CN104267605A - Smooth nonsingular terminal sliding-mode control method suitable for control system with relative degree of 1 - Google Patents

Abstract

The invention discloses a smooth nonsingular terminal sliding-mode control method suitable for a control system with a relative degree of 1, relates to a nonsingular terminal sliding-mode control method, and solves the problems that the existing nonsingular terminal sliding-mode control method exists buffeting to cause a controller not to output continuous smooth control signals and not be applied to the control system with the relative degree of 1. The smooth nonsingular terminal sliding-mode control method comprises the following steps: if the controlled system serves as a single-input single-output control system with the relative degree of 1, then acquiring the system state differential in real time, designing a nonsingular terminal sliding mode, introducing a virtual controlled quantity and utilizing an integral action to enable the actual output controlled quantity to be smooth and continuous; if the controlled system serves as a matched multi-input multi-output control system, then converting a scalar realization form of the control system to a matrix vector realization form; if the controlled system serves as a non-matching indeterminate multi-input multi-output control system, then conducting nonsingular state change for two times according to the controllability index r to resolve the system into r subsystems, introducing a reference model to eliminate time-varying uncertainty of input channels of the subsystems, and further designing a control law and a minor control law. The smooth nonsingular terminal sliding-mode control method disclosed by the invention can be applied to smooth nonsingular terminal sliding-mode control of the control system with the relative degree of 1.

Description

Be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system

Technical field

The present invention relates to and be a kind ofly applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system.

Background technology

Existing non-singular terminal sliding-mode control is only applicable to the control system of Relative order >=2, and application has certain limitation.

(1) non-singular terminal sliding-mode control is a kind of nonlinear sliding mode control method occurred in recent years, can make system state overall situation finite time convergence control, therefore have applications well prospect at high speed, high precision control field.But, because Controller gain variations still continues to use traditional Sliding Mode Control Design Method, be only applied to the Mechatronic Systems of Relative order >=2 at present, as mechanical arm, permagnetic synchronous motor, DC converter etc.

(2) there is buffeting problem in existing non-singular terminal sliding-mode control, serious its practical engineering application of restriction.

With shape as second order control be example, non-singular terminal sliding mode is typically designed to: wherein, system state known, design parameter β >0, p and q are odd number, and 1<p/q<2.Visible, the Relative order of non-singular terminal sliding mode s is 1, and namely controlled quentity controlled variable u is aobvious is contained in in.At Lyapunov stable condition under constraint, directly comprise in controlled quentity controlled variable u and switch control item sgn (.), to ensure that system state arrives and maintains on the sliding-mode surface that designs in advance, and to Parameter Perturbation and external disturbance, there is robustness.But in actual control system, the fast switching controlled frequency of theory unlimited cannot realize, and makes system pass through sliding-mode surface back and forth, produces buffeting problem, cause system that vibration and unstable, serious its practical application of restriction occur.

Be different from single input single output control system, the robust stability of strong coupling restriction multi-input multi-output system controls.Complicated relevance is there is between different input channel and input quantity, system state, output quantity, utilize existing decoupling algorithm that System Model Reduction can be made for canonical structure, namely certain controlled quentity controlled variable can according to the one or more controlled volume of a certain pass effect, from simply disturb different, the stable single-loop system of multiple script can be caused in a lot of situation of coupling to become instability.Especially, in high dimension actual control system, when there is Parameter Perturbation and external disturbance in multiple input channel, cannot by realize Reasonable between input/output variable to and internal state between decoupling zero, set up the uncertain mathematics interact relation to system input and output and internal stability, and then the robust stability realizing non-matching uncertain multi-input multi-output system controls.

Summary of the invention

The object of the invention is to make controller cannot export the problem of the control signal of continuously smooth to solve existing non-singular terminal sliding-mode control existence buffeting, and the problem that Relative order is the control system of 1 cannot be applied to.

Be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, described control method is realized by following steps:

If step one controlled system is Relative order be 1 single input single output control system A: in formula, represent the state differential signal of single input single output control system A, x represents the system state of single input single output control system A, and u represents the single output control system A controlled quentity controlled variable of input, and t represents the time, then perform the control method of step 2; If controlled system is Relative order be 1 multi-input multi-output control system: in formula, represent multi-input multi-output control system state differential signal, x _srepresent multi-input multi-output control system state, u _srepresent multi-input multi-output control system controlled quentity controlled variable, t represents the time, then perform the control method of step 3;

Step 2, described single input single output control system: control method be specially:

Step 2 one, due to single input single output control system A: relative order be 1, described system state differential signal be unknown quantity at real system, then utilize High-Order Sliding Mode robust precision differential device Real-time Obtaining

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{y}=v_0\\ v_0=v_1-{\mathrm{\&lambda;}}_0{|y-x|}^{1/2}\mathrm{sign}(y-x)\\ {\stackrel{\&CenterDot;}{v}}_1=-{\mathrm{\&lambda;}}_1\mathrm{sign}(v_1-v_0)\end{array}\right.---\left(3\right),$ In formula, λ ₀for design parameter one, λ ₁for design parameter two; Y, v ₀and v ₁for the intermediate variable of High-Order Sliding Mode robust precision differential device state;

Step 2 two, design non-singular terminal sliding mode s (t), if in formula, the design parameter c>0 of non-singular terminal sliding mode s (t); Design parameter p, q of non-singular terminal sliding mode s (t) are odd number, and meet p>q>0,1<p/q<2;

Step 2 three, introduces virtual controlling amount v (4), makes described system state x be 2 relative to the Relative order of virtual controlling amount v, add the Relative order of system; Based on Lyapunov Stability Theorem design virtual controlling amount v, ensure that system state arrives and maintains on sliding-mode surface s (t)=0 designed in advance, and to Parameter Perturbation and external disturbance, there is robustness, and due to integral action u=∫ vdt (5), make actual output controlled quentity controlled variable u smoothly continuous;

Step 3, described multi-input multi-output control system: concrete control method be:

If the controlled Relative order of step 3 one is the multi-input multi-output control system of 1: interior indeterminate or disturbance term are matching uncertainties, and namely described multi-input multi-output control system is coupling multi-input multi-output control system B:

$\stackrel{\&CenterDot;}{x}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)---\left(6\right),$ Only the scalar way of realization of single input single output control system A in step 2 one to step 2 three need be transformed to matrix vector way of realization, and level and smooth non-singular terminal sliding-mode control has unchangeability because of its robustness had to matching uncertainties disturbance;

If controlled system is Relative order be 1 non-matching uncertain multi-input multi-output control system C:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+(B+\mathrm{\&Delta;B}\left(t\right))u\left(t\right)+f\left(t\right)---\left(7\right),$ In formula, x ∈ R ⁿfor system state, u ∈ R ^mfor controlled quentity controlled variable, and 1≤m≤n; A ∈ R ^{n × n}known constant matrix, B ∈ R ^{n × m}be known constant matrix, dimension is n ₁, n ₁≤ m; (A, B) is controlled, if r is the controllability conditions of constant matrices (A, B) in non-matching uncertain multi-input multi-output control system C; Δ B (t) ∈ R ^{n × m}represent the matching uncertainties in multiple input path, that is:

Δ B (t)=Bd (t) (8), in formula, d (t) ∈ R ^{m × m}time become bounded, its scope || d (t) ||≤l _d; F (t) represents the non-matching external disturbance of non-matching uncertain multi-input multi-output control system C, is R ⁿsmooth limited function, then perform the control method of step 3 two;

Step 3 two, described non-matching multi-input multi-output control system C: concrete control method be:

Step 1, according to the controllability conditions r of non-matching multi-input multi-output control system C, first time nonsingular state transformation is done to non-matching multi-input multi-output control system C:

Y=F ₁x (9), in formula, y represents the state after first time nonsingular state transformation, and x represents the system state before first time nonsingular state transformation, F ₁∈ R ^{n × n}for transformation matrix; Then non-matching multi-input multi-output control system C is transformed to block control standard form control system D:

$\stackrel{\&CenterDot;}{y}\left(t\right)=A^{\&prime;}y\left(t\right)+(B^{\&prime;}+\mathrm{\&Delta;}{B}^{\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;}\left(t\right)---\left(10\right),$ In formula, represent coefficient matrices A through first time nonsingular state transformation after matrix; $B^{\&prime;}=F_{1}B={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of coefficients B matrix after first time nonsingular state transformation, represent the submatrix of B ', dimension is n ₁; Δ B ' (t)=B ' d (t) represents the matching uncertainties in multiple input path; F ' (t) represents the non-matching external disturbance of block control standard form control system D, and there is f ' (t)=F ₁f (t);

Step 2, block control standard form control system D step 1 obtained is written as block form, and corresponding plot control standard form control system D is decomposed into internal subsystems one and IOS one,

Internal subsystems one: ${\stackrel{\&CenterDot;}{y}}_i\left(t\right)=\underset{j=i}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{i,j}^{\&prime;}{y}_i\left(t\right)+B_{i,i-1}{y}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;}\left(t\right),i=2,...r---\left(11a\right),$

IOS one: ${\stackrel{\&CenterDot;}{y}}_1\left(t\right)=\underset{j=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,j}^{\&prime;}{y}_j\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;}\left(t\right)---\left(11b\right),$ Controlled quentity controlled variable u only appears in IOS one; Y represents the state after first time nonsingular state transformation, $y={[y_r^T,...,y_1^T]}^T,y_i\&Element;R^{n_i}$ For the component of y, dimension is n _i, i=1 ... r, n ₁+ ... + n _r=n; $f^{\&prime;}={[f_{\mathrm{ur}}^{\&prime;T},...,f_{u2}^{\&prime;T},f_m^{\&prime;T}]}^T,$ for the component of f ', for the component of f '; A ' _i,jfor the design matrix of internal subsystems one, i=2 ... r;

Step 3, the internal subsystems one of existence step 2 obtained coupling does the nonsingular state transformation of second time:

$z=F_2^{-1}y---\left(12\right),$ In formula, y is the state after first time nonsingular state transformation, and z is the state after the nonsingular state transformation of second time, F ₂for transformation matrix,

Submatrix $K_{i,i+1}=B_{i+1,i}^{+}(K_{i+1,i+2}{B}_{i+2,i+1}+A_{i+1,i+1}^{\&prime;}-N_{i+1}),i,2,...,r-1,$

Submatrix $K_{i,j}=B_{i+1,i}^{+}(K_{i+1,j}{N}_j+A_{i+1,j}^{\&prime;}+K_{i+1,j+1}{B}_{j+1,j}-\underset{k=j-1}{\overset{i+1}{\mathrm{\&Sigma;}}}{A}_{i+1,k}^{\&prime;}{K}_{k,j}),i=\mathrm{1,2},...r-2,j=i+2,i+3,...r,$ for B _{i, i-1}moore-Penrose inverse, N _ifor transformation matrix F ₂the design matrix of middle submatrix, corresponding plot control standard form control system D is converted to control system E further: $\stackrel{\&CenterDot;}{z}=A^{\&prime;\&prime;}z\left(t\right)+(B^{\&prime;\&prime;}+\mathrm{\&Delta;}{B}^{\&prime;\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;\&prime;}\left(t\right)---\left(13\right),$ In formula, A "=(F ₂) ^-1a ' F ₂represent coefficient matrices A ' through second time nonsingular state transformation after matrix; $B^{\&prime;\&prime;}={\left(F_2\right)}^{-1}{B}^{\&prime;}={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of matrix of coefficients B ' after the nonsingular state transformation of second time; Δ B " (t)=B " d (t) represents the matching uncertainties in the control system E multiple input path after the nonsingular state transformation of second time; F " (t)=(F ₂) ^-1f ' (t) represents the non-matching external disturbance of control system E after the nonsingular state transformation of second time;

Step 4, control system E step 3 obtained is write as block form, and correspondingly control system E is decomposed into internal subsystems two and IOS two:

Internal subsystems two: ${\stackrel{\&CenterDot;}{z}}_i\left(t\right)=N_i{z}_i\left(t\right)+B_{i,i}{z}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right),i=2,...r---\left(14a\right),$

IOS two: ${\stackrel{\&CenterDot;}{z}}_1\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(14b\right),$ In formula, $z={[z_r^T,...,z_1^T]}^T,z_i\&Element;R^{n_i}$ Represent the component of z; $f^{\&prime;\&prime;}={[f_{\mathrm{ur}}^{\&prime;\&prime;T},...,f_{u2}^{\&prime;\&prime;T},f_m^{\&prime;\&prime;T}]}^T,f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right)\&Element;R^{n_i}$ Represent f " component i=1 ..., r, represent f " component; for the design matrix of internal subsystems two, i=2 ... r;

Step 5, the IOS two obtained step 4 is introduced reference model one and is asked for B to avoid applying classic method in step 4 _1,0the process of d (t) inverse matrix, described reference model one is:

$\stackrel{\&CenterDot;}{\mathrm{\&xi;}}\left(t\right)\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)-\mathrm{\&epsiv;}{B}_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)+w^{\&prime;}\left(t\right)---\left(15\right),$ In formula, represent the state of reference model one, ε (R ¹represent the design parameter of reference model one, the control law of the reference model one of u ' expression design, represent the auxiliary control law of the reference model one of design.

Beneficial effect of the present invention is:

First the inventive method judges the control system that Relative order is 1, then respectively by: be the single input single output control system of 1 for Relative order, utilize High-Order Sliding Mode robust precision differential device real-time acquisition system state differential, design non-singular terminal sliding mode, introducing virtual controlling amount utilizes the method for integral action to eliminate high frequency switching control item sgn (.) completely, the non-singular terminal sliding formwork realizing continuously smooth controls, reach the effect inherently eliminating the intrinsic buffeting problem of existing non-singular terminal sliding-mode control, the control signal of direct output continuously smooth.For coupling multi-input multi-output control system B, then the scalar way of realization of single input single output control system A is transformed to matrix vector way of realization, and level and smooth non-singular terminal sliding-mode control has unchangeability because of its robustness had to matching uncertainties disturbance.In addition, to non-matching uncertain multi-input multi-output control system C, then do twice nonsingular state transformation according to controllability conditions r, control system is resolved into r subsystem, then introduce reference model one, play and avoid application classic method to ask for B _1,0the process of d (t) inverse matrix, and there is the effect of the time-varying Hurst index eliminating IOS input channel, there is short-circuit evolution computation process and eliminate the effect of the intrinsic buffeting problem of existing non-singular terminal sliding-mode control, and then the robust stability that design control law and auxiliary control law realize control system adaptively controls.Having expanded existing non-singular terminal synovial membrane control method by said method is application in the control system of 1 at Relative order, has ubiquity application.Make existing non-singular terminal sliding-mode control correctly, to be effectively applied to Relative order be in 1 control system.

Accompanying drawing explanation

Fig. 1 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system deviation differential component signal convergence process figure, ordinate expression state differential vector value, horizontal ordinate t represents the time, and unit is second (s);

Fig. 2 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system non-singular terminal sliding mode s ₁component convergence process figure, ordinate s ₁represent non-singular terminal sliding mode vector value, horizontal ordinate t represents the time, and unit is second (s);

The component convergence process figure of Fig. 3 to be embodiment 1 Relative order be non-matching uncertain multi-input multi-output control system control law u of 1, ordinate represents the vector value of control law u, and horizontal ordinate t represents the time, and unit is second (s);

Fig. 4 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system bias state e component at the process schematic of finite time convergence control to zero, ordinate represents system deviation state e vector value, horizontal ordinate t represents the time, and unit is second (s);

Fig. 5 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system reference model state differential component convergence process figure, ordinate represent reference model state differential vector value, horizontal ordinate t represents the time, and unit is second (s);

Fig. 6 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system non-singular terminal sliding mode s ₂component convergence process figure, ordinate s ₂represent non-singular terminal sliding mode vector value, horizontal ordinate t represents the time, and unit is second (s);

Fig. 7 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system assist the component convergence process figure of control law v, ordinate represents the vector value of auxiliary control law v, and horizontal ordinate t represents the time, and unit is second (s);

Fig. 8 is embodiment 1 Relative order is IOS state z in the non-matching uncertain multi-input multi-output control system of 1 ₁component is at the process schematic of finite time convergence control to zero, and ordinate represents IOS state z ₁vector value, horizontal ordinate t represents the time, and unit is second (s);

Fig. 9 to be embodiment 1 Relative order be 1 non-matching uncertain multi-input multi-output control system zero dy namics subsystem state z ₂component and z ₃convergence process figure, ordinate represents zero dy namics subsystem state z ₂vector value and z ₃convergency value, horizontal ordinate t represents the time, and unit is second (s);

The component convergence process figure of Figure 10 to be embodiment 1 Relative order be non-matching uncertain multi-input multi-output control system original state x component of 1, ordinate represents the vector value of system initial state x, and horizontal ordinate t represents the time, and unit is second (s).

Embodiment

Embodiment one: present embodiment be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, described control method is realized by following steps:

If step one controlled system is Relative order be 1 single input single output control system A: in formula, represent the state differential signal of single input single output control system A, x represents the system state of single input single output control system A, and u represents the single output control system A controlled quentity controlled variable of input, and t represents the time, then perform the control method of step 2; If controlled system is Relative order be 1 multi-input multi-output control system: in formula, represent multi-input multi-output control system state differential signal, x _srepresent multi-input multi-output control system state, u _srepresent multi-input multi-output control system controlled quentity controlled variable, t represents the time, then perform the control method of step 3;

Step 2, described single input single output control system: control method be specially:

Step 2 one, due to single input single output control system A: relative order be 1, described system state differential signal be unknown quantity at real system, then utilize High-Order Sliding Mode robust precision differential device Real-time Obtaining

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{y}=v_0\\ v_0=v_1-{\mathrm{\&lambda;}}_0{|y-x|}^{1/2}\mathrm{sign}(y-x)\\ {\stackrel{\&CenterDot;}{v}}_1=-{\mathrm{\&lambda;}}_1\mathrm{sign}(v_1-v_0)\end{array}\right.---\left(3\right),$ In formula, λ ₀for design parameter one, λ ₁for design parameter two; Y, v ₀and v ₁for the intermediate variable of High-Order Sliding Mode robust precision differential device state;

Step 2 two, design non-singular terminal sliding mode s (t), if in formula, the design parameter c>0 of non-singular terminal sliding mode s (t); Design parameter p, q of non-singular terminal sliding mode s (t) are odd number, and meet p>q>0,1<p/q<2;

Step 2 three, introduces virtual controlling amount v (4), makes described system state x be 2 relative to the Relative order of virtual controlling amount v, add the Relative order of system; Based on Lyapunov Stability Theorem design virtual controlling amount v, ensure that system state arrives and maintains on sliding-mode surface s (t)=0 designed in advance, and to Parameter Perturbation and external disturbance, there is robustness, and due to integral action u=∫ vdt (5), make actual output controlled quentity controlled variable u smoothly continuous;

Step 3, described multi-input multi-output control system: concrete control method be:

If the controlled Relative order of step 3 one is the multi-input multi-output control system of 1: interior indeterminate or disturbance term are matching uncertainties, and namely described multi-input multi-output control system is coupling multi-input multi-output control system B:

$\stackrel{\&CenterDot;}{x}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)---\left(6\right),$ Only the scalar way of realization of single input single output control system A in step 2 one to step 2 three need be transformed to matrix vector way of realization, and level and smooth non-singular terminal sliding-mode control has unchangeability because of its robustness had to matching uncertainties disturbance;

If controlled system is Relative order be 1 non-matching uncertain multi-input multi-output control system C:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+(B+\mathrm{\&Delta;B}\left(t\right))u\left(t\right)+f\left(t\right)---\left(7\right),$ In formula, x ∈ R ⁿfor system state, u ∈ R ^mfor controlled quentity controlled variable, and 1≤m≤n; A ∈ R ^{n × n}known constant matrix, B ∈ R ^{n × m}be known constant matrix, dimension is n ₁, n ₁≤ m; (A, B) is controlled, if r is the controllability conditions of constant matrices (A, B) in non-matching uncertain multi-input multi-output control system C; Δ B (t) ∈ R ^{n × m}represent the matching uncertainties in multiple input path, that is: Δ B (t)=Bd (t) (8), in formula, d (t) ∈ R ^{m × m}time become bounded, its scope || d (t) ||≤l _d; F (t) represents the non-matching external disturbance of non-matching uncertain multi-input multi-output control system C, is R ⁿsmooth limited function, then perform the control method of step 3 two;

Step 3 two, described non-matching multi-input multi-output control system C: concrete control method be:

Step 1, according to the controllability conditions r of non-matching multi-input multi-output control system C, first time nonsingular state transformation is done to non-matching multi-input multi-output control system C:

Y=F ₁x (9), in formula, y represents the state after first time nonsingular state transformation, and x represents the system state before first time nonsingular state transformation, F ₁∈ R ^{n × n}for transformation matrix; Then non-matching multi-input multi-output control system C is transformed to block control standard form control system D:

$\stackrel{\&CenterDot;}{y}\left(t\right)=A^{\&prime;}y\left(t\right)+(B^{\&prime;}+\mathrm{\&Delta;}{B}^{\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;}\left(t\right)---\left(10\right),$ In formula, represent coefficient matrices A through first time nonsingular state transformation after matrix; $B^{\&prime;}=F_{1}B={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of coefficients B matrix after first time nonsingular state transformation, represent the submatrix of B ', dimension is n ₁; Δ B ' (t)=B ' d (t) represents the matching uncertainties in multiple input path; F ' (t) represents the non-matching external disturbance of block control standard form control system D, and there is f ' (t)=F ₁f (t);

Step 2, block control standard form control system D step 1 obtained is written as block form, and corresponding plot control standard form control system D is decomposed into internal subsystems one and IOS one,

Internal subsystems one: ${\stackrel{\&CenterDot;}{y}}_i\left(t\right)=\underset{j=i}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{i,j}^{\&prime;}{y}_i\left(t\right)+B_{i,i-1}{y}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;}\left(t\right),i=2,...r---\left(11a\right),$

IOS one: ${\stackrel{\&CenterDot;}{y}}_1\left(t\right)=\underset{j=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,j}^{\&prime;}{y}_j\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;}\left(t\right)---\left(11b\right),$ Controlled quentity controlled variable u only appears in IOS one; Y represents the state after first time nonsingular state transformation, $y={[y_r^T,...,y_1^T]}^T,y_i\&Element;R^{n_i}$ For the component of y, dimension is n _i, i=1 ... r, n ₁+ ... + n _r=n; $f^{\&prime;}={[f_{\mathrm{ur}}^{\&prime;T},...,f_{u2}^{\&prime;T},f_m^{\&prime;T}]}^T,$ for the component of f ', for the component of f '; A ' _i,jfor the design matrix of internal subsystems one, i=2 ... r;

Step 3, the internal subsystems one of existence step 2 obtained coupling does the nonsingular state transformation of second time:

$z=F_2^{-1}y---\left(12\right),$ In formula, y is the state after first time nonsingular state transformation, and z is the state after the nonsingular state transformation of second time, F ₂for transformation matrix,

Submatrix $K_{i,i+1}=B_{i+1,i}^{+}(K_{i+1,i+2}{B}_{i+2,i+1}+A_{i+1,i+1}^{\&prime;}-N_{i+1}),i,2,...,r-1,$

Submatrix $K_{i,j}=B_{i+1,i}^{+}(K_{i+1,j}{N}_j+A_{i+1,j}^{\&prime;}+K_{i+1,j+1}{B}_{j+1,j}-\underset{k=j-1}{\overset{i+1}{\mathrm{\&Sigma;}}}{A}_{i+1,k}^{\&prime;}{K}_{k,j}),i=\mathrm{1,2},...r-2,j=i+2,i+3,...r,$ for B _{i, i-1}moore-Penrose inverse, N _ifor transformation matrix F ₂the design matrix of middle submatrix, corresponding plot control standard form control system D is converted to control system E further: $\stackrel{\&CenterDot;}{z}=A^{\&prime;\&prime;}z\left(t\right)+(B^{\&prime;\&prime;}+\mathrm{\&Delta;}{B}^{\&prime;\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;\&prime;}\left(t\right)---\left(13\right),$ In formula, A "=(F ₂) ^-1a ' F ₂represent coefficient matrices A ' through second time nonsingular state transformation after matrix;

$B^{\&prime;\&prime;}={\left(F_2\right)}^{-1}{B}^{\&prime;}={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of matrix of coefficients B ' after the nonsingular state transformation of second time;

Δ B " (t)=B " d (t) represents the matching uncertainties in the control system E multiple input path after the nonsingular state transformation of second time; F " (t)=(F ₂) ^-1f ' (t) represents the non-matching external disturbance of control system E after the nonsingular state transformation of second time;

Step 4, control system E step 3 obtained is write as block form, and correspondingly control system E is decomposed into internal subsystems two and IOS two:

Internal subsystems two: ${\stackrel{\&CenterDot;}{z}}_i\left(t\right)=N_i{z}_i\left(t\right)+B_{i,i}{z}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right),i=2,...r---\left(14a\right),$

IOS two: ${\stackrel{\&CenterDot;}{z}}_1\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(14b\right),$ In formula, $z={[z_r^T,...,z_1^T]}^T,z_i\&Element;R^{n_i}$ Represent the component of z; $f^{\&prime;\&prime;}={[f_{\mathrm{ur}}^{\&prime;\&prime;T},...,f_{u2}^{\&prime;\&prime;T},f_m^{\&prime;\&prime;T}]}^T,f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right)\&Element;R^{n_i}$ Represent f " component i=1 ..., r, represent f " component; for the design matrix of internal subsystems two, i=2 ... r;

Step 5, the IOS two obtained step 4 is introduced reference model one and is asked for B to avoid applying classic method in step 4 _1,0the process of d (t) inverse matrix, described reference model one is:

$\stackrel{\&CenterDot;}{\mathrm{\&xi;}}\left(t\right)\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)-\mathrm{\&epsiv;}{B}_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)+w^{\&prime;}\left(t\right)---\left(15\right),$

In formula, represent the state of reference model one, ε (R ¹represent the design parameter of reference model one, the control law of the reference model one of u ' expression design, represent the auxiliary control law of the reference model one of design.

Design control law u (t), makes IOS two finite time follow the tracks of upper reference model one;

Design assistant control law w (t), makes reference model one finite time convergence control to zero.

Embodiment two:

With embodiment one unlike, present embodiment is applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, in order to make IOS two finite time follow the tracks of upper reference model one, the design process of the control law u ' of reference model one described in the step 5 in step 3 two is specially:

First, deviation variables e=z is defined ₁– ξ, can obtain by IOS two and reference model one bias system that Relative order is 1: $\stackrel{\&CenterDot;}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}(I+\frac{d\left(t\right)}{\mathrm{\&epsiv;}+1})u^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(16\right),$ Make ε >>l _d, then || d (t)/(ε+1) || < < 1, namely has (I+d (t)/(ε+1)) ≈ I, so Relative order be 1 bias system be approximately:

$\stackrel{\&CenterDot;}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(17\right);$

Secondly, design non-singular terminal sliding mode one: $s_1\left(t\right)=e+c_1{\stackrel{\&CenterDot;}{e}}^{p_1/q_1}---\left(18\right),$ In formula, represent non-singular terminal sliding mode one; c ₁=diag (c11 ..., c1ni) and represent the design parameter of non-singular terminal sliding mode one non-singular terminal sliding mode one and c _1i>0, i=1 ... n ₁; The design parameter p of non-singular terminal sliding mode one ₁, q ₁for odd number, and p ₁>q ₁>0,1<p ₁/ q ₁<2;

Finally, based on Relative order and sliding formwork equivalent control measurements, the level and smooth control law of design robust:

$u^{\&prime;}\left(t\right)=u_{\mathrm{eq}}\left(t\right)+u_n\left(t\right)---\left(19\right),$

In formula, $u_{\mathrm{eq}}\left(t\right)=\frac{1}{\mathrm{\&epsiv;}+1}{B}_{\mathrm{1,0}}^{+}{w}^{\&prime;}\left(t\right)---\left(20\right),$

U _eqt () represents equivalent control term, u _nt () represents switching control item, then introduce virtual controlling item: based on Lyapunov Stability Theorem virtual controlling item v ₁t () is designed to:

$\left\{\begin{array}{c}{v}_1\left(t\right)=v_{1\mathrm{eq}}\left(t\right)+v_{1n}\left(t\right)\\ v_{1\mathrm{eq}}\left(t\right)=-\frac{q_1}{(\mathrm{\&epsiv;}+1)p_1}{B}_{\mathrm{1,0}}^{+}{c}_1^{-1}{\stackrel{\&CenterDot;}{e}}^{2-p_1/q_1}\\ v_{1n}\left(t\right)=-\frac{1}{\mathrm{\&epsiv;}+1}{B}_{\mathrm{1,0}}^{+}\left(\right|\left|{\stackrel{\&CenterDot;}{f}}_m^{\&prime;\&prime;}\right||+{\mathrm{\&eta;}}_1)\mathrm{sgn}\left(s_1\right)\end{array}\right.---\left(22\right),$

Namely by introducing virtual controlling item v ₁and add the relative exponent number of system, and then obtain level and smooth continuous print working control rule u ', wherein, η ₁>0 is ride gain, v _1eqand v _1nfor state variable, and convergence time is:

$t_1=\frac{p_1}{p_1-q_1}\underset{i=1,...,{\mathrm{\&eta;}}_1}{\max }\left(c_{1i}{e}_i{\left(0\right)}^{\frac{p_1}{p_1-q_1}}\right)---\left(23\right),$ In formula, e _i(0) be deviation variables e _ithe original state value of (t).

Its stability proof procedure: choose Lyapunov function and first order derivative is asked to V (t), then have:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right){=s}_1^T{\stackrel{\&CenterDot;}{s}}_1\\ =s_1^T\left(\frac{p_1}{q_1}{c}_1\mathrm{diag}\left({\stackrel{\&CenterDot;}{e}}^{p_1/q_1-1}\right)\right)(\stackrel{\&CenterDot;\&CenterDot;}{e}+\frac{q_1}{p_1}{c}_1^{-1}{\stackrel{\&CenterDot;}{e}}^{2-p/q})\end{array},$

By equivalent control term u _eqt () substitutes into the approximate formula of bias system: in, then have: $\stackrel{\&CenterDot;}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}{u}_n\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right),$

Thus, make: $\stackrel{\&CenterDot;}{V}\left(t\right)=s_1^T\left(\frac{p_1}{q_1}{c}_1\mathrm{diag}\left({\stackrel{\&CenterDot;}{e}}^{p_1/q_1-1}\right)\right)(\frac{q_1}{p_1}{c}_1^{-1}{\stackrel{\&CenterDot;}{e}}^{2-p_1/q_1}+(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}(v_{1\mathrm{eq}}+v_{1n})+{\stackrel{\&CenterDot;}{f}}_m^{\&prime;\&prime;}\left(t\right)),$ Again according to virtual controlling item v ₁t () design switches control formula, due to c ₁for diagonal matrix, therefore also be diagonal matrix,

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)=-{\mathrm{\&eta;}}_1\frac{p_1}{q_1}\left(c_1\mathrm{diag}\left({\stackrel{\&CenterDot;}{e}}^{p_1/q_1-1}\right)\right)s_1^T\mathrm{sgn}\left(s_1\right)\\ \&le;-{\mathrm{\&eta;}}_1\frac{p_1}{q_1}\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{c}_{1i}{\stackrel{\&CenterDot;}{e}}_i^{p_1/q_1-1}\left|s_{1i}\right|\&le;0\end{array}$

When system does not also reach stable, || s ₁(t) || ≠ 0, to above-mentioned inequality, and if only if and the transient state of system, explanation state can not keep.

Embodiment three:

With embodiment one or two unlike, present embodiment be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, in order to make reference model one at finite time convergence control to zero and after bias system converges to zero described in step 5, reference model one is of equal value with IOS two, be zero by described bias system approximate formula, namely $\stackrel{\&CenterDot;}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)=0,$ Then:

substitute into reference model one and obtain reference model two:

$\stackrel{\&CenterDot;}{\mathrm{\&xi;}}\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)+\frac{1}{\mathrm{\&epsiv;}+1}w\left(t\right)-\frac{\mathrm{\&epsiv;}}{\mathrm{\&epsiv;}+1}{f}_m^{\&prime;\&prime;}\left(t\right)---\left(24\right),$ Relative order due to reference model two is 1, with the level and smooth non-singular terminal sliding mode controller design process of the step 2 controlled quentity controlled variable u in step 3 two, direct Design assistant control law w ' (t), then described in step 5, the design process of reference model one control law w ' (t) is specially:

First, non-singular terminal sliding mode two is designed to:

$s_2\left(t\right)=\mathrm{\&xi;}+c_2{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}^{p_2/q_2}---\left(25\right),$

In formula, represent non-singular terminal sliding mode two, c ₂=diag (c ₂₂..., c _2ni) represent non-singular terminal sliding mode two design parameter and c _2i>0, i=1 ... n ₂; Non-singular terminal sliding mode two design parameter p ₂, q ₂for odd number and p ₂>q ₂>0,1<p ₂/ q ₂<2;

Secondly, for make reference model two system state ξ and at convergence time inside converge to zero, the Controller gain variations of reference model two is: w " (t)=w _eq(t)+w _nt () (27), in formula, equivalent control term is $w_{\mathrm{eq}}\left(t\right)=-(\mathrm{\&epsiv;}+1)\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)---\left(28\right),$ Introduce virtual controlling item to increase the relative exponent number of system, and actual switching control item w _n(t)=∫ v ₂(t) dt continuously smooth; Based on Lyapunov Stability Theorem virtual controlling item v ₂t () is designed to:

$\left\{\begin{array}{c}{v}_2\left(t\right)=w_n\left(t\right),\\ v_2\left(t\right)=v_{2\mathrm{eq}}\left(t\right)+v_{2n}\left(t\right),\\ v_{2\mathrm{eq}}\left(t\right)=-\frac{q_2}{p_2}(\mathrm{\&epsiv;}+1)c_2^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}^{2-p_2/q_2},\\ v_{2n}\left(t\right)=-\left(\mathrm{\&epsiv;}\right|\left|{\stackrel{\&CenterDot;}{f}}_m^{\&prime;\&prime;}\left(t\right)\right||+{\mathrm{\&eta;}}_2)\mathrm{sgn}\left(s_2\right),\end{array}\right.$

In formula, η ₂>0 is ride gain.The design matrix N that in embodiment one, step 4 relates to _iwith non-matching external disturbance f " _uit () impact on internal system stability is described:

As IOS state z ₁after converging to zero, internal subsystems one becomes the zero dy namics subsystem with disturbance:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_i\left(t\right)=N_i{z}_i\left(t\right)+B_{i,i-1}{z}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right),i=3,...,r\\ {\stackrel{\&CenterDot;}{z}}_2\left(t\right)=N_2{z}_2\left(t\right)+f_{u2}^{\&prime;\&prime;}\left(t\right)\end{array}\right.---\left(30\right),$ Visible, for the i-th layer subsystem, z _i-1can be considered virtual controlling amount, and B _{i, i-1}full rank, therefore the stability of zero dy namics subsystem only depends on design matrix N _iwith non-matching external disturbance f " _ui(t).

According to Lyapunov Stability Theorem, N _i, i=2 ..., the eigenwert of r should be negative entirely, to ensure that described zero dy namics subsystem is at equilibrium point z _ithe local stability of=0, designs here: $N_i=-{\mathrm{\&lambda;}}_i{I}_{n_i}---\left(31\right),$ Wherein ,-λ ₂<...<-λ _r<0, makes z ₂..., z _rorder convergence.

For described zero dy namics subsystem, there is symmetric positive definite matrix P _i, Q _i∈ R ^{ni × ni}, i=2 ..., r meets then z ₂..., z _rasymptotic convergence is to Ω _iin: ${\mathrm{\&Omega;}}_i=\{z_i\&Element;R^{n_i}:|\left|z_i\right||\&le;{\mathrm{\&tau;}}_i\}---\left(32\right),$

In formula, τ _i=2 (|| f " _ui||+τ _i-1|| B _{i, i-1}||)/μ _i, μ _i=λ _min(Q _i)/λ _max(P _i), λ _min(Q _i) and λ _max(P _i) represent Q respectively _iand P _iminimum and maximum eigenwert; τ _i-1=0.

Asymptotic convergence region Ω _iderivation be: because N _i, i=2 ..., r eigenwert is designed to bear, to any positive definite symmetrical matrix Q _i, necessarily there is positive definite symmetrical matrix P _imeet choose Lyapunov function and first order derivative is asked to it, then have

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_i\left(t\right)={\stackrel{\&CenterDot;}{z}}_i^T{P}_i{z}_i+z_i^T{P}_i{\stackrel{\&CenterDot;}{z}}_i\\ =z_i^T(N_i^T{P}_i+P_i{N}_i)z_i+2f_{\mathrm{ui}}^{\&prime;\&prime;T}{P}_i{z}_i+2z_i^T{P}_i{B}_{i,i-1}{z}_i-1\\ =-z_i^T{Q}_i{z}_i+2f_{\mathrm{ui}}^{\&prime;\&prime;T}{P}_i{z}_i+2z_i^T{P}_i{B}_{i,i-1}{z}_{i-1}\\ \&le;-{\mathrm{\&lambda;}}_{\min }\left(Q_i\right){\left|\right|z_i\left|\right|}^2+2{\mathrm{\&lambda;}}_{\max }\left(P_i\right)\left|\right|z_i\left|\right|\left|\right|f_{\mathrm{ui}}^{\&prime;\&prime;}\left|\right|+2{\mathrm{\&lambda;}}_{\max }\left(P_i\right)\left|\right|f_{\mathrm{ui}}^{\&prime;\&prime;}\left|\right|\left|\right|B_{i,i-1}\left|\right|\left|\right|z_{i-1}\left|\right|\\ =-\left|\right|z_i\left|\right|{\mathrm{\&lambda;}}_{\max }\left(P_i\right)\left({\mathrm{\&mu;}}_i\right|\left|z_i\right||-2|\left|f_{\mathrm{ui}}^{\&prime;\&prime;}\right||-2{\mathrm{\&tau;}}_{i-1}{B}_{i,i-1})\end{array}$

According to Lyapunov Stability Theorem, then when in, then expression formula is derived:

${\mathrm{\&Omega;}}_i=\{z_i\&Element;R^{n_i}:|\left|z_i\right||\&le;{\mathrm{\&tau;}}_i\};$

Because and twice nonsingular state transformation y=F ₁x and for nonsingular, thus former non-matching uncertain multi-input multi-output control system C can be obtained: state x _i, i=1 ..., the convergence range of n: ${\mathrm{\&Omega;}}_x=\{x\&Element;R^n:|\left|x\right||\&le;|\left|F_1{F}_2\right|\left|{\left(\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&tau;}}_i^2\right)}^{1/2}\right\}---\left(33\right).$

Embodiment 1:

Design shape such as Relative order is the non-matching uncertain multi-input multi-output control system C of 1:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+(B+\mathrm{\&Delta;B}\left(t\right))u\left(t\right)+f\left(t\right)$ Following expression, wherein:

$A=\left[\begin{array}{ccccccc}2& 1& 0& 0& 0& 0& 0\\ 0& 2& 0& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0& 0\\ 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right],B=\left[\begin{array}{ccc}2& 1& 1\\ 2& 1& 1\\ 1& 1& 1\\ 3& 2& 1\\ -1& 0& 0\\ 1& 0& 1\\ 1& 0& 0\end{array}\right],f\left(t\right)=\sin \left(2t\right)\left[\begin{array}{c}-0.1\\ 0.1\\ -0.1\\ 0.1\\ 0.1\\ 0.1\\ 0.1\end{array}\right],d\left(t\right)=\sin \left(2t\right)\left[\begin{array}{ccc}0.1& 0& 0\\ 0& 0.1& 0\\ 0& 0& 0.1\end{array}\right]$

System initial state x (0)=[10.94,19.09,27.03,45.56,31.39 ,-16.68 ,-14.09] ^t; Due to system controllability index r=3, therefore can be analyzed to 3 subsystem B _1,0, N ₂and N ₃be respectively:

$B_{\mathrm{1,0}}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 1\\ 1& 0& 0\end{array}\right],N_2=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right],N_3=-0.6,$

Thus, according to first time nonsingular state transformation y=F ₁x and the nonsingular state transformation of second time described formula, twice transformation matrix F ₁and F ₂for:

$F_1=\left[\begin{array}{ccccccc}0& 1& 0& -0.5& -0.5& -0.5& -0.5\\ 1& 0& -1& 0& 0& 0& -1\\ 0& 0& -2& 1& 0& 1& -2\\ 0& 0& 0& 0& 1& 0& 1\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]{,F}_2=\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ -5.2& 0& 0& 1& 0& 0& 0\\ -6.72& -3& 2.5& 4.1& 1& 0& 0\\ 8.32& 0& 0& -4.6& 0& 1& 0\\ 4.16& 0& -1.5& 2.3& 0& 0& 1\end{array}\right],$

Thus obtain the IOS after decoupling zero and be:

${\stackrel{\&CenterDot;}{z}}_1=\left[\begin{array}{ccc}5& -1.6& 1\\ 0& 5.6& 0\\ 0& 0.8& 4\end{array}\right]z_1+\left[\begin{array}{ccc}-9& 7.5& 15.66\\ 0& 0& -13.36\\ 0& -3& -6.68\end{array}\right]z_2+\left[\begin{array}{c}-17.472\\ 13.312\\ 6.656\end{array}\right]z_3+B_{\mathrm{1,0}}(I+d\left(t\right))u\left(t\right)+\sin \left(2t\right)\left[\begin{array}{c}-0.26\\ -0.54\\ 0.08\end{array}\right],$

Internal subsystems is write as block form:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_2=N_2{z}_2+\left[\begin{array}{ccc}1& 0& 2\\ 0& -1& 2\\ 0& 1& 0\end{array}\right]z_1+\sin \left(2t\right)\left[\begin{array}{c}-0.1\\ 0.2\\ -0.32\end{array}\right]\\ {\stackrel{\&CenterDot;}{z}}_3=N_3{z}_3+\begin{array}{c}\left[\begin{array}{ccc}0& 0& 0.5\end{array}\right]z_2-0.1\sin \left(2t\right)\end{array}\end{array}\right.,$

Controller gain variations parameter p ₁=p ₂=5, q ₁=q ₂=3, c ₁=diag (0.3,0.3,0.3), c ₂=diag (0.8,1,0.7), ε=10, η ₁=20, η ₂=80; High-Order Sliding Mode robust precision differential device described in step one is utilized to ask deviation with design parameter λ _0e1=22, λ _0e2=λ _0e3=15, λ _1e1=λ _1e2=λ _1e3=50; Ask reference model state differential with design parameter λ _0z1=22, λ _0z2=15, λ _0z3=13, λ _1z1=λ _1z2=λ _1z3=50, simulation result as shown in figures 1-8.

Fig. 1-4 and Fig. 5-8 adopts the control law u of level and smooth non-singular terminal sliding-mode control design and the simulation process of auxiliary control law v, and wherein Fig. 1 is system state differential , Fig. 5 is system state differential in Fig. 1, Fig. 5, solid line represents real differential signal, and dotted line represent by High-Order Sliding Mode robust precision differential device obtain differential signal, all can extract differential signal quickly and accurately; Fig. 2 and Fig. 6 is the non-singular terminal sliding mode s that the differential signal of application fetches is formed ₁and s ₂all can rapidly converge to zero; As Fig. 3 and Fig. 7 is respectively control law u and auxiliary control law v, signal smoothing is continuous, and essence eliminates high-frequency buffeting signal, and makes the IOS state z of bias state e and Fig. 8 as Fig. 4 respectively ₁equal finite time convergence control to zero, thus demonstrates correctness and the validity of put forward level and smooth non-singular terminal sliding-mode control.

Fig. 9 is zero dy namics subsystem state z ₂and z ₃convergence curve, according to expression formula utilize LMI tool box can obtain its convergence region to be || z ₂||≤0.0710, || z ₃||≤0.0533.

Figure 10 is the convergence curve of original system original state x, can obtain its maximum convergence range to be further || x||≤4.4946.The correctness that Relative order is the level and smooth non-singular terminal sliding-mode control of the control system of 1 is applicable to described in above simulation results show.

Claims (3)

1. be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, it is characterized in that: described control method is realized by following steps:

If step one controlled system is Relative order be 1 single input single output control system A: in formula, represent the state differential signal of single input single output control system A, x represents the system state of single input single output control system A, and u represents the single output control system A controlled quentity controlled variable of input, and t represents the time, then perform the control method of step 2; If controlled system is Relative order be 1 multi-input multi-output control system: in formula, represent multi-input multi-output control system state differential signal, x _srepresent multi-input multi-output control system state, u _srepresent multi-input multi-output control system controlled quentity controlled variable, t represents the time, then perform the control method of step 3;

Step 2, described single input single output control system: control method be specially:

Step 2 one, due to single input single output control system A: relative order be 1, described system state differential signal be unknown quantity at real system, then utilize High-Order Sliding Mode robust precision differential device Real-time Obtaining

$\left\{\begin{array}{c}\stackrel{.}{y}=v_0\\ v_0=v_1-{\mathrm{\&lambda;}}_0{|y-x|}^{1/2}\mathrm{sign}(y-x)\\ {\stackrel{.}{v}}_1=-{\mathrm{\&lambda;}}_1\mathrm{sign}(v_1-v_0)\end{array}\right.---\left(3\right),$

In formula, λ ₀for design parameter one, λ ₁for design parameter two; Y, v ₀and v ₁for the intermediate variable of High-Order Sliding Mode robust precision differential device state;

Step 2 two, design non-singular terminal sliding mode s (t), if in formula, the design parameter c>0 of non-singular terminal sliding mode s (t); Design parameter p, q of non-singular terminal sliding mode s (t) are odd number, and meet p>q>0,1<p/q<2;

Step 2 three, introduces virtual controlling amount v (4), makes described system state x be 2 relative to the Relative order of virtual controlling amount v, add the Relative order of system; Based on Lyapunov Stability Theorem design virtual controlling amount v, ensure that system state arrives and maintains on sliding-mode surface s (t)=0 designed in advance, and to Parameter Perturbation and external disturbance, there is robustness, and due to integral action u=∫ vdt (5), make actual output controlled quentity controlled variable u smoothly continuous;

Step 3, described multi-input multi-output control system: concrete control method be:

If the controlled Relative order of step 3 one is the multi-input multi-output control system of 1: interior indeterminate or disturbance term are matching uncertainties, and namely described multi-input multi-output control system is coupling multi-input multi-output control system B:

$\stackrel{.}{x}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)---\left(6\right),$

Only the scalar way of realization of single input single output control system A in step 2 one to step 2 three need be transformed to matrix vector way of realization, and level and smooth non-singular terminal sliding-mode control has unchangeability because of its robustness had to matching uncertainties disturbance;

If controlled system is Relative order be 1 non-matching uncertain multi-input multi-output control system C:

$\stackrel{.}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+(B+\mathrm{\&Delta;B}\left(t\right))u\left(t\right)+f\left(t\right)---\left(7\right),$

In formula, x ∈ R ⁿfor system state, u ∈ R ^mfor controlled quentity controlled variable, and 1≤m≤n; A ∈ R ^{n × n}known constant matrix, B ∈ R ^{n × m}be known constant matrix, dimension is n ₁, n ₁≤ m; (A, B) is controlled, if r is the controllability conditions of constant matrices (A, B) in non-matching uncertain multi-input multi-output control system C; Δ B (t) ∈ R ^{n × m}represent the matching uncertainties in multiple input path, that is:

ΔB(t)＝Bd(t) (8)，

In formula, d (t) ∈ R ^{m × m}time become bounded, its scope || d (t) ||≤l _d; F (t) represents the non-matching external disturbance of non-matching uncertain multi-input multi-output control system C, is R ⁿsmooth limited function, then perform the control method of step 3 two;

Step 3 two, described non-matching multi-input multi-output control system C: concrete control method be:

Step 1, according to the controllability conditions r of non-matching multi-input multi-output control system C, first time nonsingular state transformation is done to non-matching multi-input multi-output control system C:

y＝F ₁x (9)，

In formula, y represents the state after first time nonsingular state transformation, and x represents the system state before first time nonsingular state transformation, F ₁∈ R ^{n × n}for transformation matrix; Then non-matching multi-input multi-output control system C is transformed to block control standard form control system D:

$\stackrel{.}{y}\left(t\right)=A^{\&prime;}y\left(t\right)+(B^{\&prime;}+{\mathrm{\&Delta;B}}^{\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;}\left(t\right)---\left(10\right),$

In formula, A '=F ₁aF ₁ ^-1represent coefficient matrices A through first time nonsingular state transformation after matrix; $B^{\&prime;}=F_{1}B={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of coefficients B matrix after first time nonsingular state transformation, represent the submatrix of B ', dimension is n ₁; Δ B ' (t)=B ' d (t) represents the matching uncertainties in multiple input path; F ' (t) represents the non-matching external disturbance of block control standard form control system D, and there is f ' (t)=F ₁f (t);

Step 2, block control standard form control system D step 1 obtained is written as block form, and corresponding plot control standard form control system D is decomposed into internal subsystems one and IOS one,

Internal subsystems one: ${\stackrel{\&CenterDot;}{y}}_i\left(t\right)=\underset{j=i}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{i,j}^{\&prime;}{y}_i\left(t\right)+B_{i,i-1}{y}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;}\left(t\right),i=2,...r---\left(11a\right),$

IOS one: ${\stackrel{.}{y}}_1\left(t\right)=\underset{j=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,j}^{\&prime;}{y}_j\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;}\left(t\right)---\left(11b\right),$

Controlled quentity controlled variable u only appears in IOS one; Y represents the state after first time nonsingular state transformation, $y={[y_r^T,...,y_1^T]}^T,$ for the component of y, dimension is n _i, i=1 ... r, n ₁+ ... + n _r=n; $f^{\&prime;}={[f_{\mathrm{ur}}^{\&prime;T},...,f_{u2}^{\&prime;T},f_m^{\&prime;T}]}^T,$ for the component of f ', for the component of f '; A ' _{i, j}for the design matrix of internal subsystems one, i=2 ... r;

Step 3, the internal subsystems one of existence step 2 obtained coupling does the nonsingular state transformation of second time:

$z=F_2^{-1}y---\left(12\right),$

In formula, y is the state after first time nonsingular state transformation, and z is the state after the nonsingular state transformation of second time, F ₂for transformation matrix,

Submatrix $K_{i,i+1}=B_{i+1,i}^{+}(K_{i+1,i+2}{B}_{i+2,i+1}+A_{i+1,i+1}^{\&prime;}-N_{i+1}),i=\mathrm{1,2},...r-1,$

Submatrix $K_{i,j}=B_{i+1,i}^{+}(K_{i+1,j}{N}_j+A_{i+1,j}^{\&prime;}+K_{i+1,j+1}{B}_{j+1,j}-\underset{k=j-1}{\overset{i+1}{\mathrm{\&Sigma;}}}{A}_{i+1,k}^{\&prime;}{K}_{k,j}),i=\mathrm{1,2},...r-2,j=i+2,i+3,...r,$

for B _{i, i-1}moore-Penrose inverse, N _ifor transformation matrix F ₂the design matrix of middle submatrix, corresponding plot control standard form control system D is converted to control system E further: $\stackrel{.}{z}\left(t\right)=A^{\&prime;\&prime;}z\left(t\right)+(B^{\&prime;\&prime;}+{\mathrm{\&Delta;B}}^{\&prime;\&prime;}\left(t\right))u\left(t\right)+f^{\&prime;\&prime;}\left(t\right)---\left(13\right),$ In formula, A "=(F ₂) ^-1a ' F ₂represent coefficient matrices A ' through second time nonsingular state transformation after matrix; $B^{\&prime;\&prime;}={\left(F_2\right)}^{-1}{B}^{\&prime;}={\left[\begin{array}{cc}0& B_{\mathrm{1,0}}^T\end{array}\right]}^T$ Represent the matrix of matrix of coefficients B ' after the nonsingular state transformation of second time; Δ B " (t)=B " d (t) represents the matching uncertainties in the control system E multiple input path after the nonsingular state transformation of second time; F " (t)=(F ₂) ^-1f ' (t) represents the non-matching external disturbance of control system E after the nonsingular state transformation of second time;

Step 4, control system E step 3 obtained is write as block form, and correspondingly control system E is decomposed into internal subsystems two and IOS two:

Internal subsystems two: ${\stackrel{.}{z}}_i\left(t\right)=N_i{z}_i\left(t\right)+B_{i,i-1}{z}_{i-1}\left(t\right)+f_{\mathrm{ui}}^{\&prime;\&prime;}\left(t\right),i=2,...r---\left(14a\right),$

IOS two: ${\stackrel{.}{z}}_1\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)+(B_{\mathrm{1,0}}+B_{\mathrm{1,0}}d\left(t\right))u\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(14b\right),$

In formula, $z={[z_r^T,...,z_1^T]}^T,$ represent the component of z; $f^{\&prime;\&prime;}={[f_{\mathrm{ur}}^{\&prime;\&prime;T},...,f_{u2}^{\&prime;\&prime;T},f_m^{\&prime;\&prime;T}]}^T,$ represent f " component i=1 ..., r, represent f " component; for the design matrix of internal subsystems two, i=2 ... r;

Step 5, the IOS two obtained step 4 is introduced reference model one and is asked for B to avoid applying classic method in step 4 _1,0the process of d (t) inverse matrix, described reference model one is:

$\stackrel{.}{\mathrm{\&xi;}}\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)-{\mathrm{\&epsiv;B}}_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)+w^{\&prime;}\left(t\right)---\left(15\right),$

In formula, represent the state of reference model one, ε ∈ R ¹represent the design parameter of reference model one, the control law of the reference model one of u ' expression design, represent the auxiliary control law of the reference model one of design.

2. be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system according to claim 1, it is characterized in that: in order to make IOS two finite time follow the tracks of upper reference model one, the design process of the control law u ' of reference model one described in the step 5 in step 3 two is specially:

First, deviation variables e=z is defined ₁– ξ, can obtain by IOS two and reference model one bias system that Relative order is 1: $\stackrel{.}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}(I+\frac{d\left(t\right)}{\mathrm{\&epsiv;}+1})u^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(16\right),$

Make ε > > l _d, then || d (t)/(ε+1) || < < 1, namely has (I+d (t)/(ε+1)) ≈ I, so Relative order be 1 bias system be approximately:

$\stackrel{.}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)---\left(17\right);$

Secondly, design non-singular terminal sliding mode one: $s_1\left(t\right)=e+c_1{\stackrel{\&CenterDot;}{e}}^{p_1/q_1}---\left(18\right),$ In formula, represent non-singular terminal sliding mode one; c ₁=diag (c11 ..., c1ni) and represent the design parameter of non-singular terminal sliding mode one non-singular terminal sliding mode one and c _1i>0, i=1 ... n ₁; The design parameter p of non-singular terminal sliding mode one ₁, q ₁for odd number, and p ₁>q ₁>0,1<p ₁/ q ₁<2;

Finally, based on Relative order and sliding formwork equivalent control measurements, the level and smooth control law of design robust:

u′(t)＝u _eq(t)+u _n(t) (19)，

In formula, $u_{\mathrm{eq}}\left(t\right)=\frac{1}{\mathrm{\&epsiv;}+1}{B}_{\mathrm{1,0}}^{+}{w}^{\&prime;}\left(t\right)---\left(20\right),$

U _eqt () represents equivalent control term, u _nt () represents switching control item, then introduce virtual controlling item:

Based on Lyapunov Stability Theorem virtual controlling item v ₁t () is designed to:

$\left\{\begin{array}{c}{v}_1\left(t\right)=v_{1\mathrm{eq}}\left(t\right)+v_{1n}\left(t\right)\\ v_{1\mathrm{eq}}\left(t\right)=-\frac{q_1}{(\mathrm{\&epsiv;}+1)p_1}{B}_{\mathrm{1,0}}^{+}{c}_1^{-1}{\stackrel{.}{e}}^{2-p_1/q_1}\\ v_{1n}\left(t\right)=-\frac{1}{\mathrm{\&epsiv;}+1}{B}_{\mathrm{1,0}}^{+}\left(\right|\left|{\stackrel{.}{f}}_m^{\&prime;\&prime;}\right||+{\mathrm{\&eta;}}_1)\mathrm{sgn}\left(s_1\right)\end{array}\right.---\left(22\right),$

Namely by introducing virtual controlling item v ₁and add the relative exponent number of system, and then obtain level and smooth continuous print working control rule u ', wherein, η ₁>0 is ride gain, v _1eqand v _1nfor state variable, and convergence time is:

$t_1=\frac{p_1}{p_1-q_1}\underset{i=1,...,n_1}{\max }\left(c_{1i}{e}_i{\left(0\right)}^{\frac{p_1}{p_1-q_1}}\right)---\left(23\right),$

In formula, e _i(0) be deviation variables e _ithe original state value of (t).

3. according to claim 1 or 2, be applicable to the level and smooth non-singular terminal sliding-mode control that Relative order is 1 control system, it is characterized in that: in order to make reference model one at finite time convergence control to zero and after bias system converges to zero described in step 5, reference model one is of equal value with IOS two, be zero by described bias system approximate formula, namely

$\stackrel{.}{e}\left(t\right)=(\mathrm{\&epsiv;}+1)B_{\mathrm{1,0}}{u}^{\&prime;}\left(t\right)-w^{\&prime;}\left(t\right)+f_m^{\&prime;\&prime;}\left(t\right)=0,$ Then:

substitute into reference model one and obtain reference model two:

$\stackrel{.}{\mathrm{\&xi;}}\left(t\right)=\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)+\frac{1}{\mathrm{\&epsiv;}+1}w\left(t\right)-\frac{\mathrm{\&epsiv;}}{\mathrm{\&epsiv;}+1}{f}_m^{\&prime;\&prime;}\left(t\right)---\left(24\right),$

Relative order due to reference model two is 1, with the level and smooth non-singular terminal sliding mode controller design process of the step 2 controlled quentity controlled variable u in step 3 two, direct Design assistant control law w ' (t), then described in step 5, the design process of reference model one control law w ' (t) is specially:

First, non-singular terminal sliding mode two is designed to:

$s_2\left(t\right)=\mathrm{\&xi;}+c_2{\stackrel{.}{\mathrm{\&xi;}}}^{p_2/q_2}---\left(25\right),$

In formula, represent non-singular terminal sliding mode two, c ₂=diag (c ₂₂..., c _2ni) represent non-singular terminal sliding mode two design parameter and c _2i>0, i=1 ... n ₂; Non-singular terminal sliding mode two design parameter p ₂, q ₂for odd number and p ₂>q ₂>0,1<p ₂/ q ₂<2;

Secondly, for make reference model two system state ξ and at convergence time inside converge to zero, the Controller gain variations of reference model two is: w " (t)=w _eq(t)+w _nt () (27), in formula, equivalent control term is $w_{\mathrm{eq}}\left(t\right)=-(\mathrm{\&epsiv;}+1)\underset{\mathrm{\&alpha;}=1}{\overset{r}{\mathrm{\&Sigma;}}}{A}_{1,\mathrm{\&alpha;}}^{\&prime;\&prime;}{z}_{\mathrm{\&alpha;}}\left(t\right)---\left(28\right),$ Introduce virtual controlling item to increase the relative exponent number of system, and actual switching control item w _n(t)=∫ v ₂(t) dt continuously smooth; Based on Lyapunov Stability Theorem virtual controlling item v ₂t () is designed to:

$\left\{\begin{array}{c}{v}_2\left(t\right)=w_n\left(t\right),\\ v_2\left(t\right)=v_{2\mathrm{eq}}\left(t\right)+v_{2n}\left(t\right),\\ v_{2\mathrm{eq}}\left(t\right)=-\frac{q_2}{p_2}(\mathrm{\&epsiv;}+1)c_2^{-1}{\stackrel{.}{\mathrm{\&xi;}}}^{2-p_2/q_2},\\ v_{2n}\left(t\right)=-\left(\mathrm{\&epsiv;}\right|\left|{\stackrel{.}{f}}_m^{\&prime;\&prime;}\left(t\right)\right||+{\mathrm{\&eta;}}_2)\mathrm{sgn}\left(s_2\right),\end{array}---\right.\left(29\right),$

In formula, η ₂>0 is ride gain.