A Generalization of the Passivity Theorem and the Small Gain Theorem Based on $\rho $-Stability, with Application to a Parameter Adaptation Algorithm for Recursive Identification

TL;DR

This paper extends classical passivity and small gain theorems using $ ho$-stability, allowing for broader stability conditions and applying these results to a parameter adaptation algorithm in recursive identification.

Contribution

It introduces a generalized stability framework based on $ ho$-passivity, broadening the applicability of passivity and small gain theorems in control theory.

Findings

The closed-loop remains stable under $ ho$-passivity conditions.

The approach generalizes classical passivity and small gain theorems.

Application demonstrated in a parameter adaptation algorithm for recursive identification.

Abstract

The usual passivity theorem considers a closed-loop, the direct chain of which consists of a strictly passive stable operator $H_{1}$, and the feedback chain of which consists of a passive operator $H_{2}$. Then the closed-loop is stable. Let $\rho >1$. We show here that the closed-loop is still stable when the direct chain consists of a strictly $\rho ^{-1}$-passive $% \rho ^{-1}$-stable operator (a weaker condition than above) and the feedback chain consists of a $\rho $-passive operator (a stronger condition than above). Variations on the theme of the small gain theorem (incremental or not) can be made similarly. This approach explains the results obtained in a paper on identification which was recently published.

Equations

$$\gamma \left(H\right)=\inf \left\{\delta \ge 0:\exists \beta \in \mathbb{R},{\Vert Hx\Vert }_T\le \delta {\Vert x\Vert }_T+\beta \text{ for all }T\in \mathbb{T}\right\}.$$

$${\gamma }^0\left(H\right)=\inf \left\{\delta \ge 0:{\Vert Hx\Vert }_T\le \delta {\Vert x\Vert }_T\text{ for all }T\in \mathbb{T}\right\}$$

$$e_1$$

$$e_2$$

$$⟨x,{\rho }^{-1}\circ H_1\circ \rho \circ x⟩$$

$$⟨x,\rho \circ H_2\circ {\rho }^{-1}\circ x⟩$$

$${\delta }_1+{\delta }_2>0$$

$${⟨e_1,y_1⟩}_T+{⟨e_2,y_2⟩}_T$$

$${⟨e_1,y_1⟩}_T$$

$${⟨e_2,y_2⟩}_T$$

$${\delta }_1{\Vert e_1\Vert }_T^2+{\beta }_1^{\prime }+{\delta }_2\left({\Vert u_1\Vert }_T^2-2{\Vert u_1\Vert }_T{\Vert e_1\Vert }_T+{\Vert e_1\Vert }_T^2\right)+{\beta }_2^{\prime }$$

$$\left({\delta }_1+{\delta }_2\right){\Vert e_1\Vert }_T^2$$

$$⟨x,{\rho }^{-1}\circ H_1\circ \rho \circ x⟩$$

$$⟨x,\rho \circ H_2\circ {\rho }^{-1}\circ x⟩$$

$$\left(H_{1}x\right)\left(t\right)=\sum _{\tau \in \mathbb{Z}}{\mathbf{h}}_1\left(t,\tau \right)x\left(\tau \right).$$

$$\left({\rho }^{-1}\circ H_1\circ \rho \circ x\right)\left(t\right)=\sum _{\tau \in \mathbb{Z}}{\mathbf{h}}_1\left(t,\tau \right){\rho }^{\tau -t}x\left(\tau \right).$$

$${\lambda }_{\min }\left\{\frac{{\hat{h}}^T\left(\rho e^{-i\theta }\right)+\hat{h}\left(\rho e^{i\theta }\right)}{2}\right\}\ge {\delta }_1$$

$${\lambda }_{\min }\left\{\frac{{\hat{h}}^T\left(\alpha -i\omega \right)+\hat{h}\left(\alpha +i\omega \right)}{2}\right\}\ge {\delta }_1$$

$${\left({\rho }^{-1}\circ H\circ \rho -I\right)}^{-1}{\left(I+{\rho }^{-1}\circ H\circ \rho \right)}^{-1}={\rho }^{-1}\left(H-I\right)\left(I+H\right)\rho .$$

Full text

1 Introduction and preliminaries

Many stability theorems were derived for a standard closed-loop system (as depicted in, e.g., Figure III.1 of ([5], p. 37)), the direct chain of which consists of an operator $H_{1}$, with input $e_{1}$ and output $y_{1},$ and the feedback chain of which consists of an operator $H_{2},$ with input $e_{2}$ and output $y_{2}$. The interconnection equations are $e_{1}=u_{1}-y_{2},$ $e_{2}=u_{2}+y_{1}$ where $u_{1},u_{2}$ are external signals.

Let $\mathbb{T}=\mathbb{Z}$ in the discrete-time case and $\mathbb{T}=\mathbb{R}$ in the continuous-time case.  In addition, let $\mathcal{S}^{n}$ be the subspace of $\left(l^{2}\right)^{n}$ in the former case, of $\left(L^{2}\right)^{n}$ in the latter, consisting of those signals which have a left-bounded support; $\mathcal{S}^{n}$ is a Hilbert space. Let $T\in\mathbb{T}$, let $P_{T}$ be the truncation operator, such that $\left(P_{T}x\right)\left(t\right)=x\left(t\right)$ if $t\leq T$ and $\left(P_{T}x\right)\left(t\right)=0$ otherwise ([7], Sect. 2.3), and let $\mathcal{S}_{e}^{n}$ be the extended space consisting of all signals $x\in\left(\mathbb{R}^{n}\right)^{\mathbb{T}}$ such that $P_{T}x\in\mathcal{S}$ for all $T\in\mathbb{T}$. If $x,y\in\mathcal{S}_{e}^{n}$, the inner product $\left\langle P_{T}x,P_{T}y\right\rangle_{\mathcal{S}^{n}}$ is denoted by $\left\langle x,y\right\rangle_{T},$ and $\left\|x\right\|_{T}:=\sqrt{\left\langle x,y\right\rangle_{T}}.$ Let $H:\mathcal{S}_{e}^{n}\rightarrow\mathcal{S}_{e}^{n}$ be an operator. Its gain $\gamma\left(H\right)\leq+\infty$ is defined to be ([5], Sect. 3.1)

[TABLE]

We put

[TABLE]

and $H$ is said to be $\mathcal{S}$-stable if $\gamma^{0}\left(H\right)<+\infty$ ([5], Sect. 3.7).

Let $\mathbf{G}$ be the multiplicative Abelian group of all positive real numbers. This group acts on $\mathcal{S}_{e}^{n}$ as follows: if $\rho\in\mathbf{G},$ $x\in\mathcal{S}_{e}^{n},$ then $\left(\rho\circ x\right)\left(t\right)=\rho^{t}x\left(t\right).$  In the continuous-time case, let $\alpha=\ln\left(\rho\right);$ then the concept of $\rho$-stability as defined in [4] is equivalent to $\alpha$-stability as introduced in [1] and developped in [2], [3].

The following is assumed in this paper (with the above notation): if $u_{1},u_{2}\in\mathcal{S}^{n},$ then there are solutions $e_{1},e_{2}\in\mathcal{S}_{e}^{n}$ (”well-posedness” of the closed-loop).

The classical passivity (resp. small gain) theorem states that if $H_{1}$ is strictly passive and such that $\gamma^{0}\left(H_{1}\right)<+\infty$ (resp. is such that $\gamma^{0}\left(H_{1}\right)<+\infty$) and $H_{2}$ is passive (resp. is such that $\gamma^{0}\left(H_{2}\right)<+\infty$ and $\gamma^{0}\left(H_{1}\right).\gamma^{0}\left(H_{2}\right)<1$) then the operator $\left(u_{1},u_{2}\right)\mapsto\left(e_{1},e_{2},y_{1},y_{2}\right)$ is $\mathcal{S}$-stable.  This result has variants which will be mentioned below.

In what follows, using the action of $\mathbf{G}$, we relax the assumption on $H_{1}$ and strengthen the assumption on $H_{2},$ or vice-versa.

2 Extension of stability results

2.1 Extended passivity stability theorem

Consider again the closed-loop system as specified in Section 1, assumed to be well-posed, with $H_{1}$ replaced by $\rho^{-1}\circ H_{1}\circ\rho$ and $H_{2}$ replaced by $\rho\circ H_{2}\circ\rho^{-1},$ so that

[TABLE]

One passes from the original closed-loop to the new one by introducing multipliers $\rho\circ$ and $\rho^{-1}\circ.$

Theorem 1

Assume that $\gamma\left(\rho^{-1}\circ H_{1}\circ\rho\right)<+\infty$ and that there are constants $\delta_{1},\beta_{1}^{\prime},\varepsilon_{2},\beta_{2}^{\prime}$ such that

[TABLE]

for all $x\in\mathcal{S}_{e}^{n}$ and all $T\in\mathbb{T}$. If

[TABLE]

then $e_{1},e_{2},y_{1},y_{2}\in\mathcal{S}^{n}$ whenever $u_{1},u_{2}\in\mathcal{S}^{n}.$

Proof. For any $T\in\mathbb{T}$, we have that

[TABLE]

In addition,

[TABLE]

Therefore, setting $\gamma_{1}=\gamma\left(\rho^{-1}\circ H_{1}\circ\rho\right),$

[TABLE]

which implies

[TABLE]

This is the same equality as in ([5], section 6.5, (23)) (correcting an obvious misprint) and the result follows as in this reference.

Corollary 2

(Extended passivity theorem) Assume that $\gamma^{0}\left(\rho^{-1}\circ H_{1}\circ\rho\right)<+\infty$ and that there exists $\delta_{1}>0$ such that

[TABLE]

for all $x\in\mathcal{S}_{e}^{n}$ and all $T\in\mathbb{T}$.  (In this case, we will say that $H_{1}$ is $\rho^{-1}$-stable and is $\rho^{-1}$-passive, and that $H_{2}$ is $\rho$-passive.) Then the operator $\left(u_{1},u_{2}\right)\mapsto\left(e_{1},e_{2},y_{1},y_{2}\right)$ is $\mathcal{S}$-stable.

Proof. This follows from Theorem 1 by the same rationale as in the proof of Corollary 27 of ([5], Sect. 6.5).

The proof of the following is elementary:

Proposition 3

In the discrete-time case, let $H_{1}$ be the linear operator such that for any $x\in\mathcal{S}_{e}^{n}$

[TABLE]

Then

[TABLE]

In particular, if $H_{1}$ is LTI, then $\mathbf{h}_{1}\left(t,\tau\right)=h_{1}\left(t-\tau\right)$ (where $h_{1}$ is the impulse response), so that the transfer matrix of $\rho^{-1}\circ H_{1}\circ\rho$ is $z\mapsto\hat{h}_{1}\left(\rho z\right)$ where $z\mapsto\hat{h}_{1}\left(z\right)$ is the transfer matrix of $H_{1}.$

Remark 4

Likewise, in the continuous LTI case, the transfer matrix of $\rho^{-1}\circ H_{1}\circ\rho,$ with $\rho=e^{\alpha},$ is $s\mapsto\hat{h}_{1}\left(s+\alpha\right)$ where $s\mapsto\hat{h}_{1}\left(s\right)$ is the transfer matrix of $H_{1}.$

The proof of the following is easy:

Corollary 5

*Let $H_{1}$ be an LTI operator with rational transfer matrix $\hat{h}_{1}$. Then the conditions on $H_{1}$ in Corollary 2 are satisfied provided that:

  - In the discrete-time case, the transfer matrix $\hat{h}_{1}$ is analytic and bounded in $\left|z\right|>\rho,$ and for all $\theta\in\left[0,\pi\right]$*

[TABLE]

- In the continuous-time case, the transfer matrix $\hat{h}_{1}$ is analytic and bounded in $\Re\left(s\right)>\alpha,$ and for all $\omega\geq 0$

[TABLE]

2.2 Extended small gain theorem

Consider again the closed-loop system, assumed to be well-posed and defined by equations $\left(\ref{eq-connexion-1}\right),$ $\left(\ref{eq-connexion-2}\right).$

Theorem 6

*(i) Assume that $\gamma_{1}:=\gamma\left(\rho^{-1}\circ H_{1}\circ\rho\right)<+\infty,$ $\gamma_{2}:=\gamma\left(\rho\circ H_{2}\circ\rho^{-1}\right)<+\infty,$ and that $\gamma_{1}\gamma_{2}<1.$ Then, the operator $\left(u_{1},u_{2}\right)\mapsto\left(e_{1},e_{2}\right)$ has finite gain.

  (ii) Assume that $\gamma_{1}^{0}:=\gamma^{0}\left(\rho^{-1}\circ H_{1}\circ\rho\right)<+\infty,$ $\gamma_{2}^{0}:=\gamma^{0}\left(\rho\circ H_{2}\circ\rho^{-1}\right)<+\infty,$ and that $\gamma_{1}^{0}\gamma_{2}^{0}<1.$ Then, the operator $\left(u_{1},u_{2}\right)\mapsto\left(e_{1},e_{2}\right)$ is $\mathcal{S}$-stable.*

Proof. The proof is similar to that of ([5], section 3.2, Theorem 1).

Remark 7

(1) As in the usual case $\rho=1$, the extended small gain theorem and the extended passivity theorem are closely related.  Indeed, let $H:\mathcal{S}_{e}^{n}\rightarrow\mathcal{S}_{e}^{n}$ be such that $\left(I+H\right)^{-1}$ is a well-defined operator $\mathcal{S}_{e}^{n}\rightarrow\mathcal{S}_{e}^{n}.$ As easily seen, $\rho^{-1}\circ\left(I+H\right)\rho=\left(I+\rho^{-1}\circ H\circ\rho\right)^{-1},$ thus $\left(I+\rho^{-1}\circ H\circ\rho\right)^{-1}$ is well-defined.  In addition,

[TABLE]

*Therefore, putting $S:=\left(H-I\right)\left(I+H\right)^{-1},$

  (a) Condition $\left(\ref{cond-passivite}\right)$ is satisfied if and only if $\gamma^{0}\left(S\right)\leq 1.$

  (b) The following conditions (i), (ii) are equivalent: (i) there exists $\delta_{1}>0$ such that Condition $\left(\ref{cond-stricte-passivite}\right)$ is satisfied and $\gamma^{0}\left(\rho^{-1}\circ H\circ\rho\right)<+\infty;$ (ii) $\gamma^{0}\left(S\right)<1$ (see [5], section 6.10, lemma 7 for the details). Thus, one passes from Corollary 2 to statement (ii) of Theorem 6 via the usual loop transformation described in ([5], section 6.10).

  (2) A generalized version of the incremental small gain theorem ([5], section 3.3) can be obtained following the same line, and its statement is left to the reader.  The pattern of noncausal multiplier technique, as described in ([5], section 9.2), can also be extended in a similar way.*

3 Application to a parameter adaptation algorithm

We consider now the parameter adaptation algorithm (PAA) in [6].  The aim of the algorithm is to identify a discrete-time system with poles on or outside the unit circle.  The simulations in Section 4 of [6] show that this is indeed possible since the PAA is $\rho$-stable with $\rho>1.$  However, although the theorems of [6] are correct mathematically, they do not explain this result.  In the two theorems of [6], the condition that $H\left(z/\rho\right)-\frac{\lambda_{2}}{2}$ $\left(\rho>1\right)$ be strictly positive real is indeed more restrictive than the condition that $H\left(z\right)-\frac{\lambda_{2}}{2}$ be strictly positive real.  Thus, this condition must be replaced by: $H\left(\rho z\right)-\frac{\lambda_{2}}{2}$ is strictly positive real.  By Corollary 2 and Proposition 3 here above, with this change the identification algorithm of [6] converges (with degree of stability 1, not $\rho$).  This observation was the first motivatio of this paper.