Reachable set p-d controller design method for nonlinear time-varying time-delay dc motor system

Abstract

The application belongs to the technical field of DC motor system control, and relates to a reachable set P-D controller design method for a nonlinear time-varying time-delay DC motor system. The application solves the problem that the existing reachable set estimation method cannot design an effective controller with low conservativeness, easy to solve and capable of reflecting time-delay dependent characteristics for a continuous DC motor system under multiple disturbances and complex dynamic coupling. According to the dynamic relationship of the DC motor, the application constructs an open-loop singular model containing time-varying time delay and nonlinearity, and introduces a P-D controller to equivalently transform the open-loop singular model into a closed-loop normal system. A Lyapunov-Krasovskii functional is constructed and differentiated, a free weight matrix is introduced to construct a constant zero equation, and the three are superimposed to establish a nonlinear matrix inequality. After extraction and simplification of the controllable matrix inequality, the P-D controller gain and ellipsoid matrix are obtained through a solver. The application is mainly applied to the field of robust control of DC motor systems.

Description

Technical Field

[0001] This invention belongs to the field of DC motor system control technology. Background Technology

[0002] In recent years, with the development of science and technology and the improvement of living standards, the demand for various mechanical equipment and power systems has been increasing. Electric motors, as an important tool for energy conversion, are indispensable in production and daily life. Among them, DC motors are widely used due to their high reliability, low cost, and ease of management. Correspondingly, effective control of these motors is necessary and has attracted widespread attention from engineers and scholars. Since DC motor systems involve both power supply and mechanical transmission components, and numerous variables, this undoubtedly increases the difficulty of controller design, requiring comprehensive consideration. Currently, traditional control technologies for this system are relatively mature.

[0003] In fact, by appropriately modeling and processing DC motor systems, they can be described as singular systems. As one of the important research directions in singular systems, reachability set estimation is an active research area. Reachability set estimation aims to determine a closed bounded region that encompasses all possible state trajectories of the system, given initial conditions and bounded peak input conditions. In recent years, research results on reachability set estimation for singular systems have become increasingly abundant. For example, existing studies have explored the reachability set estimation problem for switching singular systems with mixed time delays, as well as the reachability set estimation problem for nonlinear singular systems with distributed time delays. Furthermore, reachability set estimation methods have provided new ideas for the control problem of singular systems. For instance, for linear time-varying time-delay singular systems, research has been conducted on the reachability set synthesis problem based on an adaptive event-triggered mechanism; other scholars have studied the real-time reachability set control problem of singular Markov jump systems under mixed time delays and uncertain parameters, fully demonstrating the potential of reachability set estimation methods in system control research.

[0004] Time delay, a ubiquitous physical phenomenon, is a significant factor leading to system performance degradation and even instability. Due to data transmission delays, inaccurate modeling, linearization approximations, unknown interferences, and measurement errors, time delay is unavoidable in various dynamic systems. Considering time delay in system models helps to more accurately reflect actual operating conditions; for example, time-delay discrete population models and time-delay network control systems have been widely incorporated into the modeling of various practical systems. Furthermore, system components may exhibit potential faults during operation; neglecting this factor in practical modeling can lead to system performance degradation. Therefore, related research has explored the problem of fault detection and diagnosis in dynamic systems, highlighting the importance of ensuring system reliability.

[0005] In summary, although the control problem of DC motor systems has a certain research foundation, existing work still has limitations. For continuous DC motor systems with both bounded peak disturbances and fault inputs, and which include nonlinear characteristics and interval time-varying delays, the control problem has not yet been effectively solved within the reachability set framework. The main difficulty lies in the coupling of system singularity, nonlinear dynamics, and interval time-varying delays, which complicates the construction of suitable Lyapunov-Krasovskii functionals. Furthermore, under the combined effects of disturbances and faults, it is difficult to obtain conditions that are both easy to solve and fully reflect the time-delay dependence characteristics while maintaining low conservatism. To address these issues, this study aims to design a controller for a class of continuous nonlinear interval time-varying delay DC motor systems, considering both bounded peak disturbances and fault inputs, using reachability set estimation and optimization methods. Summary of the Invention

[0006] The purpose of this invention is to address the problem that existing reachability set estimation methods cannot design effective controllers for continuous DC motor systems that are low in conservatism, easy to solve, and reflect time-delay dependence characteristics under multiple disturbances and complex dynamic coupling. This invention provides a method for designing reachability set PD controllers for nonlinear time-varying time-delay DC motor systems.

[0007] The design method for reachable set PD controllers for nonlinear time-varying time-delay DC motor systems includes:

[0008] S1. Based on the dynamic relationship of the open-loop DC motor system, construct an open-loop singular system model with time-varying time delay and nonlinear system function;

[0009] S2. Eliminate the singularity of the open-loop singular system model and transform it into an equivalent closed-loop normal system model.

[0010] S3. Construct the Lyapunov-Krasovsky functional for the closed-loop normal system model and solve for the derivative of the functional.

[0011] S4. Linearize the nonlinear function of the system to obtain the linearization inequality of the nonlinear function of the system.

[0012] Based on the closed-loop normal system model, multiple invertible free weight matrices are introduced to construct the constant zero equation;

[0013] S5. Superimpose the linearization inequalities of the system's nonlinear function, the derivative of the functional, and the zero-resistance equation to establish an ellipsoidal matrix containing the unknown. and the proportional gain matrix of the PD state feedback controller and differential gain matrix Nonlinear matrix inequalities;

[0014] S6. Extract the controllable matrix inequalities from the nonlinear matrix inequalities. Simplify the controllable matrix inequalities based on the established relationships between multiple invertible free weight matrices. Perform a linear transformation on the simplified controllable matrix inequalities using matrix transformations. Solve the parameters of the linearly transformed controllable matrix inequalities using a solver to obtain the proportional gain matrix. Differential gain matrix and ellipsoidal matrix .

[0015] Preferably, the expression for the open-loop singular system model is:

[0016]

[0017] in,

[0018] Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative;

[0019] System nonlinear function satisfy; , Given the first real constant, It is a constant matrix;

[0020] System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ;

[0021] It is a singular matrix and satisfies , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system. , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative of For time-varying time delay The system status, This is the control input for the system.

[0022] Preferably, in step S2, the equivalent transformation of the open-loop singular system model into a closed-loop normal system model is implemented as follows:

[0023] By introducing a PD state feedback controller into the open-loop singular system model and eliminating singularities, a closed-loop normal system model is obtained.

[0024] The expression for the closed-loop normal system model is:

[0025] ;

[0026] in,

[0027] Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative;

[0028] System nonlinear function satisfy: , Given the first real constant, It is a constant matrix;

[0029] System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ;

[0030] It is a singular matrix. As a non-singular matrix, and satisfying , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system. , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative of For time-varying time delay The system status.

[0031] Preferably, in step S3, the expression of the constructed Lyapunov-Krasovsky functional is:

[0032] ;

[0033] in,

[0034]

[0035]

[0036]

[0037]

[0038] For the expression of the Lyapunov-Krasovsky functional, to These are the first to fourth components of the Lyapunov-Krasovsky functional expression, respectively. For time, For system status, For transpose, Time-varying time delay The value of , It is a natural constant. The system state corresponding to the independent variable of the first integral term. for The derivative of Let be the independent variable of the first integral term. to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, Given a second real constant, Let be the independent variable of the second integral term. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. is the length of the time-varying time-delay interval.

[0039] Preferably, in step S4, the expression for the linearization inequality of the system's nonlinear function is:

[0040] ;

[0041] in, For the nonlinear function of the system, For system status, For time, For transpose, Given the first real constant, It is a constant matrix.

[0042] Preferably, in step S4, the expression of the constructed constant-zero equation is:

[0043] ;

[0044] in, , , and These are the first to fourth invertible free weight matrices, This is a transpose.

[0045] Preferably, in step S5, the established ellipsoidal matrix includes the unknown ellipsoidal matrix. and the proportional gain matrix of the PD state feedback controller and differential gain matrix The expression for the nonlinear matrix inequality is:

[0046]

[0047] in,

[0048] ;

[0049]

[0050]

[0051] And controllable matrix In ,satisfy As a controllable matrix inequality;

[0052] For the Lyapunov-Krasovsky functional, For functionals The derivative of For system status, for The derivative of For those with time delay The system status, For time-varying time delay The system status, For those with time delay The system status, For time, The system state corresponding to the independent variable of the first integral term. Let be the independent variable of the first integral term. For transpose, Given a second real constant, and Given the third real constant and the fourth real constant, respectively. The intermediate vector;

[0053] to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in the 4th row and columns 4, 8, 9, and 10 of the data. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , and They are respectively The elements in the 6th row and columns 1, 3, and 6. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , , They are respectively The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in columns 1, 2, 4, 8, and 10 of row 10. Time-varying time delay The possible values ​​of ;

[0054] This is an operation that adds a matrix to its transpose. It is the identity matrix. Let be the constant to be solved, and , to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, The length of the time-varying time-delay interval. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. , , and These are the first to fourth invertible free weight matrices, Given a second real constant, It is a natural constant. Time-varying time delay The upper bound of the time derivative.

[0055] Preferably, in step S6, the relationship between the multiple invertible free weight matrices is as follows:

[0056]

[0057] in, These represent the first to third coefficients, , , and These are the first to fourth invertible free weight matrices, respectively.

[0058] Preferably, in step S6, the method for performing a linear transformation on the simplified controllable matrix inequality using matrix transformation is as follows:

[0059] For the simplified controllable matrix inequality, the controllable matrix Left multiplication of the transformation matrix Right multiplication of the transformation matrix The transpose of; where,

[0060] ;

[0061] and, , , , , , , , , , , and ;

[0062] This is the fourth invertible free weight matrix. The ellipsoid matrix after linear transformation. to These are the first to third 2×2 positive definite matrices after linear transformation. and These are the fourth and fifth 2×2 positive definite matrices after linear transformation. and These are the sixth and seventh 2×2 positive definite matrices after linear transformation. and These are the proportional and differential gain matrices of the PD state feedback controller after linear transformation. For transpose, The identity matrix after linear transformation. It is a constant matrix. is the constant matrix after linear transformation.

[0063] Preferably, the expression for the controllable matrix inequality after the linear transformation in step S6 is:

[0064]

[0065]

[0066] Furthermore, the controllable matrix after linear transformation In ,satisfy

[0067] to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 4 in the dataset. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , , for The elements in the 6th row and columns 1, 3, and 6. , , for The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , and for The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in the 10th row and columns 1, 2, 4, 8, and 10.

[0068] The beneficial effects of this invention are:

[0069] This invention addresses the control challenges of nonlinear time-varying time-delay DC motor systems under multiple disturbances and complex dynamic coupling scenarios, systematically solving the problems of high conservatism, difficult solution, and insufficient characterization of time delay characteristics in existing methods.

[0070] A refined model addressing time-delay dependence significantly reduces conservatism: By establishing an open-loop singular system model incorporating time-varying delays, nonlinear functions, external disturbances, and actuator failures, and introducing a PD state feedback controller, it is equivalently transformed into a closed-loop normal system. Compared to traditional methods that ignore time-delay dynamics or simplify to steady-state delays, this invention explicitly constrains the upper bound of the time-delay derivative and designs piecewise weighted integral terms in the Lyapunov-Krasovsky functional to accurately characterize the time-varying characteristics of the time delay within the interval. This approach to time-delay dependence avoids the excessive conservatism of traditional time-delay-independent methods, making the reachability set boundary estimation closer to the actual dynamics of the system.

[0071] By integrating the free weight matrix and the zero-constancy equality, this invention enhances the solvability of controllable matrix inequalities: Addressing the challenge of coupling system state, nonlinear functions, and time-delay terms, this invention constructs a zero-constancy equality through a system model and introduces invertible free weight matrices of some terms as relaxation variables. Combining the linearization inequality of nonlinear functions, the functional derivative, nonlinear function constraints, and the zero-constancy equality are superimposed to construct a nonlinear matrix inequality including the ellipsoid matrix and controller gain. This design effectively solves the problem of positive definite matrices on the main diagonal in controllable matrix inequalities by flexibly adjusting the free weight matrices of some terms, thus avoiding the inequality being unsolvable.

[0072] Matrix transformation enables convexification of nonlinear problems, improving solution efficiency: To address the difficulty of directly solving nonlinear matrix inequalities, this invention significantly reduces the number of decision variables by setting proportional relationships between free weight matrices. Furthermore, by constructing transformation matrices to linearly transform controllable matrix inequalities, the original nonlinear matrix inequalities are converted into linear matrix inequalities, enabling efficient solutions using standard solvers and overcoming the shortcomings of poor convergence and high computational complexity of traditional iterative algorithms.

[0073] Synchronous suppression of multiple disturbances enhances system robustness: This invention explicitly incorporates energy constraints for disturbances and faults into the matrix inequality design, adjusting the disturbance suppression weights through allocation coefficients. Combined with the predictive compensation of time-varying dynamics by the differential element of the PD controller, the divergent effects of disturbances and faults on the system state are significantly reduced while ensuring the boundedness of the reachable set. Simulations show that this method can strictly constrain the system state trajectory within a smaller ellipsoidal domain, improving the control accuracy of the rotational angular velocity and armature current of the DC motor under strong disturbance conditions.

[0074] This invention establishes a comprehensive solution within the reachable set framework for the control problem of continuous nonlinear DC motor systems simultaneously involving time-varying time delays, bounded peak disturbances, and fault inputs. It also facilitates the optimization of the designed controller parameters, and computer simulations further demonstrate the feasibility of the method. It covers complex superimposed operating conditions, has a wider range of applications, and strong universality; the relevant results can be easily extended to linear DC motor systems, constant time-delay nonlinear / linear DC motor systems, etc. Attached Figure Description

[0075] Figure 1 This is a flowchart of the reachable set PD controller design method for the nonlinear time-varying time-delay DC motor system described in this invention;

[0076] Figure 2 This is a block diagram of an open-loop DC motor system.

[0077] Figure 3 The state trajectory curve of a closed-loop nonlinear time-varying time-delay DC motor system;

[0078] Figure 4 For a closed-loop nonlinear time-varying time-delay DC motor system under disturbance of The fault is The reachable state set and boundary ellipsoid diagram at time. Detailed Implementation

[0079] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0080] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0081] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0082] Specific Implementation Method 1: Combination Figure 1 and Figure 2 This embodiment describes a method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system, including:

[0083] S1. Based on the dynamic relationship of the open-loop DC motor system, construct an open-loop singular system model with time-varying time delay and nonlinear system function;

[0084] S2. Eliminate the singularity of the open-loop singular system model and transform it into an equivalent closed-loop normal system model.

[0085] S3. Construct the Lyapunov-Krasovsky functional for the closed-loop normal system model and solve for the derivative of the functional.

[0086] S4. Linearize the nonlinear function of the system to obtain the linearization inequality of the nonlinear function of the system.

[0087] Based on the closed-loop normal system model, multiple invertible free weight matrices are introduced to construct the constant zero equation;

[0088] S5. Superimpose the linearization inequalities of the system's nonlinear function, the derivative of the functional, and the zero-resistance equation to establish an ellipsoidal matrix containing the unknown. and the proportional gain matrix of the PD state feedback controller and differential gain matrix Nonlinear matrix inequalities;

[0089] S6. Extract the controllable matrix inequalities from the nonlinear matrix inequalities. Simplify the controllable matrix inequalities based on the established relationships between multiple invertible free weight matrices. Perform a linear transformation on the simplified controllable matrix inequalities using matrix transformations. Solve the parameters of the linearly transformed controllable matrix inequalities using a solver to obtain the proportional gain matrix. Differential gain matrix and ellipsoidal matrix .

[0090] This invention first constructs an open-loop singular system model with time-varying delays and nonlinear functions, incorporating multiple disturbances and dynamic couplings into the model structure. Then, the singular system is transformed into a normal system to avoid the solution obstacles caused by singularity. Next, a Lyapunov-Krasovskii functional is constructed and its derivative is calculated. The functional derivative carries the upper and lower bounds of the time delays, making the stability condition dependent on the time delay. The nonlinear terms are linearized, and multiple invertible free weight matrices are added to construct constant-zero equations. These are superimposed with the functional derivative to form a nonlinear matrix inequality containing the ellipsoidal matrix and the gain of the PD state feedback controller. Then, the controllable matrix inequality is extracted, simplified using the free weight matrix relationship, and transformed into a linear matrix inequality through matrix transformation. The proportional gain, differential gain, and ellipsoidal matrix are directly calculated using a solver. In this way, the time delay dependence is preserved through the functional and time delay boundaries, conservatism is reduced through the free weight matrix and linearization, and the solution difficulty is reduced through linear transformation. Finally, controller parameters that can be practically applied in continuous systems are provided.

[0091] Figure 2 middle, Indicates the input voltage. Indicates armature current, Indicates rotational angular velocity. Indicates armature resistance. Represents back electromotive force. and These represent electromagnetic torque and angular displacement, respectively. and These represent the moment of inertia of the motor and the moment of inertia of the load, respectively. and These represent the damping coefficient at the motor end and the damping coefficient at the load end, respectively. This indicates armature inductance, and excitation indicates the excitation circuit.

[0092] Furthermore, in step S1, the expression for the open-loop singular system model is:

[0093] (1);

[0094] in,

[0095] Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative;

[0096] System nonlinear function satisfy; , Given the first real constant, It is a constant matrix;

[0097] System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ;

[0098] It is a singular matrix and satisfies , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system. , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative of For time-varying time delay The system status, This is the control input for the system.

[0099] This preferred embodiment provides a specific mathematical expression for the open-loop singular system model, clarifying the upper and lower bounds of the time-varying time delay and its derivative upper bound, the Lipschitz-type constraints of the nonlinear function, the amplitude boundaries of disturbances and faults, and the rank condition of the singular matrix. This model integrates the electromagnetic and mechanical dynamics, time-delay effects, nonlinear characteristics, and external abnormal inputs of the DC motor system, providing a complete description of the controlled object for subsequent controller design and ensuring that the model used in the design can cover the main uncertainties in actual operation.

[0100] In step S2, the equivalent transformation of the open-loop singular system model into a closed-loop normal system model is achieved as follows:

[0101] By introducing a PD state feedback controller into the open-loop singular system model and eliminating singularities, a closed-loop normal system model is obtained.

[0102] The expression for the closed-loop normal system model is:

[0103] (2);

[0104] in,

[0105] Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative;

[0106] System nonlinear function satisfy: , Given the first real constant, It is a constant matrix;

[0107] System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ;

[0108] It is a singular matrix. As a non-singular matrix, and satisfying , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system. , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative of For time-varying time delay The system status.

[0109] This preferred embodiment provides a specific expression for the closed-loop normal system model, clarifying the structure of the system obtained by introducing a PD state feedback controller and eliminating singularities, while preserving the original constraint forms for time-varying delays, nonlinear functions, disturbances, and faults. Transforming the singular system into a normal system avoids the solution difficulties caused by matrix singularities, allowing subsequent Lyapunov-Krasovskii functional construction and matrix inequality processing to be performed within the standard system framework.

[0110] Furthermore, in step S3, the expression of the constructed Lyapunov-Krasovsky functional is:

[0111] (3);

[0112] in,

[0113] (4);

[0114] (5);

[0115] (6);

[0116] (7);

[0117] For the expression of the Lyapunov-Krasovsky functional, to These are the first to fourth components of the Lyapunov-Krasovsky functional expression, respectively. For time, For system status, For transpose, Time-varying time delay The value of , It is a natural constant. The system state corresponding to the independent variable of the first integral term. for The derivative of Let be the independent variable of the first integral term. to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, Given a second real constant, Let be the independent variable of the second integral term. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. is the length of the time-varying time-delay interval.

[0118] This preferred embodiment presents a specific construction of the Lyapunov-Krasovskii functional, comprising seven positive definite matrices and multiple integral terms, wherein the integration intervals cover the entire time delay interval and its upper and lower bounds. This construction fully utilizes the boundary information of time-varying delays, making the functional derivative more sensitive to time delay variations, thereby reducing the conservatism of reachability set estimation and enabling the final stability condition to explicitly reflect time delay dependence.

[0119] Furthermore, in step S4, the expression for the linearization inequality of the system's nonlinear function is:

[0120] (8); among which, For the nonlinear function of the system, For system status, For time, For transpose, Given the first real constant, It is a constant matrix.

[0121] This preferred method provides a specific expression for the linearized inequality of the system's nonlinear function, constraining the nonlinear term to a quadratic form, which includes given constant coefficients and a constant matrix. This approach allows the nonlinear term to be uniformly processed within the framework of linear matrix inequalities, avoiding the nonconvexity and computational difficulties caused by directly handling the nonlinear function, while preserving the original magnitude constraint information of the nonlinear function.

[0122] Furthermore, in step S4, the expression for the constructed zero-equality is:

[0123] (9);

[0124] in, , , and These are the first to fourth invertible free weight matrices, This is a transpose.

[0125] This preferred method provides a specific expression for the constant-zero equality and introduces four invertible free weight matrices. This equality holds true when both sides of the system state equation are multiplied by the free weight matrices. Adding it to the functional derivative does not change the equivalence of the original system, but increases the degrees of freedom of the optimization variables. This can expand the feasible region in subsequent matrix inequality solutions and help reduce conservatism.

[0126] Furthermore, in step S5, the established ellipsoidal matrix includes the unknown ellipsoidal matrix. and the proportional gain matrix of the PD state feedback controller and differential gain matrix The expression for the nonlinear matrix inequality is:

[0127] (10);

[0128] in,

[0129] ;

[0130]

[0131]

[0132] And controllable matrix In ,satisfy As a controllable matrix inequality;

[0133] For the Lyapunov-Krasovsky functional, For functionals The derivative of For system status, for The derivative of For those with time delay The system status, For time-varying time delay The system status, For those with time delay The system status, For time, The system state corresponding to the independent variable of the first integral term. Let be the independent variable of the first integral term. For transpose, Given a second real constant, and Given the third real constant and the fourth real constant, respectively. The intermediate vector;

[0134] to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in the 4th row and columns 4, 8, 9, and 10 of the data. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , and They are respectively The elements in the 6th row and columns 1, 3, and 6. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , , They are respectively The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in columns 1, 2, 4, 8, and 10 of row 10. Time-varying time delay The possible values ​​of ;

[0135] This is an operation that adds a matrix to its transpose. It is the identity matrix. Let be the constant to be solved, and , to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, The length of the time-varying time-delay interval. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. , , and These are the first to fourth invertible free weight matrices, Given a second real constant, It is a natural constant. Time-varying time delay The upper bound of the time derivative.

[0136] In this preferred embodiment, a complete expression for a nonlinear matrix inequality comprising an ellipsoidal matrix, a proportional gain matrix, and a differential gain matrix is ​​provided, along with a detailed list of the elements of each matrix block. This inequality unifies and integrates functional derivatives, nonlinear constraints, and the constant-zero equality, forming a coupling condition for the synchronous design of the reachable set boundary and controller parameters, thus providing a clear objective inequality form for subsequent parameter solving.

[0137] Furthermore, in step S6, the relationship between the multiple invertible free weight matrices is as follows:

[0138]

[0139] in, These represent the first to third coefficients, , , and These are the first to fourth invertible free weight matrices, respectively.

[0140] In this preferred embodiment, a linear relationship is given between four invertible free weight matrices. Three of these matrices are represented as a linear combination of the fourth matrix using three coefficients. This relationship reduces the number of independent variables, lowers the computational complexity of subsequent matrix transformations and solutions, and does not increase the conservatism of the original inequality.

[0141] Furthermore, in step S6, the method for performing a linear transformation on the simplified controllable matrix inequality using matrix transformation is as follows:

[0142] For the simplified controllable matrix inequality, the controllable matrix Left multiplication of the transformation matrix Right multiplication of the transformation matrix The transpose of; where,

[0143] ;

[0144] and, , , , , , , , , , , and ;

[0145] This is the fourth invertible free weight matrix. The ellipsoid matrix after linear transformation. to These are the first to third 2×2 positive definite matrices after linear transformation. and These are the fourth and fifth 2×2 positive definite matrices after linear transformation. and These are the sixth and seventh 2×2 positive definite matrices after linear transformation. and These are the proportional and differential gain matrices of the PD state feedback controller after linear transformation. For transpose, The identity matrix after linear transformation. It is a constant matrix. is the constant matrix after linear transformation.

[0146] In this preferred embodiment, the specific form of the transformation matrix used in the linear transformation and the corresponding relationships of the variables after the transformation are given, including the update expressions for the ellipsoid matrix, the seven positive definite matrices, the proportional gain matrix, and the differential gain matrix. By left-multiplying and right-multiplying the transformation matrix and its transpose, the original nonlinear matrix inequality is transformed into an equivalent linear matrix inequality, allowing the problem to be directly processed by the standard LMI solver, avoiding nonlinear iterative solutions.

[0147] Furthermore, the expression for the controllable matrix inequality after the linear transformation in step S6 is as follows:

[0148]

[0149]

[0150] Furthermore, the controllable matrix after linear transformation In ,satisfy ,

[0151] to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 4 in the dataset. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , , for The elements in the 6th row and columns 1, 3, and 6. , , for The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , and for The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in the 10th row and columns 1, 2, 4, 8, and 10.

[0152] Verification experiment:

[0153] Based on the above theoretical analysis, the parameters of the DC motor are selected as follows: , , , , , , and , and These represent the moment of inertia of the motor and the moment of inertia of the load, respectively. and These represent the damping coefficient at the motor end and the damping coefficient at the load end, respectively. Indicates armature resistance. and They represent the electromotive force constant and the torque constant, respectively. Indicates the transmission ratio.

[0154] Therefore, some matrices in the open-loop singular system model in formula (1) can be represented as follows:

[0155]

[0156] Furthermore, it can be assumed that

[0157]

[0158] as well as

[0159] For such a practical continuous nonlinear interval time-varying time-delay singular system, it can be determined that , , , .make and By solving the controllable matrix inequalities after linear transformation , , The relevant results can be obtained, and calculations can be performed to arrive at the conclusions. The minimum value is 0.0855. Additionally, the corresponding PD controller gain parameters are... and and ellipsoidal matrix As shown below:

[0160] .

[0161] Based on system disturbance System failure And the system's nonlinear function expression, giving parameters , and .

[0162] Figure 3 The paper describes the trajectory curve of the state of a closed-loop nonlinear time-varying time-delay DC motor system over time. It can be observed that due to the presence of bounded disturbances and faults, the system's trajectory curve oscillates within the bounded region. Then, Figure 4 The reachable state set and corresponding boundary ellipsoid of a closed-loop nonlinear time-varying time-delay DC motor system are depicted. It can be clearly observed that the reachable state set of the closed-loop DC motor system is completely enclosed by the boundary of the closed ellipsoid, indicating that the open-loop DC motor system can be controlled to operate within a safe region. Therefore, for Figure 2 In actual physical systems, the present invention can successfully estimate the reachable set of shaft speed and current in an open-loop DC motor system under closed-loop conditions after the introduction of a controller, which shows that the method of the present invention is feasible for constraining and limiting the reachable state trajectory of the system. Figure 3 and Figure 4 middle, Indicates time The changing system state Indicates time The changing rotational angular velocity is a function of time. Changing system state The first component, Indicates time The changing armature current is a function of time. Changing system state The second component, Figure 3 middle x-axis Indicates time, Figure 4 middle Based on the ellipsoid matrix Draw the reachability set boundary ellipsoid.

[0163] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system, characterized in that, include: S1. Based on the dynamic relationship of the open-loop DC motor system, construct an open-loop singular system model with time-varying time delay and nonlinear system function; S2. Eliminate the singularity of the open-loop singular system model and transform it into an equivalent closed-loop normal system model. S3. Construct the Lyapunov-Krasovsky functional for the closed-loop normal system model and solve for the derivative of the functional. S4. Linearize the nonlinear function of the system to obtain the linearization inequality of the nonlinear function of the system. Based on the closed-loop normal system model, multiple invertible free weight matrices are introduced to construct the constant zero equation; S5. Superimpose the linearization inequalities of the system's nonlinear function, the derivative of the functional, and the zero-resistance equation to establish an ellipsoidal matrix containing the unknown. and the proportional gain matrix of the PD state feedback controller and differential gain matrix Nonlinear matrix inequalities; S6. Extract the controllable matrix inequalities from the nonlinear matrix inequalities. Simplify the controllable matrix inequalities based on the established relationships between multiple invertible free weight matrices. Perform a linear transformation on the simplified controllable matrix inequalities using matrix transformations. Solve the parameters of the linearly transformed controllable matrix inequalities using a solver to obtain the proportional gain matrix. Differential gain matrix and ellipsoidal matrix .

2. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 1, characterized in that, The expression for the open-loop singular system model is: in, Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative; System nonlinear function satisfy; , Given the first real constant, It is a constant matrix; System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ; It is a singular matrix and satisfies , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative, For time-varying time delay The system status, This is the control input for the system.

3. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 1, characterized in that, In step S2, the equivalent transformation of the open-loop singular system model into a closed-loop normal system model is achieved as follows: By introducing a PD state feedback controller into the open-loop singular system model and eliminating singularities, a closed-loop normal system model is obtained. The expression for the closed-loop normal system model is: ； in, Time-varying and time-delay satisfy: and They are time-varying and time-delay respectively The lower and upper boundaries; Time-varying time delay The upper bound of the time derivative; System nonlinear function satisfy: , Given the first real constant, It is a constant matrix; System disturbance and system failure satisfy: and These are the upper bounds of the disturbance amplitude and the upper bound of the fault amplitude, respectively. , ; It is a singular matrix. As a non-singular matrix, and satisfying , To obtain the rank of the matrix, and Represent matrices respectively The rank and state dimension of the system , , , , and These are the first to sixth real constant matrices, respectively. For system status, For time, for The derivative, For time-varying time delay The system status.

4. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 2, characterized in that, In step S3, the expression of the constructed Lyapunov-Krasovsky functional is: ； in, ， ， ， ， For the expression of the Lyapunov-Krasovsky functional, to These are the first to fourth components of the Lyapunov-Krasovsky functional expression, respectively. For time, For system status, For transpose, Time-varying time delay The value of , It is a natural constant. The system state corresponding to the independent variable of the first integral term. for The derivative, Let be the independent variable of the first integral term. to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, Given a second real constant, Let be the independent variable of the second integral term. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. is the length of the time-varying time-delay interval.

5. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 1, characterized in that, In step S4, the expression for the linearization inequality of the system's nonlinear function is: ； in, For the nonlinear function of the system, For system status, For time, For transpose, Given the first real constant, It is a constant matrix.

6. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 2, characterized in that, In step S4, the expression for the constructed zero-equality is: ； in, , , and These are the first to fourth invertible free weight matrices, This is a transpose.

7. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 2, characterized in that, In step S5, the established ellipsoidal matrix contains the unknown. and the proportional gain matrix of the PD state feedback controller and differential gain matrix The expression for the nonlinear matrix inequality is: ， in, ； ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ； And controllable matrix In ,satisfy , As a controllable matrix inequality; For the Lyapunov-Krasovsky functional, For functionals The derivative, For system status, for The derivative, For those with time delay The system status, For time-varying time delay The system status, For those with time delay The system status, For time, The system state corresponding to the independent variable of the first integral term. Let be the independent variable of the first integral term. For transpose, Given a second real constant, and Given the third real constant and the fourth real constant, respectively. The intermediate vector; to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in the 4th row and columns 4, 8, 9, and 10 of the data. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , and They are respectively The elements in the 6th row and columns 1, 3, and 6. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , , They are respectively The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in columns 1, 2, 4, 8, and 10 of row 10. Time-varying time delay The value of ; This is an operation that adds a matrix to its transpose. It is the identity matrix. Let be the constant to be solved, and , to These are the first to third 2×2 positive definite matrices, respectively. and They are time-varying and time-delay respectively The lower and upper boundaries, The length of the time-varying time-delay interval. and These are the fourth and fifth 2×2 positive definite matrices, respectively. and These are the sixth and seventh 2×2 positive definite matrices, respectively. , , and These are the first to fourth invertible free weight matrices, Given a second real constant, It is a natural constant. Time-varying time delay The upper bound of the time derivative.

8. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 7, characterized in that, In step S6, the relationship between the multiple invertible free weight matrices is as follows: ； in, These represent the first to third coefficients, , , and These are the first to fourth invertible free weight matrices, respectively.

9. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 8, characterized in that, In step S6, the method for performing a linear transformation on the simplified controllable matrix inequality using matrix transformation is as follows: For the simplified controllable matrix inequality, the controllable matrix is... Left multiplication of the transformation matrix Right multiplication of the transformation matrix The transpose of; where, ； and, , , , , , , , , , , and ; This is the fourth invertible free weight matrix. The ellipsoid matrix after linear transformation. to These are the first to third 2×2 positive definite matrices after linear transformation. and These are the fourth and fifth 2×2 positive definite matrices after linear transformation. and These are the sixth and seventh 2×2 positive definite matrices after linear transformation. and These are the proportional and differential gain matrices of the PD state feedback controller after linear transformation. For transpose, The identity matrix after linear transformation. It is a constant matrix. It is the constant matrix after linear transformation.

10. The method for designing a reachable set PD controller for a nonlinear time-varying time-delay DC motor system according to claim 8, characterized in that, The expression for the controllable matrix inequality after the linear transformation in step S6 is: Furthermore, the controllable matrix after linear transformation In ,satisfy to They are respectively The elements in the first row and columns 1 to 4 of the array. for The element in the 6th column of the 1st row, They are respectively The elements in the 8th, 9th, and 10th columns of the 1st row, , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of the second row in the dataset. , , , and They are respectively The elements in the 3rd row and columns 1, 3, 5, 6, and 7 of the data. , , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 4 in the dataset. , and They are respectively The elements in the 3rd, 5th, and 7th columns of the 5th row, , , for The elements in the 6th row and columns 1, 3, and 6. , , for The elements in the 3rd, 5th, and 7th columns of the 7th row, , , , and They are respectively The elements in columns 1, 2, 4, 8, 9, and 10 of row 8. , , , and for The elements in the 9th row and columns 1, 2, 4, 8, and 9. , , , , for The elements in the 10th row and columns 1, 2, 4, 8, and 10.

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