I^{\rm ext}_{AC}\hat{D}^{C}+{\mathcal{R}}_{3}\,.$

Similar arguments to above show that the latest remainder ${\mathcal{R}}_{3}$
still has transverse order at most three. Using Equation (7.1), note that
$\hat{D}^{T}_{A}=I_{AB}^{\rm
ext}\hat{D}^{B}+\frac{1}{d+2w-3}\,X_{A}I\cdot\hat{D}^{2}$. Note that we
encounter no poles applying this identity to the above display when $h=5$.
This maneuver produces

(7.35) $I\cdot\hat{D}^{4}=-\hat{D}^{T}_{A}\circ
I\cdot\hat{D}^{2}\circ\hat{D}^{T{\hskip
0.85358pt}A}+\tfrac{1}{h-3}\hat{D}^{T}_{A}\circ I\cdot\hat{D}^{2}\circ
X_{A}I\cdot\hat{D}^{2}\\\
+\tfrac{1}{h-9}X_{A}I\cdot\hat{D}^{4}\circ\hat{D}^{T{\hskip
0.85358pt}A}-\tfrac{1}{(h-3)(h-9)}X_{A}I\cdot\hat{D}^{4}\circ
X^{A}I\cdot\hat{D}^{2}+{\mathcal{R}}_{3}\,.$

Applying identities from Lemma 7.11, for the second, third, and fourth terms
on the right hand side above, we find

$\displaystyle\hat{X}^{A}I\cdot\hat{D}^{4}\circ D_{A}^{T}$
$\displaystyle=\tfrac{h(h-7)(h-d-8)}{2(h-3)(h-4)}\,I\cdot\hat{D}^{4}+\cdots\,,$
$\displaystyle\hat{D}^{T}_{A}\circ I\cdot\hat{D}^{2}\circ
X^{A}I\cdot\hat{D}^{2}$
$\displaystyle=\tfrac{(h-5)(h-6)(h+d-10)}{2(h-4)(h-9)}\,I\cdot\hat{D}^{4}+\cdots\,,$
$\displaystyle X_{A}I\cdot\hat{D}^{4}\circ X^{A}I\cdot\hat{D}^{2}$
$\displaystyle=-\tfrac{6(h-d-8)}{h-4}I\cdot\hat{D}^{4}+\cdots\,,$

where $\cdots$ represents lower transverse order terms. Then, after collecting
all the terms in Equation (7.35) containing $I\cdot\hat{D}^{4}$ on the left
hand side, we have

$\tfrac{(2h-9)(h+d-10)}{(h-3)(h-4)(h-9)}\,I\cdot\hat{D}^{4}=-\hat{D}^{T}_{A}\circ
I\cdot\hat{D}^{2}\circ\hat{D}^{T{\hskip 0.85358pt}A}+{\mathcal{R}}_{4}\,.$

When $h=5$, the coefficient on the left hand side is $-\frac{d-5}{8}$.
Moreover, at that value of $h$ the operator $I\cdot\hat{D}^{2}$ in the first
term on the right hand side acts on tractors of weight $1-\frac{d-1}{2}$, in
which case it is tangential and equals $P_{2}^{\Sigma\hookrightarrow M}$.
Therefore we have established that

$P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\stackrel{{\scriptstyle\Sigma}}{{=}}{\hskip 0.85358pt}\tfrac{8}{d-5}{\hskip
0.85358pt}\big{(}\hat{D}^{TA}_{\scriptscriptstyle\Sigma}\circ
P_{2}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\circ\hat{D}^{T}_{{\scriptscriptstyle\Sigma}{\hskip
0.85358pt}A}-{\mathcal{R}}\big{)}\,.$

It remains to compute the operator ${\mathcal{R}}$ by evaluating
${\mathcal{R}}_{4}$ along $\Sigma$ in the limit $h\to 5$.

We must therefore now discuss the poles in ${\mathcal{R}}_{4}$. The issue is
that we cannot use Equation (7.1) to convert the operator $\hat{D}$ to
$\hat{D}^{T}$ when acting on tractors of weight $1-\frac{d-1}{2}$. However,
since we know that the limit $h\to 5$ of the operator $I\cdot\hat{D}^{4}$ is
well-defined, we may employ Equation (7.1) at general weights and apply a
limiting procedure at the end of our calculations.

Returning to ${\mathcal{R}}$, we first apply Lemma 7.33 to simplify the term
$\hat{D}_{A}I\cdot\hat{D}^{2}\hat{D}^{A}$ in $\mathcal{R}_{4}$. To compute
$\mathcal{R}_{4}$, we employ an algorithm whose starting point
${\mathcal{R}}_{4}$ is an operator of transverse order no more than three,
that acts on an arbitrary weight tractors, and is evaluated along $\Sigma$.
Moreover, ${\mathcal{R}}_{4}$ is expressed as a sum of words (each of which
has transverse order no more than three) composed of operator-valued letters
in the alphabet given by the scale $\sigma$, the scale tractor $I$, the
canonical tractor $X$, the Thomas-$D$ operator, the $W$ tractor, the
Thomas-$D$ operator acting on any of the other letters (possibly multiple
times), and rational functions of ${\sf h}$. Note that the tractor identities
derived so far can be used to simplify these words, e.g.,
$X^{A}\circ(\hat{D}_{A}I_{B})=0$. Our algorithm manipulates such words and
letters. We also introduce the distinguished letter

$\hat{y}:=-I\cdot\hat{D}\,.$

The aim of the algorithm is to iteratively convert any word of transverse
order $\ell$ into the form

${\sf Op}\circ{\hat{y}}^{\ell}\circ f({\sf h})+\cdots\,,$

where the terms $\cdots$ have transverse order lower than $\ell$ and $f({\sf
h})$ is some rational function of the operator ${\sf h}$. Here ${\sf Op}$ is
some tangential operator that may involve an additional letter
$\hat{D}^{T}\\!$. Let us first sketch main ideas of the algorithm:

Step 0 of the algorithm takes any word containing rational functions of ${\sf
h}$ and rewrites those words by shifting these operators to the right end of
the word. Then, it applies a simplification-type procedure. While in
principle, this simplification is not necessary for the algorithm to achieve
the goal desired, it significantly reduces computational complexity in the
implementation in the attached documentation [7]. This step will be repeated
after each of the following steps.

Step 1 takes any word ending in $f_{1}({\sf h})$ and rewrites it in the order

$U^{\Theta_{1}}_{1}\cdots\circ
T^{\Phi_{1}\sharp}_{1}\cdots\circ\hat{D}^{T}_{A_{1}}\cdots\circ\hat{D}_{B_{1}}\cdots\circ{\hat{y}}^{\ell}\circ
f_{2}({\sf h})\,,$

for $\\{U^{\Theta_{i}}_{i}\\}$ some set of multiplicative letters (acting by
tensor multiplication by a tractor), $\\{T^{\Phi_{j}\sharp}_{j}\\}$ some set
of tractor-valued tractor-endomorphism letters, and $f_{2}({\sf h})$ some
possibly new rational function of the letter ${\sf h}$.

Step 2 prepares to combine pairs of letters $I^{A}$ and $\hat{D}_{A}$ into
$-{\hat{y}}$ by commuting the corresponding $I$’s to the right of the
multiplicative letters, the tractor-valued tractor endomorphisms, and the
tangential Thomas-$D$ operators. That is, words containing $I^{A}$ and
$\hat{D}_{A}$ are manipulated to take the form

$U^{\Theta_{1}}_{1}\cdots\circ
T^{\Phi_{1}\sharp}_{1}\cdots\circ\hat{D}^{T}_{A_{1}}\cdots\circ
I^{B_{k}}\circ\hat{D}_{B_{1}}\cdots\hat{D}_{B_{k}}\cdots\circ{\hat{y}}^{\ell}\circ
f({\sf h})\,.$

Step 3 applies the tractor lemmas above to push every $I^{B_{k}}$ past
Thomas-$D$ operators with different indices until it is left-adjacent to its
corresponding $\hat{D}_{B_{k}}$, and then combines these terms to form
$-{\hat{y}}$.

Step 4 reapplies Step 1 (so that this newly formed ${\hat{y}}$ letter is
commuted to the right), leaving us with words of the form

$U^{\Theta_{1}}_{1}\cdots\circ
T^{\Phi_{1}\sharp}_{1}\cdots\circ\hat{D}^{T}_{A_{1}}\cdots\circ\hat{D}_{B_{1}}\cdots\circ{\hat{y}}^{\ell}\circ
f({\sf h})\,,$

where no letter $U^{\Theta_{i}}_{i}$ is a letter $I^{B_{i}}$ with
corresponding letter $\hat{D}_{B_{i}}$.

Step 5 rewrites $\hat{D}_{B_{1}}$ (which by the previous step has no
corresponding $I^{B_{1}}$) in terms of the tangential Thomas-$D$ operator,
$\hat{D}^{T}_{B_{1}}$.

Step 6 repeats the previous five steps so long as any letters $\hat{D}$
remain. The output of the algorithm is a linear combination of words of the
form

$U^{\Theta_{1}}_{1}\cdots\circ
T^{\Phi_{1}\sharp}_{1}\cdots\circ\hat{D}^{T}_{A_{1}}\cdots\circ
I\cdot\hat{D}^{\ell}\circ f({\sf h})\,.$

We now present the algorithm in full detail.

Step 0:

Step 0a:

For every letter pair of an operator $f({\sf h})$ and some tractor-valued
operator $T:\Gamma({\mathcal{T}}^{\Phi}M[w])$ $\rightarrow$
$\Gamma({\mathcal{T}}^{\Theta}M[w^{\prime}])$ with “operator weight”
$w^{\prime}-w$, replace $f({\sf h})\circ T^{\Theta}_{\Phi}$ by
$T^{\Theta}_{\Phi}\circ f({\sf h}+2(w^{\prime}-w))$.

Step 0b:

For any word beginning with multiplicative letters, combine those letters to
reduce complexity using the definitions found in 7.2, the Leibniz failure 7.1,
and Lemmas 7.5 and 7.7. For example, one may write
$X_{A}(\hat{D}^{B}\hat{D}^{C}I^{A})=-P^{BC}$ or
$I_{A}P^{AB}=-\tfrac{1}{d-2}KX^{B}$.

Step 1:

Repeat the following sub-steps until a full iteration leaves the expression
unchanged (i.e., “repeat until termination”):

Step 1a:

Rewrite all two-letter pairs $I^{A}\circ\hat{D}_{A}$ as $-{\hat{y}}$.

Step 1b:

Rewrite every instance of the letter combination $[\hat{D}_{A},\hat{D}_{B}]$
following Proposition 7.8.

Step 1c:

Repeat until termination: For every letter of the form $T^{\Phi\sharp}$ and
every multiplicative letter $U^{\Theta}\in\Gamma({\mathcal{T}}^{\Theta}M[w])$,
rewrite pairs $T^{\Phi\sharp}\circ U^{\Theta}$ as

$(T^{\Phi\sharp}U^{\Theta})+U^{\Theta}\circ T^{\Phi\sharp}\,.$

Step 1d:

Repeat until termination: For some operators ${\sf Op}_{1}$, ${\sf Op}_{2}$,
and ${\sf Op}_{3}$, apply Proposition 7.1 and Lemmas 7.7 and 7.9 to write all
words of the form ${\sf Op}_{1}\circ{\hat{y}}\circ{\sf
Op}_{2}\circ{\hat{y}}^{\ell}$ as

${\sf Op}_{1}\circ{\sf Op}_{2}\circ{\hat{y}}^{\ell+1}+{\sf
Op}_{3}\circ{\hat{y}}^{\ell}\,.$

By construction ${\sf Op}_{3}$ has fewer ${\hat{y}}$’s (including pairs
$I^{A}\circ\hat{D}_{A}$) in it than the operator ${\sf Op}_{1}\circ{\sf
Op}_{2}$. Then apply Step 0.

Step 1e:

Repeat until termination: For any operators ${\sf Op}_{1}$, ${\sf Op}_{2}$
(where ${\sf Op}_{2}$ does not contain the letter $\hat{D}$), and ${\sf
Op}_{3}$, apply Proposition 7.1 and Lemmas 7.5 and 7.9 to write all words of
the form ${\sf Op}_{1}\circ\hat{D}^{A_{1}}\circ{\sf
Op}_{2}\circ\hat{D}^{A_{2}}\cdots\hat{D}^{A_{k}}\circ{\hat{y}}^{\ell}$ as

${\sf Op}_{1}\circ{\sf
Op}_{2}\circ\hat{D}^{A_{1}}\cdots\hat{D}^{A_{k}}\circ{\hat{y}}^{\ell}+{\sf
Op}_{3}\circ\hat{D}^{A_{2}}\cdots\hat{D}^{A_{k}}\circ{\hat{y}}^{\ell}\,.$

By construction ${\sf Op}_{3}$ has no more operators $\hat{D}$ in it than the
operator ${\sf Op}_{1}$. Then apply Step 0.

Step 1f:

Repeat until termination: Use Proposition 7.1 and Lemmas 7.5 and 7.7, for
every combination of letters $\hat{D}^{T}\circ I$ to the left of every
appearance of $\hat{D}$ or ${\hat{y}}$, to faithfully replace it with

$I\circ\hat{D}^{T}+{\sf Op}_{1}^{T}\,.$

By construction the operator ${\sf Op}^{T}_{1}$ is tangential. Similarly, for
every combination of letters $\hat{D}^{T}\circ X$, replace it with

$X\circ\hat{D}^{T}+{\sf Op}_{2}^{T}\,.$

Again the operator ${\sf Op}^{T}_{2}$ must be tangential. Apply similar
identities to letters of the form $\hat{D}^{T}\circ U^{\Phi}$ where $U^{\Phi}$
is a multiplicative letter. Then apply Step 0.

Step 2:

Repeat until every word containing at least one pair $I^{A}\circ{\sf
Op}\circ\hat{D}_{A}$ is written as ${\sf Op}^{\prime}\circ
I^{A_{k}}\circ\hat{D}_{A_{1}}\cdots\hat{D}_{A_{k}}$, where ${\sf Op}$ and
${\sf Op}^{\prime}$ are some combination of letters:

Step 2a:

For some operators ${\sf Op}_{1}$ and ${\sf Op}_{2}$, and for every
multiplicative letter $U^{\Theta}\in\Gamma({\mathcal{T}}^{\Theta}M[w])$,
rewrite every word of the form ${\sf Op}_{1}\circ I^{A}\circ U^{\Phi}\circ{\sf
Op}_{2}$ as

${\sf Op}_{1}\circ U^{\Phi}\circ I^{A}\circ{\sf Op}_{2}\,.$

Step 2b:

For some operators ${\sf Op}_{1}$, ${\sf Op}_{2}$, and ${\sf Op}_{3}$, and for
each tractor-valued tractor-endomorphism letter $T^{\Phi\sharp}$, rewrite
every word of the form ${\sf Op}_{1}\circ I^{A}\circ T^{\Phi\sharp}\circ{\sf
Op}_{2}\circ\hat{D}_{A}\circ{\sf Op}_{3}$ as

${\sf Op}_{1}\circ T^{\Phi\sharp}\circ I^{A}\circ{\sf
Op}_{2}\circ\hat{D}_{A}\circ{\sf Op}_{3}-{\sf Op}_{1}\circ T^{\Phi
A}{}_{B}I^{B}\circ{\sf Op}_{2}\circ\hat{D}_{A}\circ{\sf Op}_{3}\,.$

Step 2c:

For some operators ${\sf Op}_{1}$, ${\sf Op}_{2}$, and ${\sf Op}_{3}$, apply
Lemmas 7.5 and 7.7 as well as the definition of $\hat{D}^{T}$ in Definition
7.2 to rewrite every word of the form ${\sf Op}_{1}\circ
I^{A}\circ\hat{D}^{T}\circ{\sf Op}_{2}\circ\hat{D}_{A}\circ{\sf Op}_{3}$ as

${\sf Op}_{1}\circ\hat{D}^{T}\circ I^{A}\circ{\sf
Op}_{2}\circ\hat{D}_{A}\circ{\sf Op}_{3}+{\sf Op}^{\prime}_{A}\circ{\sf
Op}_{2}\circ\hat{D}^{A}\circ{\sf Op}_{3}\,,$

for some ${\sf Op}^{\prime}_{A}$ that does not contain the letter $I^{A}$.
This step is designed to only move $I$’s to the right when they can be
contracted on to $\hat{D}$’s. Apply Step 0.

Step 3:

Repeat until each word has one fewer (or zero) pair(s) of letters $I^{A_{i}}$
and $\hat{D}_{A_{i}}$. For some operators ${\sf Op}_{1}$ and ${\sf Op}_{2}$,
apply Proposition 7.1 and Lemma 7.5 to rewrite every word of the form ${\sf
Op}_{1}\circ
I_{A_{i}}\circ\hat{D}^{A_{1}}\cdots\hat{D}^{A_{i}}\cdots\hat{D}^{A_{k}}\circ{\hat{y}}^{\ell}$
as

$-{\sf
Op}_{1}\circ\hat{D}^{A_{1}}\cdots{\hat{y}}\cdots\hat{D}^{A_{k}}\circ{\hat{y}}^{\ell}+{\sf
Op}_{2}\circ{\hat{y}}^{\ell}\,,$

where ${\sf Op}_{2}$ is some operator that does not contain $I_{A_{i}}$. Apply
Step 0.

Step 4:

Reapply Step 1.

Step 5:

In any given word, by virtue of Proposition 7.3, rewrite the left-most letter
$\hat{D}_{A}$ as
$\hat{D}^{T}_{A}-I_{A}\circ{\hat{y}}-X_{A}\circ{\hat{y}}^{2}\circ\tfrac{1}{h-3}$.

Step 6:

If any word contains the letter $\hat{D}$, repeat Steps 0 through 5.

The remainder of the calculation amounts to rewriting combinations of non-
derivative letters

$U_{1}^{\Theta_{1}}\cdots\circ T_{1}^{\Phi_{1}\sharp}\cdots$

in terms of hypersurface tractors via holographic formulæ. This process is
generally tedious and relies on dozens of identities arising from the Leibniz
failure (both on the hypersurface and in the ambient space), Lemma 7.5 (and
its direct application to the hypersurface operator $\hat{\bar{D}}$), Lemma
7.7, and the Gauss-Thomas formula (7.4). Some of these identities can be found
in [6]. Finally, we can take the limit $h\rightarrow 5$ to resolve
$\mathcal{R}$. This entire procedure was implemented using FORM in the file

anc/General-tensor/Paneitz-tensor-algorithm2.frm.

That the proposed algorithm here terminates requires no proof, since we
explicitly verified that six iterations suffices. The result obtained from
this computation is below:

$\displaystyle-\tfrac{8}{d-5}{\mathcal{R}}=$
$\displaystyle+\Big{(}\tfrac{4(2d-11)}{d-5}L^{AC}L_{C}^{B}+\tfrac{4(d-3)(d-6)}{d-5}F^{AB}-\tfrac{8}{d-5}W^{AB\,\sharp}\Big{)}\circ\hat{D}^{T}_{A}\circ\hat{D}^{T}_{B}$
$\displaystyle+\Big{(}\tfrac{4(2d-11)}{d-5}L_{BC}(\hat{\bar{D}}^{B}L^{CA})-\tfrac{2(d^{2}-8d+17)}{(d-1)(d-2)(d-5)}(\hat{\bar{D}}^{A}K)+\tfrac{4(3d-16)}{d-5}N_{B}L_{CD}W^{BCDA}$
$\displaystyle\phantom{+\Big{(}Q}-\tfrac{4(3d-16)}{d-5}N_{B}L^{A}_{C}W^{BC\sharp}+\tfrac{4(d-4)}{d-5}N_{B}N^{C}(\hat{D}_{C}{W}^{BA}{}_{{\bm{\cdot}}{\hskip
0.85358pt}{\bm{\cdot}}})^{\sharp}\Big{)}\circ\hat{D}^{T}_{A}$
$\displaystyle+\tfrac{2d^{4}-21d^{3}+95d^{2}-200d+152}{(d-1)(d-2)^{2}(d-4)^{2}}K^{2}+\tfrac{d-2}{(d-4)^{2}}L{\hskip
0.85358pt}\cdot{\hskip
0.85358pt}J-\tfrac{2(d-2)}{(d-4)^{2}}L^{4}+\tfrac{2(2d^{3}-27d^{2}+118d-172)}{(d-4)^{2}}L\cdot
F\cdot L$
$\displaystyle+\tfrac{(3d-16)(d-3)^{2}}{d-4}F^{2}-\tfrac{2(d-5)(d-6)}{(d-4)^{2}}L^{AC}L^{BD}{\bar{W}}_{ABCD}+\tfrac{(d-5)(d-8)}{(d-4)(d-7)}(\hat{\bar{D}}_{A}L_{BC})(\hat{\bar{D}}^{A}L^{BC})$
$\displaystyle+4W_{N}{}^{B\sharp}\circ
W_{NB}{}^{\sharp}-4N_{B}L_{AC}\Big{(}\hat{D}^{A}W^{BC}{}_{{\bm{\cdot}}{\hskip
0.85358pt}{\bm{\cdot}}}\Big{)}^{\sharp}\,.$

The above expression for $\mathcal{R}$ contains operators of the type
$\mathcal{W}^{A}\circ\hat{D}^{T}_{A}\circ\hat{D}^{T}_{B}$ which, strictly, are
not defined. However, because $\mathcal{W}^{A}\circ X_{A}=0$, as discussed
earlier, we have used use the notation $\hat{D}^{T}$ for the left-most
Thomas-$D$ operator. This completes the proof. ∎

###  7.3. Proofs of Results 1.1, 4.1,5.1, 5.2, and 6.2

###### Proof of Corollary 1.1.

The proof mainly amounts to an application of Theorem 7.12. Because the
operator acts on scalar densities, we may use Theorem 7.4 to convert operators
$\hat{D}^{T}$ to $\hat{\bar{D}}$ plus lower order terms. The proof then splits
into two separate computations. The first expresses $\hat{\bar{D}}^{A}\circ
P_{2}^{\scriptscriptstyle\Sigma\hookrightarrow M}\circ\hat{\bar{D}}_{A}$ in
terms of Riemannian operators, while the second similarly handles the
subleading terms. The entire computation is carried out in FORM: the first
computation can be found in the file

anc/Paneitz-scalar/DbID2Db-scalar.frm

and the second in

anc/Paneitz-scalar/Paneitz-scalar-Riemannian.frm.

The final step uses Equation (2.6) to rewrite ${\mathring{{\bf\rm IV}}{\hskip
0.56905pt}}{}_{ab}$ in terms of $C_{\hat{n}(ab)}^{\top}$ in order that the
result can be continued to $d=5$ (and subsequently used in Theorem 1.2). Well-
definedness of the final result in $d=5$ can be established by inspection. ∎

###### Proof of Theorem 4.1.

As in Corollary 1.1, this theorem is also an application of Theorem 7.12. Note
that the normal tractor has weight zero in all dimensions, while the
$P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow M}$ acts on tractors of weight
$\frac{5-d}{2}$. Moreover, to use Theorem 7.12 in five dimensions we must
compute in general $d$ and then continue to $d=5$. Thus we introduce a scalar
density $\tau$ of weight $\frac{5-d}{2}$ and instead compute
$P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow M}(N^{A}\tau)$, continue to
$d=5$, and thereafter set $\tau$ to unity. Similarly to the previous proof,
handling the term $\hat{D}^{T\,B}\circ
P_{2}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\circ\hat{D}^{T}_{B}(N^{A}\tau)$ term is challenging. It is computed in the
FORM file

anc/Paneitz-N/DtID2Dt-N.frm

The remainder of the computation is performed by the file

anc/Paneitz-N/Paneitz-N-Riemannian.frm

Finally, we again write ${\mathring{{\bf\rm IV}}{\hskip 0.56905pt}}{}_{ab}$ in
terms of $C_{\hat{n}(ab)}^{\top}$ so that the result is well-defined in $d=5$.
∎

###### Proof of Lemma 5.1.

This calculation is a significant exercise in Riemannian hypersurface
geometry. We use three facts: First, acting on a weight $w\neq 1-\frac{d}{2}$
tractor, the operator $I\cdot\hat{D}$ is given by

$I\cdot\hat{D}\stackrel{{\scriptstyle
g}}{{=}}\nabla_{n}+w\rho-\tfrac{s}{d+2w-2}(\Delta+wJ)\,.$

Second, written in terms of the canonical extension $\hskip
1.13809pt\mathring{\rm{II}}{}^{\rm e}_{ab}=Z^{A}_{a}Z^{B}_{b}P_{AB}$, we have
that $K^{\rm e}=(\hskip 1.13809pt\mathring{\rm{II}}{}^{\rm e})^{2}$. Third,
this canonical extension can be written as $\hskip
1.13809pt\mathring{\rm{II}}{}^{\rm e}_{ab}=\nabla_{a}n_{b}+sP_{ab}+\rho
g_{ab}$ (see the discussion in [6]). Using these facts, we can recast
$\dddot{K}=I\cdot\hat{D}^{3}K^{\rm e}$ as a Riemannian operator on tensors, at
which point the problem reduces to standard (albeit lengthy) hypersurface
calculations, carried out in the FORM program

anc/Riemannian-identities/Kddd.frm

In order to obtain a manifestly invariant result, we note that the operator
$L^{ab}$ given by

$\Gamma(\odot^{2}_{\circ}T^{*}\Sigma[4-d])\ni
t\mapsto\bar{\nabla}\cdot\bar{\nabla}\cdot t+\bar{P}\cdot
t\in\Gamma({\mathcal{E}}\Sigma[-d])\,,$

described in [21] can act on three independent structures appearing in
$\dddot{K}$: namely,

$\hskip 1.13809pt\mathring{\rm{II}}{}^{3}_{(ab)\circ},\;2\hskip
1.13809pt\mathring{\rm{II}}{}_{(a}\cdot{\hskip
1.70717pt}\mathring{\\!{F}}{\hskip 0.56905pt}_{b)\circ},\;K\hskip
1.13809pt\mathring{\rm{II}}{}_{ab}\in\Gamma(\odot^{2}_{\circ}T^{*}\Sigma[4-d])\,.$

The conversion (in five dimensions) of the result to expressions involving the
weight $-1$ densities $L^{ab}\hskip
1.13809pt\mathring{\rm{II}}{}^{3}_{(ab)\circ}$, $2L^{ab}(\hskip
1.13809pt\mathring{\rm{II}}{}_{(a}\cdot{\hskip
1.70717pt}\mathring{\\!{F}}{\hskip 0.56905pt}_{b)\circ})$, and $L^{ab}(K\hskip
1.13809pt\mathring{\rm{II}}{}_{ab})$ is also carried out in the above FORM
file. The tractor expression $W_{ABCN}^{\top}\hat{\bar{D}}^{A}F^{BC}$ can be
computed with standard tractor techniques and is also included in the above
FORM computation. ∎

###### Proof of Theorem 5.2.

In the proof of Theorem 4.1 above, we showed how to calculate
$P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow M}N^{A}$. As noted in Equation
(5.1), to calculate the obstruction density ${\mathcal{B}}_{\Sigma}$, we
should first compute $P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow
M}N^{A}-3I\cdot\hat{D}^{3}(X^{A}K_{\rm e})$ and in turn

$\tfrac{1}{1440}(\bar{D}_{A}\circ\top)\big{(}P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow
M}N^{A}-3I\cdot\hat{D}^{3}(X^{A}K_{\rm e})\big{)}\,.$

To do so, we start with $I\cdot\hat{D}^{3}(X^{A}K_{\rm e})$, and employ a
combination of the tractor and hypersurface calculi developed above. This is
carried out in the FORM program

anc/Paneitz-N/ID3xK.frm

Note that this computation involves $\dddot{K}$ and thus requires Lemma 5.1.
Combining this result with that for
$P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow M}N^{A}$, we can directly
evaluate the obstruction; this is carried out in the FORM program

anc/Paneitz-N/Obstruction-d5.frm

∎

###### Proof of Theorem 6.2.

Because $\Sigma$ is embedded as the conformal infinity of a Poincaré–Einstein
structure $(M,\bm{c})$, from Theorem 1.8 of [6], we have that $\hskip
1.13809pt\mathring{\rm{II}}{}_{ab}={\hskip 1.70717pt}\mathring{\\!{F}}{\hskip
0.56905pt}_{ab}=C_{\hat{n}(ab)}^{\top}=0$. Consequently, from Theorem 1.3 of
the same, $\hat{D}^{T}\mathrel{\stackrel{{\scriptstyle\Sigma{\hskip
0.85358pt}}}{{=}}}\hat{\bar{D}}$. Note that another straightforward
consequence is that
$C_{a\hat{n}\hat{n}}\mathrel{\stackrel{{\scriptstyle\Sigma{\hskip
0.85358pt}}}{{=}}}B_{\hat{n}a}\mathrel{\stackrel{{\scriptstyle\Sigma{\hskip
0.85358pt}}}{{=}}}0$ (see [25, Proposition 4.3]), which implies that
$W_{NABC}\mathrel{\stackrel{{\scriptstyle\Sigma{\hskip 0.85358pt}}}{{=}}}0$.
Thus, to prove the theorem, from Equation (6.2), it suffices to compute
$\hat{\bar{D}}^{A}\circ P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\circ\hat{\bar{D}}_{A}$. According to Theorem 7.12, in this case we have
that

$\displaystyle P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\mathrel{\stackrel{{\scriptstyle\Sigma{\hskip
0.85358pt}}}{{=}}}\tfrac{8}{d-5}\left(\hat{\bar{D}}^{A}\circ
P_{2}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\circ\hat{\bar{D}}_{A}-W^{AB\sharp}\circ\hat{\bar{D}}_{A}\circ\hat{\bar{D}}_{B}+\tfrac{d-4}{2}N_{B}N^{C}(\hat{D}_{C}W^{BA}{}_{{\bm{\cdot}}{\hskip
0.85358pt}{\bm{\cdot}}})^{\sharp}\circ\hat{\bar{D}}_{A}\right)\,.$

We explicitly compute the operator $\hat{\bar{D}}^{A}\circ
P_{4}^{\scriptscriptstyle\Sigma\hookrightarrow M}\circ\hat{\bar{D}}_{A}$ using
FORM to computing in dimension $d$, and then continue to $d=5$. First, we
compute

$(\hat{\bar{D}}^{B}\circ P_{2}^{\scriptscriptstyle\Sigma\hookrightarrow
M}\circ\hat{\bar{D}}_{B}\circ\hat{\bar{D}}_{A})f$

for $f\in\Gamma({\mathcal{E}}\Sigma[\tfrac{7-d}{2}])$ using the FORM file

anc/P6/DbP2Db-Dbf.frm

We take the resulting expression and feed it into the following file which
completes the calculation:

anc/P6/P6-PE.frm

This outputs the result displayed in the theorem. ∎

## Acknowledgements

A.W. was also supported by a Simons Foundation Collaboration Grants for
Mathematicians ID 317562 and 686131, and thanks the University of Auckland for
warm hospitality. A.W. and A.R.G. gratefully acknowledge support from the
Royal Society of New Zealand via Marsden Grants 16-UOA-051 and 19-UOA-008.

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# ADRC in Output and Error Form:
Connection, Equivalence, Performance

Rafal Madonski  Gernot Herbst  and Momir Stankovic  Rafal Madonski is with the
Energy and Electricity Research Center, Jinan University, China (e-mail:
rafal.madonski@jnu.edu.cn).Gernot Herbst is with University of Applied
Sciences Zwickau, Germany (e-mail: gernot.herbst@fh-zwickau.de).Momir
Stankovic is with the Military Academy, University of Defence, Belgrade,
Serbia (e-mail: momir_stankovic@yahoo.com).

###### Abstract

In this work, we investigate two specific linear ADRC structures, namely
output- and error-based. The former is considered a ’standard’ version of
ADRC, a title obtained primarily thanks to its simplicity and effectiveness,
which have spurred its adoption across multiple industries. The latter is
found to be especially appealing to practitioners as its feedback error-driven
structure bares similarities to conventional control solutions, like PI and
PID. In this paper, we describe newly found connections between the two
considered ADRC structures, which allowed us to formally establish conditions
for their equivalence. Furthermore, the conducted comprehensive performance
comparison between output- and error-based ADRCs has facilitated the
identification of specific modules within them, which can now be conveniently
used as building blocks, thus aiding the control designers in customizing
ADRC-based solutions and making them most suitable for their applications.

###### Index Terms:

Active disturbance rejection control (ADRC), Output-based form, Error-based
form, Equivalence condition, Comparative analysis

##  I Introduction

###  I-A Background and Related Work

Active disturbance rejection control (ADRC) is a highly practical general
purpose control methodology [1], which can be applied in a variety of ways,
depending on control requirements, availability of certain signals,
computational capabilities of the target hardware, etc. Especially its
streamlined simplified, linear version [2] has found its way into numerous
application domains [3]. There has been an intensified effort recently to
foster the adoption of ADRC in industry and to ensure its backward
compatibility with classical solutions (e.g. [4, 5, 6, 7]). This manifests
itself, for example, with finding equivalence between ADRC and various well-
established control schemes, like PID ([8, 9]). Through drawing such
connections, the goal is to support control practitioners by facilitating the
process of ADRC understanding, designing, implementing, and tuning.

In that spirit, an error-based version of ADRC was independently proposed in
[10] and [11], and was later generalized and proved in [12]. It allowed to
rewrite the entire ADRC scheme, usually expressed in an output-based form, in
a feedback error-driven form, which then enabled an immediate comparability
with existing classical control solutions.111The details of the used notation
’output-based’ and ’error-based’ in the context of ADRC are explained at the
beginning of Sect. II. It also introduced a practical solution for scenarios
where consecutive time-derivatives of the reference signal (needed to
synthesize the feedback controller) are not available, hence cannot be used in
the control law. In error-based ADRC (eADRC), they can be lumped into the
total disturbance term and conveniently estimated by the extended state
observer, without resorting to an additional tracking differentiator (e.g.
[13, 1]). The efficacy of eADRC was validated in numerous applications so far,
including mechatronic manipulators [14, 15, 16, 17], aerial robotics [18],
power electronics [19, 20], process control [21], and military applications
[22]. The error-based formulation is also how the recently introduced “ADRC
Toolbox”222https://www.mathworks.com/matlabcentral/fileexchange/102249-active-
disturbance-rejection-control-adrc-toolbox for MATLAB/Simulink is implemented
with [23].

###  I-B Motivation and Goal

One can find works, including [24, 25, 12], that show the differences between
the oADRC and eADRC schemes in terms of control performance. Those works,
however, only showed basic quantitative comparisons and did not go beyond
continuous-time domain. Therefore, the current literature lacks information
about the exact sources of these discrepancies and how they influence the ADRC
design process as well as the realization of the control objective.

Motivated to bridge the above gap, a systematic investigation is carried in
this paper with the goal of answering the question about the formal relation
between oADRC and eADRC. The investigation is performed utilizing the
following research methodology. First, the two control schemes are recalled in
continuous-time domain for the same class of controlled systems and utilizing
the same tuning methodology. Then, they are expressed as transfer functions
and their respective structures are comprehensively compared using frequency
domain. Such analysis allows to formally define similarities and differences
between the two control schemes and better understand what is the influence of
those differentiating elements on the control performance.

Additionally, inspired by [26], the investigation also covers the practically
important problem of realizability of the considered ADRC schemes in transfer
function forms. Making sure the oADRC and eADRC schemes are realizable allows
not only to conveniently analyze them in frequency domain (for example see
[27]), but also to compare them to existing classical solutions more easily,
for example by analyzing the pole/zero placement of the feedback controller
part. In return, this facilitates understanding and implementing various ADRCs
using classical feedback controller and filter structures.

###  I-C Novelty

The results of the paper contribute to the current body of knowledge related
to ADRC design. The added new information can be considered as guidelines for
control designers to choose a suitable ADRC-based structure and to better
customize it in order to meet certain requirements in terms of implementation
simplicity and control performance. The specific contributions of the paper
are:

  * i)

Establishment of equivalence condition between oADRC and eADRC. A formal
condition is given in frequency domain that shows the connection between the
two schemes. The formed condition also shows how to go from one structure to
the other, and back.

  * ii)

Quantitative comparison between oADRC and eADRC in time and frequency domains.
Its results show what are the advantages and disadvantages of using both ADRC
structures, with the special emphasis on the influence of the differentiating
elements on the control performance. The systematic comparison includes key
control performance metrics, like reference tracking accuracy, stability
margins (gain and phase margins), disturbance rejection, and measurement noise
sensitivity.

  * iii)

Establishment of conditions of realizability of ADRC transfer function
implementation in output- and error-based forms. This allows to compare the
two considered ADRCs to existing classical solutions more easily. This can be
considered as an extension of [26], where only the output-based version was
considered.

  * iv)

Qualitative comparison between oADRC and eADRC. Its outcome gives insight into
the applicability of both structures, which is also visualized with a decision
tree, dedicated to the prospective ADRC user looking for an appropriate
control structure based on the available signals and desired performance. To
further aid the control designers, three modules are identified that allow
straightforward customization of a desired ADRC scheme.

### Nomenclature

Throughout the paper, we omit writing explicit dependencies from time for
higher-order terms, meaning we only write $x^{(k)}$ instead of $x^{(k)}(t)$ to
represent $k$-th time-derivative of signal $x(t)$. We also omit writing
explicit time-dependency for dot notation for differentiation, meaning we
write $\dot{x}$, $\ddot{x}$, etc., instead of redundant $\dot{x}(t)$,
$\ddot{x}(t)$, etc. Additionally, symbol $\stackrel{{\scriptstyle!}}{{=}}$ is
the stacked operator indicating that both sides of the mathematical formula
shall be made equal.

##  II Considered ADRC Schemes

In this section, the two considered ADRC structures, output-based and error-
based, are briefly recalled. In order to allow a systematic and
straightforward comparison later, they are both derived here for the same
class of controlled systems. Hence, let us first introduce a $n$-th order
single-input single-output plant model with output $y(t)$, control input
$u(t)$, and $d(t)$ being the external disturbance input:

$y^{(n)}=\sum_{i=0}^{n-1}a_{i}\cdot y^{(i)}+b\cdot u(t)+d(t),$ (1)

where $a_{i}$ are generally unknown plant parameters and $b$ is a partially
known plant gain. Since ADRC is a general purpose methodology, it can be
designed in various ways and tailor made to a particular class of systems and
specific application. That is why, in this section, we specify which exact
ADRC variants we are considering in the study. Therefore, the following
clarifying terminology is introduced.

Output-based ADRC (oADRC). This is the commonly seen version of ADRC in the
literature, popularized by Prof. Zhiqiang Gao through his seminal work [2]. It
is often considered the classic version of ADRC. It utilizes a linear observer
(extended with one extra state variable, representing the total disturbance)
and a linear state feedback controller. It is denoted in this paper as
”output-based” since the system’s state vector in this approach, used during
the observer and controller syntheses, consists of plant’s output $y(t)$ and
its derivatives $\dot{y},\ddot{y},\ldots,y^{(n-1)}$. A block diagram of oADRC
can be seen in Fig. 1.

Error-based ADRC (eADRC). This is a version of the ADRC scheme that, similarly
to oADRC utilizes a linear observer (extended with one extra state variable,
representing the total disturbance) and a linear state feedback controller,
but the system’s state vector in this instance, used during the observer and
controller syntheses, consists of feedback error $e(t)\triangleq r(t)-y(t)$
and its consecutive time derivatives $\dot{e},\ddot{e},\ldots,e^{(n-1)}$. It
can be viewed as a special form of the classic oADRC. More detailed
information on the reasoning behind such representation and some exemplary
applications can be found in [10, 12]. A block diagram of eADRC can be seen in
Fig. 2.

###### Remark 1

The above terminology oADRC and eADRC is not widely used in the scientific
literature but is used here specifically and locally to distinguish the two
types of ADRC structures considered in this paper.

Next, the considered ADRC schemes are derived, oADRC in Sect. II-A and eADRC
in Sect. II-B, which lays the foundation of their comparative analysis in
Sect. III.

###  II-A Output-based ADRC (oADRC)

By including uncertainties of the plant gain as $b=b_{0}+\Delta{b}$, where
$b_{0}\neq 0$ is the best available estimate of the $b$ and $\Delta{b}$ is the
parametric uncertainty, the application of the ADRC methodology allows to
represent (1) in a more compact form:

$y^{(n)}=f(t)+b_{0}\cdot u(t),$ (2)

where $f(t)=\sum_{i=0}^{n-1}a_{i}\cdot y^{(i)}+\Delta{b}\cdot u(t)+d(t)$ is
denoted as the generalized (total) disturbance.

A block diagram of the continuous-time linear ADRC in its original state-space
output-based form for the plant model (2) is presented in Fig. 1. Its main
components include: plant gain inversion, disturbance rejection utilizing the
estimated total disturbance term, and an outer state-feedback controller. The
control law is formed as:

Figure 1: Continuous-time output-based ADRC (oADRC) in state-space
implementation. Here we denote the observer as output-based ESO as one of its
inputs is the system output signal $y(t)$. The second input is the control
signal $u(t)$, which is also the case for observer in Fig. 2.

$u(t)=\frac{1}{b_{0}}\cdot\left(k_{1}\cdot
r(t)+\bm{k_{\textrm{r}}}^{\mathrm{T}}\cdot\bm{r_{\textrm{d}}}-\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot\bm{\hat{x}}\right),$
(3)

with
$\bm{k_{\textrm{r}}}^{\mathrm{T}}=\begin{pmatrix}k_{2}&\cdots&k_{n}&1\end{pmatrix}$,
$\bm{k}^{\mathrm{T}}=\begin{pmatrix}k_{1}&\cdots&k_{n}\end{pmatrix}$,
and
$\bm{r_{\textrm{d}}}^{\mathrm{T}}=\begin{pmatrix}\dot{r}&\cdots&r^{(n)}\end{pmatrix}$.

The state estimation vector, defined as:

$\displaystyle\bm{\hat{x}}$
$\displaystyle=\begin{pmatrix}\hat{x}_{1}(t)&\cdots&\hat{x}_{n}(t)&\hat{x}_{n+1}(t)\end{pmatrix}^{\mathrm{T}}$
$\displaystyle=\begin{pmatrix}\hat{y}(t)&\cdots&\hat{y}^{(n-1)}&\hat{f}(t)\end{pmatrix}^{\mathrm{T}},$

can be obtained using the extended state observer (ESO):

$\bm{\dot{\hat{x}}}=\bm{A}\cdot\bm{\hat{x}}+\bm{b}\cdot
u(t)+\bm{l}\cdot\left(y(t)-\bm{c}^{\mathrm{T}}\cdot\bm{\hat{x}}\right),$ (4)
$\displaystyle\text{with}\quad\bm{A}$
$\displaystyle=\begin{pmatrix}\bm{0}^{n\times 1}&\bm{I}^{n\times n}\\\
0&\bm{0}^{1\times
n}\end{pmatrix},\quad\bm{b}=\begin{pmatrix}\bm{0}^{(n-1)\times 1}\\\ b_{0}\\\
0\end{pmatrix},$ $\displaystyle\bm{l}$
$\displaystyle=\begin{pmatrix}l_{1}&\cdots&l_{n+1}\end{pmatrix}^{\mathrm{T}},\quad\bm{c}^{\mathrm{T}}=\begin{pmatrix}1&\bm{0}^{1\times
n}\end{pmatrix}.$

The application of oADRC scheme leads to a robust feedback-linearization of
the original system dynamics (1), which is followed next by a design of the
outer-loop state-feedback with feedforward controller designed for the
resultant chain of integrators (see Fig. 1). One can notice that the
compensation of total disturbance $f(t)$ can be seen as an inner control loop
with the actual plant. The inner control loop forms a canonical plant for the
outer state-feedback controller $\bm{k}^{\mathrm{T}}$ and has a unity-gain
integrator chain behavior.

###### Remark 2

The use of feedforward term in the control law (3) allows to give the desired
steady-state value. If the reference signals are available, the generated
feedforward signal is useful in shaping the system response [28].

###### Remark 3

It should be noted that the control law (3) can also be realized without the
reference derivatives (i.e. $\bm{r_{\textrm{d}}}=0$) as shown in [2, 29]). For
the purpose of this work, such scenario, where the reference derivatives are
not available for controller synthesis, will be considered as the nominal
version of ADRC and denoted as ’case A’. Alternatively, a scenario, where all
the reference derivatives are available, will be denoted as ’case B’. It is a
deliberate choice of ours to focus on cases A and B as they represent opposite
extremes, where either none or all of the reference derivatives are available
for controller synthesis. It is intuitive that if the information about the
reference derivative is somehow available, either through a’priori knowledge
or on-line calculation using a signal differentiator (e.g. [1]), then such
information should be used as it can improve control quality. To visualize
this relation, in the Appendix, we have prepared a generic example of a third-
order system ($n=3$) to show the influence of different levels of knowledge
about the reference derivatives on the tracking quality.

#### Controller Tuning

Assuming the observer’s convergence is fast enough, the rejection of total
disturbance will be so effective that the (outer) state-feedback controller
can control canonical plant model (denoted using subscript ’CP’) with unity-
gain integrator chain behaviour, regardless of the actual plant. By defining a
system state vector as
$\bm{x}_{\mathrm{CP}}=\begin{pmatrix}y(t)&\cdots&y^{(n-1)}\end{pmatrix}$, the
state-space model of the canonical plant with unity-gain integrator chain can
be expressed as:

$\displaystyle\bm{\dot{x}}_{\mathrm{CP}}=\bm{A}_{\mathrm{CP}}\cdot\bm{x}_{\mathrm{CP}}+\bm{b}_{\mathrm{CP}}\cdot
u(t),$ (5)
$\displaystyle\text{with}\quad\bm{A}_{\mathrm{CP}}=\begin{pmatrix}\bm{0}^{(n-1)\times
1}&\bm{I}^{(n-1)\times(n-1)}\\\
0&\bm{0}^{1\times(n-1)}\end{pmatrix},\quad\bm{b}_{\mathrm{CP}}=\begin{pmatrix}\bm{0}^{(n-1)\times
1}\\\ 1\end{pmatrix}.$

Now, by applying the ‘bandwidth parameterization’ approach from [2], which
consists of placing all the poles at one common location
$\lambda=-\omega_{\mathrm{CL}}$, makes the controller tuning relatively simple
as it comes down to choosing only the desired closed-loop bandwidth
$\omega_{\mathrm{CL}}$, i.e.:

$\displaystyle\left(\lambda+\omega_{\mathrm{CL}}\right)^{n}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}\det\left(\lambda\bm{I}-\left(\bm{A}_{\mathrm{CP}}-\bm{b}_{\mathrm{CP}}\bm{k}^{\mathrm{T}}\right)\right)$
$\displaystyle=\lambda^{n}+k_{n}\lambda^{n-1}+\ldots+k_{2}\lambda+k_{1}.$ (6)

The binominal expansion of (6) leads to the controller gains:

$k_{i}=\frac{n!}{(n-i+1)!\cdot(i-1)!}\cdot\omega_{\mathrm{CL}}^{n-i+1},\quad\forall
i=1,\ldots,n.$ (7)

###### Remark 4

The use of bandwidth parameterization for tuning is a subjective choice of
ours, driven mostly by its simplicity as well as its prevalence in the
scientific literature on ADRC. A variety of other tuning methods could be
applied here as well, with handful examples shown in [30].

#### Observer Tuning

In the same manner as in the controller tuning, the observer gains can be
obtained, for example, based on the bandwidth parameterization approach ([2])
by placing the closed-loop observer poles at one common location
$\lambda=-k_{\mathrm{ESO}}\cdot\omega_{\mathrm{CL}}$, with $k_{\mathrm{ESO}}$
being the relative factor between the observer loop and the control loop
bandwidths:

$\displaystyle\left(\lambda+k_{\mathrm{ESO}}\cdot\omega_{\mathrm{CL}}\right)^{n+1}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}\det\left(\lambda\bm{I}-\left(\bm{A}-\bm{l}\bm{c}^{\mathrm{T}}\right)\right)$
$\displaystyle=\lambda^{n+1}+l_{1}\lambda^{n}+\ldots+l_{n}\lambda+l_{n+1}.$
(8)

As in the case of computing the controller gains, the binominal expansion of
(8) yields the observer gains:

$l_{i}=\frac{(n+1)!}{(n-i+1)!\cdot
i!}\cdot\left(k_{\mathrm{ESO}}\cdot\omega_{\mathrm{CL}}\right)^{i},\quad\forall
i=1,\ldots,n+1.$ (9)

###  II-B Error-based ADRC (eADRC)

In Sect. II-A, it was shown that in case B of oADRC, deriving the outer-loop
state-feedback with feedforward controller designed for the resultant chain of
integrators assumes perfect knowledge (availability) of the consecutive
reference time-derivatives up to the order of the linearized dynamics. Through
the use of eADRC, this assumption can be removed, as it allows to treat
simultaneously the reference trajectory as a source of an additional external
disturbance which has to be actively rejected [10, 11, 12].

Figure 2: Continuous-time error-based ADRC (eADRC) in state-space
implementation. Here we denote the observer as error-based ESO as one of its
inputs is the feedback error signal $e(t)$. The second input is the control
signal $u(t)$, which is also the case for observer in Fig. 1.

Following the definition of the tracking error $e(t)\triangleq r(t)-y(t)$,
system (1) can be reformulated as:

$e^{(n)}=f_{\textrm{e}}(t)-b_{0}\cdot u(t),$ (10)

where $f_{\textrm{e}}(t)=r^{(n)}-\sum_{i=0}^{n-1}a_{i}\cdot
r^{(i)}+\sum_{i=0}^{n-1}a_{i}\cdot e^{(i)}-\Delta{b}\cdot u(t)-d(t)$ is the
generalized (total) disturbance in the error domain (cf. (2)).

###### Remark 5

In (10), the initial reference trajectory tracking control task is
reformulated into a regulation control task, hence the reference signal and
its derivatives can become part of the total disturbance $f_{\textrm{e}}(t)$.

A block diagram of the eADRC structure is shown in Fig. 2. One can see that
the main difference, compared to the standard output-based form depicted in
Fig. 1, is the feedback controller, here based on the tracking error
estimation vector:

$u_{\textrm{e}}(t)=\frac{1}{b_{0}}\cdot\left(\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot\bm{\hat{x}_{\textrm{e}}}\right),$
(11) $\displaystyle\text{where}\quad\bm{\hat{x}_{\textrm{e}}}$
$\displaystyle=\begin{pmatrix}\hat{x}_{\textrm{e}1}(t)&\cdots&\hat{x}_{\textrm{e}(n)}(t)&\hat{x}_{\textrm{e}(n+1)}(t)\end{pmatrix}^{\mathrm{T}}$
$\displaystyle=\begin{pmatrix}\hat{e}(t)&\cdots&\hat{e}^{(n-1)}&\hat{f}_{\textrm{e}}(t)\end{pmatrix}^{\mathrm{T}},$

is the state estimation vector, obtained by the following error-based version
of the ESO:

$\bm{\dot{\hat{x}}_{\textrm{e}}}=\bm{A}\cdot\bm{\hat{x}_{\textrm{e}}}-\bm{b}\cdot
u_{\textrm{e}}(t)+\bm{l}\cdot\left(e(t)-\bm{c}^{\mathrm{T}}\cdot\bm{\hat{x}_{\textrm{e}}}\right).$
(12)

It can be seen that the above error-based ESO has almost the same structure as
the output-based ESO in (4), with one of the differences being the sign of the
control signal-related term $\bm{b}\cdot u_{\textrm{e}}(t)$. Also, in (4), the
input is the system output $y(t)$, here, the input is the tracking error
$e(t)$.

#### Controller and Observer Tuning

Assuming perfect rejection of the total disturbance $f_{\textrm{e}}(t)$, in
the same manner as in standard ADRC in Sect. II-A, the controller design and
tuning can be performed for a canonical plant with unity-gain integrator
chain. Then, having in mind Remark 4, the same ‘bandwidth parameterization’
approach from [2] can be applied in eADRC as well. Hence, choosing the desired
closed-loop bandwidth $\omega_{\mathrm{CL}}$ leads to the same controllers
coefficients as in (6). Similarly, following the procedure from (8), the
observer gains can be calculated as in (9). This means that the eADRC uses the
same tuning methodology as the output-based structure, which retains
simplicity of only two tuning parameters ($\omega_{\mathrm{CL}}$ and
$k_{\mathrm{ESO}}$), which is one of the selling points of ADRC to industry.

##  III Comparative Analysis

In this section, a comparison is performed between oADRC (Sect. II-A) and
eADRC (Sect. II-B) in order to answer three fundamental questions about their
relation. The first one, in Sect. III-A, is about establishing a formal
relation between the two approaches, with formal definitions of their
similarities and differences. The second question, in Sect. III-B, is about
the influence of their differentiating elements on the control performance.
The third one, in Sect. III-C, is about their realizability in practice. For
the purpose of the comparison, following assumptions are made.

###### Assumption 1

In the comparison, we consider the same class of systems defined as seen in
(1). Special focus is on first- and second-order systems due to their
prevalence in engineering practice.

###### Assumption 2

To make the comparison fair, the same observer and controller tuning
methodology is used, namely the single bandwidth parameterization approach
from [2].

###### Assumption 3

For oADRC, two practical cases are considered (see Remark 3), one assuming the
availability of the reference signal derivatives for controller synthesis
(case B) and another without it (case A). For eADRC, no extra cases need to be
defined since this scheme does not require the reference signal derivatives
for the design process. The control schemes and their specific cases used in
the upcoming analysis are listed in Table I.

TABLE I: The investigated control schemes. Control scheme | Gains | Assumption
---|---|---
oADRC (case A) | $\bm{k_{\textrm{r}}}=0$ |  $\bm{r_{\textrm{d}}}$ not available
oADRC (case B) | $\bm{k_{\textrm{r}}}\neq 0$ |  $\bm{r_{\textrm{d}}}$ available
eADRC | — |  $\bm{r_{\textrm{d}}}$ not required

###  III-A What is the Relation Between oADRC and eADRC?

Based on the preliminary information about the two considered ADRC structures
in Sect. II, one can notice that eADRC scheme does not explicitly use the
reference channel in this derivation but still bares significant similarities
with its oADRC counterpart. By visual examination of Fig. 1 and Fig. 2,
following observations can be made. From the oADRC perspective, the eADRC can
be viewed as its inner component. On the other hand, eADRC can be viewed as a
simplified, bare-bones version of oADRC—its special case when the reference
channel is not explicitly used for controller synthesis. Therefore, there
seems to be a connection between the two ADRC schemes.

To enable better understanding of the considered oADRC and eADRC structures,
and potentially find formal conditions of their equivalence, we first
introduce transfer-function representation of their continuous-time forms. We
start with the oADRC. The Laplace transform of the control signal (3) is
substituted in (4), which gives333Throughout the paper, we assume, without
loss of generality, that the initial values that appear when making Laplace
transformation are equal to zero.:

$\bm{\hat{x}}(s)=\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\cdot\left[\frac{\bm{b}}{b_{0}}\left(k_{1}\cdot
r(s)+\bm{k_{\textrm{r}}}^{\mathrm{T}}\cdot\bm{r_{\textrm{d}}}(s)\right)+\bm{l}\cdot
y(s)\right]$ (13)

where:

$\displaystyle\quad\bm{A}_{\mathrm{CL}}$
$\displaystyle=\bm{A}-\bm{l}\bm{c}^{\mathrm{T}}-\frac{1}{b_{0}}\bm{b}\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}$
(14) $\displaystyle=\begin{pmatrix}\bm{0}^{n\times 1}&\bm{I}^{n\times n}\\\
0&\bm{0}^{1\times n}\end{pmatrix}-\begin{pmatrix}\bm{l}&\bm{0}^{(n+1)\times
n}\end{pmatrix}-\begin{pmatrix}\bm{0}^{(n-1)\times(n+1)}\\\
\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\\\
\bm{0}^{1\times(n+1)}\end{pmatrix}.$

We eliminate the observer estimation vector $\hat{\bm{x}}$ by substituting
(13) in the Laplace transform of (3), which yields:

$\displaystyle u(s)$
$\displaystyle=\frac{k_{1}}{b_{0}}r(s)+\frac{1}{b_{0}}\cdot\bm{k_{\textrm{r}}}^{\mathrm{T}}\cdot\bm{r_{\textrm{d}}}(s)-\frac{1}{b_{0}}\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot$
(15)
$\displaystyle\cdot\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\cdot\left[\frac{\bm{b}}{b_{0}}\left(k_{1}r(s)+\bm{k_{\textrm{r}}}^{\mathrm{T}}\bm{r_{\textrm{d}}}(s)\right)+\bm{l}\cdot
y(s)\right].$

By introducing
$\bm{r_{\textrm{d}}}(s)=r(s)\cdot\begin{pmatrix}s&\cdots&s^{(n)}\end{pmatrix}^{\mathrm{T}}$,
the control signal (15) can be rewritten as seen in (16),

$u(s)=\frac{1}{b_{0}}\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\bm{l}\cdot\left[\left(\frac{\left(b_{0}-\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\bm{b}\right)\cdot\left(k_{1}+\bm{k_{\textrm{r}}}^{\mathrm{T}}\begin{pmatrix}s&\cdots&s^{(n)}\end{pmatrix}^{\mathrm{T}}\right)}{b_{0}\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\cdot\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\bm{l}}\right)r(s)-y(s)\right]$
(16)

which is equivalent to the standard transfer function444Due to the size of the
equation, it has been written separately on next page while keeping
chronological numbering.:

$u(s)=G_{\mathrm{FB}}(s)\cdot\left[G_{\mathrm{PF}}(s)\cdot r(s)-y(s)\right],$
(17)

where:

$G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\frac{\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}{\operatorname{det}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}\cdot\bm{l},$
(18)

is the feedback transfer function while (19)

$G_{\mathrm{PF}}(s)=\frac{\left(k_{1}+\bm{k_{\textrm{r}}}^{\mathrm{T}}\begin{pmatrix}s&\cdots&s^{(n)}\end{pmatrix}^{\mathrm{T}}\right)\left(\operatorname{det}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)-\frac{1}{b_{0}}\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\bm{b}\right)}{\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\cdot\bm{l}}$
(19)

represents the reference pre-filter transfer function, derived for the control
law (3) with the appropriate reference derivatives (case B). The pre-filter
transfer function, defined by (19), can be expressed alternatively
as${}^{\text{2}}$:

$G_{\mathrm{PF}}(s)=G_{\mathrm{R}}(s)\cdot\bar{G}_{\mathrm{PF}}(s),$ (20)

with:

$\displaystyle\bar{G}_{\mathrm{PF}}(s)=\frac{\operatorname{det}(s\bm{I}-\bm{A}_{\mathrm{CL}})-\frac{1}{b_{0}}\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\bm{b}}{\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\cdot\bm{l}},$
(21)

and
$G_{\mathrm{R}}(s)=k_{1}+\bm{k_{\textrm{r}}}^{\mathrm{T}}\begin{pmatrix}s&\cdots&s^{(n)}\end{pmatrix}^{\mathrm{T}}$,
where the latter results from summing the reference signal and its derivatives
and multiplying them with appropriate gains. The pre-filter transfer function
for the control law (3), generated without the reference derivatives (case A),
can be simply obtained from (20) by substituting $G_{\mathrm{R}}(s)=k_{1}$. It
should be noted, however, that both cases A and B have the same feedback
transfer function (18), because it does not depend on the reference
derivatives.

Now, to represent eADRC in the transfer function form, similar approach is
taken. The Laplace transform of the control signal (11) is first substituted
in (12), which gives:

$\bm{\hat{x}_{\textrm{e}}}(s)=\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)^{-1}\cdot\bm{l}\cdot
e(s).$ (22)

By substituting (22) in the Laplace transform of (12) and eliminating
$\bm{\hat{x}}_{\textrm{e}}(s)$, eADRC transfer function is:

$u(s)=\frac{1}{b_{0}}\cdot\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\frac{\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}{\operatorname{det}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}\cdot\bm{l}\cdot
e(s).$ (23)

It is evident that (23) can be represented with only the feedback transfer
function from the tracking error ($e$) to the control input ($u$):

$\frac{u(s)}{e(s)}=\frac{1}{b_{0}}\cdot\begin{pmatrix}\bm{k}^{\mathrm{T}}&1\end{pmatrix}\frac{\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}{\operatorname{det}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)}\cdot\bm{l}\equiv
G_{\textrm{FB}}(s),$ (24)

which is identical to the oADRC feedback transfer function
$G_{\textrm{FB}}(s)$ from (18).

This established relation between oADRC and eADRC control systems of the plant
model $G_{\textrm{P}}(s)$ can be visually expressed through Fig. 3. From this
figure, one can notice that the entire eADRC has the same structure as the
feedback transfer function of oADRC, or looking at it from a different
perspective, when the oADRC structure has no pre-filter (i.e.
$G_{\textrm{PF}}(s)=1$), it boils down to eADRC.

###### Remark 6

It should be noted that oADRC, in the form shown in Sect. II-A, although
(visually) has a structure of a two degrees-of-freedom (2DOF) system with
transfer functions $G_{\textrm{FB}}(s)$ and $G_{\textrm{PF}}(s)$, is not
strictly a 2DOF control scheme. That is of course if one follows the
definition from [31], which states that a controller is of 2DOF structure if
one can write it using two transfer functions that can be tuned fully
independently. The used oADRC form does not satisfy that definition because
its transfer functions are coupled with each other through the use of tuning
parameters $k_{\mathrm{ESO}}$ and $\omega_{\mathrm{CL}}$, hence it is not
possible to decouple setting of the disturbance rejection and set-point
tracking performances, which is a strict requirement given the above
definition of a 2DOF control system. The eADRC, on the other hand, has a
typical one degree-of-freedom (1DOF) control structure.

Answer. To address the question posed in the title of this subsection, the
direct comparison of output- and error-based control algorithms, conducted in
$s$-domain, shows that the pre-filter transfer function $G_{\mathrm{PF}}(s)$
is the element that differentiates the two designs (see Fig. 3). The relation
between oADRC and eADRC can be thus viewed from two perspectives. On one hand,
the eADRC can be considered as a special instance of oADRC when
$G_{\mathrm{PF}}(s)=1$. On the other hand, the oADRC can be considered as an
extended control structure, where additional component (namely
$G_{\mathrm{PF}}(s)$) is added to the eADRC. It is thus interesting to
investigate the influence of the term $G_{\mathrm{PF}}(s)$ on the system
performance using oADRC with respect to eADRC. Hence, this question is
investigated next.

Figure 3: Transfer functions of oADRC and eADRC variants.

###### Remark 7

The equivalence between oADRC and eADRC structures, established in this
subsection, is also valid in discrete time. A detailed derivation of that fact
is, however, omitted here to avoid redundancy and the necessity of having to
introduce another domain, related to discrete time.

###  III-B What is the Influence of Term $G_{\mathrm{PF}}(s)$ on the System
Performance?

It was revealed in Sect. III-A that the pre-filter transfer function
$G_{\mathrm{PF}}(s)$ is a component that fundamentally differentiates the
considered oADRC and eADRC structures from each other. Therefore, the
influence of $G_{\mathrm{PF}}(s)$ on the system stability as well as on
steady-state and transient performances, is analyzed here.

#### Stability and Steady-State Performance

Based on the previously derived transfer functions of the considered oADRC and
eADRC structures (cf. Fig. 3), it is evident that their open-loop transfer
functions are the same, i.e.:

$G_{\mathrm{OL}}(s)=G_{\mathrm{FB}}(s)G_{\textrm{P}}(s),$ (25)

which means that they have the same stability characteristics (for the same
tuning parameters), regardless of $G_{\mathrm{PF}}(s)$.

One can also see that oADRC and eADRC structures have identical transfer
functions from disturbance $d(s)$ to output $y(s)$:

$G_{\mathrm{YD}}(s)=\frac{G_{\textrm{P}}(s)}{1+G_{\mathrm{OL}}(s)},$ (26)

which means that they have the same disturbance rejection capabilities in
steady-state.

In both considers ADRC schemes, the transfer function between the measurement
noise $n(s)$ in the output signal $y(s)$ and the control signal $u(s)$ is:

$G_{\mathrm{UN}}(s)=\frac{-C_{\mathrm{FB}}(s)}{1+G_{\mathrm{OL}}(s)},$ (27)

which indicates that both considered control structures achieve the same level
of noise sensitivity.

For oADRC, the transfer function from the reference signal $r(s)$ to the
tracking error $e(s)$ is:

$G_{\mathrm{ER(o)}}(s)=1-\frac{C_{\mathrm{PF}}(s)G_{\mathrm{OL}}(s)}{1+G_{\mathrm{OL}}(s)},$
(28)

and for the eADRC is:

$G_{\mathrm{ER(e)}}(s)=1-\frac{G_{\mathrm{OL}}(s)}{1+G_{\mathrm{OL}}(s)}.$
(29)

In both above instances, there is a direct influence of $G_{\mathrm{PF}}(s)$.
Therefore, this influence is further investigated. For the purpose of the
investigation, two mathematical models of common industrial plants are
introduced, namely a first-order system model ($n=1$) with time-delay:

$G_{\textrm{P1}}(s)=\frac{1}{s+1}e^{-0.2s},$ (30)

and a normalized second-order system model ($n=2$):

$G_{\textrm{P2}}(s)=\frac{1}{s^{2}+2s+1},$ (31)

The pre-filter and feedback transfer functions for the considered first- and
second-order plant models are presented in Table II and Table III,
respectively.

TABLE II: Transfer functions for first-order plant model ($n=1$). Control scheme | Pre-filter transfer function | Feedback transfer function
---|---|---
oADRC (case A) | $G_{\mathrm{PF}}(s)=\frac{k_{1}\cdot(s^{2}+l_{1}s+l_{2})}{(k_{1}l_{1}+l_{2})\cdot s+k_{1}l_{2}}$ | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+l_{2})\cdot s+k_{1}l_{2}}{s^{2}+(k_{1}+l_{2})\cdot s}$
oADRC (case B) | $G_{\mathrm{PF}}(s)=\frac{(k_{1}+s)\cdot(s^{2}+l_{1}s+l_{2})}{(k_{1}l_{1}+l_{2})\cdot s+k_{1}l_{2}}$ | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+l_{2})\cdot s+k_{1}l_{2}}{s^{2}+(k_{1}+l_{2})\cdot s}$
eADRC | — | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+l_{2})\cdot s+k_{1}l_{2}}{s^{2}+(k_{1}+l_{2})\cdot s}$
TABLE III: Transfer functions for second-order plant model ($n=2$). Control scheme | Pre-filter transfer function | Feedback transfer function
---|---|---
oADRC (case A) | $G_{\mathrm{PF}}(s)=\frac{k_{1}\cdot(s^{3}+l_{1}s^{2}+l_{2}s+l_{3})}{(k_{1}l_{1}+k_{2}l_{2}+l_{3})\cdot s^{2}+(k_{1}l_{2}+k_{2}l_{3})\cdot s+k_{1}l_{3}}$ | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+k_{2}l_{2}+l_{3})\cdot s^{2}+(k_{1}l_{2}+k_{2}l_{3})\cdot s+k_{1}l_{3}}{s^{3}+(k_{2}+l_{1})\cdot s^{2}+(l_{2}+k_{1}+l_{1}k_{2})\cdot s}$
oADRC (case B) | $G_{\mathrm{PF}}(s)=\frac{(s^{2}+k_{2}s+k_{1})\cdot(s^{3}+l_{1}s^{2}+l_{2}s+l_{3})}{(k_{1}l_{1}+k_{2}l_{2}+l_{3})\cdot s^{2}+(k_{1}l_{2}+k_{2}l_{3})\cdot s+k_{1}l_{3}}$ | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+k_{2}l_{2}+l_{3})\cdot s^{2}+(k_{1}l_{2}+k_{2}l_{3})\cdot s+k_{1}l_{3}}{s^{3}+(k_{2}+l_{1})\cdot s^{2}+(l_{2}+k_{1}+l_{1}k_{2})\cdot s}$
eADRC | — | $G_{\mathrm{FB}}(s)=\frac{1}{b_{0}}\cdot\frac{(k_{1}l_{1}+k_{2}l_{2}+l_{3})\cdot s^{2}+(k_{1}l_{2}+k_{2}l_{3})\cdot s+k_{1}l_{3}}{s^{3}+(k_{2}+l_{1})\cdot s^{2}+(l_{2}+k_{1}+l_{1}k_{2})\cdot s}$

The comparative frequency responses of (28) (for cases A and B) and (29), with
design parameters $\omega_{\textrm{CL}}=3$rad/s and $k_{\mathrm{ESO}}=8$, are
shown in Fig. 4(a) for the first-order plant (30) and in Fig. 4(b) for the
second-order plant (31).

(a)

(b)

Figure 4: Comparative analyses of the reference tracking performances of oADRC
(cases A and B) and eADRC for plant model $G_{\textrm{P1}}(s)$ (a) and
$G_{\textrm{P2}}(s)$ (b).

(a) System response (zoomed figure) for a reference signal (dashed line) using
fixed $\omega_{\mathrm{CL}}=1.5$rad/s and linearly increasing values of
$k_{\mathrm{ESO}}$, from $k_{\mathrm{ESO}}=6$ (low color saturation) to
$k_{\mathrm{ESO}}=24$ (high color saturation).

(b) System response for a reference signal (dashed line) using fixed
$k_{\mathrm{ESO}}=8$ and linearly increasing values of $\omega_{\mathrm{CL}}$,
from $\omega_{\mathrm{CL}}=1.5$ rad/s (low color saturation) to
$\omega_{\mathrm{CL}}=4.5$ rad/s (high color saturation).

Figure 5: Influence of tuning parameters on the tracking performance of system
$G_{\textrm{P1}}(s)$ with the control structures oADRC in case B (left), oADRC
in case A (middle), and eADRC (right).

(a) System response (zoomed figure) for a reference signal (dashed line) using
fixed $\omega_{\mathrm{CL}}=3$rad/s and linearly increasing values of
$k_{\mathrm{ESO}}$, from $k_{\mathrm{ESO}}=6$ (low color saturation) to
$k_{\mathrm{ESO}}=24$ (high color saturation).

(b) System response for a reference signal (dashed line) using fixed
$k_{\mathrm{ESO}}=8$ and linearly increasing values of $\omega_{\mathrm{CL}}$,
from $\omega_{\mathrm{CL}}=3$ rad/s (low color saturation) to
$\omega_{\mathrm{CL}}=9$ rad/s (high color saturation).

Figure 6: Influence of tuning parameters on the tracking performance of system
$G_{\textrm{P2}}(s)$ with the control structures oADRC in case B (left), oADRC
in case A (middle), and eADRC (right).

The obtained results from Fig. 4(a) and Fig. 4(b) show that the oADRC with the
reference derivatives included in the control law (case B) achieves
significantly lower tracking error than the oADRC without reference
derivatives in the control law (case A) over the entire reference frequency
range of interest (lower than the tuned system closed-loop bandwidth
$\omega_{\mathrm{CL}}=3$rad/s). Also, one can notice that the eADRC provides
practically the same tracking performance as the oADRC (case B), but without
requiring the availability of the reference derivatives in the control law. It
can be explained by the fact that eADRC indirectly estimates the reference
derivatives by the ESO (12) and use them in the controller synthesis, which
can be seen in the definition of the total disturbance term
$f_{\textrm{e}}(t)$ under (10).

#### Transient Performance

The transient performances of the considered ADRC structures are analyzed in
time-domain using numerical simulations with the plant models (30) and (31).
First, the simulations are conducted with the reference signal in form of a
step signal filtered by a low-pass filter
$G_{\textrm{F}}(s)=\frac{1}{(0.1s+1)^{2}}$ and with various tuning parameters,
in two scenarios: with fixed $\omega_{\mathrm{CL}}$ and variable
$k_{\mathrm{ESO}}$, and with fixed $k_{\mathrm{ESO}}$ and variable
$\omega_{\mathrm{CL}}$. The obtained results are gathered in Fig. 5 and Fig.
6, respectively.

From the obtained results, one can notice that in all the simulated scenarios,
both oADRC structures gave responses with lower overshoots than eADRC,
especially for the second-order system $G_{\textrm{P2}}(s)$, where oADRC
structures provide practically minimal overshoot. Therefore, it can be
concluded from the performed tests that the oADRC schemes, thanks to the
presence of component $G_{\textrm{PF}}(s)$, represent a better options than
eADRC in terms the ability to shape the system response during the transient
stage. Also, one can conclude that the oADRC (case B) and eADRC enable faster
responses compared to oADRC (case A) due to the use of the reference signal
derivatives in the control law, which are interpreted as actionable
information that is consciously used by the control designer to improve the
control performance.

Additionally, comparing the responses of the first- and second-order plants
for all the considered controller structures, one can notice that in the
instance of second-order system $G_{\textrm{P2}}(s)$, increasing parameters
$k_{\textrm{ESO}}$ and $\omega_{\mathrm{CL}}$ leads to smaller overshoot. That
is due to higher observer and controller dynamics, which implies faster
convergence of the total disturbance estimation and better tracking
performance. On the other hand, in the instance of first-order system
$G_{\textrm{P1}}(s)$, increasing parameters $k_{\textrm{ESO}}$ and
$\omega_{\mathrm{CL}}$ lead to larger overshoot. That is not surprising as it
is a consequence of the presence of time-delay in the plant model.

In order to compare energy requirements of the considered controllers as well
as to show time-domain disturbance rejection performance and measurement noise
sensitivity, the simulation with reference signal in form of a step signal
filtered by a low-pass filter $G_{\textrm{F}}(s)=\frac{1}{(0.2s+1)^{2}}$ in
the presence of step disturbance from $t\geq 10$s and measurement noise from
$t\geq 15$s, are performed. The obtained system responses for plants (30) and
(30) are depicted in Fig. 7 and Fig. 8, respectively. The simulations are
performed for fixed tuning parameters $\omega_{\textrm{CL}}=3$rad/s and
$k_{\mathrm{ESO}}=8$. Please note that the overshoot effect in Fig. 7 is also
closely related with time-delay in $G_{\textrm{P1}}(s)$, which additionally
limits the upper range of a practically feasible desired closed-loop
bandwidth.

(a)

(b)

Figure 7: Comparative analyses of transient performances of oADRC (cases A and
B) and eADRC for plant model $G_{\textrm{P1}}(s)$, where (a) is the system
output and (b) is the control signal. Note: during the disturbance rejection
test ($10\textrm{s}\leq t<15\textrm{s}$) and measurement noise sensitivity
test ($15\textrm{s}\leq t\leq 20\textrm{s}$), the signals overlap. Note:
remember that $G_{\textrm{P1}}(s)$ has time-delay, which especially influences
the initial response (overshoot).

(a)

(b)

Figure 8: Comparative analyses of transient performances of oADRC (cases A and
B) and eADRC for plant model $G_{\textrm{P2}}(s)$, where (a) is the system
output and (b) is the control signal. Note: during the disturbance rejection
test ($10\textrm{s}\leq t<15\textrm{s}$) and measurement noise sensitivity
test ($15\textrm{s}\leq t\leq 20\textrm{s}$), the signals overlap.

From the figures, it can be noted that the price for the faster responses of
oADRC (case B) and eADRC is the increased level of control signal, especially
in the second-order system simulation with oADRC in Fig. 8(b). It is a
consequence of a relatively large value of the reference signal’s second time-
derivative, which is necessary for the synthesis of the control law.
Therefore, the application of the ADRC (case B) structure for higher-order
systems ($n>2$) can be limited in practice, due to a potentially large value
of the reference $n$-th derivatives. Regarding the disturbance rejection
performances, as expected, it is evident that all the considered structures
have the same disturbance rejection responses, which confirms the previously
made conclusion resulting from (26). Similarly, looking at the control signals
at time $t\geq 15$s in Fig. 7(b) and Fig. 8(b)), one can notice identical
levels of measurement noise sensitivity, which confirms conclusions made in
$s$-domain in (27).

Answer. To address the question posed in the title of this subsection, the
pre-filter transfer function $G_{\mathrm{PF}}(s)$ has, based on the conducted
tests, impact on the tracking performance during the transient control stage.
The analysis has shown that in steady-state, $G_{\mathrm{PF}}(s)$ has no
effect on the stability margins, sensitivity to measurement noise, and
disturbance rejection, as well as a negligible impact on the steady-state
reference tracking performance when it is realized by including reference
derivative terms (oADRC in case B). The inclusion of time-delay in
$G_{\textrm{P1}}(s)$ was deliberate, as this type of system is often seen in
industrial practice, but it does not change the findings or the general
conclusions from this section.

###  III-C Are the Considered ADRC Transfer Functions Realizable?

To enable realizability of the considered transfer function representations of
oADRC and eADRC, it is necessary to make all the transfer functions within
those structures realizable.

For eADRC, it requires only the feedback transfer function (18) for its
implementation (cf. Fig. 3), hence the realizability of this transfer function
has to be checked. By analyzing (18), one can see that its numerator and
denominator orders depend on matrix $\bm{A}_{\textrm{CL}}$. The fact that the
last row and rightmost column of the $\bm{A}_{\textrm{CL}}$ are zero, and by
assuming non-zero observer and controller gains, makes its rank equal to $n$.
At the same time, the order of numerator polynomial of $G_{\mathrm{FB}}(s)$ is
$n$, while its denominator polynomial is $n+1$, which is also evident from
Table II and Table III. It can be thus concluded that the denominator
polynomial order exceeds the numerator polynomial order, meaning that
$G_{\mathrm{FB}}(s)$ (and eADRC as a result) is a strictly proper and a
realizable transfer function.

For oADRC, its realizability depends on the realizability of (20). Assuming
that $G_{\mathrm{R}}(s)$ can be realized as the sum of $n+1$ inputs (i.e.
reference signal and its $n$ derivatives), multiplied by the corresponding
coefficients, the problem of realizability is focused on (21). By its
inspection, one can see that due to $A_{\textrm{CL}}$ having rank $n$, (21) is
not proper, meaning the transfer function is not realizable, because its
numerator polynomial has order $n+1$ but its denominator polynomial has order
$n$ (cf. Table II and Table III). Therefore, oADRC cannot be realized in the
form as presented in Fig. 3.

Figure 9: Proposed modular approach to designing customized ADRC-based
structures in form of realizable transfer functions, with visual cues for
obtaining nominal oADRC case A (dashed-dotted red), oADRC case B (short-long
dashed blue), and eADRC (dashed green). Interpretation of the modules: Module
1—eADRC, which is the bare-bones version of ADRC consisting just the feedback
transfer function $G_{\textrm{FB}}(s)$; Module 2—eADRC is transformed into
oADRC by adding a reference channel with a prefilter
$\tilde{G}_{\textrm{PF}}(s)$ (for shaping transient performance) and a
feedforward term $G_{\textrm{FF}}(s)$ (for making the transfer function
realizable); Module 3—it additionally equips oADRC with the derivatives of the
reference signal (assuming they are available) to improve the tracking
performance (especially helpful in motion control tasks).

Following methodology presented in [26], this problem can be solved by a
reformulation of the transfer function representation of oADRC from the
standard pre-filter expression (17) to the combination of pre-filter and
feedforward:

$\displaystyle u(s)$
$\displaystyle=G_{\mathrm{FB}}(s)\cdot\left[G_{\mathrm{R}}(s)\tilde{G}_{\mathrm{PF}}(s)r(s)-y(s)\right]+$
$\displaystyle+G_{\mathrm{R}}(s)G_{\mathrm{FF}}(s)r(s),$ (32)

with the introduced feedforward transfer function, which carries the $s^{n+1}$
numerator term of (21):

$G_{\mathrm{FF}}(s)=\frac{s^{n+1}}{b_{0}\cdot\operatorname{det}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)},$
(33)

and modified pre-filter transfer function:

$\displaystyle\tilde{G}_{\mathrm{PF}}(s)=\frac{\operatorname{det}(s\bm{I}-\bm{A}_{\mathrm{CL}})-s^{n+1}-\frac{1}{b_{0}}\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\cdot\bm{b}}{\begin{pmatrix}k^{\mathrm{T}}&1\end{pmatrix}\operatorname{adj}\left(s\bm{I}-\bm{A}_{\mathrm{CL}}\right)\cdot\bm{l}}.$
(34)

One can note that order of numerator and denominator of (33) will be $n$ due
to cancellation of its numerator with the integrator introduced by
$\operatorname{det}(s\bm{I}-\bm{A}_{\mathrm{CL}})$, which means that this
transfer function can be realized as $n$-th order high-pass filter. On the
other hand, modified pre-filter transfer function (34) has $n$-th order
denominator, same as (19), and $n$-th order numerator, what makes it proper
and realizable. An additional advantage of this structure is that integrator
component exists only in the feedback transfer function, which is important
from the practical realization point of view [26].

The graphical representation of the proposed realizable forms of the
considered oADRC (in cases A and B) and eADRC, based on a modular approach,
are shown in Fig. 9. One can see that eADRC is the least complex structure
which requires only one module (’Module 1’) for its realization. The
transition from eADRC to oADRC (case A) is possible by adding ’Module 2’. The
oADRC (case B) represents the most complex structure.

Answer. To address the question posed in the title of this subsection, the
transfer function representation of the eADRC scheme has a realizable form and
thus can be directly used. For oADRC, we have shown a simple modification to
its nominal form that allows to represent the algorithm in an realizable way
by utilizing the combination of pre-filter and feedforward concepts.

##  IV Summary

In this paper, output- and error-based versions of the ADRC scheme were
investigated. Through an analysis in $s$-domain, a condition of equivalence
between the two structures has been established. It was found that the only
difference between the two structures is the presence of a so-called pre-
filter transfer function ($G_{\textrm{PF}}(s)$). The frequency domain analysis
has shown that in steady-state, $G_{\textrm{PF}}(s)$ has no effect on the
stability margins, sensitivity to measurement noise, or disturbance rejection,
but has impact on the steady-state reference tracking performance. It was also
deduced that synthesising and utilizing $G_{\textrm{PF}}(s)$ without reference
derivatives (i.e. oADRC in case A) reduces steady-state reference tracking
performance, whereas adding reference derivatives in the oADRC (i.e. case B)
provides quantitatively better performance, practically the same as in the
eADRC control structure (where $G_{\textrm{PF}}(s)=1$ is assumed).
Additionally, the conducted investigation of the influence of
$G_{\textrm{PF}}(s)$ on basic control performance criteria in time-domain has
revealed that the ADRC with the pre-filter allows the control designer to
better shape the tracking performance in the transient stage, which can be
beneficial in certain practical scenarios.

That is why, in order to assist the control designers in choosing most
suitable ADRC-based solution for their applications, a summary of the
conducted systematic comparison has been graphically represented with a
(heuristic) decision tree, depicted in Fig. 10. In short, it can be seen that
the eADRC has a simpler, more compact design, resulting in a simpler
implementation (due to the lack of the pre-filter), but for the price of
reducing the control over transient tracking performance. The eADRC also gives
more freedom to the control designer to construct custom control solutions
since the pre-filter $G_{\textrm{PF}}(s)$ is not built-in (imposed) in the
design in advance (as it is the case in oADRC). Because there are different
control scenarios that call for a different control solution, a modular
approach has been chosen in this work. Certain modules have been identified
(see Fig. 9) that aid the process of designing the desired ADRC solution.

Finally, in order to increase the potential applicability of both considered
ADRC schemes, their realizable transfer functions forms were given.

Figure 10: Proposed heuristic decision tree for choosing ADRC structure (for
the interpretation of ’intermediate cases (A1, A2)’ — see Remark 3 and the
Appendix).

## Appendix — Influence of Reference Derivatives

In order to show the influence of various levels of knowledge about the
reference derivatives on the tracking quality in oADRC, we introduce a third-
order ($n=3$) plant model:

$G_{\mathrm{P3}}(s)=\frac{1}{s^{3}+s^{2}+s+1},$ (35)

that does not represent a physical plant and is used here solely as a toy
example which happens to be useful to us in supporting the idea we want to
convey here.

Next, a numerical simulation is performed with oADRC controllers (in
previously defined cases A and B) as well as two intermediate cases, which
include terms related to the first and second reference derivative in (3).
Those intermediate cases of oADRC are denoted as A1 and A2 and are formally
defined in Table IV. The controllers considered in the simulation are all
tuned using the ’bandwidth parameterization’ approach with the same
$\omega_{\mathrm{CL}}=2.5$rad/s and $k_{\mathrm{ESO}}=5$. The tracking
performances obtained for a reference sinusoidal signal $r(t)=\sin(t)$ are
shown in Fig. 11.

From Fig. 11, it is evident that, by including more derivatives of the
reference signal to the oADRC control law, the reference tracking performance
is improved. This makes oADRC (case B) variant especially suitable for
tracking control of dynamic signals. However, it should also be noted that
adding more reference derivatives during the controller synthesis also makes
the final controller more complex and that in industrial practice the
reference derivatives are not always available. Therefore, the bottom line is
that the selection of a specific ADRC controller structure should be done on a
case-by-case basis.

TABLE IV: Considered oADRC cases for $G_{\mathrm{P3}}(s)$ (cf. Table I). oADRC | Gains | Assumption
---|---|---
case A | $\bm{k_{\textrm{r}}}^{\mathrm{T}}=\begin{pmatrix}0&0&0\end{pmatrix}$ |  $\dot{r},\ddot{r},r^{(3)}$ not available
case A1 | $\bm{k_{\textrm{r}}}^{\mathrm{T}}=\begin{pmatrix}k_{2}&0&0\end{pmatrix}$ | only $\dot{r}$ available
case A2 | $\bm{k_{\textrm{r}}}^{\mathrm{T}}=\begin{pmatrix}k_{2}&k_{3}&0\end{pmatrix}$ | only $\dot{r},\ddot{r}$ available
case B | $\bm{k_{\textrm{r}}}^{\mathrm{T}}=\begin{pmatrix}k_{2}&k_{3}&1\end{pmatrix}$ |  $\dot{r},\ddot{r},r^{(3)}$ available

## Statements and Declarations

Figure 11: Reference tracking performances of oADRC in cases A and B, as well
as two intermediate cases A1 and A2, with reference signal and system outputs
(top) and tracking error (bottom).

  * •

R. Madonski was supported by the Fundamental Research Funds for the Central
Universities (project no. 21620335).

  * •

M. Stankovic was supported by the International Foreign Expert Project Fund of
Jinan University (project no. G2021199027L, coordinator: Dr. Hui Deng).

  * •

The authors have no competing interests to declare that are relevant to the
content of this article.

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