26 results on '"Liberzon, Daniel"'
Search Results
2. Integral-Input-to-State Stability of Switched Nonlinear Systems Under Slow Switching.
- Author
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Liu, Shenyu, Russo, Antonio, Liberzon, Daniel, and Cavallo, Alberto
- Subjects
STABILITY of nonlinear systems - Abstract
In this article we study integral-input-to-state stability (iISS) of nonlinear switched systems with jumps. We demonstrate by examples that iISS is not always preserved under slow enough dwell time switching, and then we present sufficient conditions for iISS to be preserved under slow switching. These conditions involve, besides a sufficiently large dwell time, some additional properties of comparison functions characterizing iISS of the individual modes. When the sufficient conditions that guarantee iISS are only partially satisfied, we are then able to conclude weaker variants of iISS, also introduced in this work. As an illustration, we show that switched systems with bilinear zero-input-stable modes are always iISS under sufficiently large dwell time. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Entropy and Minimal Bit Rates for State Estimation and Model Detection.
- Author
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Liberzon, Daniel and Mitra, Sayan
- Subjects
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ENTROPY , *ESTIMATION theory , *NONLINEAR systems , *QUANTIZATION (Physics) , *TRAJECTORY optimization - Abstract
We study a notion of estimation entropy for continuous-time nonlinear systems, formulated in terms of the number of system trajectories that approximate all other trajectories up to an exponentially decaying error. We also consider an alternative definition of estimation entropy, which uses approximating functions that are not necessarily trajectories of the system, and show that the two entropy notions are equivalent. We establish an upper bound on the estimation entropy in terms of the sum of the desired convergence rate and an upper bound on the matrix measure of the Jacobian, multiplied by the system dimension. A lower bound on the estimation entropy is developed as well. We then turn our attention to state estimation and model detection with quantized and sampled state measurements. We describe an iterative procedure that uses such measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy, and no other algorithm of this type can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we discuss an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of “exponential separation” of the candidate models, detection always happens in finite time. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
4. Relaxed conditions for norm-controllability of nonlinear systems.
- Author
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Muller, Matthias A., Liberzon, Daniel, and Allgower, Frank
- Abstract
In this paper, we further study the recently introduced notion of norm-controllability, which captures the responsiveness of a nonlinear system with respect to applied inputs in terms of the norm of an output map. We give sufficient conditions for this property based on higher-order lower directional derivatives, which generalize the results obtained in our earlier work and help to establish norm-controllability for systems with outputs having relative degree greater than one. Furthermore, we illustrate the obtained results by means of a chemical reaction example. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
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5. On norm-controllability of nonlinear systems.
- Author
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Muller, Matthias A., Liberzon, Daniel, and Allgower, Frank
- Abstract
In this paper, we introduce and study the notion of “norm-controllability” for nonlinear systems. This property captures the responsiveness of a system with respect to the applied inputs, which is quantified via the norm of an output map. As a main contribution, we obtain a Lyapunov-like sufficient condition for norm-controllability. Several examples illustrate the various aspects of the proposed concept, and we also further elaborate norm-controllability for the special case of linear systems. [ABSTRACT FROM PUBLISHER]
- Published
- 2011
- Full Text
- View/download PDF
6. An Asymptotic Ratio Characterization of Input-to-State Stability.
- Author
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Liberzon, Daniel and Shim, Hyungbo
- Subjects
- *
ASYMPTOTIC expansions , *STABILITY theory , *NONLINEAR systems , *LYAPUNOV functions , *EXISTENCE theorems - Abstract
For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic ratio input-to-state stability (ISS) Lyapunov function. The derivative of such a function along solutions is upper-bounded by the difference of two terms whose ratio is asymptotically smaller than 1 for large states. This asymptotic ratio condition is sometimes more convenient to check than standard ISS Lyapunov function conditions. We show that the existence of an asymptotic ratio ISS Lyapunov function is equivalent to ISS. A related notion of ISS with nonuniform convergence rate is also explored. [ABSTRACT FROM PUBLISHER]
- Published
- 2015
- Full Text
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7. Norm-Controllability of Nonlinear Systems.
- Author
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Muller, Matthias A., Liberzon, Daniel, and Allgower, Frank
- Subjects
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CONTROLLABILITY in systems engineering , *NONLINEAR systems , *HIGHER order transitions , *LYAPUNOV functions , *CHEMICAL reactors , *MATHEMATICAL models - Abstract
In this paper, we introduce and study the notion of norm-controllability for nonlinear systems. This property captures the responsiveness of a system with respect to applied inputs, which is quantified via the norm of an output. As a main contribution, we obtain several Lyapunov-like sufficient conditions for norm-controllability, some of which are based on higher-order derivatives of a Lyapunov-like function. Various aspects of the proposed concept and the sufficient conditions are illustrated by several examples, including a chemical reactor application. Furthermore, for the special case of linear systems, we establish connections between norm-controllability and standard controllability. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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8. Input/output-to-state stability and state-norm estimators for switched nonlinear systems
- Author
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Müller, Matthias A. and Liberzon, Daniel
- Subjects
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STABILITY theory , *MATRIX norms , *NONLINEAR systems , *ESTIMATION theory , *AUTOMATIC control systems , *INPUT-output analysis - Abstract
Abstract: In this paper, the concepts of input/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that when the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for. [Copyright &y& Elsevier]
- Published
- 2012
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9. Invertibility of switched nonlinear systems
- Author
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Tanwani, Aneel and Liberzon, Daniel
- Subjects
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NONLINEAR systems , *ALGORITHMS , *DATA recovery , *SIGNALS & signaling , *COMPUTER input design , *SIGNAL processing - Abstract
Abstract: This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts. [Copyright &y& Elsevier]
- Published
- 2010
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10. Lyapunov conditions for input-to-state stability of impulsive systems
- Author
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Hespanha, João P., Liberzon, Daniel, and Teel, Andrew R.
- Subjects
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LYAPUNOV functions , *NONLINEAR systems , *DYNAMICS , *DIFFERENTIAL equations , *MICROELECTROMECHANICAL systems , *SYSTEMS theory - Abstract
Abstract: This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network. [Copyright &y& Elsevier]
- Published
- 2008
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11. STABILITY ANALYSIS OF DETERMINISTIC AND STOCHASTIC SWITCHED SYSTEMS VIA A COMPARISON PRINCIPLE AND MULTIPLE LYAPUNOV FUNCTIONS.
- Author
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Chatterjee, Debasish and Liberzon, Daniel
- Subjects
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LYAPUNOV functions , *NONLINEAR systems , *STOCHASTIC differential equations , *DIFFERENTIAL equations , *MULTIPLE comparisons (Statistics) - Abstract
This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
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12. Common Lyapunov functions for families of commuting nonlinear systems
- Author
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Vu, Linh and Liberzon, Daniel
- Subjects
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LYAPUNOV functions , *NONLINEAR systems , *SYSTEMS theory , *VECTOR analysis - Abstract
Abstract: We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based on an iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individual systems. Our results extend a previously available one which relies on exponential stability of the vector fields. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
13. Supervision of integral-input-to-state stabilizing controllers
- Author
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Hespanha, João P., Liberzon, Daniel, and Morse, A. Stephen
- Subjects
- *
NONLINEAR systems , *SYSTEM analysis - Abstract
The subject of this paper is hybrid control of nonlinear systems with large-scale uncertainty. We describe a high-level controller, called a “supervisor”, which orchestrates logic-based switching among a family of candidate controllers. We show that in this framework, the problem of controller design at the lower level can be reduced to finding an integral-input-to-state stabilizing control law for an appropriate system with disturbance inputs. Employing the recently introduced “scale-independent hysteresis” switching logic, we prove that in the case of purely parametric uncertainty with unknown parameters taking values in a finite set the switching terminates in finite time and state regulation is achieved. [Copyright &y& Elsevier]
- Published
- 2002
- Full Text
- View/download PDF
14. Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation
- Author
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Liberzon, Daniel, Sontag, Eduardo D., and Wang, Yuan
- Subjects
- *
NONLINEAR systems , *LOOPS (Group theory) - Abstract
We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed. [Copyright &y& Elsevier]
- Published
- 2002
- Full Text
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15. The Bang-Bang Funnel Controller for Uncertain Nonlinear Systems With Arbitrary Relative Degree.
- Author
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Liberzon, Daniel and Trenn, Stephan
- Subjects
- *
FEEDBACK control systems , *CONTROL theory (Engineering) , *TRACKING control systems , *FEASIBILITY studies , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound—a funnel—around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
16. Input to State Stabilizing Controller for Systems With Coarse Quantization.
- Author
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Sharon, Yoav and Liberzon, Daniel
- Subjects
- *
STRUCTURAL stability , *QUANTIZATION (Physics) , *MEASURE theory , *PARAMETER estimation , *FEEDBACK control systems , *STRUCTURAL analysis (Engineering) , *NONLINEAR systems - Abstract
We consider the problem of achieving input-to-state stability (ISS) with respect to external disturbances for control systems with quantized measurements. Quantizers considered in this paper take finitely many values and have an adjustable “center” and “zoom” parameters. Both the full state feedback and the output feedback cases are considered. Similarly to previous techniques from the literature, our proposed controller switches repeatedly between “zooming out” and “zooming in.” However, here we use two modes to implement the “zooming in” phases, which allows us to attenuate an unknown disturbance while using the minimal number of quantization regions. Our analysis is trajectory-based and utilizes a cascade structure of the closed-loop hybrid system. We further show that our method is robust to modeling errors using a specially adapted small-gain theorem. The main results are developed for linear systems, but we also discuss their extension to nonlinear systems under appropriate assumptions. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
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17. On Stability of Randomly Switched Nonlinear Systems.
- Author
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Chatterjee, Debasish and Liberzon, Daniel
- Subjects
- *
STOCHASTIC processes , *PROBABILITY theory , *SYSTEM analysis , *STOCHASTIC systems , *ESTIMATION theory , *RANDOM walks , *MECHANICS (Physics) , *STATICS , *MARKOV processes , *NONLINEAR systems - Abstract
This note is concerned with stability analysis and stabilization of randomly switched systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure stability and stability in the mean using Lyapunov-based methods when individual subsystems are stable and a certain "slow switching" condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results, therefore, hold for Markovian jump systems in particular. For systems with control inputs, we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
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18. Quantization, Time Delays, and Nonlinear Stabilization.
- Author
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Liberzon, Daniel
- Subjects
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AUTOMATIC control systems , *AUTOMATION , *FEEDBACK control systems , *DISCRETE-time systems , *ROBUST control , *PROCESS control systems , *TIME delay systems , *NONLINEAR systems , *DYNAMICS - Abstract
The purpose of this note is to demonstrate that a unified study of quantization and delay effects in nonlinear control systems is possible by merging the quantized feedback control methodology recently developed by the author and the small-gain approach to the analysis of functional differential equations with disturbances proposed earlier by Teel. We prove that under the action of a robustly stabilizing feedback controller in the presence of quantization and time delays satisfying suitable conditions, solutions of the closed-loop system starting in a given region remain bounded and eventually enter a smaller region. We present several versions of this result and show how it enables global asymptotic stabilization via a dynamic quantization strategy. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
19. Stabilization of Nonlinear Systems With Limited Information Feedback.
- Author
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Liberzon, Daniel and Hespanha, João P.
- Subjects
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NONLINEAR systems , *SYSTEMS theory , *FEEDBACK control systems , *AUTOMATIC control systems , *ASYMPTOTIC theory of system theory , *AUTOMATION - Abstract
This note is concerned with the problem of stabilizing a non-linear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. The issue of relaxing the latter condition is also discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
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20. Nonlinear Norm-Observability Notions and Stability of Switched Systems.
- Author
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Hespanha, João P., Liberzon, Daniel, Angeli, David, and Sontag, Eduardo D.
- Subjects
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SWITCHING power supplies , *NONLINEAR systems , *NONLINEAR statistical models , *ELECTRONIC feedback , *ELECTRONICS , *VACUUM-tube amplifiers - Abstract
This paper proposes several definitions of "norm-observability" for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for norm-observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
21. Output-Input Stability and Minimum-Phase Nonlinear Systems.
- Author
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Liberzon, Daniel, Morse, A. Stephen, and Sontag, Eduardo D.
- Subjects
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STABILITY (Mechanics) , *NONLINEAR systems - Abstract
Studies the notion of output-input stability with the variation of minimum-phase property for general smooth nonlinear control systems. Development of a natural extension to nonlinear systems; Areas of linear system analysis and design; Duality between inputs and outputs.
- Published
- 2002
- Full Text
- View/download PDF
22. Nonlinear Feedback Systems Perturbed by Noise: Steady-State Probability Distributions and Optimal...
- Author
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Liberzon, Daniel and Brockett, Roger W.
- Subjects
- *
NONLINEAR systems , *FEEDBACK control systems , *LYAPUNOV functions - Abstract
Describes a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found. Monotemperaturic systems' relationship with Lyapunov functions; Stochastic optimal control problems in the case of quantized feedback.
- Published
- 2000
- Full Text
- View/download PDF
23. Almost Lyapunov functions for nonlinear systems.
- Author
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Liu, Shenyu, Liberzon, Daniel, and Zharnitsky, Vadim
- Subjects
- *
NONLINEAR functions , *LYAPUNOV functions , *NONLINEAR systems , *PHASE space , *VECTOR fields - Abstract
We study convergence of nonlinear systems in the presence of an "almost Lyapunov" function which, unlike the classical Lyapunov function, is allowed to be nondecreasing – and even increasing – on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
24. Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems.
- Author
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Mironchenko, Andrii, Yang, Guosong, and Liberzon, Daniel
- Subjects
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LYAPUNOV stability , *SMALL-gain theorem (Mathematics) , *NONLINEAR systems , *HYBRID systems , *DIFFERENTIAL equations - Abstract
We prove a novel Lyapunov-based small-gain theorem for networks composed of n ≥ 2 hybrid subsystems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the ISS Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
25. Robustness of Pecora–Carroll synchronization under communication constraints.
- Author
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Andrievsky, Boris, Fradkov, Alexander L., and Liberzon, Daniel
- Subjects
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NONLINEAR systems , *SYNCHRONIZATION , *SIGNAL quantization , *ROBUST statistics , *LORENZ equations - Abstract
We study synchronization of nonlinear systems with robustness to disturbances that arise when measurements sent from the master system to the slave system are affected by quantization and time sampling. Viewing the synchronization problem as an observer design problem, we invoke a recently developed theory of nonlinear observers robust to output measurement disturbances and formulate a sufficient condition for robust synchronization. The approach is illustrated by a detailed analysis of the Pecora–Carroll synchronization scheme for the Lorenz system, for which an explicit bound on the synchronization error depending on the quantizer range and sampling period is derived. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. Energy control of a pendulum with quantized feedback.
- Author
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Seifullaev, Ruslan E., Fradkov, Alexander, and Liberzon, Daniel
- Subjects
- *
QUANTIZATION (Physics) , *FEEDBACK control systems , *NONLINEAR systems , *INVARIANT manifolds , *STATE feedback (Feedback control systems) , *MATHEMATICAL bounds - Abstract
The problem of controlling a nonlinear system to an invariant manifold using quantized state feedback is considered by the example of controlling the pendulum’s energy. A feedback control law based on the speed gradient algorithm is chosen. The main result consisting in precisely characterizing allowed quantization error bounds and resulting energy deviation bounds is presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
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