| 17th March | 18th March | 19th March | 20th March | 21st March | 9:30 -- 11:00 AM (Talk, Discussion and Tea break) | A. J. Krener Slides |
P. S. Krishnaprasad Slides |
R. Brockett Slides |
P. S. Krishnaprasad | D. E. Chang Slides |
11:00 AM -- 12:30 PM (Talk and Discussion) | T. Ratiu Slides |
A. J. Krener Slides |
S. P. Bhat Slides |
W. Respondek | W. Respondek | 2:00 -- 3:00 PM (Talk, Discussion and Tea break) | Amitabh Saraf | M. V. Dhekane Slides |
Ramaprakash Bayadi Slides |
Harish K. Pillai Slides |
IIR | 3:00 -- 4:30 PM (Talk and Discussion) | S. P. Bhat Slides |
T. Ratiu | D. E. Chang Slides |
W. Respondek | IAR |
9:30 AM to 11:00 AM, Monday, 17th March by A. J. Krener.
Over the past several decades there has been tremendous progress in the development of nonlinear systems theory but implementation of these ideas have lagged behind because of the lack of effective computational tools. Generally speaking to be implementable the nonlinear systems theory requires the solution of partial differential equations in continuous time and functional equations in discrete time. Algorithms for these have been developed but further work is needed to make them broadly and easily accessible. We believe that computational nonlinear control is in a similar stage of development that computational linear control was in around early 1980s. Then there was a well developed theory of linear control but computational tools lagged behind. Soon after comprehensive tools such as MATLAB and Matrix X were developed and put to great use in implementing the linear theory.
Advancements in numerical methods together with the exponential increase in computational power has made it possible to solve complex nonlinear problems, many of which are closely related to control systems applications. Developing computational algorithms and software tools for such control systems are not only promising, but also necessary. The three topics that we will focus on are the following (1) Numerical solution of Hamillton-Jacobi-Bellman and Dynamic Programming equations, (2) Numerical calculation of optimal trajectories and (3) Numerical calculation of invariant manifolds.
11:00 AM to 12:30 PM, Monday, 17th March by T. Ratiu.
A class of optimal control problems based on the Clebsch approach to Euler-Poincare dynamics is introduced. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of N-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the N-Camassa-Holm equation and a new geodesic interpretation of its singular solutions. While these problems are fully actuated, the interest in this approach resides in the fact that mechanical problems can be formulated exclusively in a control theoretical manner. Next, we shall relax the dynamic constraint and replace it by a quadratic penalty term which leads to an optimization problem. Applications to image registration and the theory of metamorphosis are emphasized. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be diffeomorphisms, but penalizing them if they are not. The possibilities of this approach are just beginning to be developed, structure preserving algorithms associated to these problems are currently being developed.
2:00 PM to 3:00 PM, Monday, 17th March by Amitabh Saraf.
A number of challenges are faced by the flight control system designers of a modern fly-by-wire fighter aircraft as the aircraft is gradually flight tested to extreme regions of its flight envelope. This flight testing is necessary to extract maximum performance possible from the aircraft without jeopardizing its flight safety. First part of the talk gives a broad perspective on various aspects of modelling the aircraft dynamics in different regimes of flight. Specifically, various nonlinear aerodynamic phenomena observed at high angles of attack are described and special models developed using data gathered through wind tunnel and flight tests are presented. The second part of the talk focuses on various control law design strategies that have been developed to meet the challenges posed by the aircraft dynamics. These include designs that ensure that the aircraft never exceeds the specified boundaries in terms of various flight parameters like speed, angle of attack, normal accelerations etc. Flight control strategy for ski-jump take off of a naval aircraft variant is also described.
3:00 PM to 4:30 PM, Monday, 17th March by Sanjay P. Bhat (Senior Consultant, Advanced Technology Center, Tata Consultancy Services, Hyderabad).
Control Moment Gyroscopes (CMGs) are actuation devices for spacecraft attitude control that have found successful use in the space industry since the days of the Skylab. We will begin with a brief introduction to the principle of operation of a CMG and its use in attitude control. For purposes of mechanical redundancy as well as control authority, CMGs are used in arrays rather than singly. Unfortunately, every CMG array possesses singular configurations in which the array is unable to produce actuation torque along certain directions. We will briefly explain what a singular configuration is and why singular configurations are seen as being problematic for attitude control. This will lead us to pose the main question for this talk: Does the presence of singular CMG configurations pose an obstruction to global attitude controllability? We will use standard tools from nonlinear controllability theory to discover the answer to this question.
9:30 AM to 11:00 AM, Tuesday, 18th March by P. S. Krishnaprasad.
Geometric ideas enter the investigation of collective behavior from multiple vantage points: the structure of configuration space; the synthesis of control strategies; the role of symmetry and reduction in closed loop dynamics; and the analysis of empirical data from biology. In this lecture we will present an overview of recent progress in these directions. We will consider methods to assimilate sampled observations of collectives of continuous time dynamical systems (e.g. predator-prey encounters and bird flocking events), into generative models with continuous time inputs and outputs. The purpose of such assimilation is to evaluate hypotheses of interest, based on correlations, delays, and mechanisms of interaction between elementary units of the observed population. Initial ideas on the development of control strategies were strongly influenced by studies in the laboratory (with Cynthia Moss and her students), on the prey-capture behavior of echolocating bats. Strategies found in these studies have served as building blocks for rules of collective behavior. Analysis of trajectory data on large flocks of starlings provided by Andrea Cavagna has necessitated efficient reconstruction techniques. Again the data on prey-capture behavior of bats served as a testing ground for our methods of reconstruction. We will discuss some robotics experiments guided by these studies. Outline of presentation: (1) Models of individual agents (self-steering particles, particles in matrix Lie groups); (2) Dyadic interactions as building blocks (pursuit, escape, boundary tracking); (3) Models of collectives (graphs, strategies, optimality); (4) Configuration space methods (shapes, ensemble moment of inertia, energy splitting); (5) Data assimilation (optimal fitting, cross-validation, extraction of rules for natural collectives).
11:00 AM to 12:30 PM, Tuesday, 18th March by A. J. Krener.
Model Predictive Regulation (MPR) is a way to regulate a nonlinear plant in the presence of external commands or disturbances. It is extension of Model Predictive Control (MPC), an increasing popular method of stabilizing a nonlinear plant. The virtue of MPR is that it does away with need to globally solve functional equations such as the Francis-Byrnes-Isidori equation or the Bellman Dynamic Programming equation.
2:00 PM to 3:00 PM, Tuesday, 18th March by M. V. Dhekane.
Control of Launch vehicles pose several challenges, which involve modelling the dynamics, design and validation. Mathematical models of such vehicles take into account complex interactions between rigid body aerodynamics, propulsion, slosh, flexibility, actuator, sensors and coupling pitch, yaw, roll dynamics. Controller designs are a blend of classical and modern techniques to satisfy the performance specifications under the parameter perturbation. Design implementation requires scheduling with respect to time or mach number. Integrated on board softwares are validated extensively on various test beds with sensors and actuators. This talk, therefore, will briefly highlight the above aspects with specific reference to Reusable Launch Vehicles.
3:00 PM to 4:30 PM, Tuesday, 18th March by T. Ratiu.
A class of optimal control problems based on the Clebsch approach to Euler-Poincare dynamics is introduced. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of N-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the N-Camassa-Holm equation and a new geodesic interpretation of its singular solutions. While these problems are fully actuated, the interest in this approach resides in the fact that mechanical problems can be formulated exclusively in a control theoretical manner. Next, we shall relax the dynamic constraint and replace it by a quadratic penalty term which leads to an optimization problem. Applications to image registration and the theory of metamorphosis are emphasized. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be diffeomorphisms, but penalizing them if they are not. The possibilities of this approach are just beginning to be developed, structure preserving algorithms associated to these problems are currently being developed.
9:30 AM to 11:00 AM, Wednesday, 19th March by R. Brockett.
Many important engineering systems accomplish their purpose using cyclic processes whose characteristics are under feedback control. Examples involving thermodynamic cycles and electromechanical energy conversion processes are particularly noteworthy. Likewise, cyclic processes are prevalent in nature and the idea of a pattern generator is widely used to rationalize mechanisms used for orchestrating movements such as those involved in locomotion and respiration. In this paper, we develop a linkage between the use of cyclic processes and the control of nonholonomic systems, emphasizing the problem of achieving stable regulation. The discussion brings to the fore characteristic phenomena that distinguish the regulation problem for such strongly nonlinear systems from the more commonly studied linear feedback regulators. Finally, we compare this approach to controlling nonholonomic systems to another approach based on the idea of an open-loop approximate inverse as discussed in the literature.
11.30 AM to 12:30 PM, Wednesday, 19th March by Sanjay P. Bhat (Senior Consultant, Advanced Technology Center, Tata Consultancy Services, Hyderabad).
In this talk, we will examine if the presence of singular configurations in arrays of control moment gyroscopes (CMGs) poses an obstruction to the small-time local controllability (STLC) and local asymptotic stabilizability of the attitude dynamics at equilibria. Unlike the question of global controllability considered in the first talk, the questions of STLC and stabilizability will require us to examine the nature of singular configurations a little more closely. We will identify a special class of singular configurations called critically singular configurations. We will show that the linearized dynamics are controllable at configurations which are either nonsingular or singular but not critically singular, thus settling both STLC and stabilizability at such configurations. Next, we will use sufficient conditions by Bianchini and Stefani, and a neccesary condition by Stefani to provide conditions on a critically singular configuration that either ensure or rule out STLC. In the case where STLC may be shown to fail, we will illustrate how one may exploit reduction in a neat way to draw conclusions about the nature of small-time unreachable states. Finally, we illustrate how the well known Brockett condition for stabilizability may be used to rule out stabilizability at certain critically singular configurations.
2:00 PM to 3:00 PM, Wednesday, 19th March by Ramaprakash Bayadi.
We first demonstrate that the rigid body with either external or internal actuation can be stabilized about a given orientation using the same energy-like function. To construct such an energy-like function, we take motivation from the spinning top system under the influence of gravity. We show how to construct this energy function using vector observations. The resulting control law does not require the attitude matrix to be explicitly determined, which has the potential to save the computational time, if implemented on-board a spacecraft.
3:00 PM to 4:30 PM, Wednesday, 19th March by D. E. Chang.
The falling cat problem has been very popular in control, mechanics and mathematics since Kane and Sher published a paper on this topic in 1969. A cat, after released upside down, executes a 180-degree reorientation, all the while having a zero angular momentum. It makes use of the conservation of angular momentum that is induced by rotational symmetry in the dynamics. In general, however, the angular momentum is not conserved if there is a symmetry- breaking force, such as a frictional force, on the system.
Recently, we have discovered an exciting phenomenon in controlled mechanical systems with external damping forces. If a control force is activated on such a system for a while and then gets deactivated, the unactuated cyclic variables, which get excited initially from rest by the control force, eventually all converge back to their initial values as time tends to infinity, which is called a damping-induced self-recovery phenomenon.
A self-recovery phenomenon can be observed in the simple experiment with a rotating stool and a bicycle wheel which is a typical setup in physics classes to demonstrate the conservation of angular momentum. Sitting on the stool, one spins the wheel by hand while holding it horizontally. A reaction torque will be created to initiate a rotational motion of the stool in the opposite direction. After some time, if the person applies a braking force halting the wheel spin, then the stool will asymptotically return to its original position, as if it has a memory, provided that there is a viscous damping force on the rotation axis of the stool.
We have also discovered the self-recovery phenomenon in incompressible viscous fluid flows. As a corollary, we give a "dynamic" explanation of the famous experiment by G.I. Taylor on the "kinematic" reversibility of low-Reynolds-number flows.
In this talk, several videos will be displayed to show instances of damping-induced self-recovery phenomena.
9:30 AM to 11:00 AM, Thursday, 20th March by P. S. Krishnaprasad.
The self-steering particle model of agents in a planar collective gives rise to dynamics on the N-fold direct product of the rigid motion group SE(2). Assuming a connected undirected graph of interaction between the agents, one can pose a class of symmetric optimal control problems with a coupling parameter governing the strength of the interactions. Reduction of the necessary conditions for optimality to a Lie-Poisson dynamical system raises interesting questions: Are there additional (hidden) symmetries beyond the manifest ones used in reduction? Can the dynamics be explicitly integrated? In this talk we show that in the strong coupling limit the answer to these questions is affirmative. Indeed, in the strong coupling limit, a nonlinear oscillator (clock) governs all agents to steer identically. We put forward the possibility that allelomimetic behavior observed in diverse natural collectives may have evolved to meet certain optimality principles. This is joint work with Eric Justh.
11:00 AM to 12:30 PM, Thursday, 20th March by W. Respondek.
We will discuss various concepts of linearization of nonlinear control systems: via change of coordinates in the state-space, via static feedback, via orbital feedback, and via dynamic precompensation.
For each notion of linearization, we will give necessary and sufficient conditions for linearizability (if they exist) and provide methods to find linearizing transformations. Describing linearizable system will require some tools from differential geometry, like Lie brackets, vector distributions and their involutivity. We will introduce them and show that they appear in nonlinear control theory in a natural way. We will illustrate the presented results with the help of physical, mainly mechanical, examples.
2:00 PM to 3:00 PM, Thursday, 20th March by Harish K. Pillai.
One dimensional linear piecewise-smooth maps with a jump discontinuity are known to have stable periodic orbits. Further, these stable periodic orbits are known to exhibit period adding bifurcation. In this talk, I shall analyse different situations arising in linear piecewise-smooth maps with a jump discontinuity where (stable/unstable) periodic orbits exist. Using elementary mathematics, I shall connect the above problem with some interesting patterns.
9:30 AM to 11:00 AM, Friday, 21st March by D. E. Chang.
I will talk about two topics: energy shaping of mechanical systems, and quasi-linearization of the equations of motion of mechanical systems. The former is an effective method to stabilize mechanical systems, and the latter is a way to simplify the equations of motion via a linear transformation of the velocity variables. Time permitting, I will also talk about a simple proof of the Pontryagin maximum principle on manifolds.
11:00 AM to 12:30 PM, Friday, 21st March by W. Respondek.
We study the geometry of mechanical control systems. A special class is distinguished: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterize those nonlinear control systems that are state equivalent to a system from that class and describe, in terms of the original control system, the canonical mechanical structure attached to it: configuration manifold, tangent bundle structure, affine connection, and forces. We also describe equivariants of mechanical control systems.
In the second part we discuss linearization, via a diffeomorphism, of mechanical control systems and study the problem of whether both structures, linear and mechanical ones, are compatible. The first problem we consider is: given a mechanical control systems that is linearizable, can we linearize it preserving, simultaneously, its given mechanical structure. The second problem is whether a general control-affine system that is linearizable and admits a mechanical control structure can be transformed into a linear mechanical structure.
Our results are based on a joint work with Sandra Ricardo (Universita de Vila Real, Portugal).