CRVCONTROL OF ROBOT AND VIBRATION LABORATORY

Transformation of a Mismatched Nonlinear

Dynamic Systems into Strict Feedback Form

December 1, 2012

Speaker: Ittidej Moonmangmee3rd years of PhD studentLecturer at STOU

by Johanna L. Mathieu and J. Karl HedrickDepartment of Mechanical Engineering, University

of California, Berkeley, USA

Journal of Dynamic Systems, Measurement and Control,Transactions of the ASMEVol. 133, July 2011, Q2

Johanna L. Mathieu2012, PostDoc at EEH – Power Systems Laboratory, ETH Zurich 2006 – 2012, MS/PhD Student at the University of California, Berkeley, USA2006 – 2012, Affiliate at the Lawrence Berkeley National Laboratory, Berkeley, California, USA2008, Visiting researcher at the Bangladesh University of Engineering and Technology Department of Civil Engineering, Dhaka, Bangladesh2005, Research Assistant at the MIT Sea Grant College Program, Cambridge, Massachusetts, USA2004 – 2005, U.S. Peace Corps Volunteer, Tanzania2000 – 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.

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

Outline4/18

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

3. Simulate and compare two controllers performance

desired trajectory

forward velocity of the bicycle

steering angular velocityof the handle bars

5/18

Dynamic System Description6/18

MIMO System

Two inputs:

u1 forward velocity of the bicycle

u2 angular velocity of the handle bars

Two outputs:

heading angle

steering angle

Controllability7/18

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

#Relative degree = #State = 6

So, it has no zero dynamics

Minimum-phase

Control using Feedback LinearizationDynamic Extension: See [Sastry’s Nonlinear Systems, 1999]

8/18

Control using Feedback Linearization9/18

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)

10/18

Transformation into Strict Feedback Form11/18

Dynamic Surface Control (DSC)12/18

Dynamic Surface Control (DSC)13/18

Simulation and Results

x 1 e

rror x1 position error

x 2 e

rror

x2 position error

t

14/18

MATLAB ode45

Disturbances: w1 = 0.10 + 0.02r1(t) w2 = 0.15 + 0.02r2(t) w3 = 0.20 + 0.02r3(t) w4 = 0.10 + 0.02r4(t)where r i(t) (0, 1)

Uncertainty

bounds: δ1, δ2, δ4 = 0.2,

δ3 = 0.25 and δ5, δ6, δ7, δ8 are

change with the function of the state.

Simulation and Resultsu 2 (

rad/

s)u 1 (

rad/

s)u 4 (

rad/

s)

t

t

t

15/18

Control saturated: -10 to +10

Controller gains: k = [10, 10, 1, 1, 10, 10]

Filter Time Constant: τ = [0.05, 0.05, 0.05, 0.05]

From the dynamic extension:

Simulation and Results

Sliding surfaces for x2Sliding surfaces for x1

16/18

Concluding Remarks17/18

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

CRV

CONTROL OF ROBOT AND VIBRATION LABORATORY

Thank youPlease comments and suggests!