1. Nonlinear Control Systems
Lecturer: Teng-Hu Cheng
SCHEDULE: WEEK1, 11 Sept., 2017

2. oUTLINES Brief review of Linear Systems Phase Portraits
Why Nonlinear Control Systems?
Autonomous and Non-Autonomous Systems

3. Examples 𝑥 1 =− 𝑥 1 +5 𝑥 1 =− 𝑥 2 +5 𝑥 2 = 𝑥 1 𝑥 1 =− 𝑥 2 +5 𝑥 2 = 𝑥 1
Brief Review of Linear Syst.
Why Nonlinear Control Systems?
Phase Portraits
Auto./Non-Auto. Systems
Examples
𝑥 1 =− 𝑥 1 +5
𝑥 1 =− 𝑥 2 +5
𝑥 2 = 𝑥 1
𝑥 1 =− 𝑥 2 +5
𝑥 2 = 𝑥 1
𝑥 1 =− 𝑥 − 𝑥 1 −5 𝑢
𝑥 2 = 𝑥 1
𝑥 1 =− 𝑥 2 − 𝑥 1
𝑥 2 = 𝑥 1

4. LTI v.s. LTV LTI LTV 𝑥 =𝐴𝑥+𝐵𝑢, where 𝐴 and 𝐵 are time-invariant
Brief Review of Linear Syst.
Why Nonlinear Control Systems?
Phase Portraits
Auto./Non-Auto. Systems
LTI v.s. LTV
LTI
LTV
𝑥 =𝐴𝑥+𝐵𝑢, where 𝐴 and 𝐵 are time-invariant
e.g., 𝑥 1 =− 𝑥 1 +u
𝑥 =𝐴(𝑡)𝑥+𝐵(𝑡)𝑢, where 𝐴(𝑡) and 𝐵(𝑡) are time-varying
e.g., 𝑥 1 =− 𝑡 2 𝑥 1 +tu

5. Why Nonlinear Control Systems?
Brief Review of Linear Syst.
Why Nonlinear Control Systems?
Phase Portraits
Auto./Non-Auto. Systems
Why Nonlinear Control Systems?
Equilibrium points
Local/global properties

6. Linear VS NONLINEAR SYSTEMS
Brief Review of Linear Syst.
Why Nonlinear Control Systems?
Phase Portraits
Auto./Non-Auto. Systems
Linear VS NONLINEAR SYSTEMS
𝑥 1 =− 𝑥 2 − 𝑥 1
𝑥 2 = 𝑥 1
𝑥 1 = 𝑥 2
𝑥 2 =−sin⁡( 𝑥 1 )
(e.g., undamped pendulum)

7. Autonomous Systems (could be linear or nonlinear systems)
Brief Review of Linear Syst.
Why Nonlinear Control Systems?
Phase Portraits
Auto./Non-Auto. Systems
Autonomous Systems (could be linear or nonlinear systems)
Definition: A system that does NOT explicitly depend on time
Non-autonomous Systems
Definition: A system that explicitly depends on time
𝑥 =𝑓(𝑥)
Ex:
𝑥 =−𝑥,
𝑥 =− 𝑥 2
𝑥 =−cos⁡(𝑥)
𝑥 =𝑓(𝑥(𝑡))
Ex:
𝑥 =−𝑥+t,
𝑥 =−sin⁡(𝑡) 𝑥 2
𝑥 =−cos⁡(𝑡)cos⁡(𝑥)