# Research

Applied math and control theory, with applications from aeroastro to systems pathobiology:

- Systems approaches in the Human Microbiome,
- Learning (new work in this area will focus on distributed learning), and
- Fundamental Properties of Network Systems.

See below some details on the human microbiome, synchronization and learning, transients in adaptive control (learning) systems, and some old work on aircraft control.

## Systems Approaches in the Human Microbiome

One of the most exciting fields in Medicine and Biology at the moment is that of Human Microbiology. To understand the importance of the human microbiome let us begin with an anecdote. This story begins with Jane coming to the hospital because of an infection in her leg. To kill the infection she is given broad spectrum antibiotics. After a few days the infection is gone, but Jane now has severe diarrhea. The antibiotics have killed some of the healthy bacteria in her gut and now Jane has an over abundance of *Clostridium difficile*. This over abundance is termed "*Clostridium Difficile Infection*" (CDI). Ironically, the most frequently prescribed treatment for CDI is another antibiotic. This targeted antibiotic works in temporarily reducing the abundance *C. difficile*, but the CDI is recurrent in Jane's case. So with no other options, Jane asks her brother John for a fecal sample. This fecal sample is prepared and transplanted into Jane (*Fecal Microbial Transplantation* (FMT)). As if a miracle has occurred Jane is healthy again.

Ultimately we wish to make clinical predictions from patient microbial samples. Before this question can be addressed, however, we to need understand how microbes interact in different hosts, if they interact at all. This line of questioning is visualized on the right. Do two microbes interact in the same way in two otherwise healthy hosts? Our work suggests the answer is in the affermative.

With regard to FMTs we wish to be able to distinguish between the two scenarios illustrated below, before a donor is even chosen (data for these plots was obtained from this study). The figure below shows trajectories of a patient's stool microbiome, pre and post-FMT. We can see that in both cases the abundance profiles rapidly converge to the donor profile. For the patient on the left, however, this similarity in stool microbial abundance is only temporary, where as on the right, the patient's gut microbiome remains similar to the donor long after the the FMT is performed. We wish to be able to predict this outcome just from patient and donor pre-FMT samples.

- T.E. Gibson et al.,
*On the Origins and Control of Community Types in the Human Microbiome*, PLoS Comput Biol 12(2): e1004688, 2016 [PDF,Supplement,doi,arXiv pre-print] - Travis E. Gibson, “Opportunities for Systems and Control in the Human Microbiome”,
*American Control Conference,*2016 [accepted] - A. Bashan, T.E. Gibson, et al,
*Universality of human microbial dynamics*, Nature, 534, 259–262, 2016 [doi,Nature News&Views] - Travis .E. Gibson, Translate or Die, IFAC Blog Post

## Synchronization and Learning

Fundamental properties such as learning and consensus have been studied both in the adaptive control and network control literature. The use of an error feedback is essential for the realization of both properties. In adaptive control, error feedback is used to update adaptive parameters in an effort to accomplish learning and tracking. In network control, error feedback is used to achieve consensus. The two types of error feedback are seldom studied in concert. This work takes a first step towards this objective and explores the implications of concomitantly achieving consensus and learning in adaptive and networked systems. Conditions under which synchronous inputs can enhance adaptation and learning are analyzed. The tradeoff between synchronization and learning is explored both in the context of two interacting dynamical systems and a network of dynamical systems interacting over a graph.

- T.E. Gibson (submitted), “Adaptation and Synchronization over a Network: Asymptotic Error Convergence and Pinning”,
*Control Decision Conference*, 2016 [submission] - T.E. Gibson (submitted), “Adaptation and Synchronization over a Network: Stabilization without a Reference Model”,
*Control Decision Conference*, 2016 [submission] - B.M. Jenkins, A.M. Annaswamy, E. Lavretsky, T.E. Gibson (submitted),
*Convergence Properties of Adaptive Systems and the Definition of Exponential Stability*'' [arXiv per-print] - T.E. Gibson and A.M. Annaswamy, “Learning and Consensus in Adaptive and Networked Systems”,
*17th Yale Workshop on Adaptive and Learning Systems*, New Haven Connecticut, 2015 - B.M. Jenkins, T.E. Gibson, A.M. Annaswamy, and E. Lavretsky, "Uniform Asymptotic Stability and Slow Convergence in Adaptive Systems,"
*11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing*, 2013 - B.M. Jenkins, T.E. Gibson, A.M. Annaswamy, and E. Lavretsky, "Accuracy, Smoothness, and Speed in Adaptive Systems with Closed-loop Reference Models,"
*AIAA Guidance Navigation and Control Conference*2013

## Theoretical Extensions to Classical Adaptive Control

### Closed Loop Reference Models In Adaptive Control

A universal observation in all adaptive control systems is a convergent, yet oscillatory behavior in the underlying errors. These oscillations increase with adaptation gain, and as such, lead to constraints on the speed of adaptation. The main obvious challenge in quantification of transients in adaptive systems stems from their nonlinear nature. A second obstacle is the fact that most adaptive systems possess an inherent trade-off between the speed of convergence of the tracking error and the size of parametric uncertainty. In this work, we overcome these long standing obstacles by proposing an adaptive control design that judiciously makes use of an underlying linear time-varying system, analytical tools that quantify oscillatory behavior in adaptive systems, and the use of tools for decoupling speed of adaptation from parametric uncertainty.

In this work, we start with CRM adaptive systems as the design candidate, and quantify the underlying transient performance. This is accomplished by deriving L-2 bounds on key signals and their derivatives in the adaptive system. These bounds are then related to the corresponding frequency content using a Fourier analysis, thereby leading to an analytical basis for the observed reduction in oscillations with the use of CRM. It is also shown that in general, a peaking phenomenon can occur with CRM-adaptive systems, which then is shown to be minimized through an appropriate design of the CRM-parameters. Extensive simulation results are provided, illustrating the conspicuous absence of oscillations in CRM-adaptive systems in contrast to their dominant presence in ORM-adaptive systems.

- T.E. Gibson, Z. Qu, A.M. Annaswamy, E. Lavretsky,
*Adaptive Output Feedback based on Separation Principle and Closed-loop Reference Models,*IEEE Transactions on Automatic Control, 2015 [arXiv, IEEE Link, pdf] - Travis E. Gibson,
*Closed-loop Reference Model Adaptive Control: with Application to Very Flexible Aircraft*,

PhD Thesis, Massachusetts Institute of Technology, 2014 [pdf] - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*On Adaptive Control with Closed-loop Reference Models: Transients, Oscillations, and Peaking,*IEEE ACCESS 2013 [IEEE Link, arXiv pre-print] - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*Closed-Loop Reference Models in Adaptive Control: Composite Control and Observer Feedback*, 11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, 2013 - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*Output Feedback Adaptive Control with Closed-Loop Reference Models*, European Control Conference, 2013 - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*Closed-Loop Reference Models in Adaptive Control, Part 1: Transient Performance,*American Control Conference 2013 [PDF] - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*Closed-loop Reference Models for**Output Feedback Adaptive Systems*[arXiv pre-print] - T.E. Gibson, A.M. Annaswamy, E. Lavretsky,
*Closed-Loop Reference Models in Adaptive Control: Stability, Robustness and Performance*, [arXiv pre-print] - T.E. Gibson, A.M. Annaswamy, and E. Lavretsky,
*Improved Transient Response in Adaptive Control Using Projection Algorithms and Closed Loop Reference Models*, AIAA Guidance Navigation and Control Conference, 2012 [PDF]

## Flight Control Applications

### Adaptive Control of Very Flexible Aircaft

A recent area of interest is in the control of Very Flexible Aircraft (VFA). These aircraft are characterized by wings with a large aspect ratio and strong rigid flexible coupling. One example of such an aircraft is NASA's Helios, shown right. During flight the local angle of attack along the wing is expected to change signifigantly as the wing structure flexes. Further more, any uncertainty in the rigid body dynamics will be exacerbated by uncertainty in the flexible dynamics. Adaptive control has been successful in dealing with parametric uncertainty as illustrated by the above examples; however, control architectures for VFA differ from the previous applications do to the fact that not all of the states of the aircraft will be measurable, namely the flexible states. The methodologies proposed in this work look to simplify partial states accessible adaptive control technologies so that they can be applied to real aircraft with only partial measurements. Results from this work have lead to several novel extensions related to open problems in adaptive control.

Modeling for Control of VFA (AIAA conference paper)

### NASA Langley Generic Transport Model

The GTM is a dynamically scaled model of a transport aircraft for which NASA Langley has developed a high-fidelity simulink model. This simulation uses non-linear aerodynamic models extracted from wind tunnel data, and considers avionics, sensor dynamics, engine dynamics, atmospheric models, sensor noise and bias, telemetry effects, etc. This aircraft has ten controllable inputs: the elevator in the left outboard, the elevator in the right outboard, the elevator in the right inboard, the elevator in the left inboard, the left aileron, the right aileron, the upperrudder, the lower rudder, and the throttle inputs to the right and left engines. This set of inputs work synchronously under nominal operating conditions. Overall, the open-loop plant has 278 states.#### GTM Control Design

This work proposes an adaptive controller for a generic transport vehicle subject to center-of-gravity uncertainty and time-delays. The adaptive control architecture is based on a linearized model of the the aircraft dynamics. The adaptive algorithm specifically accommodates for actuator saturation and augments a baseline controller predicated on sequential loop closing techniques and integral antiwindup logic. The adaptive design is validated using the high-fidelity GTM SIMULNIK code developed at NASA Langley. The resilience of the adaptive algorithm is compared to that of the baseline controller for the uncertainties mentioned above by monitoring the structural loading and command tracking performance of the two controllers.

Adaptive Control of GTM (Active-Adaptive Controls Lab Report)

Analog time delay | 60 ms | |

Digital time delay | 105 ms | |

Left aileron | Locked -10 deg | |

Left inboard elevator | Locked 0 deg | |

Bottom rudder | Locked 0 deg | |

Top rudder | 25% effective | |

Left outboard elevator | 50% effective | |

All right elevators | 25% effective | |

Center of gravity shift | -45% MAC |

In this video the GTM simulink model is flown with a pilot in the loop. X-Plane is used for the visualization environment and a Logitech joystick is used to command the aircraft. The pilot is able to fly the airplane even though several of the control surfaces are locked or partially effective, the center of gravity is shifted back from nominal and several time delays are present in the system.

#### Internal Algorithm Monitors for Adaptive Systems

The safe flight of aircraft under abnormal and damaged flight conditions is the subject of research being conducted under NASA’s Aviation Safety Program, in the Integrated Resilient Aircraft Control (IRAC) Project. Much of this work has focused on the design of adaptive architectures as an augmentation mechanism for traditional feedback controllers. While the field of adaptive control has a long, some 60 year, history of academic analysis, the implementation of these algorithms is still a challenge. This work does not specifically address the challenges associated with the implementation of adaptive control. However, it proposes a method of monitoring the internal signals of adaptive systems so as to provide a leading indicator for unexpected behavior. This will allow for the safe testing of adaptive algorithms on real experimental testbeds.

### Adaptive Robust Control for Hypersonic Vehicles (ARCH)

The methodology proposed is a unique combination of adaptive deterministic methods and probabilistic robust control methods, thereby attaining satisfactory performance amidst various uncertainties that occur along the flight envelope. The architecture includes:- A family of adaptive control algorithms that accommodates for the mission-varying levels of uncertainty, control saturation and faults.
- Parametric safety margin analyses that help evaluate the efficacies of the adaptive controllers.
- Probabilistic assessments of the satisfaction of Figures of Merit.

#### Adaptive Proportional Integral Filter Control for Elastic HSV

This work proposes an adaptive controller for a hypersonic cruise vehicle subject to aerodynamic uncertainties, center-of-gravity movements, actuator saturation, failures, and time-delays. The adaptive control architecture is based on a linearized model of the underlying rigid body dynamics and explicitly accommodates for all uncertainties. It also includes a baseline proportional integral filter commonly used in optimal control designs. The control design is validated using a highfidelity HSV model that incorporates various effects including coupling between structural modes and aerodynamics, and thrust pitch coupling. An elaborate comparative analysis of the proposed Adaptive Robust Controller for Hypersonic Vehicles (ARCH) is carried out using a control verification methodology. In particular, we study the resilience of the controller to the uncertainties mentioned above for a set of closed-loop requirements that prevent excessive structural loading, poor tracking performance and engine stalls. This analysis enables the quantification of the improvements that result from using and adaptive controller for a typical maneuver in the V-h space under cruise conditions.