Robust Control Design For Virotherapy Model Using Successive Method

Bilinear systems is the simplest class of nonlinear systems that represent many real physical processes. As an example, chemotherapy and virotherapy in cancer cells are bilinear systems. Disturbances is a factor that can interfere the work process of the system, and it can make the output of the system to be not in accordance with the desired output. Thus a robust controller must be found to make the system produce the desired output. In designing the controller, it requires a solution to the state dependent algebraic Riccati equation (SDARE). However it is difficult to solve the SDARE. Successive method is one of the methods that can be used to solve this issue. The idea of this method is converting the bilinear systems into time-varying linear system. This method has the following steps : first, we need to obtain the robust control for the linear system by ignoring the multiplicative term of bilinear system. Second, convert the bilinear systems into the time-varying linear systems using the previous result, and then solve the SDARE by the new performance index and the associated Hamilton-Jacobi-Isaacs equation. Last, iterate the steps until the convergence of state satisfied. In this research, successive method were applied in virotherapy control problems. The virotherapy model has been widely developed, one of which is the cell cycle-specific model. This model is a bilinear systems. There are four groups in this model: quiescent cells (Q), cancer cells (S), virus (V), and infected cells (I). Virus are injected into the human body as the control input to control the amount of the cancer cells. In this case, virus can only infect the cancer cells, and the infected cells will die when the lysis process occurs. Virus, as a control, is given with the aim of minimizing the energy used in the system. In this model we consider the body's immune response as an additive disturbances to the model. Simulation show that virotherapy can reduce the number of cancer cells in the body on day 50, so the number of cancer cells in the body is only 16.6%. Based on the simulation, the next virototherapy can be done after the day 50.