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Lyapunov stability in control system pdf

Continue This article is about the asymptomatic stability of nonlineous systems. For the stability of linear systems, see . Part of the Series on Astrodynamics Orbital Settings Argument Periapsy Azimut Eccentricity Tilt Middle Anomaly Orbital Nodes Semi-Large Axis True types of anomalies of two-door orbits, by the eccentricity of the of the Elliptical Orbit Orbit Transmission of Orbit (Hohmann transmission of orbitBi-elliptical orbit of transmission of orbit) Parabolic orbit Hyperbolic orbit Radial Orbit Decay Orbital Equation Dynamic friction Escape speed Ke Kepler Equation Kepler Acts Planetary Motion Specific Orbital Vis-viva Equation Celestial Gravitational Mechanics Sphere Influence of N-Body OrbitLagrangian Dots () Lissajous Orbits The Lapun Orbit Engineering and Efficiency Of the Pre-Flight Engineering Payload ratio propellant the mass fraction of the Tsiolkovsky Missile Equation Gravity Measure help the of Vte Different types of stability can be discussed to address differential equations or differences in equations describing dynamic systems. The most important type is that of stability decisions near the equilibrium point. This may be what Alexander Lyapunov's theory says. Simply put, if solutions that start near the equilibrium point x e display style x_e remain close to x e display style x_ forever, then x e display style x_ e is the Lapun stable. More strongly, if x e displaystyle x_e) is a Lyapunov stable and all the solutions that start near x e display style x_ converge with x e display style x_, then x e displaystyle x_ is asymptotically stable. The notion of exponential stability guarantees a minimum level of decay, i.e. an assessment of how quickly solutions converge. The idea of Lyapunov's stability can be extended to infinitely dimensional diversity, where it is known as , which concerns the behavior of different but near solutions to differential equations. Entering Stability (ISS) applies the concepts of Lyapunov to login systems. In a limited problem with the three bodies of the orbit of Lyapunov are curved paths around the , which completely lie in the plane of the two primary bodies, as opposed to the halo-orbits and orbits of The Foxjuz, which also move above and below the plane. The history of Lyapunov's stability is named after Alexander Lyapunov, a Russian mathematician who defended his thesis The Common Problem of Movement Stability at Kharkiv University in 1892. A.M. Lyapunov was a pioneer in the successful development of a global approach to analyzing the stability of nonlinear dynamic systems compared to the widely disseminated Method method His work, originally published in Russian and then translated into French, has not received much attention for many years. Mathematical theory of movement stability, founded by A.M. Lyapunov, significantly foreshadowed the time of its implementation in science and technology. Moreover, Lapunov himself did not make applications in this area, his own interests were in the stability of rotating liquid masses with astronomical application. He had no doctoral students who followed the studies in the field of stability, and his own fate was terribly tragic because of the Russian Revolution of 1917. For several decades, the theory of stability has fallen into total oblivion. Russian-Soviet mathematician and mechanic Nikolai Gurevich Chetayev, who worked at the Kazan Aviation Institute in the 1930s, was the first to realize the incredible scale of the discovery made by A.M. Lyapunov. Actually, his figure as a great scientist is comparable to the figure of A.M. Lyapunov. N.G. Chetayev's contribution to the theory was so significant that many mathematicians, physicists and engineers consider him a direct successor to Lyapunov and the next scientific descendant in the creation and development of mathematical theory of stability. Interest in it suddenly increased during the Cold War, when it was found that the so-called second Lyapunov method (see below) applied to the stability of aerospace guidance systems, which usually contain strong non-linear phenomena that are not curable by other methods. A large number of publications appeared then and since then in the literature of management and systems. Recently, the concept of the exhibitor Lyapunov (associated with the first method of discussing the stability of Lyapunov) has received widespread interest in connection with the theory of chaos. Lyapunov's stability methods are also used to find equilibrium solutions to traffic-related problems. Definition for Continuous Time Systems Consider autonomous non-linear dynamic system x ̇ f (x (t) , x (x_{0} 0) where x ( t) ∈ D ⊆ R n' displaystyle x(t) in mathematics (D'subsqete) matebb (R) means system status vector, D and F : D → R n displaystyle f: Mathematics Drightarrow mathbbR continuous on D displaystyle mathematics DD Assuming that f displaystyle f has a balance in the style of x e display x_ so the so. that f (x e) displaystyle f (x_ e) then this equilibrium is said to be a stable Lyapunov, if, for each ε zgt; 0 display (epsilon zgt;0) , there is a δ qgt; 0 display (delta zgt;0) so that if ‖ x (0) x ‖ qlt; δ display x(0)-x_'e'lt'lt'delta, then for each tstyle ≥ 0 display t geq 0 we have a lt q‖ x (t) e ‖ < ε «displaystyle»x(t)- x_'e'<'epsilon . . The equilibrium of the aforementioned system is considered asymptically stable if it is stable and exists δ qgt; 0 display delta gt;0 in such a way that if ‖ x (0 ‖ qlt; δ) (0)-x_ 'e'lt'delta, then lim t → ∞ ‖ x (t) - x e ‖ 0 displaylim 't'rightarrow 'infty'x(t)-x_'e'e'0'0. The equilibrium of the above system is said to be exponentially stable if it is asymptot stable and there are α of the system, β, δ qgt; 0 display alpha zgt'0, beta zgt;0, delta zgt;0 so that if ‖ x x x ‖ δ ‖ 'display' x(0)-x_ zlt; x (t) - x e ‖ ≤ α ‖ x (0 ) - x e ‖ e - β t displaystylex (t)-x_'e'leq alpha x (0)-x_e-beta-ts For all t ≥ 0 displaystyle t'geq 0. Conceptually, the meaning of the above terms is as follows: Liapuns balance stability means that decisions beginning close enough to equilibrium (within a distance of δ delta display from it) remain close enough forever (within a distance ε displaystyle epsilon from it). Please note that this should be true for any ε displaystyle epsilon that you can choose from. Asymptomatic stability means that decisions that start close enough not only stay close enough, but ultimately converge to equilibrium. Exponential stability means that solutions not only converge, but actually converge faster than, or at least as fast as a certain known speed α ‖ x (0 ) , x e ‖ e - β i.e. display style alpha x(0)-x_e-beta-t. The trajectory x (locally) is attractive if ‖ g (t) - x ( t) ‖ → 0 displaystyley (t)-x(t → ∞) and globally attractive if this property has all trajectories. That is, if x refers to the interior of its stable variety, it is asymptically stable, if it is both attractive and stable. (There are examples showing that attractiveness does not imply asymptomatic stability. If the Jacobian dynamic system on balance turns out to be a matrix of stability (i.e. if the real part of each egenval is strictly negative), the balance is asymptically stable. The system in deviations Instead of considering an arbitrary solution φ (t) display (t) can reduce the problem to a zero solution study φ ̇ φ ̇ φ. is called a deviation system. Most of the results are formulated for such systems. The second method of stability of Lyapunov in his original work in 1892 proposed two ways of demonstrating stability. The first method developed a solution in a series that then turned out to be converged within. The second method, which is now called the Stability Criterion of Lyapunov or the Straight method, uses the Lyapunov V(x function), which has an analogy with the potential function of classical dynamics. It is introduced as follows for the system x ̇ th f (x) display style dotf(x) having a equilibrium point on x Consider function V : R n → R (display V:'mathbb (R) (R) rightarrow (r) such that V (x) 0 V (x) if and only if x ≠ 0 display xeq 0 v ̇ (x ∑) ∂ v ∂ x i f i (x) - ∇ v ⋅ f (x) ≤ 0 display (point V (x) Frak (partially va partial x_ yo-i f_ (x) abla's V'c displaystyle xeq 0 for all x ≠ 0 displaystyle xeq 0. Note: asymptotic stability requires V ̇ (x) qlt; 0 display (V point) (x) qlt;0 for x ≠ 0 displaystyle xeq 0. and the system is stable in the sense of Lyapunov (Note that V (0) , 0 display V(0)0 is required; otherwise, for example, V (x) Display style V (x)1/(1x) will prove that ̇ (t) - x displaystyle (dot) (t) - is locally stable). Achieving global stability requires an additional condition called correctness or radial limitlessness. Global asymptomatic stability (GAS) follows in a similar way. It is easier to visualize this method of analysis by thinking about the physical system (e.g. vibrating spring and mass) and taking into account the energy of such a system. If the system loses energy over time and the energy never recovers, then eventually the system must grind to a halt and reach some final state of rest. This final condition is called an amort. However, finding a function that provides the exact energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems the concept of energy may not be applicable. Lapunov realizes that stability can be proven without requiring knowledge of true physical energy, provided that the function of Lyapunov can be found to meet the above limitations. Definition for discrete time systems Definition for discrete time systems is almost identical to the definition for continuous time systems. The definition below provides for this using an alternative language commonly used in more mathematical texts. Let (X, d) be the and f : X → X . A x in X is considered to be a stable lyapun if, ∀ ε zgt; 0 ∃ δ zgt; 0 ∀ g ∈ x x, d) δ ⇒ ∀ n ∈ n d (f n) ( f n ) f n (y) q lt; ε . > > ...... Left d (x,y) qlt;delta (forall n'in mathbf N (f'n) (x), f'n(y) (right) qlt;epsilon (right). that x is asymptomatically stable if it belongs to the interior of its stable set, i.e. if, ∃ δ x (x) δ ⇒ → ∞ g (f n ( f n ( y) delta (rightarrow)(lim) (lim) to oil (f'n) , f'n (y) (right) Stability for linear models of state space Linear model of the state space x ̇ - x display style (x'textbf) where displaystyle A is the ultimate matrix, is asymptomatically stable (actually, exponentially stable) if all the real parts of eigenvalu A This condition is equivalent to the following: T M and M A (display A)T'MA) is negative for some positive defined M matrix (appropriate function of Lapunov - V (x) - x T M x (display V(x) , a time-discrete linear model of the state space x t No. 1 - x t display - asymptomatically stable (actually, exponentially stable) if all Egenvales A'displaystyle A) have modusons less than one. This latter condition was summarized for switched systems: linear switched discrete time systems (rule set of matrix - A 1 , ... , a m A_ A_{1}. x x x 1 - I t x t, I t A_ ∈ i_ 1, ... m Tyatembf (x't), four A_ i_t't (A_{1}), points, A_ - asymptomatically stable (actually, exponentially stable) if the joint spectral radius of set No. 1 , ... A_, A m A_{1}. Stability for systems with inputs System With inputs (or controls) has the form of x ̇ f(x,u) display style textbf x textbf f(x,u) where (usually depending on the time) input u(t) can be considered as control, external input, incentive, dysfunction. It has been shown that near the equilibrium point, which is the Lyapunov stable, the system remains stable with minor disturbances. For larger input violations, the study of such systems is the subject of management theory and is applied in engineering management. For login systems, you need to quantify the impact of input on system stability. The main two approaches to this analysis are the stability of BIBO (for linear systems) and stability in the state (ISS) (for non-linear systems) Example Consider an equation where compared to the equation van der Paul Paul Term changed: y y y ε (y ̇ 3 3 - y ̇) 0. Display-style ddot (y'y)y - varepsilon (left) (frak (dot)y'{3} {3}-dot (right) Here is a good example of a failed attempt to find the , which proves stability. let x 1 y, x 2 y ̇ display style x_{1} th, x_{2} point y th, so the appropriate system x ̇ 1 x 2 , x ̇ 2 x x 1 x ε (x 2 3 x 2). Display-style beginning of the aligned point (x'{1}'x_{2}), point x {2}-x_{1}-x_{1} varepsilon on the left (frak x_{2} {3}{3}-x_{2}) Balance x 1 x x 0, x 2 and 0. Display style x_{1} 0, x_{2} 0. Let's choose as a function of Lyapunov v No. 1 2 (x 1 2 x 2 2 2) display Vfrac {1}{2} (x_{1}-{2} x_{2}-{2}) which is clearly positive definite. Its derivative is v ̇ and x 1 x ̇ 1 x x 2 x ̇ 2 x 1 x 2 x 1 x 2 ε x 2 x 2 x 2 x 2 ε ε x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ε 2 2 2 2 2 2 2. «Дисплейстайл »точка »Ви x_{1}»точка (х){1}»x_{2} точка (х{2}x_{1}x_{2}-x_{1}x_{2} »варепсилон »фрак »x_{2}»{4} {3}-x_{2} (x_{2}) {2} варепсилон (фрак) x_{2} {4} {3}-варепсилон (x_{2} {2}). Кажется, что если параметр ε «дисплей стиль »varepsilon» является положительным, стабильность является асимптотической для х 2 < 3.= {\displaystyle=> <3.} but= this= is= wrong,= since= v= ˙= {\displaystyle= {\dot= {v}}}= does= not= depend= on= x= 1= {\displaystyle= x_{1}}= ,= and= will= be= 0= everywhere= on= the= x= 1= {\displaystyle= x_{1}}= axis.= the= equilibrium= is= lyapunov= stable.= barbalat's= lemma= and= stability= of= time-varying= systems= assume= that= f= is= a= function= of= time= only.= having= f= ˙= (= t= )= →= 0= {\displaystyle= {\dot= {f}}(t)\to= 0}= does= not= imply= that= f= (= t= )= {\displaystyle= f(t)}= has= a= limit= at= t= →= ∞= {\displaystyle= t\to= \infty= }= .= for= example (q f) (t) sin ⁡ ,ln'⁡ (t q q ) , q t'gt;2 0 display style f't)sin (ln(t)),;t'gt;0. Having a f (t) displaystyle f(t) approaches the limit as t → ∞ display t to infty does not mean that f ̇ (t) → 0 display (t) ( t {2}) for example, f (t) - sin ⁡ (t 2) / t, t,t t'gt;0 . having a f (t) displaystyle f't lower limit and decreasing (f ̇ ≤ 0 display (fleq 0) means that ̇ → it converges to the limit. → ∞ displaystyle t to infty. Lemma Barbalata says: If f (t) (displaystyle f't) has a final limit, as t → ∞ displaystyle t to infty and if the f ̇ display (point) is evenly continuous (or f q display ddot f is limited) , then f ̇ (t) → ̇ 0 display (point f (t) to 0 as t → ∞ ⋅ displaystyle t to infty. >w(t) g ̇ and e ⋅ w (t) . Displaystyle g-e'cdot w(t). Suppose the entry w (t) (displaystyle w(t) is limited. Taking V e 2 g 2 display V {2}g'{2} gives v ̇ 2 e 2 ≤ 0. Displaystyle (Ve point{2} 0. that V (t) ≤ V (0) (displaystyle V(t) leq V(0) under the first two conditions and therefore e displaystyle e and g display gstyle are limited. The use of the Barbagal lema: v - 4 e (e g g ⋅ w) . This is limited because e displaystyle e, g displaystyle g and w displaystyle w are limited ̇ →. 0) as t → ∞ displaystyle t to infty and therefore e → 0 displaystyle eto 0. , AM Common problem of movement stability (in Russian language), doctoral thesis, Univ. Kharkiv 1892 English translations: (1) Stability of the movement, Academic press, New York and London, 1966 (2) Common problem of movement stability, (A. T. Fuller trans.) Taylor and Francis, London 1992. Included is Smirnov's biography and an extensive bibliography of Lyapunov's work. Chetayev, N.G. On Stable Dynamics Trajectories, Kazan Unive Sci Notes, vol.4 No.1 1936; Stability of the movement, originally published in Russian language in 1946. State. a technical-theoretical. Lit., Moscow-Leningrad. Translation by Morton Nadler, Oxford, 1961, 200 pages. Letov, A.M. (1955). Stability of non-linear control systems (in Russian language). Moscow: Gostekhizdat. English tr. Princeton 1961 - Kalman, R.E.; Bertram, J. F (1960). Control system analysis and design using the second method of Lapunov: I am a continuous time system. In the journal Basic Engineering. 82 (2): 371–393. doi:10.1115/1.3662604. LaSal, J. P.; Lefshetz, S. (1961). Stability is the second method of Lyapunov with applications. New York: Academic press. Parks, P. C. (1962). The Lapunov method in the theory of automatic control. Management. I November 1962 II Dec 1962. Kalman, R.E. (1963). Lyapunov is working to solve the Lur'e problem in automatic control. Proc Natl Acad Sci USA. 49 (2): 201–205. Bibkod:1963PNAS... 49..201K. doi:10.1073/pnas.49.2.201. PMC 299777. PMID 16591048. Smith, M.J.; Visten, M.B. (1995). Continuous day-to-day traffic destination model and continuous dynamic user balance. Records Operational research. 60 (1): 59–79. doi:10.1007/BF02031940. Go, B.S. (1977). Global stability in many types of systems. American naturalist. 111 (977): 135–143. doi:10.1086/283144. - Malkin I.G. Movement , Moscow 1952 (Guesthizdat) Chap II paragraph 4 (Russian) Engl. transl, Bureau of Language Service, Vasileton AEK -tr-3352; originally About Stability in Permanent Violations of Prikl Mat 1944, vol. 8 No.3 241-245 Amer. Mat. Soc. transl. No 8 Further reading bhatia, Nam Parshad; Sege, Giorgio. Theory of dynamic systems stability. Springer. ISBN 978-3-540-42748-3. Gandolfo, Giancarlo (1996). Economic Dynamics (Third - Berlin: Springer. 407-428. ISBN 978-3-540-60988-9. Parks, P. C. (1992). The theory of stability of A.M. Lyapunov - 100 years later. IMA is a journal of mathematical control and information. 9 (4): 275–303. doi:10.1093/imamci/9.4.275. Slotin, jean-Jacques E.; Weiping Lee (1991). Applied non-linear control. NJ: Prentice Hall. Teshl, G. (2012). Conventional differential equations and dynamic systems. Providence: American Mathematical Society. ISBN 978-0- 8218-8328-0. Wiggins, S. (2003). Introduction to applied non-linear dynamic systems and chaos (2nd new York: Springer Verlag. ISBN 978-0-387-00177-7. This article includes material from the asymptotically stable at PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Extracted from the in control system pdf. lyapunov matrix equation in system stability and control

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