# A Mechanical Feedback Classification of Linear Mechanical Control Systems

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## Abstract

**:**

## 1. Introduction

- (a)
- Given two linear mechanical systems, how to determine whether they are equivalent?
- (b)
- Is there a set of complete invariants that are computable in terms of objects on the configuration space only?
- (c)
- Is there a distinguished normal (or canonical) form?

## 2. Problem Statement

#### 2.1. Linear Mechanical Control Systems

**Proposition**

**1.**

**Remark**

**1.**

#### 2.2. Linear Mechanical Feedback Transformations

**Definition**

**1.**

## 3. Classification of Controllable Systems (2)

**Lemma**

**1**

- (i)
- $\left(\mathcal{LMS}\right)$ is controllable
- (ii)
- $\mathrm{rank}\phantom{\rule{4pt}{0ex}}\left(\right)open="("\; close=")">\widehat{B},\widehat{A}\widehat{B},\dots ,{\widehat{A}}^{2n-1}\widehat{B}$ (Kalman Rank Condition)
- (iii)
- $\mathrm{rank}\phantom{\rule{4pt}{0ex}}\left(\right)open="("\; close=")">\widehat{B},{\widehat{A}}^{2}\widehat{B},\dots ,{\widehat{A}}^{2(n-1)}\widehat{B}$
- (iv)
- $\mathrm{rank}\phantom{\rule{4pt}{0ex}}\left(\right)open="("\; close=")">B,EB,\dots ,{E}^{n-1}B$ (Mechanical Kalman Rank Condition)

**Proof.**

**Remark**

**2.**

**Proposition**

**2.**

- (i)
- the sequence of indices $\mathcal{R}\left(\right)open="("\; close=")">\widehat{A},\widehat{B}$ is the doubled sequence of $\overline{\mathcal{R}}\left(\right)open="("\; close=")">E,B$, i.e., $\left(\right)open="("\; close=")">{r}_{0},{r}_{1},\dots ,{r}_{2n-1}$;
- (ii)
- the mechanical half-indices are half of the controllability indices, i.e., ${\rho}_{j}=2{\overline{\rho}}_{j}$, for $1\le j\le m$.

**Proof.**

**Theorem**

**1.**

- (i)
- Two controllable systems (2) and $\left(\tilde{\mathcal{LMS}}\right)$, represented by pairs $(E,B)$ and $(\tilde{E},\tilde{B})$, respectively, are LMF-equivalent,
- (ii)
- $\overline{\mathcal{R}}(E,B)=\overline{\mathcal{R}}(\tilde{E},\tilde{B})$,
- (iii)
- $\overline{\mathcal{P}}(E,B)=\overline{\mathcal{P}}(\tilde{E},\tilde{B})$, i.e., the mechanical half-indices coincide,
- (iv)
- $\mathcal{P}(\widehat{A},\widehat{B})=\mathcal{P}(\widehat{\tilde{A}},\widehat{\tilde{B}})$, i.e., the controllability indices coincide,

**Proof.**

**Remark**

**3.**

**Corollary**

**1.**

## 4. Classification of Uncontrollable Systems (2)

**Theorem**

**2.**

- (i)
- for a real eigenvalue ${\lambda}_{j}$ of ${E}_{d}^{J}$$$\begin{array}{c}\hfill \phantom{\rule{4.pt}{0ex}}\mathrm{either}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\begin{array}{c}\hfill {D}_{j}^{\mathbb{R}}=\left(\begin{array}{ccccc}{\lambda}_{j}& \phantom{\rule{1.em}{0ex}}& & \phantom{\rule{1.em}{0ex}}& \\ \phantom{\rule{1.em}{0ex}}& \ddots & \phantom{\rule{1.em}{0ex}}& \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& {\lambda}_{j}& \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& & {\lambda}_{j}\end{array}\right)\end{array}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}\begin{array}{c}\hfill {J}_{j}^{\mathbb{R}}=\left(\begin{array}{ccccc}{\lambda}_{j}& 1& \phantom{\rule{1.em}{0ex}}& & \\ \phantom{\rule{1.em}{0ex}}& \ddots & \ddots & \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& {\lambda}_{j}& 1\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& & {\lambda}_{j}\end{array}\right),\end{array}\end{array}$$
- (ii)
- for a complex eigenvalue ${\lambda}_{j}={\alpha}_{j}\pm \mathtt{i}{\beta}_{j}$ of ${E}_{d}^{J}$$$\begin{array}{c}\hfill \phantom{\rule{4.pt}{0ex}}\mathrm{either}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\begin{array}{c}\hfill {D}_{j}^{\mathbb{C}}=\left(\begin{array}{ccccc}{C}_{j}& \phantom{\rule{1.em}{0ex}}& & \phantom{\rule{1.em}{0ex}}& \\ \phantom{\rule{1.em}{0ex}}& \ddots & \phantom{\rule{1.em}{0ex}}& \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& {C}_{j}& \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& & {C}_{j}\end{array}\right)\end{array}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}\begin{array}{c}\hfill {J}_{j}^{\mathbb{C}}=\left(\begin{array}{ccccc}{C}_{j}& {I}_{2}& \phantom{\rule{1.em}{0ex}}& & \\ \phantom{\rule{1.em}{0ex}}& \ddots & \ddots & \\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& {C}_{j}& {I}_{2}\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& & {C}_{j}\end{array}\right),\end{array}\end{array}$$

- (i)
- (ii)
- a dynamical linear mechanical system (a system without controls) represented in $({x}_{d},{y}_{d})$-coordinates by the second-order differential equation $\ddot{x}={E}_{d}{x}_{d}$, where the matrix ${E}_{d}$ can always be transformed into the Jordan form ${E}_{d}^{J}$; the subindex “d” stands for dynamical.

**Proof.**

**Proposition**

**3.**

**Proof.**

**Example**

**1.**

## 5. Classification of Lagrangian Systems (6)

**Proposition**

**4.**

**Proposition**

**5.**

- (i)
- Any Lagrangian mechanical system (6) is LMF-equivalent to the following Lagrangian system:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{x}}_{c}& ={y}_{c}\hfill \\ \hfill {\dot{y}}_{c}& ={E}_{F}^{L}{x}_{c}+{B}_{F}u\hfill \end{array}\phantom{\rule{1.em}{0ex}}\begin{array}{cc}\hfill {\dot{x}}_{d}& ={y}_{d}\hfill \\ \hfill {\dot{y}}_{d}& ={E}_{d}^{L}{x}_{d},\hfill \end{array}\end{array}$$
- (ii)
- Any two Lagrangian systems of the form (19), with the same ${E}_{d}^{L}$ but arbitrary terms ${a}_{j}^{i}$ and ${\tilde{a}}_{j}^{i}$ (such that, for each block $1\le i\le m$, the eigenvalues ${\lambda}_{j}^{i}$ of ${L}_{i}$ are mutually distinct and so are ${\tilde{\lambda}}_{j}^{i}$ of ${\tilde{L}}_{i}$), are LMF-equivalent.

**Proof.**

**Proposition**

**6.**

## 6. Stability and Stabilization

**Lemma**

**2.**

**Proof.**

$\mathbf{\lambda}\left(\mathbf{E}\right)$ | $\mathbf{\sigma}\left(\widehat{\mathbf{A}}\right)$ | Sketch |

${\mathbb{R}}_{+}$ | $\pm a$ | |

${\mathbb{R}}_{\le 0}$ | $\pm a\mathtt{i}$ | |

$\mathbb{C}$ | $\pm (a+b\mathtt{i})$ and $\pm (a-b\mathtt{i})$ |

**Proposition**

**7.**

- (AS)
- The system $\dot{z}=\widehat{A}z$ is never asymptotically stable.
- (S)
- The following conditions are equivalent:
- (i)
- the system $\dot{z}=\widehat{A}z$ is stable,
- (ii)
- all eigenvalues ${\lambda}_{j}$ of E satisfy ${\lambda}_{j}\in {\mathbb{R}}_{\le 0}$ and, moreover, their algebraic and geometric multiplicities coincide, i.e., ${\mu}_{j}={\gamma}_{j}$, for $1\le j\le q$, where q is the number of distinct eigenvalues of E
_{d}, - (iii)
- E has a Lagrangian structure, i.e., $E={M}^{-1}P$ for some symmetric matrices M (invertible) and P and the eigenvalues of E satisfy ${\lambda}_{j}\in {\mathbb{R}}_{\le 0}$.

**Proof.**

**Theorem**

**3.**

- (AS)
- The system (2) is never asymptotically stabilizable.
- (S)
- The following conditions are equivalent:
- (i)
- (2) is stabilizable,
- (ii)
- the matrix ${E}_{d}$ has all eigenvalues ${\lambda}_{j}\in {\mathbb{R}}_{\le 0}$ and, moreover, is diagonalizable, i.e., its Jordan form ${E}_{d}^{J}$ consists of diagonal blocks ${D}_{j}^{\mathbb{R}}$ only, corresponding to ${\lambda}_{j}\in {\mathbb{R}}_{\le 0}$,
- (iii)
- all eigenvalues ${\lambda}_{j}$ of ${E}_{d}$ satisfy ${\lambda}_{j}\in {\mathbb{R}}_{\le 0}$ and, moreover, their algebraic and geometric multiplicities coincide, i.e., ${\mu}_{j}={\gamma}_{j}$, for $1\le j\le q$, where q is the number of distinct eigenvalues of ${E}_{d}$,
- (iv)
- the uncontrolled subsystem ${\dot{x}}_{d}={y}_{d}$, ${\dot{y}}_{d}={E}_{d}{x}_{d}$ is T-equivalent to a Lagrangian system,
- (v)

**Proof.**

**Corollary**

**2.**

**Proposition**

**8.**

**Proof.**

## 7. Examples

**Example**

**2**

**Example**

**3**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Nowicki, M.; Respondek, W.
A Mechanical Feedback Classification of Linear Mechanical Control Systems. *Appl. Sci.* **2021**, *11*, 10669.
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**AMA Style**

Nowicki M, Respondek W.
A Mechanical Feedback Classification of Linear Mechanical Control Systems. *Applied Sciences*. 2021; 11(22):10669.
https://doi.org/10.3390/app112210669

**Chicago/Turabian Style**

Nowicki, Marcin, and Witold Respondek.
2021. "A Mechanical Feedback Classification of Linear Mechanical Control Systems" *Applied Sciences* 11, no. 22: 10669.
https://doi.org/10.3390/app112210669