the right, we apply a positive, that is to say counterclockwise, torque. In other words, we apply proportional feedback (t) (1.3) where a is positive real number, the feedback gain ze the resulting closed-loop equation obtained when the value of the control given by(1.3)is substituted into the open-loop original equation (1.2) p(t-p(t)+oo(t)=0 If a>1, the solutions of this differential equation are all oscillatory, since the roots of the associated characteristic equation are purely imaginary z =tiVa-1. If instead a< 1, then all of the solutions except for those with 中(0)=-p(0)1-a diverge to too. Finally, if a= 1, then each set of initial values with p(0)=0 is an equilibrium point of the closed-loop system. Therefore, in none of the cases is the system guaranteed to approach the desired configuration We have seen that proportional control does not work. We proved this for the linearized model, and an exercise below will show it directly for the origin nonlinear equation(1. 1). Intuitively, the problem can be understood as follows Take first the case a 1. For any initial condition for which p(O) is small but positive and p(0)=0, there results from equation(1.4)that p(0)>0 Therefore, also and hence yp increase, and the pendulum moves away, rather than toward, the vertical position. When a>l the problem is more subtle The torque is being applied in the correct direction to counteract the natural stability of the pendulum, but this feedback helps build too much inertia In particular, when already close to (0)=0 but moving at a relatively large speed, the controller(1.3)keeps pushing toward the vertical, and overshoot and eventual oscillation result The obvious solution is to keep a> l but to modify the proportional feed- back(1.3)through the addition of a term that acts as a brake, penalizing ve- locities. In other words, one needs to add damping to the system. We arrive then at a PD, or proportional-derivative feedback law, u(t)=-ap(t)-B(t with a> I and B>0. In practice, implementing such a controller involves measurement of both the angular position and the velocity. If only the former is easily available, then one must estimate the velocity as part of the control gorithm; this will lead later to the idea of observers, which are techniques for4 1. Introduction the right, we apply a positive, that is to say counterclockwise, torque. In other words, we apply proportional feedback u(t) = −αϕ(t), (1.3) where α is some positive real number, the feedback gain. Let us analyze the resulting closed-loop equation obtained when the value of the control given by (1.3) is substituted into the open-loop original equation (1.2), that is ϕ¨(t) − ϕ(t) + αϕ(t)=0 . (1.4) If α > 1, the solutions of this differential equation are all oscillatory, since the roots of the associated characteristic equation z2 + α − 1 = 0 (1.5) are purely imaginary, z = ±i √α − 1. If instead α < 1, then all of the solutions except for those with ϕ˙(0) = −ϕ(0)√1 − α diverge to ±∞. Finally, if α = 1, then each set of initial values with ˙ϕ(0) = 0 is an equilibrium point of the closed-loop system. Therefore, in none of the cases is the system guaranteed to approach the desired configuration. We have seen that proportional control does not work. We proved this for the linearized model, and an exercise below will show it directly for the original nonlinear equation (1.1). Intuitively, the problem can be understood as follows. Take first the case α < 1. For any initial condition for which ϕ(0) is small but positive and ˙ϕ(0) = 0, there results from equation (1.4) that ¨ϕ(0) > 0. Therefore, also ˙ϕ and hence ϕ increase, and the pendulum moves away, rather than toward, the vertical position. When α > 1 the problem is more subtle: The torque is being applied in the correct direction to counteract the natural instability of the pendulum, but this feedback helps build too much inertia. In particular, when already close to ϕ(0) = 0 but moving at a relatively large speed, the controller (1.3) keeps pushing toward the vertical, and overshoot and eventual oscillation result. The obvious solution is to keep α > 1 but to modify the proportional feed￾back (1.3) through the addition of a term that acts as a brake, penalizing ve￾locities. In other words, one needs to add damping to the system. We arrive then at a PD, or proportional-derivative feedback law, u(t) = −αϕ(t) − βϕ˙(t), (1.6) with α > 1 and β > 0. In practice, implementing such a controller involves measurement of both the angular position and the velocity. If only the former is easily available, then one must estimate the velocity as part of the control algorithm; this will lead later to the idea of observers, which are techniques for