# Lyapunov stability theorem pdf

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stability the sense of Lyapunov (i.s.L.). It is p ossible to ha v e stabilit y in Ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i.s.L., as the follo wing example sho ws. An equilibrium p oin t that is not stable i.s.L. termed unstable. Lyapunov Stability Game Lyapunov Theorem and Level Curves The limiting level curve V(x 1,x 2)=V(0)=0 is 0 at the equilibrium state x e=0 The trajectory moves through the level curves by cutting them in the inward direction ultimately ending at x e=0. 59 The trajectory is moving in the direction of decreasing V Note that. 60 Level Sets The level curves can be thought of as contours of a. 2 Lyapunov-Krasovskii stability theorem for fractional systems with delay system transient response, or generally, even an instability. Numerous reports have been published on this matter, with particular emphasis on the application of Lyapunov’s second method [5, 6]. In recent years, considerable attention has been paid to control systems whose processes and/or controllers are of File Size: KB.

# Lyapunov stability theorem pdf

Skip to main content. Stability by linearization Consider the system 4. By Naveen S. Download PDF. Thang Truong. Further, it is required that the convergence in equation 4.Lyapunov Stability Game Lyapunov Theorem and Level Curves The limiting level curve V(x 1,x 2)=V(0)=0 is 0 at the equilibrium state x e=0 The trajectory moves through the level curves by cutting them in the inward direction ultimately ending at x e=0. 59 The trajectory is moving in the direction of decreasing V Note that. 60 Level Sets The level curves can be thought of as contours of a. Stability Lyapunov theorem Theorem If there is V(x) 2C1 such that it is pdin B (0) and V_ is nsdin B (0), then x = 0 isstable. If in addition V_ (x) is ndin B (0) then x = 0 isAS. Extremely useful! Not necessary to compute state trajectories: it is enough to check the sign a..) =).)) +).)) +. " # 4 = =! =: = = Nonlinear systems - Lyapunov stability theory G. Ferrari Trecate. • Lyapunov stability conditions • the Lyapunov operator and integral • evaluating quadratic integrals • analysis of ARE • discrete-time results • linearization theorem 13–1. The Lyapunov equation the Lyapunov equation is ATP +PA+Q = 0 where A, P, Q ∈ Rn×n, and P, Q are symmetric interpretation: for linear system x˙ = Ax, if V(z) = zTPz, then V˙ (z) = (Az)TPz +zTP(Az. Théorème de stabilité de apunoyL v Références: Rouvière, Petit guide du alculc di érentiel à l'usage de la licence et de l'agrgationé, 3 e édition Cassini, p. A Lyapunov global asymptotic stability theorem suppose there is a function V such that • V is positive deﬁnite • V˙ (z). Theorem gives sufficient conditions for the stability of the origin of a system. It does not, however, give a prescription for determining the Lyapunov function V (x, t). Since the theorem only gives sufficient conditions, the search for a Lyapunov function establishing stability of an equilibrium point could be arduous. However, it is a. and the other is the Lyapunov’s direct method which concerns with construction of the Lyapunov function. The stability problem has motivated the study of Lyapunov function in both finite (,  and ) and infinite dimensional ( and ) spaces. Here, the Lyapunov’s direct method is used. It is the purpose of this paper to investigate the exponential and asymptotic stabilization for. Lyapunov Stability Theorem Theorem 1 (Lyapunov Theorem). Let DˆRnbe a set containing an open neighborhood of the origin. If there exists a PD function V: D!R such that L fV is NSD (2) then the origin is stable. If in addition, L fV is ND (3) then the origin is asymptotically stable. Remarks: A PD C1 function satisfying (2) or (3) will be called a Lyapunov function Under condition (3), if V. stability the sense of Lyapunov (i.s.L.). It is p ossible to ha v e stabilit y in Ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i.s.L., as the follo wing example sho ws. An equilibrium p oin t that is not stable i.s.L. termed unstable. Lyapunov Theorem V pdf + V˙ nsdf → stable V pdf + V˙ ndf → asymptotically stable Lecture 4 – p. 10/ Lyapunov Stability Theorem Example Consider the pendulum equation with friction x ˙1 = x2 x˙2 = − g l sinx1− k m x2 V1(x) = a(1−cos(x1))+(1/2)x2 2 ⇒ Stable. V2(x) = a(1−cos(x1))+(1/2)xTPx ⇒ Asympt. Stable. Conclusion: Lyapunov’s stability conditions are only.

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2Basic Lyapunov Theory Nonlinear Systems, time: 1:18:12
Tags: Abramos el corazon pdf, Hbr september 2013 pdf, 2 Lyapunov-Krasovskii stability theorem for fractional systems with delay system transient response, or generally, even an instability. Numerous reports have been published on this matter, with particular emphasis on the application of Lyapunov’s second method [5, 6]. In recent years, considerable attention has been paid to control systems whose processes and/or controllers are of File Size: KB. Lyapunov Stability Theorems Theorem – 1 (Stability) () () Let 0 be an equilibrium point of. Let: be a continuously differentiable function such that: 0 0 0, {0} 0, {0} Then 0 is "stable". XXfXfDn VD iV ii V X in D iii V X in D X ==→ → = >− ≤− = R R. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 13 Lyapunov Stability Theorems Theorem – 2. • Lyapunov stability conditions • the Lyapunov operator and integral • evaluating quadratic integrals • analysis of ARE • discrete-time results • linearization theorem 13–1. The Lyapunov equation the Lyapunov equation is ATP +PA+Q = 0 where A, P, Q ∈ Rn×n, and P, Q are symmetric interpretation: for linear system x˙ = Ax, if V(z) = zTPz, then V˙ (z) = (Az)TPz +zTP(Az. Lyapunov Theorem V pdf + V˙ nsdf → stable V pdf + V˙ ndf → asymptotically stable Lecture 4 – p. 10/ Lyapunov Stability Theorem Example Consider the pendulum equation with friction x ˙1 = x2 x˙2 = − g l sinx1− k m x2 V1(x) = a(1−cos(x1))+(1/2)x2 2 ⇒ Stable. V2(x) = a(1−cos(x1))+(1/2)xTPx ⇒ Asympt. Stable. Conclusion: Lyapunov’s stability conditions are only. Stability Lyapunov theorem Theorem If there is V(x) 2C1 such that it is pdin B (0) and V_ is nsdin B (0), then x = 0 isstable. If in addition V_ (x) is ndin B (0) then x = 0 isAS. Extremely useful! Not necessary to compute state trajectories: it is enough to check the sign a..) =).)) +).)) +. " # 4 = =! =: = = Nonlinear systems - Lyapunov stability theory G. Ferrari Trecate.Théorème de stabilité de apunoyL v Références: Rouvière, Petit guide du alculc di érentiel à l'usage de la licence et de l'agrgationé, 3 e édition Cassini, p. Theorem gives sufficient conditions for the stability of the origin of a system. It does not, however, give a prescription for determining the Lyapunov function V (x, t). Since the theorem only gives sufficient conditions, the search for a Lyapunov function establishing stability of an equilibrium point could be arduous. However, it is a. Lyapunov Theorem V pdf + V˙ nsdf → stable V pdf + V˙ ndf → asymptotically stable Lecture 4 – p. 10/ Lyapunov Stability Theorem Example Consider the pendulum equation with friction x ˙1 = x2 x˙2 = − g l sinx1− k m x2 V1(x) = a(1−cos(x1))+(1/2)x2 2 ⇒ Stable. V2(x) = a(1−cos(x1))+(1/2)xTPx ⇒ Asympt. Stable. Conclusion: Lyapunov’s stability conditions are only. stability the sense of Lyapunov (i.s.L.). It is p ossible to ha v e stabilit y in Ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i.s.L., as the follo wing example sho ws. An equilibrium p oin t that is not stable i.s.L. termed unstable. Lyapunov Stability Theorems Theorem – 1 (Stability) () () Let 0 be an equilibrium point of. Let: be a continuously differentiable function such that: 0 0 0, {0} 0, {0} Then 0 is "stable". XXfXfDn VD iV ii V X in D iii V X in D X ==→ → = >− ≤− = R R. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 13 Lyapunov Stability Theorems Theorem – 2. Moreover, a similar Lyapunov condition is presented for exponential stability that generalizes the stability condition in Beker et al. Beker et al. (, Theorem 1) in several directions. First, our results use locally Lipschitz–Lyapunov functions, including piecewise quadratic Lyapunov functions, as opposed to differentiable Lyapunov functions that were used in Beker et al. (). Lyapunov Stability Theorem Theorem 1 (Lyapunov Theorem). Let DˆRnbe a set containing an open neighborhood of the origin. If there exists a PD function V: D!R such that L fV is NSD (2) then the origin is stable. If in addition, L fV is ND (3) then the origin is asymptotically stable. Remarks: A PD C1 function satisfying (2) or (3) will be called a Lyapunov function Under condition (3), if V. 1 Lyapunov Local Stability Theorems Consider the map x0= f(x); x e = 0 (1) De ne the Lyapunov function V: N!R by (L1) V is pd (L2) 4V = V(x0) V(x) is nsd Theorem 1 (Stability) If there exists a Lyapunov function for the system of Eq(1), then x e = 0 is stable. Theorem 2 (Asymptotic stability) If there exists a Lyapunov function for the system of Eq(1), with the additional property that (L3. Lyapunov Theorem V pdf + V˙ nsdf → stable V pdf + V˙ ndf → asymptotically stable Lecture 4 – p. 10/ Lyapunov Stability Theorem Example Consider the pendulum equation with friction x ˙1 = x2 x˙2 = − g l sinx1− k m x2 V1(x) = a(1−cos(x1))+(1/2)x2 2 ⇒ Stable. V2(x) = a(1−cos(x1))+(1/2)xTPx ⇒ Asympt. Stable. Conclusion: Lyapunov’s stability conditions are only. and the other is the Lyapunov’s direct method which concerns with construction of the Lyapunov function. The stability problem has motivated the study of Lyapunov function in both finite (,  and ) and infinite dimensional ( and ) spaces. Here, the Lyapunov’s direct method is used. It is the purpose of this paper to investigate the exponential and asymptotic stabilization for.

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