Non-linear generalized system adaptive H-infinity control method based on static output feedback

Abstract

The invention discloses a nonlinear generalized system adaptive H-infinity control method based on static output feedback, and the method comprises the following steps: S1, constructing a nonlinear generalized system with actuator saturation, and constructing a static output feedback control strategy u = u (y), enabling L2 of a closed-loop system from w to z to be smaller than gamma, and enabling the closed-loop system to be asymptotically stable when w = 0, gamma is a given disturbance attenuation level and gamma gt; 0; s2, analyzing a system stability problem; and S3, carrying out adaptive H-infinity control design on the system in the step S1 by using a static output feedback method. By designing a static output feedback control strategy, sufficient conditions for system pulse controllability are given, meanwhile, the self-adaptive H-infinity controller is designed through the asymptotic stability problem of a closed-loop system, and the effectiveness of the static output feedback self-adaptive H-infinity controller is verified and determined through analogue simulation of a nonlinear singular circuit.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear generalized systems, and in particular to an adaptive H-based nonlinear generalized system based on static output feedback. ∞ Control method. Background Art

[0002] Generalized systems can also be called differential algebra, implicit systems and singular systems. Generalized systems will inevitably produce impulses, which often have an adverse effect when studying the controller design of generalized systems.

[0003] Saturation phenomenon exists in almost all practical systems. Actuator saturation is one of the most important saturation phenomena, which can degrade the performance of closed-loop systems and even make the system unstable. Therefore, stability analysis and controller design of systems with actuator saturation have been widely studied.

[0004] After searching, the Chinese patent application number CN113504729A discloses an H∞ robust control method and system for a bionic flapping-wing aircraft based on LMI, which designs a robust controller for the dynamic model of the aircraft to achieve the stability of the aircraft system;

[0005] The Chinese patent application number CN105955025B discloses a method for interference compensation and suppression of generalized uncertain systems, which addresses the problem that generalized systems have harmonic interference and norm-bounded interference and low control accuracy.

[0006] However, considering the actual situation of generalized systems, it is impossible to find all state variables when studying the stability of the system, and the calculation cost is very high. In addition, the information of output feedback is easy to obtain, so the application of output feedback has been promoted. Feedback control is to suppress the initial disturbance of the system, which mainly includes output feedback and state feedback. Since state feedback is sometimes not directly measurable, the application of output feedback should be considered. The challenge associated with dynamic output feedback in practical systems lies in the complexity of the closed-loop system. Therefore, the application of static output feedback is worth studying. Summary of the invention

[0007] The purpose of the present invention is to solve the defects in the prior art and propose an adaptive H ∞ Control method.

[0008] In order to achieve the above object, the present invention adopts the following technical solutions:

[0009] An adaptive H-based nonlinear descriptor system based on static output feedback ∞ The control method comprises the following steps:

[0010] S1: Construct a nonlinear generalized system with actuator saturation and construct a static output feedback control strategy u=u(y) so that the L2 of the closed-loop system from w to z is less than γ and the closed-loop system is asymptotically stable when w=0, where γ is a given disturbance attenuation level and γ>0;

[0011] S2: Analyze system stability issues;

[0012] S3: The system in step S1 is adaptively adjusted using the static output feedback method. ∞ Control design, including designing adaptive control laws, designing adaptive H ∞ Controllers and Design Systems for Robust Adaptive H ∞ Control laws;

[0013] S4: Complete Adaptive H for Nonlinear Singular Systems with Actuator Saturation ∞ Control design.

[0014] Furthermore, in step S1, the nonlinear generalized system with actuator saturation is specifically:

[0015]

[0016] Where p is the unknown parameter perturbation vector, ||p||≤p0 and p0 is a known constant, is the output, is the state of the system, and It is a punishment signal and external interference.

[0017] in:

[0018] When a vector function g(x) has continuous n-order partial derivatives, g(0) = 0, then g(x) can be expressed as:

[0019] g(x)=a1(x)x1…+a n (x)x n =A(x)x (3)

[0020] In the formula, a i (x) is a vector function, i = 1, 2, ..., n., Then the system (1) can be expressed as

[0021]

[0022] in, is some vector of x, P satisfies ζ(x,0)=0.

[0023] Further, in step S3, an adaptive H is designed based on the following conditions:∞ Controller:

[0024]

[0025]

[0026] A3) There exists a matrix Make

[0027]

[0028] in, is the unknown constant vector of P,

[0029] Further, in step S3, A1 and A2 are sufficient conditions for the system pulse to be controllable, and A3 is a matching condition, wherein: system 4 can be transformed into:

[0030]

[0031] Furthermore, when A1 and A2 exist and can be established, the system (4) is defined to be pulse controllable;

[0032] Based on system (8), when A1, A2 and A3 all exist and hold, the robust adaptive H ∞ Control issues are solved by the following controllers:

[0033]

[0034] in

[0035]

[0036] is an estimate of θ, and N>0 is the adaptive gain constant matrix.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] By designing a static output feedback control strategy, sufficient conditions for the system pulse controllability are given. At the same time, an adaptive H ∞ The controller is verified and confirmed by simulating a nonlinear singular circuit. ∞ Controller effectiveness. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0040] Figure 1 A schematic diagram of a nonlinear circuit system in an embodiment of the present invention;

[0041] Figure 2 is a response diagram of state x in an open-loop system in an embodiment of the present invention;

[0042] Figure 3 is the state x and Response graph of ;

[0043] Figure 4 Response diagram of saturation control sat(u) in a closed-loop system in an embodiment of the present invention. DETAILED DESCRIPTION

[0044] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0045] Adaptive response of nonlinear descriptor systems based on static output feedback ∞ The control method comprises the following steps:

[0046] S1: Construct a nonlinear generalized system with actuator saturation and construct a static output feedback control strategy u=u(y) so that the L2 of the closed-loop system from w to z is less than γ and the closed-loop system is asymptotically stable when w=0, where γ is a given disturbance attenuation level and γ>0;

[0047] S2: Analyze system stability issues;

[0048] S3: The system in step S1 is adaptively H- ∞ Control design, including designing adaptive control laws, designing adaptive H ∞ Controllers and Design Systems for Robust Adaptive H ∞ Control laws;

[0049] S4: Complete Adaptive H for Nonlinear Singular Systems with Actuator Saturation ∞ Control design.

[0050] In a specific embodiment of the present application, the nonlinear generalized system with actuator saturation is specifically:

[0051]

[0052] Where p is the unknown parameter perturbation vector, ||p||≤p0 and p0 is a known constant, is the output, is the state of the system, and It is a punishment signal and external interference.

[0053] Where: Assume there exists a non-singular matrix P such that P T E=E T P≥0 holds.

[0054] f(x) and B(x) represent smooth vector fields of appropriate dimensions, is the input.

[0055] The saturation nonlinear function sat(·) is defined as follows:

[0056]

[0057] Preferably, the following definitions and theorems are included:

[0058] Definition 1: A control law u(x) is called an admissible control law if, for any initial condition Ex0, the resulting closed-loop generalized system has no impulsive solutions. If there exists such an admissible control law that makes the resulting closed-loop generalized system have index 1 at its equilibrium point, then the original system is called an impulsive controllable system.

[0059] Lemma 1: When a vector function g(x) has continuous n-order partial derivatives, g(0) = 0, then g(x) can be expressed as:

[0060] g(x)=a1(x)x1…+a n (x)x n =A(x)x (3)

[0061] In the formula, a i (x) is a vector function, i = 1, 2, ..., n., Then the system (1) can be expressed as

[0062]

[0063] in, is some vector of x, P satisfies ζ(x,0)=0.

[0064] In addition, the admissible stable controller for designing system (4) also includes the following lemma:

[0065] Lemma 2: Let

[0066] α=u-sat(u). (5)

[0067] Then there exists a real number β∈(0,1) such that

[0068] αT α≤βu T u (6)

[0069] in

[0070] In a specific embodiment of the present application, in step S3, in system (1), the adaptive H of the static output feedback method is used. ∞ In the control problem, for a given disturbance attenuation level γ, the following adaptive control law is designed:

[0071]

[0072] The L2 gain of the closed-loop system from w to z is less than γ, and the closed-loop system is asymptotically stable when w = 0. Furthermore, the adaptive H is designed based on the following conditions: ∞ Controller:

[0073]

[0074]

[0075] A3) There exists a matrix Make

[0076]

[0077] in, is the unknown constant vector of P,

[0078] It needs to be further explained that A1 and A2 are sufficient conditions for the system pulse to be controllable, and A3 is a matching condition, among which: System 4 can be transformed into:

[0079]

[0080] In the specific embodiment of the present application, when A1 and A2 exist and can be established, it is defined that the system (4) is pulse controllable;

[0081] Based on system (8), when A1, A2 and A3 all exist and hold, the robust adaptive H ∞ Control issues are solved by the following controllers:

[0082]

[0083] in

[0084]

[0085] is an estimate of θ, and N>0 is the adaptive gain constant matrix.

[0086] It is necessary to further explain that the above contents are proved as follows:

[0087] Substituting equations (5) and (10) into system (8) yields:

[0088]

[0089] make

[0090]

[0091] Then the system (12) can be transformed into the following form

[0092]

[0093] When A1 and A2 hold, according to Theorem 1, the closed-loop system (13) is pulse-free.

[0094] in addition,

[0095] Consider the Lyapunov function:

[0096]

[0097] According to system (13) and inequality (9), we have:

[0098]

[0099] therefore

[0100]

[0101] Right now

[0102]

[0103] The trajectory along system (13) holds, so the L2 gain of system (13) from w to z is no greater than γ.

[0104] When w = 0, we can get from system (13):

[0105]

[0106] When w = 0, the closed-loop system (13) is asymptotically stable. This proof is thus completed.

[0107] In order to better understand the technical solution of the present application, it is further explained below in conjunction with a circuit simulation system.

[0108] like Figure 1As shown in the figure, in a nonlinear circuit system, the capacitance is controlled by the charge q, and its characteristics are u1 = f1(q1), u2 = f2(q2 ) Indicates that the voltage source and current source have saturation characteristics, and i w is the disturbance signal.

[0109] According to Kirchhoff's voltage law and Kirchhoff's current law, the system can be expressed as

[0110]

[0111] Choose y = [0, q2] T , w=i w . Let f1(q1)=a1+p, R3=(1 / 2)Ω,|U s |≤2V,|I s |<0.5V. It means x=[q1,q2] T , u=[U s , I s ] T .

[0112] Then the system (13) can be written as:

[0113]

[0114] According to f(0,0)=0, we can get

[0115] It can be concluded that A1 and A3 are valid;

[0116] in θ=p, obviously, A2 also holds.

[0117] choose N=1, γ=1, β=0.05, where P T E=E T P≥0.

[0118] According to the inequality equation (9), we can get a solution

[0119]

[0120] Make

[0121] Therefore, all the conditions of Theorem 2 are satisfied. Based on Theorem 2, the SOF adaptive H of system equation (15) ∞ The controller can be designed as follows:

[0122]

[0123] In order to demonstrate the effectiveness of the adaptive H' controller (Eq. (26)), some numerical simulations were performed with the following options: Given the initial condition Ex(0) = [1, 0] T , p = 1. In order to test the robustness of the controller to external disturbances, we add an amplitude of [0, 1] to the system for a duration of [1 to 3 s]. T square wave disturbance. Figure 2-Figure 4 The status x, And the response of the control signal sat(u).

[0124] From the simulation results, we can see that the designed adaptive H ∞ The controller is very effective.

[0125] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. An adaptive H algorithm for nonlinear descriptor systems based on static output feedback ∞ The control method is characterized in that The following steps are involved: S1: Construct a nonlinear generalized system with actuator saturation and construct a static output feedback control strategy u=u(y) so that the L2 of the closed-loop system from w to z is less than γ and the closed-loop system is asymptotically stable when w=0, where γ is a given disturbance attenuation level and γ>0; S2: Analyze system stability issues; S3: The system in step S1 is adaptively adjusted using the static output feedback method. ∞ Control design, including designing adaptive control laws, designing adaptive H ∞ Controllers and Design Systems for Robust Adaptive H ∞ Control laws; S4: Complete Adaptive H for Nonlinear Singular Systems with Actuator Saturation ∞ Control design.

2. The nonlinear generalized system adaptive H based on static output feedback according to claim 1 ∞ The control method is characterized in that In step S1, the nonlinear generalized system with actuator saturation is specifically: Where p is the unknown parameter perturbation vector, ||p||≤p0 and p0 is a known constant, is the output, is the state of the system, and It is a punishment signal and external interference. in: When a vector function g(x) has continuous n-order partial derivatives, g(0) = 0, then g(x) can be expressed as: g(x)=a1(x)x1…+a n (x)x n =A(x)x (3) In the formula, a i (x) is a vector function, i = 1, 2, ..., n., Then the system (1) can be expressed as in, is some vector of x, P satisfies ζ(x,0)=0.

3. The nonlinear generalized system adaptive H based on static output feedback according to claim 2 ∞ The control method is characterized in that In step S3, an adaptive H is designed based on the following conditions: ∞ Controller: A1) A2) A3) There exists a matrix Make in, is the unknown constant vector of P, 4. The nonlinear generalized system adaptive H based on static output feedback according to claim 3 ∞ The control method is characterized in that In step S3, A1 and A2 are sufficient conditions for the system pulse to be controllable, and A3 is a matching condition, where: System 4 can be transformed into: (8)。 5. The nonlinear generalized system adaptive H based on static output feedback according to claim 4 ∞ The control method is characterized in that When A1 and A2 exist and can be established, the system (4) is defined to be pulse controllable; Based on system (8), when A1, A2 and A3 all exist and hold, the robust adaptive H ∞ Control issues are solved by the following controllers: in is an estimate of θ, and N>0 is the adaptive gain constant matrix.

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