CONVERTING TO THE STATE SPACE 2 CONVERTING TO THE STATE SPACE Differential equations can be solved analytically when the equations are linear and when the nonlinearities are relatively simple. Otherwise, the equations need to be solved numerically. To solve the equations numerically, the differential equations are converted to a standard format. The standard format is a set of 1st-order nonlinear differential equations, called stateequations. The equations are converting to a standard format to produce a corresponding numerical procedure that is standardized, as well. 1. Nonlinear State Equations The single degree-of-freedom of the pendulum is associated with the pendulum’s configuration. For this reason, the linearization described in the previous chapter is sometimes referred to as being carried out in the configuration space. In contrast, we’ll momentarily carry out the linearization in the state space. The pendulum’s state consists of the angle θ(t) and its angular rate Once θ(t) and are prescribed as initial conditions, the future “state” of the system can be predicted – which is ).(tθ&)(tθ&MAE 461: DYNAMICS AND CONTROLS
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CONVERTING TO THE STATE SPACE why the term state is used. The pendulum’s state variables are (2 – 1) )()()()(21ttxttxθθ&==From Eq. (1 – 3), the two state equations that describe the motion of the pendulum are expressed in terms of its state variables as (2 – 2) )sin(1221xLgxxx−==&&Equations (2 – 2) are two 1st-order differential equations. The first of the two equations defines x2(t) as the time derivative of x1(t) and the second of the two equations is the equation of motion coming from Eq. (1 – 3). So, one 2nd-order differential equation has been converted into two 1st-order differential equations. The benefit of the state format is a matter of standardization. Just about any differential equation or system of differential equations, not just 2nd-order differential equations, can be converted into a system of 1st-order equations. In fact, the state format is not only used for numerical integration, but in all kinds of methods of analysis and design. Control methods, filtering techniques, estimation procedures, to name a few, have been standardized for state equations. These methods are called state variable methods. Let’s now retrace our steps and re-develop the material previously covered in Chapter 1 using this new state-variable format. MAE 461: DYNAMICS AND CONTROLS