Andrew Thangaraj
Aug-Nov 2020
System state variables at time \(k=0,1,\ldots\)
\(x_k=(x_{k1},x_{k2},\ldots,x_{kn})\)
Evolution from time \(k\) to \(k+1\)
\(x_{k+1} = A x_k\), where \(A\): \(n\times n\) matrix
From time \(0\) to \(k\)
\(x_k = A^k x_0\), where \(x_0\): initial state
Bounded-input, bounded-output stable
If \(x_0\) is bounded, \(x_k\) is bounded for all \(k\).
\(\lambda\): eigenvalue of \(A\) with eigenvector \(v\)
\(Av=\lambda v\)
Initial state: \(x_0=v\)
\(x_1 = A x_0 = \lambda v\)
\(x_2 = A^2 x_0 = \lambda^2 v\)
\(\vdots\)
\(x_k = A^k x_0 = \lambda^k v\)
Unstable if \(|\lambda|>1\)
Basis of eigenvectors for \(A\): \(\{v_1,\ldots,v_n\}\)
Eigenvalues: \(\lambda_1,\ldots,\lambda_n\)
Initial state in eigenbasis: \(x_0=\tilde{x}_{01}v_1+\cdots+\tilde{x}_{0n}v_n\)
\(x_1 = Ax_0=\tilde{x}_{01}\lambda_1 v_1+\cdots+\tilde{x}_{0n}\lambda_n v_n\)
\(x_2 = A^2x_0=\tilde{x}_{01}\lambda^2_1 v_1+\cdots+\tilde{x}_{0n}\lambda^2_n v_n\)
\(\vdots\)
\(x_k = A^k x_0=\tilde{x}_{01}\lambda^k_1 v_1+\cdots+\tilde{x}_{0n}\lambda^k_n v_n\)
Stable if \(|\lambda_i|< 1\) for \(i=1,\ldots,n\)
\(A = \begin{bmatrix} \lambda&1\\ 0&\lambda \end{bmatrix}\)
Eigenvalues: \(\lambda,\lambda\); Eigenvector: \((1,0)\)
\(A^k=\begin{bmatrix} \lambda^k&k\lambda^{k-1}\\ 0&\lambda^k \end{bmatrix}\)
Proof: by induction
\(A = \begin{bmatrix} \lambda&1&0\\ 0&\lambda&1\\ 0&0&\lambda \end{bmatrix}\)
Eigenvalues: \(\lambda,\lambda,\lambda\); Eigenvector: \((1,0,0)\)
\(A^k=\begin{bmatrix} \lambda^k&k\lambda^{k-1}&\dfrac{(k-1)k}{2}\lambda^{k-2}\\ 0&\lambda^k&k\lambda^{k-1}\\ 0&0&\lambda^k \end{bmatrix}\)
Proof: by induction
\(A = \begin{bmatrix} \lambda&1&0&0&0\\ 0&\lambda&1&0&0\\ 0&0&\lambda&1&0\\ 0&0&0&\lambda&1\\ 0&0&0&0&\lambda \end{bmatrix}\)
Eigenvalues: \(\lambda,\lambda,\lambda,\lambda,\lambda\); Eigenvector: \((1,0,0,0,0)\)
\(A^k=\begin{bmatrix} \lambda^k&k\lambda^{k-1}&O(k^2)\lambda^{k-2}&O(k^3)\lambda^{k-3}&O(k^4)\lambda^{k-4}\\ 0&\lambda^k&k\lambda^{k-1}&O(k^2)\lambda^{k-2}&O(k^3)\lambda^{k-3}\\ 0&0&\lambda^k&k\lambda^{k-1}&O(k^2)\lambda^{k-2}\\ 0&0&0&\lambda^k&k\lambda^{k-1}\\ 0&0&0&0&\lambda^k\\ \end{bmatrix}\)
Proof: by induction
Jordan form for any matrix (in a suitable basis)
\(A\leftrightarrow\begin{bmatrix} A_1&0&\cdots&0\\ 0&A_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&A_m \end{bmatrix}\), \(n\times n\)
Form of \(A_i\), \(l_i\times l_i\), \(n=l_1+\cdots+l_m\)
\(\lambda_i\) (\(l_i=1\)) or \(\begin{bmatrix} \lambda_i&1&0&0&\cdots&0\\ 0&\lambda_i&1&0&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&0&\lambda_i&1&0\\ 0&0&\cdots&0&\lambda_i&1\\ 0&0&0&\cdots&0&\lambda_i \end{bmatrix}\) (\(l_i\ge2\))
Values of \(A^k\): \(O(k^{l_i-1})\lambda^{k-l_i+1}_i\)
If \(|\lambda_i|<1\), values of \(A^k\) tend to 0 as \(k\to\infty\) \(\Rightarrow\) stable