CN116679570B - Adaptive control methods for nonlinear systems with input time delay and state constraints - Google Patents

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

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CN116679570B
CN116679570B CN202310836163.2A CN202310836163A CN116679570B CN 116679570 B CN116679570 B CN 116679570B CN 202310836163 A CN202310836163 A CN 202310836163A CN 116679570 B CN116679570 B CN 116679570B
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闵惠芳
时洋洲
仲达
郝书涛
周希雅
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Nanjing University of Science and Technology
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
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Abstract

本发明提出了一种具有输入时滞和状态约束的非线性系统的自适应控制方法,属于自适应控制、状态约束和输入时滞的技术领域。该方法首先，通过结合自适应控制与参数分离法来处理系统的参数非线性。将Pade逼近法与障碍Lyapunov函数(BLF)结合在统一框架内，解决了由输入时滞与状态约束引起的不确定性。基于反步设计过程，并结合严谨的稳定性分析得到一个自适应状态反馈控制策略。经验证，闭环系统中所有信号都是一致最终有界的；系统的跟踪误差总能收敛到原点的一个小的邻域内；系统的状态从未违背约束限制。最后，将所设计的控制器应用到单连杆机械臂系统中证明控制策略的有效性。

This invention proposes an adaptive control method for nonlinear systems with input time delay and state constraints, belonging to the technical fields of adaptive control, state constraints, and input time delay. First, the method addresses the parametric nonlinearity of the system by combining adaptive control with parameter separation. Then, it integrates the Pade approximation method with the barrier Lyapunov function (BLF) within a unified framework to resolve the uncertainties caused by input time delay and state constraints. Based on a backstepping design process and rigorous stability analysis, an adaptive state feedback control strategy is derived. Verification shows that all signals in the closed-loop system are uniformly and eventually bounded; the system's tracking error always converges to a small neighborhood of the origin; and the system's state never violates the constraints. Finally, the designed controller is applied to a single-link robotic arm system to demonstrate the effectiveness of the control strategy.

Description

Adaptive control method for nonlinear system with input time lag and state constraint
Technical Field
The invention relates to the technical field of self-adaptive control, state constraint and input time lag, in particular to a self-adaptive control method of a nonlinear system with input time lag and state constraint.
Background
Since nonlinear systems are commonly existing in actual engineering systems such as chemical reaction systems, mechanical arm systems, unmanned aerial vehicle systems and the like, and the corresponding control problems thereof are always research hotspots in the control field, the nonlinear systems have attracted extensive attention and research by many students. Aiming at an uncertain nonlinear system, the controller design problem of the nonlinear system under the condition of uncertain parameters can be effectively solved by combining the self-adaptive technology, the parameter separation method and the backstepping method. In recent years, nonlinear system control has been widely studied and developed, but many nonlinear system control problems have long been pending due to constraints, time lags and other uncertainties.
The stable and safe operation of the system is one of the targets of nonlinear control. Due to the characteristics of the controlled object and the influence of the environment, the practical nonlinear system is inevitably limited by various factors, and the limitation is called a constraint control problem of the nonlinear system. Constraints of nonlinear systems include input, output, and state constraints, against which may lead to degradation of the actual system performance or even to disruption of the stability of the system, rendering the system incapable of functioning properly and resulting in some unnecessary loss. In recent years, research has been conducted into a lyapunov function (BLF) control method, which is an important and effective tool for solving the problem of nonlinear system control of output or state constraints. Since then, nonlinear constraint control problems have attracted considerable attention and have achieved a number of achievements.
On the other hand, the time lag is a phenomenon commonly existing in many practical systems, and the input time lag is one of the time lag problems, and various forms such as physical transmission delay, calculation delay and the like in the practical systems can be equivalent to the input time lag problem. Time lags often lead to reduced performance of the control system and even to a failure of the stability of the system, and also present significant difficulties in analysis and control of the system. For nonlinear systems with input skew, the Pade approximation is one of the effective methods currently dealing with the input skew problem, by which many nonlinear system control schemes are created. Although there are a great deal of literature currently researching control problems for nonlinear systems with time lags, control results for state-constrained nonlinear systems with input time lags are few. Furthermore, in addition to constraint and time-lapse problems, system parameter uncertainty is also an important factor that leads to system instability and performance degradation. The parameter separation quotients provide a new method for processing nonlinear systems with parameter uncertainties. In recent years there have been many references to combine adaptive control and BLF to address the control problem of constrained nonlinear systems with parameter uncertainty terms.
From the practical point of view, it is very interesting to consider the problem of uncertainty of parameters and design a control scheme for a nonlinear system with input time lags and state constraints. To the best of the authors, there are no relevant research results on the controller design problem of nonlinear systems with simultaneous input time lags, state constraints and parameter uncertainty terms. Inspired by the above discussion, the present invention will investigate the state feedback control problem of a class of state-constrained nonlinear systems with input time lags and parameter uncertainty terms.
Disclosure of Invention
In accordance with the above-presented series of problems with constrained nonlinear system control, the present invention presents an adaptive control method for nonlinear systems with input skew and state constraints. The technical scheme of the invention comprises the following specific steps:
1. A method for adaptive control of a nonlinear system having input skew and state constraints, the method comprising the steps of:
step 1, establishing a nonlinear system equation with input time lag and state constraint:
Wherein the method comprises the steps of As a state variable of the system,Is a real space of n dimension, andAs the first derivative of state x i,Is the first derivative of state x n; is a state vector consisting of states x 1,...,xi, Is i-dimensional real space, u is control input of the system, and Is a real number set, θ is an uncertain parameter item, and I and n both represent the order of the system for d-dimensional real spaceIs thatThe function has f 1 (0, θ) =0,The function is a function set with continuous first-order partial derivatives, d i (t) represents bounded external disturbance in the ith-order system, d n (t) represents bounded external disturbance in the nth-order system, τ >0 represents input time lag, and t represents t moment;
step 2, processing the input time lag of the nonlinear system by using a Pade approximation method, and obtaining according to the Laplace time lag theorem:
where l {.cndot. } represents the Laplace transform process and s is the Laplace variable;
subsequently, an intermediate variable x n+1 is introduced, which satisfies
From (3), it can be seen that
From the Laplace transform and (4) can be obtained
The combination of systems (1), (3) and (5) gives the following new systems
Wherein η is defined intermediate variable andF n (x, θ) is a nonlinear function of the system.
Step 3, introducing state coordinate transformation
And define the adaptive parameters as follows
Wherein z 1,...,zi,...,zn is the state of the system after coordinate transformation, Θ is the adaptive parameter, α 2,...,αn is the virtual control law to be designed, reference signal y d,I-th derivative ofBounded on Ω d, where Ω d is the set that satisfies the reference signal constraint andIs positive constant, and can be obtained according to the formulas (6) - (7)
Wherein, the As the estimated value of theta,D 2,di,dj,dn are respectively bounded external disturbances of the corresponding order system;
and 4, setting constraint conditions of the system as follows:
Where k bi is a known constant. The introduction of the obstacle Lyapunov function (BLF) according to the constraint conditions is:
Wherein k ci >0 is a positive constant and Γ >0 is an adaptive gain constant; is Θ and its estimated value Error values between the two.
Step 5, designing an adaptive controller of a nonlinear system with input time lag and state constraint as follows:
Wherein, the C n,σ0,εn,kcn >0 is a known positive constant; representing the nth derivative of the reference signal.
Compared with the prior art, the invention has the following advantages:
1. the invention firstly tries to consider the influence of input time lag, state constraint and parameter uncertainty at the same time, and solves the control problem of a more general nonlinear system.
2. By combining the Pade approximation method, BLF and self-adaptive backstepping design in a unified framework, the difficulties brought to the design of a system controller by input time lag, state constraint and parameter uncertainty are successfully compensated;
3. The invention designs a new BLF-based self-adaptive state feedback controller, which proves that all signals in a closed loop system are consistent and finally bounded, and the tracking error of the system always converges to a small neighborhood of an origin. In particular, the state of the system never violates constraint limits.
Drawings
FIG. 1 is a flow chart of a design of an adaptive control method for a nonlinear system with input skew and state constraints according to the present invention;
FIG. 2 is a schematic diagram of a single link robotic arm system;
FIG. 3 is a response curve of tracking error in the simulation of the present invention;
FIG. 4 is a response curve of the system state x 1 in the simulation of the present invention;
FIG. 5 is a response curve of the system state x 2 in the simulation of the present invention;
FIG. 6 is a response curve of adaptive parameters in the simulation of the present invention;
FIG. 7 is a response curve of the system control input in the simulation of the present invention.
Detailed Description
In order to make the design concept of the present invention more clear, the following will describe in detail the design, principle and demonstration of the state feedback controller, and the following will describe in detail the drawings in conjunction with the accompanying drawings.
As shown in fig. 1, the present invention provides an adaptive control method for a nonlinear system with input skew and state constraints. The technical scheme comprises the following specific steps:
step 1, establishing a nonlinear system equation with input time lag and state constraint:
Wherein the method comprises the steps of As a state variable of the system,Is a real space of n dimension, andAs the first derivative of state x i,Is the first derivative of state x n; is a state vector consisting of states x 1,...,xi, Is i-dimensional real space, u is control input of the system, and Is a real number set, θ is an uncertain parameter item, and I and n both represent the order of the system for d-dimensional real spaceIs thatThe function has f 1 (0, θ) =0,The function is a function set with continuous first-order partial derivatives, d i (t) represents bounded external disturbance in the ith-order system, d n (t) represents bounded external disturbance in the nth-order system, τ >0 represents input time lag, and t represents t moment;
step 2, processing the input time lag of the nonlinear system by using a Pade approximation method, and obtaining according to the Laplace time lag theorem:
where l {.cndot. } represents the Laplace transform process and s is the Laplace variable;
subsequently, an intermediate variable x n+1 is introduced, which satisfies
From (3), it can be seen that
From the Laplace transform and (4) can be obtained
The combination of systems (1), (3) and (5) gives the following new systems
Wherein η is defined intermediate variable andF n (x, θ) is a nonlinear function of the system.
Step 3, introducing state coordinate transformation
And define the adaptive parameters as follows
Wherein z 1,...,zi,...,zn is the state of the system after coordinate transformation, Θ is the adaptive parameter, α 2,...,αn is the virtual control law to be designed, reference signal y d,I-th derivative ofBounded on Ω d, where Ω d is the set that satisfies the reference signal constraint andIs positive constant, and can be obtained according to the formulas (6) - (7)
Wherein, the As the estimated value of theta,D 2,di,dj,dn are respectively bounded external disturbances of the corresponding order system;
and 4, setting constraint conditions of the system as follows:
Where k bi is a known constant. The introduction of the obstacle Lyapunov function (BLF) according to the constraint conditions is:
Wherein k ci >0 is a positive constant and Γ >0 is an adaptive gain constant; is Θ and its estimated value Error values between the two.
Step 5, designing an adaptive controller of a nonlinear system with input time lag and state constraint as follows:
Wherein, the C n,σ0,εn,kcn >0 is a known positive constant; representing the nth derivative of the reference signal.
To better illustrate the inventive technique, a designed adaptive state feedback controller will be demonstrated. The following assumptions and quotients will be used in the proving process.
Assuming 1, there is a positive constant d iM such that for i=1..n, the external disturbance satisfies |d i(t)|≤diM
Assuming a 2, correlated reference signal y d,I-th derivative thereofIn the closed setIn the inner part of the inner part,Y 1,Yi (i=2., n) is a normal number.
Lemma 1, k ci >0 for arbitrary constant and allSatisfying the requirements of |z i|≤kci,This is true.
2, For real variable x >0, y >0, there areWherein m is a real number and is more than or equal to 1.
Lemma 3 for i=1, the combination of the first and second components, n,Function ofSatisfy the following requirements
Wherein the method comprises the steps ofIs a non-negative smooth function and Θ 0 is not less than 1, which is a design constant.
4, For any k ci >0, setAnd (3) withIs an open set. Consider a systemWherein the method comprises the steps ofIs a state variable, where z= (z 1,z2,...,zn)T,With respect to t segment continuum, with respect to z i local Lipoz continuum, atAnd is consistent with the above. Assuming that there is a continuously differentiable positive definite functionAndSuch that when z i→kci or z i→-kci there is V 1→∞,γ1(||ω||)≤W(||ω||)≤γ2 (|ω||), where γ 1 and γ 2 areA function. Definition V (κ) =v 1 +w (ω), where z (0) ∈Ω 1, if inequalityAt the position ofThe above holds, where μ, Δ is a positive constant, ω is bounded and z i∈Ω1,
The proving process is as follows:
and the first step, designing a self-adaptive controller.
For the system (6), the invention uses a back-stepping method to design the controller, and the design steps are divided into n steps:
the first step is to select Lyapunov function as
Wherein k c1 >0 is a design constant, Γ >0 is an adaptive gain constant; is Θ and its estimated value Error values between the two. Obviously, V 1 is atThe inner is conductive. From (9) - (10), the derivative of V 1 is
From the quotients 2-3, it is assumed that 1-2 and (7) - (8) are known
Wherein the method comprises the steps of
Substituting (16) - (18) into (15) can result in
Wherein, the Thus, the first virtual control law is selected as follows
Substituting (20) into (19) to obtain
Wherein c 1 >0 and sigma 0 >0 are design constants, and there are
Step two, selecting Lyapunov function V 2 as
Obviously, V 2 is atAnd (3) internal conduction. Deriving the above and using (9) and (21)
From hypothesis 1-2 and lemma 2-3
Wherein ε 21＞0,ε22 >0 is the design constant;
substituting (25) - (27) into (24) to obtain
In the middle of
The virtual control laws are now selected as follows
Where c 2 >0 is the design constant. Substituting (29) into (28) to obtain
Wherein σ 2 >0 is a design constant, and has
Step i (i=3, 4.,. N-1) this step uses induction. Assume that there is a positive Lyapunov function in step i-1 asAnd has a series of virtual control laws
So that
Wherein σ i-1 is a non-negative continuous function and Δ i-1 is a positive constant.
We demonstrate that (33) is true for step i. Lyapunov function V i is defined asFrom (9) and (33), it can be seen that the derivative of V i is
For further derivation, the following inequality can be obtained by using the lemma 2-3 and hypothesis 1
Wherein the method comprises the steps ofAnd ε i >0 is a constant.
Substituting (35) - (37) into (34) and constructing alpha i+1 as
Can be made (34) to satisfy
Wherein c i >0 is a design constant;
Step one, selecting Lyapunov function V n as
Similar to the construction process of the i step, the controller and the adaptive control law are designed as follows
Finally can lead to
Where c n,Δn is a positive constant and γ n,φn is a non-negative continuous function.
And secondly, analyzing stability.
From the above derivation, the present invention gives the following theorem:
Theorem 1 considers that the system (1) under the condition of assuming 1-2, when the actual controller is designed as (41) and the adaptive controller is designed as (42), there are
(I) All signals in the closed loop system are consistent and ultimately bounded;
(ii) The tracking error converging to within a certain closed set of origin, i.e
(Iii) The system state satisfies |x i|＜kbi, i.e., the constraint is not violated.
And (3) proving:
(i) First by (43) Can be written as
Wherein the method comprises the steps ofFrom the quotation mark 1, it can be seen that
In combination (40) and (46), formula (45) can be written as
Where c=min {2c i,Γσ0, i=1,..n }.
From (40), (47) and lemma 4, it can be seen that z i (t) andIs consistent and eventually bounded, since Θ is a constant value, soAlso consistently ultimately bounded, as known from z 1＝x1-yd, state x 1 consistently ultimately bounded, as known from equation (20) a 2 consistently ultimately bounded, again as known from equation (7) x 2＝z2+α2, x 2 consistently ultimately bounded, and so on, we can obtain a 3,α4,...,αn,x3,x4,...,xn+1 and u consistently ultimately bounded. In summary, all signals in a closed loop system are consistent and ultimately bounded.
(Ii) Multiplying equation (47) by e ct and integrating the two sides of the inequality over [0, t ] to obtain
And can be seen from (40)Is obtained by arrangement
From equation (7), we can define the tracking error as y (t) -y d(t)＝z1 (t), combined with (49) to know the error forThe tracking error satisfies
That is, the tracking error remains within a small neighborhood of the origin, which is a bounded tight set
(Iii) Assume thatWherein the method comprises the steps ofIs a positive constant. From x 1＝z1+yd and hypothesis 2It can be seen thatOrder theCan obtain |x 1|＜kb1. Again byAnd (3) withIt can be seen thatOrder theLet us get |x 2|＜kb2 by analogy we can finally get |x i|＜kbi, i=1, 2.
The effectiveness of the controller designed by the invention is verified by a single-link mechanical arm system. Consider a single link mechanical arm system as in FIG. 2, which is coupled to a DC motor by a rigid link through gears, the system dynamics being expressed as
Wherein m=1 kg·m 2 is inertia, m=1 kg is link mass, q is the angular position of the link; the unit is rad/s for the angular velocity of the connecting rod; The unit is the angular acceleration of the connecting rod, rad/s 2;g＝9.8m/s2 is the gravitational acceleration, and F is the control force of the connecting rod. We regard i as a parameter uncertainty term and define the state variable x 1 = q, And a control input u=f, wherein the state constraint satisfies |x 1|＜kb1＝1.5,|x2|＜kb2 =1.8, equation (51) can be written as
Wherein f 2＝-0.5mgsin(x1) and θ=l. In the simulation, the disturbance signal is selected as d 2 (t) =0.2 sint, the reference signal is selected as y d =0.1 sint, and |d 2 (t) |is less than or equal to 0.2 from the assumption 1-2,AndIn addition, in the case of the optical fiber,Wherein Θ 0 =θ, select
The adaptive controller is designed according to the above-described controller design process as follows
Wherein the method comprises the steps of
And is also provided with
C 1,c2,ε2 >0 is a design constant.
In the simulation, assuming an input skew τ=0.01, the other parameters are chosen as ε 2＝2,Γ＝10,c1＝1,c2＝2,σ0 =1. The initial conditions of the system are chosen 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 respectivelyIt can be seen from fig. 3 that under different initial conditions the tracking error of the system can converge to a small neighborhood of the origin, i.e. the system output signal y can track the reference signal y d very well, that under different initial conditions the system state is not against the constraint, the adaptive parameters are consistent and eventually bounded, and that from fig. 7 the control input u of the system is also consistent and eventually bounded. In summary, all signals of the closed loop system are consistent and ultimately bounded.
The invention researches the tracking control problem of a state constraint nonlinear system with input time lag and parameter uncertainty. The parameter nonlinearity of the system is treated by combining the self-adaptive back-off method and the parameter separation method. Combining Pade approximation with BLF in a unified framework successfully addresses the uncertainty caused by input skew and state constraints. An adaptive state feedback controller is obtained through rigorous stability analysis. The research result shows that all signals in the closed loop system are consistent and finally bounded, the tracking error of the system always converges to a small neighborhood of the origin, and the state of the system never violates the constraint limit. Further work will focus on adaptive time-controlled assignment of constrained nonlinear systems.
The foregoing is merely illustrative of general procedures and is not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements, etc. which fall within the spirit and principles of the present invention should be included in the scope of the present invention.

Claims (4)

1.An adaptive control method of a nonlinear system with input time lag and state constraint is applied to a single-link mechanical arm system, wherein the single-link mechanical arm system is coupled to a direct current motor through a gear by a rigid link, and the dynamic expression of the system is as follows:
Wherein the method comprises the steps of Is inertial; the connecting rod is of a connecting rod mass; is the angular position of the connecting rod; Is the angular velocity of the connecting rod, and has the unit of ;Is the angular acceleration of the connecting rod, the unit is; Gravitational acceleration; Is the control force of the connecting rod; as parameter uncertainty items and defining state variables Control input;
Characterized in that the method comprises the steps of:
S1, establishing a nonlinear system equation with input time lag and state constraint:
Wherein the method comprises the steps of As a state variable of the system,Is thatReal space of dimension, andIs in state ofIs used as a first derivative of (a),Is in state ofIs the first derivative of (a); Is composed of state The state vector of the composition,Is thatA dimension real space; is a control input of the system, and ,Is a real number set; is an uncertain parameter item, and ,Is thatA dimension real space;And (3) with All represent the order of the system, forIs thatFunction and have,The function is a set of functions having successive first order partial derivatives; Represent the first A bounded external disturbance in the order system,Represent the firstBounded external disturbances in the order system; The time-lag of the input is indicated, Representation ofTime;
S2, processing input time lag of a nonlinear system by using a Pade approximation method, and obtaining according to a Laplacian time lag theorem:
Wherein, the The process of the laplace transform is represented,Is a Laplace variable;
Subsequent introduction of intermediate variables Which satisfies the following requirements
From (3), it can be seen that
From the Laplace transform and (4) can be obtained
The combination of systems (1), (3) and (5) gives the following new systems
Wherein, the Is defined as an intermediate variable and,Is a nonlinear function of the system;
S3, introducing state coordinate transformation
And define the adaptive parameters as follows
Wherein, the The state of the system after coordinate transformation; is an adaptive parameter; For the virtual control law to be designed, reference signal And to the process for preparing the sameDerivative of the orderAt the position ofUpper boundary of whichTo meet the set of reference signal constraints and,Is positive constant, and can be obtained according to the formulas (6) - (7)
Wherein, the ,Is thatIs used for the estimation of the (c),Respectively the system statesIs the first derivative of (a); respectively limiting external disturbance in a corresponding order system;
s4, setting constraint conditions of the system, and introducing a barrier Lyapunov function according to the constraint conditions;
s5, completing the design of the self-adaptive controller of the nonlinear system with input time lag and state constraint.
2. The adaptive control method of a nonlinear system with input time lag and state constraint according to claim 1, wherein the constraint condition of the system set in step S4 is:
Wherein the method comprises the steps of To meet state constraintsA set of dimensional real numbers,Is a known constant.
3. The adaptive control method of a nonlinear system with input time lag and state constraint according to claim 2, wherein the setting BLF according to state constraint in step S4 is:
Wherein the method comprises the steps of For a set obstacle Lyapunov function,Is a normal number of times, and the number of times is equal to the normal number,Is an adaptive gain constant; is that And its estimated valueError values between the two.
4. A method of adaptive control of a nonlinear system having input skew and state constraints according to claim 3, wherein said adaptive controller of step S5 is configured to:
Wherein, the ;,; Is a known positive constant; Representing reference signals And (5) an order derivative.
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Adaptive prescribed-time control for state-constrained non-linear systems with input delay;Huifang Min et al;《IMA Journal of Mathematical Control and Information》;20250407;全文 *

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