High-Gain Observers in Nonlinear Feedback Control Lecture # 1 Introduction & Stabilization High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 1/4
Brief History Linear Systems: Doyle and Stein (1979) LQR Loop Transfer Recovery Nonlinear Systems: 1990s -... French School: Gauthier, Hammouri, others (Global Lipschitz conditions; Global results; More general structures) US School: Khalil, Saberi, Teel, Praly, others (No Global Lipschitz Conditions; Semiglobal results) Esfandiari and Khalil (1992) Peaking phenomenon Saturation High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 2/4
Motivating Example Let ẋ 1 = x 2 ẋ 2 = φ(x, u, w, d) y = x 1 u = γ(x, w) stabilize the origin x = 0 of the closed-loop system ẋ 1 = x 2 ẋ 2 = φ(x, γ(x, w), w, d) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 3/4
Observer: ˆx 1 = ˆx 2 + h 1 (y ˆx 1 ) ˆx 2 = φ o (ˆx, u, w) + h 2 (y ˆx 1 ) [ ] [ ] x = x 1 x 2 = x 1 ˆx 1 x 2 ˆx 2 x 1 = h 1 x 1 + x 2 x 2 = h 2 x 1 + δ(x, x, w, d) δ(x, x, w, d) = φ(x, γ(ˆx, w), w, d) φ o (ˆx, γ(ˆx, w), w) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 4/4
x 1 = h 1 x 1 + x 2 x 2 = h 2 x 1 + δ Design h 1 and h 2 to make [ h 1 1 h 2 0 ] Hurwitz and attenuate the effect of δ on x [ 1 1 G o (s) = s 2 + h 1 s + h 2 s + h 1 ] High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 5/4
G o (s) = h 1 = α 1 ε, h 2 = α 2 ε 2 ε (εs) 2 + α 1 εs + α 2 lim G o(s) = 0 ε 0 [ η 1 = x 1 ε, η 2 = x 2 ε η 1 = α 1 η 1 + η 2 ε η 2 = α 2 η 1 + εδ ε εs + α 1 ] High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 6/4
Peaking Phenomenon: η 1 = x 1 ˆx 1 ε x 1 (0) ˆx 1 (0) η 1 (0) could be O(1/ε) The transient response could contain a term of the form (1/ε)e at/ε, a > 0 1 ε e at/ε approaches an impulse function as ε 0 The peaking phenomenon could destabilize system High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 7/4
Example ẋ 1 = x 2, ẋ 2 = x 3 2 + u, y = x 1 State feedback control: u = x 3 2 x 1 x 2 Output feedback control: u = ˆx 3 2 ˆx 1 ˆx 2 ˆx 1 = ˆx 2 + (2/ε)(y ˆx 1 ) ˆx 2 = (1/ε 2 )(y ˆx 1 ) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 8/4
x 1 0.5 0 0.5 1 1.5 SFB OFB ε = 0.1 OFB ε = 0.01 OFB ε = 0.005 x 2 2 0 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 0 100 u 200 300 400 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 9/4
ε = 0.004 x 1 0.2 0 0.2 0.4 0.6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 200 x 2 400 600 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 2000 1000 u 0 1000 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 t High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 10/4
How to deal with the peaking phenomenon? Key Idea: Make γ(ˆx, w) a globally bounded function of ˆx Can be always achieved by saturating u or ˆx outside a compact set of interest Back to our example: Closed-loop system under state feedback: [ ] 0 1 ẋ = x 1 1 }{{} A The compact set Ω c = {x T Px c} is positively invariant where P is the solution of PA + A T P = I High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 11/4
Using it can be shown that over Ω c max Lx = c LP 1/2 x T P x c x 1 0.8944 c, x 2 1.0954 c With c = 0.385, u 1 u 1.0954 c(1 + 1.1999c) u = sat( ˆx 3 2 ˆx 1 ˆx 2 ) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 12/4
x 1 0.15 0.1 0.05 SFB OFB ε = 0.1 OFB ε = 0.01 OFB ε = 0.001 x 2 0 0.05 0 1 2 3 4 5 6 7 8 9 10 0.05 0 0.05 0.1 0 1 2 3 4 5 6 7 8 9 10 0 u 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 13/4
Region of attraction under the state feedback control u = sat( x 3 2 x 1 x 2 ) 2 1 x 2 0 1 2 3 2 1 0 1 2 3 x 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 14/4
Region of attraction under the output feedback control u = sat( ˆx 3 2 ˆx 1 ˆx 2 ) 1 0.5 x 2 0 0.5 1 1.5 1 0.5 0 0.5 1 1.5 x 1 ε = 0.1 (dashed) and ε = 0.05 (dash-dot) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 15/4
The output feedback controller recovers the following properties of the (saturated) state feedback controller as ε tends to zero: Asymptotic stability of the origin Region of attraction State trajectories High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 16/4
η O(1/ε) O(ε) Ω b Ω c x High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 17/4
Reduced-Order Observer ẋ 1 = x 2 ẋ 2 = φ(x 1, x 2, u, w, d) y = x 1 ẇ = h(w + hy) + φ o (y, ˆx 2, u, w) ˆx 2 = w + hy h = α ε, η = x 2 ˆx 2 ε η = αη + εδ High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 18/4
Transfer functions from y to ˆx when φ o = 0: Full-order observer α 2 (εs) 2 + α 1 εs + α 2 [ 1 + (εα 1 /α 2 )s s ] [ 1 s ] as ε 0 Reduced-order observer: αs εs + α s as ε 0 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 19/4
What is the effect of measurement noise? y = x 1 + v ε η 1 = α 1 η 1 + η 2 (α 1 /ε)v ε η 2 = α 2 η 1 + εδ(x, x, w, d) (α 2 /ε)v v(t) µ x(t) ˆx(t) c 1 ε + c 2 µ ε, t T High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 20/4
c 1 ε + c 2 µ/ε k a µ 1/2 c a µ 1/2 ε High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 21/4
Separation Principle Teel and Praly (1994) Atassi and Khalil (1999) ẋ = Ax + Bφ(x, z, u) ż = ψ(x, z, u) y = Cx ζ = q(x, z) (A, B, C) represent a channel of r integrators ẋ i = x i+1, for 1 i r 1 ẋ r = φ(x, z, u) y = x 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 22/4
State Feedback Controller: ϑ = Γ(ϑ, x, ζ) u = γ(ϑ, x, ζ) γ and Γ are globally bounded functions of x Observer: ˆx = Aˆx + Bφ o (ˆx, ζ, u) + H(y Cˆx) H = [α 1 /ε, α 2 /ε 2,..., α r /ε r] T s r + α 1 s r 1 + + α r 1 s + α r is Hurwitz High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 23/4
The output feedback controller recovers the performance of the state feedback controller for sufficiently small ε Asymptotic stability Region of attraction If R is the region of attraction under state feedback, then for any compact set S in the interior of R and any compact set Q R r, the set S Q is included in the region of attraction under output feedback control the trajectory of (x, z, ϑ) under output feedback approaches the trajectory under state feedback as ε 0 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 24/4
Minimum Phase Systems η = f 0 (η, ξ 1,, ξ r 1, ξ r ) ξ i = ξ i+1, 1 i r 1 ξ r = b(η, ξ) + a(η, ξ)u, a( ) 0 y = ξ 1 The zero dynamics η = f 0 (η, 0) are asymptotically stable Virtual output: s = k 1 ξ 1 + k r 1 ξ r 1 + ξ r Rel Deg 1 Design k 1 to k r 1 to stabilize the zero dynamics η = f 0 (η, ξ 1,, ξ r 1, k 1 ξ 1 k r 1 ξ r 1 ) ξ i = ξ i+1, 1 i r 2 ξ r 1 = k 1 ξ 1 k r 1 ξ r 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 25/4
η = f 0 (η, ξ 1,, ξ r 1, k 1 ξ 1 k r 1 ξ r 1 ) ξ i = ξ i+1, 1 i r 2 ξ r 1 = k 1 ξ 1 k r 1 ξ r 1 If f 0 (η, ξ) is input-to-state stable (ISS) when ξ is viewed as the input, then we can design k 1 to k r 1 to stabilize the linear system ξ 1 ξ 2. ξ r 1 = 0 1 0 0 1. k 1 k 2 k r 1. ξ 1 ξ 2. ξ r 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 26/4
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 sup t 0 τ t u(τ) γ(r) belongs to class K if it is strictly increasing and γ(0) = 0. It belongs to class K if γ(r) as r β(r, s) belongs to class KL if, for each fixed s, β(r, s) is a class K function of r and, for each fixed r, β(r, s) is decreasing with respect to s and β(r, s) 0 as s High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 27/4
ISS of ẋ = f(x, u) implies BIBS stability 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 ISS is equivalent to the existence of positive definite functions V (x), W(x) and class K function ρ such that V x f(x, u) W(x), x ρ( u ) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 28/4
Robust & Practical stabilization η = f 0 (η, ξ), f 0 (0, 0) = 0 ξ i = ξ i+1, 1 i r 1 ξ r = a(η, ξ)u + δ(η, ξ, u), a(η, ξ) a o > 0 y = ξ 1 If η = f 0 (η, ξ) is ISS δ(η, ξ, u)/a(η, ξ) (ξ) + k u, for k < 1, where and k are known we can design a partial state feedback control that depends only on ξ to stabilize (or practically stabilize) the origin High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 29/4
Robust control techniques such as (continuously-implemented) Sliding mode control Lyapunov redesign High-gain feedback can achieve stabilization if δ(0, 0, 0) = 0 practical stabilization if δ(0, 0, 0) 0 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 30/4
Continuously-Implemented Sliding Mode Control s = k 1 ξ 1 + k r 1 ξ r 1 + ξ r Choose k 1 to k r 1 such that A def = 0 1 0 0 1. k 1 k 2 k r 1. is Hurwitz High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 31/4
ξ r s η = f 0 (η, ξ 1,, ξ r 1, k 1 ξ 1 k r 1 ξ r 1 + s) ξ i = ξ i+1, 1 i r 2 ξ r 1 = k 1 ξ 1 k r 1 ξ r 1 + s Establish properties of (η, ξ 1,, ξ r 1 ) for abounded s ξ 1 ξ ζ = 2., B = 0. [ ] 0, E = k 1 k 2 k r 1 1 ξ ρ 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 32/4
ζ = Aζ + Bs [ ζ ξ = Eζ + s ] PA + A T P = I, V 1 (ζ) = ζ T Pζ λ min (P) ζ 2 V 1 (ζ) λ max (P) ζ 2 V 1 = ζ T ζ + 2ζ T PBs ζ 2 + 2 PB ζ s Suppose s c and let λ 1 > 4λ max (P) PB 2. λ 1 = 4λ max(p) PB 2 θ 2 for some θ < 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 33/4
V 1 (ζ) λ 1 c 2 λ max (P) ζ 2 4λ max (P) PB 2 c 2 /θ 2 ζ 2 PB c/θ V 1 ζ 2 + 2 PB ζ s (1 θ) ζ 2 θ ζ 2 + 2 PB ζ c (1 θ) ζ 2 The set {V 1 (ζ) λ 1 c 2 } is positively invariant; that is, every trajectory starting in Ω c stays in Ω c for all t 0. Moreover, if ζ is outside {V 1 (ζ) λ 1 c 2 }, it will reach it in finite time High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 34/4
ξ = [ ζ Eζ + s ] ξ (1 + E ) ζ + s s c and ζ {V 1 (ζ) λ 1 c 2 } imply ξ (1 + E )c λ 1 /λ min (P) + c Let λ 2 = 1 + (1 + E ) λ 1 /λ min (P) ξ λ 2 c High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 35/4
Let V 0 (η) be an ISS Lyapunov function of η = f(η, ξ) α 1 ( η ) V 0 (η) α 2 ( η ) V 0 W(η), η ρ( ξ ) Define γ(r) = α 2 (ρ(r)) (γ = α 2 ρ) V 0 (η) γ(λ 2 c) α 2 ( η ) α 2 (ρ(λ 2 c)) η ρ(λ 2 c) η ρ( ξ ) V 0 W(η) The set {V 0 (η) γ(λ 2 c)} is positively invariant and if η is outside it, it will reach it in finite time High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 36/4
ṡ = k 1 ξ 2 + + k r 1 ξ r + a(η, ξ)u + δ(η, ξ, u) = r 1 i=1 k iξ i+1 + δ, (η, ξ, u) φ(ξ) + k u a ṡ = a(η, ξ)[u + (η, ξ, u)] d ( 1 2 dt s2) = sṡ sṡ = a(η, ξ)s [u + (η, ξ, u)] u = β(ξ)sat(s/µ), µ > 0 sṡ = a(η, ξ)β(ξ)s sat(s/µ) + a(η, ξ)s (η, ξ, u) High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 37/4
sṡ a(η, ξ)φ(ξ) s a(η, ξ)β(ξ)s sat(s/µ) + ka(η, ξ)β(ξ) s sat(s/µ) For s µ, s sat(s/µ) = s sṡ a(η, ξ)φ(ξ) s (1 k)a(η, ξ)β(ξ) s β(ξ) φ(ξ) + β 0 (1 k), β 0 > 0 sṡ a(η, ξ)β 0 s a 0 β 0 s High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 38/4
With c > µ, the set Ω c = {V 0 (η) γ(λ 2 c)} {V 1 (ζ) λ 1 c 2 } { s c} is positively invariant sṡ a 0 β 0 s d dt s = d s 2 = sṡ dt s a 0β 0 s(t) s(0) a 0 β 0 t s(t) reaches the set { s µ} in finite time High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 39/4
Every trajectory starting in Ω c reaches the set Ω µ = {V 0 (η) γ(λ 2 µ)} {V 1 (ζ) λ 1 µ 2 } { s µ} in finite time. The partial state feedback control achieves semiglobal practical stabilization Semiglobal because the ISS Lyapunov function V 0 (η) is radially unbounded (V 0 (η) as η ); hence the set Ω c is bounded for every c > 0. By choosing c large enough we can include any compact set inside Ω c Practical stabilization because given any neighborhood N = { (η, ζ, s) < r 0 } of the origin we can choose µ small enough that Ω µ N. Hence, all trajectories starting in Ω c reach N in finite time High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 40/4
Regional Results: Suppose the inequalities V 0 W(η), η ρ( ξ ) V 1 ζ 2 + 2 PB ζ s sṡ a 0 β 0 s hold over a domain D that contains the origin. Choose c such that the set Ω c is compact and contained in D Then, with µ < c, all trajectories starting in Ω c reach Ω µ in finite time. The partial state feedback control achieves practical stabilization with Ω c as an estimate (subset) of the region of attraction High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 41/4
Vanishing Perturbation: (0, 0, 0) = 0, φ(0) = 0 [ ] η z =, ż = F(z, s) ζ Exponentially stable zero-dynamics: If the origin of η = f 0 (η, 0) is exponentially stable and F(z, s) is continuously differentiable in the neighborhood of the origin, then the origin of ż = F(z, 0) is exponentially stable and there is a Lyapunov function V (z) that satisfies V c 1 z 2 V (z) c 2 z 2 z F(z, s) c 3 z 2 + c 4 z s High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 42/4
If φ(ξ) is locally Lipschitz, then inside the set Ω µ we have sṡ a(η, ξ)φ(ξ) s a(η, ξ)β(ξ)s sat(s/µ) + ka(η, ξ)β(ξ) s sat(s/µ) a(η, ξ)φ(ξ) s (1 k)a(η, ξ)β(ξ) s 2 /µ c 5 z s + c 6 s 2 c 7 s 2 /µ V c 3 z 2 + c 4 z s V = V (z) + 1 2 s 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 43/4
V c 3 z 2 + c 4 z s + c 5 z s + c 6 s 2 c 7 s 2 /µ Q = [ V [ z s ] T Q [ z s c 3 (c 4 + c 5 )/2 (c 4 + c 5 )/2 (c 7 /µ c 6 ) ] ] det(q) > 0 for µ < µ = c 3 c 7 c 3 c 6 + ( c 4 +c 5 2 The origin of the closed-loop system is exponentially stable and Ω c is a subset of the region of attraction ) 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 44/4