Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

Stability of Multi-Dimensional Switched Systems with an Application to Open Multi-Agent Systems

Extended from the classic switched system, themulti-dimensional switched system (MDSS) allows for subsystems(switching modes) with different state dimensions. In this work,we study the stability problem of the MDSS, whose state transi-tion at each switching instant is characterized by the dimensionvariation and the state jump, without extra constraint imposed.Based on the proposed transition-dependent average dwell time(TDADT) and the piecewise TDADT methods, along with the pro-posed parametric multiple Lyapunov functions (MLFs), sufficientconditions for the practical and the asymptotical stabilities of theMDSS are respectively derived for the MDSS in the presenceof unstable subsystems. The stability results for the MDSS areapplied to the consensus problem of the open multi-agent system(MAS) which exhibits dynamic circulation behaviors. It is shownthat the (practical) consensus of the open MAS with disconnectedswitching topologies can be ensured by (practically) stabilizingthe corresponding MDSS with unstable switching modes via theproposed TDADT and parametric MLF methods.Comment: 12 pages, 9 figure

Online Optimization of Switched LTI Systems Using Continuous-Time and Hybrid Accelerated Gradient Flows

This paper studies the design of feedback controllers that steer the output of a switched linear time-invariant system to the solution of a possibly time-varying optimization problem. The design of the feedback controllers is based on an online gradient descent method, and an online hybrid controller that can be seen as a regularized Nesterov's accelerated gradient method. Both of the proposed approaches accommodate output measurements of the plant, and are implemented in closed-loop with the switched dynamical system. By design, the controllers continuously steer the system output to an optimal trajectory implicitly defined by the time-varying optimization problem without requiring knowledge of exogenous inputs and disturbances. For cost functions that are smooth and satisfy the Polyak-Lojasiewicz inequality, we demonstrate that the online gradient descent controller ensures uniform global exponential stability when the time-scales of the plant and the controller are sufficiently separated and the switching signal of the plant is slow on the average. Under a strong convexity assumption, we also show that the online hybrid Nesterov's method guarantees tracking of optimal trajectories, and outperforms online controllers based on gradient descent. Interestingly, the proposed hybrid accelerated controller resolves the potential lack of robustness suffered by standard continuous-time accelerated gradient methods when coupled with a dynamical system. When the function is not strongly convex, we establish global practical asymptotic stability results for the accelerated method, and we unveil the existence of a trade-off between acceleration and exact convergence in online optimization problems with controllers using dynamic momentum. Our theoretical results are illustrated via different numerical examples

Second order sliding mode control of underactuated Mechanical systems I: Local stabilization with application to an inverted pendulum

International audienceSecond order sliding mode control synthesis is developed for underactuated mechanical systems, operating under uncertainty conditions. In order to locally stabilize an underactuated system around an unstable equilibrium, an output is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Then, the desired stability property of the closed-loop system is provided by applying a quasihomogeneous second order sliding mode controller, driving the system to the zero dynamics manifold in finite time. Although the present synthesis exhibits an infinite number of switches on a finite time interval, it does not rely on the generation of first order sliding modes, while providing robustness features similar to those possessed by their standard sliding mode counterparts. A second order sliding mode appears on the zero dynamics manifold which is of co-dimension greater than the control space dimension. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of a cart-Pendulum system

Modeling and Control of High-Voltage Direct-Current Transmission Systems: From Theory to Practice and Back

The problem of modeling and control of multi-terminal high-voltage direct-current transmission systems is addressed in this paper, which contains five main contributions. First, to propose a unified, physically motivated, modeling framework - based on port-Hamiltonian representations - of the various network topologies used in this application. Second, to prove that the system can be globally asymptotically stabilized with a decentralized PI control, that exploits its passivity properties. Close connections between the proposed PI and the popular Akagi's PQ instantaneous power method are also established. Third, to reveal the transient performance limitations of the proposed controller that, interestingly, is shown to be intrinsic to PI passivity-based control. Fourth, motivated by the latter, an outer-loop that overcomes the aforementioned limitations is proposed. The performance limitation of the PI, and its drastic improvement using outer-loop controls, are verified via simulations on a three-terminals benchmark example. A final contribution is a novel formulation of the power flow equations for the centralized references calculation

Results on data-driven controllers for unknown nonlinear systems

The big data revolution is deeply changing the way we understand and analyze natural phenomena around us. In the field of control engineering, data-driven control enables researchers to explore new intelligent algorithms to model and control complex dynamical systems. Data-driven control is based on the paradigm of learning controllers of an unknown dynamical system by directly using data. The underlying idea is that information about the model can be gathered from experiments, bypassing completely the identification step, which can be impractical or too costly. This thesis presents data-driven control solutions for different families of unknown dynamical systems, with a focus on both linear and special classes of nonlinear ones. In the first part of the thesis, we consider the linear quadratic regulator problem for linear time-invariant discrete-time systems. The system is assumed to be unknown and information on the system is given by a finite set of data. This allows determining the optimal control law in one shot, with no intermediate identification step. Secondly, we present an online algorithm for learning controllers applied to switched linear systems. By collecting data on the fly, the control mechanism can capture any changes in the dynamics of the plant and adapt itself accordingly to achieve stabilization of the running dynamics. Finally, we derive data-driven methods for a more general class of nonlinear systems via nonlinearity cancellation. To this end, we make use of a "dictionary" of nonlinear terms that includes the nonlinearities of the unknown system

High-feedback Operation of Power Electronic Converters

electronic