Chapter 13 Controllability II

13.1 Examples

  • Example 13.1. Consider two first order systems with the same control acting on both:

    x˙1+x1=u,x˙2+αx2=u, where α>0. This gives

    A=[100α],B=[11].

    If α=1, then it is easy to see from the definition that the system is not controllable. Choose x0=[00]. Since the equations are identical (and we have chosen identical initial conditions), we then have x1(t)=x2(t) for all t, no matter what u is. With x1=[10] we then have that x1 is not reachable from x0: assume that it is reachable in time T, then 1=x1(T)=x2(T)=0, which is a contradition.

    The Kalman controllability matrix is

    [BAB]=[111α],

    which has determinant

    α+1,

    which is zero if and only if α=1. We see that the system is controllable if and only if α1. If α=1, then the Kalman controllability matrix equals

    [1111]

    and we see that its image equals the multiples of

    [11],

    so that the reachable subspace consists of vectors in R2 which have their first and second component the same.

    The Hautus controllability matrix is

    [sIAB]=[s+1010s+α1].

    Selecting the first and last column we obtain a matrix with determinant s+1 which therefore is invertible as long as s1. Therefore the Hautus controllability matrix is surjective for s1. For s=1 the Hautus controllability matrix equals

    [00101+α1].

    Since the first column is zero, we only have to consider the last two columns. The matrix formed of the last two columns has determinant α1 and is therefore invertible if and only if α1. We then obtain (using that the neglected column is zero) that the Hautus controllability matrix is invertible for all sC if and only if α1. Therefore we again see the system is controllable if and only if α1.

    The control Lyapunov equation is (using that we know that Q is symmetric)

    [100α][Q1Q0Q0Q2]+[Q1Q0Q0Q2][100α]+[11][11]=[0000],

    which is

    [2Q1+1Q0(1α)+1Q0(1α)+12αQ2+1]=[0000].

    Solving this (we really just have three separate equations in the three unknowns Q1, Q2 and Q0) gives

    Q=[1211+α11+α12α].

    The determinant of Q equals

    14α1(1+α)2,

    which is zero if and only if (1+α)2=4α, i.e. when (1α)2=0, i.e. when α=1. So we again see that the system is controllable if and only if α1. When α=1 we have

    Q=[12121212],

    which has image the scalar multiples of

    [11],

    so that we again obtain the reachable subspace as obtained using a different method above.

  • Example 13.2. Consider the second order scalar differential equation

    q¨+2ζq˙+q=u,

    where ζ0, with state x=[qq˙]. We then have

    A=[0112ζ],B=[01].

    The Kalman controllability matrix is

    [BAB]=[0112ζ].

    The determinant equals 1 which is nonzero so that the Kalman controllability matrix is invertible and therefore surjective and therefore the system is controllable.

  • Example 13.3. We again consider the second order scalar differential equation from Example 13.2 (now with ζ>0) and aim to find the infinite-time controllability Gramian. The control Lyapunov equation (using that Q is symmetric) is

    [0112ζ][Q1Q0Q0Q2]+[Q1Q0Q0Q2][0112ζ]+[01][01]=[0000],

    which is

    [2Q0Q2Q12ζQ0Q2Q12ζQ02Q04ζQ2+1]=[0000].

    From the top-left corner we obtain Q0=0 and subsequently from the bottom-right corner we obtain Q2=14ζ and from the off-diagonal entry we obtain Q1=14ζ. Hence

    Q=14ζ[1001].

    Since this is invertible, we see (once again) that the system is controllable.

  • Example 13.4. Consider the undamped second order scalar differential equation

    q¨+q=u,

    with state x=[qq˙], so that

    A=[0110],B=[01].

    The aim is to, for a given ζ>0, find a F such that the characteristic polynomial of A equals s2+2ζs+1. From our knowledge of second order scalar differential equations, we know that u=2ζq˙ will achieve this. We want to check that Ackermann’s formula also leads to this.

    Ackermann’s formula tells us that

    F=[01][BAB]1(A2+2ζA+I),

    will achieve the objective. From Example 13.2 we have

    [BAB]=[0110],

    so that

    [BAB]1=[0110].

    We further have

    A2=[1001],

    so that Ackermann’s formula is

    F=[01][0110]([1001]+2ζ[0110]+[1001])=[01][0110]2ζ[0110]=[01]2ζ[1001]=2ζ[01], which indeed is equivalent to the formula u=2ζq˙ 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

A=[d1M101M10d2M21M2kk0],B=[1M1001M200].

We show that this system is controllable by considering the Kalman controllability matrix. We have

AB=[d1M1200d2M22kM1kM2].

We see that [BAB] already has rank 3 as the first three colums are

[1M10d1M1201M2000kM1],

which is upper-triangular with non-zero diagonal entries and therefore invertible. Hence the Kalman controllability matrix [BABA2B] is surjective.

13.3 Case study: a suspension system*

Consider the suspension system with only the control inputs. This gives an input-state system with

A=[0kus001mus00001010000],B=[01mus01ms].

We show that this system is controllable by considering the Kalman controllability matrix. We have

[BABA2BA3B]=[01mus0ksmus21mus0ksmus2001ms+1mus0ksmus21ms000].

To see that this matrix is invertible, we calculate its determinant. We develop by the last row to obtain

1msdet[kusmus0kus2mus20kusmus201mus+1ms0kusmus2],

which we develop by the middle row to obtain

kusmus2msdet[kusmuskus2mus21mus+1mskusmus2],

which gives

kusmus2ms(kus2mus3+kus2mus3+kus2mus2ms)=kus3mus4ms2,

which is nonzero. Hence the Kalman controllability matrix is surjective and therefore our system is controllable.