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33
Design of Optimal Sparse Interconnection Graphs for Synchronization of Oscillator Networks
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2014
"... We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization probl ..."
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Cited by 10 (3 self)
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We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization problems that address the tradeoff between synchronization performance and the number and strength of oscillator couplings. We promote the sparsity of the coupling network by penalizing the number of interconnection links. For identical oscillators, we establish convexity of the optimization problem and demonstrate that the design problem can be formulated as a semidefinite program. Finally, for special classes of oscillator networks we derive explicit analytical expressions for the optimal conductance values.
A Splitting Method for Optimal Control
, 2012
"... We apply an operator splitting technique to a generic linearconvex optimal control problem, which results in an algorithm that alternates between solving a quadratic control problem, for which there are efficient methods, and solving a set of singleperiod optimization problems, which can be done in ..."
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Cited by 5 (0 self)
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We apply an operator splitting technique to a generic linearconvex optimal control problem, which results in an algorithm that alternates between solving a quadratic control problem, for which there are efficient methods, and solving a set of singleperiod optimization problems, which can be done in parallel, and often have analytical solutions. In many cases the resulting algorithm is divisionfree (after some offline precomputations) and so can be implemented in fixedpoint arithmetic, for example on a fieldprogrammable gate array (FPGA). We demonstrate the method on several
A Parametric NonConvex Decomposition Algorithm for RealTime and Distributed NMPC
, 2014
"... A novel decomposition scheme to solve parametric nonconvex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of proximal linearised alternating minimisations and a dual update per time step. Hence, the proposed approach is attractive in ..."
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Cited by 5 (1 self)
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A novel decomposition scheme to solve parametric nonconvex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of proximal linearised alternating minimisations and a dual update per time step. Hence, the proposed approach is attractive in a realtime distributed context. Assuming that the Nonlinear Program (NLP) is semialgebraic and that its critical points are strongly regular, contraction of the sequence of primaldual iterates is proven, implying stability of the suboptimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimalitytracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralised NMPC problem to control a DC motor and a distributed NMPC program for collaborative tracking of unicycles, both within a realtime framework. Furthermore, an analysis of the suboptimality error as a function of the sampling period is proposed given a fixed computational power.
Convex relaxation for optimal distributed control problem
 ONLINE]. AVAILABLE: HTTP://WWW.EE.COLUMBIA.EDU/LAVAEI/DEC CONTROL 2014 PARTII.PDF
, 2014
"... This paper is concerned with the optimal distributed control (ODC) problem. The objective is to design a fixedorder distributed controller with a prespecified structure for a discretetime system. It is shown that this NPhard problem has a quadratic formulation, which can be relaxed to a semide ..."
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Cited by 4 (4 self)
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This paper is concerned with the optimal distributed control (ODC) problem. The objective is to design a fixedorder distributed controller with a prespecified structure for a discretetime system. It is shown that this NPhard problem has a quadratic formulation, which can be relaxed to a semidefinite program (SDP). If the SDP relaxation has a rank1 solution, a globally optimal distributed controller can be recovered from this solution. By utilizing the notion of treewidth, it is proved that the nonlinearity of the ODC problem appears in such a sparse way that its SDP relaxation has a matrix solution with rank at most 3. A nearoptimal controller together with a bound on its optimality degree may be obtained by approximating the lowrank SDP solution with a rank1 matrix. This convexification technique can be applied to both timedomain and Lyapunovdomain formulations of the ODC problem. The efficacy of this method is demonstrated in numerical examples.
Sparsity measures for spatially decaying systems
 iEEE Trans. Autom. Control
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Distributed MPC via dual decomposition and alternative direction method of multipliers,” arXiv:1207.3178
, 2012
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Efficient Convex Relaxation for Stochastic Optimal Distributed Control Problem
, 2014
"... This paper is concerned with the design of an efficient convex relaxation for the notorious problem of stochastic optimal distributed control (SODC) problem. The objective is to find an optimal structured controller for a dynamical system subject to input disturbance and measurement noise. With no ..."
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Cited by 3 (3 self)
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This paper is concerned with the design of an efficient convex relaxation for the notorious problem of stochastic optimal distributed control (SODC) problem. The objective is to find an optimal structured controller for a dynamical system subject to input disturbance and measurement noise. With no loss of generality, this paper focuses on the design of a static controller for a discretetime system. First, it is shown that there is a semidefinite programming (SDP) relaxation for this problem with the property that its SDP matrix solution is guaranteed to have rank at most 3. This result is due to the extreme sparsity of the SODC problem. Since this SDP relaxation is computationally expensive, an efficient twostage algorithm is proposed. A computationallycheap SDP relaxation is solved in the first stage. The solution is then fed into a second SDP problem to recover a nearglobal controller with an enforced sparsity pattern. The proposed technique is always exact for the classical H2 optimal control problem (i.e., in the centralized case). The efficacy of our technique is demonstrated on the IEEE 39bus New England power network, a massspring system, and highlyunstable random systems, for which nearoptimal stabilizing controllers with optimality degrees above 90 % are designed under a wide range of noise levels.
A Fast Algorithm for Sparse Controller Design
, 2013
"... We consider the task of designing sparse control laws for largescale systems by directly minimizing an infinite horizon quadratic cost with an `1 penalty on the feedback controller gains. Our focus is on an improved algorithm that allows us to scale to large systems (i.e. those where sparsity is m ..."
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Cited by 2 (0 self)
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We consider the task of designing sparse control laws for largescale systems by directly minimizing an infinite horizon quadratic cost with an `1 penalty on the feedback controller gains. Our focus is on an improved algorithm that allows us to scale to large systems (i.e. those where sparsity is most useful) with convergence times that are several orders of magnitude faster than existing algorithms. In particular, we develop an efficient proximal Newton method which minimizes periteration cost with a coordinate descent active set approach and fast numerical solutions to the Lyapunov equations. Experimentally we demonstrate the appeal of this approach on synthetic examples and real power networks significantly larger than those previously considered in the literature.