Cesar O. Aguilar

Dynamics of an Inverted Pendulum with Delayed Feedback Control

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos.

Graph Controllability Classes for the Laplacian Leader-Follower Dynamics

In this paper, we consider the problem of obtaining graph-theoretic characterizations of controllability for the Laplacian-based leader-follower dynamics. Our developments rely on the notion of graph controllability classes, namely, the classes of essentially controllable, completely uncontrollable, and conditionally controllable graphs. In addition to the topology of the underlying graph, the controllability classes rely on the specification of the control vectors; our particular focus is on the set of binary control vectors. The choice of binary control vectors is naturally adapted to the Laplacian dynamics, as it captures the case when the controller is unable to distinguish between the followers and, moreover, controllability properties are invariant under binary complements. We prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases. Although graph symmetries play an important role in graph-theoretic characterizations of controllability, we provide an explicit class of asymmetric graphs that are completely uncontrollable, namely the class of block graphs of Steiner triple systems. We prove that for graphs on four and five vertices, a repeated Laplacian eigenvalue is a necessary condition for complete uncontrollability but, however, show through explicit examples that for eight and nine vertices, a repeated eigenvalue is not necessary for complete uncontrollability. For the case of conditional controllability, we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two-dimensional controllable subspace. Several constructive examples demonstrate our results.

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.

SMALL-TIME LOCAL CONTROLLABILITY FOR A CLASS OF HOMOGENEOUS SYSTEMS ∗

In this paper we consider the local controllability problem for control-affine systems that are homogeneous with respect to a one-parameter family of dilations corresponding to time-scaling in the control. We construct and derive properties of a variational cone that completely characterizes local controllability for these homogeneous systems. In the process, we are able to give a bound on the order, in terms of the integers describing the dilation, of perturbations that do not alter the local controllability property. Our approach uses elementary Taylor expansions and avoids unnecessarily complicated open mapping theorems to prove local controllability. Examples are given that illustrate the main results.

Necessary conditions for controllability of nonlinear networked control systems

In this paper, we study the controllability of nonlinear networked systems. In particular, we describe how graph symmetries combined with dynamic symmetries result in a loss of controllability in nonlinear leader-follower networks. Our result generalizes those of Rahmani et al. (2009) who considered the case of a linear consensus-type dynamics, namely the unweighted Laplacian network flow. We consider several nonlinear network control systems that have been previously studied in the literature and characterize the presence of leader-follower graph symmetries that result in the lack of controllability.

Patchy solution of a Francis–Byrnes–Isidori partial differential equation

SUMMARY The solution to the nonlinear output regulation problem requires one to solve a first-order partial differential equation, known as the Francis–Byrnes–Isidori equations. In this paper, we propose a method to compute approximate solutions to the Francis–Byrnes–Isidori equations when the zero dynamics of the plant are hyperbolic and the exosystem is two dimensional. With our method, we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two-dimensional analytic center manifolds. Copyright

Almost equitable partitions and new necessary conditions for network controllability

Abstract In this paper, we consider the controllability problem for multi-agent networked control systems. The main results of the paper are new graph-theoretic necessary conditions for controllability involving almost equitable graph vertex partitions. We generalize the known results on the role of graph symmetries and uncontrollability to weighted digraphs with multiple-leaders and we also consider the broadcasted control scenario. Our results show that the internal structure of communities in a graph can induce obstructions to controllability that cannot be characterized by symmetry arguments alone and that in some cases depend on the number-theoretic properties of the communities. We show via examples that our results can be used to account for a large portion of uncontrollable inducing leader-selections that could not have otherwise been accounted for using symmetry results.

Model Predictive Regulation

Abstract We show how optimal nonlinear regulation can be achieved in a model predictive control fashion.

Local controllability of control-affine systems with quadractic drift and constant control-input vector fields

In this paper we study the small-time local controllability (STLC) property of polynomial control-affine systems whose drift vector field is a 2-homogeneous polynomial vector field and whose control-input vector fields are constant. Such systems arise in the study of controllability of mechanical control systems. Using control variations and rooted trees, we obtain a combinatorial expression for the Taylor series coefficients of a composition of flows of vector fields and use it to derive a high-order sufficient condition for STLC for these systems. The resulting condition is stated in terms of the image of the control-input subspace under the drift vector field and is therefore invariant under (linear) feedback transformations.

On almost equitable partitions and network controllability

The main contribution of this paper are two new general necessary conditions for controllability of multi-agent networked control systems involving almost equitable partitions, along with an extension of a well-known symmetry condition to weighted digraphs and multi-input broadcast control signals. The new necessary conditions identify leader selections that break all “symmetries” induced by an almost equitable partition, and in particular genuine graph symmetries, but yet induce uncontrollable dynamics. The results are illustrated on non-trivial examples whose controllability properties are fully characterized by these conditions.