基于半定规划的多项式非线性系统镇定控制研究
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摘要
多项式非线性系统在实际应用中广泛存在。尤其在运动控制系统、机电系统、过程控制、生物系统、电子电路等系统中,很多控制问题均可建模,转化或近似成多项式非线性系统。因此,如何分析及综合多项式非线性系统对非线性理论发展以及工程应用是件非常有意义的工作。近年来,通过数值计算的方法对多项式非线性系统进行研究取得了相当多的关注,尤其在半定规划方法和平方和分解方法上,并取得了一系列成果。尽管这些方法在多项式非线性系统稳定性分析上取得重大进展,控制器综合依然是个难题,需要进一步研究和探索。
本论文研究了时不变多项式非线性系统控制器综合的数值求解方法。论文主要研究工作和成果包括以下几个方面:
1.研究了多项式正定性验证的数值求解方法。我们给出三种多项式分解算法用于正定多项式验证问题和正定约束下多项式构造问题。在多项式分解算法基础上,我们开发了软件包(命名为PolyDecomp),采用矩阵不等式求解器求解所转化的问题。PolyDecomp用于解决多项式正定性验证问题,在Matlab环境下运行。许多多项式优化问题可以转化成多项式正定验证问题,可由PolyDecomp有效求解。
2.提出基于Lyapunov稳定定理的时不变多项式非线性系统镇定控制器设计方法。我们通过构造Lyapunov候选函数(CLF)并将之转成能被PolyDecomp求解的正定多项式验证问题来实现状态反馈的镇定控制器综合。由多项式全基构造的控制器在相对高阶系统中具有很多项单项式,阻碍了其实际应用性。为解决这一问题,我们提出简约控制器结构设计方法。在控制性能优化方面,我们考虑了最速控制和最小方差控制。对局部镇定控制,我们提出多Lyapunov径向膨胀控制器设计方法,在保证性能的前提下扩大闭环系统的吸引域。
3.拓展了时不变多项式非线性系统控制器设计方法。我们提出基于消去法的控制器设计方法,能用于较复杂的多项式非线性系统。消去法关键在于构造合适的CLF,以保证在控制器作用为0的子空间中闭环系统仍能稳定。我们还给出了基于特征根负定配置的二维多项式非线性系统镇定控制,给后推法控制器设计带来潜在价值。此外,我们还给出了Chua's系统和Chen系统混沌同步控制器设计。
4.提出空间刚体全局姿态控制方法。从控制角度综评方向余弦、欧拉角、四元数和Rodrigues/修正Rodrigues参数几种常用姿态描述动态方程,指出各种方法的优缺点,及相互转化。给出基于Rodrigues/修正Rodrigues模型的姿态镇定控制器设计,并取得全局收敛。我们将姿态角调整控制转化为镇定控制问题,并能实现任何姿态角的调整。在仿真中给出SAPPHIRE卫星的姿态镇定控制及姿态调整控制,取得良好效果。
Polynomial nonlinear systems appear widely in practical applications. In particular, many control problems in motion control systems, mechatronic systems, process control, biological systems, electric circuits and so on, can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Therefore, how to analyze and synthesize polynomial nonlinear systems is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial nonlinear systems in numerical approach, especially by using semidefinite programming and the sum of squares decomposition. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches; however, control synthesis still remains a stubborn problem and needs further development.
This thesis explores numerical solution for control synthesis of time-invariant polynomial nonlinear systems. The main research content and achievements are showed as follows:
1. Numerical solution for polynomial positivity validation (PPV) is explored. Three algorithms for polynomial decomposition are proposed to check PPV problem of a given polynomial or to construct a polynomial with positive constraints. Furthermore, the software named PolyDecomp is developed based on polynomial decomposition algorithms and matrix inequalities solvers. PolyDecomp is applied for PPV and implemented in Matlab environment. Many polynomial optimization problems can be formulated into PPV problems and can be further solved by PolyDecomp.
2. Stabilizing control scheme for time-invariant polynomial nonlinear systems based on Lyapunov theorem is studied. State-feedback stabilizing control of polynomial nonlinear systems is presented by constructing control Lyapunov functions (CLF) and transforming corresponding problems into the PPV type which can be solved by PolyDecomp. Controller constructed by full monomial base will be in numerous terms for relatively high-order systems and obstruct its application in practice. To overcome this problem, reduced-form controller is introduced. Control performance optimization is considered in terms of maximum convergence rate control and minimum variance control. Local stabilizing control is given in terms of designing controller to expand the domain of attraction with guaranteed performance.
3. Control scheme for polynomial nonlinear systems is expanded. A nonlinearity-cancelling control scheme is proposed, which is adaptable for more complex polynomial nonlinear systems. The key technique in this scheme is to construct an appropriate CLF to guarantee stability of the closed loop systems in some subspaces with zero control inputs. Stabilizing control scheme for 2-D polynomial nonlinear systems is considered based on negative eigenvalue placement, which has potential value for back-stepping control development. Chaos synchronization control is also developed and demonstrated in Chua's system and Lorenz system.
4. Global attitude control for space rigid body is addressed. Several common description methods for rotational kinematics are discussed in control point, such as direction cosine matrix, Euler angles, quaternions and Rodrigues/Modified-Rodrigues parameters. Transformation relationships, advantages and disadvantages among those methods are discussed. Attitude control based on Rodrigues/Modified-Rodrigues parameters description is given and achieves global convergence. Attitude regulator is established by converting the corresponding problem to stabilizing control problem in Rodrigues/Modified-Rodrigues parameters. The proposed global attitude control scheme achieves effective performance as illustrated in simulation of the SAPPHIRE satellite.
本论文研究了时不变多项式非线性系统控制器综合的数值求解方法。论文主要研究工作和成果包括以下几个方面:
1.研究了多项式正定性验证的数值求解方法。我们给出三种多项式分解算法用于正定多项式验证问题和正定约束下多项式构造问题。在多项式分解算法基础上,我们开发了软件包(命名为PolyDecomp),采用矩阵不等式求解器求解所转化的问题。PolyDecomp用于解决多项式正定性验证问题,在Matlab环境下运行。许多多项式优化问题可以转化成多项式正定验证问题,可由PolyDecomp有效求解。
2.提出基于Lyapunov稳定定理的时不变多项式非线性系统镇定控制器设计方法。我们通过构造Lyapunov候选函数(CLF)并将之转成能被PolyDecomp求解的正定多项式验证问题来实现状态反馈的镇定控制器综合。由多项式全基构造的控制器在相对高阶系统中具有很多项单项式,阻碍了其实际应用性。为解决这一问题,我们提出简约控制器结构设计方法。在控制性能优化方面,我们考虑了最速控制和最小方差控制。对局部镇定控制,我们提出多Lyapunov径向膨胀控制器设计方法,在保证性能的前提下扩大闭环系统的吸引域。
3.拓展了时不变多项式非线性系统控制器设计方法。我们提出基于消去法的控制器设计方法,能用于较复杂的多项式非线性系统。消去法关键在于构造合适的CLF,以保证在控制器作用为0的子空间中闭环系统仍能稳定。我们还给出了基于特征根负定配置的二维多项式非线性系统镇定控制,给后推法控制器设计带来潜在价值。此外,我们还给出了Chua's系统和Chen系统混沌同步控制器设计。
4.提出空间刚体全局姿态控制方法。从控制角度综评方向余弦、欧拉角、四元数和Rodrigues/修正Rodrigues参数几种常用姿态描述动态方程,指出各种方法的优缺点,及相互转化。给出基于Rodrigues/修正Rodrigues模型的姿态镇定控制器设计,并取得全局收敛。我们将姿态角调整控制转化为镇定控制问题,并能实现任何姿态角的调整。在仿真中给出SAPPHIRE卫星的姿态镇定控制及姿态调整控制,取得良好效果。
Polynomial nonlinear systems appear widely in practical applications. In particular, many control problems in motion control systems, mechatronic systems, process control, biological systems, electric circuits and so on, can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Therefore, how to analyze and synthesize polynomial nonlinear systems is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial nonlinear systems in numerical approach, especially by using semidefinite programming and the sum of squares decomposition. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches; however, control synthesis still remains a stubborn problem and needs further development.
This thesis explores numerical solution for control synthesis of time-invariant polynomial nonlinear systems. The main research content and achievements are showed as follows:
1. Numerical solution for polynomial positivity validation (PPV) is explored. Three algorithms for polynomial decomposition are proposed to check PPV problem of a given polynomial or to construct a polynomial with positive constraints. Furthermore, the software named PolyDecomp is developed based on polynomial decomposition algorithms and matrix inequalities solvers. PolyDecomp is applied for PPV and implemented in Matlab environment. Many polynomial optimization problems can be formulated into PPV problems and can be further solved by PolyDecomp.
2. Stabilizing control scheme for time-invariant polynomial nonlinear systems based on Lyapunov theorem is studied. State-feedback stabilizing control of polynomial nonlinear systems is presented by constructing control Lyapunov functions (CLF) and transforming corresponding problems into the PPV type which can be solved by PolyDecomp. Controller constructed by full monomial base will be in numerous terms for relatively high-order systems and obstruct its application in practice. To overcome this problem, reduced-form controller is introduced. Control performance optimization is considered in terms of maximum convergence rate control and minimum variance control. Local stabilizing control is given in terms of designing controller to expand the domain of attraction with guaranteed performance.
3. Control scheme for polynomial nonlinear systems is expanded. A nonlinearity-cancelling control scheme is proposed, which is adaptable for more complex polynomial nonlinear systems. The key technique in this scheme is to construct an appropriate CLF to guarantee stability of the closed loop systems in some subspaces with zero control inputs. Stabilizing control scheme for 2-D polynomial nonlinear systems is considered based on negative eigenvalue placement, which has potential value for back-stepping control development. Chaos synchronization control is also developed and demonstrated in Chua's system and Lorenz system.
4. Global attitude control for space rigid body is addressed. Several common description methods for rotational kinematics are discussed in control point, such as direction cosine matrix, Euler angles, quaternions and Rodrigues/Modified-Rodrigues parameters. Transformation relationships, advantages and disadvantages among those methods are discussed. Attitude control based on Rodrigues/Modified-Rodrigues parameters description is given and achieves global convergence. Attitude regulator is established by converting the corresponding problem to stabilizing control problem in Rodrigues/Modified-Rodrigues parameters. The proposed global attitude control scheme achieves effective performance as illustrated in simulation of the SAPPHIRE satellite.
引文
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