Continuous Time Systems

Let’s turn off the input, so \(x=0\). Then, we have \(\dot{\mathbf{q}}(t) = A\mathbf{q}(t)\). If \(\mathbf{q}(0)\) is the initial state, then taking the derivative of \(\mathbf{q}\) allows us to write:

\begin{align} \mathbf{q}(t) = e^{At}\mathbf{q}(0) \notag \end{align}

Now assume that \(A\) has distinct eigenvalues \(\lambda_i\) with corresponding eigenvectors \(\mathbf{v}_i\). Since the eigenvectors are linearly independent, they span the plane, which means we can express our initial state as a linear combination of the eigenvectors:

\begin{align} \mathbf{q}(0) &= \alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2 + \cdots \notag \\ \mathbf{q}(t) &= e^{At}(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2 + \cdots) \notag \end{align}

We can rewrite \(e^{At}\mathbf{v}_i\) as:

\begin{align} e^{At}\mathbf{v}_i &= \left(\sum_{k=0}^\infty \frac{t^k}{k!} A^k\right)\mathbf{v}_i \notag \\ &=\sum_{k=0}^\infty \frac{t^k}{k!}A^k\mathbf{v}_i \notag \\ &=\left(\sum_{k=0}^\infty \frac{(\lambda_i t)^k}{k!}\right)\mathbf{v}_i \notag \\ &= e^{\lambda_i t}\mathbf{v}_i \notag \end{align}

Thus, we can express \(\mathbf{q}(t)\) as:

\begin{align} \boxed{\mathbf{q}(t) = \alpha_1 e^{\lambda_1 t}\mathbf{v}_1 + \alpha_2 e^{\lambda_2 t}\mathbf{v}_2 + \cdots} \end{align}

Say \(\lambda\) is complex. Then, \(\lambda = \sigma + i\omega_d\), and the exponential term becomes:

\begin{align} e^{\lambda t} = e^{\sigma t}e^{i\omega_d t} \notag \end{align}

The \(e^{\sigma t}\) controls whether the signal is decaying or growing, and the \(e^{i\omega_d t}\) is the oscillatory term. The system is called internally stable if \(\sigma < 0\). Equivalently, the system is internally/asymptotically stable if and only if \(\text{Re}\{\lambda_k\} < 0\).