# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 1 Introduction  
We consider the observer design problem for a dynamic linear time-invariant system (LTIS) of a given differentiation order (DO). Assuming the knowledge at each instant of time of the pair of functions \(u,y:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}\), corresponding to the system input and output, respectively, the problem consists on finding an estimation of its internal variables (state or pseudo state) and of its parameters. To solve this problem, we propose a mixed order approach using minimal realizations, meaning that the observer and the system are defined in terms of the minimum number of parameters while the derivation orders used in the observer are arbitrarily chosen and possibly different from those of the system.  
Non-mixed order observers (NMOO) have been proposed using Luenberger observers for integer order systems (IOS) and fractional order systems (FOS) [1; 2; 3; 4; 5; 6]. Our contributions are summarized and commented in what follows.  
* 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 ofparameters to describe the system, chosen to simplify the design. It yields a different structure from Luenberger observer.
* In Sections 3 and 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 Sections 3 and 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 Sections 3-5 and a control application is developed in Section 6.  
An alternative approach to robust non-mixed order observer (NMOO) is found in [21], where prior information on the uncertainty is required. In contrast, here only upper bounds are needed. Our observers are asymptotic; a nonasymptotic NMOO has been proposed in [12] for linear Riemann-Liouville systems and a specific class of uncertainties.