控制系统建模

A system can be described as one of the followings:

• Differential equation
• Difference Equation
• Transfer Function
• State Space

Differential Equation

$$\begin{aligned} \ddot{x} + \dot{x} &= ax + b \\\\ y &= x \end{aligned}$$

Difference Equation

$$x_{t} = ax_{t-1} + b$$

Transfer Function

• Polynomial $$Y(s) = \frac{a_m s^m + ... + a_0}{b_n s^n + ... + b_0}$$

• Poles and Zeros $$Y(s) = K \frac{(s-z_m)...(s-z_0)}{(s - p_n) ... (s-p_0)}$$

State Space

$$\begin{align} \dot{x}(t) &= Ax(t) + Bu(t) \\\\ y(t) &= Cx(t) + Du(t) \end{align}$$

• x as the state vector,
• A as the system matrix (square, N x N for N states),
• B as the input matrix (N rows x 1 column for a single-input, single output (SISO) system,
• C as the output matrix (one row x N columns for a SISO system),

• D as the feedforward matrix (1 x 1 for a SISO system).

• (*) The poles of the transfer function are the eigenvalues of the system matrix A

Matlab Code

sys = ss(a,b,c,d)

sys = ss(a,b,c,d,Ts)

sys_ss = ss(sys)

Matlab functions

Transfer Function

s = tf('s')

feedback(G(s), H(s))

G = zpk(sys)

G = zpk([1],[1],[1])

Poles and Zeros

Find poles of a SISO or MIMO system: pole(sys)

pzplot(sys)

State Space

System Analysis

linearSystemAnalyzer(G,T1,T2)

step(sys)

Reference

1. Control Tutorials, Inverted Pendulum: Digital Controller Design, University of Michigan