CN105068420A - Non-affine uncertain system self-adaptive control method with range restraint - Google Patents

Abstract

The present invention relates to a non-affine uncertain system self-adaptive control method with a range restraint. In combination with the characteristic that system input and states are subject to the range restraint, based on a mean value theorem, a non-affine system is converted into a time varying system (a strict feedback system) with a linear structure, and a varying interval of time-varying uncertain parameters of the time varying system is obtained. On the basis, a parameter adaptive projection technique is utilized to carry out online estimation on bounded uncertain time-varying parameters and external disturbances, and a parameter estimation error is compensated by using the nonlinear dynamic damping technology. Based on the nonlinear mapping technology, a restraint amount is mapped to the whole real number space to solve a state restraint problem of the system, thus to ensure the state is always within a restraint range. A hyperbolic tangent function and the Nussbaum gain technology are simultaneously utilized to solve the problem that system input is subject to the range restraint or the problem of input saturation and the potential problem of singular values of a controller, and the controller is designed in combination with a method of inversion, thereby solving the problem of self-adaptive control of the non-affine uncertain system with the range restraint.

Description

A kind of nonaffine uncertain system self-adaptation control method of Operations of Interva Constraint

Technical field

The present invention inputs automation field, specifically, is a kind of nonaffine uncertain system self-adaptation control method of Operations of Interva Constraint.

Background technology

In recent years, Complex Nonlinear System control problem has become a study hotspot of Chinese scholars, and in succession proposes many control methods, as feedback linearization, self-adaptation inverting, neural network theory and fuzzy logic control etc.Can most achievement in research be all for Affine Systems, and the achievement in research of Non-Affine Systems be less.

Non-Affine Systems is not entered and the dynamic perfromance of influential system by ride gain in linear scaling mode due to its control inputs, but the mode contained with Nonlinear Implicit is to system generation effect, there is no the concept of " ride gain " in Affine Systems and " controlling party to ", its problem is more complicated and have more challenge, and the achievement in research obtained is less.It is reported, Non-Affine Systems control problem does not also form a kind of systematized design theory at present.But, many Practical Project system natures are all nonaffines, as the rudder surface control system of aircraft ^[7], biochemical system, alternating current-direct current integrated transmission system etc., their mathematical model is difficult to be expressed as affine form, moreover Non-Affine Systems is the method for expressing that control system is the most general, can say, most nonlinear system can adopt nonaffine form to describe.Therefore, the control problem specializing in Non-Affine Systems has important theoretical and practical significance.The control problem of nonaffine uncertain system, due to its theory value and practical application meaning, in recent years, becomes a problem received much concern gradually.At present, two classes can be probably divided into about its control method: indirect control theory and direct control method.

Indirect control theory: first Non-Affine Systems is converted into Affine Systems by mathematics manipulation, then adopt affine self-adaptation control method to carry out the design of controller.At present, typical affineization mathematical processing methods has: Hadamard lemma, Taylor series expansion, mean value theorem, increase integral method.Although these methods have certain feasibility, but concerning a lot of actual engineering system, also there is certain limitation: (1) affine conversion process can reduce the accuracy of system model, also may lose some important nonlinear characteristics simultaneously, the range of application of designed controller strategy is greatly reduced.Ignore higher order term as Taylor launches to get linear approximation method, along with the expansion using region, error also increases thereupon, problems such as " algebraic loops " of mean value theorem.(2) hypothesis relied on and condition are that a lot of Practical Project system cannot meet, if the controlling party of system is to known, system architecture with parameter is determined or system only Parameter uncertainties but nondeterministic function f (x) meets parameter linearization condition: f (x)=θ ψ (x).(3) when system architecture is uncertain or Uncertain nonlinear function f (x) does not meet parameter line condition, neural network and fuzzy control technology is adopted to carry out online approximating to unknown function.Neural network and fuzzy control have very strong learning ability, can approach and process the dynamic perfromance of various uncertain system.But, these two kinds of disposal routes can only approach unknown continuous function in certain compacts, and very complicated for its design process of nonlinear system of complexity, how to ensure that the real-time of system is a difficult problem, moreover neural network and fuzzy control can introduce new reconstructed error.

Direct control method: do not need Non-Affine Systems to be converted into Affine Systems in control procedure.With regard to published document and technical information, more representational method has: Inverted control system method, based on observer method with based on neural network or fuzzy control technology method.Even if but the inverse of known Non-Affine Systems exists, and utilizing implicit function theorem to solve, this is explicit inverse still very difficult and need to compensate it.The use of observer technology can introduce new variable, adds the complicacy of controller, also can introduce new evaluated error simultaneously.The introducing of fuzzy control technology also can increase the complicacy of controller, reduces the real-time of system and introduces new reconstructed error.

Strictly speaking, any system all will be subject to the constraint of various physical condition.A lot of real system is all such as inputted, export or state constraint, reaction may be physical discontinuity in physical system, saturated, dead band and in order to ensure some restrictions that system performance and safety are done, these constraints can impact the ruuning situation of system, if constraint is not being met, control performance decline, closed loop instability will be caused, even cause system destruction.The Controller gain variations of restricted problem has become an important research topic, all brings challenge to theoretical and reality.Also some noticeable achievements are created at present, as the Controller gain variations based on invariant set theory, based on the Controller gain variations of adaptive statistical filtering, instruction filter method, Model Predictive Control and the Controller gain variations etc. based on obstacle Lyapunov function method (BarrierLyapunovFunction, BLF).Although the Controller gain variations of restricted problem has got some achievements, these achievements be all propose based on linear system and affine nonlinear system and several greatly absolutely to system by state, input or output about intrafascicular a certainly to process.At present, rare with all affined control problem achievement in research of input in state about non-affine nonlinear systems.

In sum, the research of the Parameter uncertainties Non-Affine Systems adaptive control algorithm of Operations of Interva Constraint has theory significance and the using value of reality, but due to unique property and the relevant restraining factors of Non-Affine Systems, make the research of this problem face no small challenge.

Summary of the invention

The object of this invention is to provide one can make system have structure and parameter uncertainty, and state and input follow the tracks of the Non-Affine Systems self-adaptation control method of specifying reference signal under being subject to the combined influence of Operations of Interva Constraint and external disturbance.

For solving the problems of the technologies described above, invent a kind of nonaffine uncertain system self-adaptation control method of Operations of Interva Constraint, according to mean value theorem, Non-Affine Systems is converted into the time-varying system with linear structure, on this basis, change method to realize based on non-linear mapping techniques, hyperbolic tangent function and inversion technique and Non-Affine Systems composition closed-loop system, it is characterized in that comprising the following steps:

(1) under the assumed condition meeting engineering practice and the prerequisite of not losing the important nonlinear characteristic of system, based on mean value theorem, Non-Affine Systems is converted into the time-varying system (tight feedback system) with linear structure, and coupling system state and input are by the characteristic of Operations of Interva Constraint, become uncertain parameter constant interval when obtaining time-varying system based on lagrange's method of multipliers.

(2) adopt parameter adaptive shadow casting technique to carry out On-line Estimation to becoming uncertain parameter during bounded, parameter estimating error adopts Nonlinear Dynamic damping to compensate.

(3) adopt non-linear mapping techniques amount of restraint to be mapped to whole real number space and carry out the problem for the treatment of system state by Operations of Interva Constraint, adopt the input of hyperbolic tangent function and Nussbaum gain techniques disposal system by Operations of Interva Constraint problem and controller singular value problem simultaneously.

(4) on the basis of first three step, utilize and according to method of inversion CONTROLLER DESIGN, nonaffine uncertain system is controlled.The present invention compared with prior art, its remarkable advantage for:: (1) institute research object be structure and parameter all uncertain and by state and input double constraints Non-Affine Systems; (2) coupling system state and the affined characteristic of input, under the important nonlinear characteristic of system of not losing and loss model precision prerequisite, based on mean value theorem Non-Affine Systems is converted into the time-varying system and its time-varying parameter bounded with linear structure, weakens the assumed condition that Non-Affine Systems controls further; (3) nonlinear mapping method is utilized will to be mapped as whole real number space between confining region, to be tied problem with the state of disposal system, the controller singular value simultaneously utilizing hyperbolic tangent function and Nussbaum gain techniques to overcome may to exist and system input affined problem; (4) without the need to adopting any neural network, fuzzy technology and observer technology infinitely to approach and On-line Estimation unknown nonlinear function and indeterminate in system in the design of controller, without the need to multi-model switching control, controller architecture is simple.Designed controller has good robustness, adaptivity and versatility, and has certain fault-tolerant ability.

Below in conjunction with accompanying drawing, this aspect is further described.

Fig. 1 is nonlinear mapping function

Fig. 2 is the nonaffine uncertain system self-adaptation control method structural drawing of Operations of Interva Constraint of the present invention

Embodiment

As shown in Fig. 1 ~ 2, the nonaffine uncertain system self-adaptation control method of a kind of Operations of Interva Constraint of the present invention, be nonaffine uncertain system design adaptive controller according to technology such as mean value theorem, Nonlinear Mapping and invertings, its feature is comprising the following steps:

(1) Non-Affine Systems assumed condition and affine method for transformation:

A: nonaffine uncertain system assumed condition is: the state of system, input and uncertain parameter are by Operations of Interva Constraint.

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_i=x_{i+1},1\≤i\≤n-1\\ {\stackrel{\·}{x}}_n=f(x,u(\mathrm{\ν}\left)\right)+d_0\left(t\right)\\ y=x_1\end{array}\right.$

In formula, x=[x ₁, x ₂..., x _n] ^t∈ R ⁿconstraint condition is met for state vector: for known normal number; Y is that system exports; F () is all uncertain nonlinear function of structure and parameter; d ₀t indeterminate that () causes for external interference; V is actuator input; U (ν) is the output by actuator saturation properties influence; The specific descriptions of constraint function u (ν) are as follows.

$u\left(\mathrm{\&nu;}\right)=sat\left(\mathrm{\&nu;}\right)=\left\{\begin{array}{cc}-u_M,& \mathrm{\&nu;}<-u_M\\ \mathrm{\&nu;},& -u_M\&le;\mathrm{\&nu;}\&le;u_M\\ u_M,& \mathrm{\&nu;}>u_M\end{array}\right.$

U in formula _mfor constraint function boundary value, it is known normal number.The assumed condition of system is:

Suppose that 1 nonlinear function f () can lead continuously about x and u, and partial derivative bounded.

Suppose the uncertain parameter bounded of 2 system, and be positioned at known boundary, or the uncertain parameter unbounded of system, but it is realized, as sin cos functions etc. by limited function the impact of system.

Though suppose that 3 system architectures are uncertain, potential all possible structure uncertain factor can be predicted.

Suppose 4 indeterminate d ₀t () meets: | d ₀(t) |≤d ₀, d ₀for known normal number.

Suppose 5f () initial point place f (0,0 ..., 0) and meaningful, and | f (0,0 ..., 0) |≤Δ ₀, Δ ₀for known normal number.

Suppose 6 reference signal y _rt () remains in confining region, namely to arbitrarily there is positive number have

B: Non-Affine Systems affine linear method for transformation is:

$\begin{array}{c}{f}_i\left({\stackrel{\&OverBar;}{x}}_i,x_{i+1}\right)=f_i\left({\stackrel{\&OverBar;}{x}}_i,x_{i+1}\right)-f_i\left(0,x_2\mathrm{...}{x}_i,x_{i+1}\right)+\\ f_i\left(0,x_2,\mathrm{...}{x}_i,x_{i+1}\right)-f_i\left(0,0,x_3,\mathrm{..},x_i,x_{i+1}\right)+\\ .\\ .\\ .\\ f_i\left(0,\mathrm{...},0,x_{i+1}\right)-f_i-\left(0,\mathrm{...0},0\right)+f_i\left(0,0,\mathrm{...0}\right)\end{array}$

Utilize mean value theorem can be by be described as:

$\begin{array}{c}f(x,u)=\left(\frac{\&part;f(\&CenterDot;)}{\&part;x_1}{|}_{\left({\mathrm{\&gamma;}}_1,x_2,\mathrm{...},x_n,u\right)}\right)x_1+\left(\frac{\&part;f(\&CenterDot;)}{\&part;x_2}{|}_{\left(0,{\mathrm{\&gamma;}}_2,x_3,\mathrm{...},x_n,u\right)}\right)x_2+\mathrm{...}\\ +\left(\frac{\&part;f(\&CenterDot;)}{\&part;x_n}{|}_{\left(0,\mathrm{...},0,{\mathrm{\&gamma;}}_n,x_{i+1}\right)}\right)x_n+\left(\frac{\&part;f(\&CenterDot;)}{\&part;u}{|}_{\left(0,\mathrm{...},0,{\mathrm{\&gamma;}}_u\right)}\right)x_u+f\left(0\right)\end{array}$

Can be by system converting further:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_i=x_{i+1},1\&le;i\&le;n-1\\ {\stackrel{\&CenterDot;}{x}}_n=b_1\left(t\right)x_1+\mathrm{...}+b_n\left(t\right)x_n+\mathrm{\&beta;}\left(t\right)u\left(\mathrm{\&nu;}\right)+f\left(0,\mathrm{...},0\right)+d_0\left(t\right)\\ y=x_1\end{array}\right.$

In formula

$b_i\left(t\right)=\frac{\&part;f}{\&part;x_i}{|}_{(0,...,0,{\mathrm{\&gamma;}}_i,x_{i+1},\mathrm{...},x_{n}u(v\left)\right)},i=1,...,n$

$\mathrm{\&beta;}\left(t\right)=\frac{\&part;f}{\&part;u}{|}_{(0,...,0,{\mathrm{\&gamma;}}_u)},\mathrm{\&beta;}\left(t\right)\&NotEqual;0$

γ _i, i=1 ..., n is between 0 and x _ibetween certain value, γ _ufor certain value between 0 and u.F (0), b _iwith β be respectively f (0 ..., 0), b _it () and β (t) write a Chinese character in simplified form.

C: become during time-varying system uncertain parameter constant interval acquisition methods into:

A: according to hypothesis 3, although though the structure of system exists uncertain, but because its potential all possible structure uncertain factor can be predicted, therefore mathematical modeling can all be carried out to set up systems most model to often kind of structure uncertain condition, simultaneously because the energy of system is limited, so the parameter of often kind of model is known or unknown bounded.

B: according to the known time-varying parameter b of the character of mean value theorem _it () and β (t) are the function of x and u, i.e. b _it () and β (t) can be expressed as: b _i(t)=f _bi(x, u (ν)) and β (t)=f _β(x, u (ν)), according to hypothesis 1 ~ 2, function f _bi() and f _βthe equal bounded of each parameter in ().

C: because the state x of system is all tied, so b in often kind of mathematical model with input u _it between the confining region of () and β (t), Solve problems is and asks extreme-value problem under multivariate function constraint condition, can solve Extreme Value Problem of Multi-Variable Functions under constraint condition based on lagrange's method of multipliers.

D: getting the b of often kind of mathematical model _iafter between the confining region of (t) and β (t), find out b between wherein maximum confining region _i∈ (b _imin, b _imax) and β ∈ (β _min, β _max), be time-varying parameter b in system operation _ibetween the confining region that may travel through of (t) and β (t), between this confining region, meet the parameter in system cloud gray model and structural uncertainty requirement in theory.

(2) uncertain parameter method of estimation is become during bounded:

Utilize during self-adaptation shadow casting technique On-line Estimation bounded and become uncertain parameter, specific as follows:

${\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left(\&CenterDot;i\right)=\left\{\begin{array}{cc}0,& \widehat{\mathrm{\&theta;}}={\mathrm{\&theta;}}_{\max },\&CenterDot;i>0\\ 0,& \widehat{\mathrm{\&theta;}}={\mathrm{\&theta;}}_{\min },\&CenterDot;i<0\\ \&CenterDot;i,& other\end{array}\right.$

represent the estimated value of θ, represent parameter estimating error.Γ is parameter identification speed diagonal matrix, Ω _θ∈ [θ _min, θ _max].

(3) state and the affined disposal route of input:

A: adopt non-linear mapping techniques to be unconfined variable by affined condition conversion, finally obtain the unfettered Controlling model of following state:

In formula, $g\left(x_i^{*}\right)=(e^{x_i^{*}}+e^{-x_i^{*}}+2)(i=1,...,n),h\left(x_{i+1}^{*}\right)={(e^{x_{i+1}^{*}}+1)}^{-1}(i=1,...,n-1),$ Its concrete conversion process is: definition maps Η one by one: as follows:

Obvious Η is continuous elementary function, can obtain:

Differentiate, can obtain:

B: adopt the input of hyperbolic tangent function and Nussbaum gain techniques disposal system by the method for Operations of Interva Constraint and controller singular value to be:

u(v)＝sat(v)＝h(v)+d _v(t)

In formula, h (v) is hyperbolic tangent function, d _vt () is limited function.

$h\left(v\right)=u_M\&times;\tanh \left(\frac{\mathrm{\&nu;}}{u_M}\right)=u_M\frac{e^{\mathrm{\&nu;}/u_M}-e^{-\mathrm{\&nu;}/u_M}}{e^{\mathrm{\&nu;}/u_M}+e^{-\mathrm{\&nu;}/u_M}}$

|d _v(t)|≤u _M(1-tanh(1))

So constraint function can be equivalent to: sat (v)=h (ν)+d _v(t), wherein d _vt () can be considered external disturbance, therefore, compound disturbance d (t) of system is final for being expressed as:

d(t)＝d ₀(t)+βd _v(t)

According to boundedness and the d of β _vthe boundary value of (t), the composite interference of known system (8) still meets hypothesis 4, namely | and d (t) |≤d, d are known normal number.

(4) adaptive controller design method:

A: first sub-system error is defined as by i-th (i=1 ..., n) sub-system error is defined as (n+1)th sub-system error is defined as z _n+1=h (v)-α _n, then in conjunction with method of inversion design adaptive controller.

B: in the n-th step, adopts Nussbaum gain techniques processing controller singular value problem.

C: adopt hyperbolic tangent function and the process of Nussbaum gain techniques to input affined problem in the (n+1)th step, finally, utilize non-linear differential tracking technique, calculate the differential signal of virtual controlling amount, to solve differential explosion issues.Thus the control of the nonaffine uncertain system to Operations of Interva Constraint can be realized.

In order to make design process have good readability, first introducing mean value theorem and Nonlinear Mapping relevant knowledge, not being general, considering following nonaffine uncertain system:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_i=x_{i+1},1\&le;i\&le;n-1\\ {\stackrel{\&CenterDot;}{x}}_n=f(x,u(v\left)\right)+d_0\left(t\right)\\ y=x_1\end{array}\right.---\left(1\right)$

X=[x in formula ₁, x ₂..., x _n] ^t∈ R ⁿconstraint condition is met for state vector: for known normal number; Y is that system exports; F () is all uncertain nonlinear function of structure and parameter; d ₀t indeterminate that () causes for external interference; V is actuator input; U (ν) is the output by actuator saturation properties influence; The specific descriptions of constraint function u (ν) are as follows.

$u\left(\mathrm{\&nu;}\right)=sat\left(\mathrm{\&nu;}\right)=\left\{\begin{array}{cc}-u_M,& \mathrm{\&nu;}<-u_M\\ \mathrm{\&nu;},& -u_M\&le;\mathrm{\&nu;}\&le;u_M\\ u_M,& \mathrm{\&nu;}>u_M\end{array}\right.---\left(2\right)$

U in formula _mfor constraint function boundary value, it is known normal number.Before carrying out Controller gain variations, provide following assumed condition, definition and lemma.

Suppose that 1 nonlinear function f () can lead continuously about x and u, and partial derivative bounded.

Suppose the uncertain parameter bounded of 2 system, and be positioned at known boundary, or the uncertain parameter unbounded of system, but it is realized, as sin cos functions etc. by limited function the impact of system.

Though suppose that 3 system architectures are uncertain, potential all possible structure uncertain factor can be predicted.

Suppose 4 indeterminate d ₀t () meets: | d ₀(t) |≤d ₀, d ₀for known normal number.

Suppose 5f () initial point place f (0,0 ..., 0) and meaningful, and | f (0,0 ..., 0) |≤Δ ₀, Δ ₀for known normal number.

Suppose 6 reference signal y _rt () remains in confining region, namely to arbitrarily there is positive number have

If definition 1 continuous function N (κ): R → R meets following condition, then claim N (κ) for Nussbaum function.

$\left(1\right)---\underset{s\&RightArrow;+\mathrm{\&infin;}}{\lim }sup\frac{1}{s}{\&Integral;}_0^{s}N\left(\mathrm{\&kappa;}\right)d\mathrm{\&kappa;}=+\mathrm{\&infin;}$

$\left(2\right)---\underset{s\&RightArrow;+\mathrm{\&infin;}}{\lim }inf\frac{1}{s}{\&Integral;}_0^{s}N\left(\mathrm{\&kappa;}\right)d\mathrm{\&kappa;}=+\mathrm{\&infin;}$

Lemma 1 establishes V () and κ () to be defined in [0, t _f) on smooth function, meet v (t)>=0; N (κ) is a smooth Nussbaum type function, if following inequality is set up, then V (t), κ (t) and at interval [0, t _f) must bounded:

$0\&le;V\left(t\right)\&le;{\mathrm{\&sigma;}}_0+e^{-\mathrm{\&sigma;}t}{\&Integral;}_0^t\left(g\left(x\left(\mathrm{\&tau;}\right)\right)N\left(\mathrm{\&kappa;}\right)+1\right)\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}{e}^{\mathrm{\&sigma;}t}d\mathrm{\&tau;}---\left(3\right)$

In formula, σ ₀for suitable constant, σ normal number, (x (τ) is for time-varying function and in closed interval for g and interior value.

Lemma 2 is for any given normal number t _f> 0, if the solution of closed-loop system is at interval [0, t _f) upper bounded, then t _f=∞.

Because f () in system (1) is all uncertain nonlinear function of structure and parameter, the state of coupling system and the affined characteristic of input, first the time-varying system utilizing mean value theorem to be first converted into by Non-Affine Systems to have linear structure also, between the confining region simultaneously obtaining time-varying parameter, has:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_i=x_{i+1},1\&le;i\&le;n-1\\ {\stackrel{\&CenterDot;}{x}}_n=b_1\left(t\right)x_1+\mathrm{...}+b_n\left(t\right)x_n+\mathrm{\&beta;}\left(t\right)u\left(\mathrm{\&nu;}\right)+f\left(0,\mathrm{...},0\right)+d_0\left(t\right)\\ y=x_1\end{array}\right.---\left(4\right)$

In formula

$b_i\left(t\right)=\frac{\&part;f}{\&part;x_i}{|}_{(0,...,0,{\mathrm{\&gamma;}}_i,x_{i+1},\mathrm{...},x_n,x_n,u(v\left)\right)},i=1,...,n$

$\mathrm{\&beta;}\left(t\right)=\frac{\&part;f}{\&part;u}{|}_{(0,...,0,{\mathrm{\&gamma;}}_u)},\mathrm{\&beta;}\left(t\right)\&NotEqual;0$

γ _i, i=1 ..., n is between 0 and x _ibetween certain value, γ _ufor certain value between 0 and u.For convenience, below f (0 ..., 0), b _it () and β (t) are abbreviated as f (0), b respectively _iand β.Owing to needing to use b in Controller gain variations _ibetween the confining region of (t) and β (t).B _it between () and β (t) confining region, acquisition methods is described as follows:

(1) according to hypothesis 3, although though the structure of system exists uncertain, but because its potential all possible structure uncertain factor can be predicted, therefore mathematical modeling can all be carried out to set up systems most model to often kind of structure uncertain condition, simultaneously because the energy of system is limited, so the parameter of often kind of model is known or unknown bounded.

(2) according to the known time-varying parameter b of the character of mean value theorem _it () and β (t) are the function of x and u, i.e. b _it () and β (t) can be expressed as: b _i(t)=f _bi(x, u (ν)) and β (t)=f _β(x, u (ν)), according to hypothesis 1 ~ 2, function f _bi() and f _βthe equal bounded of each parameter in ().

(3) because the state x of system is all tied, so b in often kind of mathematical model with input u _it between the confining region of () and β (t), Solve problems is and asks extreme-value problem under multivariate function constraint condition.

(4) b of often kind of mathematical model is being got _iafter between the confining region of (t) and β (t), find out b between wherein maximum confining region _i∈ (b _imin, b _imax) and β ∈ (β _min, β _max), be time-varying parameter b in system operation _ibetween the confining region that may travel through of (t) and β (t), between this confining region, meet the parameter in system cloud gray model and structural uncertainty requirement in theory.

Above-mentioned b _it the concrete solution procedure between () and β (t) confining region will provide in simulation calculation.Due to b _iobtain according to the characteristic of state constraint between (t) and β (t) confining region, if system in operational process can not Guarantee Status all the time between its confining region interior change, then b _it () and β (t) can lose efficacy.Therefore, designed controller must can ensure that system state meets its constraint condition.

Because the state of system (4) is tied, if directly CONTROLLER DESIGN may cause the state of system not meet constraint condition therefore, before CONTROLLER DESIGN, nonlinear mapping function need be utilized affined variable x _ibe mapped as unconfined variable for this reason, definition maps Η one by one: as follows:

Obvious Η is continuous elementary function, as shown in Figure 1.Can obtain according to formula (5):

To formula (6) differentiate, can obtain:

Definition $g\left(x_i^{*}\right)=\left(e^{x_i^{*}}+e^{-x_i^{*}}+2\right)\left(i=1,\mathrm{...},n\right),h\left(x_{i+1}^{*}\right)={\left(e^{x_{i+1}^{*}}+1\right)}^{-1}\left(i=1,\mathrm{...},n-1\right),$ Obviously with all be greater than zero and have nothing to do with the nonlinear function f () of original system, affined for state system (1) can be converted to the equivalent system such as formula (8) by formula (7) substitution formula (4).

According to hypothesis 6, the reference signal y of system _rt () is also in interval interior value, therefore can make:

According to the character of Nonlinear Mapping Η be unconfined variable, can in whole real number space value.So far, affined system (1) can be converted to the unconstrained system such as formula (8), after mapping, the reference signal of system (8) should be to be system (8) design adaptive controller below, and before designing, following lemma need be introduced.

Set is considered in lemma 3 set and map one by one if y _r(t) ∈ C ⁿand bounded, so and bounded ^[22].

Set is considered in lemma 4 set and map one by one if so x _i(t) → q _i(t)=Η ^-1(p _i(t)) ^[22].

For convenience of calculation is symmetrical by being defined as between confining region herein, this also without loss of generality, for asymmetric interval, always can make to be symmetrical between confining region by translate coordinate system.

Because hyperbolic tangent function is to the approximate effect of constraint of saturation function, it can be adopted to carry out approximate processing to saturation function.Hyperbolic tangent function is described below:

$h\left(v\right)=u_M\&times;\tanh \left(\frac{\mathrm{\&nu;}}{u_M}\right)=u_M\frac{e^{\mathrm{\&nu;}/u_M}-e^{-\mathrm{\&nu;}/u_M}}{e^{\mathrm{\&nu;}/u_M}+e^{-\mathrm{\&nu;}/u_M}}---\left(10\right)$

Make d _v(t)=sat (v)-h (v), and d _vt () is limited function, its boundary value can represent:

|d _v(t)|≤u _M(1-tanh(1))(11)

So constraint of saturation function can be equivalent to: sat (v)=h (ν)+d _v(t), wherein d _vt () can be considered external disturbance, therefore, compound disturbance d (t) of system is final for being expressed as:

d(t)＝d ₀(t)+βd _v(t)(12)

According to boundedness and the d of β _vthe boundary value of (t), the composite interference of known system (8) still meets hypothesis 4, namely | and d (t) |≤d, d are known normal number.Design corresponding controller below in conjunction with the method for inversion, propose following specific design process.

Step1 considers the 1st subsystem, definition:

$z_1=x_1^{*}-y_r^{*}---\left(13\right)$

Select virtual controlling rule α ₁for:

C in formula ₁for positive design constant.Definition

$z_2={\mathrm{\&alpha;}}_1-h\left(x_2^{*}\right)---\left(15\right)$

Can obtain according to formula (13) ~ (15):

For Degenerate Λ type three level atom, Setpi (i=2 ... n) corresponding parameter similar designs in.Will with traditional anti-pushing manipulation be considered as virtual controlling variable different, the present invention will look smooth function it is the virtual controlling amount of i-th subsystem.

Step2 considers the 2nd subsystem, selects virtual controlling rule α ₂for:

Definition

$z_3={\mathrm{\&alpha;}}_2-h\left(x_3^{*}\right)---\left(18\right)$

According to formula (17) and (18), can obtain formula (15) differentiate:

Stepi (i=3 ... n-1), consider i-th subsystem, definition

$z_i={\mathrm{\&alpha;}}_{i-1}-h\left(x_i^{*}\right)---\left(20\right)$

Select virtual controlling rule α _ifor:

Definition

$z_{i+1}={\mathrm{\&alpha;}}_i-h\left(x_{i+1}^{*}\right)---\left(22\right)$

According to formula (21) and (22), can obtain formula (20) differentiate:

The appearance of Stepn control inputs u (v), definition

$z_n={\mathrm{\&alpha;}}_{n-1}-h\left(x_n^{*}\right)---\left(24\right)$

Order for convenience, hereinafter be abbreviated as then can be expressed as:

Convolution (12) and (25), can obtain formula (24) both sides differentiate:

In above formula select virtual controlling rule α _nfor:

In formula

Wherein χ is the normal number of design, and parameter update law is selected as follows:

Wherein represent estimated value, represent parameter estimating error.Parameter update law adopt adaptive statistical filtering Proj () to obtain, its concrete form is as follows

Wherein for parameter identification speed diagonal matrix, , for Identification of parameter (29), there is following characteristic: i.e. parameter estimation bounded, and be positioned at known boundary, its concrete proof procedure is shown in document [30].Definition

Z _n+1=h (v)-α _n(30) can obtain according to formula (26) ~ (30):

The appearance of Stepn+1 working control input v, the design of control law of system is following form:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\mathrm{\&nu;}}=-c\mathrm{\&nu;}+\mathrm{\&omega;}\\ \mathrm{\&omega;}=N\left(\mathrm{\&kappa;}\right)(c_{n+1}{z}_{n+1}-\mathrm{\&eta;}c\mathrm{\&nu;}-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_n)\\ N\left(\mathrm{\&kappa;}\right)={\mathrm{\&kappa;}}^{2}cos\left(\mathrm{\&kappa;}\right)\end{array}\right.---\left(32\right)$

Parameter update law is selected as follows:

$\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}=(c_{n+1}{z}_{n+1}-\mathrm{\&eta;}c\mathrm{\&nu;}-\stackrel{\&CenterDot;}{\mathrm{\&alpha;}})z_{n+1}---\left(33\right)$ Design parameter c > 0, c in formula _n+1> 0, for bounded time-varying function.Consideration formula (32) ~ (33), can obtain formula (30) differentiate:

$\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_{n+1}=\frac{\&part;h\left(\mathrm{\&nu;}\right)}{\&part;\mathrm{\&nu;}}\stackrel{\&CenterDot;}{\mathrm{\&nu;}}-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_n=\mathrm{\&eta;}\left(-c\mathrm{\&nu;}+\mathrm{\&omega;}\right)-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_n\\ =-c_{n+1}{z}_{n+1}+\left(\mathrm{\&eta;}N\left(\mathrm{\&kappa;}\right)+1\right)\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}/z_{n+1}\end{array}---\left(34\right)$

In Step2 ~ n+1, virtual controlling rule α need be calculated _idifferential signal this calculating is very complicated, thus causes " differential blast " problem, reduces the real-time of system.Second nonlinear Nonlinear Tracking Differentiator head it off can be adopted.Theorem 1 can be obtained according to above-mentioned design process.

Theorem 1 is for the nonaffine uncertain system of satisfied hypothesis 1 ~ 6, Controller gain variations becomes the form of (32), the parameter update law that employing formula (33) describes and shape are such as formula (14), (17), (21) and (27) virtual controlling rule, if starting condition is then there is parameter c _i, c _n+1, c, χ and , closed-loop system is met:

(1) state x and output y (t) of system will remain at set: with namely the constraint condition of system can not be run counter to;

(2) all signals of closed-loop system are all half overall ultimately uniform boundary, and tracking error is also ultimate boundness.

Prove: in traditional " cancellation pusher " method for designing, in order to cancellation exists in do not wish the z that sees _nz _n+1, need to define V _n+1for V _nwith and the quadratic sum of the evaluated error that may exist, like this at V _n+1in just there will be two Nussbaum functions.Lemma 1 ~ 2 just can not be utilized to process if there is two Nussbaum functions.For this reason, in stability proof procedure, we adopt the method for " decoupling zero pusher ".

First to n in Step1 ~ n the following lyapunov energy function of sub-Systematic selection:

$V_n=\underset{i=1}{\overset{n}{\&Sigma;}}\frac{1}{2}{z}_i^2---\left(35\right)$

Consideration formula (16), (19), (23) and (31), can obtain formula (35) differentiate:

According to formula (27) and (28) and in conjunction with inequality ab≤(1/4) a ²+ b ², formula (36) can be write as further:

In formula λ ₁=min (2c _i, 2c _n-2), i=1 ..., n-1.Make δ ₀=V _n(0)+χ/λ ₁, formula (37) both sides are multiplied by simultaneously and can obtain at [0, t] interior integration:

$V_n\left(t\right)\&le;{\mathrm{\&delta;}}_0+e^{-{\mathrm{\&lambda;}}_{1}t}{\&Integral;}_0^t\left(\stackrel{\&OverBar;}{\mathrm{\&beta;}}N\right({\mathrm{\&kappa;}}_0)+1){\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_0{e}^{{\mathrm{\&lambda;}}_1\mathrm{\&tau;}}d\mathrm{\&tau;}+Z---\left(38\right)$

Wherein if there is no Ζ item, can obtain according to lemma 1: at finite time [0, t _f) in, V _nt () and κ, as z _iwith be all bounded, and by lemma 2, know t _f=∞, therefore this bounded is uniform ultimate bounded.But due to the existence of Ζ item, lemma 1 can not directly be used, if but z _n+1can be calmed, and finite time [0, t _f) interior bounded, then Ζ item meets as lower inequality:

$Z\&le;e^{-{\mathrm{\&lambda;}}_{1}t}\underset{\mathrm{\&iota;}\&Element;\&lsqb;0,t\&rsqb;}{sup}{z}_{n+1}^2\frac{{\mathrm{\&beta;}}_m^2}{4}{\&Integral;}_0^t{e}^{{\mathrm{\&lambda;}}_1\mathrm{\&iota;}}d\mathrm{\&iota;}\&le;\frac{{\mathrm{\&beta;}}_m^2}{8}\underset{\mathrm{\&iota;}\&Element;\&lsqb;0,t\&rsqb;}{sup}{z}_{n+1}^2---\left(39\right)$

Obvious Ζ bounded, formula (38) can be described as:

$V_n\left(t\right)\&le;{\mathrm{\&lambda;}}_0+e^{-{\mathrm{\&lambda;}}_{1}t}{\&Integral;}_0^t\left(\stackrel{\&OverBar;}{\mathrm{\&beta;}}N\right({\mathrm{\&kappa;}}_0)+1){\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}}_0{e}^{{\mathrm{\&lambda;}}_1\mathrm{\&tau;}}d\mathrm{\&tau;}---\left(40\right)$

λ in formula ₀=V _n(0)+δ/λ ₁+ Ζ, therefore according to lemma 1, z _i(i=1 ..., n) and V _nboundedness also can be guaranteed.The situation of present consideration (n+1)th step, select following lyapunov energy function:

$V_{n+1}=\frac{1}{2}{z}_{n+1}^2---\left(41\right)$

Consideration formula (34), can obtain formula (41) differentiate:

${\stackrel{\&CenterDot;}{V}}_{n+1}=z_{n+1}{\stackrel{\&CenterDot;}{z}}_{n+1}=-c_{n+1}{z}_{n+1}^2+\left(\mathrm{\&eta;}N\right(\mathrm{\&kappa;})+1)\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}\&le;-{\mathrm{\&lambda;}}_2{V}_{n+1}+\left(\mathrm{\&eta;}N\right(\mathrm{\&kappa;})+1)\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}---\left(42\right)$

λ in formula ₂≤ 2c _n+1.Similarly, can obtain:

$V_{n+1}\left(t\right)\&le;\stackrel{\&OverBar;}{{\mathrm{\&lambda;}}_0}+e^{-{\mathrm{\&lambda;}}_{2}t}{\&Integral;}_0^t\left(\mathrm{\&eta;}N\right(\mathrm{\&kappa;})+1)\stackrel{\&CenterDot;}{\mathrm{\&kappa;}}{e}^{{\mathrm{\&lambda;}}_2\mathrm{\&iota;}}d\mathrm{\&iota;}---\left(43\right)$

In formula according to lemma 1, easily know V _n+1and z _n+1the closed loop solution of subsystem is all uniform ultimate bounded.Due to z _n+1bounded, backward recursion 1 step, the z in known Step1 ~ n _i(i=1 ..., n) and V _nboundedness be guaranteed, so n subsystem closed loop solution is also bounded in Step1 ~ n, and system (8) tracking error meet as lower inequality:

$\frac{1}{2}\left|\right|z_1|{|}^2\&le;V_1\&DoubleRightArrow;|\left|e\right||\&le;\sqrt{2V_n}---\left(44\right)$

Yi Zhi makes tracking error z by adjustment controller parameter ₁little as much as possible i.e. z ₁→ 0, to reach desired tracking accuracy.

Character according to Nonlinear Mapping Η can obtain:

(1) by formula (37), (40) and lemma 1 ~ 2, known κ and uniform bound, according to formula (5) and (6) known, state x _iremain at set in.

(2) due to z ₁→ 0, system (1) tracking error e=x can be obtained according to lemma 3 ₁-y _ralso e → 0 is met.Card is finished.

Claims (5)

1. the nonaffine uncertain system self-adaptation control method of an Operations of Interva Constraint, under the condition meeting engineering practice, nonaffine uncertain system is supposed, the method realizes the control to nonaffine uncertain system based on technology such as mean value theorem, adaptive control and invertings, it is characterized in that, comprise the following steps:

(1) under the assumed condition meeting engineering practice and the prerequisite of not losing the important nonlinear characteristic of system, based on mean value theorem, Non-Affine Systems is converted into the time-varying system (tight feedback system) with linear structure, and coupling system state and input are by the characteristic of Operations of Interva Constraint, become uncertain parameter constant interval when obtaining time-varying system based on lagrange's method of multipliers;

(2) adopt parameter adaptive shadow casting technique to carry out On-line Estimation to becoming uncertain parameter during bounded, parameter estimating error adopts Nonlinear Dynamic damping to compensate;

(3) adopt non-linear mapping techniques amount of restraint to be mapped to whole real number space and carry out the problem for the treatment of system state by Operations of Interva Constraint, adopt the input of hyperbolic tangent function and Nussbaum gain techniques disposal system by Operations of Interva Constraint problem and controller singular value problem simultaneously;

(4) on the basis of first three step, utilize and according to method of inversion CONTROLLER DESIGN, nonaffine uncertain system is controlled.

2. the nonaffine uncertain system self-adaptation control method of the Operations of Interva Constraint described by claim 1, is characterized in that, Non-Affine Systems assumed condition and the affine method for transformation of described step (1) are:

A: nonaffine uncertain system assumed condition is: the state of system, input and uncertain parameter are by Operations of Interva Constraint;

In formula, x=[x ₁, x ₂..., x _n] ^t∈ R ⁿconstraint condition is met for state vector: | x _i| < l _i, l _ifor known normal number; Y is that system exports; F () is all uncertain nonlinear function of structure and parameter; d ₀t indeterminate that () causes for external interference; V is actuator input; U (ν) is the output by actuator saturation properties influence; The specific descriptions of constraint function u (ν) are as follows;

U in formula _mfor constraint function boundary value, it is known normal number.The assumed condition of system is:

Suppose that 1 nonlinear function f () can lead continuously about x and u, and partial derivative bounded;

Suppose the uncertain parameter bounded of 2 system, and be positioned at known boundary, or the uncertain parameter unbounded of system, but it is realized, as sin cos functions etc. by limited function the impact of system;

Though suppose that 3 system architectures are uncertain, potential all possible structure uncertain factor can be predicted;

Suppose 4 indeterminate d ₀t () meets: | d ₀(t) |≤d ₀, d ₀for known normal number;

Suppose 5f () initial point place f (0,0 ..., 0) and meaningful, and | f (0,0 ..., 0) |≤Δ ₀, Δ ₀for known normal number;

Suppose 6 reference signal y _rt () remains in confining region, namely to arbitrary l ₁, there is positive number l in > 0 _r< l ₁, have | y _r(t) |≤l _r;

B: Non-Affine Systems affine linear method for transformation is:

Utilize mean value theorem can be by be described as:

Can be by system converting further:

In formula

i＝1,…,n

β(t)≠0

γ _i, i=1 ..., n is between 0 and x _ibetween certain value, γ _ufor certain value between 0 and u.F (0), b _iwith β be respectively f (0 ..., 0), b _it () and β (t) write a Chinese character in simplified form;

C: become during time-varying system uncertain parameter constant interval acquisition methods into:

A: according to hypothesis 3, although though the structure of system exists uncertain, but because its potential all possible structure uncertain factor can be predicted, therefore mathematical modeling can all be carried out to set up systems most model to often kind of structure uncertain condition, simultaneously because the energy of system is limited, so the parameter of often kind of model is known or unknown bounded;

B: according to the known time-varying parameter b of the character of mean value theorem _it () and β (t) are the function of x and u, i.e. b _it () and β (t) can be expressed as: b _i(t)=f _bi(x, u (ν)) and β (t)=f _β(x, u (ν)), according to hypothesis 1 ~ 2, function f _bi() and f _βthe equal bounded of each parameter in ();

C: because the state x of system is all tied, so b in often kind of mathematical model with input u _it between the confining region of () and β (t), Solve problems is and asks extreme-value problem under multivariate function constraint condition, can solve Extreme Value Problem of Multi-Variable Functions under constraint condition based on lagrange's method of multipliers;

D: getting the b of often kind of mathematical model _iafter between the confining region of (t) and β (t), find out b between wherein maximum confining region _i∈ (b _imin, b _imax) and β ∈ (β _min, β _max), be time-varying parameter b in system operation _ibetween the confining region that may travel through of (t) and β (t), between this confining region, meet the parameter in system cloud gray model and structural uncertainty requirement in theory.

3. the nonaffine uncertain system self-adaptation control method of the Operations of Interva Constraint described by claim 2, is characterized in that, become during the bounded of described step (2) uncertain parameter method of estimation into:

Utilize during self-adaptation shadow casting technique On-line Estimation bounded and become uncertain parameter, specific as follows:

represent the estimated value of θ, represent parameter estimating error.Γ is parameter identification speed diagonal matrix, Ω _θ∈ [θ _min, θ _max].

4. the nonaffine uncertain system self-adaptation control method of the Operations of Interva Constraint described by claim 3, is characterized in that, state and the affined disposal route of input of described step (3) are:

A: adopt non-linear mapping techniques to be unconfined variable by affined condition conversion, finally obtain the unfettered Controlling model of following state:

In formula, its concrete conversion process is: definition maps one by one as follows:

Obvious Η is continuous elementary function, can obtain:

Differentiate, can obtain:

B: adopt the input of hyperbolic tangent function and Nussbaum gain techniques disposal system by the method for Operations of Interva Constraint and controller singular value to be:

u(v)＝sat(v)＝h(v)+d _v(t)

In formula, h (v) is hyperbolic tangent function, d _vt () is limited function;

|d _v(t)|≤u _M(1-tanh(1))

So constraint function can be equivalent to: sat (v)=h (ν)+d _v(t), wherein d _vt () can be considered external disturbance, therefore, compound disturbance d (t) of system is final for being expressed as:

d(t)＝d ₀(t)+βd _v(t)

According to boundedness and the d of β _vthe boundary value of (t), the composite interference of known system (8) still meets hypothesis 4, namely | and d (t) |≤d, d are known normal number.

5. the nonaffine uncertain system self-adaptation control method of the Operations of Interva Constraint described by claim 4, is characterized in that, obtains controller and controls nonaffine uncertain system, be specially in described step (4):

A: first sub-system error is defined as by i-th (i=1 ..., n) sub-system error is defined as (n+1)th sub-system error is defined as z _n+1=h (v)-α _n, then in conjunction with method of inversion design adaptive controller.

B: in the n-th step, adopts Nussbaum gain techniques processing controller singular value problem;

C: adopt hyperbolic tangent function and the process of Nussbaum gain techniques to input affined problem in the (n+1)th step, finally, utilize non-linear differential tracking technique, calculate the differential signal of virtual controlling amount, to solve differential explosion issues, thus the control of the nonaffine uncertain system to Operations of Interva Constraint can be realized.