Inverted pendulum preview tracking control method and system based on double nonlinear constraints
By introducing dual nonlinear constraints and predictive control mechanisms into the inverted pendulum system, an expanded error system is constructed, and state feedback and static output feedback controllers are designed. This solves the problem of fine description of nonlinear characteristics and trajectory tracking in the inverted pendulum system, and achieves high-precision, phase-lag-free tracking control.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHANDONG JIANZHU UNIV
- Filing Date
- 2026-06-29
- Publication Date
- 2026-07-24
AI Technical Summary
Existing technologies struggle to achieve precise description of nonlinear factors and high-precision trajectory tracking in inverted pendulum systems. Traditional control methods are ineffective in handling multi-source nonlinear characteristics, and feedback control schemes suffer from phase lag.
A predictive tracking control method for an inverted pendulum based on dual nonlinear constraints is adopted. By constructing an amplified error system and combining the predictive target signal information, a state feedback and static output feedback predictive controller is designed to achieve high-precision tracking of the inverted pendulum system.
It significantly reduces the conservatism of controller design, improves system stability and tracking accuracy over a wide range, overcomes the phase lag problem of large inertial systems, and reduces hardware costs and computing resource requirements.
Smart Images
Figure CN122450168A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of robot control technology, and in particular to an inverted pendulum predictive tracking control method and system based on dual nonlinear constraints. Background Technology
[0002] The inverted pendulum system, as a typical underactuated, strongly coupled, and highly nonlinear controlled object, has long been an important platform for verifying control theory and applying it in engineering. However, precise control of the inverted pendulum faces severe challenges in real-world physical environments. On the one hand, the nonlinear factors present in the system, such as the trigonometric characteristics of the pendulum rod during large-angle swings and complex damping effects like joint friction and air resistance, often lead traditional Lipschitz-constraint-based control methods to employ a global coverage strategy, sacrificing the controller's dynamic response performance. While existing quasi-unilateral constraints (QOSL) have relaxed the restrictions to some extent, a single constraint dimension is insufficient to achieve a precise deconstruction of the "local slope" and the "global energy boundary" when dealing with systems like the inverted pendulum, which exhibit multi-source nonlinear characteristics.
[0003] On the other hand, in time-varying trajectory tracking tasks, traditional feedback control schemes suffer from significant phase lag and reduced tracking accuracy when the target signal (such as a ramp signal) changes due to the large physical inertia of the inverted pendulum.
[0004] Therefore, there is an urgent need for an inverted pendulum predictive tracking control strategy that takes into account both the detailed description of nonlinear constraints and the advance compensation for trajectory tracking. Summary of the Invention
[0005] To address the shortcomings of existing technologies, this invention provides an inverted pendulum predictive tracking control method and system based on dual nonlinear constraints; On the one hand, a predictive tracking control method for an inverted pendulum based on dual nonlinear constraints is provided; including: A dynamic model of an inverted pendulum with double nonlinear constraints is established to satisfy the double nonlinear constraint conditions. Based on the inverted pendulum dynamics model and tracking error signal, an expanded error system containing prediction information of the predicted target signal is constructed. Pre-defined unified closed-loop performance requirements serve as constraints for controller design. Based on the aforementioned amplified error system and dual nonlinear constraints, a state feedback predictive controller and a static output feedback predictive controller are designed respectively to ensure that the closed-loop system of the controller satisfies the controller design constraints, and the corresponding controller is selected according to the measurable information type of the inverted pendulum system. The system acquires sensor feedback signals in real time, reads future information of the target signal within the prediction window, calculates control commands based on the selected controller, and outputs them to the actuator to drive the inverted pendulum to complete high-precision tracking.
[0006] On the other hand, an inverted pendulum predictive tracking control system based on dual nonlinear constraints is provided; including: The nonlinear modeling module is used to establish a dynamic model of an inverted pendulum with double nonlinear constraints, which satisfies the double nonlinear constraint conditions. The nonlinear constraints simultaneously satisfy the unilateral Lipschitz condition and the quadratic inner bounded condition. The error system augmentation module is used to construct an expanded error system containing prediction information of the predicted target signal based on the inverted pendulum dynamics model and the tracking error signal; The closed-loop performance preset module is used to preset uniform closed-loop performance requirements as a design constraint for the controller. The controller selection module is used to design a state feedback predictive controller and a static output feedback predictive controller based on the amplified error system and the dual nonlinear constraints, respectively, so that the closed-loop system of the controller satisfies the controller design constraints, and selects the corresponding controller according to the measurable information type of the inverted pendulum system. The real-time drive execution module is used to acquire sensor feedback signals in real time, read future information of target signals within the prediction window, calculate control commands in combination with the selected controller, and output them to the actuator to drive the inverted pendulum to complete high-precision tracking.
[0007] The above technical solution has the following advantages or beneficial effects: This invention introduces dual nonlinear constraints—one-sided Lipschitz conditions and quadratic internal bounded conditions—to accurately preserve the nonlinear characteristics of the inverted pendulum system from both the local slope and overall energy dimensions. Compared with the traditional single Lipschitz constraint method, this significantly reduces the conservatism of controller design, enabling the system to remain stable over a wider operating range and reducing unnecessary control redundancy.
[0008] Meanwhile, this invention deeply couples the predictive control mechanism with the nonlinear system. By constructing an expanded error system that includes a target signal with a finite step size in the future, the inverted pendulum can perceive the trajectory change trend in advance and make feedforward compensation, effectively overcoming the phase lag problem of large inertial systems and achieving high-precision tracking of time-varying signals such as slopes with zero steady-state error and zero lag.
[0009] In response to common engineering scenarios where sensor configuration is limited, this invention specifically designs a static output feedback predictive controller. It can achieve control simply by acquiring a linear combination of position and velocity signals, eliminating the need to construct a complex state observer and significantly reducing hardware costs and computational resource requirements. Attached Figure Description
[0010] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.
[0011] Figure 1 This is a flowchart of the method in Embodiment 1 of the present invention.
[0012] Figure 2 This is a comparison diagram of the output response of the inverted pendulum based on state feedback and the reference trajectory in Embodiment 1 of the present invention.
[0013] Figure 3 This is a trajectory diagram of the tracking error change of the inverted pendulum based on state feedback in Embodiment 1 of the present invention.
[0014] Figure 4 This is a trajectory diagram of the inverted pendulum control input signal based on state feedback in Embodiment 1 of the present invention.
[0015] Figure 5 This is a comparison diagram of the output response of the inverted pendulum based on static output feedback and the reference trajectory in Embodiment 1 of the present invention.
[0016] Figure 6 This is a trajectory diagram of the tracking error change of the inverted pendulum based on static output feedback in Embodiment 1 of the present invention.
[0017] Figure 7 This is a trajectory diagram of the inverted pendulum control input signal based on static output feedback in Embodiment 1 of the present invention. Detailed Implementation
[0018] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.
[0019] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.
[0020] All data acquisition in this embodiment is carried out in accordance with laws and regulations and with user consent, and the data is used legally.
[0021] Example 1 like Figure 1 As shown, this embodiment provides a predictive tracking control method for an inverted pendulum based on dual nonlinear constraints; it includes the following steps: S1. Establish a dynamic model of an inverted pendulum with double nonlinear constraints, satisfying the double nonlinear constraint conditions; the nonlinear constraints simultaneously satisfy the unilateral Lipschitz condition and the quadratic inner bounded condition. S2. Based on the inverted pendulum dynamics model and tracking error signal, construct an expanded error system that includes prediction information of the predicted target signal; S3. Preset unified closed-loop performance requirements as a design constraint for the controller; S4. Based on the aforementioned amplified error system and dual nonlinear constraints, design a state feedback predictive controller and a static output feedback predictive controller respectively, so that the closed-loop system of the controller satisfies the controller design constraints, and select the corresponding controller according to the measurable information type of the inverted pendulum system. S5. Real-time acquisition of sensor feedback signals, reading future information of target signals within the prediction window, combining with the selected controller to calculate control commands, and outputting them to the actuator to drive the inverted pendulum to complete high-precision tracking.
[0022] Further, in step S1, the dynamic model of the nonlinear system is: (1) In the formula, for time; Let be the state vector of the system, and ; As the control input of the system, and ; Let be the system's output vector, and ; It is an external interference, and satisfies , External interference The derivative, It is a space of square-integrable functions; , , , , , H Given a constant matrix; Nonlinear terms that satisfy the one-sided Lipschitz (OSL) condition and the quadratic inner bounded (QIB) condition; These represent the dimensions of the vector space.
[0023] This embodiment makes the following assumptions for subsequent theoretical considerations: Assumption 1. Let satisfy (2)
[0024] (3) in It is the OSL constant. and It is the QIB constant.
[0025] Assumption 2. Let the target signal be... Piecewise continuous differentiable and satisfying ,in It is a constant vector, and The derivative satisfies Furthermore, from the current moment... Start, target signal It was foreseeable. Here, This is called the prediction length of the target signal.
[0026] To simplify the representation, symbols are introduced. express .
[0027] To facilitate subsequent analysis and proof, the following lemma is given.
[0028] Lemma 1. For a matrix of appropriate dimension... , and symmetric matrix The necessary and sufficient condition for is that one of the following conditions is true: (4) (5) Lemma 2. For a matrix of appropriate dimension and constant If the inequality (6) Then there is an inequality Established.
[0029] Furthermore, S2 specifically includes: S201. Differentiate the state equations and output equations of the nonlinear dynamic system. (7) In the formula, The derivative of the system state; The derivative for controlling the input; The derivative of the external disturbance; The derivative of the output vector; This step of differentiation is to deal with the one-sided Lipschitz nonlinear term in the original system. By applying Assumption 1 and cleverly combining the S-method, the derivative of the nonlinear term obtained after differentiation can be effectively solved, which facilitates the design of the subsequent predictive controller.
[0030] S202. Obtain the tracking error and the dynamic equation satisfied by the tracking error.
[0031] Tracking error is (8) Differentiating the tracking error, we obtain the dynamic equation that the tracking error satisfies. (9) In the formula, The derivative of the tracking error; The derivative of the target signal.
[0032] S203. Introduce auxiliary state variables, solve the state derivative equation and the error dynamic equation simultaneously, and construct the augmented error system.
[0033] Among them, auxiliary state variables are introduced. ,get (10) make By simultaneously solving the state derivative equation (7) and the error dynamic equation (10), the augmented error system is obtained. (11) In the formula, , , , For augmented matrices, Let represent the coefficient matrix of the state vector, and , The coefficient matrix represents the derivative of the control input. , The coefficient matrix represents a nonlinear vector. ; The coefficient matrix represents the interference vector. ; To augment nonlinear vector functions, and ; To augment the interference vector, and ; for p An identity matrix of order 1.
[0034] S204. Introduce the linear quadratic performance index function and performance signal, and combine the augmented error system and performance signal to obtain the expanded error system.
[0035] Assuming 2, the target signal From the current moment Beginning, Future Information within the step range This information is known in advance. This embodiment makes full use of this information and uses some clever mathematical operations to modify the system, thereby facilitating the subsequent design of the controller to introduce a predictive feedforward compensation mechanism for the target signal.
[0036] To evaluate the tracking performance of the system, a linear quadratic performance index function is introduced. : (12) In the formula, , is the augmented state weighting matrix. This is a weighted matrix for the rate of change of control inputs in the performance metrics.
[0037] Define the performance signal as: (13) In the formula, The weight matrix for the augmented state. ; The weight matrix for controlling the input derivative, and .
[0038] The performance index function can be further expressed as the square of the 2-norm of the performance signal, i.e.: (14) By combining the augmented error system and the performance signal, the expanded error system is obtained: (15) In predictive control theory, the system described above is usually referred to as the expanded error system. Therefore, the predictive control problem of the nonlinear system dynamics model is transformed into the expanded error system under the performance signal... H ∞ Control issues. Technical benefits: Transforming the original "output tracking problem" into a "stability problem" of the new system is the core step in achieving zero steady-state error tracking.
[0039] Furthermore, S3 predefines a unified closed-loop performance target to ensure the tracking effect of the inverted pendulum under complex nonlinear constraints, providing a unified stability and robustness constraint criterion for subsequent controller design; specific performance requirements include system asymptotic stability and disturbance attenuation performance. Among them, the asymptotic stability of the system requires that, without considering external disturbances (i.e., Under ideal operating conditions, the extended error closed-loop system can spontaneously tend to an equilibrium state, that is, the tracking error eventually converges to zero over time. Disturbance attenuation performance ( H ∞ Performance): The system is required to have a preset anti-interference capability in the presence of external disturbances (such as friction, air resistance, and other limited energy interference), so that the controlled output is sensitive to disturbances (i.e., Gain is suppressed at a given attenuation level. Within; that is (16) In the formula, This is to augment the interference vector. expressH ∞ Performance index (attenuation level), expressed in the above formula: if The smaller the value, the better the impact of external disturbances on system performance is suppressed, and the stronger the system robustness.
[0040] Further, in step S4, constructing the state feedback predictive controller includes: S411. Design a state feedback controller based on the state vector of predictable target signal information; (17) In the formula, The feedback controller gain matrix to be determined. A state vector that incorporates predictable information about the target signal.
[0041] S412. Substitute the state feedback controller into the amplified error system to obtain the closed-loop system; (18) At this point, the research question is transformed into the stability of the closed-loop system and H ∞ Performance analysis issues.
[0042] S413. Based on Lyapunov stability theory, unilateral Lipschitz and quadratic inner bounded constraints, derive the asymptotic stability of the closed-loop system and the condition that it satisfies... H ∞ Sufficient conditions for disturbance attenuation performance are derived and transformed into a set of linear matrix inequalities.
[0043] Specifically, Theorem 1. Assume that Assumptions 1-2 hold. For a given scalar... , , OSL-QIB constant , , and the given weight matrix , , If a matrix exists sum matrix , making Case 1: When hour (19) Case 2: When hour (20) in,
[0044] Then the closed-loop system (18) of the expanded error system (15) is asymptotically stable and satisfies H ∞ Performance index criteria (16).
[0045] Proof: Using Theorem 1 Construct the following Lyapunov function: (twenty one) To make the closed-loop system (18) asymptotically stable and have H ∞ For interference suppression performance to hold, the following inequality must be true: (twenty two) in
[0046] From hypothesis 1 and the chain rule, we obtain (twenty three) (twenty four) in , , .
[0047] Therefore, combining equations (23) and (24), equation (22) can be further expressed as: (25) in, ,
[0048] At this point, the problem is transformed into finding inequalities. Sufficient conditions for its establishment.
[0049] Applying Lemma 1, we know that Equivalent to (26) Perform a congruent transformation on the left-hand side of equation (26): that is, multiply the left-hand side by an invertible matrix. Multiply the right side by its transpose, and let... You can get (27) in
[0050]
[0051] To facilitate numerical solutions, matrix inequality (27) is based on The sign of is divided into two different cases, as described in Theorem 1. Therefore, if condition (27) is satisfied, then condition (26) holds, and thus Theorem 1 is proved.
[0052] S414. Solve the system of linear matrix inequalities using a convex optimization algorithm to obtain the optimal state / output feedback gain and predictive compensation gain matrix.
[0053] Controller gain matrix It can be by Calculated.
[0054] To more clearly illustrate the controller structure, the gain matrix of the state feedback controller will be used. Decomposed into: (28) In the formula, , is the state feedback gain matrix. , where is the gain matrix of the tracking error integral term and the prediction feedforward compensation term.
[0055] S415. The state feedback predictive controller designed based on the amplified error system is reverted to the inverted pendulum nonlinear system to obtain the final state feedback predictive controller.
[0056]
[0057] As can be seen from the right side of the above equation, the controller proposed in this invention consists of the following three parts: the first part It is the status feedback item, part two. It is the integral term of the output tracking error, used to eliminate the static error of the system, Part 3 It is a feedforward compensation term that anticipates future information of the target signal, used to improve the tracking performance of the system.
[0058] Furthermore, in step S4, constructing the static output feedback predictive controller includes: S421. Construct a new output equation and design an output feedback controller.
[0059] In order to utilize the predictable information of the reference trajectory, a new output equation is constructed. (30) In the formula, the output matrix .
[0060] Based on the new output equation, the output feedback controller is designed as follows: (31) In the formula, The output feedback controller gain matrix to be determined. The output vector is used to introduce predictable information about the target signal.
[0061] S422. Substitute the output feedback controller into the amplified error system to obtain the closed-loop system; (32) Therefore, the research question is transformed into the stability of the closed-loop system and H ∞ Performance analysis issues.
[0062] S423. Based on Lyapunov stability theory, the one-sided Lipschitz condition, and the quadratic inner bounded constraint, derive the asymptotic stability of the closed-loop system with expanded error under the static output feedback condition, while satisfying the following: H ∞ Sufficient conditions for disturbance attenuation performance are derived and transformed into a set of linear matrix inequalities.
[0063] Theorem 3. Assume that assumptions 1-2 hold. For a given scalar... , , , OSL-QIB constant , , Given a matrix , and weight matrix , , If a matrix exists and , making Case 1: When hour (33) Case 2: When hour (34) in, , ,
[0064] Then the closed-loop system (32) of the expanded error system (15) is asymptotically stable and satisfies H ∞ Performance index criteria (16).
[0065] Proof: Using Theorem 3 Construct the following Lyapunov function: (35) To make the closed-loop system (18) asymptotically stable and have H ∞ For interference suppression performance to hold, the following inequality must be true: (36) in
[0066] Combining equations (23) and (24), equation (36) can be further expressed as: (37) in
[0067] At this point, the problem is transformed into finding inequalities. Sufficient conditions for its validity. Applying Lemma 1, we know that... Equivalent to (38) Perform a congruent transformation on the left-hand side of equation (38): that is, multiply the left-hand side by an invertible matrix. Multiplying it by its transpose on the right yields the following result. (39) in
[0068] To handle the nonlinear coupling terms in inequality (39), variable substitution and auxiliary matrices are used. Let ,but Therefore, equation (39) can be further rewritten as follows: (40) in, , , .
[0069] Applying Lemma 2, we know that the sufficient condition for inequality (40) to hold is: (41) in, .
[0070] To facilitate numerical solutions, matrix inequality (41) is based on The sign of is divided into two different cases, as described in Theorem 3. Therefore, if condition (41) is satisfied, then condition (40) holds, and thus Theorem 3 is proved.
[0071] S424. Solve the system of linear matrix inequalities using a convex optimization algorithm to obtain the static output feedback gain and the predictive compensation gain.
[0072] Controller gain matrix It can be by Calculated.
[0073] To more clearly illustrate the controller structure, the output feedback controller gain matrix will be used. Decomposed into: (42) In the formula, , is the output feedback gain matrix. , where is the gain matrix of the tracking error integral term and the prediction feedforward compensation term.
[0074] S425. The static output feedback predictive controller designed based on the amplified error system is reverted to the inverted pendulum nonlinear system to obtain the final static output feedback predictive controller.
[0075] (43) As can be seen from the right side of the above equation, the first part... It is based on the system's output feedback control action, and the other items are consistent with equation (29).
[0076] This step significantly reduces the system's reliance on sensors and lowers hardware costs.
[0077] Furthermore, based on the inverted pendulum system, simulation experiments were conducted to compare the method proposed in this embodiment with the control schemes of the prior art, in order to verify the effectiveness and superiority of the control method proposed in this embodiment.
[0078] The inverted pendulum system is as follows:
[0079] In the formula, , , , .
[0080] Corresponding system (1), equivalent to , , , , ,
[0081] It has been verified that the nonlinear term The OSL-QIB conditions (2) and (3) in Assumption 1 are satisfied, where , , Select parameters , , , , Based on Theorem 1, and using the LMI toolbox in MATLAB, the gain matrix of the predictive controller based on state feedback is calculated as follows: , .
[0082] For numerical simulation purposes, let the foreseeable target signal be... , External interference is taken as The initial state of the system is taken as .
[0083] Figures 2-4 These are the system's output response, tracking error, and control input response curves under state feedback, respectively. (In summary...) Figures 2 to 4 Simulation results show that the predictive control of this invention effectively overcomes the phase lag defect in output response of traditional control, achieving accurate and proactive tracking of time-varying target signals. Its dynamic error in the main tracking interval approaches zero, significantly improving control accuracy compared to traditional control. The technical mechanism behind this superior performance lies in the fact that the predictive control utilizes future reference information to output control compensation in advance, thereby actively offsetting the system's own dynamic delay. Notably, the predictive compensation controller designed in this invention achieves proactive sensing of the target signal through a feedforward compensation mechanism, enabling the system to respond in advance to changes in the reference input. This control strategy significantly improves the system's dynamic response characteristics, achieving rapid adjustment and optimized response time.
[0084] We first select the adjustment parameters. According to Theorem 3, the output feedback gain matrix can be solved. , .
[0085] Figures 5-7 Simulation comparison results of the system under output feedback control are presented. For example... Figure 5 The output response comparison shows that the predictive control of this invention effectively overcomes the obvious phase lag of traditional control, achieving high synchronization and advance tracking with the reference trajectory; for example... Figure 6As shown in the tracking error curve, predictive control significantly reduces the peak dynamic error of traditional control from approximately 0.06 to approximately 0.013, and the steady-state error strictly converges to zero, significantly improving the overall tracking accuracy of the system; combined with Figure 7 As can be seen from the control input response, the control signal of predictive control starts earlier and transitions smoothly. While achieving advance compensation, it avoids drastic changes in control energy and effectively ensures the stable operation of the actuator in the actual physical system.
[0086] Example 2 This embodiment provides an inverted pendulum predictive tracking control system based on dual nonlinear constraints, including: The nonlinear modeling module establishes a dynamic model of an inverted pendulum with double nonlinear constraints, satisfying the double nonlinear constraint conditions; the nonlinear constraints simultaneously satisfy the unilateral Lipschitz condition and the quadratic inner bounded condition. The error system augmentation module is used to construct an expanded error system containing prediction information of the predicted target signal based on the inverted pendulum dynamics model and the tracking error signal; The closed-loop performance preset module is used to preset uniform closed-loop performance requirements as a design constraint for the controller. The controller selection module is used to design a state feedback predictive controller and a static output feedback predictive controller based on the amplified error system and the dual nonlinear constraints, respectively, so that the closed-loop system of the controller satisfies the controller design constraints, and selects the corresponding controller according to the measurable information type of the inverted pendulum system. The real-time drive execution module is used to acquire sensor feedback signals in real time, read future information of target signals within the prediction window, calculate control commands by combining the selected controller and preset closed-loop performance requirements, and output them to the actuator to drive the inverted pendulum to complete high-precision tracking.
[0087] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A predictive tracking control method for an inverted pendulum based on dual nonlinear constraints, characterized in that, include: A dynamic model of an inverted pendulum with double nonlinear constraints is established to satisfy the double nonlinear constraint conditions. Based on the inverted pendulum dynamics model and tracking error signal, an expanded error system containing prediction information of the predicted target signal is constructed. Pre-defined unified closed-loop performance requirements serve as constraints for controller design. Based on the aforementioned amplified error system and dual nonlinear constraints, a state feedback predictive controller and a static output feedback predictive controller are designed respectively to ensure that the closed-loop system of the controller satisfies the controller design constraints, and the corresponding controller is selected according to the measurable information type of the inverted pendulum system. The system acquires sensor feedback signals in real time, reads future information of the target signal within the prediction window, calculates control commands based on the selected controller, and outputs them to the actuator to drive the inverted pendulum to complete high-precision tracking.
2. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, The dual nonlinear constraints include the unilateral Lipschitz condition and the quadratic inner bounded condition.
3. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, Constructing an augmented error system that includes the predicted target signal information involves: differentiating the state equation and output equation of the nonlinear system dynamics; obtaining the tracking error and deriving the dynamic equation satisfied by the tracking error; introducing auxiliary state variables, simultaneously solving the state derivative equation and the error dynamic equation to construct the augmented error system; and introducing a linear quadratic performance index function and performance signal, simultaneously solving the augmented error system and the performance signal to obtain the augmented error system.
4. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, The closed-loop performance objectives include system asymptotic stability and disturbance attenuation performance; under ideal operating conditions without considering external disturbances, the amplified error closed-loop system can spontaneously tend towards equilibrium, and the tracking error asymptotically converges to zero; in the presence of external disturbances, the performance signal is improved from the amplified external disturbance. The gain is suppressed within a given attenuation level.
5. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, Construct a state feedback predictive controller, including: Design a state feedback controller based on the state vector of predictable target signal information; Substituting the state feedback controller into the amplified error system yields the closed-loop system; Based on Lyapunov stability theory, unilateral Lipschitz conditions, and quadratic inner bounded constraints, we derive the asymptotic stability of the closed-loop system and its satisfying condition. H ∞ Sufficient conditions for disturbance attenuation performance are derived and transformed into a set of linear matrix inequalities; The optimal state / output feedback gain and predictive compensation gain matrix are obtained by solving the system of linear matrix inequalities using a convex optimization algorithm. By reverting the state feedback predictive controller designed based on the amplified error system to the inverted pendulum nonlinear system, the final state feedback predictive controller is obtained.
6. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, Constructing the static output feedback predictive controller includes: Construct a new output equation and design an output feedback controller; Substituting the output feedback controller into the amplified error system yields the closed-loop system; Based on Lyapunov stability theory, the one-sided Lipschitz condition, and the quadratic inner bounded constraint, it is derived that the closed-loop asymptotic stability of the expanded error system under the static output feedback case satisfies the following conditions. H ∞ Sufficient conditions for disturbance attenuation performance are derived and transformed into a set of linear matrix inequalities; The static output feedback gain and the predictive compensation gain are obtained by solving the system of linear matrix inequalities using a convex optimization algorithm. By reverting the static output feedback predictive controller designed based on the amplified error system to the inverted pendulum nonlinear system, the final static output feedback predictive controller is obtained.
7. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 5, characterized in that, The state feedback predictive controller is: in, The state feedback gain matrix, The gain matrix for the tracking error integral term and the prediction feedforward compensation term; for time; Let this be the system's state vector; For tracking error; For target signal; The predicted length of the target signal.
8. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 6, characterized in that, The static output feedback predictive controller is: in, For the output feedback gain matrix; The gain matrix for the tracking error integral term and the prediction feedforward compensation term; for time; This is the system's output vector; For tracking error; For target signal; The predicted length of the target signal.
9. The inverted pendulum predictive tracking control method based on dual nonlinear constraints according to claim 1, characterized in that, If the system state is fully measurable, select the state feedback predictive controller; if the system state information is not fully measurable, select the static output feedback predictive controller.
10. A predictive tracking control system for an inverted pendulum based on dual nonlinear constraints, characterized in that, include: The nonlinear modeling module establishes a dynamic model of an inverted pendulum with double nonlinear constraints, satisfying the double nonlinear constraint conditions; the nonlinear constraints simultaneously satisfy the unilateral Lipschitz condition and the quadratic inner bounded condition. The error system augmentation module is used to construct an expanded error system containing prediction information of the predicted target signal based on the inverted pendulum dynamics model and the tracking error signal; The closed-loop performance preset module is used to preset uniform closed-loop performance requirements as a design constraint for the controller. The controller selection module is used to design a state feedback predictive controller and a static output feedback predictive controller based on the amplified error system and the dual nonlinear constraints, respectively, so that the closed-loop system of the controller satisfies the controller design constraints, and selects the corresponding controller according to the measurable information type of the inverted pendulum system. The real-time drive execution module is used to acquire sensor feedback signals in real time, read future information of target signals within the prediction window, calculate control commands by combining the selected controller and preset closed-loop performance requirements, and output them to the actuator to drive the inverted pendulum to complete high-precision tracking.