鲁棒滑模反步控制法及其在减摇鳍中的应用
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摘要
无论是自然界还是人类社会，不确定性是一个普遍存在的因素，但对工程技术而言，这种不确定性的存在一般是不允许的。不确定性的存在使实际对象与所用数学模型之间有很大差别，如果仅仅根据标称系统来设计控制器，会使大量工程控制问题无法获得满意结果，甚至严重影响系统性能。因此，在非线性控制系统的设计中系统的鲁棒性成为一个必须研究的问题，使设计出来的控制器对各种各样的不确定性具有良好的鲁棒性，这是实际控制问题的需要。
本文对一类可以代表相当广泛的非线性系统的模型给出了鲁棒反步法和动态鲁棒反步法的设计过程，并对一类二阶级联系统给出了两种控制算法。论文给出的控制方法能够克服系统的不确定性，并有很好的鲁棒性。论文所研究的非线性系统是一类广泛的半严格反馈系统，系统的不确定性包括有界的参数不确定性，有界的输入增益的不确定性以及未知但有界的非线性方程和外部干扰。
在控制方法设计中主要应用了反步法(Backstepping)和滑模控制法。反步法用来递归设计每一步中的虚拟控制器，而每一步中的虚拟控制器以及最后的实际控制器利用滑模控制实现，对系统的不确定性进行补偿。论文给出的滑模反步法适用的系统范围更广泛，利用控制参数来决定每一步中的循迹误差。而且每一步中的虚拟控制能够补偿前一个子系统传递来的误差。
在此基础上，论文给出了动态鲁棒反步法的设计过程。这种方法的主要优点在于利用过滤器避免了每一步中对虚拟控制器的求导，过滤器的应用大大简化了控制器设计步骤。而且转换后的动态系统只含有较小的不确定性，因此每一步中的控制器只需要一个较小的增益来保证子系统的稳定。
其次，对于海浪干扰下的船舶，论文利用能量等分法给出了建立随机海浪模型的方法，提出了减摇和未减摇船舶横摇运动的计算方法，并通过仿真与实船数据的比较证明了论文提出方法的可行性。
最后建立了船舶运动的非线性数学模型，利用论文给出的滑模反步法设计了鲁棒控制器的有效性，仿真结果证明了横摇控制器完全可以实现系统设
哈尔滨工程大学博士学位论文
计的要求，而且也证明了这一控制算法具有较好的鲁棒性。
关键词:不确定非线性系统;鲁棒控制;反步法(BackstePPing);滑模控制;
减摇鳍
The uncertainties exist everywhere in either the nature or the society. But the existence of the uncertainty is not permitted in the engineering. There is great difference between the actual systems and the mathematic models because of the uncertainties. So the engineering problem can not be solved satisfyingly if it is designed only based on the nominal system. All the above make the robustness of the nonlinear system a necessary problem to be researched. It's essential to make designed controller have excellent robustness to all kinds of uncertainties.In this paper the design of robust controllers for a broad class of nonlinear system is given, and two control algorithms are also discussed for a group of cascaded seconded order systems. The approach can overcome the uncertainties and has good robustness. The class of nonlinear systems considered is referred to as a semi-strict feedback system and includes parametric uncertainties, input gain uncertainties and the unknown but bounded nonlinear function and disturbance.The key components of the control design are the use of the backstepping design procedure, sliding mode control. The backstepping method is used to recursively design virtual controllers for each step in the procedure. Each virtual controller and the resulting actual controller are designed using sliding mode control to compensate for model uncertainty. The control method can be used in a broader class of nonlinear systems, and the tracking error is determined by the control parameters. The virtual controllers can compensate the error transformed from the previous subsystems.Based on the above the dynamical robust backstepping method is presented. The main feature of this approach is it alleviates the need to take derivatives of the virtual control at each step of the design. This has two positive effects. The first is that a reduction in the complexity in the controller design. The second is a drastically smaller amount of uncertainty results in the transformed dynamical systems. Therefore the controller at each step will require less gain to stabilize the
subsystem.Then the method of energy in part is given with which we can get the model of random wave, and predict the roll motion of unstabilized ship and stabilized ship using the wave model. The simulation of a ship stabilizer system is also made. The comparison of the simulation with real ship indicates that the method can be used in the calculation and prediction of roll motion of a stabilized ship in random wave.Finally the nonlinear model of ship is given. A robust controller is designed based on the sliding mode backstepping method presented in this paper to test the effectiveness of these approaches. The simulation achieves desired results and the controllers meet the requirement and have excellent robustness.
本文对一类可以代表相当广泛的非线性系统的模型给出了鲁棒反步法和动态鲁棒反步法的设计过程，并对一类二阶级联系统给出了两种控制算法。论文给出的控制方法能够克服系统的不确定性，并有很好的鲁棒性。论文所研究的非线性系统是一类广泛的半严格反馈系统，系统的不确定性包括有界的参数不确定性，有界的输入增益的不确定性以及未知但有界的非线性方程和外部干扰。
在控制方法设计中主要应用了反步法(Backstepping)和滑模控制法。反步法用来递归设计每一步中的虚拟控制器，而每一步中的虚拟控制器以及最后的实际控制器利用滑模控制实现，对系统的不确定性进行补偿。论文给出的滑模反步法适用的系统范围更广泛，利用控制参数来决定每一步中的循迹误差。而且每一步中的虚拟控制能够补偿前一个子系统传递来的误差。
在此基础上，论文给出了动态鲁棒反步法的设计过程。这种方法的主要优点在于利用过滤器避免了每一步中对虚拟控制器的求导，过滤器的应用大大简化了控制器设计步骤。而且转换后的动态系统只含有较小的不确定性，因此每一步中的控制器只需要一个较小的增益来保证子系统的稳定。
其次，对于海浪干扰下的船舶，论文利用能量等分法给出了建立随机海浪模型的方法，提出了减摇和未减摇船舶横摇运动的计算方法，并通过仿真与实船数据的比较证明了论文提出方法的可行性。
最后建立了船舶运动的非线性数学模型，利用论文给出的滑模反步法设计了鲁棒控制器的有效性，仿真结果证明了横摇控制器完全可以实现系统设
哈尔滨工程大学博士学位论文
计的要求，而且也证明了这一控制算法具有较好的鲁棒性。
关键词:不确定非线性系统;鲁棒控制;反步法(BackstePPing);滑模控制;
减摇鳍
The uncertainties exist everywhere in either the nature or the society. But the existence of the uncertainty is not permitted in the engineering. There is great difference between the actual systems and the mathematic models because of the uncertainties. So the engineering problem can not be solved satisfyingly if it is designed only based on the nominal system. All the above make the robustness of the nonlinear system a necessary problem to be researched. It's essential to make designed controller have excellent robustness to all kinds of uncertainties.In this paper the design of robust controllers for a broad class of nonlinear system is given, and two control algorithms are also discussed for a group of cascaded seconded order systems. The approach can overcome the uncertainties and has good robustness. The class of nonlinear systems considered is referred to as a semi-strict feedback system and includes parametric uncertainties, input gain uncertainties and the unknown but bounded nonlinear function and disturbance.The key components of the control design are the use of the backstepping design procedure, sliding mode control. The backstepping method is used to recursively design virtual controllers for each step in the procedure. Each virtual controller and the resulting actual controller are designed using sliding mode control to compensate for model uncertainty. The control method can be used in a broader class of nonlinear systems, and the tracking error is determined by the control parameters. The virtual controllers can compensate the error transformed from the previous subsystems.Based on the above the dynamical robust backstepping method is presented. The main feature of this approach is it alleviates the need to take derivatives of the virtual control at each step of the design. This has two positive effects. The first is that a reduction in the complexity in the controller design. The second is a drastically smaller amount of uncertainty results in the transformed dynamical systems. Therefore the controller at each step will require less gain to stabilize the
subsystem.Then the method of energy in part is given with which we can get the model of random wave, and predict the roll motion of unstabilized ship and stabilized ship using the wave model. The simulation of a ship stabilizer system is also made. The comparison of the simulation with real ship indicates that the method can be used in the calculation and prediction of roll motion of a stabilized ship in random wave.Finally the nonlinear model of ship is given. A robust controller is designed based on the sliding mode backstepping method presented in this paper to test the effectiveness of these approaches. The simulation achieves desired results and the controllers meet the requirement and have excellent robustness.
引文
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[5] 慕春隶，梅生伟，申铁龙．非线性系统鲁棒控制理论的一些新进展[J]．控制理论与应用，2001，18(1)：1-6
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[2] 褚健，俞立，苏宏业编著．鲁棒控制理论及应用．杭州：浙江大学出版社，2000
[3] 冯纯伯，费树岷编著．非线性控制系统分析与设计．北京：电子工业出版社，1998
[4] P Kokotovic, M Arcak. Constructive nonlinear control:A historical perspective. Automatica, 2001, 37(5):637-662P
[5] 慕春隶，梅生伟，申铁龙．非线性系统鲁棒控制理论的一些新进展[J]．控制理论与应用，2001，18(1)：1-6
[6] Black H S. Stabilized feedback amplifiers. U.S., Patent No.2, 102, 671, 1927
[7] Nyquist H. Regeneration theory. Bell Syst. Tech. J., 1932, 11:126-147
[8] Bode H W. Network analysis and feedback amplifier design. Princeton, NJ:Van Nostrand, 1945
[9] Horowitz I. Synthesis of feedback system. New York:Academic Press, 1963
[10] Cruz J B Ed. System sensitivity analysis. Stroudsburg, PA:Dowden, Hutchinson and Ross, 1973
[11] Davison E J. The output control of linear time invariant multivariable systems with unmeasurable arbitrary disturbances. IEEE Transaction on Automatic Control, 1971, AC-17:621-630
[12] Pearson J B, Staats P W Jr. Robust controllers for linear regulators. IEEE Transaction on Automatic Control, 1974, AC-19:231-234
[13] MacFarlane A G J and Postlethwaite I. The generalized Nyquist stability criterion and multivariable root loci. International Journal of Control, 1977, 25:81-127
[14] Youla D C. On the factorization of rational matrices. IRE Transaction on Information Theory, 1961, IT-7(3): 172-189
[15] Youla D C, Jabr H and Bongiorno J J Jr. Modern Wiener-Hopf design of optimal controllers-Part II: the multivariable case. IEEE Transactions on Automatic Control, 1976, AC-21: 319-338
[16] Zames G. Functional analysis applied to nonlinear feedback systems. IEEE Transaction on Circuit Theory, 1963, CT-10: 392-404
[17] Kalman R E. When is a linear control system optimal?. Transaction ASME, Ser. D, 1964, 86:51-60
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