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Controllability and Observability Grammians

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

Summary: (Blank Abstract)

Controllability Grammian

The finite controllability grammian at time t<∞t is defined as follows.

P⁢t=∫0teA⁢τ⁢B⁢B*⁢eA*⁢τdτ P t τ t 0 A τ B B* A* τ
(1)

This grammian has two important properties. First, P⁢t=P*⁢t≥0 P t P* t 0 . Secondly, the columns of P⁢tPt span the controllable space, i.e. im⁢P⁢t=im⁢C⁢AB im P t im C A B It can be shown that the state defined by AA and BB is controllable if, and only if, P⁢tPt is positive definite for some t>0t0.

Using the controllability grammian, we can determine how to most efficiently take a system from the zero state to a certain state x―x―. Given that x―x― is in the controllable space, there exists ξξ such that

x―=P⁢T―⁢ξ x― P T― ξ
(2)
for some T―>0T―0. In this case, the minimum energy input required to move the system from zero to x―x― is u―=B*⁢eA*⁢(T―−t)⁢ξ― 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―u― in terms of x―x―:
u― =B*⁢eA*⁢(T―−t)⁢P-1T―⁢x― u― B* A* T― t P T― x―
(3)

In general, this minimal energy is exactly equal to ξ―*⁢P⁢T―⁢ξ― ξ― * P T― ξ― . If the system is controllable, then this formula becomes

Energy⁢u―=x―*⁢P-1T―⁢x― Energy u― x― * P T― x―
(4)
If you don't want to start at the zero state, the formulas above can still be applied for taking a system at state x1x1 to a state x2x2. This holds even if x1x1 and x2x2 are not controllable; in this case, all that is necessary is for x2−x1 x2 x1 to be in the controllable space. (This makes sense if you think of x1x1 as being the zero state and x2x2 as being the general state we are trying to reach; it is the exact analog of the previous case. Using x1x1 and x2x2 is just like using 00 and xx with an appropriate offset.)

Observability Grammian

The finite observability grammian at time t<∞t is defined as

Q⁢t=∫0teA*⁢τ⁢C*⁢C⁢eA⁢τdτ Q t τ t 0 A* τ C* C A τ
(5)
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⁢CA ker Q t ker O C A

Using this grammian, we can find an expression for the energy of the output yy at time TT caused by the system's initial state xx:

Energy⁢y=x*⁢Q⁢T⁢x Energy y x* Q T x
(6)

Infinite Grammians

Consider a continuous-time linear system defined, as per normal, by the matrices AA, BB, CC, andDD. 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=∞t.

P=∫0∞eA⁢τ⁢B⁢B*⁢eA*⁢τdτ P τ 0 A τ B B* A* τ
(7)
Q=∫0∞eA*⁢τ⁢C*⁢C⁢eA⁢τdτ Q τ 0 A* τ C* C A τ
(8)
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 P P A* B B* 0
(9)
A*⁢Q+Q⁢A+C*⁢C=0 A* Q Q A C* C 0
(10)

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 xcxc is

xc*⁢P-1⁢xc xc * P xc
(11)
Similarly, the largest observation energy produced by the state xoxo is obtained for an infinite observation interval and is equal to:
xo*⁢Q⁢xo xo * Q xo
(12)

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