Self-adaptive tracking control method for state-limited disturbed nonlinear system

Abstract

The invention provides an adaptive tracking control method for a state-limited disturbed nonlinear system. The method comprises the following steps: firstly, establishing a mathematical model of a state-limited nonlinear system containing matching parameterization uncertainty and mismatching disturbance; secondly, aiming at the mismatched disturbance in the model, a disturbance observer is constructed to estimate the mismatched disturbance and the derivative thereof, so that the exponential convergence of disturbance estimation errors is realized, and the disturbance estimation speed is accelerated; secondly, state transformation is introduced, and state constraint is converted into tracking error constraint; thirdly, designing a self-adaptive controller by adopting dynamic surface control, and introducing a barrier Lyapunov function (BLF) in the last step of the dynamic surface control, so that the design process of the controller is greatly simplified; finally, the provided control method is applied to the magnetic suspension ball system, and the effectiveness of the control method is verified. The method has the advantages of being high in response speed, high in control precision, simple in control design and the like, and the problems that a traditional control method is complex in design and poor in transient performance are solved in a targeted mode.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear system control, and particularly relates to an adaptive tracking control method for a state-constrained and disturbed nonlinear system. Background Art

[0002] In the era of rapid technological development today, nonlinear systems have been widely applied in many fields, such as robot systems, flight control systems, power systems, ship navigation systems, etc. However, these nonlinear systems generally have the problems of state constraints and disturbances, which are always the key factors restricting the improvement of system performance. Traditional control methods often fail to meet the system performance requirements, and the controller design is relatively complex, which brings great challenges to the control of the system. Therefore, it is of great theoretical and practical significance to study the adaptive tracking control method for state-constrained and disturbed nonlinear systems.

[0003] State constraint is an inevitable problem in many practical systems. In practical engineering applications, the state of the system is often restricted by factors such as physical conditions and safety requirements. For example, the motion range of robot joints, the attitude angle range of aircraft, etc. If the state constraint problem is not considered, it may lead to a decline in system performance and even cause safety accidents. When dealing with the state constraint problem by traditional control methods, the design of the controller is often relatively complex. The barrier Lyapunov function (BLF), as an important tool in control theory for dealing with constrained nonlinear systems, provides an effective way to solve the state constraint problem. When the system state approaches the constraint boundary, the value of the BLF will tend to infinity. By maintaining the boundedness of the BLF in the closed-loop system, it can be ensured that the system state will not exceed the set boundary, thus avoiding constraint violation. However, the traditional BLF method is difficult to handle the cross-term between the mismatched filtering error generated in dynamic surface control and the derivative of the BLF, resulting in that it is difficult for the controller to limit the system state within the required range, which brings challenges to the adaptive control of state-constrained nonlinear systems.

[0004] External disturbances are also important factors affecting the performance of nonlinear systems. In actual operation, the system will be affected by various external disturbances and internal uncertainties, such as load changes, environmental noise, model parameter changes, etc. These factors will cause the system performance to deteriorate and even make the system lose stability. Traditional control methods often lack sufficient robustness when facing mismatched disturbances, and it is difficult to ensure good control performance of the system. As an effective tool, the disturbance observer can estimate the disturbance in the system and suppress the disturbance by introducing compensation in the control input. However, for disturbed nonlinear systems with state constraints, if the control scheme combining the traditional BLF method with the disturbance observer is adopted, then in the Lyapunov stability analysis, there will be difficult-to-handle cross terms between external disturbances and BLF derivatives, which will increase the difficulty of controller design. In the literature [BLF-based Adaptive Fuzzy DSC for a class of UncertainNonlinear Systems with Full State Constraints and Input Saturation using Disturbance Observer], BLF is placed in the first step of dynamic surface control design, which means that the cross terms between disturbance and BLF derivative need to be processed in each subsequent step of controller design, making the controller design process cumbersome and increasing the complexity of controller design.

[0005] In summary, for disturbed nonlinear systems with state constraints, providing an adaptive tracking control method with low design difficulty is a technical problem that needs to be solved urgently. Summary of the invention

[0006] In order to overcome the adverse effects of mismatched filter errors generated by dynamic surface control on state constraints, avoid the appearance of cross terms between disturbances and BLF derivatives, and solve the problem of complex controller design process, the present invention provides an adaptive tracking control method for a state-constrained disturbed nonlinear system.

[0007] To achieve the purpose of the present invention, the technical solution adopted by the present invention is as follows: a state-constrained disturbed nonlinear system adaptive tracking control method, comprising the following steps:

[0008] Step 1: Establish a mathematical model of a state-constrained nonlinear system with matched parameterized uncertainties and mismatched disturbances;

[0009] Step 2: According to the mathematical model, a disturbance observer is established for the mismatched disturbance;

[0010] Step 3: Set the state constraint conditions and reference signals of the system asymmetry, introduce state transformation, and transform the state constraint conditions into tracking error constraint conditions according to the state transformation;

[0011] Step 4: According to the mathematical model, establish the virtual controller and its corresponding first-order filter in the first n - 1 steps of the dynamic surface control;

[0012] Step 5: According to the mathematical model, design and introduce the BLF in the nth step of the dynamic surface control, and establish the adaptive controller.

[0013] Furthermore, the mathematical model of the above Step 1 is specifically as follows:

[0014]

[0015] y = x1,

[0016] where and are the state vector and control input of the system; and are known smooth functions; represents the matching parametric uncertainty; is the unknown parameter vector; is the mismatched disturbance; is the output of the system;

[0017] To achieve the design of the controller, the following assumptions are given:

[0018] Assumption 1: The unknown parameter vector θ belongs to a known compact set Θ, that is

[0019]

[0020] where is the known boundary of the parameter;

[0021] Assumption 2: The mismatched disturbance d i (x, t) is continuously differentiable and satisfies the following conditions:

[0022]

[0023] where μ i > 0 is the known boundary.

[0024] Furthermore, the disturbance observer described in the above Step 2 is designed as follows:

[0025]

[0026] where i = 2, …, n - 1; is the state vector of the disturbance observer; l i1,l i2 > 0 represents the gain of the disturbance observer;

[0027] Based on the above disturbance observer, the estimated values of the disturbance and its derivative are constructed as follows:

[0028]

[0029] where represents the estimated value of the disturbance, represents the estimated value of the disturbance derivative.

[0030] Furthermore, in the above step three, the asymmetric state constraint condition is set as:

[0031]

[0032] where Y represents the lower bound of the state x1(t), represents the upper bound of the state x1(t), and

[0033] The reference signal is set as x d , and x d satisfies the following conditions:

[0034] The reference signal x d and its first derivative exist and are available, and there exist known constants X d and such that

[0035] Furthermore, in the above step three, the following state transformation is introduced:

[0036] e1 = x1 - x d ,

[0037] e i = x i - y (i-1)f , i = 2, …, n,

[0038] where y (i-1)f is the filtered signal;

[0039] Based on the above state transformation, the state constraint condition can be transformed into the following tracking error constraint condition:

[0040] k b ≤ e1 ≤ k a ,

[0041] where k b = Y - X d ,

[0042] Furthermore, the virtual controller and its corresponding first-order filter in the first n - 1 steps of the dynamic surface control described in Step 4 above are designed as follows:

[0043]

[0044] where α1, α i (i = 2, …, n - 1) are the virtual controllers; k1, k i (i = 2, …, n - 1) > 0 are the virtual controller gain parameters; y 1f , y if (i = 2, …, n - 1) are the filtered signals of the virtual controllers after passing through the first-order filters; τ1, τ i (i = 2, …, n - 1) > 0 are the filtering time constants.

[0045] Furthermore, in Step 5 above, the following BLF is designed in the nth step of the dynamic surface control:

[0046]

[0047] where the function q(e1) is designed as follows:

[0048]

[0049] Based on the designed BLF, the adaptive controller is designed as follows:

[0050]

[0051] where k n > 0 is the controller gain parameter; Γ = diag[γ1, …, γ p (γ j > 0, j = 1, …, p) is the adaptive law gain matrix; represents the estimated value of θ; the function represents the value after passing through the projection algorithm, and the projection algorithm is designed as follows:

[0052]

[0053] where δ j > 0 is the design parameter.

[0054] Furthermore, the design of the disturbance observer gain and the controller gain should satisfy the following conditions:

[0055]

[0056] where k1, k j , ρ11 , ρ j1 , ρ j2 , ρ j3 , ρ j4 , l j1 , l j2 > 0, j = 2, …, n - 1.

[0057] Compared with the prior art, the advantages of the present invention are as follows:

[0058] 1. Existing research usually only considers matching disturbances and matching parametric uncertainties simultaneously. Since the disturbances of such systems are in the space spanned by the control inputs, the processing methods are relatively easy. However, for systems with mismatched disturbances and matching parametric uncertainties, when designing a controller, cross - terms of mismatched filtering errors and the derivative of the BLF, as well as mismatched disturbances and the derivative of the BLF, will appear in the expression of the derivative of the Lyapunov function. The difficulty in solving this problem lies in that when performing Lyapunov stability analysis, mathematical tools need to be used to skillfully scale and compensate the cross - terms in each step, thus increasing the complexity of controller design. The mathematical model provided by the present invention simultaneously considers mismatched disturbances and matching parametric uncertainties, and realizes the stable tracking control of the system through an improved method based on traditional adaptive control.

[0059] 2. An adaptive tracking control method proposed based on the mathematical model provided by the present invention has the following design core:

[0060] (1) The disturbance observer provided by the present invention can achieve exponential convergence of the disturbance estimation error, greatly accelerating its convergence speed, and its convergence speed can be reasonably adjusted according to the disturbance observer gain;

[0061] (2) Different from existing methods that introduce the BLF in the first step of dynamic surface control design, the present invention introduces the BLF in the nth step (the last step) of dynamic surface control, avoiding the appearance of cross - terms of mismatched filtering errors and the derivative of the BLF, as well as mismatched disturbances and the derivative of the BLF in the first n - 1 steps of control design, making the design of the controller reduce the process of scaling and compensating the cross - terms, and thus greatly simplifying the controller design process;

[0062] (3) In the control method provided by the present invention, the tracking error of the system can be reasonably adjusted according to the desired performance requirements through the parameters of the adaptive controller, thereby improving the control accuracy.

[0063] 3. The adaptive controller designed based on the disturbance observer and the improved BLF method provided by the present invention can effectively suppress the adverse effects brought by unmatched disturbances and matched parametric uncertainties to the system, while ensuring the satisfaction of state constraints; by comprehensively regulating the controller gains and observer gains in the control protocol, the exponential convergence of the disturbance estimation error can be achieved, the system response speed can be accelerated, the transient performance of the system can be improved, the robustness of the system can be enhanced, the tracking error can be effectively reduced, and the control accuracy can be improved. Description of the Drawings

[0064] Figure 1 is the control block diagram of an adaptive tracking control method for a state-constrained and disturbed nonlinear system of the present invention;

[0065] Figure 2 is the structure diagram of the magnetic levitation ball system;

[0066] Figure 3 is the curve of the system state x1(t) in the simulation of the present invention;

[0067] Figure 4 is the curve of the tracking error in the simulation of the present invention;

[0068] Figure 5 is the curve of the disturbance observation error in the simulation of the present invention;

[0069] Figure 6 is the curve of the parameter estimation error in the simulation of the present invention;

[0070] Figure 7 is the curve of the control signal in the simulation of the present invention. Detailed Embodiment

[0071] The following details the embodiments of the present invention. To facilitate the understanding of the present invention, the following further explains the present invention with specific embodiments in conjunction with the drawings.

[0072] The design idea of the present invention is as follows: First, establish a mathematical model of a class of state-constrained nonlinear systems containing matched parametric uncertainties and unmatched disturbances; second, for the unmatched disturbances in the model, design a disturbance observer to estimate the disturbances and their derivatives; then, introduce a state transformation to convert the state constraints into tracking error constraints; finally, derive an adaptive controller based on dynamic surface control. By placing the BLF at the last step of the dynamic surface control, the design process of the controller is greatly simplified.

[0073] See Figure 1 , an adaptive tracking control method for a state-constrained and disturbed nonlinear system provided by the present invention, includes the following steps:

[0074] Step 1: Establish a mathematical model of a class of state-constrained nonlinear systems with matched parametric uncertainties and unmatched disturbances, as follows:

[0075]

[0076] y = x1,

[0077] where and are the state vector and control input of the system; and are known smooth functions; represents the matched parametric uncertainty; is the vector of unknown parameters; is the unmatched disturbance; is the output of the system;

[0078] To achieve the design of the controller, the following assumptions are given:

[0079] Assumption 1: The vector of unknown parameters θ belongs to a known compact set Θ, i.e.,

[0080]

[0081] where is the known bound of the parameter;

[0082] Assumption 2: The unmatched disturbance d i (x, t) is continuously differentiable and satisfies the following conditions:

[0083]

[0084] where μ i > 0 is the known bound.

[0085] Step 2: According to the mathematical model, for the unmatched disturbance, establish a disturbance observer as follows:

[0086]

[0087] where i = 2, …, n - 1; is the state vector of the disturbance observer; l i1 , l i2 > 0 represents the gain of the disturbance observer;

[0088] Based on the above disturbance observer, construct the estimated values of the disturbance and its derivative as follows:

[0089]

[0090] where, represents the estimated value of the disturbance, Represents the estimated value of the perturbation derivative.

[0091] Furthermore, based on the perturbation estimation value, the perturbation estimation error is defined as follows:

[0092]

[0093] Differentiating the perturbation estimation error, the dynamic system of the perturbation estimation error can be obtained as follows:

[0094]

[0095] The above dynamic system of the perturbation estimation error can be described in the following compact form:

[0096]

[0097] where the form of each vector is:

[0098]

[0099] Since E i is a Hurwitz matrix, therefore, for any given positive definite symmetric matrix there exists a positive definite matrix P i , such that the Lyapunov equation holds;

[0100] Define the Lyapunov candidate function Differentiating it gives:

[0101]

[0102] where λ min (Q i ) represents the minimum eigenvalue of Q i , λ max (P i ) represents the maximum eigenvalue of P i ; it can be seen from the above formula that the perturbation observation error is exponentially convergent and uniformly ultimately bounded.

[0103] Step 3: Set the state constraint conditions and reference signals of the system asymmetry, introduce a state coordinate transformation, and transform the state constraint conditions into tracking error constraint conditions according to the state coordinate transformation;

[0104] Set the state constraint conditions of the system asymmetry as:

[0105]

[0106] where Y represents the lower bound of the state x1(t), Denote the upper bound of the state \(x_1(t)\), and

[0107] Set the reference signal as \(x\) d , and \(x\) d satisfies the following conditions:

[0108] The reference signal \(x\) d and its first derivative exist and are available, and there exist known constants X d and such that

[0109] Introduce the state transformation as follows:

[0110] \(e_1 = x_1 - x\) d ,

[0111] \(e\) i \(= x\) i \(- y\) (i-1)f , \(i = 2, \ldots, n\),

[0112] where \(y\) (i-1)f is the filtered signal;

[0113] Based on the above state transformation, the state constraint conditions can be transformed into the following tracking error constraint conditions:

[0114] \(k\) b \(\leq e_1 \leq k\) a ,

[0115] where \(k\) b \(=\) Y \(-\) X d ,

[0116] Step 4: According to the mathematical model, establish the virtual controller and its corresponding first-order filter in the first \(n - 1\) steps of the dynamic surface control;

[0117] The steps to design the controller using the dynamic surface control method are as follows:

[0118] (1) Differentiate the variable \(e_1\), and its dynamic system can be obtained as:

[0119]

[0120] Design the virtual controller for the state \(x_2\) as follows:

[0121]

[0122] where \(k_1 > 0\) is the virtual controller gain parameter;

[0123] Design a first-order filter as follows to filter the virtual controller α1:

[0124]

[0125] where τ1 > 0 is the filtering time constant, and y 1f is the filtered signal of the virtual controller α1 after passing through the first-order filter;

[0126] Define the filtering error y1 = y 1f −α1 and the candidate Lyapunov function Differentiating V1 gives:

[0127]

[0128] Use Young's inequality to scale the cross-term in the above equation:

[0129]

[0130] where ρ 11 > 0 is a design parameter; furthermore satisfies:

[0131]

[0132] (2) Differentiating the variable e2, its dynamic system can be obtained as:

[0133]

[0134] Design the virtual controller for the state x3 as follows:

[0135]

[0136] where k2 > 0 is the virtual controller gain parameter;

[0137] Design a first-order filter as follows to filter the virtual controller α2:

[0138]

[0139] where τ2 > 0 is the filtering time constant, and y 2f is the filtered signal of the virtual controller α2 after passing through the first-order filter;

[0140] Define the filtering error y2 = y 2f −α2 and the candidate Lyapunov function Differentiating V2 gives:

[0141]

[0142] Use Young's inequality to scale the cross - term in the above formula:

[0143]

[0144] where ρ 21 , ρ 21 , ρ 22 , ρ 23 , ρ 24 > 0 are design parameters; furthermore satisfies:

[0145]

[0146] (3) Differentiate the variable e i (i = 3, …, n - 1), and its dynamic system can be obtained as:

[0147]

[0148] For the state x i+1 design the virtual controller as follows:

[0149]

[0150] where k i > 0 is the virtual controller gain parameter;

[0151] Design the following first - order filter to filter the virtual controller α i :

[0152]

[0153] where τ i > 0 is the filtering time constant, and y if is the filtered signal of the virtual controller α i after passing through the first - order filter;

[0154] Define the filtering error y i = y if - α i and the candidate Lyapunov function Differentiating V i yields:

[0155]

[0156] Use Young's inequality to scale the cross - term in the above formula:

[0157]

[0158] where ρ i1 , ρ i1 , ρi2 , ρ i3 , ρ i4 > 0 is a design parameter; furthermore Satisfy:

[0159]

[0160] Step Five: According to the mathematical model, introduce the BLF in the nth (last) step of the dynamic surface control to establish an adaptive controller.

[0161] Differentiate the variable e n to obtain its dynamic system as:

[0162]

[0163] To ensure that the tracking error constraint holds, design the BLF as follows in the nth (last) step of the dynamic surface control:

[0164]

[0165] where the function q(e1) is designed as follows:

[0166]

[0167] To ensure that the parameter estimate value is within the given bounds, define the following parameter estimation error vector:

[0168]

[0169] where, represents the estimated value of θ, represents the value after the projection algorithm, and the projection algorithm is designed as follows:

[0170]

[0171] where δ j > 0 is a design parameter;

[0172] According to the above parameter estimation error vector, define the function V θ as follows:

[0173]

[0174] where γ j > 0 is a design parameter;

[0175] Based on the above-designed V b and V θ , define the global candidate Lyapunov function whose derivative is calculated as follows:

[0176]

[0177] Design the adaptive controller for the state - constrained and disturbed nonlinear system as follows according to the above formula:

[0178]

[0179] where \(k\) n \(> 0\) is the controller gain parameter, \(\Gamma=\text{diag}[\gamma_1,\cdots,\gamma\) p is the adaptive - law gain matrix;

[0180] By placing \(V\) b in the \(n\) - th (last) step of the dynamic - surface control, the cross - terms of the mismatched filtering error and the derivative of the BLF, as well as the mismatched disturbance and the derivative of the BLF, are avoided, greatly simplifying the controller design process;

[0181] Substitute the above - designed adaptive controller into to obtain:

[0182]

[0183] Since Therefore

[0184] From the expression of \(\eta\), the tracking error of the system can be reasonably adjusted according to the parameters \(\rho\) j1 and \(\rho\) j4 ;

[0185] In the above controller, the design of the disturbance - observer gain and the controller gain should satisfy the following conditions:

[0186]

[0187] where \(k_1,k\) j ,\(\rho\) 11 ,\(\rho\) j1 ,\(\rho\) j2 ,\(\rho\) j3 ,\(\rho\) j4 ,\(l\) j1 ,\(l\) j2 \(> 0,j = 2,\cdots,n - 1\); at this time, Satisfy:

[0188]

[0189] Because \(\rho\) j1 ,\(\rho\) j2 ,\(\rho\) j3 ,\(\rho\) j4 and \(\mu\) j are positive constants, so \(\eta\) is bounded; and because the lumped filtering error is bounded. Therefore, it can be seen from the above equation that the closed-loop system is uniformly ultimately bounded. Thus, V b is bounded; from the boundedness of V b it can be seen that the tracking error constraint is satisfied, and furthermore, the state constraint is satisfied.

[0190] To verify the effectiveness of the adaptive tracking control method for a state-constrained and disturbed nonlinear system of the present invention, taking the magnetic levitation ball system as an example, the control algorithm design and MATLAB simulation verification are carried out as follows:

[0191] As Figure 2 shown, the magnetic levitation ball system consists of a laser sensor, a small ball, an electromagnet, a drive circuit, and an A / D converter. Its working principle is as follows: when the small ball moves vertically up and down under the magnet, the laser sensor at the base detects the distance from the bottom of the small ball to the surface of the laser sensor and generates a corresponding voltage. The voltage signal is fed back to the controller, and the control amount is calculated to control the current of the electromagnet coil, thereby generating a magnetic force to balance the gravity of the small ball, so that the small ball stably floats at any position within a certain area below the magnet.

[0192] According to the working principle of the magnetic levitation ball system, its mathematical model is established:

[0193]

[0194] where X represents the air-gap distance between the center of mass of the small ball and the magnetic pole of the electromagnet, F(i,x) is the electromagnetic attraction force function, m is the mass of the small ball, g is the acceleration due to gravity, K is the magnetic constant, i is the coil current, R is the nominal resistance of the coil, L is the inductance of the coil, U is the input voltage, R Δ represents the perturbation of the coil resistance value caused by coil heating during the operation of the system, f d represents the influence of external wind disturbance on the force acting on the small ball in the vertical direction;

[0195] Introduce the coordinate transformation After this coordinate transformation, the above magnetic levitation system can be expressed as the following state-space model:

[0196]

[0197] where d = f d / m.

[0198] The model parameters of the system are shown in Table 1.

[0199] Table 1 Model Parameters

[0200]

[0201] Since the magnetic levitation system does not satisfy the strict-feedback type system, the state transformation is introduced After this state transformation, the above-mentioned maglev system is transformed into a strict-feedback type system as follows:

[0202] z1 = z2,

[0203]

[0204] where

[0205] In the simulation settings, the disturbance signal is set to The reference signal is set to x d = 0.01 + 0.001sin(0.5πt) m, and the system state constraint is set to Y = 0.007 m,

[0206] An adaptive controller is designed for the above system by using the method provided by the present invention. The initial values of the system and the controller are selected as x1(0) = 0.01, x2(0) = 0, x3(0) = 0.1, p 11 (0) = 0.015, p 12 (0) = 0, y 1f (0) = 0, y 2f (0) = -9.3. The filtering time constants are selected as τ1 = τ2 = 0.001, the projection parameter is selected as δ = 0.01, the gain parameters are selected as k1 = k2 = k3 = 80, l 21 = 100, l 22 = 20, γ = 0.015.

[0207] The simulation curves are as Figures 3 - 7 shown.

[0208] Figure 3 is the curve of the system state x1(t) under the action of the controller. It can be seen from the figure that the controller can achieve disturbance suppression, the state x1(t) can track the reference signal well, and x1(t) always remains within the required boundaries; Figure 4 is the curve of the tracking error under the action of the controller. It can be seen from the figure that the controller shows good tracking performance, the tracking error can quickly converge to the vicinity of the origin, and the tracking accuracy can be improved to 10 -4 m; Figure 5 is the curve of the disturbance observation error. It can be seen that the disturbance observer proposed by the present invention can reach the steady state within 0.05 s, quickly estimate the disturbance, the disturbance estimation error converges in an exponential form, and is uniformly ultimately bounded; Figure 6This is the parameter estimation situation. It can be seen from the figure that the adaptive law proposed by the present invention can achieve fast and accurate estimation of unknown parameters, and the parameter estimation error converges to zero exponentially with time; Figure 7 shows the variation curve of the control signal with time. From Figure 7 it can be seen that when the system reaches the steady state, the input voltage remains at about 10V.

[0209] It should be noted that the above embodiments are only an application case of the present invention and do not limit the protection scope of the present invention. Any equivalent replacement or substitution based on the above technical solutions shall fall within the protection scope of the present invention.

Claims

1. An adaptive tracking control method for a state-constrained and disturbed nonlinear system, characterized in that: Including the following steps Step 1: Establish a mathematical model of a state-constrained nonlinear system with matched parametric uncertainties and unmatched disturbances; Step 2: According to the mathematical model, construct a disturbance observer for the unmatched disturbances; Step 3: Set asymmetric state constraint conditions and a reference signal, introduce a state transformation, and transform the state constraint conditions into tracking error constraint conditions according to the state transformation; Step 4: According to the mathematical model, establish virtual controllers and their corresponding first-order filters in the first n - 1 steps of dynamic surface control; Step 5: According to the mathematical model, design and introduce a BLF in the nth step of dynamic surface control to establish an adaptive controller.

2. The adaptive tracking control method for a state-constrained and disturbed nonlinear system according to claim 1, characterized in that: The mathematical model in Step 1 is as follows: y = x1, where and are the state vector and control input of the system; and are known smooth functions; represents the matching parametric uncertainty; is the unknown parameter vector; is the mismatched disturbance; is the output of the system; To realize the design of the controller, the following assumption conditions are given: Assumption 1: The unknown parameter vector θ belongs to a known compact set Θ, that is wherein are known boundaries of the parameters; Hypothesis 2: Mismatch perturbation d i (x, t) is continuously differentiable and satisfies the following conditions: where μ i > 0 is a known boundary.

3. An adaptive tracking control method for a state-constrained and disturbed nonlinear system according to claim 1, characterized in that: The disturbance observer described in Step 2 is designed as follows: where \(i = 2,\ldots,n - 1\); is the state vector of the disturbance observer; \(l\) i1 , \(l\) i2 > 0 represents the gain of the disturbance observer; Based on the above disturbance observer, the estimated values of the disturbance and its derivative are constructed as follows: wherein represents an estimated value of the perturbation, represents an estimated value of the perturbation derivative.

4. An adaptive tracking control method for a state-constrained and perturbed nonlinear system according to claim 1, characterized in that: In Step 3, the asymmetric state constraint conditions are set as: wherein Y represents the lower bound of the state x1(t), represents the upper bound of the state x1(t), and Set the reference signal as x d , and x d satisfies the following conditions: Reference signal x d and its first derivative exist and are available, and there exist known constants X d and such that 5. The adaptive tracking control method for a state-constrained and disturbed nonlinear system according to claim 1, characterized in that: In Step 3, the state transformation is introduced as follows: e1 = x1 - x d , e i = x i - y (i-1)f , i = 2, …, n, where y (i-1)f is the filtered signal; Based on the above state transformation, the state constraint conditions can be transformed into the following tracking error constraint conditions: k b ≤e1≤k a , where k b = Y - X d , 6. An adaptive tracking control method for a state-constrained and disturbed nonlinear system according to claim 1, characterized in that: The virtual controllers and their corresponding first-order filters in the first n - 1 steps of dynamic surface control described in Step 4 are designed as follows: where α1, α i (i = 2, …, n - 1) are virtual controllers; k1, k i (i = 2, …, n - 1) > 0 are virtual controller gain parameters; y 1f 、y if (i = 2, …, n - 1) are the filtered signals of the virtual controllers after passing through a first-order filter; τ1, τ i (i = 2, …, n - 1) > 0 are the filtering time constants.

7. An adaptive tracking control method for a state-constrained perturbed nonlinear system according to claim 1, characterized in that: In Step 5, the following BLF is designed in the nth step of dynamic surface control: Among them, the function q(e1) is designed as follows: Based on the designed BLF, the adaptive controller is designed as follows: where k n > 0 is the controller gain parameter; Γ = diag[γ1,…,γ p (γ j > 0, j = 1,…,p) is the adaptive law gain matrix; denotes the estimated value of θ; the function denotes the value after passing through the projection algorithm, and the projection algorithm is designed as follows: where δ j > 0 is a design parameter.

8. An adaptive tracking control method for a state-constrained and perturbed nonlinear system according to claim 1, characterized in that: The design of the disturbance observer gain and the controller gain should meet the following conditions: where k1, k j , ρ 11 , ρ j1 , ρ j2 , ρ j3 , ρ j4 , l j1 , l j2 > 0, j = 2, …, n - 1.

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