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72
On the Construction of Lyapunov Functions using the Sum of Squares Decomposition
, 2002
"... A relaxation of Lyapunov's direct method has been proposed recently that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In this paper, the above technique ..."
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Cited by 89 (17 self)
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A relaxation of Lyapunov's direct method has been proposed recently that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In this paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain nonpolynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain nonpolynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.
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.
A framework for worstcase and stochastic safety verification using barrier certificates
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2007
"... This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do ..."
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Cited by 50 (1 self)
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This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method.
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
Pre and Postprocessing Sumofsquares Programs in Practice
"... Abstract—Checking nonnegativity of polynomials using sumofsquares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often beco ..."
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Cited by 34 (2 self)
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Abstract—Checking nonnegativity of polynomials using sumofsquares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often become increasingly illconditioned. To alleviate these problems, it is important to exploit properties of the analyzed polynomial, and postprocess the obtained solution. This paper describes how the sumofsquares module in the MATLAB toolbox YALMIP handles these issues. Index Terms—Optimization methods, Polynomials, Software packages. I.
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
Safety verification using barrier certificates
 In HSCC, volume 2993 of LNCS
, 2004
"... Abstract — We develop a new method for safety verification of stochastic systems based on functions of states termed barrier certificates. Given a stochastic continuous or hybrid system and sets of initial and unsafe states, our method computes an upper bound on the probability that a trajectory of ..."
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Cited by 26 (8 self)
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Abstract — We develop a new method for safety verification of stochastic systems based on functions of states termed barrier certificates. Given a stochastic continuous or hybrid system and sets of initial and unsafe states, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, both the upper bound and its corresponding barrier certificate can be computed using convex optimization, and hence the method is computationally tractable. I.
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.
Stability region analysis using polynomial and composite polynomial Lyapunov functions and sumofsquares programming
 IEEE Transactions on Automatic Control
, 2008
"... We propose using (bilinear) sumofsquares programming for obtaining inner bounds of regionsofattraction for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions, comprised of pointwise maximums of polynomial functions. Results for sever ..."
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Cited by 22 (3 self)
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We propose using (bilinear) sumofsquares programming for obtaining inner bounds of regionsofattraction for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions, comprised of pointwise maximums of polynomial functions. Results for several examples from the literature are presented using the proposed methods and the PENBMI solver. I.
Stability analysis for sampleddata systems with a timevarying period
 48TH IEEE CONFERENCE ON DECISION AND CONTROL, CDC 2009, SHANGAI: CHINA
, 2009
"... This paper proposes a novel stability analysis of linear systems with sampleddata inputs. Inspired by the inputdelay approach and the stability of impulsive systems, this method provides novel sufficient stability conditions. The stability analysis concerns both constant and timevarying samplin ..."
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Cited by 21 (9 self)
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This paper proposes a novel stability analysis of linear systems with sampleddata inputs. Inspired by the inputdelay approach and the stability of impulsive systems, this method provides novel sufficient stability conditions. The stability analysis concerns both constant and timevarying sampling periods. The delaydependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency and the limitation of such stability criteria.