Proof of Theorem 5.7.1
The proof uses a back-stepping procedure and an inductive argument. Thus, under the assumptions (i), (ii) and (iv), a global change of coordinates for the system exists such that the system is in the strict-feedback form [140, 153, 199, 212]. By augmenting the ρ linearly independent set
z1=h(x),z2=Lfh(x),…,zρ=Lρ−1fh(x)
with an arbitrary n−ρ linearly independent set zρ+1 = ψρ+1(x),…, zn = ψn with ψi(0) = 0, 〈dψi, Gρ−1〉 = 0, ρ + 1 ≤ i ≤ n. Then the state feedback
u=1Lg2Lρ−1fh(x)(υ−Lρfh(x))
globally transforms the system into the form:
z˙i=zi+1+ΨTi(z1,…,zi)w 1≤i≤ρ−1,z˙ρ=υ+ΨTρ(z1,…zρ)wz˙μ=ψ(z)+ΞT(z)w,y=z1,⎫⎭⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ |
(A.1) |
where zμ = (zρ+1,…, zn). Moreover, in the z-coordinates we have
Gj=span{∂∂zρ−j,…,∂∂zρ},0≤j≤ρ−1,
so that by condition (iii) the system equations (A.1) can be represented as
z˙i=zi+1+ΨTi(z1,…,zi,zμ)w 1≤i≤ρ−1,z˙ρ=υ+ΨTρ(z1,…zρ,zμ)wz˙μ=ψ(z1,zμ)+ΞT(z1,zμ)w.⎫⎭⎬⎪⎪⎪⎪ |
(A.2) |
Now, define
z⋆2=−z1−14λz1(1+ΨT1(z1,zμ)Ψ1(z1,zμ))
and set z2=z⋆2(z1,zμ,λ) in (A.2). Then consider the time derivative of the function:
V1=12z21
along the trajectories of the closed-loop system:
V˙1=−z21−14λz21(1+ΨT1Ψ1)+z1ΨT1w=−z21−λ[14z21(1+ΨT1Ψ1)+z1ΨT1λ−z1λΨT1w+(ΨT1w)2λ2(1+ΨT1Ψ1)]+(ΨT1w)2λ(1+ΨT1λΨ1)=−z21−λ(12z11+ΨT1Ψ1−−−−−−−−√−ΨT1wλ1+ΨT1Ψ1√)2+(ΨT1w)2λ(1+ΨT1Ψ1)≤−z21+(ΨT1w)2λ(1+ΨT1Ψ1)≤−z21+1λ∥∥ΨT1∥∥21+∥∥ΨT1∥∥2∥w∥2≤−z21+1λ∥w∥2. |
(A.3) |
Now if ρ = 1, then we set v=z⋆2(z1,zμ,λ) and integrating (A.3) from t0 with z(t0) = 0 to some t, we have
V1(x(t))−V1(0)≤−∫ttoy2(τ)dτ+1λ∫tt0∥w(t)∥2dτ
and since V1(0) = 0, V1(x) ≥ 0, the above inequality implies that
∫tt0y2(τ)dτ≤1λ∫tt0∥w(τ)∥2dτ.
For ρ > 1, we first prove the following lemma.
Lemma A.0.1 Suppose for some index i, 1 ≤ i ≤ ρ, for the system
z˙1=z2+ΨT1(z1,zμ)w ⋮z˙i=zi+1+ΨTi(z1,…,zi,zμ)w⎫⎭⎬⎪⎪⎪⎪⎪⎪, |
(A.4) |
there exist i functions
z⋆j=z⋆j(z1,…,zj−1,λ), z⋆j=z⋆(0,…,0,λ)=0, 2≤j≤i≤+1,
such that the function
Vi=12∑j=1iz˜2j,
where
z˜1=z1, z˜j=zj−z⋆j(z1,…,zj−1,zμ,λ), 2≤j≤i,
has time derivative
V˙i=−∑j=1iz˜2j+cλ∥w∥2
along the trajectories of (A.4) with zi+1=z⋆i+1 and for some real number c > 0.
Then, for the system
z˙1=z2+ΨT1(z1,zμ)w ⋮z˙i=zi+2+ΨTi+1(z1,…,zi+1,zμ)w⎫⎭⎬⎪⎪⎪⎪⎪⎪, |
(A.5) |
there exists a function
z⋆i+2(z1,…,zi+1,zr,λ), z⋆i+2(0,…,λ)=0
such that the function
Vi+1=12∑j=1i+1z˜2j
where
z˜j=zj−z⋆j(z1,…,zi,zμ,λ), 1≤j≤i+1,
has time derivative
V˙i=−∑j=1i+1z˜2j+c+1λ∥w∥2
along the trajectories of (A.5) with zi+2=z⋆i+2.
Proof: Consider the function
Vi+1=12∑j=1i+1z˜2j
with zi+2=z⋆i+2(z1,…,zi+1,zμ,λ) in (A.4), and using the assumption in the lemma, we have
V˙i+1≤−∑j=1iz˜2j+cλ∥w∥2+z˜i+1(z˜i+ΨTi+1w−∑j=1i∂z⋆i+1∂zj(zj+1+ΨTjw)− ∂z⋆i+1∂zμ(ψ+ΞTw)+z⋆i+2). |
(A.6) |
Let
α1(z1,…,zi+1,zμ)=z˜i−∑j=1i∂z⋆i+1∂zjzj+1−∂z⋆i+1∂zμψ,α2(z1,…,zi+1,zμ)=Ψi+1−∑j=1i∂z⋆i+1∂zjΨj−Ξ(∂z⋆i+1∂zμψ)T,z⋆i+2(z1,…,zi+1,zμ)=−α1−z˜i+1−14λz˜i+1(1+αT2α2)
and subsituting in (A.6), we get
V˙i+1≤−∑ji+1z˜2j+z˜i+1αT2w−14λz˜2i+1(1+αT2α2)+cλ∥w∥2=−∑j=1i+1z˜2j−λ(14z˜2i+1(1+αT2α2)−z˜i+1λαT2w+(αT2w)2λ2(1+αT2α2))+ (αT2w)2λ(1+αT2α2)+cλ∥w∥2=−∑j=1i+1z˜2j−λ⎛⎝⎜12z˜i+11+αT2α2−−−−−−−−√−αT2wλ1+αT2α2−−−−−−−−√⎞⎠⎟2+(αT2w)2λ(1+αT2α2)+ cλ∥w∥2≤−∑j=1i+1z˜2j+c+1λ∥w∥2
as claimed. □
To conclude the proof of the theorem, we note that the result of the lemma holds for ρ = 1 with c = 1. We now apply the result of the lemma (ρ − 1) times to arrive at the feedback contol
υ=z⋆ρ+1(z1,…,zρ,zμ,λ)
and the function
Vρ=12∑j=1ρz˜2j
has time derivative
V˙ρ=−∑j=1ρz˜2j+ρλ∥w∥2 |
(A.7) |
along the trajectories of (A.4) with zi+1=z⋆i+1(z1,…,zρ,zμ,λ), 1≤i≤ρ.
Finally, integrating (A.7) from t0 and z(t0) = 0 to some t, we get
V(x(t))−V(x(t0))≤−∫tt0y2(τ)dτ−∑j=2ρ∫tt0z˜2j(τ)dτ+ρλ∫tt0∥w(τ)∥2dτ,
and since Vρ(0)=0,Vρ(x)≥0, we have
∫tt0y2(τ)dτ≤ρλ∫tt0∥w(τ)∥2dτ,
and the L2-gain can be made arbitrarily small since λ is arbitrary. This concludes the proof of the theorem. □