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In this work, we leverage the 2-contraction theory, which extends the capabilities of classical contraction theory, to develop a global stability framework. Coupled with powerful geometric tools such as the Poincare index theory, the 2-contraction theory enables us to analyze the stability of planar nonlinear systems without relying on local equilibrium analysis. By utilizing index theory and 2-contraction results, we efficiently characterize the nature of equilibrium points and delineate regions in 2-dimensional state space where periodic solutions, closed orbits, or stable dynamics may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may occur, as well as 2-contraction regions that guarantee the nonexistence of such solutions. Additionally, we address a critical problem in engineering the determination of the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria identifying candidate BOAs becomes highly nontrivial. We propose a novel methodology leveraging the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple stable equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the practical utility of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing their efficacy in an opinion dynamics problem., Comment: 14 pages, 10 figures

This work addresses the problem of finding a consensus point in the state space ($\mathbb{R}^2$) for a multi-agent system that is comprised of $N$ identical double integrator agents. It is assumed that each agent operates under constrained control input (i.e., $|u_i(t)| \leq 1$ $\forall i = 1, \hdots N$). Further, a fixed fuel budget is also assumed i.e., the total amount of cumulative input that can be expended is limited by $\int_0^{t_f}|u(t)|dt \le \beta$. First, the attainable set $\mathcal{A}(t,x_0,\beta)$ at time $t$, which is the set of all states that an agent can attain starting from initial conditions $x_0$ under the fuel budget constraints at time $t$ is computed for every agent. This attainable set is a convex set for all $t\ge0$. Then the minimum time to consensus is the minimum time $\bar{t}$ at which attainable sets of all agents intersect, and the consensus point is the point of intersection. A closed-form expression for the minimum time consensus point is provided for the case of three agents. Then, using Helly's theorem, the intersection will be non-empty at a time when all the $N \choose 3$ triplets of agents have non-empty intersection. The computation of minimum time consensus for all $N \choose 3$ triplets is performed independently and can be distributed among all the $N$ agents. Finally, the overall minimum time to consensus is given by the triplet that has the highest minimum time to consensus. Further, the intersection of all the attainable sets of this triplet gives the minimum time consensus point for all $N$ agents.

Understanding conditions for the synchronization of a network of interconnected oscillators is a challenging problem. Typically, only sufficient conditions are reported for the synchronization problem. Here, we adopted the Lyapunov-Floquet theory and the Master Stability Function approach in order to derive the synchronization conditions for a set of coupled nonlinear oscillators. We found that the positivity of the coupling constant is a necessary and sufficient condition for synchronizing linearly full-state coupled identical oscillators. Moreover, in the case of partial state coupling, the asymptotic convergence of volume in state space is ensured by a positive coupling constant. The numerical calculation of the Master Stability Function for a benchmark two-dimensional oscillator validates the synchronization corresponding to the positive coupling. The results are illustrated using numerical simulations and experimentation on benchmark oscillators., Comment: 6 pages, 7 figures, Journal

In this letter, the incremental stability of interconnected systems is discussed. In particular, we consider an interconnection of switched nonlinear systems. The incremental stability of the switched interconnected system is a stronger property as compared to conventional stability. Guaranteeing such a notion of stability for an overall interconnected nonlinear system is challenging, even if individual subsystems are incrementally stable. Here, preserving incremental stability for the interconnection is ensured with a set of sufficient conditions. Contraction theory is used as a tool to achieve incremental convergence. For the feedback interconnection, the small gain characterisation is presented for the overall system's incremental stability. The results are derived for a special case of feedback, i.e., cascade interconnection. Matrix-measure-based conditions are presented, which are computationally tractable. Two numerical examples are demonstrated and supported with simulation results to verify the theoretical claims.

Physical Human-Robot Interaction (pHRI) task involves tight coupling between safety constraints and compliance with human intentions. In this paper, a novel switched model reference admittance controller is developed to maintain compliance with the external force while upholding safety constraints in the workspace for an n-link manipulator involved in pHRI. A switched reference model is designed for the admittance controller to generate the reference trajectory within the safe workspace. The stability analysis of the switched reference model is carried out by an appropriate selection of the Common Quadratic Lyapunov Function (CQLF) so that asymptotic convergence of the trajectory tracking error is ensured. The efficacy of the proposed controller is validated in simulation on a two-link robot manipulator.

In this paper an artificial delay based impedance controller is proposed for robotic manipulators with uncertainty in dynamics. The control law unites the time delayed estimation (TDE) framework with a second order switching controller of super twisting algorithm (STA) type via a novel generalized filtered tracking error (GFTE). While time delayed estimation framework eliminates the need for accurate modelling of robot dynamics by estimating the uncertain robot dynamics and interaction forces from immediate past data of state and control effort, the second order switching control law in the outer loop provides robustness against the time delayed estimation (TDE) error that arises due to approximation of the manipulator dynamics. Thus, the proposed control law tries to establish a desired impedance model between the robot end effector variables i.e. force and motion in presence of uncertainties, both when it is encountering smooth contact forces and during free motion. Simulation results for a two link manipulator using the proposed controller along with convergence analysis are shown to validate the proposition.

While the Artificial Pancreas is effective in regulating the blood glucose in the safe range of 70-180 mg/dl in type 1 diabetic patients, the high intra-patient variability, as well as exogenous meal disturbances, poses a serious challenge. The existing control algorithms thus require additional safety algorithms and feed-forward actions. Moreover, the unavailability of insulin sensors in Artificial Pancreas makes this task more difficult. In the present work, a subcutaneous model of type 1 diabetes (T1D) is considered for observer-based controller design in the framework of contraction analysis. A variety of realistic multiple-meal scenarios for three virtual T1D patients have been investigated with +30 % and -30 % of parametric variability. The average time spent by the three T1D patients is found to be 77 %, 73 % and 76 %, respectively. A significant reduction in the time spent in hyperglycemia (>180 mg/dl) is achieved without any feed-forward action for meal compensation.

A problem of computing time-fuel optimal control for state transfer of a single input linear time invariant (LTI) system to the origin is considered. The input is assumed to be bounded. Since, the optimal control is bang-off-bang in nature, it is characterized by sequences of +1 , 0 and -1 and the corresponding switching time instants. All (candidate) sequences satisfying the Pontryagin's maximum principle (PMP) necessary conditions are characterized. The number of candidate sequences is obtained as a function of the order of system and a method to list all candidate sequences is derived. Corresponding to each candidate sequence, switching time instants are computed by solving a static optimization problem. Since the candidate control input is a piece-wise constant function, the time-fuel cost functional is converted to a linear function in switching time instants. By using a simple substitution of variables, reachability constraints are converted to polynomial equations and inequalities. Such a static optimization problem can be solved separately for each candidate sequence. Finally, the optimal control input is obtained from candidate sequences which gives the least cost. For each sequence, optimization problem can be solved by converting it to a generalized moment problem (GMP) and then solving a hierachical sequence of semidefinite relaxations to approximate the minima and minimizer [1]. Lastly, a numerical example is presented for demonstration of method.

In this paper, detection of deception attack on deep neural network (DNN) based image classification in autonomous and cyber-physical systems is considered. Several studies have shown the vulnerability of DNN to malicious deception attacks. In such attacks, some or all pixel values of an image are modified by an external attacker, so that the change is almost invisible to the human eye but significant enough for a DNN-based classifier to misclassify it. This paper first proposes a novel pre-processing technique that facilitates the detection of such modified images under any DNN-based image classifier as well as the attacker model. The proposed pre-processing algorithm involves a certain combination of principal component analysis (PCA)-based decomposition of the image, and random perturbation based detection to reduce computational complexity. Next, an adaptive version of this algorithm is proposed where a random number of perturbations are chosen adaptively using a doubly-threshold policy, and the threshold values are learnt via stochastic approximation in order to minimize the expected number of perturbations subject to constraints on the false alarm and missed detection probabilities. Numerical experiments show that the proposed detection scheme outperforms a competing algorithm while achieving reasonably low computational complexity., Comment: 10 pages, 3 figures, 3 algorithms, 8 tables. Extension of the Conference paper:- Kartik Mundra, Rahul Modpur, Arpan Chattopadhyay, and Indra Narayan Kar. Adversarial image detection in cyber-physical systems. In Proceedings of the 1st ACM Workshop on Autonomous and Intelligent Mobile Systems, pages 1-5, 2020. Can be found at https://dl.acm.org/doi/abs/10.1145/3377283.3377285

In this paper, the tracking control problem of an Euler-Lagrange system is addressed with regard to parametric uncertainties, and an adaptive-robust control strategy, christened Time-Delayed Adaptive Robust Control (TARC), is presented. TARC approximates the unknown dynamics through the time-delayed estimation, and the adaptive-robust control provides robustness against the approximation error. The novel adaptation law of TARC, in contrast to the conventional adaptive-robust control methodologies, requires neither complete model of the system nor any knowledge of predefined uncertainty bounds to compute the switching gain, and circumvents the over- and underestimation problems of the switching gain. Moreover, TARC only utilizes position feedback and approximates the velocity and acceleration terms from the past position data. The adopted state-derivatives estimation method in TARC avoids any explicit requirement of external low pass filters for the removal of measurement noise. A new stability notion in continuous-time domain is proposed considering the time delay, adaptive law, and state-derivatives estimation which in turn provides a selection criterion for gains and sampling interval of the controller.

This work proposes a new adaptive-robust control (ARC) architecture for a class of uncertain Euler-Lagrange (EL) systems where the upper bound of the uncertainty satisfies linear in parameters (LIP) structure. Conventional ARC strategies either require structural knowledge of the system or presume that the overall uncertainties or its time derivative are norm bounded by a constant. Due to unmodelled dynamics and modelling imperfection, true structural knowledge of the system is not always available. Further, for the class of systems under consideration, prior assumption regarding the uncertainties (or its time derivative) being upper bounded by a constant, puts a restriction on states beforehand. Conventional ARC laws invite overestimation-underestimation problem of switching gain. Towards this front, Adaptive Switching-gain based Robust Control (ASRC) is proposed which alleviates the overestimation-underestimation problem of switching gain. Moreover, ASRC avoids any presumption of constant upper bound on the overall uncertainties and can negotiate uncertainties regardless of being linear or nonlinear in parameters. Experimental results of ASRC using a wheeled mobile robot notes improved control performance in comparison to adaptive sliding mode control., Comment: 10 pages, 9 figures

In this note, analysis of time delay systems using Lambert W function approach is reassessed. A common canonical form of time delay systems is defined. We extended the recent results of [6] for second order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above said method. A state variable transformation is used to transform the proposed class of systems into the common canonical form. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is analyesed by the help of Nyquist plot. The approach is validated through an example.

Convergence of controller parameters in standard model reference adaptive control (MRAC) requires the system states to be persistently exciting (PE), a restrictive condition to be verified online. A recent data-driven approach, concurrent learning, uses information-rich past data concurrently with the standard parameter update laws to guarantee parameter convergence without the need of the PE condition. This method guarantees exponential convergence of both the tracking and the controller parameter estimation errors to zero, whereas, the classical MRAC merely ensures asymptotic convergence of tracking error to zero. However, the method requires knowledge of the state derivative, at least at the time instances when the state values are stored in memory. The method further assumes knowledge of the control allocation matrix. This paper addresses these limitations by using a memory-based finite-time system identifier in conjunction with a data-driven approach, leading to convergence of both the tracking and the controller parameter estimation errors without the PE condition and knowledge of the system matrices and the state derivative. A Lyapunov based stability proof is included to justify the validity of the proposed data-driven approach. Simulation results demonstrate the efficacy of the suggested method.

In this paper, the tracking control problem of a class of uncertain Euler-Lagrange systems subjected to unknown input delay and bounded disturbances is addressed. To this front, a novel delay dependent control law, referred as Adaptive Robust Outer Loop Control (AROLC) is proposed. Compared to the conventional predictor based approaches, the proposed controller is capable of negotiating any input delay, within a stipulated range, without knowing the delay or its variation. The maximum allowable input delay is computed through Razumikhin-type stability analysis. AROLC also provides robustness against the disturbances due to input delay, parametric variations and unmodelled dynamics through switching control law. The novel adaptive law allows the switching gain to modify itself online in accordance with the tracking error without any prerequisite of the uncertainties. The uncertain system, employing AROLC, is shown to be Uniformly Ultimately Bounded (UUB). As a proof of concept, experimentation is carried out on a nonholonomic wheeled mobile robot with various time varying as well as fixed input delay, and better tracking accuracy of the proposed controller is noted compared to predictor based methodology.

Two different aspects of formation control of multiple agents subjected to linear transformation have been addressed in this paper. We consider a set of complex single integrator systems so that the dimension of the system reduces to half as opposed to the vector representation in Cartesian coordinate system. We first design a stable formation controller in an attempt to solve the formation control turned to stabilization problem and then find a collision avoidance controller in the transformed domain, respectively. Different linear transformations are used to facilitate the formation control task in a different way. For example Jacobi transformation is used to separate the shape control and trajectory control. The inverse of the transformation must have nonzero eigenvalues with both positive and negative real parts which may lead the system to instability. If the inverse of the transformation appears in closed loop then a diagonal stabilizing matrix is required to reassign the eigenvalues of the inverse of transformation in the right half of complex plane. The algorithm to find such stabilizing matrix is provided. We then define a matrix of potential in the actual domain which is the stepping stone to find a matrix of potential in the transformed domain. Thus collision avoidance controller can be designed directly in the transformed domain. The mathematical proof is given that both the actual and transformed system behaves identically. Simulation results are provided to support our claim.

Different geometric patterns and shapes are generated using groups of agents, and this needs formation control. In this paper, Centroid Based Transformation (CBT), has been applied to decompose the combined dynamics of nonholonomic Wheeled Mobile Robots (WMRs) into three subsystems: intra and inter group shape dynamics, and the dynamics of the centroid. The intra group shape dynamics can further be partitioned into the shape dynamics of each group, giving the notion of multiple group. Thus separate controllers have been designed for each subsystem. The gains of the controllers are such chosen that the overall system becomes singularly perturbed system, and different subsystems converge to their desired values at different times. Then multi-time scale convergence analysis has been carried out in this paper. Negative gradient of a potential based function has been added to the controller to ensure collision avoidance among the robots. Simulation results have been provided to demonstrate the effectiveness of the proposed controller., Comment: arXiv admin note: text overlap with arXiv:1412.6164

Formation control of multiple groups of agents finds application in large area navigation by generating different geometric patterns and shapes, and also in carrying large objects. In this paper, Centroid Based Transformation (CBT) \cite{c39}, has been applied to decompose the combined dynamics of wheeled mobile robots (WMRs) into three subsystems: intra and inter group shape dynamics, and the dynamics of the centroid. Separate controllers have been designed for each subsystem. The gains of the controllers are such chosen that the overall system becomes singularly perturbed system. Then sliding mode controllers are designed on the singularly perturbed system to drive the subsystems on sliding surfaces in finite time. Negative gradient of a potential based function has been added to the sliding surface to ensure collision avoidance among the robots in finite time. The efficacy of the proposed controller is established through simulation results., Comment: 8 pages, 5 figures