A finite-time robust stabilization control method for nonlinear singular time-delay systems

Abstract

The application provides a finite time robust stabilization control method for a nonlinear singular time-delay system, comprising the following steps: selecting a Hamilton function for the nonlinear singular time-delay system; converting the Hamilton function of the nonlinear singular time-delay system into an equivalent differential algebraic form by combining a state decomposition method; designing a robust controller; substituting the robust controller into the equivalent differential algebraic form to obtain a strictly equivalent dissipative Hamilton form of the system; and selecting a Lyapunov function to verify the accuracy of the robust controller based on the strictly equivalent dissipative Hamilton form. The application provides a finite time robust stabilization control method for a nonlinear singular time-delay system, studies a more complex nonlinear singular system than a linear singular system, considers the influence of time delay on the system, and gives a new finite time robust stabilization control method for the system under the condition of adding external disturbance.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of finite time robust control, in particular to a finite time robust stabilization control method for a nonlinear singular time-delay system. BACKGROUND

[0002] A singular system can more accurately describe some real physical systems than a general normal system, and a singular system model widely exists in various fields of social production, such as a power grid system, a circuit, a chemical process and the like. In addition, time delay also affects the stability of a system. Therefore, a singular time-delay system plays an important role in a mathematical model about a control problem.

[0003] Chinese patent CN108241297B discloses a method for determining the asymptotic stability of a singular time-delay system, and a method for maintaining the asymptotic stability of the singular time-delay system. The method for determining the asymptotic stability of the singular time-delay system includes: analyzing the time-delay stability of the singular time-delay system by using a mathematical decomposition method to obtain a condition for maintaining the stability of a singular time-delay dependent system; and determining the stability of the singular time-delay dependent system according to the condition for maintaining the stability of the singular time-delay dependent system. The method for maintaining the asymptotic stability of the singular time-delay system includes: using the condition for maintaining the stability of the singular time-delay dependent system to improve the asymptotic stability of the singular time-delay system.

[0004] The prior art cannot accurately determine the stability of a singular time-delay system, and cannot effectively improve the stability of a nonlinear singular time-delay system. SUMMARY

[0005] In view of the problems in the prior art, the present application provides a finite time robust stabilization control method for a nonlinear singular time-delay system, studies a nonlinear singular system which is more complex than a linear singular system, considers the influence of time delay on the system, and provides a new finite time robust stabilization control method for the system in the presence of external disturbance.

[0006] The technical solution adopted by the present application is as follows:

[0007] A finite time robust stabilization control method for a nonlinear singular time-delay system, characterized in that the method comprises the following steps:

[0008] Selecting a Hamilton function for the nonlinear singular time-delay system;

[0009] Converting the Hamilton function of the nonlinear singular time-delay system into an equivalent differential algebraic form by using a state decomposition method;

[0010] A robust controller is designed, and the robust controller is substituted into the equivalent differential algebraic form to obtain a strictly equivalent dissipative Hamilton form of the system;

[0011] A Lyapunov function is selected, and the accuracy of the robust controller is verified based on the strictly equivalent dissipative Hamilton form.

[0012] The present application has the following beneficial effects:

[0013] Unlike the infinite-time results on nonlinear singular time-delay systems, the present application studies the finite-time problem and provides a finite-time control result for a class of nonlinear singular time-delay Hamilton systems. The present application obtains a strictly equivalent dissipative Hamilton form of the studied system by using a state decomposition method, and studies the finite-time robust stabilization problem based on the strictly equivalent dissipative Hamilton form. Compared with the infinite-time results, the finite-time control method provided by the present application has faster convergence, better robustness and anti-interference. Unlike the existing results based on the linearization method, the present application studies the nonlinear singular time-delay system from the nonlinear perspective, which means that the method provided by the present application is more difficult and has a wider application background. Unlike the existing finite-time control results on time-delay-free singular systems, the present application proposes a control scheme for nonlinear time-delay singular systems by using the state decomposition method. BRIEF DESCRIPTION OF DRAWINGS

[0014] Figure 1 A finite-time robust stabilization control method flowchart of a nonlinear singular time-delay system;

[0015] Figure 2 A nonlinear circuit system structure schematic diagram of the present application embodiment;

[0016] Figure 3 A state response curve schematic diagram of the present application embodiment;

[0017] Figure 4 A control input signal schematic diagram of the present application embodiment;

[0018] Figure 5 A state response curve schematic diagram of the prior art. DETAILED DESCRIPTION

[0019] The present application will be further described below in conjunction with the accompanying drawings and embodiments: In order to clearly illustrate the technical features of the present application, the present application will be described in detail below with specific embodiments and in conjunction with the accompanying drawings. The disclosure below provides many different embodiments or examples for implementing the different structures of the present application. In order to simplify the disclosure of the present application, the components and settings of specific examples are described below. In addition, the present application can repeatedly refer to the same reference numerals and / or letters in different examples. Such repetition is for the purpose of simplification and clarity, and does not indicate the relationship between the various embodiments and / or settings being discussed. It should be noted that the components illustrated in the drawings are not necessarily drawn to scale. The present application omits the description of well-known components and processing techniques and processes to avoid unnecessarily limiting the present application.

[0020] As shown in Figure 1 , the present application provides a finite time robust stabilization control method for nonlinear singular time-delay systems- an energy-based method. By converting the general nonlinear singular time-delay Hamilton system into a strictly equivalent dissipative form, a suitable controller is designed to solve its finite time robust control problem, and a good robust stabilization effect is obtained.

[0021] Step 1: Select Hamilton function for nonlinear singular time-delay system;

[0022] Step 2: Combine state decomposition method to convert the general nonlinear singular time-delay Hamilton system into equivalent differential algebraic form;

[0023] Step 3: Design a robust controller, and substitute the robust controller into the equivalent differential algebraic form to obtain the strictly equivalent dissipative Hamilton form of the system;

[0024] Step 4: Select Lyapunov function to verify the accuracy of the robust controller based on the strictly equivalent dissipative Hamilton form.

[0025] Step 5: Taking a charging circuit as an example, the effectiveness of the method proposed in the present application is verified.

[0026] The equation form of the nonlinear singular time-delay Hamilton system of the present application is as follows:

[0027]

[0028] wherein is the state of the system (1), is the time delay of the system, is a singular matrix and is the control input, is the external disturbance, is a known constant matrix, is a known constant matrix, and is a constant matrix, respectively, are the penalty signal and the output of the system; H(x) is an energy function, which takes its minimum value at x = 0, and satisfies A(x) + A T (x) < 0 for all x e Ω (Ω denotes the definition domain of the system state x, which means x is a real number greater than zero), and

[0029] The Hamilton function (energy function) selected in step 1 is as follows:

[0030]

[0031] where φ i is a row unit vector with the i-th component being 1, and a is a real number greater than 1. In order to obtain the strictly equivalent dissipative Hamilton form in step 2, the following relations need to be satisfied for the matrices E, A, T and g1

[0032] deg(det(sE-A)) = rank(E) (3)

[0033]

[0034] If (3) is satisfied, it means that x(t) has no impulse perturbation, i.e., the system (1) has no impulse solution. On this basis, if |sE-A|≠0, the matrix pair (E, A) is admissible, and there always exist two non-singular matrices M, such that

[0035]

[0036] Proof: Since the matrix pair (E, A) is admissible, there exists s such that det(sE-A)≠0, i.e., the matrix sE-A is invertible. Let there exists a non-singular matrix T such that where is an invertible matrix, is a zero power matrix, is an invertible matrix.

[0037] Let then

[0038]

[0039]

[0040] The proof is complete.

[0041] Let

[0042]

[0043]

[0044] where

[0045] In system (1)

[0046]

[0047] Under (5)(6)(7) system (1) can be expressed as:

[0048]

[0049] It can be seen that system (8) and system (1) are equivalent, A1=A1(x) can be expressed as A1=J-R0, where is an anti-symmetric matrix, is a symmetric matrix.

[0050] If the index of the system at the equilibrium point is 1, the controller is designed as follows

[0051]

[0052] where is a gain matrix, and the equilibrium point refers to the critical value within a certain range that the system tends to be stable.

[0053] Substitute the controller (9) into system (8) to obtain the equivalent form of system (1):

[0054]

[0055] where R=R(x) is a positive definite matrix, Λ is a weight matrix with full column rank.

[0056] From formula (4) we can get

[0057]

[0058] That is

[0059] rank[sI r -A1 g 11 ]=r

[0060] Therefore, there is always a matrix such that and The eigenvalues of can be arbitrarily configured. Therefore, we choose the matrix K such that It can be seen that (10) and (1) are strictly equivalent, according to the implicit function existence theorem, if the system has an index of 1 at the equilibrium point, there exists a function f(·) such that

[0061] In the above step 3, the robust control problem of system (10) can be completed by the following control rate

[0062]

[0063] Where γ>0 is the disturbance rejection level, v1 and v2 are two reference inputs.

[0064] Substituting (12) into (11), the strictly equivalent dissipative Hamilton form of system (1) is as follows

[0065]

[0066] In the above step 4, the Lyapunov function selected is as follows:

[0067]

[0068] System (12) satisfies the Hamiltonian-Jacobian inequality

[0069]

[0070] Then the L2 gain (from w to z) of the system does not exceed γ, and when w=0, the system (12) is globally finite time stable.

[0071] In the above step 5, the present application proposes an embodiment of a nonlinear charging circuit to illustrate the effectiveness of the finite time robust stabilization controller of the singular nonlinear time-delay system, as shown in Figure 2 The capacitor is controlled by the charge q, and the inductor is controlled by the magnetic flux linkage Ψ; their characteristics can be represented as u1=f1(q1(t))+f1(q1(t-h)), u2=f2(q2(t))+f2(q2(t-h)), i3=f3(ψ3(t))+f3(ψ3(t-h)), i w Is the disturbance signal.

[0072] According to the Kirchhoff's current law, the Kirchhoff's voltage law and considering the influence of the time delay term on the system, the above system can be represented as:

[0073]

[0074] Where y and z=[z1,z2] T Are the output and penalty signal of the system respectively.

[0075] R4 = 2Ω, and let x = [ψ3, q1, q2] T , w = i w , u = [U s , I s ] T .

[0076] Then the system (14) can be expressed as follows:

[0077]

[0078] There exists a Hamiltonian function where α = 2, such that the system (15) has a constant Hamiltonian realization as follows:

[0079]

[0080] Next, we transform the system (16) into an equivalent nonlinear differential algebraic system by a nonsingular transformation. Let us choose

[0081]

[0082] Then the system (16) can be transformed into the following equivalent form:

[0083]

[0084] where By state feedback

[0085]

[0086] The dissipative Hamiltonian form of the system (16) can be obtained as follows:

[0087]

[0088] Let us choose the gain matrix the weight matrix Then the H ∞ controller of the system (16) can be designed as:

[0089]

[0090] To verify the effectiveness of the H ∞ controller, we give a numerical simulation, where we let γ = 1, and the initial value of the system Ex(0) = [5, -8, 2.4] T . To verify the robustness of the controller to external disturbances, we apply a disturbance of size [6, 6] to the system in the time interval 0-0.12 s.T The simulation results are shown in Figure 3 and Figure 4 , where Figure 3 is the state response curve, Figure 4 is the control signal used. It can be seen from Figure 3 that the system is damped at 0.12 seconds due to external disturbance and the system state tends to be stable within 1.5 seconds. The simulation results show that the H ∞ controller is very effective and has good robustness to external disturbance.

[0091] In order to highlight the effectiveness of the method proposed in the present application, a simulation result using an infinite-time controller is given below for comparison, and the infinite-time Hamilton function is The system (16) is simulated, and the same initial value, disturbance parameter and time delay parameter are selected, and the simulation results are shown in Figure 5 This figure is used for comparison with Figure 3 , and mainly compares the influence of the two controllers on the system state under the same initial value, time delay and disturbance. Figure 3 In the system, the three states tend to be stable at about 2 seconds, Figure 5 In the system, the three states tend to be stable at about 3 seconds, and Figure 3 In the system, the amplitude of the state is smaller than that in Figure 5 , which shows that the controller used in Figure 3 has better anti-disturbance and robustness, and faster convergence. Obviously, the control method proposed in the present application has faster convergence speed, better robustness and anti-disturbance.

[0092] Finally, it should be noted that: the above examples are only used to illustrate the technical solutions of the present application and not to limit it, although the present application has been described in detail with reference to the above examples, those skilled in the art should understand that: the specific embodiments of the present application can still be modified or replaced by the same, without departing from the spirit and scope of the present application. Any modification or equivalent replacement, which should be covered within the protection scope of the claims of the present application.

Claims

1. A finite-time robust stabilization control method for a nonlinear singular time-delay system, characterized in that, The method includes the following steps: Step 1: Select the Hamiltonian function for the nonlinear singular time-delay system; The equations for the nonlinear singular time-delay Hamiltonian system are as follows: ⑴ in It is the state of system (1). It is the system's time delay. It is a singular matrix and , It is a control input. It's external interference. It is a known constant matrix. It is a known constant matrix. and It is a constant matrix. These are the system's penalty signal and output, respectively; H(x) is the energy function, which reaches its minimum value at x=0. And for all All satisfied , and The Ω refers to the domain of the system state x, indicating that x is a real number greater than zero. The Hamiltonian function selected is as follows: ⑵ in Let i be the row unit vector with the i-th component being 1. It is a real number greater than 1; Step 2: Combining the state decomposition method, the general nonlinear singular time-delay Hamiltonian system is transformed into an equivalent differential algebraic form; The specific implementation method is as follows: First, for matrices E, A, T, and... The following relationship must be satisfied. ⑶ ⑷ If (3) holds, it means that x(t) has no impulse perturbation, that is, the system (1) has no impulse solution. Based on this, if If the matrix pair (E, A) is admissible, then there always exist two nonsingular matrices. Make ⑸ Proof: Since matrix pairs (E, A) are admissible, there exists s such that det(sE-A) ≠ 0, i.e., matrix sE-A is invertible. Let Then there exists a non-singular matrix T such that , ,in It is an invertible matrix. It is a zero-power matrix. , It is an invertible matrix; make , Then there is Proof complete; make ⑹ in , , , ; ； In system (1) ⑺ Under (5)(6)(7), system (1) can be represented as: ⑻ System (8) and System (1) are equivalent. Represented as ,in It is an antisymmetric matrix. It is a symmetric matrix; If the system's exponent at the equilibrium point is 1, the controller design is as follows. ⑼ in It is a gain matrix, and the equilibrium point refers to the critical value at which the system tends to stabilize within a certain range; Substituting the controller (9) into system (8), we obtain the equivalent form of system (1): ⑽ Where R = R(x) is a positive definite matrix. , It is a weight matrix with full column rank; Step 3: Design a robust controller and substitute the robust controller into the equivalent differential algebra form to obtain the strictly equivalent dissipative Hamiltonian form of the system; The specific implementation method is as follows: The robust control problem of system (10) can be solved by the following control laws. ⑾ in It is the level of interference suppression. and There are two reference inputs; Substituting (11) into (10), we obtain the strictly equivalent dissipative Hamiltonian form of system (1) as follows: ⑿ Step 4: Select the Lyapunov function and verify the accuracy of the robust controller based on the strictly equivalent dissipative Hamiltonian form; In step 4, the Lyapunov function selected is: ⒀ Step 5: Apply the above method to the nonlinear charging circuit; The specific implementation method is as follows: The nonlinear charging circuit system structure is as follows: the capacitor is controlled by charge q, and the inductor is controlled by magnetic flux linkage Ψ. Their characteristics are expressed as follows: , It is an interference signal; According to Kirchhoff's current law, Kirchhoff's voltage law, and considering the effect of time delay on the system, the nonlinear charging circuit system is expressed as: ⒁ Where y and These are the system output and penalty signal, respectively. make , , , , , , And record , ; The nonlinear charging circuit system is described as follows: ⒂ There exists a Hamiltonian function ,in This allows the nonlinear charging circuit system to have a constant Hamilton value, implemented as follows: ⒃ The nonlinear charging circuit system is represented as an equivalent nonlinear differential-algebraic system through a nonsingular transformation. The nonlinear charging circuit system can then be transformed into the following equivalent form: ⒄ in , ; Through status feedback ⒅ The Hamiltonian form of the dissipation of the nonlinear charging circuit system is obtained as follows: ⒆ Select the gain matrix Weight matrix Then the nonlinear charging circuit system The controller is designed as follows: 。

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