Stability and Robustness Formal Guarantees and Proof

Chapter 4: Analysis, Proofs, and Implementation of Neural-Fly

4.5 Stability and Robustness Formal Guarantees and Proof

We divide the proof of (3.12) into two steps. First, in Theorem 4.5.1, we show that the combined composite velocity tracking error and adaptation error, k [𝑠; ˜𝑎] k, ex- ponentially converges to a bounded error ball. This implies the exponential con- vergence of 𝑠. Then in Corollary 4.5.1.1 we show that when 𝑠 is exponentially bounded,𝑞˜ is also exponentially bounded. Combining the exponential bound from Theorem 4.5.1 and the ultimate bound from Corollary 4.5.1.1 proves Theorem 3.4.1.

Before discussing the main proof, let us consider the robustness properties of the feedback controller without considering any specific adaptation law. Taking the dy- namics (3.1), control law (3.7), the composite velocity error definition (3.10), and the parameter estimation error𝑎˜ =𝑎ˆ−𝑎, we find

𝑀𝑠¤+ (𝐶+𝐾)𝑠=−𝜙𝑎˜+𝑑 (4.6) We can use the Lyapunov functionV =𝑠^>𝑀 𝑠under the assumption of bounded𝑎˜ to show that

𝑡→∞lim k𝑠k ≤ sup𝑡k𝑑−𝜙𝑎˜k𝜆_max(𝑀)

𝜆_min(𝐾)𝜆_min(𝑀) (4.7) Taking this results alone, one might expect that any online estimator or learning algorithm will lead to good performance. However, the boundedness of 𝑎˜ is not guaranteed; Slotine and Li discuss this topic thoroughly [9]. In the full proof below, we show the stability and robustness of the Neural-Fly adaptation algorithm.

First, we introduce the parameter measurement noise𝜖¯, where𝜖¯ = 𝑦− 𝜙𝑎. Thus,

¯

𝜖 = 𝜖 + 𝑑 and k𝜖¯k ≤ k𝜖k + k𝑑k by the triangle inequality. Using the above closed loop dynamics (4.6), the parameter estimation error 𝑎˜, and the adaptation

law (3.8) and (3.9), the combined velocity and parameter-error closed-loop dynam- ics are given by

"

𝑀 0

0 𝑃⁻¹

# "

¤ 𝑠

¤˜ 𝑎

# +

"

𝐶+𝐾 𝜙

−𝜙^𝑇 𝜙^>𝑅⁻¹𝜙+𝜆 𝑃⁻¹

# "

𝑠

˜ 𝑎

#

=

"

𝑑

𝜙^>𝑅⁻¹𝜖¯−𝑃⁻¹𝜆𝑎−𝑃⁻¹𝑎¤

#

(4.8) 𝑑

𝑑 𝑡

𝑃⁻¹

=−𝑃⁻¹𝑃 𝑃¤ ⁻¹=𝑃⁻¹

2𝜆 𝑃−𝑄+𝑃 𝜙^>𝑅⁻¹𝜙 𝑃

𝑃⁻¹ (4.9) For our stability proof, we rely on the fact that𝑃⁻¹is both uniformly positive definite and uniformly bounded, that is, there exists some positive definite, constant matri- ces 𝐴and𝐵such that 𝐴 𝑃⁻¹ 𝐵. Dieci and Eirola [7] show the slightly weaker result that 𝑃 is positive definite and finite when 𝜙is bounded under the looser as- sumption𝑄 0. Following the proof from [7] with the additional assumption that 𝑄is uniformly positive definite, one can show the uniform definiteness and uniform boundedness of 𝑃. Hence, 𝑃⁻¹ is also uniformly positive definite and uniformly bounded.

Theorem 4.5.1. Given dynamics that evolve according to (4.8) and (4.9), uniform positive definiteness and uniform boundedness of𝑃⁻¹, the norm of

"

𝑠

˜ 𝑎

#

exponentially converges to the bound given in (4.10) with rate𝛼.

lim

𝑡→∞

"

𝑠

˜ 𝑎

#

≤ 1

𝛼𝜆_min(M)

sup

𝑡

k𝑑k +sup

𝑡

( k𝜙^>𝑅⁻¹𝜖¯k) +𝜆_max(𝑃⁻¹)sup

𝑡

( k𝜆𝑎+ ¤𝑎k)

(4.10) where𝛼andM are functions of𝜙, 𝑅, 𝑄 , 𝐾 , 𝑀 and𝜆, and 𝜆_min(·)and𝜆_max(·) are the minimum and maximum eigenvalues of(·)over time, respectively. Given Corol- lary 4.5.1.1 and (4.10), the bound in (3.12) is proven. Note𝜆_max(𝑃⁻¹) =1/𝜆_min(𝑃) and a sufficiently large value of𝜆_min(𝑃)will make the RHS of (4.10) small.

Proof. Now consider the Lyapunov functionV given by

V =

"

𝑠

˜ 𝑎

#> "

𝑀 0

0 𝑃⁻¹

# "

𝑠

˜ 𝑎

#

(4.11)

This Lyapunov function has the derivative V¤ =2

"

𝑠

˜ 𝑎

#> "

𝑀 0

0 𝑃⁻¹

# "

¤ 𝑠

¤˜ 𝑎

# +

"

𝑠

˜ 𝑎

#> "

𝑀¤ 0

0 _{𝑑 𝑡}^𝑑 𝑃⁻¹

# "

𝑠

˜ 𝑎

#

(4.12)

=−2

"

𝑠

˜ 𝑎

#>"

𝐶+𝐾 𝜙

−𝜙^𝑇 𝜙^>𝑅⁻¹𝜙+𝜆 𝑃⁻¹

# "

𝑠

˜ 𝑎

# +2

"

𝑠

˜ 𝑎

#>"

𝑑

𝜙^>𝑅⁻¹𝜖¯−𝑃⁻¹𝜆𝑎−𝑃⁻¹𝑎¤

#

+

"

𝑠

˜ 𝑎

#>"

𝑀¤ 0

0 _{𝑑 𝑡}^𝑑 𝑃⁻¹

# "

𝑠

˜ 𝑎

#

(4.13)

=−2

"

𝑠

˜ 𝑎

#>"

𝐾 𝜙

−𝜙^𝑇 𝜙^>𝑅⁻¹𝜙+𝜆 𝑃⁻¹

# "

𝑠

˜ 𝑎

# +2

"

𝑠

˜ 𝑎

#> "

𝑑

𝜙^>𝑅⁻¹𝜖¯−𝑃⁻¹𝜆𝑎−𝑃⁻¹𝑎¤

#

+

"

𝑠

˜ 𝑎

#>"

0 0

0 2𝜆 𝑃⁻¹−𝑃⁻¹𝑄 𝑃⁻¹+𝜙^>𝑅⁻¹𝜙

# "

𝑠

˜ 𝑎

#

(4.14)

=−

"

𝑠

˜ 𝑎

#>"

2𝐾 0

0 𝜙^>𝑅⁻¹𝜙+𝑃⁻¹𝑄 𝑃⁻¹

# "

𝑠

˜ 𝑎

# +2

"

𝑠

˜ 𝑎

#> "

𝑑

𝜙^>𝑅⁻¹𝜖¯−𝑃⁻¹𝜆𝑎−𝑃⁻¹𝑎¤

#

(4.15) where we used the fact 𝑀¤ −2𝐶 is skew-symmetric. As𝐾, 𝑃⁻¹𝑄 𝑃⁻¹, 𝑀, and𝑃⁻¹ are all uniformly positive definite and uniformly bounded, and 𝜙^>𝑅⁻¹𝜙 is positive semidefinite, there exists some𝛼 >0such that

−

"

2𝐾 0

0 𝜙^>𝑅⁻¹𝜙+𝑃⁻¹𝑄 𝑃⁻¹

#

−2𝛼

"

𝑀 0

0 𝑃⁻¹

#

(4.16) for all𝑡.

Define an upper bound for the disturbance term𝐷as

𝐷 =sup

𝑡

"

𝑑

𝜙^>𝑅⁻¹𝜖¯−𝑃⁻¹𝜆𝑎−𝑃⁻¹𝑎¤

#

(4.17) and define the functionM,

M =

"

𝑀 0

0 𝑃⁻¹

#

(4.18) By (4.16), the Cauchy-Schwartz inequality, and the definition of the minimum eigen- value, we have the following inequality forV¤:

V ≤ −2¤ 𝛼V +2 s

V

𝜆_min(M)𝐷 (4.19)

Consider the related systems,WwhereW = V,2WW¤ =V¤, and the following three equations hold

2WW ≤ −2¤ 𝛼W²+ 2𝐷W p

𝜆_min(M) (4.20)

W ≤ −¤ 𝛼W + 𝐷 p

𝜆_min(M) (4.21)

By the Comparison Lemma [10],

√

V =W ≤e^−𝛼𝑡 W (0) − 𝐷 𝛼

p

𝜆_min(M)

!

+ 𝐷

𝛼 p

𝜆_min(M) (4.22) and the stacked state exponentially converges to the ball

𝑡→∞lim

"

𝑠

˜ 𝑎

#

≤ 𝐷

𝛼𝜆_min(M) (4.23)

This completes the proof.

Next, we present a corollary which shows the exponential convergence of𝑞˜when𝑠 is exponentially stable.

Corollary 4.5.1.1. Ifk𝑠(𝑡) k ≤ 𝐴exp(−𝛼𝑡) +𝐵/𝛼for some constants 𝐴, 𝐵, and𝛼, and𝑠=𝑞¤˜+Λ𝑞˜, then

k𝑞˜k ≤ e^{−𝜆min(Λ)𝑡}k𝑞˜(0) k +

∫ 𝑡 0

e^{−𝜆min(Λ) (𝑡−𝜏)}𝐴e^−𝛼𝜏d𝜏+

∫ 𝑡 0

e^{−𝜆min(Λ) (𝑡−𝜏)}𝐵 𝛼d𝜏

(4.24) thusk𝑞˜k exponentially approaches the bound

𝑡→∞lim k𝑞˜k ≤ 𝐵

𝛼𝜆_min(Λ) (4.25)

Proof. From the Comparison Lemma [10], we can easily show (4.24). This can be further reduced as follows.

k𝑞˜k ≤ e^{−𝜆min(Λ)𝑡}k𝑞˜(0) k +𝐴e^{−𝜆min(Λ)𝑡}

∫ ^𝑡

0

e^{(𝜆min(Λ)−𝛼)𝜏}d𝜏+

∫ ^𝑡

0

e^{−𝜆min(Λ) (𝑡−𝜏)}𝐵 𝛼d𝜏 (4.26)

≤ e^{−𝜆min(Λ)𝑡}k𝑞˜(0) k +𝐴

e^−𝛼𝑡−e^{−𝜆min(Λ)𝑡} 𝜆_min(Λ) −𝛼

+ 𝐵

1−e^{−𝜆min(Λ)𝑡}

𝛼𝜆_min(Λ) (4.27)

Taking the limit, we arrive at (4.25)

With the following corollary, we will justify that 𝛼 is strictly positive even when 𝜙 ≡ 0, and thus the adaptive control algorithm guarantees robustness even in the absence of persistent excitation or with ineffective learning. In practice, we expect some measurement information about all the elements of 𝑎, that is, we expect a non-zero𝜙.

Corollary 4.5.1.2. If𝜙≡ 0, then the bound in (4.10) can be simplified to lim

𝑡→∞

"

𝑠

˜ 𝑎

#

≤ supk𝑑k +𝜆_max(𝑃⁻¹)sup( k𝜆𝑎+ ¤𝑎k)

min(𝜆, 𝜆_min(𝐾)/𝜆_max(𝑀))𝜆_min(M) (4.28) Proof. Assuming𝜙≡0immediately leads to𝛼of

𝛼=min 1

2𝜆_min(𝑃⁻¹𝑄),

𝜆_min(𝐾) 𝜆_max(𝑀)

(4.29) 𝜙 ≡ 0also simplifies the 𝑃¤ equation to a stable first-order differential matrix equa- tion. By integrating this simplified 𝑃¤ equation, we can show 𝑃exponentially con- verges to the value𝑃 = ₂^𝑄

𝜆. This leads to bound in (4.28).

We now introduce another corollary for the Neural-Fly-Constant, when 𝜙 = 𝐼. In this case, the regularization term is not needed, as it is intended to regularize the linear coefficient estimate in the absence of persistent excitation, so we set𝜆 = 0. This corollary also shows that Neural-Fly-Constant is sufficient for perfect tracking control when 𝑓 is constant; though in this case, even the nonlinear baseline controller with integral control will converge to perfect tracking. In practice for quadrotors, we only expect 𝑓 to be constant when the drone air-velocity is constant, such as in hover or steady level flight with constant wind velocity.

Corollary 4.5.1.3. If 𝜙 ≡ 𝐼,𝑄 = 𝑞 𝐼, 𝑅 =𝑟 𝐼,𝜆 = 0, and𝑃(0) = 𝑝₀𝐼 is diagonal, where𝑞,𝑟and𝑝₀are strictly positive scalar constants, then the bound in (4.10) can be simplified to

lim

𝑡→∞

"

𝑠

˜ 𝑎

#

≤ 1+𝑟⁻¹

sup_𝑡k𝑓 −𝑎k+𝜖/𝑟

𝜆_max(𝑀)

𝜆_min(𝐾)𝜆_min(M) (4.30) Proof. Under these assumptions, the matrix differential equation for𝑃is reduced to the scalar differential equation

𝑑 𝑝 𝑑 𝑡

=𝑞− 𝑝²/𝑟 (4.31)

where 𝑃(𝑡) = 𝑝(𝑡)𝐼. This equation can be integrated to find that 𝑝 exponentially converges to 𝑝 = √

𝑞𝑟. Then by (4.16), 𝛼 ≤ p

𝑞/𝑟 and 𝛼 ≤ 𝜆_min(𝐾)/𝜆_max(𝑀). If we choose 𝑞 and𝑟 such that p

𝑞/𝑟 = 𝜆_min(𝐾)/𝜆_max(𝑀), then we can take𝛼 = 𝜆_min(𝐾)/𝜆_max(𝑀). Then, the error bound reduces to

lim

𝑡→∞

"

𝑠

˜ 𝑎

#

≤ 𝐷 𝜆_max(𝑀)

𝜆_min(𝐾)𝜆_min(M) (4.32) Take𝑎as a constant. Then𝑎¤ =0,𝑑 = 𝑓 −𝑎, and𝐷is bounded by

𝐷 ≤

1+𝑟⁻¹

sup

𝑡

k𝑓 −𝑎k +𝜖/𝑟 (4.33)