Non-linear system tracking control method with segmented discontinuous output constraint

Abstract

The invention provides a nonlinear system tracking control method with segmented discontinuous output constraints, and aims to solve the problem that a constraint boundary function is discontinuous but bounded at a certain moment within a period of time when a system starts to run and solve the problem that system output is not limited by constraints at a certain moment. Different from most existing methods requiring consecutive and bounded constrained boundary functions and high-order derivatives thereof, the method provided by the invention only needs consecutive and bounded converted constrained boundary functions and first-order derivatives thereof, so that the requirements on the constrained boundary functions are relaxed, the practicability of a control strategy is greatly improved, and the complexity and calculation burden of a controller are reduced. Besides, by changing discontinuous moments without changing the structure and design parameters of the controller, the proposed control strategy can also be suitable for other constraint situations, such as continuous constraint conditions, discontinuous constraint conditions at specific moments and unconstrained limitation of system output at a certain moment, so that the practicability of the control strategy is further enhanced.

Description

Technical Field

[0001] The present invention relates to the technical field of constraint control of nonlinear systems, and in particular to a tracking control method of a nonlinear system with piecewise discontinuous output constraints. Background Art

[0002] In actual engineering applications, the system may be subject to output constraints due to various factors such as the surrounding environment or user needs. If these constraints are not considered, the system may lose stability. It is worth pointing out that many existing related control strategies are only for the case where the constraint boundary function is continuous. In fact, the constraint boundary function is often discontinuous. For example, a drone may transition from entering one tunnel to another tunnel and then enter an open space. In this case, the traditional method is no longer applicable. In recent years, although some research results have designed control strategies for discontinuous constraint boundary functions, such as solving the problem of system output being constrained for a period of time by designing transfer functions, the related results only study the problem of the system output transitioning between being constrained and unconstrained, and still do not solve the problem that the constraint boundary function is discontinuous but bounded at a certain moment during a period of time when the system starts running. In addition, in related studies, the requirements for the constraint boundary function are high. Therefore, studying a nonlinear system tracking control method with piecewise discontinuous output constraints still has potential application value. Summary of the invention

[0003] The present invention aims at the output constraint control problem of a class of uncertain nonlinear systems, and proposes a set of neural network adaptive control strategies to solve the problem that the constraint boundary function is discontinuous but bounded at a certain moment within a period of time when the system starts to run, and solves the problem that the system output is not constrained at a certain moment. In addition, unlike most existing methods that require the constraint boundary function and its high-order derivatives to be continuous and bounded, the present invention only requires the converted constraint boundary function and its first-order derivative to be continuous and bounded, relaxes the requirements for the constraint boundary function, greatly improves the practicality of the control strategy, and reduces the complexity and computational burden of the controller. In addition, by changing the discontinuous moment, there is no need to change the controller structure and design parameters. The proposed control strategy can also be applied to other constraint situations, such as continuous constraints, discontinuous constraints at specific moments, and unconstrained system output at a certain moment, further enhancing the practicality of the control strategy.

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

[0005] A nonlinear system tracking control method with piecewise discontinuous output constraints of the present invention comprises the following steps:

[0006] Step 1, establish a class of uncertain nonlinear system dynamics model;

[0007] Step 2, constructing a transfer function, and combining the constraint boundary function to construct a connection function, so as to transform the discontinuous constraint boundary function into a continuous function;

[0008] Step 3, construct another transfer function and combine it with the tracking error to perform error transformation, so as to transform the output constrained situation into an unconstrained situation;

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

[0010] In step 5, the designed control scheme is applied to the scenario where the constraints are continuous, the constraints are discontinuous at a specific moment, and the system output is not constrained at a certain moment by changing the discontinuous moment.

[0011] Furthermore, in step 1, a class of uncertain nonlinear system dynamics models is established as follows:

[0012]

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

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

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

[0016]

[0017] Among them, △1=l1(T1-t)>0, l1>0 is a constant that can be selected by the user, and T1>0 is a given discontinuous moment.

[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) at t∈[T1,+∞) is always 1;

[0021] ③Φ(t) and 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 It means that in the two stages t∈[0,T1) and t∈[T1,+∞), the tracking error z1=x1-y r The constraint boundary function, y r is the known expected output trajectory, means |y r The maximum value of |, k c1 With k c2 Then they represent the constraint boundary function of the system output x1 at 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. 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) is valid when t∈[T1,+∞);

[0028] ③ It is continuous and bounded at t∈[0,+∞), that is, there exists a positive number K max , so that Established.

[0029] Furthermore, in step 3, another transfer function is constructed and combined with the tracking error for error transformation to transform the output constrained situation into an unconstrained situation, as follows:

[0030] Constructed transfer function for:

[0031]

[0032] Among them, △2=l2(t-T2)>0, l2>0 is a constant that can be selected by the user. Further, the transfer function constructed in step 3 is It has the following properties:

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

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

[0035] ③ and It is continuous and bounded at t∈[0,+∞).

[0036] For the convenience of representation, the present invention will omit the independent variables of some functions. Further, in step 4, a neural network adaptive control scheme based on output constraints is designed to relax the continuous and bounded conditions of the higher-order derivatives of the constraint boundary function in the traditional control method, reduce the computational burden and controller complexity, ensure that the output of the system does not violate the piecewise discontinuous constraint conditions, and the output of the system closely tracks the expected trajectory, specifically including the following steps:

[0037] Step 4.1: To facilitate the design of the controller, define the tracking error z1 and the virtual error z i , as follows:

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

[0039] z i =x i -α if (6)

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

[0041]

[0042] Among them, ε i >0 is the design parameter, α i-1 (i=2,...,n) is the virtual control variable. Next, define the filtering error:

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

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

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

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

[0047]

[0048] In order to solve the problem that the system output is not constrained when t ≥ T2, the tracking error z1 is transformed:

[0049]

[0050] Step 4.2, design the controller with the help of the approximation principle of neural network and backstepping technology. The specific steps are as follows:

[0051] S1: z1 = x1 - y r , we can deduce:

[0052]

[0053] Thus, we can further conclude that:

[0054]

[0055] The Lyapunov function is selected as follows:

[0056]

[0057] in, is the parameter estimation error, is the estimated value of the virtual parameter b1, and β1>0 is the design parameter. According to formula (13), it is easy to deduce that the derivative of V1 with respect to time t is:

[0058]

[0059] Where γ1=K 2 >0, Define ρ1=K 2 -η 2 , according to Young's inequality, we can easily get:

[0060]

[0061] Next, define the unknown smooth function Using the radial basis function neural network to approximate the unknown smooth function F1, we can get:

[0062]

[0063] in, is the input signal of the neural network, w1∈R s and χ1(ξ1)∈R s Represent the ideal weight matrix and basis function of the neural network respectively, s is the number of neurons, δ1(ξ1) represents the approximation error of the neural network and satisfies in is an unknown constant. From Young's inequality, we can get:

[0064]

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

[0066] Substituting formula (19) into formula (15), we can deduce:

[0067]

[0068] Furthermore, the virtual control variable α1 is designed to be consistent with the adaptive law as follows:

[0069]

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

[0071]

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

[0073]

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

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

[0076]

[0077]

[0078] Among them, c i >0,β i >0 and τ i >0 are 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 that of 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 in Equation (30) and the designed parameter adaptation laws in Equations (22), (27), and (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 concluded 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 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 constraints in the interval \(t\in[T2,+\infty)\).

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

[0099] Furthermore, in step 5, by changing the discontinuous time, apply the designed control scheme 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 concluded 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 is 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 is 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 is 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, constraints that are discontinuous at specific moments, and situations 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 constraint boundary function and its higher-order derivatives to exist and be bounded, the present invention only requires the transformed constraint boundary function and its first-order derivative to be 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. Under the premise that the controller structure and design parameters remain unchanged, the control strategy proposed in the present 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, which enhances the practical applicability of the control strategy and improves the feasibility of the algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] Figure 1 It is a control block diagram of the nonlinear system tracking control based on piecewise discontinuous output constraints of the present invention. DETAILED DESCRIPTION

[0113] In industrial applications, the state of a nonlinear system is measured by a sensor, and the measured state is transmitted to a neural network adaptive controller, where the difference between the first measured state and the expected signal, combined with the transfer function after error transformation, is transmitted to the controller together with the transfer function and the tracking error. The controller processes the received signal according to the control law, and then controls the nonlinear system through the actuator.

[0114] A nonlinear system tracking control method with piecewise discontinuous output constraints according to the present invention is as follows: Figure 1 As shown, the specific steps are as follows:

[0115] Step 1: Establish a class of uncertain nonlinear system dynamics model:

[0116]

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

[0118] Step 2: construct a transfer function and build a connection function in combination with the constraint boundary function to transform the discontinuous constraint boundary function into a continuous function.

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

[0120]

[0121] Among them, △1=l1(T1-t)>0, l1>0 is a constant that can be selected by the user, and T1>0 is a given discontinuous moment.

[0122] It is worth pointing out 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) at t∈[T1,+∞) is always 1;

[0125] ③Φ(t) and 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 It means that in the two stages t∈[0,T1) and t∈[T1,+∞), the tracking error z1=x1-y r The constraint boundary function, y r is the known expected output trajectory, means |y r The maximum value of |, k c1 With k c2 They represent the constraint boundary function of the system output x1 at the two stages t∈[0,T1) and t∈[T1,T2) respectively, T2 is the given moment when the system output changes from constrained to 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) is valid when t∈[T1,+∞);

[0133] ③ It is continuous and bounded at t∈[0,+∞), that is, there exists a positive number K max , so that Established.

[0134] Step 3, construct another transfer function and combine it with the tracking error to perform error transformation, so as to transform the output constrained situation into an unconstrained situation;

[0135] Constructed transfer function for:

[0136]

[0137] Among them, △2=l2(t-T2)>0, l2>0 is a constant that can be selected by the user. It is worth pointing out that the constructed transfer function It has the following properties:

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

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

[0140] ③ and It is continuous and bounded at t∈[0,+∞).

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

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

[0143] Step 4.1, to facilitate the design of the controller, define the tracking error z1 and the virtual error z i , as follows:

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

[0145] z i =x i -α if (6)

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

[0147]

[0148] Among them, ε i >0 is the design parameter, α i-1 (i=2,...,n) is the virtual control variable. Next, define the filtering error:

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

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

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

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

[0153]

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

[0155]

[0156] Step 4.2, design a controller by means of the approximation principle of neural network 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 always 1 within t ∈ [0, T2), it can be obtained that z1 is always 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, we can easily get:

[0166]

[0167] Next, define the unknown smooth function Using radial basis function neural network to approximate the unknown smooth function F1, we can get:

[0168]

[0169] in, is the input signal of the neural network, w1∈R s and χ1(ξ1)∈R s Represent the ideal weight matrix and basis function of the neural network respectively, s is the number of neurons, δ1(ξ1) represents the approximation error of the neural network and satisfies in is an unknown constant. From Young's inequality, we can get:

[0170]

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

[0172] Substituting formula (19) into formula (15), we can deduce:

[0173]

[0174] Therefore, the virtual control variable α1 is designed to be consistent with the adaptive law as follows:

[0175]

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

[0177]

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

[0179]

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

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

[0182]

[0183]

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

[0185] By choosing the Lyapunov function as:

[0186]

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

[0188]

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

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

[0191]

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

[0193] By choosing the Lyapunov function as:

[0194]

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

[0196]

[0197] in, 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 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.

[0199] 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.

[0200] From Eq. (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, it is easy to obtain that by choosing appropriate design parameters, the system output can closely track the desired trajectory. 易得通过选取合适的设计参数,系统输出可以紧密跟踪期望轨迹。

[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 that are 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 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] Transfer function The transfer function \(\varPhi(t)\) is consistent with the structure of Equation (2). Therefore, the structure of the connection function \(K(t)\) is consistent with the structure of Equation (3), and further it can be 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] Transfer function is consistent with the structure of Equation (4), the transfer function \(\varPhi(t)=1\). Therefore, the connection function \(K(t)=k_2(t)\), and further it can be 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 transfer function Φ(t)=1, so the connection function K(t)=k2(t), and it can be concluded that the constraint boundary function is uniformly continuous and bounded.

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

[0220] In summary, by changing the discontinuous moment, without changing the design parameters and controller structure, the designed control strategy can be used for continuous constraints, constraints that are discontinuous at a specific moment, and situations where the output is not restricted by the 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 rather than to limit it, and cannot be used to limit the protection scope of the present invention. Based on the technical idea of ​​the present invention, technical modifications or equivalent substitutions should all be included in the scope of the claims of the present invention.

Claims

1. A nonlinear system tracking control method with piecewise discontinuous output constraints, characterized in that: It includes the following steps: Step 1, establish a class of uncertain nonlinear system dynamic models; Step 2, construct a transfer function, and combine it with a constraint boundary function to construct a connection function, turning the discontinuous constraint boundary function into a continuous function; Step 3, construct another transfer function, and combine it with the tracking error for error transformation, turning the situation where the output is constrained into an unconstrained situation; Step 4, design a neural network adaptive control scheme based on the output constraint conditions. By selecting appropriate design parameters, make the designed control scheme ensure that all signals in the closed-loop system are bounded, ensure that the system output does not violate the piecewise discontinuous constraint conditions, and the system output closely tracks the desired trajectory; Step 5, by changing the discontinuous time, apply the designed control scheme to continuous constraints, constraints that are discontinuous at specific times, and scenarios where the system output is unconstrained at a certain time.

2. A nonlinear system tracking control method with piecewise discontinuous output constraints according to claim 1, characterized in that: A class of uncertain nonlinear system dynamic models established in Step 1 is as follows: in, u∈R and y∈R are the state, input, and output variables of the system respectively. and They represent the unknown smooth nonlinear function and the unknown control gain respectively, where i=1,...,n-1, n represents the order of the system, and R represents a set of real numbers.

3. A nonlinear system tracking control method with piecewise discontinuous output constraints according to claim 2, characterized in that: Step 2 specifically includes: Step 2.1, the constructed transfer function Φ(t) is: where, △1 = l1(T1 - t) > 0, l1 > 0 is a constant selected by the user, and T1 > 0 is the given discontinuous time; The transfer function Φ(t) has the following specific properties: ① Φ(t) is strictly monotonically increasing for t ∈ [0, T1); ② The value of Φ(t) is constantly 1 for t ∈ [T1, +∞); ③Φ(t) and its first-order derivative are both continuous and bounded; Step 2.2, based on the transfer function Φ(t), construct the connection function K(t) to turn the discontinuous constraint boundary function into a continuous function, specifically as follows: Based on the transfer function Φ(t), the constructed connection function K(t) is: Where k1(t)=k c1 (t)-k c2 (t) and It means that in the two stages t∈[0,T1) and t∈[T1,+∞), the tracking error z1=x1-y r The constraint boundary function, y r is the known expected output trajectory, means |y r The maximum value of |, k c1 With k c2 Then they represent the constraint boundary function of the system output x1 at 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; The connection function K(t) has the following properties: ① K(t) > 0 is continuous and bounded for t ∈ [0, +∞); ② K(t) ≡ k2(t) holds for t ∈ [T1, +∞); ③The first-order derivative of K(t) It is continuous and bounded at t∈[0,+∞), that is, there exists a positive number K max , so that Established.

4. A nonlinear system tracking control method with piecewise discontinuous output constraints according to claim 3, characterized in that: The transfer function constructed in step 3 As shown below: where, △2 = l2(t - T2) > 0, l2 > 0 is a constant that the user can choose; Transfer function It has the following properties: ① The value in t∈[0,T2) is always 1; ② is strictly monotonically decreasing at t∈[T2,+∞), and ③ The first derivative of and The second derivative of It is continuous and bounded at t∈[0,+∞).

5. A nonlinear system tracking control method with piecewise discontinuous output constraints according to claim 4, characterized in that: Step 4 includes: Step 4.1, define tracking error z1 and virtual error z i , as follows: z1=x1-y r (5) z i =x i -α if (6) Among them, i=2,...,n,α if is the output of a first-order filter: Among them, ε i >0 is the design parameter, α i-1 is a dummy control variable; Define the filtered error: y i =a if -a i-1 , i=2,...,n; (8) Derived from Equation (6) and Equation (8): x i =z i +y i +α i-1 , i=2,...,n(9) From equations (7) and (8), we can deduce α if The first derivative of : To solve the problem that the system output is unconstrained when t ≥ T2, transform the tracking error z1: Step 4.2, design a controller by means of the approximation principle of neural networks and the backstepping technique. The specific steps are as follows: S1: z1 = x1 - y r , derive the first-order derivative of z1: Represents y r The first derivative of ; Further obtain: Respectively The first derivative of ; Select the Lyapunov function as follows: in, is the parameter estimation error, is the estimated value of the virtual parameter b1, and β1>0 is the design parameter, g1 represents the lower bound of g1; According to formula (13), the first-order derivative of V1 with respect to time t is deduced as: Where γ1=K 2 >0, express The first derivative of; define ρ1=K 2 -η 2 , according to Young's inequality: Step 4.3, define the unknown smooth function Using radial basis function neural network to approximate the unknown smooth function F1, we get: in, is the input signal of the neural network, w1∈R s and χ1(ξ1)∈R s Represent the ideal weight matrix and basis function of the neural network respectively, s is the number of neurons, δ1(ξ1) represents the approximation error of the neural network and satisfies in is an unknown constant; from Young's inequality: in, r1>0 represents unknown virtual parameters, non-negative scalar functions and design parameters respectively; Step 4.4, substitute Equation (19) into Equation (15) and derive: Design virtual control variable α1 and adaptive law as follows: where, τ1 > 0 and c1 > 0 are both design parameters; from the Young's inequality and Equation (9), we get: Substitute Equations (21)-(24) into Equation (20) and derive: in, is a constant, is an unknown constant term, and Step 4.5, in step S i In which i = 2, ..., n-1, the designed virtual controller α i Adaptive Law as follows: Among them, c i >0,β i >0 and τ i >0 are design parameters selected by the user; In step S n In the design, the virtual controller u and the adaptive law as follows: Among them, c n >0,β n >0 and τ n >0 are design parameters selected by the user; Consider 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 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 closely track the desired trajectory, and the system output does not violate the piecewise discontinuous constraint conditions; Furthermore, due to and are all compact sets, so There is a constant ζ j >0, so that |M j | ≤ζ j holds, where j=1,...,n-1; further we get: Among them, Therefore, we get V n (t) ≤ V n (0)e -μt + H n / μ, that is, V n ∈ l ∞ , so that under the condition that the initial condition satisfies |z1(0)| < K(0), we have z1 ∈ l ∞ , y j ∈ l ∞ , z j ∈ l ∞ , where j = 1,..., n - 1, i = 1,..., n; Furthermore, due to roll out because Easy to get Since x1=z1+y r , and we can infer that x1∈l ∞ , so α1∈l ∞ , and α if ∈l ∞ , so x2∈l ∞ Through a similar analysis process, it is proved that all internal signals of the closed-loop system are bounded, and at the same time, It is easy to obtain that by selecting appropriate design parameters, the system output closely tracks the expected trajectory; Furthermore, based on the structure 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, specifically as follows: Since in the interval t∈[0,T2), It is easy to get η = z l , thus we can conclude that: Therefore, in the interval 0 ≤ t < T1, we get that is, |x1| < k c1 , in the interval T1 ≤ t < T2, we obtain that is, |x1| < k c2 , which means that the system output is not restricted in the intervals t ∈ [0, T1) and t ∈ [T1, T2); Next, since Φ(t)=1, K(t)=k2(t) in the interval t∈[T2,+∞), it is easy to deduce because It can be further deduced that |x1|<∞, that is, the system output is not constrained in the interval t∈[T2,+∞).

6. A nonlinear system tracking control method with piecewise discontinuous output constraints according to claim 5, characterized in that: In Step 5, by changing the discontinuous time, apply the designed control scheme to continuous constraints, constraints that are discontinuous at specific times, and scenarios where the system output is unconstrained at a certain time, specifically as follows: ① When T1 > 0, T2 = +∞, we get: Transfer function The transfer function Φ(t) is consistent with the structure of formula (2), so the connection function K(t) is consistent with the structure of formula (3), which leads to the conclusion that the constraint boundary function is bounded, but discontinuous at t = T1; ② When T1 = 0, 0 < T2 < +∞, we get: Transfer function Keeping the same structure as formula (4), the transfer function Φ(t) = 1, so the connection function K(t) = k2(t), further deriving the constraint boundary function is continuous in the interval t∈[0,T2), and the system output is not constrained at t = T2; ③When T1=0,T2=+∞,then we get: Transfer function The transfer function Φ(t)=1, so the connection function K(t)=k2(t), and it can be concluded that the constraint boundary function is uniformly continuous and bounded.

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