# Toward Avoiding Unmeasured Premise Signals in TS Discrete Time Observer Design: State Extension With Nonlinear Output Injection

Dalil Ichalal, Thierry-Marie Guerra

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Dalil Ichalal, Thierry-Marie Guerra. Toward Avoiding Unmeasured Premise Signals in TS Discrete Time Observer Design: State Extension With Nonlinear Output Injection. 20th IEEE International Conference on Networking, Sensing and Control (ICNSC 2023), Oct 2023, Marseille, France. pp.1-6, 10.1109/ICNSC58704.2023.10318970 . hal-04362576

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# Toward Avoiding Unmeasured Premise Signals in TS Discrete Time Observer Design: State Extension With Nonlinear Output Injection

Dalil Ichalal

IBISC-Lab

Univ. Evry, Paris-Saclay University,
40, rue de Pelvoux, 91020 Evry Courcouronnes, France

dalil.ichalal@univ-evry.fr

Thierry-Marie Guerra

Université Polytechnique Haut de France (UPHF)

CNRS, UMR 8201 - LAMIH

F-59313, Valenciennes, France

thierry.guerra@uphf.fr

***Abstract***—This paper addresses an enhancement of the recently proposed algorithm to overcome the unmeasured premise variables in discrete-time Takagi-Sugeno (TS) systems for observers design [8]. The approach is based on immersion and state extension for tow main objectives: the first one is that the new state vector of the target system after immersion should contain all the state vector of the original system in order to avoid the inverse transformation. Secondly, express the nonlinear system in an adequate form for its transformation in TS form without any unmeasured premise variable. This state extension is performed by introducing simple functions depending on measured variables and their delayed measures. Examples are presented to illustrate the proposed approach.

## I. INTRODUCTION

Linear dynamical systems have been studied since a long time and even in the present and motivate huge research topics. In the field of observation and observers design, the problem is solved by Kalman [11] and Luenberger [16] who proposed interesting results in the state space framework. Since a state-space model is not unique, important researches have been conducted in order to characterize the canonical observable form (commonly called Brunovsky observable form). This form is interesting in the sense that the state space system is expressed with a state matrix having a special structure describing a chain of integrators and the coefficients of these matrices are nothing than the coefficients of the characteristic equation of the state matrix. This considerably simplifies the computation of gains of the linear observers since it avoids the computation of the characteristic equation by considering only the coefficients of the state space matrix and then construct the characteristic equation. From the development of the Linear Matrix Inequality framework [3], computing the gains of an observer is performed by solving a set of LMIs without transforming, necessarily, the system in observable canonical form. LMI framework tends to become a standard in linear control theory. Now, let us go to the nonlinear systems. This topic is more complex than its linear counterpart due to the loss of the linearity property. From Poincaré to now, the study of nonlinear dynamical systems goes through the transformation of the system in a linear one

via a set of coordinate transformations. These lasts aim to transform the original system into an equivalent one (at least locally) which has a linear structure in canonical observable form with nonlinear output injections. This reasoning aims to express the state estimation error dynamics in a linear form which allow to use the rich tools of the linear framework. A lot of researches have been reported in that sense and the state transformation (dipheomorphism) is introduced in [12]. Necessary and sufficient conditions for the existence of such coordinate transformations have been established. Thereafter, for the systems that are not transformable in a linearizable error, dynamics transformations have been proposed in [15] and generalized in the immersion theory that aims to immerse a nonlinear system into a linear one with higher dimension. The state transformation and immersion approaches transform the nonlinear system into a linear canonical observable form.

In contrary to the coordinate transformation approaches, several techniques have been proposed aiming to design observers in the original coordinates without transformations. One can cite the Lipschitz approaches, Polytopic approaches including LPV, TS and switched systems. These techniques are interesting in the sense that thy do not need to state transformation and exploit the LMI formalism which ease the design of observers. However, the LMIs provide, generally, only sufficient but not necessary conditions for ensuring the state estimation convergence. Recently, Sufficient LMI conditions have been proposed for polytopic systems (LPV and TS) in both continuous and discrete time cases by using the Polya's theorem and the window time variation of the Lyapunov function [17], [13]. The obtained LMI conditions are sufficient and asymptotic necessary. However, these approaches assume that the premise variables (or parameters in LPV systems) are measured. In the case of unmeasured premise variables, the LMIs are only sufficient and may be conservative due to the unknown premise variables.

From the work of Takagi-Sugeno in [18] provided an alternative method to deal with nonlinear systems. The simple mathematical formulation of the TS structure allows to capture the nonlinear terms of the original model in a polytopic
way [19]. In addition, it avoids the search of coordinate transformations. Some tools and theories (Lyapunov theory, LMIs, control and observation) developed for linear systems can be extended straightforwardly for this class of nonlinear systems thanks of the convex sum property.

One of the ways to obtain a TS model from a nonlinear one is the use of the sector nonlinear transformation. This approach may provide either a TS model with premise variables depending on measured signals or TS models with premise variables depending on partly or completely unmeasured states of the system. The first class of TS models is largely studied in the literature [1], [14], [9] with the Lipschitz property. Since this last introduce some conservatism in the design of the observer, other solutions have been proposed in [7], [6], by using the differential mean value theorem. Always in the purpose of reducing the conservatism, a new approach exploiting the Input-to-State Stability (ISS) Property is proposed in [10] and [20].

For discrete-time TS systems with premise variables depending on the state of the system, only few results can be found in the literature which motivates the present work.

The observer design for TS models with unmeasured premise signals is more complex. The question asked in this paper is: Is it more interesting to combine between state extension (immersion) techniques and TS models? The objective in this question is to search if it possible to extended the dynamics of the system in order to obtain an equivalent TS model with measured premise variables. The answer to this question is of course yes as shown in the first work [8]. The present paper provides an enhanced approach by using dynamic state extension inspired from the immersion techniques (see [2] and references therein) with adding dynamic output injections. The proposed algorithm does not need inverse transformation computation.

The objective of this paper is to introduce an algorithm aiming to obtain a TS model with measured premise variables by using state extension dynamics and output injections. The output injection can be used at current sample or the past samples. In addition, the target system is no longer linear with a particular structure, as in the classic coordinate transformation techniques, but a general system where the only purpose is the obtain a nonlinear system whose TS equivalent model has measured premise variables.

The contributions of the paper are:

* Transform a nonlinear system leading to TS systems with unmeasured premise variables into an equivalent nonlinear system whose TS transformation is free from any unmeasured premise variable.

* The state of the equivalent target system contains the state of the original system which does not need any inverse transformation (or at least a simple non-singular transformation).

* Exploit the results of observer design for TS systems with measured premise variables.

Recently, the first author proposed an immersion approach in [8] that met the above objectives, however, it presents some

limitations on the structure of the studied systems. Indeed, for some systems, the algorithm fails to obtain a TS system with unmeasurable premise variables due to the infinite iterations in the algorithm. This fact is illustrated in the following motivating examples.

### A. Motivating examples

#### a) Example 1: Let us consider the system

$$x_1^+ = x_2, \quad x_2^+ = x_2^2, \quad y = x_1 \quad (1)$$

By applying the algorithm presented in [8], the following new states are obtained

$$z_3 = z_2^2, \quad z_4 = z_2^4, \quad z_5 = z_2^8, \quad \dots \quad (2)$$

It can be seen that the algorithm presents an infinite number of iteration and then fails to provide a solution that met the requirements of TS system with measured premise variables.

#### b) Example 2: Let us consider the nonlinear system in continuous time

$$\dot{x}_1 = x_2, \quad \dot{x}_2 = x_2^2, \quad y = x_1 \quad (3)$$

After Euler discretization with a sampling time T, one obtains the discrete-time system

$$x_1^+ = x_1 + Tx_2, \quad x_2^+ = x_2 + Tx_2^2, \quad y = x_1 \quad (4)$$

Applying the approach of [8] leads to infinite iterations.

The following section states an extended algorithm of the one proposed in [8], that solves the problem of infinite iteration and enlarges the class of systems for which there exist state extensions for obtaining TS systems with measurable premise variables.

**Definition 1:** Let us consider variables $s \in \mathbb{R}^{n_s}$ and $x \in \mathbb{R}^{n_x}$. A nonlinear and sufficiently smooth function $\varphi(s, x): \mathbb{R}^{n_x+n_s} \to \mathbb{R}$ is called affine with respect to $x$ if it can be expressed as follows

$$\varphi(s, y) = \varphi_1(s)x_1 + \varphi_2(s)x_2 + \dots \varphi_{n_x}(s)x_{n_x}$$

## II. MAIN RESULT

Consider the nonlinear system expressed by

$$x^+ = f(x) + g(x)u, \quad y = Cx \quad (5)$$

where $x \in \mathbb{R}^n$, $u \in \mathbb{R}^{n_u}$ and $y \in \mathbb{R}^{n_y}$ represents, respectively, the state, the input and the output vectors. $f(x)$ and $g(x)$ are nonlinear vector functions depending on the state vector where $f(0) = 0$. $C$ is a known constant matrix with appropriate dimension.

Classically, the sector nonlinear transformation is used to obtain a TS model, in a compact set $\Omega$ of the state space, in the following form [19]

$$x^+ = \sum_{i=1}^{r} h_i(\xi) (A_i x + B_i u), \quad y = Cx \quad (6)$$

where $\xi_k$ is the vector of premise variables and the matrices $A_i$ and $B_i$ are with appropriate dimensions and correspond to the $i^{th}$ sub-model of the TS system. $r$ is the number of sub-models
and the functions $h_i(\xi_k)$ are the weighting functions depending on the premise variables $\xi_k$ and satisfying the convex sum property

$$ \sum_{i=1}^{r} h_i(\xi) = 1, \quad 0 \le h_i(\xi_k) \le 1, \forall k, i = 1, \dots, r \quad (7) $$

This transformation often leads to a TS model having unmeasured premise variables $\xi_k$. In order to avoid this difficulty, the idea is then to transform, equivalently, the original nonlinear system in the following form

$$ z^+ = A(y, u)z + B(y, u)u, \quad y = Cz \quad (8) $$

where $z \in \mathbb{R}^N$ is the resulted state vector of dimension $N \ge n$. Notice that the components of $x$ are contained directly in $z$. Consequently, when $z$ is estimated the original state $x$ is directly estimated. The system (8) can then be expressed in TS model by using the nonlinear sector transformation. It follows

$$ z^+ = \sum_{i=1}^{r'} h_i(y, u) (A_i z + B_i u), \quad y = C z \quad (9) $$

where the matrices $A_i, B_i$ and $C$ are constant with appropriate dimensions. The weighting functions depend only on the measured variables $y_k$ and $u_k$. $r'$ is the number of the sub-models of the new TS model. In such a case the proportional observer takes the form

$$ \hat{z}^+ = \sum_{i=1}^{r'} h_i(y, u) (A_i \hat{z} + B_i u + L_i (y - \hat{y})), \quad \hat{y} = C \hat{z} \quad (10) $$

Let us define the state estimation error $e = z - \hat{z}$, its dynamics is governed by the difference equation

$$ e^+ = \sum_{i=1}^{r'} h_i(y, u) (A_i - L_i C) e \quad (11) $$

The stability of the state estimation error dynamics is largely studied and interesting results are provided terms of LMIs in order to design the gains $L_i$ [5], [4].

### A. System transformation by extended state extension

In this section, the approach to transform the system (5) into the form (8) will be given.

1) **Step 1:** Initialize the first new variables from the state vector $z_i = x_i$, $i = 1, \dots, n$.

2) **Step 2:** For each new variable, compute its value at $k+1$ (exactly the same equations as the original system (5)) and separate all the functions depending only on measured variables $y$ and $u$ and define the remaining nonlinear functions with adequate nonlinear output injections as new variables $z_l, l > n$. By computing the values of these variables at $k+1$ it follows

$$ z_l^+ = \sum_{i=1}^{l} a_{l,i}(y, u)z_i + \sum_{i=l+1}^{s} a_{l+1,i}(y, u)z_i + \varphi_l(y, u, y) \quad (12) $$

where $z_i, i = l+1, \dots, s$ denote other defined new variables. The functions $a_{l,i}, a_{l+1,i}$ and $\varphi_l(y_k, u_k)$ are scalar and depend only on measured variables. The step 2 is iterated for all new defined variables and the parameter $s$ converges to a finite number, if a transformation exists, to $N$ the dimension of the new state vector $z$.

3) **Step 3:** The algorithm stops when the time derivative of the $l^{th}$ new state is free from nonlinear functions depending on the unknown states.

By using this procedure, the system described in equation (8) is obtained and by using the sector nonlinear transformation in a compact set, one obtains the TS system (9) having only measurable premise variables. Then the design of the observer follows the classical approaches.

**Remark 1:** Notice that in the above algorithm, it is possible to use a nonlinear parametrization of the new state variables by using the current outputs and their past samples (For more details see the examples).

### B. Examples

Let us go back to the motivating examples and apply the new algorithm.

a) **Example 1 (Cont.):** Consider the equivalent system by adding and subtracting the same function $\alpha x_1^\beta$

$$ x_1^+ = x_2, \quad x_2^+ = x_2^2 + \alpha x_1^\beta - \alpha x_1^\beta, \quad y = x_1 \quad (13) $$

where $\alpha$ and $\beta$ are real constants to be determined. The first step is to define $z_1 = x_1$ and $z_2 = x_2$. Now, if we consider the state extension with $z_3 = z_2^2 + \alpha z_1^\beta$, its dynamics is given by

$$ z_3^+ = z_2^4 + \alpha z_2^\beta \quad (14) $$

Then by choosing $\alpha = -1$ and $\beta = 4$, one obtains

$$ z_3^+ = 0 \quad (15) $$

The system is then immersed into the following new equivalent system

$$ z_1^+ = z_2, \quad z_2^+ = z_3 + z_1^4, \quad z_3^+ = 0, \quad y = z_1 \quad (16) $$

where the nonlinearity depends only on the measured variable $y = z_1$. Then two possibilities to design an observer: the first one leads to a linear state estimation error by expressing the system as

$$ z^+ = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} z + \begin{bmatrix} 0 \\ y^4 \\ 0 \end{bmatrix} $$

and the second one in TS form with measured premise variables by transforming the following quasi-LPV system in polytopic form

$$ z^+ = \begin{bmatrix} 0 & 1 & 0 \\ y^3 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} z $$

Of course, the first solution is better for this example since it is not needed to bound the output in order to obtain the TS system.
b) Example 2 (Cont.): For the second example, if we consider the equivalent system

$$
\left\{
\begin{array}{l}
x_1^+ = x_1 + Tx_2 \\
x_2^+ = x_2 + Tx_2^2 + \sum_{s=1}^3 \alpha_s (x_1 - x_1^-)^{\beta_s} \\
\quad - \sum_{s=1}^3 \alpha_s (x_1 - x_1^-)^{\beta_s} \\
y = x_1
\end{array}
\right.
$$

Then by considering the new state $z_3 = Tx_2^2 + \sum_{s=1}^{3} \alpha_s (x_1 - x_1^-)^{\beta_s}$, its dynamics is given by

$$
\begin{align}
z_3^+ &= (x_2 + Tx_2^2)^2 + \sum_{s=1}^3 \alpha_s (Tx_2)^{\beta_s} \tag{17} \\
&= x_2^2 + 2Tx_2^3 + T^2x_2^4 \nonumber \\
&\quad + \alpha_1 T^{\beta_1} x_2^{\beta_1} + \alpha_2 T^{\beta_2} x_2^{\beta_2} + \alpha_3 T^{\beta_3} x_2^{\beta_3} \tag{18}
\end{align}
$$

Finally, it is clear that if one choose $\beta_1 = 2$, $\alpha_1 = -\frac{1}{T^2}$, $\beta_2 = 3$, $\alpha_2 = -\frac{2}{T^2}$, $\beta_3 = 4$, $\alpha_3 = -\frac{1}{T^2}$, the following system is obtained

$$
\left\{
\begin{array}{l}
z_1^+ = z_1 + Tz_2 \\
z_2^+ = z_2 + z_3 - \sum_{s=1}^3 \alpha_s (z_1 - z_1^-)^{\beta_s} \\
z_3^+ = 0 \\
y = z_1
\end{array}
\right.
$$

or in matrix form

$$
\begin{align*}
z^+ &= \begin{bmatrix} 1 & T & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} z + \begin{bmatrix} 0 \\ -\sum_{s=1}^3 \alpha_s (y-y^-)^{\beta_s} \\ 0 \end{bmatrix} \\
y &= \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} z
\end{align*}
$$

which can be handled exactly as in the previous example.

c) Example 3 (Nonlinear output): In the above examples, the output equation is assumed linear. In order to illustrate the approach for nonlinear output equation, let us consider the following system borrowed from [21]:

$$
x_1^+ = x_2, \quad x_2^+ = x_1 + x_2^2, \quad y = \sin(x_1) \quad (19)
$$

defined in the open subset $U_0 = \{(x_1, x_2) : |x_1| < 1, |x_2| < 1\}$. The system is equivalent to the following one

$$
\left\{
\begin{array}{l}
z_1^+ = z_2 \\
z_2^+ = z_1 + z_2^2 + \underbrace{\alpha (\arcsin(y^-) + \arcsin(y))^2}_{\varphi(y^-, y)} - \varphi(y^-, y) \\
y = \sin(z_1)
\end{array}
\right.
$$

where $y^- = y(k-1)$, which leads to

$$
\left\{
\begin{array}{l}
z_1^+ = z_2 \\
z_2^+ = z_1 + z_2^2 + \alpha (z_1^- + z_1^2)^\beta - \alpha (z_1^- + z_1^2)^\beta \\
y = \sin(z_1)
\end{array}
\right.
$$

Now by considering the new state variable $z_3 = z_2^2 + \alpha (z_1^- + z_1)^\beta$, its dynamics is given by

$$
z_3^+ = (z_1 + z_2^2)^2 + \alpha (z_1 + z_2^2)^\beta
$$

Then, by fixing the parameters $\alpha = -1$ and $\beta = 2$, one obtains the system

$$
\left\{
\begin{array}{l}
z_1^+ = z_2 \\
z_2^+ = z_1 + z_3 + \varphi(y^-, y) \\
z_3^+ = 0 \\
y = \sin(z_1)
\end{array}
\right.
$$

A sector nonlinear transformation leads to a TS system with measurable premise variables.

### III. FURTHER DISCUSSIONS

In this paper an algorithm of state extension is proposed in order to avoid the unmeasured premise variables in the obtained TS model, the state of the initial system is contained in the new state (no need of the inverse transformation). The complexity of observer design for TS with unmeasured premise variables is then reduced to the classical observer design for TS systems with measured premise variables. Two points should be discussed:

* In some situations, it is interesting to reduce the number of variables in the LMIs and then reduce the dimension of the target system. It is then possible to relax the algorithm by avoiding the second objective: the state of the initial system is contained in the target one with simple state transformation. To illustrate this fact, let us consider the system

$$
\left\{
\begin{array}{l}
x_1^+ = x_2 \\
x_2^+ = x_3 \\
x_3^+ = x_1 + x_2 x_3 \\
y = x_1
\end{array}
\right.
$$

for this example, let us define the following equivalent system

$$
\left\{
\begin{array}{l}
x_1^+ = x_2 \\
x_2^+ = x_3 + \alpha x_1 x_2 - \alpha x_1 x_2 \\
x_3^+ = x_1 + x_2 x_3 \\
y = x_1
\end{array}
\right.
$$

Then by considering the state transformation

$$
z_1 = x_1, \quad z_2 = x_2, \quad z_3 = x_3 + \alpha x_1 x_2
$$

The new system becomes (with $\alpha = -1$)

$$
\left\{
\begin{array}{l}
z_1^+ = z_2 \\
z_2^+ = z_3 + yz_2 \\
z_3^+ = z_1 \\
y = x_1
\end{array}
\right.
$$

or in matrix form

$$
\left\{
\begin{array}{l}
z^+ = \begin{bmatrix} 0 & 1 & 0 \\ 0 & y & 1 \\ 1 & 0 & 0 \end{bmatrix} z \\
y = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} z
\end{array}
\right.
$$
which leads to a TS system with premise variables depending only on $y$. After estimating the state $z$, the original state $x$ is obtained by a simple inversion as follows

$$x_1 = z_1, \quad x_2 = z_2, \quad x_3 = z_3 - z_1 z_2$$

* The construction of the nonlinear injections is particular for each system and each considered nonlinearity. Then, constructing these injections can be done by combining the first algorithm given in [8] and the one presented in this paper. Indeed, applying the algorithm of [8] can give an idea on the form of the nonlinear injection aiming to cancel the unmeasured nonlinear term that cause the unmeasurable premise variables.

* Notice also that the nonlinear output injection functions may depend on unmeasured states but this should be done in an affine way (see Definition 1) with respect to the unmeasured states (see the example of the second point above).

* Notice that even if the major part of the examples given in this paper present polynomial nonlinearities, the approach works also for other type of nonlinearities. For example, let us consider the system described by the model

$$x_1^+ = x_2, \quad x_2^+ = \sin(x_2), \quad y = x_1$$

By using the approach of [8], the algorithm does not converge and presents an infinite number of iterations. But if we consider the equivalent system

$$\begin{cases} x_1^+ = x_2 \\ x_2^+ = \sin(x_2) + \alpha \sin(\sin(y)) - \alpha \sin(\sin(y)) \\ y = x_1 \end{cases}$$

The new state is $z_3 = \sin(z_2) + \alpha \sin(\sin(z_1))$ with the dynamics $z_3^+ = \sin(\sin(z_2)) + \alpha \sin(\sin(z_2))$. Then by fixing $\alpha = -1$, one obtains the following system

$$\begin{cases} z_1^+ = z_2 \\ z_2^+ = z_3 + \sin(\sin(y)) \\ z_3^+ = 0 \\ y = z_1 \end{cases}$$

or in LPV form

$$\begin{cases} z^+ = \begin{bmatrix} 0 & 1 & 0 \\ \frac{\sin(\sin(y))}{y} & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} z \\ y = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} z \end{cases}$$

## IV. ACKNOWLEDGMENT

This research is supported by the French ARTISMO (Intelligent Estimation Algorithms for Mobility) financed by ANR (National Research Agency).

## V. CONCLUSIONS

This paper introduces a new direction in designing observers for nonlinear systems which lead, classically, to a TS system with state dependent unmeasured premise variables. The first approach given in this paper exploits the notions of immersion,

state extension and state transformation of the original nonlinear system in order to obtain a new equivalent nonlinear system which leads to a TS system with measured premise variables. In addition, the proposed state transformation aims to obtain a new equivalent system whose state contains the original state of the system which avoids the inverse transformation, or at least a simple inversion. It is illustrated that this research direction reduces the complexity of classical seeking state transformations and inversions and relaxes the structure of the target system which is not necessarily in canonical form (Linear or LPV). Future works will concern the characterization of the conditions that ensure the existence of such coordinate transformations. In addition, the proposed approach will be studied in the presence of modeling uncertainties and unknown inputs.

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