线性切换系统极小测度的估计
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摘要
切换系统是一类重要的混杂系统，它由若干个子系统和一个协调子系统切换的切换法则构成。由于切换信号的引入，使得切换系统不仅可以保持各个子系统部分的动态性能，还表现出子系统所不具有的复杂动态。因此，切换系统模型可用来精确描述各种复杂的非线性动态过程。在过去的几十年中，切换系统的研究以其广泛的应用背景和重要的理论研究意义，成为控制领域的研究热点之一。
对一个动态系统来说，系统的最大Lyapunov指数是系统状态轨线最差的收敛速率，刻画了系统的稳定性能。众所周知，连续时间线性系统的最大Lyapunov指数等于系统的谱坐标（所有特征值的最大实部）；离散时间线性系统的最大Lyapunov指数等于系统谱半径的自然对数。在过去的近二十年，线性切换系统最大Lyapunov指数的研究也取得了长足进展。离散时间线性切换系统的最大Lyapunov指数等于系统的（联合/广义）谱半径的自然对数。其中，系统的（联合/广义）谱半径和子系统矩阵对所有范数的极小共同诱导范数相等。不同于离散时间的线性切换系统，连续时间的线性切换系统没有对应于线性系统的结论。它的最大Lyapunov指数一般不等于系统的谱坐标。最近，Sun证明了连续时间的线性切换系统的最大Lyapunov指数和所有子系统的极小（共同）测度相等。关于最大Lyapunov指数的计算问题，对离散时间的线性切换系统，其实就是（联合/广义）谱半径的数值计算问题。目前常用的方法有椭圆范数法、半正定过程法、平方和多项式技术法等。对连续时间的线性切换系统，其实就是极小（共同）测度的计算问题。目前关于该问题的研究只集中在一些具有特殊结构的线性切换系统，很少有对一般线性切换系统进行讨论。
本学位论文将针对极小（共同）测度的计算问题展开讨论，给出逼近连续时间线性切换系统的最大Lyapunov指数的数值算法。本学位论文的主要研究问题和主要贡献如下：
1.对临界稳定的线性切换系统，存在一个由Barabanov范数定义的球面，使得从该球面上出发的最不稳定轨线一直保持在球面上。特别地，对二阶和三阶的含有两个子系统的线性切换系统，最不稳定轨线是一条周期闭轨。从而，1是系统在半个周期的状态转移矩阵的一个特征值。当线性切换系统稳定时，经过两次切换的状态转移矩阵的所有实特征根都大于1，当系统不稳定时，经过两次切换的状态转移矩阵存在小于1的实特征根。基于此几何性质，设计一个数值算法可任意精确逼近系统的极小测度。但是当系统阶数高于三阶时，系统临界稳定时的最不稳定轨线不一定是闭的周期轨线，算法失效。
2.对一般的线性切换系统，根据矩阵（集合）测度和极小测度的定义，矩阵（集合）测度依赖于选择的向量范数，极小测度是矩阵测度在所有向量范数下的极小值。平方和多项式能够任意逼近任一向量范数，因此可用平方和多项式去逼近系统在任一范数下的测度，其可转化为一组线性矩阵不等式。考虑到Rn上所有的齐次平方和多项式集合是一个凸锥，因此可在凸集内搜索平方和多项式逼近系统矩阵集合的共同极小范数。基于该极小范数，可得到系统极小测度的一个近似值。通过这一过程，将线性切换系统的极小测度的逼近问题转化为一组线性矩阵不等式的广义特征值问题，可通过MATLAB的LMI工具箱求解。该算法的精度依赖于线性切换系统的特征值，特征值越集中，离虚轴越远，精度越高。当特征值比较分散，或离虚轴较近时，该算法得到的估计值比较保守。
3.考虑到基于平方和多项式技术的逼近算法的精度依赖于系统矩阵，当系统的特征值比较靠近虚轴时，该算法比较保守。对此类系统，需要考虑新的算法估计系统的极小测度。由于稳定的线性切换系统一定存在一个凸的分段二次型的Lyapunov函数，而凸的分段二次型Lyapunov函数可诱导一个分段二次型向量范数。在此分段二次型范数下，容易计算对应的矩阵（集合）的测度。从而可将线性切换系统的极小测度的估计问题弱化为线性切换系统在所有分段二次型范数下的极小测度问题。结合S–procedure引理，该问题可转化为（一组）双线性矩阵不等式的广义特征值问题，其可以用网格法搜索最优值。针对含有两个元的分段二次型范数，通过变量替换可减少双线性矩阵不等式中的参数，有效提高了求解最优值的效率和精度。
对于上述三种算法，都通过数值仿真验证了算法的有效性。
最后，总结全文，并对一些有待研究的问题进行展望。
Switched systems are a class of hybrid systems consisting of a family of subsystemsand a switching law that orchestrates the switching among the subsystems. For switchedsystems, switching plays a nontrivial role which not only makes the systems remain somedynamical performances of the subsystems, but also introduces strong complexities. Inthis manner, switched systems are used to model a variety of complex nonlinear systems.Due to the practical importance and theoretical challenging, switched systems have beenattracting much attention in the last several decades.
For a dynamical system, the largest divergence rate is the worst-case convergence rateof the state trajectories, which characterizes the system performance. For linear systems,as we all know, the largest divergence rate is equal to spectral abscissa (maximal realpart of the eigenvalues) in continuous time and the logarithm of the spectral radius indiscrete time. During the last two decades, a considerable progress has been made in thestudy of the largest divergence rate of switched linear systems. For discrete-time switchedlinear systems, the rate coincides with the logarithm of the (joint/generalized) spectralradius. The spectral radius is known to be equal to least common norm of the subsystemmatrices among all the matrix norm. For continuous-time switched linear systems, there isno counterpart of the spectral radius, as spectral abscissa in linear case. Recently, Sun hasproved that the largest divergence rate is equal to the least common matrix measure of thesubsystem matrices. From the computational point of view, several approaches have beendeveloped to approximate the spectral radius, for instance, the ellipsoid norm approach,the semi-defnite lifting procedure, the sum-of-squares technique and so on. While, forthe continuous-time case, a few algorithms have been introduced to approximate the ratefor switched linear systems with special structures. However, there are few computationalalgorithms for general systems.
The purpose of this dissertation is to develop algorithms that approximate the leastcommon matrices measure of switched linear systems. The main contents and contribu-tions of this dissertation are summarized in the following:
1. For marginally stable switched linear systems, there exists a unit sphere defned by Barabanov norm, such that the worst-case trajectory will stay on the sphere ifthe initial state is on it. Especially, the worst-case trajectory is a closed periodicorbit for2nd-or3rd-order switched linear system with two subsystems. Thus,1is an eigenvalue of the transition matrix in half-period. Furthermore, all thereal eigenvalues of the transition matrix in half-period are greater than1if thesystem is stable, while there exists a real eigenvalue is less than1if the system isinstable. From this geometrical property, a bisection algorithm could be designedto approximate the least common matrices measure for switched linear systems inany accuracy. However, this algorithm is no longer applicable to the systems whentheir orders are higher than three for the worst-case trajectories that might not beclosed periodic orbits. Thus the algorithm is no longer applicable.
2. For general switched linear systems, from the defnitions of matrix (set) measureand least measure, the matrix (set) measure is dependent on the vector norm andthe least measure is the least value of the matrix (set) measure for all possible vectornorms. As any vector norm could be bounded by a homogeneous sum-of-squarespolynomial, it will allow us to approximate the matrix (set) measure by sum-of-squares polynomials. This problem can be transformed into a series of linear matrixinequalities. Considering the set of homogeneous polynomials in Rnis a convex cone,thus we could search a sum-of-squares polynomial to approximate the least commonnorm. Based on this polynomial, a approximation of the least measure is derived.Through this procedure, the approximation problem is transformed into generaleigenvalue problem, which could be solved via the LMI toolbox in MATLAB. Theaccuracy of this algorithm is dependent on the system matrices. If the eigenvaluesof the system are far away from the imaginary axis and concentrated, the algorithmhas a high precision. Otherwise, the algorithm is relatively conservative.
3. As the algorithm based on the sum-of-squares technique is conservative when theeigenvalues are closed to the imaginary axis. Thus, a new algorithm is needed forthis case. For any stable switched linear system, there exists a convex piecewiseLyapunov function. This Lyapunov function can induce a piecewise quadratic vectornorm. It is easy to calculate the matrix (set) measure under this piecewise quadraticvector norm. Thus the least measure of all possible vector norm could be relaxedas the least measure for all possible piecewise quadratic vector norms. By the S- procedure lemma, the relaxed problem can be transformed into a general eigenvalueproblem of bilinear matrix inequalities which could be solved by grid method. Forthe case of the two-term piecewise quadratic vector norm, the parameters could bereduced by half through a variable substitution. It efectively improves the efciencyand accuracy of the algorithm.
For the aforementioned three algorithms, there are numerical examples to illustratethe efectiveness.
Finally, the conclusion and the prospects of future research are given at the end ofthis dissertation.
对一个动态系统来说，系统的最大Lyapunov指数是系统状态轨线最差的收敛速率，刻画了系统的稳定性能。众所周知，连续时间线性系统的最大Lyapunov指数等于系统的谱坐标（所有特征值的最大实部）；离散时间线性系统的最大Lyapunov指数等于系统谱半径的自然对数。在过去的近二十年，线性切换系统最大Lyapunov指数的研究也取得了长足进展。离散时间线性切换系统的最大Lyapunov指数等于系统的（联合/广义）谱半径的自然对数。其中，系统的（联合/广义）谱半径和子系统矩阵对所有范数的极小共同诱导范数相等。不同于离散时间的线性切换系统，连续时间的线性切换系统没有对应于线性系统的结论。它的最大Lyapunov指数一般不等于系统的谱坐标。最近，Sun证明了连续时间的线性切换系统的最大Lyapunov指数和所有子系统的极小（共同）测度相等。关于最大Lyapunov指数的计算问题，对离散时间的线性切换系统，其实就是（联合/广义）谱半径的数值计算问题。目前常用的方法有椭圆范数法、半正定过程法、平方和多项式技术法等。对连续时间的线性切换系统，其实就是极小（共同）测度的计算问题。目前关于该问题的研究只集中在一些具有特殊结构的线性切换系统，很少有对一般线性切换系统进行讨论。
本学位论文将针对极小（共同）测度的计算问题展开讨论，给出逼近连续时间线性切换系统的最大Lyapunov指数的数值算法。本学位论文的主要研究问题和主要贡献如下：
1.对临界稳定的线性切换系统，存在一个由Barabanov范数定义的球面，使得从该球面上出发的最不稳定轨线一直保持在球面上。特别地，对二阶和三阶的含有两个子系统的线性切换系统，最不稳定轨线是一条周期闭轨。从而，1是系统在半个周期的状态转移矩阵的一个特征值。当线性切换系统稳定时，经过两次切换的状态转移矩阵的所有实特征根都大于1，当系统不稳定时，经过两次切换的状态转移矩阵存在小于1的实特征根。基于此几何性质，设计一个数值算法可任意精确逼近系统的极小测度。但是当系统阶数高于三阶时，系统临界稳定时的最不稳定轨线不一定是闭的周期轨线，算法失效。
2.对一般的线性切换系统，根据矩阵（集合）测度和极小测度的定义，矩阵（集合）测度依赖于选择的向量范数，极小测度是矩阵测度在所有向量范数下的极小值。平方和多项式能够任意逼近任一向量范数，因此可用平方和多项式去逼近系统在任一范数下的测度，其可转化为一组线性矩阵不等式。考虑到Rn上所有的齐次平方和多项式集合是一个凸锥，因此可在凸集内搜索平方和多项式逼近系统矩阵集合的共同极小范数。基于该极小范数，可得到系统极小测度的一个近似值。通过这一过程，将线性切换系统的极小测度的逼近问题转化为一组线性矩阵不等式的广义特征值问题，可通过MATLAB的LMI工具箱求解。该算法的精度依赖于线性切换系统的特征值，特征值越集中，离虚轴越远，精度越高。当特征值比较分散，或离虚轴较近时，该算法得到的估计值比较保守。
3.考虑到基于平方和多项式技术的逼近算法的精度依赖于系统矩阵，当系统的特征值比较靠近虚轴时，该算法比较保守。对此类系统，需要考虑新的算法估计系统的极小测度。由于稳定的线性切换系统一定存在一个凸的分段二次型的Lyapunov函数，而凸的分段二次型Lyapunov函数可诱导一个分段二次型向量范数。在此分段二次型范数下，容易计算对应的矩阵（集合）的测度。从而可将线性切换系统的极小测度的估计问题弱化为线性切换系统在所有分段二次型范数下的极小测度问题。结合S–procedure引理，该问题可转化为（一组）双线性矩阵不等式的广义特征值问题，其可以用网格法搜索最优值。针对含有两个元的分段二次型范数，通过变量替换可减少双线性矩阵不等式中的参数，有效提高了求解最优值的效率和精度。
对于上述三种算法，都通过数值仿真验证了算法的有效性。
最后，总结全文，并对一些有待研究的问题进行展望。
Switched systems are a class of hybrid systems consisting of a family of subsystemsand a switching law that orchestrates the switching among the subsystems. For switchedsystems, switching plays a nontrivial role which not only makes the systems remain somedynamical performances of the subsystems, but also introduces strong complexities. Inthis manner, switched systems are used to model a variety of complex nonlinear systems.Due to the practical importance and theoretical challenging, switched systems have beenattracting much attention in the last several decades.
For a dynamical system, the largest divergence rate is the worst-case convergence rateof the state trajectories, which characterizes the system performance. For linear systems,as we all know, the largest divergence rate is equal to spectral abscissa (maximal realpart of the eigenvalues) in continuous time and the logarithm of the spectral radius indiscrete time. During the last two decades, a considerable progress has been made in thestudy of the largest divergence rate of switched linear systems. For discrete-time switchedlinear systems, the rate coincides with the logarithm of the (joint/generalized) spectralradius. The spectral radius is known to be equal to least common norm of the subsystemmatrices among all the matrix norm. For continuous-time switched linear systems, there isno counterpart of the spectral radius, as spectral abscissa in linear case. Recently, Sun hasproved that the largest divergence rate is equal to the least common matrix measure of thesubsystem matrices. From the computational point of view, several approaches have beendeveloped to approximate the spectral radius, for instance, the ellipsoid norm approach,the semi-defnite lifting procedure, the sum-of-squares technique and so on. While, forthe continuous-time case, a few algorithms have been introduced to approximate the ratefor switched linear systems with special structures. However, there are few computationalalgorithms for general systems.
The purpose of this dissertation is to develop algorithms that approximate the leastcommon matrices measure of switched linear systems. The main contents and contribu-tions of this dissertation are summarized in the following:
1. For marginally stable switched linear systems, there exists a unit sphere defned by Barabanov norm, such that the worst-case trajectory will stay on the sphere ifthe initial state is on it. Especially, the worst-case trajectory is a closed periodicorbit for2nd-or3rd-order switched linear system with two subsystems. Thus,1is an eigenvalue of the transition matrix in half-period. Furthermore, all thereal eigenvalues of the transition matrix in half-period are greater than1if thesystem is stable, while there exists a real eigenvalue is less than1if the system isinstable. From this geometrical property, a bisection algorithm could be designedto approximate the least common matrices measure for switched linear systems inany accuracy. However, this algorithm is no longer applicable to the systems whentheir orders are higher than three for the worst-case trajectories that might not beclosed periodic orbits. Thus the algorithm is no longer applicable.
2. For general switched linear systems, from the defnitions of matrix (set) measureand least measure, the matrix (set) measure is dependent on the vector norm andthe least measure is the least value of the matrix (set) measure for all possible vectornorms. As any vector norm could be bounded by a homogeneous sum-of-squarespolynomial, it will allow us to approximate the matrix (set) measure by sum-of-squares polynomials. This problem can be transformed into a series of linear matrixinequalities. Considering the set of homogeneous polynomials in Rnis a convex cone,thus we could search a sum-of-squares polynomial to approximate the least commonnorm. Based on this polynomial, a approximation of the least measure is derived.Through this procedure, the approximation problem is transformed into generaleigenvalue problem, which could be solved via the LMI toolbox in MATLAB. Theaccuracy of this algorithm is dependent on the system matrices. If the eigenvaluesof the system are far away from the imaginary axis and concentrated, the algorithmhas a high precision. Otherwise, the algorithm is relatively conservative.
3. As the algorithm based on the sum-of-squares technique is conservative when theeigenvalues are closed to the imaginary axis. Thus, a new algorithm is needed forthis case. For any stable switched linear system, there exists a convex piecewiseLyapunov function. This Lyapunov function can induce a piecewise quadratic vectornorm. It is easy to calculate the matrix (set) measure under this piecewise quadraticvector norm. Thus the least measure of all possible vector norm could be relaxedas the least measure for all possible piecewise quadratic vector norms. By the S- procedure lemma, the relaxed problem can be transformed into a general eigenvalueproblem of bilinear matrix inequalities which could be solved by grid method. Forthe case of the two-term piecewise quadratic vector norm, the parameters could bereduced by half through a variable substitution. It efectively improves the efciencyand accuracy of the algorithm.
For the aforementioned three algorithms, there are numerical examples to illustratethe efectiveness.
Finally, the conclusion and the prospects of future research are given at the end ofthis dissertation.
引文
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