Convex Relaxation of Bilinear Matrix Inequalities Part II: Applications to Optimal Control Synthesis

Kheirandishfard, Mohsen; Zohrizadch, Fariba; Adil, Muhammad; Madani, Ramtin

doi:10.1109/cdc.2018.8619762

Cited by 13 publications

(12 citation statements)

References 54 publications

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“…Applying Theorem 2 leads to solve the BMI feasibility problem of its statement (i). To achieve this, an adapted version of the algorithm proposed in [28] is used (Algorithm 1). Starting from an arbitrary initial pointQ inv init , chosen here as the initial guess, it proceeds by solving a sequence of LMI optimization problems.…”

Section: B Iterative Algorithm Based On Lmi Relaxationsmentioning

confidence: 99%

“…Again, this leads to solve an LMI problem. Finally,Q inv k−1 is updated in preparation of the next iteration (line 8 of Algorithm 1), based on the Nesterov's acceleration parameter η k := k−1 k+2 (cf [28] and the reference therein for further details). Notice that the algorithm is stopped if a feasible solution of the original BMI problem is found or if the number of iterations exceeds max iter.…”

Section: B Iterative Algorithm Based On Lmi Relaxationsmentioning

confidence: 99%

See 1 more Smart Citation

Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach

Perodou

Korniienko

Scorletti

et al. 2021

IEEE Trans. Circuits Syst. I

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This paper deals with the frequency design of lossless passive electronic filters under magnitude constraints. With the huge increase in design complexity for mobile applications, new systematic and efficient methods are required. This paper focuses on the direct synthesis approach, an historical design approach that has not been recently updated. It consists in directly synthesizing the LC values of a pre-specified circuit until the spectral mask is satisfied. While beneficial in practice, this approach typically leads to an important computational time and requires an initial guess to reduce it. Based on recent developments of the System and Control community, that led to efficient methods for system design, the direct synthesis approach is revisited. To achieve this, the port-Hamiltonian Differential Algebraic Equation (pHDAE) representation, that particularly fits the design problem, is introduced. A synthesis method is then developed, leading to solve an optimization problem of moderate complexity. For particular cases, this complexity happens to be remarkably low. Based on this observation, a second method reveals how to obtain such complexity for the more general case, using an original combination between the pHDAE and the LFT representations. Finally, a numerical example shows the validity and illustrates the benefits of this work.

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Section: B Iterative Algorithm Based On Lmi Relaxationsmentioning

confidence: 99%

Section: B Iterative Algorithm Based On Lmi Relaxationsmentioning

confidence: 99%

Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach

Perodou

Korniienko

Scorletti

et al. 2021

IEEE Trans. Circuits Syst. I

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show abstract

“…The bisection operation in Algorithm 2 induces -in the worst casean exponential blow-up in the number of branches. In practice, one can prune branches inducing only negative objective values, via, e.g., convex relaxation [26].…”

Section: Finding the Initial Solutionmentioning

confidence: 99%

Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming

Wang

Chen

Bai

et al. 2021

Lecture Notes in Computer Science

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A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over the infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is by far the least conservative one on barrier certificates, and can be shown as the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition—thereby synthesizing invariant barrier certificates—can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, in the sense that a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmark examples demonstrate the effectiveness and efficiency of our approach.

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“…Several works extend sampling-based methods from single to multi-robot, but suffer from extremely slow convergence despite theoretical guarantee of asymptotical optimality [27]- [31]. Our method tackles the computational complexity of planning in the joint-space by leveraging recently developed Parabolic relaxation [34]- [36]. Since it does not rely on discretizing or decoupling the state-space, trajectories generated are both optimal and dynamically feasible.…”

Section: A Related Workmentioning

confidence: 99%

Optimal Multi-Robot Motion Planning via Parabolic Relaxation

Choi¹,

Adil²,

Rahmani³

et al. 2021

Preprint

Self Cite

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Multi-robot systems offer enhanced capability over their monolithic counterparts, but they come at a cost of increased complexity in coordination. To reduce complexity and to make the problem tractable, multi-robot motion planning (MRMP) methods in the literature adopt de-coupled approaches that sacrifice either optimality or dynamic feasibility. In this paper, we present a convexification method, namely "parabolic relaxation", to generate optimal and dynamically feasible trajectories for MRMP in the coupled joint-space of all robots. We leverage upon the proposed relaxation to tackle the problem complexity and to attain computational tractability for planning over one hundred robots in extremely clustered environments. We take a multistage optimization approach that consists of i) mathematically formulating MRMP as a non-convex optimization, ii) lifting the problem into a higher dimensional space, iii) convexifying the problem through the proposed computationally efficient parabolic relaxation, and iv) penalizing with iterative search to ensure feasibility and recovery of feasible and near-optimal solutions to the original problem. Our numerical experiments demonstrate that the proposed approach is capable of generating optimal and dynamically feasible trajectories for challenging motion planning problems with higher success rate than the state-of-the-art, yet remain computationally tractable for over one hundred robots in a highly dense environment.

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Convex Relaxation of Bilinear Matrix Inequalities Part II: Applications to Optimal Control Synthesis

Cited by 13 publications

References 54 publications

Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach

Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach

Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming

Optimal Multi-Robot Motion Planning via Parabolic Relaxation

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