# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems

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## Abstract

**:**

## 1. Introduction

- In Section 2 we provide a mathematical framework for designing observer for LTIS, which includes systems defined by any generic fractional order derivative (GFOD) e.g., Caputo, Riemann–Liouville, etc. This is a subtle issue given the variety of fractional order derivatives (FOD) existent [7] and the fact that NMOO have been designed for specific type of derivatives [6,8,9,10,11,12]. In this framework, the concept of initial conditions (IC) is unambiguously defined and linear properties (superposition and separation) are easily obtained. Moreover, a minimal realization (MR) structure is chosen in the sense that the dimension of the internal variables is minimal and it is used the minimum number of parameters to describe the system, chosen to simplify the design. It yields a different structure from Luenberger observer.
- In Section 3 and Section 4 convergence and robustness conditions are provided for non-adaptive and adaptive minimal realization observers (MRO) for commensurate systems. The conditions obtained meet those of the integer order observers (IOO) [4] when particularized to them, and constitutes a contribution in the non-mixed fractional order observers (FOO) literature for the adaptive case (cf. [8,9,13,14]), the latter being especially challenging, since currently there is not a fractional Lasalle theorem [15] available. The importance of this contribution is supported by the capability of fractional order systems (FOS) to model complex phenomena [10,16,17], whereby observers are needed since the internal variables are usually not accessible.
- In Section 3 and Section 4 robust convergence conditions for adaptive and non-adaptive mixed-order observers (MOO) are stated, allowing in particular to design FOO for integer order systems (IOS) under the same assumptions than those of IOO. The difficulty is now to prove convergence and robustness for a system which is composed of subsystems with different DO. This novel idea makes possible (a) to dispose fractional calculus and their capabilities to model long memory effects [18] to observe real processes approximated by integer order (IO) linear systems, and (b) to have extra degrees of freedom (the derivation orders) to optimize criteria such as transient behavior, robustness, disturbance rejection, etc., which are relevant in current observer designs [19,20]. These advantages and some generalizations are qualitatively discussed in Section 3, Section 4 and Section 5 and a control application is developed in Section 6.

## 2. Linear Systems Preliminaries

**P1:**The superposition of responses to a linear combination of inputs holds when the IC are null.

## 3. Non-Adaptive Mixed Order Observer (NAMOO)

**Theorem**

**1.**

**(a)**- If $u,y$ have additive bounded uncertainties, i.e., $u\left(t\right)+{\Delta}_{u}\left(t\right)$ and $y\left(t\right)+{\Delta}_{y}\left(t\right)$, the estimates of internal variables and the output error remain bounded, having also high frequency rejection.
**(b)**- If parameter vector p (i.e., the elements of A and b) is in fact $p+{\Delta}_{p}$, where parameter disturbances n${\Delta}_{p},u$ and y are bounded functions, then $x-\widehat{x}$ and $y-\widehat{y}$ remain bounded.

**Proof.**

**Remark**

**1.**

- (i)
- A benefit of the MO approach to design observers is that the transient behavior of the estimate $\widehat{x}$ can be regulated by the choice of the FO β. A NMFOO where $\beta =\alpha $—like the Luenberger observer (11)—can only regulate the speed of convergence through the choice of F.
- (ii)
- Unlike Luenberger observer (11), there is no need of an output error feedback term $F\left(\widehat{y}-y\right)$. Since this term is essential in the stability proof of the Luenberger observer, the introduction of this term in our designs can give some freedom to further improvements.

## 4. Adaptive Mixed Order Observer (AMOO)

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

- (i)
- All the variables of the system remain bounded, $\left(\widehat{y}-y\right)$ has bounded γ-integral and its $RMS$ value converges to zero. These claims remain true if a bounded additive perturbation of quadratic bounded $\gamma -$integral is added on the input and/or output of the system (6).
- (ii)
- If the spectral measure of u is not concentrated on $k<2n$ points, then $\widehat{p}$ converges to the true parameters of the plant p and $\widehat{x}$ converges to x as $t\to \infty $.

**Proof.**

- (i)
- From (9) to (16), it follows that$$\begin{array}{c}\hfill \widehat{x}\left(t\right)-x\left(t\right)=\xi \left(t;{\widehat{\psi}}_{0},F,\beta \right)-\xi \left(t;{\psi}_{0},F,\alpha \right)-\left[{\eta}_{1}\dots {\eta}_{2n}\right]\left(t\right)\left(\widehat{p}\left(t\right)-p\right).\end{array}$$Hence,$$\begin{array}{cc}\hfill {c}^{T}\left(\widehat{x}\left(t\right)-x\left(t\right)\right)& ={c}^{T}\left(\xi \left(t;{\widehat{\psi}}_{0},F,\beta \right)-\xi \left(t;{\psi}_{0},F,\alpha \right)\right)+{c}^{T}\left[{\eta}_{1}\dots {\eta}_{2n}\right]\left(t\right)\left(\widehat{p}\left(t\right)-p\right)\hfill \\ \hfill e\left(t\right)& :=\widehat{y}\left(t\right)-y\left(t\right)=\nu \left(t\right)+{W}^{T}\left(t\right)\tilde{p}\left(t\right)\hfill \end{array}$$Without loss of generality, we assume $\kappa =1$ (otherwise, we always can redefine W as $\kappa W$). Let $V:={\tilde{p}}^{T}\tilde{p}+C-1/2{I}^{\gamma}{\nu}^{2}$. Note that V is non-negative, since $\nu $ has bounded quadratic $\gamma -$integral. By ([25], Theorem 3) and continuity of the solutions, $\left[{\phantom{\rule{0.222222em}{0ex}}}_{{t}_{0}}{D}^{\gamma}{\tilde{p}}^{T}\tilde{p}\right]\left(t\right)\le 2{\tilde{p}}^{T}\left(t\right){\phantom{\rule{0.222222em}{0ex}}}_{{t}_{0}}{D}^{\gamma}\tilde{p}\left(t\right)$ [26,27]. Using that ${\phantom{\rule{0.222222em}{0ex}}}_{{t}_{0}}{D}^{\gamma}\tilde{p}={\phantom{\rule{0.222222em}{0ex}}}_{{t}_{0}}{D}^{\gamma}\widehat{p}$, since p is constant, together with (17) and (19), we obtain$$\begin{array}{cc}\hfill {D}^{\gamma}V& \le -2{\left({\tilde{p}}^{T}W\right)}^{2}-2{\tilde{p}}^{T}W\nu -1/2{\nu}^{2}\hfill \\ & =-2{\left({\tilde{p}}^{T}W+1/2\nu \right)}^{2}\le 0.\hfill \end{array}$$By $\gamma -$integration of (20), we obtain $V\left(t\right)\le V\left(0\right)$. That is, ${\tilde{p}}^{T}\tilde{p}+C-1/2{I}^{\gamma}{\nu}^{2}$ is bounded. Since $C>1/2{I}^{\gamma}$, it follows that $\widehat{p}$ is bounded. Using the BIBO stability property of system (8) due to the choice of F, it follows that W is bounded whenever $y,u$ are bounded. Then, e is bounded. By $\gamma -$integration of (20), we also conclude that ${e}^{2}={\left({\tilde{p}}^{T}W+1/2\nu \right)}^{2}$ has bounded $\gamma $-integral. Thus, ${e}_{RMS}$ converges to zero ([28], Proposition 1).Using expression (15) and redefining $\nu $, we have also proved that if a bounded perturbation $\Delta $ with bounded quadratic $\gamma -$fractional integral, is added to the model input and/or output, all the above claims remain true.
- (ii)
- If the spectral measure of u is not concentrated on $k<2n$ points, $W={\eta}_{1}$ is a persistently exciting (PE) function ([28], Property 11). Using this and the fact that $\nu $ converges to zero, it follows that $\tilde{p}$ converges to zero ([24], Section 3) i.e., $\widehat{p}$ converges to p. Since $u,y$ are bounded, ${\eta}_{i}$ is bounded for $i=1,\dots ,2n$. Therefore, using (18), $x-\widehat{x}$ converges to zero.

**Remark**

**2.**

## 5. Discussion

#### 5.1. Non-Conmensurate Systems

#### 5.2. Adaptive Observer for Unstable Plants or Non-Caputo Systems

#### 5.3. A Non-Local Observer for a Local System?

#### 5.4. Unknown Initial Time

#### 5.5. Non-Linear Systems

#### 5.6. Other Assumptions

## 6. Application

#### Simulation Example

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Evolution of the control error for a step reference of magnitude 5, using different values of the FO $\beta $ for the observer.

**Figure 2.**Evolution of the norm of the estimation error for a step reference of magnitude 5, using different valued of the FO $\beta $ for the observer.

**Figure 3.**Evolution of the control error for a step reference of magnitude 5 and an external disturbance of 10 % magnitude added to the plant output, using different values of the FO $\beta $ for the observer.

**Figure 4.**Evolution of the norm of the estimation error for a step reference of magnitude 5 and an external disturbance of 10 % magnitude added to the plant output, using different values for the FO $\beta $ for the observer.

**Figure 5.**Evolution of the control error for a step reference of magnitude 5 and white noise present in the plant output, using different values of the FO $\beta $ for the observer.

**Figure 6.**Evolution of the norm of the estimation error for a step reference of magnitude 5 and white noise present in the plant output, using different values of the FO $\beta $ for the observer.

β = 0.7 | β = 0.9 | β = 1 | β = 1.1 | β = 1.3 | |
---|---|---|---|---|---|

$IS{E}_{c}$ | $1.74\times {10}^{3}$ | $1.57\times {10}^{3}$ | $1.61\times {10}^{3}$ | $1.73\times {10}^{3}$ | $2.76\times {10}^{3}$ |

$IS{E}_{e}$ | $10.66$ | $3.1693$ | $3.36\times {10}^{-3}$ | $3.20$ | $9.91$ |

β = 0.7 | β = 0.9 | β = 1 | β = 1.1 | β = 1.3 | |
---|---|---|---|---|---|

$IS{E}_{c}$ | $3.27\times {10}^{3}$ | $2.58\times {10}^{3}$ | $2.31\times {10}^{3}$ | $2.26\times {10}^{3}$ | $3.02\times {10}^{3}$ |

$IS{E}_{e}$ | $55.82$ | $48.40$ | $45.61$ | $44.46$ | $48.46$ |

β = 0.7 | β = 0.9 | β = 1 | β = 1.1 | β = 1.3 | |
---|---|---|---|---|---|

$IS{E}_{c}$ | $3.69\times {10}^{3}$ | $3.40\times {10}^{3}$ | $3.38\times {10}^{3}$ | $3.47\times {10}^{3}$ | $4.51\times {10}^{3}$ |

$IS{E}_{e}$ | $64.55$ | $62.63$ | $61.84$ | $61.93$ | $67.07$ |

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**MDPI and ACS Style**

Duarte-Mermoud, M.A.; Gallegos, J.A.; Aguila-Camacho, N.; Castro-Linares, R. Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems. *Algorithms* **2018**, *11*, 136.
https://doi.org/10.3390/a11090136

**AMA Style**

Duarte-Mermoud MA, Gallegos JA, Aguila-Camacho N, Castro-Linares R. Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems. *Algorithms*. 2018; 11(9):136.
https://doi.org/10.3390/a11090136

**Chicago/Turabian Style**

Duarte-Mermoud, Manuel A., Javier A. Gallegos, Norelys Aguila-Camacho, and Rafael Castro-Linares. 2018. "Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems" *Algorithms* 11, no. 9: 136.
https://doi.org/10.3390/a11090136