SYSTEM IDENTIFICATION ALGORITHM FOR ...

Appl. Comput. Math., V.14, N.3, Special Issue, 2015, pp.336-348

SYSTEM IDENTIFICATION ALGORITHM FOR NEGATIVE IMAGINARY SYSTEMS MOHAMED A. MABROK1,2 , MOHAMED A. HAGGAG3 , IAN R. PETERSEN1 Abstract. In this paper, we present a modified subspace system identification algorithm that ensures the negative imaginary property in the identified model. This is achieved by imposing constraints that guarantee negative imaginariness and stability of the identified model. The importance of having negative imaginary model comes from the fact that the negative imaginary property occurs in many important applications. For instance, flexible structure systems with collocated force actuators and position sensors are negative imaginary systems. Obtaining a mathematical model for this class of systems using system identification methods often results in inaccurate models that poorly reflect the negative imaginary property. This paper aims to overcome this problem. As an application of these results, an example of modelling a flexible system with a piezoelectric actuator and position sensor is presented. Keywords: Negative Imaginary Systems, Flexible Structures, Subspace System Identification. AMS Subject Classification: 49K30.

1. Introduction A common solution to issues of robustness, stability, and performance in the control of highly resonant flexible structures is to use force actuators combined with collocated measurements of velocity, position, or acceleration [12,29,31]. Here, collocation refers to the fact that the sensors and actuators have the same location and the same direction [30]. Collocated control with velocity measurements, known as negative-velocity feedback, can be used to directly increase the effective damping in the system, thereby facilitating the design of controllers that can guarantee closed-loop stability in the presence of parameter variations and unmodeled plant dynamics [30]. Similarly, a class of collocated actuators with position measurements, known as positiveposition feedback controllers, where velocity sensors are replaced with position sensors, can also be used to increase damping in flexible systems as discussed in [12]. Also, positive-position feedback controllers are robust against uncertainties in resonant frequencies as well as unmodeled plant dynamics, in a similar way to negative-velocity feedback controllers [12, 18, 29]. Structure with collocated force actuators and position measurements are known to be negative imaginary systems [18, 29]. The negative imaginary (NI) property is defined, in the single input single output (SISO), by considering the properties of the imaginary part of the system frequency response G(jω) and requiring the condition j (G(jω) − G(jω)∗ ) ≥ 0 for all ω ∈ (0, ∞). Systems with flexible structure dynamics such as flexible robot manipulators [38], ground and aerospace vehicles [15], atomic force microscopes (AFMs) [4, 24] often have colocated force actuators and position 1

School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy Canberra ACT 2600, Australia e-mail: [email protected],[email protected] 2 Mathematics Department, Faculty of Science, Suez Canal University, Egypt 3 Electrical Power and Machines Engineering, Mansoura University, Egypt Currently at Intelligent Systems and Informatics Lab, Graduate School of Information Science and Technology, the University of Tokyo e-mail: [email protected]. This research was supported by the Australian Research Council Manuscript received 02 September 2015. 336

M.A. MABROK et al.: SYSTEM IDENTIFICATION ALGORITHM...

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sensors and hence, can be modeled as NI systems. Another area where the underlying system dynamics are NI, are nano-positioning systems; see e.g., [4, 8–11, 25, 26, 32, 33, 36]. Also, the positive-position feedback control scheme in [12, 13], can be considered using the NI framework. Furthermore, other control methodologies in the literature such as integral resonant control (IRC) [28] and resonant feedback control [14, 23], fit into the NI framework and their stability robustness properties can be explained by NI systems theory. One important property of NI systems is the stability robustness of interconnected NI systems. This property has been studied in [18, 29]. It has been shown that a necessary and sufficient condition for the internal stability of a positive-feedback control system (see Fig. 1) consisting of an NI plant with a transfer function matrix of G(s) and a strictly negative imaginary (SNI) ¯ controller with a transfer function matrix of G(s) is given by the DC gain condition ¯ λmax (G(0)G(0)) < 1, (1) where the notation λmax (·) denotes the maximum eigenvalue of a matrix with only real eigenvalues. This stability result has been used in a number of practical applications [1, 4, 5, 7, 9, 24].

w1 +

u1

y2

G(s)

G(s)

y1

u2

+

w2

Figure 1. A negative-imaginary feedback control system. If the plant transfer function matrix G(s) is NI and ¯ the controller transfer function matrix G(s) is SNI, then the positive-feedback interconnection is internally ¯ stable if and only if the DC gain condition, λmax (G(0)G(0)) < 1, is satisfied.

In [4, 24], the NI stability result is applied to nano-positioning in an atomic force microscope. In [9], an IRC scheme based on the stability results provided in [18, 29] is used to design an active vibration control system for the mitigation of human induced vibrations in light-weight civil engineering structures, such as floors and footbridges via proof-mass actuators. In [7], the NI stability result is applied to the problem of decentralized control of large vehicle platoons. A positive position feedback control scheme based on the NI stability result provided in [18, 29] is used to design a novel compensation method for a coupled fuselage-rotor mode of a rotary wing unmanned aerial vehicle in [1]. In addition, it is shown in [35] that the class of linear systems having NI transfer function matrices is closely related to the class of linear Hamiltonian inputoutput systems. Also, an extension of the NI systems theory to infinite-dimensional systems is presented in [27]. A mathematical model is a description of a system using mathematical concepts. There different types of mathematical models, such as statistical models, differential equations, and dynamical systems (to which we will limit our discussion). In addition to being of great importance in the natural sciences, mathematical models are of prime importance in Engineering as well. Engineers analyze the system to be controlled or optimized almost exclusively in terms of one of its mathematical models. There are two main approaches for constructing a mathematical model of a particular (dynamic) system: the analytical approach, and system identification. In the first approach, also called white-box modelling, the model is developed from the first principles (physical laws) that govern the workings of the systems. However, it is not always possible to have a transparent system (i.e., a system whose internals are fully-known), and in many cases the developed models will be overly complex to be practical, or even impossible to obtain at all. In the second approach, the model of the dynamical system is developed from experimental

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data: the behaviour of the system in response to external stimulation is measured (inputs and outputs), then the mathematical relation between them is determined, without going into the details of what is actually happening inside the system. There are two types of models in the field of system identification: grey-box model, and black-box models. A black-box model is the complete opposite to a white-box model; there is no knowledge of any of the internal workings of the system, and therefore, the model must be developed solely based on input-output data. The grey-box model lies somewhere in between the two previous types of models; although it is not fully known what is going on inside the system, a certain model with free parameters is constructed based on both insight into the system and measurements. These parameters are then estimated using system identification. On important class of grey-box model identification algorithms is subspace identification. Subspace system identification algorithms have many advantages over classical identification algorithms. In addition to completely avoiding convergence and parametrization-related problems in the classical approach, it is more user friendly (there is only one parameter to be specified by the user, the order of the system) and they provide a unified geometric framework in which seemingly different models (deterministic, stochastic, and combined deterministicstochastic models) are treated in a unified manner. System identification methods are widely used in the field of control system design, signal processing and many other areas [20, 34, 37]. However, the resulting model may not exactly describe the true dynamics of the underlying system, especially under noisy measurements due to sensitivity to boundary conditions and environmental effects. In such cases, the identified system models can sometimes lead to mathematical models that do not reflect the actual characteristics of the underlying system. For example, the process of system identification when applied to linear time-invariant (LTI) systems which are known to be NI from physical conditions might lead to a model which is not NI. In such cases, the system model should be perturbed to enforce the underling NI dynamics. This problem has been addressed in [5, 22]. In [22] a first-order perturbation method is proposed for iteratively collapsing the frequency bands where the negative imaginary property is violated and finally displacing the eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the negative imaginary dynamics. However, the technique presented in [22] is found to be only successful where the NI violated region is very small. A system identification algorithm which enforces the NI constraint is proposed in [5] for estimating model parameters. However, in [5], the NI enforcement comes only in identifying the matrices B and D in a state space model. In other words, no stability constraint is enforced, which is one of the requirements of a system to be NI. In this paper, we impose constraints that guarantee negative imaginariness and stability of the identified model. This has been achieved by modifying the subspace system identification algorithm by adding two constraints that guarantee the NI property. Unlike the method presented in [5], the NI property is considered when estimating the full state space model at the beginning of the estimation process. Also, [5] considers the frequency domain approach to a subspace system identification whereas we consider the time domain approach. A preliminary conference version of this paper presented in [21]. This paper is further organized as follows: Section 2 recalls the definitions of NI systems. Section 3, introduces the subspace system identification method. In Section 4, we present a modified subspace system identification method that guarantees the NI property. In Section 5.1, an example is presented to illustrate the results of the paper. Also, a practical example of modeling a flexible system with a collocated force actuator and position sensor is given in Section 5.2. Section 6 concludes the paper.


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2. Negative imaginary systems Negative imaginary systems theory was introduced by Lanzon and Petersen in [17,18]. In the single-input single-output (SISO) case, such systems are defined by considering the properties of the imaginary part of the frequency response G(jω) and requiring the condition j (G(jω) − G(jω)∗ ) ≥ 0 for all ω ∈ (0, ∞). Roughly speaking, SISO NI systems are stable systems having a phase lag between 0 and −π for all ω > 0. That is, their Nyquist plot lies below the real axis; i.e., see Fig. 2 when the frequency varies in the open interval (0, ∞) (for strictly negative-imaginary systems, the Nyquist plot should not touch the real axis except at zero frequency and/or at infinity). This is similar to PR systems where the Nyquist plot is constrained to lie in the right half of the complex plane [3, 6].

Nyquist Diagram

−0.6

−0.4

−0.2

0 Real Axis

0.2

0.4

0.6

0.8

1

Figure 2. Positive-frequency Nyquist plot of a negative-imaginary system. A single-input, single-output negative-imaginary transfer function has no poles in the closed right half of the complex plane and has a frequency response with negative imaginary part for all frequencies. Consequently, the Nyquist plot for ω > 0 is contained in the lower half of the complex plane.

The first definition of negative imaginary systems is introduced in [17, 18], which is given as follows: Definition 1. A square transfer function matrix G(s) is NI if the following conditions are satisfied: 1. All of the poles of G(s) lie in the open left half plane; 2. For all ω > 0, such that jω is not a pole of G(s), and j (G(jω) − G(jω)∗ ) ≥ 0. A generalized definition for NI systems has been introduced by Xiong et. al. in [39, 40]. This generalization allows for simple poles on the imaginary axis of the complex plane except at the origin. Definition 2. [40] A square transfer function matrix G(s) is NI if the following conditions are satisfied: 1. G(s) has no pole at the origin and in Re[s] > 0; 2. For all ω > 0, such that jω is not a pole of G(s), j (G(jω) − G(jω)∗ ) ≥ 0; 3. If jω0 , is a pole of G(jω), it is at most a simple pole and the residue matrix K0 = lims→jω0 (s − jω0 )sG(s) is positive semidefinite Hermitian.

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2.1. Negative Imaginary Lemma. A state-space characterization of NI systems in terms of a pair of linear matrix inequalities (LMIs) has been given in [17, 18]. This result is analogous to the positive-real lemma [3, 6] and thus is referred to as the negative imaginary lemma. This result also generalized in [40][ to include ] poles on the imaginary axis except at the origin. A B Lemma 1 (See [40]). Let be a minimal state space realization of a transfer function C D matrix G(s). Then G(s) is NI if and only if det(A) ̸= 0, D = DT and there exists a real matrix P > 0 such that AP + P A∗ ≤ 0, (2) and B = −AP C ∗ .

(3)

2.2. Strict Negative Imaginary Lemma. One of the motivations of NI theory is to derive a stability condition for the positive feedback interconnection between subsystems that satisfy the NI property. To present such results on the robust stability of positive-position feedback and related control schemes, the following definition of strictly negative imaginary (SNI) systems has been introduced in [17, 18, 40]. Definition 3. A square transfer function matrix G(s) is SNI if the following conditions are satisfied: (1) G(s) has no pole in Re[s] ≥ 0; (2) For all ω > 0, j (G(jω) − G(jω)∗ ) > 0. A linear time-invariant system is SNI if its transfer function matrix is SNI. The following two results from [29] give sufficient conditions for the SNI property in term of a state space characterization. [ ] A B Lemma 2 (See [29]). Let be a minimal state space realization of a transfer function C D matrix G(s). Then G(s) is SNI if the following conditions are satisfied. (1) All eigenvalues of A are in OLHP; (2) D = DT ; (3) There exists a real matrix Yˆ > 0 and [ ]α, ϵ such that −α is not an [ positive] numbers [ ] A 0 ˆ = B , C −I satisfy eigenvalue of A and the matrices Aˆ = ,B ϵI 0 −αI ˆ = −AˆYˆ Cˆ ∗ . AˆYˆ + Yˆ Aˆ∗ ≤ 0 and B (4) ] A B be a minimal state space realization of a transfer function Lemma 3 (See [29]). Let C D matrix G(s). Then G(s) is SNI if the following conditions are satisfied: (1) All eigenvalues of A are in the open left half of the complex plane (OLHP), (2) D = DT , (3) There exists a real matrix Yˆ > 0 and positive numbers α, ϵ and  β such that α ̸= β, A 0 0 ˆ = 0 ,B −α, −β are not an eigenvalues of A and the matrices Aˆ =  0 −αI 0 0 −βI   B [ ] ϵI  , C −I −I satisfy ϵI [

ˆ = −AˆYˆ Cˆ ∗ . AˆYˆ + Yˆ Aˆ∗ ≤ 0 and B

(5)

Another SNI lemma has been given in [19]. This lemma gives a necessary and sufficient condition for the strong SNI property.


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[

] A B Lemma 4 (See [19]). Let a state space realization of a transfer function matrix C D G(s). Suppose G(s) + G(−s)T has normal rank m and (C, A) is observable. Then, A is Hurwitz and G(s) is SNI with lim jω(G(jω) − G(jω)∗ ) > 0

jω→∞

and 1 lim j (G(jω) − G(jω)∗ ) > 0, jω→0 ω if and only if D = DT and there exists a matrix Y > 0 such that AY + Y A∗ < 0 and B = −AY C ∗ .

(6)

3. Subspace system identification methods In this section, we will give a brief introduction to the subspace system identification method. The discrete time system can be described as follows, x(k + 1) = Ad x(k) + B d u(k),

(7)

y(k) = C d x(k) + Dd u(k),

(8)

where Ad ∈ Rn×n , B d ∈ Rn×m , C d ∈ Rm×n , and Dd ∈ Rm×m . The main idea of the time domain m×q and output Y ∈ Rm×q system identification method uses set of time domain [ d input ]U ∈ R d A B measurements to estimate the system matrices , where m is the number of the C d Dd inputs and outputs and q is the number of measurements. [ d ] According to [37], there are two main A Bd methods of estimating the system matrices . These two methods are illustrated in C d Dd Fig. 3. Input-Output Data

Orthogonal or oblique projection

State Sequences

Least squares

System Matrices

Classical Identification

System Matrices

Kalman filter

Kalman States

Figure 3. Two types of system identification methods, the left hand side explains the subspace system identification method. The right hand side explains the classical method of system identification.

In this paper, we focus in the subspace system identification approach. In subspace system identification, there are two major steps. Step one: In this step, the input u(k : k + q) and output y(k : k + q) measurements are used to estimate the states of the system X(k : k + q) ∈ Rn×q . This is done using block Hankel matrices formed from the input u(k : k + q) and output y(k : k + q).

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Step two: In this step, the estimated states X(k : k + q) are used to setup a least squares problem the state space matrices [ d to dobtain ] A B . This least squares problem can be formed as follows; C d Dd

([ ]

X(k + 1 : k + q)

W min −

y(k : k + q − 1) Ad ,B d ,C d ,D d [ d ][ ]) 2

A B d X(k : k + q − 1) ˆ , − W (9) d d

u(k : k + q − 1) C D ˆ are weighting matrices. Note that u(k : k + q) and y(k : k + q) are the given where W and W input and output and X(k : k + q) is the estimated states from step one.

4. Negative imaginary subspace system identification In this section, we will introduce a subspace system identification algorithm with NI property constraints as given in the NI lemma, Lemma 1. The NI lemma is formulated in continuous time. Therefore, a bilinear transformation [2] in the following form, 1 (I + Ad )−1 (Ad − 1), T 1 B = √ (I + Ad )−1 B d , T 1 d C = √ C (I + Ad )−1 , T d D = D − C d (I + Ad )−1 B d , A=

(10)

will be used to transform the conditions (2) and (3) into corresponding discrete time conditions. The LMI (2) is transformed from continuous to a discrete time form as follows: AP + P AT ≤ 0 ⇔(I + Ad )−1 (Ad − 1)P + P (Ad − 1)(I + Ad )−1 ≤ 0, T

T

T

T

⇔(Ad P − P )(I + Ad ) + (P + Ad P )(Ad − I) ≤ 0, T

T

T

⇔Ad P + Ad P Ad − P − P Ad + P Ad + Ad P Ad

T

− P − Ad P ≤ 0, T

T

⇔Ad P Ad − P + Ad P Ad − P ≤ 0, T

⇔Ad P Ad − P ≤ 0.

(11)

Also, the equality (3) can be transformed to the following corresponding discrete time form: B = −AP C T 1 T T ⇔ B d = − (Ad − I)P (I + Ad )−1 C d , T

(12)

where the Lyapunov inequality in the discrete form (11) and the equality (12) are non-convex conditions in the state space matrices and P , which is a computational issue. We start by replacing the matrix inequality (11) by a strict inequality. In fact, we restrict the identified model to be a strict stable NI system. In other words, we use the SNI lemma, Lemma 4, where


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the Lyapunov inequality is AP + P AT < 0 rather than Lemma 1. As in [16], we assume that there is exist an α > 0 such that P ≥ αI, T

Ad P Ad − P ≤ −αI. Using Schur complements, (13) and (14) can be written as follows; [ ] P − αI Ad P ≥ 0, T P Ad P ] [ P − αI Q ≥ 0, ⇔ QT P

(13) (14)

(15)

where Q = Ad P . The LMI (15) is now convex in the variables Q and P . Furthermore, the matrix Ad can be recovered from Ad = QP −1 . Now, the optimization problem (9) can be separated into two sub-optimization problems (see [16] for more explanations of why this is possible). The first sub-optimization problem is to solve for the matrices (Ad , B d ) as follows;

([ ] min W1

X(k + 1 : k + q) − Ad ,B d

[ ]) 2

[ d ] X(k : k + q − 1) d ˆ . − A B W

u(k : k + q − 1)

(16)

The second sub-optimization problem is to solve for the matrices (C d , Dd ) as follows;

([ ] min

W2 y(k : k + q − 1) − C d ,D d

−

[

Cd

Dd

]) 2 [

] X(k : k + q − 1) ˆ . W

u(k : k + q − 1)

(17)

By choosing the weighting matrix W in (16) as follows; [ ]T X(k : k + q − 1) ˆ W = × u(k : k + q − 1) ([ ][ ]T )† X(k : k + q − 1) X(k : k + q − 1) × × u(k : k + q − 1) u(k : k + q − 1) [ ] P R1 0 × . (18) 0 R2 [ ] X(k : k + q − 1) Note that is assumed full row rank (as q > n + 1) and this assumption u(k : k + q − 1) reasonable when sufficient data is collected. Also, R1 and R2 are new weighting matrices that can be chosen by the designers to tune the optimisation. This implies that the sub-optimization problem (16) can be combined with the LMI (15) constraint as follows;

( [ ] [ ]) 2

ˆ − QR1 B d R2 min W1 X(k + 1 : k + q) W

,

P,Q,B

subject to [ ] P − αI Q ≥ 0. QT P

(19)

(20)

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By solving this optimization problem, we obtain the matrices (Ad , B d , P ). Then, using (12), we can calculate the matrix C d . The matrix Dd , can be obtained by solving the optimization problem (17). Then the corresponding continuous time state space models are obtained using (10). 5. Examples 5.1. Example 1. In this example, a fourth order transfer function in the form; 2s4 + 1.2s3 + 78.18s2 + 22.6s + 728 (21) s4 + 0.6s3 + 38.09s2 + 11.4s + 345 is used to generate a set of outputs y(tk ) corresponding to a random set of inputs u(tk ). Then, the inputs u(tk ) and the outputs y(tk ) is used to test our proposed NI subspace system identification algorithm and compare the algorithm with the standard subspace system identification algorithm presented in [2]. It is shown in Fig. 5.1 that the NI constrained subspace system identification algorithm (green line) gives a better fit to the data generated from the transfer function (21) (red dashed line) compared with the standard subspace system identification algorithm [2] (blue line) for the same model order. Also, the NI model obtained from the constrained subspace method (green line) satisfies the NI property. This fact can be easily verified from the phase plot in Fig. 5.1, since the phase of this model lies between [0, −180o ], see, [18, 29]. However the system model obtained from the standard subspace method (blue line) does not satisfy the NI property. Bode Diagram

Un−constrained algorithm NI cinstrained algorithm Data

0

1

10

10

2

10

Frequency (rad/s)

Figure 4. Bode plot of models obtained using the NI constrained subspace system identification algorithm (green line), standard subspace system identification algorithm [2] (blue line) and the original transfer function given in (21).

5.2. Example 2. This section presents an application of modeling a flexible system with two colocated piezoelectric patches, see Fig. 5 and Fig. 6. Here, one piezoelectric patches acts as an actuator while the other acts as a sensor. The system has one input and one output: the input is the voltage Va applied to the piezoelectric actuator, whereas the output is the voltage Vs produced by the piezoelectric sensor. As this system involves collocated force actuators and position sensors, its transfer function should satisfy the NI property; e.g., see [29]. A DSPACE system is used to generate a random signal as an input u(t) signal to the PZTactuator through a high voltage amplifier. Then, the PZT-sensor is used to measure the output y(t) signal as shown in Fig. 5. The signals (u(t), y(t)) are used in the identification process. In this example, we employ both the standard and the NI constrained subspace system identification algorithms to identify the transfer function of the given system. The weighting matrices


345

W1 , R1 , R2 are chosen to be identity matrices. A fifth order model is identified using both algorithms. It is shown in Fig. 7 that the NI constrained subspace system identification algorithm (green line) gives a better fit to the data (red dashed line) compared with the standard subspace system identification algorithm [2] (blue line) for the same model order. Also, the NI constrained subspace method (green line) satisfies the NI property. This fact can be easily verified using the phase plot in Fig. 7, since the phase of this model lies between [0, −180o ], see, [18, 29], whereas the standard subspace system method (blue line) does not satisfy NI property.

A/D

PZT-Sensor

PZT-Actuator DSPACE Random Signal

D/A

Voltage Amp.

Figure 5. Flexible beam with piezoelectric actuator and piezoelectric sensor.

Figure 6. Flexible beam with piezoelectric actuator and piezoelectric sensor.

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Bode Diagram Un−constrained algorithm NI cinstrained algorithm Data

1

10

Frequency (rad/s)

2

10

Figure 7. Bode plot of models obtained by NI constrained subspace system identification algorithm (green line), standard subspace system identification algorithm [2] (blue line) and the measured data.

6. Conclusion In this paper, we presented a subspace system identification algorithm is for negative imaginary systems. The modified algorithm guarantees the negative imaginary property in the identified model. This subspace system identification algorithm can be used in the systems where the dynamics are known to be negative imaginary. An example of modelling a system with collocated force actuator and position sensor is presented. In this example, it has been shown that the modified algorithm gives a closer fit compared with the standard subspace system identification algorithm.

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Mohamed A. Mabrok - is currently working as a Postdoctoral fellow at Capability Systems Centre, University of New South Wales, Canberra Australia, at the Australian Defence Force Academy. He is also a lecturer at the Mathematics Department, Faculty of Science, Suez Canal University, Egypt. His main research includes Control Systems, Systems Engineering, System Design and Complex Systems. Dr. Mabrok received his B.Sc. in 2003 and M.Sc in 2009 in Applied Mathematics from the University of Suez Canal, Egypt. In 2013, he received his Ph.D. in Electrical Engineering from School of Engineering and Information Technology, University of New South Wales Canberra

Mohamed A. Haggag - is currently a research student at Intelligent Systems and Informatics Lab, Graduate School of Information Science and Technology, the University of Tokyo. Mr. Haggag holds a bachelor degree in Electrical Engineering from Electrical Power and Machines Engineering, Mansoura University, Egypt. His main research interests are in Robotics, Automations and Artificial Intelligence.

Ian R. Petersen - was born in Victoria, Australia. He received the Ph.D degree in electrical engineering from the University of Rochester, Rochester, NY, in 1984. From 1983 to 1985, he was a Postdoctoral Fellow at the Australian National University. In 1985, he joined the University of New South Wales at the Australian Defence Force Academy, Canberra, Australia, where he is currently a Scientia Professor and an Australian Research Council Federation Fellow in the School of Information Technology and Electrical Engineering. He has served as an Associate Editor for the IEEE transactions on automatic control, Systems and Control Letters, Automatica, and SIAM Journal on Control and Optimization. Currently he is an Editor for Automatica. His main research interests are in robust control theory, quantum control theory, and stochastic control theory.