Historical Data-Based Prescribed Performance Optimal Control for PMSM Systems with Disturbance

Authors: st bowen zhang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-16 11:35:23

Abstract: This paper proposed a prescribed performance optimal control method for permanent magnet synchronous motor (PMSM) systems with nonlinear disturbance. Firstly, a novel prescribed performance approach is proposed to constrain both the state and error of PMSM systems in order to obtain high precision control performance. Then, an adaptive dynamic programming method based on historical data is designed to solve the optimal controller. Furthermore, an identifier based on neural network is constructed to approximate the system disturbances. Finally, a simulation example is implemented to validate the efficacy of the proposed control method.

Main paper: 
 i. introduction: 
 Permanent Magnet Synchronous Motors (PMSM) are widely applied in robotics and industrial fields due to their advantages such as higher power density, higher torque-inertia ratio, and higher efficiency [1] . In the current context of energy shortages, how to effectively enhance the working efficiency of PMSM and thereby conserve more energy has emerged as a primary concern. Optimal control theory is often utilized to conserve control resources by minimizing performance index functions. [2] et al. propose a nonlinear optimal controller and observer schemes based on a θ-D approximation approach for PMSM systems, which can effectively elevate the operational efficiency of PMSM control systems. However, existing methods need repeated checks of the adaptive updating law to ensure the satisfaction of the persistent excitation (PE) condition. [3] et al. introduced an adaptive dynamic programming approach based on historical data to obtain the optimal solution for the system, effectively eliminating the need for PE signals. Currently, designing an optimal control method for PMSM systems that can dispense with the PE condition poses a challenging task.Existing optimal control strategies for PMSM systems often struggle to meet high-precision control requirements. Prescribed performance control represents an effective approach to achieving high-performance control demands by constraining system performance. [4] et al. proposed a prescribed performance optimal control method to enhance control performance. Existing prescribed performance methods typically impose constraints only on system errors. [5] et al. introduced a low-complexity prescribed performance approach that can simultaneously constrain both system states and errors. This method is equally applicable to PMSM systems. However, this method lacks discussion of system disturbance issues. Therefore, our current primary research motivation is to develop an anti-disturbance prescribed performance method for implementing optimal control of PMSM systems.Taking inspiration from the above, this paper proposed a prescribed performance optimal control method for permanent magnet synchronous motor (PMSM) systems with nolinear disturbance. The main contributions are listed below:1) A novel prescribed performance approach is proposed to constrain both the state and error of PMSM systems in order to obtain high precision control performance. System state and system error can be constrained simultaneously. 2) An adaptive dynamic programming method based on historical data is designed to solve the optimal controller. The PE conditions required by most adaptive dynamic programming methods can be eliminated. 3) An identifier based on neural network is constructed to approximate the system disturbances.In the field-oriented control, the PMSM has the following physical model in the d-q frame:     ω = -B J ω -T L J + 1.5npψ F J + Dτ J i d = -R S L d i d + n p ωi q + 1 L d u d + D d i q = -n p ωi d -R S Lq i d - npψ F Lq ω + 1 Lq u q + D q (1)where ω, n p , T L and J are the angular velocity, the number of pole pairs, load torque, and rotor inertia. i q , i d , u q and u d denote the stator current and voltage of q-axis and daxis, respectively. L d is the stator inductance and R S is the stator resistance. B and ψ F stand for the viscous frictional coefficient and the rotor flux linkage. The model ( 1) can be rewritten asEQUATIONwhere X = [ω, i d , i q ] T is the system state,f (X(t)) =    -B J ω -T L J + 1.5npψ F J -R S L d i d + n p ωi q -n p ωi d -R S Lq i d - npψ F Lq ω   , g =   0 0 1 Lq 0 0 1 L d   , u(t) = [u d , u q ] Trepresents the voltage of q-axis and d-axis,d(t) = [ Dτ J , D d , D q ] T is the system disturbance.A NN identifier is introduced to estimate the PMSM system with disturbance, with the aim of mitigating the negative impacts of disturbances on the system.The function d(t) can be approximated by the NN in the following:EQUATIONwhereW * i ∈ R n ef * 3 is ideal weight matrix, σ i ∈ R n efis the NNs activation function, i is bounded approximation error, n ef represents the number of neurons in the NNs.Let Ŵi denote the estimation, the adaptive identifier is built asEQUATIONwhere X(t) is the identifier state, X(t) = X(t) -X(t) is the identification error, k a is a positive parameter, and Wi is the estimation error of Ŵi . The updating law of Ŵi is designed as ẆiEQUATIONwhere T ∈ R n ef * n ef is the positive definite gain matrix and k b is the positive design parameter.This section considers the situation where both the tracking error and all system states are constrained simultaneously. Before a discussion, the tracking error of PMSM is defined asEQUATIONwhere X d (t) is the reference signal.The proposed control scheme needs to satisfy simultaneous constraints on both the tracking error and all system states of the PMSM system, namely, meeting the following conditions:• Prescribed performance constraints:EQUATION• Full-state constraints:EQUATIONwhere X 1 (t), X 2 (t) and X 3 (t) represent the states ω, i d , and i q of the PMSM system, respectively, and α ci indicates the maximum value that the system states can withstand.To implement the performance constraints in the previous section, the prescribed performance function is defined asEQUATIONwhere i (t) is the shifting function, is chosen asEQUATIONı i = exp( α ci 2α bi)+exp(-α ci 2α bi ) exp( α ci 2α bi))-exp(-α ci 2α bi)) where α ci and α bi is a positive design constant.To achieve the prescribed performance (7), a mapping error transformation is introduced asEQUATION), e ci = e ti α ci .Then, we introduce an error transformation related to ξ iEQUATION)where η i (t) is another transformation error. In order to guarantee η i (t) ∈ (-1, 1) for all t ≥ 0. An error transformation is given asEQUATIONThrough a series of error transformations (9)-( 12), the tracking problem of the original constrained PMSM system is transformed into a stabilization one of the following unconstrained system:EQUATIONwhereA a =    µ1αc1ν1 ω 2 1 0 0 0 µ2αc2ν2 ω 2 2 0 0 0 µ3αc3ν3 ω 2 3    and A b =     ξ1µ1 1 ω 2 1 ξ2µ2 2 ω 2 2 ξ3µ3 3 ω 2 3     .ξ i , ν i and ω i are all readily computable variables asEQUATIONEQUATIONν i = 2ξ i (exp( eci 2 ) + exp(-eci 2 )) 2 .(16)The performance index function is defined as follows:EQUATION)where R(ε(t), u(t)) = ε(t) T Ωε(t) + u(t) T u(t), ρ > 0 represents the discount factor, and Ω is a positive defined matrix.Then, the optimal value function is given byEQUATIONwhere u * (t) is the optimal controller. For simplicity, V * (ε(t), u * (t), t) is written as V * .Based on Bellman's optimality principle, the following HJB equation is derived:EQUATION)where V * ε = ∂V * /∂ε. By using the stationarity condition ∂H(ε(t), u * (t), V * ε , t)/∂u(t) = 0, we have the optimal controller, formulated asEQUATION)By inserting (20) into (19), we find that the HJB equation is able to be rewritten asEQUATION)In order to obtain the optimal controller of PMSM system, it is necessary to obtain the gradient term a of the optimal performance index, which is expected to be obtained by solving the HJB equation ( 21). However, due to unknown disturbances and inherent nonlinearity, it is impossible to solve the HJB equation by analytical method. Therefore, an ADP method combining historical data is proposed to solve the HJB equation (21) in the next section.An NN is introduced to approximateEQUATIONwhere W ci ∈ R ñc denotes the weight vector, σ ci ∈ R ñc is the basic function vector, ñc represents the number of neurons, and ε ci is a constant that denotes the approximation error. Then, the estimated ûi of optimal controller is obtained asEQUATIONSubstituting ( 22) and ( 23) into the Bellman residual error e Bi yieldse Bi = Ĥi ( i , Ĵ i , ûi , ωi ) -H * i ( i , J * i , u * i , ω * i ). Then we define the term φ i = σ ci A a (f (X(t)) + d(t) - Ẋd (t)) -A b .To mark sure that the Bellman residual error e Bi decays to 0, the gradient descent method is generally applied to the function 1 2 e T Bi e Bi by adjusting the updating law of the weight matrix Ŵci . Historical state data is introduced into the following objective function in order to avoid PE conditionsEQUATIONwhere φ pi and e Bpi for p = 1, 2, . . . N o (note: N o ≤ ñc, ñc stands for the number of neurons used in the NN) are defined as below:φ pi = σ ci A a (f (X(t p )) + d(t p ) -Ẋd (t p )) -A b e Bpi = ŴT ci φ pi + ε(t p ) T Ωε(t p ) + u(t p ) T u(t p ).The gradient descent method is applied to the objective function E. The updating law of Ŵci is obtained as follow with t p ∈ [t j , t j+1 ] denoting the index to mark the historical state timeEQUATIONwhere (1 + φ T i φ i ) -2 and (1 + φ T pi φ pi ) -2 are the normalization terms and l ci > 0 is the design parameter.Define the weight estimation error as Wci = W * ci -Ŵci , then based on (26), one hasEQUATIONwhereϕ i = φ i (1 + φ T i φ i ) 2 , ϕ pi = φ pi (1 + φ T pi φ pi ) 2 , ε Hi = ε ci A a (f (X(t p )) + d(t p ) -Ẋd (t p )) -A b , ε Hpi = ε ci A a (f (X(t)) + d(t) -Ẋd (t)) -A b .In order to verify the proposed control method, a simulation example based on PMSM is implemented. Figure .1 shows the PMSM system status tracking diagram. It can be seen that the control method proposed in this paper can effectively make the PMSM system state track the reference trajectory. This paper have proposed a prescribed performance optimal control method for permanent magnet synchronous motor (PMSM) systems with nolinear disturbance. Firstly, a novel prescribed performance approach has been proposed to constrain both the state and error of PMSM systems in order to obtain high precision control performance. Then, an adaptive dynamic programming method based on historical data has been designed to solve the optimal controller. Furthermore, an identifier based on neural network has been constructed to approximate the system disturbances. Finally, a simulation example has been implemented to validate the efficacy of the proposed control method. 
 
 Appendix:  



Reviews: review: title: title: Review Comments for Manuscript No. 194
main_review: main_review: 1.The manuscript lacks proper logical transitions between sections. The narrative should be improved to ensure that each section flows naturally into the next.

2.There are noticeable errors in the references, which need to be corrected. Additionally, there are several apparent typographical or formatting mistakes within the manuscript. The authors should thoroughly proofread the manuscript and make the necessary corrections.

3.The manuscript focuses on the Historical Data-Based Method and PP-based PMSM control. The authors should address the following points:

3.1. Is the Historical Data-Based Method an original contribution by the authors? The manuscript mentions that historical state data is introduced to avoid persistent excitation (PE) conditions. However, it should clarify whether the information provided by historical data is equivalent to or surpasses the information gained through PE. Does this method impose any specific requirements on the duration of data collection?

3.2. Apart from the Historical Data-Based Method, how does the theoretical derivation in the remaining parts of the manuscript compare to and relate to the backstepping-based nonlinear control method? The authors should clarify the distinctions and connections between these approaches.

4.The titles of all the figures in the simulation section are identical, which reduces clarity. Each figure should have a distinct and descriptive title. Additionally, Figure 3 is not cited in the manuscript and should be properly referenced.
title: title: Lack of innovation
main_review: main_review: 1. The main controbation of this paper is not clear.
2. The literature review in the Introduction surveys some previous work on the solutions and methods to address prescribed performance
optimal control, while comparation among these methods which is the most important part is neglected.
title: title: Lack of innovation in research
main_review: main_review: 1. The current work does not demonstrate sufficient novelty or advancement over existing literature to merit publication.
2. The writing quality falls short of the standards. The reviewers suggest significant revisions to improve clarity and conciseness.