Zhendong SUN

Key Lab of Systems&Control,Academy of Mathematics&Systems Science,Chinese Academy of Sciences,Beijing 100190,China;

School of Mathematical Science,University of Chinese Academy of Sciences,Beijing 100049,China

Recent advances on analysis and design of switched linear systems

Zhendong SUN

Key Lab of Systems&Control,Academy of Mathematics&Systems Science,Chinese Academy of Sciences,Beijing 100190,China;

School of Mathematical Science,University of Chinese Academy of Sciences,Beijing 100049,China

A switched linear system is a special hybrid system that consists of a set of linear continuous-time/discrete-time subsystems and a rule that orchestrates the switching among them.The two-level(execution-supervision)structure makes the switched system theoretically interesting and practically attractive.Under active investigation for more than three decades,huge progress has been made in understanding the dynamical behavior of switched systems.In particular,it has been well recognized that,a switched linear system could produce highly nonlinear&complex behaviors,for instance,controllability might not imply stabilizability,and stabilizability might not imply the existence of convex(control-)Lyapunov function.Through properly utilizing the rich dynamical behavior,it is possible to improve the system’s performance(controllability,stabilizability,adaptability,optimality,among many others)by means of systematic control/switching design.Meanwhile,many powerful tools,such as the common Lyapunov method and the logic-based switching design,have been developed for analysis and control of switched systems,which are also widely applied to other system frameworks like multi-agent systems and cyber-physical systems.

In this short note,we report recent advances on performance analysis and optimization of switched linear systems.The material is mainly based on my group’s research work,and is organized in a highly prejudiced manner.

A key index to measure the worst case asymptotic performance is the so-called largest Lyapunov exponent,which is also known as the largest divergence rate or worst-case divergence rate.It is well known that the computation of the value is notoriously challenging for nonlinear systems.In our earlier work,we found a connection between the rate and an algebraic index of a matrix set for switched linear systems under arbitrary switching.To be more precisely,we extend the notion of spectral abscissa for a matrix to that for a set of matrices,and prove the equity between the convergence rate and the abscissa of the matrix set composed by the subsystems’matrices.Furthermore,the abscissa is shown to be equal to the least common measure of the matrices.It is clear that,the Lyapunov exponent is defined by means of the dynamical solutions,while the least common measure is a static index that could be approximated by means of searching properly among certain sets of norm-induced matrix measures.Based on this connection,we propose several approaches that estimate the least measure of a matrix set.The first is the piecewisequadratic-induced-norm approach,which approximate a vector norm by means of a class of piecewise quadratic functions.The least common measure could be computed by solving an optimization problem with non-convex constraints.By introducing the S-procedure,the constraints could be expressed in bilinear matrix inequalities form,where BMI computational softwares might apply.The second is the sum-of-square(SOS)approach that approximate a vector norm by SOS polynomials.Through the square matrical representation method,a computational algorithm is developed that transforms the computing of the least common measure into a series of linear matrical inequalities which could be solved as a generalized eigenvalue problem.The algorithm provides two-sided bounds of the leastcommon measure.The third is the algebraic transformation approach,which utilizes(non-square)coordinate change to transform the original system into a new one with possibly larger system dimension.An advantage of this approach is that the ?1common measure of the transformed system could be used to approximate the least measure of the original system,thus there is no need to search for proper vector norms.Nevertheless,the main load is moved to finding proper coordinate transformations.For this,we propose several algorithms that update the coordinate transformations in an iterative manner,and the resultant sequence of ?1measures is convergent whose limit is used to approximate the spectral abscissa.

The above performance analysis applies not only to switched linear systems,but also a wider framework that we term as multi-linear dynamical systems.A multi-linear system is a dynamical system which evolves along several linear directions and their combinations.Classical multi-linear frameworks include differential/difference inclusions,linear parameter-varying systems,switched linear systems,piecewise linear systems,and jump linear systems.As a historical fact,these systems were investigated almost independently by different communities with various backgrounds and motivations.Our performance analysis could provide tight estimate of the worst-case divergence rates for linear differential inclusions and polytopic-type linear parameter-varying systems,and also provide upperbound estimates for piecewise linear systems and jump linear systems.In particular,the classical open problem of absolute stability could be seen as a special case.Indeed,any Lurie system with known sector-bounds admits two extreme subsystems each along one boundary of the sector,and absolute stability is equivalent to a negative least common measure of the switched system composed of the two subsystems.Therefore,it allows to verify absolute stability via computing the least common measure of the corresponding switched linear system.

While switching among multiple models is proven to be effective in improving the asymptotic performance of complex systems,there is also the possibility of improving the transient performance of switched systems via proper control/switching design.The motivations are two folds:first,while the asymptotic performance could be improved by means of mode switching,switching among subsystems might degenerate the transient performance of the overall systems,which should be avoided whenever possible;second yet more importantly,we believe that the twolevel structure of switched systems also provides a strong potential in improving the transient performance while keeping a satisfactory asymptotic behavior.In classical control theory,many indexes for characterizing the transient dynamics,for instance,overshoot,settling time,gain/phase margin,etc.,were defined and discussed in the context of graphical analysis such as Nyquist plot and Bode plot.Some indexes are straightforwardly/easily extended to the context of switched linear systems,others might be harder to extend due to the inapplicability of the frequencydomain method to switched systems.Nevertheless,for switched linear systems,performance improvement via control/switching is a very challenging topic,and we focus on the nonovershoot stabilization problem that finds proper control/switching laws to achieve low overshoot when nonovershooting is unavoidable.The nonovershoot problem seeks control/switching laws to achieve output nonovershooting,which means that the output is signinvariant and monotonely contractive.It turns out that nonovershooting requires either at least one subsystem admits relative degree one or a control-independent matrix set is negative semi-definite,which implies that the design of control law and of the switching law is separable.When taking stabilization into account,the nonovershoot stabilization problem is much involved as the separation principle is no longer applicable.For systems with specific structures or properties,it is still possible to interactively design control/switching laws that solve the problem.In particular,we propose two control/switching design schemes for two types of switched linear systems,respectively.While the design ideas are simple and heuristic,they demonstrate some potential benefits by appropriately switching among subsystems to achieve better stability/nonovershoot performance.In addition,for discrete-time forced-free switched linear systems,some properties of minimum overshooting were revealed both for the finite time and infinite time horizons.By properly modifying the path-wise state-feedback switching law developed in our earlier work,we obtain new switching law that achieves exponential stability with sub-minimum system overshoot.A computational procedure is also proposed to calculate the system overshoot.

Note that the above progress is preliminary,and both the asymptotic performance estimation problem and the transient performance design problem are still largely open.For the former problem,while the developed algorithms could possibly approximate the worst-case divergence rate in theory,how to estimate the accuracy of the approximation is a key issue to be addressed.Another issue is that the computational complexity is quite high,which makes the algorithms applicable only to low-dimensional systems(fifth-order or less).For the latter problem,a general interactive control/switching design scheme is still missing,not to mention how to tradeoff between overshooting and speed of response to achieve satisfactory transient dynamics.

As a final remark,we stress that the switching mechanism plays a central rule in the analysis and design of switched linear systems,yet it is not well understood in contrast with the counterpart of continuous dynamics.In the new era that autonomous systems are more and more demanding,the simple hierarchical architecture of switched linear systems provide a potential to achieve high autonomy by incorporating supervised switching into the dynamics.On the one hand,when facing with significant control/decision uncertainties,accurate estimate of worst-case performance is needed.On the other hand,intelligent decision-making algorithms should be developed to generate control actions so that a satisfactory performance is maintained,where the transient performance design problem could possibly provide a partial yet powerful solution.

E-mail:Zhendong.Sun@amss.ac.cn.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Zhendong SUNis currently a Researcher.His research interests include analysis and design of hybrid dynamical systems,control and optimization of M/NEMS systems,and data analysis of public health models.E-mail:Zhendong.Sun@amss.ac.cn.