Controllability Grammian The finite controllability grammian at time t is defined as follows. P t τ t 0 A τ B B* A* τ This grammian has two important properties. First, P t P* t 0 . Secondly, the columns of Pt span the controllable space, i.e. im P t im C A B It can be shown that the state defined by A and B is controllable if, and only if, Pt is positive definite for some t0. Using the controllability grammian, we can determine how to most efficiently take a system from the zero state to a certain state x―. Given that x― is in the controllable space, there exists ξ such that x― P T― ξ for some T―0. In this case, the minimum energy input required to move the system from zero to x― is u― B* A* T― t ξ― If the controllability matrix is invertible, we can use the relation equation between ξ and certain state to put u― in terms of x―: u― B* A* T― t P T― x― In general, this minimal energy is exactly equal to ξ― * P T― ξ― . If the system is controllable, then this formula becomes Energy u― x― * P T― x― If you don't want to start at the zero state, the formulas above can still be applied for taking a system at state x1 to a state x2. This holds even if x1 and x2 are not controllable; in this case, all that is necessary is for x2 x1 to be in the controllable space. (This makes sense if you think of x1 as being the zero state and x2 as being the general state we are trying to reach; it is the exact analog of the previous case. Using x1 and x2 is just like using 0 and x with an appropriate offset.)

Observability Grammian The finite observability grammian at time t is defined as Q t τ t 0 A* τ C* C A τ Parallel to the finite controllability grammian, the kernel of finite observability grammian is equal to the kernel of the observability matrix. (This relationship holds for positive time only.) ker Q t ker O C A Using this grammian, we can find an expression for the energy of the output y at time T caused by the system's initial state x: Energy y x* Q T x

Infinite Grammians Consider a continuous-time linear system defined, as per normal, by the matrices A, B, C, andD. Assuming that this system is stable (i.e. all of its eigenvalues have negative real parts), both the controllability and observability grammians are defined for t. P τ 0 A τ B B* A* τ Q τ 0 A* τ C* C A τ These are called the infinite controllability and infinite observability grammians, respectively. These grammians satisfy the linear matrix equations known as the Lyapunov equations. A P P A* B B* 0 A* Q Q A C* C 0 In the case of infinite grammians, the equations for minimal energy state transfer and observation energy drop their dependence on time. Assuming stability and complete controllability, the minimal energy required to transfer from zero to state xc is xc * P xc Similarly, the largest observation energy produced by the state xo is obtained for an infinite observation interval and is equal to: xo * Q xo