1.2. Proportional-Derivative Control reliably performing such an estimation. We assume here that y can indeed be easured. Consider then the resulting closed-loop system p(t)+B(t)+(a-1)y(t)=0 The roots of its associated characteristic equation z2+Bz+a-1=0 2 both of which have negative real parts. Thus all the solutions of(1. 2)converge to zero. The system has been stabilized under feedback. This convergence may be oscillatory, but if we design the controller in such a way that in addition t the above conditions on a and B it is true that then all of the solutions are combinations of decaying exponentials and no os- cillation results We conclude from the above discussion that through a suitable choice of the gains a and B it is possible to attain the desired behavior, at least for the linearized model. That this same design will still work for the original nonlinear model, and, hence, assuming that this model was accurate, for a real pendulum is due to what is perhaps the most important fact in control theory -and for that matter in much of mathematics- namely that first-order approximations re sufficient to characterize local behavior. Informally, we have the following linearization principle Designs based on linearizations work locally for the original system The term "local"refers to the fact that satisfactory behavior only can be ex- pected for those initial conditions that are close to the point about which the linearization was made. Of course, as with any "principle, this is not a theorem. It can only become so when precise meanings are assigned to the various terms and proper technical assumptions are made. Indeed, we will invest some effort this text to isolate cases where this principle may be made rigorous. One of these cases will be that of stabilization, and the theorem there will imply that if we can stabilize the linearized system (1.2) for a certain choice of parameters a, B in the law(1.6), then the same control law does bring initial conditions of (1.1)that start close to 0=T, 6=0 to the vertical equilibrium Basically because of the linearization principle, a great deal of the literature in control theory deals exclusively with linear systems. From an engineering t of view, local solutions to control problems are often enough; when they not, ad hoc methods sometimes may be used in order to "patch"together h local solutions, a procedure called gain scheduling. Sometimes, one may1.2. Proportional-Derivative Control 5 reliably performing such an estimation. We assume here that ˙ϕ can indeed be measured. Consider then the resulting closed-loop system, ϕ¨(t) + βϕ˙(t)+(α − 1)ϕ(t)=0 . (1.7) The roots of its associated characteristic equation z2 + βz + α − 1 = 0 (1.8) are −β ± pβ2 − 4(α − 1) 2 , both of which have negative real parts. Thus all the solutions of (1.2) converge to zero. The system has been stabilized under feedback. This convergence may be oscillatory, but if we design the controller in such a way that in addition to the above conditions on α and β it is true that β2 > 4(α − 1), (1.9) then all of the solutions are combinations of decaying exponentials and no os￾cillation results. We conclude from the above discussion that through a suitable choice of the gains α and β it is possible to attain the desired behavior, at least for the linearized model. That this same design will still work for the original nonlinear model, and, hence, assuming that this model was accurate, for a real pendulum, is due to what is perhaps the most important fact in control theory —and for that matter in much of mathematics— namely that first-order approximations are sufficient to characterize local behavior. Informally, we have the following linearization principle: Designs based on linearizations work locally for the original system The term “local” refers to the fact that satisfactory behavior only can be ex￾pected for those initial conditions that are close to the point about which the linearization was made. Of course, as with any “principle,” this is not a theorem. It can only become so when precise meanings are assigned to the various terms and proper technical assumptions are made. Indeed, we will invest some effort in this text to isolate cases where this principle may be made rigorous. One of these cases will be that of stabilization, and the theorem there will imply that if we can stabilize the linearized system (1.2) for a certain choice of parameters α,β in the law (1.6), then the same control law does bring initial conditions of (1.1) that start close to θ = π, ˙ θ = 0 to the vertical equilibrium. Basically because of the linearization principle, a great deal of the literature in control theory deals exclusively with linear systems. From an engineering point of view, local solutions to control problems are often enough; when they are not, ad hoc methods sometimes may be used in order to “patch” together such local solutions, a procedure called gain scheduling. Sometimes, one may