1.2. Proportional-Derivative Control where m is the mass, g the acceleration due to gravity, and u(t) the value of the external torque at time t(counterclockwise being positive). We call u( the input or control function. To avoid having to keep track of constants, let us assume that units of time and distance have been chosen so that m=g=l The vertical stationary position(0=T, 0=0) is an equilibrium when no control is being applied(u E 0), but a small deviation from this will result in n unstable motion. Let us assume that our objective is to apply torques as needed to correct for such deviations. For small 8-T, (6-丌)+o(6-丌) Here we use the standard"little-o"notation: o(z)stands for some function g(a) for which g(a) Since only small deviations are of interest, we drop the nonlinear part repre- sented by the term o(0-T). Thus, with o: =0-T as a new variable, we replace equation(1. 1) by the linear differential equation ()-g(t)=u(t) as our object of study. (See Figure 1.2.) Later we will analyze the effect of the ignored nonlinearity u 1.2: Inverted pendulum Our objective then is to bring y and y to zero, for any small nonzero initial values p(0), p(O) in equation(1.2), and preferably to do so as fast as possible, with few oscillations, and without ever letting the angle and velocity become too large. Although this is a highly simplified system, this kind of "servo oblem illustrates what is done in engineering practice. One typically wants to achieve a desired value for certain variables, such as the correct idling spe in an automobile's electronic ignition system or the position of the read write head in a disk drive controller a naive first attempt at solving this control problem would be as follows: If re are to the left of the vertical, that is, if p=0-t>0, then we wish to move to the right, and therefore, we apply a negative torque. If instead we are to1.2. Proportional-Derivative Control 3 where m is the mass, g the acceleration due to gravity, and u(t) the value of the external torque at time t (counterclockwise being positive). We call u(·) the input or control function. To avoid having to keep track of constants, let us assume that units of time and distance have been chosen so that m = g = 1. The vertical stationary position (θ = π, ˙ θ = 0) is an equilibrium when no control is being applied (u ≡ 0), but a small deviation from this will result in an unstable motion. Let us assume that our objective is to apply torques as needed to correct for such deviations. For small θ − π, sin θ = −(θ − π) + o(θ − π). Here we use the standard “little-o” notation: o(x) stands for some function g(x) for which limx→0 g(x) x = 0 . Since only small deviations are of interest, we drop the nonlinear part repre￾sented by the term o(θ−π). Thus, with ϕ := θ−π as a new variable, we replace equation (1.1) by the linear differential equation ϕ¨(t) − ϕ(t) = u(t) (1.2) as our object of study. (See Figure 1.2.) Later we will analyze the effect of the ignored nonlinearity. u φ Figure 1.2: Inverted pendulum. Our objective then is to bring ϕ and ˙ϕ to zero, for any small nonzero initial values ϕ(0), ϕ˙(0) in equation (1.2), and preferably to do so as fast as possible, with few oscillations, and without ever letting the angle and velocity become too large. Although this is a highly simplified system, this kind of “servo” problem illustrates what is done in engineering practice. One typically wants to achieve a desired value for certain variables, such as the correct idling speed in an automobile’s electronic ignition system or the position of the read/write head in a disk drive controller. A naive first attempt at solving this control problem would be as follows: If we are to the left of the vertical, that is, if ϕ = θ −π > 0, then we wish to move to the right, and therefore, we apply a negative torque. If instead we are to