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| On the exponential stability and instability analyses of switched second- and higher-order linear systems via a novel application of differential inequalities: part 1 (theory) |
| Y.V.Venkatesh1,2 |
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| (1 (Formerly) Division of Electrical Sciences, Indian Institute of Science, Sir C. V. Raman Road, Bangalore, Karnataka 560012, India
2 Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge Crescent, Singapore 119260, Singapore) |
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| 摘要: |
| Differential inequalities generated in an extended Lyapunov framework are employed in the stability and instability analyses
of a class of switched continuous-time second- and higher order linear systems with an arbitrary number of switching
matrices. The exponential stability and instability (ESI) conditions so obtained involve the supremum and infimum of ratios
of certain quadratic forms of the matrices, leading to global time-averages of their activity intervals. Further, motivated by
linear switching system examples of (i) instability with stable matrices and (ii) stability with unstable matrices (found in the
literature primarily for second-order systems), the proposed framework is generalized to establish ESI conditions that include
both the activity intervals of the matrices and their switching rates, the latter being governed by a certain logarithmic measure
of the normalized magnitudes of discontinuities caused by switching. In effect, (the new, globally averaged) dwell-time is
flexibly traded, apparently for the first time, but under specific conditions (related, in part, to the eigenvalues of the matrices),
for switching discontinuity-based conditions. Two further novel aspects of the proposed approach are: (i) For second-order
matrices, switching lines in phase space can be chosen for periodic switching to stabilize or destabilize the system, and even generate oscillations, depending on the eigenvalues of the system matrices. But for third- (and higher) order matrices, such an analytically tractable (and controlled) periodical switching entails solution of an explicit non-convex multi-parameter optimization problem for which a stochastic optimization algorithm from the literature can be invoked. (ii) Lower and upper bounds on the solutions of the system equations can be quantified to reflect the stability/instability/oscillatory property of the system. Illustrative examples, which demonstrate the novelty of the derived stability and instability conditions, are presented in part 2 which is advisedly to be read along with this part 1 for a coherent merging of theory with practice. |
| 关键词: Differential inequalities · Dwell time · Exponential stability and instability · Quadratic Lyapunov functions · Switched linear second- and higher order systems |
| DOI：https://doi.org/10.1007/s11768-025-00245-x |
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| 基金项目: |
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| On the exponential stability and instability analyses of switched second- and higher-order linear systems via a novel application of differential inequalities: part 1 (theory) |
| Y. V. Venkatesh1,2 |
| (1 (Formerly) Division of Electrical Sciences, Indian Institute of Science, Sir C. V. Raman Road, Bangalore, Karnataka 560012, India
2 Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge Crescent, Singapore 119260, Singapore) |
| Abstract: |
| Differential inequalities generated in an extended Lyapunov framework are employed in the stability and instability analyses
of a class of switched continuous-time second- and higher order linear systems with an arbitrary number of switching
matrices. The exponential stability and instability (ESI) conditions so obtained involve the supremum and infimum of ratios
of certain quadratic forms of the matrices, leading to global time-averages of their activity intervals. Further, motivated by
linear switching system examples of (i) instability with stable matrices and (ii) stability with unstable matrices (found in the
literature primarily for second-order systems), the proposed framework is generalized to establish ESI conditions that include
both the activity intervals of the matrices and their switching rates, the latter being governed by a certain logarithmic measure
of the normalized magnitudes of discontinuities caused by switching. In effect, (the new, globally averaged) dwell-time is
flexibly traded, apparently for the first time, but under specific conditions (related, in part, to the eigenvalues of the matrices),
for switching discontinuity-based conditions. Two further novel aspects of the proposed approach are: (i) For second-order
matrices, switching lines in phase space can be chosen for periodic switching to stabilize or destabilize the system, and even generate oscillations, depending on the eigenvalues of the system matrices. But for third- (and higher) order matrices, such an analytically tractable (and controlled) periodical switching entails solution of an explicit non-convex multi-parameter optimization problem for which a stochastic optimization algorithm from the literature can be invoked. (ii) Lower and upper bounds on the solutions of the system equations can be quantified to reflect the stability/instability/oscillatory property of the system. Illustrative examples, which demonstrate the novelty of the derived stability and instability conditions, are presented in part 2 which is advisedly to be read along with this part 1 for a coherent merging of theory with practice. |
| Key words: Differential inequalities · Dwell time · Exponential stability and instability · Quadratic Lyapunov functions · Switched linear second- and higher order systems |
