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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.
Complexity of ten decision problems in continuous time dynamical systems
"... Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomi ..."
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Cited by 6 (3 self)
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Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomial time) algorithm unless P=NP: local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, invariance of a ball, invariance of a quartic semialgebraic set under linear dynamics, local collision avoidance, and existence of a stabilizing control law. We also extend our earlier NPhardness proof of testing local asymptotic stability for polynomial vector fields to the case of trigonometric differential equations of degree four. I.
On the difficulty of deciding asymptotic stability of cubic homogeneous vector fields
 In Proceedings of the 2012 American Control Conference
, 2012
"... Abstract — It is wellknown that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next inte ..."
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Cited by 3 (3 self)
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Abstract — It is wellknown that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NPhard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NPhardness result, we obtain a Lyapunovinspired technique for proving positivity of forms. We also show that for asymptotically stable homogeneous cubic vector fields in as few as two variables, the minimum degree of a polynomial Lyapunov function can be arbitrarily large. Finally, we show that there is no monotonicity in the degree of polynomial Lyapunov functions that prove AS; i.e., a homogeneous cubic vector field with no homogeneous polynomial Lyapunov function of some degree d can very well have a homogeneous polynomial Lyapunov function of degree less than d. A. Background I.
Stability of polynomial differential equations: Complexity and converse lyapunov questions
 CoRR
"... Stability analysis of polynomial differential equations is a central topic in nonlinear dynamics and control which in recent years has undergone major algorithmic developments due to advances in optimization theory. Notably, the last decade has seen a widespread interest in the use of sum of squares ..."
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Cited by 2 (0 self)
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Stability analysis of polynomial differential equations is a central topic in nonlinear dynamics and control which in recent years has undergone major algorithmic developments due to advances in optimization theory. Notably, the last decade has seen a widespread interest in the use of sum of squares (sos) based semidefinite programs that can automatically find polynomial Lyapunov functions and produce explicit certificates of stability. However, despite their popularity, the converse question of whether such algebraic, efficiently constructable certificates of stability always exist has remained elusive. Naturally, an algorithmic question of this nature is closely intertwined with the fundamental computational complexity of proving stability. In this paper, we make a number of contributions to the questions of (i) complexity of deciding stability, (ii) existence of polynomial Lyapunov functions, and (iii) existence of sos Lyapunov functions. (i) We show that deciding local or global asymptotic stability of cubic vector fields is strongly NPhard. Simple variations of our proof are shown to imply strong NPhardness of several other decision problems: testing local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, invariance of the unit ball, boundedness of trajectories, conver
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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Cited by 1 (1 self)
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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.
Some Applications of Polynomial Optimization in Operations Research and RealTime Decision Making
"... We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed ..."
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We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing realtime certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (iii) designing a nonlinear hovering controller for a quadrotor UV, which has recently been used for load transportation. On our smallerscale applications, we apply the sum of squares (SOS) relaxation and solve the underlying problems with semidefinite programming. On the largerscale or realtime applications, we use our recently introduced “SDSOS Optimization ” techniques which result in second order cone programs. To the best of our knowledge, this is the first study of realtime applications of sum of squares techniques in optimization and control. No knowledge in dynamics and control is assumed from the reader. 1
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
, 2014
"... We develop a general framework for obtaining upper bounds on the “practical ” computational complexity of stability problems, for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems in firstorder theories over the real numbers, an ..."
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We develop a general framework for obtaining upper bounds on the “practical ” computational complexity of stability problems, for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems in firstorder theories over the real numbers, and reduce stability problems to the δdecision problems of their descriptions. The framework allows us to give a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the δstability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we measure their degrees of unsolvability.