Observer-Based Sampled-Data Stabilisation of Switched Systems with Lipschitz Nonlinearities and Dwell-Time

Antonio Russo; Gian Paolo Incremona; Giulia Giordano; Patrizio Colaneri; Rami Katz

arxiv: 2511.01672 · v2 · submitted 2025-11-03 · 🧮 math.OC

Observer-Based Sampled-Data Stabilisation of Switched Systems with Lipschitz Nonlinearities and Dwell-Time

Rami Katz , Antonio Russo , Gian Paolo Incremona , Patrizio Colaneri , Giulia Giordano This is my paper

Pith reviewed 2026-05-18 01:40 UTC · model grok-4.3

classification 🧮 math.OC

keywords switched systemssampled-data stabilizationstate observerLipschitz nonlinearitiesdwell timeLyapunov-Metzler inequalitieslinear matrix inequalitiesoutput feedback

0 comments

The pith

A sampled-data observer and Lyapunov-Metzler inequalities stabilize switched systems with Lipschitz nonlinearities under dwell-time constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a switching law for systems that alternate between different modes while containing uncertain nonlinear terms that satisfy a Lipschitz condition. Decisions to switch are made using only sampled output measurements fed to a state observer, with the law constructed so that Lyapunov-Metzler inequalities hold and produce time-dependent linear matrix inequality conditions. These conditions certify global asymptotic stability when there are no affine terms, or ultimate boundedness otherwise, and they also supply an estimate of average quadratic cost together with a bound on its deviation. The feasibility discussion shows how observer gains enter the inequalities, how reduced-order versions can be derived, and how time dependence can be removed by checking a finite grid of instants.

Core claim

We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability (or, in the presence of switching affine terms, ultimate boundedness) of the resulting closed-loop system. We obtain an estimate of the average quadratic cost and a bound on its maximum deviation from the actual cost. Moreover, we discuss the feasibility of the derived LMIs. Specifically, we show how the observer gains can be incorporated into the matrix inequalities, provide equivalent reduced-order LMI conditions, and prove that the time dependence of the LMIs can be removed by discretising on a

What carries the argument

Lyapunov-Metzler inequalities modified to incorporate sampled-data output measurements and observer gains, which design the switching law and generate the time-dependent LMI conditions that certify closed-loop stability or boundedness.

If this is right

An estimate of the average quadratic cost together with a bound on its deviation from the true cost follows directly from the design.
Observer gains enter the matrix inequalities directly, and equivalent reduced-order LMI conditions are available.
Time dependence in the LMIs can be eliminated by checking the conditions on a finite discretization grid.
The resulting controller applies to practical power-system examples and outperforms an existing output-feedback method under sampled measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduced-order LMIs may lower the computational load when the method is implemented on embedded hardware with limited resources.
The cost bound supplies a quantitative performance metric that could guide selection of dwell time in design iterations.
Grid-based removal of time dependence suggests a systematic numerical procedure that could be automated for families of similar systems.

Load-bearing premise

Suitable observer gains exist that keep the Lyapunov-Metzler inequalities feasible once the sampled-data measurements and Lipschitz nonlinearities are taken into account.

What would settle it

A simulation or experiment in which the closed-loop trajectories diverge or escape the claimed bound even though the time-dependent LMIs hold for the chosen observer and switching law would disprove the stability guarantee.

Figures

Figures reproduced from arXiv: 2511.01672 by Antonio Russo, Gian Paolo Incremona, Giulia Giordano, Patrizio Colaneri, Rami Katz.

read the original abstract

We investigate the stabilisation of nominally linear-affine switched systems with uncertain Lipschitz nonlinearities under dwell-time constraints, using a sampled-data switching law based on a state observer. We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability (or, in the presence of switching affine terms, ultimate boundedness) of the resulting closed-loop system. We obtain an estimate of the average quadratic cost and a bound on its maximum deviation from the actual cost. Moreover, we discuss the feasibility of the derived LMIs. Specifically, we show how the observer gains can be incorporated into the matrix inequalities, provide equivalent reduced-order LMI conditions, and prove that the time dependence of the LMIs can be removed by discretising on a finite grid. Numerical examples, including practical applications to real-world engineering scenarios in power systems, illustrate our theoretical results and compare them with an existing approach for output-feedback stabilisation of switched systems, subject to sampled-data measurements

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit time-dependent LMI conditions plus reduced-order forms for observer-based sampled-data stabilization of switched Lipschitz systems under dwell time, with cost bounds included.

read the letter

The main point is that this work produces time-dependent LMI conditions for designing both an observer and a switching law that stabilize switched systems with Lipschitz nonlinearities, using only sampled output data and enforcing a dwell time. It also supplies reduced-order equivalent LMIs, a way to remove time dependence via discretization, and bounds on the quadratic cost and its deviation from the true value. The authors compare against an earlier output-feedback approach and test the results on power-system examples that show both global asymptotic stability and ultimate boundedness cases. That combination of observer design, sampled-data switching, and Lyapunov-Metzler inequalities in one LMI setting is the concrete advance over prior work. The derivations follow standard steps but are carried through carefully enough that the discretization argument and the incorporation of observer gains look traceable. The numerical examples are concrete and directly illustrate the claims against the cited baseline. The soft spot is the feasibility discussion. The paper shows how observer gains enter the inequalities and notes that feasibility can be checked, yet it stops short of giving explicit a priori relations between the Lipschitz constant, the maximum sampling interval, and the dwell time that would guarantee solutions exist. Without those bounds a designer can still run into parameter sets where the LMIs have no feasible gains even if the underlying system is stabilizable. That leaves the method conditional on the specific numbers rather than fully predictive. This is written for control theorists who already work with LMI methods for switched or sampled-data systems. A reader who needs a ready framework for observer-based switching with nonlinear terms and performance bounds will get direct value from the reduced-order forms and the examples. The paper is coherent on its own terms and supplies enough new explicit conditions and checks to merit a full referee report rather than a desk rejection. I would send it for peer review; the targeted extension and the concrete examples are worth the feedback, especially on tightening the feasibility guarantees.

Referee Report

1 major / 0 minor

Summary. The manuscript develops an observer-based sampled-data switching strategy for stabilizing switched linear-affine systems subject to Lipschitz nonlinearities and dwell-time constraints. It adapts Lyapunov-Metzler inequalities to incorporate sampled-data output measurements, derives time-dependent LMI conditions that certify global asymptotic stability (or ultimate boundedness when affine terms are present), supplies an estimate of the average quadratic cost together with a bound on its deviation, and discusses LMI feasibility by showing how observer gains enter the inequalities, providing equivalent reduced-order LMIs, and proving that time dependence can be eliminated via discretization on a finite grid. Numerical examples, including power-system applications, illustrate the approach and compare it with prior output-feedback methods.

Significance. If the derived LMIs are feasible, the work supplies a concrete design procedure that extends Lyapunov-Metzler theory to sampled-data observer-based switching while handling Lipschitz nonlinearities and providing explicit cost bounds. The reduced-order reformulation and discretization argument are technically useful for implementation, and the power-system examples demonstrate relevance to engineering practice. The absence of a priori feasibility bounds, however, limits the immediate applicability of the certification to systems for which suitable observer gains can be found.

major comments (1)

[Feasibility discussion] Feasibility discussion (abstract and corresponding section): the manuscript shows how observer gains are incorporated into the matrix inequalities, supplies reduced-order LMI equivalents, and proves that time dependence can be removed by finite-grid discretization, yet it does not furnish explicit a priori bounds relating the Lipschitz constant, maximum sampling interval, and dwell time that guarantee existence of feasible gains. Because the central stability claim relies on the LMIs being feasible for the given system data, this omission is load-bearing for the practical utility of the certification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comments and positive evaluation of our manuscript. We provide a point-by-point response to the major comment below.

read point-by-point responses

Referee: [Feasibility discussion] Feasibility discussion (abstract and corresponding section): the manuscript shows how observer gains are incorporated into the matrix inequalities, supplies reduced-order LMI equivalents, and proves that time dependence can be removed by finite-grid discretization, yet it does not furnish explicit a priori bounds relating the Lipschitz constant, maximum sampling interval, and dwell time that guarantee existence of feasible gains. Because the central stability claim relies on the LMIs being feasible for the given system data, this omission is load-bearing for the practical utility of the certification.

Authors: We thank the referee for this insightful comment. While explicit a priori bounds on the parameters (Lipschitz constant, sampling interval, dwell time) that ensure LMI feasibility would indeed be beneficial for immediate applicability without numerical optimization, obtaining such bounds in closed form is not straightforward for the general class of systems considered. The feasibility depends on the specific system matrices, the choice of observer gains, and the interplay between the time-dependent Lyapunov functions and the switching. Our paper focuses on deriving the LMI conditions and providing practical means to verify them, including the reduced-order equivalents and the finite-grid discretization method, which enable efficient numerical checks. This computational approach is standard in LMI-based designs and is illustrated in our examples, including the power system application. We argue that this provides sufficient practical utility, as the designer can tune the parameters and check feasibility directly. revision: no

Circularity Check

0 steps flagged

Derivation is self-contained from standard Lyapunov-Metzler theory and sampled-data LMI techniques with no reduction to fitted inputs or self-citation loops.

full rationale

The paper starts from established Lyapunov-Metzler inequalities for switched systems and extends them to incorporate sampled-data observers and Lipschitz nonlinearities, deriving time-dependent LMIs that are then discretized. No step equates a claimed prediction or stability certificate to a parameter fitted from the target result itself, nor does any load-bearing premise reduce to a self-citation whose validity depends on the present work. Feasibility of observer gains is discussed via reduced-order LMIs, but this is presented as an analysis step rather than a definitional closure. The approach remains externally falsifiable through the LMI feasibility for given system parameters, sampling periods, and dwell times, consistent with standard control-theoretic derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard Lyapunov stability theory and the feasibility of modified Metzler inequalities for sampled measurements; no free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the provided text.

axioms (2)

domain assumption Existence of observer gains making the sampled-data Lyapunov-Metzler inequalities feasible
Invoked when the switching law is designed and LMI conditions are stated for stability
domain assumption Lipschitz continuity of the uncertain nonlinearities
Used to bound the nonlinear terms in the closed-loop analysis

pith-pipeline@v0.9.0 · 5724 in / 1367 out tokens · 48007 ms · 2026-05-18T01:40:31.379818+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Ψ_i(t) given in (12) ... guarantees ... globally asymptotically stable

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Albea, C., Garcia, G., Hadjeras, S., Heemels, W.P.M.H., and Zaccarian, L. (2019). Practical stabilization of switched affine systems with dwell-time guarantees. IEEE Trans. on Automatic Control, 64(11), 4811--4817

work page 2019
[2]

Albea, C., Serieye, M., Seuret, A., and Jungers, M. (2024). Stabilization of aperiodic sampled-data switched affine systems to hybrid limit cycles. European Journal of Control, 79, 101094

work page 2024
[3]

and Seuret, A

Albea, C. and Seuret, A. (2021). Time-triggered and event-triggered control of switched affine systems via a hybrid dynamical approach. Nonlinear Analysis: Hybrid Systems, 41, 101039

work page 2021
[4]

Blanchini, F., Colaneri, P., and Valcher, M.E. (2012). Co-positive L yapunov functions for the stabilization of positive switched systems. IEEE Trans. on Automatic Control, 57(12), 3038--3050

work page 2012
[5]

and Spinelli, W

Bolzern, P. and Spinelli, W. (2004). Quadratic stabilization of a switched affine system about a nonequilibrium point. In Proc. American Control Conference, volume 5, 3890--3895

work page 2004
[6]

and Yang, Q

Boyd, S. and Yang, Q. (1989). Structured and simultaneous L yapunov functions for system stability problems. International Journal of Control, 49(6), 2215--2240

work page 1989
[7]

and Bernussou, J

Daafouz, J. and Bernussou, J. (2001). Parameter dependent L yapunov functions for discrete time systems with time varying parametric uncertainties. Systems & Control Letters, 43(5), 355--359

work page 2001
[8]

Deaecto, G.S. (2016). Dynamic output feedback H _ control of continuous-time switched affine systems. Automatica, 71, 44--49

work page 2016
[9]

and Daiha, H.R

Deaecto, G.S. and Daiha, H.R. (2020). LMI conditions for output feedback control of switched systems based on a time-varying convex L yapunov function. Journal of the Franklin Institute, 357(15), 10513--10528

work page 2020
[10]

Deaecto, G.S., Hirata, R.A., and Teixeira, M.C. (2023). Static output feedback global asymptotic stability of limit cycles for discrete-time switched affine systems. IFAC-PapersOnLine, 56(2), 4132--4137

work page 2023
[11]

Decarlo, R., Branicky, M., Pettersson, S., and Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7), 1069--1082

work page 2000
[12]

and Deaecto, G.S

Egidio, L.N. and Deaecto, G.S. (2019). Novel practical stability conditions for discrete-time switched affine systems. IEEE Trans. on Automatic Control, 64(11), 4705--4710

work page 2019
[13]

and Deaecto, G.S

Egidio, L.N. and Deaecto, G.S. (2021). Dynamic output feedback control of discrete-time switched affine systems. IEEE Trans. on Automatic Control, 66(9), 4417--4423

work page 2021
[14]

Faron, E. (1996). Quadratic stability of switched systems via state and output feedback. Ph.D. thesis, Lab. Inf. Decis. Syst. Massachusetts Inst. Technol (MIT)

work page 1996
[15]

Fridman, E. (2014). Introduction to time-delay systems. Analysis and Control. Birkh \"a user , 75

work page 2014
[16]

and Colaneri, P

Geromel, J.C. and Colaneri, P. (2006 a ). Stability and stabilization of discrete time switched systems. International Journal of Control, 79(7), 719--728

work page 2006
[17]

and Colaneri, P

Geromel, J.C. and Colaneri, P. (2006 b ). Stability and stabilization of continuous--time switched linear systems. SIAM Journal on Control and Optimization, 45(5), 1915--1930

work page 2006
[18]

Geromel, J.C., Colaneri, P., and Bolzern, P. (2008). Dynamic output feedback control of switched linear systems. IEEE Trans. on Automatic Control, 53(3), 720--733

work page 2008
[19]

Hespanha, J. (2004). Uniform stability of switched linear systems: extensions of LaSalle 's I nvariance P rinciple. IEEE Trans. on Automatic Control, 49(4), 470--482

work page 2004
[20]

and Fridman, E

Hetel, L. and Fridman, E. (2013). Robust sampled–data control of switched affine systems. IEEE Trans. on Automatic Control, 58(11), 2922--2928

work page 2013
[21]

and Johnson, C.R

Horn, R.A. and Johnson, C.R. (2012). Matrix analysis. Cambridge University Press

work page 2012
[22]

and Rantzer, A

Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic L yapunov functions for hybrid systems. IEEE Trans. on Automatic Control, 43(4), 555--559

work page 1998
[23]

Liberzon, D. (2003). Switching in Systems and Control. Birkh \"a user, Boston, MA

work page 2003
[24]

and Morse, A

Liberzon, D. and Morse, A. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19(5), 59--70

work page 1999
[25]

Russo, A., Incremona, G.P., and Colaneri, P. (2024). Stabilization of switched affine systems with dwell-time constraint. arXiv:2411.12939

work page arXiv 2024
[26]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sent...

work page
[27]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page

[1] [1]

Albea, C., Garcia, G., Hadjeras, S., Heemels, W.P.M.H., and Zaccarian, L. (2019). Practical stabilization of switched affine systems with dwell-time guarantees. IEEE Trans. on Automatic Control, 64(11), 4811--4817

work page 2019

[2] [2]

Albea, C., Serieye, M., Seuret, A., and Jungers, M. (2024). Stabilization of aperiodic sampled-data switched affine systems to hybrid limit cycles. European Journal of Control, 79, 101094

work page 2024

[3] [3]

and Seuret, A

Albea, C. and Seuret, A. (2021). Time-triggered and event-triggered control of switched affine systems via a hybrid dynamical approach. Nonlinear Analysis: Hybrid Systems, 41, 101039

work page 2021

[4] [4]

Blanchini, F., Colaneri, P., and Valcher, M.E. (2012). Co-positive L yapunov functions for the stabilization of positive switched systems. IEEE Trans. on Automatic Control, 57(12), 3038--3050

work page 2012

[5] [5]

and Spinelli, W

Bolzern, P. and Spinelli, W. (2004). Quadratic stabilization of a switched affine system about a nonequilibrium point. In Proc. American Control Conference, volume 5, 3890--3895

work page 2004

[6] [6]

and Yang, Q

Boyd, S. and Yang, Q. (1989). Structured and simultaneous L yapunov functions for system stability problems. International Journal of Control, 49(6), 2215--2240

work page 1989

[7] [7]

and Bernussou, J

Daafouz, J. and Bernussou, J. (2001). Parameter dependent L yapunov functions for discrete time systems with time varying parametric uncertainties. Systems & Control Letters, 43(5), 355--359

work page 2001

[8] [8]

Deaecto, G.S. (2016). Dynamic output feedback H _ control of continuous-time switched affine systems. Automatica, 71, 44--49

work page 2016

[9] [9]

and Daiha, H.R

Deaecto, G.S. and Daiha, H.R. (2020). LMI conditions for output feedback control of switched systems based on a time-varying convex L yapunov function. Journal of the Franklin Institute, 357(15), 10513--10528

work page 2020

[10] [10]

Deaecto, G.S., Hirata, R.A., and Teixeira, M.C. (2023). Static output feedback global asymptotic stability of limit cycles for discrete-time switched affine systems. IFAC-PapersOnLine, 56(2), 4132--4137

work page 2023

[11] [11]

Decarlo, R., Branicky, M., Pettersson, S., and Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7), 1069--1082

work page 2000

[12] [12]

and Deaecto, G.S

Egidio, L.N. and Deaecto, G.S. (2019). Novel practical stability conditions for discrete-time switched affine systems. IEEE Trans. on Automatic Control, 64(11), 4705--4710

work page 2019

[13] [13]

and Deaecto, G.S

Egidio, L.N. and Deaecto, G.S. (2021). Dynamic output feedback control of discrete-time switched affine systems. IEEE Trans. on Automatic Control, 66(9), 4417--4423

work page 2021

[14] [14]

Faron, E. (1996). Quadratic stability of switched systems via state and output feedback. Ph.D. thesis, Lab. Inf. Decis. Syst. Massachusetts Inst. Technol (MIT)

work page 1996

[15] [15]

Fridman, E. (2014). Introduction to time-delay systems. Analysis and Control. Birkh \"a user , 75

work page 2014

[16] [16]

and Colaneri, P

Geromel, J.C. and Colaneri, P. (2006 a ). Stability and stabilization of discrete time switched systems. International Journal of Control, 79(7), 719--728

work page 2006

[17] [17]

and Colaneri, P

Geromel, J.C. and Colaneri, P. (2006 b ). Stability and stabilization of continuous--time switched linear systems. SIAM Journal on Control and Optimization, 45(5), 1915--1930

work page 2006

[18] [18]

Geromel, J.C., Colaneri, P., and Bolzern, P. (2008). Dynamic output feedback control of switched linear systems. IEEE Trans. on Automatic Control, 53(3), 720--733

work page 2008

[19] [19]

Hespanha, J. (2004). Uniform stability of switched linear systems: extensions of LaSalle 's I nvariance P rinciple. IEEE Trans. on Automatic Control, 49(4), 470--482

work page 2004

[20] [20]

and Fridman, E

Hetel, L. and Fridman, E. (2013). Robust sampled–data control of switched affine systems. IEEE Trans. on Automatic Control, 58(11), 2922--2928

work page 2013

[21] [21]

and Johnson, C.R

Horn, R.A. and Johnson, C.R. (2012). Matrix analysis. Cambridge University Press

work page 2012

[22] [22]

and Rantzer, A

Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic L yapunov functions for hybrid systems. IEEE Trans. on Automatic Control, 43(4), 555--559

work page 1998

[23] [23]

Liberzon, D. (2003). Switching in Systems and Control. Birkh \"a user, Boston, MA

work page 2003

[24] [24]

and Morse, A

Liberzon, D. and Morse, A. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19(5), 59--70

work page 1999

[25] [25]

Russo, A., Incremona, G.P., and Colaneri, P. (2024). Stabilization of switched affine systems with dwell-time constraint. arXiv:2411.12939

work page arXiv 2024

[26] [26]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sent...

work page

[27] [27]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page