Keywords: Robotics, Mechatronics and Mechanical Systems
Abstract: A harsh robotic environment is defined as one where it is difficult to sense, communicate, control, move around or otherwise operate a cyberphysical system – for which robots are one example. The harshest environments include underwater and space (Earth orbit). Traditional control theory constructs a plant model to design, develop and test a proposed control methodology. Uncertainties in the plant are always acknowledged as caveats when qualifying the performance and efficacy of a methodology. The robot’s interaction with an unstructured dynamic environment precludes a well-defined plant. This presentation will look at the use cases of underwater and space environments and note similarities and differences in their harsh environment control challenges. Finally, promising research directions and examples are presented.

Keywords: Time-domain Methods, Approximation Methods
Abstract: There are a number of recent results aimed at developing finite necessary and sufficient stability tests for linear time-invariant time delay systems within the framework of Lyapunov-Krasovskii functionals with prescribed derivative approach. These tests are based on the concept of Lyapunov matrix. They usually require verification of positive definiteness of a certain block matrix whose blocks correspond to the Lyapunov matrix evaluated at several discretization points. In this work, we present a new test of this kind which employs a combination of the discretized Lyapunov functionals method of K. Gu and the functionals with prescribed derivative approach. Unlike existing combinations of those techniques, we avoid discretization of the functional's derivative and use the methodology of the necessary and sufficient stability tests development instead. We show a connection between our test and some of the existing results.

Keywords: Infinite-dimensional Systems and Delays, Time-domain Methods, Approximation Methods
Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters.

Keywords: Time-domain Methods
Abstract: Presenting a stability finite criteria of systems with delays, which are infinite-dimensional, expressed in terms of the delay Lyapunov matrix was a milestone. However, the computational burden of the stability test is often heavy. In this note, we introduce some slight modifications to reduce this burden.

Keywords: Time-domain Methods, Delays in Biological Systems
Abstract: Integral delay systems with piecewise constant kernel are studied. The delay Lyapunov matrix for the case of commensurate delays is computed by solving an auxiliary boundary value problem of delay-free linear matrix differential equations. The relation between the solution of the auxiliary system and the delay Lyapunov matrix is discussed. It is also shown that if the integral delay system satisfies the Lyapunov condition, then the delay Lyapunov matrix is unique.

Keywords: Time-domain Methods, Infinite-dimensional Systems and Delays
Abstract: We present a necessary and sufficient test for the stability of neutral type linear time-delay systems. Our approach combines the Lyapunov functional expressed in terms of the delay Lyapunov matrix for this class of systems with a piecewise linear approximation of the functional argument. As a result, the functional is approximated via quadratic form whose matrix provides the stability test for a system. A bound on the discretization step required for the test to be necessary and sufficient is given.

Keywords: Sampled-Data Control, Stabilization, Time-varying Delays
Abstract: This work reviews some recent results about event-triggered control of time-delay systems, following a Lyapunov-based approach. Event-based control aims at saving computational resources and controlling the system whenever it really needs attention, and has not been exhaustively explored so far in the case of fully nonlinear systems affected by time delays. Particular attention is given in this paper to the design of event-triggered controllers in the context of sampled-data stabilization of nonlinear systems with state delays.

Keywords: Infinite-dimensional Systems and Delays, Approximation Methods
Abstract: In recent years we provided numerical methods based on pseudospectral collocation for computing the Floquet multipliers of different types of delay equations, with the goal of studying the stability of their periodic solutions. The latest work of the series concerns the extension of these methods to a piecewise approach, in order to take the properties of numerically computed solutions into account. We present this method and its MATLAB implementation with practical usage examples, including instances where the piecewise approach is essential and the novel application to neutral renewal equations.

Keywords: Infinite-dimensional Systems and Delays, Approximation Methods, Delays in Biological Systems
Abstract: Via pseudospectral discretization, a delay equation can be approximated with a system of ODEs, whose bifurcation properties can be studied with existing software for ODEs, like MatCont for MATLAB. We discuss the method highlighting in particular the parallelism between the approximation of retarded functional differential equations and renewal equations, summarizing the convergence results and the strengths of the method in applications.

Keywords: Infinite-dimensional Systems and Delays, Approximation Methods
Abstract: An effective and versatile numerical approach for the bifurcation analysis of a delay equation consists in deriving a system of ordinary differential equations and investigating its dynamics by using existing software, e.g., MatCont. We present an extension of MatCont implementing the method and providing a user-friendly graphical interface for the analysis of delay equations.

Keywords: Frequency-domain Methods, Stabilization
Abstract: The aim of this paper is to give a presentation of the Python toolbox YALTAPy dedicated to the stability study of standard and fractional delay systems as well as its online version YALTAPy_Online. Both toolboxes are derived from YALTA whose functionalities will be recalled here. Examples will be given to show how these toolboxes may be used.

Keywords: Infinite-dimensional Systems and Delays, Approximation Methods, Predictor-based Control
Abstract: Modeling, prediction, and control of time-delay systems have been attracting researchers for decades. Most of the designed controllers in the literature have an FIR structure, which needs to be approximated by a finite-dimensional system. Many methods have been proposed to reduce the order of models, which approximates the original system under specific criteria. This paper proposes another model reduction method, which aims to minimize the H-infinity norm of the difference between the reduced model and the original system. An FIR structure obtained for a generalized Smith Predictor controller design is used to compare the results of the proposed method with one of the most used model order reduction methods in the literature.

Keywords: Approximation Methods, Time-domain Methods, Stabilization
Abstract: In this paper, we apply the semi-discretization method to controllable single-input linear systems with input delay and fractional-order feedback. This requires the fractional derivative to be discretized in a way that the resulting discrete map is linear and time-invariant. To this end, three different techniques are investigated, namely, the short-memory principle, the application of adaptive time steps, and the exponential approximation of the derivative weights. For all three approaches, we construct a linear map that describes the dynamics of the approximate semi-discrete system. The matrices of these linear maps can be used to investigate the stability of the system. As an example, stability charts are determined for the fractional-order proportional-derivative delayed control of the inverted pendulum, and the numerically obtained stability charts are compared with the exact stability boundaries.

Keywords: Structural Properties, Complex Systems, Uncertain Systems - SSSC
Abstract: This work presents a graph theoretic approach to the investigation of stability and stabilizability of discrete-time structured linear systems - i.e., discrete-time dynamical systems defined by linear maps whose entries are only known to be either zero or nonzero (unknown) values. The main result consists in a necessary and sufficient condition for each element of the family of systems represented by a given discrete-time structured linear system to be asymptotically stable. In particular, under the stated condition, convergence to zero of the free state evolution of each system of the family is shown to be achieved in a finite number of steps, through what will be referred to as a dead-beat behavior. The notions of essential state feedback and essential output injection are then introduced and a sufficient condition for stabilizability by essential state feedback and by essential output injection, respectively, is given. An obstruction to stabilizability by essential state feedback or by essential output injection, respectively, is also pointed out.

Keywords: Infinite-dimensional Systems and PDEs Models, Structural Properties, Time-invariant Systems
Abstract: The paper considers the problem of gramians computation for linear hyperbolic distributed parameter systems. Two different cases are considered: vibrating string and beam systems. The presented approach is based on directly deriving the equations solutions by using time - space separation of variables and the Fourier series representation method. The initial problem framework is based on the state space formulation for infinite dimensional systems. This framework uses operators defined over Hilbert spaces and implements the concept of a C_0 – strongly continuous semigroup generated by bounded system operator. The solution of the hyperbolic partial differential equations is divided in two parts. The zero input part is due to the initial conditions and participates in obtaining the observability gramian of the system. The zero state part is a consequence of the input signal effect and is used to compute the controllability gramian.

Keywords: Infinite-dimensional Systems and PDEs Models, Feedback Control, Drilling Systems, Vibration and Control
Abstract: In this paper, we give explicit criteria to guarantee the robust stabilization of simple first-order transport systems coupled with finite-dimensional dynamics, using Proportional or Proportional Integral boundary output feedback controllers. Using a frequency-sweeping approach developed for Time-Delay Systems, we characterize the closed-loop stability intervals in the parameter space for several interconnected partial differential equations and ordinary differential equations configurations. The performances of the proposed boundary feedback controllers are compared with other classical approaches in simulation. The proposed methodology is a necessary step before a detailed comparison between proportional integral controllers with low computational complexity and other control approaches for infinite-dimensional systems.

Keywords: Time-invariant Systems, Structural Properties
Abstract: In this paper, we analyse the positive invariance of polyhedral sets with respect to the trajectories of linear dynamical systems represented in terms of delay-differential equations. An appropriate model transformation is employed, together with a matrix parametrization which allows exploiting system's structure by decoupling delay-dependent modes from delay-independent ones. Then, positive invariance conditions are obtained with explicit dependence on the delay, seen as a parameter. Connections and equivalence of positive invariance between the original and the transformed systems are established. The derived conditions are used to tackle the problem of computing a feedback control law which makes a given polyhedron positively invariant with respect to an input-delayed linear system. Illustrative examples complete the presentation.

Keywords: Feedback Control, Regulation, Disturbance Rejection
Abstract: Extending the Remaining Useful Life (RUL) of dynamic systems functioning in closed loop in accordance with damage progression dynamics is a challenging task. Such target combines the challenges emanating from the domain of Prognostic Health Management (PHM) and engineering of the control theory. The main contribution of the paper consists in the synthesis and the analysis of two control reconfiguration strategies in order to achieve such objective.

This paper presents two control strategies, one reconfigures the controls input and the other reconfigures the setpoint. The first structure modifies the controls input sent to the system using a modulation parameter. The second structure proposes a modification to the operational setpoint of the system's control loop. These modulations are obtained from an optimization algorithm making it possible to achieve a trade-off between the dynamic performance requirement and the RUL criteria.

The optimization algorithm is based on the prediction of the RUL, the estimation of the deterioration. A numerical example illustrates the use of each of these two strategies, through the results of estimating the deterioration, predicting RUL, and obtaining the modulation parameter from the optimization. The RUL extension and its impact on performance trade-off are illustrated to evaluate the performances of both strategies.

Keywords: Delays in Biological Systems
Abstract: We consider networks of oscillator nodes with time delayed, global circulant coupling. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of phase-locked oscillations. We apply the theory to a network inspired by biological neural networks in the cortex. Each node consists of two populations: one excitatory and one inhibitory. There is non-delayed coupling between excitatory and inhibitory populations in the same node and time-delayed coupling between excitatory population in different nodes.

Keywords: Aerospace Dynamics, Stability and Stabilization
Abstract: We first give a brief review of classical Gain-Scheduled (GS) controllers with our design example of Stability/Control Augmentation System (S/CAS) for Quad-Tilt-Wing UAV (QTWUAV), and we clarify the issues to be overcome from the S/CAS design example. Then, a brief review of Linear Parameter-Varying (LPV) GS controller design, which has been proposed to overcome the issues in classical GS controller design, is given from the viewpoint of the practicality. Finally, several methods, which have been proposed by the author, are briefly reviewed with some verification results using JAXA's research airplane MuPAL-alpha. This paper ends with concluding remarks and future research topics related to LPV GS controllers.

Keywords: Complex Systems, Other Applications of Linear and Fractional Systems
Abstract: In this discussion paper, we survey recent research surrounding robustness of machine learning models. As learning algorithms become increasingly more popular in data-driven control systems, their robustness to data uncertainty must be ensured in order to maintain reliable safety-critical operations. We begin by reviewing common formalisms for such robustness, and then move on to discuss popular and state-of-the-art techniques for training robust machine learning models as well as methods for provably certifying such robustness. From this unification of robust machine learning, we identify and discuss pressing directions for future research in the area.

Keywords: Delay Compensation, Distributed Delays, Control Design
Abstract: In this note, we present an effective solution to the stabilization of linear input delay systems subject to dissipative constraints while all the effect of input delay is compensated by a controller with novel structure. The method is inspired by the recent development in the mathematical treatment of distributed delays and predictor controllers, which are critical for the derivation of the solution. An important conceptual innovation is the use of a parameterized dynamical state feedback controller (DSFC), where the dimension of the controller equals the dimension of the control input. A sufficient condition for the existence of a dissipative DSFC is obtained via the Krasovskii functional approach, where the condition includes a bilinear matrix inequality (BMI). To solve the BMI, we apply an inner convex approximation algorithm which can be initialized based on an explicit construction of a predictor controller gain. The proposed DSFC can be considered as an extension of the classical predictor controller, thereby capable of compensating all the effects of the pointwise input delay while satisfying dissipative constraints. A numerical example is given to illustrate the effectiveness of our proposed methodology.

Keywords: Time-varying Delays, Robust Analysis, Time-domain Methods
Abstract: The generalized Myshkis problem consists in analysis of robust stability of time-delay system under variation of its parameters within a set in an infinite-dimensional space. In our paper, we consider a linear system, where the delay is a bounded function under an additional restriction on its growth rate. Some sufficient conditions of solvability of the generalized Myshkis problem are expressed in the form of quadratic convex optimization task. The new approach is based on a modification of the Lyapunov-Razumikhin method.

Keywords: PID Control, Control Design, Delay Estimation
Abstract: This paper discusses the feasible region of PI control parameters for first-order nonlinear stochastic time-delay systems. The system can robust asymptotically track a set-point under any PI control parameters within the feasible region. First, the first-order nonlinear stochastic time-delay systems is transformed into a linear stochastic time-delay systems with the uncertainty. Second, for the fixed PI control parameters, the maximal allowable time delay under PI control is obtained by using the genetic algorithm (GA) and the generalized eigenvalue problem. Third, a new algorithm is proposed to compute the feasible region of PI control parameters for a fixed allowable time delay. Moreover, the improved algorithm is also proposed to reduce the time complexity. In particular, the stochastic disturbance and time delay disappear, the feasible region of PI control parameters obtained by the algorithm is consistent with the previous work. Finally, two simulation examples show the feasibility and effectiveness of the proposed two algorithms, respectively.

Keywords: Infinite-dimensional Systems and Delays, Control Design, Other Applications of Time Delay Systems
Abstract: This paper presents the design and analysis of the extremum seeking for static maps with input governed by a parabolic partial differential equation (PDE) of the diffusion type defined on a time-varying spatial domain described by an ordinary differential equation (ODE). This is the first effort to pursue an extension of extremum seeking from the heat PDE to the Stefan PDE. We compensate the average-based actuation dynamics by a controller via backstepping transformation for the moving boundary, which is utilized to transform the original coupled PDE-ODE into a target system whose exponential stability is proved. The local exponential convergence to a small neighborhood of the optimal point is proven by means of backstepping methodology, Lyapunov functional and averaging in infinite dimensions. The extension for the delay-compensated extremum seeking control of the Stefan problem is also discussed.

Keywords: Frequency-domain Methods, Infinite-dimensional Systems and Delays
Abstract: This paper applies a recent characterization of exponential stability for linear periodic difference delay systems to the stability of one-dimensional hyperbolic systems of PDE, and elaborates on the case where the delays are rationally independent.

Keywords: Delays and Vibration Control, Stabilization, Mechatronics and Mechanical Systems
Abstract: The dynamics of pressure relief valves are highly influenced by the environment of the valve. This model describes the dynamics of the valve mounted on a vessel equipped with a downstream pipe. The frequencies of the self-excited vibrations and the stability boundaries in the dimensionless plane of the pipe length and vessel volume are calculated in closed form. The aim of this study is to find the vessel volume and pipe length combination within the stable domain corresponding to the largest decay rate.

Keywords: Distributed Delays, Stabilization, Frequency-domain Methods
Abstract: Pole placement represents a classical method for controlling finite-dimensional linear time-invariant systems, largely covered in the open literature. Basically, it consists of placing the poles of the closed-loop system in some predetermined loci in the complex plane. This paper discusses some of the extensions of this method to linear systems described by delay-differential equations. Among others, the finite spectrum assignment (FSA), the continuous pole placement (CPP) and the partial pole placement (PPP) approaches are presented and illustrated through some simple low-order dynamical systems.

Keywords: Frequency-domain Methods, Robust Analysis
Abstract: The objective of this paper is to present a MATLAB-based toolbox called CSA-T-TDS (acronym for Complete Stability Analysis Toolbox for Time-Delay Systems), developed by the authors. By using this toolbox, one can easily find the whole stability delay-set for a linear system including commensurate delays. For a better understanding of the methodology at the origin of the software, the theoretical core of CSA-T-TDS (the frequency-sweeping approach together with the auxiliary characteristic function) is discussed. Then, the application of the toolbox will be demonstrated along with some numerical examples.

Keywords: Frequency-domain Methods, Stabilization, PID Control
Abstract: This paper presents the software entitled "Partial Pole Placement via Delay Action", or "P3delta" for short. P3delta is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays for dynamical systems. After recalling the theoretical foundation of the so-called "Partial Pole Placement" methodology we propose as well the main features of the current version of P3delta. We illustrate its use in feedback stabilization of several control systems operating under time delays.

Keywords: Control Design, Stabilization
Abstract: A new Matlab-based software called DrStabilization is introduced. The main purpose of the software is to stabilize multi-input multi-output linear time-invariant systems which may have an arbitrary number of discrete time-delays. The stabilization problem is formulated as the minimization of spectral abscissa function over the controller parameters. By modifying this problem, it is also possible to consider strong stability, i.e., stability achieved by a stable controller. The software allows designing structured dynamic output feedback controllers. By structuring the controller matrices, one can design decentralized controllers and/or well-known servocompansators such as PI/PID/PIR. Furthermore, the software also allows designing time-delay controllers in which case an arbitrary number of artificial time-delays can be introduced both to the I/O channels and the dynamics of the controller itself. The utilization of the software is illustrated through examples.

Keywords: Stabilization, Control Design, Robust Control
Abstract: TDS-CONTROL is an integrated MATLAB package for the analysis and controller-design of continuous-time, linear time-invariant state-space systems with discrete delays. The code can deal with a wide range of systems including retarded and neutral systems. Delays can be present in the state, input, output, and direct feed-through terms. Also systems described by Delay Differential Algebraic Equations (DDAE) can be considered. Firstly, TDS-CONTROL provides functionality for computing the (strong) spectral abscissa, the (strong) H-infinity norm and the pseudo-spectral abscissa with respect to structured, real-valued uncertainties that can affect both the system matrices and the delays. Secondly, TDS-CONTROL contains controller design methods. These controller design algorithms are based on optimizing one (or a combination) of the (performance) measures mentioned above, with respect to the controller parameters.

Keywords: Approximation Methods, Frequency-domain Methods, Time-domain Methods
Abstract: We present SSD, Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend the frequency-limited balanced reduction to delay systems first time, propose a computational algorithm and give its implementation. We collected two sets of benchmark problems on H2 norm computation and model reduction. SSD is publicly available in GitHub. Our reproducible paper and two benchmark collections are shared as executable notebooks.

Keywords: Uncertain Systems - LPVS, Observer Design, Robotics
Abstract: The information about vehicle side-slip angle is essential given its relation to lateral stability and its importance for active safety control systems. Since direct measurement of the side-slip angle is expensive, developing efficient indirect techniques to extract this information from available sensor measurements is essential. This paper proposes a novel, cost-effective indirect strategy for the robust estimation of the side-slip angle using a state observer that relies on a discrete-time Linear Parameter Varying (LPV) lateral vehicle dynamic model. The proposed algorithm relies on a finite-time interval LMI-based observer, which accounts for the uncertainties on the model parameters (cornering stiffness) with known upper and lower uncertainty bounds. The simulation results show the effective and robust estimation performance of the proposed observer under various changes in system parameters. Moreover, comparison results with H∞ set-membership observer for discrete-time LPV systems show the effectiveness of the developed observer

Keywords: Automotive Dynamics, Observer Design, Diagnosis and Control
Abstract: This paper proposes an H-infinity Non Linear Parameter Varying (NLPV) observer for fault estimation in semi-active Electro-Rheological (ER) suspensions. The damper fault (a loss-of-efficiency factor) is modeled as a lost force of unknown/free dynamics to be estimated. Thanks to the parameter-dependent descriptor-form system modeling, there is no assumption made on the fault dynamics, thus making this method applicable to all considered types of damper faults. The nonlinearity in the damper model is bounded by its Lipschitz property, while the road disturbance and the measurement noise are handled using the H-infinity condition. The observer is parameterized and then designed by solving Linear Matrix Inequalities (LMIs) and is implemented in a polytopic gain scheduling approach. Synthesis results including Bode plots and simulations illustrate the method in both the frequency and the time domains.

Keywords: Automotive Dynamics, Hinf Control
Abstract: This paper proposes a control design framework with guarantees for systems, which contains learning-based control elements. The framework is based on a supervisory control structure, which contains a supervisor, a robust Linear Parameter Varying (LPV) controller and the learning-based control elements. This paper presents the design of a lateral path following control for driver assistance systems, which is aided with a learning-based agent. In the paper the formulation of the supervisor, the design method of the robust LPV controller are provided, while the learning-based agent via an imitation learning process is considered to be given. The effectiveness of the method on driver-in-the-loop simulation scenario is demonstrated. It is shown that the proposed control system is able to provide guarantee on the limitation of the path following error, and to guarantee transition between the driver steering actuation and the automated control intervention.

Keywords: Automotive Dynamics, Uncertain Systems - LPVS, Predictive Control
Abstract: This paper proposes a control architecture for autonomous lane-keeping by a vehicle. In this paper, the vehicle dynamics consist of two parts: lateral and longitudinal dynamics. Therefore, the control architecture comprises two subsequent controllers. A longitudinal model predictive control (MPC) makes the vehicle track the desired longitudinal speeds that are assumed to be generated by a speed planner. The longitudinal speed is then passed to a lateral MPC for lane-keeping. Due to the dependence of the lateral dynamics on the longitudinal speed, they are represented in a linear parameter-varying (LPV) form, where its scheduling parameter is the longitudinal speed of the vehicle. In order to deal with the non precise information of the future longitudinal speed (the scheduling parameter), a nominal trajectory of the future longitudinal velocity is considered with a bound of uncertainty around it. Then, a tube-based LPV-MPC is adopted to control the lateral dynamics for attaining the lane-keeping goal. Moreover, a method for computing parameterized homothetic tubes with fewer number of vertices is proposed. In the end, by carrying out simulation tests, the effectiveness of the proposed methods is illustrated.

Keywords: Stability and Stabilization, Hinf Control, Automotive Dynamics
Abstract: This work presents a two-step LMI-based approach for the synthesis of State-Feedback and Static-Output-Feedback controllers for discrete-time Linear Parameter Varying systems. The benefit of the proposed approach lies on the fact that the two-step LMI method allows to enforce a common parametric structure on the controller for all the parameter space. This results in a final controller implementation that does not requires any interpolation step and thus, is of easy implementation. An illustrative example is given for an autonomous vehicle steering control design case.