Observer and descriptor satisfying incremental quadratic constraint for class of chaotic systems and its applications in a quadrotor chaotic system.

A drone based on four rotors is considered in this research paper. Its chaotic solution is shown bounded in an inscribed sphere whose vertices are tangent to faces of octahedron. Based on concept of constrained optimization; Linear Matrix Inequalities (LMIs) satisfying quadratic constraint increment multiplier matrix σm , state observers and descriptors with estimated parameter is calculated. Moreover, an image file is decrypted by designing description for mentioned chaotic system and then encrypted on its receiver end. Furthermore, an electric circuit is designed for chaotic quadrotor using LTspice and is fitted into wireless flying robot to observe its dynamics in bounded rectangular region.

Introduction

A scenario in which trajectories of dynamical systems blows-up due to sudden change in its initial value is known as Chaos. This phenomenon is developed for the first time by Lorenz in 1963. Due to which many new chaotic systems including; Double convection Rucklidge [1], Chen [2], Liu [3], Finance [4], Quadrotor [5] and much more are designed in last six decades. The scope of chaos in various fields especially engineering technologies [6], [7], [8], [9], [10] and medical [11], [12], [13], [14], [15] is beyond our limitation.

However, apart from applications of chaos in various fields, it has a lot of disadvantages in this era, especially on nature and living things. Therefore, control of such unpredictable systems caught attention of researcher since 1990s. Runzi Luo [16] investigated control of unpredictable behavior zhang hyperchaotic system with model uncertainties, external noise and anonymous parameters. In 2019, Omid Mofid et. al [17] used fractional-order chaotic system and controlled its chaos using adaptive controller with the help of Lyapunov stability theory. The problem of controlling three dimensional perturbed chaotic system with partially known dynamics in the existence of sensor noise is done by Aghababa [18]. A control input based on adaptive sliding technique is applied on magnet synchronous motor oscillator [19], where nonlinear terms and upper bound of mentioned dynamical system are taken as anonymous in whole process. In 2018, Luo et. al [20] considered fractional-order chaotic brushless DC motor system and used finite-time echo state controller to compensate uncertainties therein. An active control based strategy is designed in [21] for two identical discrete chaotic systems. Based on error dynamics and sequential control procedure, three controllers [22] are designed for Rucklidge chaotic system for the purpose of analysis whereas, oscillatory dynamics of a chaotic system including forcing forcing term [23] is studied and controlled for particular value of t. Recently, in 2020 an optimal controller based on sequential, parameter uncertainties is derived [24] for electromechanical system. Peng et. al proposed (ANIC) control strategy for robotic system along with consideration of external disturbance.

Pecora and Carrol [25], [26], [27], [28] introduced a concept termed as synchronization, in which one system overlaps other and adobe its characteristics with time. Synchronization is a milestone achieved in 1990s after which many chaotic systems are being synchronized. This concept is further divided into two types; identical and non-identical synchronization. Dongmo et. al [29] worked on different type of synchronization using active backstepping input for hyperchaotic systems. In 2015, two non-identical integer and fractional order chaotic systems are considered for synchronization [30] using sliding mode controller. Muñoz-Vázquez et. al [31] worked on synchronization of robotic system based on uncertain parameter values. In, 2012, network models are studied by Ramirez et. al [32] for dynamical behavior and synchronization with uncertainties in their nodes.

It is important to mention that techniques used for control and synchronization are same, while the difference in these two concepts is: in control theory, trajectory of dynamical system approaches to its stable equilibrium point as time approaches to infinity and in synchronization, trajectories on one dynamical system termed as slave or derive system approaches to its original dynamical system termed as master system.

Control theory is linked to Optimization on the basis of energy type function. As control theory is based on minimization of Lyapunov function. Therefore, equal and unequal constraints are solved for many chaotic systems to get its minimized energy function. Nowadays, solving LMIs are used as constraints for Lyapunov function to be minimized has got attention in the field of control theory. A leader-follower based algorithm is proposed by Huang [33] in 2019 for designing observer for quadrotor using consensus static controller. Zhao et. al [34] worked on incremental quadratic constraints for observer based secure communication scheme whereas, amendment in increment multiplier matrix is carried out by Zhang et. al [35]. An adaptive observer [36], satisfying Lipschitz inequality, based on state estimation is derived using linear matrix inequalities. Apart from continuous time chaotic systems, Zhang et. al [37] worked on discrete-time systems and introduced generalized Luenberger like observer. In 2020, Kaviarasan [38] used fuzzy modelling approach for robust observer based control of multi-weighted complex network systems. Moreover, multiple sensors are used in work of Wen [39] for transmission of data between heterogeneous nonlinear systems. Zhang et. al considered exponential reduced order observer [40] for chaotic systems satisfying incremental quadratic restrictions and its improved version of exponential reduced observer is given in [41]. The issue of transferring data in noise based public Chanel is observed in [42], while problem of unmatched observer is solved in [43] by Yang.

Mostly, humans are interested in robot based technology and the first thing to notice is its dynamics. Besides, many disadvantages of chaos in real life, it has advantages in robotics engineering. A contraction transformation algorithm is proposed for Arnold dynamical system to complete coverage trajectory for mobile robots [44]. In 2016, Li [45] considered Lorenz chaotic system and worked on path planning of robot in bounded region. Nasar presented mobile based robot [46] used for mission of field exploration in 2019, whereas Volos derived a method which is based on microcontroller [47] for fast path coverage of discrete-time chaotic system.

Quadrotor is type of aerial robots and is source of interesting research for engineers since many decades [48], [49]. These robots can perform tracking of moving objects [50], checking of complex power transmission lines [51], collecting media information [52], [53], providing medical equipments in tragedies such as floods, earthquakes and tsunamis [54], [55], [56]. These type of robots can be helpful in providing food supplements to people who are infected from coronavirus and in quarantine due to epidemic disease (CoV-19). Moreover, we will consider following quadrotor chaotic system [5] throughout our work. where system (1) is chaotic for moment of inertias along and axes ; drag coefficients; total rotational moment; rotational speed of propeller; and can be seen in Fig. 1 .

Fig. 1

Phase portrait of system (1) with initial value .

Therefore, upto our knowledge gained from above research papers and reviewing literature from books, we got that none of the researcher worked on design of system description including anonymous parameter as well as nonlinear output. Hence, we designed such full-order observer, descriptor and applied it for the encryption of image file to assure its application in secure communication. A circuit is also designed for mentioned chaotic system which is used for the analysis of dynamics of wireless aerial quadrotor robot for the first time.

This paper is arranged in the following pattern; In Section 2 basic definitions and lemmas are given, which can be helpful in rest of work. Generalized full-order state observer for chaotic systems is derived in Section 3, whereas its implementation in transmission of Gray scaled image file with safety is given in Section 5. Furthermore, a software for electrical circuits; LTspice is used in Section 6, for designing circuit of Quadrotor chaotic system and is fitted into wireless aerial robot to follow its path in Section 7. Concluding remarks are given in Section 8.

Mathematical preliminaries

Here, fundamental preliminaries are provided, that will be used in the main part of this work. The symmetric matrix;is written in mentioned form for our convenience and can be used through out our work.

[57] A symmetric matrix σ is an incremental multiplier matrix for if it satisfies the following incremental quadratic constraint (ICQ);where and .

[41] The Lipschitz nonlinearity satisfies the ICQ (3) and is given as;with γ > 0 a positive scalar. The above is equation equivalent towhich can be written in the form of matrix (3);for any β > 0.

[41] The one-sided Lipschitz nonlinearity is given byfor also satisfies the ICQ (3). This can be shown by rewriting the above inequality asHence, (3) is satisfied by consideringfor any β > 0.

In Gupta et al.[58] the following generalized monotone nonlinearity is considered;It is easy to see that this nonlinearity satisfies the ICQ condition (3) for

[41] Any nondecreasing nonlinearity satisfies the ICQ (3). For any nondecreasing f satisfyingthe inequality (3) is satisfied by consideringfor any β > 0.

[35] There exist a matrix with appropriate dimension satisfies the following matrix equation;

Lastly, the following standard Lemmas will be useful in the proofs of the following section.

Let be a symmetric matrix. The following properties are equivalent;

.

Let with ℘ symmetric matrix, it holds where λ min(℘), λ max(℘) are the minimum and maximum eigenvalues of ℘.

Designing observer and descriptor

In this section, we consider any dynamical system of the form;where and is any positive real number. Then, we construct the full-order observer for system (16) of the form;where is estimated value of whereas, and are gain matrices, based on Lyapunov stability theory [61], which can be calculated using Algorithm 1 given in our research paper.

Algorithm 1

Algorithm for full-order observer.

Suppose that system (16) satisfies Assumption 3 and quadratic constraint (3) with known increment multiplier matrix σ and and a symmetric matrix ℘ with suitable dimensions such that; and are inequality and equality constraints, respectively are satisfied, with and then, system (17) is suitable observer for system (16) by satisfying adaption observer law.

We begin proof of given theorem using energy type function; where e is error term of state trajectories of system (16) and (17) such that . Error dynamical system is obtained by taking derivative of error term;where and . For and ΔΦ, equation (3) can be rewritten in terms of e and ΔΦ as;where Υ is given in (20). As, energy function is selected as then the time derivative of along trajectory e is;For to be negative definite, we design adaptation observer law using (19) and (23);where Γ is any positive matrix of appropriate dimension. Putting (24) into (23) yields;However, Eq. (25) can be rewritten in matrix form as;Multiplying by the both sides of (18) and then adding to the left side of (22) gives; which is required proof of our theorem □

Our next theorem is about convergence of error dynamical system because of its importance in achieving stability of energy type function.

Norm of error dynamical system (21) is bounded, convergent and maps to a scalar valued function φ(t).

We start proof using (25) in Theorem 3.1 and assumption 3 in [34];Hence, solving in view of (28), we get;We obtain following equation using (3.4) and Lemma 2.8 on energy function;Taking square root on all sides of inequality 30;From (31), it is confirmed that error term (21) is bounded for all t. For convergence, we take on (31) which confirms;convergence of error term. It is also confirmed from (32) that error term maps to a scalar function φ(t).

Linear matrix inequalities

In this subsection, we are able to design LMIs for gain matrices and a symmetric matrix ℘ on the basis of Theorems 3.1 and 3.3 that are given and proved in Section 3. Hence, addition of all the terms in (22) using matrices Υ and σ gives another matrix inequality of the form;We use Shur’s Lemma 2.7 for the transformation of inequality (33) into following matrix inequality;which helps to find gain matrices and to assure that solution achieved is in feasible region, where .

If we consider equals to zero in (16) , we get observer given in

If original system 16 has only linear output, then we get observer given in

Corollaries 3.6 and 3.7 show generalization of observer given in our work.

Algorithm

A symmetric matrix ℘ and gain matrices have important role in numerical simulations. Therefore, the following algorithm is used to find such matrices;

Increment multiplier matrix σ for quadrotor

This section includes calculation of increment multiplier matrix for mentioned chaotic system. Algorithm for calculation of such matrices can be found in [34], [35]. Therefore, quadrotor chaotic system [5] can be rewritten in transformed form as;where ϖ = is constant matrix,Hence, using we get faces of octahedron as;where and can be calculated using radius of sphere inside octahedron with faces θ for . According to [5], estimated bound of system (1) is;We obtain the bounded sphere is inside the octahedron, when the faces become tangent to sphere;

In Fig. 2 , one can observe that the whole sphere containing chaotic solution is inside octahedron. Therefore, all required conditions for increment multiplier matrix (3) are satisfied accordingly. Hence, using θ, and Algorithm 1 in [34], we get the required matrix σ for quadrotor system as;This matrix plays leading role in finding all gain matrices for observer system and can be used in Section 5.

Fig. 2

Chaotic quadrotor inside in-sphere.

Secure communication

In this section, a secure communication scheme is proposed for transferring of image file in secure way. Therefore, for master system we inject unknown parameter, into original quadrotor chaotic system [5];where I 1 and k, are parameter values, given below (1), for which (38) is chaotic. We consider dynamical system (38) as master system which can be rewritten in form of (16);where nonlinear functions Φ1 and Φ2 are;Linear outputs, which are also shown in Figs. 4, 5, 6, and nonlinear outputs of system 38 are;andrespectively, where ξ(ν) is taken as cubic function and φ is unknown control input used as transmitter during sending image file. Observer system (17) of chaotic system (38) consist of gain matrices and a symmetric matrix ℘. Hence, for all these matrices, we follow steps of Algorithm 1 given in Section 3.8.

Fig. 4

First linear output .

Fig. 5

Second linear output .

Fig. 6

Third linear output .

The nonlinear part of given system satisfies constraint of increment multiplier matrix. Therefore, a symmetric matrix σ is calculated in Section 4 and is given in (37). Then, using adaptation law (24) anonymous parameter matrix having suitable order is calculated;to verify negative defitness of energy function . Here, we select having 1 × 1 dimension. After that, all unknown matrices are obtained from LMIs (34) using SciLab 6.0;Now, we start following the block diagram 3 of secure communication to transfer an image file, which can be seen in Fig. 7 with dimensions 128 × 128 having 8 bit depth is considered. First, the original image file is converted into digital signals which can be difficult to recover it, if reaches in bad hands before reaching to receiving end. Although, we have selected an image file, but one can also use same for text messages. For each digital bit, sequence of pulses Fig. 8 in the form of long row of a matrix having dimension 128 × 128 is considered. Meanwhile, observer system (17) synchronizes with original system (16), but this all is possible to set pulses of sequences;zero in first few seconds. The signal φ(t) is used as transmitter with scaling factor of amplitude γ and is introduced into nonlinear output . Next, the linear and nonlinear outputs are transmitted to receivers end passing through noise free public channel in which original file can be obtained using; Figs. 9 , 10 , 11 show that each trajectory of observer system is approaching towards original system with error e(t) and anonymous parameter approaches to original as time goes to infinity.

Fig. 3

Block diagram for sending an image file in noise free medium using (17).

Fig. 7

Gray image file for secure communication.

Fig. 8

Sequence of pulses.

Fig. 9

First trajectory of observer system approaches to original system.

Fig. 10

Second trajectory of observer system approaches to original system.

Fig. 11

Third trajectory of observer system approaches to original system.

Next step is shown in Fig. 12 , which clarifies if norm of error;maps to scalar valued function φ, meanwhile, sequence of pulses shown in Fig. 12 can be recovered with the help of (47) using 0, 1 bit integrator;which is used to help in the recovery of image at receivers end. This all process is validated from Fig. 13 and assures safety of communication.

Fig. 12

Norm of error and recovered signals.

Fig. 13

Signals transmitter vs signals reviver.

Circuit design of quadrotor using LTspice

In this section, we designed circuit diagram for quadrotor chaotic system using OP - Amp in LTspice software. First, we transformed variables of system (quadrotor) into electric system using;the current flowing through capacitors is equals to current flowing through resistors. Negative sign appears because of negative integrators. The following system of electric circuit differential equation is derived using (49);where ϖ are system parameters and R and C show resistors and capacitors, respectively. In Fig. 14 , we can observe that there are three multipliers U5, U6 and U7 named as . These multipliers are used to get nonlinear terms including in our chaotic system.

Fig. 14

Circuit design of Quadrotor using LTspice VII.

There are eight nodes in each multiplier whereas, all nodes are defined. Graphical explanation of all nodes are given in Fig. 15 in which 1st and 2nd nodes are for first variable depending upon sign of included variables. Similar is in the case of other variable Y and can be seen in 3rd and 4th nodes, 5th and 8th nodes for negative and positive voltage connectors, 6th node is to ground and 7th node is for the output of multiplied input variables. It is also important to mention here, that output result of multiplier must be divided by 100. Figs. 16 , 17 , 18 are transient output of circuit (50) which also exhibit chaotic behavior for specific values of included resistors and capacitors. Chaotic circuits have importance in specifying path for robot, which can be seen in Section 7.

Fig. 15

Explanation of multiplier AD633-JT.

Fig. 16

chaotic trajectory of i1.

Fig. 17

chaotic trajectory of i2.

Fig. 18

chaotic trajectory of i3.

Dynamics of wireless quadrotor robot

This section includes another application of quadrotor chaotic system [5] related to dynamic of wireless robots due to unpredictable behavior in mentioned dynamical system. The robotic system with linear velocity A and angular velocity A is;where and d is taken as distance among their four rotors. Moreover, state variables (ζ 1, ζ 2) for (51) present horizontal and vertical positions lying on bounded rectangular region. Furthermore, if we use the chaotic circuit (50) and fit it into robot model (51), then original system can be converted into six dimensional dynamical system;System (52) is used to show movement of quadrotor chaotic system in specific rectangular bounded area. We had done several simulation using MATLAB and obtained the following figures;

The above Fig. 19 is our first simulation whose initial values and (X, Y, θ) are taken as (0.001,0.001,0.001) and (1,11,0), respectively for 8 minutes. We observed from this figure that in last 3 minutes our robot is showing same dynamics but in specific subregion. Our next simulation can be seen in Fig. 20 which is simulated in same region and for same time as already done in first simulation, but initiated with slightly different initial values; and . This time we can observe from Fig. 20 that for first 100 seconds, robot is rotating in circular region and then for rest of time its movement becomes smooth.

Fig. 19

First simulation of wireless robot.

Fig. 20

Second simulation of wireless robot.

In comparison of both figures, we can say that second simulation 20 shows better results as compared to first 19. Because, in first figure dynamics of robot is started in smooth manner but after some time its movement is restricted to specific subregion, while in second result one can observe that for first 2 minutes its dynamics is restricted to specific subregion.

Conclusion

In this paper three dimensional, autonomous, nonlinear quadrotor chaotic system is considered. First of all, a state observer is derived for class of chaotic systems satisfying incremental quadratic constraints which is verified with example based on secure communication given in Section 5. It is concluded from corollaries 3.6 and 3.7 that observer (17) we derived is generalized form of observers designed in [34], [35]. Furthermore, with the aid of LTspice, a circuit is designed for quadrotor chaotic system which we used in path tracking of robot. After performing several simulations on considered chaotic system, we concluded from Figs. 19 and 20 that second simulation show better results in bounded rectangular region. In future, our target is to design an observer for hyperchaotic systems and will try to implement on RBG images, while in robotics our aim is to implement such targets on real based models which might be helpful in worst conditions such as CoV.

Authors’ contribution

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Ethical approval

This paper does not contain any studies with human participants or animals performed by any of the author.

CRediT authorship contribution statement

Muhammad Sabir: Conceptualization, Methodology, Software, Writing - original draft. Muhammad Marwan: Conceptualization, Methodology, Software, Writing - original draft. Salman Ahmad: Supervision, Visualization, Investigation. Muhammad Fiaz: Conceptualization, Software, Validation. Farhan Khan: Software, Methodology, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper