Seminar2012 d

Transformation of a Mismatched
Nonlinear Dynamic Systems into
Strict Feedback Form
by Johanna L. Mathieu and J. Karl Hedrick
Department of Mechanical Engineering, University of
California, Berkeley, USA

Journal of Dynamic Systems, Measurement and Control,
Transactions of the ASME
Vol. 133, July 2011, Q2
CONTROL OF ROBOT AND VIBRATION LABORATORY
Speaker: Ittidej Moonmangmee
3rd years of PhD student
Lecturer at STOU
December 1, 2012

Johanna L. Mathieu
2012, PostDoc at EEH – Power Systems Laboratory, ETH Zurich
2006 – 2012, MS/PhD Student at the University of California,
Berkeley, USA
2006 – 2012, Affiliate at the Lawrence Berkeley National
Laboratory, Berkeley, California, USA
2008, Visiting researcher at the Bangladesh University of
Engineering and Technology Department of Civil Engineering,
Dhaka, Bangladesh
2005, Research Assistant at the MIT Sea Grant College
Program, Cambridge, Massachusetts, USA
2004 – 2005, U.S. Peace Corps Volunteer, Tanzania
2000 – 2004, BS Student at the Massachusetts Institute of
Technology, Cambridge, Massachusetts, USA

J. Karl Hedrick (born 1944) is an American control
theorist and a Professor in the Department of
Mechanical Engineering at the University of California,
Berkeley. He has made seminal contributions in
nonlinear control and estimation. Prior to joining the
faculty at the University of California, Berkeley he was a
professor at the Massachusetts Institute of Technology
from 1974 to 1988. Hedrick received a bachelor's degree in
Engineering Mechanics from the University of Michigan
(1966) and a M.S. and Ph.D from Stanford University
(1970, 1971). Hedrick is the head of the Vehicle Dynamics
Laboratory at UC Berkeley.
In 2006, he was awarded the Rufus Oldenburger Medal
from the American Society of Mechanical Engineers.

4/18
Outline

1. Objective
2. Dynamic System Description & Controllability
 A bicycle example
3. Control Using Feedback Linearization
4. Dynamic Surface Control (DSC)
 Transformation into Strict Feedback Form
 Sliding Surface & Control law
5. Simulation & Results
6. Conclusions

5/18
Objective

1. Transform a mismatched nonlinear system into a strict
feedback form (also with a mismatched)
2. Design two controllers via
(i) Feedback Linearization method
(ii) Dynamic Surface Control method
to the bicycle tracks a desired trajectory
steering angular velocity
of the handle bars

desired trajectory

forward velocity of the bicycle

3. Simulate and compare two controllers performance

6/18
Dynamic System Description
steering angle

heading angle

MIMO System

Two inputs:
u1 forward velocity of the bicycle
u2 angular velocity of the handle bars
Two outputs:

7/18
Controllability

See [Daizhan, C., Xiaoming, H, and Tielong, S., Analysis
and Design of Nonlinear Control Systems, 2010]

8/18
Control using Feedback Linearization
Dynamic Extension:
See [Sastry’s Nonlinear
Systems, 1999]

#Relative degree = #State = 6
So, it has no zero dynamics
 Minimum-phase

9/18
Control using Feedback Linearization

10/18
Transformation into Strict Feedback Form

Goal:
Extended state equation  Strict feedback form
(available for Dynamic Surface Control (DSC) design)

Design a controller by
Dynamic Surface Control (DSC)

11/18
Transformation into Strict Feedback Form

12/18

13/18

14/18
Simulation and Results

x1 error
x1 position error

x2 error
x2 position error

t

MATLAB ode45
Disturbances: Uncertainty bounds:
w1 = 0.10 + 0.02r1(t) δ1, δ2, δ4 = 0.2,
w2 = 0.15 + 0.02r2(t) δ3 = 0.25
w3 = 0.20 + 0.02r3(t) and δ5, δ6, δ7, δ8 are
w4 = 0.10 + 0.02r4(t) change with the
where r i(t)  1)
(0, function of the state.

15/18
Control saturated:
-10 to +10
u2 (rad/s)

Controller gains:
k = [10, 10, 1, 1, 10, 10]
Filter Time Constant:
t τ = [0.05, 0.05, 0.05, 0.05]
u4 (rad/s)

From the dynamic extension:

t
u1 (rad/s)

t

16/18

Sliding surfaces for x1 Sliding surfaces for x2

17/18
Concluding Remarks

 A new method of defining states was presented for transform
a nonlinear mismatched system to the strict feedback form
 Two controller techniques were designed
 Feedback linearization (FL) with dynamic extension
 Dynamic surface control (DSC)
 In the disturbance-free case,
both FL & DSC performed tracking a desired trajectory
 In the present of disturbances,
the DSC was better to reject it than the FL
 Tracking performance of the DSC can be designed
by using the 1st order filter
 However, more control effort required for DSC

Thank you
Please comments and suggests!

CONTROL OF ROBOT AND VIBRATION LABORATORY

Seminar2012 d

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