Martin Corless | Purdue University (original) (raw)

Papers by Martin Corless

Research paper thumbnail of A Distributed Algorithm for Least Square Solutions of Linear Equations

arXiv (Cornell University), Sep 28, 2017

A distributed discrete-time algorithm is proposed for multi-agent networks to achieve a common le... more A distributed discrete-time algorithm is proposed for multi-agent networks to achieve a common least squares solution of a group of linear equations, in which each agent only knows some of the equations and is only able to receive information from its nearby neighbors. For fixed, connected, and undirected networks, the proposed discrete-time algorithm results in each agents solution estimate to converging exponentially fast to the same least squares solution. Moreover, the convergence does not require careful choices of time-varying small step sizes.

Research paper thumbnail of Analysis and control of polytopic uncertain/nonlinear systems in the presence of bounded disturbance inputs

Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), 2000

For a general input-output system subjected t.o disturbance inputs whose magnitudes are uniformly... more For a general input-output system subjected t.o disturbance inputs whose magnitudes are uniformly bounded, we introduce the notion of Loo stability with level of performance % We present a sufficient condition which guarantees Loo stability of this system. Furthermore, for a specific class of uncertain/nonlinear systems, characterized by a polytopic condition, this condition results in a bunch of matrix inequalities. An iterative algorithm based on LMIs (Linear Matrix Inequalities) is proposed for minimizing the level of performance. We also investigate the use of linear state feedback controllers to minimize the level of performance and illustrate our results with a nontrivial example.

Research paper thumbnail of A discrete time population genetic model for X-linked recessive diseases

The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a ... more The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a discretetime, structured, mathematical model. The model accounts for both de novo mutations and different reproduction rates of procreating couples depending on their health conditions. Relying on Lyapunov theory, asymptotic stability properties of equilibrium points of the model are demonstrated. The model describes the spread over time in the population of any recessive genetic disorder transmitted through the Xchromosome. Keywords—Population genetic dynamics, nonlinear dynamic analysis, stability, epidemiology.

Research paper thumbnail of Suboptimal Control of Rigid Body Motion with a Quadratic Cost

IFAC Proceedings Volumes, 1995

This paper considers the problem of controlling the rotational motion of a rigid body using three... more This paper considers the problem of controlling the rotational motion of a rigid body using three independent control torques. Given a quadratic cost we seek stabilizing state feedback controllers which guarantee that all motions starting within a speci ed bounded set satisfy a speci ed bound on a quadratic performance index or cost. For a special class of cost functions, we present explicit expressions for the optimal stabilizing controllers. For the general case, we present su cient conditions which guarantee the existence of linear, suboptimal, stabilizing controllers.

Research paper thumbnail of Improved robustness bounds using covariance matrices

Proceedings of the 28th IEEE Conference on Decision and Control

Many types of performance and stability robustness are shown to be directly related to properties... more Many types of performance and stability robustness are shown to be directly related to properties of the closed loop covariance matrix. Covariance controllers assign a specified covariance matrix to the closed loop system. Hence they can deliver a prescribed degree of robustness with respect to disturbance rejection, structured and unstructured parameter changes. New, (less conservative) bounds are derived for these robustness problems: structured and unstructured parameter variations and disturbance robustness (L, bounds) for continuous systems. The purpose of this paper is to contribute several different new robustness results. Both structured and unstructured parameter bounds are improved for continuous systems, and L, bounds on output are given for continuous time systems subject to disturbance inputs. These results are all explicitly related to the state covariance matrix. Hence we describe here the robustness properties of any covariance controller [20] that gives a specified state covariance to the closed loop system. There are. at least two parameterizations of the class of all stabilizing controllers. The early work of Youla [I71 parameterized the class of all stabilizing controllers in the frequency domain. A state space parameterization of these controllers is also available. The work in [20,21] characterizes the set of all state covariance mamces that a linear closed loop system may possess, and gives (in closed form) the class of all controllers assigning a specified state covariance. Since all stable systems have a finite state covariance, one may look upon [20,21] as another parameterization of all stabilizing controllers.

Research paper thumbnail of Some results in the control of flexible mechanical systems

Modelling and Control of Mechanical Systems, 1997

In recent years there has been considerable interest in the control of aerospace/mechanical syste... more In recent years there has been considerable interest in the control of aerospace/mechanical systems containing exible elements. We present some of the results in this area and attempt to place them in a common framework.

Research paper thumbnail of Invariant manifold techniques for attitude control of symmetric spacecraft

Proceedings of 32nd IEEE Conference on Decision and Control

In this paper we consider the problem of attitude stabilization of an axially symmetric spacecraf... more In this paper we consider the problem of attitude stabilization of an axially symmetric spacecraft using two pairs of gas jet actuators to generate control torques about two axes spanning the two-dimensional plane orthogonal to the axis of symmetry. Under the assumption that the initial spin rate about the symmetry axis is zero, and using a new kinematic formulation, we construct an invariant manifold for the closed loop system with a specific feedback law. Using this manifold we derive a stabilizing control law which achieves arbitrary reorient,ation of the spacecraft.

Research paper thumbnail of A new nonlinear model predictive control algorithm using differential transformation with application to interplanetary low-thrust trajectory tracking

2009 American Control Conference, 2009

A nonlinear model predictive control algorithm (NMPC) is proposed that incorporates a novel diffe... more A nonlinear model predictive control algorithm (NMPC) is proposed that incorporates a novel differential transformation (DT) based algorithm for the repeated computation of the open-loop optimal control problem. This DT method converts the two-point boundary value problem from the indirect optimization approach into a set of algebraic equations, which is a less computationally expensive way of solving the open-loop problem. The new NMPC algorithm is applied to an interplanetary low-thrust trajectory tracking problem.

Research paper thumbnail of Symmetrical reconfiguration of tensegrity structures

International Journal of Solids and Structures, 2002

In this article we first present a mathematical model which describes the nonlinear dynamics of t... more In this article we first present a mathematical model which describes the nonlinear dynamics of tensegrity structures. For certain tensegrity structures a particular class of motions, coined symmetrical motions, is defined. The corresponding equations of motion are derived and the conditions under which symmetrical motions occur are established. Reconfiguration procedures through symmetrical motions are proposed and examples are given.

Research paper thumbnail of Equilibrium and stability analysis of X-chromosome linked recessive diseases model

2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012

ABSTRACT We present a mathematical model describing the population distribution of genetic diseas... more ABSTRACT We present a mathematical model describing the population distribution of genetic diseases related to X chromosomes. The model captures the disease spread within a population according to the relevant inheritance mechanisms; moreover it allows to include de novo mutations (i.e., affected siblings born to unaffected parents). The resulting dynamic system is nonlinear, discrete time and positive. Among our contributions, we consider the analytical study of model's equilibrium point, that is the distribution of the population among healthy, carrier and affected subjects, and the proof of the stability properties of the equilibrium point through Lyapunov second method. In particular global exponential stability was demonstrated in the presence of significant mutation rates and global asymptotic stability for negligible mutation rates.

Research paper thumbnail of The prestressability problem of tensegrity structures: some analytical solutions

International Journal of Solids and Structures, 2001

In this paper we formulate the general prestressability conditions for tensegrity structures. The... more In this paper we formulate the general prestressability conditions for tensegrity structures. These conditions are expressed as a set of nonlinear equations and inequalities on the tendon tensions. Several examples of tensegrity structures for which the prestressability conditions can be analytically solved are then presented.

Research paper thumbnail of A Problem in Positive Systems Stability Arising in Topology Control

Lecture Notes in Control and Information Sciences, 2009

Research paper thumbnail of A result on implicit consensus with application to emissions control

IEEE Conference on Decision and Control and European Control Conference, 2011

Research paper thumbnail of The Stability and Control of the Single Track Vehicles

Bicycles, motorcycles and scooters are all examples of the single track vehicles. The dynamics of... more Bicycles, motorcycles and scooters are all examples of the single track vehicles. The dynamics of the single track vehicle involve many degrees of freedom and various modes which govern its performance, making a complicated and interesting research topic. Motorcycle in motion can roll, yaw, and steer about the steering axis. It has three main modes which determine the motion and stability of it, they are weave, capsize, and wobble. The motorcycle performance is limited by the behavior of its modes, and if even one of the modes becomes unstable, the vehicle will roll over, and crash. The goal of this research is to analyze these modes and determine what parameters affect them the most, and by redesigning the vehicle, improve the performance of it. This research uses a basic bicycle/motorcycle model developed by Whipple and adds more realistic effects such as tire side slip, lateral force delays and steering damper into mathematical model. The vehicle motion is simulated in the real time with a steering torque input from the joystick to analyze the human control ability. Some of the results are expected and intuitive, some are not. Additional work is required. It is necessary to include frame flexibility and body lean input effects to get accurate results at higher speeds. The multi-body simulation code is required to obtain a non-linear model which takes into account aerodynamics, nonlinear behavior of the tires, and other features.

Research paper thumbnail of Decentralised Control Design of Lateral and Vertical Vehicle Dynamics using Passivity

PAMM, 2008

In this paper, we present a novel methodology to design block-diagonal stable decentralised contr... more In this paper, we present a novel methodology to design block-diagonal stable decentralised controllers based on the Kalman-Yakubovich-Popov lemma. The approach yields closed-loop systems that are robust with respect to parameter uncertainty and structural perturbations. We demonstrate the methodology by designing controllers for the lateral and vertical subsystems of a vehicle emulator; namely, a test vehicle that can mimic the behaviour of a wide range of vehicles.

Research paper thumbnail of Adaptive disturbance transmission decoupling in active vehicle suspensions

PAMM, 2008

With active vehicle suspension, one can tailor a vehicles response to load and inertial without a... more With active vehicle suspension, one can tailor a vehicles response to load and inertial without affecting the vehicle response to road disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured signals. These filters require exact knowledge of certain vehicle parameters including vehicle mass to achieve the desired decoupling. Here we propose a parameter adaptive version of these filters which does not require knowledge of vehicle parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of Padé Approximations of and preservation of quadratic Lyapunov functions

PAMM, 2008

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximation... more We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of Guaranteeing Ultimate Boundedness and Exponential Rate of Convergence for a Class of Nominally Linear Uncertain Systems

Journal of Dynamic Systems, Measurement, and Control, 1989

On s'interesse a la facon de concevoir en une seule etape les parties lineaire et non-lineair... more On s'interesse a la facon de concevoir en une seule etape les parties lineaire et non-lineaire afin de garantir non seulement la limite ultime de stabilite mais aussi un taux de convergence aussi proche que celui du systeme nominal desire

Research paper thumbnail of Robust stabilization of singularly perturbed nonlinear systems

International Journal of Robust and Nonlinear Control, 1993

Consider a singularly perturbed nonlinear system endowed with a control input and suppose a stati... more Consider a singularly perturbed nonlinear system endowed with a control input and suppose a static nonlinear feedback law is designed. We show that the operations ‘compute the reduced order system’ (i.e., let the singular perturbation parameter μ = 0) and ‘close the feedback loop’ commute, i.e. the closed loop reduced‐order system is unambiguously determined. We then show that, if the reduced order system associated with the original system is stabilizable or has uncertainties matched with the input (a condition frequently used in the design of robust controllers), then the closed loop reduced‐order system enjoys the same property. As shown by an example, this result can be used to simplify the structure of a composite controller.

Research paper thumbnail of An ℒ2 disturbance attenuation solution to the nonlinear benchmark problem

International Journal of Robust and Nonlinear Control, 1998

In this paper, we use the theory of L2 disturbance attenuation for linear (H1) and nonlinear syst... more In this paper, we use the theory of L2 disturbance attenuation for linear (H1) and nonlinear systems to obtain solutions to the Nonlinear Benchmark Problem (NLBP) proposed in the paper by Bupp et. al. 1. By considering a series expansion solution to the Hamilton-Jacobi-Isaacs Equation associated with the nonlinear disturbance attenuation problem, we obtain a series expansion solution for a nonlinear controller. Numerical simulations compare the performance of the third order approximation of the nonlinear controller with its rst order approximation (which is the same as the linear H1 controller obtained from the linearized problem.

Research paper thumbnail of A Distributed Algorithm for Least Square Solutions of Linear Equations

arXiv (Cornell University), Sep 28, 2017

A distributed discrete-time algorithm is proposed for multi-agent networks to achieve a common le... more A distributed discrete-time algorithm is proposed for multi-agent networks to achieve a common least squares solution of a group of linear equations, in which each agent only knows some of the equations and is only able to receive information from its nearby neighbors. For fixed, connected, and undirected networks, the proposed discrete-time algorithm results in each agents solution estimate to converging exponentially fast to the same least squares solution. Moreover, the convergence does not require careful choices of time-varying small step sizes.

Research paper thumbnail of Analysis and control of polytopic uncertain/nonlinear systems in the presence of bounded disturbance inputs

Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), 2000

For a general input-output system subjected t.o disturbance inputs whose magnitudes are uniformly... more For a general input-output system subjected t.o disturbance inputs whose magnitudes are uniformly bounded, we introduce the notion of Loo stability with level of performance % We present a sufficient condition which guarantees Loo stability of this system. Furthermore, for a specific class of uncertain/nonlinear systems, characterized by a polytopic condition, this condition results in a bunch of matrix inequalities. An iterative algorithm based on LMIs (Linear Matrix Inequalities) is proposed for minimizing the level of performance. We also investigate the use of linear state feedback controllers to minimize the level of performance and illustrate our results with a nontrivial example.

Research paper thumbnail of A discrete time population genetic model for X-linked recessive diseases

The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a ... more The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a discretetime, structured, mathematical model. The model accounts for both de novo mutations and different reproduction rates of procreating couples depending on their health conditions. Relying on Lyapunov theory, asymptotic stability properties of equilibrium points of the model are demonstrated. The model describes the spread over time in the population of any recessive genetic disorder transmitted through the Xchromosome. Keywords—Population genetic dynamics, nonlinear dynamic analysis, stability, epidemiology.

Research paper thumbnail of Suboptimal Control of Rigid Body Motion with a Quadratic Cost

IFAC Proceedings Volumes, 1995

This paper considers the problem of controlling the rotational motion of a rigid body using three... more This paper considers the problem of controlling the rotational motion of a rigid body using three independent control torques. Given a quadratic cost we seek stabilizing state feedback controllers which guarantee that all motions starting within a speci ed bounded set satisfy a speci ed bound on a quadratic performance index or cost. For a special class of cost functions, we present explicit expressions for the optimal stabilizing controllers. For the general case, we present su cient conditions which guarantee the existence of linear, suboptimal, stabilizing controllers.

Research paper thumbnail of Improved robustness bounds using covariance matrices

Proceedings of the 28th IEEE Conference on Decision and Control

Many types of performance and stability robustness are shown to be directly related to properties... more Many types of performance and stability robustness are shown to be directly related to properties of the closed loop covariance matrix. Covariance controllers assign a specified covariance matrix to the closed loop system. Hence they can deliver a prescribed degree of robustness with respect to disturbance rejection, structured and unstructured parameter changes. New, (less conservative) bounds are derived for these robustness problems: structured and unstructured parameter variations and disturbance robustness (L, bounds) for continuous systems. The purpose of this paper is to contribute several different new robustness results. Both structured and unstructured parameter bounds are improved for continuous systems, and L, bounds on output are given for continuous time systems subject to disturbance inputs. These results are all explicitly related to the state covariance matrix. Hence we describe here the robustness properties of any covariance controller [20] that gives a specified state covariance to the closed loop system. There are. at least two parameterizations of the class of all stabilizing controllers. The early work of Youla [I71 parameterized the class of all stabilizing controllers in the frequency domain. A state space parameterization of these controllers is also available. The work in [20,21] characterizes the set of all state covariance mamces that a linear closed loop system may possess, and gives (in closed form) the class of all controllers assigning a specified state covariance. Since all stable systems have a finite state covariance, one may look upon [20,21] as another parameterization of all stabilizing controllers.

Research paper thumbnail of Some results in the control of flexible mechanical systems

Modelling and Control of Mechanical Systems, 1997

In recent years there has been considerable interest in the control of aerospace/mechanical syste... more In recent years there has been considerable interest in the control of aerospace/mechanical systems containing exible elements. We present some of the results in this area and attempt to place them in a common framework.

Research paper thumbnail of Invariant manifold techniques for attitude control of symmetric spacecraft

Proceedings of 32nd IEEE Conference on Decision and Control

In this paper we consider the problem of attitude stabilization of an axially symmetric spacecraf... more In this paper we consider the problem of attitude stabilization of an axially symmetric spacecraft using two pairs of gas jet actuators to generate control torques about two axes spanning the two-dimensional plane orthogonal to the axis of symmetry. Under the assumption that the initial spin rate about the symmetry axis is zero, and using a new kinematic formulation, we construct an invariant manifold for the closed loop system with a specific feedback law. Using this manifold we derive a stabilizing control law which achieves arbitrary reorient,ation of the spacecraft.

Research paper thumbnail of A new nonlinear model predictive control algorithm using differential transformation with application to interplanetary low-thrust trajectory tracking

2009 American Control Conference, 2009

A nonlinear model predictive control algorithm (NMPC) is proposed that incorporates a novel diffe... more A nonlinear model predictive control algorithm (NMPC) is proposed that incorporates a novel differential transformation (DT) based algorithm for the repeated computation of the open-loop optimal control problem. This DT method converts the two-point boundary value problem from the indirect optimization approach into a set of algebraic equations, which is a less computationally expensive way of solving the open-loop problem. The new NMPC algorithm is applied to an interplanetary low-thrust trajectory tracking problem.

Research paper thumbnail of Symmetrical reconfiguration of tensegrity structures

International Journal of Solids and Structures, 2002

In this article we first present a mathematical model which describes the nonlinear dynamics of t... more In this article we first present a mathematical model which describes the nonlinear dynamics of tensegrity structures. For certain tensegrity structures a particular class of motions, coined symmetrical motions, is defined. The corresponding equations of motion are derived and the conditions under which symmetrical motions occur are established. Reconfiguration procedures through symmetrical motions are proposed and examples are given.

Research paper thumbnail of Equilibrium and stability analysis of X-chromosome linked recessive diseases model

2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012

ABSTRACT We present a mathematical model describing the population distribution of genetic diseas... more ABSTRACT We present a mathematical model describing the population distribution of genetic diseases related to X chromosomes. The model captures the disease spread within a population according to the relevant inheritance mechanisms; moreover it allows to include de novo mutations (i.e., affected siblings born to unaffected parents). The resulting dynamic system is nonlinear, discrete time and positive. Among our contributions, we consider the analytical study of model's equilibrium point, that is the distribution of the population among healthy, carrier and affected subjects, and the proof of the stability properties of the equilibrium point through Lyapunov second method. In particular global exponential stability was demonstrated in the presence of significant mutation rates and global asymptotic stability for negligible mutation rates.

Research paper thumbnail of The prestressability problem of tensegrity structures: some analytical solutions

International Journal of Solids and Structures, 2001

In this paper we formulate the general prestressability conditions for tensegrity structures. The... more In this paper we formulate the general prestressability conditions for tensegrity structures. These conditions are expressed as a set of nonlinear equations and inequalities on the tendon tensions. Several examples of tensegrity structures for which the prestressability conditions can be analytically solved are then presented.

Research paper thumbnail of A Problem in Positive Systems Stability Arising in Topology Control

Lecture Notes in Control and Information Sciences, 2009

Research paper thumbnail of A result on implicit consensus with application to emissions control

IEEE Conference on Decision and Control and European Control Conference, 2011

Research paper thumbnail of The Stability and Control of the Single Track Vehicles

Bicycles, motorcycles and scooters are all examples of the single track vehicles. The dynamics of... more Bicycles, motorcycles and scooters are all examples of the single track vehicles. The dynamics of the single track vehicle involve many degrees of freedom and various modes which govern its performance, making a complicated and interesting research topic. Motorcycle in motion can roll, yaw, and steer about the steering axis. It has three main modes which determine the motion and stability of it, they are weave, capsize, and wobble. The motorcycle performance is limited by the behavior of its modes, and if even one of the modes becomes unstable, the vehicle will roll over, and crash. The goal of this research is to analyze these modes and determine what parameters affect them the most, and by redesigning the vehicle, improve the performance of it. This research uses a basic bicycle/motorcycle model developed by Whipple and adds more realistic effects such as tire side slip, lateral force delays and steering damper into mathematical model. The vehicle motion is simulated in the real time with a steering torque input from the joystick to analyze the human control ability. Some of the results are expected and intuitive, some are not. Additional work is required. It is necessary to include frame flexibility and body lean input effects to get accurate results at higher speeds. The multi-body simulation code is required to obtain a non-linear model which takes into account aerodynamics, nonlinear behavior of the tires, and other features.

Research paper thumbnail of Decentralised Control Design of Lateral and Vertical Vehicle Dynamics using Passivity

PAMM, 2008

In this paper, we present a novel methodology to design block-diagonal stable decentralised contr... more In this paper, we present a novel methodology to design block-diagonal stable decentralised controllers based on the Kalman-Yakubovich-Popov lemma. The approach yields closed-loop systems that are robust with respect to parameter uncertainty and structural perturbations. We demonstrate the methodology by designing controllers for the lateral and vertical subsystems of a vehicle emulator; namely, a test vehicle that can mimic the behaviour of a wide range of vehicles.

Research paper thumbnail of Adaptive disturbance transmission decoupling in active vehicle suspensions

PAMM, 2008

With active vehicle suspension, one can tailor a vehicles response to load and inertial without a... more With active vehicle suspension, one can tailor a vehicles response to load and inertial without affecting the vehicle response to road disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured signals. These filters require exact knowledge of certain vehicle parameters including vehicle mass to achieve the desired decoupling. Here we propose a parameter adaptive version of these filters which does not require knowledge of vehicle parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of Padé Approximations of and preservation of quadratic Lyapunov functions

PAMM, 2008

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximation... more We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of Guaranteeing Ultimate Boundedness and Exponential Rate of Convergence for a Class of Nominally Linear Uncertain Systems

Journal of Dynamic Systems, Measurement, and Control, 1989

On s'interesse a la facon de concevoir en une seule etape les parties lineaire et non-lineair... more On s'interesse a la facon de concevoir en une seule etape les parties lineaire et non-lineaire afin de garantir non seulement la limite ultime de stabilite mais aussi un taux de convergence aussi proche que celui du systeme nominal desire

Research paper thumbnail of Robust stabilization of singularly perturbed nonlinear systems

International Journal of Robust and Nonlinear Control, 1993

Consider a singularly perturbed nonlinear system endowed with a control input and suppose a stati... more Consider a singularly perturbed nonlinear system endowed with a control input and suppose a static nonlinear feedback law is designed. We show that the operations ‘compute the reduced order system’ (i.e., let the singular perturbation parameter μ = 0) and ‘close the feedback loop’ commute, i.e. the closed loop reduced‐order system is unambiguously determined. We then show that, if the reduced order system associated with the original system is stabilizable or has uncertainties matched with the input (a condition frequently used in the design of robust controllers), then the closed loop reduced‐order system enjoys the same property. As shown by an example, this result can be used to simplify the structure of a composite controller.

Research paper thumbnail of An ℒ2 disturbance attenuation solution to the nonlinear benchmark problem

International Journal of Robust and Nonlinear Control, 1998

In this paper, we use the theory of L2 disturbance attenuation for linear (H1) and nonlinear syst... more In this paper, we use the theory of L2 disturbance attenuation for linear (H1) and nonlinear systems to obtain solutions to the Nonlinear Benchmark Problem (NLBP) proposed in the paper by Bupp et. al. 1. By considering a series expansion solution to the Hamilton-Jacobi-Isaacs Equation associated with the nonlinear disturbance attenuation problem, we obtain a series expansion solution for a nonlinear controller. Numerical simulations compare the performance of the third order approximation of the nonlinear controller with its rst order approximation (which is the same as the linear H1 controller obtained from the linearized problem.