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Nonlinear Control Lecture 7: Stability Analysis IV

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Nonlinear Control Lecture 7: Stability Analysis IV Outline Input-to-State Stability Input-Output Stability Absolute Stability Nonlinear Control Lecture 7: Stability Analysis IV Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2009 Farzaneh Abdollahi Nonlinear Control Lecture 7 1/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability Stability of Cascade System Input-Output Stability L Stability of State Models Absolute Stability Positive Real Systems Circle Criterion Popov Criterion Farzaneh Abdollahi Nonlinear Control Lecture 7 2/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability I Consider the system x_ = f (t; x; u) (1) where f : [0; 1) × Rm −! Rn is piecewise continuous, bounded fcn 8t ≥ 0 I Suppose the Equ. pt. of the unforced system below is g.u.a.s. x_ = f (t; x; 0) (2) I What can be said about the behavior of the forced system in the presence of a bounded input u(t). I For an LTI system: x_ = Ax + Bu where A is Hurwitz, the solution satisfies: kkBk −λ(t−t0) kx(t)k ≤ ke kx(t0)k + sup ku(τ)k λ t0≤τ≤t I Zero-input response decays to zero I Zero-state response remains bounded for bounded input Farzaneh Abdollahi Nonlinear Control Lecture 7 3/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability I Can this conclusion be extended to nonlinear system (1)? I The answer in general is no, for instance: x_ = −3x + (1 + 2x 2)u when u = 0, the origin is g.e.s t t I However, with x(0) = 2 and u(t) ≡ 1, x(t) = (3 − e )=(3 − 2e ) is unbounded and have a finite scape time. I View the systemx _ = f (t; x; u) as a perturbation of the unforced system x_ = f (t; x; 0). I Suppose there exists a Lyap. fcn for the unforced system and calculate V_ in the presence of u I Since u is bounded, it may be possible to show that V_ is n.d. outside of a ball with radius µ where µ depends on sup kuk. I This is possible, for instance if the function f (t; x; u) is Lip. in u, i.e. kf (t; x; u) − f (t; x; 0)k ≤ Lkuk Farzaneh Abdollahi Nonlinear Control Lecture 7 4/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability I Having shown V_ is negative outside of a ball, ultimate boundedness theorem can be used, i.e. I kx(t)k is bounded by a class KL fcn β(kx(t0)k; t − t0) over [t0; t0 + T ] −1 and by a class K fcn α (α2(µ)) for t ≥ t0 + T −1 I Hence, kx(t)k ≤ β(kx(t0)k; t − t0) + α (α2(µ)) 8 t ≥ t0 I Definition: The system (1) is said to be input-to-state stable if there exist a class KL fcn β and a class K fcn γ s.t. for any initial state x(t0) and any bounded input u(t), the solution x(t) exists for all t ≥ t0 and satisfies: kx(t)k ≤ β(kx(t0)k; t − t0) + γ sup ku(τ)k t0≤τ≤t I If u(t) converges to zero as t −! 1, so does x(t). I with u(t) ≡ 0, the above equation reduces to: kx(t)k ≤ β(kx(t0)k; t − t0) implying the origin of unforced system is g.u.a.s. Farzaneh Abdollahi Nonlinear Control Lecture 7 5/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability I Sufficient condition for input-to-state stability: n I Theorem: Let V : [0; 1) × R −! R be a cont. diff. fcn. s.t. α1(kxk) ≤ V (t; x) ≤ α2(kxk) @V @V + f (t; x; u) ≤ −W (x); 8 kxk ≥ ρ(kuk) > 0 @t @x 3 n m 8(t; x; u) 2 [0; 1) × R × R where α1 and α2 are class K1 fcns and n W3(x) is a cont. p.d. fcn. on R . Then, the system (1) is input-to-state stable with −1 γ = α1 ◦ α2 ◦ ρ Farzaneh Abdollahi Nonlinear Control Lecture 7 6/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability I Lemma: Suppose f (t; x; u) is cont. diff. and globally Lip. in (x; u), uniformly in t. If the unforced system has a globally exponentially stable Equ. pt. at the origin, then the system (1) is input-to-state stable (ISS). I Proof: I View the forced system as a perturbation to unforced system I The converse theorem implies that the unforced system has a Lyap. fcn satisfying the g.e.s conditions. I The perturbation terms satisfies the Lip. cond. 8 t ≥ 0 and 8 (x; u). I Hence, V_ along the trajectories of forced system (1): @V @V @V V_ = + f (t; x; 0) + [f (t; x; u) − f (t; x; 0)] @t @x @x 2 2 2 ≤ −c3kxk + c4kxkLkuk = −c3(1 − θ)kxk − c3θkxk + c4kxkLkuk; 0 < θ < 1 Farzaneh Abdollahi Nonlinear Control Lecture 7 7/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability _ 2 c4Lkuk )V ≤ −c3(1 − θ)kxk 8 kxk ≥ ; 8 (t; x; u) c3θ I The conditions of previous theorem are satisfied with: 2 2 α1(r) = c1r ; α2(r) = c2r ; ρ(r) = (c4L=c3θ)r p I ) The system is input-to-state stable with γ(r) = c2=c1(c4L=c3θ)r I The previous lemma relies on globally Lip. fcn f and global exponential stability of the origin of the unforced system for ISS. Farzaneh Abdollahi Nonlinear Control Lecture 7 8/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability 2 I x_ = −3x + (1 + x )u does not satisfy the global Lip. cond. x I Example 7.1: x_ = − + u = f (x; u) 1 + x2 @f 1−x2 @f has a globally Lip. f since @x = − (1+x2)2 and @u = 1 and are globally bounded. x I The origin of unforced systemx _ = − 1+x2 is 2 x2 g:a:s(V = x =2 =) v_ = − 1+x2 n.d. for all x) I The system is locally e.s because of the linearized systemx _ = −x I However, the system is not g.e.s u ≡ 1; f (x; u) ≥ 1=2 =) x(t) ≥ x(t0) + t=2 8 t ≥ t0 I If g.e.s. and globally Lip. conds. are not satisfied, then we can use previous theorem to show ISS (i.e. find a region kxk ≥ µ in which V_ < 0) Farzaneh Abdollahi Nonlinear Control Lecture 7 9/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-to-State Stability 3 I Example 7.2: x_ = −x + u has a g.a.s. Equ. pt. at the origin when u = 0. 2 I Let v = x =2, then V_ can be written as: V_ = −x4 + ux = − (1 − θ)x4 − θx4 + xu 1=3 4 juj ≤ − (1 − θ)x ; 8jxj ≥ θ 0 < θ < 1. 1=3 I The system is ISS with γ(r) = (r/θ) 3 2 2 I Example 7.3: x_ = f (x; u) = −x − 2x + (1 + x )u has a g.e.s. Equ. pt. at the origin when u = 0. 2 I However, f is not globally Lip. Let v = x =2, then: V_ = −x2 − 2x4 + x(1 + x2)u2 ≤ − x4; 8 jxj ≥ u2 2 I The system is ISS with γ(r) = r Farzaneh Abdollahi Nonlinear Control Lecture 7 10/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Stability of Cascade System I Consider the cascade system n1 n2 n1 x_1 = f1(t; x1; x2); f1 : [0; 1) × R × R −! R (3) n2 n2 x_2 = f2(t; x2); f2 : [0; 1) × R −! R (4) x1 where f1 and f2 are p.c. in t and locally Lip. in x = . x2 I Supposex _1 = f1(t; x1; 0) and (4) both have g.u.a.s. Equ. pt. at x1 = 0 and x2 = 0. I Under what condition the origin of the cascade system is also g.u.a.s.? I The condition is that (3) should be ISS with x2 viewed as input. I Lemma: Under the assumption given above, if the system (3) with x2 as input, is ISS and the origin of (4) is g.u.a.s., then the origin of the cascade system (3) and (4) is g.u.a.s. Farzaneh Abdollahi Nonlinear Control Lecture 7 11/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-Output Stability I The foundation of input-output (I/O) approaches to nonlinear systems can be found in 1960's by Sandberg and Zames I An input-output model relates output to input with no knowledge of the internal structure (state equation). y = Hu; u : [0; 1) !Rm I The norm function kuk should satisfy the three properties 1. kuk = 0 iff u = 0 and it is strictly positive otherwise 2. scaling property 8a > 0; u ) kauk = akuk 3. triangular inequality: 8u1; u2; ku1 + u2k ≤ ku1k + ku2k kuk m = sup kuk < 1 Example: L1 I t≥0 sZ 1 T kuk m = u (t)u(t)dt < 1 L2 0 Z 1 p 1=p kuk m = kuk dt) < 1; 1 ≤ p < 1 Lp 0 Farzaneh Abdollahi Nonlinear Control Lecture 7 12/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-Output Stability I Stable system: any "well-behaved"input generate a "well-behaved" output m m I Extended space: Le = fujuτ 2 L ; 8τ 2 [0; 1)g u(t) 0 ≤ t ≤ τ where u is a truncation of u: u (t) = I τ τ 0 t > τ I It allows us to deal with unbounded "ever-growing" signals I Example: u(t) = t 2= L1 but uτ (t) 2 L1e m q I Casuality: mapping H : Le !Le is causal if the output (Hu)(t) at any time t depends only on the value of the input up to time t (Hu)τ = (Huτ )τ Farzaneh Abdollahi Nonlinear Control Lecture 7 13/58 Outline Input-to-State Stability Input-Output Stability Absolute Stability Input-Output Stability m q I Definition: A mapping H : Le !Le is L stable if there exist a class K function α, defined on [0; 1) and a nonneg const.
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