Solving LTI Systems

Solving LTI Systems

The closed-form solution of a continuous-time LTI system $\dot{x}(t)=Ax(t)+Bu(t)$ with initial condition $x(0)$ is
$x(t)=\underset{\text{homogeneous}}{\underbrace{e^{At}x(0)}}+\underset{\text{particular (convolution)}}{\underbrace{{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau }}.$
The first term is the homogeneous or zero-input response — how the system evolves from initial conditions alone. The second is the particular or zero-state response — the convolution of the input with the system’s impulse response. The matrix $e^{At}$ is the state transition matrix, defined by its power series
$e^{At}=I+At+\frac{(At{)}^2}{2!}+\frac{(At{)}^3}{3!}+\cdots ,$
which converges for all $t$ because $A$ is bounded.

If $A$ has a full set of eigenvectors (i.e., is diagonalizable), $A=V\Lambda V^{-1}$ with $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, then
$e^{At}=V{e}^{\Lambda t}{V}^{-1},e^{\Lambda t}=\mathrm{diag}(e^{{\lambda }_{1}t},\dots ,e^{{\lambda }_{n}t}).$
This eigendecomposition is the standard computational tool: instead of summing the power series, diagonalize $A$, exponentiate the eigenvalues, and transform back. It also exposes the system’s modes — the components $e^{{\lambda }_{i}t}{v}_i$, each evolving independently along an eigenvector direction.

In discrete time, $x[i+1]=Ax[i]+Bu[i]$ has solution
$x[i]=A^{i}x[0]+{\sum }_{k=0}^{i-1}{A}^{i-1-k}Bu[k],$
with natural response $A^i=V{\Lambda }^i{V}^{-1}$. Continuous-time natural responses are linear combinations of exponentials $e^{{\lambda }_{j}t}$; discrete-time responses are linear combinations of powers ${\lambda }_j^i$.

Non-Diagonalizable $A$ If $A$ has repeated eigenvalues without a full eigenvector set (i.e., Jordan blocks), the closed form involves $t^k{e}^{\lambda t}$ terms. The matrix exponential is still well-defined and computable, but the cleanly separable modes above do not apply. For physics problems with distinct resonances, $A$ is almost always diagonalizable over $\mathbb{C}$.

Intuition

The state transition matrix generalizes the scalar formula $x(t)=e^{at}x(0)$ for $\dot{x}=ax$. Replacing the scalar $a$ with a matrix $A$ produces $e^{At}$ — no longer a single number, but a matrix that evolves the entire state vector forward in time. Because $e^{At}$ is defined by the same power series as the scalar exponential, it inherits many of the same algebraic properties: $e^{A\cdot 0}=I$, $e^{A{t}_1}{e}^{A{t}_2}=e^{A(t_1+t_2)}$, $\frac{d}{dt}{e}^{At}=A{e}^{At}$.

Eigendecomposition reveals the system’s modes. Each eigenvalue ${\lambda }_i$ is the growth (or decay) rate of a mode, and each eigenvector $v_i$ is the state-space direction along which that mode evolves. A system’s trajectory is a linear combination of mode trajectories; the initial condition $x(0)=\sum c_i{v}_i$ determines which modes get excited. In the coordinates of the eigenbasis (i.e., $z=V^{-1}x$), the system diagonalizes into $n$ independent scalar equations ${\dot{z}}_i={\lambda }_i{z}_i$, each with the familiar exponential solution.

The convolution integral for the particular response is the linear-system analog of “add up the response to each impulse in the input, scaled by the input’s value at that time.” The kernel $e^{A(t-\tau )}B$ is the impulse response matrix: if the input is an instantaneous kick $u(\tau )=\delta (\tau )$ in a given direction, the system evolves from that kick via $e^{A(t-\tau )}B$. Integrating over all $\tau$ superposes the responses to the full input history — which works because the system is linear.

Properties

State Transition Matrix

(Definition) $e^{At}={\sum }_{k=0}^{\infty }\frac{(At{)}^k}{k!}$, convergent for all $t$.

The power series always converges because $\Vert (At{)}^k/k!\Vert \le (\Vert A\Vert t{)}^k/k!$, which is the Taylor series for $e^{\Vert A\Vert t}$. The series is the formal definition; practical computation uses eigendecomposition or specialized routines (Padé approximation, scaling and squaring).

(Semigroup property) $e^{A(t_1+t_2)}=e^{A{t}_1}{e}^{A{t}_2}$, and $e^{A\cdot 0}=I$.

These identities encode time-invariance: propagating the state forward by $t_1+t_2$ equals propagating by $t_1$ and then by $t_2$, and “no time” means “no change.” They require $A$ to be constant; for time-varying $A(t)$, the analogous property involves a time-ordered product.

(Derivative) $\frac{d}{dt}{e}^{At}=A{e}^{At}=e^{At}A$.

Differentiating the power series term-by-term gives this identity. The commutativity $A{e}^{At}=e^{At}A$ follows because $A$ commutes with its own powers. It does not hold for $e^{A+B}$ when $A$ and $B$ fail to commute — a common trap.

(Invertibility) $e^{At}$ is always invertible, with $(e^{At}{)}^{-1}=e^{-At}$.

LTI dynamics are time-reversible from a mathematical standpoint: given $x(t)$, you can always recover $x(0)$. Physical dissipation enters through the signs of eigenvalues (decaying modes), not through loss of invertibility.

Eigendecomposition

(Diagonalization) If $A$ has $n$ linearly independent eigenvectors, $A=V\Lambda V^{-1}$.

Columns of $V$ are the eigenvectors; $\Lambda$ is the diagonal matrix of eigenvalues. The condition “full eigenvector set” is generic — random matrices almost always satisfy it — but fails for specific matrices with repeated eigenvalues (e.g., Jordan blocks).

(Modal coordinates) In the eigenbasis $z=V^{-1}x$, the system decouples into $n$ independent scalar equations.

Substituting $x=Vz$ into $\dot{x}=Ax$ gives $\dot{z}=\Lambda z$, so ${\dot{z}}_i={\lambda }_i{z}_i$. Each mode evolves independently as $z_i(t)=e^{{\lambda }_{i}t}{z}_i(0)$. Transforming back, $x(t)=V\mathrm{diag}(e^{{\lambda }_{i}t})V^{-1}x(0)$.

(Mode excitation) Mode $i$ is excited by $x(0)$ in proportion to $(V^{-1}x(0){)}_i$.

Initial conditions aligned with a single eigenvector excite only that mode; random initial conditions excite all modes simultaneously. A stable system’s slow modes (smallest $|\mathrm{Re}(\lambda )|$) dominate the long-time response.

Real Systems with Complex Eigenvalues

(Complex conjugate pairs) Real $A$ with complex eigenvalues has them in conjugate pairs $\lambda ,\bar{\lambda }$ with conjugate eigenvectors.

The entries of $A$ are real, so the characteristic polynomial has real coefficients, so non-real roots come in pairs. The corresponding modes $e^{\lambda t}v$ and $e^{\bar{\lambda }t}\bar{v}$ combine to produce a real-valued oscillation $e^{\sigma t}(\alpha \cos \omega t+\beta \sin \omega t)$ where $\lambda =\sigma +i\omega$.

(Decay and oscillation) $\lambda =\sigma +i\omega$ produces $e^{\sigma t}\cos (\omega t+\varphi )$ — decay rate $\sigma$, oscillation frequency $\omega$.

$\sigma <0$ gives damped oscillation; $\sigma =0$ gives steady oscillation; $\sigma >0$ gives unstable growing oscillation. The scalar analog is the response of a second-order RLC circuit or a damped harmonic oscillator.

Discrete Time

(Power form) Discrete-time solution uses $A^i$ instead of $e^{At}$: $x[i]=A^{i}x[0]+{\sum }_{k=0}^{i-1}{A}^{i-1-k}Bu[k]$.

The convolution sum replaces the continuous integral. Modes evolve as ${\lambda }^i$ (powers) instead of $e^{\lambda t}$ (exponentials). Stability condition changes from "$\mathrm{Re}(\lambda )<0$" to "$|\lambda |<1$" — see 7.5. Stability.

Examples

Scalar First-Order System

Solve $\dot{x}=-2x+u$ with $x(0)=1$ and constant input $u(t)=1$.

Solution $A=-2$, $B=1$. $e^{At}=e^{-2t}$.

Scalar case:

Homogeneous: $e^{-2t}(1)=e^{-2t}$.

Particular: ${\int }_0^t{e}^{-2(t-\tau )}\cdot 1d\tau =\frac{1-e^{-2t}}{2}$.

Total: $x(t)=e^{-2t}+\frac{1-e^{-2t}}{2}=\frac{1}{2}+\frac{1}{2}{e}^{-2t}$. The state decays from $1$ toward the steady state $\frac{1}{2}$.

Undamped Harmonic Oscillator

Mass-spring system with $k/m=1$: $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$. Compute $e^{At}$ and the trajectory from $x(0)=(0,1{)}^T$ with $u=0$.

Solution ${\lambda }_{1,2}=\pm i$, eigenvectors $v_{1,2}=(1,\pm i{)}^T$.

Eigenvalues:

$V=\left[\begin{array}{cc}1& 1\\ i& -i\end{array}\right],\Lambda =\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right],V^{-1}=\frac{1}{2}\left[\begin{array}{cc}1& -i\\ 1& i\end{array}\right].$

$e^{At}=V{e}^{\Lambda t}{V}^{-1}=\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$ (after simplification using Euler’s formula).

Trajectory: $x(t)=e^{At}\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}\sin t\\ \cos t\end{array}\right]$. Position oscillates as $\sin t$, velocity as $\cos t$ — consistent with an undamped oscillator.

Stable System with Step Input

System $A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ with $x(0)=0$ and unit-step input $u(t)=1$ for $t\ge 0$. Find the steady-state response.

Solution $A$: characteristic polynomial ${\lambda }^2+3\lambda +2=(\lambda +1)(\lambda +2)$, so $\lambda =-1,-2$. Both have negative real parts, so the system is stable and the transient decays to zero.

Eigenvalues of

Steady state: ${\dot{x}}_{ss}=0$ gives $0=A{x}_{ss}+B$, so $x_{ss}=-A^{-1}B=\left[\begin{array}{c}1/2\\ 0\end{array}\right]$.

The full solution $x(t)={\int }_0^t{e}^{A(t-\tau )}Bd\tau$ can be computed via eigendecomposition, but the long-time behavior is simply $x_{ss}$.

See Also

• 7.1. State-Space Representation — the system being solved
• 7.4. Phase Portraits — visualizing the eigenvalue-eigenvector structure in 2D
• 7.5. Stability — when the homogeneous response decays vs grows
• eigenvalues and eigenvectors — the linear-algebra primitive behind modal decomposition