Keywords: Stability, learning based control
Abstract: Model-based control has witnessed tremendous progress over the last 100 years. In the era of artificial intelligence and autonomous systems, traditional model-based control theoretical methods are insufficient to addressing emerging control applications tied to networks of super autonomous systems involving V2X communications and operating in complex dynamically changing environments. Learning-based control is a new topic aimed at developing computationally simple, analytically tractable (reinforcement) learning algorithms. These algorithms yield direct adaptive optimal controllers from data collected online in real time. In this talk, I will first review recent developments in adaptive dynamic programming (ADP) for adaptive optimal control of unknown dynamical systems. Then, I will present robust adaptive dynamic programming (RADP) for continuous-time linear and nonlinear systems with dynamic uncertainties. The effectiveness of RADP as a new framework for data-driven adaptive and optimal nonlinear control design is demonstrated via its applications to autonomous vehicles and biological motor control.

Keywords: Input-To-State Stability, Lyapunov Methods, Hybrid Nonlinear Control Systems
Abstract: Time-varying ISS-Lyapunov functions for impulsive systems provide a necessary and sufficient condition for ISS. This property makes them a more powerful tool for stability analysis than classical candidate ISS-Lyapunov functions providing only a sufficient ISS condition. Moreover, time-varying ISS-Lyapunov functions cover systems with simultaneous instability in continuous and discrete dynamics for which candidate ISS-Lyapunov functions remain inconclusive. The present paper links these two concepts by suggesting a method of constructing time-varying ISS-Lyapunov functions from candidate ISS-Lyapunov functions, thereby effectively combining the ease of construction of candidate ISS-Lyapunov functions with the guaranteed existence of time-varying ISS-Lyapunov functions.

Keywords: Input-To-State Stability, Stability, Lyapunov Methods
Abstract: For attracting closed sets that are not compact we consider different formulations of input-to-state stability properties and related Lyapunov criteria. In the compact case, there exists a well-established hierarchy of such properties and some formulations have been quickly discarded as equivalent to input-to-state stability. In this paper we show for the noncompact case, several new phenomena appear. In particular, input-to-state stability (ISS) does not imply integral input-to-state stability, and ISS is not equivalent to integral-input-to-integral-state stability. The criteria are formulated in terms of measurement functions, which allows a uniform presentation of a number of related results.

Keywords: Lyapunov Methods
Abstract: This paper discusses the use of the so-called congelation of variables method in the reference tracking problem for a class of nonlinear systems with matched, time-varying, parametric uncertainty. It is shown that the tracking problem using the proposed framework can be decomposed into a zero-regulation problem and a disturbance rejection problem. To reduce the size of the disturbance term, a dynamical system with known input is incorporated into the parameter update law to generate a time-varying nominal parameter. To reject the remaining disturbance term, a sign-like function is exploited, which achieves an adjustable tracking performance in an average sense, while guaranteeing boundedness of all trajectories of the closed-loop system. The theory is illustrated by solving a path-tracking problem and it is shown that the proposed control scheme outperforms the classical scheme in the presence of time-varying parameters.

Keywords: Lyapunov Methods, Numerical Methods, Small Gain Theorems
Abstract: In this paper, we consider nonlinear control systems and discuss the existence of a separable control Lyapunov function. To this end, we assume that the system can be decomposed into subsystems and formulate conditions such that a weighted sum of Lyapunov functions of the subsystems yields a control Lyapunov function of the overall system. Since deep neural networks are capable of approximating separable functions without suffering from the curse of dimensionality, we can thus identify systems where an efficient approximation of a control Lyapunov function via a deep neural network is possible. A corresponding network architecture and training algorithm are proposed. Further, numerical examples illustrate the behavior of the algorithm.

Keywords: Lyapunov Methods, Stabilization, Stability
Abstract: We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additional term that vanishes in a neighborhood of the origin. The augmented controller outperforms the original controller in terms of the estimate of the region of attraction. The paper translates the results discussed in Braun et al. (2022a,b), from the continuous-time setting to the discrete-time setting, and numerically verifies that the results derived for continuous-time systems are recovered if the discrete-time system is obtained through an Euler discretization of a continuous-time system with a sufficiently small sampling rate.

Keywords: Optimal Control, Mechatronic Systems, Algebraic Methods
Abstract: In this paper, a new approach to tuning and optimisation of controlled systems regarding the tracking behaviour is presented. This approach can be understood as an extension to the iterative feedback tuning (IFT) approach known from the literature. Motivated by the sensitivity concept, the IFT algorithm is extended to both linear and nonlinear systems in state-space description with static state-feedback control, resulting in the proposed iterative state-feedback tuning (ISFT) method. The main contribution consists of the derivation of first- and second-order sensitivity functions for quadratic cost functions, which is typical for control optimisation problems. The proposed approach results in a gradient-based iterative algorithm for control parameter adaptation. Finally, exemplary simulation results will be presented for a nonlinear system, demonstrating the usability of this new approach.

Keywords: Optimal Control, Optimization and Scheduling, Feasibility and Stability Issues
Abstract: In this paper, we investigate the properties of competitive equilibriums for dynamic multi-agent systems (MAS) over an infinite horizon. When there is no external resource supply, a group of dynamic agents with distributed resource allocations form a transactive market to share their resources at a specific price. Each agent makes decisions on the locally consumed resource and the traded resource. The system reaches a competitive equilibrium when each agent’s payoff is maximized and the tradings are balanced. Firstly, we consider general utility functions and show that under feasibility assumptions, any competitive equilibrium maximizes the social welfare. Secondly, we prove that for sufficiently small initial conditions, the social welfare maximization solution constitutes a competitive equilibrium with zero price. We also prove for general feasible initial conditions, there exists a time instant after which the optimal price, corresponding to a competitive equilibrium, becomes zero. Finally, we specifically focus on quadratic MAS for which the system-level social welfare optimization becomes a classical constrained linear quadratic regulator (CLQR) problem. We construct explicitly a feasible set of initial conditions under which the price under a competitive equilibrium is zero for all time.

Keywords: Optimal Control, Robotics, Time Optimal Control
Abstract: We investigate the problem of finding paths that enable a robot modeled as a Dubins car (i.e., a constant-speed finite-turn-rate unicycle) to escape from a circular region of space in minimum time. This minimum-time escape problem arises in marine, aerial, and ground robotics in situations where a safety region has been violated and must be exited before a potential negative consequence occurs (e.g., a collision). Using the tools of nonlinear optimal control theory, we show that a surprisingly simple closed-form feedback control law solves this minimum-time escape problem, and that the minimum-time paths have an elegant geometric interpretation.

Keywords: Optimal Control, Time Optimal Control, Model Based Control
Abstract: Optimal control problems with free terminal time present many challenges including nonsmooth and discontinuous control laws, irregular value functions, many local optima, and the curse of dimensionality. To overcome these issues, we propose an adaptation of the model-based actor-critic paradigm from the field of Reinforcement Learning via an exponential transformation to learn an approximate feedback control and value function pair. We demonstrate the algorithm's effectiveness on prototypical examples featuring each of the main pathological issues present in problems of this type.

Keywords: Robustness, Stabilization, Optimization and Scheduling
Abstract: This manuscript proposes a finite-time extremum-seeking control system for a class of unknown nonlinear dynamical systems. A dual-mode approach is considered. The dual-mode control action combines a proportional feedback operating in the time-scale of the system and a slow perturbation-based extremum-seeking gradient descent. The resulting two time-scale extremum seeking control formulation is shown to achieve practical finite-time stability of the system to the optimum equilibrium conditions for the state variables and the input variables. A brief simulation study is presented to demonstrate the effectiveness of the proposed technique.

Keywords: Applications of Observer Design, Observer and Filter Design By Observer Error Linearization, Fault Detection
Abstract: A systematic methodology for designing of linear unknown input observers (UIO) for fault estimation in nonlinear systems is presented. Necessary and sufficient conditions for the existence of linear UIOs for nonlinear systems are derived and as long as these conditions are satisfied, we obtain explicit design formulas for the observer. The method is applied to a non-isothermal chemical reactor (CSTR) subject to a concentration sensor fault.

Keywords: Filter Design, State Estimation and Applications
Abstract: A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement update steps. In addition, a numerically robust square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems, including the state estimation for a challenging non-linear system.

Keywords: Observability and Observer Design
Abstract: KKL observer design consists in transforming the system dynamics into a filter of the output, which admits a trivial observer, and left-inverting the transformation to recover an estimate of the state in the system coordinates. This left-inversion is typically guaranteed under a backward-distinguishability condition. In this paper, instead, we demonstrate how this KKL approach may also be applied without any such observability assumption. We show that there exist appropriate choices of the filter such that any filter solution asymptotically contains the full information about the state indistinguishable class, namely the set of points generating the same output. Then, we investigate the existence of a set-valued left-inverse allowing to estimate asymptotically this indistinguishable class, in the Hausdorff sense. We prove that the estimate tends to be included asymptotically in the indistinguishable classes of the limit points of the system solution. Finally, we provide a numerical example illustrating this convergence.

Keywords: Observability and Observer Design, State Estimation and Applications, Filter Design
Abstract: Low-power multi high gain observers (LP MHGO) are proven to be effective in reducing the peaking of state estimation of nonlinear systems to an arbitrarily small magnitude. Moreover, they reduce the sensitivity of estimates to measurement noise. They also relax the numerical implementation problem of high gain observers by using gains powered up to the order of 2 instead of n. In this paper, we aim to further reduce the noise sensitivity of these observers by employing low-pass filters in the observer dynamics. The main results establish the convergence of the estimation error to zero with an arbitrarily small decay rate in the absence of noise, as well as an input to state stability feature when the noise is present. We also demonstrate in the linear case that the proposed observer improves the upper bound on the estimates. Simulation results compare the performance of the proposed observer with similar works and show the effectiveness of the proposed method.

Keywords: State Estimation and Applications, Filter Design, System Structure Identification
Abstract: State estimation when only a partial model of a considered system is available remains a major challenge in many engineering fields. This work proposes a joint, square-root unscented Kalman filter to estimate states and model uncertainties simultaneously by linear combinations of physics-motivated library functions. Using a sparsity promoting approach, a selection of those linear combinations is chosen and thus an interpretable model can be extracted. Results indicate a small estimation error compared to a traditional square-root unscented Kalman filter and exhibit the enhancement of physically meaningful models.

Keywords: Bilinear Systems, Linear Algebraic Methods, Geometric Methods
Abstract: In this paper, we address the problem of model reduction by moment matching for quadratic-bilinear time-delay systems. The moment of such systems is characterized by a power series representation, which can be computed by solving an infinite-dimensional system of Sylvester-like equations. A family of parameterized reduced-order models that preserves the quadratic-bilinear time-delay structure is derived through matching an approximate moment defined by truncating the power series. The developed theory is demonstrated on a numerical example.

Keywords: Fault Detection, Variable Structure Control and Sliding Mode, Applications of Observer Design
Abstract: In this paper, we study event-triggered fault reconstruction using sliding mode observer (SMO) for a linear time-invariant system. Here, the SMO is designed with an event condition in which output measurements are transmitted over the network. The observer provides the state estimates using the sampled output values in the presence of fault signals. It is shown that with a suitable (switching) gain and the triggering parameter, the observer states converge to the true states (in the practical sense), and thus, the fault signals can be reconstructed with an accuracy dependent on the steady-state error bound. Finally, simulation results are shown to illustrate fault detection employing the proposed approach.

Keywords: Numerical Methods, Nonlinear Model Predictive Control Theory and Applications, Parameter Estimation
Abstract: This paper proposes a new type of subspace state-space system identification method for nonlinear dynamical systems, which generates a model consisting of a state estimator and a predictor that can be directly used for model predictive control (MPC). The main feature of the proposed method is that it uses a neural network with a bottleneck layer between the state estimator and predictor to represent the input-output dynamics, and it is proven that the state of the dynamical system can be extracted from the bottleneck layer based on the observability of the target system. The training of the network is shown to be a natural nonlinear extension of the subspace state-space system identification method established for linear dynamical systems. This correspondence gives interpretability and optimality to the resulting model based on linear control theory. The usefulness of the proposed method and the interpretability of the model are demonstrated through an illustrative example of MPC.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Stability, Lyapunov Methods
Abstract: This paper studies the nonlinear systems obtained by considering a wave equation in closed loop with a nonlinear dynamical boundary controller. The controller is subject to a magnitude limitation and modeled by a linear ordinary differential equation with a saturation map in the input. The well-posedness of the obtained infinite-dimensional system is first studied and then two stability results are given. These two stability results apply for two cascade cases and give sufficient conditions for the asymptotic stability of the equilibrium. The well-posedness is proven by using nonlinear semigroups techniques, whereas the global asymptotic stability results are obtained by Lyapunov-based arguments in infinite-dimensional state space.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Passivity, Dissipativity
Abstract: Different from mechanic or reversible systems, such as port-Hamiltonian systems (PHS), which preserves the energy or the Hamiltonian, irreversible PHS (IPHS) also satisfy the second law of Thermodynamics, i.e., the internal entropy production of the system is always greater or equal to zero. Hence, when considering the interconnection of IPHS, it must be so that the first and second laws of Thermodynamics are satisfied, implying that the interconnection must be power-preserving and entropy-increasing. In this work the conditions for a thermodynamic admissible interconnection of IPHS have been studied and characterized. The interconnection law is given by a state modulated input-output feedback, in which each modulating function is related and defined by the corresponding irreversible thermodynamic driving force induced by the interconnection. The interconnection law also encompasses the reversible interactions, and can be interpreted as a generalization of a power-preserving interconnection to deal with thermodynamic systems. The result has been illustrated on the abstract interconnection of purely thermodynamic and thermo-mechanic systems, and on the examples of an ideal heat-exchanger and a gas-piston system.

Keywords: Geometric Methods, Hybrid Nonlinear Control Systems, Automotive Systems Marine Systems
Abstract: In this paper, we develop multiple synergistic hybrid feedback control laws for mechanical systems on matrix Lie groups with left-invariant metrics. With the goal of globally asymptotically tracking a desired reference trajectory, we propose a hybrid proportional-derivative (PD) type control law and an output feedback version which only utilizes configuration measurements. Moreover, to ensure global asymptotic tracking in the presence of a constant and unknown disturbance in the system dynamics, we introduce two novel proportional-integral-derivative (PID) type control laws with slightly different properties in terms of gain selection and integral action. The theoretical developments are validated through numerical simulation of an underwater vehicle.

Keywords: Optimal Control, Computational Efficiency
Abstract: Reachability analysis is a powerful tool when it comes to capturing the behaviour, thus verifying the safety, of autonomous systems. However, general-purpose methods, such as Hamilton-Jacobi approaches, exhibit exponential computational complexity with respect to the state dimension. In this paper, we show that supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate under- and over-approximating reachable sets for nonlinear systems, and based on this, we develop a scheme for approximating reachable sets of linear time-invariant systems via ellipsoids with polynomial computational complexity.

Keywords: Control of Mechanical, Electrical and Process Systems, Robotics
Abstract: High performance trajectory tracking control of quadrotor vehicles is an important challenge in aerial robotics. Symmetry is a fundamental property of physical systems and offers the potential to provide a tool to design high-performance control algorithms. We propose a design methodology that takes any given symmetry, linearises the associated error in a single set of coordinates, and uses LQR design to obtain a high performance control; an approach we term Equivariant Regulator design. We show that quadrotor vehicles admit several different symmetries: the direct product symmetry, the extended pose symmetry and the pose and velocity symmetry, and show that each symmetry can be used to define a global error. We compare the linearised systems via simulation and find that the extended pose and pose and velocity symmetries outperform the direct product symmetry in the presence of large disturbances. This suggests that choices of equivariant and group affine symmetries have improved linearisation error.

Keywords: Geometric Methods, Control of Sampled Data Systems
Abstract: In this contribution, we present a constructive method to derive flat sampled-data models for continuous-time flat systems through an implicit Euler-discretization. We show how the sampled-data model can be used subsequently for a flatness-based controller design, and illustrate our results with the well-known planar VTOL aircraft example.

Keywords: Geometric Methods, Numerical Methods, Stabilization
Abstract: Feedback linearization (FL) is a powerful tool to enable control synthesis for nonlinear systems. Since FL is a local construction, an important notion is the domain of validity of the procedure, and in particular, quantitative estimates of the domain of applicability of the procedure. Such quantification is missing in the existing literature and this article fills in this niche using the mathematical tool of the implicit function theorem. The results provide lower bounds on the region in which a given discrete-time system is feedback linearizable.

Keywords: Variable Structure Control and Sliding Mode, Lyapunov Methods
Abstract: Recently, passivity-based sliding mode control has been proposed for mechanical port-Hamiltonian systems. This controller has properties of both sliding mode control and passivity-based control, and Lyapunov stability is ensured by a Hamiltonian function with a non-smooth potential function. In the authors' previous study, a special form of non-smooth potential function is considered and there are few parameters to adjust the controller for various control objectives. This paper proposes a new passivity-based sliding mode controller based on a wider class of potential functions. This approach enables us to reduce chattering and improve the behavior in reaching mode by adjusting parameters. The effectiveness of the proposed method is demonstrated by a numerical simulation.

Keywords: Feasibility and Stability Issues, Disturbance Atténuation, Nonlinear Model Predictive Control Theory and Applications
Abstract: This paper addresses offset-free reference tracking of asymptotically constant reference signals and disturbances using Model Predictive Control with a Long Short-Term Memory Network model. Basically, an error free output estimation and an error free nominal tracking for time-varying references are proposed for an Incremental Input-to-State Stable Long Short-Term Memory Network. This implies an offset-free Model Predictive Control algorithm.

Keywords: Optimal Control, Dynamical Systems Techniques, Aerospace and Marine Applications
Abstract: In this paper, we present an application of reinforcement learning for automatic ship's berthing system. In the proposed method, the model-based reinforcement learning is used to generate non-linear feedback controller. The proposed system is composed of an Actor-Critic algorithm suitable for continuous state and a function approximator by RBF networks. To evaluate the performance of the proposed system, extensive computer simulations and actual sea tests are carried out using small training ship Shioji-Maru III under various conditions. As a result, we can see that the proposed non-linear optimal feedback controller by reinforcement learning is useful for designing automatic berthing system for the ship.

Keywords: Optimal Control, Dynamical Systems Techniques, Bilinear Systems
Abstract: While Koopman-based techniques like extended Dynamic Mode Decomposition are nowadays ubiquitous in the data-driven approximation of dynamical systems, quantitative error estimates were only recently established. To this end, both sources of error resulting from a finite dictionary and only finitely-many data points in the generation of the surrogate model have to be taken into account. We generalize the rigorous analysis of the approximation error to the control setting while simultaneously reducing the impact of the curse of dimensionality by using a recently proposed bilinear approach. In particular, we establish uniform bounds on the approximation error of state-dependent quantities like constraints or a performance index enabling data-based optimal and predictive control with guarantees.