Optimal Stabilization of Periodic Orbits

In this contribution the optimal stabilization problem of periodic orbits is studied in a symplectic geometry setting. For this, the stable manifold theory for the point stabilization case is generalized to the case of periodic orbit stabilization. Sufficient conditions for the existence of a \gls{nhim} of the Hamiltonian system are derived. It is shown that the optimal control problem has a solution if the related periodic Riccati equation has a unique periodic solution. For the analysis of the stable and unstable manifold a coordinate transformation is used which is moving along the orbit. As an example, an optimal control problem is considered for a spring mass oscillator system, which should be stabilized at a certain energy level.Comment: Submitted for IFAC World Congress 202

Systematic Controller Design for Dynamic 3D Bipedal Robot Walking.

Virtual constraints and hybrid zero dynamics (HZD) have emerged as a powerful framework for controlling bipedal robot locomotion, as evidenced by the robust, energetically efficient, and natural-looking walking and running gaits achieved by planar robots such as Rabbit, ERNIE, and MABEL. However, the extension to 3D robots is more subtle, as the choice of virtual constraints has a deciding effect on the stability of a periodic orbit. Furthermore, previous methods did not provide a systematic means of designing virtual constraints to ensure stability. This thesis makes both experimental and theoretical contributions to the control of underactuated 3D bipedal robots. On the experimental side, we present the first realization of dynamic 3D walking using virtual constraints. The experimental success is achieved by augmenting a robust planar walking gait with a novel virtual constraint for the lateral swing hip angle. The resulting controller is tested in the laboratory on a human-scale bipedal robot (MARLO) and demonstrated to stabilize the lateral motion for unassisted 3D walking at approximately 1 m/s. MARLO is one of only two known robots to walk in 3D with stilt-like feet. On the theoretical side, we introduce a method to systematically tune a given choice of virtual constraints in order to stabilize a periodic orbit of a hybrid system. We demonstrate the method to stabilize a walking gait for MARLO, and show that the optimized controller leads to improved lateral control compared to the nominal virtual constraints. We also describe several extensions of the basic method, allowing the use of a restricted Poincaré map and the incorporation of disturbance rejection metrics in the optimization. Together, these methods comprise an important contribution to the theory of HZD.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113370/1/bgbuss_1.pd

Port-Hamiltonian Modeling for Control

This article provides a concise summary of the basic ideas and concepts in port-Hamiltonian systems theory and its use in analysis and control of complex multiphysics systems. It gives special attention to new and unexplored research directions and relations with other mathematical frameworks. Emergent control paradigms and open problems are indicated, including the relation with thermodynamics and the question of uniting the energy-processing view of control, as emphasized by port-Hamiltonian systems theory, with a complementary information-processing viewpoint.</p

Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76737/1/AIAA-41565-705.pd

Robust stability and stabilization of discrete singular systems: An equivalent characterization

This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.published_or_final_versio

Hybrid Zero Dynamics of Planar Biped Walkers

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set—a two-dimensional zero dynamics submanifold of the full hybrid model—whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees

Path Following for Mechanical Systems Applied to Robotic Manipulators

Many applications in robotics require faithfully following a prescribed path. Tracking controllers may not be appropriate for such a task, as there is no guarantee that the robot will stay on the path. The objective of this thesis is to develop a control design method which makes the “output” of a robot get to, and move along the prescribed path without leaving the path. We consider the class of mechanical systems, which encompasses robotics. Various techniques exist for designing pah following controllers. We base our approach on a technique called “transverse feedback linearization”. Using this technique, if feasible, we decompose the dynamics of a mechanical system into a transversal subsystem and a tangential subsystem using a coordinate and feedback transformation. The transversal subsystem is linear, time-invariant and decoupled from the tangential subsystem. Stabilizing the origin of the transversal subsystem is equivalent to stabilizing a set corresponding to the output of the mechanical system being on the desired path, thereby partly achieving the control objective. Given a mechanical system and a path, we provide conditions under which this is possible. The tangential subsystem describes all of the motions of the mechanical system, when the output is on the path. Some tangential dynamics may move the output along the path, and thereby meet the design objective. In order to move the output of the mechanical system along the path, we further decompose the tangential subsystem into a subsystem which moves the output along the path, and a subsystem which does not, if feasible, using partial feedback linearization. The subsystem which governs output motions along the path is linear, time-invariant and decoupled. The remaining tangential dynamics have no special structure. We provide conditions under which such a decomposition of the tangential dynamics is possible. We show that a five-bar robotic manipulator has dynamics which may be transversely feedback linearized, and the tangential dynamics may be partially linearized. Given a circular path, we experimentally implement our path following design, and observe that our control objective is indeed met. Inherent advantages of path following over trajectory tracking are illustrated. Standard feedback linearization of a five-bar robotic manipulator with a flexible link has been shown to fail. We show that this system is transversely feedback linearizable, and its tangential dynamics may be partially linearized, under mild restrictions. Simulations illustrate path following applied to this complex system

Distributed optimal and predictive control methods for networks of dynamic systems

Several recent approaches to distributed control design over networks of interconnected dynamic systems rely on certain assumptions, such as identical subsystem dynamics, absence of dynamical couplings, linear dynamics and undirected interaction schemes. In this thesis, we investigate systematic methods for relaxing a number of simplifying factors leading to a unifying approach for solving general distributed-control stabilization problems of networks of dynamic agents. We show that the gain-margin property of LQR control holds for complex multiplicative input perturbations and a generic symmetric positive definite input weighting matrix. Proving also that the potentially non-simple structure of the Laplacian matrix can be neglected for stability analysis and control design, we extend two well-known distributed LQR-based control methods originally established for undirected networks of identical linear systems, to the directed case. We then propose a distributed feedback method for tackling large-scale regulation problems of a general class of interconnected non-identical dynamic agents with undirected and directed topology. In particular, we assume that local agents share a minimal set of structural properties, such as input dimension, state dimension and controllability indices. Our approach relies on the solution of certain model matching type problems using local linear state-feedback and input matrix transformations which map the agent dynamics to a target system, selected to minimize the joint control effort of the local feedback-control schemes. By adapting well-established distributed LQR control design methodologies to our framework, the stabilization problem of a network of non-identical dynamical agents is solved. We thereafter consider a networked scheme synthesized by multiple agents with nonlinear dynamics. Assuming that agents are feedback linearizable in a neighborhood near their equilibrium points, we propose a nonlinear model matching control design for stabilizing networks of multiple heterogeneous nonlinear agents. Motivated by the structure of a large-scale LQR optimal problem, we propose a stabilizing distributed state-feedback controller for networks of identical dynamically coupled linear agents. First, a fully centralized controller is designed which is subsequently substituted by a distributed state-feedback gain with sparse structure. The control scheme is obtained byoptimizing an LQR performance index with a tuning parameter utilized to emphasize/deemphasize relative state difference between coupled systems. Sufficient conditions for stability of the proposed scheme are derived based on the inertia of a convex combination of two Hurwitz matrices. An extended simulation study involving distributed load frequency control design of a multi-area power network, illustrates the applicability of the proposed method. Finally, we propose a fully distributed consensus-based model matching scheme adapted to a model predictive control setting for tackling a structured receding horizon regulation problem