Exact Exponential Convergence and Perturbation Bounds for Higher-Order Serial Differentiators: A Strict Lyapunov Analysis

1. Introduction

The higher-order serial differentiator (HOSD) reconstructs the derivatives of a measured signal with a cascade of identical first-order stages. It was introduced as a first-order switching differentiator in [1] and extended to the serial cascade in [2]. Each stage contains a stable linear filter and an integral relay: the filter state tracks the stage input, and the relay, acting through an integrator, drives the stage output toward the derivative of that input. A derivative of any order is therefore obtained by appending stages with the same two-gain structure. Robust exact differentiators based on sliding modes, by contrast, attain finite-time exactness under suitable signal regularity through fractional-power output-injection terms [3,4], and arbitrary-order constructions with convergence uniform in the initial differentiation error are also available [5]. The convergence of the HOSD has since been analyzed with increasing strength [6,7], as reviewed below, and a forward-Euler discretization with an explicit steady-state error bound has been given for digital implementation [8].

A differentiator usually appears inside a control scheme rather than as an end in itself, so its accuracy and convergence rate enter the closed-loop guarantee directly. The HOSD has served in this role in a line of controllers for uncertain nonlinear systems: a differentiator-based adaptive feedback linearization scheme [9] and differentiator-based uncertainty cancelling (DBUC) controllers in continuous time with output feedback [10] and in discrete time with state and output feedback [11], in which HOSD stages estimate the lumped uncertainty and, under output feedback, the unmeasured derivatives of the tracking error. It has also been adopted outside this line of work, in observer-based anti-swing controllers for crane systems [12,13], observer-based trajectory-tracking controllers for wheeled mobile robots [14,15], a finite-time sliding-mode controller for perturbed second-order systems [16], and online stiffness estimation for variable-stiffness actuators [17].

This paper focuses on verifiable sufficient conditions for exponential convergence via Lyapunov method. In addition to an acceleration bound, the analysis uses a jerk budget and relates it explicitly to the two differentiator gains. The resulting nominal contraction is then transferred to perturbed inputs and propagated through serial interconnections. In a serial cascade, every stage after the first is driven by the output of the preceding stage, so its input contains the preceding estimation error. That error is Lipschitz but generally not differentiable, and it is known only through an amplitude bound. The same information pattern arises when a serial differentiator reconstructs unmeasured derivatives inside a control loop, as in the differentiator-based adaptive feedback linearization scheme of [9] and the differentiator-based uncertainty cancelling scheme of [10]. Both certify quasi-exponential convergence of the tracking error, and in both the tolerance is inherited from the estimator rather than from the control law; Remark 6 returns to this point.

The contributions are threefold. First, under bounds on the acceleration of the input and on its rate, a self-contained Lyapunov condition on the two gains certifies exponential convergence of the estimation error to zero, with no residual neighborhood, uniformly on prescribed initial energy sublevels. Second, the same Lyapunov function transfers that estimate to a continuous input perturbation known only through its amplitude: a persistent amplitude leaves a residual of the order of its fourth root, an exponentially vanishing one leaves none, and neither a derivative of the perturbation nor a smallness condition on its amplitude is required. Third, because this is exactly the information available about the nonsmooth interstage error, the estimates propagate through any fixed finite serial cascade, exactly when the first input is exact. The cascade results hold for a fixed number of stages whose nominal derivatives satisfy the corresponding bounds, the certified rates and residual exponents are conservative, and the constants are not uniform in the stage index.

The architecture analyzed here has been studied in a sequence of earlier works, against which the present contribution is best stated. The switching differentiator of [1] was shown to be asymptotically convergent for a single stage, and [2] extended the asymptotic analysis to the higher-order serial cascade by successive application of a generalized Barbalat lemma; neither argument certifies a convergence rate. A rate was obtained in [6], where the estimation errors of the cascade were shown to converge exponentially fast to an arbitrarily small neighborhood of zero for an exact input, so a residual neighborhood remained and a corrupted measurement was not covered. The perturbed case was treated in [7] under a uniform amplitude bound $\overline{d}$ and a non-accumulation condition on the switching instants, where the estimation error was shown to be quasi-exponentially convergent: for every $\epsilon >0$ a bound $\mu e^{-\lambda t}+\epsilon$ holds with $\mu$ and $\lambda$ depending on $\epsilon$, and the perturbed bound carries a steady-state term proportional to $\overline{d}$. Each of these analyses assumes a bound on the acceleration alone.

The present paper differs in three respects. First, the convergence certified here is exponential in the ordinary sense, with one decay rate and with convergence to zero rather than to a neighborhood: a strict Lyapunov function, the energy of the error system plus a quadratic correction, yields an explicit rate together with a verifiable gain condition. The price is one further regularity requirement, a jerk bound, which that condition ties to both differentiator gains. Second, the analysis is carried out for maximal Filippov solutions throughout, so neither a case distinction on sliding nor a non-accumulation condition on the switching instants is imposed; the thin layer in which the relay sign can be corrupted is bounded directly instead. Third, the perturbation is required only to be continuous and amplitude-bounded. It is never differentiated and its amplitude is not assumed small, which is exactly the information pattern of the nonsmooth interstage error in a serial cascade. The residual exponents certified here are conservative, and no optimality is claimed for them.

The remainder of the paper is organized as follows. Section 2 introduces the Filippov notation. Section 3 treats the first-order estimator without perturbation and under uniformly bounded, mixed, and exponentially vanishing amplitude-only perturbations. Section 4 treats the serial cascade with a bounded and with an exact first input, and Section 5 concludes.

2. Preliminaries

For a vector, $|·|$ denotes the Euclidean norm; for a scalar it denotes absolute value. Discontinuous dynamics are understood in the Filippov sense, with

$$\mathrm{Sgn}\left(r\right)=\left\{\begin{array}{cc}\left\{1\right\},\hfill & r>0,\hfill \\ [-1,1],\hfill & r=0,\hfill \\ \{-1\},\hfill & r<0.\hfill \end{array}\right.$$

(1)

The corresponding single-valued sign is

$$\mathrm{Sgn}\left(r\right)=\left\{\begin{array}{cc}1,\hfill & r>0,\hfill \\ 0,\hfill & r=0,\hfill \\ -1,\hfill & r<0.\hfill \end{array}\right.$$

3. First-Order Derivative Estimator

Consider the first-order serial differentiator

$$\begin{array}{cc}\hfill \dot{\alpha }& =\kappa (a-\alpha )+\sigma ,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{\sigma }& \in L\mathrm{Sgn}(a-\alpha ),\hfill \end{array}$$

(3)

where $\kappa ,L>0$. Define the tracking and derivative-estimation errors by

$$e=a-\alpha ,w=\dot{a}-\sigma .$$

Whenever $\dot{a}$ is locally absolutely continuous, these errors satisfy

$$\dot{e}=-\kappa e+w,\dot{w}\in \ddot{a}-L\mathrm{Sgn}\left(e\right).$$

(4)

Our first objective is a verifiable rate-dependent condition for exponential convergence. Our second objective is to transfer this estimate to an additive input perturbation without differentiating that perturbation. The same Lyapunov function, combined with an integral bound over the thin layer in which the relay sign can be corrupted, gives a practical exponential estimate for a persistent amplitude of any prescribed size and exact exponential transfer for an exponentially vanishing amplitude.

3.1. No Input Perturbation

We impose bounds on both the acceleration magnitude and its temporal rate. The following condition is sufficient; it is not claimed to be necessary or sharp.

Lemma 1

(Exponential sufficient condition). Suppose a has a locally absolutely continuous second derivative and constants $D_a,N_a\ge 0$ satisfy

$$\left|\ddot{a}\left(t\right)\right|\le D_a,\left|\stackrel{⃛}{a}\left(t\right)\right|\le N_a$$

(5)

almost everywhere. If

$$L>D_a,\kappa >{\displaystyle \frac{N_a}{L-D_a}},$$

(6)

then every solution of e and w from an arbitrary finite initial state exponentially converges to zero.

Proof. 

Step 1: Lyapunov function. Under $L>D_a$, the second inequality in (6) is equivalent to $\kappa (L-D_a)>N_a$, which is the form used in the later results. Put

$$b=L-D_a,\mu ={\displaystyle \frac{1}{2}}\left(\kappa +{\displaystyle \frac{N_a}{b}}\right),q={\displaystyle \frac{\mu (\kappa -\mu )}{2}}.$$

(7)

By (6), $b>0$ and $N_a/b<\kappa$, so $\mu$ is the midpoint of the interval $(N_a/b,\kappa )$ and $q>0$. With $\Delta =\kappa b-N_a>0$, the midpoint choice gives

$$\kappa -\mu ={\displaystyle \frac{\Delta }{2b}},\mu b-N_a={\displaystyle \frac{\Delta }{2}}.$$

(8)

With $A=\ddot{a}$, define

$$V_a=\frac{1}{2}{p}^2+L\left|e\right|-Ae+q{e}^2,p=w-(\kappa -\mu )e.$$

(9)

Expanding $p^2$ and using $2q=\mu (\kappa -\mu )$ shows that $V_a=H+Q$, where

$$H=\frac{1}{2}{w}^2+L\left|e\right|-Ae,Q=-(\kappa -\mu )ew+\frac{1}{2}\kappa (\kappa -\mu )e^2.$$

Thus $V_a$ is the energy H of the error system (4) plus a quadratic correction Q in $(e,w)$.

Step 2: dissipation. Let s be the measurable selection with $\dot{w}=A-Ls$ and $s\in \mathrm{Sgn}\left(e\right)$. Then ${\displaystyle \frac{d}{dt}}\left|e\right|=s\dot{e}$ almost everywhere: this is the chain rule where $e\ne 0$, and on every positive-measure subset on which $e=0$, absolute continuity gives $\dot{e}=0$ almost everywhere. Let $\nu =w-\kappa e=\dot{e}$. Using (4) and $se=\left|e\right|$, direct differentiation gives

$$\begin{array}{cc}\hfill \dot{H}& =-\kappa \left(L\right|e|-Ae)-\dot{A}e,\hfill \\ \hfill \dot{Q}& =-(\kappa -\mu ){\nu }^2+(\kappa -\mu )\left(L\right|e|-Ae),\hfill \end{array}$$

and therefore

$${\dot{V}}_a=-(\kappa -\mu ){\nu }^2-\mu \left(L\right|e|-Ae)-\dot{A}e.$$

(10)

The energy H alone dissipates only through $\left|e\right|$. The correction Q transfers the part $\kappa -\mu$ of that dissipation to the term $-(\kappa -\mu ){\nu }^2$, which makes $V_a$ a strict Lyapunov function. By (5) and (8),

$$\begin{array}{c}{\dot{V}}_a\le -(\kappa -\mu ){\nu }^2-(\mu b-N_a)\left|e\right|\le -c_a({\nu }^2+\left|e\right|),\\ c_a={\displaystyle \frac{\Delta }{2\max \{b,1\}}}.\end{array}$$

(11)

In particular, $V_a$ is nonincreasing. Since $L\left|e\right|-Ae\ge b\left|e\right|$ and $q>0$,

$$V_a\ge \frac{1}{2}{p}^2+b\left|e\right|.$$

(12)

Step 3: sublevel-uniform estimate. The error inclusion (4) has compact convex values and linear growth, so its maximal Filippov solutions are complete. Fix a level $\ell >0$ and consider a solution with $V_a\left(0\right)\le \ell$. Since $V_a$ is nonincreasing, (12) gives $\left|e\right|\le \ell /b$ and hence $e^2\le (\ell /b)\left|e\right|$. Since $p=\nu +\mu e$ gives ${\displaystyle \frac{1}{2}}{p}^2\le {\nu }^2+{\mu }^2{e}^2$,

$$V_a\le C_{\ell }({\nu }^2+|e\left|\right),C_{\ell }=\max \left\{1,L+D_a+({\mu }^2+q){\displaystyle \frac{\ell }{b}}\right\}.$$

Combining this estimate with (11) yields

$${\dot{V}}_a\le -{\rho }_{\ell }{V}_a,{\rho }_{\ell }={\displaystyle \frac{c_a}{C_{\ell }}},$$

(13)

and therefore $V_a\left(t\right)\le \ell e^{-{\rho }_{\ell }t}$. By (12),

$$\left|e\left(t\right)\right|\le {\displaystyle \frac{\ell }{b}}{e}^{-{\rho }_{\ell }t},\left|p\left(t\right)\right|\le \sqrt{2\ell }{e}^{-{\rho }_{\ell }t/2}.$$

Since $w=p+(\kappa -\mu )e$,

$$\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M_{\ell }{e}^{-{\lambda }_{\ell }t},t\ge 0,$$

(14)

with

$${\lambda }_{\ell }={\displaystyle \frac{{\rho }_{\ell }}{2}},M_{\ell }=\sqrt{2\ell }+{\displaystyle \frac{(1+\kappa -\mu )\ell }{b}}.$$

The constants $M_{\ell }$ and ${\lambda }_{\ell }$ depend only on $\kappa ,L,D_a,N_a$, and ℓ, so (14) holds with the same constants for every solution in the sublevel $V_a\left(0\right)\le \ell$.

Step 4: conclusion. For an arbitrary finite initial state $\left(e\right(0),w(0\left)\right)$, $V_a\left(0\right)$ is finite. Taking any $\ell >0$ with $\ell \ge V_a\left(0\right)$ in (14) shows that e and w converge to zero exponentially. □

Remark 1

(Rate budget). Bounds (5) and condition (6) couple the differentiator gains to both the acceleration budget $D_a$ and the jerk budget $N_a$. In (10), the term $-\dot{A}e$ must be dominated by the restoring margin $\mu b\left|e\right|$, while the dissipation of ${\dot{e}}^2$ requires $\mu <\kappa$. A constant μ with both properties exists precisely under (6). The midpoint choice (7) splits the margin $\Delta =\kappa b-N_a$ equally between the two dissipation terms, as (8) shows. This explains why a jerk budget enters the sufficient condition. The condition need not be necessary or sharp and may therefore be conservative.

3.2. Uniformly Bounded Input Perturbation

We drive the differentiator by $f=a+d$:

$$\begin{array}{cc}\hfill \dot{\alpha }& =\kappa (f-\alpha )+\sigma ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \dot{\sigma }& \in L\mathrm{Sgn}(f-\alpha ).\hfill \end{array}$$

(16)

With $e=a-\alpha$ and $w=\dot{a}-\sigma$, the errors satisfy

$$\dot{e}=-\kappa e+w-\kappa d,\dot{w}\in \ddot{a}-L\mathrm{Sgn}(e+d),$$

(17)

which reduces to (4) for $d\equiv 0$. The perturbation enters the relay argument. It can change the relay sign only where the signs of e and $e+d$ differ, that is, in the thin layer $0<\left|e\right|\le \left|d\right|$. The analysis below bounds the effect of this layer by a time integral and requires no derivative of d. Only an amplitude bound is assumed.

The following lemma is the only place where the layer enters. It bounds the time integral of $\left|w\right|$ over the layer by a quantity of order $\sqrt{E}$, although $\left|w\right|$ itself need not be small there.

Lemma 2

(Layer integral). Let J be an interval of length T, and let $B>0$ and $E\ge 0$. Suppose that, on J, e is $C^1$, w is Lipschitz with $\left|\dot{w}\right|\le B$ almost everywhere, and $\dot{e}=w+\zeta$ with $\left|\zeta \right|\le 2\kappa E$ whenever $\left|e\right|\le E$. Then

$${\int }_J\left|w\right|{\mathbf{1}}_{\left\{\right|e|\le E\}}dt\le 4\widehat{W}\left(T+{\displaystyle \frac{\widehat{W}}{B}}\right),\widehat{W}=\sqrt{BE}+4\kappa E.$$

(18)

Proof. 

If $E=0$, the integrand vanishes. Otherwise put $\tau =\widehat{W}/B$ and split J into at most $T/\tau +1$ consecutive subintervals I of length at most $\tau$. If $\left|w\right|\le 2\widehat{W}$ on I, the contribution of I is at most $2\widehat{W}\tau$. Otherwise $|w\left(t^{\prime }\right)|>2\widehat{W}$ for some $t^{\prime }\in I$, and $|\dot{w}|\le B$ gives $|w-w\left(t^{\prime }\right)|\le \widehat{W}$ on I. Hence w has a fixed sign S and $\left|w\right|>\widehat{W}$ on I. Where in addition $\left|e\right|\le E$, $S\dot{e}\ge \left|w\right|-2\kappa E\ge \left|w\right|/2>0$, because $\widehat{W}\ge 4\kappa E$. Thus $Se$ strictly increases whenever it lies in $[-E,E]$, so it can neither re-enter $[-E,E]$ through E nor leave it through $-E$. The set $I\cap \left\{\right|e|\le E\}$ is therefore an interval on which $\left|w\right|\le 2S\dot{e}$, and its contribution is at most $2·2E=4E$. Since $\widehat{W}\tau ={\widehat{W}}^2/B\ge E$, every subinterval contributes at most $4\widehat{W}\tau$, and summing gives (18). □

Theorem 1

(Bounded amplitude-only transfer). Suppose $\ddot{a}$ is locally absolutely continuous and constants $D_a,N_a\ge 0$ satisfy

$$|\ddot{a}\left(t\right)|\le D_a,\left|\stackrel{⃛}{a}\left(t\right)\right|\le N_a,\kappa (L-D_a)>N_a$$

(19)

almost everywhere. Fix $\overline{D}\ge 0$. If d is continuous and $\left|d\right(t\left)\right|\le D$ for all $t\ge 0$ with $D\in [0,\overline{D}]$, then every solution of e and w from an arbitrary finite initial state satisfies

$$\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M{e}^{-\lambda t}+c_1{D}^{1/4}+c_2{D}^{1/2},t\ge 0,$$

(20)

where the constants $M,\lambda ,c_1,c_2>0$ depend on the initial errors but not on $D\in [0,\overline{D}]$.

Proof. 

We prove an estimate that is uniform on prescribed sets of initial errors; the later results use it in this form. Fix $D\ge 0$ and $R>0$, let d be continuous with

$$\left|d\right(t\left)\right|\le D,t\ge 0,$$

(21)

and consider solutions with

$$|a\left(0\right)-\alpha \left(0\right)|+|\dot{a}\left(0\right)-\sigma \left(0\right)|\le R.$$

(22)

Steps 1–4 show that every such solution satisfies

$$\begin{array}{c}\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M_R{e}^{-{\lambda }_{R}t}+c_R\left(D\right),t\ge 0,\\ c_R\left(D\right)=c_{1,R}{D}^{1/4}+c_{2,R}{D}^{1/2},{\lambda }_R=\frac{1}{2}{\rho }_R,\end{array}$$

(23)

with the explicit constants $M_R,{\rho }_R,c_{1,R},c_{2,R}>0$ given below. They depend only on $\kappa ,L,D_a,N_a,D$, and R, and not on the particular perturbation d or on the nominal input a beyond the bounds $(D_a,N_a)$. Step 5 shows that these constants are monotone in D, and Step 6 concludes.

The inclusion (17) has compact convex values and linear growth, so every maximal Filippov solution is complete. Let b, $\mu$, and q be given by (7), and set

$$B=L+D_a,c=\min \left\{\frac{1}{2}(\kappa -\mu ),\mu b-N_a\right\},$$

which is positive by (8).

Step 1: Lyapunov identity. Let $p=w-(\kappa -\mu )e$, $\nu =p-\mu e=w-\kappa e$, $A=\ddot{a}$, and

$$V=\frac{1}{2}{p}^2+L\left|e\right|-Ae+q{e}^2.$$

In the unperturbed case $\nu =\dot{e}$, and V coincides with $V_a$ in (9). The function V is locally absolutely continuous and satisfies $V\ge {\displaystyle \frac{1}{2}}{p}^2+b\left|e\right|$. Let s be the measurable selection with $\dot{w}=A-Ls$ and $s\in \mathrm{Sgn}(e+d)$, and let $\varsigma =\mathrm{Sgn}\left(e\right)$ where $e\ne 0$ and $\varsigma =s$ where $e=0$. Then ${\displaystyle \frac{d}{dt}}\left|e\right|=\varsigma \dot{e}$ almost everywhere, because $\dot{e}=0$ almost everywhere on $\{e=0\}$. Moreover, $\varsigma \ne s$ only where $0<\left|e\right|\le \left|d\right|$, since $\left|e\right|>\left|d\right|$ implies $\mathrm{Sgn}(e+d)=\mathrm{Sgn}\left(e\right)$. Using $\dot{e}=p-\mu e-\kappa d$, direct differentiation gives

$$\begin{array}{cc}\hfill \dot{V}=& -(\kappa -\mu ){\nu }^2-\mu \left(L\right|e|-Ae)-\dot{A}e\hfill \\ \hfill & +L(\varsigma -s)p+\kappa d\left[(\kappa -\mu )\nu -L\varsigma +A\right].\hfill \end{array}$$

(24)

For $d\equiv 0$ this reduces to (10). Young’s inequality on $\kappa (\kappa -\mu )d\nu$ yields

$$\dot{V}\le -c({\nu }^2+\left|e\right|)+\beta +\gamma ,$$

(25)

with

$$\begin{array}{cc}\hfill \beta & =L|\varsigma -s|\left|p\right|\le 2L\left|p\right|{\mathbf{1}}_{\left\{\right|e|\le |d\left|\right\}},\hfill \\ \hfill \gamma & =\kappa B\left|d\right|+\frac{1}{2}{\kappa }^2(\kappa -\mu )d^2.\hfill \end{array}$$

The dissipation in (25) grows with V. Indeed, $p=\nu +\mu e$ gives ${\displaystyle \frac{1}{2}}{p}^2\le {\nu }^2+{\mu }^2{e}^2$, so

$$V\le {\nu }^2+B\left|e\right|+({\mu }^2+q)e^2.$$

(26)

With $B_1=\max \{1,B\}$, $Q={\mu }^2+q$, and $\varrho ={\nu }^2+\left|e\right|$, this gives $V\le B_1\varrho +Q{\varrho }^2$, so $V\le 2B_1\varrho$ or $V\le 2Q{\varrho }^2$. Hence $\varrho \ge \varphi \left(V\right)$, where

$$\varphi \left(v\right)=\min \left\{{\displaystyle \frac{v}{2B_1}},\sqrt{{\displaystyle \frac{v}{2Q}}}\right\}$$

(27)

is nondecreasing and unbounded, and (25) yields

$$\dot{V}\le -c\varphi \left(V\right)+\beta +\gamma .$$

(28)

Step 1 uses only the continuity of d, not the bound (21).

Step 2: window bound. Fix $\overline{E}\ge 0$, and let J be an interval of length at most 1 on which $\left|d\right|\le E$ for some $E\in [0,\overline{E}]$. On the support of $\beta$, $\left|p\right|\le \left|w\right|+(\kappa -\mu )E$. Moreover, $\dot{e}=w+\zeta$ with $\zeta =-\kappa (e+d)$, so $\left|\zeta \right|\le 2\kappa E$ wherever $\left|e\right|\le E$, and $\left|\dot{w}\right|\le B$. The right-hand side of $\dot{e}=-\kappa e+w-\kappa d$ is continuous, because w is Lipschitz and d is continuous, so e is $C^1$. Lemma 2, applied with $\widehat{W}=\sqrt{BE}+4\kappa E\le c_W\sqrt{E}$ and $c_W=\sqrt{B}+4\kappa \sqrt{\overline{E}}$, therefore gives

$${\int }_J(\beta +\gamma )dt\le K\sqrt{E},$$

(29)

where

$$\begin{array}{cc}\hfill K=& 8L{c}_W\left(1+{\displaystyle \frac{c_W\sqrt{\overline{E}}}{B}}\right)+\left(2L(\kappa -\mu )+\kappa B\right)\sqrt{\overline{E}}\hfill \\ \hfill & +\frac{1}{2}{\kappa }^2(\kappa -\mu ){\overline{E}}^{3/2}.\hfill \end{array}$$

(30)

Every term of (30) is nondecreasing in $\overline{E}$, so K is nondecreasing in $\overline{E}$. In this proof $\overline{E}=D$, so (29) holds with $E=D$ on every interval of length at most 1.

Step 3: a priori bound. The initial set gives $\left|p\right(0\left)\right|\le (1+\kappa )R$, hence

$$V\left(0\right)\le V_0:=\frac{1}{2}{(1+\kappa )}^2{R}^2+BR+q{R}^2.$$

(31)

Put

$$\begin{array}{c}I=K\sqrt{D},V_{*}=\max \left\{{\displaystyle \frac{4B_{1}I}{c}},{\displaystyle \frac{8Q{I}^2}{c^2}}\right\},\\ R_V=\max \{V_0,V_{*}\}+I.\end{array}$$

(32)

By (29), $\beta +\gamma$ integrates to at most I over every interval of length at most 1, and $\dot{V}\le \beta +\gamma$ by (28). Hence V increases by at most I on such an interval. Moreover, (27) gives $c\varphi \left(V_{*}\right)\ge 2I$. Put $m=\max \{V_0,V_{*}\}$, and suppose that $V\left(t_2\right)>m+I$ for some $t_2>0$. Let $t_1=\sup \{t\in [0,t_2]:V\left(t\right)\le m\}$. Then $V\left(t_1\right)=m$, $V>m$ on $(t_1,t_2]$, and $t_2-t_1>1$ because $V\left(t_2\right)-V\left(t_1\right)>I$. On $[t_1,t_1+1]$ we have $V\ge m\ge V_{*}$, so integrating (28) gives

$$V(t_1+1)\le m-c\varphi \left(V_{*}\right)+I\le m-I\le m,$$

which contradicts $V(t_1+1)>m$. Therefore $V\le R_V$ for all $t\ge 0$. Consequently $\left|e\right|\le R_V/b$, hence $e^2\le (R_V/b)\left|e\right|$, and (26) gives $V\le C_V({\nu }^2+\left|e\right|)$ with

$$C_V=\max \{1,B+Q{R}_V/b\},{\rho }_R=c/C_V.$$

(33)

Thus (25) yields

$$\dot{V}\le -{\rho }_{R}V+\beta +\gamma \mathrm{for}\mathrm{almost}\mathrm{every}t\ge 0.$$

(34)

Step 4: decay. Integrating (34) gives

$$V\left(t\right)\le e^{-{\rho }_{R}t}{V}_0+{\int }_0^t{e}^{-{\rho }_R(t-s)}(\beta +\gamma )\left(s\right)ds.$$

Split $[0,t]$ into the windows $J_k=[k,k+1]$, $k=0,\dots ,\lfloor t\rfloor -1$, and the final piece $[\lfloor t\rfloor ,t]$ of length at most 1. On $J_k$ the kernel is at most $e^{-{\rho }_R(t-k-1)}$, and on the final piece it is at most 1, so (29) gives

$$V\left(t\right)\le e^{-{\rho }_{R}t}{V}_0+K\sqrt{D}{\displaystyle \sum _{k=0}^{\lfloor t\rfloor }}{\omega }_k,{\omega }_k=\left\{\begin{array}{cc}{e}^{-{\rho }_R(t-k-1)},\hfill & k<\lfloor t\rfloor ,\hfill \\ 1,\hfill & k=\lfloor t\rfloor .\hfill \end{array}\right.$$

(35)

Since $t-k-1\ge \lfloor t\rfloor -k-1\ge 0$ for $k<\lfloor t\rfloor$,

$$\sum _{k<\lfloor t\rfloor }{\omega }_k\le {\displaystyle \sum _{j=0}^{\infty }}{e}^{-{\rho }_{R}j}={\displaystyle \frac{1}{1-e^{-{\rho }_R}}}\le 1+{\displaystyle \frac{1}{{\rho }_R}},$$

and therefore

$$V\left(t\right)\le V_0{e}^{-{\rho }_{R}t}+K_D\sqrt{D},K_D=K\left(2+{\displaystyle \frac{1}{{\rho }_R}}\right).$$

Finally, $\left|e\right|\le V/b$ and $\left|w\right|\le \left|p\right|+(\kappa -\mu )\left|e\right|\le \sqrt{2V}+(\kappa -\mu )V/b$. Using $\sqrt{x+y}\le \sqrt{x}+\sqrt{y}$ and $e^{-{\rho }_{R}t}\le e^{-{\rho }_{R}t/2}$, this gives (23) with ${\lambda }_R={\rho }_R/2$ and

$$\begin{array}{c}{M}_R=\sqrt{2V_0}+{\displaystyle \frac{(1+\kappa -\mu )V_0}{b}},\\ c_{1,R}=\sqrt{2K_D},c_{2,R}={\displaystyle \frac{(1+\kappa -\mu )K_D}{b}}.\end{array}$$

Step 5: dependence on D. The constant $M_R$ depends on $V_0$ only and not on D. Since K is nondecreasing in $\overline{E}=D$, the quantity $I=K\sqrt{D}$ is nondecreasing in D, and so are $V_{*}$, $R_V$, and $C_V$. Hence ${\rho }_R=c/C_V$ and ${\lambda }_R={\rho }_R/2$ are nonincreasing in D, and $K_D=K(2+1/{\rho }_R)$, $c_{1,R}$, and $c_{2,R}$ are nondecreasing in D. If $0\le D\le \overline{D}$, the constants obtained for $\overline{D}$ therefore satisfy ${\lambda }_R\left(\overline{D}\right)\le {\lambda }_R\left(D\right)$ and $c_{i,R}\left(\overline{D}\right)\ge c_{i,R}\left(D\right)$, so the right-hand side of (23) for D is bounded by the same expression with the constants for $\overline{D}$. Thus (23) holds with the constants obtained for $\overline{D}$ for every $D\in [0,\overline{D}]$.

Step 6: conclusion. For an arbitrary finite initial state, take any $R\ge \left|e\right(0\left)\right|+\left|w\right(0\left)\right|$, so that (22) holds. By Step 5, (23) with the constants obtained for $\overline{D}$ gives (20) with $M=M_R$, $\lambda ={\lambda }_R$, $c_1=c_{1,R}$, and $c_2=c_{2,R}$, none of which depends on $D\in [0,\overline{D}]$. □

Remark 2

(Amplitude range). No smallness of the persistent amplitude is required, and no derivative of d is used. The a priori bound in Step 3 of the proof of Theorem 1 uses the dissipation $c\varphi \left(V\right)$, which grows with V, whereas the perturbation injects at most I into V per unit time. The constants deteriorate as D grows. By the monotonicity in Step 5, the constants obtained for a prescribed bound $\overline{D}$ are uniform on $[0,\overline{D}]$, so with these fixed constants the residual $c_R\left(D\right)$ tends to zero with D. No linear gain is asserted, and no optimality of the exponent $1/4$ is claimed.

3.3. Mixed Bounded and Exponentially Vanishing Perturbations

In a serial cascade, the input perturbation of every stage after the first contains a decaying transient in addition to a persistent part. We therefore extend Theorem 1 to an amplitude envelope with an exponentially vanishing part.

Theorem 2

(Mixed amplitude-only transfer). Suppose $\ddot{a}$ is locally absolutely continuous and (19) holds almost everywhere. Fix $C_{\mathrm{e}}\ge 0$, ${\gamma }_{\mathrm{e}}>0$, and $\overline{D}\ge 0$. If d is continuous and

$$\left|d\left(t\right)\right|\le D+C_{\mathrm{e}}{e}^{-{\gamma }_{\mathrm{e}}t},t\ge 0,$$

(36)

with $D\in [0,\overline{D}]$, then every solution of e and w from an arbitrary finite initial state satisfies

$$\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M{e}^{-\lambda t}+c_1{D}^{1/4}+c_2{D}^{1/2},t\ge 0,$$

(37)

where the constants $M,\lambda ,c_1,c_2>0$ depend on the initial errors but not on $D\in [0,\overline{D}]$.

Proof. 

As in the proof of Theorem 1, fix $R>0$ and consider solutions satisfying (22). Steps 2–4 below show that every such solution satisfies

$$\begin{array}{c}\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M_R{e}^{-{\lambda }_{R}t}+c_R\left(D\right),t\ge 0,\\ {\lambda }_R=\frac{1}{4}\min \{{\rho }_R,{\gamma }_{\mathrm{e}}/2\},\end{array}$$

(38)

with $c_R\left(D\right)=c_{1,R}{D}^{1/4}+c_{2,R}{D}^{1/2}$ and the explicit constants $M_R,{\rho }_R,c_{1,R},c_{2,R}>0$ given below. They depend only on $\kappa ,L,D_a,N_a,C_{\mathrm{e}},{\gamma }_{\mathrm{e}},\overline{D}$, and R, and not on the particular value of $D\in [0,\overline{D}]$, on the particular perturbation d, or on the nominal input a beyond the bounds $(D_a,N_a)$. For an arbitrary finite initial state, taking any $R\ge \left|e\right(0\left)\right|+\left|w\right(0\left)\right|$ in (38) then gives (37) with $M=M_R$, $\lambda ={\lambda }_R$, $c_1=c_{1,R}$, and $c_2=c_{2,R}$.

Completeness and Step 1 of the proof of Theorem 1 use only the continuity of d, so (25) and (28) hold. Let b, $\mu$, q, B, c, $B_1$, and Q be as in that proof, put $\overline{E}=\overline{D}+C_{\mathrm{e}}$, and let K be given by (30) with this $\overline{E}$.

Step 2: window bounds. Since $\left|d\right|\le \overline{E}$ for all $t\ge 0$, (29) with $E=\overline{E}$ shows that $\beta +\gamma$ integrates to at most $K\sqrt{\overline{E}}$ over every interval of length at most 1. Moreover, let $J_k=[k,k+1]$ and $E_k=D+C_{\mathrm{e}}{e}^{-{\gamma }_{\mathrm{e}}k}$, so that $\left|d\right|\le E_k\le \overline{E}$ on $J_k$. Hence (29) gives

$${\int }_{J_k}(\beta +\gamma )dt\le K\sqrt{E_k}\le K\left(\sqrt{D}+\sqrt{C_{\mathrm{e}}}{e}^{-{\gamma }_{\mathrm{e}}k/2}\right).$$

(39)

Step 3: a priori bound. Step 3 of the proof of Theorem 1, with $I=K\sqrt{\overline{E}}$ in (32) and the same definitions of $V_0$, $V_{*}$, $R_V$, $C_V$, and ${\rho }_R=c/C_V$, shows that $V\le R_V$ and that (34) holds for almost every $t\ge 0$.

Step 4: decay. Integrating (34), splitting $[0,t]$ as in (35), and using (39),

$$V\left(t\right)\le e^{-{\rho }_{R}t}{V}_0+{\displaystyle \sum _{k=0}^{\lfloor t\rfloor }}{\omega }_{k}K\left(\sqrt{D}+\sqrt{C_{\mathrm{e}}}{e}^{-{\gamma }_{\mathrm{e}}k/2}\right),$$

with the weights ${\omega }_k$ of (35). Since ${\omega }_k\le e^{-{\rho }_R(t-k-1)}$ for every $k\le \lfloor t\rfloor$, with ${\lambda }_{*}=\min \{{\rho }_R,{\gamma }_{\mathrm{e}}/2\}$ each vanishing term is at most $K\sqrt{C_{\mathrm{e}}}{e}^{{\rho }_R}{e}^{-{\lambda }_{*}t}$, and $1+t\le (1+2/{\lambda }_{*})e^{{\lambda }_{*}t/2}$. As in the proof of Theorem 1, ${\sum }_k{\omega }_k\le 2+1/{\rho }_R$, so the persistent terms sum to at most $K_D\sqrt{D}$ with $K_D=K(2+1/{\rho }_R)$. Therefore

$$\begin{array}{c}V\left(t\right)\le M_V{e}^{-{\lambda }_{*}t/2}+K_D\sqrt{D},\\ M_V=V_0+K\sqrt{C_{\mathrm{e}}}{e}^{{\rho }_R}\left(1+{\displaystyle \frac{2}{{\lambda }_{*}}}\right).\end{array}$$

Finally, as in the proof of Theorem 1, $\left|e\right|\le V/b$ and $\left|w\right|\le \sqrt{2V}+(\kappa -\mu )V/b$ give (38) with ${\lambda }_R={\lambda }_{*}/4$ and

$$\begin{array}{c}{M}_R=\sqrt{2M_V}+{\displaystyle \frac{(1+\kappa -\mu )M_V}{b}},\\ c_{1,R}=\sqrt{2K_D},c_{2,R}={\displaystyle \frac{(1+\kappa -\mu )K_D}{b}}.\end{array}$$

For $C_{\mathrm{e}}=0$, $\overline{E}=\overline{D}$ and $M_V=V_0$, so all constants except ${\lambda }_R$ coincide with those in (23), from the proof of Theorem 1, for $D=\overline{D}$. □

Remark 3

(Role of the vanishing part). The exponentially vanishing part produces no separate nonzero residual term; the residual vanishes when $D=0$. The certified constants nevertheless depend on $C_{\mathrm{e}}$ through $\overline{E}=\overline{D}+C_{\mathrm{e}}$, and $M_R$ also depends on ${\gamma }_{\mathrm{e}}$. No smallness of $\overline{D}$ or $C_{\mathrm{e}}$ and no derivative of d are assumed.

3.4. Exponentially Vanishing Input Perturbation

The case $D=0$ of Theorem 2 is the one used in the exact serial cascade of Section 4.2.

Corollary 1

(Exact amplitude-only exponential transfer). Suppose $\ddot{a}$ is locally absolutely continuous and (19) holds almost everywhere. If d is continuous and, for some $C_d,{\gamma }_d>0$,

$$\left|d\left(t\right)\right|\le C_d{e}^{-{\gamma }_{d}t},t\ge 0,$$

(40)

then every solution of e and w from an arbitrary finite initial state converges exponentially to zero.

Proof. 

Apply the proof of Theorem 2 with $\overline{D}=D=0$ and $(C_{\mathrm{e}},{\gamma }_{\mathrm{e}})=(C_d,{\gamma }_d)$. Since $c_R\left(0\right)=0$, (38) shows that, for every $R>0$, every solution satisfying (22) satisfies

$$\begin{array}{c}\left|e\left(t\right)\right|+\left|w\left(t\right)\right|\le M_R{e}^{-{\lambda }_{R}t},t\ge 0,\\ {\lambda }_R=\frac{1}{4}\min \{{\rho }_R,{\gamma }_d/2\},\end{array}$$

(41)

where $M_R$ and ${\rho }_R$ depend only on $\kappa ,L,D_a,N_a,C_d,{\gamma }_d$, and R. Taking any $R\ge \left|e\right(0\left)\right|+\left|w\right(0\left)\right|$ gives the claim. □

The rate in (41) satisfies ${\lambda }_R\le {\gamma }_d/8$. Any rate below ${\displaystyle \frac{1}{2}}\min \{{\rho }_R,{\gamma }_d/2\}$ is also admissible at the cost of a larger gain $M_R$; the stated form is chosen for explicit constants.

4. Higher-Order Derivative Estimator

For a serial implementation, let ${\sigma }_0$ denote the signal applied to the first stage and write stage $j\ge 1$ as

$$\begin{array}{cc}\hfill {\dot{\alpha }}_j& ={\kappa }_j({\sigma }_{j-1}-{\alpha }_j)+{\sigma }_j,\hfill \\ \hfill {\dot{\sigma }}_j& \in L_j\mathrm{Sgn}({\sigma }_{j-1}-{\alpha }_j).\hfill \end{array}$$

(42)

Thus its input is ${\sigma }_{j-1}$, and ${\sigma }_j$ is intended to estimate $a^{\left(j\right)}$. Relative to the nominal stage input $a^{(j-1)}$, define the interstage perturbation and the stage errors

$$\begin{array}{c}{d}_j={\sigma }_{j-1}-a^{(j-1)},\\ e_j=a^{(j-1)}-{\alpha }_j,w_j=a^{\left(j\right)}-{\sigma }_j,\end{array}$$

(43)

so that the relay argument is ${\sigma }_{j-1}-{\alpha }_j=e_j+d_j$ and $d_{j+1}=-w_j$. Stage j is an instance of (15) and (16) under

$$(\kappa ,L,a,d,\alpha ,\sigma )\mapsto ({\kappa }_j,L_j,a^{(j-1)},d_j,{\alpha }_j,{\sigma }_j),$$

and its errors $(e,w)$ become $(e_j,w_j)$. In particular, $\left|{\dot{\sigma }}_j\right|\le L_j$ almost everywhere, so ${\sigma }_j$ is Lipschitz, but it is generally not $C^1$: its derivative is relay-driven and may switch between $\pm L_j$. The interstage perturbation $d_{j+1}$ is therefore continuous but generally not differentiable, and a transfer theorem that differentiates it does not apply. The amplitude-only results of SubSection 3.2, Section 3.3 and Section 3.4 provide the required alternative.

Fix the number of stages $n\ge 1$. For a continuous first input ${\sigma }_0$, the cascade (42), $j=1,\dots ,n$, is a differential inclusion with compact convex values and linear growth in ${({\alpha }_j,{\sigma }_j)}_{j=1}^n$, so every maximal Filippov solution is complete. Throughout this section, assume that, for $j=1,\dots ,n$, $a^{(j+1)}$ is locally absolutely continuous and, almost everywhere,

$$\begin{array}{c}|a^{(j+1)}|\le D_{a,j},\left|a^{(j+2)}\right|\le N_{a,j},\\ {\kappa }_j(L_j-D_{a,j})>N_{a,j},\end{array}$$

(44)

The estimates in the proofs of Section 3 are trajectory-wise: they use only the stated bounds along the solution under consideration, and their constants do not depend on the particular nominal input or perturbation beyond these bounds. They therefore apply to the interstage perturbations $d_j$ generated by the cascade itself.

4.1. Uniformly Bounded Perturbation at the First Input

Theorem 3

(Practical exponential cascade). Let ${\sigma }_0=a+d$, where d is continuous and $\left|d\right(t\left)\right|\le D$ for $t\ge 0$, and suppose (44) holds for $j=1,\dots ,n$. Fix $\overline{D}\ge 0$. If $D\in [0,\overline{D}]$, then every solution of the cascade from an arbitrary finite initial state satisfies, for $j=1,\dots ,n$,

$$\begin{array}{cc}\hfill & |a^{(j-1)}\left(t\right)-{\alpha }_j\left(t\right)|+|a^{\left(j\right)}\left(t\right)-{\sigma }_j\left(t\right)|\hfill \\ \hfill & \le M_j{e}^{-{\lambda }_{j}t}+P_j{D}^{4^{-j}},t\ge 0,\hfill \end{array}$$

(45)

where the constants $M_j,{\lambda }_j,P_j>0$ depend on the initial errors but not on $D\in [0,\overline{D}]$.

Proof. 

Completeness was noted above. For an arbitrary finite initial state, fix radii $R_j>0$ with

$$|a^{(j-1)}\left(0\right)-{\alpha }_j\left(0\right)|+|a^{\left(j\right)}\left(0\right)-{\sigma }_j\left(0\right)|\le R_j,j=1,\dots ,n.$$

(46)

We show that there exist constants $M_j,{\lambda }_j,c_{1,j},c_{2,j}>0$, $j=1,\dots ,n$, that depend on $\overline{D}$ and the radii but not on D, such that, for every $D\in [0,\overline{D}]$ and $j=1,\dots ,n$,

$$\begin{array}{cc}\hfill & |a^{(j-1)}\left(t\right)-{\alpha }_j\left(t\right)|+|a^{\left(j\right)}\left(t\right)-{\sigma }_j\left(t\right)|\hfill \\ \hfill & \le M_j{e}^{-{\lambda }_{j}t}+{\delta }_{j+1},t\ge 0,\hfill \end{array}$$

(47)

where ${\delta }_1=D$ and

$${\delta }_{j+1}=c_j\left({\delta }_j\right),c_j\left(r\right)=c_{1,j}{r}^{1/4}+c_{2,j}{r}^{1/2}.$$

(48)

The first-stage rate is ${\lambda }_1={\rho }_1/2$, where ${\rho }_1$ is the rate ${\rho }_R$ in (33), from the proof of Theorem 1, for stage one with $D=\overline{D}$ and $R=R_1$, and, for $j\ge 2$,

$${\lambda }_j=\frac{1}{4}\min \{{\rho }_j,{\lambda }_{j-1}/2\},$$

(49)

where ${\rho }_j$ is the rate ${\rho }_R$ in (38), from the proof of Theorem 2, for stage j. Consequently,

$${\lambda }_j\le \frac{1}{8}{\lambda }_{j-1},{\lambda }_n\le 8^{-(n-1)}{\lambda }_1.$$

(50)

Finally, we bound ${\delta }_{j+1}$ by $P_j{D}^{4^{-j}}$, which gives (45).

We first fix all stage constants, before D is chosen. For stage one, apply the estimate (23) in the proof of Theorem 1 with $D={\overline{D}}_1=\overline{D}$ and radius $R_1$. It gives $M_1$, ${\rho }_1$, $c_{1,1}$, and $c_{2,1}$, and the rate ${\lambda }_1={\rho }_1/2$; by Step 5 of that proof, these constants serve for every amplitude in $[0,{\overline{D}}_1]$. Put ${\overline{D}}_2=c_1\left({\overline{D}}_1\right)$. For $j\ge 2$, put $(C_j,{\gamma }_j)=(M_{j-1},{\lambda }_{j-1})$ and apply the estimate (38) in the proof of Theorem 2 to stage j with $(C_{\mathrm{e}},{\gamma }_{\mathrm{e}})=(C_j,{\gamma }_j)$, amplitude bound ${\overline{D}}_j$, and radius $R_j$. It gives $M_j$, ${\rho }_j$, $c_{1,j}$, and $c_{2,j}$, and the rate (49); put ${\overline{D}}_{j+1}=c_j\left({\overline{D}}_j\right)$. None of these constants depends on D.

Fix $D\in [0,\overline{D}]$. Since each $c_j$ is increasing, ${\delta }_j\le {\overline{D}}_j$ for $j=1,\dots ,n$. For stage one, $d_1=d$ satisfies $\left|d_1\right|\le {\delta }_1$ with ${\delta }_1\le {\overline{D}}_1$, so (23) with the stage-one constants gives (47) for $j=1$. For $j\ge 2$, suppose the input perturbation of stage j is continuous and satisfies

$$\left|d_j\left(t\right)\right|\le {\delta }_j+C_j{e}^{-{\gamma }_{j}t},t\ge 0.$$

(51)

Since ${\delta }_j\le {\overline{D}}_j$, (38) with the stage-j constants gives (47) for stage j. In either case, $d_{j+1}=-w_j$, so the bound for stage j shows that $d_{j+1}$ satisfies (51) with $C_{j+1}=M_j$ and ${\gamma }_{j+1}={\lambda }_j$, and $d_{j+1}$ is continuous because ${\sigma }_j$ is Lipschitz and $a^{\left(j\right)}$ is continuous. This closes the induction. The recursion (49) gives ${\lambda }_j\le {\displaystyle \frac{1}{4}}·{\displaystyle \frac{1}{2}}{\lambda }_{j-1}$, and (50) follows.

Finally, put $K_j=c_{1,j}+c_{2,j}{\overline{D}}_j^{1/4}$. Since $r^{1/2}\le {\overline{D}}_j^{1/4}{r}^{1/4}$ for $0\le r\le {\overline{D}}_j$, ${\delta }_j\le {\overline{D}}_j$ gives ${\delta }_{j+1}=c_j\left({\delta }_j\right)\le K_j{\delta }_j^{1/4}$ for $j=1,\dots ,n$. Iterating this inequality from ${\delta }_1=D$ gives

$${\delta }_{j+1}\le P_j{D}^{4^{-j}},P_j={\displaystyle \prod _{k=1}^j}{K}_k^{4^{-(j-k)}},j=1,\dots ,n.$$

(52)

Together with (47), this gives (45). The constants $M_j$, ${\lambda }_j$, and $P_j$ depend on the radii $R_j$, and hence on the initial errors, but not on $D\in [0,\overline{D}]$. □

Remark 4

(Deterioration of the residual exponent). For each fixed n as $D\downarrow 0$, Theorem 3 shows that the persistent perturbation entering stage n is $O\left(D^{4^{-(n-1)}}\right)$, while the persistent part of the stage-n output error is $O\left(D^{4^{-n}}\right)$. The exponent approaches zero rapidly with the stage number, and the constants depend on the stage index and on $\overline{D}$. No smallness of $\overline{D}$ is required. The result is therefore a fixed-finite-stage practical estimate and does not provide stage-uniform robustness.

4.2. Special Case: Exact First Input

If $d\equiv 0$, Theorem 3 with $\overline{D}=D=0$ shows that every stage converges exponentially to zero. The following theorem states this special case directly. Its first stage uses Lemma 1 in place of Theorem 1, and its later stages use Corollary 1.

Theorem 4

(Exact exponential cascade). Let ${\sigma }_0=a$, and suppose (44) holds for $j=1,\dots ,n$. Then, for $j=1,\dots ,n$, the stage errors $a^{(j-1)}-{\alpha }_j$ and $a^{\left(j\right)}-{\sigma }_j$ of every solution from an arbitrary finite initial state converge exponentially to zero.

Proof. 

Completeness was noted above. For an arbitrary finite initial state, fix radii $R_j>0$ such that (46) holds. We show by induction that there exist constants $M_j,{\lambda }_j>0$, $j=1,\dots ,n$, such that

$$|a^{(j-1)}\left(t\right)-{\alpha }_j\left(t\right)|+|a^{\left(j\right)}\left(t\right)-{\sigma }_j\left(t\right)|\le M_j{e}^{-{\lambda }_{j}t},t\ge 0.$$

(53)

The first-stage rate is ${\lambda }_1={\rho }_1/2$, where ${\rho }_1$ is the rate ${\rho }_{\ell }$ in (13), from the proof of Lemma 1, for stage one on the sublevel (54) below. For $j\ge 2$, ${\lambda }_j$ is given by (49), where ${\rho }_j$ now denotes the rate ${\rho }_R$ in (41), from the proof of Corollary 1, for stage j; hence (50) holds. The constants depend on the stage data ${({\kappa }_j,L_j,D_{a,j},N_{a,j},R_j)}_{j=1}^n$ and are not uniform in the stage index.

Since ${\sigma }_0=a$, we have $d_1=0$, and stage one is (2) and (3) with nominal input a. Let ${\mu }_1$ and $q_1$ be given by (7) for stage one. The initial bound (46) gives $\left|p\left(0\right)\right|=|w_1\left(0\right)-({\kappa }_1-{\mu }_1)e_1\left(0\right)|\le (1+{\kappa }_1)R_1$, and therefore

$$V_{a,1}\left(0\right)\le {\ell }_1:=\frac{1}{2}{(1+{\kappa }_1)}^2{R}_1^2+(L_1+D_{a,1})R_1+q_1{R}_1^2.$$

(54)

The sublevel estimate (14) in the proof of Lemma 1, with $\ell ={\ell }_1$, gives (53) for $j=1$ with ${\lambda }_1={\rho }_1/2$ and ${\rho }_1={\rho }_{{\ell }_1}$.

Suppose $j\ge 2$ and (53) holds for stage $j-1$. Since $d_j=-w_{j-1}$, this bound gives $\left|d_j\left(t\right)\right|\le M_{j-1}{e}^{-{\lambda }_{j-1}t}$, and $d_j$ is continuous because ${\sigma }_{j-1}$ is Lipschitz and $a^{(j-1)}$ is continuous. Stage j is an instance of (15)–() with nominal input $a^{(j-1)}$, whose second derivative $a^{(j+1)}$ is locally absolutely continuous and satisfies (44). The estimate (41) in the proof of Corollary 1, with $(C_d,{\gamma }_d)=(M_{j-1},{\lambda }_{j-1})$ and radius $R_j$, therefore gives (53) for stage j, with the rate (49). This closes the induction without differentiating $d_j$ or ${\sigma }_{j-1}$. The rate bounds (50) follow as in the proof of Theorem 3. □

Remark 5

(Regularity and conservatism). Applying Theorems 3 and 4 through stage n requires, at the highest stage, that $a^{(n+1)}$ be locally absolutely continuous and that $a^{(n+2)}$ have the required essential bound, together with the corresponding intermediate acceleration, jerk, and gain conditions. The certified rates (50) are conservative. Any rate below ${\displaystyle \frac{1}{2}}\min \{{\rho }_j,{\lambda }_{j-1}/2\}$ is admissible at stage $j\ge 2$ at the cost of a larger $M_j$.

Remark 6

(Quasi-exponential closed-loop results). The control schemes of [9,10] use this differentiator and conclude quasi-exponential convergence of the tracking error. There the tolerance enters only through the estimation error, which reaches the tracking error through a bound on an exponentially stable linear system; that bound returns whatever convergence class the estimator supplies. Replacing it by Theorem 4 therefore upgrades those conclusions to exact exponential convergence, provided the differentiated signal admits the bounds of Remark 5 along closed-loop solutions and the gains satisfy (44). Beyond the acceleration bound used there, this asks for one further bounded derivative; checking it is specific to each scheme and is not done here.

5. Conclusions

This paper analyzed a serial differentiator by means of a strict Lyapunov function, the energy of the error system plus a quadratic correction. When the acceleration of the input is locally absolutely continuous and both its magnitude and its rate are bounded, an explicit condition relating the two differentiator gains to those bounds certifies exponential convergence of the estimation error to zero, with no residual neighborhood, uniformly on every prescribed initial energy sublevel. Earlier analyses of this architecture certified asymptotic convergence, or exponential convergence only to an arbitrarily small neighborhood, or a quasi-exponential bound whose rate degrades as that neighborhood is tightened. The condition obtained here is sufficient rather than necessary or sharp, and it is paid for by the additional bound on the rate of the acceleration.

The same Lyapunov function transfers this estimate to a measured input corrupted by a continuous perturbation that is known only through its amplitude. The perturbation is never differentiated and its amplitude is not assumed small; the argument needs only an integral bound over the thin layer in which the relay sign can be corrupted, together with the growth of the dissipation with the Lyapunov level. A persistent amplitude leaves a residual, which vanishes with the perturbation, so exact exponential convergence is recovered in its absence. Because this is precisely the information available about the nonsmooth interstage error of a serial cascade, the estimates propagate through any fixed finite number of stages, exactly when the first input is exact and with a residual otherwise whose certified exponent deteriorates rapidly with the number of stages. The certified exponents and rates are conservative and are not claimed to be optimal, and the constants need not be uniform in the stage index. Sharper residual exponents, less conservative stagewise conditions, and the effects of measurement noise and discretization remain topics for future work.