Tunnel preset performance self-adaptive output feedback control method under saturation constraint

Abstract

The invention discloses a tunnel preset performance self-adaptive output feedback control method under saturation constraint, and the method comprises the steps: designing a tunnel specified performance function and novel error transformation of preset time, not only reducing the conservative property of a conventional preset performance scheme, but also simplifying the control design, and achieving completely asymmetric performance constraint; a non-measurable state is estimated by using a radial basis function neural network state observer, and the limitation of a non-strict feedback structure is solved; and designing a first-order filter and a saturation compensation first-order auxiliary system, and designing a controller by using a backstepping method and an adaptive control method to realize output feedback tracking control. Compared with the prior art, the method has the advantages of simple structure and energy consumption reduction, and can also ensure that the tracking error is always limited in the predetermined boundary of the specified performance function of the tunnel.

Description

Technical Field

[0001] The present invention belongs to the technical field of automatic control, relates to the control technology related to preset performance, and particularly relates to an adaptive output feedback control method for tunnel preset performance under saturation constraints. Background Art

[0002] With the continuous improvement of the complexity of modern industrial systems, the demand for high-precision control of nonlinear systems is becoming increasingly urgent. Especially in the fields of aerospace, robot control, intelligent manufacturing, etc., the system often has constraints such as unstructured uncertainties, unmeasurable states, external disturbances, and input saturation. These factors not only affect the steady-state performance of the system, but also significantly deteriorate the transient response characteristics, resulting in problems such as overshoot and uncontrollable convergence time, seriously restricting the practical applications in high-dynamic performance scenarios. Therefore, it is necessary to improve the control quality of high-order nonlinear systems to meet the actual engineering requirements.

[0003] In the existing Prescribed Performance Control (PPC), most of them impose constraints on nonlinear systems by using a decaying exponential type of prescribed performance function to achieve the prescribed performance control of the output state. However, its convergence mechanism depends on infinite time and cannot achieve precise convergence within a preset time. Moreover, the symmetric envelope design is prone to overshoot, and the error transformation method is complex, making it difficult to be compatible with the requirements of asymmetric constraints.

[0004] In high-order systems, the backstepping method is commonly used to design high-order controllers. However, the traditional backstepping method requires taking derivatives of the virtual control laws step by step, leading to an "explosion of complexity", especially when combined with neural network adaptive technology, the computational burden is even more severe.

[0005] The control input saturation constraint is a potential problem in most industrial control systems, and there is a certain contradiction between the performance requirements of the prescribed performance control method and the actual input saturation constraint. Therefore, it is necessary to design a control strategy to alleviate this contradiction and reduce the impact of saturation constraints on the system.

[0006] Therefore, it is of great research value and application prospect to develop an output feedback control method for non-strict feedback nonlinear systems that has preset time convergence, asymmetric error constraints, low computational complexity, and can be compatible with input saturation constraints. Summary of the Invention

[0007] In order to solve the above technical problems, the present invention provides an adaptive output feedback control method for tunnel preset performance under saturation constraints, which can not only ensure the convergence of the tracking error within a preset time, but also achieve the transient performance without overshoot.

[0008] The present invention adopts the following technical solution: A tunnel preset performance adaptive output feedback control method under saturation constraints, comprising the following steps: Step 1: Establish a non-strict feedback nonlinear system model with input saturation constraints; Step 2: Construct an improved preset time tunnel preset performance function, establish preset performance constraints, and perform error transformation; Step 3: Design a state observer based on a radial basis neural network; Step 4: Introduce a coordinate transformation, and design a first-order auxiliary system to dynamically compensate for the saturation deviation for the subsequent design of the controller; Step 5: Recursively design an adaptive output feedback control scheme through the backstepping method and the dynamic surface control technique; Step 6: Based on the Lyapunov theory, prove the stability of the system according to the adaptive output feedback control scheme and the parameter update law.

[0009] Further, the non-strict feedback nonlinear system model with input saturation constraints is:

[0010] Wherein, and are the state and output variables of the system respectively, and only the output of the system is measurable. represents an unknown continuous and differentiable nonlinear function; represents an unknown time-varying and continuous control gain coefficient, satisfying , where the positive constant ; represents a time-varying external disturbance, satisfying , where is a positive constant. is the actuator input signal to be designed, represents that the actuator has an asymmetric saturation output characteristic.

[0011] The asymmetric saturation constraint is described as:

[0012] Wherein, and represent the magnitudes of the upper and lower bounds of the actuator output respectively.

[0013] Let the tracking error be , where the desired trajectory , and satisfy being continuous and bounded.

[0014] Furthermore, the improved preset performance function of the tunnel with preset time convergence is as follows:

[0015]

[0016]

[0017] where , , , , and , , is the artificially set time, represents the initial error of the system at the moment of represents the sign function.

[0018] Define the normalized error and the new error transformation as:

[0019]

[0020]

[0021] Furthermore, the designed state observer based on the radial basis neural network is specifically:

[0022] Furthermore, the result of the coordinate transformation is:

[0023] where is the artificially designed virtual control intermediate variable, is the output of the first-order filter with as the input, is an auxiliary variable used to solve the input saturation problem, is the artificially designed parameter, and its update law is:

[0024] where the constant , and .

[0025] Furthermore, recursively design the virtual control law, the adaptive output feedback control law, and the parameter update law as:

[0026]

[0027]

[0028] Among them, , , , and are positive constants to be designed, , is a constant to be designed.

[0029] Compared with the prior art, the present invention has the following beneficial effects: 1) The present invention proposes a more concise tunnel preset performance control method, in which the convergence time and convergence accuracy can be preset artificially, and a completely asymmetric error constraint is achieved through a simple control structure, which means that more diverse performance requirements can be achieved according to task requirements; 2) The RBF neural network is used to process unknown non-linear functions , and the output of the state observer based on the RBF neural network is obtained to estimate the unmeasurable state. Moreover, by utilizing the characteristics of the Gaussian basis function of the neural network, the algebraic loop problem caused by the non-strict feedback structure is cleverly avoided; 3) By adopting the dynamic surface control technology, the cumbersome analytical calculation of the time derivative of the virtual control law in the backstepping design process is avoided, and in the design of the virtual error in the last step of the backstepping method, a first-order auxiliary system is constructed to compensate for the influence of asymmetric input saturation.

[0030] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present disclosure. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is the flowchart of the method design of the present invention; Figure 2 is the comparison schematic diagram of the tunnel preset performance control of the present invention and the traditional preset performance control; Figure 3 is the schematic diagram of the completely asymmetric error constraint effect of the present invention; Figure 4 is the change curve of the tracking error under the tunnel preset performance constraint in the embodiment of the present invention; Figure 5 is the system state in the embodiment of the present invention and the observer estimated state change curve; Figure 6 is the system state in the embodiment of the present invention and the observer estimated state change curve; Figure 7 is the neural network weight parameter in the embodiment of the present invention and variation curve; Figure 8 In the embodiment of the present invention, the design control input and the saturation constraint control input variation curve. Specific implementation manner

[0032] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and the detailed implementation manner and specific operation process are given, but the protection scope of the present invention is not limited to the following embodiments.

[0033] As Figure 1 shown, the present invention relates to the design of a state observer based on a radial basis neural network for a non-strict feedback nonlinear system, the design of a transformation based on a tunnel preset performance constraint, a first-order filtering based on a backstepping method, and the design of a tracking controller for a saturation compensation auxiliary system. The method includes the following steps: Step 1: Establish a non-strict feedback nonlinear system model with input saturation constraints; Step 2: Construct an improved preset time tunnel preset performance function, establish a preset performance constraint, and perform error transformation; Step 3: Design a state observer based on a radial basis neural network; Step 4: Introduce a coordinate transformation, and design a first-order auxiliary system to dynamically compensate for the saturation deviation for the subsequent design of the controller; Step 5: Recursively design an adaptive output feedback control scheme by combining the backstepping method and the dynamic surface control technique; Step 6: Based on the Lyapunov theory, prove the stability of the system according to the adaptive output feedback control scheme and the parameter update law.

[0034] In step 1, the non-strict feedback nonlinear system model with input saturation constraints is:

[0035] wherein, and are the state and output variables of the system respectively, represents the set of n-dimensional vectors, represents the ith system state first derivative, and only the output of the system is measurable. represents an unknown continuous and differentiable nonlinear function; represents an unknown time-varying and continuous control gain coefficient, satisfying , where the positive constant ; represents the time-varying external disturbance, satisfying , where is a positive constant. is the input signal of the actuator to be designed, indicates that the actuator has an asymmetric saturation output characteristic.

[0036] The asymmetric saturation constraint is described as:

[0037] where and represent the upper and lower bounds of the actuator output respectively.

[0038] Let the tracking error be , where the desired trajectory , and are continuous and bounded.

[0039] In step 2, construct an improved preset-time tunnel preset performance function, establish a preset performance constraint, and perform error transformation. The specific steps are as follows:

[0040] Step 2-1: Design the preset-time tunnel preset performance function,

[0041]

[0042]

[0043] where , , , , and , , is the artificially set time, represents the initial error of the system at time represents the sign function.

[0044] Step 2-2: Impose the following preset performance inequality constraint on the tracking error :

[0045] It should be noted that according to the definitions of equations (14) and (15), the definition of Tunnel Prescribed Performance Control (TPPC) is in a concise form. There is only one set of expressions for the performance boundary, and the convergence time can be predefined according to the actual task requirements. Therefore, TTPC can reduce the complexity of controller design and stability analysis. To illustrate the difference between TPPC and existing traditional PPC technologies, a comparison chart of TPPC and traditional PPC is given, as Figure 2 shown. Compared with PPC, the shape of the performance boundary of TPPC has changed significantly and is similar to a "tunnel" with a narrower space. The boundaries of TPPC are distributed on the same side, which makes TPPC have a stronger ability to constrain the tracking error. By selecting appropriate TPPC parameters, transient performance control without overshoot can be achieved.

[0046] Step 2 - 3: Based on the predefined performance function of the time tunnel, the tracking control problem under constraints is transformed into an unconstrained stable control problem by using the error transformation method, and the normalized error is defined as:

[0047]

[0048]

[0049] It should be explained that by adopting the above - mentioned new error transformation, the inequality constraint of equation (16) is transformed into an equation form to solve the control problem of the predefined performance. Compared with the existing error transformation technologies, the transformation method proposed in the present invention is in a more concise form and is convenient for subsequent controller design.

[0050] It should be noted that compared with the existing PPC technologies, the new error transformation method of the present invention enables the performance functions of the upper and lower envelopes to take completely different construction forms, that is, this TPPC scheme can achieve completely asymmetric error constraints through a simple control structure. For example, Figure 3 as shown, taking , , , where .

[0051] Taking the derivative of equation (17), we can get

[0052] where, , .

[0053] Since the mapping before and after the transformation is a homeomorphic mapping, it is easy to obtain that if , and there exists a positive constant , such that for any , there is , that is, the transformation function is bounded, then the tracking error will always satisfy the inequality (15), and , there is .

[0054] In step 3, design a state observer based on a radial basis neural network. The construction of the observer is as follows,

[0055] where, is the estimated value of , is the estimation error, ; and represent the ideal weight vector and basis function vector respectively, and there exists an arbitrary constant , such that the approximation error satisfies ; is the estimated value of , is the estimation error, ; are the coefficients of the Hurwitz polynomial.

[0056] In step 4, introduce a coordinate transformation and design a first-order auxiliary system to dynamically compensate for the saturation deviation for the subsequent design of the controller.

[0057] To avoid repeatedly calculating the derivative of the virtual control and making the design too complex, the dynamic surface control method is adopted, and the following coordinate transformation is carried out:

[0058] where, is an artificially designed intermediate variable of the virtual control, is the output of a first-order filter with as the input. In each step of the backstepping process, the first-order filter is required to filter the virtual control , thus avoiding the complex calculation of its differentiation in the backstepping process. The design of the first-order filter is:

[0059] where is a positive constant to be designed.

[0060] Design a first-order auxiliary system to solve the input saturation problem. is an auxiliary variable, It is a human-designed parameter, and its update law is as follows:

[0061] Among them, the constant , and .

[0062] In step 5, an adaptive output feedback control scheme is recursively designed through the backstepping method and the dynamic surface control technique.

[0063] The adaptive virtual control law and the actual control law are as follows:

[0064] Among them, , , , and are positive constants to be designed.

[0065] The parameter update law is as follows:

[0066] Among them, , is a constant to be designed.

[0067] In step 6, based on the Lyapunov theory, the stability of the system according to the adaptive output feedback control scheme and the parameter update law is proved. The specific steps are as follows:

[0068] Step 6-1, first design a Lyapunov function for the observer estimation error system: , where the positive definite symmetric matrix is the solution that satisfies the Lyapunov equation , is a given positive definite symmetric matrix.

[0069] Step 6-2, use the backstepping method for analysis. There are n steps in total. Consider the first Lyapunov function as ; the i-th Lyapunov function is , and the n-th Lyapunov function is . Take the derivatives of the above Lyapunov functions respectively, use lemmas such as the Young's inequality for scaling, and then substitute the designed virtual control law and actual control law for operations. Finally, it is obtained that

[0070] Among them, and are positive constants.

[0071] From this, it can be obtained that , so the closed-loop signal , and are both bounded. When is bounded, the tracking error is constrained within the specified performance envelope, that is, for , there is , so from it can be obtained that is also bounded. Through the standard analysis process, it is easy to obtain , , , , and the control input are all bounded. Since and are bounded, so is bounded. Therefore, there exists a constant such that .

[0072] Step 6-3, for the auxiliary variable construct a positive definite Lyapunov function as , and by taking the derivative, it can be obtained that

[0073] From equation (26), it can be obtained that as long as , then . Therefore, will converge stably to the set and further it can be obtained that is bounded. So far, all internal signals in all closed-loop systems are semi-globally uniformly ultimately bounded, and the tracking error can converge to the specified accuracy at the specified convergence rate within the preset time , that is, the practical tracking performance of the preset time is achieved.

[0074] The convergence time and convergence accuracy of the present invention can be artificially preset. A more concise tunnel preset performance control method is also proposed, and a completely asymmetric error constraint is realized through a simple control structure, which means that more diverse performance requirements can be achieved according to task requirements.

[0075] The RBF neural network is used to process the unknown nonlinear function , and the output of the state observer based on the RBF neural network is obtained to estimate the unmeasurable state, and the characteristic of the Gaussian basis function of the neural network is used to cleverly avoid the algebraic loop problem caused by the non-strict feedback structure.

[0076] By adopting the dynamic surface control technology, the cumbersome analytical calculation of the time derivative of the virtual control law in the backstepping design process is avoided, and in the design of the virtual error in the last step of the backstepping method, a first-order auxiliary system is constructed to compensate for the influence of asymmetric input saturation. The following are two specific embodiments.

[0077] Embodiment 1: A simulation case is given to describe in detail the technical effects of the present invention.

[0078] Consider the following second-order time-varying nonlinear system for numerical simulation.

[0079] where , the unknown nonlinear function and the unknown time-varying control gain are respectively , , , the external disturbance term , .

[0080] The simulation conditions are as follows: The control input of the system is subject to saturation constraints, and the upper and lower bound values are respectively and , the desired reference trajectory is , and the initial state . For the nonlinear functions and , an RBF neural network is used for approximation, and the parameter selection is as follows: The number of neural network nodes is 15 for both, the central domains of the Gaussian functions are respectively and , and the radial basis widths are respectively and . In the simulation, the parameters of the tunnel performance function are selected as: , , , , , , . The remaining parameters of the controller to be designed are taken as follows: , , , , , , , , , , , , , and the initial values of all state estimates and parameter estimates are set to zero.

[0081] Simulation results: The simulation results are as follows Figures 4 - 8 shown. Figure 4 The output tracking error curve and output tracking process curve of the system are depicted. It can be seen from the figure that within the preset time Within seconds, the tracking error converged to the predetermined accuracy range. , so that the control algorithm ensures practical tracking performance within the preset time. Figure 5 and Figure 6 The true and estimated value curves of the system status and are shown respectively. Figure 7 It is the evolution curve of the adaptive weight parameters of the neural network. Figure 8 The adaptive input signal and input saturation The simulation results show that the control scheme proposed in this invention is effective and achieves the control target well.

[0082] In summary, the algorithm proposed in the present invention is effective. It not only optimizes the preset performance control method, but also avoids the tedious calculations in the adaptive backstepping design process. It realizes the convergence of the tracking error to a neighborhood predetermined by the performance function within a preset time, and all signals of the entire closed-loop system are bounded.

[0083] Embodiment 2: When the control object is an industrial robot arm having multiple joints, the robot arm is controlled.

[0084] Assume that a robotic arm with multiple joints needs to be controlled to move along a reference trajectory in a time-varying environment. Only the position information of the end effector of the robotic arm can be directly measured, and it is constrained by the actuator control torque. Unknown time-varying functions and disturbance parameters in the robotic arm dynamics system may be caused by environmental factors such as external load changes or friction. In order to achieve high-precision trajectory tracking, it is necessary to overcome the influence of unknown nonlinear dynamics, input saturation and unknown disturbances. This embodiment adopts the aforementioned tunnel preset performance adaptive output feedback control method.

[0085] First, according to the steps of the method, the dynamic system of the robot arm is represented as a non-strict feedback nonlinear system; Then, a preset tunnel performance function with a preset time convergence is designed, a performance constraint is imposed on the tracking error signal, and then an error transformation is performed; Use radial basis function neural network approximation algorithm to process unknown nonlinear functions and design a state observer based on radial basis function neural network; Introduce coordinate transformation and design a first-order auxiliary system to dynamically compensate for saturation deviation to facilitate the design of subsequent controllers; Finally, by using the backstepping method and the dynamic surface control technology, an adaptive output feedback control method is designed to enable the end of the robotic arm to move along the reference trajectory.

[0086] From these two embodiments, the advantages of the tunnel preset performance adaptive output feedback control method in terms of convergence time and ensuring system performance can be seen. This method can be widely applied to the control of various complex nonlinear systems, such as autonomous vehicles, unmanned aerial vehicles, and industrial automation equipment, etc.

[0087] The above is a specific description of the preferred embodiments of the present invention, but the present invention is not limited to the described embodiments. Those skilled in the art can make various equivalent deformations or substitutions without departing from the spirit of the present invention, and these equivalent deformations or substitutions are all included in the protection scope determined by the claims of this application.

Claims

1. An adaptive output feedback control method for the preset performance of a tunnel under saturation constraints, characterized in that, The method includes the following steps: Step 1: Establish a non-strict feedback nonlinear system model with input saturation constraints; Step 2: Construct an improved preset-time tunnel preset performance function, establish a preset performance constraint, and perform error transformation; Step 3: Design a state observer based on a radial basis neural network; Step 4: Introduce a coordinate transformation and design a first-order auxiliary system to dynamically compensate for the saturation deviation for the subsequent design of the controller; Step 5: Recursively design an adaptive output feedback control scheme through the backstepping method and the dynamic surface control technique; Step 6: Based on the Lyapunov theory, prove the stability of the system according to the adaptive output feedback control scheme and the parameter update law.

2. The adaptive output feedback control method for tunnel preset performance under saturated constraints according to claim 1, wherein The non-strict feedback nonlinear system model with input saturation constraints includes: wherein, and are the state and output variables of the system respectively, and only the output of the system is measurable; represents an unknown continuous and differentiable non - linear function; represents an unknown time - varying and continuous control gain coefficient, satisfying , where the positive constant ; represents a time - varying external disturbance, satisfying , where is a positive constant; is the input signal of the actuator to be designed, represents that the actuator has an asymmetric saturated output characteristic; The asymmetric saturation constraint is described as: Among them, and respectively represent the magnitudes of the upper and lower boundary values of the actuator output.

3. A tunnel preset performance adaptive output feedback control method under saturation constraints according to claim 1, characterized in that The constructed improved preset-time tunnel preset performance function is: Among them , , , , and , , is a manually set time represents the system initial error at the moment represents the sign function; Define the standard error and the new error is transformed to: Adopt a new error transformation method to transform the tracking control problem under constraints into an unconstrained stable control problem.

4. A method for adaptively outputting feedback control of tunnel preset performance under saturation constraint according to claim 1, characterized in that The radial basis neural network state observer is: Among them, is the estimated value, and is the estimation error; and represent the ideal weight vector and basis function vector respectively, and there exists an arbitrary constant such that the approximation error satisfies ; is the estimated value of and is the Hurwitz polynomial coefficient.

5. The adaptive output feedback control method for tunnel preset performance under saturation constraint according to claim 4, wherein The coordinate transformation is: Among them, is a virtual control intermediate variable designed artificially, is the output of a first-order filter with as the input, is an auxiliary variable used to solve the input saturation problem, is a parameter designed artificially, and its update law is: Among them, the constant , and .

6. The adaptive output feedback control method for tunnel preset performance under saturation constraint according to claim 5, characterized in that The virtual control law, the adaptive output feedback control law, and the parameter update law are: Among them, , , , and are positive constants to be designed, , are constants to be designed.

7. A method for adaptively outputting feedback control of tunnel preset performance under saturation constraints according to claim 6, characterized in that In Step 6, it is proved that all signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the practical tracking performance of the preset time is achieved.

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