Chapter 13 Controllability II
13.1 Examples
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Example 13.1. Consider two first order systems with the same control acting on both:
where . This givesIf
, then it is easy to see from the definition that the system is not controllable. Choose . Since the equations are identical (and we have chosen identical initial conditions), we then have for all , no matter what is. With we then have that is not reachable from : assume that it is reachable in time , then , which is a contradition.The Kalman controllability matrix is
which has determinant
which is zero if and only if
. We see that the system is controllable if and only if . If , then the Kalman controllability matrix equalsand we see that its image equals the multiples of
so that the reachable subspace consists of vectors in
which have their first and second component the same.The Hautus controllability matrix is
Selecting the first and last column we obtain a matrix with determinant
which therefore is invertible as long as . Therefore the Hautus controllability matrix is surjective for . For the Hautus controllability matrix equalsSince the first column is zero, we only have to consider the last two columns. The matrix formed of the last two columns has determinant
and is therefore invertible if and only if . We then obtain (using that the neglected column is zero) that the Hautus controllability matrix is invertible for all if and only if . Therefore we again see the system is controllable if and only if .The control Lyapunov equation is (using that we know that
is symmetric)which is
Solving this (we really just have three separate equations in the three unknowns
, and ) givesThe determinant of
equalswhich is zero if and only if
, i.e. when , i.e. when . So we again see that the system is controllable if and only if . When we havewhich has image the scalar multiples of
so that we again obtain the reachable subspace as obtained using a different method above.
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Example 13.2. Consider the second order scalar differential equation
where
, with state . We then haveThe Kalman controllability matrix is
The determinant equals
which is nonzero so that the Kalman controllability matrix is invertible and therefore surjective and therefore the system is controllable.
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Example 13.3. We again consider the second order scalar differential equation from Example 13.2 (now with
) and aim to find the infinite-time controllability Gramian. The control Lyapunov equation (using that is symmetric) iswhich is
From the top-left corner we obtain
and subsequently from the bottom-right corner we obtain and from the off-diagonal entry we obtain . HenceSince this is invertible, we see (once again) that the system is controllable.
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Example 13.4. Consider the undamped second order scalar differential equation
with state
, so thatThe aim is to, for a given
, find a such that the characteristic polynomial of equals . From our knowledge of second order scalar differential equations, we know that will achieve this. We want to check that Ackermann’s formula also leads to this.Ackermann’s formula tells us that
will achieve the objective. From Example 13.2 we have
so that
We further have
so that Ackermann’s formula is
which indeed is equivalent to the formula that we obtained above.
13.2 Case study: control of a tape drive*
We consider the tape drive with only the control input. This gives an input-state system with
We show that this system is controllable by considering the Kalman controllability matrix. We have
We see that
which is upper-triangular with non-zero diagonal entries and therefore invertible. Hence the Kalman controllability matrix
13.3 Case study: a suspension system*
Consider the suspension system with only the control inputs. This gives an input-state system with
We show that this system is controllable by considering the Kalman controllability matrix. We have
To see that this matrix is invertible, we calculate its determinant. We develop by the last row to obtain
which we develop by the middle row to obtain
which gives
which is nonzero. Hence the Kalman controllability matrix is surjective and therefore our system is controllable.