Extremum Seeking Control in Biomedical Applications

American Controls Conference 2020 Workshop

Held on June 30, 2020

Workshop Overview

Biomedical systems are notoriously difficult to model. This difficulty stems from the variation in physiology between subjects. Furthermore, an individual subject will often vary over the course of a day, a week, etc. This difficulty in modeling makes it difficult to implement optimal control solutions. Extremum Seeking Control (ESC) is a method of model-free adaptive control that modifies the arguments of a cost function to guide them to a local maximum or minimum. The versatility and model-free nature of ESC makes them very well suited for biomedical control applications.

We presented ten recent results in applying ESC to a wide variety of biomedical problems, including powered prosthetics and orthotics, medication delivery, rehabilitation therapy, and assistive heart pumps. We sought to highlight the strengths of ESC in biomedical applications and spur further research and development in the community who may not have considered this powerful approach.

Target audience

We had two target audiences in mind: researchers in biomedical control applications who are not familiar with ESC, and controls researchers in ESC who have not previously worked in biomedical applications. By bringing these groups together and sharing our experiences, we hoped to foster future research and collaborations among attendees.

Slides provided by the speakers are linked in the slides titles.

If you are interested in the outcomes or more information about the presentations, feel free to reach out to the speakers or organizers below. Nick Gans is the point of contact at ngans@uta.edu.

Agenda and List of Presentations

Click on each listing to see an abstract and presenter biosketch

8:45 AM – 9:00 AM: Welcome & Introductions

9:00 AM – 9:30 AM: Martin Guay - An introduction to Extremum Seeking Control

Abstract: The Introduction Session focuses on the basic principles of ESC, in particularly the adaptive ESC. A simplest possible ESC will be presented into details. The attendee will come to understand

  • The basic problem formulation of ESC

  • The link of ESC to optimization algorithms

  • The performance of ESC

  • The design trade-off of ESC

Biosketch: Martin Guay is a Professor in the Department of Chemical Engineering at Queen's University in Kingston, Ontario, Canada. He received his PhD from Queen's University in 1996. Dr. Guay is Senior Editor for the IEEE CSS Letters. He is deputy Editor-in-Chief of the Journal of Process Control. He is also an associate editor for Automatica, Canadian Journal of Chemical Engineering and Nonlinear Analysis & hybrid Systems. He was the recipient of the Syncrude Innovation award from the Canadian Society of Chemical Engineers. He also received the Premier Research Excellence award. His research interests are in the area of nonlinear control systems including extremum-seeking control, nonlinear model predictive control, adaptive estimation and control, and geometric control.

Abstract: Rehabilitation robots and motorized exercise machines have been coupled with functional electrical stimulation (FES) to develop therapies for people with neurological conditions. An outstanding challenge for the design of optimal therapies is how to determine the intensity level of a rehabilitation task since the movement ability of a person recovering from injury is time-varying and hard to quantify. This project implements an extremum seeking controller to compute speed and torque trajectories to maximize a rider’s power output during FES-induced cycling. The objective is to deviate from the use of predetermined cycling trajectories that usually yield suboptimal training or require burdensome manual tuning of the desired trajectories to ensure task completion. Switched controllers are designed to command stimulation inputs to the lower-limb muscles and activate an electric motor that assists the rider while pedaling. Lyapunov- and passivity-based methods are used to ensure stability of the muscle and motor subsystems. The cycling results demonstrate the effectiveness to concurrently adjust the speed and torque demands based on the rider’s performance.

Biosketch: Victor Duenas received his Ph.D. in 2018 from the Department of Mechanical and Aerospace Engineering from the University of Florida, Gainesville, FL, USA. In fall 2018, he joined the department of Mechanical and Aerospace Engineering at Syracuse University, Syracuse, NY, as an assistant professor. His current research interests include nonlinear and adaptive control for rehabilitation robotics, neuromuscular control, and human-robot interaction.

10:00 AM – 10:30 AM: Hosam Fathy - Can the Solution Structure of a Biomedical Optimal Control Application Problem be Exploited for Faster Extremum-Seeking?

Abstract: This presentation will begin with the well-recognized observation that many optimal control problems in both the biomedical domain and beyond possess a periodic solution. Extremum-seeking control is attractive for the online solution of these problems partly because it provides a pathway for adaptation in the presence of parametric uncertainties. One challenge with such an approach is the slow convergence speed potentially associated with extremum-seeking in problems with large numbers of unknown parameters. For some of these problems, the problem structure is such that the family of solution trajectories can be described in terms of a lower-dimensional parameter space. This leads to the presentation’s overarching question: to what extent can this problem structure be exploited for faster extremum-seeking control?

Biosketch: Hosam K. Fathy earned his B.Sc., M.S. and Ph.D. degrees, all in Mechanical Engineering, from the American University in Cairo (1997), Kansas State University (1999), and University of Michigan (2003), respectively. His primary area of expertise is optimal control/estimation, with application to energy systems and a strong interest in biomedical applications. He is a Mechanical Engineering Professor at The University of Maryland.

10:30 AM – 11:00 AM Coffee Break

Abstract: We present an extremum seeking controller (ESC) for simultaneously tuning the feedback control gains of a knee-ankle powered prosthetic leg using continuous-phase controllers. Gains of the continuous-phase controller for each joint are traditionally tuned manually by trial-and-error, which can be time consuming and limits top performance to a single walking speed. We propose a convex objective function, which incorporates trajectory tracking and user comfort. We then propose an ESC controller to minimize this cost function, and prove the stability of error dynamics of the continuous-phase controlled powered prosthetic leg along with the ESC dynamics. The optimum of the objective function shifts at different walking speeds, and our algorithm is suitably fast to track these changes, providing real-time adaptation for different walking conditions. Benchtop and walking experiments verify the effectiveness of the proposed ESC across various walking speeds.

Biosketch: Nicholas Gans earned his Ph.D. in Systems and Entrepreneurial Engineering from the University of Illinois Urbana-Champaign in 2005. He is currently Division Head of Autonomy and Intelligent Systems at the University of Texas at Arlington Research Institute. Prior to this position, he was a professor in the department of Electrical and Computer Engineering at The University of Texas at Dallas. His research interests are in the fields of robotics, nonlinear and adaptive control, machine vision, and autonomous vehicles. He is a senior member of IEEE, and an Associate Editor for the IEEE Transaction on Robotics.

11:30 AM – 12:00 PM : Robert Gregg - Presentation: Extremum Seeking Control for Stiffness Auto-Tuning of Pseudo-Passive Ankle Foot Orthosis

Abstract: Recently, it has been shown that lightweight ankle-foot orthoses (AFO) with a spring attached in parallel to the calf muscle, can reduce the muscular effort needed for human walking. In order to achieve maximum muscular effort reduction, the stiffness of the spring must be properly tuned. The optimal stiffness can vary across users and can even change for an individual user as a result of variations in walking conditions. Existing passive devices have a fixed stiffness during operation, preventing the AFO from responding to changes in walking conditions. We are developing a pseudo-passive ankle foot orthotic with a variable stiffness mechanism that allows the stiffness to change during use, providing a platform where real-time optimization of the stiffness in response to changes in muscular effort is possible. An extremum seeking controller (ESC) was used to tune the spring stiffness online by using EMG feedback. Preliminary results indicate that ESC can automatically find the optimal spring stiffness as the walking conditions change.

Biosketch: Robert D. Gregg IV received the B.S. degree in electrical engineering and computer sciences from the University of California, Berkeley in 2006 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana-Champaign in 2007 and 2010, respectively. He joined the University of Michigan as Associate Professor of Electrical and Computer Engineering and the Robotics Institute in Fall 2019. Prior to joining U-M, he was an Assistant Professor in the Departments of Bioengineering and Mechanical Engineering at the University of Texas at Dallas with an adjunct appointment at the UT Southwestern Medical Center. Dr. Gregg directs the Locomotor Control Systems Laboratory, which conducts research on the control mechanisms of bipedal locomotion with applications to wearable and autonomous robots.

12:00 PM – 01:30 PM : Lunch

Abstract: In this presentation, we propose an extremum-seeking regulator approach for a class of autonomous nonlinear systems with unknown dynamics subject to unknown exosystem dynamics. The proposed extremum-seeking regulator technique is based on a Lie bracket averaging technique. The Lie bracket averaging framework is advantageous and reliable for the analysis and design of ESC systems. It exploits a property of nonlinear systems subject to highly oscillatory control signals that yields approximations of the system's trajectories by a Lie bracket averaged system. In biomedical applications, the control system is subject to many restrictions such as actuator and sensor limitations. In the context of ESC, these limitations can severely hinder the design and performance of ESC. In addition, such applications are often subject to time-varying optimal solutions that arise due to the effect of exogenous disturbances.

Biosketch: Martin Guay is a Professor in the Department of Chemical Engineering at Queen's University in Kingston, Ontario, Canada. He received his PhD from Queen's University in 1996. Dr. Guay is Senior Editor for the IEEE CSS Letters. He is deputy Editor-in-Chief of the Journal of Process Control. He is also an associate editor for Automatica, Canadian Journal of Chemical Engineering and Nonlinear Analysis & hybrid Systems. He was the recipient of the Syncrude Innovation award from the Canadian Society of Chemical Engineers. He also received the Premier Research Excellence award. His research interests are in the area of nonlinear control systems including extremum-seeking control, nonlinear model predictive control, adaptive estimation and control, and geometric control.

2:00 PM – 2:30 PM : Saurav Kumar - Time Invariant ESC for Biomedical Applications

Abstract: Conventional perturbation-based extremum seeking control (ESC) employs a slow time-dependent periodic signal to find an optimum of an unknown plant. To ensure stability of the overall system, the ESC parameters are selected such that there is sufficient time-scale separation between the plant and the ESC dynamics. This approach is suitable when the plant operates at a fixed time-scale. Many biomedical systems have time-varying time scales. For example the gait cycle time scale will vary if a person changes walking speed, and the cardiopulmonary system time scale changes for different levels of activity. If the plant slows down, the time-scale separation can be violated for a constant frequency of dither, and the stability and performance of the overall system can no longer be guaranteed. We recently proposed an ESC for periodic systems, where the external time-dependent dither signal in conventional ESC is replaced with the periodic signals present in the plant, thereby making ESC time-invariant in nature. The advantage of using a state-based dither is that it inherently contains the information about the rate of the rhythmic task under control. Thus, in addition to maintaining time-scale separation at different plant speeds, the adaptation speed of a time-invariant ESC automatically changes, without changing the ESC parameters. We will discuss the proposed time-invariant ESC in the context of biomedical systems.

Biosketch: Saurav Kumar received the M.Tech degree (2015) in Robotics from Indian Institute of Information Technology. He is currently pursuing a Ph.D. in Electrical Engineering at the University of Texas at Dallas. His research interests include nonlinear and adaptive control, extremum seeking control and wearable robotics. He is a student member of IEEE and Control Systems Society.

Abstract: Extremum Seeking (ES) approach is employed to adapt the gains of a Proportional-Integral-Derivative (PID) control law for functional neuromuscular electrical stimulation (NMES). The proposed scheme is applied to control the position of the arm of stroke patients so that coordinated movements of flexion/extension for their elbow can be performed. This approach eliminates the initial tuning tests with patients since the controller parameters are automatically computed in real time. The PID parameters are updated by means of a discrete version of multivariable deterministic/stochastic ES in order to minimize a cost function which brings the desired performance requirements. Experimental results with stroke patients show the usual specifications commonly considered in physiotherapy for functional rehabilitation are eventually satisfied in terms of steady-state error, settling time, and percentage overshoot. Quantitative results show a reduction of 65% in terms of the root-mean-square error (RMSE) when comparing the tracking curves of the last cycle to the first cycle in the experiments with all subjects. The proposed ES-based PID is a self-tuning, fatigue resistant control method for NMES-based therapies.

Biosketch: Tiago Roux Oliveira joined State University of Rio de Janeiro (UERJ), Brazil, as an Associate Professor in 2010. He served as a Visiting Scholar at the University of California, San Diego (UCSD), USA, in 2014. His interests include extremum seeking, sliding mode control, partial differential equations and time-delay systems. He was a recipient of the CAPES National Award of Best Thesis in Electrical Engineering in 2011, and the FAPERJ Young Researcher Award, in 2012, 2015, and 2018. In 2017, he was nominated as an Affiliate Member of the Brazilian Academy of Sciences. In 2018, he was elevated to the grade of IEEE Senior Member of the Control Systems Society (CSS). Dr. Oliveira has served as a member for the IFAC Technical Committees: Adaptive and Learning Systems (TC 1.2) and Control Design (TC 2.1), as well as the Technical Committee on Variable Structure and Sliding Mode Control of the IEEE CSS. He has served also as an Associate Editor for the Journal of the Franklin Institute (JFI), the Journal of Control, Automation, and Electrical Systems (JCAE), the IEEE Latin America Transactions (IEEE-LAT) and Systems & Control Letters (SCL).

3:00 PM – 3:30 PM : Coffee Break

3:30 PM – 4:00 PM : Ying Tan - Presentation: Personalized Kinematic Synergies for Human-Prosthesis Interfaces Using Model-Guided Extremum Seeking

Abstract: Synergies have been adopted in prosthetic limb applications to reduce complexity of design, but typically involve a single synergy setting for a population and ignore individual preference or adaptation capacity. However, personalization of the synergy setting is necessary for the effective operation of the prosthetic device. Two major challenges hinder the personalization of synergies in human-prosthesis interfaces. The first is related to the process of human motor adaptation and the second to the variation in motor learning dynamics of individuals. This work proposes a systematic personalization of kinematic synergies for human-prosthesis interfaces using on-line measurements from each individual. The task of reaching using the upper limb is described by an objective function and the interface is parameterized by a kinematic synergy. Consequently, personalizing the interface for a given individual can be formulated as finding an optimal personalized parameter. The common knowledge of individual of human motor learning is identified and utilized in an on-line optimization scheme to identify the synergies for an individual. Such information enables on-line adaptation of the human-prosthesis interface to happen concurrently to human motor adaptation without the need to re-tune the personalization algorithm for each individual. Human-in-the-loop experimental results with able-bodied subjects, performed in a virtual reality environment to emulate amputation and prosthesis use, show that the proposed personalization algorithm was effective in obtaining optimal synergies with a fast uniform convergence speed across a group of individuals.

Biosketch: Ying Tan is an Associate Professor and Reader in the Department of Mechanic Engineering at the University of Melbourne, Australia. She received her Bachelor degree from Tianjin University, China, in 1995, and her PhD from the Department of Electrical and Computer Engineering, National University of Singapore in 2002. She joined McMaster University in 2002 as a postdoctoral fellow in the Department of Chemical Engineering. Since 2004, she has been with the University of Melbourne. She was awarded an Australian Postdoctoral Fellow (2006-2008) and a Future Fellow (2009-2013) by the Australian Research Council. Her research interests are in intelligent systems, nonlinear control systems, real time optimisation, sampled-data distributed parameter systems and formation control.

4:00 PM – 4:30 PM : Peiman Naseradinmousavi and Miroslav Krstic - Extremum Seeking Control Design for Left Ventricular Assist Device

Abstract: In this effort, we first present a linear 6th-order time-varying model of left ventricular assistive device (LVAD). LVADs are rotary pumps attached directly to the heart and major blood vessels in order to boost systemic blood flow and peripheral tissue oxygenation. Traditional control of LVADs has been carried out through a constant velocity of pump leading to backflow due to critical pressure difference across the aortic valve; the left ventricular and aortic pressures. Therefore, we formulate a computationally energy-efficient real-time extremum seeking control scheme to regulate the pump speed helping to keep the pressure difference within a safe domain.


Peiman Naseradinmousavi is Assistant Professor of Mechanical Engineering Department at Dynamic Systems and Control Laboratory (DSCL) of San Diego State University. He received his Ph.D. and B.Sc. degrees in mechanical engineering (dynamics and control) from Villanova and Tabriz universities, in 2012 and 2002, respectively. His research interests include robotics, smart flow distribution network, nonlinear dynamics, control theory, optimization, magnetic bearings, and mathematical modeling. He serves as Associate Editor for the Journal of Vibration and Control (JVC).

Miroslav Krstić is Distinguished Professor of Mechanical and Aerospace Engineering, holds the Alspach endowed chair, and is the founding director of the Cymer Center for Control Systems and Dynamics at UC San Diego. He also serves as Senior Associate Vice Chancellor for Research at UCSD.

4:30 PM – 5:00 PM : Panel Discussion

Workshop Organizers

Nicholas Gans

University of Texas at Arlington

Robert Gregg

University of Michigan

Saurav Kumar

University of Texas at Dallas