A nonlinear system tracking control algorithm with state constraints and unknown disturbances

Abstract

The present application relates to state observer design, state constrained system transformation design and tracking controller design of nonstrict feedback nonlinear system, and discloses a tracking control algorithm for nonstrict feedback nonlinear system with asymmetric dynamic full state constraint and unknown disturbance.The technology includes state observer design, state constrained system transformation design and tracking controller design.The present application designs a state observer based on fuzzy logic system theory for the problem of unmeasurable state in the system, designs a state constrained system transformation based on asymmetric logarithmic nonlinear mapping barrier function for the problem of asymmetric dynamic full state constraint in the nonlinear system, and designs a tracking controller based on backstepping method with improved command filtering and compensation system for realizing tracking control.The present application can effectively solve the tracking control problem of nonstrict feedback nonlinear system under asymmetric dynamic full state constraint and unknown disturbance.

Description

TECHNICAL FIELD

[0001] The present application relates to the design of state observer, state-constrained system transformation and tracking controller for a class of non-holonomic nonlinear systems with asymmetric dynamic full-state constraints and unknown disturbances. BACKGROUND

[0002] In recent decades, due to the wide application of nonlinear systems in various industrial fields such as robotic manipulators, unmanned vehicles, etc., people's attention to nonlinear systems has increased. Among them, the non-holonomic form is a common case, which has attracted the attention of many scholars. However, the non-holonomic form of nonlinear systems faces many obstacles. On the one hand, the nonlinear characteristics of the system and the unmeasurable state bring difficulties to the system synthesis. On the other hand, if the constraints of the system state are violated, it may lead to performance degradation or even system instability. Even worse, the computational explosion is also a big problem and obstacle from theory to practice.

[0003] For nonstrict-feedback systems with output constraints, an event-triggered adaptive output feedback tracking control scheme was developed, where a fuzzy logic system was used to approximate the more general state estimation error. [“Event-triggered adaptive fuzzy output-feedback control for nonstrict-feedback nonlinear systems with asymmetric output constraint” (A. Wang, L. Liu, J. Qiu and G. Feng, IEEE Transactions on Cybernetics, vol. 52, no. 1, pp. 712-722, Jan. 2022.)] For nonstrict-feedback nonlinear systems with asymmetric time-varying full state constraints, an adaptive fuzzy control algorithm was designed, where a fractional nonlinear mapping was adopted to transform the original system into an affine system. [“Adaptive fuzzy control for nonstrict-feedback systems under asymmetric time-varying full state constraints without feasibility condition” (Y. Liu, H. Zhang, Y. Wang and S. Sun, IEEE Transactions on Fuzzy Systems, vol. 29, no. 5, pp. 976-985, May 2021.)] For nonstrict-feedback nonlinear systems with constrained states and input saturation, a command-filter-based adaptive finite-time control algorithm was designed. [“Command-filter-based adaptive fuzzy finite-time output feedback control for state-constrained nonlinear systems with input saturation,” (W. Wei and W. Zhang, IEEE Transactions on Fuzzy Systems, vol. 30, no. 10, pp. 4044-4056, Oct. 2022.)]

[0004] However, to date, the tracking control design for nonstrict-feedback nonlinear systems has not been fully investigated, as it is more challenging to simultaneously consider unmeasurable states, external disturbances, and asymmetric dynamic full state constraints without feasibility conditions and computational explosion. SUMMARY

[0005] The problem to be solved by the present application is the unmeasurable state estimation, dynamic state constraint guarantee and tracking control problem in nonstrict feedback nonlinear system.

[0006] The algorithm adopted by the present application to solve the problem is that, for the unmeasurable state problem in the system, the present application designs a state observer based on fuzzy logic system theory; for the non-symmetrical dynamic full state constraint problem in the nonlinear system, a state constraint system transformation based on non-symmetrical logarithmic nonlinear mapping barrier function is designed; in order to realize tracking control, a tracking controller based on backstepping method with improved command filtering and compensation system is designed. The present application can effectively solve the tracking control problem of nonstrict feedback nonlinear system under non-symmetrical dynamic full state constraint and unknown disturbance.

[0007] The fuzzy state observer design is designed as follows:

[0008]

[0009] And

[0010]

[0011] And the following adaptive law is satisfied

[0012]

[0013] Wherein is the estimated value of x i , and

[0014] The non-symmetrical logarithmic state constraint system transformation design is designed as follows:

[0015]

[0016] Wherein represents the variable list of dynamic constraint, and indicates that the dynamic constraint studied in this paper is not only related to time but also related to all states, and it can be seen that ξ i is completely dependent on the system state, if the initial condition satisfies -ο 11 (0)<x1(0)<ο 12 (0) or Then ξ i →±∞ when and only when x1→-ο 11 (var) / ο 12 (var) or Therefore, as long as the boundedness of ξ i is guaranteed, the condition of maintaining full state constraint will also be guaranteed.

[0017] The tracking controller design based on backstepping method, firstly designs the filter system as follows

[0018]

[0019] Wherein represents the input, and respectively represent the zero-order output and the first-order output of the command filter, and τ i1 and τ i2 are parameters to be designed. In addition, given a positive number δ0, there is

[0020] sat(x)=x / (|x|+δ0)

[0021] Secondly, since the filter error (α i,c -α i ) cannot be ignored, the following compensation system is designed

[0022]

[0023] Wherein, is the i-th compensation signal, and all satisfy the initial condition

[0024] Finally, the following controller is designed

[0025]

[0026] And

[0027]

[0028] Wherein, c n1 >0, c n2 ≥0.5, κ>0, l n1 >0, γ>0, ε>0 are parameters to be designed. BRIEF DESCRIPTION OF DRAWINGS

[0029] Figure 1 It is the structural schematic diagram of the control system of the application. DETAILED DESCRIPTION

[0030] The technical scheme of the application is described in detail below with the non-strict feedback nonlinear system with asymmetric dynamic full-state constraints and unknown disturbance as an example.

[0031] As Figure 1As shown, the present application relates to the design of state observer based on fuzzy logic system theory for non-affine nonlinear systems, the design of state constrained system transformation based on asymmetric logarithmic nonlinear mapping barrier function, the design of tracking controller with improved command filter and compensation system based on backstepping method.

[0032] Consider the following non-affine nonlinear system

[0033]

[0034] where u e R and y e R represent the actual control input and measurable output of the system, respectively, x e Rnrepresents the state vector of the system, and f(x) and g(x) represent nonlinear functions, both of which are related to all states, d i (t), i = 1...n represent unknown external disturbances.

[0035] The non-affine nonlinear system (1) satisfies the following assumptions: (1) The smooth functions and are positive, and their derivatives up to the nth order are continuous and bounded; (2) The reference signal y r and its derivatives up to the nth order are continuous and bounded, and for any and there exist normal numbers O0, and O i (i = 1...n) such that and (3) The disturbance d i (t) is unknown but bounded by a normal number , i.e.

[0036] Generally, a fuzzy logic system is used to estimate any nonlinear function f(x): Ω → R, i.e. where represents the fuzzy basis vector function, Y *T represents the optimal weight parameter vector, and ε * > 0 represents the upper bound of the estimation error.

[0037] Design of state observer based on fuzzy logic system theory

[0038] A state space expression of the system (1) is

[0039]

[0040] where

[0041]

[0042]

[0043] Definitions A fuzzy state observer is established to estimate the unmeasured states

[0044]

[0045] where is the estimate of x i and

[0046] and define and The error system of state estimation can be obtained as follows

[0047]

[0048] where b = [b1, b2, …, b n ] T , and have b i = b i ′ + d i and where is the estimate of the optimal weight , and is the known normal quantity limit.

[0049] State-constrained system transformation design based on asymmetric logarithmic nonlinear mapping barrier function

[0050] In order to avoid violating the asymmetric dynamic full state constraints, this paper introduces an asymmetric logarithmic nonlinear mapping barrier function (NMBF).

[0051]

[0052] where represents the variable list of dynamic constraints, which means that the dynamic constraints studied in this paper are not only related to time but also related to all states. It can be seen that ξ i is completely dependent on the system state. If the initial condition satisfies -ο 11 (0)<x1(0)<ο 12 (0) or then ξ i →±∞ when and only when x1→-ο 11 (var) / ο 12 (var) or Therefore, as long as the boundedness of ξ i is guaranteed, the condition of maintaining full state constraints will also be guaranteed.

[0053] Then, the inverse mapping can be obtained by the following equation

[0054]

[0055] And the relationship between y r and ξ r can be obtained

[0056]

[0057] Taking the derivative of ξ i (i = 1...n) with respect to time, the system (1) is converted to an affine form,

[0058]

[0059] where

[0060] where ξ = [ξ1...ξ n ] T ,

[0061] o1(var) = [o 11 (var), o 21 (var),... o n1 ... on1(var)], o2(var) = [o 12 (var), o 22 (var),... o n2 (var)] and

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069]

[0070] It can be seen that, through the NMBF technique, the original state-constrained non-strict feedback nonlinear system is converted to a new stateless constrained affine nonlinear system.

[0071] Tracking controller design based on backstepping method

[0072] Define new auxiliary variables: where α i-1,c is the output signal of the command filter system, and is the compensation signal of the compensation system, the specific command filter and compensation system is as follows.

[0073] An improved command filter with saturation function is proposed, which is more general than the sign function, and the specific design of the filter system is as follows

[0074]

[0075] where denotes the input, and denote the zero-order output and first-order output of the command filter respectively, and τ i1 and τ i2 are parameters to be designed. In addition, given a positive number δ0, there is

[0076] sat(x) = x / (|x| + δ0)

[0077] Since the filter error (α i,c - α i ) cannot be ignored, an additional compensation system is designed as follows

[0078]

[0079] where, is the i-th compensation signal, and all satisfy the initial condition

[0080] Then according to the backstepping method, the specific design of the virtual control input and the actual control input is as follows

[0081] Step 1: Differentiate the auxiliary variable z1, get

[0082]

[0083] Let so that where

[0084] Design the virtual control input α1 and the parameter adaptation law θ1 as follows:

[0085]

[0086]

[0087] where c 11 > 0, c 12 ≥ 0.5, l11 > 0, κ1> 0 are parameters to be designed,

[0088] Step i (2≤i≤n-1): Differentiate z i Differentiate, we get

[0089]

[0090] Let So that Where

[0091] Design the virtual control input α i And the parameter adaptive law θ i As follows:

[0092]

[0093]

[0094] Where c i1 > 0, c i2 ≥ 0.5, l i1 > 0, κ i > 0 are parameters to be designed, S i = (ξ1, ξ2, …, ξ i , ξ r ) T .

[0095] Step n: Differentiate z n Differentiate, we get

[0096]

[0097] Let So that Where

[0098] Design the actual control input α n = u and the parameter adaptive law θ n And As follows:

[0099]

[0100]

[0101]

[0102] Where c n1 > 0, c n2 ≥ 0.5, κ n > 0, κ > 0, ln1 > 0 are parameters to be designed, S n = (ξ, ξ r ) T .

Claims

1. A tracking control strategy for a non-strictly feedback nonlinear system, characterized in that The application relates to a state observer design, a state limited system transformation design and a tracking controller design. The state observer design based on a fuzzy logic system comprises modeling of a non-strict feedback nonlinear system with asymmetric dynamic full state constraints and unknown disturbances, fuzzy estimation of a nonlinear function; the state limited system transformation design is specifically a state limited system transformation method based on an asymmetric logarithmic nonlinear mapping barrier function; The tracking controller design comprises filter and compensation system design and tracking controller design based on a backstepping method; The non-strict feedback nonlinear system with asymmetric dynamic full state constraints and unknown disturbances is described as follows: for a nonlinear system in the following non-strict feedback form where u e R and y e R represent the actual control input and measurable output of the system, respectively, x e Rnrepresents the state vector of the system, and f(x, u) represents a nonlinear function, all of which are related to all states, d i (t), i = 1... n represent unknown external disturbances; The state observer design based on the fuzzy logic system is that for any real continuous function f(x): Omega-> R, there is a fuzzy logic system such that wherein and Y * represent the fuzzy basis vector function and the optimal weight parameter vector, respectively, T denotes the transpose of a vector or matrix, ε * > 0 represents an upper bound of the optimal estimation error; The state space model of the system is where x n+1 = u, N = [n1...n n ] T D＝[1…0…0]. Definitions Establishing a fuzzy state observer to estimate the state of a system wherein is an estimate of x i and The state limited system transformation design introduces an asymmetric logarithmic nonlinear mapping barrier function where ξ represents the list of variables subject to dynamic constraints i Depends on the system state, if the initial condition satisfies 11 (0) < x1(0) < 0 12 (0) or then ξ i → ±∞ if and only if x1→ -0 11 (var) / 0 12 (var) or Therefore, as long as the boundedness of ξ i is guaranteed, the condition of state constraints will also be satisfied; ξ i (i = 1...n) are differentiated with respect to time, combined with a nonlinear mapping barrier function technique, the system will be converted into an affine form without state constraints The filter and compensation system design and the tracking controller design based on the backstepping method design the following controller And where c n1 > 0, c n2 ≥ 0.5, κ > 0, l n1 > 0, γ > 0, ε > 0 are parameters to be designed, is an auxiliary variable, where α n-1 is the (n-1)th input signal, α n-1,c is the (n-1)th filtered output signal, is the nth compensation signal, the filtering and compensation system is designed as follows: Firstly, the following filter system is designed where denotes the input, and denote the zeroth and first order output of the filter system, respectively, τ i1 and τ i2 are the parameters to be designed, and, in addition, given a positive number δ0, there are sat(x)=x / (|x|+delta0) Secondly, the following compensation system is designed sat(x)=x / (|x|+delta0) wherein is the i-th compensated signal, satisfying the initial condition

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