In Chapter 4, we have shown how to design controllers for linear systems. However, in practice, the systems are rarely linear. Nevertheless, if their state vector remains localized in a small zone of the state space, the system may be considered linear and the techniques developed in Chapter 4 can then be used. We will first show how to linearize a nonlinear system around a given point of the state space. We will then discuss how to stabilize these nonlinear systems.
Let f: be a differentiable function. In the neighborhood of a point , the first-order Taylor development of f around gives us:
This matrix is called the Jacobian matrix. Very often, in order to linearize a function, we use formal calculus for calculating the Jacobian matrix, then we instantiate this matrix around . When we differentiate by hand, we avoid ...