Stability margin measuring method based on data driving

Abstract

The invention relates to a stability margin measuring method based on data driving, which adopts a small gain theorem to ensure the stability of a closed loop of a system and defines the stability margin of the system based on the stability margin. The defined gain margin and the time delay margin can be obtained through a data-driven experimental measurement method. According to the method, a class of gain margins and time-delay margins for the multiple-input multiple-output linear system are established through the small gain theorem, so that the defect that the small gain theorem is only suitable for amplitude margin analysis is overcome. And a frequency sweeping method is established to obtain the gain margin and the time delay margin through an experimental method, so that the uncertainty and unmodeled dynamics in practice can be considered, and the obtained stability margin is closer to the stability margin of an actual system.

Description

Technical Field

[0001] The invention relates to a stability margin analysis and measurement method, in particular to the analysis and measurement of the stability margin of a multi-input multi-output aircraft, and belongs to the field of stability analysis. Background Art

[0002] The concept of stability margin plays a very important role in control engineering, as it can intuitively capture system uncertainties such as delays or unmodeled higher-order dynamics. Stability margin can be used for hazard monitoring, safety assessment, controller debugging, filter and compensator design, etc., and is widely used in control engineering, power electronics field and other fields. For single-input single-output linear time-invariant systems, the classical stability margin can be well defined and understood. It is defined in the frequency domain, can be easily obtained through Bode plots or Nyquist plots, and is simple and intuitive with clear physical meaning. Therefore, the classical stability margin is popular and widely adopted by control engineers.

[0003] However, for multi-input multi-output systems, the classical stability margin is no longer applicable. The graphical method suitable for evaluating the stability margin of a single-input single-output linear time-invariant system is difficult to extend to a multi-input multi-output system. The first reason is the complexity of the matrix perturbation; the second reason is that there are multiple loops in a multi-input multi-output system, and due to the interaction of the loops, the analysis method of a single loop at a time may lead to unreliable stability margins. Although some robust stability margin analysis methods are popular in multivariable systems, most of them are model-based and difficult to accurately model in engineering, which makes these methods not easy to use. The study of the stability margin of multi-input multi-output systems has always been one of the challenging problems in control theory and engineering applications. Summary of the invention

[0004] The technical problems to be solved by the present invention are:

[0005] In order to avoid the shortcomings of the prior art, the present invention provides a stability margin measurement method based on data driving.

[0006] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0007] A data-driven stability margin measurement method, characterized by the following steps:

[0008] Step 1: According to the loop transformation of the MIMO system, establish R i (s) link, its transfer function is as follows:

[0009] R i (s)=(I m +(H(s)KG(s)) -1 )-1

[0010] Among them, I m ∈R m×m is the identity matrix, H(s) is the frequency domain corrector, K is the feedback matrix, and G(s) is the transfer function expression of the multi-input multi-output system;

[0011] Step 2: Collect n pairs of input and output data by frequency sweeping method, where the input is R i (s) The sweep frequency signal at the input point of the link, the output is R i (s) Frequency response signal at the link output point;

[0012] Step 3: Calculate H ∞ Norm ‖R i (s)‖ and‖sR i (s)‖, i.e. and k=1,2,…,n; where ‖·‖ is H ∞ norm, ω k represents the input frequency, σ max represents the largest singular value, Indicates that at the input frequency ω k Take the upper bound of the function value within the range of variation, s is a complex variable;

[0013] Step 4: Calculate the gain margin γ according to the following formula m and the delay margin τ m :

[0014] γ m =(1-ε1)‖R i (s)‖ -1

[0015] τ m =(1-ε2)‖sR i (s)‖ -1

[0016] Among them, 0<ε1<1 is sufficiently small, and 0<ε2<1 is sufficiently small.

[0017] A further technical solution of the present invention: Step 2 is specifically as follows: first, R i (s) The input and output signals of the link are used to calculate R i (jω k ) and jω k R i (jω k ), then the maximum singular value σ can be calculated max (R i (jω k )) and σ max (jωk R i (jω k )), finally at the frequency ω of the input signal k Take the upper bound of the maximum singular value in the range and

[0018] A computer system, characterized in that it comprises: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the above-mentioned method.

[0019] A computer-readable storage medium, characterized in that it stores computer-executable instructions, which are used to implement the above method when executed.

[0020] The beneficial effects of the present invention are:

[0021] The present invention provides a data-driven stability margin measurement method, which has the following beneficial effects compared with the prior art:

[0022] (1) The present invention provides the time delay margin of the MIMO system through the small gain theorem. The defined gain margin and time delay margin can be determined by the small gain theorem, which solves the defect that the small gain theorem is only applicable to amplitude margin analysis.

[0023] (2) The present invention develops a frequency sweeping method, which is used to obtain the gain margin and the time delay margin through experiments. By considering the system uncertainty and the unmodeled dynamics, the stability margin obtained by this method is closer to the actual system.

[0024] (3) This is a data-driven frequency domain response method that can obtain the stability margin of a multi-input multi-output system without a model. It is direct, effective, and convenient for engineers to apply. In addition, there is no need to disconnect the closed-loop system in order to excite the corresponding open-loop system, which is usually adopted by the classical stability margin of a single-input single-output linear time-invariant system. This advantage makes online stability margin measurement possible. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The drawings are only for the purpose of illustrating particular embodiments and are not to be considered limiting of the present invention. Like reference symbols denote like components throughout the drawings.

[0026] Figure 1 The following are the closed-loop block diagram of the system and the loop transformation diagram of the closed-loop system: (a) is the closed-loop connection with input interference; (b) is the loop transformation of Figure (a);

[0027] Figure 2It is a flow chart for system stability margin analysis and measurement;

[0028] Figure 3 Is the F450 quadcopter built R i Schematic diagram. DETAILED DESCRIPTION

[0029] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0030] In order to give a simple and practical definition of stability margin for a MIMO linear system, the present invention adopts the small gain theorem to ensure the stability of the system closed loop L2, and defines the system stability margin based on this. The defined gain margin γ m and the delay margin τ m It can be obtained through data-driven experimental measurement methods. This method establishes a type of gain margin and time lag margin for multi-input multi-output linear systems through the small gain theorem, which solves the defect that the small gain theorem is only applicable to amplitude margin analysis. And a frequency sweep method is established to obtain the gain margin and time lag margin through experimental methods, so that the uncertainty and unmodeled dynamics in practice can be taken into account, and the stability margin obtained is closer to the stability margin of the actual system.

[0031] The invention defines the L2 gain margin and L2 time delay margin of a linear multiple-input multiple-output system, and proposes a corresponding stability margin measurement method by using loop transformation, small gain theorem and frequency sweeping method.

[0032] This is achieved through the following steps:

[0033] (a) Consider the following type of multi-input multi-output linear system

[0034]

[0035] Among them, A∈R n×n , B∈R n×m , C∈R p×n is a constant matrix, x(t)∈R n is the state vector, y(t)∈R p is a measurable output vector, μ(t)∈R m is the actual control input. The transfer function from μ to y of system (1) is expressed as G(s)=C(sI n -A) -1 B.

[0036] By adding a complex diagonal multiplicative perturbation Ι at the system input m +Δ(s) to define the multivariable gain margin and lag margin, that is,

[0037] μ(s)=(Ι m +Δ(s))u(s) (2)

[0038] Among them, I m ∈R m×m is the identity matrix, u(s)∈R m is the control input of the design; Δ(s) is the amplitude perturbation or time-delay perturbation. For the convenience of subsequent derivation, we use the system representation of the frequency domain method, and s is a complex variable.

[0039] Assume that the following stabilization controller is designed for system (1):

[0040] u(t)=L -1 (-H(s)Ky(s)) (3)

[0041] So that when Δ(s)≡0, under any initial condition x0, Where L -1 represents the inverse Laplace transform, H(s) is a stable transfer function, and the feedback matrix K∈R m×p The closed-loop system consisting of (1), (2) and (3) is as follows: Figure 1 The main contents of the present invention are discussed under the condition of no noise.

[0042] Assumption 1: m +(H(s)KG(s)) -1 ) -1 : That is, the designed controller (3) can ensure that the closed-loop system is finite gain L2 stable in the absence of uncertainty.

[0043] Assumption 2: The uncertainty under consideration satisfies Δ(s):

[0044] Definition 2: Given Δ(s) = Δ γ ≡diag{γ1,γ2,…,γ m},|γ i |≤γ m , i = 1, 2, 3…m, if the state of the closed-loop system composed of (1), (2) and (3) satisfies any initial condition x0, Then the closed-loop system corresponding to Δ(s)≡0 is said to have an L2 gain margin γ m ; Given i=1,2,3…m, if the state of the closed-loop system composed of (1), (2) and (3) satisfies any initial condition x0, Then the closed-loop system corresponding to Δ(s)≡0 is said to have an L2 delay margin τ m .

[0045] (b) Gain margin and lag margin analysis

[0046] The L2 gain margin and L2 lag margin defined in Definition 2 are used to measure the relative stability of a system.

[0047] First, the closed-loop system composed of (1), (2) and (3) is transformed, as follows: Figure 1 (b), where the uncertainty Δ(s) is related to the following transfer function R i (s) Internet

[0048] R i (s)=(I m +(H(s)KG(s)) -1 ) -1 (4)

[0049] Based on the interconnect structure and the small gain theorem, the following conclusions about stability can be obtained.

[0050] For systems (1), (2) and controller (3), if R i (s): Δ(s): and

[0051] ‖Δ(s)R i (s)‖<1 (5)

[0052] Then under the action of controller (3), the state of system (1) satisfies In formula (5), ‖·‖ is H ∞ norm, H of any system G(s) ∞ The norm is ω represents frequency, σ max represents the largest singular value, It means taking the upper bound of the function value within the range of frequency ω.

[0053] Further conclusions can be drawn about the stability margin.

[0054] Stability margin: For the system (1) and controller (3) with Δ = 0,

[0055] L2 gain margin of the closed-loop system γ m ∈R + is such that for any |γ i |≤γm ,have

[0056] ‖Δ γ (s)R i (s) ‖<1 (6)

[0057] where Δ γ (s)≡diag{γ1,γ2,…,γ m}.

[0058] L2 delay margin τ of the closed-loop system m ∈R + is such that for any 0≤τ i ≤τ m ,have

[0059] ‖Δ τ (s)R i (s)‖<1 (7)

[0060] in

[0061] (c) Data driven gain margin and skew margin measurements

[0062] For L2 gain margin, if

[0063] ‖Δ γ (s)R i (s)‖<1 (8)

[0064] Then γ i ∈[0,γ m ] can be set to ‖Δ(s)R i (s)‖<1 yes

[0065] γ m =(1-ε1)‖R i (s)‖ -1 (9)

[0066] Among them, 0<ε1<1 is sufficiently small.

[0067] For the L2 lag margin, since

[0068]

[0069] and

[0070] If there is

[0071] τ m <‖sR i (s)‖ -1 (11)

[0072] Then τ i∈[0,τ m ] can be set to ‖Δ(s)R i (s)‖<1, yes

[0073] τ m =(1-ε2)‖sR i (s)‖ -1 (12)

[0074] Among them, 0<ε2<1 is sufficiently small.

[0075] In addition, the present invention provides a method for obtaining L2 gain margin and L2 lag margin by using a frequency sweep method, and the specific steps are as follows:

[0076] Step 1: According to Figure 1 The loop transformation shown in , establishes R i (s) link, its transfer function is as shown in formula (4);

[0077] Step 2: Collect n pairs of input and output data by frequency sweeping method, where the input is R i (s) Sweep frequency signal at the link input point (the frequency of the input signal ω k , k=1,2,…,n should cover the working frequency of the actual object), the output is R i (s) Frequency response signal at the link output point;

[0078] Step 3: Using the R collected in step 2 i (s) Input and output data calculation H ∞ Norm ‖R i (s)‖ and‖sR i (s)‖, i.e. and (First, you can use R i (s) The input and output signals of the link are used to calculate R i (jω k ) and jω k R i (jω k ), then the maximum singular value σ can be calculated max (R i (jω k )) and σ max (jω k R i (jω k )), finally at the frequency ω of the input signal k Take the upper bound of the maximum singular value in the range and )

[0079] Step 4: Determine the gain margin γ according to equations (9) and (12) mand the delay margin τ m .

[0080] Embodiment 1:

[0081] The present invention uses loop transformation and small gain theorem to propose corresponding stability margin analysis and measurement methods for quadrotors, which are implemented by the following steps:

[0082] (a) Generally speaking, for the attitude loop of a quadrotor, the roll channel and the pitch channel are coupled. Therefore, considering the roll channel and the pitch channel, the quadrotor dual-channel model is:

[0083]

[0084] Where x = [φ θ] T , φ is the roll angle, θ is the pitch angle, A∈R 2×2 , B∈R 2×2 , C∈R 2×2 is a constant matrix. Control input u=[τ φ τ θ ] T represents the control torque of the roll channel and the pitch channel, which is a dual-input and dual-output coupled system. In addition, the PX4 autopilot is used to provide attitude control. The experimental goal is to obtain the L2 gain margin and L2 lag margin of the closed-loop quadrotor system when the attitude controller is turned on.

[0085] (b) Data-driven stability margin measurement

[0086] Step 1: Create Figure 3 R i (s);

[0087] Step 2: Collect n pairs of input and output data by frequency sweeping method, where the frequency range of the input frequency sweeping signal is 0.3-60rad / s;

[0088] Step 3: After performing the frequency sweep experiment and collecting the measured data, the frequency response comprehensive identification software package CIFER and MATLAB can be used to simplify the data processing process. i (jω) and jωR i The peak value of the maximum singular value of (jω) is H ∞ Norm ‖R i (s)‖ and‖sR i (s)‖;

[0089] Step 4: γ m and τ m It can be obtained from equations (9) and (12), specifically γ m =0.3608 and τ m=0.0209s.

[0090] Thus, it can be stated that for this quadrotor, when there is any uncertainty in the feedback connection Δ(s)≡diag{γ1,γ2} where |γ i |≤0.3608, or Where 0≤τ i When ≤0.0209s, the closed-loop quadrotor system can still remain stable.

[0091] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should be included in the protection scope of the present invention.

Claims

1. A data-driven stability margin measurement method, characterized in that Here are the steps: Step 1: According to the loop transformation of the MIMO system, establish R i (s) link, its transfer function is as follows: R i (s)=(I m +(H(s)KG(s)) -1 ) -1 Among them, I m ∈R m×m is the identity matrix, H(s) is the frequency domain corrector, K is the feedback matrix, and G(s) is the transfer function expression of the multi-input multi-output system; Step 2: Collect n pairs of input and output data by frequency sweeping method, where the input is R i (s) The sweep frequency signal at the input point of the link, the output is R i (s) Frequency response signal at the link output point; Step 3: Calculate H ∞ Norm ‖R i (s)‖ and‖sR i (s)‖, i.e. and k=1,2,…,n; where ‖·‖ is H ∞ norm, ω k represents the input frequency, σ max represents the largest singular value, Indicates that at the input frequency ω k Take the upper bound of the function value within the range of variation, s is a complex variable; Step 4: Calculate the gain margin γ according to the following formula m and the delay margin τ m : c m =(1-ε1)‖R i (s)‖ -1 τ m =(1-ε2)‖sR i (s)‖ -1 Among them, 0<ε1<1 is sufficiently small, and 0<ε2<1 is sufficiently small.

2. The data-driven stability margin measurement method according to claim 1, characterized in that: Step 2 is as follows: First, you can use R i (s) The input and output signals of the link are used to calculate R i (jω k ) and jω k R i (jω k ), then the maximum singular value σ can be calculated max (R i (jω k )) and σ max (jω k R i (jω k )), finally at the frequency ω of the input signal k Take the upper bound of the maximum singular value in the range and 3. A computer system, characterized in that include: One or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors are enabled to implement the method of claim 1.

4. A computer-readable storage medium, characterized in that Computer executable instructions are stored, and when the instructions are executed, they are used to implement the method of claim 1.

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