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, z, u) ż = ψ(x, z, u) y = Cx ζ = q(x, z) u R p is the control input y R m and ζ R s are the measured outputs x R r and z R l are the state variables φ, ψ and q are locally Lipschitz φ(0, 0, 0) = 0, ψ(0, 0, 0) = 0, q(0, 0) = 0 (A, B, C) represent m chains of integrators with r i integrators in the ith chain High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 2/4
The r r matrix A, the r m matrix B, and the m r matrix C are block diagonal matrices with m diagonal blocks, given by A i = 0 1 0 0 0 1 0.. 0 0 1 0 0 C i = [ r i r i 1 0 0 where 1 i m and r = r 1 + + r m, B i = ] 1 r i 0 0. 0 1 r i 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 3/4
Sources of the Model: Normal form of a system having vector relative degree (r 1,, r m ) The system χ = f(χ) + g(χ)u y = h(χ) where u, y R m has vector relative degree (r 1,, r m ) if for i = 1,..., m, r i is the least number one has to differentiate the i-th output y i so that at least one of the m inputs u 1,..., u m appears explicitly. Moreover the m m matrix G(χ), whose (i, j) element is the coefficient of the control u j as it appears in y (r i) i, is nonsingular High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 4/4
The system can be transformed into the normal form (e.g. Isidori s book) ẋ = Ax + B[f 1 (x, z) + g 1 (x, z)u] ż = ψ(x, z, u) y = Cx where g 1 (x, z) is nonsingular In this case y is the only measured output (no ζ measurement) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 5/4
SISO system represented by the nth-order differential equation y (n) = F (y, y (1),, y (n 1), u, u (1),, u (m)) Extend the dynamics of the system by adding a series of m integrators at the input side and define v = u (m) as the control input of the extended system v = u (m) u (m 1) u (1) u y Plant High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 6/4
x = y y (1). y (n 1), z = u u (1). u (m 1 ) ẋ = Ax + BF(x, z, v) ż = A c z + B c v y = Cx ζ = z (A, B, C) represent a chain of n integrators and (A c, B c ) is a controllable canonical form representing a chain of m integrators High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 7/4
Uniformly observable SISO systems (Teel & Praly (1994)) y (n y+1) χ = h = α (y, y (1),, y (n y), u, u (1),, u (n u) ) (χ, u, u (1),, u (m u) ) χ, u and y are the state, input, and output, respectively Extend the dynamics of the system by adding a series of l u = max{n u, m u } integrators at the input side and define v = u (l u+1) as the control input of the extended system High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 8/4
x = y y (1). y (n y), z = u u (1). u (l u) ẋ = Ax + Bα(h(x, z), z) ż = A c z + B c v y = Cx ζ = z (A, B, C) represent a chain of n y integrators and (A c, B c ) is a controllable canonical form representing a chain of l u integrators High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 9/4
Models of mechanical and electromechanical systems, where displacement variables are measured while their derivatives (velocities, accelerations, etc.) are not measured Example: Magnetic suspension system Controller m Light source High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 10/4
ẏ = v v = g k m v L 0ai 2 2m(a + y) 2 [ 1 i = Ri + L ] 0avi L(y) (a + y) 2 + u y is the ball position, v is its velocity, and i is the electromagnet current [ ] y x =, z = i, ζ = z v High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 11/4
State Feedback Control: ϑ = Γ(ϑ, x, ζ) u = γ(ϑ, x, ζ) γ and Γ are locally Lipschitz γ(0, 0, 0) = 0 and Γ(0, 0, 0) = 0 γ and Γ are globally bounded functions of x; that is, γ(ϑ, x, ζ) k, x R r, (ϑ, ζ) Compact set Special Case: u = γ(x, ζ) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 12/4
Design the state feedback control such that the origin of the closed-loop system X = f(x), X = (x, z, ϑ) is asymptotically stable and other design requirements are satisfied, such as Region of attraction Transient response specifications state and/or control constraints High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 13/4
Output Feedback Control: ϑ = Γ(ϑ, ˆx, ζ) u = γ(ϑ, ˆx, ζ) ˆx = Aˆx + Bφ 0 (ˆx, ζ, u) + H(y Cˆx) H is block diagonal with m diagonal blocks, given by H i = α i 1 /ε α i 2 /ε2. α i r i 1 /εr i 1 α i r i /ε r i r i 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 14/4
ε is a positive constant to be specified and the positive constants α i j are chosen such that the roots of s r i + α i 1 sr i 1 + + α i r i 1 s + αi r i = 0 are in the open left-half plane, for all i = 1,..., m φ 0 (x, ζ, u) is a nominal model of φ(x, z, u), which is required to be locally Lipschitz and globally bounded in x. Moreover, φ 0 (0, 0, 0) = 0 It is allowed to take φ 0 = 0, in which the case the observer is linear High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 15/4
Separation Theorem: Let R be the region of attraction of the origin of X = f(x), S be any compact set in the interior of R, and Q be any compact subset of R r. Then, there exists ε 1 > 0 such that, for every 0 < ε ε 1, the solutions (X(t), ˆx(t)) of the closed-loop system, starting in S Q, are bounded for all t 0 given any µ > 0, there exist ε 2 > 0 and T 2 > 0, both dependent on µ, such that, for every 0 < ε ε 2, the solutions of the closed-loop system, starting in S Q, satisfy X(t) µ and ˆx(t) µ, t T 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 16/4
given any µ > 0, there exists ε 3 > 0, dependent on µ, such that, for every 0 < ε ε 3, the solutions of the closed-loop system, starting in S Q, satisfy X(t) X s (t) µ, t 0 where X s is the solution of X s = f(x s ), X s (0) = X(0) if the origin of X = f(x) is exponentially stable, then there exists ε 4 > 0 such that, for every 0 < ε ε 4, the origin of the closed-loop system is exponentially stable and S Q is a subset of its region of attraction High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 17/4
Proof: Represent the closed-loop system in the singularly perturbed form - Example: Single-output with r = 3 ẋ 1 = x 2 ˆx1 = ˆx 2 + (α 1 /ε)(x 1 ˆx 1 ) ẋ 2 = x 3 ˆx 2 = ˆx 3 + (α 2 /ε 2 )(x 1 ˆx 1 ) ẋ 3 = φ ˆx3 = φ 0 + (α 3 /ε 3 )(x 1 ˆx 1 ) η 1 = x 1 ˆx 1 ε 2, η 2 = x 2 ˆx 2, η 3 = x 3 ˆx 3 ε ε η 1 = 1 ε [x 2 ˆx 2 (α 1 /ε)(x 1 ˆx 1 )] = α 1 η 1 + η 2 ε η 2 = x 3 ˆx 3 (α 2 /ε 2 )(x 1 ˆx 1 ) = α 2 η 1 + η 3 ε η 3 = ε[φ φ 0 (α 3 /ε 3 )(x 1 ˆx 1 )] = α 3 η 1 + ε(φ φ 0 ) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 18/4
ε η = ε η 1 = α 1 η 1 + η 2 ε η 2 = α 2 η 1 + η 3 ε η 2 = α 3 η 1 + ε(φ φ 0 ) α 1 0 1 α 2 0 1 α 3 0 0 }{{} A 0 The characteristic equation of A 0 is A 0 is Hurwitz by design 0 0 η + (φ φ 0 ) 1 }{{} B s 3 + α 1 s 2 + α 2 s + α 3 = 0 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 19/4
For multi-output systems η ij = x ij ˆx ij ε r i j, for 1 i m and 1 j r i η = [η 11,..., η 1r1,..., η m1,..., η mrm ] T D(ε) = diag[ε r 1 1,..., 1,......, ε r m 1,..., 1] ˆx = x D(ε)η X = F(X, D(ε)η) ε η = A 0 η + εb (X, D(ε)η) where F(X, 0) = f(x) and A 0 is Hurwitz Slow Model: X = f(x), Fast Model: ε η = A 0 η High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 20/4
η O(1/ε l ) l = max{r 1,..., r m } 1 Σ O(ε) Ω b Ω c X High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 21/4
Converse Lyapunov Function: Since the origin of X = f(x) is asymptotically stable and R is its region of attraction, there is a smooth, positive definite function V (X) and a continuous, positive definite function U(X), both defined for all X R, such that V X V (X) as X R f(x) U(X), X R and for any c > 0, {V (X) c} is a compact subset of R High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 22/4
The compact set S is in the interior of R. Choose positive constants b and c such that c > b > max X S V (X). Then S Ω b = {V (X) b} Ω c = {V (X) c} R W(η) = η T P 0 η, P 0 A 0 + A T 0 P 0 = I λ 1 η 2 W(η) λ 2 η 2 W η A 0η η 2 Σ = {W(η) ε 2 }, Λ = Ω c Σ High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 23/4
Due to the global boundedness of F and in ˆx, there exist positive constants k 1 and k 2, independent of ε, such that F(X, D(ε)η) k 1, (X, D(ε)η) k 2 X Ω c and η R r For any 0 < ε < 1, there is L 1, independent of ε, such that for every 0 < ε ε, we have F(X, D(ε)η) F(X, 0) L 1 η, (X, η) Λ Step 1: show that there exist positive constants and ε 1 such that Λ is positively invariant for every 0 < ε ε 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 24/4
V = V F(X, D(ε)η) X = V V F(X, 0) + [F(X, D(ε)η) F(X, 0)] X X V X L 2, F(X, D(ε)η) F(X, 0) L 1 η η Σ λ 1 η 2 W(η) ε 2 η ε /λ 1 k 3 = L 1 L 2 /λ1 V U(X) + εk 3 Let β = min V (X)=c U(X) and ε 1 = β/k 3. For ε ε 1 V 0, (X, η) {V (X) = c} Σ High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 25/4
Ẇ = W η [ ] 1 ε A 0η + B 1 ε η 2 + 2η T P 0 B 1 2ε η 2 1 2ε η 2 + 2λ 2 k 2 η Take = 16k 2 2 P 0 3 Ẇ 1 2ε η 2, for W(η) ε 2 { V 0 (X, η) Ω c Σ Ẇ 0 (X, η) Ω c Σ } Λ is positively invariant High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 26/4
Step 2: Show that for all initial states (X(0), ˆx(0)) S Q, the trajectories enter the set Λ in finite time. X(0) S X(0) Ω b (X(0), ˆx(0)) S Q η(0) k/ε l where l = max{r 1,..., r m } 1 Because Ω b is in the interior of Ω c and X(t) X(0) k 1 t as long as X(t) Ω c, there exists a finite time T 0, independent of ε, such that X(t) Ω c for all t [0, T 0 ] High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 27/4
For W(η) ε 2 Ẇ 1 2ε η 2 1 2ελ 2 W(η) def = σ 1 ε W(η) where σ 1 = 1 2λ 2 W(η(t)) W(η(0)) exp ( σ 1 t/ε) η(0) k ε l W(η(0)) λ 2k 2 where σ 2 = λ 2 k 2 ε 2l def = σ 2 ε 2l W(η(t)) σ 2 ε 2l exp ( σ 1t/ε) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 28/4
W(η(t)) σ 2 ε 2l exp ( σ 1t/ε) W(η(t)) reaches ε 2 within the interval [0, T(ε)], where σ 2 ε 2l exp ( σ 1T(ε)/ε) = ε 2 T(ε) = ε σ 1 ln lim T(ε) = 0 ε 0 ( σ2 ε 2(l+1) Choose ε 2 > 0 small enough that T(ε) 1 2 T 0 for all 0 < ε ε 2. Taking ε 1 = min { ε, ε 1, ε 2 } guarantees that, for every 0 < ε ε 1, the trajectory (X(t), η(t)) enters Λ during the interval [0, T(ε)] and remains there for all t T(ε). This completes the proof of the first bullet ) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 29/4
Step 3: Show that there is a class K function γ such that X enters the set {V (X) γ(ε)} in finite time For all (X, η) Λ, we have Let V U(x) + εk 3 = 1 2 U(X) 1 2 U(X) + εk 3 c 0 (ε) = max {V (X)} U(X) 2εk 3 c 0 (ε) is a nondecreasing function of ε and c 0 (0) = 0. Hence, we can find a class K function γ such that c 0 (ε) γ(ε) {U(X) 2εk 3 } {V (X) c 0 (ε)} {V (X) γ(ε)} High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 30/4
{U(X) 2εk 3 } {V (X) γ(ε)} V (X) γ(ε) U(X) 2εk 3 V 1 2 U(X) 1 2 U(X) + εk 3 V (X) γ(ε) V 1 2 U(X) X(t) enters the set {V (X) γ(ε)} in finite time X(t) {V (X) γ(ε)}, η(t) {W(η) ε 2 } High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 31/4
Given µ > 0 there is ε 3 > 0 such that 0 < ε ε 3 {V (X) γ(ε)} { X µ 2 } {W(η) ε 2 } { η µ 2 There is T 1 > 0 such that for all t T 1 and every 0 < ε ε 2 = min{ε 1, ε 3} X(t) µ 2, η(t) µ 2 ˆx(t) = x(t) D(ε)η(t) x(t) + η(t) µ This completes the proof of the second bullet High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 32/4
To prove the third bullet, we divide the interval [0, ) into three intervals [0, T(ε)], [T(ε), T 2 ], and [T 2, ), where T 2 will be determined From the ultimate boundedness of X(t), shown in in the second bullet, and the asymptotic stability of the origin of X = f(x), we conclude that there exists a finite time T 2 T(ε), independent of ε, such that, for every 0 < ε ε 2, we have X(t) X s (t) µ, t T 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 33/4
On the interval [0, T(ε)], we have X(t) X(0) k 1 t, X s (t) X(0) k 1 t X(t) X s (t) 2k 1 T(ε), t [0, T(ε)] Since T(ε) 0 as ε 0, there exists 0 < ε 4 ε 2 such that, for every 0 < ε ε 4 X(t) X s (t) µ, t [0, T(ε)] High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 34/4
Over the interval [T(ε), T 2 ], the solution X(t) satisfies X = F(X, D(ε)η(t)), with X(T(ε)) X s (T(ε)) δ 1 (ε) where D(ε)η is O(ε), δ 1 (ε) 0 as ε 0, and F(X, 0) = f(x) By the continuous dependence of the solutions of differential equations on initial conditions and right-hand-side functions, there exists 0 < ε 5 ε 2 such that, for every 0 < ε ε 5 X(t) X s (t) µ, t [T(ε), T 3 ] This completes the proof of the third bullet High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 35/4
If the origin of X = f(x) is exponentially stable, there exists a continuously differentiable Lyapunov function V 1 (X) which satisfies b 1 X 2 V 1 (X) b 2 X 2 V 1 X F(X, 0) b 3 X 2 V 1 X b 4 X over the ball B r0 R for some positive constants r 0, b 1, b 2, b 3, and b 4 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 36/4
V 2 (X, η) = V 1 (X) + W(η) V 2 b 3 X 2 + 2β 1 X η ((1/ε) β 2 ) η 2 for some nonnegative constants β 1 and β 2 V 2 Y T QY where Q = [ b 3 β 1 β 1 (1/ε) β 2 ], Y = [ X η ] High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 37/4
Q = [ b 3 β 1 β 1 (1/ε) β 2 ] Q will be positive definite for sufficiently small ε. Hence, there is a neighborhood N of the origin, independent of ε, and ε 6 > 0 such that for every 0 < ε ε 6, the origin is exponentially stable and every trajectory in N converges to the origin as t There exists ε 7 > 0 such that for every 0 < ε ε 7, {V (X) γ(ε)} {W(η) ε 2 } N Hence, all solutions starting in S Q enter N and converge to the origin as t Done High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 38/4
Example: stabilize the pendulum ml 2 θ + mgl sin θ + k 0 l 2 θ = u at (θ = π, θ = 0). Let x 1 = θ π, x 2 = θ ẋ 1 = x 2 ẋ 2 = g l sin x 1 k 0 m x 2 + 1 ml 2u State Feedback: s = a 1 x 1 + x 2 ṡ = a 1 x 2 + g l sin x 1 k 0 m x 2 + 1 ml 2u u = β sat ( ) a1 x 1 + x 2 µ = β sat ( a1 (θ π) + θ ) µ High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 39/4
Output Feedback: Only θ is measured Nonlinear Observer: ˆθ = ˆω + (2/ε)(θ ˆθ) ˆω = φ 0 (ˆθ, u) + (1/ε 2 )(θ ˆθ) where φ 0 = â sin ˆθ + ĉu is a nominal model of φ = (g/l)sin θ (k 0 /m) θ + (1/ml 2 )u Linear Observer: ˆθ = ˆω + (2/ε)(θ ˆθ) ˆω = (1/ε 2 )(θ ˆθ) High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 40/4
Observer eigenvalues are ( 1/ε, 1/ε) Simulation Parameters: m = 0.15, l = 1.05, k 0 = 0.02 m = 0.15, l = 1.05 a 1 = 1, β = 4, µ = 1 â = g l, ĉ = 1 ml 2 (Nonlinear - actual) m = 0.1, l = 1 (Nonlinear - nominal) ε = 0.05, 0.01 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 41/4
3.5 (a) 2 (b) 3 2.5 1 θ 2 ω 0 1.5 1 SFB OFB ε = 0.05 OFB ε = 0.01 1 0.5 0 2 4 6 8 10 2 0 2 4 6 8 10 3.5 (c) 3.5 (d) 3 3 θ 2.5 2 1.5 θ 2.5 2 1.5 1 1 0.5 0.5 0 2 4 6 8 10 Time 0 0 2 4 6 8 10 Time High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 42/4
Figures (a) and (b) show θ and ω = θ for a nonlinear high-gain observer with nominal m and l Figure (c) shows θ for a nonlinear high-gain observer with actual m and l Figure (d) shows θ for a linear high-gain observer Remark: When ε is relatively large, we see an advantage for including φ 0 in the observer when it is a good model of φ. However, if the model is not that good, a linear observer may perform better. The important thing to notice here is that the differences between the three observers diminish as ε decreases. This is expected, because decreasing ε rejects the effect of the uncertainty in modeling φ High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 43/4
What if the origin of X = f(x) is asymptotically, but not exponentially, stable? Case 1: No modeling error φ = φ 0 (Teel & Praly (1994)) X = F(X, D(ε)η) ε η = A 0 η + εb (X, D(ε)η) V X F(X, 0) U(X), W η A 0η η 2 F(X, D(ε)η) F(X, 0) L 1 η = φ(x, ζ, u) φ(ˆx, ζ, u) L 3 η High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 44/4
V 3 (X, η) = V (X) + W(η) V 3 = V V F(X, 0) + [F(X, D(ε)η) F(X, 0)] X X + 1 [ ] 2 W 1 W η ε A 0η + B (X, D(ε)η) U(X) + β 1 η β 2 ε η for some positive constants β 1 and β 2 V 3 U(X) β 2 2ε η for sufficiently small ε. The origin is asymptotically stable High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 45/4
Case 2: φ φ 0 The origin of the closed-loop system is asymptotically stable under additional conditions that restrict the modeling error φ φ 0 (see Atassi and Khalil (1999)) Example: ẋ 1 = x 2, ẋ 2 = h(x 1 ) + u, y = x 1 x 1 h(x 1 ) > 0, x 1 0, State Feedback: u = x 2 V = 1 2 x2 1 + x 1x 2 + x 2 2 + 2 x1 0 h(z) dz V = x 1 h(x 1 ) x 2 2 The origin is globally asymptotically stable High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 46/4
If h (0) = 0 (e.g., h(x 1 ) = x 3 1 ) the origin is not exponentially stable because the linearization is not Hurwitz Output Feedback: ẋ = [ 0 1 0 1 ] x ˆx 1 = ˆx 2 + (2/ε)(x 1 ˆx 1 ) ˆx 2 = h 0 (ˆx 1 ) + (1/ε 2 )(x 1 ˆx 1 ) u = ˆx 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 47/4
If h 0 (x 1 ) = x 1, the closed-loop system is given by ẋ 1 = x 2 ẋ 2 = h(x 1 ) x 2 + η 2 ε η 1 = 2η 1 + η 2 ε η 2 = (1 + ε)η 1 + ε[ h(x 1 ) + x 1 ] Linearization at the origin yields the linear system ẋ 1 0 1 0 0 ẋ 2 η 1 = 0 1 0 1 0 0 2/ε 1/ε η 2 1 0 (1 + ε)/ε 0 x 1 x 2 η 1 η 2 For ε (0, 1), there is a positive eigenvalue; hence the origin is unstable High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 48/4