Adaptive control method for nonlinear systems with input time-delay and state constraints

Abstract

The application provides an adaptive control method for a nonlinear system with input time delay and state constraints, and belongs to the technical field of adaptive control, state constraints and input time delay. The method firstly combines adaptive control with a parameter separation method to process parameter nonlinearity of the system; combines a Pade approximation method with a barrier Lyapunov function (BLF) in a unified framework to solve the uncertainty caused by the input time delay and the state constraints; obtains an adaptive state feedback control strategy based on a backstepping design process and in combination with rigorous stability analysis; and proves that all the signals in the closed-loop system are uniformly ultimately bounded; the tracking error of the system can always converge to a small neighborhood of the origin; and the state of the system never violates the constraint limit. Finally, the designed controller is applied to a single-link manipulator system to prove the effectiveness of the control strategy.

Description

Technical Field

[0001] This invention relates to the technical fields of adaptive control, state constraints, and input time delay, and specifically to an adaptive control method for a nonlinear system with input time delay and state constraints. Background Technology

[0002] Nonlinear systems are prevalent in practical engineering systems such as chemical reaction systems, robotic arm systems, and unmanned aerial vehicle systems, and their corresponding control problems have always been a research hotspot in the field of control, thus attracting widespread attention and research from many scholars. For uncertain nonlinear systems, a combination of adaptive techniques, parameter separation methods, and backstepping can effectively solve the controller design problem under conditions of parameter uncertainty. In recent years, nonlinear system control has been extensively studied and developed; however, due to the existence of constraints, time delays, and other uncertainties, many control problems of nonlinear systems have remained unresolved for a long time.

[0003] The stable and safe operation of a system is one of the goals of nonlinear control. Due to the inherent characteristics of the controlled object and the influence of the environment, real-world nonlinear systems are inevitably subject to various constraints, which we call the constraint control problem of nonlinear systems. Constraints in nonlinear systems include input, output, and state constraints. Violating these constraints may lead to a decline in the performance of the actual system or even damage to its stability, causing the system to malfunction and resulting in unnecessary losses. In recent years, research has introduced the barrier Lyapunov function (BLF) control method, which has become an important and effective tool for solving control problems of output- or state-constrained nonlinear systems. Since then, the nonlinear constraint control problem has attracted widespread attention and yielded many results.

[0004] On the other hand, time delay is a common phenomenon in many practical systems. Input time delay, as a type of time delay problem, takes various forms, such as physical transmission delay and computation delay in practical systems, which can be equivalent to input time delay problems. Time delay often leads to a decrease in control system performance or even destroys system stability, while also bringing great difficulties to system analysis and control. For nonlinear systems with input time delay, the Pade approximation method is one of the effective methods for handling input time delay problems, and many control schemes for nonlinear systems have been generated using this method. Although a large number of papers have studied the control problems of nonlinear systems with time delay, there are few results on the control of state-constrained nonlinear systems with input time delay. Moreover, in addition to constraints and time delay problems, parameter uncertainty is also an important factor leading to system instability and performance degradation. The parameter separation lemma provides a new method for handling nonlinear systems with parameter uncertainty. In recent years, many papers have combined adaptive control and BLF to handle the control problems of constrained nonlinear systems with parameter uncertainty.

[0005] From a practical application perspective, considering the parameter uncertainties and designing control schemes for nonlinear systems with input time delays and state constraints is highly meaningful. To the author's knowledge, there are currently no research results on the controller design problem for nonlinear systems simultaneously exhibiting input time delays, state constraints, and parameter uncertainties. Inspired by the above discussion, this invention will investigate the state feedback control problem for a class of state-constrained nonlinear systems with input time delays and parameter uncertainties. Summary of the Invention

[0006] Based on the aforementioned problems in the control of constrained nonlinear systems, this invention proposes an adaptive control method for nonlinear systems with input time delay and state constraints. The specific steps of the technical solution of this invention are as follows:

[0007] 1. An adaptive control method for a nonlinear system with input time delay and state constraints, characterized in that the method includes the following steps:

[0008] Step 1: Establish the equations for the nonlinear system with input time delay and state constraints:

[0009]

[0010] in For the system's state variables, Let be an n-dimensional real space, and For state x i The first derivative, For state x n The first derivative; It consists of states x1,...,x i The state vector formed Let be an i-dimensional real space; u is the control input of the system, and Let be the set of real numbers; θ is an uncertain parameter term, and Let i be a d-dimensional real space; i and n both represent the order of the system. for The function has f1(0,θ)=0, The function is the set of functions with continuous first-order partial derivatives; d i (t) represents a bounded external disturbance in the i-th order system, d n (t) represents a bounded external disturbance in the nth-order system; τ>0 represents the input time delay, and t represents time t;

[0011] Step 2: Use the Pade approximation method to handle the input time delay of the nonlinear system. According to Laplace's time delay theorem, the following is derived:

[0012]

[0013] Where l{·} represents the Laplace transform process, and s is the Laplace variable;

[0014] Then, an intermediate variable x is introduced. n+1 , its satisfaction

[0015]

[0016] From (3), we can know

[0017]

[0018] According to the Laplace transform and (4), we can obtain

[0019]

[0020] Combining systems (1), (3), and (5), we can obtain the following new system.

[0021]

[0022] Where η is a defined intermediate variable and f n (x,θ) is a nonlinear function of the system.

[0023] Step 3: Introduce state coordinate transformation

[0024]

[0025] And define the adaptive parameters as follows:

[0026]

[0027] Where z1,...,z i ,...,z n The coordinate transformation represents the state of the system; Θ represents the adaptive parameters; α2,...,α n The virtual control law to be designed; reference signal y d , and its i-th derivative In Ω d Bounded above, where Ω d For a set that satisfies the reference signal constraints and Since it is a positive constant, according to formulas (6)-(7), we can obtain

[0028]

[0029] in, This is an estimate of Θ. These are the system states z1,...,z i ,...,z n First derivative; d2,d i ,d j ,d n These are the bounded external disturbances of the corresponding order system;

[0030] Step 4: Set the system constraints as follows:

[0031]

[0032] Where k bi The constant is known. The Lyapunov function (BLF) for introducing obstacles based on the constraints is:

[0033]

[0034] Where k ci >0 represents a positive constant, and Γ>0 represents the adaptive gain constant; It is Θ and its estimated value The error value between them.

[0035] Step 5: The adaptive controller design for a nonlinear system with input time delay and state constraints is as follows:

[0036]

[0037]

[0038] in, c n ,σ0,ε n ,k cn >0 is a known positive constant; It represents the nth derivative of the reference signal.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] 1. This invention is the first to attempt to simultaneously consider the effects of input time delay, state constraints, and parameter uncertainty, thus solving the control problem of more general nonlinear systems.

[0041] 2. By combining the Pade approximation method, BLF, and adaptive backstepping design within a unified framework, the difficulties brought about by input time delay, state constraints, and parameter uncertainty to the design of the system controller were successfully compensated.

[0042] 3. This invention designs a novel adaptive state feedback controller based on BLF, proving that all signals in the closed-loop system are uniformly eventually bounded, and the system's tracking error always converges to a small neighborhood of the origin. Specifically, the system's state never violates the constraints. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the design of an adaptive control method for a nonlinear system with input time delay and state constraints, as per the present invention.

[0044] Figure 2 This is a schematic diagram of a single-link robotic arm system;

[0045] Figure 3 This is the response curve of the tracking error in the simulation of this invention;

[0046] Figure 4 This is the response curve of system state x1 in the simulation of this invention;

[0047] Figure 5 This is the response curve of system state x2 in the simulation of this invention;

[0048] Figure 6 This is the response curve of the adaptive parameters in the simulation of this invention;

[0049] Figure 7 This is the response curve of the system control input in the simulation of this invention. Detailed Implementation

[0050] To make the design concept of this invention clearer, the following will provide a detailed description of the design, principle, and proof of the state feedback controller, with reference to the accompanying drawings.

[0051] like Figure 1 As shown, this invention provides an adaptive control method for a nonlinear system with input time delay and state constraints. The specific steps of the technical solution are as follows:

[0052] Step 1: Establish the equations for the nonlinear system with input time delay and state constraints:

[0053]

[0054] in For the system's state variables, Let be an n-dimensional real space, and For state x i The first derivative, For state x n The first derivative; It consists of states x1,...,x i The state vector formed Let be an i-dimensional real space; u is the control input of the system, and Let be the set of real numbers; θ is an uncertain parameter term, and Let i be a d-dimensional real space; i and n both represent the order of the system. for The function has f1(0,θ)=0, The function is the set of functions with continuous first-order partial derivatives; d i (t) represents a bounded external disturbance in the i-th order system, d n (t) represents a bounded external disturbance in the nth-order system; τ>0 represents the input time delay, and t represents time t;

[0055] Step 2: Use the Pade approximation method to handle the input time delay of the nonlinear system. According to Laplace's time delay theorem, the following is derived:

[0056]

[0057] Where l{·} represents the Laplace transform process, and s is the Laplace variable;

[0058] Then, an intermediate variable x is introduced. n+1 , its satisfaction

[0059]

[0060] From (3), we can know

[0061]

[0062] According to the Laplace transform and (4), we can obtain

[0063]

[0064] Combining systems (1), (3), and (5), we can obtain the following new system.

[0065]

[0066] Where η is a defined intermediate variable and f n (x,θ) is a nonlinear function of the system.

[0067] Step 3: Introduce state coordinate transformation

[0068]

[0069] And define the adaptive parameters as follows:

[0070]

[0071] Where z1,...,z i ,...,z n The coordinate transformation represents the state of the system; Θ represents the adaptive parameters; α2,...,α n The virtual control law to be designed; reference signal y d , and its i-th derivative In Ω d Bounded above, where Ω d For a set that satisfies the reference signal constraints and Since it is a positive constant, according to formulas (6)-(7), we can obtain

[0072]

[0073] in, This is an estimate of Θ. These are the system states z1,...,z i ,...,z n First derivative; d2,d i ,d j ,d n These are the bounded external disturbances of the corresponding order system;

[0074] Step 4: Set the system constraints as follows:

[0075]

[0076] Where k bi The constant is known. The Lyapunov function (BLF) for introducing obstacles based on the constraints is:

[0077]

[0078] Where k ci >0 represents a positive constant, and Γ>0 represents the adaptive gain constant; It is Θ and its estimated value The error value between them.

[0079] Step 5: The adaptive controller design for a nonlinear system with input time delay and state constraints is as follows:

[0080]

[0081]

[0082] in, c n ,σ0,ε n ,k cn >0 is a known positive constant; It represents the nth derivative of the reference signal.

[0083] To better illustrate the technology of this invention, the designed adaptive state feedback controller will be proven. The following assumptions and lemmas will be used in the proof.

[0084] Assumption 1: There exists a positive constant d. iM Such that for i = 1, ..., n, the external disturbance satisfies |d i (t)|≤d iM

[0085] Assumption 2, relevant reference signal y d , and its i-th derivative In closed set Inside, Y1,Y i (i = 2, ..., n) are positive constants.

[0086] Lemma 1: For any constant k ci >0 and all Satisfy |z i |≤k ci , Established.

[0087] Lemma 2. For real variables x≥0, y>0, we have Where m≥1 is a real number.

[0088] Lemma 3. For i = 1, ..., n, function satisfy

[0089]

[0090] in It is a non-negative smooth function and Θ0≥1 is a design constant.

[0091] Lemma 4. For any k ci >0, let and It is an open set. Consider the system. in Let z be the state variable, where z = (z1, z2, ..., zn) n ) T , Regarding piecewise continuity of t, regarding z i Local Lipschitz continuity, in Consistent with the above. Assume there exists a continuously differentiable positive definite function. and So that when z i →k ci Or z i →-k ci When V1→∞, γ1(||ω||)≤W(||ω||)≤γ2(||ω||), where γ1 and γ2 are Function. Define V(κ):=V1+W(ω), where z(0)∈Ω1, if the inequality exist If the above holds true, where μ and Δ are positive constants, then ω is bounded and z is z. i ∈Ω1,

[0092] The proof is as follows:

[0093] Step 1: Adaptive controller design.

[0094] For system (6), this invention uses the backstepping method to design the controller, and the design steps are divided into n steps:

[0095] Step 1: Select the Lyapunov function as

[0096]

[0097] Where, k c1 >0 is a design constant; Γ>0 is an adaptive gain constant; It is Θ and its estimated value The error value between them. Clearly, V1 is in The interior is differentiable. From (9)-(10), we know that the derivative of V1 is...

[0098]

[0099] By using Lemma 2-3, Assumption 1-2, and (7)-(8), we know that

[0100]

[0101]

[0102]

[0103] in

[0104] Substituting (16)-(18) into (15) yields

[0105]

[0106] in, Therefore, the first virtual control law is selected as follows:

[0107]

[0108] Substituting (20) into (19) yields

[0109]

[0110] Where c1>0 and σ0>0 are design constants, and we have

[0111]

[0112] Step 2: Select the Lyapunov function V2 as...

[0113]

[0114] Obviously, V2 is It is intrinsically differentiable. Differentiating the above equation and using (9) and (21), we can obtain...

[0115]

[0116] From Assumption 1-2 and Lemma 2-3, we know

[0117]

[0118]

[0119]

[0120] Where, ε 21 >0,ε 22 >0 represents a design constant;

[0121] Substituting (25)-(27) into (24) yields

[0122]

[0123] In the formula

[0124] Now, the virtual control law is selected as follows:

[0125]

[0126] Where c2 > 0 is the design constant. Substituting (29) into (28) yields...

[0127]

[0128] Where σ²>0 is the design constant, and we have

[0129]

[0130] Step i (i = 3, 4, ..., n-1): This step uses induction. Assume there exists a positive definite Lyapunov function in step i-1. And there is a series of virtual control laws

[0131]

[0132] Make

[0133]

[0134] Where, σ i-1 It is a non-negative continuous function, Δ i-1 It is a positive number.

[0135] We will now prove that (33) holds true for the i-th step. Let the Lyapunov function V... i Defined as From (9) and (33), we can know that V i The derivative is

[0136]

[0137] To further derive this, by using Lemma 2-3 and Assumption 1, we can obtain the following inequality.

[0138]

[0139]

[0140]

[0141] in And ε i >0 is a constant.

[0142] Substitute (35)-(37) into (34) and construct α i+1 for

[0143]

[0144] This allows (34) to be satisfied.

[0145]

[0146] Among them, c i >0 is a design constant;

[0147] Step 1: Select the Lyapunov function V n for

[0148]

[0149] Similar to the construction process in step i, the controller and adaptive control law are designed as follows:

[0150]

[0151]

[0152] Ultimately, it can make

[0153]

[0154] Where c n ,Δ n It is a positive constant; γ n ,φ n It is a non-negative continuous function.

[0155] Step 2: Stability analysis.

[0156] Based on the above derivation, the present invention presents the following theorem:

[0157] Theorem 1 Considering system (1) under the conditions of assumptions 1-2, when the actual controller is designed as (41) and the adaptive controller is designed as (42), we have

[0158] (i) All signals in the closed-loop system are uniformly eventually bounded;

[0159] (ii) The tracking error converges to a closed set at the origin, i.e.

[0160]

[0161] (iii) The system state satisfies |x i|<k bi This means that the constraints were not violated.

[0162] prove:

[0163] (i) First by (43) can be written as

[0164]

[0165] in According to Lemma 1, we know that

[0166]

[0167] Combining (40) and (46), equation (45) can be written as

[0168]

[0169] Where c = min{2c} i ,Γσ0,i=1,...,n}.

[0170] According to (40), (47) and Lemma 4, we know that z i (t) and It is uniformly bounded, since Θ is a constant. It is also uniformly bounded; from z1 = x1 - y d It can be seen that state x1 is uniformly bounded; according to equation (20), α2 is also uniformly bounded; and from equation (7), x2 = z2 + α2, we know that x2 is also uniformly bounded; and so on, we can obtain α3, α4, ..., α n ,x3,x4,...,x n+1 Furthermore, u is uniformly and eventually bounded. In summary, all signals in a closed-loop system are uniformly and eventually bounded.

[0171] (ii) Multiply equation (47) by e ct Integrating both sides of the inequality over [0, t) yields

[0172]

[0173] And from (40) we can know By organizing, we can obtain

[0174]

[0175] From equation (7), we can define the tracking error as y(t) - y d (t)=z1(t), and combining with (49), we can know that for Tracking error satisfies

[0176]

[0177] In other words, the tracking error remains within a small neighborhood of the origin, which is a bounded compact set.

[0178]

[0179] (iii) Assumption in It is a positive constant. From x1 = z1 + y d Compared with hypothesis 2 It can be known make We can obtain |x1| < k b1 And from and It can be known make We can obtain |x2|<k b2 Following this logic, we can eventually obtain |x i |<k bi i = 1, 2, ..., n, meaning the state does not violate the constraints.

[0180] The effectiveness of the controller designed in this invention will be verified below using a single-link robotic arm system. Consider the attached diagram. Figure 2 The single-link robotic arm system in the image consists of a rigid link coupled to a DC motor via gears. The dynamic expression of this system is:

[0181]

[0182] Where M = 1 kg·m 2 For inertia; m = 1 kg is the mass of the connecting rod; q is the angular position of the connecting rod; The angular velocity of the connecting rod is expressed in rad / s. It is the angular acceleration of the connecting rod, measured in rad / s². 2 g = 9.8 m / s 2 Let l be the acceleration due to gravity; F is the control force of the link. We treat l as an uncertain parameter and define the state variable x1 = q. And the control input u = F, where the state constraints satisfy |x1| < k b1 =1.5,|x2|<k b2 =1.8. Equation (51) can be written as

[0183]

[0184] Where f2 = -0.5mgsin(x1) and θ = l. In the simulation, the disturbance signal is chosen as d2(t) = 0.2sint; the reference signal is chosen as yd =0.1sint; From assumption 1-2, we know |d2(t)|≤0.2, as well as also, Where Θ0 = θ, select

[0185] Based on the controller design process described above, an adaptive controller is designed as follows:

[0186]

[0187]

[0188]

[0189] in

[0190] and

[0191] c1, c2, ε2 > 0 are design constants.

[0192] In the simulation, the input time delay τ = 0.01 is assumed, and other parameters are chosen as ε2 = 2, Γ = 10, c1 = 1, c2 = 2, σ0 = 1. The initial conditions of the system are selected as [x1(0), x2(0)]. T =[0.3,0.2] T [x1(0),x2(0)] T =[1.2,0.2] T [x1(0),x2(0)] T = [-1.1, 0.2] T and from Figure 3 It can be seen that under different initial conditions, the system's tracking error can converge to a small neighborhood of the origin, meaning that the system's output signal y can track the reference signal y very well. d ;from Figure 4-6 It can be seen that under different initial conditions, the system state does not violate the constraints, and the adaptive parameters are consistently and eventually bounded; from Figure 7 This shows that the system's control input u is also uniformly and eventually bounded. In conclusion, all signals in the closed-loop system are uniformly and eventually bounded.

[0193] This invention studies the tracking control problem of a class of state-constrained nonlinear systems with input time delay and parameter uncertainty. The parametric nonlinearity of the system is addressed by combining the adaptive backstepping method and the parameter separation method. The Pade approximation method and the Black-Flush Function (BLF) are integrated within a unified framework to successfully resolve the uncertainties caused by input time delay and state constraints. An adaptive state feedback controller is derived through rigorous stability analysis. The results show that all signals in the closed-loop system are uniformly eventually bounded; the tracking error of the system always converges to a small neighborhood of the origin; and the system state never violates the constraints. Further work will focus on adaptive specified-time control of constrained nonlinear systems.

[0194] The above description is merely a general procedure of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An adaptive control method for a nonlinear system with input time-delay and state constraints, which is applied to a single-link robot arm system, the single-link robot arm system is coupled to a DC motor through a rigid link by a gear, and the system dynamics expression is: in For inertia; For the mass of the connecting rod; It refers to the angular position of the connecting rod; The angular velocity of the connecting rod is expressed in units of ω. ; It is the angular acceleration of the connecting rod, in units of . ; It is the acceleration due to gravity; It is the control force of the linkage; As an uncertain parameter, and defined as a state variable and control input ; characterized in that The method includes the following steps: S1. Establish the equations for a nonlinear system with input time delay and state constraints: in For the system's state variables, for A real space of dimension , and For state The first derivative, For state The first derivative; It is a state The state vector formed for 3D real space; As the control input of the system, and , It is the set of real numbers; For uncertain parameter terms, and , for 3D real space; and Both represent the order of the system, for for Function and have , A function is a set of functions with continuous first-order partial derivatives; Indicates the first Bounded external disturbances in an order system Indicates the first Bounded external disturbances in an order system; Indicates input delay. express time; S2. Using the Pade approximation method to handle the input time delay of nonlinear systems, based on Laplace's time delay theorem, we can derive: wherein represents a Laplace transform process, is a Laplace variable; Subsequently, an intermediate variable is introduced which satisfies From (3), we can know According to the Laplace transform and (4), we can obtain Combining systems (1), (3), and (5), we can obtain the following new system. wherein is a defined intermediate variable and , is a nonlinear function of the system; S3. Introducing State Coordinate Transformation And define the adaptive parameters as follows: wherein is the state of the system after coordinate transformation; is the adaptive parameter; is the virtual control law to be designed; reference signal and derivative is bounded on wherein is the set satisfying the reference signal constraint and , is a positive constant, according to formulas (6)-(7), it can be obtained wherein , are estimated values of are the first derivatives of the system state ; and are bounded external disturbances in the corresponding order system; S4. Set the system constraints and introduce obstacle Lyapunov functions based on the constraints; S5. Complete the design of an adaptive controller for a nonlinear system with input time delay and state constraints.

2. The adaptive control method of a nonlinear system with input time delay and state constraints according to claim 1, characterized in that, In step S4, the system constraints are set as follows: wherein to satisfy the state constraints a set of real numbers, is a known constant.

3. The adaptive control method of a nonlinear system with input time delay and state constraints according to claim 2, characterized in that, In step S4, the BLF is set according to the state constraints as follows: in The Lyapunov function is set as a barrier. For positive integers, It is the adaptive gain constant; yes Its estimated value The error value between them.

4. The adaptive control method of a nonlinear system with input time delay and state constraints according to claim 3, characterized in that, The adaptive controller described in step S5 is designed as follows: wherein ; , ; is a known constant; denotes the derivative of the reference signal.

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