Fall 2001 16.3112-1 Controllability Definition: An LTI system is controllable if, for every a*(t d every T>0, there exists an input function u(t),0<t<T, such that the system state goes from (0)=0 to a T)=I' Starting at 0 is not a special case- if we can get to any state in finite time from the origin, then we can get from any initial condition to that state in finite time as well Need only consider the forced solution to study controllability Bu(r)dr Change of variables T2=t-T, dr =-dr2 gives a(t)= eAry Bu(t-T2)dT2 0 This definition of observability is consistent with the notion we used before of being able to "influence"all the states in the system in the decoupled examples we looked at before ROT: For those decoupled examples, if part of the state cannot be"influenced"by ult), then it would be impossible to move that part of the state from 0 to *kFall 2001 16.31 12—1 Controllability • Definition: An LTI system is controllable if, for every x?(t) and every T > 0, there exists an input function u(t), 0 < t ≤ T, such that the system state goes from x(0) = 0 to x(T) = x?. — Starting at 0 is not a special case — if we can get to any state in finite time from the origin, then we can get from any initial condition to that state in finite time as well. — Need only consider the forced solution to study controllability. x(t) = Z t 0 eA(t−τ ) Bu(τ )dτ — Change of variables τ2 = t − τ , dτ = −dτ2 gives x(t) = Z t 0 eAτ2Bu(t − τ2)dτ2 • This definition of observability is consistent with the notion we used before of being able to “influence” all the states in the system in the decoupled examples we looked at before. — ROT: For those decoupled examples, if part of the state cannot be “influenced” by u(t), then it would be impossible to move that part of the state from 0 to x?