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89
Safety Verification of Hybrid Systems Using Barrier Certificates
 In Hybrid Systems: Computation and Control
, 2004
"... This paper presents a novel methodology for safety verification of hybrid systems. For proving that all trajectories of a hybrid system do not enter an unsafe region, the proposed method uses a function of state termed a barrier certificate. The zero level set of a barrier certificate separates ..."
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Cited by 89 (6 self)
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This paper presents a novel methodology for safety verification of hybrid systems. For proving that all trajectories of a hybrid system do not enter an unsafe region, the proposed method uses a function of state termed a barrier certificate. The zero level set of a barrier certificate separates the unsafe region from all possible trajectories starting from a given set of initial conditions, hence providing an exact proof of system safety. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes nonlinearity, uncertainty, and constraints can be handled directly within this framework.
Feedback Control of Quantum State Reduction
, 2004
"... Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for th ..."
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Cited by 73 (6 self)
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Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
LQRTrees: Feedback motion planning via sums of squares verification
 International Journal of Robotics Research
, 2010
"... Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree ..."
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Cited by 68 (21 self)
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Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQRstabilized trajectories. The region of attraction of this nonlinear feedback policy “probabilistically covers ” the entire controllable subset of the state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic nonlinear feedback design algorithm on simple nonlinear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm. 1
Nonlinear Control Synthesis by Convex Optimization
"... A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space an ..."
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Cited by 35 (4 self)
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A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories towards the equilibrium. The new
LMI techniques for optimization over polynomials in control: a survey
 IEEE Transactions on Automatic Control
"... Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the ..."
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Cited by 34 (17 self)
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Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya’s theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, timedelay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.
Semidefinite Programming Relaxations and Algebraic Optimization in Control
 EUROPEAN JOURNAL OF CONTROL (2003)9:307321
, 2003
"... We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting deve ..."
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Cited by 33 (5 self)
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We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting developments which have occured in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sumofsquares. These developments are illustrated with examples of applications to control systems.
Approximate Bisimulations for Nonlinear Dynamical Systems
, 2005
"... The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic n ..."
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Cited by 31 (6 self)
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The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunovlike inequalities, which for deterministic systems can be computed using recent sumofsquares techniques. Our framework is illustrated on a safety verification example.
Some controls applications of sum of squares programming
 In Proceedings of the Conference on Decision and Control
, 2003
"... Summary. We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachable set. The second is the synthesis of a polynomi ..."
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Cited by 30 (7 self)
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Summary. We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachable set. The second is the synthesis of a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the state feedback synthesis for handling systems with input saturation. Both classes of problems are demonstrated using twostate nonlinear systems. 1
Methodological Frameworks for Largescale Network Analysis and Design
 ACM SIGCOMM Computer Communications Review
, 2004
"... This paper emphasizes the need for methodological frameworks for analysis and design of large scale networks which are independent of specific design innovations and their advocacy, with the aim of making networking a more systematic engineering discipline. Networking problems have largely confounde ..."
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Cited by 24 (6 self)
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This paper emphasizes the need for methodological frameworks for analysis and design of large scale networks which are independent of specific design innovations and their advocacy, with the aim of making networking a more systematic engineering discipline. Networking problems have largely confounded existing theory, and innovation based on intuition has dominated design. This paper will illustrate potential pitfalls of this practice. The general aim is to illustrate universal aspects of theoretical and methodological research that can be applied to network design and verification. The issues focused on will include the choice of models, including the relationship between flow and packet level descriptions, the need to account for uncertainty generated by modelling abstractions, and the challenges of dealing with network scale. The rigorous comparison of proposed schemes will be illustrated using various abstractions. While standard tools from robust control theory have been applied in this area, we will also illustrate how networkspecific challenges can drive the development of new mathematics that expand their range of applicability, and how many enormous challenges remain.
Advanced methods and algorithms for biological networks analysis
 Proceedings of the IEEE
, 2006
"... Modeling and analysis of complex biological networks presents a number of mathematical challenges. For the models to be useful from a biological standpoint, they must be systematically compared with data. Robustness is a key to biological understanding and proper feedback to guide experiments, incl ..."
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Cited by 22 (5 self)
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Modeling and analysis of complex biological networks presents a number of mathematical challenges. For the models to be useful from a biological standpoint, they must be systematically compared with data. Robustness is a key to biological understanding and proper feedback to guide experiments, including both the deterministic stability and performance properties of models in the presence of parametric uncertainties and their stochastic behavior in the presence of noise. In this paper, we present mathematical and algorithmic tools to address such questions for models that may be nonlinear, hybrid, and stochastic. These tools are rooted in solid mathematical theories, primarily from robust control and dynamical systems, but with important recent developments. They also have the potential for great practical relevance, which we explore through a series of biologically motivated examples. Keywords—Biological networks, model invalidation, robust stability, sum of squares based software tools (SOSTOOLS), stochastic analysis. I.