Observer design for piecewise smooth and switched systems via contraction theory

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of piecewise smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.Comment: Preprint accepted to IFAC World Congress 201

Switching control for incremental stabilization of nonlinear systems via contraction theory

In this paper we present a switching control strategy to incrementally stabilize a class of nonlinear dynamical systems. Exploiting recent results on contraction analysis of switched Filippov systems derived using regularization, sufficient conditions are presented to prove incremental stability of the closed-loop system. Furthermore, based on these sufficient conditions, a design procedure is proposed to design a switched control action that is active only where the open-loop system is not sufficiently incrementally stable in order to reduce the required control effort. The design procedure to either locally or globally incrementally stabilize a dynamical system is then illustrated by means of a representative example.Comment: Accepted to ECC 201

Pinning dynamic systems of networks with Markovian switching couplings and controller-node set

In this paper, we study pinning control problem of coupled dynamical systems with stochastically switching couplings and stochastically selected controller-node set. Here, the coupling matrices and the controller-node sets change with time, induced by a continuous-time Markovian chain. By constructing Lyapunov functions, we establish tractable sufficient conditions for exponentially stability of the coupled system. Two scenarios are considered here. First, we prove that if each subsystem in the switching system, i.e. with the fixed coupling, can be stabilized by the fixed pinning controller-node set, and in addition, the Markovian switching is sufficiently slow, then the time-varying dynamical system is stabilized. Second, in particular, for the problem of spatial pinning control of network with mobile agents, we conclude that if the system with the average coupling and pinning gains can be stabilized and the switching is sufficiently fast, the time-varying system is stabilized. Two numerical examples are provided to demonstrate the validity of these theoretical results, including a switching dynamical system between several stable sub-systems, and a dynamical system with mobile nodes and spatial pinning control towards the nodes when these nodes are being in a pre-designed region.Comment: 9 pages; 3 figure

Incremental stability of hybrid dynamical systems

International audienceThe analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump-and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout the paper, including an event-triggered control application and a bouncing ball system with Zeno behaviour, that illustrate incremental stability with respect to continuous time or discrete time, respectively

Backstepping controller synthesis and characterizations of incremental stability

Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in either parametric-strict-feedback or strict-feedback form, and they require these control systems to be smooth. In this paper, we propose a design technique that is applicable to larger classes of (not necessarily smooth) control systems. Moreover, we propose a recursive way of constructing contraction metrics (for smooth control systems) and incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems, the approximation of stochastic hybrid systems, source-code model checking for nonlinear dynamical systems and so on. The effectiveness of the proposed results in this paper is illustrated by synthesizing a controller rendering a non-smooth control system incrementally stable as well as constructing its finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure

Convex Formulation of Controller Synthesis for Piecewise-Affine Systems

This thesis is divided into three main parts. The contribution of the first part is to present a controller synthesis method to stabilize piecewise-affine (PWA) slab systems based on invariant sets. Inspired by the theory of sliding modes, sufficient stabilization conditions are cast as a set of Linear Matrix Inequalities (LMIs) by proper choice of an invariant set which is a target sliding surface. The method has two steps: the design of the attractive sliding surface and the design of the controller parameters. While previous approaches to PWA controller synthesis are cast as Bilinear Matrix Inequalities (BMIs) that can, in some cases, be relaxed to LMIs at the cost of adding conservatism, the proposed method leads naturally to a convex formulation. Furthermore, the LMIs obtained in this work have lower dimension when compared to other methods because the dimension of the closed-loop state space is reduced. In the second part of the thesis, it is further shown that the proposed approach is less conservative than other approaches. In other words, it will be shown that for every solution of the LMIs resulting from previous approaches, there exists a solution for the LMIs obtained from the proposed method. Furthermore, it will be shown that while previous convex controller synthesis methods have no solutions to their LMIs for some examples of PWA systems, the approach proposed in this thesis yields a solution for these examples. The contribution of the last part of this thesis is to formulate the PWA time-delay synthesis problem as a set of LMIs. In order to do so, we first define a sliding surface, then control laws are designed to approach the specified sliding surface and ensure that the trajectories will remain on that surface. Then, using Lyapunov-Krasovskii functionals, sufficient conditions for exponential stability of the resulting reduced order system will be obtained. Several applications such as pitch damping of a helicopter (2nd order system), rover path following example (3rd order system) and active flutter suppression (4th order system) along with some other numerical examples are included to demonstrate the effectiveness of the approaches

Hybrid Systems, Iterative Learning Control, and Non-minimum Phase

Hybrid systems have steadily grown in popularity over the last few decades because they ease the task of modeling complicated nonlinear systems. Legged locomotion, robotic manipulation, and additive manufacturing are representative examples of systems benefiting from hybrid modeling. They are also prime examples of repetitive processes; gait cycles in walking, product assembly tasks in robotic manipulation, and material deposition in additive manufacturing. Thus, they would also benefit substantially from Iterative Learning Control (ILC), a class of feedforward controllers for repetitive systems that achieve high performance in output reference tracking by learning from the errors of past process cycles. However, the literature is bereft of ILC syntheses from hybrid models. The main thrust of this dissertation is to provide a broadly applicable theory of ILC for deterministic, discrete-time hybrid systems, i.e. piecewise defined (PWD) systems. A type of ILC called Newton ILC (NILC) serves as the foundation for this mission due to its admittance of an unusually broad range of nonlinearities. Preventing the synthesis of NILC from hybrid models is the fact that contemporary hybrid modeling frameworks do not admit closed-form function composition of a single state transition formula capturing the complete hybrid system dynamics. This dissertation offers a new, closed-form PWD modeling framework to solve this problem. However, NILC itself is not without flaw. This dissertation's research reveals that it generally fails to converge when synthesized from models with unstable inverses (i.e. non-minimum phase (NMP) models), a class that includes flexible-link robotic manipulators. Thus, to fulfill the goal of providing the most broadly applicable control theory possible, improvement to NILC must be made to avoid the operation that causes divergence when applied to NMP systems (a particular matrix inversion). Stable inversion---a technique for generating stable state trajectories from unstable systems by decoupling their stable and unstable modes---is identified as a valuable tool in this endeavor. This concept is well-explored for linear time invariant systems, but stable inversion for hybrid systems has not been explored by the prior art. Thus, to focus the research, this dissertation specifically examines piecewise affine (PWA) systems (a subset of PWD systems) for the study of NMP hybrid system control. For PWA systems (and their PWD superset), in addition to a lack of stable inversion, a general, closed-form solution to the conventional inversion problem is also absent from the literature. Having a closed-form conventional inverse model is a prerequisite for stable inversion, but inversion of PWA models is nontrivial because the uniqueness of PWA system inverses is not guaranteed as it is for ordinary affine systems. Therefore, to achieve the first ILC of a hybrid system with an unstable inverse, theory for both conventional inversion and stable inversion must be delivered for PWA systems. In summary, the three main gaps addressed by this dissertation are (1) the lack of compatibility between existing hybrid modeling frameworks and ILC synthesis techniques, (2) the failure of NILC for NMP systems, and (3) the lack of inversion and stable inversion theory for PWA systems. These issues are addressed by (1) developing a closed-form representation for PWD systems, (2) developing a new ILC framework informed by NILC but free of matrix inversion, and (3) deriving conventional and stable model inversion theories for PWA systems.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/167929/1/ispiegel_1.pd