K i2 (y d ,t)) is a smooth and strictly increasing function.

[0018] Yes i The derivative can be obtained

[0019]

[0020] in:

[0021]

[0022] The original system is converted into the following form

[0023]

[0024] in:

[0025]

[0026] definition:

[0027]

[0028] Introducing coordinate transformation

[0029]

[0030] Among them It is given by the following equation:

[0031]

[0032] In the above-mentioned time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, in order to eliminate the error The impact on the system, construct the following error compensation signal:

[0033]

[0034] where k i is the design constant, χ i (0)=0, i=1,…,n-1.

[0035] In the above-mentioned time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, the control law and adaptive rate are designed by using the improved dynamic surface control method as follows:

[0036]

[0037]

[0038] In the above-mentioned time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, in step S4, when the method is verified by example simulation, the specific method is: consider the following nonlinear system

[0039]

[0040] Where:

[0041]

[0042] η(t)=[2tsin(t),3-cos(t)]

[0043]

[0044] The expected tracking trajectory is:

[0045] y d=0.35sin(t),

[0046] The extended time-varying constraint function is defined as:

[0047] K 11 (y d ,t)=2sin(-0.3y d )+e -2t +0.5

[0048]

[0049]

[0050] The error compensation signal is designed as:

[0051]

[0052] The design parameters are:

[0053] k 1 =k 2 =1,σ 1 =1,σ 2 =2

[0054] β 1 =β 2 =0.01, a 1 =a 2 =50, l 2 =0.02

[0055] The initial values ​​are:

[0056] x 1 (0) = 0.1, x 2 (0) = 0.5

[0057] χ 1 (0) = 0.5

[0058] θ 1 (0) = 0.5, θ 2 (0) = 0.4

[0059] Virtual Controller for Application Design α 1 The system is simulated with the actual controller u.

[0060] Compared with the prior art, the present invention has the following advantages:

[0061] 1. An improved dynamic surface control method is proposed. This method transforms a parametric system with time-varying parameters and extended full-state constraints into an unconstrained nonlinear system through a constructed nonlinear mapping (NM). A compensation signal is introduced to offset the error caused by replacing the virtual control derivative with the output of a linear filter. Finally, a simulation example is used to verify the controller design scheme.

[0062] 2. Through the above method, the difficulty of controller design and the complexity of stability analysis can be reduced. The proposed control algorithm can ensure that the system achieves better tracking performance without violating the extended full-state constraint, and all signals are semi-globally consistently terminally bounded, and all states satisfy the extended full-state constraint. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 is the expected trajectory y of the system in the present invention d , output y and state constraints K 11 (y d ,t),K 12 (y d ,t) schematic diagram.

[0064] Figure 2 This is a control u curve diagram in the system of the present invention.

[0065] Figure 3 It is the state diagram of the system of the present invention.

[0066] Figure 4 This is a diagram of system constraints of the present invention.

[0067] Figure 5 is the adaptive law θ in the system of the present invention 1 and θ 2 Graph of the curve.

[0068] Figure 6 is the state x in the system of the present invention 1 and x 2 Phase plane diagram. DETAILED DESCRIPTION

[0069] Reference Figure 1-6 , a time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, where the control goal is to design an improved dynamic surface controller u so that the system output can track the expected trajectory well and all states do not violate the predefined constraints, and all signals are semi-globally uniformly terminally bounded. The method used is as follows:

[0070] S1: A parametric system with time-varying parameters and extended full-state constraints is transformed into an unconstrained nonlinear system through a constructed nonlinear mapping;

[0071] S2: Introducing a compensation signal to offset the error caused by replacing the virtual control derivative with the output of the linear filter;

[0072] S3: Design control laws and adaptive laws for the improved dynamic surface control method.

[0073] S4: Verify the improved dynamic surface control method and simulate the method through examples.

[0074] In step S1, consider the nonlinear extended constrained nonlinear system

[0075]

[0076] in is a known control gain function, η i (t)∈R ι is an unknown time-varying bounded parameter, is a known smooth function, x is the system state, u is the system input, and y is the system output.

[0077] Assume that the expected trajectory vector is continuous and available, where is a known compact set.

[0078] Assume 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.

[0079] Controller Design Based on Improved Dynamic Surface Method

[0080] In step S1, a nonlinear mapping is introduced:

[0081]

[0082] where F i (x i ,K i1 (y d ,t),K i2 (y d ,t)) is a smooth and strictly increasing function.

[0083] Yes i The derivative can be obtained

[0084]

[0085] in

[0086]

[0087] The original system is converted into the following form

[0088]

[0089] in,

[0090]

[0091] definition:

[0092]

[0093] Introducing coordinate transformation

[0094]

[0095] Among them It is given by the following equation:

[0096]

[0097] At the same time, in order to eliminate the error The impact on the system, construct the following error compensation signal:

[0098]

[0099] where k i is the design constant, χ i (0)=0, i=1,…,n-1.

[0100] The control law and adaptive rate are designed using the improved dynamic control method as follows:

[0101]

[0102] Theorem Description

[0103] For the parameterized nonlinear system with extended constraints described by equation (1), under the virtual control actual control law and adaptive control rate, all signals of the closed-loop system are semi-globally uniformly bounded, and all states do not violate the pre-defined constraints.

[0104] Proof: Definition

[0105] Where d is a positive constant,

[0106] Choose the following Lyapunov function:

[0107]

[0108] in:

[0109]

[0110] Taking the derivative of F with respect to time t, we get:

[0111]

[0112] in:

[0113]

[0114] According to Young's inequality, we can get

[0115]

[0116] thereby

[0117]

[0118] Furthermore,

[0119]

[0120] in:

[0121] b=min{a 1 β 1 ,…,a n β n}

[0122]

[0123] According to the Lyapunov stability principle, the closed-loop system is semi-globally uniformly bounded and the full-state constraints are not triggered.

[0124] Simulation Study

[0125] Consider the following nonlinear system

[0126]

[0127] Where:

[0128]

[0129] η(t)=[2tsin(t),3-cos(t)]

[0130]

[0131] The expected tracking trajectory is:

[0132] y d =0.35sin(t)

[0133] The extended time-varying constraint function is defined as:

[0134] K 11 (y d ,t)=2sin(-0.3y d )+e -2t +0.5

[0135] K 12 (y d ,t)=2sin(-0.1y d )+e -3t +0.45

[0136]

[0137] The error compensation signal is designed as:

[0138]

[0139] The design parameters are:

[0140] k 1 =k 2 =1,σ 1 =1,σ 2 =2

[0141] β 1 =β 2 =0.01, a 1 =a 2 =50, l 2 =0.02

[0142] The initial values ​​are:

[0143] x 1 (0) = 0.1, x 2 (0) = 0.5

[0144] χ 1 (0) = 0.5

[0145] θ 1 (0) = 0.5, θ 2 (0) = 0.4

[0146] The expected trajectory of the system y d , output y and state constraints K 11 (y d ,t),K 12 (y d ,t) Figure 1 As shown, the control u curve is as follows Figure 2 As shown, the state and its constraints are as follows Figure 3 and Figure 4 As shown, Figure 5 The adaptive law θ is shown 1 and θ 2 The curve of state x 1 and x 2 The phase plane is Figure 6 As shown in Figure 2. From the simulation results, it can be seen that this design method has achieved a relatively ideal control effect.

[0147] It is known from common technical knowledge that the present invention can be implemented by other embodiments that do not deviate from its spirit or essential features. Therefore, the above disclosed embodiments are only illustrative in all respects and are not exclusive. All changes within the scope of the present invention or within the scope equivalent to the present invention are included in the present invention.

Claims

1. A time-varying parameterized nonlinear system dynamic surface control and simulation method with extended time-varying constraints, characterized in that: The goal is to design an improvement to the dynamic surface controller so that the system output can track the expected trajectory well and all states do not violate the predefined constraints, and all signals are semi-globally consistent and bounded. The specific steps include: S1: A parametric system with time-varying parameters and extended full-state constraints is transformed into an unconstrained nonlinear system through a constructed nonlinear mapping; S2: Introducing a compensation signal to offset the error caused by replacing the virtual control derivative with the output of the linear filter; S3: Design control law and adaptive law for the improved dynamic surface control method; S4: Verify the improved dynamic surface control method and simulate the method through examples.

2. The method for controlling and simulating a nonlinear system dynamic surface with time-varying parameterization and extended time-varying constraints according to claim 1, characterized in that: In step S1, the nonlinear system is set as: in: is a known control gain function, η i (t)∈R ι is an unknown time-varying bounded parameter, is a known smooth function, x=[x1,…,x n ] T is the system status, u is the system input, y is the system output; And set the desired trajectory vector is continuous and available, where is a known compact set, and set 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.

3. The method for controlling and simulating a time-varying parameterized nonlinear system dynamic surface with extended time-varying constraints according to claim 1, characterized in that: In step S1, a nonlinear mapping is introduced: in: F i (x i ,K i1 (y d ,t),K i2 (y d ,t)) is a smooth and strictly increasing function; Yes i Taking the derivative we get: in: The original system is converted into the following form: in: definition: Introduce coordinate transformation: Among them It is given by the following equation:

4. The method for controlling and simulating a time-varying parameterized nonlinear system dynamic surface with extended time-varying constraints according to claim 1, characterized in that: To eliminate the error The impact on the system, construct the following error compensation signal: where k i is the design constant; x i (0)=0, i=1,…,n-1.

5. The method for controlling and simulating a time-varying parameterized nonlinear system dynamic surface with extended time-varying constraints according to claim 1, characterized in that: The control law and adaptive rate designed by the improved dynamic surface control method are as follows:

6. The method for controlling and simulating a time-varying parameterized nonlinear system dynamic surface with extended time-varying constraints according to claim 1, characterized in that: In step S4, when performing an example simulation to verify the method, the specific method is: consider the following nonlinear system: Where: η(t)=[2t sin(t),3-cos(t)] The expected tracking trajectory is: y d =0.35sin(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 +0.45 The error compensation signal is designed as: The design parameters are: k1=k2=1,σ1=1,σ2=2 β1=β2=0.01, a1=a2=50, l2=0.02 The initial values ​​are: x1(0)=0.1, x2(0)=0.5 x1(0)=0.5 θ1(0)=0.5,θ2(0)=0.4 The system is simulated using the designed virtual controller α1 and the actual controller u.

Images