BIO

The current researches of this group are as follows:

- The research about finding an algorithmic method for analysis of equilibrium point of quasi-homogeneous systems as an important part of autonomous systems, related to Hilbert’s 16
^{th}problem and Arnold’s question, associated with the indirect method of Lyapunov, - To design robust controllers based on linear and nonlinear H_infinity methods for industrial systems such as a half-car active suspension system with hydraulic actuators and RTAC (TORA), and in addition to this, to design the controllers for the surgical robots having haptic technology, by a state-feedback control law that minimizes an infinite/finite horizon cost function within the framework of linear/nonlinear matrix inequalities, and especially, simultaneously establishes robust stability conditions presented by a Lyapunov function,
- To develop and tailor the techniques of state-dependent Riccati equation (SDRE) filtering so as to rigorously estimate states and parameters of the nonlinear systems with uncertainty, exposed to unknown disturbance inputs, and also to apply SDRE filter such as robust SDRE filter based on the H-infinity norm minimization criterion to stochastic systems,
- To study Neuroscience as a multidisciplinary field, which we believe it will be significantly influenced by dynamical systems theory. We have been actively engaging in the research about modeling of the neurons, especially located at V1, and optic nerves and also investigating methods for stimulation and recording of the nerve cells, including noninvasive and invasive methods, such as Steady State Visually Evoked Potential (SSVEP) stimulation, Transcranial Magnetic Stimulation (TMS), microelectrode stimulation, Optogenetics stimulation, Electroencephalogram (EEG) recording, Electrocorticography (ECoG) recording, Local Field Potential (LPF) recoding, and Spikes recording.

ARAS Group helps me increase my self confidence to face with different challenges.

**Nonlinear Analysis**

We apply a wide variety of mathematical methods to analyzing system nonlinear behaviors. Moreover, we try to present new useful definitions and theorems which can assist us to accomplish the goal. Additionally, we are enthusiastic about pathological behaviors which common methods fail to analyze them such as Perron effect in the largest Lyapunov exponent sign, global attractivity without stability in Liénard type systems, and aperiodic long-term behavior of a non-hyperbolic strange attractor.

Most our efforts has been around determining invariant sets and their stability. In the first place, we investigate not only various definitions of invariant sets, but also different definitions of stability so as to find the definitions which can be appropriate to the needs of the analysis. In the second place, we conduct assorted methods such as topological, geometric, and algebraic methods which have the ability to be used for studying ordinary and partial differential equations, especially in state space. Particularly, we pursue to devise algorithms in order for finding the invariant sets of polynomial systems and solving their stability problem, specifically in the sense of V. Arnold.

Figure 1: Phase portrait of a system with two centers

**Control Methods**

A wide variety of control methods are considered in the group for control analysis and synthesis of dynamical systems. Among them robust control synthesis for delayed systems is one of the focuses of our research group. Time-delay appear in many systems and processes and is usually a source of instability. We have proposed an PD/PI controller synthesis for linear systems with uncertain input delay which may leads to a time-delay system of retarded or neutral type. In a neutral type system, in which delay appears both in its state and the derivatives of state, the resulting delay coefficients depend on the controller gains and makes the controller synthesis more challenging. To tackle this problem a new bounded real lemma (BRL) is introduced and the design of an output feedback PD controller for a system with uncertain time-invariant input delay is addressed. This method is further extended for designing a PI controller by augmenting an integrator to the system model.

H_infinity robust control synthesis is extensively applied in many practical systems of interest in the group. H_infinity based robust torque control of harmonic drive systems, identification of RTAC (TORA) and Robust H_infinity Control synthesis, Impedance control of a flight simulator yoke, Decentralized Robust Controller design for a half-car active suspension system with hydraulic actuators, Robust Linear Controller, Composite QFT, Nonlinear H_infinity , Robust H_2 /H_infinity and Mixed Sensitivity Approaches, for flexible joint robots with Phase Uncertainty, may be named among them.

H_infinity robust control synthesis is also applied on an MPC algorithm for non-linear discrete-time systems. The systems are composed of a linear constant part perturbed by an additive state-dependent non-linear term. The control objective is to design a state-feedback control law that minimizes an infinite horizon cost function within the framework of linear matrix inequalities. In particular, it is shown that the solution of the optimization problem can stabilize the non-linear plants. Three extensions, namely, application to systems with input delay, non-linear output tracking and using output-feedback, are followed naturally from the proposed formulation.

Figure 2: Gershgorin disks for a randomly generated 4 by 4 complex matrix

**Observer Design and Identification Methods**

State-dependent Riccati equation (SDRE) filtering techniques have been extensively used for nonlinear state/parameter estimation within a wide range of applications. The standard SDRE filter, which is set up by direct SDC parameterization, demands complete knowledge of the system model, and the disturbance inputs characteristics, which severely degrade its performance in practical applications. We proposed a robust SDRE filter based on the H_infinity norm minimization criterion, to effectively estimate the states of nonlinear uncertain systems exposed to unknown disturbance inputs. Extension of this work in considered for exponential stability, and in stochastic domain.

Currently, for highly uncertain models the proposed robust SDRE is combined with a switching algorithm. The theoretical development of this filter and its robustness analysis is reviewed and its implementation on the uncertain nonlinear model of Lithium-Ion battery is considered. The stimulation and the implementation results verify the efficiency of the extended filter, which is quite promising to be implemented on other nonlinear and uncertain plants.

Figure 3: Generic functional block diagram of SDRE Architecture with Built-in Parameter Estimator

**Brain Computer Interface (BCI)**

Brain-computer interface (BCI) establish a connection between the brain activities and robotic systems through converting the brain signals into perceptible control signals for machines. Such a system has been a specific area of interest in order to provide interaction of people with disabilities with the surrounding environment. Using the system, the brain patterns for a certain type of behavior is obtained and the related control commands is produced. This control signal represents a specific neuro-behavior of the brain and can be used in BCI systems. From the EEG signals, SSVEP may be retracted as a specific type of control signal which is produced at the occipital lobe of the brain in response to an external oscillating stimulus. The frequency of this signal is matched with the frequency of the input stimuli that can be identified using EEG test, which is the focus of the current research.

As the brain signals are the neurological reaction of the individuals to stimulus signals it is crucial to design a suitable stimulus system as well as investigation of its effectiveness. In our research group, a suitable visual stimulus system is designed and implemented, and its effectiveness is proved through experiments. Various statistical and pattern identification methods such as CCA, MSI and MEC for coding the main characteristics of the SSVEP signals are implemented in the test bed by recording data of several individuals. For having a physical perception of the BCI system a robotic arm is used. An integrated system comprising the robotic arm and commercial Emotiv Brainwear^{®} is developed and the final implementation is performed on the robotic arm. The experimental results on different subjects shoes the promising horizons of using such technology for disabled people.

Figure 4 : Control of the Avatar by EEG signal