The Research Of L₀ Norm In Optimal Control Problem

With the arrival of the era of big data,the applications of l₀ norm is becoming more important.In the paper,we mainly consider the application of a class of l₀ norm optimization methods to optimal control problems.For the problem of linear quadratic optimal control for large interconnected systems,we use the l₀ norm minimization problem to indicate the sparsity of the static feedback matrix.And then,an algorithmic framework is constructed to design controllers that provide an ideal trade-off between system performance and the sparsity of the static feedback matrix.Since the l₀ norm is a non-convex problem and discontinuous,which makes solving the l₀ norm problem an NP problem(that is,a classical combinatorial optimization problem).In general,in order to obtain sparse solutions,l1 norm is often used to approximate l₀ norm.Although the l1 norm is convex,and it is nondifferentiable,which makes it not applicable in many practical problems.In order to reduce the computational cost caused by the non-convex and discontinuous properties of l₀ norm.In this paper,the Moreau envelope function with continuity and differentiability is used to smoothly approximate the l₀ norm.Moreover,the continuity and differentiability of Moreau envelope and proximal mapping are also analyzed in the paper.And then,the multiplier direction alternation method(ADIMM)is constructed to solve the approximately optimized l₀ norm problem.In numerical experiments,with competitive sparsity and communication cost,we find that our algorithm is much more efficient than the algorithm proposed by leong et al.,which is based on the penalty method.And in some cases,e.g.,?=0.0001,our algorithm is not only efficient but also has lower communication cost than the algorithm based on the penalty method.