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Control contraction metrics (CCMs) are a new approach to nonlinear control design based on contraction theory. The resulting design problems are expressed as pointwise linear matrix inequalities and are and well-suited to solution via convex optimization. In this paper, we extend the theory on CCMs by showing that a pair of “dual” observer and controller problems can be solved using pointwise linear matrix inequalities, and that when a solution exists a separation principle holds. That is, a stabilizing output-feedback controller can be found. The procedure is demonstrated using a benchmark problem of nonlinear control: the Moore-Greitzer jet engine compressor model.

We introduce the concept of a control contraction metric, extending contraction analysis to constructive nonlinear control design. We derive sufficient conditions for exponential stabilizability of all trajectories of a nonlinear control system. The conditions have a simple geometrical interpretation, can be written as a convex feasibility problem, and are invariant under coordinate changes. We show that these conditions are necessary and sufficient for feedback linearizable systems, and also derive novel convex criteria for exponential stabilization of a nonlinear submanifold of state space. We illustrate the benefits of convexity by constructing a controller for an unstable polynomial system that combines local optimality and global stability, using a metric found via sum-of-squares programming., Comment: accepted for publication in IEEE Transactions on Automatic Control

This paper introduces new techniques for using convex optimization to fit input-output data to a class of stable nonlinear dynamical models. We present an algorithm that guarantees consistent estimates of models in this class when a small set of repeated experiments with suitably independent measurement noise is available. Stability of the estimated models is guaranteed without any assumptions on the input-output data. We first present a convex optimization scheme for identifying stable state-space models from empirical moments. Next, we provide a method for using repeated experiments to remove the effect of noise on these moment and model estimates. The technique is demonstrated on a simple simulated example.

In this paper, we derive differential conditions guaranteeing the orbital stability of nonlinear hybrid limit cycles. These conditions are represented as a series of pointwise linear matrix inequalities (LMI), enabling the search for stability certificates via convex optimization tools such as sum-of-squares programming. Unlike traditional Lyapunov-based methods, the transverse contraction framework developed in this paper enables proof of stability for hybrid systems, without prior knowledge of the exact location of the stable limit cycle in state space. This methodology is illustrated on a dynamic walking example., Comment: Conference submission

This paper addresses the problems of stabilization, robust control, and observer design for nonlinear systems. We build upon recently a proposed method based on contraction theory and convex optimization, extending the class of systems to which it is applicable. We prove converse results for mechanical systems and feedback-linearizable systems. Next we consider robust control, and give a simple construction of a controller guaranteeing an L2-gain condition, and discuss connections to nonlinear H-infinity control. Finally, we discuss a "duality" result between nonlinear stabilization problems and observer construction, in the process constructing globally stable reduced-order observers for a class of nonlinear systems., Comment: Conference submission and working paper

Control contraction metrics (CCMs) are a new approach to nonlinear control design based on contraction theory. The resulting design problems are expressed as pointwise linear matrix inequalities and are and well-suited to solution via convex optimization. In this paper, we extend the theory on CCMs by showing that a pair of "dual" observer and controller problems can be solved using pointwise linear matrix inequalities, and that when a solution exists a separation principle holds. That is, a stabilizing output-feedback controller can be found. The procedure is demonstrated using a benchmark problem of nonlinear control: the Moore-Greitzer jet engine compressor model., Comment: Conference submission

In this paper we introduce the concept of universal stabilizability: the condition that every solution of a nonlinear system can be globally stabilized. We give sufficient conditions in terms of the existence of a control contraction metric, which can be found by solving a pointwise linear matrix inequality. Extensions to approximate optimal control are straightforward. The conditions we give are necessary and sufficient for linear systems and certain classes of nonlinear systems, and have interesting connections to the theory of control Lyapunov functions., Comment: Conference submission

A new framework for nonlinear system identification is presented in terms of optimal fitting of stable nonlinear state space equations to input/output/state data, with a performance objective defined as a measure of robustness of the simulation error with respect to equation errors. Basic definitions and analytical results are presented. The utility of the method is illustrated on a simple simulation example as well as experimental recordings from a live neuron., National Science Foundation (U.S.). (Grant number 0835947)

Reinforcement Learning offers a very general framework for learning controllers, but its effectiveness is closely tied to the controller parameterization used. Especially when learning feedback controllers for weakly stable systems, ineffective parameterizations can result in unstable controllers and poor performance both in terms of learning convergence and in the cost of the resulting policy. In this paper we explore four linear controller parameterizations in the context of REINFORCE, applying them to the control of a reaching task with a linearized flexible manipulator. We find that some natural but naive parameterizations perform very poorly, while the Youla Parameterization (a popular parameterization from the controls literature) offers a number of robustness and performance advantages., National Science Foundation (U.S.) (Award IIS-0746194)

We propose a constructive control design for stabilization of non-periodic trajectories of underactuated robots. An important example of such a system is an underactuated “dynamic walking” biped robot traversing rough or uneven terrain. The stabilization problem is inherently challenging due to the nonlinearity, open-loop instability, hybrid (impact) dynamics, and target motions which are not known in advance. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents “transversal” dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design, providing exponential orbital stability of the target trajectory of the original nonlinear system. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to a wide variety of hybrid nonlinear systems., National Science Foundation (U.S.) (Grant 0746194)

A motion planning algorithm is described for bounding over rough terrain with the LittleDog robot. Unlike walking gaits, bounding is highly dynamic and cannot be planned with quasi-steady approximations. LittleDog is modeled as a planar five-link system, with a 16-dimensional state space; computing a plan over rough terrain in this high-dimensional state space that respects the kinodynamic constraints due to underactuation and motor limits is extremely challenging. Rapidly Exploring Random Trees (RRTs) are known for fast kinematic path planning in high-dimensional configuration spaces in the presence of obstacles, but search efficiency degrades rapidly with the addition of challenging dynamics. A computationally tractable planner for bounding was developed by modifying the RRT algorithm by using: (1) motion primitives to reduce the dimensionality of the problem; (2) Reachability Guidance, which dynamically changes the sampling distribution and distance metric to address differential constraints and discontinuous motion primitive dynamics; and (3) sampling with a Voronoi bias in a lower-dimensional “task space” for bounding. Short trajectories were demonstrated to work on the robot, however open-loop bounding is inherently unstable. A feedback controller based on transverse linearization was implemented, and shown in simulation to stabilize perturbations in the presence of noise and time delays., United States. Defense Advanced Research Projects Agency. Learning Locomotion Program (AFRL contract # FA8650-05-C-7262)

We propose an algorithm for design of optimal inputs for system identification when amplitude constraints on the input and output are imposed. In contrast to input design with signal power constraints, this problem is non-convex and non-smooth. We propose an iterative solution: in the first step, a convex optimization problem is solved for input design under power constraints. In subsequent steps, the constraints considered are the p-norms of the input and output signals, p increases for each iteration step. This is an adaptation of the classical Poà ¿lya algorithm for function approximation, which has previously been used for the related problem of signal crest-factor optimization. Although the difficulty of the problem prevents a proof of optimality, the performance of the algorithm is discussed with reference to a simple example., Swedish Research Council (grant 2006-7551), Swedish Foundation for Strategic Research (grant SM07-0035)

We propose a constructive control design for stabilization of non-periodic trajectories of underactuated mechanical systems. An important example of such a system is an underactuated ``dynamic walking'' biped robot walking over rough terrain. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to higher degree-of-freedom robots over arbitrary terrain and other impulsive mechanical systems.

Accurate estimates of the outflow resistance of the human cerebrospinal fluid system are important for the diagnosis of a medical condition known as hydrocephalus. In this paper we design a nonlinear observer which provides on-line estimates of the outflow resistance, to the best of our knowledge the first method to do so. The output of the observer is proven to globally converge to an unbiased estimate. Its performance is experimentally verified using the same apparatus used to perform actual patient diagnoses and a specially-designed physical model of the human cerebrospinal fluid system.

The problem is to create a hybrid periodic motion, reminiscent of walking, for amodel of an underactuated biped robot. We show how to construct a transverse linearization analytically and how to use it for stability analysis and for design of an exponentially orbitally stabilizing controller. In doing so, we extend a technique recently developed for continuous-time controlled mechanical systems with degree of underactuation one. All derivations are shown on an example of a three-link walking robot, modeled as a system with impulse effects.

This article presents the identification and control of an electro-hydraulic crane. The crane is of the type used on forestry vehicles known as forwarders, which travel off-road collecting logs cut by the harvesters. The dynamics identified include significant frictional forces, dead zones, and structural and hydraulic vibrations. The control algorithm proposed, comprised of a linear controller and a compensator for nonlinearities, is able to accurately track a reference trajectory for the end effector, despite uncertainties in the arm mechanics and hydraulic system dynamics. A further control design is presented which uses an inner loop to compensate for vibrations in the hydraulic system, and its performance is experimentally verified.

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we consider planning periodic motions and designing feedback controllers for orbital stabilization. We review classical and recent design methods based on the Poincaré first-return map and the transverse linearization. We begin with general nonlinear systems and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically.

This paper introduces a method for analytical planning of feasible hybrid periodic trajectories in non-feedback-linearizable impulsive mechanical systems with control inputs. For a planned motion, a procedure for computation of a transverse linearization for a class of hybrid controlled mechanical systems with underactuation one is presented. The resulting linear comparison system can be used for stability analysis and for design of orbitally stabilizing controllers.

This paper presents a vision-based wheeled-robot navigation technique, termed circular navigation guidance (CNG), for the interception of a moving target from a specific angle. The guidance law is not split into path-planning and path-following stages, but is continuously updated based on immediately available information, making it useful even against a manoeuvring target, or where only incomplete information is available to the robot. This is achieved by converting a finite-time navigation problem into one of maintaining a certain geometric condition at all times. Detailed theoretical analysis provides a closed-form solution for the shape of the path the robot takes against a moving target. Experimental results are presented which show that CNG is robust and accurate under real-world conditions against a manoeuvring target.