Backward Reachability Analysis of Perturbed Continuous-Time Linear Systems Using Set Propagation

Backward reachability analysis computes the set of states that reach a target set under the competing influence of control input and disturbances. Depending on their interplay, the backward reachable set either represents all states that can be steered into the target set or all states that cannot avoid entering it -- the corresponding solutions can be used for controller synthesis and safety verification, respectively. A popular technique for backward reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales exponentially with the state dimension due to gridding the state space. In this work, we instead use set propagation techniques to design backward reachability algorithms for linear time-invariant systems. Crucially, the proposed algorithms scale only polynomially with the state dimension. Our numerical examples demonstrate the tightness of the obtained backward reachable sets and show an overwhelming improvement of our proposed algorithms over state-of-the-art methods regarding scalability, as systems with well over a hundred states can now be analyzed.Comment: 16 page

Kendali Optimal pada Model Ekstraksi Lipid Mikroalga Menggunakan Metode LQR (Linear Quadratic Regulator) dan Metode PMP (Pontryagin Maximum Principle)

Tanaman yang berpotensi cukup baik sebagai bahan baku biodiesel adalah alga atau ganggang. Chlorella Vulgaris merupakan salah satu ganggang yang paling ekonomis untuk menghasilkan biodiesel, sebab ganggang hijau ini kaya karbohidrat, tak memerlukan perawatan khusus, dan mudah tumbuh. Minyak alga yang diperoleh untuk produksi biodiesel diperoleh melalui proses yang cukup panjang, salah satunya adalah proses ekstraksi lipid. Dalam penelitian ini ditunjukkan verifikasi model matematika ekstraksi lipid mikroalga, agar didapatkan nilai yang optimal pada variabel state yang selanjutnya dilakukan kendali optimal. Selanjutnya dilakukan kendali optimal dengan tujuan memaksimalkan konsentrasi lipid dalam aliran pelarut (Cs), dan dalam partikel mikroalga (Cp) dengan meminimalkan volume pelarut (v), yang dimasukkan menggunakan metode LQR dan metode PMP. Dengan menggunakan software Matlab, didapat bahwa untuk metode LQR, sebelum dilakukan kontrol saat t = 1800 (Cs, Cp, Cl) = (0.502, 0.299, 0.1009) dan setelah dilakukan control Cs, Cp, Cl) = (7.2785, 3.1913, 0.1063). Selanjutnya untuk metode PMP sebelum dilakukan kontrol didapat bahwa (Cs, Cp, Cl) = (0, 0.3046, 0.1009) dan setelah dilakukan kontrol menjadi (Cs, Cp, Cl) = (7.4138, 9.0401, 0.1709). Sehingga dari simulasi yang diakukan didapat bahwa metode PMP lebih optimal pada sistem ini dibandingkan dengan metode LQR. ====================================================== Plants that are quite good as raw materials of biodiesel are algae. Chlorella Vulgaris is one of the most economical algae to produce biodiesel, because these green algae are rich in carbohydrates, require no special care, and are easy to grow. Algae oil obtained for biodiesel production is obtained through a fairly long process, one of which is the process of lipid extraction. In this study, we discussed the model of checking on mathematical model of lipid extraction of microalgae. Furthermore, optimal control is performed with the aim of maximizing lipid concentration in the solvent flow (Cs), and in the microalgae particles (Cp) by minimizing the volume of solvent (v), which is incorporated using the LQR and PMP methods. By using software Matlab, it is found that for the LQR method, before the current control at t = 1800, (Cs, Cp, Cl) = (0.502, 0.299, 0.1009) and after controlling Cs, Cp, Cl) = (7.2785, 3.1913, 0.1063). Furthermore, for the PMP method before the control, it is found that (Cs, Cp, Cl) = (0, 0.3046, 0.1009) and after the control is (Cs, Cp, Cl) = (7, 4138, 9, 0401, 0.1709). So that from the simulation, it was found that the PMP method is more optimal in this system compared to the LQR method

Kendali Optimal Produksi Lipid pada Mikroalga dengan Keterbatasan Nutrisi dan Karbondioksida Menggunakan Metode Linear Quadratik Regulator

Mikroalga memiliki kemampuan yang tinggi untuk memproduksi lipid. Semakin cepat proses pembentukan lipid oleh mikroalga, semakin cepat juga minyak yang dihasilkan. Produksi lipid pada mikroalga dipengaruhi oleh nutrisi, karbon dioksida dan proses fotosintesis. Dalam penelitian ini akan ditunjukkan verifikasi model produksi lipid pada mikroalga, agar didapatkan nilai yang optimal pada variabel state. Dengan kendali optimal produksi lipid pada mikroalga dengan faktor pengendalinya adalah substrat nutrisi dan karbon dioksida. Verifikasi model menunjukkan bahwa model produksi lipid pada mikroalga bersifat safe dengan spesifikasi yang telah ditentukan. Metode Linear Quadratic Regulator menghasilkan konsentrasi mikroalga sebesar X=33,8 mg/CL dan Ql=78,5 mg/CL ================================================================================================= The greater produced of lipids by microalgae, will increasing biomass production. The lipids produced in the microalgae with influenced by nutrients, carbon dioxide and photosynthetic process. In this research, will be proposed using model verification to get a value from the state space, thats influence in the optimal control problem. The optimal control problem of the lipids production in the microalgae will be proposed using the optimal control method. The objective function is to maximize the lipids production while the control is substrate of nutrients and carbon dioxide. The model verification can be present, its was safety system using specification before. Furthermore, the results from the Linear Quadratic Method was the microalgae can produced X=33,8 mg/CL and Ql=78,5 mg/CL