CN117506896A - Control method for single-connecting-rod mechanical arm embedded with direct-current motor - Google Patents

Abstract

The invention discloses a control method of a single-link mechanical arm embedded with a direct-current motor, which is used for controlling a single-link mechanical arm system with an output constraint direct-current motor and comprises the following steps: firstly, expanding a traditional low-order sliding mode dynamics system to a high order based on an exponentiation integration technology, and providing a high-order sliding mode algorithm with stable fixed time; constructing a tangential barrier Lyapunov function and a fixed time disturbance observer in consideration of the problems of unknown angular positions of the connecting rod and the motor shaft and limited output voltage of the motor armature; deducing a design controller by using a similar backstepping method, and verifying according to a Lyapunov second method stability criterion and a fixed time stability theoremAnd a deterministic expression of a fixed time is obtained. The invention can ensure that the system state can realize stable convergence independently of the initial condition of the system under the condition of limited output, and the stable time can not infinitely increase the performance index, thereby improving the predictability and accuracy of the mechanical arm control systemControl capability, robustness and adaptive characteristics of the system are enhanced.

Description

Control method for single-connecting-rod mechanical arm embedded with direct-current motor

Technical Field

The invention relates to the field of control of single-link mechanical arms of embedded direct-current motors, in particular to a control method of the single-link mechanical arms of the embedded direct-current motors based on a high-order fixed-time disturbance observer and a high-order sliding mode.

Background

The embedded direct current motor single-link mechanical arm has wide application in the field of industrial automation. In order to achieve accurate and stable control, researchers are continually exploring a variety of advanced control methods. The fixed time disturbance observer and the high order sliding mode control technology are of great interest in the field of mechanical arm control due to their excellent robustness and adaptability. The invention introduces a control method of the embedded direct current motor single-link mechanical arm based on a fixed time disturbance observer and a high-order sliding mode, and discusses potential application of the control method in solving the problem of real control. One of the key challenges in robotic arm control is achieving accurate trajectory tracking and stable motion control. The traditional PID control and other methods are difficult to deal with the problems of nonlinearity, time-varying property, external disturbance and the like, so that the performance and control effect of the mechanical arm system are limited; the control system is more prone to losing stability and performance, especially in the presence of external disturbances and parameter variations.

A Fixed time disturbance observer (Fixed-Time Disruption Observer, FTDO) is an advanced control technique for estimating disturbance variables in a system. It is independent of the system model and enables an online estimation of the magnitude and influence of the disturbance, allowing the control system to remain stable under the influence of the disturbance. By introducing a fixed-time disturbance observer into the controller, the system can correct the influence of disturbance on the control effect in real time, and the robustness of the control system is improved.

High order sliding mode control (HOSM) is a control strategy that applies sliding mode transformations aimed at directing the system state onto the sliding mode face, enabling accurate tracking and control of the system state. The high-order sliding mode control has good performance in the aspect of processing nonlinear and time-varying systems, and has important significance for systems such as mechanical arms and the like which need high-precision motion control.

Disclosure of Invention

The invention aims to: aiming at the problems in the background art, the invention provides the control method for the single-link mechanical arm of the embedded direct-current motor, which can improve the control precision and predictability of the single-link mechanical arm of the embedded direct-current motor, weaken the system uncertainty caused by unknown angular positions and strengthen the anti-interference capability of the system. And a method of using a tangent barrier Lyapunov function on the basis of combining a nonlinear fixed time disturbance observer to inhibit disturbance, weakening a high-order sliding mode control strategy to influence buffeting to improve control accuracy and a similar backstepping deducing controller is used for ensuring stable convergence of constraint conditions such as angles and angular speeds in the movement of the mechanical arm and fixed time, so that the robust performance of the mechanical arm embedded with the direct-current motor single connecting rod is further improved.

The technical scheme is as follows: the invention discloses a control method of a single-connecting-rod mechanical arm embedded with a direct-current motor, which comprises the following steps:

step 1: obtaining a dynamic equation of a single-link mechanical arm of the high-order sliding mode embedded direct-current motor with output constraint and non-matching terms by considering output limited expansion;

step 2: the anti-interference characteristic of the system is improved by adopting a multi-layer integral exponentiation integral method, and an n-order fixed time interference observer for estimating non-matching term interference is constructed in combination with the high-precision control requirement of the system;

step 3: recursively designing a virtual controller and an actual control input by using a class backstepping method;

step 4: constructing and deriving a tangent barrier Lyapunov function meeting the output constraint condition, substituting the virtual controller and the actual control input in the step 3 into the derivative of the tangent barrier Lyapunov function, verifying whether the control law can enable the closed-loop system to gradually converge, if so, continuing the step 4, and if not, returning to the step 3 to redesign the virtual controller and the actual control input;

step 5: and carrying out Lyapunov stability analysis on the closed loop system, wherein the stability analysis proves that the output constraint condition is not violated, and a fixed-time deterministic expression is obtained, so that the control performance index is ensured to meet the design requirement.

Further, the model of the nonlinear higher order sliding mode system with output constraints and non-matching terms is as follows:

wherein s is _i E R, i=1..n is the output, i.e. the sliding mode variable, u e R is the control input; p is p ₀ (t, x) and q ₀ (t, x) is an unknown continuous microtransaction; x is the system state, t is the time, f _i (s _i ) I=1,..n-1 is a non-matching term, c _i (t), i=1,.. _i The relative order with respect to the control input u is n, i.e.,

system model rewrite as

Further, a nonlinear high order sliding mode system with output constraints and non-matching terms has the following assumptions:

suppose 1: there is a known normal numberAnd a known positive definite function->So that the following conditions are satisfied:

suppose 2: non-matching term f _i (s _i ) And the derivative of the non-matching perturbation termIs bounded, i.e. a bounded derivative function ρ can be found _i (s _i ) < M, M > 0 and a constant N such that:

wherein the method comprises the steps ofn is the system order;

suppose 3: in addition to the output s _i Satisfying one constraint is:

|s _i |＜Δ,i＝1,...,n。

where Δ > 0, is a positive constant greater than zero.

Further, the form of the disturbance observer designed in the step 2 is as follows:

wherein,is the observed state s _i E R, i=1,..n is the output, i.e. the sliding mode variable,is an estimate of the disturbance, v _i I=1,.. _i The following relationship is satisfied: />Is an arbitrarily small constant;

κ _i ,ω _i i=1,..n is chosen such that the matrices are:

is Hurwitz; and the fixed time estimate is:

wherein,m＝α-1，/>

λ _min (Q ₁ ) > 0 is the matrix Q ₁ Is greater than 0 and less than or equal to lambda _min (P ₁ )，Q ₁ ,Q ₂ ∈R ^n×n Is a symmetrical positive definite matrix, and matrix P ₁ ,P ₂ The following equation is satisfied:

the fixed time form of the first observer is therefore:

wherein the method comprises the steps of

The fixed time form of the second observer is:

wherein the method comprises the steps of

From this recurrence, the fixed time version of the n-1 th observer is:

wherein the method comprises the steps of

Let the observer variable and its derivative have the following errors:

the observer estimation error system has the following form

The fixed time of n-1 observers is accumulated to obtain:

further, the step 3 of designing the virtual controller and the actual control input comprises the following steps:

step 3.1: selecting a Lyapunov function capable of meeting output constraint, wherein the expression is as follows:

wherein p, r ₁ Tau is p.gtoreq.r ₁ Real numbers > 0 and τ > 0, lyapunov function V ₁ (s ₁ ) Defined in the area D ₁ ＝{s ₁ :|s ₁ I < delta, delta is a positive constant greater than zero;

step 3.2: based on the observed errorPerforming coordinate transformation on the system model to obtain a new system model:

wherein s is _i E R, i=1..n is the output, i.e. the sliding mode variable, u e R is the control input; p is p ₀ (t, x) and q ₀ (t, x) is an unknown continuous microtransaction; x is the system state, t is the time, f _i (s _i ) I=1,..n-1 is a non-matching term, c _i (t), i=1,..;

a discontinuous HOSM controller is designed to fix the time stabilization system, since at a fixed time T ₀ The internal observation error will eventually converge to zero, such that forThe system converts to the following form:

(33) The following virtual control (3) (4) and actual control input (5) are applied to ensure that semi-global consistency and limitation of system signals are realized under the condition of limited output of a closed-loop system, and the control performance meets the fixed time stability T _max ：

Wherein s is ₂ ^* For the first virtual controller to be a first virtual controller,for the kth virtual controller, u is a control input, l, mu is a positive constant greater than 0, and a satisfies the condition that p is greater than or equal to a is greater than or equal to r ₁ ；β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ Is a continuous microletterNumber ρ ₁ As a bounded derivative function, beta _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v ++ (n-1+k) l is a continuous microtransaction;

wherein,β _n ≥c _n1 +c _n2 +c _n3 +l, γ as a continuous microcompatible function _n Is a normal number of times, and the number of times is equal to the normal number,q ₀ is of normal number>Is a positive definite function; />

Is a fixed constant->

n is the system order.

Further, the n-order expression of the Lyapunov function designed in the step 3 is as follows:

wherein,k is an integral variable, ">w _i For the ith exponentiation integrator, the strong stability of the controller is achieved by superimposing the i exponentiation integrators.

The beneficial effects are that:

the invention introduces a non-matching item to solve the problem that the traditional high-order sliding mode transfers uncertainty into a control channel through a direct derivative method, so that the control gain is overlarge; considering the unmatched disturbance and introducing a Fixed Time Disturbance Observer (FTDO) estimated disturbance, the resistance of the system to external disturbance can be enhanced, and the robustness of the control system is improved in accordance with more common working conditions;

the invention solves the problems of easy saturation of the actuator, unstable system and the like in the single-link mechanical arm embedded in the direct-current motor, utilizes the output limitation to restrain the actuator and ensures that the output constraint condition is not violated; the high-order sliding mode control is adopted, and the control performance is optimized by designing more control parameters so as to realize higher control precision;

in order to reduce the influence of the shake of the second-order sliding mode control on the system, the invention adopts a multi-layer integral exponentiation integral method to alleviate the shake problem of the low-order sliding mode control so as to improve the anti-interference capability, thereby realizing the accurate control of the system state of the single-link mechanical arm embedded in the direct-current motor and weakening the serious shake situation brought by the system.

Drawings

FIG. 1 is a perspective view of a single-link flexible mechanical arm embedded with a DC motor according to the present invention;

FIG. 2 is a schematic diagram of a single link flexible mechanical arm for controlling a DC motor according to the present invention;

FIG. 3 is a flow chart of the steps performed in the present invention.

Detailed Description

The technical scheme of the invention is further described below with reference to the accompanying drawings.

In this embodiment, taking a schematic circuit diagram embedded in a single-link flexible mechanical arm of a direct-current motor as an example, a controller is designed to obtain a dynamic model of the system as follows:

wherein u (t) represents an input voltage, i (t) represents an armature current, u ^* (t) represents the armature voltage, Q ₁ (t) ∈R represents the angular position of the connecting rod, Q ₂ (t) ∈R denotes the angular position of the motor shaft, m denotes the link mass, d denotes the position of the center of gravity of the link, g denotes the gravitational acceleration, J ₁ Representing moment of inertia of the connecting rod, J ₂ Represents the rotational inertia of a motor shaft, L represents inductance, R represents armature resistance, F ₁ Representing viscous friction parameters of the connecting rod, F ₂ Represents the viscous friction parameter of the motor shaft, K is the spring parameter, K _b Is a back electromotive force constant, K _t Is a torque constant, and N is a gear rotation ratio.

And selecting a reasonable sliding variable and deriving, and combining the higher-order sliding mode dynamics form of the lower triangular structure non-matching item to obtain the following dynamics equation:

in connection with the condition of hypothesis 2, the following function may be chosen:

in order to improve the control accuracy, the invention introduces disturbance quantity on the basis of the system (2), and establishes a state space model containing disturbance as follows:

the method for controlling the single-link mechanical arm of the embedded direct-current motor based on the fixed-time disturbance observer and the high-order sliding mode comprises the following specific steps:

consider the following uncertain nonlinear high order sliding mode system model study with non-matching perturbations:

wherein s is _i E R, i=1,..n is the output (sliding mode variable), u e R is the control input; p is p ₀ (t, x) and q ₀ (t, x) is an unknown smoothing function; c _i (t), i=1,.. _i (t), i=1,... All sliding mode variables are limited to an open set D _i ＝{s _i :|s _i In | < Δ >, Δ > 0, is a positive constant greater than zero, where i=1, 2, & gt, n-1. Due to the fixed time T ₀ The internal observation error will eventually converge to zero.

The control objective of the invention is to design a control scheme based on a multiple exponentiation integral method for a nonlinear high-order sliding mode system (4) with non-matching disturbance and output constraint, and fix a time disturbance observer and the output constraint, so that the control system meets the following control objective:

target 1: all sliding mode variables of the system do not violate the output constraint, i.e. guarantee |s _i I < Δ, i=1, 2,..n-1, Δ > 0, is a positive constant greater than zero.

Target 2: the fixed time disturbance observer compensates external disturbance of the system, so that the observer error system can converge to 0 in fixed time, and the plurality of exponentiation integrators are overlapped, so that the stability of the constructed fixed time controller is improved.

Target 3: according to Lyapunov stability theorem, it is proved that all closed-loop system variables are finally consistent and bounded, so that n sliding-mode variables converge to 0.

To achieve the above control object, the following assumption condition is imposed on the system (3). A nonlinear high order sliding mode system with output constraints has the following assumptions:

suppose 1: there is a known normal numberAnd a known positive definite function->So that the following conditions are satisfied:

suppose 2: non-matching term f _i (s _i ) And the derivative of the non-matching perturbation termIs bounded, i.e. a bounded derivative function ρ can be found _i (s _i ) < M, M > 0 and a constant N such that:

wherein the method comprises the steps ofn is the system order.

Suppose 3: in addition to the output s _i Satisfying one constraint is:

|s _i |＜Δ,i＝1,...,n。

where Δ > 0, is a positive constant greater than zero.

The form of the disturbance observer is designed as follows:

wherein,is the observed state s _i E R, i=1,..n is the output, i.e. the sliding mode variable,is an estimate of the disturbance, v _i I=1,.. _i The following relationship is satisfied: />Is an arbitrarily small constant.

κ _i ,ω _i I=1,..n is chosen such that the matrices are:

is Hurwitz; n takes the value of 5, which meets the actual control requirement; and the fixed time estimate is:

wherein,m＝α-1，/>λ _min (Q ₁ ) > 0 is a momentArray Q ₁ Is greater than 0 and less than or equal to lambda _min (P ₁ )，Q ₁ ,Q ₂ ∈R ^n×n Is a symmetrical positive definite matrix, and matrix P ₁ ,P ₂ The following equation is satisfied:

the fixed time form of the first observer is therefore:

wherein the method comprises the steps of

The fixed time form of the second observer is:

wherein the method comprises the steps of

From this recurrence, the fixed time version of the n-1 th observer is:

wherein the method comprises the steps of

Let the error of observer variable and derivative thereof be

The observer estimation error system has the following form

The fixed time of n-1 observers is accumulated to obtain:

conclusion 1: the following virtual control (6) (7) and actual control input (8) are applied to ensure that the semi-global consistency and the limitation of the system signals are realized under the condition that the output of a closed-loop system is limited, and the control performance meets the fixed time stability T _max ：

Wherein s is ₂ ^* For the first virtual controller to be a first virtual controller,for the kth virtual controller, u is a control input, l, mu is a positive constant greater than 0, and a satisfies the condition that p is greater than or equal to a is greater than or equal to r ₁ ；β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ As a continuously differentiable function ρ ₁ As a bounded derivative function, beta _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v ++ (n-1+k) l is continuousCan be micro-functional.

Wherein,β _n ≥c _n1 +c _n2 +c _n3 +l, γ as a continuous microcompatible function _n Is a normal number of times, and the number of times is equal to the normal number,q ₀ is of normal number>Is a positive definite function; /> Is a fixed constant-> n is the system order.

The following is a specific proof process for combining multiple exponentiation integral technology with a fixed time disturbance observer, and proving that the tracking control performance of a closed-loop system meets the fixed time stability aiming at an uncertain nonlinear high-order sliding mode system model with output constraint and non-matching terms.

The first step: for a high-order sliding mode system containing n-1 non-matching perturbations, the fixed time form of an observer for observing the 1 st perturbation is as follows:

wherein the method comprises the steps ofThe observer fixed time form for observing the 2 nd disturbance is:

wherein the method comprises the steps ofSimilarly, the observer fixed time form for observing the n-1 st disturbance is:

wherein the method comprises the steps of

Obtaining an accumulated observer fixed time expression according to the fixed time expressions of the n-1 observers

And a second step of: selecting Lyapunov function

V taking ₁ (s ₁ ) Is obtained by time derivative of (2)

/>

Wherein the method comprises the steps ofs ₂ ^* Is a virtual control law to be designed, and according to the assumption, the virtual control law can be known,

at the same time, defineThe virtual controller can then be designed to

Wherein beta is ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ Is a smooth function, and can be obtained by combining the above

And a third step of: selecting Lyapunov function

Wherein,k is an integral variable, ">w _i For the ith exponentiation integrator, the strong stability of the controller is achieved by superimposing the i exponentiation integrators. Then define the variable +.>For V ₂ The differential along the system is available:

next, the respective estimates (15) will be madeAnd->Three items; first, according to the quotation->Is available in the form of

Substitution intoAnd apply the lemma->Get the first item->Is estimated by (a):

wherein,is a fixed constant. />

Continuing to estimate the next termFor->Has the following components

Note thatAndTherefore->Can be estimated as

And also hasBut->So that

Wherein the method comprises the steps of

A combination (19) (20) with

Wherein the method comprises the steps ofCombining (18) (21) to obtain the second item +.>Is the final expression of (2): />

Wherein the method comprises the steps ofEstimationLast item->There is->

Therefore there are

According to hypothesis 2 and using the lemmaObtaining

Wherein the method comprises the steps of

A combination (23) (24) with

Substituting (17) (22) (25) three estimates to obtain:

using virtual controllersEliminating the remainder to obtain

And (k) step: selecting Lyapunov function

/>

Similar to the first step, differentiation is performed along the system to obtain

The three expressions are also estimated, and the first term is not described here:

the second item:

third item:

substituting (30) (31) (32),

there is the k+1th virtual control law:

substituting the virtual control law eliminates the remainder to be available,

and (n) step:

differentiating along system functions

Wherein,

according to the backcasting-like concept, eliminating the unknown term results in a fixed time stable form, namely the following actual control input form:

is carried in to obtain

And because of

Wherein the method comprises the steps of

Further, obtainExpression of (2)

Andexpression of (2)

/>

Finally, according to the stable form of the fixed time, the method can obtain

Wherein the method comprises the steps of

Obtaining an expression of a fixed time

From the fixed time stability theorem, it is known that if there is a continuous positive definite function V (x): R ⁿ →R ₊ U {0} satisfies:

wherein alpha, beta, p and q are all normal numbers and p < 1, q > 1, the original system is stable in fixed time, and the convergence time T (x ₀ ) The method meets the following conditions:

taking into account the existence of a fixed-time disturbance observer, a system-wide fixed-time expression is obtained:

in order to prove that the system (4) can be stable for a fixed time, a convergence analysis is required; under the condition of assumption 2, byAnd ρ(s) ₁ ) < M, knowing the non-matching term f ₁ (s ₁ ) And non-matching perturbation term c _i (t), i=1,.. ₀ ]The inner is bounded. Due to s ₁ ,...,s _n-1 ，ρ(s ₁ ) Are all bounded and are not difficult to obtain a virtual controllerBeta of (B) ₁ ,...,β _k And c in control input u _n1 ,c _n2 ,c _n3 Is also bounded.

Let x be ₁ ,...,x _n Can be extended infinitely in time s ₁ (t,x),...,s _n-1 (t, x) is a function of t and x by designing a suitable slip-form surface s ₁ ,...,s _n-1 So that s ₁ ,...,s _n-1 ,p ₀ (t,x),q ₀ (T, x) is greater than or equal to T at T ₀ The upper agreement is bounded.

Therefore, it is necessary to prove that when t.epsilon.0, T ₀ ],Is bounded; when t is E [0, T ₀ ]When there is some positive constant k _a ,k _b ,k _c ,k _d ,k _e ,k _f ,k _g So that s ₁ ,...,s _n-1 ≤k _a ,/> q ₀ (t,x)≤|q ₀ (t,x)|≤k _d ,|β _i (s ₁ )|≤k _e ,i＝1,...,k|β _n (s _n-1 )|≤k _f ,|c _i (t)|≤k _g ,i＝1,...,n-1。

Constructing an auxiliary functionDeriving the Q(s) edge system

Wherein the method comprises the steps of

Wherein the method comprises the steps ofDue to->And->Is s ₁ ,...,s _n-1 C ₁ ,...,c _n-1 And at a fixed time T ₀ Internal observation error e ₁ ,...,e _n-1 Will converge to 0. Obviously e ₁ ,...,e _n-1 At t E [0, T ₀ ]The inner is also bounded:

|e _i |≤|e _{i max} |≤e _max ,i＝1,...,n-1,e _max > 0. Bounded availability of observer-based (for example, observer estimating first disturbance) errors

Similarly, for an n-order system, there are

Assume thatObtain->

Q(s) is estimated as

Wherein the method comprises the steps of

Is a positive constant.

On the other hand, ifThere is->Wherein D is a normal amount. Binding (38) is available->Solving the differential equation into

So thatIs bounded, meaning that the state of the system (4) will not be at a fixed time T ₀ The fixed time escape phenomenon occurs in the system, and meanwhile, the fact that the actual system (3) can be stable in fixed time under the action of n-1 observers and controllers (41) is also described.

The invention researches a single-link mechanical arm control method of an embedded direct current motor based on a fixed time disturbance observer and a high-order sliding mode, the fixed time disturbance observer is used for estimating disturbance in real time under the conditions of non-matching disturbance and uncertainty of a system, and tracking errors meet performance indexes in fixed time.

The foregoing embodiments are merely illustrative of the technical concept and features of the present invention, and are intended to enable those skilled in the art to understand the present invention and to implement the same, not to limit the scope of the present invention. All equivalent changes or modifications made according to the spirit of the present invention should be included in the scope of the present invention.

Claims (6)

1. The control method of the single-connecting-rod mechanical arm embedded with the direct-current motor is characterized by comprising the following steps of:

step 1: obtaining a dynamic equation of a single-link mechanical arm of the high-order sliding mode embedded direct-current motor with output constraint and non-matching terms by considering output limited expansion;

step 2: the anti-interference characteristic of the system is improved by adopting a multi-layer integral exponentiation integral method, and an n-order fixed time interference observer for estimating non-matching term interference is constructed in combination with the high-precision control requirement of the system;

step 3: recursively designing a virtual controller and an actual control input by using a class backstepping method;

step 4: constructing and deriving a tangent barrier Lyapunov function meeting the output constraint condition, substituting the virtual controller and the actual control input in the step 3 into the derivative of the tangent barrier Lyapunov function, verifying whether the control law can enable the closed-loop system to gradually converge, if so, continuing the step 4, and if not, returning to the step 3 to redesign the virtual controller and the actual control input;

step 5: and carrying out Lyapunov stability analysis on the closed loop system, wherein the stability analysis proves that the output constraint condition is not violated, and a fixed-time deterministic expression is obtained, so that the control performance index is ensured to meet the design requirement.

2. The method for controlling a single-link mechanical arm embedded in a direct current motor according to claim 1, wherein a model of a nonlinear high-order sliding mode system with output constraint and non-matching terms is as follows:

wherein s is _i E R, i=1..n is the output, i.e. the sliding mode variable, u e R is the control input; p is p ₀ (t, x) and q ₀ (t, x) is an unknown continuous microtransaction; x is the system state, t is the time, f _i (s _i ) I=1,..n-1 is a non-matching term, c _i (t), i=1,.. _i The relative order with respect to the control input u is n, i.e.,

system model rewrite as

3. The method of claim 1, wherein the nonlinear high-order sliding mode system with output constraint and non-matching term has the following assumption:

suppose 1: there is a known normal numberAnd a known positive definite function->So that the following conditions are satisfied:

suppose 2: non-matching term f _i (s _i ) And the derivative of the non-matching perturbation termIs bounded, i.e. a bounded derivative function ρ can be found _i (s _i ) < M, M > 0 and a constant N such that:

wherein the method comprises the steps ofn is the system order;

suppose 3: in addition to the output s _i Satisfying one constraint is:

|s _i |＜Δ,i＝1,...,n。

where delta is a positive constant greater than zero.

4. The method for controlling the single-link mechanical arm embedded with the direct-current motor according to claim 1, wherein the disturbance observer designed in the step 2 is in the form of:

wherein,is the observed state s _i E R, i=1..n is the output, i.e. the sliding mode variable, +.>Is an estimate of the disturbance, v _i I=1,..The intermediate variables of the detector, L and theta are the gain of the observer, and the times alpha _i The following relationship is satisfied: alpha _i ＝iα-(i-1),i＝2,...,n,/>Is an arbitrarily small constant;

κ _i ,ω _i i=1,..n is chosen such that the matrices are:

is Hurwitz; and the fixed time estimate is:

wherein,m＝α-1，/>

λ _min (Q ₁ ) > 0 is the matrix Q ₁ Is greater than 0 and less than or equal to lambda _min (P ₁ )，Q ₁ ,Q ₂ ∈R ^n×n Is a symmetrical positive definite matrix, and matrix P ₁ ,P ₂ The following equation is satisfied:

the fixed time form of the first observer is therefore:

wherein the method comprises the steps of

The fixed time form of the second observer is:

wherein the method comprises the steps of

From this recurrence, the fixed time version of the n-1 th observer is:

wherein the method comprises the steps of

Let the observer variable and its derivative have the following errors:

the observer estimation error system has the following form

The fixed time of n-1 observers is accumulated to obtain:

5. the method for controlling the single-link mechanical arm embedded in the direct-current motor according to claim 4, wherein the step 3 of designing the virtual controller and the actual control input comprises the following steps:

step 3.1: selecting a Lyapunov function capable of meeting output constraint, wherein the expression is as follows:

wherein p, r ₁ Tau is p.gtoreq.r ₁ Real numbers > 0 and τ > 0, lyapunov function V ₁ (s ₁ ) Defined in the area D ₁ ＝{s ₁ :|s ₁ |＜Δ}；

Step 3.2: based on the observed errorPerforming coordinate transformation on the system model to obtain a new system model:

wherein s is _i E R, i=1..n is the output, i.e. the sliding mode variable, u e R is the control input; p is p ₀ (t, x) and q ₀ (t, x) is an unknown continuous microtransaction; x is the system state, t is the time, f _i (s _i ) I=1,..n-1 is a non-matching term, c _i (t), i=1,..;

a discontinuous HOSM controller is designed to fix the time stabilization system, since at a fixed time T ₀ The internal observation error will eventually converge to zero, such that forThe system converts to the following form:

(33) The following virtual control (3) (4) and actual control input (5) are applied to ensure that semi-global consistency and limitation of system signals are realized under the condition of limited output of a closed-loop system, and the control performance meets the fixed time stability T _max ：

Wherein s is ₂ ^* For the first virtual controller to be a first virtual controller,for the kth virtual controller, u is a control input, l, mu is a positive constant greater than 0, and a satisfies the condition that p is greater than or equal to a is greater than or equal to r ₁ ；β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ As a continuously differentiable function ρ ₁ As a bounded derivative function, beta _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v ++ (n-1+k) l is a continuous microtransaction;

wherein,β _n ≥c _n1 +c _n2 +c _n3 +l, γ as a continuous microcompatible function _n Is a normal number of times, and the number of times is equal to the normal number,q ₀ is of normal number>Is a positive definite function; /> Is a fixed constant-> n is the system order.

6. The method for controlling a single-link mechanical arm embedded in a direct current motor according to any one of claims 1 to 5, wherein the n-order expression of the Lyapunov function designed in the step 3 is as follows:

wherein,k is an integral variable, ">w _i For the ith exponentiation integrator, the strong stability of the controller is achieved by superimposing the i exponentiation integrators.