ECE311-DynamicSystemsandControlLinearizationof.pdfVIP

ECE311 - Dynamic Systems and Control Linearization of Nonlinear Systems Objective This handout explains the procedure to linearize a nonlinear system around an equilibrium point. An example illustrates the technique. 1 State-Variable Form and Equilibrium Points A system is said to be in state-variable form if its mathematircal model is described by a system of n ?rst-order di?erential equations and an algebraic output equation: x˙ 1 = f1(x1, . . . , xn, u) x˙ 2 = f2(x1, . . . , xn, u) ··· x˙ n = fn(x1, . . . , xn, u) y = h(x1, . . . , xn, u). (1) The column vector x = [x1, . . . , xn]? is called the state of the system. The scalars u and y are called the control input and the system output, respectively. Denoting ?? f1(x1, . . . , xn, u) f (x, u) = ? ? f2(x1, ? ? ? . .. ... , xn, u)?? ? ? ? , ?? fn(x1, . . . , xn, u) we concisely rewrite (1) as x˙ = f (x, u) y = h(x, u). (2) When f and h are nonlinear functions of x and u, then we say that the system is nonlinear. In this course we will work exclusively with linear systems, i.e., systems for which (2) becomes x˙ = Ax + Bu y = Cx + Du, (3) Last revised: January 20, 2007 1 ECE311-Dynamic Systems and Control 1 State-Variable Form and Equilibrium Points where A is n×n, B is n×1, C is 1×n, and D is a scalar. Sometimes, physical systems are described by nonlinear models such as (2), and the tools we will learn in this course can not be employed to design controllers. However, if a nonlinear system operates around an equilibrium point, i.e., around a con?guration where the system is at rest, then it is possible to study the behavior of the system in a neighborhood of such point. Example 1 (A simple pendulum). Consider the dynamics of the pendulum depicted below, where u denotes an input torque provided by a DC motor. u θ l Mg The equation of motion for this system is I d2θ dt2 + M gl sin θ = u y = θ, (4) where I is the moment of inertia of the pendulum around the pivot point, and y is the output of the system, i.e., the va