A Solution Of Lyapunov Equations On The GPU Platform

Lyapunov equation plays an important role in various branches of themodern control theory. How to solve the Lyapunov equation has attracted greatattention, especia lly to the large size of Lyapunov equation. Previous worksma inly focus on increasing the number of multi-core CPU cores or based on CPU,which has some limitations for hand ling the large-scale data. The design conceptof GPU is different with CPU, which s how the advantage on the paralle lcomp utatio n compared with the CPU, and it is wide ly used in many fie lds.Meanwhile, the Newton iterative algorithm can be used for paralle l computing.So it is feasib le to adopt the Newton iterative algorithm to solve the Lyapunovequation on the GPU and it is of great significant.In this paper, they that the different orders of the matrix’s multip ly and thematrix’s inversio n are realized based on the GPU paralle l co mputing. The resultshows the advantage of the para llel co mputing in comparison to the paralle lcomp uting, when increasing the order of matrix, this finding confirms the strongability of GPU’s paralle l computing in deal with the large order of matrix. The n,the algorithms for the paralle l computing of the Lyapunov equation and theGPU-based paralle l computing of the Newton iterative algorithm that is realizedbased on CPU and GPU. It is shown that dealing with the4096order matrixoperations, the computation times of those two algorithms on the GPU withrespect the CPU speed-up ratio are34and57, the results evidence that theGPU-based parallel computing takes less time and has the high efficiency.Second, the algorithm is used to solve another important equation in controltheory called the Ricatti equation. In the same manner, it is designed andimp leme nted for the Ricatti equation based on CPU platform and GPU platform.In particular, it shows that when the order of matrix is4096, the ratio ofcomp utatio n time between the GPU and the CPU is280. This shows that it isfeasib le and effic ient to s love Ricatti equation by Newton iterative algorithm onthe GPU.Fina lly, a new method is realized based on GPU p latform that combines theprecis ion linear search and Newton iterative. It is shown that the new method ishigh-effic iency, when the order of matrix is4096, the ratio of computation time between the GPU and the CPU is300. Moreover, the new Newton iterative isbetter than the old one in dealing with the large size matrix. All these resultsshow the great advantage of the GPU parallel computing.