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30
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using SumofSquares Optimization
, 2003
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A tutorial on sum of squares techniques for system analysis
 In Proceedings of the American control conference, ASCC
, 2005
"... Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential e ..."
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Cited by 17 (1 self)
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Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential equations or differential algebraic equations, hybrid systems with nonlinear subsystems and/or nonlinear switching surfaces, and timedelay systems described by nonlinear functional differential equations. We will also discuss how different analysis questions such as model validation and safety verification can be answered for uncertain nonlinear and hybrid systems. I.
Searching for Control Lyapunov Functions using Sums of Squares Programming
"... Construction of a Control Lyapunov Function (CLF) for a nonlinear system is generally a difficult problem, but once a CLF is found, stabilization of the system is straightforward. In this paper, we present an algorithm that searches for CLFs for polynomial systems that are affine in control using s ..."
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Cited by 14 (2 self)
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Construction of a Control Lyapunov Function (CLF) for a nonlinear system is generally a difficult problem, but once a CLF is found, stabilization of the system is straightforward. In this paper, we present an algorithm that searches for CLFs for polynomial systems that are affine in control using sums of squares programming. We also present an algorithm for searching local CLFs for the same class of nonlinear system when global asymptotic stabilization is not possible. 1
Control design along trajectories with sums of squares programming
, 2013
"... Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance ..."
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Cited by 13 (9 self)
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Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities. These certificates, together with our proposed polynomial controllers, can be efficiently obtained via semidefinite optimization. Our approach can handle timevarying dynamics resulting from tracking a given trajectory, input saturations (e.g. torque limits), and can be extended to deal with uncertainty in the dynamics and state. The resulting controllers can be used by spacefilling feedback motion planning algorithms to fill up the space with significantly fewer trajectories. We demonstrate our approach on a severely torque limited underactuated double pendulum (Acrobot) and provide extensive simulation and hardware validation.
Robust Linear Filter Design via LMIs and Controller Design with Actuator Saturation via SOS Programming
, 2004
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Nonlinear Region of Attraction Analysis for Flight Control Verification and Validation
"... Current practice for flight control validation relies heavily on linear analyses and nonlinear, highfidelity simulations. This process would be enhanced by the addition of nonlinear analyses of the flight control system. This paper demonstrates the use of region of attraction estimation for studyi ..."
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Cited by 4 (3 self)
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Current practice for flight control validation relies heavily on linear analyses and nonlinear, highfidelity simulations. This process would be enhanced by the addition of nonlinear analyses of the flight control system. This paper demonstrates the use of region of attraction estimation for studying nonlinear effects. A nonlinear polynomial model is constructed for the longitudinal dynamics of NASA's Generic Transport Model aircraft. A polynomial model for the short period dynamics is obtained by decoupling this mode from the nonlinear longitudinal model. Polynomial optimization techniques are applied to estimate region of attractions around trim conditions.
Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures
"... Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) ..."
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Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) and provide finite dimensional approximations of this LP in terms of semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS. In contrast to traditional Lyapunov based approaches which are nonconvex and require feasible initialization, our approach is convex and does not require any form of initialization. The resulting timevarying controllers and approximated reachable sets are wellsuited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on four nonlinear systems. I.
An offline MPC strategy for nonlinear systems based on SOS programming
 Nonlinear Model Predictive Control, volume 384 of Lecture Notes in Control and Information Sciences
, 2009
"... A novel moving horizon control strategy for inputsaturated nonlinear polynomial systems is proposed. The control strategy makes use of the so called sumofsquares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions on the positiveness of polynomials. The compl ..."
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A novel moving horizon control strategy for inputsaturated nonlinear polynomial systems is proposed. The control strategy makes use of the so called sumofsquares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions on the positiveness of polynomials. The complexity of SOSbased numerical methods is polynomial in the problem size and, as a consequence, computationally attractive. SOS programming is used here to derive an “offline ” model predictive control (MPC) scheme and analyze in depth his properties. Such an approach may lead to less conservative MPC strategies than most existing methods based on the global linearization approach. An illustrative example is provided to show the benefits of the proposed SOSbased algorithm.
Flying Between Obstacles with an Autonomous KnifeEdge Maneuver
, 2012
"... We develop an aircraft and control system that is capable of repeatedly performing a high speed (7m/s or 16 MPH) “knifeedge ” maneuver through a gap that is smaller than the aircraft’s wingspan. The maneuver consists of flying towards a gap, rolling to a significant angle, accurately navigating bet ..."
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We develop an aircraft and control system that is capable of repeatedly performing a high speed (7m/s or 16 MPH) “knifeedge ” maneuver through a gap that is smaller than the aircraft’s wingspan. The maneuver consists of flying towards a gap, rolling to a significant angle, accurately navigating between the obstacles, and rolling back to horizontal. The speed and rollrate required demand a control system capable of highly precise, repeatable maneuvers. We address the necessary control theory, path planning, and hardware requirements for such a maneuver, and give a proposal for a new system that may improve upon the existing techniques.
Control and verification of highdimensional systems via DSOS and SDSOS optimization
 In Proceedings of the 53rd IEEE Conference on Decision and Control
, 2014
"... Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the c ..."
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Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the control and optimization communities, scalability has been a key challenge due to its reliance on semidefinite programming (SDP) as its main computational engine. While SDPs have many appealing features, current SDP solvers do not approach the scalability or numerical maturity of LP and SOCP solvers. Our approach is based on the recent work of Ahmadi and Majumdar [1], which replaces the positive semidefiniteness constraint inherent in the SOS approach with stronger conditions based on diagonal dominance and scaled diagonal dominance. This leads to the DSOS and SDSOS cones of polynomials, which can be optimized over using LP and SOCP respectively. We demonstrate this approach on four high dimensional control problems that are currently well beyond the reach of SOS programming: computing a region of attraction for a 22 dimensional system, analysis of a 50 node network of oscillators, searching for degree 3 controllers and degree 8 Lyapunov functions for an Acrobot system (with the resulting controller validated on a hardware platform), and a balancing controller for a 30 state and 14 control input model of the ATLAS humanoid robot. While there is additional conservatism introduced by our approach, extensive numerical experiments on smaller instances of our problems demonstrate that this conservatism can be small compared to SOS programming. I.