A Dynamic Surface Control and Simulation Method for a Nonlinear System with Time-Varying Parameters and Extended Time-Varying Constraints

Abstract

The present invention discloses a dynamic surface control and simulation method for a time-varying parameterized nonlinear system with extended time-varying constraints, which relates to the field of nonlinear system stability and control. The method includes: during the controller design process, introducing a tangent function as a nonlinear mapping to transform a class of nonlinear parameter systems with time-varying parameters and extended full-state time-varying constraints into a new class of unconstrained parameterized systems. In addition, an improved DSC with an error compensation signal is used to propose a new adaptive control technique for the transformed nonlinear system. The proposed design scheme avoids assuming that the virtual control coefficients and their upper bounds are known, and proves that all signals in the closed-loop system are semi-globally ultimately uniformly bounded. The simulation results verify the effectiveness of the method.

Description

Technical Field

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

[0002] The 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 a constant parameter system with a strict feedback form and the event-based adaptive control problem of a parameter nonlinear system with time-varying parameters have been studied respectively. And through the BLF and backstepping design methods, an adaptive inversion control method based on a nonlinear mapping has been proposed for a nonlinear system with output constraints, and a pure feedback system with full state constraints has been transformed into a new pure feedback system without constraints based on a one-to-one corresponding nonlinear mapping.

[0003] The constraint conditions considered in the above prior art in the field of adaptive control of nonlinear systems are either constants or functions of time, and there are few functions of the desired trajectory. It is difficult to handle this extended full state constraint problem with traditional barrier Lyapunov functions. Moreover, there are relatively few studies on the stability control of nonlinear systems considering both time-varying parameters and extended constraints. Based on this, the present invention designs a dynamic surface control and simulation method for a time-varying parameterized nonlinear system with extended time-varying constraints. Summary of the Invention

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

[0005] A dynamic surface control and simulation method for a time-varying parameterized nonlinear system with extended time-varying constraints aims 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 constraint conditions, and all signals are semi-globally uniformly ultimately bounded.

[0006] It includes the following steps:

[0007] S1: Transform the parameter system with time-varying parameters and extended full state constraints into an unconstrained nonlinear system through a constructed nonlinear mapping;

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

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

[0010] S4: Verify the improved dynamic surface control method and conduct simulation verification on the method through examples.

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

[0012]

[0013] where, is a known control gain function, and 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 the desired trajectory vector is set to be continuous and available, where, is a known compact set, and K i1 (y d , t) > 0 and K i2 (y d , t) > 0 are known bounded smooth functions, i = 1, …, n, and its partial derivatives with respect to y d and t are also bounded..

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

[0016]

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

[0018] Taking the derivative of ξ i yields

[0019]

[0020] where:

[0021]

[0022] Then the original system is transformed into the following form

[0023]

[0024] where:

[0025]

[0026] Definition:

[0027]

[0028] Introduce coordinate transformation

[0029]

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

[0031]

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

[0033]

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

[0035] In the above dynamic surface control and simulation method for time-varying parameterized nonlinear systems with extended time-varying constraints, adopt the improved dynamic surface control method to design the control law and adaptive rate as follows:

[0036]

[0037]

[0038] In the above dynamic surface control and simulation method for time-varying parameterized nonlinear systems with extended time-varying constraints, in step S4, when conducting instance simulation verification on the method, the specific method is: consider the following nonlinear system

[0039]

[0040] In the formula:

[0041]

[0042] η(t) = [2tsin(t), 3 - cos(t)]

[0043]

[0044] The desired 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] K 12 (y d , t) = 2sin(-0.1y d ) + e -3t + 0.45,

[0049] K 21 (y d , t) = cos(5t) + e -3yd + 1K 22 (y d , t) = cos(t) + e -3yd + 0.6,

[0050] The error compensation signal is designed as:

[0051]

[0052] The design parameters are:

[0053] k1 = k2 = 1, σ1 = 1, σ2 = 2

[0054] β1 = β2 = 0.01, a1 = a2 = 50, l2 = 0.02

[0055] The initial values are:

[0056] x1(0) = 0.1, x2(0) = 0.5

[0057] χ1(0) = 0.5

[0058] θ1(0) = 0.5, θ2(0) = 0.4

[0059] The designed virtual controller α1 and the actual controller u are applied to conduct a simulation experiment on the system.

[0060] Compared with the existing technologies, the advantages of the present invention are:

[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), introduces a compensation signal to offset the error caused by replacing the virtual control derivative with the output of a linear filter, and finally verifies the controller design scheme through a simulation example.

[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 constraints, and all signals are semi-globally uniformly ultimately bounded, and all states satisfy the extended full-state constraints. Brief Description of the Drawings

[0063] Figure 1 The desired trajectory y of the system in the present invention d , the output y and the state constraint condition K 11 (y d , t), K 12 (y d , t) schematic diagram.

[0064] Figure 2 The control u curve graph of the system in the present invention.

[0065] Figure 3 The state graph of the system in the present invention.

[0066] Figure 4 The constraint condition graph of the system in the present invention.

[0067] Figure 5 The curve graphs of the adaptation laws θ1 and θ2 in the system of the present invention.

[0068] Figure 6 The phase plane graph of the states x1 and x2 in the system of the present invention. Detailed Embodiment

[0069] Referring to Figure 1-6 , a dynamic surface control and simulation method for a nonlinear system with time-varying parameters and extended time-varying constraints, wherein the control objective is to design an improved dynamic surface controller u based on the system, so that the system output can track the expected trajectory well and all states do not violate the predefined constraint conditions, and all signals are semi-globally uniformly ultimately bounded. The method is as follows:

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

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

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

[0073] S4: Verify the improved dynamic surface control method and conduct simulation verification of the method through examples.

[0074] In step S1, consider the nonlinear extended strict-feedback nonlinear system

[0075]

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

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

[0078] Assume that K i1 (y d , t) > 0 and K i2 (y d , t) > 0 are the known bounded smooth functions, i = 1, …, n, and its partial derivatives with respect to y d and t are also bounded.

[0079] Controller design based on the improved dynamic surface method

[0080] In step S1, introduce the nonlinear mapping:

[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] Taking the derivative of ξ i yields

[0084]

[0085] where

[0086]

[0087] Then the original system is transformed into the following form

[0088]

[0089] Among them,

[0090]

[0091] Definition:

[0092]

[0093] Introduce a coordinate transformation

[0094]

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

[0096]

[0097] Meanwhile, in order to eliminate the influence of the error on the system, construct the following error compensation signal:

[0098]

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

[0100] Adopt an improved dynamic control method to design the control law and adaptive rate as follows:

[0101]

[0102] Theorem description

[0103] For the parametric 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 ultimately uniformly bounded, and all states do not violate the predefined constraint conditions.

[0104] Proof: Define

[0105] where d is a positive constant,

[0106] Select the following Lyapunov function:

[0107]

[0108] where:

[0109]

[0110] Taking the derivative of F with respect to time t gives:

[0111]

[0112] where:

[0113]

[0114] According to Young's inequality, we have

[0115]

[0116] Thus

[0117]

[0118] Furthermore, we have

[0119]

[0120] where:

[0121] b = min{a1β1,…,a n β n}

[0122]

[0123] According to the Lyapunov stability principle, the closed-loop system is semi-globally ultimately uniformly bounded, and the full-state constraint condition is 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 desired 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] k1 = k2 = 1, σ1 = 1, σ2 = 2

[0141] β1 = β2 = 0.01, a1 = a2 = 50, l2 = 0.02

[0142] The initial values are:

[0143] x1(0) = 0.1, x2(0) = 0.5

[0144] χ1(0) = 0.5

[0145] θ1(0) = 0.5, θ2(0) = 0.4

[0146] The desired trajectory y of the system d , the output y and the state constraint condition K 11 (y d , t), K 12 (y d , t) are as Figure 1 shown, the control u curve is as Figure 2 shown, the state and its constraint conditions are as Figure 3 and Figure 4 shown, Figure 5 shows the curves of the adaptation laws θ1 and θ2, and the phase plane of the states x1 and x2 is as Figure 6 shown. It can be seen from the simulation results that the proposed design method has achieved an ideal control effect.

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

Claims

1. A dynamic surface control and simulation method for a time-varying parameterized nonlinear system with extended time-varying constraints, characterized in that, The goal is to improve the dynamic surface controller so that the system output can track the desired trajectory well and all states do not violate the predefined constraints, and all signals are semi-globally uniformly ultimately bounded. The specific steps are as follows: S1: Transform the parametric system with time-varying parameters and extended full-state constraints into an unconstrained nonlinear system through a constructed nonlinear mapping; S2: Introduce a compensation signal to cancel the error caused by replacing the virtual control derivative with the output of a linear filter; S3: Design the control law and adaptive law for the improved dynamic surface control method; S4: Verify the improved dynamic surface control method and conduct simulation verification of the method through examples; In step S1, the nonlinear system is set as: Where: 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 state,​ 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 are known bounded smooth functions, i = 1, …, n, and its partial derivatives with respect to y d and t are also bounded; In step S1, introduce the nonlinear mapping: Where: F i (x i ,K i1 (y d ,t),K i2 (y d ,t)) is a smooth and strictly increasing function; Derivative with respect to ξ i Taking the derivative gives: Where: Then the original system is transformed into the following form: Where: Define: Introduce the coordinate transformation: wherein the is given by the following equation:

2. A dynamic surface control and simulation method for a time-varying parametric nonlinear system with extended time-varying constraints according to claim 1, characterized in that: To eliminate the error that affects the system, construct the following error compensation signal: where k i is a design constant; χ i (0) = 0, i = 1, …, n - 1.

3. A dynamic surface control and simulation method for a time-varying parameterized nonlinear system with extended time-varying constraints according to claim 1, characterized in that: Design the control law and adaptive rate using the improved dynamic surface control method as follows:

4. A dynamic surface control and simulation method for a time-varying parametric nonlinear system with extended time-varying constraints according to claim 1, characterized in that: In step S4, when conducting example simulation verification of the method, the specific method is: Consider the following nonlinear system: In the formula: η(t) = [2tsin(t), 3 - cos(t)] The desired tracking trajectory is: y d = 0.35sin(t) The extended time-varying constraint function is defined as: K 11 (y d ,t) = 2sin(-0.3y d ) + e -2t + 0.5 K 12 (y 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 θ1(0) = 0.5, θ2(0) = 0.4 Apply the designed virtual controller α1 and the actual controller u to conduct simulation experiments on the system.

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