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

A control method for a single-link robotic arm embedded with a DC motor

Technical Field

The present invention relates to the field of control of a single-link robot arm with an embedded DC motor, and in particular to a control method for a single-link robot arm with an embedded DC motor based on a high-order fixed-time disturbance observer and a high-order sliding mode. The method utilizes a high-order sliding mode algorithm with disturbance compensation and a fixed-time stable control using a multiple power integrator technology to improve the predictability, robustness, adaptability and precise control characteristics of the single-link robot arm with an embedded DC motor.

Background Art

Embedded DC motor single-link robotic arms are widely used in the field of industrial automation. In order to achieve precise and stable control, researchers are constantly exploring various advanced control methods. Fixed-time disturbance observer and high-order sliding mode control technology have aroused great interest in the field of robotic arm control due to their excellent robustness and adaptability. This paper will introduce this embedded DC motor single-link robotic arm control method based on fixed-time disturbance observer and high-order sliding mode, and explore its potential application in solving real-world control problems. One of the key challenges of robotic arm control is to achieve accurate trajectory tracking and stable motion control. Traditional PID control and other methods often have difficulty in dealing with problems such as nonlinearity, time-varying and external disturbances, which limits the performance and control effect of the robotic arm system; especially in the presence of external disturbances and parameter changes, the control system is more likely to lose stability and performance.

Fixed-Time Disruption Observer (FTDO) is an advanced control technique used to estimate disturbances in a system. It does not rely on the system model and can estimate the size and impact of disturbances online, allowing the control system to remain stable under the influence of disturbances. By introducing a fixed-time disturbance observer in the controller, the system can correct the impact of disturbances on the control effect in real time, improving the robustness of the control system.

Higher-Order Sliding Mode Control (HOSM) is a control strategy that applies sliding mode transformation, aiming to guide the system state to the sliding surface and achieve accurate tracking and control of the system state. High-order sliding mode control has good performance in dealing with nonlinear and time-varying systems, and is of great significance for systems that require high-precision motion control, such as robotic arms.

Summary of the invention

Purpose of the invention: In view of the problems mentioned in the background technology, the present invention provides a control method for a single-link manipulator with an embedded DC motor, which can improve the control accuracy and predictability of the single-link manipulator with an embedded DC motor, weaken the system uncertainty caused by the unknown angular position, and enhance the anti-interference ability of the system. The nonlinear fixed-time disturbance observer is combined to suppress interference, the high-order sliding mode control strategy is used to weaken the influence of jitter and improve the control accuracy, and the method of using the tangent barrier Lyapunov function on the basis of the backstepping-like derivation controller ensures the stable convergence of the angle, angular velocity and other constraints and fixed time in the movement of the manipulator, so as to further improve the robust performance of the single-link manipulator with an embedded DC motor.

Technical solution: A control method of a single-link mechanical arm embedded with a DC motor of the present invention comprises the following steps:

Step 1: Consider the output-constrained expansion to obtain the high-order sliding mode embedded DC motor single-link manipulator dynamic equation with output constraints and non-matching terms;

Step 2: The anti-interference characteristics of the system are improved by adopting the power integration method of multi-layer integration. Combined with the high-precision control requirements of the system, an n-order fixed-time interference observer is constructed to estimate the non-matching interference.

Step 3: Recursively design the virtual controller and actual control input using a backstepping-like method;

Step 4: Construct a tangent barrier Lyapunov function that satisfies the output constraint and derive it. Substitute the virtual controller and actual control input in step 3 into the derivative of the tangent barrier Lyapunov function to verify whether the control law can make the closed-loop system converge asymptotically. If so, continue with step 4. If not, return to step 3 to redesign the virtual controller and actual control input.

Step 5: Perform Lyapunov second method stability analysis on the closed-loop system to prove that the output constraints are not violated and obtain a deterministic expression for a fixed time to ensure that the control performance indicators meet the design requirements.

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

Among them, s _i ∈R, i = 1, ..., n is the output, that is, the sliding mode variable, u ∈ R is the control input; p ₀ (t, x) and q ₀ (t, x) are unknown continuous differentiable functions; x is the system state, t is the time, _fi (s _i ), i = 1, ..., n-1 is the non-matching term, c _i (t), i = 1, ..., n-1 is the non-matching disturbance term, and the relative order of the sliding mode variable s _i relative to the control input u is n, that is,

The system model is rewritten as

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

Assumption 1: There exists a known positive constant and known positive definite functions The following conditions are established:

Assumption 2: The derivatives of the non-matching term _fi (s _i ) and the non-matching disturbance term is bounded, that is, we can find a bounded derivative function ρ _i (s _i )＜M,M＞0 and a constant N such that:

in n is the system order;

Assumption 3: In addition, the output _si satisfies a constraint:

|s _i |＜Δ,i＝1,...,n.

Where Δ＞0 is a positive constant greater than zero.

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

in, is the observed state, s _i ∈R, i＝1,...,n is the output, i.e., the sliding mode variable, is the estimated value of the disturbance, v _i ,i＝1,...,n-1 is the intermediate variable of the observer, L and θ are the observer gains, and the order α _i satisfies the following relationship: is an arbitrarily small constant;

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

is Hurwitz's; and the fixed time estimate is:

in, m＝α-1，

λ _min (Q ₁ )＞0 is the minimum eigenvalue of the matrix Q ₁ , 0＜γ≤λ _min (P ₁ ), Q ₁ ,Q ₂ ∈R ^n×n are symmetric positive definite matrices, and the matrices P ₁ ,P ₂ satisfy the following equations:

So the fixed-time form of the first observer is:

in

The fixed-time form of the second observer is:

in

Recursively speaking, the fixed time form of the n-1th observer is:

in

Let the error of the observer variable and its derivative be:

Then the observer estimation error system has the following form

The fixed-time accumulation of n-1 observers yields:

Furthermore, the step 3 of designing a virtual controller and actual control input includes the following steps:

Step 3.1: Select a Lyapunov function that satisfies the output constraint. Its expression is as follows:

Where p, r ₁ , τ are real numbers satisfying p ≥ r ₁ > 0 and τ > 0, the Lyapunov function V ₁ (s ₁ ) is defined in the region D ₁ = {s ₁ : |s ₁ | < Δ}, Δ is a positive constant greater than zero;

Step 3.2: Based on the observation error Perform coordinate transformation on the system model to obtain a new system model:

Among them, s _i ∈R, i = 1, ..., n is the output, i.e., the sliding mode variable, u ∈ R is the control input; p ₀ (t, x) and q ₀ (t, x) are unknown continuous differentiable functions; x is the system state, t is the time, _fi (s _i ), i = 1, ..., n-1 is the non-matching term, and c _i (t), i = 1, ..., n-1 is the non-matching disturbance term;

A discontinuous HOSM controller is designed to stabilize the system in fixed time. Since the observation error will eventually converge to zero within the fixed time T ₀ , The system is converted into the following form:

(33) Apply the following virtual control (3) (4) and actual control input (5) to ensure that the system signal is semi-globally uniformly bounded under the condition that the closed-loop system output is limited, and the control performance satisfies the fixed time stability T _max :

Among them, s ₂ ^* is the first virtual controller, is the kth virtual controller, u is the control input, l, μ are positive constants greater than 0, a satisfies the condition p≥a≥r ₁ ; β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ is a continuously differentiable function, ρ ₁ is a bounded derivative function, β _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v +(n-1+k)l is a continuously differentiable function;

in, β _n ≥c _n1 +c _n2 +c _n3 +l, is a continuously differentiable function, γ _n is a positive constant, q ₀ is a positive constant, is a positive definite function;

is a fixed constant,

n is the system order.

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

in, K is the integration variable, w _i is the i-th power integrator, and the strong stability characteristics of the controller are achieved by superimposing i power integrators.

Beneficial effects:

The present invention introduces non-matching items to solve the problem that the traditional high-order sliding mode transfers uncertainty to the control channel through the direct derivative method, thereby causing the control gain to be too large; considering the non-matching disturbance and introducing the fixed time disturbance observer (FTDO) to estimate the disturbance can enhance the system's resistance to external disturbances, adapt to more common working conditions and improve the robustness of the control system;

The present invention solves the problems of actuator saturation and system instability that are easy to occur in a single-link mechanical arm with an embedded DC motor. The actuator is constrained by output limitation and the output constraint is ensured not to be violated. High-order sliding mode control is adopted to optimize the control performance by designing more control parameters to achieve higher control accuracy.

In order to reduce the impact of the jitter of the second-order sliding mode control on the system, the present invention adopts a multi-layer integral power integration method to alleviate the jitter problem of the low-order sliding mode control to improve the anti-interference ability, thereby achieving precise control of the state of the single-link robotic arm system embedded with a DC motor and weakening the serious jitter caused by the system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a perspective view of a single-link flexible mechanical arm with an embedded DC motor according to the present invention;

FIG2 is a schematic diagram of a single-link flexible mechanical arm controlled by a DC motor according to the present invention;

FIG3 is a flowchart of the implementation steps of the present invention.

DETAILED DESCRIPTION

The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

In this embodiment, the circuit schematic diagram of a single-link flexible mechanical arm with an embedded DC motor is taken as an example to design a controller, and the dynamic model of the system is obtained as follows:

Among them, u(t) represents the input voltage, i(t) represents the armature current, u ^* (t) represents the armature voltage, _Q1 (t)∈R represents the angular position of the connecting rod, _Q2 (t)∈R represents the angular position of the motor shaft, m represents the mass of the connecting rod, d represents the position of the center of gravity of the connecting rod, g represents the acceleration of gravity, _J1 represents the moment of inertia of the connecting rod, _J2 represents the moment of inertia of the motor shaft, L represents the inductance, R represents the armature resistance, _F1 represents the viscous friction parameter of the connecting rod, _F2 represents the viscous friction parameter of the motor shaft, K is the spring parameter, _Kb is the back electromotive force constant, _Kt is the torque constant, and N is the gear ratio.

Select reasonable sliding variables and take derivatives, and combine the high-order sliding mode dynamics form of the non-matching terms of the lower triangular structure to obtain the following dynamic equation:

Combined with the conditions of Assumption 2, the following function can be selected:

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

The control method of a single-link manipulator with an embedded DC motor based on a fixed-time disturbance observer and a high-order sliding mode according to the present invention is adopted, and the specific steps are as follows:

Consider the following uncertain nonlinear high-order sliding mode system model with unmatched disturbances:

In the formula, s _i ∈R, i = 1, ..., n is the output (sliding mode variable), u ∈R is the control input; p ₀ (t, x) and q ₀ (t, x) are unknown smooth functions; c _i (t), i = 1, ..., n-1 is the non-matching term, and c _i (t), i = 1, ..., n-1 is the non-matching disturbance term. All sliding mode variables are restricted to an open set D _i = {s _i : |s _i | < Δ}, Δ > 0, which is a positive constant greater than zero, where i = 1, 2, ..., n-1. Since the observation error will eventually converge to zero within a fixed time T ₀ .

The control objective of the present invention is to design a control scheme based on a multiple power integral method, a fixed time disturbance observer and an output constraint for a nonlinear high-order sliding mode system (4) with non-matching disturbances and output constraints, so that the control system satisfies the following control objectives:

Objective 1: All sliding mode variables of the system do not violate the output constraints, that is, ensure that |s _i |＜Δ, i＝1,2,...,n-1, Δ＞0, which is a positive constant greater than zero.

Objective 2: The fixed-time disturbance observer compensates for the external disturbance of the system to ensure that the observer error system can converge to 0 within a fixed time. Multiple power integrators are superimposed to improve the stability of the constructed fixed-time controller.

Objective 3: Based on the Lyapunov stability theorem, prove that all closed-loop system variables are ultimately uniformly bounded, so that the n sliding mode variables converge to 0.

In order to achieve the above control objectives, the following assumptions are imposed on system (3). The nonlinear high-order sliding mode system with output constraints has the following assumptions:

Assumption 1: There exists a known positive constant and known positive definite functions The following conditions are established:

Assumption 2: The derivatives of the non-matching term _fi (s _i ) and the non-matching disturbance term is bounded, that is, we can find a bounded derivative function ρ _i (s _i )＜M,M＞0 and a constant N such that:

in n is the system order.

Assumption 3: In addition, the output _si satisfies a constraint:

|s _i |＜Δ,i＝1,...,n.

Where Δ＞0 is a positive constant greater than zero.

The form of the designed disturbance observer is as follows:

in, is the observed state, s _i ∈R, i＝1,...,n is the output, i.e., the sliding mode variable, is the estimated value of the disturbance, v _i ,i＝1,...,n-1 is the intermediate variable of the observer, L and θ are the observer gains, and the order α _i satisfies the following relationship: is an arbitrarily small constant.

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

is Hurwitz's; n is 5, which meets the actual control requirements; and the fixed time estimate is:

in, m＝α-1， λ _min (Q ₁ )＞0 is the minimum eigenvalue of the matrix Q ₁ , 0＜γ≤λ _min (P ₁ ), Q ₁ ,Q ₂ ∈R ^n×n are symmetric positive definite matrices, and the matrices P ₁ ,P ₂ satisfy the following equations:

So the fixed-time form of the first observer is:

in

The fixed-time form of the second observer is:

in

Recursively speaking, the fixed time form of the n-1th observer is:

in

Let the error of the observer variable and its derivative be

Then the observer estimation error system has the following form

The fixed-time accumulation of n-1 observers yields:

Conclusion 1: Applying the following virtual control (6) (7) and actual control input (8) ensures that the system signal is semi-globally uniformly bounded under the condition that the closed-loop system output is limited, and the control performance satisfies the fixed time stability T _max :

Among them, s ₂ ^* is the first virtual controller, is the kth virtual controller, u is the control input, l, μ are positive constants greater than 0, a satisfies the condition p≥a≥r ₁ ; β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ is a continuously differentiable function, ρ ₁ is a bounded derivative function, β _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v +(n-1+k)l, is a continuously differentiable function.

in, β _n ≥c _n1 +c _n2 +c _n3 +l, is a continuously differentiable function, γ _n is a positive constant, q ₀ is a positive constant, is a positive definite function; is a fixed constant, n is the system order.

The following is a specific proof process of combining the multiple power integration technique with the fixed-time disturbance observer to prove that the closed-loop system tracking control performance satisfies the fixed-time stability for an uncertain nonlinear high-order sliding mode system model with output constraints and non-matching terms.

Step 1: For a high-order sliding mode system with n-1 non-matching disturbances, the fixed-time form of the observer observing the first disturbance is:

in The fixed-time form of the observer observing the second disturbance is:

in By analogy, the fixed-time form of the observer observing the n-1th disturbance is:

in

According to the fixed time expressions of n-1 observers, the cumulative observer fixed time expression is obtained

Step 2: Select the Lyapunov function

Taking the time derivative of V ₁ (s ₁ ) yields

in s ₂ ^* is the virtual control law to be designed. According to the assumption,

At the same time, define The virtual controller can then be designed as

Among them, β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ is a smooth function. Combining the above, we can get

Step 3: Select the Lyapunov function

in, K is the integration variable, w _i is the i-th power integrator, and the strong stability of the controller is achieved by superimposing i power integrators. Then define the variable Differentiating _V2 along the system yields:

Next, we will estimate the and Three items; first, according to the lemma Available

Substitution And apply the lemma Get the first item Estimates:

in, is a fixed constant.

Continue to estimate the next item for have

Notice as well as so can be estimated as

There are and so

in

Combining (19) and (20), we have

in Combining (18) and (21), we get the second term The final expression of is:

in Estimate the last item have

Therefore

According to Assumption 2, and using Lemma get

in

Combining (23) and (24), we have

Substituting the three estimates into (17)(22)(25), we get:

Using Virtual Controllers Eliminating the remainder, we can get

Step k: Select the Lyapunov function

Similar to the first step, differentiating along the system yields

The three expressions are estimated in the same way. I will not go into details here. The first term is:

Item 2:

Item 3:

Substitute (30)(31)(32),

The k+1th virtual control law is:

Substituting into the virtual control law and eliminating the remainder, we can obtain:

Step n:

Differentiate along the system function

in,

Based on the idea of backstepping, the unknown terms are eliminated to obtain a fixed-time stable form, which is the actual control input form as follows:

After bringing in

Also because

in

Further, we get The expression

and The expression

Finally, according to the fixed time stable form, we can get

in

Get the expression of fixed time

From the fixed-time stability theorem, we know that if there exists a continuous positive definite function V(x):R ⁿ →R ₊ ∪{0} that satisfies:

Where α, β, p, q are all positive numbers, and satisfy p＜1, q＞1, then the original system is fixed-time stable, and the convergence time T(x ₀ ) satisfies:

Considering the existence of the fixed-time disturbance observer, the fixed-time expression of the whole system is obtained:

In order to prove that system (4) can be stable in fixed time, convergence analysis is required; under the condition of assumption 2, and ρ(s ₁ )＜M, we can see that the non-matching term f ₁ (s ₁ ) and the non-matching disturbance term c _i (t), i＝1,...,n-1 are bounded in the time interval [0,T ₀ ]. Since s ₁ ,...,s _n-1 , ρ(s ₁ ) are all bounded, it is not difficult to obtain the virtual controller β ₁ ,...,β _k in and c _n1 , c _n2 , c _n3 in the control input u are also bounded.

Assume that x ₁ ,...,x _n can extend infinitely in time, s ₁ (t,x),...,s _n-1 ( _t ,x) are functions of t and x, and by designing a suitable sliding surface s ₁ ,...,s n _{- 1} ,p ₀ (t,x),q ₀ (t,x) are uniformly bounded on t≥T ₀ .

Therefore, we need to prove that when t∈[0,T ₀ ], is bounded; when t∈[0,T ₀ ], there exist some positive constants _ka , k _b , k _c , k _d , k _e , k _f , k _g such that s ₁ ,...,s _{n-1 ≤ka} , 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.

Constructor Helper Function Taking the derivative of Q(s) along the system, we have

in

in because and are the observed values of s ₁ ,...,s _n-1 and c ₁ ,...,c _n-1 , and the observation errors e ₁ ,...,en _-1 converge to 0 within a fixed time T _0. Obviously, e ₁ ,..., _en-1 is also bounded in t∈[0,T ₀ ]:

|e _i |≤|e _{i max} |≤e _max ,i＝1,...,n-1,e _max ＞0. Based on the boundedness of the observer error (taking the observer estimating the first disturbance as an example), we can get

Similarly, for an n-order system, we have

Assumptions get

Q(s) is estimated as

in

Is a normal number.

On the other hand, if Then there is Where D is a normal quantity. Combining (38) we can get Solving the differential equation is

so is bounded, which means that the state of system (4) will not escape in fixed time within fixed time T _0. It also shows that the actual system (3) can be fixed time stable under the action of n-1 observers and controller (41).

The present invention studies a control method for a single-link manipulator with an embedded DC motor based on a fixed-time disturbance observer and a high-order sliding mode. A fixed-time disturbance observer is used to estimate the disturbance in real time under conditions of non-matching disturbance and system uncertainty, and the tracking error meets the performance index within a fixed time. By superimposing multiple power integrators and combining the tangent barrier Lyapunov function with a high-order sliding mode algorithm, a high-order sliding mode is used to further weaken the influence of jitter on the system under the condition of satisfying the output constraints, thereby improving the predictability and precise control capability of the manipulator control system and enhancing the robustness and adaptability of the system.

The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with the technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. Any equivalent transformation or modification made according to the spirit of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A control method for a single-link mechanical arm embedded with a DC motor, characterized in that it comprises the following steps: Step 1: Consider the output-constrained expansion to obtain the high-order sliding mode embedded DC motor single-link manipulator dynamic equation with output constraints and non-matching terms; Step 2: The anti-interference characteristics of the system are improved by adopting the power integration method of multi-layer integration. Combined with the high-precision control requirements of the system, an n-order fixed-time interference observer is constructed to estimate the non-matching interference. Step 3: Recursively design the virtual controller and actual control input using a backstepping-like method; Step 4: Construct a tangent barrier Lyapunov function that satisfies the output constraint and derive it. Substitute the virtual controller and actual control input in step 3 into the derivative of the tangent barrier Lyapunov function to verify whether the control law can make the closed-loop system converge asymptotically. If so, continue with step 4. If not, return to step 3 to redesign the virtual controller and actual control input. Step 5: Perform Lyapunov second method stability analysis on the closed-loop system to prove that the output constraints are not violated and obtain a deterministic expression for a fixed time to ensure that the control performance indicators meet the design requirements. 2. According to the control method of a single-link manipulator with an embedded DC motor in claim 1, the model of the nonlinear high-order sliding mode system with output constraints and non-matching terms is as follows: Among them, s _i ∈R, i = 1, ..., n is the output, that is, the sliding mode variable, u ∈ R is the control input; p ₀ (t, x) and q ₀ (t, x) are unknown continuous differentiable functions; x is the system state, t is the time, _fi (s _i ), i = 1, ..., n-1 is the non-matching term, c _i (t), i = 1, ..., n-1 is the non-matching disturbance term, and the relative order of the sliding mode variable s _i relative to the control input u is n, that is, The system model is rewritten as 3. A control method for a single-link manipulator with an embedded DC motor according to claim 1, characterized in that the nonlinear high-order sliding mode system with output constraints and non-matching terms has the following assumptions: Assumption 1: There exists a known positive constant and known positive definite functions The following conditions are established: Assumption 2: The derivatives of the non-matching term _fi (s _i ) and the non-matching disturbance term is bounded, that is, we can find a bounded derivative function ρ _i (s _i )＜M,M＞0 and a constant N such that: in n is the system order; Assumption 3: In addition, the output _si satisfies a constraint: |s _i |＜Δ,i＝1,...,n. Where Δ is a positive constant greater than zero. 4. The control method of a single-link mechanical arm embedded with a DC motor according to claim 1, characterized in that the form of the disturbance observer designed in step 2 is as follows: in, is the observed state, s _i ∈R, i＝1,...,n is the output, i.e., the sliding mode variable, is the estimated value of the disturbance, v _i ,i＝1,...,n-1 is the observer intermediate variable, L and θ are the observer gains, and the order α _i satisfies the following relationship: α _i ＝iα-(i-1),i＝2,...,n, is an arbitrarily small constant; κ _i ,ω _i ,i＝1,...,n are chosen so that the matrices: is Hurwitz's; and the fixed time estimate is: in, m＝α-1， λ _min (Q ₁ )＞0 is the minimum eigenvalue of the matrix Q ₁ , 0＜γ≤λ _min (P ₁ ), Q ₁ ,Q ₂ ∈R ^n×n are symmetric positive definite matrices, and the matrices P ₁ ,P ₂ satisfy the following equations: So the fixed-time form of the first observer is: in The fixed-time form of the second observer is: in Recursively speaking, the fixed time form of the n-1th observer is: in Let the error of the observer variable and its derivative be: Then the observer estimation error system has the following form The fixed-time accumulation of n-1 observers yields: 5. The control method of a single-link mechanical arm embedded with a DC motor according to claim 4, characterized in that the step 3 of designing a virtual controller and actual control input comprises the following steps: Step 3.1: Select a Lyapunov function that satisfies the output constraint. Its expression is as follows: Where p, r ₁ , τ are real numbers satisfying p ≥ r ₁ > 0 and τ > 0, and the Lyapunov function V ₁ (s ₁ ) is defined in the region D ₁ = {s ₁ : |s ₁ | <Δ}; Step 3.2: Based on the observation error Perform coordinate transformation on the system model to obtain a new system model: Among them, s _i ∈R, i = 1, ..., n is the output, i.e., the sliding mode variable, u ∈ R is the control input; p ₀ (t, x) and q ₀ (t, x) are unknown continuous differentiable functions; x is the system state, t is the time, _fi (s _i ), i = 1, ..., n-1 is the non-matching term, and c _i (t), i = 1, ..., n-1 is the non-matching disturbance term; A discontinuous HOSM controller is designed to stabilize the system in fixed time. Since the observation error will eventually converge to zero within the fixed time T ₀ , The system is converted into the following form: (33) Apply the following virtual control (3) (4) and actual control input (5) to ensure that the system signal is semi-globally uniformly bounded under the condition that the closed-loop system output is limited, and the control performance satisfies the fixed time stability T _max : Among them, s ₂ ^* is the first virtual controller, is the kth virtual controller, u is the control input, l, μ are positive constants greater than 0, a satisfies the condition p≥a≥r ₁ ; β ₁ ≥l+μ|ξ ₁ | ^v +ρ ₁ is a continuously differentiable function, ρ ₁ is a bounded derivative function, β _k ≥c _k1 +c _k2 +c _k3 +μ|ξ _k | ^v +(n-1+k)l is a continuously differentiable function; in, β _n ≥c _n1 +c _n2 +c _n3 +l, is a continuously differentiable function, γ _n is a positive constant, q ₀ is a positive constant, is a positive definite function; is a fixed constant, n is the system order. 6. A control method for a single-link manipulator with an embedded DC motor according to any one of claims 1 to 5, characterized in that the n-order expression of the Lyapunov function designed in step 3 is as follows: in, K is the integration variable, w _i is the i-th power integrator, and the strong stability characteristics of the controller are achieved by superimposing i power integrators.