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This article proposes a virtual sensor for piecewise linear systems based on observability analysis that is in function of a commutation law related with the system's outpu. This virtual sensor is also known as a state estimator. Besides, it presents a detector of active mode when the commutation sequences of each linear subsystem are arbitrary and unknown. For the previous, this article proposes a set of virtual estimators that discern the commutation paths of the system and allow estimating their output. In this work a methodology in order to test the observability for piecewise linear systems with discrete time is proposed. An academic example is presented to show the obtained results.

The automatization of industrial processes currently makes the application of based-model control schemes be more complex, resulting in the need to model this type of systems using piecewise linear systems. In order to solve this problem of complex systems modeling, nowadays new techniques have emerged in order to represent them like piecewise linear systems [

The study and analysis of piecewise linear systems have a strong impact for large-scale systems or systems which naturally exhibit continuous and discrete dynamical behaviors (

Complex systems treated as piecewise linear systems are approached from the design of robust filters for singular [

Discrete-time piecewise linear systems with unknown active mode require tools to solve the problems of observability and state estimation [

In order to explain the observability property with known and unknown modes for piecewise linear systems, there are some works that establish the conditions to prove observability, which address the observability problem of discrete-time and continuous-time piecewise linear systems. In [

Once the methodology to prove observability in discrete-time piecewise linear systems is established, in this work we propose the design of a piecewise linear observer assuming that initially the system is fully observable under arbitrary commutation, which governs the system dynamics at all time. The advantage of a proposed observer is its capacity to work with unknown system modes and under arbitrary commutation that govern all system dynamics when this is an output function of a discrete-time piecewise linear system. However, other articles only give evidence of state estimation by assuming that the system modes are known and periodically switched [

The search and the bibliographical review in the piecewise linear system context, particularly in schemes of analysis and observation of observability, sets the guidelines to propose new observation schemes when the modes of the system are unknown, commuted with the commutation law in function of the output and time, in a piecewise linear system on a discrete time. In this context, this paper proposes a methodology to probe the observability and a new approach to estimate the states of a piecewise linear system under the conditions of unknown commutation modes depending on the system's output.

According the structure of a piecewise linear system in a discrete time:
_{k}^{n}_{k}^{m}_{k}^{p}_{k}_{k}

In order to define the piecewise linear system's active mode in a discrete time _{k}_{k}_{i}_{i}_{i}_{1}, _{2, …}_{k}_{0} = 0 and the discrete state of the system as _{k}_{i}_{i}_{i}_{i}_{+1} with

To delimit the reach of the investigation, we will work only with autonomous piecewise linear systems, assuming that the time of permanence of each one of the system's modes in active mode is enough to guarantee the mode's observability and that there is no separation time between the transfer of a mode to another, that is to say, it does not represents discontinuousness in the evolution path. The structure of the autonomous piecewise linear system on a discrete time is the following:
_{k}^{n}_{k}^{p}_{k}

In order to define the observability of piecewise linear systems in a discrete time, a finite number of output measurements of the system affected by the commutation law is required. This commutation law allows one to establish commutation sequences for the system's modes, known or unknown, periodical or arbitrary, for the studied case it focuses on commutation arbitrary and unknown sequences. From that, some definitions of observability of linear systems [

A linear system is observable in a time _{0} if its vector of state on that time _{0}, can be determined from the output function _{“}_{t}_{0}, _{t}_{1”} (or output consequence), where _{0} ≤ _{1} in some finite time. If this is accomplished for everything _{0}, the system is completely observable.

Piecewise observability. The set of pairs (

The previous definitions allow one to extend the analysis of observability to piecewise linear systems with an arbitrary and unknown commutation sequence, which eases establishment of the next proposition:

In time invariant linear systems the estimation of states exists if and only if the observability matrix fulfills the complete rank condition, however, for piecewise linear systems this does not apply, because of their different nature, that is to say, it has two inputs: commutation law and control input.

The proposition 1 extended to piecewise linear system guarantees that the observability property is a necessary condition for the system to be observable, the difficulty with this, is that the active mode in a finite time is unknown. In order to resolve this complication it is required that the modes of the system be discerned at all times, so if there are discerned modes the definition 2 can be applied in order to prove the observability in piecewise linear system on a discrete time with a commutation sequence of unknown modes. The use of the definition 2 in the context of piecewise linear systems allows us to establish the next proposition when the system's modes are unknown.

In piecewise linear systems invariable in time the estimation of states is done if the matrix of observability fulfills the complete rank condition.

In order to prove the affirmation of the preposition 2 to be true, we need to prove that we count with a sufficient number of measurements of the active mode to guarantee discernibility of mode and observability of state. For that it is assumed that all the system's modes can be known or unknown, depending the type, conditions and system's inputs. In these affirmations, the analysis of observability s presented for the next defined cases based on the amount of measurements of each one of the active modes in a time _{k}_{1} = _{2} = … = _{s}_{1} ≠ _{2} = … ≠ _{s}_{i}_{j}_{i}_{j}

The three previous cases establish that not all the times of permanence of the active mode will be equal. However, it is assumed that you have a time of permanence sufficient to guarantee the state observability. Besides, it is established that the

The dynamic evolution of the subsystem linear group guided by a commutation signal divides the dynamic path of the system in different longitudes, defined by the group of commutation times {_{1}, _{2}, …, _{k}_{1}, for the second mode beginning in _{1} it is denoted by _{2} and so on until _{s}

Every permanence time of the linear subsystem in active mode allows having a historical view of the evolution of output of the piecewise linear system from the _{=1} until an instant _{s}_{1} represented as:

From

Finally, the output vector is defined in the following way:

The results of the previous analysis give as product the construction of the observability matrix and the understanding of how the commutation law affects the system's dynamic behavior. From that analytic procedure, the next lemma is proposed:

If the observability matrix of piecewise linear system defined in

To prove the lemma 1, it is required that assumption 1 be fulfilled, which is:

If enough measurements are at hand in order to guarantee the observability and discernibility of the mode, so the piecewise linear system is fulfilled so it is observable and mode discernible.

By accomplishing the assumption 1 the observability and discernibility of mode can be guaranteed, so the sequence of mode gets determined in any instant of time

If a system is piecewise observable with index _{s}_{s}

The design of estimators in piecewise linear systems in a discrete time represents a complex challenge, due in the first place to the fact you must guarantee that the system be completely observable, it is to say, that lemma 1 seen in the previous section is fulfilled. In this context a piecewise linear system taking the structure of

If the piecewise linear system is discernible of mode, and observable in states, so an arbitrary and unknown observer can be designed, so it can estimate the state system. To prove hypothesis 1, it is assumed that the piecewise linear system is discernible in mode and observable in state from a finite amount of measurements in the active mode. This is _{k}_{i}, B_{i}, C_{i}_{k}_{k}_{k}^{n}_{k}^{m}_{k}^{p}_{k}

In order to detect the system's active mode a comparison between the real systems' output minus the output of every one of the linear subsystem is realized on _{k}

If the outputs are available on each step, it is possible to design an observer on piecewise linear systems in order to estimate a _{k}_{k}_{k}_{→ ∞} ‖_{k}_{k}

To prove the hypothesis 2 proves that the operation mode represented by the linear mode is observable, for that the use of the methodology proposed in the

Based on the analysis of observability presented in the previous section and the detector of active mode of the _{k}_{k}_{k}_{k}

The objective of the piecewise linear observer that is to be build is to reconstruct the estimate state _{k}_{k}_{k}_{k}_{1}, …,_{k}_{1}, …, _{k}_{k}_{→ ∞} ‖_{k}_{k}

In the

The piecewise linear observer converges if lim_{k}_{→ ∞} ‖_{k}_{k}

Based on Definition 3 we assume that we have _{k}_{→ ∞} ‖_{k}_{k}

From that structure of a piecewise linear observer of _{k}_{k}

In order to prove the theoretical results obtained a numerical example must be presented. For that, let us consider the next piecewise linear system as described in _{i}_{i}_{i}

The set of linear models of the

Given an initial time _{0}, a signal of commutation _{0}, _{1}) with _{0} < _{1} < ∞, _{θ}_{0}, _{1}) ↦

The validation in simulation of the results with parameters of sequence design of mode unknown reign by the commutation law in function of the output and the time is done first with the test of observability of the linear system by pieces based on

To build the observability matrix of piecewise linear system of

Once

The observer gains are calculated by the pole placement technique, in order to improve the performance and response of the system in comparison with other sophisticated techniques. In order to show the effectiveness of methodology proposed, this is compared with the results in [

The _{e}_{r}_{e}_{θ̂k}_{θ̂k}

The _{e}_{r}_{e}

To concatenate each one of the paths of the linear subsystems and implement a piecewise linear observer, it was necessary to estimate the active mode, by using the

The proposed technique can be upgraded, for example, to propose schemes of estimation of the more sophisticated modes and assume that the set of all estimated states are enclosed by the commutation law. This will imply a better refinement to the estimation technique in the evolution of states before the change of operation mode, and also a better convergence of the piecewise linear observer.

To quantitatively measure the precision of the proposed approach the mean squared error (MSE) is used here. MSE of an estimator is one of many ways to quantify the difference between values implied by an estimator and the true values of the quantity being estimated. This MSE measures the average of the squares of the “errors”. The error is the amount by which the value implied by the estimator differs from the quantity to be estimated. The results are exposed in

The output error of different approaches is approximately to zero. This means that the proposed approach provides satisfactory results, although the observer works with unknown and arbitrary switching sequences.

This article presented the analysis to prove the observability in a piecewise linear system in a discrete time represented systems ruled by an arbitrary and unknown commutation sequence. Besides, a proposal of a virtual estimator known as piecewise linear observer was presented, that solved the problem with commutation by integrating it an active mode detector. These types of systems are very helpful in the design of new control schemes, or detection of faults based on observers, among others. The obtained results in the estimation of states and the output of the system still can be upgraded by implementing a new detector more sophisticated of mode or by proposing another more elaborate estimator structure. The results encountered in this work can be extended to the design of diagnosis and fault-detection schemes based on observers, where this new design can be implemented in a fault-tolerant control for a piecewise linear system.

The authors of this paper thank CONACYT and CENIDET for the support and aid given for the termination of this investigation.

Methodology to prove the observability in piecewise linear systems.

Virtual estimator for piecewise linear systems.

Output of the observer in piecewise a linear system, unknown and arbitrary commutation sequence.

Output of the observer in piecewise a linear system, known and arbitrary commutation sequence.

Commutation signal: estimated

Output error of piecewise linear system.

Active mode of known system | 0.0069 |

Active mode of unknown system | 0.0966 |