Many analysis and synthesis tools for control systems are based on positive polynomial optimizations. In this talk, we will introduce positive polynomials and explain how analysis and synthesis problems in nonlinear control systems can be formulated in terms of positivity conditions of polynomial functions. We will show how positive polynomials can be used to compute extremal values of system trajectories starting from a given set of initial conditions, and to design polynomial-type controllers by synthesizing Lyapunov functions while minimizing some cost function (e.g. maximum amplitude of system trajectories). Time-domain constraints can be taken into account in the controller synthesis problem. Computationally, the positive polynomial conditions can be transformed into a set of linear matrix inequalities (LMI) which are readily solvable, or bilinear matrix inequalities (BMI) which can be solved by iterative methods. The talk will focus on nonlinearities of polynomial type, but extension to non-polynomial nonlinearities will be discussed.
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