Distributed Stabilization of Nonlinear Multi-Agent Systems

The study of multi-agent systems (MASs) is focused on systems in which many autonomous agents interact and operate within a limited communication environment. The general goal of the MAS research is to design interconnection control laws such that all the dynamic agents in the group are synchronized to a desired common trajectory by exchanging information with adjacent agents over certain constrained communication networks. Based on the review and modification of existing results concerning the consensus control of linear heterogeneous MASs in Moreau (2004) [21], Scardovi and Sepulchre (2009) [25], Wieland et al (2011) [30], and Alvergue et al. (2013) [1], this thesis investigates the distributed stabilization of the heterogeneous MAS, consisting of N different continuous-time nonlinear dynamic systems, under connected communication graphs. The conditions for a nonlinear dynamic agent to be feedback equivalent to a strictly passive system are derived along with the feedback law. A distributed stabilization control protocol using state feedback is then proposed under the idea of feedback connection of two passive systems. It proves to be sufficient for only one or a few agents to have access to the reference signal for the MAS to achieve stability, which lowers the communication overhead from the reference to different agents. The result can be interpreted as an extension of the stabilizing law for linear MASs introduced in [1], and considered as a fundamental preliminary for the consensus research for nonlinear MASs in the future

On the stability, stabilizability and control of certain classes of Positive Systems

In this thesis stability, stabilizability and other control issues for certain classes of Positive Systems are investigated. In the first part, the focus is on Compartmental Systems: we start from Compartmental Switched Systems and show that, with respect to the general class of Positive Switched Systems, a much clearer picture of stability under arbitrary switching, stability under persistent switching, and stabilizability (where the control action may either pertain the switching function or involve the design of feedback controllers) can be drawn. Secondly, for the class of Compartmental Multi-Input Systems the problem of designing a state-feedback matrix that preserves the compartmental property of the resulting closed-loop system, meanwhile achieving asymptotic stability is addressed. Such an analysis finally leads to the development of an algorithm that allows to assess problem solvability and provides a possible solution whenever it exists. The second part of the thesis is devoted to the Positive Consensus Problem: for a homogeneous Positive Multi-Agent System we investigate the problem of determining a state-feedback law that can be individually implemented by each agent, preserves the positivity of the overall system, and leads to the achievement of consensus. Finally, for a particular class of Positive Bilinear Systems that arises in drugs concentration design for HIV treatment, we address the problem of determining an optimal constant input that stabilizes the system while maximizing its robustness against the presence of the external disturbance

Quasi-optimal robust stabilization of control systems

In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise

On the synthesis of switched output feedback controllers for linear, time-invariant systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 189-193).The theory of switching systems has seen many advances in the past decade. Its beginnings were founded primarily due to the physical limitations in devices to implement control such as relays, but today there exists a strong interest in the development of switching systems where switching is introduced as a means of increasing performance. With the newer set of problems that arise from this viewpoint comes the need for many new tools for analysis and design. Analysis tools which include, for instance, the celebrated work on multiple Lyapunov functions are extensive. Tools for the design of switched systems also exist, but, in many cases, the method of designing stabilizing switching laws is often a separate process from the method which is used to determine the set of vector fields between which switching takes place. For instance, one typical method of designing switching controllers for linear, time-invariant (LTI) systems is to first design a set of stabilizing LTI controllers using standard LTI methods, and then design a switching law to increase performance. While such design algorithms can lead to increases in performance, they often impose restrictions that do not allow the designer to take full advantage of the switching architecture being considered.(cont.) For instance, if one switches between controllers that are individually stabilizing (without any switching), then, effectively, one is forced to switch only between stable systems and, hence, cannot take advantage of the potential benefits of switching between unstable systems in a stable way. It is, therefore, natural to wonder whether design algorithms can be developed which simultaneously design both the set of controllers to be switched and a stabilizing switching law. The work investigated here attempts to take a small step in the above direction. We consider a simple switching architecture that implements switched proportional gain control for second order LTI systems. Examination of this particular structure is motivated by its mathematical simplicity for ease of analysis (and, hence, as a means of gaining insight into the problem-at-large), but, as we will see, the design techniques investigated here can be extended to a larger class of (higher order, potentially non-linear and/or time-varying) systems using standard tools from robust control. The overall problem we investigate is the ability to create algorithms to simultaneously determine a set of switching gains and an associated switching law for a particular plant and performance objective.(cont.) After determining a set of necessary and sufficient conditions for a given second order plant to be stabilizable via the given switching architecture, we synthesize an algorithm for constructing controllers for which the corresponding closed-loop system dynamics are finite L, gain stable. Also, in an effort to demonstrate that the the given structure can, in fact. he used to increase performance. we consider a step-tracking design problem for a class of plants, where we use overshoot and settling time of the output step response to measure performance. We compare the results obtained using our switching architecture to the performance that can be obtained via two other LTI controller architectures to illustrate some of the performance benefits.by Keith Robert Santarelli.Ph.D

Stability analysis of switched systems

Ph.DDOCTOR OF PHILOSOPH

Stabilization of cascaded nonlinear systems under sampling and delays

Over the last decades, the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model. Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system

Composite control Lyapunov functions for robust stabilization of constrained uncertain dynamical systems

This work presents innovative scientific results on the robust stabilization of constrained uncertain dynamical systems via Lyapunov-based state feedback control. Given two control Lyapunov functions, a novel class of smooth composite control Lyapunov functions is presented. This class, which is based on the R-functions theory, is universal for the stabilizability of linear differential inclusions and has the following property. Once a desired controlled invariant set is fixed, the shape of the inner level sets can be made arbitrary close to any given ones, in a smooth and non-homothetic way. This procedure is an example of ``merging'' two control Lyapunov functions. In general, a merging function consists in a control Lyapunov function whose gradient is a continuous combination of the gradients of the two parents control Lyapunov functions. The problem of merging two control Lyapunov functions, for instance a global control Lyapunov function with a large controlled domain of attraction and a local one with a guaranteed local performance, is considered important for several control applications. The main reason is that when simultaneously concerning constraints, robustness and optimality, a single Lyapunov function is usually suitable for just one of these goals, but ineffective for the others. For nonlinear control-affine systems, both equations and inclusions, some equivalence properties are shown between the control-sharing property, namely the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given control Lyapunov functions, and the existence of merging control Lyapunov functions. Even for linear systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. For the class of linear differential inclusions, linear programs and linear matrix inequalities conditions are given for the the control-sharing property to hold. The proposed Lyapunov-based control laws are illustrated and simulated on benchmark case studies, with positive numerical results

Control Of Nonh=holonomic Systems

Many real-world electrical and mechanical systems have velocity-dependent constraints in their dynamic models. For example, car-like robots, unmanned aerial vehicles, autonomous underwater vehicles and hopping robots, etc. Most of these systems can be transformed into a chained form, which is considered as a canonical form of these nonholonomic systems. Hence, study of chained systems ensure their wide applicability. This thesis studied the problem of continuous feed-back control of the chained systems while pursuing inverse optimality and exponential convergence rates, as well as the feed-back stabilization problem under input saturation constraints. These studies are based on global singularity-free state transformations and controls are synthesized from resulting linear systems. Then, the application of optimal motion planning and dynamic tracking control of nonholonomic autonomous underwater vehicles is considered. The obtained trajectories satisfy the boundary conditions and the vehicles\u27 kinematic model, hence it is smooth and feasible. A collision avoidance criteria is set up to handle the dynamic environments. The resulting controls are in closed forms and suitable for real-time implementations. Further, dynamic tracking controls are developed through the Lyapunov second method and back-stepping technique based on a NPS AUV II model. In what follows, the application of cooperative surveillance and formation control of a group of nonholonomic robots is investigated. A designing scheme is proposed to achieves a rigid formation along a circular trajectory or any arbitrary trajectories. The controllers are decentralized and are able to avoid internal and external collisions. Computer simulations are provided to verify the effectiveness of these designs