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MEE5114 Advanced Control for Robotics
Lecture 1: Linear Differential Equations and
Matrix Exponential
Prof. Wei Zhang
SUSTech Insitute of Robotics
Department of Mechanical and Energy Engineering
Southern University of Science and Technology, Shenzhen, China
Advanced Control for Robotics Wei Zhang (SUSTech) 1 / 19
Outline
• Linear System Model
• Matrix Exponential
• Solution to Linear Differential Equations
Outline Advanced Control for Robotics Wei Zhang (SUSTech) 2 / 19
Linear Differential Equations (Autonomous)
Linear Differential Equations: ODEs that are linear wrt variables
e.g.:
x ˙
1
(t) + x
2
(t) = 0
x˙
2
(t) + x
1
(t) + x
2
(t) = 0
y ¨(t) + z(t) = 0
z˙(t) + y(t) = 0
State-space form (1st-order ODE with vector variables):
Linear Systems Advanced Control for Robotics Wei Zhang (SUSTech) 4 / 19
General Linear Control Systems
General (Autonomous) Dynamical Systems: x˙(t) = f (x(t))
- x(t) ∈ R
n
: state vector, f : R
n
→ R
n
: vector field
Non-autonomous: x˙(t) = f (x(t), t)
Control Systems: x˙(t) = f (x(t), u(t))
- vector field f : R
n
× R
m
depends on external variable u(t) ∈ R
m
General Linear Control Systems:
x ˙(t) = Ax(t) + Bu(t), with x(0) = x
0
y(t) = Cx(t) + Du(t)
- x ∈ R
n
: system state, u ∈ R
m
: control input, y ∈ R
p
: system output
- A, B, C, D are constant matrices with appropriate dimensions
Linear Systems Advanced Control for Robotics Wei Zhang (SUSTech) 5 / 19
Existence and Uniqueness of Linear Systems
Corollary: Linear system
x ˙(t) = Ax(t) + Bu(t)
has a unique solution for any piecewise continuous input u(t)
Homework: Suppose A becomes time-varying A(t), can you derive conditions
to ensure existence and uniqueness of x˙(t) = A(t)x(t) + Bu(t)?
Linear Systems Advanced Control for Robotics Wei Zhang (SUSTech) 7 / 19
Outline
• Linear System Model
• Matrix Exponential
• Solution to Linear Differential Equations
Matrix Exponential Advanced Control for Robotics Wei Zhang (SUSTech) 8 / 19
What is the ”Euler’s Number” e?
Consider a scalar linear system: z(t) ∈ R and a ∈ R is a constant
z ˙(t) = az(t), with initial condition z(0) = z
The above ODE has a unique solution:
What is the number “e”?
Matrix Exponential Advanced Control for Robotics Wei Zhang (SUSTech) 10 / 19
Complex Exponential
For real variable x ∈ R, Taylor series expansion for e
x
around x = 0:
e
x
X
k=
x
k
k!
= 1 + x +
x
x
This can be extended to complex variables:
e
z
X
k=
z
k
k!
= 1 + z +
z
z
This power series is well defined for all z ∈ C
In particular, we have e
jθ
= 1 + jθ −
θ
2
− j
θ
3
Comparing with Taylor expansions for cos(θ) and sin(θ) leads to the Euler’s
Formula
Matrix Exponential Advanced Control for Robotics Wei Zhang (SUSTech) 11 / 19
Some Important Properties of Matrix Exponential
Ae
A
= e
A
A
e
A
e
B
= e
A+B
if AB = BA
If A = P DP
, then e
A
= P e
D
P
For every t, τ ∈ R, e
At
e
Aτ
= e
A(t+τ )
e
A
= e
−A
Matrix Exponential Advanced Control for Robotics Wei Zhang (SUSTech) 13 / 19
Outline
• Linear System Model
• Matrix Exponential
• Solution to Linear Differential Equations
Linear Systems Solution Advanced Control for Robotics Wei Zhang (SUSTech) 14 / 19
Computation of Matrix Exponential (1/2)
Directly from definition
For diagonalizable matrix:
Linear Systems Solution Advanced Control for Robotics Wei Zhang (SUSTech) 16 / 19
Computation of Matrix Exponential (2/2)
Using Laplace transform
Linear Systems Solution Advanced Control for Robotics Wei Zhang (SUSTech) 17 / 19
More Discussions
Linear Systems Solution Advanced Control for Robotics Wei Zhang (SUSTech) 19 / 19
