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On Existence of Solutions to Structured Lyapunov Inequalities [article]

Aviar Sootla, James Anderson
2016 arXiv   pre-print
In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist.  ...  We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality.  ...  Hence, major questions still remain concerning existence theorems and scalable computation of block-diagonal solutions to Lyapunov inequalities.  ... 
arXiv:1603.07686v1 fatcat:qvzdpt4azvcyrpe6n4ozprca54

Page 507 of American Society of Civil Engineers. Collected Journals Vol. 123, Issue ST4 [page]

1997 American Society of Civil Engineers. Collected Journals  
If tmin < 0, there exists a solution of the Lyapunov inequalities, whereas t,i, = O means that the solution does not exist.  ...  Note that Lyapunov matrix inequalities, (14), are linear matrix inequalities. Khalil (1992) has presented the con- ditions for the existence of the solution of simultaneous Lya- punov inequalities.  ... 

Forwarding Design With Prescribed Local Behavior

M Sofiane Benachour, Vincent Andrieu, Laurent Praly, Hassan Hammouri
2013 IEEE Transactions on Automatic Control  
Among the non-linear control techniques, some Lyapunov design methods (Forwarding/Backstepping) take advantage of the structure of the system (Feedforward-form/Feedback-form) to formulate a continuous  ...  In addition to stabilization, we focus on the local behaviour of the closed loop system, providing conditions under which we can predetermine the behaviour around the origin for Feedforward systems.  ...  Indeed, this one is of rank . In order to apply Theorem 1, we need to find solution to the weak Lyapunov inequality (10) and (5) .  ... 
doi:10.1109/tac.2013.2277632 fatcat:5n5zyotfqffzre5oc6vojrvt7u

A necessary and sufficient minimality condition for uncertain systems

C.L. Beck, J. Doyle
1999 IEEE Transactions on Automatic Control  
This condition is based on the existence of a rank-deficient solution to either of a pair of linear matrix inequalities which generalize Lyapunov equations; the notion of Gramians is thus also generalized  ...  Subsequent results, such as the definition of and rank tests on structured controllability and observability matrices are also given.  ...  all operators in a neighborhood of zero if and only if there exist singular structured solutions to the Lyapunov inequalities.  ... 
doi:10.1109/9.793720 fatcat:ylyxgokov5h2ri7jnbclumveiq

Nonlinear Stabilization via Control Lyapunov Measure

Umesh Vaidya, Prashant G. Mehta, Uday V. Shanbhag
2010 IEEE Transactions on Automatic Control  
Using this approach, the codesign problem leads to a finite number of linear inequalities whose solutions define the feasible set of stabilizing controllers.  ...  We provide a proof of existence for a stochastic version of such a controller while the deterministic restriction is posed as the solution of a related integer programming problem.  ...  Existence of a Stochastic Controller The existence of a stochastic controller is linked to whether a solution exists to the set of inequalities given by ∑ i,a R ia P a i j < α ∑ a R ja , j = 1, . . . ,  ... 
doi:10.1109/tac.2010.2042226 fatcat:pd7nu7zovffyzctsymynujgbry

Nonlinear stabilization via control-Lyapunov measure

Umesh Vaidya, Prashant G. Mehta, Uday V. Shanbhag
2007 2007 46th IEEE Conference on Decision and Control  
Using this approach, the codesign problem leads to a finite number of linear inequalities whose solutions define the feasible set of stabilizing controllers.  ...  We provide a proof of existence for a stochastic version of such a controller while the deterministic restriction is posed as the solution of a related integer programming problem.  ...  Existence of a Stochastic Controller The existence of a stochastic controller is linked to whether a solution exists to the set of inequalities given by ∑ i,a R ia P a i j < α ∑ a R ja , j = 1, . . . ,  ... 
doi:10.1109/cdc.2007.4434956 dblp:conf/cdc/VaidyaMS07 fatcat:uydy2haurfg6ndznaewts6oj3a

Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

V. Nosov, J. A. Meda-Campaña, J. C. Gomez-Mancilla, J. O. Escobedo-Alva, R. G. Hernández-García
2015 Mathematical Problems in Engineering  
The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check.  ...  It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.  ...  The trivial solution ( ) ≡ 0 (or origin) of switched system (1)-(2) is said to be stable if for any > 0 there exists a = ( ) such that the inequality ‖ ( , 0 , 0 )‖ < is satisfied for any time > 0 whenever  ... 
doi:10.1155/2015/502475 fatcat:flbykh4uc5dwxmrg7yqwleej5i

A General Projection Neural Network for Solving Monotone Variational Inequalities and Related Optimization Problems

Y. Xia, J. Wang
2004 IEEE Transactions on Neural Networks  
In addition to its simple structure and low complexity, the proposed neural network includes existing neural networks for optimization, such as the projection neural network, the primal-dual neural network  ...  Furthermore, several improved stability criteria on two special cases of the general projection neural network are obtained under weaker conditions.  ...  This property shows that the existence of the equilibrium point of (1) is equivalent to the one of the solutions of GVI.  ... 
doi:10.1109/tnn.2004.824252 pmid:15384525 fatcat:rbik6qliznasjaktezv6wberzq

A Characterization of Lyapunov Inequalities for Stability of Switched Systems

Raphael M. Jungers, Amir Ali Ahmadi, Pablo A. Parrilo, Mardavij Roozbehani
2017 IEEE Transactions on Automatic Control  
Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.  ...  For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability.  ...  Suppose there exist Lyapunov functions V 1 , . . . , V k , one per node of G, that satisfy the inequalities imposed by the edges of G with γ > 1.  ... 
doi:10.1109/tac.2017.2671345 fatcat:oe547cqhvzct5epyjf44x3gkdq

Coulomb Control of Nonequilibrium Fixed Shape Triangular Three-Vehicle Cluster

Shuquan Wang, Hanspeter Schaub
2011 Journal of Guidance Control and Dynamics  
Theorem 2 proves the existence of solutions to the inequalities in Egs. (41) and (42) unless the condition in Eq. (48) occurs.  ...  If the following two inequalities and one equation do not happen at the same time a,b, <0 (48a) arb, <0 (48b) ayb, = a,b, (48c) then for any c € R, there always exists a solution of the matrix [A] that  ... 
doi:10.2514/1.47835 fatcat:oqsqct3ejbfj5hlv7boamdzlda

Page 3733 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
Necessary and sufficient con- ditions are derived which require the Lyapunov function to be nonincreasing only along one subsequence of the ‘switching’.”  ...  Summary: “We describe a new proof of the well-known Lyapunov matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane.  ... 

Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities

K. Kiriakidis
2001 IEEE transactions on fuzzy systems  
Simulation studies, which include an algorithm for solution of bilinear matrix inequalities, demonstrate the proposed method.  ...  It is well known, however, that there exist stable differential inclusions, hence T-S fuzzy models whose stability is unprovable by a globally quadratic Lyapunov function.  ...  To obtain a synthesis result, the state-dependent gain , which, in general, is discontinuous on , , requires additional structure as well.  ... 
doi:10.1109/91.919248 fatcat:fikjh6ficnbijkg432txzxq234

LaSalle-Yoshizawa Corollaries for Nonsmooth Systems

Nicholas Fischer, Rushikesh Kamalapurkar, Warren E. Dixon
2013 IEEE Transactions on Automatic Control  
Lyapunov-based analysis methods are developed using differential inclusions to achieve asymptotic convergence when the candidate Lyapunov derivative is upper bounded by a negative semi-definite function  ...  In this note, two generalized corollaries to the LaSalle-Yoshizawa Theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right-hand sides.  ...  ACKNOWLEDGMENT The authors would like to express their gratitude to Professor A. Teel for his constructive comments during the development of this work.  ... 
doi:10.1109/tac.2013.2246900 fatcat:f4prpdx2graz3mnqf7oarze52q

Synthesis of Lyapunov Functions using Formal Verification [article]

Lukas Munser, Grigory Devadze, Stefan Streif
2021 arXiv   pre-print
Recent employments of SMT solvers within the Lyapunov function synthesis provided effective tools for automated construction of Lyapunov functions alongside with sound computer-assisted certificates.  ...  Additionally, we address constructions of Lyapunov functions for state-dependent switching systems. We illustrate our approach by means of various examples from the control systems literature.  ...  If the inequality system has no solution, the output of the check is unsat and no Lyapunov function with the given structure exists.  ... 
arXiv:2112.01835v1 fatcat:y4erhzxp55cndhodhvl6nudx7q

Stability of fractional-order nonlinear systems by Lyapunov direct method [article]

H.T. Tuan, Hieu Trinh
2018 arXiv   pre-print
Second, we establish fractional Lyapunov functions to fractional-order systems without the assumption on the global existence of solutions.  ...  First, we prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption on the existence of derivative of pseudo-states.  ...  We also do not require the condition on the existence of derivative to pseudo-states in the inequality concerning the fractional derivatives of convex Lyapunov functions.  ... 
arXiv:1712.02921v2 fatcat:poyrtferrra2pc4ddhkwwrtaeu
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