A data-driven event-triggered control method for an inverted pendulum system

Abstract

This invention discloses a data-driven event-triggered control method for an inverted pendulum system. The method establishes a data-based system representation of the inverted pendulum system, then designs the solution conditions for the data-based control gain K of the inverted pendulum system, and finally constructs two data-based event triggering conditions for the inverted pendulum system and demonstrates the effectiveness of the proposed method, effectively alleviating the problem of data-driven event-triggered control for inverted pendulum systems that cannot be modeled.

Description

Technical Field

[0001] This invention relates to a control method for an inverted pendulum system, specifically to a data-driven event-triggered control method for an inverted pendulum system. Background Technology

[0002] Inverted pendulum systems have important applications in robotics, aerospace, and intelligent equipment. However, due to their inherent strong nonlinearity and underactuated characteristics, as well as unavoidable unmodeled dynamics and parameter uncertainties, accurately establishing a model of this system is often extremely difficult. How to design a stable and high-performance controller relying solely on measurable input state data has become a highly valuable problem. Based on this idea, researchers have conducted in-depth research on data-driven controller design and achieved many representative results.

[0003] However, existing research on the control of inverted pendulum systems has limitations due to insufficient consideration of nonlinear characteristics and bandwidth constraints. Therefore, this paper proposes a data-driven event-triggered control method to improve system stability and control performance while reducing resource consumption. Summary of the Invention

[0004] The purpose of this invention is to propose a data-driven event-triggered control method for inverted pendulum systems, which can effectively solve the problem of data-driven event-triggered control for inverted pendulum systems that cannot be modeled.

[0005] The specific technical solution of the present invention is as follows: A data-driven event-triggered control method for an inverted pendulum system, characterized by comprising the following steps:

[0006] A data-based system representation of the inverted pendulum system was established, and the specific steps are as follows:

[0007] (1) For the inverted pendulum system, its state-space model is:

[0008]

[0009] In the formula, x1 is the angular velocity, x2 is the swing angle, u is the thrust, and g is the gravitational acceleration. It is the distance from the base to the center of mass of the object, and μ is the coefficient of rotational friction. It is a quality that needs to be balanced;

[0010] (2) Applying Taylor expansion to the above inverted pendulum system yields:

[0011]

[0012] In the formula,

[0013] (3) Collect datasets by running the inverted pendulum system offline.

[0014]

[0015] In the formula, κ is time, τ is the sampling interval, and T is the total number of samples, satisfying T≥3;

[0016] (4) Based on the above dataset Choose data matrices U0 = [u(0) u(τ) … u((T-1)τ)], X0 = [x(0)x(τ) … x((T-1)τ)] and These matrices satisfy the following equation: X1 = AX0 + BU0 + R0

[0017] In the formula, R0 is a data matrix formed by samples r(x, u);

[0018] (5) Select the event trigger control rate u = Kx(κ) l ), κ∈[κ l κ l+1 ), where K is the control gain, κ l and κ l+1 It is a trigger sequence; substituting the event trigger control law into (2) yields the data-based system representation of the inverted pendulum system, as follows:

[0019]

[0020] In the formula, e = x(κ) l -x is the trigger error, and G and L are the equations... and With any solution obtained, the system representation of the inverted pendulum system based on the data is now complete;

[0021] The solution conditions for the control gain K of the inverted pendulum system based on data were designed, and the specific steps are as follows:

[0022] (1) Choose a known matrix Δ such that it satisfies the condition. Then if there exist parameters ∈1>0 and a matrix Y such that the following equation holds:

[0023]

[0024] X0Y>0

[0025] In the formula, Sym(X1Y)=X1Y+Y T X1 T , * is the symmetric element of the block matrix implied by symmetry, I is the identity matrix, and Ω>0 is an arbitrary positive definite matrix; then, in the control gain K=U0Y(X0Y) -1 The lower controller u = Kx can stabilize the inverted pendulum system at the origin. The proof is as follows:

[0026] A001: According to Shulbu, the above conditions are equivalent to

[0027] A002: Using Petersen's lemma, we can obtain Sym(-Ry)≤1ΔΔ T +1 -1 Y T Y;

[0028] A003: According to A002, for all satisfying RR T ≤ΔΔ T Since R, Sym((X1-R)Y)+Ω≤0 holds, therefore, the inequality also holds for R=R0;

[0029] A004: Choose P = (X0Y) -1 Multiplying both sides of Sym((X1-R)Y)+Ω≤0 by P, we get Sym(P(X1-R0)YP)+PΩP≤0; by defining G=YP, Sym(P(X1-R0)YP)+PΩP≤0 indicates that (X1-R0)G is Hurwitz, therefore, in the control gain K=U0Y(X0Y) -1 The lower controller u = Kx can stabilize the inverted pendulum system at the origin, thus concluding the proof;

[0030] Two data-driven event triggering conditions were constructed for the inverted pendulum system, and the effectiveness of the proposed method was demonstrated. The specific steps are as follows:

[0031] (1) Select a data-based static triggering mechanism as follows:

[0032]

[0033] In the formula, σ and β are positive parameters to be solved, and ξ is a known positive parameter;

[0034] Choose a known matrix Δ such that it satisfies the condition Then if there exist parameters σ>0, ε2>0, β>0, and λ1>0 such that the following equation holds:

[0035]

[0036] Therefore, in the control gain K = U0Y(X0Y) -1 Under the static triggering mechanism, the controller u = Kx can make the inverted pendulum system locally asymptotically stable and eliminate the Zeno phenomenon. The proof is as follows:

[0037] B001: Select the Lyapunov function as:

[0038] V(x)=x T Px

[0039] In the formula, P = (X0Y) -1 ;

[0040] B002: Taking the derivative of equation B001 with respect to κ, we get:

[0041]

[0042] B003: During the trigger interval, we have:

[0043]

[0044] B004: According to A004, we can obtain:

[0045] 2x T G T (X1-R0) T Px=x T Sym(P(X1-R0)G)x

[0046] ≤-x T PΩPx

[0047] B005: Using Yang's inequality, we have:

[0048]

[0049] B006: Substituting B004 and B005 into B003, we get:

[0050]

[0051] In the formula,

[0052] B007: The proof of Φ1≤0 is given next;

[0053] B008: Applying Schur's complement to the proposed linear matrix inequality conditions, we have:

[0054]

[0055] In the formula, Θ=[L 0] T And Ψ = [0 - PI];

[0056] B009: Because so:

[0057]

[0058] B010: Combining B008 and B009, we can obtain:

[0059]

[0060] B011: According to B010, Φ1≤0;

[0061] B012: Substituting B010 into B006, we have:

[0062]

[0063] B013: Because Therefore, we can conclude that:

[0064]

[0065] B014: According to B013, as long as x satisfies So Therefore, the inverted pendulum system is locally asymptotically stable.

[0066] B015: The following provides proof that the Zeno phenomenon has been ruled out;

[0067] B016: Using the data-based representation of the inverted pendulum system, we have:

[0068]

[0069] B017: Further scaling down B016 yields:

[0070]

[0071] In the formula, C1=||X1G||+||Δ|| ||G||, C2=||X1L||+||Δ|| ||L|| and

[0072] B018: Using the Cauchy-Schwarz inequality, we have:

[0073]

[0074] B019: According to B018, we can obtain:

[0075]

[0076] B020: Because Therefore:

[0077]

[0078] B021: Dividing both sides of B017 by ||x||, we get:

[0079]

[0080] B022: According to B014, we have:

[0081]

[0082] In the formula,

[0083] B023: Combining B022 and B020, we get:

[0084]

[0085] In the formula,

[0086] B024: Solving for B023, we have:

[0087]

[0088] B025: Because Therefore, the Zeno phenomenon is ruled out, and the proof is complete.

[0089] (2) Select a data-based dynamic triggering mechanism as shown below:

[0090]

[0091] In the formula, γ1 and γ2 are the positive parameters to be solved. It is a known positive parameter;

[0092] Choose a known matrix Δ such that it satisfies the condition Then if there exist parameters γ1>0, γ2>0, ε3>0, and λ2>0 such that the following equation holds:

[0093]

[0094] Therefore, in the control gain K = U0Y(X0Y) -1 Under the dynamic triggering mechanism, the controller u = Kx can make the inverted pendulum system locally asymptotically stable and eliminate the Zeno phenomenon. The proof is as follows:

[0095] C001: Select the Lyapunov function as:

[0096] V(x)=x T Px+h(κ)

[0097] In the formula, P = (X0Y) -1 ;

[0098] C002: Taking the derivative of equation C001 with respect to κ, we get:

[0099]

[0100] C003: According to A004, we can obtain:

[0101] 2x T G T (X1-R0) T Px=x T Sym(P(X1-R0)G)x≤-x T PΩPx

[0102] C004: Using Yang's inequality, we have:

[0103]

[0104] C005: Because the trigger interval h(κ)≥0, based on this relationship, substituting C004 and C003 into C002, we get:

[0105]

[0106] In the formula,

[0107] C006: The proof of Φ2≤0 is given next;

[0108] C007: Applying Schur's complement to the proposed linear matrix inequality conditions, we have:

[0109]

[0110] C008: Because so:

[0111]

[0112] C009: Combining C008 and C007, we get:

[0113]

[0114] C010: According to C009, Φ2≤0;

[0115] C011: Substituting C009 into C005, we have:

[0116]

[0117] C012: According to C011, as long as x satisfies So Therefore, the inverted pendulum system is locally asymptotically stable.

[0118] C013: The following is proof that the Zeno phenomenon has been ruled out;

[0119] C014: Using the proposed dynamic triggering conditions, we have:

[0120]

[0121] In the formula, It is Z T The upper bound of ΞZ;

[0122] C015: Integrating C014, we get:

[0123]

[0124] C016: Because And h(κ)=0, we have:

[0125]

[0126] C017: According to C016, we can obtain:

[0127]

[0128] C018: Because and have Therefore, the Zeno phenomenon is ruled out, and the proof is complete.

[0129] Thus, the construction of two event triggering conditions for the inverted pendulum system based on data and the demonstration of the effectiveness of the proposed method are completed. Attached Figure Description

[0130] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0131] Figure 2 The state curve diagram used in this embodiment is based on the static triggering mechanism proposed in this invention.

[0132] Figure 3 The release interval diagram used in this embodiment employs the static triggering mechanism proposed in this invention;

[0133] Figure 4 The example uses a state curve diagram under the dynamic triggering mechanism proposed in this invention.

[0134] Figure 5 The release interval diagram is based on the dynamic triggering mechanism proposed in this invention, as shown in the embodiment. Detailed Implementation

[0135] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0136] like Figure 1 As shown, a data-driven event-triggered control method for an inverted pendulum system includes the following steps:

[0137] Step 1: Randomly select u(κ) from [-1, 1] and feed it into the system. Collect data offline to obtain the data matrix U0 = [u(0) u(τ) … u((T-1)τ)], X0 = [x(0) x(τ) … x((T-1)τ)] and

[0138] Step 2: Choose δ(x) = 2|sinx1-x1|, and let δ max For the maximum value of δ(x) during offline data acquisition, T = 5, Ω = 1, using the data matrix and Solve for parameters ∈1 And matrix Y;

[0139] Step 3: Use K = U0Y(X0Y) -1 Configure the controller;

[0140] Step 4: Set ξ = 0.01 and Verify the two trigger conditions; if they are met, transmit data to the controller.

[0141] Step 5: Update the input u(κ) using the triggered data to control the inverted pendulum system:

[0142] Step 6: Repeat step 3 to enter the next cycle.

[0143] An embodiment of the present invention is described below;

[0144] Consider the event-triggered control problem of a data-driven inverted pendulum system that cannot be modeled. Its state-space model is as follows:

[0145]

[0146] Figure 1 This is a flowchart of a method according to an embodiment of the present invention; applying the proposed method, Figures 2 to 5 The diagrams show the state curves and release intervals after the proposed method is applied to the system. These diagrams demonstrate that the proposed method achieves satisfactory event-triggered control performance without requiring system model information.

[0147] References

[0148] [1]De Persis C,Tesi P.Formulas for data-driven control:Stabilization,optimality,and robustness[J].IEEE Transactions on Automatic Control,2019,65(3):909-924.

[0149] [2]Berberich J, J,Müller M A,et al.Data-driven model predictivecontrol with stability and robustness guarantees[J].IEEE TransaetionsonAutomatic Control,2020,66(4):1702-1717。

Claims

1. A data-driven event-triggered control method for an inverted pendulum system, characterized in that, Includes the following steps: The parameter representation of the inverted pendulum system is constructed using an offline-collected data matrix. The specific steps are as follows: For the inverted pendulum system, its state-space model is: In the formula, It's angular velocity. It's a swing angle. It is thrust. It is gravitational acceleration. It is the distance from the base to the center of mass of the object. It is the coefficient of rotational friction. It is a quality that needs to be balanced; Applying Taylor expansion to the above inverted pendulum system yields: In the formula, , , , ; Offline operation of the inverted pendulum system to collect datasets : In the formula, It is time. It is the sampling interval. It is the total number of samples and satisfies ; Based on the above dataset Select data matrix , and These matrices satisfy the following equation: In the formula, It is a sample The resulting data matrix; Select event trigger control rate ,in It is about controlling the gain. and It is a trigger sequence; substituting the event trigger control law into the Taylor expansion yields the data-based system representation of the inverted pendulum system, as follows: In the formula, It's a triggering error. and They are equations and With any solution obtained, the system representation of the inverted pendulum system based on the data is now complete; Design an inverted pendulum system based on data-driven control gain. The solution conditions and specific steps are as follows: Choose a known matrix To make it meet the conditions If parameters exist sum matrix Makes the following equation true: In the formula, , It is the symmetric element of the block matrix implicitly determined by symmetry. It is the identity matrix. It is an arbitrary positive definite matrix; therefore, in the control gain Lower controller The inverted pendulum system can be stabilized at the origin, as proven below: A001: According to Shulbu, the above conditions are equivalent to ; A002: Using Petersen's lemma, we can obtain ; A003: According to A002, for all satisfying of , Established, therefore, for This inequality also holds true; A004: Select ,exist Both sides ride together , can be obtained ; By definition , show It is Hurwitz, therefore in controlling the gain Lower controller The inverted pendulum system can be stabilized at the origin, thus concluding the proof. The inverted pendulum system was constructed based on two event triggering conditions, and the effectiveness of the proposed method was demonstrated. The specific steps are as follows: The static triggering mechanism designed using offline collected data is shown below: In the formula, , , and These are the positive parameters to be solved. It is a known positive parameter; Choose a known matrix To make it meet the conditions If parameters exist , , and Makes the following equation true: So, in controlling the gain Under the static triggering mechanism, the controller This can make the inverted pendulum system locally asymptotically stable and eliminate the Zeno phenomenon. The proof is as follows: B001: Select the Lyapunov function as: B002: Solve equation B001 From the derivative, we can obtain: B003: During the trigger interval, we have: B004: According to A004, we can obtain: B005: Using Yang's inequality, we have: B006: Substituting B004 and B005 into B003, we get: In the formula, ; B007: The following is given Proof; B008: Applying Schur complement to the proposed linear matrix inequality conditions, we have: In the formula, and ; B009: Because ,so: B010: Combining B008 and B009, we can obtain: B011: According to B010, we have ; B012: Substituting B010 into B006, we have: B013: Because ,in, and It is a positive scalar. express Therefore, we can obtain: In the formula, yes The smallest eigenvalue; B014: According to B013, as long as satisfy ,So Therefore, the inverted pendulum system is locally asymptotically stable; B015: The following provides proof that the Zeno phenomenon has been ruled out; B016: Using the data-based representation of the inverted pendulum system, we have: B017: Further scaling down B016 yields: In the formula, , and ; B018: Using the Cauchy-Schwarz inequality, we have: B019: According to B018, we can obtain: B020: Because Therefore, we have: B021: Divide both sides of B017. We can obtain: B022: According to B014, we have: In the formula, ; B023: Combining B022 and B020, we can obtain: In the formula, ; B024: Solving for B023, we have: B025: Because Therefore, the Zeno phenomenon is ruled out, and the proof is complete. The dynamic triggering mechanism designed using offline collected data is shown below: In the formula, , and These are the positive parameters to be solved. It is a known positive parameter; Choose a known matrix To make it meet the conditions If parameters exist , , and Makes the following equation true: So, in controlling the gain Under the dynamic triggering mechanism, the controller This can make the inverted pendulum system locally asymptotically stable and eliminate the Zeno phenomenon. The proof is as follows: C001: Select the Lyapunov function as: C002: Solve equation C001 From the derivative, we can obtain: C003: According to A004, we can obtain: C004: Using Yang's inequality, we have: C005: Because of the trigger interval Based on this relationship, substituting C004 and C003 into C002, we get: In the formula, ; C006: The following is given Proof; C007: Applying Schur's complement to the proposed linear matrix inequality conditions, we have: C008: Because ,so: C009: Combining C008 and C007, we get: C010: According to C009, there is ; C011: Substituting C009 into C005, we have: C012: According to C011, as long as satisfy ,So Therefore, the inverted pendulum system is locally asymptotically stable; C013: The following is proof that the Zeno phenomenon has been ruled out; C014: Using the proposed dynamic triggering conditions, we have: In the formula, yes The boundary; C015: Integrating C014, we get: In the formula, It is an exponential function; C016: Because and ,have: C017: According to C016, we can obtain: C018: Because and ,have Therefore, the Zeno phenomenon is ruled out, and the proof is complete. Thus, the construction of two event triggering conditions for the inverted pendulum system based on data and the demonstration of the effectiveness of the proposed method are completed.

Images