A nonlinear system tracking control method with piecewise discontinuous output constraint

Abstract

The application provides a nonlinear system tracking control method with segmented discontinuous output constraint, to solve the problem that the constraint boundary function is discontinuous at a certain moment but bounded in a certain period of time when the system starts to run, and solve the problem that the system output is not limited by the constraint at a certain moment. Unlike the requirement of continuity and boundedness of the constraint boundary function and its high-order derivative in most existing methods, the application only requires the continuity and boundedness of the converted constraint boundary function and its first-order derivative, relaxes the requirement of the constraint boundary function, greatly improves the practicability of the control strategy, and reduces the complexity and computational burden of the controller. In addition, by changing the discontinuous time, without changing the controller structure and design parameters, the proposed control strategy can also be applied to other constraint conditions, such as continuous constraint conditions, discontinuous constraint conditions at a certain moment, and system output not limited by the constraint at a certain moment, further enhancing the practicability of the control strategy.

Description

Technical Field

[0001] This invention relates to the field of constraint control technology for nonlinear systems, and in particular to a tracking control method for nonlinear systems with piecewise discontinuous output constraints. Background Technology

[0002] In practical engineering applications, systems may be subject to output constraints due to various factors such as the surrounding environment or user requirements. If these constraints are not considered, the system may lose stability. It is worth noting that many existing control strategies only address cases where the constraint boundary function is continuous. In reality, constraint boundary functions are often discontinuous. For example, a drone may transition from entering one tunnel to another and then into open space; in such cases, traditional methods are no longer applicable. In recent years, although some research has designed control strategies for discontinuous constraint boundary functions, such as using transfer functions to address the problem of system output being constrained over a period of time, these studies only address the transition of system output between constrained and unconstrained states. They do not address the issue that the constraint boundary function is discontinuous but bounded at a certain point in the system's initial operation. Furthermore, the requirements for the constraint boundary function in these studies are relatively high. Therefore, researching a tracking control method for nonlinear systems with piecewise discontinuous output constraints still has potential application value. Summary of the Invention

[0003] This invention addresses the output constraint control problem of a class of uncertain nonlinear systems by proposing a neural network adaptive control strategy. This strategy solves the problem that the constraint boundary function is discontinuous but bounded at a certain moment during the initial period of system operation, and also addresses the problem that the system output is unconstrained at a certain moment. Furthermore, unlike most existing methods that require the constraint boundary function and its higher-order derivatives to be continuous and bounded, this invention only requires the transformed constraint boundary function and its first-order derivative to be continuous and bounded. This relaxes the requirements on the constraint boundary function, greatly improving the practicality of the control strategy and reducing the complexity and computational burden of the controller. Moreover, by changing the discontinuous moments, the proposed control strategy can be applied to other constraint conditions without altering the controller structure and design parameters, such as continuous constraints, constraints that are discontinuous at specific moments, and situations where the system output is unconstrained at a certain moment, further enhancing the practicality of the control strategy.

[0004] The purpose of this invention is to provide a tracking control method for a class of uncertain nonlinear systems with piecewise discontinuous output constraints. This method ensures that the system output does not violate the piecewise discontinuous constraint conditions and that the system output can closely track the desired trajectory.

[0005] The present invention provides a tracking control method for a nonlinear system with piecewise discontinuous output constraints, comprising the following steps:

[0006] Step 1: Establish a dynamic model for a class of uncertain nonlinear systems;

[0007] Step 2: Construct the transfer function and combine it with the constraint boundary function to construct the connection function, transforming the discontinuous constraint boundary function into a continuous function;

[0008] Step 3: Construct another transfer function and perform an error transformation based on the tracking error to change the constrained output to an unconstrained output.

[0009] Step 4: Design a neural network adaptive control scheme based on output constraints. Relax the condition that the higher-order derivatives of the constraint boundary function are continuous and bounded in the traditional control method to ensure that the system output does not violate the piecewise discontinuous constraint and that the system output closely tracks the desired trajectory.

[0010] Step 5: By changing the discontinuous moments, the designed control scheme is applied to continuous constraints, constraints that are discontinuous at specific moments, and scenarios where the system output is not constrained at a certain moment.

[0011] Furthermore, in step 1, a dynamic model for an uncertain nonlinear system is established as follows:

[0012]

[0013] in, u∈R and y∈R are the system's state, input, and output variables, respectively. and Let i represent an unknown smooth nonlinear function and an unknown control gain, respectively, where i = 1, ..., n-1.

[0014] Furthermore, the transfer function and connection function constructed in step 2 are expressed as follows:

[0015] The constructed transfer function Φ(t) is:

[0016]

[0017] Where △1=l1(T1-t)>0, l1>0 is a constant that the user can choose, and T1>0 is a given discontinuous time.

[0018] Furthermore, the transfer function Φ(t) constructed in step 2 has the following specific properties:

[0019] ①Φ(t) is strictly monotonically increasing in t∈[0,T1);

[0020] ②The value of Φ(t) is always 1 in t∈[T1,+∞);

[0021] ③Φ(t) and All are continuous and bounded.

[0022] Next, based on the transfer function Φ(t), a connection function K(t) is constructed to transform the discontinuous constraint boundary function into a continuous function, as follows:

[0023] Based on the transfer function Φ(t), the constructed connection function K(t) is:

[0024]

[0025] Where, k1(t)=k c1 (t)-k c2 (t) and These represent the tracking errors z1 = x1 - y in the two stages t ∈ [0, T1) and t ∈ [T1, +∞), respectively. r The constraint boundary function, y r Given the desired output trajectory, Represents |y r The maximum value of |k c1 With k c2 This represents the constraint boundary function of the system output x1 in the two stages t∈[0,T1) and t∈[T1,T2), respectively. T2 is the given time when the system output changes from constrained to unconstrained, and T2>T1>0. Furthermore, the connection function K(t) constructed in step 2 has the following properties:

[0026] ①K(t)>0 is continuous and bounded in t∈[0,+∞);

[0027] ②K(t)≡k2(t) holds true for t∈[T1,+∞);

[0028] ③ The condition is continuous and bounded in t∈[0,+∞), meaning there exists a positive number K. max , making Established.

[0029] Furthermore, in step 3, another transfer function is constructed, and an error transformation is performed using the tracking error to change the constrained output to an unconstrained output, as detailed below:

[0030] Constructed transfer function for:

[0031]

[0032] Where △2=l2(t-T2)>0, l2>0 is a constant that the user can choose. Furthermore, the transfer function constructed in step 3... It has the following properties:

[0033] ① The value of t in [0, T2) is always 1;

[0034] ② It is strictly monotonically decreasing in t∈[T2,+∞), and

[0035] ③ and The condition is continuous and bounded in t∈[0,+∞).

[0036] For ease of representation, the independent variables of certain functions will be omitted in this invention. Furthermore, step 4 designs a neural network adaptive control scheme based on output constraints, relaxing the condition that the higher-order derivatives of the constraint boundary functions are continuous and bounded in traditional control methods, reducing computational burden and controller complexity, ensuring that the system output does not violate the piecewise discontinuity constraint condition, and that the system output closely tracks the desired trajectory. Specifically, this includes the following steps:

[0037] Step 4.1, to facilitate controller design, define the tracking error z1 and the virtual error z2. i The details are as follows:

[0038] z1 = x1 - y r (5)

[0039] z i =x i -α if (6)

[0040] Among them, i=2,...,n,α if It is the output of a first-order filter:

[0041]

[0042] Where, ε i >0 is a design parameter, α i-1 (i = 2, ..., n) are virtual control variables. Next, the filtering error is defined:

[0043] y i =α if -α i-1 ,i=2,...,n.(8)

[0044] From equations (6) and (8), we can deduce that:

[0045] x i =zi +y i +α i-1 ,i=2,...,n.(9)

[0046] From equations (7) and (8), we can deduce that:

[0047]

[0048] To address the issue of unconstrained system output when t≥T2, the tracking error z1 is transformed:

[0049]

[0050] Step 4.2: Design the controller using the approximation principle of neural networks and backstepping techniques. The specific steps are as follows:

[0051] S1: From z1 = x1 - y r It can be deduced that:

[0052]

[0053] Therefore, we can further conclude that:

[0054]

[0055] The Lyapunov function is selected as follows:

[0056]

[0057] in, It is the parameter estimation error. This is an estimate of the virtual parameter b1, and β1>0 is a design parameter. According to equation (13), the derivative of V1 with respect to time t is easily derived as:

[0058]

[0059] Where γ1=K 2 >0, Define ρ1 = K 2 -η 2 According to Young's inequality, it is easy to obtain:

[0060]

[0061] Next, we define a function for the unknown smoothness. By approximating the unknown smooth function F1 using a radial basis function neural network, we can obtain:

[0062]

[0063] in, It is the input signal of the neural network, w1∈R s and χ1(ξ1)∈R s Let represent the ideal weight matrix and basis functions of the neural network, respectively, where s is the number of neurons, and δ1(ξ1) represents the approximation error of the neural network, satisfying . in Let be some unknown constant. From Young's inequality, we can obtain:

[0064]

[0065] in, r1>0 represents unknown virtual parameters, non-negative scalar functions, and design parameters, respectively.

[0066] Substituting equation (19) into equation (15), we can derive:

[0067]

[0068] Furthermore, the virtual control variable α1 and the adaptive law are designed. as follows:

[0069]

[0070] Where τ1>0 and c1>0 are both design parameters. From Young's inequality and equation (9), we can obtain:

[0071]

[0072] Substituting equations (21)-(24) into equation (20), we can derive:

[0073]

[0074] in, It is a constant. It is an unknown constant term, and

[0075] Next, in step S i In the diagram, where i = 2, ..., n-1, the designed virtual controller α... i With Adaptive Law as follows:

[0076]

[0077]

[0078] Among them, c i >0, β i >0 and τ i >0 represents the design parameters selected by the user.

[0079] Next, in step S n , the designed virtual controller u and the adaptation law are as follows:

[0080]

[0081] where c n > 0, β n > 0 and τ n > 0 are all design parameters selected by the user.

[0082] By choosing the Lyapunov function as:

[0083]

[0084] Using a derivation process similar to S1, it can be deduced that:

[0085]

[0086] where is an unknown constant term.

[0087] Considering the uncertain nonlinear system (1), if the initial condition satisfies |η(0)| < K(0), and there exists a constant B > 0 such that V n (0) ≤ B, then under the action of the designed controller as in Eq. (30), and the designed parameter adaptation laws as in Eqs. (22), (27), (31), by choosing appropriate design parameters, the designed control strategy can make all signals of the closed-loop system bounded, the system output can closely track the desired trajectory, and the system output does not violate the piecewise discontinuous constraint conditions.

[0088] Furthermore, since and are both compact sets, there exists a constant ζ in j > 0 such that |M j | ≤ ζ j holds, where j = 1,..., n - 1. It can be further obtained that:

[0089]

[0090] where Therefore, it can be obtained that V n (t) ≤ V n (0)e -μt + H n / μ, that is Thus, under the condition that the initial condition satisfies |z1(0)| < K(0), there is

[0091] where \(j = 1,\cdots,n - 1\) and \(i = 1,\cdots,n\).

[0092] Furthermore, since it can be deduced that Since it is easy to obtain Since \(x1 = z1 + y\) r , it can be deduced that Thus and Therefore Through a similar analysis process, it can be proved that all internal signals of the closed-loop system are bounded, and at the same time, from it is easy to obtain that the system output can closely track the desired trajectory.

[0093] Furthermore, based on the structures of the connection function (3) constructed in step 2 and the transfer function (4) constructed in step 3, next, analyze that the system output does not violate the piecewise discontinuous constraint conditions, as follows:

[0094] Since in the interval \(t\in[0,T2)\), it is easy to obtain \(\eta = z\) l , thus it can be obtained that:

[0095]

[0096] Therefore, in the interval \(0\leq t < T1\), it is easy to obtain That is In the interval \(T1\leq t < T2\), it can be obtained that That is That is, the system output is not restricted by the constraints in the intervals \(t\in[0,T1)\) and \(t\in[T1,T2)\).

[0097] Next, since in the interval \(t\in[T2,+\infty)\), \(\varPhi(t)=1\), \(K(t)=k2(t)\), it is easy to deduce Since it can be further deduced that \(|x1|<\infty\), that is, the system output is not restricted by the constraints in the interval \(t\in[T2,+\infty)\).

[0098] In summary, it can be obtained that the system output does not violate the piecewise discontinuous constraint conditions.

[0099] Furthermore, in step 5, by changing the discontinuous time, the designed control scheme is applied to continuous constraints, constraints that are discontinuous at specific times, and scenarios where the system output is not restricted by constraints at a certain time, as follows:

[0100] ① When \(T1>0\) and \(T2 = +\infty\), it can be obtained that:

[0101] Transfer function The transfer function Φ(t) is consistent with the structure of Equation (2). Therefore, the structure of the connection function K(t) is consistent with Equation (3), and it can be further obtained that the constraint boundary function is bounded, but discontinuous at the moment t = T1.

[0102] ② When T1 = 0 and 0 < T2 < +∞, it can be obtained that:

[0103] Transfer function is consistent with the structure of Equation (4), and the transfer function Φ(t) = 1. Therefore, the connection function K(t) = k2(t), and it can be further obtained that the constraint boundary function is continuous in the interval t ∈ [0, T2), and the system output is not restricted by the constraint at the moment t = T2.

[0104] ③ When T1 = 0 and T2 = +∞, it is easy to obtain that:

[0105] Transfer function The transfer function Φ(t) = 1. Therefore, the connection function K(t) = k2(t), and it can be further obtained that the constraint boundary function is always continuous and bounded.

[0106] Furthermore, similar to the design and analysis process in Step 4, the feasibility and effectiveness of the designed control strategy can be proved.

[0107] In summary, by changing the discontinuous moment, without changing the design parameters and the controller structure, the designed control strategy can be used for continuous constraints, discontinuous constraints at specific moments, and the case where the system output is not restricted by constraints at a certain moment.

[0108] Beneficial effects

[0109] 1. A nonlinear system tracking control method with piecewise discontinuous output constraints proposed by the present invention can not only solve the problem that the constraint boundary function is discontinuous but bounded at a certain moment within a period when the system starts to operate, but also solve the problem that the system output is not restricted by constraints at a certain moment, and can ensure that the system output closely tracks the desired trajectory, thereby improving the practicability of the control strategy.

[0110] 2. Different from most existing methods that require the existence and boundedness of the constraint boundary function and its higher-order derivatives, the present invention only requires that the transformed constraint boundary function and its first-order derivative are continuous and bounded, thereby relaxing the restrictions on the constraint boundary function, significantly simplifying the design process of the controller, and greatly reducing the computational burden, which is more conducive to the implementation of the controller.

[0111] 3. The control strategy proposed in this invention can also be applied to other output constraint scenarios, such as traditional continuous constraints, discontinuous constraints at a certain moment, and unconstrained system output at a certain point in time, while keeping the controller structure and design parameters unchanged. This enhances the practical applicability of the control strategy and improves the feasibility of the algorithm. Attached Figure Description

[0112] Figure 1 This is the control block diagram of the nonlinear system tracking control based on piecewise discontinuous output constraints of the present invention. Detailed Implementation

[0113] In industrial applications, the state of a nonlinear system is measured by sensors. The measured state is then transmitted to a neural network adaptive controller. The difference between the measured state and the desired signal, combined with the transformation variable obtained after error transformation of the transfer function, along with the transfer function and tracking error, are all transmitted to the controller. The controller processes the received signals according to the control law and then uses actuators to control the nonlinear system.

[0114] This invention provides a tracking control method for a nonlinear system with piecewise discontinuous output constraints, such as... Figure 1 As shown, the specific steps are as follows:

[0115] Step 1: Establish a dynamic model for a class of uncertain nonlinear systems:

[0116]

[0117] in, u∈R and y∈R are the system's state, input, and output variables, respectively. and Let i represent an unknown smooth nonlinear function and an unknown control gain, respectively, where i = 1, ..., n-1.

[0118] Step 2: Construct the transfer function and combine it with the constraint boundary function to build the connection function, transforming the discontinuous constraint boundary function into a continuous function.

[0119] The constructed transfer function Φ(t) is:

[0120]

[0121] Where △1=l1(T1-t)>0, l1>0 is a constant that the user can choose, and T1>0 is a given discontinuous time.

[0122] It is worth noting that the constructed transfer function Φ(t) has the following specific properties:

[0123] ①Φ(t) is strictly monotonically increasing in t∈[0,T1);

[0124] ②The value of Φ(t) is always 1 in t∈[T1,+∞);

[0125] ③Φ(t) and All are continuous and bounded.

[0126] Next, based on the transfer function Φ(t), a connection function K(t) is constructed to transform the discontinuous constraint boundary function into a continuous function:

[0127] Based on the transfer function Φ(t), the constructed connection function K(t) is:

[0128]

[0129] Where, k1(t)=k c1 (t)-k c2 (t) and These represent the tracking errors z1 = x1 - y in the two stages t ∈ [0, T1) and t ∈ [T1, +∞), respectively. r The constraint boundary function, y r Given the desired output trajectory, Represents |y r The maximum value of |k c1 With k c2 Then, these represent the constraint boundary functions of the system output x1 in the two stages t∈[0,T1) and t∈[T1,T2), respectively. T2 is the given moment when the system output changes from being constrained to being unconstrained, and T2>T1>0.

[0130] It is worth noting that the constructed connection function K(t) has the following properties:

[0131] ①K(t)>0 is continuous and bounded in t∈[0,+∞);

[0132] ②K(t)≡k2(t) holds true for t∈[T1,+∞);

[0133] ③ The condition is continuous and bounded in t∈[0,+∞), meaning there exists a positive number K. max , making Established.

[0134] Step 3: Construct another transfer function and perform an error transformation based on the tracking error to change the constrained output to an unconstrained output.

[0135] Constructed transfer function for:

[0136]

[0137] Where △2=l2(t-T2)>0, l2>0 is a constant that the user can choose. It is worth noting that the constructed transfer function... It has the following properties:

[0138] ① The value of t in [0, T2) is always 1;

[0139] ② It is strictly monotonically decreasing in t∈[T2,+∞), and

[0140] ③ and The condition is continuous and bounded in t∈[0,+∞).

[0141] Step 4: Based on steps 2 and 3, design a neural network adaptive control scheme based on output constraints. Relax the condition that the higher-order derivatives of the constraint boundary functions are continuous and bounded in the traditional control method, reduce the computational burden and controller complexity, ensure that the system output does not violate the piecewise discontinuous constraint condition, and ensure that the system output closely tracks the desired trajectory.

[0142] The design of a controller based on piecewise discontinuous output constraints includes the following steps:

[0143] Step 4.1, to facilitate controller design, define the tracking error z1 and the virtual error z2. i The details are as follows:

[0144] z1 = x1 - y r (5)

[0145] z i =x i -α if (6)

[0146] Among them, i=2,...,n,α if It is the output of a first-order filter:

[0147]

[0148] Where, ε i >0 is a design parameter, α i-1 (i = 2, ..., n) are virtual control variables. Next, the filtering error is defined:

[0149] y i =α if -α i-1 ,i=2,...,n.(8)

[0150] It can be deduced from equations (6) and (8) that:

[0151] x i = z i + y i + α i-1 , i = 2, ..., n. (9)

[0152] It can be deduced from equations (7) and (8) that:

[0153]

[0154] To solve the problem that when t ≥ T2, the system output is not restricted by constraints, the tracking error z1 is transformed:

[0155]

[0156] Step 4.2, design a controller by means of the approximation principle of neural networks and backstepping technology. The specific steps are as follows: S1: From z1 = x1 - y r , it can be deduced that:

[0157]

[0158] Thus, it can be further obtained that:

[0159]

[0160] Select the Lyapunov function as follows:

[0161]

[0162] where is the parameter estimation error, is the estimated value of the virtual parameter b1, and β1 > 0 is the design parameter. Note that when the initial condition satisfies |η(0)| = |z1(0)| < K(0), if V 10 is bounded, then |η| < K and Therefore, according to the Lyapunov function theory, if the initial condition satisfies |z1(0)| < K(0), by designing the controller to ensure that V 10 is bounded, it can be ensured that |η| < K and z1 is bounded within t ∈ [0, ∞). In addition, since is constantly 1 within t ∈ [0, T2), it can be obtained that z1 is restricted within the interval |z1| < K during the time t ∈ [0, T2).

[0163] According to equation (13), it is easy to deduce that the derivative of V1 with respect to time t is:

[0164]

[0165] Where γ1=K 2 >0, Define ρ1 = K 2 -η 2 According to Young's inequality, it is easy to obtain:

[0166]

[0167] Next, we define a function for an unknown smoothness. By approximating the unknown smooth function F1 using a radial basis function neural network, we can obtain:

[0168]

[0169] in, It is the input signal of the neural network, w1∈R s and χ1(ξ1)∈R s Let represent the ideal weight matrix and basis functions of the neural network, respectively, where s is the number of neurons, and δ1(ξ1) represents the approximation error of the neural network, satisfying . in Let be some unknown constant. From Young's inequality, we can obtain:

[0170]

[0171] in, r1>0 represents unknown virtual parameters, non-negative scalar functions, and design parameters, respectively.

[0172] Substituting equation (19) into equation (15), we can derive:

[0173]

[0174] Therefore, design the virtual control variable α1 and the adaptive law as follows:

[0175]

[0176] Where τ1>0 and c1>0 are both design parameters. From Young's inequality and equation (9), we can obtain:

[0177]

[0178] Substituting equations (21)-(24) into equation (20), we can derive:

[0179]

[0180] in, It is a constant. It is an unknown constant term, and

[0181] Next, in step S i In the diagram, where i = 2, ..., n-1, the designed virtual controller α... i With Adaptive Law as follows:

[0182]

[0183]

[0184] Among them, c i >0, β i >0 and τ i >0 represents the design parameters selected by the user.

[0185] By selecting the Lyapunov function as:

[0186]

[0187] Using a derivation process similar to S1, we can deduce that:

[0188]

[0189] in, It is a constant. It is an unknown constant term, and

[0190] Next, in step S n In the design of the virtual controller u and the adaptive law as follows:

[0191]

[0192] Among them, c n >0, β n >0 and τ n >0 represents the design parameters selected by the user.

[0193] By selecting the Lyapunov function as:

[0194]

[0195] Using a derivation process similar to S1, we can deduce that:

[0196]

[0197] in, It is an unknown constant term.

[0198] Consider the uncertain nonlinear system (1). If the initial condition satisfies |η(0)| < K(0), and there exists a constant B such that V n (0) ≤ B, then under the designed controller as in Equation (30), and the designed parameter adaptation laws as in Equations (22), (27), (31), by selecting appropriate design parameters, the designed control strategy can make all signals of the closed-loop system bounded, the system output can closely track the desired trajectory, and the system output does not violate the piecewise discontinuous constraint conditions.

[0199] Furthermore, since and are both compact sets, there exists a constant ζ > 0 in j such that |M j | ≤ ζ j holds, where j = 1,..., n - 1.

[0200] From Equation (33), it can be further obtained that:

[0201]

[0202] where, Therefore, it can be obtained that V n (t) ≤ V n (0)e -μt + H n / μ, that is, Thus, under the condition that the initial condition satisfies |z1(0)| < K(0), there is

[0203] where j = 1,..., n - 1, i = 1,..., n.

[0204] Furthermore, since it can be deduced that Since it is easy to obtain Since x1 = z1 + y r , it can be deduced that Thus and Therefore Through a similar analysis process, it can be proved that all internal signals of the closed-loop system are bounded. At the same time, from it is easy to obtain that by selecting appropriate design parameters, the system output can closely track the desired trajectory.

[0205] Furthermore, based on the structures of the connection function (3) constructed in Step 2 and the transfer function (4) constructed in Step 3, next, analyze that the system output does not violate the piecewise discontinuous constraint conditions, as follows:

[0206] Since in the interval \(t\in[0,T_2)\), it is easy to obtain \(\eta = z\) l , thus it can be concluded that:

[0207]

[0208] Therefore, in the interval \(0\leq t<T_1\), it is easy to obtain that is In the interval \(T_1\leq t<T_2\), it can be obtained that that is that is, the system output is not restricted by constraints in the intervals \(t\in[0,T_1)\) and \(t\in[T_1,T_2)\).

[0209] Next, since in the interval \(t\in[T_2,+\infty)\), \(\varPhi(t)=1\), \(K(t)=k_2(t)\), it is easy to deduce Since it can be further deduced that \(|x_1|<\infty\), that is, the system output is not restricted by constraints in the interval \(t\in[T_2,+\infty)\).

[0210] In summary, it can be concluded that the system output does not violate the piecewise discontinuous constraint conditions.

[0211] Step 5, by changing the discontinuous moment, apply the designed control scheme to continuous constraints, constraints discontinuous at specific moments, and scenarios where the system output is not restricted by constraints at a certain moment.

[0212] Without changing the design parameters and the controller structure, the designed control strategy is still applicable to other situations, specifically as follows:

[0213] ① When \(T_1>0\), \(T_2 = +\infty\), it can be obtained that:

[0214] The transfer function The transfer function \(\varPhi(t)\) is consistent with the structure of Equation (2), so the structure of the connection function \(K(t)\) is consistent with the structure of Equation (3), and further it is concluded that the constraint boundary function is bounded, but discontinuous at \(t = T_1\).

[0215] ② When \(T_1 = 0\), \(0<T_2<+\infty\), it can be obtained that:

[0216] The transfer function is consistent with the structure of Equation (4), the transfer function \(\varPhi(t)=1\), so the connection function \(K(t)=k_2(t)\), and further it is concluded that the constraint boundary function is continuous in the interval \(t\in[0,T_2)\), and the system output is not restricted by constraints at \(t = T_2\).

[0217] ③ When \(T_1 = 0\), \(T_2 = +\infty\), it is easy to obtain:

[0218] Transfer function The transition function Φ(t) = 1, therefore the connection function K(t) = k2(t), and thus it can be concluded that the constraint boundary function is continuous and bounded.

[0219] Furthermore, a design and analysis process similar to step 4 can be used to demonstrate the feasibility and effectiveness of the designed control strategy.

[0220] In summary, by changing the discontinuous moments, without altering the design parameters and controller structure, the designed control strategy can be used for continuous constraints, constraints that are discontinuous at specific moments, and situations where the output is not limited by constraints at a certain moment.

[0221] Finally, it should be noted that the above content is only used to illustrate the technical solution of the present invention and is not intended to limit it. It should not be used to limit the scope of protection of the present invention. Any modifications or equivalent substitutions made based on the technical concept of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A method of nonlinear system tracking control with piecewise discontinuous output constraints, characterized by, The method comprises the following steps: Step 1, a class of uncertain nonlinear system dynamics model is established; Step 2, a transfer function is constructed, and a connection function is constructed in combination with a constraint boundary function, so that the discontinuous constraint boundary function is changed into a continuous function, and the specific steps include: Step 2.1, constructing the transfer function is: （2） wherein, , is a constant chosen by the user, is a given discontinuous instant; transfer function with the following specific properties: In Strictly monotonically increasing; In the value of k is always 1 ; and its first derivative are both continuous and bounded; Step 2.2, based on the transfer function , construct the connection function , make the discontinuous constraint boundary function into a continuous function, as follows: Based on the transfer function , the constructed connection function is: （3） in, and They represent in and Two stages, tracking error The constraint boundary function, Given the desired output trajectory, express The maximum value, and Then they respectively represent in and Two stages, system output The constraint boundary function, It is the given moment when the system output changes from being constrained to being unconstrained. ; Connection function has the following properties: 1) In is continuous and bounded; 2) In is true; 3) the first derivative of K(t) In is continuous and bounded, i.e. there exists a positive number such that holds. Step 3, another transfer function is constructed, and an error transformation is performed in combination with a tracking error, so that the situation that the output is subjected to constraints is changed into the situation that the output is not subjected to constraints; Transfer function of the configuration As shown below: （4） wherein , is a constant selectable by the user; transfer function has the following properties: In the value within the inner loop is always 1; In strictly monotonically decreasing, and ; , first derivative of and second derivative of in are both continuous and bounded; Step 4, a neural network adaptive control scheme based on an output constraint condition is designed, appropriate design parameters are selected, all signals of the closed-loop system are bounded by the designed control scheme, the output of the system is ensured to not violate the piecewise discontinuous constraint condition, and the output of the system is closely tracked to the expected trajectory; Step 5, the designed control scheme is applied to the continuous constraint, the discontinuous constraint at a specific time, and the scenario that the output of the system is not subjected to constraint limitation at a certain time by changing the discontinuous time.

2. The nonlinear system tracking control method with piecewise discontinuous output constraint according to claim 1, characterized in that, The class of uncertain nonlinear system dynamics model established in step 1 is as follows: （1） where , and are the state, input, and output variables of the system, respectively, and denote unknown smooth nonlinear functions and unknown control gains, respectively, where n denotes the order of the system, and R denotes the set of real numbers.

3. The nonlinear system tracking control method with piecewise discontinuous output constraint according to claim 2, characterized in that, Step 4 includes: Step 4.1, defining tracking error and virtual error as follows: （5） （6） wherein , is the output of the first order filter: （7） wherein is a design parameter, is a virtual control variable; The filter error is defined as: （8） It is derived from formula (6) and formula (8) that: （9） The first derivative of formula (7) and formula (8) is the first derivative: （10） To solve the problem that the system output is not limited by the constraints at time, the tracking error is converted: （11） Step 4.2, the controller is designed by means of the approximation principle of the neural network and the backstepping technology, and the specific steps are as follows: : by , it is deduced first derivative: （12） representing a first derivative; It is further derived that: （13） , denote the first derivative of , , respectively; The Lyapunov function is selected as follows: （14） where , is the parameter estimation error, is the virtual parameter estimate, and is the design parameter, denotes a lower bound; according to equation (13), it follows that the first order derivative of with respect to time （15） wherein , , denotes the first derivative of ; the definition , according to Young's inequality, is: （16） (17) Step 4.3, define the unknown smooth function , use a radial basis function neural network to approximate the unknown smooth function , we get: （18） where is the input signal of the neural network, and denote the ideal weight matrix and the basis function of the neural network, respectively, is the number of neurons, denotes the approximation error of the neural network and satisfies where is some unknown constant; by Young's inequality: （19） wherein , , respectively denote unknown virtual parameters, non-negative scalar functions and design parameters; Step 4.4, formula (19) is substituted into formula (15), and it is derived that: （20） Designing virtual control variables with adaptive law as follows: （21） （22） wherein and are design parameters; from Young's inequality and equation (9): （23） （24） Formula (21)-(24) are substituted into formula (20), and it is derived that: （25） wherein , is a constant, is an unknown constant term, and ; Step 4.5, in step wherein a virtual controller with an adaptive law as Next: （26） （27） wherein, , and are user-selected design parameters; In step the designed virtual controller with the adaptive law is as follows: （30） （31） wherein, , and are user-selected design parameters; Consider the uncertain nonlinear system (1), if the initial condition satisfies and there exists a constant such that then under the designed controller as in (30) and the designed parameter adaptive law as in (22), (27), (31), by choosing appropriate design parameters, the designed control strategy can achieve that all the signals in the closed-loop system are bounded, the system output closely tracks the desired trajectory, and the system output does not violate the piecewise discontinuous constraint condition; Furthermore, due to and Both are compact sets, therefore There exists a constant in , making Established, among which Further, we obtained: （34） wherein , ; thus obtaining i.e. , under the condition that the initial conditions satisfy there is , , , , wherein , ; Further, since , it is concluded that , since , it is easy to obtain , since , it is concluded that , thus , and , therefore Through similar analysis process, it is proved that all internal signals of the closed-loop system are bounded, and , it is easy to obtain that the system output closely tracks the desired trajectory by selecting appropriate design parameters; Further, based on the structure of the connection function (3) constructed in step 2 and the transfer function (4) constructed in step 3, the output of the system is analyzed to not violate the piecewise discontinuous constraint condition, and the specific steps are as follows: Since in interval, , it is easy to get , so we get: （35） Thus, in the interval , we have , in the interval , we have , i.e. , i.e. the system output is not constrained in the intervals and ; Next, since in the interval, , , it is easy to deduce , since , further deduce , that the system output is not constrained within the interval .

4. The nonlinear system tracking control method with piecewise discontinuous output constraint according to claim 3, characterized in that, In step 5, the designed control scheme is applied to the continuous constraint, the discontinuous constraint at a specific time, and the scenario that the output of the system is not subjected to constraint limitation at a certain time by changing the discontinuous time, and the specific steps are as follows: 1) when , it follows that: transfer function , transfer function is consistent with the structure of equation (2), so the connection function is consistent with the structure of equation (3), and it is concluded that the constraint boundary function is bounded, but discontinuous at moment 2) when , it follows that: transfer function In accordance with the structure of equation (4), the transfer function Therefore, the connection function Further, it is derived that the constraint boundary function is continuous in the interval and the system output is not limited by the constraint at the time ; 3) when , it follows that: transfer function , transfer function , therefore the connection function , and it follows that the constraint boundary function is continuous and bounded.

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