Quantization control method for nonlinear system, computer equipment and storage medium

Abstract

The invention relates to a quantitative control method for a nonlinear system, computer equipment and a storage medium. The method comprises the following steps of: performing sector bounded processing on the output of a static logarithmic quantizer to establish an expression for quantizing control input; constructing a nonlinear system with disturbance, time lag and parameter uncertainty, and constructing a T-S fuzzy linear model of the nonlinear system according to the expression; constructing a target function according to the T-S fuzzy linear model, and constructing an optimization problem on an infinite time domain by taking a system state constraint, a control input constraint and a disturbance constraint as conditions so as to realize minimum quantization control input under maximum disturbance; and performing stability analysis on the T-S fuzzy linear model by constructing an LKF, and solving an optimization problem under the condition of meeting the stability analysis to obtain quantitative control input. By adopting the method, stable control on a nonlinear system can be effectively realized.

Description

Technical Field

[0001] The present application relates to the field of quantitative control technology, and in particular to a quantitative control method, computer device, and storage medium for nonlinear systems. Background Art

[0002] As automated control systems become increasingly complex, Model Predictive Control (MPC), an optimization control method based on system models, which explicitly considers constraints and optimizes control inputs online, has been widely studied and applied in industrial control, networked systems, and intelligent manufacturing due to its good robustness and constraint processing capabilities.

[0003] In recent years, researchers have proposed various improved MPC algorithms to address the common problems of time-varying delays, data loss, and uncertain disturbances in complex systems, such as robust MPC (RFMPC) and stability analysis strategies based on the Lyapunov method. However, traditional methods often require the introduction of the Lyapunov-Razumikhin functional (LRF) when dealing with time-varying delays. While this theoretically guarantees stability, its complex derivation and generally conservative nature limit its engineering feasibility. Furthermore, in networked control systems, packet loss may occur during data communication. Traditional MPC methods often assume complete data transmission, making them less adaptable to the control performance degradation caused by actual communication anomalies. Some literature has extended RFMPC to a probability density function (PDF) control framework using uncertainty modeling methods to achieve fault-tolerant control in environments with both parameter disturbances and packet loss. While these methods exhibit a certain degree of robustness, they still rely heavily on resource allocation in scenarios where communication bandwidth is limited or the system requires long-term stable operation.

[0004] To further optimize the comprehensive performance of control systems in scenarios with limited computing and communication resources, quantitative control theory has emerged. Quantized control reduces bandwidth requirements and computational load by introducing finite precision expressions into control signals or state signals. Traditional quantitative control methods often use static logarithmic quantizers, supplemented by sector boundary conditions, to convert quantization errors into robustness issues. In recent years, some studies have also introduced Lyapunov-Krasovskii Functional (LKF) and dynamic event triggering mechanisms to improve the dynamic performance and stability assurance capabilities of quantitative systems. However, existing research has mostly focused on control design for linear systems or under known parameter conditions. For complex systems with nonlinear characteristics, quantization errors, parameter uncertainties, and state lags, there is still a lack of a systematic and structured optimal control method. Therefore, for nonlinear systems with the above-mentioned complex working conditions, it is urgent to design an optimal control strategy that is highly robust, has high resource utilization efficiency, and is suitable for actual network communication environments, in order to effectively solve the key technical bottlenecks of traditional methods in dealing with complex nonlinear system control problems, such as modeling difficulties, conservative stability analysis, and low resource allocation efficiency. Summary of the Invention

[0005] Based on this, it is necessary to provide a quantitative control method, computer equipment and storage medium for nonlinear systems to address the above technical problems.

[0006] A quantitative control method for a nonlinear system, the method comprising:

[0007] The output of the static logarithmic quantizer is sector-bounded and the expression of the quantized control input is established;

[0008] Constructing a nonlinear system with disturbance, time lag and parameter uncertainty, and constructing a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input;

[0009] According to the TS fuzzy linear model, an objective function including system state, control input and disturbance factor is constructed, with the purpose of achieving minimum quantitative control input under maximum disturbance, and an optimization problem in infinite time domain is constructed with system state constraint, control input constraint and disturbance constraint as constraint conditions;

[0010] Performing stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, solving the optimization problem to determine a feedback control law under the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis;

[0011] The quantitative control input is updated according to the feedback control law to realize the quantitative control of the nonlinear system.

[0012] A quantitative control device for a nonlinear system, comprising:

[0013] The control input quantization module is used to sector-bound the output of the static logarithmic quantizer and establish an expression for the quantized control input;

[0014] A model building module is used to build a nonlinear system with disturbance, time lag and parameter uncertainty, and to build a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input;

[0015] A problem construction module is used to construct an objective function including system state, control input and disturbance factors according to the TS fuzzy linear model, with the goal of achieving minimum quantitative control input under maximum disturbance, and to construct an optimization problem on an infinite time domain with system state constraints, control input constraints and disturbance constraints as constraints;

[0016] a problem-solving module, configured to perform stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, and solve the optimization problem to determine a feedback control law under the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis;

[0017] The result output module is used to update the quantitative control input according to the feedback control law to achieve quantitative control of the nonlinear system.

[0018] A computer device includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the following steps are implemented:

[0019] The output of the static logarithmic quantizer is sector-bounded and the expression of the quantized control input is established;

[0020] Constructing a nonlinear system with disturbance, time lag and parameter uncertainty, and constructing a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input;

[0021] According to the TS fuzzy linear model, an objective function including system state, control input and disturbance factor is constructed, with the purpose of achieving minimum quantitative control input under maximum disturbance, and an optimization problem in infinite time domain is constructed with system state constraint, control input constraint and disturbance constraint as constraint conditions;

[0022] Performing stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, solving the optimization problem to determine a feedback control law under the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis;

[0023] The quantitative control input is updated according to the feedback control law to realize the quantitative control of the nonlinear system.

[0024] A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the following steps:

[0025] The output of the static logarithmic quantizer is sector-bounded and the expression of the quantized control input is established;

[0026] Constructing a nonlinear system with disturbance, time lag and parameter uncertainty, and constructing a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input;

[0027] According to the TS fuzzy linear model, an objective function including system state, control input and disturbance factor is constructed, with the purpose of achieving minimum quantitative control input under maximum disturbance, and an optimization problem in infinite time domain is constructed with system state constraint, control input constraint and disturbance constraint as constraint conditions;

[0028] Performing stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, solving the optimization problem to determine a feedback control law under the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis;

[0029] The quantitative control input is updated according to the feedback control law to realize the quantitative control of the nonlinear system.

[0030] The aforementioned quantitative control method, computer device, and storage medium for nonlinear systems first sector-bounds the static logarithmic quantizer output to establish an expression for the quantitative control input. By combining a sector constraint method, offline parameter setting and online quantization are seamlessly integrated, ensuring quantization accuracy while endowing the control strategy with universality and scalability. Next, a TS fuzzy linear model for a nonlinear system with multiple complex characteristics is constructed. This model effectively linearizes the nonlinear dynamic characteristics of the system. Based on this model, an objective function is constructed and constraints are set. An optimization problem is constructed over an infinite time domain, aiming to minimize the quantitative control input under maximum disturbance. System state, control input, and disturbance factors are comprehensively considered to ensure system performance. Stability analysis is performed by constructing a Lyapunov-Krasovskii functional. The feedback control law is determined by solving the optimization problem under multiple conditions, providing ISS guarantees for the system and addressing multi-dimensional challenges. Finally, the quantitative control input is updated according to the feedback control law to achieve quantitative control. This embodiment of the present invention ensures system convergence and recursive feasibility, ensuring a safety margin in the actual control signal and effectively achieving stable control of the nonlinear system. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 1 is a flow chart of a quantitative control method for a nonlinear system according to an embodiment;

[0032] Figure 2 Schematic diagram of the dynamic characteristics of the membership function of the TS fuzzy system in one embodiment;

[0033] Figure 3 Schematic diagram of the evolution of various dimensions of the state vector of the system during online control in one embodiment;

[0034] Figure 4 Schematic diagram of dynamic characteristics of control input during online control of a system in one embodiment;

[0035] Figure 5 is a structural block diagram of a quantization control device for a nonlinear system in one embodiment;

[0036] Figure 6 FIG. 1 is a diagram showing the internal structure of a computer device in one embodiment. DETAILED DESCRIPTION

[0037] In order to make the purpose, technical solutions and advantages of this application more clearly understood, the present application is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0038] In one embodiment, Figure 1As shown, a quantitative control method for a nonlinear system is provided, comprising the following steps:

[0039] Step 102: Sector-bound the output of the static logarithmic quantizer and establish an expression for the quantization control input.

[0040] Firstly, a system analysis is carried out on the static logarithmic quantizer, and the analytical expression of the quantization control input is derived. The quantization mechanism is explicitly constructed for the static logarithmic quantizer described by the sector constraint method.

[0041] Step 104: construct a nonlinear system with disturbance, time lag and parameter uncertainty, and construct a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input.

[0042] Consider a discrete-time nonlinear system described by the following dynamical equations:

[0043] x + =f(x,x d ,u,w)

[0044] in represents the state vector, x d (k) = x(kd) is the delay state (d represents the number of delay steps), is the control input, and represents the external disturbance, and the nonlinear system is expressed in the following formula (1).

[0045] TS fuzzy system:

[0046] Consider the nonlinear system described by equation (1), assuming that it can be represented by the TS fuzzy model. The i-th rule of the fuzzy model is expressed as follows:

[0047] Fuzzy rule R i :If ν1 belongs to …, ν z belong but

[0048]

[0049] where ν j are premise variables, j=1,...,z, To describe the fuzzy set of rule premises, the TS fuzzy model is expressed in the following formula (2). and It can be expressed as:

[0050]

[0051] Among them A i ,B iis the nominal matrix, ΔA i ,ΔB i represent additive parameter uncertainties, and satisfy:

[0052] ΔA i =TF(k)N Ai ,ΔB i =TF(k)N Bi

[0053] in The real-time system matrix can be obtained by weighted summation of the membership function:

[0054]

[0055] where h i (x) is the normalized membership function, satisfying as well as

[0056] Step 106, construct an objective function including system state, control input and disturbance factors based on the TS fuzzy linear model, with the goal of achieving minimum quantitative control input under maximum disturbance, and construct an optimization problem in infinite time domain with system state constraints, control input constraints and disturbance constraints as constraints.

[0057] Based on the constructed TS fuzzy linear model, an objective function is constructed to measure system performance. This objective function comprehensively considers factors such as the cost of the system state deviating from the desired state, the consumption of control inputs, and the impact of disturbances. Furthermore, the objective function considers system state constraints, control input constraints, and disturbance constraints, which define the operating range of the system.

[0058] For a discrete-time dynamic system described by equation (1), under the control strategy u(x), if the set Satisfy the inclusion condition: F(x,x d ,u(x),w)∈Ω, Then Ω is called a Robust Positively Invariant Set (RPISet). The system state constraint ensures that the system state is in a reasonable area and does not exceed a specific range to maintain stable operation.

[0059] Control input constraints ensure that the quantized control inputs are within the executable range; disturbance constraints limit the impact of external disturbances. The optimization problem constructed over an infinite time horizon aims to find a control strategy that optimizes the objective function to address various scenarios during the system's long-term operation.

[0060] Step 108 , performing stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, solving the optimization problem to determine the feedback control law while satisfying the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis.

[0061] The ISS-Lyapunov function (Lyapunov-Krasovskii functional, LKF) is constructed to analyze the stability of the TS fuzzy linear model. For system (1), if the continuously differentiable function V: If the following conditions are met, it is called an ISS-Lyapunov candidate function: Class functions α1, α2, α3 and Class function ρ(·) such that for all and integer k≥0 have

[0062] 1. Positive definiteness conditions:

[0063] α1(‖x(k)‖)≤V(x(k))≤α2(‖x(k)‖)

[0064] This condition ensures that V(x) is a positive definite and radially unbounded Lyapunov function;

[0065] 2. Decreasing conditions with interference attenuation characteristics:

[0066] V(x(k+1))-V(x(k))≤-α3(‖x(k)‖)+ρ(‖d(k)‖)

[0067] This formula shows that under the action of external interference, V(x) exhibits attenuation characteristics along the system trajectory and is robust to external inputs.

[0068] Among them, the continuous function K: The generic definition of is: -like function: if it is strictly increasing and satisfies K(0)=0; -Class function: If it belongs to -class, and further meet

[0069] In addition, the continuous function K: Known as -like function if and only if it satisfies the following dual property: for any fixed parameter s ≥ 0, the function belong -class; for any fixed parameter r≥0, the function Strictly decreasing and satisfying

[0070] By analyzing the changes of LKF along the system trajectory, the input-to-state stability conditions, optimality conditions, control input constraints and recursive feasibility conditions are derived. Class function β(·,·) and Class function γ(·), such that for any initial state x0, input signal u(k) and external disturbance w(k), the system state satisfies The system is called ISS (Input-to-State Stability). is a set of non-negative real numbers. The input-to-state stability condition ensures that the system's state remains bounded under perturbations and that the system produces bounded outputs for bounded inputs. The optimality condition ensures that the feedback control law optimizes the objective function; the control input constraint ensures that the control input meets the actual execution requirements; and the recursive feasibility condition ensures that the optimization problem has a feasible solution at each sampling time.

[0071] In the derivation process, Lemma 1 is used to transform the inequalities involving matrices and time-varying matrices so that they meet the requirements of convex optimization and other solutions, and then solve the optimization problem to determine the feedback control law. Lemma 1 is: Let a real matrix P with appropriate dimensions = P T , W, G and satisfy the spectrum constraint condition X T (t)X(t)≤I, then the matrix inequality P+WX(t)G + G T X T (t)W T The necessary and sufficient condition for P to be globally valid in the admissible parameter space of X is that there exists a scalar ∈>0 such that P+∈WW T +∈ -1 G T G<0.

[0072] When solving optimization problems, Lemma 1 provides the necessary and sufficient conditions for processing matrix inequalities under specific conditions, which is used to convert complex stability conditions, constraints, etc. into a solvable form.

[0073] Step 110: Update the quantized control input according to the feedback control law to achieve quantized control of the nonlinear system.

[0074] The feedback control law is derived based on the analysis and solution in the previous steps. It calculates the appropriate control input adjustment according to the current state of the system, so that the system can operate towards the desired performance while meeting various stability and constraint conditions. This ensures that the system can effectively respond to complex factors such as disturbances, time delays, and parameter uncertainties in actual operation, and maintain a stable and optimized operating state.

[0075] In the aforementioned quantitative control method for nonlinear systems, the static logarithmic quantizer output is first sector-bounded to establish an expression for the quantitative control input. By combining a sector constraint method, offline parameter setting and online quantization processes are seamlessly integrated, ensuring quantization accuracy while endowing the control strategy with universality and scalability. Next, a TS fuzzy linear model for a nonlinear system with various complex characteristics is constructed. This model effectively linearizes the nonlinear dynamic characteristics of the system. Based on this model, an objective function is constructed and constraints are set. An optimization problem is constructed over an infinite time domain, aiming to minimize the quantitative control input under maximum disturbance. System state, control input, and disturbance factors are comprehensively considered to ensure system performance. Stability analysis is performed by constructing a Lyapunov-Krasovskii functional. The feedback control law is determined by solving the optimization problem under multiple conditions, providing ISS guarantees for the system and addressing multi-dimensional challenges. Finally, the quantitative control input is updated according to the feedback control law to achieve quantitative control. This embodiment of the present invention ensures system convergence and recursive feasibility, ensuring a safety margin in the actual control signal and effectively achieving stable control of the nonlinear system.

[0076] In one embodiment, sector-bounding the output of the static logarithmic quantizer to establish an expression for the quantization control input includes mapping the continuous signal output by the static logarithmic quantizer to a discrete value set using a sector-bounding method, and establishing an expression for the quantization control input as follows:

[0077]

[0078] Among them, u(k) is the quantized control input at time k, n u is the number of components of the quantized control input, υ(k) is the original input signal without quantization, f(·) is the quantization function of the static logarithmic quantizer, D(k) is the binary diagonal matrix, is the transpose of the matrix.

[0079] In this embodiment, a static logarithmic quantizer is used to implement signal quantization processing, and its mathematical description is as follows:

[0080]

[0081] in By quantization density Based on the sector bounding method proposed by Minyue Fu and Lihua Xie.Thesector bound approach to quantized feedback control.IEEE Transactions on Automatic Control, 50(11):1698–1711, 2005.doi:10.1109 / TAC.2005.858689, the quantization function can be expressed as f(υ(k))=(1+τ)υ, where τ∈[-σ,σ]. The quantized control input expression can be obtained as follows:

[0082]

[0083] In the formula It is a binary diagonal matrix whose elements are in {1-σ,1+σ}.

[0084] In one embodiment, the optimization problem is:

[0085]

[0086] stx(k+i+1|k)=f(x(k+i|k),u(k+i|k),w(k+i|k))

[0087]

[0088] in, is the objective function, x(k+i|k) is the system state at the i-th moment in the k-time prediction domain, u(k+i|k) is the quantized control input at the i-th moment in the k-time prediction domain, and w(k+i|k) is the disturbance at the i-th moment in the k-time prediction domain. and denote the state space, the admissible control input space, and the bounded disturbance space, respectively.

[0089] In this embodiment, for the constructed TS fuzzy system (2), the objective function corresponding to the optimal control problem is first defined:

[0090]

[0091] Then we establish the optimization problem in infinite time domain, in particular, define and

[0092] To facilitate subsequent analysis, the ellipsoid set Ω(k) is defined as follows:

[0093]

[0094] At every moment, the current state and delayed state of the system are both located in the ellipsoid set.

[0095] In one embodiment, the input state stability condition is used to ensure that the system state can remain within a bounded range when the nonlinear system is disturbed; the input state stability condition is:

[0096]

[0097] Among them, V(x(k)) is the LKF function value corresponding to the system at time k, Q and R are positive definite matrices, u(k) is the quantized control input at time k, w(k) is the disturbance at time k, κ is the decay rate, and x(k) is the system state at time k. is the transpose of the matrix.

[0098] In one embodiment, the optimality condition is used to ensure that the determined feedback control law can optimize the objective function; the optimality condition is:

[0099]

[0100] Among them, x(k) is the system state at time k, Y j is a positive definite matrix, s=1,...,d, d is the delay order, ζ is a positive scalar, x(ks) is the system delay state, Θ j is a positive definite matrix, is the transpose of the matrix.

[0101] In one embodiment, the control amplitude constraint is used to ensure that the quantized control input is within a practical executable range; the control input constraint is:

[0102]

[0103] Among them, u(k) is the quantized control input at time k, D is the binary diagonal matrix, u max Enter the maximum value for quantization control, is the transpose of the matrix.

[0104] In one embodiment, the recursive feasibility condition is used to ensure that a feasible solution exists for the optimization problem at each sampling time. The recursive feasibility condition is:

[0105]

[0106] Where ζ is a positive scalar, x is the current system state, (·) + is the parameter of the next moment, x(ks) is the system lag state, β is the weight, Y μ is a positive definite matrix, Θ μ is a positive definite matrix, λ is a positive scalar, w is the perturbation, is the transpose of the matrix.

[0107] In order to ensure that the control strategy is stable and feasible under the influence of disturbances, nonlinearity and quantization errors, Theorem 1 is proposed to uniformly model the key design requirements such as state stability, optimality, control amplitude limit and recursive feasibility into a set of linear matrix inequalities (LMI). By verifying the feasibility of the LMI problem, theoretical support and feasibility guarantee can be provided for the entire control architecture. Theorem 1: For the TS fuzzy system considered in the present invention as shown in formula (2), if there exists a positive definite matrix Φ i ,Ψ i (and the corresponding matrix Φ below j ,Ψ j and Φ l ), matrices Λ, Γ, and positive scalars ζ, β, η ijl ,ξ ijl , which makes the following constrained optimization problem feasible:

[0108]

[0109]

[0110] in, and Under this condition, the feedback control law can be analytically determined as u(k)=Kx(k), where K=ΛΓ -1 By continuously solving this optimization problem, the TS fuzzy system can ensure the realization of ISS.

[0111] prove:

[0112] The LKF is constructed as follows:

[0113] V(x(k))=V1(x(k))+V2(x(k))

[0114] The various parts are:

[0115]

[0116] Next, the proof is provided from four aspects.

[0117] Input to the state stability condition:

[0118] To ensure the ISS of system (2), both the positive definiteness condition and the decreasing condition with interference attenuation characteristics must be satisfied. With the inherent positive definiteness of the LKF and the boundedness constraints imposed on V1 and V2, the positive definiteness condition can be naturally satisfied in this theoretical framework. As for the decreasing condition with interference attenuation characteristics, the following inequality must be ensured:

[0119]

[0120] Where κ is a preset positive scalar, representing the decay rate. Then the above can be equivalently written as:

[0121]

[0122] By substituting the TS fuzzy model given by formula (2) into the above Lyapunov function decreasing condition, the above problem can be reconstructed into the following form:

[0123]

[0124] Through defuzzification, the above formula can be guaranteed as follows:

[0125]

[0126] remember Then the above inequality condition can be rewritten as:

[0127]

[0128] By applying Schuler's patch continuously, the following results can be derived:

[0129]

[0130] Let Y = ζΦ -1 and Θ=ζΨ -1 Substitute the above matrix inequality conditions and multiply both sides of the above formula by the diagonal matrix Available

[0131]

[0132] Multiply both sides of the above equation by and its transpose can be obtained

[0133]

[0134] because and The above formula can be guaranteed by the following:

[0135]

[0136] Equivalently, we can get:

[0137]

[0138] Then, based on Lemma 1, the above condition can be reconstructed as:

[0139]

[0140] make Then the above formula is equivalent to:

[0141]

[0142] Using Schur complement, we can see that the above formula can be guaranteed by (1).

[0143] Optimality conditions:

[0144] Consider the ellipsoid set Ω(k). After defuzzification, the state in Ω(k) satisfies

[0145]

[0146] Equivalently,

[0147]

[0148] Since ζ>0, we set Y=ζΦ -1 and Θ=ζΨ -1 Substituting this into the equation above and dividing both sides by ζ yields:

[0149]

[0150] By applying Schur's complement, the above formula can be derived:

[0151]

[0152] Control amplitude constraints:

[0153] Control amplitude constraint |u s |≤u smax ,s=1,...,n u can be expressed as:

[0154]

[0155] in according to It can be seen that a sufficient condition for the above formula to be satisfied is:

[0156]

[0157] The above formula can be guaranteed by the following:

[0158]

[0159] Eliminate from both sides and its transpose can be obtained

[0160]

[0161] Equivalently,

[0162]

[0163] By using Schur's complement, the above formula can be transformed into

[0164]

[0165] Applying Schur complement again, the above formula can be further transformed into:

[0166]

[0167] By multiplying both sides of the above equation by and its transpose, we can derive the following inequality conditions:

[0168]

[0169] because The above conditions are given by the formula mentioned above Satisfaction guaranteed.

[0170] RPI conditions:

[0171] To ensure the recursive feasibility of RFMPC, the ellipsoid set Ω(k) must satisfy the RPI condition. In this case, if a solution exists at the beginning, it should still exist for all subsequent values of x. Therefore, we should have

[0172]

[0173] at the same time

[0174]

[0175] By using the S-process, the sufficient condition for the ellipsoid set Ω(k) to become the RPI set is:

[0176]

[0177] in is a manually selected constant. By expanding the state term at the next moment and defuzzifying it, the above conditions can be guaranteed as follows:

[0178]

[0179] By pairing vector groups Perform integration and implement Schur complement operation, while introducing variable substitution Y = ζΦ -1 and Θ=ζΨ -1 , the above expression can be transformed into:

[0180]

[0181] By multiplying both sides of the equation by the matrix and its transposed matrix, and using the inequality and And a similar process using Lemma 1 as described in the input to state stability condition, we can finally get the condition:

[0182]

[0183] Due to the presence of the (1-β)G and (1-β)H terms in the above condition, this constraint becomes a bilinear matrix inequality constraint, requiring a non-convex optimization solver such as PENBMI. By fixing β to a preset constant, the condition can be transformed into a form that can be solved using conventional convex optimization tools such as SDPT3 and Mosek.

[0184] The -κζI term in condition (1) can be integrated into the -θI term, thereby treating θ as an auxiliary free variable. This method automatically adjusts the parameters through the optimization framework, eliminating the need for manual parameter adjustment of κ.

[0185] In one embodiment, updating the quantized control input according to the feedback control law includes: substituting the current system state into the feedback control law, calculating the adjustment amount of the control input, and updating the quantized control input according to a preset update strategy.

[0186] In a specific embodiment, to solve the optimal control problem for nonlinear systems with quantization effects, bounded disturbances, parameter uncertainty, and state lag, the proposed method first conducts system analysis on a static logarithmic quantizer to derive an analytical expression for the quantized control input. Then, leveraging the TS fuzzy modeling framework, a structured decomposition method is used to map the system's nonlinear characteristics into a combination of local linear subsystems. Furthermore, a stability analysis based on the LKF is performed to ensure system convergence and robustness under complex operating conditions characterized by disturbances, parameter uncertainty, and lag effects. Targeting the practical control needs of industrial systems such as robots, drones, and unmanned ground platforms, this algorithm innovatively addresses the limitations of traditional algorithms in key areas such as modeling strong nonlinear systems, compensating for lag effects, and robustness to parameter perturbations. Furthermore, by integrating a quantitative control strategy, resource optimization is achieved for the control process. Taking the implementation of a vehicle path tracking control system as an example, a commonly used two-degree-of-freedom bicycle model is selected, assuming that the vehicle travels on a dry and flat road, and that actuator dynamics such as braking and acceleration, as well as the vehicle's vertical motion, are negligible:

[0187]

[0188] Φd Represents the system matrix related to the time-delay state, which is used to describe the impact of the time-delay state signal on the system due to factors such as signal transmission delay. x Indicates the longitudinal speed of the vehicle; and denote the cornering stiffness of the front and rear wheels respectively, where and And Δc f ,Δc r Represent an unmeasurable bounded variable; l f With l r Represents the distance from the front and rear axles to the vehicle's center of mass, respectively; β represents the vehicle's side slip angle, θ represents the vehicle's heading angular velocity, and d is the vehicle's preview distance. The symbol ω represents the curvature of the road corresponding to the vehicle's preview point, which is defined as the inverse of the road's curvature radius at that point. θ represents the relative yaw angle between the vehicle heading and the reference path, e y represents the lateral deviation of the vehicle relative to the reference path; δ is the steering angle of the front wheel. Select the sampling time T s After that, the above continuous-time system can be discretized to obtain the path tracking control system in discrete time.

[0189] During vehicle dynamic driving, the tire cornering stiffness parameter It is inevitable that changes will occur. Without loss of generality, it is reasonable to assume that there is a known upper bound on the amount of change. At the same time, the vehicle's front wheel steering angle is limited by the physical constraints of the vehicle's mechanical structure (e.g., the maximum is usually no more than 40°), and the road path curvature can also be assumed to be within a specific change range. In addition, the real-time control of the vehicle communication system during the path tracking process is interfered by the historical state three sampling periods ago. This scheme presets the logarithmic quantizer parameters is a constant value. By using the method proposed in Theorem 1, the current optimal steering control signal can be solved online at each sampling moment, thereby achieving accurate tracking control of the target path.

[0190] The effectiveness of the proposed method is verified through numerical examples:

[0191] Consider a discrete-time nonlinear system with time-delay characteristics as shown in Equation (1), where the time-delay step is d = 4. The constraints on the control input and external disturbance are |u(k)|≤7 and The logarithmic quantizer parameters are set to Based on this, we can calculate that the value of parameter σ is 0.11. Therefore, the diagonal matrix D(k) is selected as a scalar constant, and its modulus is 1-σ=0.89.

[0192] Assuming that the above system can be expressed as a TS fuzzy model according to formula (2), the corresponding state matrix is as follows:

[0193]

[0194] A d1 =0.05A1,A d2 =0.04A1,A d3 =0.06A3,

[0195]

[0196] N A1 =[0.1 0.2 0.1],N A2 =[0.1 0.1 0.2],

[0197] N A3 =[0.2 0.1 0.1],

[0198] N B1 =0.06,N B2 =0.04,N B3 =0.05.

[0199] At the same time, the corresponding membership value function is defined as follows:

[0200]

[0201] Their function distribution is as follows Figure 2 shown. Figure 3 The evolution of the system state trajectory over time during online operation is shown, revealing its asymptotic convergence characteristics in various dimensions. The changes in the control input signal are further analyzed in Figure 4 In this scenario, the quantitative control behavior is clearly always within the preset operating boundaries. It is worth noting that the actual control signal has a certain margin relative to the predefined boundary constraints. This phenomenon stems from the adopted Min-Max optimization framework, which fully considers the worst-case disturbances and parameter uncertainties. This conservatism introduces a safety margin for the system. In practical application scenarios, exogenous disturbances and endogenous uncertainties are usually not always so significant, so the stability of the system can generally be maintained with moderate and small control efforts without frequently saturating the actuators. Overall, these results strongly confirm the effectiveness and robustness of the proposed method.

[0202] It should be understood that although Figure 1The steps in the flowchart are shown in sequence as indicated by the arrows, but these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise specified in this document, there is no strict order restriction for the execution of these steps, and these steps can be executed in other orders. In addition, Figure 1 At least part of the steps may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed in turn or alternately with other steps or at least part of the sub-steps or stages of other steps.

[0203] In one embodiment, Figure 5 As shown, a quantitative control device for a nonlinear system is provided, comprising:

[0204] The control input quantization module is used to sector-bound the output of the static logarithmic quantizer and establish an expression for the quantized control input;

[0205] Model building module, used to build nonlinear systems with disturbances, time delays and parameter uncertainties, and to construct TS fuzzy linear models of nonlinear systems based on the expressions of quantitative control inputs;

[0206] The problem construction module is used to construct an objective function including system state, control input and disturbance factors based on the TS fuzzy linear model. The goal is to achieve the minimum quantitative control input under the maximum disturbance. The optimization problem in the infinite time domain is constructed with the system state constraint, control input constraint and disturbance constraint as the constraint conditions.

[0207] The problem-solving module is used to perform stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional. Under the conditions of input-to-state stability, optimality, control amplitude constraint, and recursive feasibility obtained from the stability analysis, the optimization problem is solved to determine the feedback control law.

[0208] The result output module is used to update the quantitative control input according to the feedback control law to achieve quantitative control of the nonlinear system.

[0209] The specific limitations of the quantitative control device for nonlinear systems can be found in the limitations of the quantitative control method for nonlinear systems described above and will not be further elaborated here. Each module in the aforementioned quantitative control device for nonlinear systems can be implemented in whole or in part via software, hardware, or a combination thereof. Each of the aforementioned modules can be embedded in or independent of a processor in a computer device in hardware form, or can be stored in a memory in the computer device in software form, so that the processor can call and execute the corresponding operations of each of the aforementioned modules.

[0210] In one embodiment, a computer device is provided. The computer device may be a terminal, and its internal structure diagram may be as follows: Figure 6 As shown. The computer device includes a processor, a memory, a network interface, a display screen and an input device connected via a system bus. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The network interface of the computer device is used to communicate with an external terminal via a network connection. When the computer program is executed by the processor, a quantitative control method for a nonlinear system is implemented. The display screen of the computer device can be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer device can be a touch layer covering the display screen, or a key, trackball or touchpad provided on the computer device housing, or an external keyboard, touchpad or mouse.

[0211] Those skilled in the art will understand that Figure 6 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0212] In one embodiment, a computer device is provided, including a memory and a processor. The memory stores a computer program, and the processor implements the steps of the method in the above embodiment when executing the computer program.

[0213] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the steps of the method in the above embodiment are implemented.

[0214] Those skilled in the art will appreciate that all or part of the processes in the above-mentioned embodiments can be implemented by instructing the relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, storage, database or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM) or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link (Synchlink) DRAM (SLDRAM), memory bus (Rambus) direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).

[0215] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0216] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and such modifications and improvements are intended to fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A quantitative control method for nonlinear systems, characterized in that: The method comprises: The output of the static logarithmic quantizer is sector-bounded and the expression of the quantized control input is established; Constructing a nonlinear system with disturbance, time lag and parameter uncertainty, and constructing a TS fuzzy linear model of the nonlinear system according to the expression of the quantized control input; According to the TS fuzzy linear model, an objective function including system state, control input and disturbance factor is constructed, with the purpose of achieving minimum quantitative control input under maximum disturbance, and an optimization problem in infinite time domain is constructed with system state constraint, control input constraint and disturbance constraint as constraint conditions; Performing stability analysis on the TS fuzzy linear model by constructing a Lyapunov-Krasovskii functional, solving the optimization problem to determine a feedback control law under the input-to-state stability condition, optimality condition, control amplitude constraint condition, and recursive feasibility condition obtained from the stability analysis; The quantitative control input is updated according to the feedback control law to realize the quantitative control of the nonlinear system.

2. The method according to claim 1, characterized in that The sector-bounding of the output of the static logarithmic quantizer and the establishment of an expression for the quantization control input include: The continuous signal output by the static logarithmic quantizer is mapped to a discrete value set through the sector bounding method, and the expression of the quantized control input is established as: Among them, u(k) is the quantized control input at time k, n u is the number of components of the quantized control input, υ(k) is the original input signal without quantization, f(·) is the quantization function of the static logarithmic quantizer, D(k) is the diagonal matrix of binarization, (·) T is the transpose of the matrix.

3. The method according to claim 1, characterized in that The optimization problem is: stx(k+i+1|k)=f(x(k+i|k),u(k+i|k),w(k+i|k)) in, is the objective function, x(k+i|k) is the system state at the i-th moment in the k-time prediction domain, u(k+i|k) is the quantized control input at the i-th moment in the k-time prediction domain, and w(k+i|k) is the disturbance at the i-th moment in the k-time prediction domain. and denote the state space, the admissible control input space, and the bounded disturbance space, respectively.

4. The method according to claim 1, wherein The input-to-state stability condition is used to ensure that the system state of the nonlinear system can remain within a bounded range when the nonlinear system is disturbed; the input-to-state stability condition is: V(x(k+1))-V(x(k))≤-x T (k)Qx(k)-u T (k)Ru(k)+κw T (k)w(k) Where V(x(k)) is the LKF function value of the system at time k, Q and R are positive definite matrices, u(k) is the quantized control input at time k, w(k) is the disturbance at time k, κ is the decay rate, x(k) is the system state at time k, (·) T is the transpose of the matrix.

5. The method according to claim 1, wherein The optimality condition is used to ensure that the determined feedback control law can optimize the objective function; the optimality condition is: Among them, x(k) is the system state at time k, Y j is a positive definite matrix, s=1,...,d, d is the delay order, ζ is a positive scalar, x(ks) is the system delay state, Θ j is a positive definite matrix, (·) T is the transpose of the matrix.

6. The method according to claim 1, characterized in that The control amplitude constraint condition is used to ensure that the quantized control input is within the practical executable range; the control input constraint condition is: Among them, u(k) is the quantized control input at time k, D is the binary diagonal matrix, u max is the maximum value of the quantized control input, (·) T is the transpose of the matrix.

7. The method according to claim 1, characterized in that The recursive feasibility condition is used to ensure that at each sampling moment, there is a feasible solution to the optimization problem; the recursive feasibility condition is: Where ζ is a positive scalar, x is the current system state, (·) + is the parameter of the next moment, x(ks) is the system lag state, β is the weight, Y μ is a positive definite matrix, Θ μ is a positive definite matrix, λ is a positive scalar, w is the perturbation, (·) T is the transpose of the matrix.

8. The method according to claim 1, characterized in that Updating the quantized control input according to the feedback control law includes: Substitute the current system state into the feedback control law, calculate the adjustment amount of the control input, and update the quantitative control input according to the preset update strategy.

9. A computer device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 8 are implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.

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