Cascade Synthesis of Differentiators with Piecewise Linear Correction Signals

Abstract

Based on the state observer theory of dynamic plants operating under uncertainty, we propose a method for reconstructing high-order derivatives of an online signal (for example, a reference action in a tracking system). The method requires neither numerical differentiation nor the presence of an analytical description of the signal. The dynamic differentiator is constructed as a replica of the virtual canonical model with an unknown but bounded input. The use of bounded correction actions and a special structure of the differentiator permit one to reduce the outliers of the resulting estimates at the beginning of a transient compared with a linear differentiator with high-gain coefficients. By way of application, we consider the problem of tracking a spatial trajectory by the center of mass of an unmanned aerial vehicle and present simulation results.

APPENDIX

Proof of the Theorem. Taking into account the specific features of the problem to be solved, without loss of generality we take the initial conditions in system (2.11) in the form (2.18); namely,

$$ \big |\varepsilon _{1} (0)\big |\le \delta ,\quad \big |\varepsilon _{i} (0)\big |\le 2G_{i} ,\quad 2G_{i} \gg \delta ,\quad i={2,\ldots ,n+1}.$$

(A.1)

We divide the time interval $[0; T]$ into $2n$ intervals using the points $0=t_{0} <t_{1} <t_{2} <\ldots <t_{2n-1} <t_{2n} =T$ and formalize in time the desired behavior of observation errors and their derivatives ensuring the validity of (2.19),

$$ \begin {aligned} \big |\varepsilon _{1}(t)\big |&\le \delta ,\;t\ge t_{0},\; \big |\dot {\varepsilon }_{1}(t)\big |\le \Delta _{1,1},\;t\ge t_{1},\; \big |\ddot {\varepsilon }_{1}(t)\big |\le \Delta _{1,2},\; t\ge t_{2},\;\ldots ,\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\big |\varepsilon _{1}^{(n)} (t)\big |\le \Delta _{1,n},\;t\ge t_{n},\\ \big |\varepsilon _{2}(t)\big |&\le \delta ,\;t\ge t_{2},\; \big |\dot {\varepsilon }_{2}(t)\big |\le \Delta _{2,1},\;t\ge t_{3},\;\big |\ddot {\varepsilon }_{2}(t)\big |\le \Delta _{2,2},\; t\ge t_{4},\;\ldots ,\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\big |\varepsilon _{2}^{(n-1)}(t)\big |\le \Delta _{2,n-1},\;t\ge t_{n + 1},\\ \big |\varepsilon _{3}(t)\big |&\le \delta ,\;t\ge t_{4},\; \big |\dot {\varepsilon }_{3}(t)\big |\le \Delta _{3,1},\;t\ge t_{5},\; \big |\ddot {\varepsilon }_{3}(t)\big |\le \Delta _{3,2},\;t\ge t_{6},\;\ldots ,\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\big |\varepsilon _{3}^{(n-2)}(t)\big |\le \Delta _{3,n-2},\;t\ge t_{n+2},\\ &\dots \\ \big |\varepsilon _{n-1}(t)\big |&\le \delta ,\;t\ge t_{2(n-1)-2},\; \big |\dot {\varepsilon }_{n-1}(t)\big |\le \Delta _{n-1,1},\; t\ge t_{2(n-1)-1},\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\big |\ddot {\varepsilon }_{n-1}(t)\big |\le \Delta _{n-1,2},\;t\ge t_{2(n-1)},\\ \big |\varepsilon _{n}(t)\big |&\le \delta ,\;t\ge t_{2n-2},\; \big |\dot {\varepsilon }_{n}(t)\big |\le \Delta _{n,1},\;t\ge t_{2n-1},\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}\Delta _{i,j}<\delta ,\; i={1,\ldots ,n},\;j={1,\ldots ,n+1-i};\\ \big |\varepsilon _{n + 1}(t)\big |&\le \delta ,\;t\ge t_{2n}. \end {aligned} $$

(A.2)

Taking into account (2.13) and (2.15), we specify the first inequalities in the rows of (A.2) assuming that the indicated relations were satisfied on the preceding intervals,

$$ \begin {aligned} &\big |\varepsilon _{1} (t)\big |\le 1/l_{1} \le \delta ,\quad t\ge t_{0} , \\ &\big |v_{i-1} (t)-z_{i} (t)\big |=\big |\varepsilon _{i} (t)-\dot {\varepsilon }_{i-1} (t)\big |\le 1/l_{i} \Leftrightarrow \big |\varepsilon _{i} (t)\big |\le 1/l_{i} +\Delta _{i-1,1} \le \delta ,\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {}{t\ge t_{2i-2} ,\quad i={2,\ldots ,n+1}.} \end {aligned} $$

(A.3)

The convergence of the arguments of the correction signals into the linear zone (A.3) is ensured by the choice of amplitudes $p_{i} >0$, while the dimensions of the linear zones and the validity of the other, auxiliary inequalities in (A.2) are ensured by the choice of the high-gain coefficients $l_{i} >0$, $i={1,\ldots ,n+1}$.

As was already noted, $\left |\varepsilon _{1} (0)\right |\le \delta$ and $\mathrm {sgn}\,v_{1} (t))= \mathrm {sgn\,(}\varepsilon _{1} (t))$, $t\ge 0$ in system (2.11)–(2.12) by construction; it is required to ensure the first inequality in (A.3) by the choice of $p_{1} >0$ that coincides with the primary adjustment (the first inequality in (2.16)); namely,

$$ \varepsilon _{1} \dot {\varepsilon }_{1} =\varepsilon _{1} \big (g_{2} -p_{1} \mathrm {sgn\,(}\varepsilon _{1} )\big )\le \left |\varepsilon _{1} \right |(G_{2} -p_{1} )<0\Rightarrow p_{1} >G_{2} . $$

(A.4)

In the other equations of system (2.11), the coincidence of the signs $\mathrm {sgn\,(}v_{i} (t))= \mathrm {sgn\,(}\varepsilon _{i} (t))$, $i={2,\ldots ,n+1}$, may not occur for ${0\le t\le t_{2(i-1)-1}}$ and is guaranteed only for ${t>t_{2(i-1)-1}}$ outside the neighborhood $\left |\varepsilon _{i}\right |\le \Delta _{i-1,1}$. It follows that the values of $\varepsilon _{i} (t)$ in the general case increase in absolute value on the interval $[0;t_{2(i-1)-1} ]$; it is required to ensure their convergence in domain (A.3) in time $t_{2(i-1)} - t_{2(i-1)-1}$ by the choice of $p_{i}$, $i={2,\ldots ,n+1}$. Let us elaborate the primary adjustment of amplitudes taking into account the initial conditions (A.1) and the prescribed convergence time,

$$ \begin {gathered} p_{i} \ge \frac {\left |\varepsilon _{i} (t_{2(i-1)-1} )\right |}{t_{2(i-1)} -t_{2(i-1)-1} } +G_{i+1} , \\[.3em] \left |\varepsilon _{i} (t_{2(i-1)-1} )\right |\le 2G_{i} +(G_{i+1} +p_{i} )t_{2(i-1)-1} ,\\[.3em] i={2,\ldots ,n+1},\quad t\ge 0. \end {gathered}$$

Hence we have

$$ p_{i} \ge \frac {2G_{i} +(G_{i+1} +p_{i} )t_{2(i-1)-1} }{t_{2(i-1)} -t_{2(i-1)-1} } +G_{i+1} \Rightarrow p_{i} \ge \frac {2G_{i} +G_{i+1} t_{2(i-1)} }{t_{2(i-1)} -2t_{2(i-1)-1} } ,\quad i={2,\ldots ,n+1}, $$

(A.5)

where

$$ {2t_{2(i-1)-1} <t_{2(i-1)} .} $$

Let us set, for example, all odd time intervals the same,

$$ {\Delta t=t_{2i-1} -t_{2i-2} >0},\quad {i={1,\ldots ,n}},$$

and for the even ones we set

$$ {\Delta t=t_{2(i-1)} -2t_{2(i-1)-1} },\quad { i={2,\ldots ,n+1}}; $$

then

$$ t_{2(i-1)} - t_{2(i-1)-1} =(3\cdot 2^{i-2} -1)\Delta t,\quad i={2,\ldots ,n+1},$$

(A.6)

and we have an upper bound for the choice of $\Delta t>0$,

$$ t_{2n} =3(1+2+2^{2} +2^{3} +\ldots +2^{n-1} )\Delta t\le T\Rightarrow 0<\Delta t\le \frac {T}{3(2^{n} -1)} . $$

(A.7)

Taking into account (A.4)–(A.7), we have lower bounds for the choice of amplitudes ensuring the convergence of the arguments of correction signals into the linear zones (A.3) within a given time,

$$ \eqalign { p_{1}^{*} &=G_{2} ,\cr p_{i}^{*} &=\frac {2G_{i} +G_{i+1} 3(2^{i-1} -1)\Delta t}{\Delta t} =\frac {2G_{i} }{\Delta t} +3(2^{i-1} -1)G_{i+1} ,\quad i={2,\ldots ,n+1}.}$$

(A.8)

Note one more time that, unlike (3.26), the amplitudes in the differentiator (2.10), (2.12) are selected independently of each other.

To adjust the high-gain coefficients, we consider the equations in system (2.11)–(2.12) with allowance for (2.13) in the linear zones where they fall into the indicated time intervals,

$$ \begin {aligned} &{\dot {\varepsilon }_{1} =g_{2} -p_{1} l_{1} \varepsilon _{1} ,\quad \left |\varepsilon _{1} \right |\le 1/l_{1} ,\quad t\ge 0,} \\ &{\dot {\varepsilon }_{i} =g_{i+1} -p_{i} l_{i} (\varepsilon _{i} -\dot {\varepsilon }_{i-1} ),\quad \left |\varepsilon _{i} -\dot {\varepsilon }_{i-1} \right |\le 1/l_{i} ,\quad t\ge t_{2i-2} ,\quad i={2,\ldots ,n+1}.} \end {aligned}$$

(A.9)

Based on the sufficient convergence conditions $\varepsilon _{i}\dot {\varepsilon }_{i} <0$, we find lower bounds for the selection of $l_{i}>0$, $i={1,\ldots ,n+1}$, ensuring the prescribed estimation accuracy (2.19) as well as establish the accuracy that needs to be ensured when stabilizing the first derivatives of estimation errors by dividing the given value $\delta$ into two parts, for example, in halves,

$$\begin {aligned} \big |\varepsilon _{1} (t)\big |&\le \delta ( t\ge 0)\Rightarrow l_{1} \ge \frac {G_{2} }{p_{1} \delta } , \\ \big |\varepsilon _{i} (t)\big |&\le \underbrace {\frac {G_{i+1} }{p_{i} l_{i} } }_{\delta /2} +\underbrace {\left |\dot {\varepsilon }_{i-1} \right |}_{\delta /2} \le \delta (t\ge t_{2i-2} )\Rightarrow l_{i} \ge \frac {2G_{i+1} }{p_{i} \delta } , \\ \big |\dot {\varepsilon }_{i-1} \big |&\le \Delta _{i-1,1} =\frac {\delta }{2} ,\quad i={2,\ldots ,n+1}. \end {aligned}$$

(A.10)

Further, to ensure the stabilization of the derivatives of the observation errors (A.2) with a prescribed accuracy in a given time, we consider an iteration procedure consisting of $n$ steps, where $n$ is the maximum order of derivatives taken into account.

Step 1. By selecting $l_{i}$, $i={1,\ldots ,n}$, one should also ensure the convergence of the first derivatives of the observation errors $\dot {\varepsilon }_{i} (t)$ into the established domains (A.10) in time $t_{2i-1} -t_{2i-2} =\Delta t$ (A.2) from the initial conditions $\left |\dot {\varepsilon }_{i} (t_{2i-2} )\right |=G_{i+1} +p_{i}$ (A.9). To this end, on the indicated intervals we estimate the solutions of the auxiliary system

$$ \ddot {\varepsilon }_{1} =g_{3} -p_{1} l_{1} \dot {\varepsilon }_{1} ;\quad \ddot {\varepsilon }_{i} =g_{i+2} -p_{i} l_{i} (\dot {\varepsilon }_{i} -\ddot {\varepsilon }_{i-1} ),\quad i={2,\ldots ,n}, $$

(A.11)

and establish the accuracy to be ensured when stabilizing the second derivatives of estimation errors by dividing the values $\Delta _{i,1} =\delta /2$ into parts, for example, as follows:

$$ \begin {aligned} \big |\dot {\varepsilon }_{1} (t)\big |&\le \underbrace {(G_{2} +p_{1} )e^{-p_{1} l_{1} \Delta t} }_{\delta /4} +\underbrace {\frac {G_{3} }{p_{1} l_{1} } }_{\delta /4} \le \Delta _{1,1} \\ &=\frac {\delta }{2} (t\ge t_{1} )\Rightarrow l_{1} \ge \frac {1}{p_{1} } \max \left \{\frac {4G_{3} }{\delta } ; \frac {1}{\Delta t} \ln \frac {4(G_{2} +p_{1} )}{\delta } \right \}, \\ \big |\dot {\varepsilon }_{i} (t)\big |&\le \underbrace {(G_{i+1} +p_{i} )e^{-p_{i} l_{i} \Delta t} }_{\delta /8} +\underbrace {\frac {G_{i+2} }{p_{i} l_{i} } }_{\delta /8} +\underbrace {\left |\ddot {\varepsilon }_{i-1} \right |}_{\delta /4} \le \Delta _{i,1} =\frac {\delta }{2} (t\ge t_{2i-1} ) \\ &\quad {}\Rightarrow {} l_{i} \ge \frac {1}{p_{i} } \max \left \{\frac {8G_{i+2} }{\delta } ; \frac {1}{\Delta t} \ln \frac {8(G_{i+1} +p_{i} )}{\delta } \right \}, \\ \big |\ddot {\varepsilon }_{i-1} \big |&\le \Delta _{i-1,2} =\frac {\delta }{4} ,\quad i=2,\dots ,n. \end {aligned}$$

(A.12)

Step 2. By selecting $l_{i}$, $i={1,\ldots ,n-1}$, one should also ensure the convergence of the second derivatives of the observation errors $\ddot {\varepsilon }_{i} (t)$ into the established domains (A.12) in time $t_{2i} - t_{2i-1} =(3\cdot 2^{i-1} -1)\Delta t$ (A.2), (A.6) from the initial conditions ${\left |\ddot {\varepsilon }_{i} (t_{2i-1} )\right |=G_{i+2} +p_{i}}$ (A.11). To this end, on the indicated intervals we estimate the solutions of the auxiliary system

$$ \stackrel {\ldots }{\varepsilon }_{1} =g_{4} -p_{1} l_{1} \ddot {\varepsilon }_{1} ;\quad \stackrel {\ldots }{\varepsilon }_{i} =g_{i+3} -p_{i} l_{i} (\ddot {\varepsilon }_{i} -\stackrel {\ldots }{\varepsilon }_{i-1} ),\quad i={2,\ldots ,n-1},$$

and establish the accuracy to be ensured when stabilizing the third derivatives of the estimation errors by dividing the values $\Delta _{i,2} =\delta /4$, for example, by analogy with (A.12),

$$ \begin {aligned} \big |\ddot {\varepsilon }_{1} (t)\big |&\le \underbrace {(G_{3} +p_{1} )e^{-p_{1} l_{1} 2\Delta t} }_{\delta /8} +\underbrace {\frac {G_{4} }{p_{1} l_{1} } }_{\delta /8} \le \frac {\delta }{4} (t\ge t_{2} )\Rightarrow l_{1} \ge \frac {1}{p_{1} } \max \left \{\frac {8G_{4} }{\delta } ; \frac {1}{2\Delta t} \ln \frac {8(G_{3} +p_{1} )}{\delta } \right \}, \\ \big |\ddot {\varepsilon }_{i} (t)\big |&\le \underbrace {(G_{i+2} +p_{i} )e^{-p_{i} l_{i} (3\cdot 2^{i-1} -1)\Delta t} }_{\delta /16} +\underbrace {\frac {G_{i+3} }{p_{i} l_{i} } }_{\delta /16} +\underbrace {\left |\stackrel {\ldots }{\varepsilon }_{i-1} \right |}_{\delta /8} \le \Delta _{i,2}=\frac {\delta }{4} (t\ge t_{2i} ) \\ &\quad {}\Rightarrow l_{i} \ge \frac {1}{p_{i} } \max \left \{\frac {16G_{i+3} }{\delta } ; \frac {1}{(3\cdot 2^{i-1} -1)\Delta t} \ln \frac {16(G_{i+2} +p_{i} )}{\delta } \, \right \},\\ \big |\stackrel {\ldots }{\varepsilon }_{i-1} \big |&\le \Delta _{i-1,3} =\frac {\delta }{8} ,\quad i={2,\ldots ,n-1}, \end {aligned}$$

and so on. At each step, the number of considered high-gain coefficients and the dimension of auxiliary systems decrease by unity. Thus, at the last $n$th step, by selecting $l_{1}$ one should ensure the convergence of $\varepsilon _{1}^{(n)} (t)$ into the domain established at the previous step, for example, in the above-indicated manner ${\Delta _{1,n} =\delta /2^{n}}$ in time $t_{n} -t_{n-1}$ (A.2) from the initial conditions $|\varepsilon _{1}^{(n)} (t_{n-1} )|=G_{n+1} +p_{1}$. Estimating the solution of the auxiliary equation

$$ \varepsilon _{1}^{(n+1)} =g_{n+2} -p_{1} l_{1} \varepsilon _{1}^{(n)}$$

yields the following result:

$$ \begin {aligned} \left |\varepsilon _{1}^{(n)} (t)\right |&\le \underbrace {(G_{n+1} +p_{1} )e^{-p_{1} l_{1} (t_{n} -t_{n-1} )\Delta t} }_{\delta /2^{n+1} } +\underbrace {\frac {G_{n+2} }{p_{1} l_{1} } }_{\delta /2^{n+1} } \\ &\le \frac {\delta }{2^{n} } \Rightarrow l_{1} \ge \frac {1}{p_{1} } \max \left \{\frac {2^{n+1} G_{n+2} }{\delta } ; \frac {1}{(t_{n} -t_{n-1} )\Delta t} \ln \frac {2^{n+1} (G_{n+1} +p_{1} )}{\delta } \right \}. \end {aligned}$$

Here if $n$ is odd, then $t_{n} -t_{n-1} =\Delta t$, and if $n$ is even, then $t_{n} - t_{n-1} =(3\cdot 2^{n/2-1} -1)\Delta t$.

Taking into account the fact that the logarithmic function increases very slowly, the factor multiplying $\Delta t$ and determining the length of the even interval (A.6) is a positive integer, and $1/\Delta t>1/((3\cdot 2^{i-1}-1)\Delta t)$, we can simplify the definitive result by assuming this factor to be equal to unity in the formulas obtained at even steps.

The inequalities for the selection of the high-gain coefficients (A.10) and those of the type (A.12) obtained at various steps of the procedure must hold simultaneously. Considering the indicated simplification, we merge them to obtain the ultimate lower bounds under which problem (2.19) is ensured with allowance for rapid motions and errors in the static equations,

$$ \begin {aligned} l_{n+1}^{*} & = \frac {2G_{n+2} }{p_{n+1} \delta } ,\\ l_{n}^{*} &= \frac {1}{p_{n} } \max \left \{ \frac {2G_{n+1} }{\delta } ; \frac {8G_{n+2} }{\delta } ; \frac {1}{\Delta t} \ln \frac {8(G_{n+1} + p_{n} )}{\delta } \right \}, \\ l_{n-1}^{*} & = \frac {1}{p_{n-1} } \max \left \{\frac {2G_{n} }{\delta } ; \frac {8G_{n+1} }{\delta } ; \frac {16G_{n+2} }{\delta } ; \frac {1}{\Delta t} \ln \frac {8(G_{n} +p_{n-1} )}{\delta } ;\frac {1}{\Delta t} \ln \frac {16(G_{n+1} + p_{n-1} )}{\delta } \right \} , \\ & {\dots } \\ l_{3}^{*} & = \frac {1}{p_{3} } \max \left \{\frac {2G_{4} }{\delta } ; \frac {2^{3} G_{5} }{\delta } ; \frac {2^{4} G_{6} }{\delta } ; \ldots ; \right . \frac {2^{n} G_{n+2} }{\delta } ; \\ &\qquad \qquad \qquad {}{\left . \frac {1}{\Delta t} \ln \frac {2^{3} (G_{4} + p_{3} )}{\delta } ; \frac {1}{\Delta t} \ln \frac {2^{4} (G_{5} + p_{3} )}{\delta } ; \ldots ; \frac {1}{\Delta t} \ln \frac {2^{n} (G_{n+1} + p_{3} )}{\delta } \right \} ,} \\ l_{2}^{*} & = \frac {1}{p_{2} } \max \left \{\frac {2G_{3} }{\delta } ; \frac {2^{3} G_{4} }{\delta } ; \frac {2^{4} G_{5} }{\delta } ; \ldots ; \right . \frac {2^{n+1} G_{n+2} }{\delta } ; \\ &\qquad \qquad \qquad {}{\left . \frac {1}{\Delta t} \ln \frac {2^{3} (G_{3} + p_{2} )}{\delta } ; \frac {1}{\Delta t} \ln \frac {2^{4} (G_{4} + p_{2} )}{\delta } ; \ldots ; \frac {1}{\Delta t} \ln \frac {2^{n+1} (G_{n+1} + p_{2} )}{\delta } \right \} ,} \\ l_{1}^{*} & = \frac {1}{p_{1} } \max \left \{\frac {G_{2} }{\delta } ; \frac {2^{2} G_{3} }{\delta } ; \frac {2^{3} G_{4} }{\delta } ; \right . \ldots ; \frac {2^{n+1} G_{n+2} }{\delta } ; \\ &\qquad \qquad \qquad {}{\left . \frac {1}{\Delta t} \ln \frac {2^{2} (G_{2} + p_{1} )}{\delta } ; \frac {1}{\Delta t} \ln \frac {2^{3} (G_{3} + p_{1} )}{\delta } ; \ldots ; \frac {1}{\Delta t} \ln \frac {2^{n+1} (G_{n+1} + p_{1} )}{\delta } \right \} .} \end {aligned}$$

(A.13)

Thus, there exist $p_{i}^{*}$ (A.8) and $l_{i}^{*}$ (A.13) such that for any $p_{i} ,\;l_{i}$: $p_{i} >p_{i}^{*}$, $l_{i} >l_{i}^{*}$, $i={1,\ldots ,n+1}$, inequalities (2.19) are satisfied. The proof of the Theorem is complete. $\quad \blacksquare$

It should be noted that the estimates for the selection of the high-gain coefficients (A.13) obtained from the sufficient conditions may prove rather conservative, especially, for systems (2.11) of large dimension. When practically applying this procedure, it is recommended to rely on the given values of $G_{i}$, $i={2,\ldots ,n+2}$, $\delta >0$, $T>0$, and if it is necessary reduce the design estimates:

• – Take into account the factors multiplying $\Delta t$ in the formulas obtained at the even steps of the procedure.

• – Use another method of partitioning $\Delta _{i,j}$ ( $i={1,\ldots ,n}$, $j={1,\ldots ,n+1-i}$) into parts, allotting a smaller share in the estimation of the damped proper motions of the derivatives.

Note that the introduced trick with partitioning the domains of convergence of the derivatives of observation errors has allowed one to make the choice of the high-gain coefficients (A.13) independent of each other. We can use another, coupled adjustment procedure in which, from the sufficient conditions, we consecutively (bottom to top) fix the values of $l_{i}^{*}$, $i={n+1,\ldots ,1}$; in view of these values, the domains of convergence of the highest derivatives are determined by the residual principle. This procedure will be more time consuming but may lead to less conservative design estimates.