# FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm
## 3 Proposed Proxy-Based Super-Twisting Algorithm (PSTA)
Since \(\mu\) is always and simulatively subjected to (7b) and (7c), (8) can be equivalently rewritten as the following system:  
\[\dot{s}_{2}\in[-F_{0},F_{0}]+[g_{m},g_{M}]\mu-\dot{s}_{1},\,\mu\in -F\operatorname{sgn}(s_{2}), \tag{9a}\] \[\mu=-\kappa_{1}\lfloor s_{1}\rceil^{1/2}+\upsilon,\,\upsilon\in -\kappa_{2}\operatorname{sgn}(s_{1}), \tag{9b}\]  
from which one can see that the control input \(\mu\) is simultaneously subjected to constraints of the first-order SMC and the second-order STA and is independent on the unknown model (4). Alternatively, one can say that the input \(\mu\) computed by the STA always satisfies \(|\mu|\leq F\), which is constraint by the first-order SMC. The closed-loop system (9) has the stability property shown in Theorem 1. **Theorem 1**. : _Consider the dynamical system (5) with the uncertain smooth functions \(f(t,x)\) and \(g(t,x)\) satisfying \(\sup|f(t,x)|\leq F_{0}\) and \(0<g_{m}\leq g(t,x)\leq g_{M},\,\forall t\geq t_{0}\geq 0,x\in\mathbb{R}\). If \(F>F_{0}/g_{m}\), \(\kappa_{2}\geq F_{1}/g_{m}\), \(\kappa_{1}>\sqrt{\kappa_{2}+F_{1}}/g_{m}\), and \(\kappa_{3}>0\) are satisfied for all \(t\geq 0\), then, with bounded initial condition \(s_{2}(t_{0})+s_{1}(t_{0})=\sigma(t_{0})\), the equilibrium of the sliding variable \(\sigma=s_{1}+s_{2}=0\) of the closed-loop system (5) interconnected with (7) is asymptotically attained._  
Proof. : Here, only the sketch of the proof is provided, because both the stabilities of first-order SMC and the STA are well studied in the literature. Because the closed-loop system (5) interconnected with (7) can be equivalently expressed as (9), one can only check the stability of (9). Let us first check the stability of (9a):  
\[\dot{s}_{2}\in[-F_{0},F_{0}]+[g_{m},g_{M}]\mu-\dot{s}_{1},\,\mu\in-F \operatorname{sgn}(s_{2}), \tag{10}\]  
of which the stability can be easily recognizable when \(F>F_{0}/g_{m}\) and \(|\dot{s}_{1}|\) remains bounded, because it is the standard first-order SMC. The asymptotical satiiability of (10) is straightforward to show by defining the Lyapunov function \(V_{1}(s_{2})=s_{2}^{2}/2\) according to ([26], Prop. 1), when only (10) is considered and \(\dot{s}_{1}\) is considered as a bounded disturbance. If the assumptions in (6) and \(F>F_{0}/g_{m}\) hold, there exists a finite time \(t_{1}\geq 0\) (depending on initial conditions \(s_{2}(t_{0})\)), \(\forall t\geq t_{1}\), \(\dot{s}_{2}=s_{2}=0\) ([26], Prop. 1). This convergence period \(t\in[t_{0},t_{1})\) of (10) can be considered as the first stage of (9). Therefore, in the fist stage, both \(|s_{2}|\) and \(|\dot{s}_{2}|\) are shrinking if \(|\dot{s}_{1}|\) is bounded. Then, for \(t\geq t_{1},s_{2}=\dot{s}_{2}=0\), i.e., in the second stage, from (9), due to \(\sigma=s_{1}+s_{2}\) and \(\sigma=s_{1}+\dot{s}_{2}\), one has the following expressions:  
\[\dot{s}_{1}\in[-F_{0},F_{0}]+[g_{m},g_{M}]\mu=f(t,x)+g(t,x)\mu \tag{11a}\] \[\mu=-\kappa_{1}\lfloor s_{1}\rceil^{1/2}+\upsilon,\,\upsilon\in- \kappa_{2}\operatorname{sgn}(s_{1}), \tag{11b}\]  
which can be further rewritten as follows:  
\[\mu \in-F\,\mathrm{sgn}(s_{2}),\quad s_{2}=\sigma-s_{1},\]0 \[\mu \in-F\,\mathrm{sgn}(s_{2}),\quad s_{2}=\sigma-s_{1},\]1  
with definition \(z_{1}:=f(t,x)+g(t,x)\). Note from (11) to (12), \(\dot{g}(t,x)=0\) has been used, because the input gain \(0<g_{m}\leq g(t,x)\leq g_{M}\) in (6) assumes to be a constant or slowly varying but bounded parameter. The system (12) is finite-time stable if the above gain conditions satisfied, if \(\kappa_{2}\geq F_{1}/g_{m}\), \(\kappa_{1}>\sqrt{\kappa_{2}+F_{1}}/g_{m}\), and \(\kappa_{3}>0\) hold [37]. The above proof is based on the assumption that \(|\dot{s}_{1}|\) is bounded. To demonstrate that \(|\dot{s}_{1}|\) and \(|\dot{s}_{1}|\) are shrinking and bounded, one can equivalently rewrite (10) as the dynamics about \(s_{1}\) with the control law \(\mu\) replaced by the STA (7c) and consider \(\dot{s}_{2}\) as the disturbance.