CN105159065A - Stability determination method of non-linear active-disturbance-rejection control system - Google Patents

Abstract

The invention discloses a stability determination method of a non-linear active -disturbance-rejection control system. The method comprises steps of non-linear auto-disturbance-rejection control system establishment, system conversion, and stability determination of an indirect lurie system based on a robustness Popov criterion. The method has the following beneficial effects: the non-linear auto-disturbance-rejection control system is converted into an indirect lurie system and the system stability can be determined by using the robustness Popov criterion, and thus stability of a nominal system as well as robustness stability of the parameter shooting system can be determined, so that the operation becomes convenient; and the original complex non-linear extended state observer is changed into a variable-gain linear extended state observer and then stability is determined based on a routh criterion, so that the operation becomes simple and convenient.

Description

A kind of non-linear Active Disturbance Rejection Control system stability determination methods

Technical field

The invention belongs to technical field of automation, relate to a kind of non-linear Active Disturbance Rejection Control system stability determination methods.

Background technology

Active Disturbance Rejection Control is a kind of new practical control method that Chinese Academy of Sciences Han Jingqing researcher proposes, and the control theory of its uniqueness has obtained the accreditation of people, and the control performance of its excellence is by theoretical analysis and practical application is widely confirmed.The non-linear Active Disturbance Rejection Control that Han Jing proposes clearly, owing to employing nonlinear function in extended state observer and design of control law, therefore has certain advantage in antijamming capability, control accuracy etc.But the introducing of nonlinear function makes, and system motion becomes more complicated, stability and performance evaluation more difficult, the stability analysis of nonlinear extension state observer and non-linear Active Disturbance Rejection Control system is a difficult point always.

Utilize the Active Disturbance Rejection Control system Frequency-domain Stability containing single and two nonlinear elements in describing function method research extended state observer, but it is limited owing to considering non-linear number, the limitation that graph transformation is comparatively complicated and describing function method is intrinsic, is difficult to be generalized in general nonlinearity active disturbance rejection system.The time-domain stability of single-input single-output and multiple-input, multiple-output Active Disturbance Rejection Control system is have studied based on Lyapunov theorem of stability, propose some adequate condition of stability, but due to too much restrictive condition, comparatively complicated derivation, processing general Active Disturbance Rejection Control system has certain difficulty.On the whole, the stability analysis of non-linear Active Disturbance Rejection Control system also needs more direct, easy analytical approach.

Summary of the invention

Technical matters to be solved by this invention is to provide a kind of simple and easy to do non-linear Active Disturbance Rejection Control system stability determination methods.

For solving the problems of the technologies described above adopted technical scheme be: a kind of non-linear Active Disturbance Rejection Control system stability determination methods, comprises the steps:

(1) set up non-linear Active Disturbance Rejection Control system, it comprises controlled device and non-linear automatic disturbance rejection controller; Described non-linear automatic disturbance rejection controller comprises Nonlinear Tracking Differentiator, nonlinear extension state observer and nonlinearity erron Feedback Control Laws u;

Described controlled device is

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=x_2\\ .\\ .\\ .\\ {\stackrel{\·}{x}}_n=a_n{x}_1+a_{n-1}{x}_2+\mathrm{...}{a}_1{x}_n+bu\\ y=x_1\end{array}\right.$ (formula 1)

In (formula 1), x _ifor the state of controlled device, a _ifor the gain of controlled device corresponding state, the corresponding first order derivative of state for controlled device, wherein i=1,2 ..., n, n be greater than 1 positive integer; Y is that controlled device exports, and nonlinearity erron Feedback Control Laws u is the control inputs amount of controlled device, and b is control channel gain;

Described Nonlinear Tracking Differentiator is

$\left\{\begin{array}{c}{\stackrel{\·}{v}}_1=v_2\\ {\stackrel{\·}{v}}_2=v_2\\ .\\ .\\ .\\ {\stackrel{\·}{v}}_{n+1}=v_n\\ {\stackrel{\·}{v}}_n={\mathrm{\λ}}^n\mathrm{\ψ}(v_1-r,\frac{v_2}{\mathrm{\λ}},\mathrm{...},\frac{v_n}{{\mathrm{\λ}}^{n-1}})\end{array}\right.$ (formula 2)

In (formula 2), r is the input of described Nonlinear Tracking Differentiator, v _i(i=1,2 ..., n) be the output of described Nonlinear Tracking Differentiator, λ is the adjustable speed factor, for quick tracing function;

Described nonlinear extension state observer is

(formula 3)

In (formula 3), the b times of gain of controlled device output y and nonlinearity erron Feedback Control Laws u is respectively the input signal of described nonlinear extension state observer; z _i(i=1,2 ..., n+1) be the output of described nonlinear extension state observer; β _0i(i=1,2 ..., n+1) and be the gain of described nonlinear extension state observer; E represents the output y of controlled device and the output z of nonlinear extension state observer ₁between deviation; In described nonlinear extension state observer usually following nonlinear function is got:

(formula 4)

Wherein, a ₁(i=1,2 ..., n+1) and δ be normal number, sgn () represents sign function;

The expression formula of described nonlinearity erron Feedback Control Laws u is as follows:

$u=\frac{u_0-z_{n+1}}{b}$ (formula 5)

Wherein,

$u_0=\underset{i=1}{\overset{n}{\Σ}}{k}_{i}fal(v_i-z_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})$ (formula 6)

In (formula 5) and (formula 6), k _ifor gain coefficient, a ' _i(i=1,2 ..., n) be normal number; B is control channel gain; u ₀feedback of status part, i.e. state feedback control law in nonlinearity erron Feedback Control Laws u;

The output z of described nonlinear extension state observer _i(i=1,2 ..., n) with the output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) do subtraction relatively after as described state feedback control law u ₀input e _i; The b times of gain of described nonlinearity erron Feedback Control Laws u is as the first input signal of described nonlinear extension state observer; The output y of described controlled device is as the second input signal of described nonlinear extension state observer;

(2) system conversion is carried out;

Suppose that the input r of Nonlinear Tracking Differentiator described in A1 is 0, the output v of so described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) be 0;

Make e _i=v _i-z _i; Then to the fal (v in (formula 6) _i-z _i, a ' _i, δ) do as down conversion:

$fal(v_i-z_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=\frac{fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}{e}_i;$

Order

$\frac{fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}={\mathrm{\λ}}_i\left(e_i\right)$

Then

Fal (v _i-z _i, a ' _i, δ) and=λ _i(e _i) e _i(formula 7)

By λ _i(e _i) be abbreviated as λ _i,

Obtained by (formula 6) and (formula 7):

$u_0=\underset{i=1}{\overset{n}{\Σ}}{k}_i{\mathrm{\λ}}_i{e}_i;$ (formula 8)

Make X=[x ₁, x ₂..., x _n] ^t, Z=[z ₁, z ₂..., z _n] ^t, obtained by (formula 1), (formula 5) and (formula 8):

$\left\{\begin{array}{c}\stackrel{\·}{X}=A_{11}X+A_{12}Z+A_{12}{z}_{n+1}\\ y=x_1\end{array}\right.$

(formula 9)

Wherein,

$A_{11}=\left[\begin{array}{ccccc}0& 1& 0& \mathrm{...}& 0\\ 0& 0& 1& \mathrm{...}& 0\\ & .& & .& \\ & .& & .& \\ & .& & .& \\ 0& 0& \mathrm{...}& 0& 1\\ a_n& a_{n-1}& \mathrm{...}& a_2& a_1\end{array}\right],A_{12}=\left[\begin{array}{cccc}0& \mathrm{...}& 0& 0\\ & .& & \\ & .& & \\ & .& & \\ 0& \mathrm{...}& 0& 0\\ -{\mathrm{\λ}}_1{k}_1& -{\mathrm{\λ}}_2{k}_2& \mathrm{...}& {\mathrm{\λ}}_n{k}_n\end{array}\right],$

Order

(formula 10)

(formula 10) represents and works as corresponding when obtaining maximal value wherein i=1,2 ..., n, n+1;

Described nonlinear extension state observer (formula 3) is done following distortion:

(formula 11)

Order

(formula 12)

Then

(formula 13)

Obtained by (formula 5), (formula 8) and (formula 13):

(formula 14)

Wherein,

$A_{21}=\left[\begin{array}{ccccc}0& 1& 0& \mathrm{...}& 0\\ 0& 0& 1& \mathrm{...}& 0\\ & .& & .& \\ & .& & .& \\ & .& & .& \\ 0& 0& \mathrm{...}& 0& 1\\ -{\mathrm{\λ}}_1{k}_1& -{\mathrm{\λ}}_2{k}_2& \mathrm{...}& -{\mathrm{\λ}}_{n-1}{k}_{n-1}& -{\mathrm{\λ}}_n{k}_n\end{array}\right],$

$A_{22}={\[{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01},{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02},...,{\widetilde{\mathrm{\λ}}}_{0n}{\mathrm{\β}}_{0n}\]}^T.$

Obtained by (formula 9) and (formula 14):

(formula 15)

Wherein,

c ₁＝[-1，０，…，０］ ^Ｔ∈R ⁿ，c ₂＝[1，０，…，０］ ^Ｔ∈R ⁿ；

Make Y=A ₁₁x+A ₁₃z _n+1, e=σ, then

( formula 16)

(formula 16) is expressed as further:

(formula 17)

Wherein,

$\tilde{x}={\left[\begin{array}{cc}Y& Z\end{array}\right]}^T,A=\left[\begin{array}{cc}{A}_{11}& A_{11}{A}_{12}\\ 0& A_{21}\end{array}\right],b=\left[\begin{array}{c}{A}_{13}{\mathrm{\β}}_{0(n+1)}\\ A_{22}\end{array}\right],c=\[\begin{array}{ccc}{e_1}^T& {A_{11}}^{-1}& {C_2}^T\end{array}\]$

$p=-{c_1}^T{A_{11}}^{-1}{A}_{13}{\mathrm{\β}}_{0(n+1)}=-\frac{{\widetilde{\mathrm{\λ}}}_{0(n+1)}{\mathrm{\β}}_{0(n+1)}}{a_n}.$

(formula 17) is indirect Lurie's systems, and the expression formula of forward path function G (s) of described indirect Lurie's systems is:

G (s)=c ^t(sI-A) ^-1b+ ρ/s; (formula 18)

(3) robust Popov criterion is utilized to carry out judgement of stability to indirect Lurie's systems;

First, the interval transport function G of indirect Lurie's systems is defined _ias follows:

$G_I=\{G\left(s\right):G\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)},N\left(s\right)\∈N_I,D\left(s\right)\∈D_I\}$ (formula 19);

Wherein,

$\begin{array}{c}{N}_I=\{N\left(s\right):N\left(s\right)=\underset{i=0}{\overset{m}{\Σ}}{b}_i{s}^i,b_i\∈\[{\underset{\‾}{b}}_i,{\stackrel{\‾}{b}}_i\],i=0,\mathrm{...},m\}\\ D_I=\{D\left(s\right):D\left(s\right)=s^n+\underset{i=0}{\overset{n-1}{\Σ}}{c}_i{s}^i,c_i\∈\[{\underset{\‾}{c}}_i,{\stackrel{\‾}{c}}_i\],i=0,\mathrm{...},n-1\}\end{array}$

M, n are respectively integer;

And then be defined as follows a point son vertex polynomial expression N _kas follows:

N _k={ N ₁(s), N ₂(s), N ₃(s), N ₄(s) } (formula 20) wherein,

$\begin{array}{c}{N}_1\left(s\right)={\stackrel{\‾}{b}}_0+{\stackrel{\‾}{b}}_{1}s+{\underset{\‾}{b}}_2{s}^2+{\underset{\‾}{b}}_3{s}^3+{\stackrel{\‾}{b}}_4{s}^4+{\stackrel{\‾}{b}}_5{s}^5+\mathrm{...},\\ N_2\left(s\right)={\stackrel{\‾}{b}}_0+{\underset{\‾}{b}}_{1}s+{\underset{\‾}{b}}_2{s}^2+{\stackrel{\‾}{b}}_3{s}^3+{\stackrel{\‾}{b}}_4{s}^4+{\underset{\‾}{b}}_5{s}^5+\mathrm{...},\\ N_3\left(s\right)={\underset{\‾}{b}}_0+{\underset{\‾}{b}}_{1}s+{\stackrel{\‾}{b}}_2{s}^2+{\stackrel{\‾}{b}}_3{s}^3+{\underset{\‾}{b}}_4{s}^4+{\underset{\‾}{b}}_5{s}^5+\mathrm{...},\\ N_4\left(s\right)={\underset{\‾}{b}}_0+{\stackrel{\‾}{b}}_{1}s+{\stackrel{\‾}{b}}_2{s}^2+{\underset{\‾}{b}}_3{s}^3+{\underset{\‾}{b}}_4{s}^4+{\stackrel{\‾}{b}}_5{s}^5+\mathrm{...},\end{array}$

be respectively real number;

Equally, denominator summit polynomial expression D is defined _kas follows:

D _k={ D ₁(s), D ₂(s), D ₃(s), D ₄(s) } (formula 21)

Wherein,

$\begin{array}{c}{D}_1\left(s\right)={\stackrel{\‾}{c}}_0+{\stackrel{\‾}{c}}_{1}s+{\underset{\‾}{c}}_2{s}^2+{\underset{\‾}{c}}_3{s}^3+{\stackrel{\‾}{c}}_4{s}^4+{\stackrel{\‾}{c}}_5{s}^5+\mathrm{...},\\ D_2\left(s\right)={\stackrel{\‾}{c}}_0+{\underset{\‾}{c}}_{1}s+{\underset{\‾}{c}}_2{s}^2+{\stackrel{\‾}{c}}_3{s}^3+{\stackrel{\‾}{c}}_4{s}^4+{\underset{\‾}{c}}_5{s}^5+\mathrm{...},\\ D_3\left(s\right)={\underset{\‾}{c}}_0+{\underset{\‾}{c}}_{1}s+{\stackrel{\‾}{c}}_2{s}^2+{\stackrel{\‾}{c}}_3{s}^3+{\underset{\‾}{c}}_4{s}^4+{\underset{\‾}{c}}_5{s}^5+\mathrm{...},\\ D_4\left(s\right)={\underset{\‾}{c}}_0+{\stackrel{\‾}{c}}_{1}s+{\stackrel{\‾}{c}}_2{s}^2+{\underset{\‾}{c}}_3{s}^3+{\underset{\‾}{c}}_4{s}^4+{\stackrel{\‾}{c}}_5{s}^5+\mathrm{...},\end{array}$

be respectively real number;

Define the transport function collection G of indirect Lurie's systems _kas follows:

$G_K=\{G\left(s\right):G\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)},N\left(s\right)\∈N_K,D\left(s\right)\∈D_K\}$ (formula 22)

From (formula 20), (formula 21) and (formula 22), the transport function collection G of indirect Lurie's systems _kcomprise following 16 forward paths function G (s):

$\begin{array}{c}{G}_K=\{G\left(s\right):G\left(s\right)=\frac{N_1\left(s\right)}{D_1\left(s\right)},\frac{N_2\left(s\right)}{D_1\left(s\right)},\frac{N_3\left(s\right)}{D_1\left(s\right)},\frac{N_4\left(s\right)}{D_1\left(s\right)},\\ \frac{N_1\left(s\right)}{D_2\left(s\right)},\frac{N_2\left(s\right)}{D_2\left(s\right)},\frac{N_3\left(s\right)}{D_2\left(s\right)},\frac{N_4\left(s\right)}{D_2\left(s\right)},\frac{N_1\left(s\right)}{D_3\left(s\right)},\frac{N_2\left(s\right)}{D_3\left(s\right)},\\ \frac{N_3\left(s\right)}{D_3\left(s\right)},\frac{N_4\left(s\right)}{D_3\left(s\right)},\frac{N_1\left(s\right)}{D_4\left(s\right)}\frac{N_2\left(s\right)}{D_4\left(s\right)},\frac{N_3\left(s\right)}{D_4\left(s\right)},\frac{N_4\left(s\right)}{D_4\left(s\right)}\}\end{array}$

(formula 23)

As G (s) ∈ G _ktime, if there is an arithmetic number θ all meet Popov stability condition, then forward path transport function G (s) is interval transport function G formula (19) Suo Shi _i, i.e. G (s) ∈ G _iindirect Lurie's systems be absolute stable.

Described nonlinear extension state observer is third-order non-linear extended state observer, i.e. n=2; Its expression formula is as follows:

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-{\mathrm{\β}}_{01}\·e\\ {\stackrel{\·}{z}}_2=z_3-{\mathrm{\β}}_{02}\·fal(e,{\mathrm{\α}}_2,\mathrm{\δ})+bu\\ {\stackrel{\·}{z}}_3=-{\mathrm{\β}}_{03}\·fal(e,{\mathrm{\α}}_3,\mathrm{\δ})\end{array}\right.$ (formula 24)

Wherein, z ₁, z ₂, z ₃be respectively the output of described third-order non-linear extended state observer; β ₀₁, β ₀₂, β ₀₃be respectively the gain of described third-order non-linear extended state observer;

Order

$fal(e,{\mathrm{\α}}_2,\mathrm{\δ})=\frac{fal(e,{\mathrm{\α}}_2,\mathrm{\δ})}{e}e={\mathrm{\λ}}_{02}\left(e\right)e,fal(e,{\mathrm{\α}}_3,\mathrm{\δ})=\frac{fal\left(e,{\mathrm{\α}}_3,\mathrm{\δ}\right)}{e}e={\mathrm{\λ}}_{03}\left(e\right)e;$

Then obtain variable-gain linear extended state observer, its expression formula is as follows:

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-{\mathrm{\β}}_{01}\·e\\ {\stackrel{\·}{z}}_2=z_3-\left({\mathrm{\β}}_{02}\·{\mathrm{\λ}}_{02}\left(e\right)\right)e+bu\\ {\stackrel{\·}{z}}_3=-\left({\mathrm{\β}}_{03}\·{\mathrm{\λ}}_{03}\left(e\right)\right)e\end{array}\right.$ (formula 25)

By λ ₀₂(e), λ ₀₃e () is abbreviated as λ respectively ₀₂, λ ₀₃;

Order

f(l)＝αx ₁+α _n-1x ₂+…α ₁x _n，

Then the transport function of described variable-gain linear extended state observer is:

$\begin{array}{c}{z}_1=\frac{{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}y+\frac{s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}bu\\ z_2=\frac{\left({\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}\right)s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}y+\frac{\left(s+{\mathrm{\β}}_{01}\right)s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}bu\\ z_3=\frac{{\mathrm{\λ}}_{02}{\mathrm{\β}}_{03}{s}^{2}y-{\mathrm{\λ}}_{03}bu}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}-\frac{{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}f\left(s\right)\end{array}$

(formula 26)

According to Routh Criterion method, the stable necessary and sufficient condition of described variable-gain linear extended state observer is:

λ ₀₂β ₀₁β ₀₂> λ ₀₃β ₀₃; (formula 27) is further, if α ₂=α ₃, due to λ ₀₂, λ ₀₃all the function of tracking error e, then λ ₀₂=λ ₀₃, therefore, the stable necessary and sufficient condition of described variable-gain linear extended state observer is:

β ₀₁β ₀₂> β ₀₃; (formula 28)

Illustrate: formula (27) is the stable general condition of third-order non-linear extended state observer, and formula (28) is meeting α ₂=α ₃the specific condition that in situation, third-order non-linear extended state observer is stable.

The invention has the beneficial effects as follows: (1) is by being converted to interval Lurie's systems by non-linear Active Disturbance Rejection Control system, robust Popov criterion is utilized to judge the stability of system, both the stability of nominal system can have been judged, also the robust stability of Systems With Paramater Perturbation can be it is determined that the presence of, very convenient; (2) originally complicated nonlinear extension state observer stability analysis, by being converted into the linear extended state observer of variable-gain, recycling Routh Criterion can judge its stability, very simply, conveniently.

Accompanying drawing explanation

Fig. 1 is typical non linear Active Disturbance Rejection Control system architecture diagram.

Fig. 2 is indirect Lurie's systems structured flowchart.

Fig. 3 stability analysis schematic diagram.

Embodiment

Below in conjunction with Fig. 1-Fig. 3 and embodiment, the invention will be further described.

Suppose that the input r of Nonlinear Tracking Differentiator described in A1 is 0, the output v of so described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) be 0;

The second nonlinear Active Disturbance Rejection Control system stability analysis formed for second order controlled device and non-linear automatic disturbance rejection controller, sets forth the present invention's application process in practice.

Based on the nonlinear extension state observer stability analysis of Routh Criterion.

For the permanent controlled device of certain second-order linearity, its mathematical model is as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=-3x_1-5x_2+u\\ y=x_1\end{array}\right.$

(formula 29)

Wherein, x ₁, x ₂be represented as the state of controlled device, represent the first order derivative of corresponding state, y is that controlled device exports, and nonlinearity erron Feedback Control Laws u is the control inputs amount of controlled device.

The design of non-linear automatic disturbance rejection controller:

In view of Nonlinear Tracking Differentiator is a relatively independent structure, the not stability of influential system, and according to hypothesis A1 the output v of Nonlinear Tracking Differentiator _i(i=1,2 ..., n) be 0, not bamboo product Nonlinear Tracking Differentiator.

In view of controlled device is second-order system, design a typical third-order non-linear extended state observer as follows:

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-{\mathrm{\β}}_{01}\·e\\ {\stackrel{\·}{z}}_2=z_3-{\mathrm{\β}}_{02}\·fal(e,{\mathrm{\α}}_2,\mathrm{\δ})+bu\\ {\stackrel{\·}{z}}_3=-{\mathrm{\β}}_{03}\·fal(e,{\mathrm{\α}}_3,\mathrm{\δ})\end{array}\right.$ (formula 30)

Nonlinearity erron Feedback Control Laws u is designed to following form:

u＝[k ₁fal(v ₁-z ₁，α′ ₁，δ)+k ₂fal(v ₂-z ₂,α′ ₂,δ)-z ₃]/b

(formula 31)

Herein, according to the output v supposing Nonlinear Tracking Differentiator described in A1 _i(i=1,2 ..., n) be 0, then v ₁, v ₂be zero.

Determine non-linear automatic disturbance rejection controller parameter: make ω _o=20, ω _c=10, β ₀₁=3 ω ₀, k ₂=2 ω _c, δ=0.01, α ₂=0.5, α ₃=0.5, α ' ₁=0.75, α ' ₂=1.5.

Non-linear Active Disturbance Rejection Control system stability analysis based on robust Popov criterion:

Second nonlinear Active Disturbance Rejection Control system forward channel transfer function expression formula is obtained according to (formula 17) and (formula 18):

$G\left(s\right)=\frac{{\stackrel{\‾}{B}}_0{s}^4+{\stackrel{\‾}{B}}_1{s}^3+{\stackrel{\‾}{B}}_2{s}^2+{\stackrel{\‾}{B}}_{3}s+{\stackrel{\‾}{B}}_4}{{\tilde{A}}_0{s}^5+{\tilde{A}}_1{s}^4+{\tilde{A}}_2{s}^3+{\tilde{A}}_3{s}^2+{\tilde{A}}_{4}s+{\tilde{A}}_5}$ (formula 32)

Molecule, each term coefficient of denominator of (formula 32) are as follows:

${\tilde{A}}_0=1,{\tilde{A}}_1={\mathrm{\λ}}_2,k_2-a_1,{\tilde{A}}_2={\mathrm{\λ}}_1{k}_1-a_2-a_1{\mathrm{\λ}}_2{k}_2$

${\tilde{A}}_3=-a_1{\mathrm{\λ}}_1{k}_1-a_2{\mathrm{\λ}}_2{k}_2{,\tilde{A}}_4=-a_2{\mathrm{\λ}}_1{k}_1,{\tilde{A}}_5=0$

${\tilde{B}}_0={\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}$

${\tilde{B}}_1={\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02}-a_1{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}+{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}{\mathrm{\λ}}_2{k}_2$

${\tilde{B}}_2={\widetilde{\mathrm{\λ}}}_{03}{\mathrm{\β}}_{03}-a_1{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02}-a_2{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}+{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}{\mathrm{\λ}}_1{k}_1+{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02}{\mathrm{\λ}}_2{k}_2-a_1{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}{\mathrm{\λ}}_2{k}_2$

${\tilde{B}}_3={\widetilde{\mathrm{\λ}}}_{03}{\mathrm{\β}}_{03}{\mathrm{\λ}}_2{k}_2-a_2{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02}+{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02}{\mathrm{\λ}}_1{k}_1-a_2{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01}{\mathrm{\λ}}_2{k}_2$

${\tilde{B}}_4={\widetilde{\mathrm{\λ}}}_{03}{\mathrm{\β}}_{03}{\mathrm{\λ}}_1{k}_1$

By ω _o=20, ω _c=10, β ₀₁=3 ω ₀, k ₂=2 ω _c, δ=0.01, α ₂=0.5, α ₃=0.5, α ' ₁=0.75, α ' ₂=1.5, a ₁=-5, a ₂=-3, setting

E _i∈ [0,1] (i=1,2), e ∈ [0,1], substitutes into (formula 32), can obtain

${\tilde{A}}_0=1,{\tilde{A}}_1=\[7,25\],{\tilde{A}}_2\∈\[113,419\],{\tilde{A}}_3\∈\[506,1640\],{\tilde{A}}_4\∈\[300,949\],{\tilde{A}}_{05}=0$

${\tilde{B}}_0=\∈\[6,60\],{\tilde{B}}_1\∈\[282,1740\],{\tilde{B}}_2\∈\[3.25\×{10}^3,3.20\×{10}^4\],$

${\tilde{B}}_3\∈\[2.38\×{10}^5,2.94\×{10}^5\],{\tilde{B}}_4\∈\[8.89\×{10}^4,2.81\×{10}^5\]$

It is as follows according to 16 transport functions that robust Popov criterion is determined,

$\begin{array}{c}{G}_K=\{G\left(s\right):G\left(s\right)\}=\\ \frac{60s^4+282s^3+3.25\×{10}^3{s}^2+2.94\×{10}^{5}s+2.81\×{10}^5}{s^5+25s^4+113s^3+506s^2+949s},\\ \frac{60s^4+1740s^3+3.25\×{10}^3{s}^2+2.38\×{10}^{5}s+2.81\×{10}^5}{s^5+25s^4+113s^3+506s^2+949s},\\ \frac{6s^4+1740s^3+3.20\×{10}^4{s}^2+2.38\×{10}^{5}s+8.89\×{10}^4}{s^5+25s^4+113s^3+506s^2+949s},\\ \frac{6s^4+282s^3+3.20\×{10}^4{s}^2+2.94\×{10}^{5}s+8.89\×{10}^4}{s^5+25s^4+113s^3+506s^2+949s},\\ \frac{60s^4+1740s^3+3.25\×{10}^3{s}^2+2.38\×{10}^{5}s+2.81\×{10}^5}{s^5+25s^4+419s^3+506s^2+300s},\\ \frac{60s^4+282s^3+3.25\×{10}^3{s}^2+2.94\×{10}^{5}s+2.81\×{10}^5}{s^5+25s^4+419s^3+506s^2+300s},\\ \frac{6s^4+1740s^3+3.20\×{10}^4{s}^2+2.38\×{10}^{5}s+8.89\×{10}^4}{s^5+25s^4+419s^3+506s^2+300s},\\ \frac{6s^4+282s^3+3.20\×{10}^4{s}^2+2.94\×{10}^{5}s+8.89\×{10}^4}{s^5+25s^4+419s^3+506s^2+300s},\end{array}$

$\begin{array}{c}\frac{6s^4+1740s^3+3.20\×{10}^4{s}^2+2.38\×{10}^{5}s+8.89\×{10}^4}{s^5+7s^4+419s^3+1640s^2+300s},\\ \frac{60s^4+282s^3+3.25\×{10}^3{s}^2+2.94\×{10}^{5}s+2.81\×{10}^5}{s^5+7s^4+419s^3+1640s^2+300s},\\ \frac{60s^4+1740s^3+3.25\×{10}^3{s}^2+2.38\×{10}^{5}s+2.81\×{10}^5}{s^5+7s^4+419s^3+1640s^2+300s},\\ \frac{6s^4+282s^3+3.20\×{10}^4{s}^2+2.94\×{10}^{5}s+8.89\×{10}^4}{s^5+7s^4+419s^3+1640s^2+300s},\\ \frac{6s^4+282s^3+3.20\×{10}^4{s}^2+2.94\×{10}^{5}s+8.89\×{10}^4}{s^5+7s^4+506s^3+1640s^2+949s},\\ \frac{60s^4+282s^3+3.25\×{10}^3{s}^2+2.94\×{10}^{5}s+2.81\×{10}^5}{s^5+7s^4+506s^3+1640s^2+949s},\\ \frac{60s^4+1740s^3+3.25\×{10}^3{s}^2+2.38\×{10}^{5}s+2.81\×{10}^5}{s^5+7s^4+506s^3+1640s^2+949s},\\ \frac{6s^4+1740s^3+3.20\×{10}^4{s}^2+2.38\×{10}^{5}s+8.89\×{10}^4}{s^5+7s^4+506s^3+1640s^2+949s}\}\end{array}$

(formula 33)

And then make corresponding Popov curve as shown in Figure 3 according to (formula 33).As shown in Figure 3, this group Popov curve all meets Popov criterion, therefore this system is stable.

Nonlinear extension state observer stability analysis based on Routh Criterion:

Due to α ₂=α ₃, therefore (formula 28) condition is set up, namely

β ₀₁β ₀₂> β ₀₃(formula 34)

Therefore, the nonlinear extension state observer that (formula 30) represents is stable.

The above embodiment is only the preferred embodiments of the present invention, and and non-invention possible embodiments exhaustive.For persons skilled in the art, to any apparent change done by it under the prerequisite not deviating from the principle of the invention and spirit, all should be contemplated as falling with within claims of the present invention.

Claims (2)

1. a non-linear Active Disturbance Rejection Control system stability determination methods, is characterized in that comprising the steps:

(1) set up non-linear Active Disturbance Rejection Control system, it comprises controlled device and non-linear automatic disturbance rejection controller; Described non-linear automatic disturbance rejection controller comprises Nonlinear Tracking Differentiator, nonlinear extension state observer and nonlinearity erron Feedback Control Laws u;

Described controlled device is

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=x_3\\ .\\ .\\ .\\ {\stackrel{\·}{x}}_n=a_n{x}_1+a_{n-1}{x}_2+\mathrm{...}{a}_1{x}_n+bu\\ y=x_1\end{array}\right.$

(formula 1)

In (formula 1), x _ifor the state of controlled device; a _ifor the gain of controlled device corresponding state; the corresponding first order derivative of state for controlled device; Wherein i=1,2 ..., n, n be greater than 1 positive integer; Y is that controlled device exports; Nonlinearity erron Feedback Control Laws u is the control inputs amount of controlled device; B is control channel gain;

Described Nonlinear Tracking Differentiator is

$\left\{\begin{array}{c}{\stackrel{\·}{v}}_1=v_2\\ {\stackrel{\·}{v}}_2=v_2\\ .\\ .\\ .\\ {\stackrel{\·}{v}}_{n-1}=v_n\\ {\stackrel{\·}{v}}_n={\mathrm{\λ}}^n\mathrm{\ψ}(v_1-r,\frac{v_2}{\mathrm{\λ}},\mathrm{...},\frac{v_n}{{\mathrm{\λ}}^{n-1}})\end{array}\right.$ (formula 2)

In (formula 2), r is the input of described Nonlinear Tracking Differentiator, v _i(i=1,2 ..., n) be the output of described Nonlinear Tracking Differentiator, λ is the adjustable speed factor, for quick tracing function;

Described nonlinear extension state observer is

(formula 3)

In (formula 3), the b times of gain of controlled device output y and nonlinearity erron Feedback Control Laws u is respectively the input signal of described nonlinear extension state observer; z _i(i=1,2 ..., n+1) be the output of described nonlinear extension state observer; β _0i(i=1,2 ..., n+1) and be the gain of described nonlinear extension state observer; E represents the output y of controlled device and the output z of nonlinear extension state observer ₁between deviation; In described nonlinear extension state observer usually following nonlinear function is got:

(formula 4)

Wherein, α _t(i=1,2 ..., n+1) and δ be normal number, sgn () represents sign function;

The expression formula of described nonlinearity erron Feedback Control Laws u is as follows:

$u=\frac{u_0-z_n+1}{b}$ (formula 5)

Wherein,

$u_0=\underset{i=1}{\overset{n}{\Σ}}{k}_{i}fal(v_i-z_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})$ (formula 6)

In (formula 5) and (formula 6), k _ifor gain coefficient, α ' _i(i=1,2 ..., n) be normal number; B is control channel gain; u ₀feedback of status part, i.e. state feedback control law in nonlinearity erron Feedback Control Laws u;

The output z of described nonlinear extension state observer _i(i=1,2 ..., n) with the output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) do subtraction relatively after as described state feedback control law u ₀input e _i; The b times of gain of described nonlinearity erron Feedback Control Laws u is as the first input signal of described nonlinear extension state observer; The output y of described controlled device is as the second input signal of described nonlinear extension state observer;

(2) system conversion is carried out;

Suppose that the input r of Nonlinear Tracking Differentiator described in A1 is 0, the output v of so described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) be 0;

Make e _i=v _i-z _i; Then to the fal (v in (formula 6) _i-z _i, α ' _i, δ) do as down conversion:

$fal(v_i-z_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=\frac{fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}{e}_i;$

Order

$\frac{fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}={\mathrm{\λ}}_i\left(e_i\right)$

Then

Fal (v _i-z _i, α ' _i, δ) and=λ _i(e _i) e _i(formula 7)

By λ _i(e _i) be abbreviated as λ _i, obtained by (formula 6) and (formula 7):

$u_0=\underset{i=1}{\overset{n}{\Σ}}{k}_i{\mathrm{\λ}}_i{e}_i;$ (formula 8)

Make X=[x ₁, x ₂..., x _n] ^t, Z=[z ₁, z ₂..., z _n] ^t, obtained by (formula 1), (formula 5) and (formula 8):

$\left\{\begin{array}{c}\stackrel{\·}{X}=A_{11}X+A_{12}Z+A_{13}{z}_{n+1}\\ y=x_1\end{array}\right.$

(formula 9)

Wherein,

$A_{11}=\left[\begin{array}{ccccc}0& 1& 0& \mathrm{...}& 0\\ 0& 0& 1& \mathrm{...}& 0\\ & .& & .& \\ & .& & .& \\ & .& & .& \\ 0& 0& \mathrm{...}& 0& 1\\ a_n& a_{n-1}& \mathrm{...}& a_2& a_1\end{array}\right],A_{12}=\left[\begin{array}{cccc}0& \mathrm{...}& 0& 0\\ & .& & \\ & .& & \\ & .& & \\ 0& \mathrm{...}& 0& 0\\ -{\mathrm{\λ}}_1{k}_1& -{\mathrm{\λ}}_2{k}_2& \mathrm{...}& {\mathrm{\λ}}_n{k}_n\end{array}\right],$

Order

(formula 10)

(formula 10) represents and works as when obtaining maximal value wherein i=1,2 ..., n, n+1; Described nonlinear extension state observer (formula 3) is done following distortion:

(formula 11)

Order

(formula 12)

Then

(formula 13)

Obtained by (formula 5), (formula 8) and (formula 13):

(formula 14)

Wherein,

$A_{21}=\left[\begin{array}{ccccc}0& 1& 0& \mathrm{...}& 0\\ 0& 0& 1& \mathrm{...}& 0\\ & .& & .& \\ & .& & .& \\ & .& & .& \\ 0& 0& \mathrm{...}& 0& 1\\ -{\mathrm{\λ}}_1{k}_1& -{\mathrm{\λ}}_2{k}_2& \mathrm{...}& -{\mathrm{\λ}}_{n-1}{k}_{n-1}& -{\mathrm{\λ}}_n{k}_n\end{array}\right],$

$A_{22}={\[{\widetilde{\mathrm{\λ}}}_{01}{\mathrm{\β}}_{01},{\widetilde{\mathrm{\λ}}}_{02}{\mathrm{\β}}_{02},...,{\widetilde{\mathrm{\λ}}}_{0n}{\mathrm{\β}}_{0n}\]}^T.$

Obtained by (formula 9) and (formula 14):

(formula 15)

Wherein,

c ₁＝[-1，0，…，0] ^T∈R ⁿ，c ₂＝[1，0，…，0] ^T∈R ⁿ；

Make Y=A ₁₁x+A ₁₃z _n+1, e=σ, then

(formula 16)

(formula 16) is expressed as further:

(formula 17)

Wherein,

$\tilde{x}={\left[\begin{array}{cc}Y& Z\end{array}\right]}^T,A=\left[\begin{array}{cc}{A}_{11}& A_{11}{A}_{12}\\ 0& A_{21}\end{array}\right],b=\left[\begin{array}{c}{A}_{13}{\mathrm{\β}}_{0(n+1)}\\ A_{22}\end{array}\right],c=\[\begin{array}{ccc}{e_1}^T& {A_{11}}^{-1}& {C_2}^T\end{array}\]$

$\mathrm{\ρ}=-{c_1}^T{A_{11}}^{-1}{A}_{13}{\mathrm{\β}}_{0(n+1)}=-\frac{{\widetilde{\mathrm{\λ}}}_{0(n+1)}{\mathrm{\β}}_{0(n+1)}}{a_n}.$

(formula 17) is indirect Lurie's systems, and the expression formula of forward path transport function G (s) of described indirect Lurie's systems is:

G (s)=c ^t(sI-A) ^-1b+ ρ/s; (formula 18)

(3) robust Popov criterion is utilized to carry out judgement of stability to indirect Lurie's systems;

First, the interval transport function G of indirect Lurie's systems is defined _ias follows:

$G_I=\{G\left(s\right):G\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)},N\left(s\right)\∈N_I,D\left(s\right)\∈D_I\}$ (formula 19);

Wherein,

$N_I=\{N\left(s\right):N\left(s\right)=\underset{i=0}{\overset{m}{\Σ}}{b}_i{s}^i,b_i\∈\[{\underset{\‾}{b}}_i,{\stackrel{\‾}{b}}_i\],i=0,\mathrm{...},m\}$

$D_I=\{D\left(s\right):D\left(s\right)=s^n\underset{i=0}{\overset{n-1}{\Σ}}{c}_i{s}^i,c_i\∈\[{\underset{\‾}{c}}_i,{\stackrel{\‾}{c}}_i\],i=0,\mathrm{...},n-1\}$

M, n are respectively integer;

And then be defined as follows a point son vertex polynomial expression N _kas follows:

N _k={ N ₁(s), N ₂(s), N ₃(s), N ₄(s) } (formula 20)

Wherein,

$N_1\left(s\right)={\stackrel{\‾}{b}}_0+{\stackrel{\‾}{b}}_{1}s+{\underset{\‾}{b}}_2{s}^2+{\underset{\‾}{b}}_3{s}^3+{\stackrel{\‾}{b}}_4{s}^4+{\stackrel{\‾}{b}}_5{s}^5+\mathrm{...},$

$N_2\left(s\right)={\stackrel{\‾}{b}}_0+{\underset{\‾}{b}}_{1}s+{\underset{\‾}{b}}_2{s}^2+{\stackrel{\‾}{b}}_3{s}^3+{\stackrel{\‾}{b}}_4{s}^4+{\underset{\‾}{b}}_5{s}^5+\mathrm{...},$

$N_3\left(s\right)={\underset{\‾}{b}}_0+{\underset{\‾}{b}}_{1}s+{\stackrel{\‾}{b}}_2{s}^2+{\stackrel{\‾}{b}}_3{s}^3+{\underset{\‾}{b}}_4{s}^4+{\underset{\‾}{b}}_5{s}^5+\mathrm{...},$

$N_4\left(s\right)={\underset{\‾}{b}}_0+{\stackrel{\‾}{b}}_{1}s+{\stackrel{\‾}{b}}_2{s}^2+{\underset{\‾}{b}}_3{s}^3+{\underset{\‾}{b}}_4{s}^4+{\stackrel{\‾}{b}}_5{s}^5+...,$

be respectively real number;

Equally, denominator summit polynomial expression D is defined _kas follows:

D _k={ D ₁(s), D ₂(s), D ₃(s), D ₄(s) } (formula 21)

Wherein,

$D_1\left(s\right)={\stackrel{\‾}{c}}_0+{\stackrel{\‾}{c}}_{1}s+{\underset{\‾}{c}}_2{s}^2+{\underset{\‾}{c}}_3{s}^3+{\stackrel{\‾}{c}}_4{s}^4+{\stackrel{\‾}{c}}_5{s}^5+\mathrm{...},$

$D_2\left(s\right)={\stackrel{\‾}{c}}_0+{\underset{\‾}{c}}_{1}s+{\underset{\‾}{c}}_2{s}^2+{\stackrel{\‾}{c}}_3{s}^3+{\stackrel{\‾}{c}}_4{s}^4+{\underset{\‾}{c}}_5{s}^5+\mathrm{...},$

$D_3\left(s\right)={\underset{\‾}{c}}_0+{\underset{\‾}{c}}_{1}s+{\stackrel{\‾}{c}}_2{s}^2+{\stackrel{\‾}{c}}_3{s}^3+{\underset{\‾}{c}}_4{s}^4+{\underset{\‾}{c}}_5{s}^5+\mathrm{...},$

$D_4\left(s\right)={\underset{\‾}{c}}_0+{\stackrel{\‾}{c}}_{1}s+{\stackrel{\‾}{c}}_2{s}^2+{\underset{\‾}{c}}_3{s}^3+{\underset{\‾}{c}}_4{s}^4+{\stackrel{\‾}{c}}_5{s}^5+...,$

be respectively real number;

Define the transport function collection G of indirect Lurie's systems _kas follows:

$G_K=\{G\left(s\right):G\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)},N\left(s\right)\∈N_K,D\left(s\right)\∈D_K\}$ (formula 22)

From (formula 20), (formula 21) and (formula 22), the transport function collection G of indirect Lurie's systems _kcomprise following 16 forward paths function G (s):

$\begin{array}{c}{G}_K=\{G\left(s\right):G\left(s\right)=\frac{N_1\left(s\right)}{D_1\left(s\right)},\frac{N_2\left(s\right)}{D_1\left(s\right)},\frac{N_3\left(s\right)}{D_1\left(s\right)},\frac{N_4\left(s\right)}{D_1\left(s\right)},\\ \frac{N_1\left(s\right)}{D_2\left(s\right)},\frac{N_2\left(s\right)}{D_2\left(s\right)},\frac{N_3\left(s\right)}{D_2\left(s\right)},\frac{N_4\left(s\right)}{D_2\left(s\right)},\frac{N_1\left(s\right)}{D_3\left(s\right)},\frac{N_2\left(s\right)}{D_3\left(s\right)},\\ \frac{N_3\left(s\right)}{D_3\left(s\right)},\frac{N_4\left(s\right)}{D_3\left(s\right)},\frac{N_1\left(s\right)}{D_4\left(s\right)}\frac{N_2\left(s\right)}{D_4\left(s\right)},\frac{N_3\left(s\right)}{D_4\left(s\right)},\frac{N_4\left(s\right)}{D_4\left(s\right)}\}\end{array}$

(formula 23)

As G (s) ∈ G _ktime, if there is an arithmetic number θ all meet Popov stability condition, then forward path transport function G (s) is interval transport function G formula (19) Suo Shi _i, i.e. G (s) ∈ G _iindirect Lurie's systems be absolute stable.

2. one according to claim 1 non-linear Active Disturbance Rejection Control system stability determination methods, is characterized in that: described nonlinear extension state observer is third-order non-linear extended state observer, i.e. n=2; Its expression formula is as follows:

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-{\mathrm{\β}}_{01}\·e\\ {\stackrel{\·}{z}}_2=z_3-{\mathrm{\β}}_{02}\·fal(e,{\mathrm{\α}}_2,\mathrm{\δ})+bu\\ {\stackrel{\·}{z}}_3=-{\mathrm{\β}}_{03}\·fal(e,{\mathrm{\α}}_3,\mathrm{\δ})\end{array}\right.$ (formula 24)

Wherein, z ₁, z ₂, z ₃be respectively the output of described third-order non-linear extended state observer; β ₀₁, β ₀₂, β ₀₃be respectively the gain of described third-order non-linear extended state observer;

Order

$fal(e,{\mathrm{\α}}_2,\mathrm{\δ})=\frac{fal(e,{\mathrm{\α}}_2,\mathrm{\δ})}{e}e={\mathrm{\λ}}_{02}\left(e\right)e,fal(e,{\mathrm{\α}}_3,\mathrm{\δ})=\frac{fal\left(e,{\mathrm{\α}}_3,\mathrm{\δ}\right)}{e}e={\mathrm{\λ}}_{03}\left(e\right)e;$

Then obtain variable-gain linear extended state observer, its expression formula is as follows:

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-{\mathrm{\β}}_{01}\·e\\ {\stackrel{\·}{z}}_2=z_3-\left({\mathrm{\β}}_{02}\·{\mathrm{\λ}}_{02}\left(e\right)\right)e+bu\\ {\stackrel{\·}{z}}_3=-\left({\mathrm{\β}}_{03}\·{\mathrm{\λ}}_{03}\left(e\right)\right)e\end{array}\right.$ (formula 25)

By λ ₀₂(e), λ ₀₃e () is abbreviated as λ respectively ₀₂, λ ₀₃;

Order

f(t)＝a _nx ₁+α _n-1x ₂+…a ₁x _n，

Then the transport function of described variable-gain linear extended state observer is:

$z_1=\frac{{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}y+\frac{s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}bu$

$z_2=\frac{\left({\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}\right)s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}y+\frac{\left(s+{\mathrm{\β}}_{01}\right)s}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}bu$

$z_3=\frac{{\mathrm{\λ}}_{02}{\mathrm{\β}}_{03}{s}^2+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}bu}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}-\frac{{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}{s^3+{\mathrm{\β}}_{01}{s}^2+{\mathrm{\λ}}_{02}{\mathrm{\β}}_{02}s+{\mathrm{\λ}}_{03}{\mathrm{\β}}_{03}}f\left(s\right)$

(formula 26)

According to Routh Criterion method, the stable necessary and sufficient condition of described variable-gain linear extended state observer is:

λ ₀₂β ₀₁β ₀₂> λ ₀₃β ₀₃; (formula 27)

Further, if α ₂=α ₃, due to λ ₀₂, λ ₀₃all the function of tracking error e, then λ ₀₂=λ ₀₃, therefore, the stable necessary and sufficient condition of described variable-gain linear extended state observer is:

β ₀₁β ₀₂＞β ₀₃。(formula 28)