Linear Time-Invariant Systems

Linear time-invariant (LTI) systems form the foundation of modern control theory and optimal control. This document covers the mathematical representation, solution methods, and key properties of LTI systems.

State-Space Representation

The standard form of a linear time-invariant system is expressed as:

State equation:

$$\dot{x}(t) = Ax(t) + Bu(t)$$

Output equation:

$$y(t) = Cx(t) + Du(t)$$

Where:

• $x(t) \in \mathbb{R}^n$ is the state vector
• $u(t) \in \mathbb{R}^m$ is the input vector
• $y(t) \in \mathbb{R}^p$ is the output vector
• $A \in \mathbb{R}^{n \times n}$ is the system matrix
• $B \in \mathbb{R}^{n \times m}$ is the input matrix
• $C \in \mathbb{R}^{p \times n}$ is the output matrix
• $D \in \mathbb{R}^{p \times m}$ is the feedthrough matrix

Matrix Exponential

Scalar Case

For a scalar differential equation:

$$\dot{x} = ax \implies x(t) = x(0)e^{at}$$

where $a$ is a scalar constant.

Matrix Case

For the matrix differential equation:

$$\dot{x} = Ax \implies x(t) = x(0)e^{At}$$

where $A$ is a matrix and $e^{At}$ is the matrix exponential.

Taylor Series Expansion

The matrix exponential is defined using the Taylor series:

Scalar exponential:

$$e^{at} = 1 + at + \frac{1}{2!}(at)^2 + \frac{1}{3!}(at)^3 + \cdots$$

Matrix exponential:

$$e^{At} = I + At + \frac{1}{2!}(At)^2 + \frac{1}{3!}(At)^3 + \cdots$$

Derivative of Matrix Exponential

The derivative of the matrix exponential is:

$$\frac{d}{dt}e^{At} = 0 + A + \frac{2}{2!}A^2t + \frac{3}{3!}A^3t^2 + \cdots = Ae^{At}$$

Properties of Matrix Exponential

• $e^{A \cdot 0} = I$ (identity matrix)
• $\frac{d}{dt}e^{At} = Ae^{At} = e^{At}A$
• $(e^{At})^{-1} = e^{-At}$
• $e^{A(t_1+t_2)} = e^{At_1}e^{At_2}$ (when $A$ commutes with itself)

Solution of State-Space Equations

Method 1: Laplace Transform Approach

Applying the Laplace transform to the state equation:

$$sX(s) - x(0) = AX(s) + BU(s)$$

Rearranging:

$$(sI - A)X(s) = x(0) + BU(s)$$

Solving for $X(s)$:

$$X(s) = (sI - A)^{-1}x(0) + (sI - A)^{-1}BU(s)$$

Taking the inverse Laplace transform:

$$x(t) = \mathcal{L}^{-1}[X(s)] = e^{At}x(0) + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau$$

where:

• $e^{At} = \mathcal{L}^{-1}[(sI-A)^{-1}]$ (state transition matrix)
• The convolution integral represents the forced response

Limitation

The Laplace transform method assumes zero initial time ($t_0 = 0$) due to the differential properties of the Laplace transform.

Method 2: Direct Integration

Multiply both sides of the state equation by $e^{-At}$:

$$e^{-At}\frac{d}{dt}x(t) = e^{-At}Ax(t) + e^{-At}Bu(t)$$

Rearranging:

$$e^{-At}\frac{d}{dt}x(t) - e^{-At}Ax(t) = e^{-At}Bu(t)$$

Using the product rule, the left side becomes:

$$\frac{d}{dt}(e^{-At}x(t)) = e^{-At}Bu(t)$$

Integrating from $t_0$ to $t$:

$$\int_{t_0}^t \frac{d}{d\tau}(e^{-A\tau}x(\tau))d\tau = \int_{t_0}^t e^{-A\tau}Bu(\tau)d\tau$$

This gives:

$$e^{-At}x(t) - e^{-At_0}x(t_0) = \int_{t_0}^t e^{-A\tau}Bu(\tau)d\tau$$

Final solution:

$$x(t) = e^{A(t-t_0)}x(t_0) + \int_{t_0}^t e^{A(t-\tau)}Bu(\tau)d\tau$$

Solution Components

The general solution consists of two parts:

1. Zero-input response (homogeneous solution):

   $$x_{zi}(t) = e^{A(t-t_0)}x(t_0)$$

2. Zero-state response (particular solution):

   $$x_{zs}(t) = \int_{t_0}^t e^{A(t-\tau)}Bu(\tau)d\tau$$

State Transition Matrix

The matrix $\Phi(t,t_0) = e^{A(t-t_0)}$ is called the state transition matrix and represents how the state evolves from time $t_0$ to time $t$ in the absence of inputs.

Key Properties

Stability

The system is asymptotically stable if and only if all eigenvalues of matrix $A$ have negative real parts.

Controllability

The system is completely controllable if the controllability matrix:

$$\mathcal{C} = [B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B]$$

has full rank $n$.

Observability

The system is completely observable if the observability matrix:

$$\mathcal{O} = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}$$

has full rank $n$.

Applications in Optimal Control

LTI systems are fundamental to optimal control because:

1. Linear Quadratic Regulator (LQR) problems are naturally formulated for LTI systems
2. Model Predictive Control (MPC) often uses linearized models
3. Dynamic programming solutions have closed-form expressions for LTI systems
4. Kalman filtering is optimal for LTI systems with Gaussian noise

References

1. Wang, T., & Huang, J. (2023). 控制之美（卷2）—最优化控制MPC与卡尔曼滤波器. Tsinghua University Press.
2. Kawada, M. MATLAB/Simulinkによる制御工学入門. Morikita Publishing.
3. Franklin, G. F., Powell, J. D., & Emami-Naeini, A. Feedback Control of Dynamic Systems (7th ed.).