CN105563489A - Flexible manipulator control method based on non-linear active disturbance rejection control technique - Google Patents

Abstract

A flexible manipulator control method based on the non-linear active disturbance rejection control technique includes the steps of building a flexible manipulator system module, initializing the system state and controller parameters, designing a high-order tracking differentiator, designing a non-linear extended state observer, determining the observer parameters through a pole assignment mode and adding non-linear feedback. The extended state observer is designed and used for estimating the system state and external disturbance. The observer gain parameter is determined through the pole assignment mode. A non-linear feedback control law is designed, the effect that the system tracking error is rapid and stable and converges to the null point is guaranteed, and finally rapid and stable control over a flexible manipulator system is realized. By means of the flexible manipulator control method based on the non-linear active disturbance rejection control technique, the problem that the internal state of the system and external disturbance can not be observed is solved, the influences of non-linear links and uncertain items existing in the system are made up, the problem of a common PID control method is relieved, and the system can rapidly and stably track expected signals.

Description

Based on the flexible mechanical arm control method of non-linear Auto Disturbances Rejection Control Technique

Technical field

The present invention relates to a kind of flexible mechanical arm control method based on non-linear Auto Disturbances Rejection Control Technique, be applicable to the Position Tracking Control of the flexible mechanical arm servo-drive system with indeterminate.

Background technology

Along with robot system is towards high speed, heavy duty, high accuracy development, this must make the elastic deformation of component increase, cause the contradiction with its robot motion's stationarity, accuracy, mechanical arm must be connected by some tiny cross portions, easily cause mechanical oscillation, thus make the control of its end position more difficult.Therefore, the research method of traditional Rigid Robot Manipulator can not directly apply to flexible mechanical arm research.For this reason, people are by mechanism dynamic branch---and-elastodynamics is introduced wherein, creates a kind of emerging robot: flexible robot.For Rigid Robot Manipulator, flexible mechanical arm has the more free degree, and has more nonlinear element in flexible mechanical arm.But the system mode of flexible mechanical arm is difficult to observation, is difficult to apply effective controlled quentity controlled variable, accurate control is realized to it.Therefore, how realizing controlling the accurate tracking of flexible mechanical arm system, is one of flexible mechanical arm system key issue urgently to be resolved hurrily.

Active Disturbance Rejection Control is the Inheritance＆ development controlled classical PID.By introducing " transient process or Nonlinear Tracking Differentiator ", " design extended state observer " and " nonlinear Feedback Control rule " in original PID framework, enable the external disturbance of system real-time tracking and system indeterminate, and by feedback rate control, disturbance and indeterminate compensate to external world, improve the control efficiency of system, make system have good tracking effect.Therefore, Auto Disturbances Rejection Control Technique is applicable to having the flexible mechanical arm system that Nonlinear uncertainty and system mode are difficult to measure very much.

But so far, the parameter of extended state observer is mainly selected based on engineering experience.Method of Pole Placement (PoleAssignment) is a kind of Synthesis principle that the feedback of passing ratio link moves on to the limit of Linear Time-Invariant System in precalculated position, its essence is the freely-movable pattern removing to change original system by Proportional Feedback, with the requirement of satisfied design.Therefore, the parameter of extended state observer can be determined by Method of Pole Placement.

Summary of the invention

In order to the components of system as directed state and disturbance that overcome prior art can not be surveyed, and linear Feedback Control restrains the problems such as gain is larger, the present invention proposes a kind of control method of the flexible mechanical arm system based on non-linear Auto Disturbances Rejection Control Technique, adopt extended state observer (ExtendedStateObserver, ESO) estimating system state and external disturbance etc. can not survey item, and the higher differentiation of input signal is obtained by Nonlinear Tracking Differentiator, the control method simultaneously adopting nonlinear feedback to add LINEARIZATION WITH DYNAMIC COMPENSATION draws controlled quentity controlled variable, follow the tracks of desired signal with achieving system fast and stable.

In order to the technical scheme solving the problems of the technologies described above proposition is as follows:

Based on a flexible mechanical arm control method for non-linear Auto Disturbances Rejection Control Technique, comprise the following steps:

Step 1: set up such as formula the Equation of Motion shown in (1):

$\left\{\begin{array}{c}I\stackrel{\·\·}{q_1}+MgLsin\left(q_1\right)+K(q_1-q_2)=0\\ J\stackrel{\·\·}{q_2}-K(q_1-q_2)=u\end{array}\right.---\left(1\right)$

Wherein, q ₁for input end of motor rotational angle, for input end of motor angular acceleration, for input end of motor angle acceleration, q ₂for motor output end rotational angle, for motor output end angular acceleration, for motor output end angle acceleration, I is for connecting inertia, and J is motor inertia, and K is mechanical arm rigidity, and u is input torque, M and L is load quality and loading moment length;

Step 2: definition status variable: formula (1) is rewritten as

$\left\{\begin{array}{c}\stackrel{\·}{x_1}=x_2\\ \stackrel{\·}{x_2}=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}(x_1-x_2)\\ \stackrel{\·}{x_3}=x_4\\ \stackrel{\·}{x_4}=\frac{K}{J}(x_1-x_3)+\frac{1}{J}*u\end{array}\right.---\left(2\right)$

Mapped by differomorphism and formula (2) is written as

$\left\{\begin{array}{c}{z}_1=x_1\\ z_2=x_2\\ z_3=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ z_4=\frac{-MgL}{I}*\cos \left(x_1\right)*x_2-\frac{K}{I}(x_2-x_4)\end{array}\right.---\left(3\right)$

Finally obtaining the system state equation after converting is:

$\left\{\begin{array}{c}\stackrel{\·}{z_1}=z_2\\ \stackrel{\·}{z_2}=z_3\\ \stackrel{\·}{z_3}=z_4\\ \stackrel{\·}{z_4}=a\left(z\right)+b*u\end{array}\right.---\left(4\right)$

Wherein, $a\left(z\right)=\frac{MgL}{I}*sin\left(z_1\right)*(z_2^2-\frac{K}{J})-(\frac{MgL}{I}*cos(z_1)+\frac{K}{J}+\frac{K}{I})*z_3,b=\frac{K}{I*J};$

Step 3: design quadravalence Nonlinear Tracking Differentiator

$\left\{\begin{array}{c}\stackrel{\·}{z_1^{*}}=z_2^{*}\\ \stackrel{\·}{z_2^{*}}=z_3^{*}\\ \stackrel{\·}{z_3^{*}}=z_4^{*}\\ \stackrel{\·}{z_4^{*}}=f\\ f=-r\left(r\right(r\left(r\left(z_1^{*}-v\right)+4z_2^{*}\right)+6z_3^{*})+4z_4^{*})\end{array}\right.---\left(5\right)$

Wherein, be respectively i-th-1 order derivative of input signal v, r>0 is velocity factor, and v is input signal;

Step 4, design nonlinear extension state observer;

4.1 make a (x)=a ₀+ Δ a, b=b ₀+ Δ b, d=Δ a+ Δ bu, wherein b ₀and a ₀be respectively the optimal estimation value of b and a (x), given according to system architecture; Based on the design philosophy of expansion observer, definition expansion state z ₅=d, then formula (4) is rewritten as following equivalents:

$\left\{\begin{array}{c}\stackrel{\·}{z_1}=z_2\\ \stackrel{\·}{z_2}=z_3\\ \stackrel{\·}{z_3}=z_4\\ \stackrel{\·}{z_4}=a_0+b_0*u+d\\ \stackrel{\·}{z_5}=h\end{array}\right.---\left(6\right)$

Wherein, $h=\stackrel{\·}{d};$

4.2 make w _i, i=1,2,3,4,5 are respectively state variable z in formula (5) _iobservation, definition tracking error wherein for desired signal, observation error is e _oi=z _i-w _i, then designing nonlinear extension state observer expression formula is:

$\left\{\begin{array}{c}\stackrel{\·}{w_1}=w_2+{\mathrm{\β}}_1{g}_1\left(e_{o1}\right)\\ \stackrel{\·}{w_2}=w_3+{\mathrm{\β}}_2{g}_2\left(e_{o1}\right)\\ \stackrel{\·}{w_3}=w_4+{\mathrm{\β}}_3{g}_3\left(e_{o1}\right)\\ \stackrel{\·}{w_4}=w_5+{\mathrm{\β}}_4{g}_4\left(e_{o1}\right)+a_0+b_0*u\\ \stackrel{\·}{w_5}={\mathrm{\β}}_5{g}_5\left(e_{o1}\right)\end{array}\right.---\left(7\right)$

Wherein, β ₁, β ₂, β ₃, β ₄, β ₅for observer gain parameter, need determine with Method of Pole Placement, g _j(e ₀₁) be

$g_j\left(e_{o1}\right)=\left\{\begin{array}{cc}{\left|e_{o1}\right|}^{{\mathrm{\&alpha;}}_j}sign\left(e_{o1}\right),& \left|e_{o1}\right|>\mathrm{\&theta;},\\ \frac{e_{o1}}{{\mathrm{\&delta;}}^{1-{\mathrm{\&alpha;}}_j}},& \left|e_{o1}\right|<\mathrm{\&theta;}\end{array}\right.,j=1,2,3,4,5---\left(8\right)$

Wherein, α _j=[1,0.5,0.25,0.125,0.0625], θ=1.

Step 5, uses Method of Pole Placement determination observer gain parameter beta ₁, β ₂, β ₃, β ₄, β ₅value;

5.1 make δ x ₁=z ₁-w ₁, δ x ₂=z ₂-w ₂, δ x ₃=z ₃-w ₃, δ x ₄=z ₄-w ₄, δ x ₅=h-w ₅, then formula (6) deducts formula (7)

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_1*g_1\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2*g_2\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}={\mathrm{\&delta;x}}_4-{\mathrm{\&beta;}}_3*g_3\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}={\mathrm{\&delta;x}}_5-{\mathrm{\&beta;}}_4*g_4\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}=h-{\mathrm{\&beta;}}_5*g_5\left({\mathrm{\&delta;x}}_1\right)\end{array}\right.---\left(9\right)$

If h bounded, and g (e _o1) be smooth, g (0)=0, g ' (e _o1) ≠ 0, according to Taylor's formula, formula (9) is written as

Order then formula (10) is written as following state space equation form

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}\end{array}\right]=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\&delta;x}}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right]h---\left(11\right)$

5.2 design compensation matrixes

$A=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right],\mathrm{\&delta;}x=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]$

Then formula (11) is written as

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}x}=A*\mathrm{\&delta;}x+Eh---\left(12\right)$

So far, parameter beta _idetermination be converted into l _idetermination, make formula (12) asymptotically stable necessary condition under the effect of disturbance h be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (12) is born fully, thus, according to Method of Pole Placement, the selected limit p expected _i(i=1,2,3), make parameter l _imeet

$\left|sI-A\right|={\&Pi;}_{i=1}^5(s-p_i);---\left(13\right)$

Wherein, I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter l respectively ₁, l ₂, l ₃, l ₄, l ₅value, thus the expression formula obtaining extended state observer is

Step 6, based on Auto-disturbance-rejection Control design nonlinear feedback LINEARIZATION WITH DYNAMIC COMPENSATION controller u;

6.1, design nonlinear feedback:

$fal(\mathrm{\&epsiv;},\mathrm{\&alpha;},\mathrm{\&delta;})=\left\{\begin{array}{cc}{\left|\mathrm{\&epsiv;}\right|}^{\mathrm{\&alpha;}}sign\left(\mathrm{\&epsiv;}\right),& \left|\mathrm{\&epsiv;}\right|>\mathrm{\&delta;}\\ \frac{\mathrm{\&epsiv;}}{{\mathrm{\&delta;}}^{1-\mathrm{\&alpha;}}},& \left|\mathrm{\&epsiv;}\right|\&le;\mathrm{\&delta;}\end{array}\right.---\left(15\right)$

Wherein, $\mathrm{\&epsiv;}=z_i^{*}-w_i,\mathrm{\&delta;}=1;$ δ＝1；

6.2, as follows according to the thought design automatic disturbance rejection controller of LINEARIZATION WITH DYNAMIC COMPENSATION:

$\begin{array}{c}u=\frac{1}{b_0}\&lsqb;-a_0+\stackrel{\&CenterDot;}{z_4^{*}}+k_1(z_1^{*}-w_1)+k_2(z_2^{*}-w_2)+k_3(z_3^{*}-w_3)+\\ k_4(z_4^{*}-w_4)-w_5\&rsqb;;---\left(16\right)\end{array}$

Wherein, k ₁, k ₂, k ₃, k ₄for controller parameter.

6.3, use Method of Pole Placement determination observer gain parameter k ₁, k ₂, k ₃, k ₄value:

After bringing formula (16) into formula (4), have

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=\stackrel{\&CenterDot;}{z_4^{*}}+k_1(\stackrel{\&CenterDot;}{z_1^{*}}-w_1)+k_2(\stackrel{\&CenterDot;}{z_2^{*}}-w_2)+k_3(\stackrel{\&CenterDot;}{z_3^{*}}-w_3)+k_4(\stackrel{\&CenterDot;}{z_4^{*}}-w_4)\end{array}\right.---\left(17\right)$

Section 4 in formula (17) is rewritten as and obtains

Wherein, $e^{\left(4\right)}={\stackrel{\&CenterDot;}{Z}}_4-{\stackrel{\&CenterDot;}{Z}}_4^{*},$ For four subderivatives of e, $e=Z_1^{*}-W_1.$

Order

$\left\{\begin{array}{c}{D}_1=e\\ D_2=\stackrel{\&CenterDot;}{e}\\ D_3=\stackrel{\&CenterDot;\&CenterDot;}{e}\\ D_4=\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}\end{array}\right.---\left(18\right)$

Then formula (18) is written as

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{D_1}=D_2\\ \stackrel{\&CenterDot;}{D_2}=D_3\\ \stackrel{\&CenterDot;}{D_3}=D_4\\ \stackrel{\&CenterDot;}{D_4}=k_1{D}_1+k_2{D}_2+k_3{D}_3+k_4{D}_4\end{array}\right.---\left(19\right)$

Formula (19) can be written as matrix form wherein

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ k_1& k_2& k_3& k_4\end{array}\right]$

Make formula (19) asymptotically stable necessary condition be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (19) is born fully, thus, according to Method of Pole Placement, and the selected limit p expected _i(i=1,2,3,4), make parameter k _imeet

$\left|sI-A\right|={\&Pi;}_{i=1}^4(s-p_i);---\left(20\right)$

Wherein, I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter k respectively ₁, k ₂, k ₃, k ₄value.

Technical conceive of the present invention is: the flexible mechanical arm system can not surveying and exist external disturbance for part system state, devises a kind of control method based on non-linear auto-disturbance rejection technology, eliminates external disturbance as much as possible to the impact of Systematical control.By setting up new expansion state, design extended state observer estimating system indeterminate and external disturbance, and adopt the parameter of Method of Pole Placement determination extended state observer, use nonlinear Feedback Control rule simultaneously, realize controlling the fast and stable of flexible mechanical arm system.

Advantage of the present invention is: the present invention is by using extended state observer, effectively can observe flexible mechanical arm system mode and external disturbance, the Nonlinear control law adopted improves the control efficiency of system, achieves and controls the accurate tracking of flexible mechanical arm system.

Accompanying drawing illustrates:

Fig. 1 is system mode z ₂with its observation response curve;

Fig. 2 is system mode z ₂observation error e _o2response curve;

Fig. 3 is system mode z ₃with its observation response curve;

Fig. 4 is system mode z ₃observation error e _o3response curve;

Fig. 5 is system mode z ₄with its observation response curve;

Fig. 6 is system mode z ₄observation error e _o4response curve;

Fig. 7 is system mode z ₅with its observation response curve;

Fig. 8 is system mode z ₅observation error e _o5response curve;

Fig. 9 is that system exports the response curve with desired signal;

Figure 10 is system tracking error e _c1response curve;

Figure 11 is the response curve of system control signal u;

Figure 12 is the basic flow sheet of the servo system self-adaptive sliding-mode control based on extended state observer of the present invention.

Detailed description of the invention:

Below in conjunction with accompanying drawing, the present invention will be further described.

With reference to Fig. 1-Fig. 9, a kind of servo system self-adaptive sliding-mode control based on extended state observer, comprises the steps:

Step 1: set up such as formula the Equation of Motion shown in (1):

$\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q_1}+MgLsin\left(q_1\right)+K(q_1-q_2)=0\\ J\stackrel{\&CenterDot;\&CenterDot;}{q_2}-K(q_1-q_2)=u\end{array}\right.---\left(1\right)$

Wherein, q ₁for input end of motor rotational angle, for input end of motor angular acceleration, for input end of motor angle acceleration, q ₂for motor output end rotational angle, for motor output end angular acceleration, for motor output end angle acceleration, I is for connecting inertia, and J is motor inertia, and K is mechanical arm rigidity, and u is input torque, M and L is load quality and loading moment length;

Step 2: definition status variable: formula (1) is rewritten as

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x_1}=x_2\\ \stackrel{\&CenterDot;}{x_2}=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}(x_1-x_2)\\ \stackrel{\&CenterDot;}{x_3}=x_4\\ \stackrel{\&CenterDot;}{x_4}=\frac{K}{J}(x_1-x_3)+\frac{1}{J}*u\end{array}\right.---\left(2\right)$

Mapped by differomorphism and formula (2) is written as

$\left\{\begin{array}{c}{z}_1=x_1\\ z_2=x_2\\ z_3=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ z_4=\frac{-MgL}{I}*\cos \left(x_1\right)*x_2-\frac{K}{I}(x_2-x_4)\end{array}\right.---\left(3\right)$

Finally obtaining the system state equation after converting is:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=a\left(z\right)+b*u\end{array}\right.---\left(4\right)$

Wherein, $a\left(z\right)=\frac{MgL}{I}*sin\left(z_1\right)*(z_2^2-\frac{K}{J})-(\frac{MgL}{I}*cos(z_1)+\frac{K}{J}+\frac{K}{I})*z_3,b=\frac{K}{I*J};$

Step 3: design quadravalence Nonlinear Tracking Differentiator

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1^{*}}=z_2^{*}\\ \stackrel{\&CenterDot;}{z_2^{*}}=z_3^{*}\\ \stackrel{\&CenterDot;}{z_3^{*}}=z_4^{*}\\ \stackrel{\&CenterDot;}{z_4^{*}}=f\\ f=-r\left(r\right(r\left(r\left(z_1^{*}-v\right)+4z_2^{*}\right)+6z_3^{*})+4z_4^{*})\end{array}\right.---\left(5\right)$

Wherein, be respectively i-th-1 order derivative of input signal v, r>0 is velocity factor, and v is input signal;

Step 4, design nonlinear extension state observer;

4.1 make a (x)=a ₀+ Δ a, b=b ₀+ Δ b, d=Δ a+ Δ bu, wherein b ₀and a ₀be respectively the optimal estimation value of b and a (x), given according to system architecture; Based on the design philosophy of expansion observer, definition expansion state z ₅=d, then formula (4) is rewritten as following equivalents:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=a_0+b_0*u+d\\ \stackrel{\&CenterDot;}{z_5}=h\end{array}\right.---\left(6\right)$

Wherein, $h=\stackrel{\&CenterDot;}{d};$

4.2 make w _i, i=1,2,3,4,5 are respectively state variable z in formula (5) _iobservation, definition tracking error wherein for desired signal, observation error is e _oi=z _i-w _i, then designing nonlinear extension state observer expression formula is:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{w_1}=w_2+{\mathrm{\&beta;}}_1{g}_1\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_2}=w_3+{\mathrm{\&beta;}}_2{g}_2\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_3}=w_4+{\mathrm{\&beta;}}_3{g}_3\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_4}=w_5+{\mathrm{\&beta;}}_4{g}_4\left(e_{o1}\right)+a_0+b_0*u\\ \stackrel{\&CenterDot;}{w_5}={\mathrm{\&beta;}}_5{g}_5\left(e_{o1}\right)\end{array}\right.---\left(7\right)$

Wherein, β ₁, β ₂, β ₃, β ₄, β ₅for observer gain parameter, need determine with Method of Pole Placement, g _j(e _o1) be

$g_j\left(e_{o1}\right)=\left\{\begin{array}{cc}{\left|e_{o1}\right|}^{{\mathrm{\&alpha;}}_j}sign\left(e_{o1}\right),& \left|e_{o1}\right|>\mathrm{\&theta;},\\ \frac{e_{o1}}{{\mathrm{\&delta;}}^{1-{\mathrm{\&alpha;}}_j}},& \left|e_{o1}\right|<\mathrm{\&theta;}\end{array}\right.,j=1,2,3,4,5---\left(8\right)$

Wherein, α _j=[1,0.5,0.25,0.125,0.0625], θ=1.

Step 5, uses Method of Pole Placement determination observer gain parameter beta ₁, β ₂, β ₃, β ₄, β ₅value;

5.1 make δ x ₁=z ₁-w ₁, δ x ₂=z ₂-w ₂, δ x ₃=z ₃-w ₃, δ x ₄=z ₄-w ₄, δ x ₅=h-w ₅, then formula (6) deducts formula (7)

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_1*g_1\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2*g_2\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}={\mathrm{\&delta;x}}_4-{\mathrm{\&beta;}}_3*g_3\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}={\mathrm{\&delta;x}}_5-{\mathrm{\&beta;}}_4*g_4\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}=h-{\mathrm{\&beta;}}_5*g_5\left({\mathrm{\&delta;x}}_1\right)\end{array}\right.---\left(9\right)$

If h bounded, and g (e _o1) be smooth, g (0)=0, g ' (e _o1) ≠ 0, according to Taylor's formula, formula (9) is written as

Order then formula (10) is written as following state space equation form

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}\end{array}\right]=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\&delta;x}}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right]h---\left(11\right)$

5.2 design compensation matrixes

$A=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right],\mathrm{\&delta;}x=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]$

Then formula (11) is written as

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}x}=A*\mathrm{\&delta;}x+Eh---\left(12\right)$

So far, parameter beta _idetermination be converted into l _idetermination, make formula (12) asymptotically stable necessary condition under the effect of disturbance h be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (12) is born fully, thus, according to Method of Pole Placement, the selected limit p expected _i(i=1,2,3), make parameter l _imeet

$\left|sI-A\right|={\&Pi;}_{i=1}^5(s-p_i);---\left(13\right)$

Wherein, I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter l respectively ₁, l ₂, l ₃, l ₄, l ₅value, thus the expression formula obtaining extended state observer is

Step 6, based on Auto-disturbance-rejection Control design nonlinear feedback LINEARIZATION WITH DYNAMIC COMPENSATION controller u;

6.1, design nonlinear feedback:

$fal(\mathrm{\&epsiv;},\mathrm{\&alpha;},\mathrm{\&delta;})=\left\{\begin{array}{cc}{\left|\mathrm{\&epsiv;}\right|}^{\mathrm{\&alpha;}}sign\left(\mathrm{\&epsiv;}\right),& \left|\mathrm{\&epsiv;}\right|>\mathrm{\&delta;}\\ \frac{\mathrm{\&epsiv;}}{{\mathrm{\&delta;}}^{1-\mathrm{\&alpha;}}},& \left|\mathrm{\&epsiv;}\right|\&le;\mathrm{\&delta;}\end{array}\right.---\left(15\right)$

Wherein, $\mathrm{\&epsiv;}=z_i^{*}-w_i,\mathrm{\&delta;}=1;$ δ＝1；

6.2, as follows according to the thought design automatic disturbance rejection controller of LINEARIZATION WITH DYNAMIC COMPENSATION:

$\begin{array}{c}u=\frac{1}{b_0}\&lsqb;-a_0+\stackrel{\&CenterDot;}{z_4^{*}}+k_1(z_1^{*}-w_1)+k_2(z_2^{*}-w_2)+k_3(z_3^{*}-w_3)+\\ k_4(z_4^{*}-w_4)-w_5\&rsqb;;---\left(16\right)\end{array}$

Wherein, k ₁, k ₂, k ₃, k ₄for controller parameter.

6.3, use Method of Pole Placement determination observer gain parameter k ₁, k ₂, k ₃, k ₄value:

After bringing formula (16) into formula (4), have

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=\stackrel{\&CenterDot;}{z_4^{*}}+k_1(\stackrel{\&CenterDot;}{z_1^{*}}-w_1)+k_2(\stackrel{\&CenterDot;}{z_2^{*}}-w_2)+k_3(\stackrel{\&CenterDot;}{z_3^{*}}-w_3)+k_4(\stackrel{\&CenterDot;}{z_4^{*}}-w_4)\end{array}\right.---\left(17\right)$

Section 4 in formula (17) is rewritten as and obtains $e^{\left(4\right)}=k_{1}e+k_2\stackrel{\&CenterDot;}{e}+k_3\stackrel{\&CenterDot;\&CenterDot;}{e}+k_4\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e};$

Wherein, $e^{\left(4\right)}={\stackrel{\&CenterDot;}{Z}}_4-{\stackrel{\&CenterDot;}{Z}}_4^{*},$ For four subderivatives of e, $e=Z_1^{*}-W_1.$

Order

$\left\{\begin{array}{c}{D}_1=e\\ D_2=\stackrel{\&CenterDot;}{e}\\ D_3=\stackrel{\&CenterDot;\&CenterDot;}{e}\\ D_4=\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}\end{array}\right.---\left(18\right)$

Then formula (18) is written as

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{D_1}=D_2\\ \stackrel{\&CenterDot;}{D_2}=D_3\\ \stackrel{\&CenterDot;}{D_3}=D_4\\ \stackrel{\&CenterDot;}{D_4}=k_1{D}_1+k_2{D}_2+k_3{D}_3+k_4{D}_4\end{array}\right.---\left(19\right)$

Formula (19) can be written as matrix form wherein

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ k_1& k_2& k_3& k_4\end{array}\right]$

Make formula (19) asymptotically stable necessary condition be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (19) is born fully, thus, according to Method of Pole Placement, and the selected limit p expected _i(i=1,2,3,4), make parameter k _imeet

$\left|sI-A\right|={\&Pi;}_{i=1}^4(s-p_i);---\left(20\right)$

Wherein I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter k respectively ₁, k ₂, k ₃, k ₄value.

For validity and the superiority of checking institute extracting method, carry out emulation experiment, the primary condition in emulation experiment and partial parameters are set, that is: MgL=10, K=100, I=1, J=1 in system equation.Controller parameter is k ₁=2500, k ₂=850, k ₃=280, k ₄=40; , in addition, calculated each gain parameter in extended state observer by Method of Pole Placement, get l respectively ₁=5*41, l ₂=10*41 ², l ₃=10*41 ³, l ₄=5*41 ⁴, l ₅=1*41 ⁵.The each state initial value of system, the initial value of Nonlinear Tracking Differentiator, extended state observer state initial value, controller u initial value, expansion state d initial value is all set to 0.

Fig. 1-Fig. 6 represents each state observation effect of system and observation error respectively, Fig. 7 and Fig. 8 represents observation effect and the observation error of expansion state respectively.As can be seen from Figure 2, observation error tends towards stability after 1 second, and maintains in the error range of 2%.Can find out from Fig. 4, Fig. 6, Fig. 8, along with the order of system mode improves, observation error increases to some extent, but observation error can reduce fast and remain in the error range of 10% in 2 seconds.As can be seen from simulation result, extended state observer can effective observation system state, estimates indeterminate and external disturbance.

Fig. 9 gives the contrast effect of desired signal and real output signal, can find out, system has just followed the tracks of desired signal after 0.5s.And as can be seen from Figure 10 the tracking error of system keeps, within 0.05, having good tracking accuracy.Figure 11 display be the controlled quentity controlled variable of system, as can be seen from Figure 11, system control amount can reduce after 2 seconds and tend towards stability, and maintains in the scope of [-15,15].

From simulation result, method of the present invention can effectively be estimated and the indeterminate of bucking-out system existence and external disturbance, follows the tracks of desired signal with making system energy fast and stable.Obvious the present invention is not just limited to above-mentioned example, and basis of the present invention also can accurately control other similar systems.

Claims (1)

1. based on a flexible mechanical arm control method for non-linear Auto Disturbances Rejection Control Technique, it is characterized in that: this control method comprises the following steps:

Step 1: set up such as formula the Equation of Motion shown in (1):

$\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q_1}+MgL\sin \left(q_1\right)+K\left(q_1-q_2\right)=0\\ J\stackrel{\&CenterDot;\&CenterDot;}{q_2}-K\left(q_1-q_2\right)=u\end{array}\right.---\left(1\right)$

Wherein, q ₁for input end of motor rotational angle, for input end of motor angular acceleration, for input end of motor angle acceleration, q ₂for motor output end rotational angle, for motor output end angular acceleration, for motor output end angle acceleration, I is for connecting inertia, and J is motor inertia, and K is mechanical arm rigidity, and u is input torque, M and L is load quality and loading moment length;

Step 2: definition status variable: x ₁=q ₁, x ₃=q ₂, formula (1) is rewritten as

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x_1}=x_2\\ \stackrel{\&CenterDot;}{x_2}=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}\left(x_1-x_3\right)\\ \stackrel{\&CenterDot;}{x_3}=x_4\\ \stackrel{\&CenterDot;}{x_4}=\frac{K}{J}\left(x_1-x_3\right)+\frac{1}{J}*u\end{array}\right.---\left(2\right)$

Mapped by differomorphism and formula (2) is written as

$\left\{\begin{array}{c}{z}_1=x_1\\ z_2=x_2\\ z_3=\frac{-MgL}{I}*\sin \left(x_1\right)-\frac{K}{I}(x_1-x_3)\\ z_4=\frac{-MgL}{I}*\cos \left(x_1\right)*x_2-\frac{K}{I}(x_2-x_4)\end{array}\right.---\left(3\right)$

Finally obtaining the system state equation after converting is:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=a\left(z\right)+b*u\end{array}\right.---\left(4\right)$

Wherein, $a\left(z\right)=\frac{MgL}{I}*sin\left(z_1\right)*(z_2^2-\frac{K}{J})-(\frac{MgL}{I}*cos(z_1)+\frac{K}{J}+\frac{K}{I})*z_3,b=\frac{K}{I*J};$

Step 3: design quadravalence Nonlinear Tracking Differentiator

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1^{*}}=z_2^{*}\\ \stackrel{\&CenterDot;}{z_2^{*}}=z_3^{*}\\ \stackrel{\&CenterDot;}{z_3^{*}}=z_4^{*}\\ \stackrel{\&CenterDot;}{z_4^{*}}=f\\ f=-r\left(r\left(r\left(r\left(z_1^{*}-v\right)+4z_2^{*}\right)+6z_3^{*}\right)+4z_4^{*}\right)\end{array}\right.---\left(5\right)$

Wherein, be respectively i-th-1 order derivative of input signal v, r>0 is velocity factor, and v is input signal;

Step 4, design nonlinear extension state observer;

4.1 make a (x)=a ₀+ Δ a, b=b ₀+ Δ b, d=Δ a+ Δ bu, wherein b ₀and a ₀be respectively the optimal estimation value of b and a (x), given according to system architecture; Based on the design philosophy of expansion observer, definition expansion state z ₅=d, then formula (4) is rewritten as following equivalents:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=a_0+b_0*u+d\\ \stackrel{\&CenterDot;}{z_5}=h\end{array}\right.---\left(6\right)$

Wherein, $h=\stackrel{\&CenterDot;}{d};$

4.2 make w _i, i=1,2,3,4,5 are respectively state variable z in formula (5) _iobservation, definition tracking error wherein for desired signal, observation error is e _oi=z _i-w _i, then designing nonlinear extension state observer expression formula is:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{w_1}=w_2+{\mathrm{\&beta;}}_1{g}_1\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_2}=w_3+{\mathrm{\&beta;}}_2{g}_2\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_3}=w_4+{\mathrm{\&beta;}}_3{g}_3\left(e_{o1}\right)\\ \stackrel{\&CenterDot;}{w_4}=w_5+{\mathrm{\&beta;}}_4{g}_4\left(e_{o1}\right)+a_0+b_0*u\\ \stackrel{\&CenterDot;}{w_5}={\mathrm{\&beta;}}_5{g}_5\left(e_{o1}\right)\end{array}\right.---\left(7\right)$

Wherein, β ₁, β ₂, β ₃, β ₄, β ₅for observer gain parameter, need determine with Method of Pole Placement, g _j(e _o1) be

$g_j\left(e_{o1}\right)=\left\{\begin{array}{cc}|e_{o1}{|}^{{\mathrm{\&alpha;}}_j}sign\left(e_{o1}\right),& \left|e_{o1}\right|>\mathrm{\&theta;},\\ \frac{e_{o1}}{{\mathrm{\&delta;}}^{1-{\mathrm{\&alpha;}}_j}},& \left|e_{o1}\right|<\mathrm{\&theta;}\end{array}\right.,j=1,2,3,4,5---\left(8\right)$

Wherein, α _j=[1,0.5,0.25,0.125,0.0625], θ=1;

Step 5, uses Method of Pole Placement determination observer gain parameter beta ₁, β ₂, β ₃, β ₄, β ₅value;

5.1 make δ x ₁=z ₁-w ₁, δ x ₂=z ₂-w ₂, δ x ₃=z ₃-w ₃, δ x ₄=z ₄-w ₄, δ x ₅=h-w ₅, then formula (6) deducts formula (7)

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_1*g_1\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2*g_2\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}={\mathrm{\&delta;x}}_4-{\mathrm{\&beta;}}_3*g_3\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}={\mathrm{\&delta;x}}_5-{\mathrm{\&beta;}}_4*g_4\left({\mathrm{\&delta;x}}_1\right)\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}=h-{\mathrm{\&beta;}}_5*g_5\left({\mathrm{\&delta;x}}_1\right)\end{array}\right.---\left(9\right)$

If h bounded, and g (e _o1) be smooth, g (0)=0, g ' (e _o1) ≠ 0, according to Taylor's formula, formula (9) is written as

Order then formula (10) is written as following state space equation form

$\left[\begin{array}{c}\stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_1}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_2}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_3}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_4}\\ \stackrel{\&CenterDot;}{{\mathrm{\&delta;x}}_5}\end{array}\right]=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right]h---\left(11\right)$

5.2 design compensation matrixes

$A=\left[\begin{array}{ccccc}-l_1& 1& 0& 0& 0\\ -l_2& 0& 1& 0& 0\\ -l_3& 0& 0& 1& 0\\ -l_4& 0& 0& 0& 1\\ -l_5& 0& 0& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right],\mathrm{\&delta;}x=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\\ {\mathrm{\&delta;x}}_4\\ {\mathrm{\&delta;x}}_5\end{array}\right]$

Then formula (11) is written as

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}x}=A*\mathrm{\&delta;}x+Eh---\left(12\right)$

So far, parameter beta _idetermination be converted into l _idetermination, make formula (12) asymptotically stable necessary condition under the effect of disturbance h be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (12) is born fully, thus, according to Method of Pole Placement, the selected limit p expected _i(i=1,2,3), make parameter l _imeet

$|sI-A|={\&Pi;}_{i=1}^5(s-p_i);---\left(13\right)$

Wherein, I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter l respectively ₁, l ₂, l ₃, l ₄, l ₅value, thus the expression formula obtaining extended state observer is

Step 6, based on Auto-disturbance-rejection Control design nonlinear feedback LINEARIZATION WITH DYNAMIC COMPENSATION controller u;

6.1, design nonlinear feedback:

$fal(\mathrm{\&epsiv;},\mathrm{\&alpha;},\mathrm{\&delta;})=\left\{\begin{array}{cc}|\mathrm{\&epsiv;}{|}^{\mathrm{\&alpha;}}\mathrm{sgn}\left(\mathrm{\&epsiv;}\right),& \left|\mathrm{\&epsiv;}\right|>\mathrm{\&delta;}\\ \frac{\mathrm{\&epsiv;}}{{\mathrm{\&delta;}}^{1-\mathrm{\&alpha;}}},& \left|\mathrm{\&epsiv;}\right|\&le;\mathrm{\&delta;}\end{array}\right.---\left(15\right)$

Wherein, $\mathrm{\&epsiv;}=z_i^{*}-w_i,$ δ＝1；

6.2, as follows according to the thought design automatic disturbance rejection controller of LINEARIZATION WITH DYNAMIC COMPENSATION:

$\begin{array}{c}u=\frac{1}{b_0}\&lsqb;-a_0+\stackrel{\&CenterDot;}{z_4^{*}}+k_1(z_1^{*}-w_1)+k_2(z_2^{*}-w_2)+k_3(z_3^{*}-w_3)+\\ k_4\left(z_4^{*}-w_4\right)-w_5\&rsqb;;\end{array}---\left(16\right)$

Wherein, k ₁, k ₂, k ₃, k ₄for controller parameter;

6.3, use Method of Pole Placement determination observer gain parameter k ₁, k ₂, k ₃, k ₄value:

After bringing formula (16) into formula (4), have

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{z_1}=z_2\\ \stackrel{\&CenterDot;}{z_2}=z_3\\ \stackrel{\&CenterDot;}{z_3}=z_4\\ \stackrel{\&CenterDot;}{z_4}=\stackrel{\&CenterDot;}{z_4^{*}}+k_1\left(\stackrel{\&CenterDot;}{z_1^{*}}-w_1\right)+k_2\left(\stackrel{\&CenterDot;}{z_2^{*}}-w_2\right)+k_3\left(\stackrel{\&CenterDot;}{z_3^{*}}-w_3\right)+k_4\left(\stackrel{\&CenterDot;}{z_4^{*}}-w_4\right)\end{array}\right.---\left(17\right)$

Section 4 in formula (17) is rewritten as and obtains $e^{\left(4\right)}=k_{1}e+k_2\stackrel{\&CenterDot;}{e}+k_3\stackrel{\&CenterDot;\&CenterDot;}{e}+k_4\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e};$

Wherein, $e^{\left(4\right)}=\stackrel{\&CenterDot;}{z}-\stackrel{\&CenterDot;}{z_4^{*}},$ For four subderivatives of e, $e=z_1^{*}-w_1;$

Order

$\left\{\begin{array}{c}{D}_1=e\\ D_2=\stackrel{\&CenterDot;}{e}\\ D_3=\stackrel{\&CenterDot;\&CenterDot;}{e}\\ D_4=\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}\end{array}\right.---\left(18\right)$

Then formula (18) is written as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{D}}_1=D_2\\ {\stackrel{\&CenterDot;}{D}}_2=D_3\\ {\stackrel{\&CenterDot;}{D}}_3=D_4\\ {\stackrel{\&CenterDot;}{D}}_4=k_1{D}_1+k_2{D}_2+k_3{D}_3+k_4{D}_4\end{array}\right.---\left(19\right)$

Formula (19) is written as matrix form wherein

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ k_1& k_2& k_3& k_4\end{array}\right]$

Make formula (19) asymptotically stable necessary condition be that the characteristic value of compensation matrix A all drops on the Left half-plane of complex plane, namely the limit of formula (19) is born fully, thus, according to Method of Pole Placement, and the selected limit p expected _i, i=1,2,3,4, make parameter k _imeet

$|sI-A|={\&Pi;}_{i=1}^4(s-p_i);---\left(20\right)$

Wherein I is unit matrix, makes the right and left equal about polynomial each term coefficient of s, then obtains parameter k respectively ₁, k ₂, k ₃, k ₄value.