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Function in control theory
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V ( x ) {\displaystyle V(x)} to systems with control
Control-Lyapunov_function
Concept in the analysis of dynamical systems
ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory
Lyapunov_function
Property of a dynamical system where solutions near an equilibrium point remain so
as a control, external input, stimulus, disturbance, or forcing function. It has been shown that near to a point of equilibrium which is Lyapunov stable
Lyapunov_stability
Control theory for nonlinear or time-variant systems
linear for purposes of control design: Feedback linearization And Lyapunov based methods: Lyapunov redesign Control-Lyapunov function Nonlinear damping Backstepping
Nonlinear_control
Optimization for dynamical systems
refers to the use of a Lyapunov function to optimally control a dynamic system. Lyapunov functions are used extensively in control theory to ensure different
Lyapunov_optimization
Equation from stability analysis
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical
Lyapunov_equation
Method for controlling a prosthesis
developed called Rapid Exponentially Stabilizing Control Lyapunov Functions(RES-CLF). Control Lyapunov function are used to stabilize a nonlinear system to
Robotic_prosthesis_control
Russian mathematician (1857–1918)
Lyapunov equation Lyapunov exponent Lyapunov fractal Lyapunov function Lyapunov stability Lyapunov time Lyapunov's central limit theorem Lyapunov's condition
Aleksandr_Lyapunov
Rate of separation of infinitesimally close trajectories
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the exponential rate
Lyapunov_exponent
Concept in theory of differential equations
Haddad, W.M.; Chellaboina, VS (2008). Nonlinear Dynamical Systems and Control, a Lyapunov-based approach. Princeton University Press. ISBN 9780691133294. Teschl
LaSalle's invariance principle
LaSalle's_invariance_principle
Branch of engineering and mathematics
generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called
Control_theory
Theorem in control theory
differentiable control-Lyapunov function if and only if it admits a regular stabilizing feedback u(x), that is a locally Lipschitz function on Rn\{0}. The
Artstein's_theorem
Topics referred to by the same term
geometry, a synonym for dimension function; in control theory and dynamical systems, a synonym for Lyapunov candidate function; in gauge theory, a synonym for
Gauge_function
Topics referred to by the same term
mechanician and physicist Lyapunov equation, used in many branches of control theory, such as stability analysis and optimal control Lyapunov fractal, bifurcational
Lyapunov_theorem
time derivative of the system's Lyapunov function is negative definite at that point. In other words, even if the control input is arbitrarily small, a
Small_control_property
Regulation of nonlinear systems
performance and robustness. Single Quadratic Lyapunov Function (SQLF) Parameter Dependent Quadratic Lyapunov Function (PDQLF) to bound the achievable level of
Linear parameter-varying control
Linear_parameter-varying_control
Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem
Lyapunov–Schmidt_reduction
Generalization of finite measure to Banach spaces
lemma, which has been viewed as a discrete analogue of Lyapunov's theorem. Bochner measurable function Bochner integral – Concept in mathematics Bochner space –
Vector_measure
In nonlinear control, the technique of Lyapunov redesign refers to the design where a stabilizing state feedback controller can be constructed with knowledge
Lyapunov_redesign
Fundamental theorem in probability theory and statistics
the rate of growth of these moments is limited by the Lyapunov condition given below. Lyapunov CLT—Suppose { X 1 , … , X n , … } {\textstyle \{X_{1},\ldots
Central_limit_theorem
Numerical optimization process
across a variety of areas, including control theory (in particular, for searching for polynomial Lyapunov functions for dynamical systems described by polynomial
Sum-of-squares_optimization
Approach to controller design that explicitly deals with uncertainty
control (LQG) control. Other robust techniques includes quantitative feedback theory (QFT), passivity based control, Lyapunov-based control, etc
Robust_control
Maximized objective function of an optimization problem
online closed-loop approximate optimal control, the value function is also a Lyapunov function that establishes global asymptotic stability of the closed-loop
Value_function
Robust nonlinear control to achieve exponential stabilization
controllers. The control design is underpinned by a Lyapunov stability analysis that utilizes an auxiliary function, often referred to as the P-function, to establish
RISE_controllers
Stability notion for nonlinear control systems with external inputs
and state-space methods, widely used within the control systems community. ISS unified the Lyapunov and input-output stability theories and revolutionized
Input-to-state_stability
Examining complex systems as a whole
shown to exhibit stable behavior given a suitable Lyapunov control function by Aleksandr Lyapunov in 1892. Thermodynamic systems were treated as early
Systems_thinking
dynamics of proteins under the effect of optical tweezers. Lyapunov function — a function whose existence guarantees stability Mohammadi, A.; Spong, Mark
Chetaev_instability_theorem
Technique in nonlinear control theory
assumed that a Lyapunov function V x {\displaystyle V_{x}} for this stable subsystem is known. Backstepping provides a way to extend the controlled stability
Backstepping
Generalization of signum function to matrices
by J.D. Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccati equation in a technical report of Cambridge University
Matrix_sign_function
Method in nonlinear control theory
other continuous control designs. The following theorems form the foundation of variable structure control. Consider a Lyapunov function candidate where
Sliding_mode_control
theory and nonlinear control, Massera's lemma, named after José Luis Massera, deals with the construction of the Lyapunov function to prove the stability
Massera's_lemma
Method to analyze non-binary inputs
Haber, Rodolfo; Haber-Guerra, Rodolfo E.; Reyes, Fernando (1999). "Lyapunov Stable Control of Robot Manipulators: A Fuzzy Self-Tuning Procedure". Intelligent
Fuzzy_control_system
Mathematical function, inverse of an exponential function
information conveyed by any one such message is quantified as log2 N bits. Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical
Logarithm
a diagonal quadratic Lyapunov function, which makes these systems more numerical tractable in the context of Lyapunov analysis. It is also important
Positive_systems
Overview of and topical guide to control engineering
Krener's theorem Least squares Lyapunov stability Minor loop feedback Perceptual control theory State observer Vector control Labview Matlab Simulink Embedded
Outline of control engineering
Outline_of_control_engineering
systems as Lyapunov stability, uniform asymptotic stability etc. Let C ( X , Y ) {\displaystyle C(X,Y)} be a space of continuous functions acting from
Comparison_function
Argentine American mathematician
(ISS), a stability theory notion for nonlinear systems, and control-Lyapunov functions. Many of the subsequent results were proved in collaboration with
Eduardo_D._Sontag
Mathematical model of a system in control engineering
which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability. The zeros found in the numerator of G ( s ) {\displaystyle \mathbf
State-space_representation
Property of differential equations describing physical phenomena
generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters. Inverse
Well-posed_problem
Part of mathematics that addresses the stability of solutions
involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria
Stability_theory
Concept in control theory mathematics
take a closer look at the Controllability Gramian. The controllability Gramian can be found as the solution of the Lyapunov equation given by A W c +
Controllability_Gramian
Technique for teaching a computer or a robot new behaviors
invariant systems to control robotic motions. However, this is restricted to dynamical systems with only quadratic Lyapunov functions. The new approach Tau-SEDS
Programming_by_demonstration
Lebanese-Greek-American mathematician
optimal control, backstepping control, disturbance rejection control, and robust control via fixed and parameter-dependent Lyapunov functions for nonlinear
Wassim_Michael_Haddad
Zinober, Alan S.I., ed. (1994). Variable Structure and Lyapunov Control. Lecture Notes in Control and Information Sciences. Vol. 193. London: Springer-Verlag
Variable_structure_control
system analysis and control theory. The eminent researchers (born after 1920) include the winners of at least one award of the IEEE Control Systems Award,
List of people in systems and control
List_of_people_in_systems_and_control
Algorithm in queueing theory
achieves maximum network throughput, which is established using concepts of Lyapunov drift. Backpressure routing considers the situation where each job can
Backpressure_routing
System where changes of output are not proportional to changes of input
in Hamiltonian systems Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities Linearization via Taylor expansion
Nonlinear_system
Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press. ISBN 9781400841042.
Popov_criterion
Type of artificial neural network
conditions is known as the Lyapunov exponent. We assume the output of the logistic map can be manipulated through a control parameter c [ x ( t ) , t ]
Radial_basis_function_network
positive Lyapunov exponents. Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent
Hyperchaos
the desired u x {\displaystyle u_{x}} control. The control design is based on the augmented Lyapunov function candidate V 1 ( x , z 1 ) = V x ( x ) +
Strict-feedback_form
Property of functions which is weaker than continuity
measures is compact. More generally, many functionals of interest—such as Lyapunov exponents, dimension spectra, or return time statistics—are semicontinuous
Semi-continuity
Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that robustly
Barrier_certificate
Branch of ordinary differential equations
exponents are Lyapunov exponents. The zero solution is asymptotically stable if all Floquet exponents have negative real part. It is Lyapunov stable if all
Floquet_theory
Graphical method of determining the stability of a dynamical system
graph of a nonlinear operator. Additionally, other stability criteria like Lyapunov methods can also be applied for non-linear systems. Although Nyquist is
Nyquist_stability_criterion
Automation and Remote Control. 49: 152–157. Barabanov, N. E. (1988). "Lyapunov indicators of discrete nestings II". Automation and Remote Control. 49: 283–287
Joint_spectral_radius
Field of mathematics and science based on non-linear systems and initial conditions
scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: Chaotic electrical circuits, about 1 millisecond;
Chaos_theory
generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called
Jan_Camiel_Willems
oscillatory, wave-like, character. Chetaev’s method of constructing Lyapunov functions as a coupling (combination) of first integrals. The previous result
Nikolay_Gur'yevich_Chetaev
Hungarian-American mathematician (1930–2016)
Kalman, R. E.; Bertram, J. E. (1960). "Control system analysis and design Via the "second method" of Lyapunov: I — Continuous-time systems". Journal of
Rudolf_E._Kálmán
Space of all possible states that a system can take
(trajectories) on the phase diagram. A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. However the
Phase_space
Aleksandr Lyapunov, founder of stability theory, author of the Lyapunov's central limit theorem, Lyapunov equation, Lyapunov fractal, Lyapunov time etc
List of Russian mathematicians
List_of_Russian_mathematicians
American control theorist
parameters, using Lyapunov, swapping, and passive estimators traffic flow stabilization control of ARZ PDEs (with Yu) additive manufacturing control of Stefan
Miroslav_Krstić
Method of analysing a dynamical system
measures with the positive maximal Lyapunov exponent is not as easy as stated, but even more complex (to calculate the Lyapunov exponent from an RP, the whole
Recurrence quantification analysis
Recurrence_quantification_analysis
Sequence of data points over time
Correlation dimension Recurrence plots Recurrence quantification analysis Lyapunov exponents Entropy encoding Time-series metrics or features that can be
Time_series
Fundamental theorem in condensed matter physics
times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied to
Bloch's_theorem
Thermodynamically open system which is not in equilibrium
dynamical system, the role of Lyapunov functions. Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear
Dissipative_system
Decision-making approach
adaptation of adaptive collaborative control in 2010 was used to design a fault tolerant control system using a Lyapunov function based analysis. The simuland
Adaptive collaborative control
Adaptive_collaborative_control
Nonlinear equation which arises on linear optimal control problems
inside the unit circle. Lyapunov equation Schur decomposition Sylvester equation Chow, Gregory (1975). Analysis and Control of Dynamic Economic Systems
Algebraic_Riccati_equation
Type of functional equation (mathematics)
equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the
Differential_equation
Type of differential equation
Tsuyoshi; Sasaki, Toru; Takeuchi, Yasuhiro (2012-08-01). "Construction of Lyapunov functionals for delay differential equations in virology and epidemiology"
Delay_differential_equation
Visual representation used in non-linear control system analysis
equation. The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional
Phase_plane
Matrix equation in control theory
applications, the derived Sylvester equation has a closed form solution. Lyapunov equation, a special case of the Sylvester equation Algebraic Riccati equation
Sylvester_equation
Iterative method for solving the Sylvester matrix equations
B} are normal matrices. These assumptions are met, for example, by the Lyapunov equation A X + X A ∗ = C {\displaystyle AX+XA^{*}=C} when A {\displaystyle
Alternating-direction implicit method
Alternating-direction_implicit_method
Area of mathematics
identified as the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can
Catastrophe_theory
Mathematical Theory
the squares of all queue sizes at time t, and is called a Lyapunov function. The Lyapunov drift is defined: Δ L ( t ) = L ( t + 1 ) − L ( t ) {\displaystyle
Drift_plus_penalty
Probabilistic problem-solving algorithm
1016/S0304-4149(99)00094-0. Del Moral, Pierre (2003). "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups"
Monte_Carlo_method
Romanian systems theorist
nonlinear feedback control systems. He continued this work and obtained the equivalence between the state space (Lyapunov function based) approach and
Vasile_M._Popov
Continuous-time linear system with only negative real parts
In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles
Exponential_stability
Feature of systems that defy description
inductive Turing machines can decrease even the complexity class of a function, language or set (Burgin 2005). This shows that tools of activity can be
Complexity
Cybernetics), was launched with Lyapunov as its editor. For the 1960 First International Federation of Automatic Control, Wiener came to Russia to lecture
Cybernetics in the Soviet Union
Cybernetics_in_the_Soviet_Union
(6): 1304–1307. Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control" (PDF). Proceedings of the National Academy
Kalman–Yakubovich–Popov_lemma
Machine learning technique where agents learn from demonstrations
hdl:10059/2298. ISSN 0360-0300. Wei, Yusi; Arvin, Farshad; Hu, Junyan (2026). "Lyapunov-Constrained Behavior Cloning for Robust Trajectory Tracking in Autonomous
Imitation_learning
Indian electrical engineer
systems. He is known to have introduced Lyapunov's method in researches on power system stability and energy functions. Introduction of computer applications
Mangalore_Anantha_Pai
Topics referred to by the same term
dynamical systems Asymptotic stability Exponential stability Linear stability Lyapunov stability Marginal stability Orbital stability Structural stability Stability
Stability
financial derivatives pricing and risk management. By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses
Deep backward stochastic differential equation method
Deep_backward_stochastic_differential_equation_method
Numerical method for solving physical or engineering problems
Bessel function To complete the discretization, we must select a basis of V {\displaystyle V} . In the one-dimensional case, for each control point x
Finite_element_method
terminating at 1? Lyapunov function: Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Partial differential equation with nonlinear terms
to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Branch of mathematical biology
exhibiting both Hopfield and Attractor-like network dynamics. The Lyapunov function is a nonlinear technique used to analyze the stability of the zero
Dynamical_neuroscience
Soviet and Russian mathematician
publication Problems of Cybernetics, with Lyapunov as its first editor-in-chief. Yablonsky succeeded Lyapunov as the editor-in-chief of Problems of Cybernetics
Sergey_Yablonsky
System of equations in mathematics
is not completely solvable for the derivatives of all components of the function x because these may not all appear (i.e. some equations are algebraic);
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Methods of mathematical approximation
Eigenvalue perturbation Homotopy perturbation method Interval finite element Lyapunov stability Method of dominant balance Order of approximation Perturbation
Perturbation_theory
Average value of a random variable
special case that f(x) = |x|t/s for positive numbers s < t, one obtains the Lyapunov inequality ( E | X | s ) 1 / s ≤ ( E | X | t ) 1 / t . {\displaystyle
Expected_value
Simple polynomial map exhibiting chaotic behavior
logistic map Logistic function, solution of the logistic map's continuous counterpart: the Logistic differential equation. Lyapunov stability#Definition
Logistic_map
Nonlinear two-terminal fundamental circuit element
properties in common with a Hopfield network, such as the existence of Lyapunov functions and classical tunnelling phenomena. In the context of memristive networks
Memristor
Soviet military analyst, engineer, mathematician and information scientist
1962. P. 357–362. Berg A.I., Kitov A.I., Lyapunov A.A. On the Possibilities of the Automation of Control in the National Economy // Soviet Computer
Anatoly_Kitov
Zinober, Alan S.I., ed. (1994). Variable Structure and Lyapunov Control. Lecture Notes in Control and Information Sciences. Vol. 193. London: Springer-Verlag
Variable_structure_system
Lyapunov Asymptotic Exponential Series solutions Rate of convergence Asymptotic series Special functions Numerical integration Dirac delta function Solution
List of named differential equations
List_of_named_differential_equations
Differential equation containing derivatives with respect to only one variable
DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with
Ordinary differential equation
Ordinary_differential_equation
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION
CONTROL LYAPUNOV-FUNCTION