13.2: Stability of Linear Systems

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We may apply the preceding definitions to the LTI case by considering a system with a diagonalizable \(A\) matrix (in our standard notation) and \(\mathbf{u} \equiv 0\). The unique equilibrium point is at \(x = 0\), provided \(A\) has no eigenvalue at 0 (respectively 1) in the CT (respectively DT) case. (Otherwise every point in the entire eigenspace corresponding to this eigenvalue is an equilibrium.) Now

\[\begin{aligned}
\dot{x}(t) &=e^{A t} x(0) \\
&=V\left[\begin{array}{ccc}
e^{\lambda_{1} t} & \\
& \ddots & \\
& & e^{\lambda_{n} t}
\end{array}\right] W x(0) \quad(C T) \ (13.4)
\end{aligned} \nonumber\]

\[\begin{aligned}
x(k) &=A^{k} x(0) \\
&=V\left[\begin{array}{ccc}
\lambda_{1}^{k} & \\
& \ddots & \\
& & \lambda_{n}^{k}
\end{array}\right] W x(0) \quad(D T) \ (13.5)
\end{aligned} \nonumber\]

Hence, it is clear that in continuous time a system with a diagonalizable \(A\) is asymptotically stable iff

\[\mathcal{R} e\left(\lambda_{i}\right)<0, \quad i \in\{1, \ldots, n\} \ \tag{13.6}\]

while in discrete time the requirement is that

\[\left|\lambda_{i}\right|<1 \quad i \in\{1, \ldots, n\} \ \tag{13.7}\]

Note that if \(\mathcal{R} e\left(\lambda_{i}\right)=0(\mathrm{CT})\) or \(\left|\lambda_{i}\right|=1(\mathrm{DT})\), the system is not asymptotically stable, but is marginally stable.

Exercise

For the nondiagonalizable case, use your understanding of the Jordan form to show that the conditions for asymptotic stability are the same as in the diagonalizable case. For marginal stability, we require in the CT case that \(\mathcal{R} e\left(\lambda_{i}\right) \leq 0\), with equality holding for at least one eigenvalue; furthermore, every eigenvalue whose real part equals 0 should have its geometric multiplicity equal to its algebraic multiplicity, i.e., all its associated Jordan blocks should be of size 1. (Verify that the presence of Jordan blocks of size greater than one for these imaginary-axis eigenvalues would lead to the state variables growing polynomially with time.) A similar condition holds for marginal stability in the DT case.

Stability of Linear Time- Varying Systems

Recall that the general unforced solution to a linear time-varying system is

\[x(t)=\Phi\left(t, t_{0}\right) x\left(t_{0}\right)\nonumber\]

where \(\Phi(t, \tau)\) is the state transition matrix. It follows that the system is

stable i.s.L. at \(\bar{x}=0 \text { if } \sup _{t}\left\|\Phi\left(t, t_{0}\right)\right\|=m\left(t_{0}\right)<\infty\).
asymptotically stable at \(\bar{x}=0 \text { if } \lim _{t \rightarrow \infty}\left\|\Phi\left(t, t_{0}\right)\right\| \rightarrow 0, \forall t_{0}\).

These conditions follow directly from Definition 13.1