Robust stability of switched systems with arbitrarily fast switchings by generalized matrix measure

Li, Z.G.; Chen, Wen; Soh, Yeng Chai; Xie, Wei

doi:10.1109/acc.2000.878692

Cited by 9 publications

(10 citation statements)

References 8 publications

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“…For this purpose, a planar uncertain switched linear system constituted by three subsystems, and a three‐dimensional uncertain switched linear system with four subsystems are investigated for the robust exponential stability. The results are compared with [22, 24]. For fair comparisons to other approaches, all numerical simulations are conducted in the same simulation environment, using YALMIP toolbox of MATLAB [37].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Moreover, if the switched linear system (24) is considered as a switched linear inclusion system, the stability bound on parameter uncertainties will be 16.14% of the nominal values, which shows that the guaranteed stability bound has not significantly been improved, even though the complexity has been increased rather than the result of Theorem 2. Besides, if one ignores the information about the structure of uncertainties, and uses [22] to find a stability margin, locally asymptotically stability will be concluded for the uncertainties that satisfy

δ q_{1} E_{1}^{k} + δ q_{2} E_{2}^{k} \leq 0.1250 (][, \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})

for

k = 1, 2, 3

, which is equal to 8.19% variation in the uncertain parameters

a

and

b

. Example 2 Consider a parameterised three‐dimensional switched linear system with the subsystem matrices

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A_{1} (a, b, c) & = [\begin{matrix} 1em4pt \\ - 0.1 a - 1.7 & 0.2 a + 0.2 c + 2 & - 0.2 b - 1.8 \\ - 0.1 b - 0.3 c + 1.7 & - 0.2 a - 2.4 & 0.3 a + 0.2 b + 3.9 \\ 0.1 a + 0.2 b - 0.2 c - 1.1 & 0.3 c - 0.1 b - 3.5 & 0.1 c - 0.1 a - 0.9 \end{matrix}], \end{matrix}

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A_{2} (a, b, c) & = [\begin{matrix} 1em4pt \\ 0.1 c - 3.2 & 0.2 a + 0.4 & 0.1 a + 0.1 b - 4.4 \\ 0.2 b + 1.8 \end{matrix}] \end{matrix}

…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In [22], some sufficient conditions were proposed for robust local stability of arbitrary switched non‐linear systems with unstructured uncertainties. The results of [22] are applicable for robust stability analysis of switched linear systems, when the Hermitian part of

A_{k}

is negative definite for each

k \in M

, assuming that

A_{k}

denotes the

k

th subsystem matrix (i.e. eigenvalues of

false(A_{k} + A_{k}^{'} false)

are negative).…”

Section: Introductionmentioning

confidence: 99%

“…In [26–28], stabilisation of uncertain switched linear systems via appropriate switching signals was investigated, while the stabilisation via multiple state feedback gains and the appropriate switching signal was addressed in [29–31]. In [22–31], unstructured uncertainties were considered, which did not possess obvious physical interpretation. Moreover, when uncertain systems with parametric uncertainties are investigated, theorems for robust stability of systems with unstructured uncertainties will impose unnecessary conservations.…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that unnecessary conservations will be imposed to the investigations, if the known information about the parametric uncertainties is ignored to use the previously published theorems for stability analysis of switched linear systems with unstructured uncertainties. (ii) The investigated model for parametric uncertainties in this article describes the variations of physical parameters in real‐world applications of switched linear systems. (iii) The proposed theorems for stability of switched linear systems are applicable for the class of switched linear systems in which

V false(x false) = x^{'} P x

is a CQLF for the nominal switched linear system. While it was assumed in [22] that all eigenvalues of

false(A_{k} + A_{k}^{'} false)

were negative for each subsystem

k \in M

, which is equivalent to existence of CQLF

V false(x false) = x^{'} x

for all subsystems and does not usually hold for many quadratically stable switched linear systems. (iv) In the third proposed theorem, it is not needed to identify the active subsystem and switching signal. Therefore, the designed controller will be easily realised on switched linear systems with arbitrary switching signals.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching

Bagherzadeh

Ghaisari

Askari

2016

IET Control Theory & Applications

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Section: Numerical Examplesmentioning

confidence: 99%

δ q_{1} E_{1}^{k} + δ q_{2} E_{2}^{k} \leq 0.1250 (][, \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})

for

k = 1, 2, 3

, which is equal to 8.19% variation in the uncertain parameters

a

and

b

. Example 2 Consider a parameterised three‐dimensional switched linear system with the subsystem matrices

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A_{1} (a, b, c) & = [\begin{matrix} 1em4pt \\ - 0.1 a - 1.7 & 0.2 a + 0.2 c + 2 & - 0.2 b - 1.8 \\ - 0.1 b - 0.3 c + 1.7 & - 0.2 a - 2.4 & 0.3 a + 0.2 b + 3.9 \\ 0.1 a + 0.2 b - 0.2 c - 1.1 & 0.3 c - 0.1 b - 3.5 & 0.1 c - 0.1 a - 0.9 \end{matrix}], \end{matrix}

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A_{2} (a, b, c) & = [\begin{matrix} 1em4pt \\ 0.1 c - 3.2 & 0.2 a + 0.4 & 0.1 a + 0.1 b - 4.4 \\ 0.2 b + 1.8 \end{matrix}] \end{matrix}

…”

Section: Numerical Examplesmentioning

confidence: 99%

A_{k}

is negative definite for each

k \in M

, assuming that

A_{k}

denotes the

k

th subsystem matrix (i.e. eigenvalues of

false(A_{k} + A_{k}^{'} false)

are negative).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

V false(x false) = x^{'} P x

is a CQLF for the nominal switched linear system. While it was assumed in [22] that all eigenvalues of

false(A_{k} + A_{k}^{'} false)

were negative for each subsystem

k \in M

, which is equivalent to existence of CQLF

V false(x false) = x^{'} x

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching

Bagherzadeh

Ghaisari

Askari

2016

IET Control Theory & Applications

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Reducing conservatism in robust stability analysis and stabilisation of arbitrary switched linear systems

Bagherzadeh

Ghaisari

2022

IET Control Theory & Appl

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Parametric modelling of uncertainties has been considered for switched linear systems in the past few years to improve the efficiency of robust stability analysis. The main goal of this paper is simultaneously alleviating conservatism and computational effort for robust stability analysis and stabilisation of arbitrary switched linear systems with uncertain parameters. For this purpose, stability conditions are revised and some constraints are relaxed. First, an improved weighted stability bound is proposed which is larger than previous results. Since the obtained stability bound depends on the chosen Lyapunov function, an optimisation problem is also given to determine its maximum value. Then, using the stability results, robust stabilisation of parametric uncertain switched linear systems under arbitrary switching is investigated. In this regard, a design method is presented to offer a static state feedback control law. The conservatism of the proposed control law is reduced by relaxing the constraint on upper bound of a matrix variable. Finally, illustrative examples are included to demonstrate the effectiveness of the proposed method.

show abstract

Stability criteria for linear periodic switched systems

Schwartz

Haddad

Proceedings of the 2003 American Control Conference, 2003.

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Robust stability of switched systems with arbitrarily fast switchings by generalized matrix measure

Cited by 9 publications

References 8 publications

Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching

Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching

Reducing conservatism in robust stability analysis and stabilisation of arbitrary switched linear systems

Stability criteria for linear periodic switched systems

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