ISS and ESS Hybrid Control Systems

Chapter VI: Estimate-to-State Stability

6.3 ISS and ESS Hybrid Control Systems

To model an amputee-prosthesis system, we employ a hybrid control system. From the prosthesis’ perspective, the human is uncontrollable, hence we consider the zero dynamics in our hybrid system to represent the human. Since the human expects a certain control action from the prosthesis, when the prosthesis control law is based on estimated dynamics, its deviation from the nominal acts as a disturbance to the human. We show there exists conditions such that the human is e-ISS to these input

disturbances. Because of the discrete dynamics, these conditions are only defined locally, so we additionally show the prosthesis stays within this bounded region.

Hybrid Control Systems

Consider a hybrid control system,

ℋ𝒞𝜂_¯𝑧¯ =





















¤¯

𝜂 = 𝑓¯(𝜂,¯ 𝑧¯) +𝑔¯(𝜂,¯ 𝑧¯)𝑢¯+𝑔¯_𝐹(𝜂,¯ 𝑧¯)𝐹(𝜂,¯ 𝑧¯)

¤¯

𝑧 =Ψ¯(𝜂,¯ 𝑧¯) +Ψ¯𝑑(𝜂,¯ 𝑧¯)𝑑_𝑧¯

ifΦ⁻¹(𝜂,¯ 𝑧¯) ∈D \¯ 𝑆¯

¯

𝜂⁺ = Δ_N¯(𝜂¯⁻,𝑧¯⁻)

¯

𝑧⁺ = Δ_𝑍¯(𝜂¯⁻,𝑧¯⁻) ifΦ⁻¹(𝜂¯⁻,𝑧¯⁻) ∈𝑆,¯

(6.8)

where ¯𝜂 ∈N ⊂¯ R^𝑛¯ are controlled (output) states, ¯𝑧 ∈ 𝑍¯ ⊂ R^𝑛¯𝑧 uncontrolled states,

¯

𝑈 ⊂ R^𝑚¯ is a set of admissible control inputs for ¯𝑢, and 𝑑_𝑧¯ ∈ R^𝑛𝑑 is an input disturbance in the uncontrolled dynamics. The functions ¯𝑓 , 𝑔,¯ 𝑔¯_𝐹, 𝐹 , Ψ¯, Ψ¯𝑑, Δ_N¯, and Δ𝑍_¯ are locally Lipschitz in their arguments. There exists a constant𝑐_¯

Ψ𝑑 > 0 such that ∥Ψ¯𝑑(0,𝑧¯) ∥ ≤ 𝑐_¯

Ψ𝑑, ∀¯𝑧 ∈ 𝑍¯. The domain ¯D is a closed subset of ¯N ×𝑍¯, the guard or switching surface ¯𝑆 ⊂ D¯ is a co-dimension one submanifold of ¯D, respectively defined as

D¯ ={(𝜂,¯ ∈¯N ׯ 𝑍¯ : ¯ℓ(𝜂,¯ 𝑧¯) ≥0},

¯

𝑆 ={(𝜂,¯ 𝑧¯) ∈N ׯ 𝑍¯ : ¯ℓ(𝜂,¯ 𝑧¯) =0,ℎ¤(𝜂,¯ 𝑧¯) <0}, (6.9) where the continuously differentiable function ¯ℓ : ¯N ×𝑍¯ → Ryields𝐿_𝑔¯ℓ¯= 𝐿_𝑔¯

𝑑

ℓ¯= 0. We assume ¯𝑓(0,𝑧¯) = 𝑔¯(0,𝑧¯) = 𝑔¯_𝐹(0,𝑧¯) = Δ_N¯(0,𝑧¯) = 0 such that the surface

¯

𝑍 defined by ¯𝜂 = 0 with 𝑑_𝑧¯ = 0, 𝑧¤¯ = Ψ¯(0,𝑧¯), is invariant for the continuous and discrete dynamics. The hybrid system for the hybrid zero dynamics is,

ℋ|_𝑍¯ =

( 𝑧¤¯=Ψ(0¯ ,𝑧¯) if ¯𝑧 ∈𝑍¯ \ (𝑆¯∩𝑍¯)

¯

𝑧⁺ = Δ_𝑍¯(0,𝑧¯⁻) if ¯𝑧⁻ ∈𝑆¯∩𝑍 .¯

(6.10)

RES-CLFs. As discussed in 3.1 a RES-CLF, developed in [199], guarantees stability of a hybrid periodic orbit of the zero dynamics in the full-order dynamics. For the continuous output dynamics of (6.8), the continuously differentiable function 𝑉_𝜀 : R^𝑛¯ → R≥0 with constants 𝑐₁, 𝑐₂, 𝑐₃ > 0 is a RES-CLF, such that for all 0< 𝜀 < 1 and(𝜂,¯ 𝑧¯) ∈N ׯ 𝑍¯,

𝑐₁∥𝜂¯∥² ≤𝑉_𝜀(𝜂¯) ≤ 𝑐₂ 𝜀²

∥𝜂¯∥² inf

¯ 𝑢∈𝑈¯

[𝐿 _¯

𝑓𝑉_𝜀(𝜂,¯ 𝑧¯) +𝐿_𝑔¯𝑉_𝜀(𝜂,¯ 𝑧¯)𝑢¯+𝐿_𝑔¯

𝐹𝐹𝑉_𝜀(𝜂,¯ 𝑧¯)] ≤ 𝑐₃ 𝜀

𝑉_𝜀(𝜂¯).

(6.11)

The following set contains all control inputs that satisfy (6.11), 𝐾_𝜀(𝜂,¯ 𝑧¯) ={𝑢¯ ∈𝑈¯ : 𝐿_¯

𝑓𝑉_𝜀(𝜂,¯ 𝑧¯) +𝐿_𝑔¯𝑉_𝜀(𝜂,¯ 𝑧¯)𝑢¯+𝐿_𝑔¯

𝐹𝐹𝑉_𝜀(𝜂,¯ 𝑧¯) ≤ −𝑐₃ 𝜀

𝑉_𝜀(𝜂¯)}. Note while this formulation appears slightly different from that proposed in [199]

which did not have ¯𝑔_𝐹(𝜂,¯ 𝑧¯)𝐹(𝜂,¯ 𝑧¯) in the system dynamics, we can equivalently write the formulation given here with ˜𝑓(𝜂,¯ 𝑧¯) := 𝑓¯(𝜂,¯ 𝑧¯) +𝑔¯_𝐹(𝜂,¯ 𝑧¯)𝐹(𝜂,¯ 𝑧¯), resulting in the same form as in [199].

A ¯𝑢_𝜀(𝜂,¯ 𝑧¯) ∈ 𝐾_𝜀(𝜂,¯ 𝑧¯) for all (𝜂,¯ 𝑧¯) ∈N ׯ 𝑍¯, with zero disturbance input (𝑑_𝑧¯ = 0), gives the closed-loop hybrid system of (6.8):

ℋ𝜂_¯𝑧,𝜀¯ =





















¤¯

𝜂 = 𝑓¯(𝜂,¯ 𝑧¯) +𝑔¯(𝜂,¯ 𝑧¯)𝑢¯_𝜀(𝜂,¯ 𝑧¯) +𝑔¯_𝐹(𝜂,¯ 𝑧¯)𝐹(𝜂,¯ 𝑧¯)

¤¯

𝑧 =Ψ(¯ 𝜂,¯ 𝑧¯) ifΦ⁻¹(𝜂,¯ 𝑧¯) ∈D \¯ 𝑆¯

¯

𝜂⁺ = Δ_N¯(𝜂¯⁻,𝑧¯⁻)

¯

𝑧⁺ = Δ_𝑍¯(𝜂¯⁻,𝑧¯⁻) ifΦ⁻¹(𝜂¯⁻,𝑧¯⁻) ∈𝑆 .¯ (6.12) Re-ESS-CLF.We assumed we have a RES-CLF controller ¯𝑢_𝜀acting on our control- lable dynamics. However we cannot perfectly determine a ¯𝑢 ∈ 𝐾_𝜀(𝜂,¯ 𝑧¯) when we only have access to ˆ𝐹 and not𝐹(𝜂,¯ 𝑧¯). Hence, we employ the e-ESS constructions of Section 6.2 to construct arapidly exponentially estimate-to-state stabilizing CLF (Re-ESS-CLF).

Corollary 1: The continuously differentiable function𝑉_𝜀,𝜀¯ : R^𝑛¯ → R≥0defined for constants𝑐₁, 𝑐₂, 𝑐₃ > 0,

𝑐₁∥𝜂¯∥² ≤𝑉_𝜀,𝜀¯(𝜂¯) ≤ 𝑐₂ 𝜀²

∥𝜂¯∥² inf

¯ 𝑢∈R^𝑚¯

[𝐿 _¯

𝑓𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯) +𝐿_𝑔¯𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝑢¯+𝐿_𝑔¯

𝐹

𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝐹ˆ]

≤ −𝑐₃ 𝜀

𝑉_𝜀,𝜀¯(𝜂¯) − 1

¯ 𝜀

𝐿_𝑔¯

𝐹

𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝐿_𝑔¯

𝐹

𝑉 𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)^𝑇,

(6.13)

is Re-ESS-CLF of the continuous𝜂¯dynamics of (6.8), and a control input𝑢¯in the class,

𝐾_𝜀,𝜀¯(𝜂,¯ 𝑧,¯ 𝐹ˆ) ={𝑢¯ ∈R^𝑚¯ : 𝐿_¯

𝑓𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯) +𝐿_𝑔¯𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝑢¯ +𝐿_𝑔¯

𝐹𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝐹ˆ ≤ −𝑐₃ 𝜀

𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯) − 1

¯ 𝜀

𝐿_𝑔¯

𝐹𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)𝐿_𝑔¯

𝐹𝑉_𝜀,𝜀¯(𝜂,¯ 𝑧¯)^𝑇},

(6.14)

converges to a set proportional to√

¯

𝜀such that decreasing𝜀¯decreases the set size

¯

𝜂converges to.

Proof: The first part of the proof is similar to that of Theorem 5 resulting in 𝑉¤_𝜀,𝜀¯(𝜂,¯ 𝑧,¯ 𝑢¯) ≤ −^𝑐3

𝜀𝑉_𝜀,𝜀¯(𝜂¯)for,

∥𝜂¯∥ ≥ 𝜀 2

√︂ 𝜀𝜀¯

𝑐₂𝑐₃(1−𝜆)∥Δ𝐹∥_∞ :=𝛿_𝜂¯(𝜀,𝜀¯) ∥Δ𝐹∥_∞, (6.15) with𝜆∈ (0,1)and constant𝛿_𝜂¯ > 0, dependent on𝜀,𝜀¯. □ We give the convergence rate of ¯𝜂for future use,

∥𝜂¯∥ ≤ 1 𝜀

√︂𝑐₂ 𝑐₁

𝑒⁻

𝑐3 2𝜀𝜆𝑡

∥𝜂¯(0) ∥. (6.16)

Note, to have exponential stability closer to the origin, we can decrease𝛿_𝜂¯in (6.15) by decreasing ¯𝜀without affecting the convergence rate here.

Periodic Orbits. Let the periodic flow of the continuous dynamics of (6.12) be 𝜑_𝑡(𝜂,¯ 𝑧¯). We assume the fixed point(𝜂¯^∗,𝑧¯^∗)is in the switching surface,(𝜂¯^∗,𝑧¯^∗) ∈ 𝑆¯. We consider the flow𝜑_𝑡 to be hybrid periodic with period𝑇 >0 if 𝜑_𝑇(Δ(𝜂¯^∗,𝑧¯^∗))= (𝜂¯^∗,𝑧¯^∗), with Δ𝜂_¯𝑧¯(𝜂,¯ 𝑧¯) = (Δ_N¯(𝜂,¯ 𝑧¯),Δ_𝑍¯(𝜂,¯ 𝑧¯)). Let 𝒪 be the associated periodic orbit where𝒪 = {𝜑_𝑡(Δ(𝜂¯^∗,𝑧¯^∗)) : 0 ≤ 𝑡 ≤ 𝑇}. Corresponding to this periodic orbit an considering ¯𝑆 as the Poincaré section, we have the Poincaré map𝑃 : ¯𝑆 → 𝑆¯, a partial function:

𝑃(𝜂,¯ 𝑧¯) =𝜑_𝑇

𝑃(𝜂,¯𝑧)¯ (Δ𝜂_¯𝑧¯(𝜂,¯ 𝑧¯)). Here the time-to-impact function𝑇_𝑃 : ¯𝑆 → D¯ is,

𝑇_𝑃(𝜂,¯ 𝑧¯)=inf{𝑡 ≥ 0 :𝜑_𝑡(Δ𝜂_¯𝑧¯(𝜂,¯ 𝑧¯)) ∈𝑆¯}.

By the implicit function theorem, this function𝑇_𝑃is well-defined in a neighborhood of(𝜂¯^∗,𝑧¯^∗)[199] and hence𝑇_𝑃(𝜂¯^∗,𝑧¯^∗) =𝑇 and𝑃(𝜂¯^∗,𝑧¯^∗) =(𝜂¯^∗,𝑧¯^∗). Since𝜑_𝑡(𝜂,¯ 𝑧¯)is Lipschitz continuous, so is𝑇_𝑃. We can divide the Poincaré map into the ¯𝜂-component P𝜂_¯ and the ¯𝑧-componentP𝑧_¯, i.e. P= (P𝜂_¯,P𝑧_¯).

We similarly define the periodic flow of the continuous zero dynamics of (6.10) as 𝜑^𝑧

𝑡 and its corresponding hybrid periodic orbit as𝒪_𝑍¯. We call the associated Poincaré map 𝜌 : ¯𝑆 ∩𝑍¯ → 𝑆¯∩ 𝑍¯ the restricted Poincaré map. Here this partial function is,

𝜌(𝑧¯) =𝜑^𝑧

𝑇𝜌(𝑧)¯ (Δ_𝑍¯(0,𝑧¯)). (6.17) Here𝑇_𝜌(𝑧¯) is the restricted time-to-impact function, defined as𝑇_𝜌(𝑧¯) := 𝑇_𝑃(0,𝑧¯).

The period is 𝑇^∗ = 𝑇_𝜌(0). Because we assume the zero dynamics surface ¯𝑍 is

invariant, for a periodic orbit for the zero dynamics𝒪_𝑍¯ there exists a corresponding periodic orbit for the full-order dynamics,𝒪 =𝜄₀(𝒪_𝑍¯). Here𝜄₀ : ¯𝑍 →N ׯ 𝑍¯ is the canonical embedding 𝜄₀(𝑧¯) = (0,𝑧¯). From this we assume ¯𝑥^∗ = 0 and without loss of generality we assume ¯𝑧^∗ =0 too.

We assume the norm on ¯N ×𝑍¯is constructed as∥ (𝜂,¯ 𝑧¯) ∥ =∥𝜂¯∥ + ∥𝑧¯∥without losing generality. Then the distance from a periodic orbit𝒪to a point(𝜂,¯ 𝑧¯)is,

∥ (𝜂,¯ 𝑧¯) ∥_𝒪 = inf

(𝑥^′,𝑧¯^′)∈𝒪

∥ (𝜂,¯ 𝑧¯) − (𝑥^′,𝑧¯^′) ∥

= inf

¯ 𝑧^′∈𝒪𝑍_¯

∥𝑧¯−𝑧¯^′∥ + ∥𝑥−0∥ =∥𝑧¯∥_𝒪¯

𝑍 + ∥𝜂¯∥.

Zero Dynamics Lyapunov Function. To establish e-ISS of the full hybrid system (6.12), we construct a Lyapunov function for the stable hybrid periodic orbit 𝒪_𝑍¯ of the zero dynamics. By [272], since 𝒪_𝑍¯ is exponentially stable, there exists a Lyapunov function𝑉_𝑧¯ : ¯𝑍 → R≥0 for a neighborhood \𝐵_𝑟(𝒪_𝑍¯) with 𝑟 > 0 of𝒪_𝑍¯ such that,

𝑐_1,𝑧¯∥𝑧¯∥²

𝒪𝑍_¯

≤𝑉_𝑧¯(𝑧¯) ≤𝑐_2,𝑧¯∥𝑧¯∥²

𝒪𝑍_¯

(6.18)

𝜕𝑉_𝑧¯

𝜕𝑧¯

Ψ(¯ 0,𝑧¯) ≤ −𝑐_3,𝑧¯∥𝑧¯∥²

𝒪𝑍¯

(6.19)

𝜕𝑉_𝑧¯

𝜕𝑧¯

≤ 𝑐_4,𝑧¯∥𝑧¯∥_𝒪¯

𝑍

, (6.20)

with constants𝑐_1,𝑧¯, 𝑐_2,𝑧¯, 𝑐_3,𝑧¯, 𝑐_4,𝑧¯ > 0. Since ¯Ψ𝑑(0,𝑧¯)is upper bounded by𝑐_Ψ𝑑, then with a disturbance𝑑_𝑧¯, our system has an e-ISS-Lyapunov function,

𝜕𝑉_𝑧¯

𝜕𝑧¯

(Ψ(0¯ ,𝑧¯) +Ψ¯𝑑(0,𝑧¯)𝑑_𝑧¯)

≤ −𝑐_3,𝑧¯∥𝑧¯∥²

𝒪_𝑍¯ + 𝜕𝑉_𝑧¯

𝜕𝑧¯

Ψ¯𝑑(0,𝑧¯)𝑑_𝑧¯

≤ −𝑐_3,𝑧¯∥𝑧¯∥²

𝒪𝑍_¯

+𝑐_4,𝑧¯∥𝑧¯∥_𝒪¯

𝑍

𝑐_Ψ¯

𝑑∥𝑑_𝑧¯∥_∞.

Similarly to Theorem 5, we use𝜆_𝑧¯ ∈ (0,1) to establish exponential convergence at a rate−^𝑐3,𝑧¯

2 𝜆_𝑧¯for,

∥𝑧¯∥_𝒪¯

𝑍 ≥

𝑐_4,𝑧¯𝑐_¯

Ψ𝑑

𝑐_3,𝑧¯(1−𝜆_𝑧¯)∥𝑑_𝑧¯∥_∞ :=𝛿_𝑧¯∥𝑑_𝑧¯∥_∞. (6.21) Main Result

We now establish the main system result of the chapter: guaranteeing e-ISS of the hybrid periodic orbit of the zero dynamics𝒪_𝑍¯ with our Re-ESS-CLF of (6.13). To

establish bounds on the Poincaré maps and their time-to-impact functions, we give a proof sketch for a lemma due to space constraints. A similar proof can be found in Lemma 1 of [199] and Lemma 2 of [270]. Following, a theorem proves the main result. The basic method for the proof follows closely to that of Theorem 2 of [199]

and Theorem 2 of [270]. These proofs are unique since they are developed for the hybrid system (6.8) which has a disturbance𝑑_𝑧¯with input matrix ¯Ψ𝑑(𝜂,¯ 𝑧¯)in the zero dynamics and whose control input ¯𝑢 may not be a RES-CLF controller in𝐾_𝜀(𝜂,¯ 𝑧¯) but rather depends on force estimate ˆ𝐹. We highlight where the differences for our system come into these proofs.

Lemma 3: Given the hybrid system(6.8)with input disturbance and𝑑_𝑧¯, control input

¯

𝑢(𝜂,¯ 𝑧¯) ∈ 𝐾_𝜀,𝜀¯(𝜂,¯ 𝑧,¯ 𝐹ˆ) (6.14), and periodic orbit𝒪_𝑍¯ of the hybrid zero dynamics ℋ|_𝑍¯ (6.10) transverse to 𝑆¯ ∩ 𝑍¯, for 𝑟 > 0 such that (𝜂,¯ 𝑧¯) ∈ \𝐵_𝑟(0,0), and

∥𝜂¯∥ ≥ 𝛿_𝜂¯(𝜀,𝜀¯) ∥Δ𝐹∥_∞ (6.15), there exists finite constants 𝐴_𝑇𝜂¯, 𝐴_{𝑇 𝑑}

¯

𝑧, 𝐴_𝑃𝜂¯, 𝐴_{𝑃 𝑑}

¯ 𝑧 > 0 such that,

∥𝑇_𝑃(𝜂,¯ 𝑧¯) −𝑇_𝜌(𝑧¯) ∥ ≤ 𝐴_𝑇𝜂¯∥𝜂¯∥ + 𝐴_{𝑇 𝑑}

¯

𝑧∥𝑑_𝑧¯∥_∞, (6.22)

∥P𝑧_¯(𝜂,¯ 𝑧¯) −𝜌(𝑧¯) ∥ ≤ 𝐴_𝑃𝜂¯∥𝜂¯∥ +𝐴_{𝑃 𝑑}

𝑥∥𝑑_𝑧¯∥_∞. (6.23) Proof Sketch: We construct an auxiliary time-to-impact function, 𝑇_𝐵, to relate to both 𝑇_𝜌 and 𝑇_𝑃, such that we can then relate 𝑇_𝜌 to 𝑇_𝑃. The difference between 𝑇_𝐵 and𝑇_𝜌 is bounded by a Lipschitz constant. Bounds of𝑇_𝑃 are found which are valid for(𝜂,¯ 𝑧¯) ∈\𝐵_𝑟(0,0). For a given solution,𝑇_𝐵 =𝑇_𝑃 because these are locally unique solutions in ¯𝑆. To bound a solution𝑥(𝑡)of𝜑_𝑡(Δ(𝜂,¯ 𝑧¯)), the initial condition is bounded with the Lipschitz constant ofΔ_N¯ which is used in the bound of (6.16) with the𝑇_𝑃 bounds and ∥𝜂¯∥. Note this bound only holds for∥𝜂¯∥ ≥ 𝛿_𝜂¯∥Δ𝐹∥_∞, a specific element of this proof. Using a Gronwall-Bellman argument similar to that in [199], we bound the difference between a solution ¯𝑧(𝑡)for the full-order dynamics and zero dynamics. This bound again includes∥𝜂¯∥as well as∥𝑑_𝑧¯∥_∞, another unique element of this proof. Grouping terms gives (6.22). Using the maximum of ¯Ψ(0,𝑧¯(𝑡))for a given solution ¯𝑧(𝑡) and (6.22), we arrive at (6.23). □ Theorem 6: Given the hybrid system(6.8)with input disturbance𝑑_𝑧¯, control input

¯

𝑢 ∈ 𝐾_𝜀,𝜀¯(𝜂,¯ 𝑧,¯ 𝐹ˆ), and periodic orbit 𝒪𝑍_¯ of the hybrid zero dynamicsℋ|_𝑍¯ (6.10) transverse to 𝑆¯∩𝑍¯, for𝑟 > 0such that (𝜂,¯ 𝑧¯) ∈\𝐵_𝑟(0,0), there exists𝛿 > 0such that for all∥𝑑_𝑧¯∥_∞ < 𝛿 the periodic orbit𝒪 =𝜄₀(𝒪_𝑍¯) is e-ISS.

Proof: Since the ISS stability of a hybrid periodic orbit can be analyzed via its Poincaré map [269], we seek to establish e-ISS of the Poincaré map. We aim to find a Lyapunov function𝑉_𝑃(𝜂,¯ 𝑧¯) for the discrete dynamics of the Poincaré map𝑃that satisfies this discrete-time e-ISS Lyapunov condition with𝜄 ∈𝐾_∞:

𝑉_𝑃(P (𝜂,¯ 𝑧¯)) −𝑉_𝑃(𝜂,¯ 𝑧¯)

≤ −𝜅( ∥𝜂¯∥²+ ∥𝑧¯∥²) +𝜄( ∥𝑑_𝑧¯∥_∞).

(6.24)

For the zero dynamics on the switching surface ¯𝑆, we establish a Lyapuonv function.

There exists an 𝑟_𝑧¯ > 0 such that 𝜌 : 𝐵_𝑟

¯

𝑧(0) ∩ (𝑆¯ ∩𝑍¯) → 𝐵_𝑟

¯

𝑧(0) ∩ (𝑆¯∩ 𝑍¯) is well defined for all𝑧 ∈ 𝐵_𝑟

¯

𝑧(0) ∩ (𝑆¯∩𝑍¯) and ¯𝑧_k+1 = 𝜌(𝑧¯_k)is locally exponentially stable. By the converse Lyapunov theorem for discrete-time systems, there exists a Lyapunov function𝑉_𝜌 defined on 𝐵_𝑟

¯

𝑧(0) ∩ (𝑆¯∩𝑍¯) for some𝑟_𝑧¯ > 𝜅_𝑘∥𝑑_𝑧¯∥_∞ and constants𝑐_{1, 𝜌}, 𝑐_{2, 𝜌}, 𝑐_{3, 𝜌}, 𝑐_{4, 𝜌} > 0 such that,

𝑐_{1, 𝜌}∥𝑧¯∥²≤ 𝑉_𝜌(𝑧¯) ≤ 𝑐_{2, 𝜌}∥𝑧¯∥² 𝑉_𝜌(𝜌(𝑧¯)) −𝑉_𝜌(𝑧¯) ≤ −𝑐_{3, 𝜌}∥𝑧¯∥²

|𝑉_𝜌(𝑧¯) −𝑉_𝜌(𝑧¯^′) | ≤ 𝑐_{4, 𝜌}∥𝑧¯−𝑧¯^′∥ ( ∥𝑧¯∥ + ∥𝑧¯^′∥).

To obtain a Lyapunov function for the ¯𝜂 dynamics in ¯𝑆, we define our Re-ISS- CLF𝑉_𝜀,𝜀¯ restricted to the switching surface by,𝑉^𝑆¯

𝜀,𝜀¯(𝜂¯) :=𝑉_𝜀,𝜀¯|_𝑆¯(𝜂¯). We define a composite Lyapunov function on\𝐵_𝑟(0,0) ∩𝑆¯,

𝑉_𝑃(𝜂,¯ 𝑧¯) =𝑉_𝜌(𝑧¯) +𝜎𝑉^𝑆¯

𝜀,𝜀¯(𝜂¯),

with constant𝜎 > 0, which we define later, lower bound min{𝑐_{1, 𝜌}, 𝜎 𝑐₁}∥ (𝜂,¯ 𝑧¯) ∥², and upper bound max{𝑐_{2, 𝜌}, 𝜎^𝑐2

𝜀²}∥ (𝜂,¯ 𝑧¯) ∥². To satisfy (6.24), we first establish, 𝑉_𝜌(P𝑧_¯(𝜂,¯ 𝑧¯)) −𝑉_𝜌(𝑧¯)

=𝑉_𝜌(P𝑧_¯(𝜂,¯ 𝑧¯) −𝑉_𝜌(𝜌(𝑧¯)) +𝑉_𝜌(𝜌(𝑧¯)) −𝑉_𝜌(𝑧¯)

≤ 𝑐_{4, 𝜌}∥P𝑧_¯(𝜂,¯ 𝑧¯) − 𝜌(𝑧¯) ∥ ( ∥P𝑧_¯(𝜂,¯ 𝑧¯)) ∥ + ∥𝜌(𝑧¯) ∥) −𝑐_{3, 𝜌}∥𝑧¯∥².

(6.25)

Using (6.23) and the Lipschitz constant 𝐿_𝜌 of𝜌(𝑧¯), (6.17), gives,

∥P𝑧_¯(𝜂,¯ 𝑧¯) ∥ = ∥P𝑧_¯(𝜂,¯ 𝑧¯) − 𝜌(𝑧¯) +𝜌(𝑧¯) −𝜌(0) ∥

≤ 𝐴_𝑃𝜂¯∥𝜂¯∥ +𝐴_{𝑃 𝑑}

¯

𝑧∥𝑑_𝑧¯∥_∞+𝐿_𝜌∥𝑧¯∥

∥𝜌(𝑧¯) ∥ = ∥𝜌(𝑧¯) −𝜌(0) ∥ ≤ 𝐿_𝜌∥𝑧¯∥,

yielding known finite bounds along with those of (6.23) for (6.25). Because (6.23) is a unique development of this work from Lemma 3, this theorem proof is unique since it uses (6.23) and carries out the following steps with it. We next establish for

∥𝜂¯∥ ≥ 𝛿_𝜂¯∥Δ𝐹∥_∞, a unique bound in this proof, 𝑉^𝑆¯

𝜀,𝜀¯(P𝜂_¯(𝜂,¯ 𝑧¯)) −𝑉^𝑆¯

𝜀,𝜀¯(𝜂¯)

≤ 𝑒⁻

𝑐3 𝜀𝜆𝑡

𝑉_𝜀,𝜀¯(Δ_N¯(𝜂,¯ 𝑧¯)) −𝑐₁∥𝜂¯∥²

≤ 𝑐₂ 𝜀²

𝐿²

Δ_N¯𝑒⁻

𝑐3 𝜀𝜆𝑐

𝑇𝑇^∗

| {z }

𝐴_𝑉𝜂¯(𝜀)

−𝑐₁

∥𝜂¯∥²,

(6.26)

where 𝐴_𝑉𝜂¯ is a constant dependent on 𝜀. Since 𝐴_𝑉𝜂¯(0⁺) := lim𝜀→0⁺𝐴_𝑉𝜂¯(𝜀) = 0, there exists an ˜𝜀 such that 𝐴_𝑉𝜂¯(𝜀) < 𝑐₁, ∀0 < 𝜀 < 𝜀˜ such that (6.26) is negative definite.

The discrete-time Lyapunov condition (6.24) becomes, 𝑉_𝜌(P𝑧_¯(𝜂,¯ 𝑧¯)) −𝑉_𝜌(𝑧¯) +𝜎(𝑉^𝑆¯

𝜀,𝜀¯(P𝜂_¯(𝜂,¯ 𝑧¯)) −𝑉^𝑆¯

𝜀,𝜀¯(𝜂¯))

≤ 𝑐_{4, 𝜌}(𝐴_𝑃𝜂¯∥𝜂¯∥ +𝐴_{𝑃 𝑑}

¯

𝑧∥𝑑_𝑧¯∥_∞) (𝐴_𝑃𝜂¯∥𝜂¯∥ +𝐴_{𝑃 𝑑}

¯ 𝑧∥𝑑_𝑧¯∥_∞ +2𝐿_𝜌∥𝑧¯∥) −𝑐_{3, 𝜌}∥𝑧¯∥²+𝜎(𝐴_𝑉𝜂¯(𝜀) −𝑐₁) ∥𝜂¯∥²

=−h

∥𝜂¯∥ ∥𝑧¯∥ i

Λ(𝜀)

"

∥𝜂¯∥

∥𝑧¯∥

# + 𝐴²

𝑃 𝑑_𝑧¯∥𝑑_𝑧¯∥²_∞ +2𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐴_{𝑃 𝑑}

¯

𝑧∥𝜂¯∥ ∥𝑑_𝑧¯∥_∞+2𝐴_{𝑃 𝑑}

¯

𝑧𝐿_𝜌∥𝑧¯∥ ∥𝑑_𝑧¯∥_∞, where,

Λ(𝜀)=

"

−𝑐_{4, 𝜌}𝐴²

𝑃𝜂¯+𝜎(𝑐₁− 𝐴_𝑉𝜂¯(𝜀)) −𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐿_𝜌

−𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐿_𝜌 𝑐_{3, 𝜌}

# .

To yield positive definiteness of Λ(𝜀), we choose 𝜎 > 0 such that for𝜎 > 𝜎 and 𝜀 < 𝜀˜, det(Λ(𝜀)) > 0:

det(Λ(𝜀)) =−𝑐_{4, 𝜌}𝐴²

𝑃𝜂¯+𝜎(𝑐₁− 𝐴_𝑉𝜂¯(𝜀))𝑐_{3, 𝜌}−2𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐿_𝜌. We select,

𝜎 :=

𝑐_{4, 𝜌}𝐴²

𝑃𝜂¯+2𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐿_𝜌 (𝑐₁−𝐴_𝑉𝜂¯(𝜀))𝑐_{3, 𝜌}

,

where𝜎 >0 for all 0 < 𝜀 <𝜀˜. We choose𝜅=𝜆_min(Λ(𝜀)), the minimum eigenvalue ofΛ(𝜀). As in Theorem 5, we again split the derivative with𝜆_𝑃 ∈ (0,1),

𝑉_𝑃(P𝑧_¯(𝜂,¯ 𝑧¯)) −𝑉_𝑃(𝜂,¯ 𝑧¯)

≤ −𝜆_min(Λ(𝜀))𝜆_𝑃∥ (𝜂,¯ 𝑧¯) ∥²−𝜆_min(Λ(𝜀)) (1−𝜆_𝑃) ∥ (𝜂,¯ 𝑧¯) ∥² + 𝐴_𝜂¯𝑧¯∥ (𝜂,¯ 𝑧¯) ∥ ∥𝑑_𝑧¯∥_∞+𝐴²

𝑃 𝑑_𝑧¯∥𝑑_𝑧¯∥²_∞,

(6.27)

with 𝐴_𝜂¯𝑧¯ = max{2𝑐_{4, 𝜌}𝐴_𝑃𝜂¯𝐴_{𝑃 𝑑}

¯ 𝑧,2𝐴_{𝑃 𝑑}

¯

𝑧𝐿_𝜌}. This satisfies the discrete time e-ISS Lyapunov condition (6.24). Setting the last 3 terms of (6.27) ≤ 0 and solving for the positive root of the resultant quadratic equation, we establish exponential convergence at a rate of−𝜆_min(Λ(𝜀))𝜆_𝑃 to the set,

∥ (𝜂,¯ 𝑧¯) ∥ ≤

𝐴_𝜂¯𝑧¯ +√︃

𝐴²

¯

𝜂𝑧¯+4𝜆_min(Λ(𝜀)) (1−𝜆_𝑃)𝐴²

𝑃 𝑑𝑧_¯

2𝜆_min(Λ(𝜀)) (1−𝜆_𝑃) ∥𝑑_𝑧¯∥_∞. We require this bound to be less than𝑟_𝑧¯, yielding,

𝛿_𝑃 := 2𝑟_𝑧¯𝜆_min(Λ(𝜀)) (1−𝜆_𝑃) 𝐴_𝜂¯𝑧¯+√︃

𝐴²

¯

𝜂𝑧¯ +4𝜆_min(Λ(𝜀)) (1−𝜆_𝑃)𝐴²

𝑃 𝑑_¯𝑧

.

To also ensure the continuous dynamics of ¯𝑧 remain bounded by 𝑟_𝑧¯, we require

∥𝑑_𝑧¯∥_∞ < 𝛿:=min{𝛿_𝑃,

𝑟_𝑧¯

𝛿_𝑧¯}, unique to this proof, and hence establishing e-ISS of𝒪

for ∥𝑑_𝑧¯∥_∞ < 𝛿. □