Finite-time control method and system for a non-linear two-machine interconnected system

Abstract

The present invention discloses a finite-time control method for a non-linear dual-machine interconnected system, including establishing a dynamic mathematical model of the dual-machine interconnected system; establishing an object rule base based on the T-S fuzzy model method, and approximating the dynamic mathematical model of the dual-machine interconnected system with multiple linear local system models; taking into account the faults of intermittent sensors and actuators in each local system model; designing a dynamic output feedback controller and incorporating the T-S fuzzy model of the system, assuming that the controller gain matrix is known, to obtain the mathematical expression of the closed-loop system; giving a sufficient condition for the finite-time control of the closed-loop system by designing a time-delay-dependent Lyapunov function; solving the sufficient condition to obtain the controller gain matrix required for the dynamic output feedback controller to achieve finite-time control; bringing the calculated controller gain matrix into the controller to complete the finite-time control of the non-linear dual-machine interconnected system. The present invention improves the stability of control.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear system control, and particularly to a finite-time control method and system for a nonlinear dual-machine interconnected system. Background Art

[0002] In large-scale systems such as nonlinear dual-machine interconnected systems, the stable operation of the system is very important. However, there are many factors that can affect the stability of the system. For example, when transmitting system signals, there may sometimes be a delay phenomenon, or even multiple delays. At the same time, intermittent faults also have a great impact on the stability of the system.

[0003] On the other hand, the Takagi-Sugeno (T-S) fuzzy model is one of the very famous landmarks in the history of fuzzy control systems. It can be regarded as an approximately piecewise linear model. This model is equivalent to dividing the input space into several fuzzy subspaces. First, a local linear model is established in each fuzzy subspace, and then the local models are smoothly connected using membership functions to form a global fuzzy model with a nonlinear function. Summary of the Invention

[0004] The present invention aims to propose a finite-time control method for a nonlinear dual-machine interconnected system for large-scale systems, and uses a dynamic output feedback controller and a multi-time-delay-dependent Lyapunov function with low conservatism to improve the stability of the nonlinear dual-machine interconnected system affected by delays and intermittent faults.

[0005] The finite-time control method for the nonlinear dual-machine interconnected system in the present invention includes

[0006] Step 1: Establish a dynamic mathematical model of the dual-machine interconnected system;

[0007] Step 2: Based on the T-S fuzzy model method, establish an object rule base, and approximate the dynamic mathematical model of the dual-machine interconnected system with multiple linear local system models; intermittent sensor and actuator faults are taken into account in each local system model;

[0008] Step 3: Design a dynamic output feedback controller and incorporate the T-S fuzzy model of the system. Assuming that the controller gain matrix is known, obtain the mathematical expression of the closed-loop system;

[0009] Step 4: By designing a time-delay-dependent Lyapunov function, give a sufficient condition for the finite-time control of the closed-loop system;

[0010] Step 5: Solve this sufficient condition to obtain the controller gain matrix required for the dynamic output feedback controller to achieve finite-time control;

[0011] Step 6: Bring the calculated controller gain matrix into the controller to complete the finite-time control of the nonlinear two-machine interconnected system.

[0012] Further, in Step 1, the dynamic mathematical model of the two-machine interconnected system is as follows:

[0013]

[0014]

[0015] In the formula, and respectively represent the absolute rotor angle and angular velocity of the th machine, is the inertia coefficient, is the damping coefficient, is the internal voltage, is the th and the modulus of the transfer admittance between the

[0016] Further, in Step 2, each rule in the object rule base is formulated in the following form:

[0017] The th rule of the th subsystem is: "If is and …, and is then there is:

[0018]

[0019]

[0020]

[0021] "

[0022] Where is the rule number,

[0023]

[0024] represents the antecedent variable system of the th subsystem,

[0025]

[0026] is the fuzzy set based on the membership function of the th subsystem;

[0027] L is the number of subsystems, Indicates the state of the th subsystem, Indicates the control input of the th subsystem, Indicates the actuator fault in the th subsystem, Indicates the external disturbance in the th subsystem. Indicates the measurement output of the th subsystem, Indicates the sensor fault in the th subsystem, Indicates the controlled output of the th subsystem;

[0028] The matrix Indicates the interconnection term between the th and the l-th local models;

[0029] The matrix and Satisfy:

[0030]

[0031]

[0032]

[0033]

[0034]

[0035] Where and Represent known constant matrices, and Represent the parameter uncertainties, satisfying:

[0036]

[0037]

[0038]

[0039] Where the constant matrices and Are known. The time-varying matrices and Are unknown, indicating:

[0040]

[0041] and indicates the faults of intermittent sensors and actuators, and results in the random occurrence of fault phenomena. The occurrence probabilities of the faults are as follows:

[0042]

[0043]

[0044] where represents a known scalar, that is, and

[0045] represents the time-varying state delay introduced in the interconnection, satisfying where represents the minimum value of, τ and respectively represent and the maximum values of, t ∈ [-τ, 0] represents this initial continuously differentiable function.

[0046] Furthermore, in step two, a total of nine rules are established as follows:

[0047] Object rule 1:

[0048] If and then,

[0049]

[0050]

[0051]

[0052]

[0053] Object rule 2:

[0054] If and x 2 (1, t) ≈ 0, then,

[0055]

[0056]

[0057]

[0058]

[0059] Object Rule 3:

[0060] If and then,

[0061]

[0062]

[0063]

[0064]

[0065] Object Rule 4:

[0066] If x 1 (1, t) ≈ 0, and then,

[0067]

[0068]

[0069]

[0070]

[0071] Object Rule 5:

[0072] If x 1 (1, t) ≈ 0, and x 2 (1, t) ≈ 0, then,

[0073]

[0074]

[0075]

[0076]

[0077] Object Rule 6:

[0078] If x 1 (1, t) ≈ 0, and then,

[0079]

[0080]

[0081]

[0082]

[0083] Object Rule 7:

[0084] If and then,

[0085]

[0086]

[0087]

[0088]

[0089] Object Rule 8:

[0090] If and x 2 (1, t) ≈ 0, then,

[0091]

[0092]

[0093]

[0094]

[0095] Object Rule 9:

[0096] If and then,

[0097]

[0098]

[0099]

[0100]

[0101] Furthermore, in Step 2, let the model of the complete

[0102]

[0103]

[0104]

[0105]

[0106] Among them,

[0107]

[0108]

[0109] denotes the membership function in

[0110] is the number of object rules.

[0111] Furthermore, in the step, the expression of the dynamic output feedback controller is as follows:

[0112]

[0113]

[0114]

[0115] Among them, is the controller state, and represent the controller gain matrix to be determined, is the simplified representation of

[0116] Furthermore, in step three, it denotes

[0117] incorporated into the T-S model, the fuzzy closed-loop mathematical expression of the dynamic output feedback controller can be described as follows:

[0118]

[0119]

[0120]

[0121] Among them,

[0122]

[0123]

[0124]

[0125]

[0126]

[0127] Furthermore, sufficient conditions for finite-time control are given by the following Lyapunov function, and the gain matrices of each solution controller are obtained:

[0128]

[0129]

[0130]

[0131]

[0132]

[0133] Another object of the present invention is to provide a finite-time control system for a non-linear two-machine interconnected system, which controls the non-linear two-machine interconnected system according to the aforementioned method.

[0134] Advantages of the present invention:

[0135] Under the premise of considering multiple time delays and intermittent sensor and actuator failures, the T-S fuzzy model is used to describe the non-linear two-machine interconnected system and the corresponding dynamic output feedback controller is designed to make the system bounded in finite time. Then, an augmented closed-loop model is constructed. Finally, sufficient conditions for finite-time control of the system are given by using a multi-time-delay related Lyapunov function with low conservatism, and the controller gain matrix is obtained and substituted into the controller to realize the real-time finite control of the two-machine interconnected system, reducing the conservatism of the system and improving the stability of the system. Description of the drawings

[0136] Figure 1 It is a design strategy diagram of the controller in the embodiment of the present invention.

[0137] Figure 2 It is a schematic diagram of the membership function established in the embodiment of the present invention.

[0138] Figure 3 It is a curve graph of x(t) of the state response of the non-linear two-machine interconnected system obtained by simulation in the embodiment of the present invention.

[0139] Figure 4 It is a curve graph of y(t) of the output response of the non-linear two-machine interconnected system obtained by simulation in the embodiment of the present invention.

[0140] Figure 5 It is a curve graph of ζ(t) of the state response of the controller obtained by simulation in the embodiment of the present invention.

[0141] Figure 6The controlled output response y of the non-linear two-machine interconnected system obtained through simulation in the embodiments of the present invention z (t) curve graph.

[0142] Figure 7 The curve graph of the spectral norm ||u(t)|| of the control input obtained through simulation in the embodiments of the present invention.

[0143] Figure 8 The curve graph of the spectral norm of the augmented state obtained through simulation in the embodiments of the present invention. Specific implementation manners

[0144] The following further describes the present invention in combination with the accompanying drawings and embodiments. In this embodiment, the dynamic mathematical model of the original two-machine interconnection is first established as follows:

[0145]

[0146]

[0147] In the formula, and respectively represent the absolute rotor angle and angular velocity of the th machine, is the inertia coefficient, is the damping coefficient, is the internal voltage, is the th and the transfer admittance modulus between the lth machines.

[0148] In this embodiment, the parameters of this system are set as: M 1 = 1.03, M 2 = 1.25, D 1 = 0.8, D 2 = 1.2,

[0149] E 1 = 1.017, E 2 = 1.005, Y 21 = Y 12 = 1.98, θ 21 = θ 12 = -1.5,

[0150] Considering that this system is subject to interference such as interconnection time delay, parameter uncertainty, intermittent actuators and intermittent sensors, and disturbances. By linearizing the non-linear system near and taking this as the origin, this model is described as a T-S fuzzy model.

[0151] For a multi-time-delay nonlinear large-scale system with intermittent faults, it can be described by the T-S fuzzy model as follows:

[0152] First, the form of the object rules is as follows:

[0153] The rule of the th subsystem: "If is and …, and is then there is:

[0154]

[0155]

[0156]

[0157] "

[0158] where is the rule number,

[0159]

[0160] represents the antecedent variable system of the th subsystem,

[0161]

[0162] is the fuzzy set based on the membership function of the th subsystem.

[0163] L is the number of subsystems, represents the state of the th subsystem, represents the control input of the th subsystem, represents the actuator fault in the th subsystem, represents the external disturbance in the th subsystem. represents the measured output of the th subsystem, represents the sensor fault in the th subsystem, represents the controlled output of the th subsystem.

[0164] The matrix represents the interconnection term between the th and the lth local models;

[0165] Matrix and satisfy:

[0166]

[0167]

[0168]

[0169]

[0170]

[0171] where and represent known constant matrices, and represent the parameter uncertainties, satisfying:

[0172]

[0173]

[0174]

[0175] where the constant matrices and are known. The time-varying matrices and are unknown, denoting:

[0176]

[0177] and represent the faults of intermittent sensors and actuators, and result in the random occurrence of fault phenomena. Assuming that these intermittent sensor and actuator faults follow a Bernoulli distribution and take values in {0, 1}, the fault probability is as follows:

[0178]

[0179]

[0180] where represents a known scalar, i.e., and

[0181] represent the time-varying state delays introduced in the interconnection, satisfying where represents the minimum value, τ and respectively represent and the maximum value, t ∈ [-τ, 0] represents this initial continuously differentiable function.

[0182] Then, combining with Figure 2 the membership functions given in, the following nine specific object rules can be formulated for this model with two subsystems:

[0183] Object Rule 1:

[0184] If and Then,

[0185]

[0186]

[0187]

[0188]

[0189] Object Rule 2:

[0190] If and x 2 (1, t) ≈ 0, then,

[0191]

[0192]

[0193]

[0194]

[0195] Object Rule 3:

[0196] If and Then,

[0197]

[0198]

[0199]

[0200]

[0201] Object Rule 4:

[0202] If x1 (1, t) ≈ 0, and Then,

[0203]

[0204]

[0205]

[0206]

[0207] Object Rule 5:

[0208] If x 1 (1, t) ≈ 0, and x 2 (1, t) ≈ 0, then,

[0209]

[0210]

[0211]

[0212]

[0213] Object Rule 6:

[0214] If x 1 (1, t) ≈ 0, and Then,

[0215]

[0216]

[0217]

[0218]

[0219] Object Rule 7:

[0220] If And Then,

[0221]

[0222]

[0223]

[0224]

[0225] Object Rule 8:

[0226] If and x 2 (1,t)≈0, then,

[0227]

[0228]

[0229]

[0230]

[0231] Object Rule 9:

[0232] If and then,

[0233]

[0234]

[0235]

[0236]

[0237] According to the aforementioned dynamic model of dual-machine interconnection, the matrix of parameters can be determined as follows:

[0238]

[0239]

[0240]

[0241]

[0242]

[0243]

[0244]

[0245] Other parameters are set by yourself according to the actual situation. In this embodiment, they are set as

[0246] Therefore, let In summary, the th complete T-S fuzzy large-scale subsystem is represented as follows:

[0247]

[0248]

[0249]

[0250]

[0251] Among them,

[0252]

[0253]

[0254] denote in the membership function,

[0255] where the number of object rules

[0256] Assume that for each there is:

[0257]

[0258] Therefore,

[0259]

[0260] For the sake of simplicity, hereinafter use to replace

[0261] Then, in this embodiment, for the above T-S fuzzy large-scale system model according to the design strategy shown in Figure 1 a dynamic output feedback controller is designed, and the controller can be expressed as follows:

[0262]

[0263]

[0264]

[0265] Among them, is the controller state. and represent the controller gain matrices to be determined. Then, in the following steps, it is necessary to solve for and for the above large-scale system model.

[0266] Then here it represents

[0267] Incorporating the aforementioned T-S fuzzy large-scale system model, the fuzzy closed-loop model of the aforementioned controller can be described as follows:

[0268]

[0269]

[0270]

[0271] wherein,

[0272]

[0273]

[0274]

[0275]

[0276]

[0277] while can be approximated by and and the error brought by the approximation can be calculated as follows:

[0278]

[0279]

[0280] wherein,

[0281]

[0282] Thus, based on this approximation, the fuzzy closed-loop model of the controller can be further described as follows:

[0283]

[0284]

[0285]

[0286] Define:

[0287]

[0288]

[0289]

[0290]

[0291]

[0292] a(t) = diag{a 1 (t), …, a L (t)},

[0293] s(t) = diag{s 1 (t), …, s L (t)},

[0294]

[0295] where

[0296] Assumption 1: Suppose there is a time scalar T > 0, a scalar and a matrix F 1 > 0. For a specified time interval [0, T], there is:

[0297]

[0298] Assumption 2: where the scalar

[0299] Assumption 3: For the scalar there is

[0300] Assumption 4: For the scalar there is

[0301] Definition 1: Suppose there is a positive definite matrix R = diag{R 1 , …, R L}, positive constants c 1 , c 2 and T. If it produces

[0302]

[0303] then the above model is applicable to t ∈ [0, T] and can be called the finite-time boundedness of (c 1 , c 2 , T, R).

[0304] Definition 2: If the model can be finite-time bounded (c 1 , c 2 , T, R), and if for the following Assumption 1 there is:

[0305]

[0306] where \(V(\cdot)\) represents a function with \(V(0,0) = 0\).

[0307] Then, for the assigned positive scalars \(\gamma\) and \(\upsilon\), the model can satisfy the robust finite-time \(H_{\infty}\) performance with a disturbance attenuation level of \(\gamma\).

[0308] Based on the above model, definitions, and assumptions, in this embodiment, the sufficient conditions for finite-time control are given by the following Lyapunov function, and the gain matrices of each solution controller are obtained:

[0309]

[0310]

[0311]

[0312]

[0313]

[0314] It can be seen that the time-delay-dependent Lyapunov function designed in this embodiment takes into account both the influence of large-scale subsystems and the influence of membership functions and multiple time delays, reducing the conservatism of the method.

[0315] Finally, the gain matrix of the controller is calculated and substituted into the controller to complete the finite-time control of the nonlinear two-machine interconnected system.

[0316] Next, simulations are carried out using Matlab to verify the effectiveness of the fuzzy control method for controlling the nonlinear two-machine interconnected system in this embodiment.

[0317] Except for the system fixed parameter settings, other parameters are set as

[0318] Assume \(\tau\) 21 (t)=\(\tau\) 12 (t) = 0, \(c\) 1 = 2, \(c\) 2 = 10, \(T = 5\), \(F\) 1 = \(I\), \(R = I\), and select \(\gamma = 0.01\), \(\upsilon = 0.001\), Then, by using the MATLAB-LMI toolbox to solve the linear matrix inequality, the gain matrix of the fuzzy controller designed in this embodiment can be obtained

[0319] The final simulation results are as Figures 3-7 shown. Among them, Figure 3 the curve graph of the state response \(x(t)\) of the nonlinear two-machine interconnected system,Figure 4 It is a curve graph of the output response y(t) of the non - linear two - machine interconnected system. Figure 5 It is a curve graph of the state response ζ(t) of the controller. Figure 6 It is for the controlled output response y z (t) of the non - linear two - machine interconnected system. Figure 7 It is a curve graph of the spectral norm ||u(t)|| of the control input. Figure 8 It is a curve graph of the spectral norm of the augmented state obtained by simulation in the embodiment of the present invention. It can be seen from the simulation graph that the decentralized fuzzy controller designed in this embodiment can not only ensure the finite - time stability of the closed - loop system, but also ensure the stability of the original non - linear system (the original two - machine interconnected system). In addition, the state - limited and input - limited conditions are satisfied.

[0320] Then, after computer - coding the method in this example and installing it on a control terminal such as an industrial control computer, and connecting the corresponding control actuators and sensors, a finite - time control system of the linear two - machine interconnected system can be obtained.

[0321] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the purpose and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A finite-time control method for a non-linear two-machine interconnected system, characterized in that, it includes Step 1: Establish the dynamic mathematical model of the two-machine interconnected system; Step 2: Based on the T-S fuzzy model method, establish an object rule base, and approximate the dynamic mathematical model of the two-machine interconnected system with multiple linear local system models; the faults of intermittent sensors and actuators are taken into account in each local system model; Step 3: Design a dynamic output feedback controller and incorporate the T-S fuzzy model of the system. Assuming that the controller gain matrix is known, obtain the mathematical expression of the closed-loop system; Step 4: By designing a time-delay dependent Lyapunov function, give the sufficient conditions for the finite-time control of the closed-loop system; Step 5: Solve the sufficient conditions to obtain the controller gain matrix required for the dynamic output feedback controller to achieve finite-time control; Step 6: Substitute the calculated controller gain matrix into the controller to complete the finite-time control of the non-linear two-machine interconnected system; In Step 1, the dynamic mathematical model of the two-machine interconnected system is as follows, In the formula, and respectively represent the absolute rotor angle and angular velocity of the th machine, is the inertia coefficient, is the damping coefficient, is the internal voltage, is the modulus of the transfer admittance between the th and the lth machines; In Step 2, each rule in the object rule base is formulated in the following form: The rules of the subsystem are: "If is and …, and is then there is: wherein is the rule number, represents the antecedent variable of the th subsystem, is the fuzzy set for the membership function based on the th subsystem; Let \(L\) be the number of subsystems, denote the status of the \(i\)-th subsystem, denote the control input of the \(i\)-th subsystem, denote the actuator fault in the \(i\)-th subsystem, denote the exogenous disturbance in the \(i\)-th subsystem; Matrix represents the interconnection term between the Matrix and Satisfy: wherein and represent known constant matrices, and represent parameter uncertainties, satisfying: Among them, the constant matrices and are known. The time-varying matrices and are unknown, indicating that: and indicate faults of intermittent sensors and actuators, and result in random occurrences of fault phenomena, and the occurrence probabilities of the faults are as follows: wherein represents a known scalar, i.e., and represents the time-varying state delay introduced in the interconnection, satisfying where represents the minimum value of, τ and respectively represent and the maximum values of, t ∈ [-τ, 0] represents this initial continuously differentiable function; In Step 3, the expression of the dynamic output feedback controller is as follows: Among them, is the controller state, and represent the controller gain matrix to be determined, is a simplified representation of.

2. The method according to claim 1, characterized in that, in Step 2, a total of nine rules are established as follows: Object Rule 1: If And Then Object Rule 2: If and x 2 (1,t)≈0, then Object Rule 3: If And Then Object Rule 4: If x 1 (1, t) ≈ 0, and Then, Object Rule 5: If x 1 (1, t) ≈ 0, and x 2 (1, t) ≈ 0, then, Object Rule 6: If x 1 (1,t) ≈ 0, and then, Object Rule 7: If And Then Object Rule 8: If and x 2 (1,t)≈0, then Object Rule 9: If And Then, 3. The method according to claim 2, characterized in that, In Step 2, let The model of the complete T-S subsystem is represented as follows: wherein, representation in membership function of is the number of object rules.

4. The method according to claim 3, characterized in that, In step three, it means Incorporating the T-S model, the fuzzy closed-loop mathematical expression of the dynamic output feedback controller can be described as follows: wherein, 5. The method according to claim 4, characterized in that, in Step 5, the sufficient conditions for finite-time control are given through the following Lyapunov function, and the solution controller gain matrix is obtained:

6. A finite-time control system for a linear two-machine interconnected system, characterized in that, this system implements the control of the linear two-machine interconnected system according to the method described in any one of claims 1-5.

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