Introduction to the Modeling and Analysis of Complex Systems

100 CHAPTER 6. CONTINUOUS-TIME MODELS I: MODELING6.2 Classifications of Model EquationsDistinctions between linear and nonlinear systems as well as autonomous and non-autonomoussystems, which we discussed in Section 4.2, still apply to continuous-time models. Butthe distinction between first-order and higher-order systems are slightly different, as follows.First-order system A differential equation that involves first-order derivatives of statevariables ( dxdt ) only.Higher-order system A differential equation that involves higher-order derivatives ofstate variables ( d2 xdt , d3 x2 dt , etc.).3Luckily, the following is still the case for continuous-time models as well:Non-autonomous, higher-order differential equations can always be converted intoautonomous, first-order forms by introducing additional state variables.Here is an example:d 2 θdt = − g sin θ (6.3)2 LThis equation describes the swinging motion of a simple pendulum, which you might haveseen in an intro to physics course. θ is the angular position of the pendulum, g is thegravitational acceleration, and L is the length of the string that ties the weight to thepivot. This equation is obviously nonlinear and second-order. While we can’t removethe nonlinearity from the model, we can convert the equation to a first-order form, byintroducing the following additional variable:ω = dθdt(6.4)Using this, the left hand side of Eq. (6.3) can be written as dω/dt, and therefore, theequation can be turned into the following first-order form:dθdt = ω (6.5)dωdt = − g sin θL(6.6)