非线性系统的安全分析与控制: 障碍函数方法

陈杰; 吕梓亮; 黄鑫源; 洪奕光

doi:10.16383/j.aas.c220888

非线性系统的安全分析与控制: 障碍函数方法

doi: 10.16383/j.aas.c220888

陈杰^{1, 2, 3,},
吕梓亮^{1, 2, 3,},
黄鑫源^{1, 2, 3,},
洪奕光^{1, 2, 3,}

1.
同济大学电子与信息工程学院上海 200092
2.
自主智能无人系统全国重点实验室上海 201210
3.
同济大学上海自主智能无人系统科学中心上海 201210

基金项目: 国家自然科学基金(61903027), 上海市重大专项(2021SHZDZX0100), 上海市科技成果转化和产业化项目(1951113210, 19511132101)资助

详细信息

作者简介:
陈杰：中国工程院院士, 同济大学教授, 北京理工大学自动化学院教授, 自主智能无人系统全国重点实验室教授. 1986年、1996年和2001年分别获得北京理工大学控制理论与应用专业学士学位、硕士学位和博士学位. 主要研究方向为复杂系统智能控制与优化, 多智能体协同控制. E-mail: chenjie@bit.edu.cn

吕梓亮：同济大学电子与信息工程学院博士研究生. 2017年和2020年分别获得广东工业大学学士和硕士学位. 主要研究方向为非线性系统安全分析与控制. E-mail: zlyu@tongji.edu.cn

黄鑫源：同济大学电子与信息工程学院博士研究生. 2019年获得同济大学控制科学与工程专业学士学位. 主要研究方向为安全控制和基于形式化方法的控制. E-mail: xy_huang@tongji.edu.cn

洪奕光：同济大学电子与信息工程学院教授, 自主智能无人系统全国重点实验室教授. 1987年和1990年分别获得北京大学力学系学士和硕士学位, 1993年获得中科院系统所博士学位. 主要研究方向为复杂系统, 控制理论, 人工智能. 本文通信作者. E-mail: yghong@tongji.edu.cn

计量
- 文章访问数: 5709
- HTML全文浏览量: 789
- PDF下载量: 1126
- 被引次数: 0
出版历程
- 收稿日期: 2022-11-13
- 网络出版日期: 2023-02-06
- 刊出日期: 2023-03-20

Safety Analysis and Safety-critical Control of Nonlinear Systems: Barrier Function Approach

CHEN Jie^{1, 2, 3
,},
LYU Zi-Liang^{1, 2, 3
,},
HUANG Xin-Yuan^{1, 2, 3
,},
HONG Yi-Guang^{1, 2, 3
,}

1.
College of Electronic and Information Engineering, Tongji University, Shanghai 200092
2.
National Key Laboratory of Autonomous Intelligent Unmanned Systems, Shanghai 201210
3.
Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai 201210

Funds: Supported by National Natural Science Foundation of China (61903027), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100), and Shanghai Municipal Commission of Science and Technology (1951113210, 19511132101)

More Information

Author Bio:
CHEN Jie　Academician of Chinese Academy of Engineering, professor at Tongji University, School of Automation, Beijing Institute of Technology, and National Key Laboratory of Autonomous Intelligent Unmanned Systems. He received his bachelor, master, and Ph.D. degrees in control science and application from Beijing Institute of Technology in 1986, 1996, and 2001, respectively. His research interest covers intelligent control and optimization of complex systems, and cooperative control of multi-agent systems

LYU Zi-Liang　Ph.D. candidate at the College of Electronic and Information Engineering, Tongji University. He received his bachelor and master degrees from Guangdong University of Technology in 2017 and 2020, respectively. His research interest covers safety analysis and safety-critical control of nonlinear systems

HUANG Xin-Yuan　Ph.D. candidate at the College of Electronic and Information Engineering, Tongji University. He received his bachelor degree in control science and engineering from Tongji University in 2019. His research interest covers safety control and formal methods-based control

HONG Yi-Guang　Professor at College of Electronic and Information Engineering, Tongji University and National Key Laboratory of Autonomous Intelligent Unmanned Systems. He received his bachelor and master degrees from Department of Mechanics of Peking University in 1987 and 1990, respectively, and his Ph.D. degree from Institute of Systems Science of Chinese Academy of Sciences in 1993. His research interest covers complex systems, control theory, and artificial intelligence. Corresponding author of this paper

摘要

摘要: 近年来, 非线性系统的安全分析与控制已成为控制领域中的热门研究方向, 而障碍函数则是该方向的一种重要工具. 基于障碍函数的安全分析与控制方法具有计算效率高、鲁棒性强等优点. 本文首先从多个角度介绍了基于障碍函数的非线性系统安全性分析的理论成果, 并进一步综述了障碍函数方法在非线性系统安全控制中的最新进展. 最后, 简要地介绍了当前基于障碍函数的安全分析与控制理论中一系列尚未解决的问题, 并指出了未来可能发展的一些研究方向.
- 安全性分析 /
- 安全控制 /
- 障碍函数 /
- 非线性系统
Abstract: Safety analysis and safety-critical control of nonlinear systems have been important issues in the control society. The barrier function is a promising tool for the safety analysis and safety-critical control of nonlinear systems. This methodology has the advantages of high computation efficiency and strong robustness. In this paper, we first review the theoretical results on barrier function based safety analysis from different viewpoints, and then summarize some recent advances of the barrier function theory in the safety-critical control of nonlinear systems. Finally, we outline a series of open questions in the barrier function theory and point out some possible research directions.
- Safety analysis /
- safety-critical control /
- barrier functions /
- nonlinear systems
注释:

1) ¹ 扩展$ K $类函数是传统$ K $类函数^[2]在安全分析与控制上的推广. 对于任意函数$ \alpha: {\bf{R}}\rightarrow {\bf{R}} $, 我们说它是一个扩展$ K $类函数, 如果该函数在$ {\bf{R}} $上连续、严格递增并满足$ \alpha(0)=0 $; 特别地, 如果$ \alpha(s) $分别随着$s $趋于$ +\infty $和$ -\infty $而趋于$ +\infty $和$ -\infty $, 则该函数是一个扩展$ K_\infty $类函数.

2) ² 这里的相对阶是把$ h(x) $当做某种意义下的输出.

3) ³ 在控制领域, 组合系统有时也被称作互联系统(Interconnected system).

HTML全文

参考文献(95)

[1]	Isidori A. Nonlinear Control Systems. London: Springer-Verlag, 1995.
[2]	Khalil H K. Nonlinear Systems. Upper Saddle River: Prentice Hall, 2002.
[3]	洪奕光, 程代展. 非线性系统的分析与控制. 北京: 科学出版社, 2005. Hong Yi-Guang, Cheng Dai-Zhan. Analysis and Design of Nonlinear Systems. Beijing: Science Press, 2005.
[4]	Aubin J P. Viability Theory. Boston: Birkhäuser, 1991.
[5]	Nagumo M. Über die lage der integralkurven gewöhnlicher differentialgleichungen. Proceedings of the Physico-Mathematical Society of Japan, 1942, 24(6), 551–559.
[6]	Clarke E M, Grumberg O, Peleg D. Model Checking. Cambridge: MIT Press, 1999.
[7]	Baier C, Katoen J P. Principles of Model Checking. Cambridge: MIT Press, 2008.
[8]	Tomlin C, Pappas G J, Sastry S. Conflict resolution for air traffic management: A study in multiagent hybrid systems. IEEE Transactions on Automatic Control, 1998, 43(4): 509–521. doi: 10.1109/9.664154
[9]	Mitchell I M, Bayen A M, Tomlin C J. A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Transactions on Automatic Control, 2005, 50(7): 947–957. doi: 10.1109/TAC.2005.851439
[10]	Gao Y, Johansson K H, Xie L. Computing probabilistic controlled invariant sets. IEEE Transactions on Automatic Control, 2021, 66(7): 3138–3151. doi: 10.1109/TAC.2020.3018438
[11]	Prajna S. Optimization-based Methods for Nonlinear and Hybrid Systems Verification [Ph.D. dissertation], California Institute of Technology, USA, 2005
[12]	Ames A D, Grizzle J W, Tabuada P. Control barrier function based quadratic programs with application to adaptive cruise control. In: Proceedings of the 53rd IEEE Conference on Decision and Control (CDC). Los Angeles, CA, USA: IEEE, 2014. 6271–6278
[13]	Xu X, Tabuada P, Grizzle J W, Ames A D. Robustness of control barrier functions for safety-critical control. IFAC-PapersOnLine, 2015, 48(27): 54–61. doi: 10.1016/j.ifacol.2015.11.152
[14]	Ames A D, Xu X, Grizzle J W, Tabuada P. Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 2017, 62(8), 3861–3876. doi: 10.1109/TAC.2016.2638961
[15]	Annichini A, Bouajjani A, Sighireanu M. TReX: A tool for reachability analysis of the complex systems. In: Proceedings of the International Conference on Computer Aided Verification (CAV). Paris, France: Springer, 2001. 368−372
[16]	Mitchell I M. A toolbox of level set methods [Online], available: https://www.cs.ubc.ca/~mitchell/ToolboxLS, September 3, 2019
[17]	Fisac J F, Chen M, Tomlin C J, Sastry S S. Reach-avoid problems with time-varying dynamics, targets and constraints. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control (HSCC). New York, USA: ACM, 2015. 11−20
[18]	Herbert S L, Chen M, Han S, Bansal S, Fisac J F, Tomlin C J. FaSTrack: A modular framework for fast and guaranteed safe motion planning. In: Proceedings of the 56th IEEE Conference on Decision and Control (CDC). Melbourne, Australia: IEEE, 2017. 1517−1522
[19]	Bajcsy A, Bansal S, Bronstein E, Tolani V, Tomlin C J. An efficient reachability-based framework for provably safe autonomous navigation in unknown environments. In: Proceedings of the 58th IEEE Conference on Decision and Control (CDC). Nice, France: IEEE, 2019. 1758−1765
[20]	Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004.
[21]	Glavaski S, Papachristodoulou A, Ariyur K. Safety verification of controlled advanced life support system using barrier certificates. In: Proceedings of the International Workshop on Hybrid Systems: Computation and Control (HSCC). Zurich, Switzerland: Springer, 2005. 306−321
[22]	Wang L, Ames A D, Egerstedt M. Safety barrier certificates for collisions-free multirobot systems. IEEE Transactions on Robotics, 2017, 33(3): 661–674. doi: 10.1109/TRO.2017.2659727
[23]	Xu X, Grizzle J W, Tabuada P, Ames A D. Correctness guarantees for the composition of lane keeping and adaptive cruise control. IEEE Transactions on Automation Science and Engineering, 2017, 15(3): 1216–1229.
[24]	Prajna S, Jadbabaie A. Safety verification of hybrid systems using barrier certificates. In: Proceedings of the International Workshop on Hybrid Systems: Computation and Control (HSCC). Philadelphia, PA, USA: Springer, 2004. 477−492
[25]	Prajna S, Jadbabaie A, Pappas G J. A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Transactions on Automatic Control, 2007, 52(8): 1415–1428. doi: 10.1109/TAC.2007.902736
[26]	Prajna S, Rantzer A. Convex programs for temporal verification of nonlinear dynamical systems. SIAM Journal on Control and Optimization, 2007, 46(3): 999–1021. doi: 10.1137/050645178
[27]	Wongpiromsarn T, Topcu U, Lamperski A. Automata theory meets barrier certificates: Temporal logic verification of nonlinear systems. IEEE Transactions on Automatic Control, 2015, 61(11): 3344–3355.
[28]	Wisniewski R, Sloth C. Converse barrier certificate theorems. IEEE Transactions on Automatic Control, 2015, 61(5): 1356–1361.
[29]	Ratschan S. Converse theorems for safety and barrier certificates. IEEE Transactions on Automatic Control, 2018, 63(8): 2628–2632. doi: 10.1109/TAC.2018.2792325
[30]	Liu J. Converse barrier functions via lyapunov functions. IEEE Transactions on Automatic Control, 2021, 67(1): 497–503.
[31]	Maghenem M A, Sanfelice R G. On the converse safety problem for differential inclusions: Solutions, regularity, and time-varying barrier functions. IEEE Transactions on Automatic Control, 2022: 1–16 doi: 10.1109/TAC.2022.3148226, to be published.
[32]	Jankovic M. Robust control barrier functions for constrained stabilization of nonlinear systems. Automatica, 2021, 96(x): 359–367.
[33]	Ngo K B, Mahony R, Jiang Z P. Integrator backstepping using barrier functions for systems with multiple state constraints. In: Proceedings of the 44th IEEE Conference on Decision and Control (CDC). Seville, Spain: IEEE, 2005. 8306−8312
[34]	Tee K P, Ge S S, Tay E H. Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica, 2009, 45(4): 918–927. doi: 10.1016/j.automatica.2008.11.017
[35]	Wang X, Lyu Z, Dong Y. A unified approach for safety critical control problem via output regulation theory and barrier function. In: Proceedings of the 41st Chinese Control Conference (CCC). Hefei, China: IEEE, 2022. 833−837
[36]	Kong H, He F, Song X, Hung W N, Gu M. Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In: Proceedings of the International Conference on Computer Aided Verification (CAV). Saint Petersburg, Russia: Springer, 2013. 242−257
[37]	Lin Y, Sontag E D, Wang Y. A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization, 1996, 34(1): 124–160. doi: 10.1137/S0363012993259981
[38]	Li A, Wang L, Pierpaoli P, Egerstedt M. Formally correct composition of coordinated behaviors using control barrier certificates. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Madrid, Spain: IEEE, 2018. 3723−3729
[39]	Glotfelter P, Cortés J, Egerstedt M. Nonsmooth barrier functions with applications to multi-robot systems. IEEE Control Systems Letters, 2017, 1(2): 310–315. doi: 10.1109/LCSYS.2017.2710943
[40]	Ahmadi M, Singletary A, Burdick J W, Ames A D. Safe policy synthesis in multi-agent pomdps via discrete-time barrier functions. In: Proceedings of the 58th IEEE Conference on Decision and Control (CDC). Nice, France: IEEE, 2019. 4797−4803
[41]	Sontag E D, Wang Y. On characterizations of the input-to-state stability property. Systems & Control Letters, 1995, 24(5): 351–359.
[42]	Romdlony M Z, Jayawardhana B. On the new notion of input-to-state safety. In: Proceedings of the 55th IEEE Conference on Decision and Control (CDC). Las Vegas, NV, USA: IEEE, 2016. 6403−6409
[43]	Romdlony M Z, Jayawardhana B. Robustness analysis of systems' safety through a new notion of input-to-state safety. International Journal of Robust and Nonlinear Control, 2019, 29(7): 2125–2136. doi: 10.1002/rnc.4482
[44]	Kolathaya S, Ames A D. Input-to-state safety with control barrier functions. IEEE Control Systems Letters, 2018, 3(1): 108–113.
[45]	Lyu Z, Xu X, Hong Y. Small-gain theorem for safety verification of interconnected systems. Automatica, 2022, 139: Article No. 110178
[46]	Krstic M. Inverse optimal safety filters [Online], available: https://arxiv.org/abs/2112.08225, February 20, 2023
[47]	Taylor A J, Ong P, Cortés J, Ames A D. Safety-critical event triggered control via input-to-state safe barrier functions. IEEE Control Systems Letters, 2020, 5(3): 749–754.
[48]	Long L, Wang J. Safety-critical dynamic event-triggered control of nonlinear systems. Systems & Control Letters, 2022, 162: Article No. 105176
[49]	Li X, Yin X, Li S. Cooperative event triggered control for multi-robot systems with collision avoidance. In: Proceedings of the 40th Chinese Control Conference (CCC). Shanghai, China: IEEE, 2021. 5460−5465
[50]	Cortez W S, Dimarogonas D V. Correct-by-design control barrier functions for Euler-Lagrange systems with input constraints. In: Proceedings of the American Control Conference (ACC). Denver, CO, USA: IEEE, 2020. 950−955
[51]	Shi L, Singh S K. Decentralized adaptive controller design for large-scale systems with higher order interconnections. IEEE Transactions on Automatic Control, 1992, 37(8): 1106–1118. doi: 10.1109/9.151092
[52]	Lyu Z, Xu X, Hong Y. Small-gain theorem for safety verification under high-relative-degree constraints [Online], available: https://arxiv.org/abs/2204.04376, February 20, 2023
[53]	Nguyen Q, Sreenath K. Exponential control barrier functions for enforcing high relative-degree safety-critical constraints. In: Proceedings of the American Control Conference (ACC). Boston, MA, USA: IEEE, 2016. 322−328
[54]	Xu X. Constrained control of input-output linearizable systems using control sharing barrier functions. Automatica, 2018, 87(x): 195–201.
[55]	Xiao W, Belta C. High order control barrier functions. IEEE Transactions on Automatic Control, 2021: 1–8 doi: 10.1109/TAC.2021.3105491, to be published.
[56]	Tan X, Cortez W S, Dimarogonas D V. High-order barrier functions: Robustness, safety, and performance-critical control. IEEE Transactions on Automatic Control, 2021, 67(6): 3021–3028.
[57]	Parrilo P A. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization [Ph.D. dissertation], California Institute of Technology, USA, 2000
[58]	Prajna S, Papachristodoulou A, Parrilo P A. Introducing SOSTOOLS: A general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control (CDC). Las Vegas, NV, USA: IEEE, 2002. 741–746
[59]	Papachristodoulou A, Prajna S. A tutorial on sum of squares techniques for systems analysis. In: Proceedings of the American Control Conference (ACC). Portland, OR, USA: IEEE, 2005. 2686−2700
[60]	Seiler P. SOSOPT: A toolbox for polynomial optimization [Online], available: https://arxiv.org/abs/1308.1889, February 20, 2023
[61]	Willems J C. Dissipative dynamical systems part i: General theory. Archive for Rational Mechanics and Analysis, 1972, 45(5): 321–351. doi: 10.1007/BF00276493
[62]	Desoer C A, Vidyasagar M. Feedback Systems: Input-Output Properties. New York: Academic Press, 1975.
[63]	Hill D J. A generalization of the small-gain theorem for nonlinear feedback systems. Automatica, 1991, 27(6): 1043–1045. doi: 10.1016/0005-1098(91)90140-W
[64]	Jiang Z P, Teel A R, Praly L. Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems, 1994, 7(2): 95–120. doi: 10.1007/BF01211469
[65]	Teel A R. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control, 1996, 41(9): 1256–1270. doi: 10.1109/9.536496
[66]	Huang X, Lyu Z, Hong Y. Safety verification of large-scale nonlinear systems: A cyclic-small-gain approach. In: Proceedings of the 17th International Conference on Control & Automation (ICCA). Naples, Italy: IEEE, 2022. 459−462
[67]	Jagtap P, Swikir A, Zamani M. Compositional construction of control barrier functions for interconnected control systems. In: Proceedings of the 23rd International Conference on Hybrid Systems: Computation and Control (HSCC). New York, USA: ACM, 2000. 1−11
[68]	Yin X, Li S. Recent advances on formal methods for safety and security of cyber-physical systems. Control Theory and Technology, 2020, 18(4): 459–461. doi: 10.1007/s11768-020-00008-w
[69]	Belta C, Yordanov B, Gol E A. Formal Methods for Discrete-Time Dynamical Systems. Berlin: Springer, 2017.
[70]	Tian D, Fang H, Yang Q, Wei Y. Decentralized motion planning for multiagent collaboration under coupled LTL task specifications. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2022, 52(6): 3602–3611. doi: 10.1109/TSMC.2021.3073105
[71]	Wieland P, Allgöwer F. Constructive safety using control barrier functions. IFAC Proceedings Volumes, 2007, 40(12): 462–467. doi: 10.3182/20070822-3-ZA-2920.00076
[72]	Artstein Z. Stabilization with relaxed controls. Nonlinear Analysis: Theory, Methods & Applications, 1983, 7(11): 1163–1173.
[73]	Sontag E D. A ùniversal' construction of Artstein's theorem on nonlinear stabilization. Systems & Sontrol Letters, 1989, 13(2): 117–123.
[74]	Hsu S C, Xu X, Ames A D. Control barrier function based quadratic programs with application to bipedal robotic walking. In: Proceedings of the American Control Conference (ACC). Chicago, IL, USA: IEEE, 2015. 4542−4548
[75]	Krstic M, Kokotovic P V, Kanellakopoulos I. Nonlinear and Adaptive Control Design. New York: Wiley and Sons, 1995.
[76]	Romdlony M Z, Jayawardhana B. Uniting control Lyapunov and control barrier functions. In: Proceedings of the 53rd IEEE Conference on Decision and Control (CDC). Los Angeles, CA, USA: IEEE, 2014. 2293−2298
[77]	Wu C, Fang H, Yang Q, Zeng X, Wei Y, Chen J. Distributed cooperative control of redundant mobile manipulators with safety constraints. IEEE Transactions on Cybernetics, 2022: 1–13 doi: 10.1109/TCYB.2021.3104044, to be published.
[78]	Raman V, Donzé A, Maasoumy M, Murray R M. Model predictive control with signal temporal logic specifications. In: Proceedings of the 53rd IEEE Conference on Decision and Control (CDC). Los Angeles, CA, USA: IEEE, 2014. 81−87
[79]	Wu S, Liu T, Jiang Z P. Continuous safety control of mobile robots in cluttered environments. IEEE Robotics and Automation Letters, 2022, 7(3): 8012–8019. doi: 10.1109/LRA.2022.3187492
[80]	Nguyen Q, Sreenath K. Safety-critical control for dynamical bipedal walking with precise footstep placement. IFAC-PapersOnLine, 2015, 48(27): 147–154. doi: 10.1016/j.ifacol.2015.11.167
[81]	Khan M, Zafar M, Chatterjee A. Barrier functions in cascaded controller: Safe quadrotor control. In: Proceedings of the American Control Conference (ACC). Denver, CO, USA: IEEE, 2020. 1737−1742
[82]	Wang L, Theodorou E A, Egerstedt M. Safe learning of quadrotor dynamics using barrier certificates. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA). Brisbane, Australia: IEEE, 2018. 2460−2465
[83]	Wu G, Sreenath K. Safety-critical control of a 3D quadrotor with range-limited sensing. In: Proceedings of the ASME Dynamic Systems and Control Conference (DSCC). Minneapolis, MN, USA: ASME, 2016. 1−11
[84]	Xiao W, Mehdipour N, Collin A, Bin-Nun A Y, Frazzoli E, Tebbens R D, et al. Rule-based optimal control for autonomous driving. In: Proceedings of the ACM/IEEE 12th International Conference on Cyber-Physical Systems (ICCPS). New York, NY, USA: ACM/IEEE, 2021. 143−154
[85]	Glotfelter P, Cortés J, Egerstedt M. Boolean composability of constraints and control synthesis for multi-robot systems via nonsmooth control barrier functions. In: Proceedings of the IEEE Conference on Control Technology and Applications (CCTA). Copenhagen, Denmark: IEEE, 2018. 897−902
[86]	Glotfelter P, Cortés J, Egerstedt M. A nonsmooth approach to controller synthesis for boolean specifications. IEEE Transactions on Automatic Control, 2020, 66(11): 5160–5174.
[87]	Lindemann L, Dimarogonas D V. Control barrier functions for signal temporal logic tasks. IEEE Control Systems Letters, 2018, 3(1): 96–101.
[88]	Manna Z, Pnueli A. Temporal Verification of Reactive Systems: Safety. New York: Springer Verlag, 1995.
[89]	Tian D, Fang H, Yang Q, Guo Z, Cui J, Liang W, Wu Y. Two-phase motion planning under signal temporal logic specifications in partially unknown environments. IEEE Transactions on Industrial Electronics, 2022: 1–10 doi: 10.1109/TIE.2022.3203752, to be published.
[90]	Li X, Serlin Z, Yang G, Belta C. A formal methods approach to interpretable reinforcement learning for robotic planning. Science Robotics, 2019, 4(37): 1–15.
[91]	Gundana D, Kress-Gazit H. Event-based signal temporal logic synthesis for single and multi-robot tasks. IEEE Robotics and Automation Letters, 2021, 6(2): 3687–3694. doi: 10.1109/LRA.2021.3064220
[92]	Maler O, Nickovic D. Monitoring temporal properties of continuous signals. Formal Techniques, Modelling and Analysis of Timed and Fault-Tolerant Systems, 2004: 152−166
[93]	Huang X, Li L, Chen J. Multi-agent system motion planning under temporal logic specifications and control barrier function. Control Theory and Technology, 2020, 18(3): 269–278. doi: 10.1007/s11768-020-0110-6
[94]	Jagtap P, Soudjani S, Zamani M. Formal synthesis of stochastic systems via control barrier certificates. IEEE Transactions on Automatic Control, 2020, 66(7): 3097–3110.
[95]	Srinivasan M, Coogan S. Control of mobile robots using barrier functions under temporal logic specifications. IEEE Transactions on Robotics, 2020, 37(2): 363–374.