In particular, propose the polynomial systems to be represented as a state-dependent linear form, and a state-dependent Lyapunov function is proposed to be in terms of polynomial vector fields. The recent results in the framework of state feedback control synthesis for polynomial systems which utilize SOS decomposition method can be referred in. We will further describe the details of the SOS decomposition method. This is the main advantage of using the SOS decomposition approach because the solution is indeed tractable. However, if these nonnegativity conditions are replaced by the SOS conditions, then not only testing the Lyapunov function conditions, but also constructing the Lyapunov function can be done effectively using SDP. This is probably due to the lack of algorithmic constructions of Lyapunov functions. However, when the vector field of the system f ( x ) and the Lyapunov function candidate V ( x ) are in polynomial forms, then the Lyapunov conditions are essentially polynomial nonnegativity conditions, which can be NP-hard to test. The difficulty of this construction is solely dependent upon the analytical skills of the researcher. Notice that, for small systems, the construction of the Lyapunov function can be done manually. Generally, the most interesting and important point that was never seen until recently is that the amount of proving the certificates of the Lyapunov function V ( x ) and − V ˙ ( x ) can be reduced to the SOS. The popularity of this method grew quickly among the community of control researchers because the algorithmic analysis of nonlinear systems can be delivered using the most popular Lyapunov method (as discussed earlier). Since the SOS decomposition technique introduced about 10 years ago, the system analysis for polynomial systems can be performed more efficiently because it helps to answer many difficult questions on system analysis that were hard to answer before. The associated Lyapunov function is given by V ( x ) = x T P x. Meanwhile, for discrete-time systems x ( k + 1 ) = f ( x ( k ) ), we need to search for a positive definite Lyapunov function V ( x ) defined in some region of the state space containing the equilibrium point whose difference of the Lyapunov function, Δ V = x T ( k + 1 ) P x ( k + 1 ) − x T ( k ) P x ( k ), is negative semidefinite along the system trajectories. Taking the linear case, for instance, x ˙ = A x, these conditions amount to finding a positive definite matrix P such that A T P + P A is negative definite then the associated Lyapunov function is given by V ( x ) = x T P x. It is clear that to find a stability using the Lyapunov method, we need to find a positive definite Lyapunov function V ( x ) defined in some region of the state space containing the equilibrium point whose derivative V ˙ = d v d x f ( x ) is negative semidefinite along the system trajectories. Let us consider the problem of solving the stability for an equilibrium of a dynamical system x ˙ = f ( x ) using the Lyapunov function method. The Lyapunov stability theory can be generalized as follows. However, it has become of equal importance for control designs over the last decades. The Lyapunov stability is a method that was developed for analysis purposes. This success is owed to its simplicity, generality, and usefulness. Although this method was introduced more than hundred years ago, it remains popular among control researchers. It is well known that Lyapunov's stability theory is one of the most fundamental pillars in control theory. Alireza Nasiri, in Analysis and Synthesis of Polynomial Discrete-Time Systems, 2017 1.3.1.1 On the literature on controller synthesis for polynomial systems: the Lyapunov method and SOS decomposition approach
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