State-Space Models
x ˙ = A x + B u , y = C x + D u \dot{x} = Ax + Bu, \qquad y = Cx + Du x ˙ = A x + B u , y = C x + D u The transfer function is
G ( s ) = C ( s I − A ) − 1 B + D G(s) = C(sI - A)^{-1}B + D G ( s ) = C ( s I − A ) − 1 B + D A change of basis x = T z x = Tz x = T z
A ~ = T − 1 A T , B ~ = T − 1 B , C ~ = C T , D ~ = D \tilde{A} = T^{-1}AT, \qquad \tilde{B} = T^{-1}B, \qquad \tilde{C} = CT, \qquad \tilde{D} = D A ~ = T − 1 A T , B ~ = T − 1 B , C ~ = CT , D ~ = D Controllable Canonical Form
For
G ( s ) = b n − 1 s n − 1 + ⋯ + b 1 s + b 0 s n + a n − 1 s n − 1 + ⋯ + a 1 s + a 0 + d G(s) = \frac{b_{n-1}s^{n-1} + \cdots + b_1 s + b_0}{s^n + a_{n-1}s^{n-1} + \cdots + a_1 s + a_0} + d G ( s ) = s n + a n − 1 s n − 1 + ⋯ + a 1 s + a 0 b n − 1 s n − 1 + ⋯ + b 1 s + b 0 + d a realization is
A = [ 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 − a 0 − a 1 − a 2 ⋯ − a n − 1 ] , B = [ 0 ⋮ 0 1 ] , C = [ b 0 b 1 ⋯ b n − 1 ] , D = d A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & & & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{bmatrix}, \qquad B = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}, \qquad C = \begin{bmatrix} b_0 & b_1 & \cdots & b_{n-1} \end{bmatrix}, \qquad D = d A = 0 0 ⋮ 0 − a 0 1 0 0 − a 1 0 1 0 − a 2 ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 ⋮ 1 − a n − 1 , B = 0 ⋮ 0 1 , C = [ b 0 b 1 ⋯ b n − 1 ] , D = d with det ( λ I − A ) = λ n + a n − 1 λ n − 1 + ⋯ + a 1 λ + a 0 \det(\lambda I - A) = \lambda^n + a_{n-1}\lambda^{n-1} + \cdots + a_1\lambda + a_0 det ( λ I − A ) = λ n + a n − 1 λ n − 1 + ⋯ + a 1 λ + a 0 companion matrix ).
Solution
e A t = I + A t + A 2 t 2 2 ! + A 3 t 3 3 ! + ⋯ , d d t e A t = A e A t e^{At} = I + At + \frac{A^2 t^2}{2!} + \frac{A^3 t^3}{3!} + \cdots, \qquad \frac{d}{dt} e^{At} = A e^{At} e A t = I + A t + 2 ! A 2 t 2 + 3 ! A 3 t 3 + ⋯ , d t d e A t = A e A t The state transition matrix is Φ ( t ) = e A t = L − 1 { ( s I − A ) − 1 } \Phi(t) = e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\} Φ ( t ) = e A t = L − 1 {( s I − A ) − 1 }
x ( t ) = e A t x ( 0 ) + ∫ 0 t e A ( t − τ ) B u ( τ ) d τ , y ( t ) = C x ( t ) + D u ( t ) x(t) = e^{At}x(0) + \int_0^t e^{A(t - \tau)} B u(\tau)\, d\tau, \qquad y(t) = Cx(t) + Du(t) x ( t ) = e A t x ( 0 ) + ∫ 0 t e A ( t − τ ) B u ( τ ) d τ , y ( t ) = C x ( t ) + D u ( t ) Computing the Matrix Exponential
Diagonalizable A If A v i = λ i v i Av_i = \lambda_i v_i A v i = λ i v i T = [ v 1 ∣ ⋯ ∣ v n ] T = [v_1 | \cdots | v_n] T = [ v 1 ∣ ⋯ ∣ v n ] A T = T Λ AT = T\Lambda A T = T Λ
A = T Λ T − 1 , A k = T Λ k T − 1 , e A t = T e Λ t T − 1 , e Λ t = d i a g ( e λ 1 t , … , e λ n t ) A = T\Lambda T^{-1}, \qquad A^k = T\Lambda^k T^{-1}, \qquad e^{At} = T e^{\Lambda t} T^{-1}, \qquad e^{\Lambda t} = \mathrm{diag}(e^{\lambda_1 t}, \ldots, e^{\lambda_n t}) A = T Λ T − 1 , A k = T Λ k T − 1 , e A t = T e Λ t T − 1 , e Λ t = diag ( e λ 1 t , … , e λ n t ) Defective A If an eigenvalue has algebraic multiplicity larger than geometric multiplicity, use chains of generalized eigenvectors, ( A − λ i I ) v i , 2 = v i , 1 (A - \lambda_i I) v_{i,2} = v_{i,1} ( A − λ i I ) v i , 2 = v i , 1 T = [ v 1 , 1 ∣ v 1 , 2 ∣ ⋯ ] T = [v_{1,1} | v_{1,2} | \cdots] T = [ v 1 , 1 ∣ v 1 , 2 ∣ ⋯ ] A = T J T − 1 A = TJT^{-1} A = T J T − 1 J J J Jordan form .
Complex eigenvalues For real A ∈ R 2 × 2 A \in \mathbb{R}^{2 \times 2} A ∈ R 2 × 2 λ = σ ± j ω \lambda = \sigma \pm j\omega λ = σ ± jω v = R e ( v ) + j I m ( v ) v = \mathrm{Re}(v) + j\,\mathrm{Im}(v) v = Re ( v ) + j Im ( v ) A v = λ v Av = \lambda v A v = λ v
A [ R e ( v ) I m ( v ) ] = [ R e ( v ) I m ( v ) ] [ σ ω − ω σ ] A\begin{bmatrix} \mathrm{Re}(v) & \mathrm{Im}(v) \end{bmatrix} = \begin{bmatrix} \mathrm{Re}(v) & \mathrm{Im}(v) \end{bmatrix} \begin{bmatrix} \sigma & \omega \\ -\omega & \sigma \end{bmatrix} A [ Re ( v ) Im ( v ) ] = [ Re ( v ) Im ( v ) ] [ σ − ω ω σ ] so with T = [ R e ( v ) ∣ I m ( v ) ] T = [\mathrm{Re}(v) \,|\, \mathrm{Im}(v)] T = [ Re ( v ) ∣ Im ( v )]
A = T Λ T − 1 , Λ = [ σ ω − ω σ ] , e Λ t = e σ t [ cos ω t sin ω t − sin ω t cos ω t ] A = T\Lambda T^{-1}, \qquad \Lambda = \begin{bmatrix} \sigma & \omega \\ -\omega & \sigma \end{bmatrix}, \qquad e^{\Lambda t} = e^{\sigma t} \begin{bmatrix} \cos \omega t & \sin \omega t \\ -\sin \omega t & \cos \omega t \end{bmatrix} A = T Λ T − 1 , Λ = [ σ − ω ω σ ] , e Λ t = e σ t [ cos ω t − sin ω t sin ω t cos ω t ]