1. Nonlinear Control Via Approximation Input-Output Linearization: The Ball and Beam Example
A paper1 by John Hauser, Shankar Sastry, and Petar Kokotovic
Presented by Christopher K. Johnson
MEAM 613
25 April 2002
1IEEE Transactions on Automatic Control, Vol. 37, No. 3, March 1992

2. Introduction
Presentation of a method for constructing approximate systems for nonlinear systems that do not have a well-defined relative degree. The approximate system can be used to control the original nonlinear system.
Outline:
Ball and Beam example
Approximate I/O linearization of the Ball and Beam
Presentation of simulation results
Analysis of Approximate Linearization

3. System Dynamics Ball and Beam r   Define states Define new input
(invertible)

4. Standard Input-Output Linearization
Usual Procedure for Linearization
Relative degree is undefined when r = 0 or d/dt = 0
b(x)
a(x)u

5. Approximate I/O Linearization
Achieve approximate I/O linearization by choosing a control law to exactly linearize a system that is close to the true N.L. system
Change coordinates =(x) and choose input to make original system look like the approximate system perturbed by higher order terms():
1/s 4 3 2 1 v y
small N.L. terms ()

6. Approximation 1 (modification of f)
 (ignore)
Neglect higher-order centrifugal force term
Well-defined relative degree
Choice of what to neglect determines .

7. SIMULINK® Simulation Model

8. Simulation Results: Approx #1
Small tracking error (increases nonlinearly with A)
0.2 rad = 11.5 deg
Successful tracking control

9. Approximation 2 (modification of g)
 (ignore)
Only neglect terms that prevent a well-defined relative deg.
Well-defined relative degree of 4

10. Simulation Results: Approx #2
Smaller tracking errors than Approx #1
0.2 rad = 11.5 deg
Very successful tracking control

11. Approximation 3 (Jacobian)
 (ignore)
b(x)+a(x)
Neglect terms higher order than O(x) or O(). (Standard “Jacobian” linearization). Note , not u.
Purpose is for a baseline comparison.
Well-defined relative degree of 4

12. Simulation Results: Jacobian Approx
Large Error for A = 1,2
Unstable for A = 3
Relatively poor control scheme
Larger scale than previous graphs

13. Maximum Error e=(yd-x1)
Approx #1
Approx #2
Approx #3
(Jacobian) 1
3.1 x 10-4
1.3 x 10-4
2.6 x 10-2 2
2.5 x 10-3
1.0 x 10-3
4.6 x 10-1 3
8.4 x 10-3
3.6 x 10-3
Unstable 6
7.0 x 10-2
3.3 x 10-2

14. Analysis of Approximate Linearization
For systems where the relative degree fails to exist:
Choose function to approximate the output.
O(x)2
Differentiate along trajectories
All O(x) and higher from Lg and (if desired) any O(x)2 or higher from Lf
Stop when Lg term is O(1) a(x) is O(1).
ROBUST RELATIVE DEGREE 

15. Analysis of Approximate Linearization
Nonlinear System:
Linearized System: (x=u=0)
Characterize the robust relative degree:
Thm 4.1: The robust relative degree of the nonlinear system is equal to the relative degree of the Jacobian linearized system whenever either is defined.
Corollary 4.2: The robust relative degree of a nonlinear system is invariant under a state-dependent change of control coordinates of the form u(x,v) = a(x) + b(x)v [a,b smooth and a(0)=0 and b(0)<>0].

16. Analysis of Approximate Linearization
Show that neglecting i produces an approximation to the true system
fi(.) can be used as part of a local change of coordinates
Frobenius Thm, we can complete the N.L change of coordinates with a set of functions hi(x), i = 1,…,n-g s.t. Lghi(x)=0
New Coordinates
Normal Form + Perturbations I of O(x,u)2

17. Analysis of Approximate Linearization
This choice of u will linearize the nonlinear system from v to y up to terms O(x,u)2 [Byrnes and Isidori].
If the relative degree = n, then the system is approximately full-state linearizable.
Previous work by Krener shows that with the satisfaction of a controllability condition and with D order r involutive, there exists an output function h(x) w.r.t which the system has robust relative degree n and such that the remainder functions are O(x,u)r.
This paper is different in that it constructs an approximate system.
For the ball and beam, involutivity is satisfied with r=3 and the system can be input-output and full-state linearized up to terms of order 3.

18. Analysis of Approximate Linearization
Choice of terms included from Lf can be used to improve the approximation. No choice with terms from Lg.
We cannot guarantee that we can get an approximation to an arbitrary degree.
“In specific applications, [this method] may produce better approximations than the Jacobian approximation. Furthermore, the resulting approximations may be valid on larger domains than the Jacobian approximation.”
The following slides make this notion more precise…

19. Analysis of Approximate Linearization
Remark: Since the largest ball that fits in Uε is Bε, the set Uε must get smaller in at least one direction as ε is decreased
The functions ψi(x,u) that are omitted in the approximation are O(x,u)2 in the neighborhood of the origin. To extend the approx to higher regions, the following definition is used:
Remark: If ψ(x,u) is uniformly higher order on Uε x Bε, then it is O(x,u)2

20. Analysis of Approximate Linearization
If our approximate system is exponentially minimum phase and the (ignored) ψi terms are uniformly higher order on Uε x Bε, we use the stable tracking control law for the approximate system (with Hurwitz alphas):

21. Analysis of Approximate Linearization
Proof of Theorem:
(Outline)
Remarks:
The actual restriction on the class of trajectories that can be tracked is related to how large the functions ψi are when the approximate state ξi is close to the desired trajectory/derivative y(i)
In certain cases where the ψi functions depend only on the derivative of the output, the main restriction is on the derivatives of the desired trajectory rather than it value