Linear state space equations and system stability

Andrew Thangaraj

Aug-Nov 2020

Recap

  • Vector space \(V\) over a scalar field \(F\)
    • \(F\): real field \(\mathbb{R}\) or complex field \(\mathbb{C}\) in this course
  • \(m\times n\) matrix A represents a linear map \(T:F^n\to F^m\)
    • dim null \(T\) + dim range \(T\) = dim \(V\)
  • Linear equation: \(Ax=b\)
    • Solution (if it exists): \(u+\) null\((A)\)
  • Four fundamental subspaces of a matrix
    • Column space, row space, null space, left null space
  • Eigenvalue \(\lambda\) and Eigenvector \(v\): \(Tv=\lambda v\)
    • Distinct eigenvalues have independent eigenvectors
    • Basis of eigenvectors results in a diagonal matrix for \(T\)
    • Every linear map has an upper triangular matrix representation

System state and evolution

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 \(k\).

Eigenvalues and instability

\(\lambda\): 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 \(|\lambda|>1\)

\(A\): diagonalizable

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 \(i=1,\ldots,n\)

\(A\): non-diagonalizable - \(2\times 2\), \(3\times 3\) examples

\(A = \begin{bmatrix} \lambda&1\\ 0&\lambda \end{bmatrix}\)

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

\(A\): non-diagonalizable - \(5\times 5\) example

\(A = \begin{bmatrix} \lambda&1&0&0&0\\ 0&\lambda&1&0&0\\ 0&0&\lambda&1&0\\ 0&0&0&\lambda&1\\ 0&0&0&0&\lambda \end{bmatrix}\)

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

\(A\): non-diagonalizable - general case

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