A Luenberger-Like Observer for Multistable Kapitaniak Chaotic System

The objective of this paper is to estimate the unmeasurable variables of a multistable chaotic system using a Luenberger-like observer. First, the observability of the chaotic system is analyzed. Next, a Lipschitz constant is determined on the attractor of this system. Then, the methodology proposed by Raghavan and the result proposed by Thau are used to try to ﬁnd an observer. Both attempts are unsuccessful. In spite of this, a Luenberger-like observer can still be used based on a proposed gain. The performance of this observer is tested by numerical simulation showing the convergence to zero of the estimation error. Finally, the chaotic system and its observer are implemented using 32-bit microcontrollers. The experimental results conﬁrm good agreement between the responses of the implemented and simulated observers.


Introduction
Due to the absence or high cost of sensors, some of the variables associated with the dynamics of a system could not be available for measurement. Nevertheless, if the system must be monitored or a state feedback controller implemented, such unmeasurable variables should be estimated. Consequently, an aggregated dynamic system (observation scheme) based on the system model and the measurable states must be incorporated to reconstruct the unavailable variables. In 1996, Luenberger proposed for the first time, an observer and a design methodology for linear systems [1]. Currently, the problem of state estimation for a linear system is well understood, and the solution is well established. However, the nonlinear case is more challenging. For this case, a first proposal of a solution was provided by au in [2] in which structure of the Luenberger observer was applied to Lipschitz nonlinear systems. In this context, the structure is known as the Luenberger-like observer. Based on this observer, au presented sufficient conditions to guarantee the asymptotic convergence to zero of the estimation error. However, no design procedure to find the observer gain was provided. In [3], Xia and Gao showed a necessary condition for the existence of an exponential observer. Tsinias provided sufficient conditions and a simple approach for the observer design [4]. In fact, this approach was a direct extension of the observer design in the linear case. Based on the off-line solution of an algebraic Riccati equation, Raghavan and Hedrick proposed an iterative procedure of observer design for a class of Lipschitz nonlinear systems [5]. In [6][7][8][9][10][11], different kinds of observers were studied for chaotic systems. e main use of an observer in chaotic systems is for synchronization [12][13][14][15][16][17][18][19].
A chaotic system is a dynamical system with the following properties: (1) high sensitivity to initial conditions, (2) dense periodic orbits, and (3) topological mixing. Consequently, it is impossible to carry out accurate predictions about its long-term dynamic behavior [20][21][22][23]. In spite of that, the boundedness of its states can be guaranteed. Chaotic systems can be classified according to the nature of its equilibrium points as (a) no-equilibrium systems; in this kind of systems, there are no real equilibrium points, (b) stable equilibrium systems [24]; in this case, the real parts of all eigenvalues associated with the equilibrium point are negative, (c) line equilibrium systems [25]; there is an infinite number of equilibrium points along a straight line, and (d) curve equilibrium systems; the equilibrium points form a locus such as a circle [26], square [27], and three-leaved clover [28]. All these aforementioned systems belong to the general class of chaotic systems with hidden attractors.
In this paper, the attention is focused on the state estimation of a multistable chaotic system proposed by Kapitaniak and coauthors in [45] using a Luenberger-like observer. e system has two attractors for the same set of parameters. e main contribution of this paper is as follows: for the first time, the Kapitaniak system has (a) its basins of attraction thoroughly studied, (b) its observability analyzed, (c) an observer is proposed for it, and (d) the system and its corresponding observer are implemented using 32-bit microcontrollers.

Multistable Kapitaniak Chaotic System
A three-dimensional chaotic system with a fixed point attractor and a hidden strange attractor was presented in [45]. Each one of these attractors can be reached depending on the selected initial condition. e mathematical model of the system is given as follows: where x 1 , x 2 , and x 3 are the system's states. By using Wolf's algorithm, the Lyapunov exponents of system (1) can be calculated as L 1 � 0.1501,L 2 � 0, and L 3 � −1.1501. As L 1 > 0, it can be confirmed that system (1) is chaotic. Additionally, the Kaplan-Yorke dimension can be determined as D KY � 2.1305. By inspection, the equilibrium point is given by (0, −0.2, 0). e eigenvalues associated with it are λ 1 � −0.9835 and λ 2,3 � −0.0082 ± 2.2547i. us, it can be concluded that this equilibrium is stable. is equilibrium point can be reached, for example, by taking the initial condition as (2, 3, 0). With respect to the strange attractor, this can be reached, for example, by using the initial condition (5.4, −1.8, 3.3) (see [45]). e projections of this attractor on x 1 − x 2 ,x 2 − x 3 , and x 1 − x 3 planes are presented in Figure 1. e attractor was obtained by simulation of equation (1) using Simulink ® with solver ode45 (Dormand-Prince), relative tolerance 1e − 6, absolute tolerance 1e − 7, start time 0, and stop time 500 sec. If the stop time is increased to 1,000,000 sec, it can be determined that the chaotic states of system (1) belong to the following set: (2) us, the following bounds for x 1 , x 2 , and x 3 can be established: Finally, it is important to mention that system (1) can briefly be represented as where 2 Complexity

Basin of Attraction
e basin of attraction of an attractor comprises the set of points in the state space that leads to the attractor [46,47].
e current system (1) has two attractors including a fixed point attractor and a chaotic hidden strange attractor. e basin of attraction of both attractors is shown in Figure 2 on the x 1 -x 2 plane (Figure 2(a)) and on the x 1 -x 3 plane ( Figure 2(b)). In the first case, the plane is chosen such that x 3 � 0. Orbits are started at every pixel in a region of interest and followed until they return to the Poincare section or diverge to infinity, and the corresponding initial point is colored accordingly: red, if it identifies the chaotic attractor, light green, if it identifies the fixed point attractor, and white, otherwise. Similarly, for completeness, we choose the plane containing the equilibrium x 2 � −0.2 and perform similar operations as before. e resultant basin of attraction can be appreciated, as shown in Figure 2 Moreover, it is important to classify and quantify the basin of attraction based on the work reported in [48]. According to the results, a probability function considered at large distances is the basis for classifying and quantifying chaotic attractors' basins. e function at large distances, has power law scaling: where P(r) is the probability that an initial condition at a distance r from the attractor lies within the basin of attraction, and P 0 and c are the classification and quantification parameters. Based on these parameters, the basin of a chaotic system can be grouped into one of four classes [48]. As Figure 3 shows, system (1) has a class 3 fractal basin with noninteger power law scaling. e basin of attraction of the chaotic attractor extends to infinity, but since the codimension of the basin is almost 2, the basin most likely has a narrow width.

Problem Formulation
Let us focus on the case when not all the states of system (1) are available for measurement because the corresponding sensors do not exist or they are very expensive. In this case, we can represent system (4) as where y is the output vector and C is the constant output matrix with appropriate dimension. roughout this work, we consider that the output is simply given by is means that To reproduce the unavailable states x 2 and x 3 , we need to use a dynamic system known as an observer. For the linear case, Luenberger proposed a well-known structure of an observer. For the nonlinear case, we could use a Complexity 3 generalization of this structure called the Luenberger-like observer. For system (7), this observer is given by where x is the observer's state and L is a constant gain matrix. e Luenberger-like observer is formed by the model of the original system (with the true state x replaced by the estimated state x), plus a linear correction term.
e difference between the states of the observer (10) and the system (7) is called the estimation error which is defined as e problem of state estimation consists of finding an appropriate gain matrix L in such a way that lim t⟶∞ e � 0, that is, e converges asymptotically to zero.

Observability Analysis
Before attempting to find the observer gain, or in general, to use any observation scheme, a fundamental question must be resolved. at is, what are the conditions under which the reconstruction problem of the unmeasurable states of a system has a solution?
Definition 1 (see [49,50]). System (7) is said to be observable over the time interval [t 0 , t 1 ],t 1 > t 0 , if the knowledge of the output y over [t 0 , t 1 ] suffices to uniquely determine the initial state x(t 0 ) completely. e observability analysis for linear systems is a wellunderstood problem. However, the case for nonlinear systems is subtler and more complicated. Results on the observability of nonlinear systems are discussed in [51] and references therein. In this work, these results are summarized for an unforced system with a unique output like system (7).
Consider the extended output vector as e observability matrix for the nonlinear case is defined as 4 Complexity Corollary 1 (see [50]). System (7) is locally observable in a neighbourhood of the point Remark 1. Although the Kalman condition for observability of linear systems is necessary and sufficient, condition expressed in (14) is only sufficient.
To begin with the observability analysis for system (7), the extended output vector is calculated as Next, the observability matrix can be determined as It is easy to show that us, system (7) loses its observability only when x 1 � −5.

Raghavan Observer
In this section, observer gain L is tried to be determined using the Raghavan procedure. First, the Lipschitz constant of system (7) must be found.

Lipschitz Constant Determination
Definition 2 (see [49,52]). A function f(x): R n ⟶ R n is said to be locally Lipschitz on Ω ⊂ R n if there exists a constant c (known as the Lipschitz constant) such that for all x 1 , x 2 ∈ Ω, the following inequality holds: Finally, f is said to be globally Lipschitz if it satisfies (18) with Ω � R n .
Lemma 1 (see [49,52]). If a function f: R n ⟶ R n is continuously differentiable on a set Ω ⊂ R n , then it is locally Lipschitz on Ω.
Taking into account Lemma 1, Khalil [52] proposed a procedure to calculate the Lipschitz constant c [53]. Although this procedure produces conservative results, it is enough for the purpose of this work. First, for the function f given in (5), let us calculate its Jacobian matrix as Let us define the matrix F as where us, F is a matrix whose elements are the maximum absolute values of each corresponding element in the Jacobian matrix (19) on the set Ω. Next, F can be determined as Finally, the Lipschitz constant for f on the set Ω is given by where ‖F‖ denotes the two-norm of F, that is,

Raghavan Design
Procedure. The procedure proposed by Raghavan is based on the following result.
Theorem 1 (see [5]). Given system (7) and its observer (10), if there exists ε > 0 such that the following algebraic Riccati equation (ARE) has a symmetric, positive definite solution P: then, by selecting the observer gain as L � (1/2ε)PC T , the estimation error e � x − x converges asymptotically to zero for all f with a Lipschitz constant c.

Remark 2.
A necessary condition for the existence of a symmetric, positive definite solution P is that the pair (A, C) be detectable.

Complexity 5
Let us apply Algorithm 1 to our problem. We must verify that (A, C) is detectable. Let us define the observability matrix O using (6) and (9) as e rank of O is 3, that is, the pair (A, C) is observable, a stronger condition than detectability. Now, by setting ε � 10.9227, we use Algorithm 1 implemented in Matlab ® with command "are" for the solution of equation (24). e algorithm gives a result until the 14th iteration when ε � 1/1500. With this value, the corresponding solution of equation (24) However, as can be easily verified, P is not a symmetric matrix. Besides, smaller values for ε do not produce a symmetric matrix either. us, the algorithm has failed.

Scaling of System. If Algorithm 1 does not work, a possible solution could be to scale the chaotic system (1).
Consider the variables z 1 , z 2 , and z 3 defined as where k 1 , k 2 , and k 3 are the positive constants. By taking the first derivative of (27) with respect to time and substituting (1), after some algebraic operations, can be found that Succinctly, system (28) can be represented as where z :� z 1 z 2 z 3 T , and y 1 is the output of system (29), that is, y 1 � z 1 .
To find the Lipschitz constant of scaled system (29), the Jacobian matrix of f is determined as and the matrix F for (31), that is, F, is given by us, the scaled Lipschitz constant can be calculated as To try to find a positive result for Algorithm 1, it is important to reduce c. is can be achieved by reducing k 1 and k 2 and increasing k 3 . e minimum value for c is 3.3. Although different values for k 1 , k 2 , k 3 , and ε are tested, Algorithm 1 does not work on the scaled system (28). us, it can be inferred that there does not exist a Raghavan observer for Kapitaniak system (1).

Thau Observer
In [2], au provided a sufficient condition to guarantee the asymptotic convergence to zero of the estimation error. However, a systematic procedure of design was not provided. at is, to use this result, first, the user must propose a value of gain for the observer by trial and error. 6 Complexity Theorem 2 (see [2,49]). Given system (7), the corresponding observer (10), a symmetric positive definite matrix Q 1 , and an observer gain L, proposed by the user, such that A − LC is Hurwitz, if the following Lyapunov equation: has a symmetric positive definite solution P 1 which satisfies the following inequality: where λ min (Q 1 ) and λ max (P 1 ) are the minimum eigenvalues of Q 1 and the maximum eigenvalue of P 1 , respectively; then, the estimation error e converges asymptotically to zero.
It should be noted that relation in (35) can be increased if the minimum eigenvalue of Q 1 is increased and/or the maximum eigenvalue of P 1 is reduced. According to [54,55], the ratio in (35) can be maximized if Q 1 � I. To apply eorem 2 to our problem, several values for the gain observer L are proposed. In spite of the exhaustive search, the inequality (35) cannot be satisfied. In this case, the scaled system (28) is considered. Several combinations of values for k 1 , k 2 , k 3 , and observer gain are tested. However, attempts are again unsuccessful. us, it can be inferred that the au observer does not exist for Kapitaniak system (1).

Luenberger-Like Observer
In spite of the negative results of the previous sections, it is important to take into account that eorems 1 and 2 provide only sufficient conditions. If these conditions are not satisfied, this does not mean the nonexistence of the Luenberger-like observer. In fact, in this section, the existence of a Luenberger-like observer (10) for Kapitaniak chaotic system (1) is verified by numerical simulation. Consider the following value for the observer gain L: Once the gain L has been proposed, the observer (10) can easily be simulated. Models for the chaotic system (7) and for the corresponding observer (10) are built on Simulink®. e initial condition for the chaotic system is again (5.4, −1.8, 3.3) as in Section 2. As the states are not available, it is reasonable to propose the initial condition of the observer simply as (0, 0, 0). For a fair comparison with respect to the experimental results of the following section, the simulation is accomplished using solver ode4 (Runge-Kutta) with a fixed step size of 0.0004 sec, a start time 0, and a stop time 10 sec. In Figure 4, the observation process is shown, whereas in Figure 5, the signal of the estimation error e is presented.
As can be appreciated from Figure 5, the estimation error converges asymptotically to zero.

Microcontroller Implementation and
Experimental Results e implementation is accomplished using two Teensy USB Development Board based on the 72 MHz Cortex-M4 3.3 V signal microcontroller.
e chaotic system (7) is implemented on the first board, whereas the corresponding Luenberger-like observer (10) is implemented on the second board. A block diagram of this implementation is shown in Figure 6.
Both systems are implemented using solver ode4 (Runge-Kutta) with a fixed step size of 0.0004 sec. Since the observer requires signal x 1 as an input, a unidirectional communication between the microcontrollers must be set. e communication is achieved using an analog to digital conversion based on PWM and a digital to analog converter built on the microcontroller. e following process is accomplished: (1) the first microcontroller with the chaotic system provides the states x 1 ,x 2 , and x 3 , (2) the states are sent by the PWM port (10-bit resolution and 5 Mhz frequency), and also the states are scaled, (3) the PWM signals go to a low-pass RC filter with a cutoff frequency of 30 Hz, (4) the output of this filter is taken by an analog to digital converter of the second microcontroller. is converter has a 10-bit resolution and a sample frequency of 5 MHz, (5) the digital signal is scaled to the original range, (6) the second microcontroller with the Luenberger-like observer produces the states x 1 ,x 2 , and x 3 , (7) the observer states are scaled, (8) these states are sent to the PWM port, and (9) such signals go to a second low-pass filter. e states of the chaotic systems and the states of the observer are sent by serial communication to a computer for visualization (see Figure 7). At the same time, the outputs of both low-pass filters are sent to an oscilloscope to verify the measurements.
To facilitate the implementation process and the reproduction of our results, the pseudocodes for the master microcontroller and the slave microcontroller are as follows: Pseudocode of the master microcontroller: BEGIN Initial conditions of states define step time Step 1. Set ε to a positive value.
Step 3. If P is symmetric and positive definite, then L � (1/2ε)PC T and the process is terminated.
Step 4. If not, set ε � ε/2, and go to Step 2. If ε is below some precision value, abandon the method. ALGORITHM 1: Procedure to obtain the observer gain [5].      { Read values of the real system through the analog to digital converter and remove the offset and the scale factor x1_r ⟵ (analog input/scale factor)-offset x2_r ⟵ (analog input/scale factor)-offset x3_r ⟵ (analog input/scale factor)-offset Calculate observer states, using the x1_r state and the Runge-Kutta method { x1_O x2_O x3_O } Calculate error between the real or original system and the observer system Applied offset to observer states to get positives values in the PWM port and the scale factor for distributing the values in all range of the PWM port Sx1_O = (x1_O + offset) * scale factor Sx2_O = (x2_O + offset) * scale factor Sx3_O = (x3_O + offset) * scale factor Send values through the PWM port PWM output ⟵ (Sx1_O) PWM output ⟵ (Sx2_O) PWM output ⟵ (Sx3_O) Send states of real system, states of observer system, and error of systems through the serial/USB port Calculate time delay for the sample time } In Figures 8 and 9, the experimental results are presented. e observation process is shown in Figure 8 for the first states, second states, and third states, respectively, of the implemented systems. Finally, the estimation error between the implemented chaotic system and the implemented observer can be appreciated in Figure 9.
Data Availability e experimental microcontroller data used to support this study are available from the second author upon request to the e-mail: jacobmarcoap@gmail.com.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. system with line equilibrium and its application to secure communications using a descriptor observer," Technologies, vol. 7, pp. 1-16, 2019.  10 Complexity