Chapter 2 Stability

  • Definition 2.1.  Let \(A\in \mC ^{n\times n}\). The matrix \(A\) is called asymptotically stable if all solutions of \(\dot {x}=Ax\) satisfy \(\lim _{t\to \infty }x(t)=0\).

  • Definition 2.2.  Let \(p\) be a polynomial with complex coefficients. The polynomial \(p\) is called stable if all its roots have negative real part.

  • Lemma 2.3.  A matrix \(A\) is asymptotically stable if and only if its characteristic polynomial \(\det (sI-A)\) is stable.

  • Remark 2.4.  We note that a neccessary (but not in general sufficient) condition for a polynomial with real coefficients to be stable is that all coefficients have the same sign.

The following result is called the Routh-Hurwitz stability criterion (note that we assume that the coeficients of the polynomial are real, which will be true for the characteristic polynomial of \(A\) if \(A\) is real).

  • Theorem 2.5.  Let \(p\) be given by

    \[ p(s)=\sum _{j=0}^na_js^j=a_0+a_1s+\ldots +a_{n-1}s^{n-1}+a_ns^n, \]

    where \(a_0,\ldots ,a_n\in \mathbb {R}\) with \(a_n>0\). Let \(H\) be the \(n\times n\) matrix given by

    \[ H_{ij}=\begin {cases} a_{n-(2i-j)}&0\leq 2i-j\leq n,\\ 0&\text {otherwise}, \end {cases} \]

    i.e.,

    \[ H=\bbm { a_{n-1}&a_n&0&0&\cdots &0&0&0 \\a_{n-3}&a_{n-2}&a_{n-1}&a_n&\cdots &0&0&0 \\a_{n-5}&a_{n-4}&a_{n-3}&a_{n-2}&\cdots &0&0&0 \\\vdots &\vdots &\vdots &\vdots &\ddots &\vdots &\vdots &\vdots \\0&0&0&0&\cdots &a_2&a_3&a_4 \\0&0&0&0&\cdots &a_0&a_1&a_2 \\0&0&0&0&\cdots &0&0&a_0 }. \]

    Define \(H_k\) for \(k=1,\ldots ,n\) as the principal subdeterminants of \(H\):

    \begin{gather*} H_1:=a_{n-1},\qquad H_2:=\det \bbm {a_{n-1}&a_n\\a_{n-3}&a_{n-2}},\\ H_3:=\det \bbm {a_{n-1}&a_n&0\\a_{n-3}&a_{n-2}&a_{n-1}\\a_{n-5}&a_{n-4}&a_{n-3}},\, \ldots ,\, H_n:=\det (H)=a_0H_{n-1}. \end{gather*} The polynomial \(p\) is stable if and only if \(H_k>0\) for all \(k\in \{1,\ldots ,n\}\).

  • Example 2.6.  Consider the general second degree polynomial

    \[ p(s)=a_2s^2+a_1s+a_0, \]

    with \(a_2>0\). The corresponding matrix is

    \[ H=\bbm {a_1&a_2\\0&a_0}, \]

    and the principal subdeterminants are

    \[ H_1=a_1,\qquad H_2=a_0H_1. \]

    The necessary and sufficient condition for stability therefore is:

    \[ a_1>0\qquad a_0>0. \]

  • Example 2.7.  Consider the general third degree polynomial

    \[ p(s)=a_3s^3+a_2s^2+a_1s+a_0, \]

    with \(a_3>0\). The corresponding matrix is

    \[ H=\bbm {a_2&a_3&0\\a_0&a_1&a_2\\0&0&a_0}, \]

    and the principal subdeterminants are

    \[ H_1=a_2,\qquad H_2=\det \bbm {a_2&a_3\\a_0&a_1}=a_2a_1-a_0a_3,\qquad H_3=a_0H_2. \]

    The necessary and sufficient condition for stability therefore is:

    \[ a_2>0\qquad a_0>0,\qquad a_2a_1-a_0a_3>0. \]

  • Example 2.8.  Consider the general fourth degree polynomial

    \[ p(s)=a_4s^4+a_3s^3+a_2s^2+a_1s+a_0, \]

    with \(a_4>0\). The corresponding matrix is

    \[ H=\bbm {a_3&a_4&0&0\\a_1&a_2&a_3&a_4\\0&a_0&a_1&a_2\\0&0&0&a_0}, \]

    and the principal subdeterminants are

    \begin{gather*} H_1=a_3,\qquad H_2=\det \bbm {a_3&a_4\\a_1&a_2}=a_3a_2-a_1a_4,\\ H_3=\det \bbm {a_3&a_4&0\\a_1&a_2&a_3\\0&a_0&a_1}=a_1H_2-a_3^2a_0,\qquad H_4=a_0H_3. \end{gather*} The condition that \(H_1,H_2,H_3,H_4>0\) can be rewritten in a more convenient way using the necessary condition that all coefficients of the polynomial must be positive. The condition that \(H_3>0\) combined with the condition that \(a_0,a_1>0\) gives that \(H_2>0\). Therefore the necessary and sufficient condition for stability is:

    \[ a_3,a_1,a_0>0,\qquad a_3a_2a_1-a_4a_1^2-a_3^2a_0>0. \]

2.1 Examples

2.1.1 First order systems

For a first order system we have \(A=\frac {-1}{T}\), which is clearly asymptotically stable (recall that \(T>0\)).

2.1.2 Second order systems

For second order systems (of both types), we have

\[ A=\bbm {0&1\\-\omega _0^2&-2\zeta \omega _0}, \]

which has characteristic polynomial (note that we can obtain this without computation from the second order form (1.5))

\[ s^2+2\zeta \omega _0s+\omega _0^2. \]

The Routh–Hurwitz condition for stability is that \(2\zeta \omega _0>0\) and \(\omega _0^2>0\). Since \(\omega _0>0\) by assumption, this is true if and only if \(\zeta >0\). So if the damping ratio \(\zeta \) is positive, then the second order system is asymptotically stable, but if it is zero (i.e. the system is undamped), then the second order system is not asymptotically stable.

2.2 Case study: a suspension system

We will show that the matrix

\[ A_\cl =\bbm {0&k_{us}&0&0\\ \frac {-1}{m_{us}}&\frac {-d}{m_{us}}&\frac {-k_s}{m_{us}}&\frac {d}{m_{us}}\\ 0&1&0&-1\\ 0&\frac {d}{m_s}&\frac {k_s}{m_s}&\frac {-d}{m_s} }, \]

obtained from the fixed structure suspension system is asymptotically stable (provided of course that all physical parameters are positive).

The characteristic polynomial of \(A_\cl \) multiplied by \(m_sm_{us}\) is:

\[ m_sm_{us}s^4+d[m_s+m_{us}]s^3+[k_sm_s+k_sm_{us}+k_{us}m_s]s^2+dk_{us}s+k_sk_{us}. \]

We apply the Routh–Hurwitz criterion for the degree four case derived above to this polynomial. Since we assume that all physical parameters are positive, we clearly have that all coefficients are positive. Therefore the only condition to check is \(a_3a_2a_1-a_4a_1^2-a_3^2a_0>0\). We have

\[ \begin {aligned} a_3a_2a_1&=d^2[m_s+m_{us}][k_sm_s+k_sm_{us}+k_{us}m_s]k_{us}\\ &=d^2k_{us}\left ([m_s+m_{us}]^2k_s+m_s^2k_{us}+m_sm_{us}k_{us}\right ), \end {aligned} \]

and

\[ \begin {aligned} a_4a_1^2&=m_sm_{us}d^2k_{us}^2\\ a_3^2a_0&=d^2k_{us}[m_s+m_{us}]^2k_s, \end {aligned} \]

so that cancelling like terms we obtain the condition

\[ d^2k_{us}^2m_s^2>0, \]

which of course is true. Therefore, the matrix \(A_\cl \) is asymptotically stable.

The matrix

\[ A=\bbm {0&k_{us}&0&0\\ \frac {-1}{m_{us}}&0&0&0\\ 0&1&0&-1\\ 0&0&0&0 }, \]

for the general design of a suspension system is not asymptotically stable. Since it has a zero column, \(0\) is an eigenvalue of the matrix and therefore the matrix is not asymptotically stable.

2.3 Case study: control of a tape drive

The matrix

\[ A=\bbm { -\frac {d_1}{M_1}&0&\frac {1}{M_1}\\ 0&-\frac {d_2}{M_2}&-\frac {1}{M_2}\\ -k&k&0 }, \]

is asymptotically stable. Its characteristic polynomial is

\begin{align*} \det \bbm { s+\frac {d_1}{M_1}&0&-\frac {1}{M_1}\\ 0&s+\frac {d_2}{M_2}&\frac {1}{M_2}\\ k&-k&s } &=\left (s+\frac {d_1}{M_1}\right )\det \bbm {s+\frac {d_2}{M_2}&\frac {1}{M_2}\\-k&s} -\frac {1}{M_1}\det \bbm {0&s+\frac {d_2}{M_2}\\k&-k} \\ &=\left (s+\frac {d_1}{M_1}\right )\left (s^2+\frac {d_2}{M_2}s+\frac {k}{M_2}\right ) +\frac {k}{M_1}\left (s+\frac {d_2}{M_2}\right ) \\ &=s^3+\left (\frac {d_1}{M_1}+\frac {d_2}{M_2}\right )s^2+\left (\frac {k}{M_1}+\frac {k}{M_2}+\frac {d_1d_2}{M_1M_2}\right )s+\frac {(d_1+d_2)k}{M_1M_2}. \end{align*} Multiplying though by \(M_1M_2\) gives

\[ M_1M_2s^3+(d_1M_2+d_2M_1)s^2+(kM_2+kM_1+d_1d_2)s+(d_1+d_2)k. \]

We apply the Routh–Hurwitz criterion for the degree three case derived above to this polynomial. Since we assume that all physical parameters are positive, we clearly have that all coefficients are positive. Therefore the only condition to check is that \(a_2a_1-a_0a_3>0\). This is

\[ (d_1M_2+d_2M_1)(kM_2+kM_1+d_1d_2)-(d_1+d_2)kM_1M_2>0. \]

Multiplying out the left-hand side, this is

\[ d_1kM_2^2+d_1kM_1M_2+d_1^2d_2M_2+d_2kM_1M_2+d_2kM_1^2+d_1d_2^2M_1-d_1kM_1M_2-d_2kM_1M_2>0, \]

which upon cancelling like terms is

\[ d_1kM_2^2+d_1^2d_2M_2+d_2kM_1^2+d_1d_2^2M_1>0, \]

which is true.

2.4 Problems

Let \(m,k>0\) and \(d\geq 0\). Determine whether or not the matrix

\[ A=\bbm {0&1\\\frac {-k}{m}&\frac {-d}{m}}, \]

is asymptotically stable.

  • Solution. The characteristic polynomial is

    \[ \det (sI-A)=\det \bbm {s&-1\\\frac {k}{m}&s+\frac {d}{m}}=s^2+\frac {d}{m}s+\frac {k}{m}. \]

    We see that if \(d=0\), then the matrix is not asymptotically stable (since one of the coefficients is zero) whereas for \(d>0\) the matrix is asymptotically stable (since all coefficients are positive, which for the degree two case is sufficient).  □