Abstract
This study presents a command filtered control scheme for multi-input multi-output (MIMO) strict feedback nonlinear unmodeled dynamical systems with its applications to power systems. To deal with dynamic uncertainties, a dynamic signal is introduced, together with radial basis function neural networks (RBFNNs) to overcome the influences of the dynamic uncertainties. Command filters (CFs) are used to prevent the explosion of complexity, where the compensating signals can eliminate the effect of filter errors. Compared with single-input single-output strict feedback nonlinear systems, the method proposed in this study has more suitability. In the end, the simulation experiments are carried out by applying the developed algorithm to power systems, where the simulation results verify the efficacy of the approach proposed.
1 Introduction
In recent years, adaptive control has become a hotspot because of its strong disturbance-rejection property. Related theories, such as model reference control, robust adaptive control, and adaptive dynamic programming (Mukherjee et al., 2017; Yang et al., 2021b; Han and Liu, 2020; Yang et al., 2021d; L’Afflitto, 2018; Yang et al., 2021e), have been applied to many fields, including power systems, wind energy systems, and multi-agent systems (Li et al., 2020; Xu et al., 2018; Wu et al., 2017; Ghaffarzdeh and Mehrizi-Sani, 2020; Zou et al., 2020b; Ghosh and Kamalasadan, 2017; Namazi et al., 2018; Zou et al., 2020a). Moreover, applications of adaptive control on energy systems are also widely reported (Deese and Vermillion, 2021; Quan et al., 2020; Liu et al., 2022; Nascimento Moutinho et al., 2008; Liu et al., 2021). Among them, backstepping is a powerful tool since many energy systems can essentially be modeled as strict feedback systems, which can be analyzed through the backstepping technique.
The main idea of backstepping is to divide the whole system into a series of subsystems so that they can be analyzed individually. In this way, the control design and stability analysis can both be simplified, especially for large-scale systems (Yang et al., 2021a). Meanwhile, for unmodeled dynamical systems, if the unmodeled dynamics are ignored, the disturbance from dynamic uncertainties may result in unbounded evolution. Therefore, the dynamic uncertainties need to be paid enough attention, which is not considered in the aforementioned literatures. Zhao J. et al. (2021) presented a fuzzy adaptive control approach with an observer design for unmodeled dynamical systems. Xia et al. developed an output feedback controldesign with quantized performance for dynamic uncertainties in Xia and Zhang (2018). Wang et al. (2017)investigated nonstrict feedback systems with unmodeled dynamics and dead zones through output feedback-based control methods. Although the aforementioned results can successfully tackle dynamic uncertainties, they are not able to deal with the explosion of complexity and avoid the influences of filter errors.
In the backstepping process, the explosion of complexity often occurs because the virtual control is repeatedly differentiated. Meanwhile, the computational complexity increases significantly, which results in the presented design not being suitable for applications (Yang et al., 2020). To deal with this issue, the dynamic surface control method is proposed (Wang and Huang, 2005). The dynamic surface control method uses first-order filters, where the virtual control is replaced by the filter states in each subsystem (Yang et al., 2021c). In this way, the repeated differentiation issue can be evaded. However, filter errors are introduced simultaneously, which degrades the control precision. Thus, command filters (CFs) are developed (Farrell et al., 2009). Based on the dynamic surface control approach, CFs additionally introduce compensating signals to compensate for the loss caused by filter errors, which further improves the control accuracy compared with the dynamic surface control method. Owing to this advantage, CFs are widely applied to many systems. For example, Zhu et al. (2018)investigated a command filtered robust adaptive neural network (NN) control for strict feedback nonlinear systems with input saturation. Zhao L. et al. (2021)presented an adaptive finite-time tracking control design with CFs. The adaptive fuzzy backstepping control approach of uncertain strict feedback nonlinear systems is developed by Wang et al. (2016). However, the applications of the backstepping technique in energy systems are not taken into consideration in these works. In addition, the systems of interest in these works are single-input single-output systems, which may give conservative results. Therefore, in this study, for multi-input multi-output (MIMO) strict feedback nonlinear unmodeled dynamical systems, a command filtered control method is developed and applied to energy systems.
The contributions of this study are two-fold. First, this study designs an adaptive backstepping control scheme for MIMO strict feedback nonlinear unmodeled dynamical systems with CFs, the compensating signal design and controller design are improved such that they can get higher tracking precision. Second, this study investigates the applications of the presented CF-based adaptive backstepping control approach on power systems, and a MIMO circuit system is used in the simulation experiments to verify the effectiveness of the method developed.
The rest of this article is organized as follows. Section 2 provides the problem formulation and necessary assumptions. In Section 3, the control design is proposed. The stability analysis of the system with the presented design is carried out in Section 4. In Section 5, a voltage source converter-high voltage direct current transmission system is used to verify the efficacy of the proposed method. The conclusion is made in Section 6.
2 Problem Formulation
In this study, the circuit system under consideration is modeled aswhere , , and are the system state, output, and the control input, respectively. is a known continuous function, is an unknown continuous function, Gi ≠ 0 is a known constant, is an unknown constant vector, , is the unmeasured portion of the state, and is the unmodeled dynamics.
In this study, the following assumptions are needed.
Assumption 1Jiang and Praly (1998): The dynamic uncertainty Δi in Eq. 1 is assumed to satisfywith unknown smooth functions and . In addition, is assumed to be strictly increasing.
Assumption 2Jiang and Praly (1998): There exists an input-to-state practically stable Lyapunov function for in Eq. 1 such thatwith ω1 and ω2 belonging to class functions,, and c0 and d0 being positive constants.To deal with the dynamic uncertainty, a dynamic signal is designed with the following dynamics,where , , and c0 > 0 and r0 are constants.
Lemma 1Hardy et al. (1952): For any ξ0 > 0, one haswhere χ > 0 is a constant.
Lemma 2Jiang and Praly (1998):For the unmeasured partial state with initial state ς0, given in Assumption 2, the dynamic signal r(t) in Eq. 4, and all t ≥ 0, there is a non-negative function such thatIn addition, there is a limited time such that for all t≥ T0.With no loss of generality, choose as . Accordingly, the dynamic signal r(t) is designed asThe control objective of this study can be formulated as follows.
Consider the reference output
Xdsatisfying
are bounded. Under Assumptions 1–2, design a neuro-adaptive controller for the system (1), such that,
1. the system output X1 can track the reference Xd asymptotically, and
2. all signals in the closed-loop system keep bounded.
3 Neuro-Adaptive Controller Design
First, the tracking errors Ei, filter errors Zi, and the compensated tracking errors Λi are defined for each subsystem aswhere Ai is the filter state, A0 = Xd, Si is the virtual control, and Bi is the compensating signal.
For the subsequent design and analysis, denote with being the ideal weight vector of the RBFNNs. In addition, denote as the estimation of Θi with an estimation error .
3.1 Adaptive Backstepping Design
3.1.1 Step 1
Based on Eqs 1, 7, taking a derivative of E1 yields
For the first subsystem, the virtual control S1 is designed aswith is a positive definite matrix, and η1 > 0. To avoid repeated differentiation of the virtual control, a CF is designed aswith a positive constant τ1. To eliminate the effect of filter errors, the compensating signal is developed as
To compensate for the unknown dynamics, the adaptive law for Θ1 is presented aswhere γ1 > 0 is a constant.
3.1.2 Step
From Eqs 1, 7, differentiating Ei leads to
The virtual control design Si is developed aswhere is a positive definite matrix, and ηi > 0. To obviate repeated differentiation of the virtual control Si, a CF is given aswith a positive design parameter τi. To diminish the influences of filter errors, the compensating signal is proposed as
To deal with the parameter estimation, the adaptive law to estimate Θi is designed aswith a constant γi > 0.
3.1.3 Step n
According to Eqs 1, 7, the differentiation of En can be transformed as
The controller design is given aswith design parameters is a positive definite matrix, and ηn > 0. The compensating signal for this step is presented as
The adaptive law is developed aswhere γn > 0 is a constant.
4 Stability Analysis
In this section, we analyze the stability of the closed-loop system (Eq. 1) with the presented design of the virtual control (Eqs 9, 14), controller (Eq. 19), adaptive laws (Eqs 12, 17, 21), CFs (Eq. 10) and (15), and compensating signals (Eqs 11, 16, 20).
4.1 Step 1
Inserting Eq. 9 into Eq. 8, we obtain
From the aforementioned equation and Eq. 11, one has
The Lyapunov function is defined as . From Assumption 1, the term satisfies
For the term in the aforementioned equation, based on Lemma 1, one haswith and ɛ11 being positive constants and
Consider the term in Eq. 24, according to Lemma 2, we have
It is to be noted that ϕ12(⋅) is strictly increasing and non-negative from Assumption 1, together with the fact that , one has
From Lemma 1, we can obtainwhere and ε12 are positive constants, andwhere . From Eqs 23–29, the derivative of V1 can be expressed as
Using RBFNNs satisfieswhere . It is to be noted that is an unknown function. Then, according to the universal approximation theory, the unknown function can be approximated by the RBFNNs in the following form,with being the ideal weight vector defined aswhere and are compact regions for W1 and Y1, respectively. The corresponding approximation error is defined aswith and a positive constant ε1.
Based on the definition of Θ1, combining with Young’s inequality, we have
Inserting Eq. 33 into Eq. 31 yields
4.2 Step
Inserting the virtual control design Eq. 14 into Eq. 13, we have
On the basis of Eq. 16 and the aforementioned equation, one can obtain
To analyze the stability of the i-th subsystem through the Lyapunov theory, define the Lyapunov function for Λi and as . Based on Assumption 1, the term satisfies
Consider the term in Eq. 37, on account of Lemma 1, one haswith , ɛi1 > 0, and
For the term in (37), according to Lemma 2, we can obtain
Since ϕi2 is strictly increasing and non-negative from Assumption 1, based on the fact , one has
On the basis of Lemma 1, we can obtainwith , ɛi2 > 0, and
Using Young’s inequality, we havewhere .
From Eqs 36–42, the derivative of Vi becomes
Applying RBFNNs yieldswhere . The unknown function can be approximated in the following form:where is the ideal weight vector defined aswith and being compact regions for Wi and Yi, respectively. The approximation error is defined aswhere and ɛi > 0.
Based on the definition of Θi, using Young’s inequality, one has
Inserting Eq. 46 into Eq. 44, one can obtain
4.3 Step n
Inserting Eq. 19 into Eq. 18 results in
Based on the aforementioned equation and Eq. 20, we have
To investigate system stability through the Lyapunov theory, the Lyapunov function is defined for Λn and as . According to Assumption 1, the term satisfies
For the term in Eq. 50, one can obtainwith and εn1 being positive constants and
For the term , from Lemma 2, we have
Based on the facts that ϕn2(⋅) is strictly increasing and non-negative from Assumption 1 and , one has
From Lemma 1, we can obtainwhere and εn2 > 0 are constants and
Applying Young’s inequality, we havewith . From Eqs 48–55, the derivative of Vn becomes
Inserting Eqs 19, 51, 52 into Eq. 56 results inwhere . The unknown function can be estimated aswith being the ideal weight vector defined aswhere and are compact regions for Wn and Yn, respectively, with the approximation error defined aswith satisfying and a positive constant ɛn.
From the definition of Θn, combining with Young’s inequality, we can obtain
Applying Young’s inequality, substituting Eqs 21, 59 into Eq. 57 yields
Under Assumptions 1–2, with the virtual control (
Eqs 9,
14), the CF design (
Eqs 10,
15), the adaptive laws (
Eqs 12,
17,
21), the compensating signals (
Eqs 11,
16,
20), and the controller (
Eq. 19), the following facts hold.
1. The tracking errors will converge to the neighborhood of the origin asymptotically.
2. The boundedness of all signals in the closed-loop system (Eq. 1) can be guaranteed.
ProofDefine , applying Young’s inequality yieldsBased on Eqs 34, 47, 60, the overall Lyapunov function satisfieswhere Im is the m-dimension identity matrix,Therefore, Λi, , and are bounded. Next, we investigate the boundedness of Zi, and the dynamics of the filter error Zi can be expressed aswhereis continuous on the compact set withand R0 > 0, Ri > 0. Thus, is bounded, which derives that Zi is also bounded from Eq. 61. According to Eqs 11, 16, 20, Bi is bounded. Thus, Ei, Ai, Si, U, and Xi are all bounded, which invokes ς, Δ, and r to be bounded based on Lemma 2 and Eq. 6. In the end, we can conclude that the boundedness of all the signals in the closed-loop system can be guaranteed. This completes the proof.
5 Simulation Study
The system considered in this section is a voltage source converter-high voltage direct current transmission system with the following dynamics (Hu et al. (2020)).where L1 and L2 are the electrical inductances, and C1 and C2 are the capacitances. Applying variable transformation , , , , , and , the aforementioned equation becomes
By applying the presented control scheme, the control design is developed aswith the compensating signal design
In addition, the CF design and adaptive law design are the same as Eqs 10, 11, 15, 16, 20.
The design parameters are given as L1 = 4 mH, L2 = 8 mH, C2 = 0.1μF, , ω = 100π rad/s, , , , γ1 = 0.00085, γ2 = 0.00066, γ3 = 0.00059, η1 = 0.00005, η2 = 0.000003, η3 = 0.000004.
The RBFNNs are chosen in typical Gaussian form. To be specific, the RBFNN contains 32 nodes with the center and width being [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] and 2, respectively. RBFNN contains 128 nodes and the center and width are distributed in [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] and 2. RBFNN contains 512 nodes with the center and width selected as [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] × [−2, 2] and 2, respectively.
The simulation results are shown in Figure 1. From Figure 1, it can be observed that the output tracking objective can be achieved and the system output can track the reference output asymptotically. The dynamic uncertainties can also converge with the convergence of system states.
FIGURE 1
Output tracking performance and evolution of dynamic uncertainties.
6 Conclusion
In this study, a control approach for MIMO strict feedback nonlinear unmodeled dynamical systems with CFs is developed. The dynamic signal design introduced together with RBFNNs can efficiently prevent the effect of the dynamic uncertainties. The CFs employed in the controller design can not only prevent the explosion of complexity, but can also eliminate the effect of filter errors through the compensating signal design. Compared with single-input single-output strict feedback nonlinear systems, the approach proposed in this study is suitable for more general cases. Finally, in the simulation experiments, the presented method is applied to power systems, where the simulation results validate the effect of the scheme proposed.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
XF, LS, and YZ contributed to conception and design of this study. XF investigated the theoretical analysis for the command filter design. LS performed the simulation study with application to an energy system. YZ organized the writing of the manuscript. XF, LS, and YZ collaborated to write all the sections of the manuscript. All authors contributed to manuscript revision, and read and approved the submitted version.
Funding
This work was supported by the Youth Innovation Promotion Association of Chinese Academy of Sciences under Grant 2020134.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
power system, dynamic uncertainty, command filter, MIMO system, strict feedback nonlinear system
Citation
Feng X, Shi L and Zhang Y (2022) Intelligent Command Filter Design for Strict Feedback Unmodeled Dynamic MIMO Systems With Applications to Energy Systems. Front. Energy Res. 10:899732. doi: 10.3389/fenrg.2022.899732
Received
19 March 2022
Accepted
11 April 2022
Published
24 May 2022
Volume
10 - 2022
Edited by
Yushuai Li, University of Oslo, Norway
Reviewed by
Liqiang Tang, University of Science and Technology Beijing, China
Yongshan Zhang, University of Macau, China
Updates
Copyright
© 2022 Feng, Shi and Zhang.
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*Correspondence: Yumeng Zhang, zhangym203409@aircas.ac.cn
This article was submitted to Smart Grids, a section of the journal Frontiers in Energy Research
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.