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, γ, and T? Let ˆψ(x), ˆγ(x), and ˆT(x) be nominal models of ψ(x), γ(x), and T(x) u = ˆγ 1 (x)[ ˆψ(x) K ˆT(x)] Closed-loop system: ż = (A BK)z +B (z)
ż = (A BK)z +B (z) ( ) V(z) = z T Pz, P(A BK)+(A BK) T P = I Lemma 9.1 Suppose (*) is defined in D z R n If (z) k z z D z, k < 1/(2 PB ), then the origin of (*) is exponentially stable. It is globally exponentially stable if D z = R n If (z) k z +δ z D z and B r D z, then there exist positive constants c 1 and c 2 such that if δ < c 1 r and z(0) {z T Pz λ min (P)r 2 }, z(t) will be ultimately bounded by δc 2. If D z = R n, z(t) will be globally ultimately bounded by δc 2 for any δ > 0
Example 9.4 (Pendulum Equation) ẋ 1 = x 2, ẋ 2 = sin(x 1 +δ 1 ) bx 2 +cu ( ) 1 u = [sin(x 1 +δ 1 ) (k 1 x 1 +k 2 x 2 )] c [ ] 0 1 A BK = k 1 (k 2 +b) u = ( 1 ĉ ) [sin(x 1 +δ 1 ) (k 1 x 1 +k 2 x 2 )] ẋ 1 = x 2, ẋ 2 = k 1 x 1 (k 2 +b)x 2 + (x) ( ) c ĉ (x) = [sin(x 1 +δ 1 ) (k 1 x 1 +k 2 x 2 )] ĉ
(x) k x +δ, x ( k = c ĉ ) ĉ 1+ k1 2 +k2 2, δ = c ĉ ĉ sinδ 1 [ ] P(A BK)+(A BK) T p11 p P = I, P = 12 p 12 p 22 1 k < 2 p 2 12 +p2 22 sinδ 1 = 0 & k < 1 GUB 2 p 2 12 +p 2 22 GES
b = 0, ĉ = m o /ˆm ] k = m [1+ k1 2 +k2 2, δ = m sinδ 1, m = ˆm m m K = [ σ 2 2σ ], σ > 0, λ(a BK) = σ, σ 1 k < 2 2σ 3 m < p 2 12 +p2 22 [1+σ σ 2 +4] 4σ 2 +(σ 2 +1) 2 RHS 0.4 0.3951 0.3 0.2 0.1 0 0 5 10 σ
Is feedback linearization a good idea? Example 9.5 ẋ = ax bx 3 +u, a,b > 0 u = (k +a)x+bx 3, k > 0, ẋ = kx bx 3 is a damping term. Why cancel it? u = (k +a)x, k > 0, ẋ = kx bx 3 Which design is better?
Example 9.6 ẋ 1 = x 2, ẋ 2 = h(x 1 )+u h(0) = 0 and x 1 h(x 1 ) > 0, x 1 0 Feedback Linearization: u = h(x 1 ) (k 1 x 1 +k 2 x 2 ) With y = x 2, the system is passive with V = x1 0 h(z) dz + 1 2 x2 2 V = h(x 1 )ẋ 1 +x 2 ẋ 2 = yu
The control u = σ(x 2 ), σ(0) = 0, x 2 σ(x 2 ) > 0 x 2 0 creates a feedback connection of two passive systems with storage function V V = x 2 σ(x 2 ) x 2 (t) 0 ẋ 2 (t) 0 h(x 1 (t)) 0 x 1 (t) 0 Asymptotic stability of the origin follows from the invariance principle Which design is better?
The control u = σ(x 2 ) has two advantages: It does not use a model of h The flexibility in choosing σ can be used to reduce u However, u = σ(x 2 ) cannot arbitrarily assign the rate of decay of x(t). Linearization of the closed-loop system at the origin yields the characteristic equation s 2 +σ (0)s+h (0) = 0 One of the two roots cannot be moved to the left of Re[s] = h (0)
Partial Feedback Linearization Consider the nonlinear system ẋ = f(x)+g(x)u [f(0) = 0] Suppose there is a change of variables [ ] [ ] η T1 (x) z = = T(x) = ξ T 2 (x) defined for all x D R n, that transforms the system into η = f 0 (η,ξ), ξ = Aξ +B[ψ(x)+γ(x)u] (A,B) is controllable and γ(x) is nonsingular for all x D
u = γ 1 (x)[ ψ(x)+v] η = f 0 (η,ξ), ξ = Aξ +Bv v = Kξ, where (A BK) is Hurwitz
Lemma 9.2 The origin of the cascade connection η = f 0 (η,ξ), ξ = (A BK)ξ is asymptotically (exponentially) stable if the origin of η = f 0 (η,0) is asymptotically (exponentially) stable Proof With b > 0 sufficiently small, V(η,ξ) = bv 1 (η)+ ξ T Pξ, (asymptotic) V(η,ξ) = bv 1 (η)+ξ T Pξ, (exponential)
If the origin of η = f 0 (η,0) is globally asymptotically stable, will the origin of η = f 0 (η,ξ), ξ = (A BK)ξ be globally asymptotically stable? In general No Example 9.7 The origin of η = η is globally exponentially stable, but η = η +η 2 ξ, ξ = kξ, k > 0 has a finite region of attraction {ηξ < 1+k}
Example 9.8 η = 1 2 (1+ξ 2)η 3, ξ1 = ξ 2, ξ2 = v The origin of η = 1 2 η3 is globally asymptotically stable [ v = k 2 def 0 1 ξ 1 2kξ 2 = Kξ A BK = k 2 2k The eigenvalues of (A BK) are k and k ] e (A BK)t = (1+kt)e kt k 2 te kt te kt (1 kt)e kt
Peaking Phenomenon: max{k 2 te kt } = k t e as k ξ 1 (0) = 1, ξ 2 (0) = 0 ξ 2 (t) = k 2 te kt η = 1 2 ( 1 k 2 te kt) η 3, η(0) = η 0 η 2 (t) = η 2 0 1+η 2 0[t+(1+kt)e kt 1] If η 2 0 > 1, the system will have a finite escape time if k is chosen large enough
Lemma 9.3 The origin of η = f 0 (η,ξ), ξ = (A BK)ξ is globally asymptotically stable if the system η = f 0 (η,ξ) is input-to-state stable Proof Apply Lemma 4.6
u = γ 1 (x)[ ψ(x) KT 2 (x)] What is the effect of uncertainty in ψ, γ, and T 2? Let ˆψ(x), ˆγ(x), and ˆT 2 (x) be nominal models of ψ(x), γ(x), and T 2 (x) u = ˆγ 1 (x)[ ˆψ(x) K ˆT 2 (x)] η = f 0 (η,ξ), ξ = (A BK)ξ +B (z) ( ) = ψ +γˆγ 1 [ ˆψ K ˆT 2 ]+KT 2
η = f 0 (η,ξ), ξ = (A BK)ξ +B (z) ( ) Lemma 9.4 If η = f 0 (η,ξ) is input-to-state stable and (z) δ for all z, for some δ > 0, then the solution of (*) is globally ultimately bounded by a class K function of δ Lemma 9.5 If the origin of η = f 0 (η,0) is exponentially stable, (*) is defined in D z R n, and (z) k z +δ z D z, then there exist a neighborhood N z of z = 0 and positive constants k, δ, and c such that for k < k, δ < δ, and z(0) N z, z(t) will be ultimately bounded by cδ. If δ = 0, the origin of (*) will be exponentially stable