The Valse team studies the estimation and control problems arising in the analysis and the design of distributed, uncertain, and interconnected dynamical systems:

The Valse team works in the domains of control science: dynamical systems,
stability analysis, estimation, and automatic control. Our developments
are focused on the theoretical and applied aspects related to the control
and estimation of large-scale multi-sensor and multi-actuator systems
based on the use of the theories of finite-time/fixed-time/hyperexponential
convergence and homogeneous systems. The Lyapunov function method
and other methods of analysis of dynamical systems form a basis for
the studies in the Valse team.

The key idea of the research program for the team is that a fast (non-asymptotic) convergence of the regulation and estimation errors increases the reliability of intelligent distributed actuators and sensors in complex scenarios, such as interconnected cyber-physical systems (CPSs).

The expertise of Valse's members in theoretical developments of control and estimation theory (finite-time control and estimation algorithms in centralized context 84, 70, 81, 80, 77, homogeneity framework for differential equations 85, 72, 71, 73, 75, 86, 82, time-delay systems 74, 76, 89, distributed systems 83 and algebraic-based methods for estimation 87, 88) is an essential ingredient to achieve our objective.

The generic chart of different goals and tasks included in the scientific work program of Valse and interrelations between them are presented in Fig. 1. We have selected three main objectives to pursue with the related tasks to fulfill:

All these objectives are interconnected: from a particular problem
in an IoT application, it is planned to design control or estimation
algorithms, leading to the development of theoretical tools; and vice
versa, a new theoretical advance can provide a possibility for the development
of novel tools which can be used in applications.

To explain our motivation: why use finite time? Applying any method for control/estimation
has a price in terms of its advantages and disadvantages. There is
no universal framework that is the best always and everywhere. Finite-time
may appear as a luxurious property for a physical system, requiring
nonlinear tools. Of course, if an asymptotic convergence
and a linear model are enough for solving a given problem, then there
is no reason to develop something else. However, most of the present
problems in CPS and IoT are nonlinear (i.e., they have various
local behaviors that cannot be collected in only one linear model).
Design and analysis of various local linearized models and solutions are luxurious, too. The theory of homogeneity can go beyond linearity, offering many new features while not appearing as severe as other nonlinear tools and having almost all hints of the linear framework. Suppose that, thanks to the homogeneity theory, finite-time/fixed-time can be obtained with little difficulty while adding the bonuses of stronger robustness and faster convergence compared to the linear case? We are convinced that the price of going beyond linear control and estimation can be strongly dropped by maturing the theory of homogeneity and finite/fixed-time convergence. We are also convinced that it will be compensated in terms of robustness and speed, which can be demanded in the new areas of application such as IoT for example.

An objective of the team is the application of the developed control and estimation algorithms for different scenarios in IoT or CPSs. Participation in various potential applications allows the Valse team to better understand the features of CPSs and their required performances, and to properly formulate the control and estimation problems that must be solved. Here is a list of ongoing, past and potential applications addressed by the team:

It is worth highlighting a widespread distribution of various scientific
domains in the list of applications for the team given above. Such
interdisciplinarity for Valse is unsurprising since control theory is a science of systems whose interest today is, by
nature, to interface with other disciplines and their fields of application.
This is also well aligned with the domain of CPSs, which by its origin
requires multidisciplinary competencies.

Activities of the team related with social responsibility:

Activities of the team related with environmental responsibility:

Overall, social and environmental responsibility for researchers in the team involves using mathematical expertise to address societal issues, promote inclusivity, and contribute to environmental sustainability through research, collaboration, and outreach efforts.

HCS Toolbox for MATLAB ver. 0.1

This is the first release of HCS Toolbox for MATLAB. The list of MATLAB functions provided for homogeneous control systems design: (Homogeneous Objects) hnorm - computation of homogeneous norm hproj - computation of homogeneous projection hcurve - generation of points of a homogeneous curve hsphere - generation of a random grid on a homogeneous sphere (Homogeneous Control Design) hpc_design - Homogeneous Proportional Control (HPC) design hpci_design - Homogeneous Proportional-Integral Control (HPIC) design hsmc_design - Homogeneous Sliding Mode Controller (HSMC) design hsmci_design - design of HSMC with Integral action fhpc_design - Fixed-time HPC design fhpic_design - Fixed-time HPIC design lpc2hpc - upgrading Linear Proportional Control (LPC) to HPC lpic2hpc - upgrading Linear PI control (LPIC) to HPIC (Discretization of Homogeneous Control) e_hpc - explicit discretization of HPC e_hpc - semi-implicit discretization of HPC c_hpc - consistent discretization of HPC e_hpic - explicit discretization of HPIC e_hsmc - explicit discretization of HSMC si_hsmc - semi-implicit explicit discretization of HSMC e_hsmci - explicit discretization of HSMC with Integral action e_fhpc - explicit discretization of Fixed-time HPC si_fhpc - semi-implicit discretization of Fixed-time HPC e_fhpic - explicit discretization of Fixed-time HPIC (Homogeneous Observer Design) ho_design - Homogeneous Observer (HO) design fho_design - Fixed-time HO design lo2ho - upgrading Linear Observer (LO) to HO (Discretization of Homogeneous Observer) e_ho - explicit Euler discretization of HO e_fho - explicit Euler discretization of FHO si_ho - semi-implicit discretization of HO si_fho - semi-implicit discretization of FHO (Block forms ) block_con - transformation to block controlability form bloc_obs - transformation to block bservability form trans_con - transformation to partial block controlability form trans_con - transformation to partial block observability form output_form - transformation to reduced order output control system (Examples) demo_hnorm - demo of computation of a homogeneous norm demo_hsphere - plot of homogeneous balls in 2D demo_hpc - demo of HPC design and simulation demo_hpic - demo of HPIC design and simulation demo_hsmc - demo of HSMC design and simulation demo_hsmci - demo of HSMCI design and simulation demo_fhpc - demo of FHPC design and simulation demo_fhpic - demo of FHPIC design and simulation demo_lpc2hpc - demo of upgrading LPC to HPC/FHPC demo_lpic2hpic - demo of upgrading LPIC to HPIC/FHPIC demo_ho - demo of HO design and simulation demo_fho - demo of FHO design and simulation demo_lo2ho - demo of upgrading LO to HO/FHO For more details please read the documentation: HCS_doc.pdf

Input-to-state stability (ISS) is one of the most utilizable robust stability properties for nonlinear dynamical systems, while (nearly) fixed-time convergence is a kind of decay for trajectories of disturbance-free systems that is independent in initial conditions. The presence of both these features for a system can be checked by existence of a proper Lyapunov function. The objective of 10 was to provide the conditions for a converse result that (nearly) fixed-time input-to-state stable systems admit a respective Lyapunov function. Similar auxiliary results for uniform finite-time stability and uniform (nearly) fixed-time stability are obtained.

Usually, singularly perturbed models are used to justify the decomposition of the interconnected systems into the Main Dynamics (MD) and the Parasitic Dynamics (PD). In 20, the effect of a homogeneous PD on the stability of a homogeneous MD, when homogeneity degrees are possibly different, is studied via ISS approach in the framework of singular perturbations. Proposed analysis discovers three kinds of stability in the behavior of such an interconnection by assuming that both, regulated MD and unforced PD, are globally asymptotically stable.

The paper 33 develops control algorithms for a class of affine nonlinear systems using the so-called canonical homogeneous representation. It is demonstrated that such a representation exists for any homogeneous vector field bounded on the unit sphere. It is shown that canonical homogeneous representation is useful for LMI-based control design and stability analysis of nonlinear systems.

Non-overshooting stabilization is a form of safe control where the setpoint chosen by the user is at the boundary of the safe set. Exponential non-overshooting stabilization, including suitable extensions to systems with deterministic and stochastic disturbances, has been solved previously. In 25, we develop homogeneous feedback laws for fixed-time nonovershooting stabilization of nonlinear systems that are input-output linearizable with a full relative degree, i.e., for systems that are diffeomorphically equivalent to the chain of integrators. These homogeneous feedback laws can also assume the secondary role of 'fixed-time safety filters' (FxTSf filters) which keep the system within the closed safe set for all time but, in the case where the user's nominal control commands approach to the unsafe set, allow the system to reach the boundary of the safe set no later than a desired time that is independent of nominal control and independent of the value of the state at the time the nominal control begins to be overridden.

The problem of finite-time stabilization of a linear plant with an optimization of both a settling time and an averaged/weighted control energy is studied using the concept of generalized homogeneity in 23. It is shown that the optimal finite-time stabilizing control in this case can be designed by means of solving a simple linear algebraic equation. Robustness of the obtained control law is studied.

The paper 11 addresses the problem of input-to-state stabilization for heat equation with external input via boundary control strategy. Following our previous results, based on the backstepping approach, a switching boundary control law depending on the system state is designed. By estimating the upper bound of kernel functions, switching levels are determined and commutation law is further constructed. For the chosen switched control, the well-posedness property is verified. It is proved that the resulting system is ISS, and the solutions of the system will not exceed the highest admissible level dependent on the disturbance amplitude. Meanwhile, a stronger result is also obtained, that is the finite-time stability for the disturbance-free system.

For a class of nonlinear systems with homogeneous right-hand sides of non-zero degree and distributed delays, the problem of stability robustness of the zero solution with respect to time-varying perturbations multiplied by a nonlinear functional gain is studied in 7. It is assumed that the disturbance-free and delay-free system (that results after substitution of non-delayed state for the delayed one) is globally asymptotically stable. First, it is demonstrated that in the disturbance-free case the zero solution is either locally asymptotically stable or practically globally asymptotically stable, depending on the homogeneity degree of the delay-free counterpart. Second, using averaging tools several variants of the time-varying perturbations are considered and the respective conditions are derived evaluating the stability margins in the system. The results are obtained by a careful choice and comparison of Lyapunov-Krasovskii and Lyapunov-Razumikhin approaches. Finally, the obtained theoretical findings are illustrated on two mechanical systems.

For input-affine nonlinear dynamical systems, an ISS analysis with respect to (weighted) average values of exogenous perturbations is proposed in 9. The time-delay method is used to represent the system in a suitable form for investigation: a kind of neutral-type differential equation. The introduced approach allows the asymptotic gains with respect to zero-mean periodic signals to be evaluated for nonlinear systems (an analogue of Bode magnitude plot), as well as for integral ISS (iISS) systems with periodic inputs. The results are illustrated on the class of homogeneous systems.

The paper 22 proposes a discretization (sampled-time implementation) algorithm for a class of homogeneous controllers for linear time-invariant systems preserving the finite-time and nearly fixed-time stability properties. The sampling period is assumed to be constant. Both single-input and multiple-input cases are considered. The robustness (ISS) of the obtained sampled-time control system is studied as well.

The paper 21 is dedicated to the experimental analysis of discrete-time differentiators implemented in closed-loop control systems. To this end, two laboratory setups, namely an electro-pneumatic system and a rotary inverted pendulum have been used to implement 25 different differentiators. Since the selected laboratory setups behave differently in the case of dynamic response and noise characteristics, it is expected that the results remain valid for a wide range of control applications. The validity of several theoretical results, which have been already reported in the literature using mathematical analysis and numerical simulations, has been investigated, and several comments are provided to allow one to select an appropriate differentiation scheme in practical closed-loop control systems.

In 18, the leader-following consensus problem for multi-agent systems is considered. Each agent is assumed to be modeled by a linear multi-input system. A novel (generalized homogeneous) consensus control protocol is designed under the assumption that there are some uncertainties in the dynamic of the leader. Some LMIs are derived to select the control parameters in order to ensure the ISS and global finite-time stability of the consensus errors with desired homogeneity degrees.

The methodology of the unit sliding mode control design (known since 1970s) for linear systems is revised based on the concept of the generalized homogeneity in 24. The restriction about a consistency of the number of control inputs with the dimension of the sliding surface is eliminated. A simple procedure of control tuning based on a known maximal magnitude of matched perturbations is developed. To deal with perturbations of unknown magnitude, a homogeneous sliding mode control with integral action is designed as well.

A generalized homogeneous control with integral action for a multiple-input plant operating under uncertainty conditions is designed in 31. The stability analysis is essentially based on a special version of the non-smooth Lyapunov function theorem for differential equations with discontinuous right-hand sides. A Lyapunov function for analysis of the closed-loop system is presented. For negative homogeneity degree, this Lyapunov function becomes a strict Lyapunov function allowing an advanced analysis to be provided. In particular, the maximum control magnitude and the settling-time of the closed-loop system are estimated and a class of disturbances to be rejected by the control law is characterized. The control parameters are tuned by solving a system of LMIs, whose feasibility is proved at least for small (close to zero) homogeneity degrees.

Generalized Persidskii systems (or Lur'e systems) represent the dynamics described by the superposition of a linear part with multiple sector nonlinearities and exogenous perturbations. They can be used to model many physical and engineering phenomena.

The paper 19 studies the trajectory behavior evaluation for generalized Persidskii systems with an essentially bounded input on a finite time interval. Also, the notions of annular settling and output annular settling for general nonlinear systems are introduced, and the respective stability conditions are porposed. These conditions are based on the verification of linear matrix inequalities. An application to recurrent neural networks illustrates the usefulness of the proposed notions and conditions.

The paper 12 considers the state estimation problem for a class of non-autonomous nonlinear systems. We propose conditions on the existence and stability of a nonlinear observer based on the invariant manifold approach in both continuous and discrete-time scenarios. The requirements are formulated using linear matrix equalities and LMIs. We present two possible applications of the result: a reduced-order observer (e.g., an observer for unmeasured states) and regression in linear and nonlinear, continuous-and discretetime settings. With nonlinear regression being a sophisticated case, the parameter estimation problem for a particular output equation (when the fusion of linear and nonlinear sensors is weighted) is investigated.

The article 27 deals with the problem of time-varying parameter identification in dynamical regression models affected by disturbances. The disturbances comprise time-dependent external perturbations and nonlinear unmodeled dynamics. With this aim in mind, we propose a robust nonlinear adaptive observer. The algorithm ensures the asymptotic convergence of the parameter identification error to an acceptably small region around the origin in the presence of disturbances. The synthesis of the adaptive observer is given in terms of linear matrix inequalities since the error dynamics is reduced to Persidskii form, providing a constructive design method.

The paper 32 deals with the tracking problem for the unicycle mobile robot with slippage effects. A homogeneous controller is developed based on a particular cascade control strategy. The robustness of the closed-loop system is studied. The design is essentially based on the canonical homogeneous norm being a Lyapunov function of the system. The (finite-time or exponential) convergence rate of the homogeneous controller can be tuned by a proper selection of the so-called homogeneity degree. Some experimental results illustrate the performance of the proposed homogeneous control in the UMR QBot2 by Quanser.

The paper 28 contributes to the design of a second order sliding-mode controller for the trajectory tracking problem in perturbed unicycle mobile robots. The proposed strategy takes into account the design of two particular sliding variables, which ensure the convergence of the tracking error to the origin in a finite time despite the effect of some external perturbations. The straightforward structure of the controller is simple to tune and implement. The global, uniform and finite-time stability of the closed-loop tracking error dynamics is demonstrated by means of Lyapunov functions. Furthermore, the performance of the proposed approach is validated through some experiments using a QBot2 unicycle mobile robot.

In the paper 30, an approach is proposed for the remote observation of a dynamical system through a data-rate constrained communication channel. The focus is put on discrete-time systems with a Lipschitz nonlinearity, driven by an external signal, and subject to bounded state perturbation and measurement error. The problem at hand is providing estimates of system's state at a remote location, which is connected via a channel, which can only sent limited numbers of bits per unit of time. A solution, named observation scheme, is proposed in the form of several interacting agents. This solution is designed such that the maximum observation error is upper-bounded by a computable quantity dependent on system constants and selectable parameters. The scheme is designed in an event-triggered fashion, such that the actual communication rate is sometimes much lower than the theoretically evaluated maximum one.

We present new results for global boundedness of state periodic systems in 29. Thereby, we address both the case of systems, whose dynamics is periodic with respect to a part of the state vector and the case of systems, whose dynamics is periodic with respect to all state variables. Both classes of systems are of high relevance in diverse and timely applications. To derive the results the notion of strong Leonov functions is refined. The main results are complemented by a number of relaxations based on the concpet of weak Leonov functions.

In the paper 8, we study the problem of robust stabilization of affine nonlinear multistable systems in the presence of exogenous disturbances. The results are based on the theory of ISS and iISS for systems with multiple invariant sets. The notions of ISS and iISS control Lyapunov functions (CLFs) and the small control property are extended within the multistability framework. Such properties are also complemented by the concept of a weak iISS CLF and corresponding small control property. It is verified that the universal control formula can be applied to yield the ISS (iISS) property for the closed-loop system. The efficiency of the extended CLF framework in the multistable sense is illustrated for a Duffing system and in application to a noise-induced transition in a semiconductor-gas-discharge gap system.

When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended ones. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, was exploited in 16.

A discrete-time version of the blended dynamics theorem is proposed in 13 that can be used for design of distributed computation algorithms. The blended dynamics theorem enables to predict the behavior of heterogeneous multi-agent systems. Therefore, once we get a blended dynamics for a particular computational task, design idea of node dynamics for individual heterogeneous agents can easily occur. In the continuous-time case, prediction by blended dynamics was enabled by high coupling gain among neighboring agents. In the discrete-time case, we propose an equivalent action.

The paper 17 examines how weak synaptic coupling can achieve rapid synchronization in heterogeneous networks. The assumptions aim at capturing the key mathematical properties that make this possible for biophysical networks. In particular, the combination of nodal excitability and synaptic coupling are shown to be essential to the phenomenon.

The paper 15 investigates the finite-time stability and nearly fixed-time stability of nonlinear impulsive systems with destabilizing impulses. A Lyapunov inequality with linear terms has been used to derive sufficient conditions based on the dwell time (DT) and average dwell time (ADT) properties of impulsive sequences to ensure stability of the system. The main results of this paper are applied to the synchronization problem of impulsive neural networks with destablizing impulses.

Leonid Fridman, UNAM, until Jan 2023

The members of the team serve as reviewers for major journals and conferences in the field of the theory of control and automation.

The members of the team participated in numerous juries for Ph.D. defense.