Double-event-triggered optimal control method for nonlinear systems with multi-source disturbances

Abstract

The present invention discloses a dual-event triggering optimal control method for a nonlinear system with multi-source disturbances, comprising the following steps: 1) first constructing the state variables, control variables, dynamic model and performance index of the system, and obtaining the control strategy expression for achieving Nash equilibrium; 2) constructing a nonlinear disturbance observer to offset the same-channel disturbance; 3) designing a dual-event triggering mechanism, and updating the system control law according to the "first-arrived-same-triggering" triggering mechanism; 4) constructing a single evaluation neural network, and designing a control strategy and a weight update strategy. Compared with the existing optimal control method, the present invention can save communication resource consumption while processing multi-source disturbances while ensuring the optimal control performance of the system.

Description

Technical Field

[0001] The invention relates to a control method, in particular to a double-event triggering optimal control method for a nonlinear system with multi-source disturbances. Background Art

[0002] In complex nonlinear systems, the system will be affected by many disturbances. Different types of disturbances make the controlled system unstable in a changing environment, which in turn affects the control accuracy of the system. The Disturbance Observer-Based Control (DOBC) method has gradually been extended to systems with multi-source disturbances. For example, the composite DOBC and H ∞ Control has been widely used in the fields of robotic arm systems, drone systems and multi-agent systems. When dealing with the interference problems of these complex nonlinear systems, the adaptive dynamic programming (ADP) algorithm based on neural networks has unique advantages. Combining ADP and zero-sum game theory, researchers have proposed new algorithms to solve control problems with external disturbances. Through online learning strategies, these methods can effectively suppress interference while ensuring system performance.

[0003] System interference has the characteristics of multiple sources and multiple types, and it is impossible to resist the impact of disturbances on the system through a single method. Therefore, a nonlinear system anti-interference optimization control method with multi-source disturbances is designed. At the same time, while ensuring the control performance of the system, it reduces unnecessary resource consumption and designs a dual-event dual-event trigger mechanism to reduce communication consumption. Summary of the invention

[0004] The purpose of the present invention is to overcome the defects of the prior art and provide a dual-event triggered optimal control method for a nonlinear system with multi-source disturbances, which can significantly reduce the consumption of communication resources while achieving the anti-interference optimal control of the nonlinear system.

[0005] The object of the present invention is achieved by: a double event triggered optimal control method for a nonlinear system with multi-source disturbances, comprising the following steps:

[0006] Step 1) establishing a nonlinear system model with multi-source disturbances, including state variables, control variables, and a dynamic model of the system;

[0007] Step 2) define the performance indicators of the system and find the optimal control to achieve the Nash equilibrium solution;

[0008] Step 3) construct a nonlinear disturbance observer to offset the same-channel disturbance and form a composite controller with the optimal control strategy;

[0009] Step 4) Design a dual event trigger mechanism and update the system control law according to the "first-come, first-served" trigger mechanism;

[0010] Step 5) Construct a single evaluation neural network, design control and weight update strategies, and obtain optimal control.

[0011] As a further limitation of the present invention, the step 1) comprises: establishing a nonlinear system model with multi-source disturbances:

[0012]

[0013] in, is the system state vector, is the control input, is the disturbance input, is a nonlinear perturbation with w∈L2(0,∞), is a smooth nonlinear dynamics; assuming that the initial state of the system is x(0) = x0, the system controller u r is the optimal control to be designed, is the estimated value of the interference d.

[0014] As a further limitation of the present invention, the step 2) includes: defining the performance index of the system as:

[0015]

[0016] in and is a positive definite matrix, γ is the predetermined interference attenuation level;

[0017] According to the nonlinear zero-sum differential game, the optimal performance index is obtained:

[0018]

[0019] in, and is a positive definite matrix, γ is the predetermined interference attenuation level; V * (x) is the optimal performance index of the system. The above optimal performance index has a unique solution. According to the Hamiltonian function, the optimal control and the worst perturbation w * :

[0020]

[0021] in, g=g(x), k=k(x).

[0022] As a further limitation of the present invention, the step 3) comprises: designing a nonlinear disturbance observer to estimate the disturbance d

[0023]

[0024] in, is the estimated value of the disturbance, z is the estimate of the internal state of the nonlinear disturbance observer, p(x) is the nonlinear vector-valued function to be designed, l(x) is the gain of the nonlinear disturbance observer, and the disturbance estimation error is defined as

[0025] As a further limitation of the present invention, the step 4) includes: designing a dual event trigger mechanism, and updating the system control according to a first-come-same-trigger mechanism;

[0026]

[0027] in, x(t) represents the current state, Indicates the updated status of the controller and the controlled object. is the triggering moment of the trigger mechanism, and the positive definite matrix R is decomposed into R = SS T , L is a positive real constant; is the triggering moment of the trigger mechanism, and δ∈(0,1) is a designed constant.

[0028] As a further limitation of the present invention, step 5 includes: constructing a single evaluation neural network, designing the control of step 2) and weight update strategy; the performance index based on the neural network is expressed as:

[0029]

[0030] in, To evaluate the ideal weights of the neural network, To evaluate the activation function of the neural network, N is the number of hidden neurons, is the evaluation network approximation error. is the ideal weight W c The estimated value of , the actual output of the evaluation neural network is:

[0031]

[0032] Then, we get and w * Approximate value of

[0033]

[0034] Yes V * The approximate value of (x), yes The approximate value of Yes * The approximate value of the neural network approximation error ||ε H ‖≤ε Hmax , ε Hmax is a positive scalar constant;

[0035] The weight update rate is:

[0036]

[0037] in, α>0 represents the learning rate.

[0038] The present invention adopts the above technical solution, and compared with the prior art, the beneficial effects are as follows: the present invention selects a nonlinear system with multi-source disturbances, designs a nonlinear disturbance observer to compensate for the external disturbances of the system, and at the same time expresses the problem as a two-person zero-sum game to reduce the impact of another type of disturbance on the system performance. Based on the double trigger mechanism of "first-come, same-trigger", the communication load is reduced while ensuring the optimal performance of the system. A single evaluation network is used to approximate the optimal performance index, ensuring that the computational burden is reduced and the complexity is reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 Flow chart of the control method of the present invention.

[0040] Figure 2 It is a schematic diagram of the dual-event triggered optimal control principle of a nonlinear system with multi-source disturbances according to the present invention. DETAILED DESCRIPTION

[0041] like Figure 1 A dual-event triggered optimal control method for a nonlinear system with multi-source disturbances is shown, comprising the following steps:

[0042] Step 1) establishing a nonlinear system model with multi-source disturbances, including state variables, control variables, and a dynamic model of the system;

[0043] Model a nonlinear system with multiple sources of disturbances:

[0044]

[0045] in, is the system state vector, is the control input, is the disturbance input, is a nonlinear perturbation with w∈L2(0,∞), is a smooth nonlinear dynamics; assuming that the initial state of the system is x(0) = x0, the system controller u r is the optimal control to be designed, is the estimated value of the interference d.

[0046] Step 2) define the performance indicators of the system and find the optimal control to achieve the Nash equilibrium solution;

[0047] The performance indicators of the system are defined as:

[0048]

[0049] in and is a positive definite matrix, γ is the predetermined interference attenuation level;

[0050] According to the nonlinear zero-sum differential game, there exists an optimal performance index V that satisfies the following formula: * (x):

[0051]

[0052] Then the above problem has a unique solution, and the Hamiltonian function is defined as

[0053] H(x,u r ,w)=r(x,u r ,w)+▽V T (f+gu r +kw)

[0054] in, Optimal performance index V * (x) satisfies the HJI equation

[0055]

[0056] To achieve optimal control and the worst perturbation w * :

[0057]

[0058] Therefore, the Hamilton-Jacobi-Isaac (HJI) equation can be written as

[0059]

[0060] in, f=f(x), g=g(x), k=k(x).

[0061] Step 3) construct a nonlinear disturbance observer to offset the same-channel disturbance and form a composite controller with the optimal control strategy;

[0062] According to the system in step 1), a nonlinear disturbance observer is designed to estimate the disturbance d:

[0063]

[0064] in, is the estimated value of the disturbance, z is the estimate of the internal state of the nonlinear disturbance observer, p(x) is the nonlinear vector-valued function to be designed, l(x) is the gain of the nonlinear disturbance observer, and the disturbance estimation error is defined as

[0065] Step 4) According to Figure 2 As shown in Figure 1, a dual event trigger mechanism is designed, and the system control is updated according to the "first-come, first-served" trigger mechanism. The evaluation network is designed to obtain the optimal control strategy of the system. and the worst perturbation strategy w * , optimal control The disturbance observer designed in step 3 constitutes a composite controller After the event is triggered, the discrete signal is converted into a continuous signal through a zero-order holder and finally acts on the nonlinear system.

[0066] On the one hand, for optimal control Design the following trigger conditions to determine the update of data:

[0067]

[0068] in, x(t) represents the current state, Indicates the updated status of the controller and the controlled object. is the triggering moment of the trigger mechanism, the positive definite matrix R can be decomposed into R = SS T , L is a positive real constant.

[0069] On the other hand, according to step 3), the interference estimation The trigger conditions are designed as follows:

[0070]

[0071] in, is the triggering moment of the trigger mechanism, δ∈(0,1) is a designed constant; according to the principle of “first come, same trigger”, the following dual event trigger mechanism is designed:

[0072]

[0073] Among them, t∈[t j ,t j+1 ).

[0074] The lower bound of the minimum trigger time interval is expressed as:

[0075]

[0076] in,

[0077]

[0078] K is a normal number, η is a very small normal number, The minimum interval for status event triggering. The minimum interval between triggering events is the interference estimation; in the above two cases, no matter which variable is triggered first, the minimum time interval between triggering is always greater than zero. Therefore, the system can avoid the Zeno phenomenon.

[0079] Step 5) Construct a single evaluation neural network, design control and weight update strategies, and obtain optimal control.

[0080] Construct a single-evaluation neural network and design the control of step 2) and weight update strategy; the performance index based on the neural network is expressed as:

[0081]

[0082] in, To evaluate the ideal weights of the neural network, To evaluate the activation function of the neural network, N is the number of hidden neurons, To evaluate the network approximation error. The derivative of V(x) with respect to x is:

[0083] ▽V(x)=▽φ T (x)W c +▽ε

[0084] Using the neural network approximation performance indicator, the Hamiltonian function can be changed to:

[0085]

[0086] Among them, ε H is the approximation error of the neural network, ε H Locally bounded, as the number of hidden layer units increases, ε H converges to zero; that is, there exists ε Hmax > 0, so that ||ε H ||≤ε Hmax .

[0087] set up is the ideal weight W c The estimated value of , the actual output of the evaluation neural network is:

[0088]

[0089] Then, we can get and w * Approximate value of

[0090]

[0091] Yes V * The approximate value of (x), yes The approximate value of Yes * Approximate value of .

[0092] The approximate Hamiltonian function is:

[0093]

[0094] Where e is the residual equation error; let E = 1 / 2e T e. Choose the appropriate To minimize E, the weight update law of the evaluation neural network is:

[0095]

[0096] in, α>0 represents the learning rate.

[0097] According to the weight update law, define the weight estimation error of the evaluation neural network

[0098]

[0099] Using the designed neural network and dual event trigger mechanism, the controller is designed as follows:

[0100]

[0101] The designed controller includes approximate optimal control and disturbance compensation under zero-sum game; the designed dual-event trigger mechanism can reduce the transmission update of the controller and save communication resources.

[0102] The present invention is not limited to the above-mentioned embodiments. On the basis of the technical solution disclosed in the present invention, technicians in this field can make some substitutions and deformations to some technical features therein according to the disclosed technical content without creative labor, and these substitutions and deformations are all within the protection scope of the present invention.

Claims

1. A dual-event triggered optimal control method for a nonlinear system with multi-source disturbances, characterized in that: The following steps are involved: Step 1) establishing a nonlinear system model with multi-source disturbances, including state variables, control variables, and a dynamic model of the system; Step 2) define the performance indicators of the system and find the optimal control to achieve the Nash equilibrium solution; Step 3) construct a nonlinear disturbance observer to offset the same-channel disturbance and form a composite controller with the optimal control strategy; Step 4) Design a dual event trigger mechanism and update the system control law according to the "first-come, first-served" trigger mechanism; The step 4) includes: designing a dual event trigger mechanism, and updating the system control according to the first-come-first-served trigger mechanism; in, x(t) represents the current state, Indicates the updated status of the controller and the controlled object. is the triggering moment of the trigger mechanism, and the positive definite matrix R is decomposed into R = SS T , L is a positive real constant; is the triggering moment of the trigger mechanism, δ∈(0,1) is a designed constant; γ is the predetermined interference attenuation level; is the estimated value of interference d; is the optimal control, w * is the worst disturbance; is a positive definite matrix; Step 5) Construct a single evaluation neural network, design control and weight update strategies, and obtain optimal control.

2. The dual-event triggered optimal control method for a nonlinear system with multi-source disturbances according to claim 1, characterized in that: The step 1) includes: establishing a nonlinear system model with multi-source disturbances: in, is the system state vector, is the control input, is the disturbance input, is a nonlinear perturbation with w∈L2(0,∞), is a smooth nonlinear dynamics; assuming that the system initial state is x(0) = x0, the system controller u r is the optimal control to be designed, is the estimated value of the interference d.

3. The dual-event triggered optimal control method for a nonlinear system with multi-source disturbances according to claim 2, characterized in that: The step 2) includes: defining the performance index of the system as: in and is a positive definite matrix, γ is the predetermined interference attenuation level; According to the nonlinear zero-sum differential game, the optimal performance index is obtained: in, and is a positive definite matrix, γ is the predetermined interference attenuation level; V * (x) is the optimal performance index of the system. The above optimal performance index has a unique solution. According to the Hamiltonian function, the optimal control and the worst perturbation w * : in, g=g(x), k=k(x).

4. The dual-event triggered optimal control method for a nonlinear system with multi-source disturbances according to claim 3, characterized in that: The step 3) includes: designing a nonlinear disturbance observer to estimate the disturbance d in, is the estimated value of the disturbance d, z is the estimate of the internal state of the nonlinear disturbance observer, p(x) is the nonlinear vector-valued function to be designed, l(x) is the gain of the nonlinear disturbance observer, and the disturbance estimation error is defined as 5. The dual-event triggered optimal control method for a nonlinear system with multi-source disturbances according to claim 4, characterized in that: The step 5) includes: constructing a single evaluation neural network, designing the control of step 2) and weight update strategy; the performance index based on the neural network is expressed as: in, To evaluate the ideal weights of the neural network, To evaluate the activation function of the neural network, N is the number of hidden neurons, To evaluate the network approximation error; is the ideal weight W c The estimated value of , the actual output of the evaluation neural network is: Then, we get and w * Approximate value of Yes V * The approximate value of (x), yes The approximate value of Yes * The approximate value of the neural network approximation error ||ε H ||≤ε Hmax , ε Hmax is a positive scalar constant; The weight update rate is: in, α>0 represents the learning rate.

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