We have determined conditions under which we may compute the complete system state x(t) from the output history y(t) and the input history u(t). Next, we seek conditions under which we may choose the input history to effect some desired state trajectory x(t). Definition. A LTI system is controllable if, for every x1 and every time T> 0, there exists an input history u(t) defined on the interval 0 ≤ t ≤ T which drives the system from x(0) = 0 to x(T) = x1. Note: Different textbooks will define controllability in different ways. Some state that the system can be driven from any initial state to any final state in an arbitrarily short time. Others say that the system can be driven from any initial state to zero in an arbitrarily short time. In fact, because of the principle of superposition, all of these definitions are equivalent. We will use Bélanger’s definition  to emphasize that controllability is essentially a question about a system’s zero state response (i.e., its response to forcing). In contrast, observability is essentially a question about a system’s zero input response (i.e., its initial condition response). Definition. A state x ∗ � = 0 is uncontrollable if the zero-state response x(t) is orthogonal to x ∗ for all t> 0 and all input functions. This means that, regardless of the input, the zero state response never develops any component in the “x ∗ direction ” because 1 0 = x ∗ · x(t) = x ∗
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