1. Introduction even be lucky and find a way to transform the problem of interest into one that is globally linear; we explain this later using again the pendulum as an example In many other cases, however, a genuinely nonlinear approach is needed, and much research effort during the past few years has been directed toward that goal. In this text, when we develop the basic definitions and results for the linear theory we will always do so with an eye toward extensions to nonlinear obal. results An Exercise As remarked earlier, proportional control (1. 3) by itself is inadequate for the original nonlinear model. Using again =8-T, the closed-loop equatio p(t)-sin(t)+ap(t)=0 (1.10) The next exercise claims that solutions of this equation typically will not ap proach zero, no matter how the feedback gain a is picked Exercise 1.2.1 Assume that a is any fixed real number, and consider the("en- ergy")function of two real variables COSa 1+(ax2+y2) (1.11) Show that V((t),p(t) is constant along the solutions of (1.10). Using that v(a, O)is an analytic function and therefore that its zero at z=0 is isolated conclude that there are initial conditions of the type P(0)=E,(0)=0, with e arbitrarily small, for which the corresponding solution of (1.10) does not satisfy that p(t)→0andp(t)→0ast→∝ 1. 3 Digital Control The actual physical implementation of (1.6)need not concern us here, but some remarks are in order. Assuming again that the values p(t)and p(t), or equiva- lently g(t)and 0(t), can be measured, it is necessary to take a linear combina- tion of these in order to determine the torque u(t) that the motor must apply Such combinations are readily carried out by circuits built out of devices called perational amplifiers. Alternatively, the damping term can be separately im- plemented directly through the use of an appropriate device(a"dashpot"), and he torque is then made proportional to p(t) A more modern alternative, attractive especially for larger systems, is to convert position and velocity to digital form and to use a computer to calcu- late the necessary controls. Still using the linearized inverted pendulum as an illustration, we now describe some of the mathematical problems that this leads6 1. Introduction even be lucky and find a way to transform the problem of interest into one that is globally linear; we explain this later using again the pendulum as an example. In many other cases, however, a genuinely nonlinear approach is needed, and much research effort during the past few years has been directed toward that goal. In this text, when we develop the basic definitions and results for the linear theory we will always do so with an eye toward extensions to nonlinear, global, results. An Exercise As remarked earlier, proportional control (1.3) by itself is inadequate for the original nonlinear model. Using again ϕ = θ − π, the closed-loop equation becomes ϕ¨(t) − sin ϕ(t) + αϕ(t)=0 . (1.10) The next exercise claims that solutions of this equation typically will not ap￾proach zero, no matter how the feedback gain α is picked. Exercise 1.2.1 Assume that α is any fixed real number, and consider the (“en￾ergy”) function of two real variables V (x,y) := cos x − 1 + 1 2 (αx2 + y2). (1.11) Show that V (ϕ(t), ϕ˙(t)) is constant along the solutions of (1.10). Using that V (x, 0) is an analytic function and therefore that its zero at x = 0 is isolated, conclude that there are initial conditions of the type ϕ(0) = ε, ϕ˙(0) = 0, with ε arbitrarily small, for which the corresponding solution of (1.10) does not satisfy that ϕ(t) → 0 and ˙ϕ(t) → 0 as t → ∞. ✷ 1.3 Digital Control The actual physical implementation of (1.6) need not concern us here, but some remarks are in order. Assuming again that the values ϕ(t) and ˙ϕ(t), or equiva￾lently θ(t) and ˙ θ(t), can be measured, it is necessary to take a linear combina￾tion of these in order to determine the torque u(t) that the motor must apply. Such combinations are readily carried out by circuits built out of devices called operational amplifiers. Alternatively, the damping term can be separately im￾plemented directly through the use of an appropriate device (a “dashpot”), and the torque is then made proportional to ϕ(t). A more modern alternative, attractive especially for larger systems, is to convert position and velocity to digital form and to use a computer to calcu￾late the necessary controls. Still using the linearized inverted pendulum as an illustration, we now describe some of the mathematical problems that this leads to