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 state equations. 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 CONTROLSCONVERTING 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 CONTROLSCONVERTING TO THE STATE SPACE 2. Equilibrium First rewrite the state equations (2 – 1) in the general functional form (2 – 2) ),,()( ),,()( 21222111txxftxtxxftx==&& In the case of the pendulum, the right sides of Eq. (2 – 2) are (2 – 3) 12122211sin),,(),,( xLgtxxfxtxxf −== In terms of the state variables, the equilibrium state is found by substituting 0 and 021== xx&& into Eq. (2 – 2) to get (2 – 4) ),,(0 ),,(0 )(2)(12)(2)(11txxftxxfrrrr== where denote the r-th equilibrium state (r = 1, 2). The pendulum’s two equilibrium states are (r)2)(1 and xxr (2 – 5) 0:200 :1)1(2)2(1)1(2)1(1====xxxxπ 3. Linearization Next, let’s linearize the state equations (2 – 2) about each of the equilibrium states. Expanding the functions in Taylor series about each of 21 and ffMAE 461: DYNAMICS AND CONTROLSCONVERTING TO THE STATE SPACE the equilibrium states (r = 1, 2), and retaining only linear terms, yields (2 – 6) ).()()()(),,(),()()()(),,()(22022)(11012212)(22021)(11011211rrrrxxxfxxxftxxfxxxfxxxftxxf−∂∂+−∂∂=−∂∂+−∂∂= Let’s also redefine the state variables to be relative to each of the equilibrium states (r = 1, 2). We define the new state variables (2 – 7) )(222)(111)()()()(rrxtxtyxtxty −=−= Next, substitute the Taylor series approximations into Eq. (2 – 2), evaluate the partial derivatives at the first equilibrium state, and obtain the linearized state equations (2 – 8) ).()( ),()(1221tyLgtytyty −==&& Equations (2 – 8) describe the pendulum’s state in the neighborhood of the first equilibrium state. Next, evaluate the partial derivatives at the second equilibrium state and obtain the linearized state equations (2 – 9) ).()( ),()(1221tyLgtytyty +==&& Equations (2 - 9) describe the pendulum’s state in the neighborhood of the second equilibrium state. MAE 461: DYNAMICS AND CONTROLSCONVERTING TO THE STATE SPACE The n-th Order Differential Equation Let’s see how an nth-order differential equation is converted to state equations. The nth-order differential equation is (2 – 10) ),,...,,(1122tdtxddtxddtdxxfdtxdnnnn−−= in which f is any function of x, its time derivatives up to the n-1-th time derivative, and time. In order to predict x(t), initially, the position x(0) and its time derivatives 1122)0(,...)0(,)0(−−nndtxddtxddtdx are needed. To convert Eq. (2 – 10) and the initial conditions into state equations and an initial state, introduce the state variables (2 – 11) 1122321,−−====nnndtxdxdtxdxdtdxxxx L The initial state variables are (2 – 12) 1122321)0()0(,)0()0()0()0()0()0(−−====nnndtxdxdtxdxdtdxxxx L Differentiate Eq. (2 – 11) with respect to time, considering Eq. (2 – 10), to get the state equations (2 – 13) ),,,(2113221txxxfxxxxxxxnnnnL&&M&&====−MAE 461: DYNAMICS AND CONTROLSCONVERTING TO THE STATE SPACE 4. The Euler Method (1st–Order Numerical Integration) The 1st–order numerical integrator is developed below, first, for one state equation and then for n state equations. The implementation of the integrator in a computer program is discussed in section 6. One State Equation The single state equation is (2 – 14) ),()(txftx=& Let’s predict x(t + ΔT) from x(t) in which ΔT is a time step. To make the prediction, a Taylor series expansion of x(t) is performed. (2 – 15) L&&&+Δ+Δ+=Δ+!2)()()()()(2TtxTtxtxTtx The first term on the right side is called a 0th–order term, the next term is a 1st –order term, the next is a 2nd-order term, etc. As a simple approximation, we retain the first two terms on the right side and neglect the others. This is called a 1st-order approximation. By neglecting the 2nd- and higher-order terms, we get (2 – 16) xtxTtxTfxΔ+=Δ+Δ⋅=Δ)()( Equation (2 – 16) is the 1st – order numerical integrator for Eq. (2 – 14). It’s called the Euler algorithm. Equation (2 – 16) is