Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems

Size: px
Start display at page:

Download "Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems"

Transcription

1 p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems

2 p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D. The origin is an equilibrium point at t = 0 if f(t, 0) = 0, t 0 While the solution of the autonomous system ẋ = f(x), x(t 0 ) = x 0 depends only on (t t 0 ), the solution of may depend on both t and t 0 ẋ = f(t, x), x(t 0 ) = x 0

3 p. 3/5 Comparison Functions A scalar continuous function α(r), defined for r [0, a) is said to belong to class K if it is strictly increasing and α(0) = 0. It is said to belong to class K if it defined for all r 0 and α(r) as r A scalar continuous function β(r, s), defined for r [0, a) and s [0, ) is said to belong to class KL if, for each fixed s, the mapping β(r, s) belongs to class K with respect to r and, for each fixed r, the mapping β(r, s) is decreasing with respect to s and β(r, s) 0 as s

4 p. 4/5 Example α(r) = tan 1 (r) is strictly increasing since α (r) = 1/(1 + r 2 ) > 0. It belongs to class K, but not to class K since lim r α(r) = π/2 < α(r) = r c, for any positive real number c, is strictly increasing since α (r) = cr c 1 > 0. Moreover, lim r α(r) = ; thus, it belongs to class K α(r) = min{r, r 2 } is continuous, strictly increasing, and lim r α(r) =. Hence, it belongs to class K

5 p. 5/5 β(r, s) = r/(ksr + 1), for any positive real number k, is strictly increasing in r since β r = 1 (ksr + 1) 2 > 0 and strictly decreasing in s since β s = kr2 (ksr + 1) 2 < 0 Moreover, β(r, s) 0 as s. Therefore, it belongs to class KL β(r, s) = r c e s, for any positive real number c, belongs to class KL

6 p. 6/5 Definition: The equilibrium point x = 0 of ẋ = f(t, x) is uniformly stable if there exist a class K function α and a positive constant c, independent of t 0, such that x(t) α( x(t 0 ) ), t t 0 0, x(t 0 ) < c uniformly asymptotically stable if there exist a class KL function β and a positive constant c, independent of t 0, such that x(t) β( x(t 0 ), t t 0 ), t t 0 0, x(t 0 ) < c globally uniformly asymptotically stable if the foregoing inequality is satisfied for any initial state x(t 0 )

7 p. 7/5 exponentially stable if there exist positive constants c, k, and λ such that x(t) k x(t 0 ) e λ(t t 0), x(t 0 ) < c globally exponentially stable if the foregoing inequality is satisfied for any initial state x(t 0 )

8 p. 8/5 Theorem: Let the origin x = 0 be an equilibrium point for ẋ = f(t, x) and D R n be a domain containing x = 0. Suppose f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and x D. Let V (t, x) be a continuously differentiable function such that (1) W 1 (x) V (t, x) W 2 (x) (2) V t + V x f(t, x) 0 for all t 0 and x D, where W 1 (x) and W 2 (x) are continuous positive definite functions on D. Then, the origin is uniformly stable

9 p. 9/5 Theorem: Suppose the assumptions of the previous theorem are satisfied with V t + V x f(t, x) W 3(x) for all t 0 and x D, where W 3 (x) is a continuous positive definite function on D. Then, the origin is uniformly asymptotically stable. Moreover, if r and c are chosen such that B r = { x r} D and c < min x =r W 1 (x), then every trajectory starting in {x B r W 2 (x) c} satisfies x(t) β( x(t 0 ), t t 0 ), t t 0 0 for some class KL function β. Finally, if D = R n and W 1 (x) is radially unbounded, then the origin is globally uniformly asymptotically stable

10 p. 10/5 Theorem: Suppose the assumptions of the previous theorem are satisfied with k 1 x a V (t, x) k 2 x a V t + V x f(t, x) k 3 x a for all t 0 and x D, where k 1, k 2, k 3, and a are positive constants. Then, the origin is exponentially stable. If the assumptions hold globally, the origin will be globally exponentially stable.

11 p. 11/5 Example: ẋ = [1 + g(t)]x 3, g(t) 0, t 0 V (x) = 1 2 x2 V (t, x) = [1 + g(t)]x 4 x 4, x R, t 0 The origin is globally uniformly asymptotically stable Example: ẋ 1 = x 1 g(t)x 2 ẋ 2 = x 1 x 2 0 g(t) k and ġ(t) g(t), t 0

12 p. 12/5 V (t, x) = x [1 + g(t)]x2 2 x x2 2 V (t, x) x2 1 + (1 + k)x2 2, x R2 V (t, x) = 2x x 1x 2 [2 + 2g(t) ġ(t)]x g(t) ġ(t) 2 + 2g(t) g(t) 2 V (t, x) 2x x 1x 2 2x 2 2 = xt [ The origin is globally exponentially stable ] x

13 p. 13/5 Perturbed Systems Nominal System: ẋ = f(x), f(0) = 0 Perturbed System: ẋ = f(x) + g(t, x), g(t, 0) = 0 Case 1: The origin of the nominal system is exponentially stable c 1 x 2 V (x) c 2 x 2, V x c 4 x V x f(x) c 3 x 2

14 p. 14/5 Use V (x) as a Lyapunov function candidate for the perturbed system Assume that V (t, x) = V x f(x) + V x g(t, x) g(t, x) γ x, γ 0 V (t, x) c 3 x 2 + V x g(t, x) c 3 x 2 + c 4 γ x 2

15 p. 15/5 γ < c 3 c 4 V (t, x) (c 3 γc 4 ) x 2 The origin is an exponentially stable equilibrium point of the perturbed system

16 p. 16/5 Example ẋ 1 = x 2 ẋ 2 = 4x 1 2x 2 + βx 3 2, β 0 A = [ ẋ = Ax + g(x) ] 0 1, g(x) = 4 2 [ 0 βx 3 2 ] The eigenvalues of A are 1 ± j 3 PA + A T P = I P =

17 p. 17/5 V (x) = x T Px, V x Ax = xt x c 3 = 1, c 4 = 2 P = 2λ max (P) = = g(x) = β x 2 3 g(x) satisfies the bound g(x) γ x over compact sets of x. Consider the compact set Ω c = {V (x) c} = {x T Px c}, c > 0 k 2 = max x T P x c x 2 x 3 2 k2 2 x 2

18 p. 18/5 k 2 = max x T P x c [0 1]x = max y T y c = c [0 1]P 1/2 = c c [0 1]P 1/2 y g(x) β c (1.8194) 2 x, x Ω c g(x) γ x, x Ω c, γ = β c (1.8194) 2 γ < c 3 c 4 β < (1.8194) 2 c 0.1 c β < 0.1/c V (x) (1 10βc) x 2 Hence, the origin is exponentially stable and Ω c is an estimate of the region of attraction

19 p. 19/5 Alternative Bound on β V (x) = x 2 + 2x T Pg(x) x βx3 2 ([2 5]x) x βx2 2 x 2 Over Ω c, x 2 2 (1.8194)2 c ( ) V (x) β(1.8194)2 c ( = 1 βc ) x x 2 If β < 0.448/c, the origin will be exponentially stable and Ω c will be an estimate of the region of attraction

20 Remark: The inequality β < 0.448/c shows a tradeoff between the estimate of the region of attraction and the estimate of the upper bound on β p. 20/5

21 p. 21/5 Application to Linearization ẋ = f(x) = [A + G(x)]x A = f x (x), G(x) 0 as x 0 x=0 Theorem: The origin of ẋ = f(x) is exponentially stable if and only if A is Hurwitz

22 p. 22/5 Proof Of sufficiency: Suppose A is Hurwitz. Choose Q = Q T > 0 and solve the Lyapunov equation PA + A T P = Q for P Use V (x) = x T Px as a Lyapunov function candidate for ẋ = f(x) V (x) = x T Pf(x) + f T (x)px = x T P[A + G(x)]x + x T [A T + G T (x)]px = x T (PA + A T P)x + 2x T PG(x)x = x T Qx + 2x T PG(x)x

23 p. 23/5 V (x) x T Qx + 2 P G(x) x 2 For any γ > 0, there exists r > 0 such that G(x) < γ, x < r x T Qx λ min (Q) x 2 x T Qx λ min (Q) x 2 Choose V (x) < [λ min (Q) 2γ P ] x 2, x < r γ < λ min(q) 2 P The origin of ẋ = f(x) is exponentially stable

24 p. 24/5 Proof of Necessity: Suppose the origin of ẋ = f(x) is exponentially stable. View the system ẋ = Ax = f(x) G(x)x as a perturbation of ẋ = f(x). Recall that G(x) < γ, x < r Because the origin of ẋ = f(x) is exponentially stable, let V (x) be the function provided by the converse Lyapunov theorem over a domain { x < r 0 }. Use V (x) as a Lyapunov function candidate for ẋ = Ax.

25 p. 25/5 In the domain { x < min{r 0, r}}, we have V x V V Ax = f(x) x x G(x)x c 3 x 2 + c 4 γ x 2 = (c 3 c 4 γ) x 2 Take γ < c 3 /c 4, and set λ = (c 3 c 4 L) > 0 V x Ax λ x 2, x < min{r 0, r} The origin of ẋ = Ax is exponentially stable

26 p. 26/5 Case 2: The origin of the nominal system is asymptotically stable V (t, x) = V x f(x)+ V x g(t, x) W 3(x)+ V g(t, x) x Under what condition will the following inequality hold? V g(t, x) x < W 3(x) Special Case: Quadratic-Type Lyapunov function V x f(x) c 3φ 2 (x), V x c 4φ(x)

27 p. 27/5 V (t, x) c 3 φ 2 (x) + c 4 φ(x) g(t, x) If g(t, x) γφ(x), with γ < c 3 c 4 V (t, x) (c 3 c 4 γ)φ 2 (x)

28 p. 28/5 Example ẋ = x 3 + g(t, x) V (x) = x 4 is a quadratic-type Lyapunov function for the nominal system ẋ = x 3 V x ( x3 ) = 4x 6, V x = 4 x 3 φ(x) = x 3, c 3 = 4, c 4 = 4 Suppose g(t, x) γ x 3, x, with γ < 1 V (t, x) 4(1 γ)φ 2 (x) Hence, the origin is a globally uniformly asymptotically stable

29 p. 29/5 Remark: A nominal system with asymptotically, but not exponentially, stable origin is not robust to smooth perturbations with arbitrarily small linear growth bounds Example ẋ = x 3 + γx The origin is unstable for any γ > 0

30 p. 30/5 Ultimate Boundedness Definition: The solutions of ẋ = f(t, x) are uniformly bounded if c > 0 and for every 0 < a < c, β = β(a) > 0 such that x(t 0 ) a x(t) β, t t 0 0 uniformly ultimately bounded with ultimate bound b if b and c and for every 0 < a < c, T = T(a, b) 0 such that x(t 0 ) a x(t) b, t t 0 + T Globally if a can be arbitrarily large Drop uniformly if ẋ = f(x)

31 p. 31/5 Lyapunov Analysis: Let V (x) be a cont. diff. positive definite function and suppose that the sets Ω c = {V (x) c}, Ω ε = {V (x) ε}, Λ = {ε V (x) c} are compact for some c > ε > 0 Λ Ω ε Ω c

32 p. 32/5 Suppose V (t, x) = V x f(t, x) W 3(x), x Λ, t 0 W 3 (x) is continuous and positive definite Ω c and Ω ε are positively invariant k = min x Λ W 3(x) > 0 V (t, x) k, x Λ, t t 0 0 V (x(t)) V (x(t 0 )) k(t t 0 ) c k(t t 0 ) x(t) enters the set Ω ε within the interval [t 0, t 0 + (c ε)/k]

33 p. 33/5 Suppose V (t, x) W 3 (x), µ x r, t 0 Choose c and ε such that Λ {µ x r} B µ Ω ε Ω c B r

34 p. 34/5 Let α 1 and α 2 be class K functions such that α 1 ( x ) V (x) α 2 ( x ) V (x) c α 1 ( x ) c x α 1 1 (c) What is the ultimate bound? c = α 1 (r) Ω c B r x µ V (x) α 2 (µ) ε = α 2 (µ) B µ Ω ε V (x) ε α 1 ( x ) ε x α 1 1 (ε) = α 1 1 (α 2(µ))

35 p. 35/5 Theorem (special case of Thm 4.18): Suppose V α 1 ( x ) V (x) α 2 ( x ) x f(t, x) W 3(x), x µ > 0 t 0 and x r, where α 1, α 2 K, W 3 (x) is continuous & positive definite, and µ < α 1 2 (α 1(r)). Then, for every initial state x(t 0 ) { x α 1 2 (α 1(r))}, there is T 0 (dependent on x(t 0 ) and µ) such that x(t) α 1 1 (α 2(µ)), t t 0 + T If the assumptions hold globally and α 1 K, then the conclusion holds for any initial state x(t 0 )

36 p. 36/5 Remarks: The ultimate bound is independent of the initial state The ultimate bound is a class K function of µ; hence, the smaller the value of µ, the smaller the ultimate bound. As µ 0, the ultimate bound approaches zero

37 p. 37/5 Example ẋ 1 = x 2, ẋ 2 = (1 + x 2 1 )x 1 x 2 + M cos ωt, M 0 With M = 0, ẋ 2 = (1 + x 2 1 )x 1 x 2 = h(x 1 ) x 2 V (x) = x T V (x) = x T x x1 0 x x4 1 (y + y 3 ) dy (Example 4.5) def = x T Px x4 1

38 p. 38/5 λ min (P) x 2 V (x) λ max (P) x x 4 α 1 (r) = λ min (P)r 2, α 2 (r) = λ max (P)r r4 V = x 2 1 x4 1 x2 2 + (x 1 + 2x 2 )M cos ωt x 2 x M 5 x = (1 θ) x 2 x 4 1 θ x 2 + M 5 x (0 < θ < 1) (1 θ) x 2 x 4 1, x M 5/θ def = µ The solutions are GUUB by b = α 1 1 (α λ max (P)µ 2 + µ 4 /2 2(µ)) = λ min (P)

39 p. 39/5 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ = f(x) + g(t, x), g(t, 0) 0 Case 1: The origin of ẋ = f(x) is exponentially stable c 1 x 2 V (x) c 2 x 2 V x f(x) c 3 x 2, x B r = { x r} V x c 4 x

40 p. 40/5 Use V (x) to investigate ultimate boundedness of the perturbed system Assume V (t, x) = V x g(t, x) δ, V (t, x) c 3 x 2 + f(x) + V x V x g(t, x) t 0, x B r g(t, x) c 3 x 2 + c 4 δ x = (1 θ)c 3 x 2 θc 3 x 2 + c 4 δ x 0 < θ < 1 (1 θ)c 3 x 2, x δc 4 /(θc 3 ) def = µ

41 p. 41/5 Apply Theorem 4.18 x(t 0 ) α 1 2 (α 1(r)) x(t 0 ) r c1 µ < α 1 2 (α 1(r)) δc 4 c1 < r δ < c 3 c1 θr θc 3 c 2 c 4 c 2 b = α 1 c2 1 (α 2(µ)) b = µ b = δc 4 c2 c 1 θc 3 c 1 For all x(t 0 ) r c 1 /c 2, the solutions of the perturbed system are ultimately bounded by b c 2

42 p. 42/5 Example ẋ 1 = x 2, ẋ 2 = 4x 1 2x 2 + βx d(t) β 0, d(t) δ, t 0 V (x) = x T Px = x T x V (t, x) = x 2 + 2βx 2 ( x 1x ) 16 x d(t) ( 1 8 x x ) δ x βk2 2 x 2 + x 8

43 p. 43/5 k 2 = max x T P x c x 2 = c Suppose β 8(1 ζ)/( 29k2 2 ) (0 < ζ < 1) V (t, x) ζ x δ 8 x (1 θ)ζ x 2, x 29δ 8ζθ def = µ (0 < θ < 1) If µ 2 λ max (P) < c, then all solutions of the perturbed system, starting in Ω c, are uniformly ultimately bounded by 29δ λ max (P) b = 8ζθ λ min (P)

44 p. 44/5 Case 2: The origin of ẋ = f(x) is asymptotically stable α 1 ( x ) V (x) α 2 ( x ) V x f(x) α 3( x ), V x k x B r = { x r}, α i K, i = 1, 2, 3 V (t, x) α 3 ( x ) + V x g(t, x) α 3 ( x ) + δk (1 θ)α 3 ( x ) θα 3 ( x ) + δk (1 θ)α 3 ( x ), x α < θ < 1 ( δk θ ) def = µ

45 p. 45/5 Apply Theorem 4.18 µ < α 1 2 (α 1(r)) α 1 3 ( ) δk θ < α 1 2 (α 1(r)) δ < θα 3(α 1 2 (α 1(r))) k Example V (x) = x 4 Compare with δ < c 3 c1 θr c 4 c 2 ẋ = x 1 + x 2 V [ x ] x 1 + x 2 = 4x4 1 + x 2 α 1 ( x ) = α 2 ( x ) = x 4 ; α 3 ( x ) = 4 x x 2; k = 4r3

46 p. 46/5 The origin is globally asymptotically stable θα 3 (α 1 2 (α 1(r))) k = θα 3(r) k rθ 1 + r 2 0 as r ẋ = x 1 + x 2 + δ, δ > 0 δ > 1 2 lim t x(t) = = rθ 1 + r 2

47 p. 47/5 Input-to-State Stability (ISS) Definition: The system ẋ = f(x, u) is input-to-state stable if there exist β KL and γ K such that for any initial state x(t 0 ) and any bounded input u(t) ( ) x(t) β( x(t 0 ), t t 0 ) + γ ISS of ẋ = f(x, u) implies BIBS stability sup t 0 τ t u(τ) x(t) is ultimately bounded by a class K function of sup t t0 u(t) lim t u(t) = 0 lim t x(t) = 0 The origin of ẋ = f(x, 0) is GAS

48 p. 48/5 Theorem (Special case of Thm 4.19): Let V (x) be a continuously differentiable function such that V α 1 ( x ) V (x) α 2 ( x ) x f(x, u) W 3(x), x ρ( u ) > 0 x R n, u R m, where α 1, α 2 K, ρ K, and W 3 (x) is a continuous positive definite function. Then, the system ẋ = f(x, u) is ISS with γ = α 1 1 α 2 ρ Proof: Let µ = ρ(sup τ t0 u(τ) ); then V x f(x, u) W 3(x), x µ

49 p. 49/5 Choose ε and c such that V x f(x, u) W 3(x), x Λ = {ε V (x) c} Suppose x(t 0 ) Λ and x(t) reaches Ω ε at t = t 0 + T. For t 0 t t 0 + T, V satisfies the conditions for the uniform asymptotic stability. Therefore, the trajectory behaves as if the origin was uniformly asymptotically stable and satisfies x(t) β( x(t 0 ), t t 0 ), for some β KL For t t 0 + T, x(t) α 1 1 (α 2(µ))

50 p. 50/5 x(t) β( x(t 0 ), t t 0 ) + α 1 1 (α 2(µ)), t t 0 x(t) β( x(t 0 ), t t 0 ) + γ ( sup u(τ) τ t 0 ), t t 0 Since x(t) depends only on u(τ) for t 0 τ t, the supremum on the right-hand side can be taken over [t 0, t]

51 p. 51/5 Example ẋ = x 3 + u The origin of ẋ = x 3 is globally asymptotically stable V = 1 2 x2 V = x 4 + xu = (1 θ)x 4 θx 4 + xu (1 θ)x 4, x ( u θ ) 1/3 0 < θ < 1 The system is ISS with γ(r) = (r/θ) 1/3

52 p. 52/5 Example ẋ = x 2x 3 + (1 + x 2 )u 2 The origin of ẋ = x 2x 3 is globally exponentially stable V = 1 2 x2 V = x 2 2x 4 + x(1 + x 2 )u 2 = x 4 x 2 (1 + x 2 ) + x(1 + x 2 )u 2 x 4, x u 2 The system is ISS with γ(r) = r 2

53 p. 53/5 Example ẋ 1 = x 1 + x 2 2, ẋ 2 = x 2 + u Investigate GAS of ẋ 1 = x 1 + x 2 2, ẋ 2 = x 2 V (x) = 1 2 x x4 2 V = x x 1x 2 2 x4 2 = (x x2 2 )2 ( ) x 4 2 Now u 0, V = 1 2 (x 1 x 2 2 )2 1 2 (x2 1 + x4 2 ) + x3 2 u 1 2 (x2 1 + x4 2 ) + x 2 3 u V 1 2 (1 θ)(x2 1 + x4 2 ) 1 2 θ(x2 1 + x4 2 ) + x 2 3 u (0 < θ < 1)

54 p. 54/5 1 2 θ(x2 1 + x4 2 ) + x 2 3 u 0 if x 2 2 u or x 2 2 u and x 1 θ θ if x 2 u ( ) 2 u θ θ ρ(r) = 2r ( ) 2r θ θ ( 2 u θ ) 2 V 1 2 (1 θ)(x2 1 + x4 2 ), x ρ( u ) The system is ISS

Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability

Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability p. 1/1 Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability p. 2/1 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ

More information

Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems

Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin

More information

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1 Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium

More information

Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems

Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin

More information

Nonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points

Nonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence

More information

Lecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.

Lecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University. Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ

More information

Nonlinear Control Lecture 5: Stability Analysis II

Nonlinear Control Lecture 5: Stability Analysis II Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41

More information

High-Gain Observers in Nonlinear Feedback Control

High-Gain Observers in Nonlinear Feedback Control High-Gain Observers in Nonlinear Feedback Control Lecture # 1 Introduction & Stabilization High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 1/4 Brief History Linear

More information

Solution of Additional Exercises for Chapter 4

Solution of Additional Exercises for Chapter 4 1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the

More information

Lyapunov Stability Theory

Lyapunov Stability Theory Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous

More information

Converse Lyapunov theorem and Input-to-State Stability

Converse Lyapunov theorem and Input-to-State Stability Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts

More information

Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis

Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of

More information

Nonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points

Nonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points Nonlinear Control Lecture # 2 Stability of Equilibrium Points Basic Concepts ẋ = f(x) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f( x) = 0 Characterize and

More information

Input-to-state stability and interconnected Systems

Input-to-state stability and interconnected Systems 10th Elgersburg School Day 1 Input-to-state stability and interconnected Systems Sergey Dashkovskiy Universität Würzburg Elgersburg, March 5, 2018 1/20 Introduction Consider Solution: ẋ := dx dt = ax,

More information

High-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle

High-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle High-Gain Observers in Nonlinear Feedback Control Lecture # 2 Separation Principle High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 1/4 The Class of Systems ẋ = Ax + Bφ(x,

More information

Input to state Stability

Input to state Stability Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part III: Lyapunov functions and quantitative aspects ISS Consider with

More information

EN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015

EN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions

More information

2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ

2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ 1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.

More information

Lecture 8. Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control. Eugenio Schuster.

Lecture 8. Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control. Eugenio Schuster. Lecture 8 Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture

More information

Chapter III. Stability of Linear Systems

Chapter III. Stability of Linear Systems 1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,

More information

CDS Solutions to the Midterm Exam

CDS Solutions to the Midterm Exam CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2

More information

Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS

Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns

More information

EG4321/EG7040. Nonlinear Control. Dr. Matt Turner

EG4321/EG7040. Nonlinear Control. Dr. Matt Turner EG4321/EG7040 Nonlinear Control Dr. Matt Turner EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt

More information

Outline. Input to state Stability. Nonlinear Realization. Recall: _ Space. _ Space: Space of all piecewise continuous functions

Outline. Input to state Stability. Nonlinear Realization. Recall: _ Space. _ Space: Space of all piecewise continuous functions Outline Input to state Stability Motivation for Input to State Stability (ISS) ISS Lyapunov function. Stability theorems. M. Sami Fadali Professor EBME University of Nevada, Reno 1 2 Recall: _ Space _

More information

Nonlinear Control Systems

Nonlinear Control Systems Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes

More information

1 Lyapunov theory of stability

1 Lyapunov theory of stability M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

More information

Nonlinear Control. Nonlinear Control Lecture # 6 Passivity and Input-Output Stability

Nonlinear Control. Nonlinear Control Lecture # 6 Passivity and Input-Output Stability Nonlinear Control Lecture # 6 Passivity and Input-Output Stability Passivity: Memoryless Functions y y y u u u (a) (b) (c) Passive Passive Not passive y = h(t,u), h [0, ] Vector case: y = h(t,u), h T =

More information

Georgia Institute of Technology Nonlinear Controls Theory Primer ME 6402

Georgia Institute of Technology Nonlinear Controls Theory Primer ME 6402 Georgia Institute of Technology Nonlinear Controls Theory Primer ME 640 Ajeya Karajgikar April 6, 011 Definition Stability (Lyapunov): The equilibrium state x = 0 is said to be stable if, for any R > 0,

More information

STABILITY ANALYSIS OF DYNAMIC SYSTEMS

STABILITY ANALYSIS OF DYNAMIC SYSTEMS C. Melchiorri (DEI) Automatic Control & System Theory 1 AUTOMATIC CONTROL AND SYSTEM THEORY STABILITY ANALYSIS OF DYNAMIC SYSTEMS Claudio Melchiorri Dipartimento di Ingegneria dell Energia Elettrica e

More information

Modeling & Control of Hybrid Systems Chapter 4 Stability

Modeling & Control of Hybrid Systems Chapter 4 Stability Modeling & Control of Hybrid Systems Chapter 4 Stability Overview 1. Switched systems 2. Lyapunov theory for smooth and linear systems 3. Stability for any switching signal 4. Stability for given switching

More information

High-Gain Observers in Nonlinear Feedback Control. Lecture # 3 Regulation

High-Gain Observers in Nonlinear Feedback Control. Lecture # 3 Regulation High-Gain Observers in Nonlinear Feedback Control Lecture # 3 Regulation High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 1/5 Internal Model Principle d r Servo- Stabilizing u y

More information

Stability of Nonlinear Systems An Introduction

Stability of Nonlinear Systems An Introduction Stability of Nonlinear Systems An Introduction Michael Baldea Department of Chemical Engineering The University of Texas at Austin April 3, 2012 The Concept of Stability Consider the generic nonlinear

More information

L 2 -induced Gains of Switched Systems and Classes of Switching Signals

L 2 -induced Gains of Switched Systems and Classes of Switching Signals L 2 -induced Gains of Switched Systems and Classes of Switching Signals Kenji Hirata and João P. Hespanha Abstract This paper addresses the L 2-induced gain analysis for switched linear systems. We exploit

More information

7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system

7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system 7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically

More information

IN [1], an Approximate Dynamic Inversion (ADI) control

IN [1], an Approximate Dynamic Inversion (ADI) control 1 On Approximate Dynamic Inversion Justin Teo and Jonathan P How Technical Report ACL09 01 Aerospace Controls Laboratory Department of Aeronautics and Astronautics Massachusetts Institute of Technology

More information

Nonlinear Control Lecture # 14 Input-Output Stability. Nonlinear Control

Nonlinear Control Lecture # 14 Input-Output Stability. Nonlinear Control Nonlinear Control Lecture # 14 Input-Output Stability L Stability Input-Output Models: y = Hu u(t) is a piecewise continuous function of t and belongs to a linear space of signals The space of bounded

More information

Prof. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait

Prof. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use

More information

Solution of Linear State-space Systems

Solution of Linear State-space Systems Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state

More information

Introduction to Nonlinear Control Lecture # 4 Passivity

Introduction to Nonlinear Control Lecture # 4 Passivity p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive

More information

Convergence Rate of Nonlinear Switched Systems

Convergence Rate of Nonlinear Switched Systems Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the

More information

ẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.

ẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D. 4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R

More information

MCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory

MCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A

More information

LMI Methods in Optimal and Robust Control

LMI Methods in Optimal and Robust Control LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear

More information

arxiv: v3 [math.ds] 22 Feb 2012

arxiv: v3 [math.ds] 22 Feb 2012 Stability of interconnected impulsive systems with and without time-delays using Lyapunov methods arxiv:1011.2865v3 [math.ds] 22 Feb 2012 Sergey Dashkovskiy a, Michael Kosmykov b, Andrii Mironchenko b,

More information

1 The Observability Canonical Form

1 The Observability Canonical Form NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)

More information

Using Lyapunov Theory I

Using Lyapunov Theory I Lecture 33 Stability Analysis of Nonlinear Systems Using Lyapunov heory I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline Motivation Definitions

More information

Nonlinear Control Systems

Nonlinear Control Systems Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f

More information

Stability in the sense of Lyapunov

Stability in the sense of Lyapunov CHAPTER 5 Stability in the sense of Lyapunov Stability is one of the most important properties characterizing a system s qualitative behavior. There are a number of stability concepts used in the study

More information

Input to state Stability

Input to state Stability Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part IV: Applications ISS Consider with solutions ϕ(t, x, w) ẋ(t) =

More information

Hybrid Systems - Lecture n. 3 Lyapunov stability

Hybrid Systems - Lecture n. 3 Lyapunov stability OUTLINE Focus: stability of equilibrium point Hybrid Systems - Lecture n. 3 Lyapunov stability Maria Prandini DEI - Politecnico di Milano E-mail: prandini@elet.polimi.it continuous systems decribed by

More information

Dynamical Systems & Lyapunov Stability

Dynamical Systems & Lyapunov Stability Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence

More information

Nonlinear Control Systems

Nonlinear Control Systems Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear

More information

MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems MCE693/793: Analysis and Control of Nonlinear Systems Lyapunov Stability - I Hanz Richter Mechanical Engineering Department Cleveland State University Definition of Stability - Lyapunov Sense Lyapunov

More information

Chapter 2 Mathematical Preliminaries

Chapter 2 Mathematical Preliminaries Chapter 2 Mathematical Preliminaries 2.1 Introduction In this chapter, the fundamental mathematical concepts and analysis tools in systems theory are summarized, which will be used in control design and

More information

Stability theory is a fundamental topic in mathematics and engineering, that include every

Stability theory is a fundamental topic in mathematics and engineering, that include every Stability Theory Stability theory is a fundamental topic in mathematics and engineering, that include every branches of control theory. For a control system, the least requirement is that the system is

More information

1 Relative degree and local normal forms

1 Relative degree and local normal forms THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a

More information

Nonlinear Control Lecture # 14 Tracking & Regulation. Nonlinear Control

Nonlinear Control Lecture # 14 Tracking & Regulation. Nonlinear Control Nonlinear Control Lecture # 14 Tracking & Regulation Normal form: η = f 0 (η,ξ) ξ i = ξ i+1, for 1 i ρ 1 ξ ρ = a(η,ξ)+b(η,ξ)u y = ξ 1 η D η R n ρ, ξ = col(ξ 1,...,ξ ρ ) D ξ R ρ Tracking Problem: Design

More information

Nonlinear Systems Theory

Nonlinear Systems Theory Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we

More information

Nonlinear Control Lecture 4: Stability Analysis I

Nonlinear Control Lecture 4: Stability Analysis I Nonlinear Control Lecture 4: Stability Analysis I Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 4 1/70

More information

Stability of Stochastic Differential Equations

Stability of Stochastic Differential Equations Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

More information

Lyapunov Theory for Discrete Time Systems

Lyapunov Theory for Discrete Time Systems Università degli studi di Padova Dipartimento di Ingegneria dell Informazione Nicoletta Bof, Ruggero Carli, Luca Schenato Technical Report Lyapunov Theory for Discrete Time Systems This work contains a

More information

3 rd Tutorial on EG4321/EG7040 Nonlinear Control

3 rd Tutorial on EG4321/EG7040 Nonlinear Control 3 rd Tutorial on EG4321/EG7040 Nonlinear Control Lyapunov Stability Dr Angeliki Lekka 1 1 Control Systems Research Group Department of Engineering, University of Leicester arch 9, 2017 Dr Angeliki Lekka

More information

Nonlinear Control Lecture # 36 Tracking & Regulation. Nonlinear Control

Nonlinear Control Lecture # 36 Tracking & Regulation. Nonlinear Control Nonlinear Control Lecture # 36 Tracking & Regulation Normal form: η = f 0 (η,ξ) ξ i = ξ i+1, for 1 i ρ 1 ξ ρ = a(η,ξ)+b(η,ξ)u y = ξ 1 η D η R n ρ, ξ = col(ξ 1,...,ξ ρ ) D ξ R ρ Tracking Problem: Design

More information

Stabilization in spite of matched unmodelled dynamics and An equivalent definition of input-to-state stability

Stabilization in spite of matched unmodelled dynamics and An equivalent definition of input-to-state stability Stabilization in spite of matched unmodelled dynamics and An equivalent definition of input-to-state stability Laurent Praly Centre Automatique et Systèmes École des Mines de Paris 35 rue St Honoré 7735

More information

Event-based Stabilization of Nonlinear Time-Delay Systems

Event-based Stabilization of Nonlinear Time-Delay Systems Preprints of the 19th World Congress The International Federation of Automatic Control Event-based Stabilization of Nonlinear Time-Delay Systems Sylvain Durand Nicolas Marchand J. Fermi Guerrero-Castellanos

More information

Modeling and Analysis of Dynamic Systems

Modeling and Analysis of Dynamic Systems Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability

More information

An asymptotic ratio characterization of input-to-state stability

An asymptotic ratio characterization of input-to-state stability 1 An asymptotic ratio characterization of input-to-state stability Daniel Liberzon and Hyungbo Shim Abstract For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic

More information

EML5311 Lyapunov Stability & Robust Control Design

EML5311 Lyapunov Stability & Robust Control Design EML5311 Lyapunov Stability & Robust Control Design 1 Lyapunov Stability criterion In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems.

More information

Computation of Local ISS Lyapunov Function Via Linear Programming

Computation of Local ISS Lyapunov Function Via Linear Programming Computation of Local ISS Lyapunov Function Via Linear Programming Huijuan Li Joint work with Robert Baier, Lars Grüne, Sigurdur F. Hafstein and Fabian Wirth Institut für Mathematik, Universität Würzburg

More information

Stability of Parameter Adaptation Algorithms. Big picture

Stability of Parameter Adaptation Algorithms. Big picture ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about

More information

Hybrid Control and Switched Systems. Lecture #7 Stability and convergence of ODEs

Hybrid Control and Switched Systems. Lecture #7 Stability and convergence of ODEs Hybrid Control and Switched Systems Lecture #7 Stability and convergence of ODEs João P. Hespanha University of California at Santa Barbara Summary Lyapunov stability of ODEs epsilon-delta and beta-function

More information

State-norm estimators for switched nonlinear systems under average dwell-time

State-norm estimators for switched nonlinear systems under average dwell-time 49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA State-norm estimators for switched nonlinear systems under average dwell-time Matthias A. Müller

More information

Nonlinear Control. Nonlinear Control Lecture # 24 State Feedback Stabilization

Nonlinear Control. Nonlinear Control Lecture # 24 State Feedback Stabilization Nonlinear Control Lecture # 24 State Feedback Stabilization Feedback Lineaization What information do we need to implement the control u = γ 1 (x)[ ψ(x) KT(x)]? What is the effect of uncertainty in ψ,

More information

BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

More information

Passivity-based Stabilization of Non-Compact Sets

Passivity-based Stabilization of Non-Compact Sets Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained

More information

From convergent dynamics to incremental stability

From convergent dynamics to incremental stability 51st IEEE Conference on Decision Control December 10-13, 01. Maui, Hawaii, USA From convergent dynamics to incremental stability Björn S. Rüffer 1, Nathan van de Wouw, Markus Mueller 3 Abstract This paper

More information

Quadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University

Quadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University .. Quadratic Stability of Dynamical Systems Raktim Bhattacharya Aerospace Engineering, Texas A&M University Quadratic Lyapunov Functions Quadratic Stability Dynamical system is quadratically stable if

More information

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct

More information

A small-gain type stability criterion for large scale networks of ISS systems

A small-gain type stability criterion for large scale networks of ISS systems A small-gain type stability criterion for large scale networks of ISS systems Sergey Dashkovskiy Björn Sebastian Rüffer Fabian R. Wirth Abstract We provide a generalized version of the nonlinear small-gain

More information

Ordinary Differential Equations II

Ordinary Differential Equations II Ordinary Differential Equations II February 9 217 Linearization of an autonomous system We consider the system (1) x = f(x) near a fixed point x. As usual f C 1. Without loss of generality we assume x

More information

TTK4150 Nonlinear Control Systems Solution 6 Part 2

TTK4150 Nonlinear Control Systems Solution 6 Part 2 TTK4150 Nonlinear Control Systems Solution 6 Part 2 Department of Engineering Cybernetics Norwegian University of Science and Technology Fall 2003 Solution 1 Thesystemisgivenby φ = R (φ) ω and J 1 ω 1

More information

Hybrid Systems Course Lyapunov stability

Hybrid Systems Course Lyapunov stability Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata OUTLINE Focus: stability

More information

Calculating the domain of attraction: Zubov s method and extensions

Calculating the domain of attraction: Zubov s method and extensions Calculating the domain of attraction: Zubov s method and extensions Fabio Camilli 1 Lars Grüne 2 Fabian Wirth 3 1 University of L Aquila, Italy 2 University of Bayreuth, Germany 3 Hamilton Institute, NUI

More information

EE363 homework 7 solutions

EE363 homework 7 solutions EE363 Prof. S. Boyd EE363 homework 7 solutions 1. Gain margin for a linear quadratic regulator. Let K be the optimal state feedback gain for the LQR problem with system ẋ = Ax + Bu, state cost matrix Q,

More information

Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers

Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers 28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC15.1 Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers Shahid

More information

1. Find the solution of the following uncontrolled linear system. 2 α 1 1

1. Find the solution of the following uncontrolled linear system. 2 α 1 1 Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +

More information

Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.

Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R

More information

Bounded uniformly continuous functions

Bounded uniformly continuous functions Bounded uniformly continuous functions Objectives. To study the basic properties of the C -algebra of the bounded uniformly continuous functions on some metric space. Requirements. Basic concepts of analysis:

More information

Formula Sheet for Optimal Control

Formula Sheet for Optimal Control Formula Sheet for Optimal Control Division of Optimization and Systems Theory Royal Institute of Technology 144 Stockholm, Sweden 23 December 1, 29 1 Dynamic Programming 11 Discrete Dynamic Programming

More information

ASTATISM IN NONLINEAR CONTROL SYSTEMS WITH APPLICATION TO ROBOTICS

ASTATISM IN NONLINEAR CONTROL SYSTEMS WITH APPLICATION TO ROBOTICS dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1, 1997 Electronic Journal, reg. N P23275 at 07.03.97 http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru Control problems in nonlinear systems

More information

Switched systems: stability

Switched systems: stability Switched systems: stability OUTLINE Switched Systems Stability of Switched Systems OUTLINE Switched Systems Stability of Switched Systems a family of systems SWITCHED SYSTEMS SWITCHED SYSTEMS a family

More information

Nonlinear Control Lecture # 1 Introduction. Nonlinear Control

Nonlinear Control Lecture # 1 Introduction. Nonlinear Control Nonlinear Control Lecture # 1 Introduction Nonlinear State Model ẋ 1 = f 1 (t,x 1,...,x n,u 1,...,u m ) ẋ 2 = f 2 (t,x 1,...,x n,u 1,...,u m ).. ẋ n = f n (t,x 1,...,x n,u 1,...,u m ) ẋ i denotes the derivative

More information

Set-Valued and Convex Analysis in Dynamics and Control

Set-Valued and Convex Analysis in Dynamics and Control Set-Valued and Convex Analysis in Dynamics and Control Rafal Goebel November 12, 2013 Abstract Many developments in convex, nonsmooth, and set-valued analysis were motivated by optimization and optimal

More information

Convergent systems: analysis and synthesis

Convergent systems: analysis and synthesis Convergent systems: analysis and synthesis Alexey Pavlov, Nathan van de Wouw, and Henk Nijmeijer Eindhoven University of Technology, Department of Mechanical Engineering, P.O.Box. 513, 5600 MB, Eindhoven,

More information

Automatic Control 2. Nonlinear systems. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2. Nonlinear systems. Prof. Alberto Bemporad. University of Trento. Academic year Automatic Control 2 Nonlinear systems Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 1 / 18

More information

Global output regulation through singularities

Global output regulation through singularities Global output regulation through singularities Yuh Yamashita Nara Institute of Science and Techbology Graduate School of Information Science Takayama 8916-5, Ikoma, Nara 63-11, JAPAN yamas@isaist-naraacjp

More information

Handout 2: Invariant Sets and Stability

Handout 2: Invariant Sets and Stability Engineering Tripos Part IIB Nonlinear Systems and Control Module 4F2 1 Invariant Sets Handout 2: Invariant Sets and Stability Consider again the autonomous dynamical system ẋ = f(x), x() = x (1) with state

More information

Trajectory Tracking Control of Bimodal Piecewise Affine Systems

Trajectory Tracking Control of Bimodal Piecewise Affine Systems 25 American Control Conference June 8-1, 25. Portland, OR, USA ThB17.4 Trajectory Tracking Control of Bimodal Piecewise Affine Systems Kazunori Sakurama, Toshiharu Sugie and Kazushi Nakano Abstract This

More information

Nonlinear Dynamical Systems Lecture - 01

Nonlinear Dynamical Systems Lecture - 01 Nonlinear Dynamical Systems Lecture - 01 Alexandre Nolasco de Carvalho August 08, 2017 Presentation Course contents Aims and purpose of the course Bibliography Motivation To explain what is a dynamical

More information

Nonlinear systems. Lyapunov stability theory. G. Ferrari Trecate

Nonlinear systems. Lyapunov stability theory. G. Ferrari Trecate Nonlinear systems Lyapunov stability theory G. Ferrari Trecate Dipartimento di Ingegneria Industriale e dell Informazione Università degli Studi di Pavia Advanced automation and control Ferrari Trecate

More information