Nonlinear Systems Control
Using a Subset Stabilization Approach
Except where reference is made to the work of others, the work described in this dissertation
is my own or was done in collaboration with my advisory committee. This dissertation does
not include proprietary or classified information.
Adam T. Simmons
Certificate of Approval:
Jitendra K. Tugnait
Professor
Department of Electrical and Computer
Engineering
A. Scottedward Hodel, Chair
Associate Professor
Department of Electrical and Computer
Engineering
John Y. Hung
Associate Professor
Department of Electrical and Computer
Engineering
Stephen L. McFarland
Acting Dean
Graduate School
Nonlinear Systems Control
Using a Subset Stabilization Approach
Adam T. Simmons
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 11, 2006
Nonlinear Systems Control
Using a Subset Stabilization Approach
Adam T. Simmons
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at their expense. The author
reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Adam Terry Simmons was born March 10, 1978, the second child of D. Terry and Beth
Simmons. In his hometown, Rome, Georgia, he attended St. Mary?s Catholic School through
the third grade. From fourth grade until High School Graduation, he attended Pepperell Pub-
lic Schools located in Lindale, Georgia. At Pepperell, Adam graduated with Honors and an
Advanced Technical/Science degree.
After graduating from High School, he attended his grandfather?s alma mater, Mercer
University, a small Baptist University located in Macon Georgia. At Mercer he was a member
of Phi Delta Theta, placed on the Freshman Advisory Board and given membership to several
Honor Societies (i.e. Order of Omega, Phi Sigma Tau). After graduating from Mercer with a
B.S. in Engineering, he worked in a small software firm for 6 months, after which he decided to
attend Auburn University to continue his education.
iv
Dissertation Abstract
Nonlinear Systems Control
Using a Subset Stabilization Approach
Adam T. Simmons
Doctor of Philosophy, May 11, 2006
(M.S., Auburn University, 2003)
(B.S., Mercer University, 2001)
95 Typed Pages
Directed by A. Scottedward Hodel
Numerous problems in nonlinear control systems are difficult or impossible to solve with
commonly proposed methods of design. For such systems, we propose a two-loop control strat-
egy that we call the subset stabilization approach (SSA), that combines the strengths of non-
linear control design with the flexibility of robust control design. We present several control
system examples for which standard control system techniques either fail or perform poorly
with respect to regions of convergence. We then apply our proposed method to these systems
and compare results. These simulation examples establish that the SSA method provides a
relatively simple but powerful method for the design of nonlinear control systems. Further,
these observations are bolstered by a theoretical analysis that establishes the principles upon
with the success of the SSA is based.
v
Acknowledgments
I must first give my thanks to Jesus Christ for all that He has done and will continue to
do in my life.
My undying gratitude and love go to my parents, Terry and Beth Simmons for their
financial and emotional support throughout my life. Without their religious guidance and
continuous love, I would not be the person I am today.
I would like to thank my committee. They have been patient and helpful. I especially
would like to thank Dr. A. Scottedward Hodel, my graduate advisor and mentor; his vast
experience and knowledge, along with a truly devout religious character were instrumental in
my completion of this research. His understanding over the past 5 years has been limitless and
he has my profound gratitude. Dr. John Y. Hung also deserves my thanks due to his constant
push for me to approach problems in unique ways, and this research would not have existed
without the persistence of such a teacher, researcher and mentor.
The completion of my degree would have been infinitely more difficult without the prayers,
support and advice of my sister, Tracy, and her husband, Chris Small.
Finally, good friends provided much needed prayers, encouragement and accountability,
like Rob Daily, Matthew Hollimon, Kim Jackson, Joe Jacobs, Karen Miller, Jill Moore, Will &
Kim Travis and others.
This work was funded by a NASA Graduate Student Research Fellowship from NASA-
NGT8-52904.
vi
Style manual or journal used Journal of Approximation Theory (together with the style
known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file auphd.sty.
vii
Table of Contents
List of Figures x
1 Introduction 1
2 Nonlinear Control 2
2.1 Lyapunov Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Input-Output Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Input-State Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Ball On Beam Example 20
3.1 Review of the Linear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Lyapunov Control Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Input-Output Linearization Method . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Input-State Linearization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Underlying Issue: Poorly Defined Relative degree . . . . . . . . . . . . . . . . . . 26
3.6 Dynamic Analysis of the Ball On Beam System . . . . . . . . . . . . . . . . . . . 27
3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 New Subset Stabilization Approach (SSA) Control Design 33
4.1 Subset Stabilization Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Five Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.2 System 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.3 System 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.4 System 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.5 System 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Conclusions 55
5.1 Design Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 SSA for Nonlinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Future Work 57
6.1 Experimental Validation of the Ball-on-beam Control Effort . . . . . . . . . . . . 57
6.1.1 Subset Stabilization Approach to [21] . . . . . . . . . . . . . . . . . . . . 59
6.1.2 Comparison of State Equations and Control Solution . . . . . . . . . . . . 61
6.2 Analytical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
viii
Bibliography 64
Appendices 66
A Five Systems Examples,Code: 67
B Code for Five System Phase Portrait 76
C Ball On Beam Example Code/Diagrams 79
C.1 Block Diagram For Ball-on-beam Model . . . . . . . . . . . . . . . . . . . . . . . 79
C.2 S-Function for Nonlinear Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
C.3 Feedback M File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
ix
List of Figures
2.1 Inner loop of input-output linearizing control, and equivalent model . . . . . . . 8
2.2 Input-output relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Inner loop of input-state linearizing control, and equivalent model . . . . . . . . 11
2.4 Integrator Backstepping Example Block Diagram . . . . . . . . . . . . . . . . . . 17
2.5 Integrator Backstepping Example Block Diagram, Part 2 . . . . . . . . . . . . . . 18
3.1 Ball on Beam Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Comparison of a Modified System with the Original System . . . . . . . . . . . . 28
3.3 Block Diagram of Simplified ball-on-beam System . . . . . . . . . . . . . . . . . 28
3.4 Stability of x3 and x4 confirmed . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 x1 Response with nonlinear control, unl . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 x3 and x4 Responses with nonlinear control, unl . . . . . . . . . . . . . . . . . . . 31
3.7 x1 Response with unl and u = ?Kx control inputs . . . . . . . . . . . . . . . . . 32
3.8 x1 Response with unl and Hauser?s control inputs . . . . . . . . . . . . . . . . . . 32
4.1 Example System 1: State responses to LQR feedback and SSA control . . . . . . 38
4.2 Example System 2: State responses to LQR feedback and SSA control . . . . . . 40
4.3 Example System 3: State responses to LQR feedback and SSA control . . . . . . 43
4.4 Example System 4: State responses to LQR feedback and SSA control . . . . . . 45
4.5 Example 5 (ball-on-beam control): State responses to LQR feedback and SSA
control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6 Subset Stabilization Approach Block Diagram . . . . . . . . . . . . . . . . . . . . 49
x
4.7 Subspace stabilized system phase portrait. The integer parameter n in equation
(4.15) is varied from 0 (a) to 1 (b) to 2 (c). Observe that with increased values
of n the phase portrait tends to more closely approach the null subset manifold
x2 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.8 Region of Convergence, Comparison of SSA-based control and LQR control ef-
forts for System 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1 Uran and Jezernik Ball on Beam Apparatus . . . . . . . . . . . . . . . . . . . . . 58
6.2 Uran and Jezernik System Simulation with SSA control Solution . . . . . . . . . 61
xi
Chapter 1
Introduction
All engineering disciplines model the dynamics of complex systems through mathematical
equations. These models represent a system through a set of either linear or nonlinear dif-
ferential equations. Linear models approximate simple dynamic characteristics and are widely
used in undergraduate engineering classes. Conversely, nonlinear models approximate complex
dynamic characteristics. In either representation, using the most accurate model of a system
will provide the most complete understanding of that system; system understanding is the
cornerstone of control systems engineering.
Control of nonlinear systems has found increased usage within industrial electronics appli-
cations such as robotics, electric motors, and power electronics. Some of the most well-known
techniques include Lyapunov stability design [13], input-output linearization [11], input-state
linearization [2], and integrator backstepping [1]. For some nonlinear models these techniques
are either excessively difficult to apply or result in an infinite control effort at specific operating
points, i.e. singularities. These techniques are further limited by structural model requirements.
In this dissertation we present a brief overview of nonlinear control techniques, several
example control systems, and a new approach to nonlinear systems control. In Chapter 2 four
traditional nonlinear control techniques are reviewed: Lyapunov control design, input-output
linearization, input-state Linearization, and integrator backstepping. A well-known example,
the ball-on-beam system, is presented in Chapter 3. In Chapter 4, five other example systems
are used to present a new two-loop control strategy that we call the subset stabilization approach
(SSA). Finally, conclusions and future work are presented in Chapters 5 and 6, respectively.
1
Chapter 2
Nonlinear Control
In this chapter we provide background related to nonlinear systems and nonlinear systems
control. We present four common nonlinear control techniques: Lyapunov control design, input-
output linearization, input-state linearization, and integrator backstepping. Nonlinear control
techniques are typically used to achieve better closed loop system properties, e.g., region of
stability, performance, robustness, etc., than that achievable through the use of linear control
techniques. Linear control design techniques do not directly address the nonlinearities of a
plant, and good performance is restricted to a region about the nominal design condition.
Nonlinear techniques utilize more mathematical information about the modeled plant and,
therefore, provide a control solution with a larger region of asymptotic stability.
Our analysis will require the following definitions.
Definition 1 We say that a point ?x ? IRn is an equilibrium point of the system ?x = f(x) if
f(?x) = 0. Without loss of generality, we assume the equilibrium point ?x is at the origin.
Definition 2 We say that ?x is a stable (attractive) equilibrium point for ?x = f(?x) if, for each
small value of epsilon1 > 0 there is a ? = ?(epsilon1) > 0 such that ||?x|| < ? ? ||x(t)|| < epsilon1,?t ? 0. We
can further say ?x is asymptotically stable if it is stable and the value of ? is chosen such that
||?x||<? ? limt??x(t) = 0. If ?x is not stable, then ?x is an unstable equilibrium point.
Definition 3 A vector field is a construction in vector calculus that associates a vector with
every point in a Euclidean space.
2
Definition 4 Consider functions h: D ? IR and f: D ? IRn. The Lie Derivative of h with
respect to f at a point x?D is defined by
Lfh(x) = ?xhTf(x)
The Lie derivative is analogous to the derivative of h along the trajectories of the system
?x = f(x).
2.1 Lyapunov Control Design
We here consider the application of Lyapunov?s stability theory in the design of control laws
for nonlinear systems. Recently, Lyapunov control design methods have been used to explore
stabilizing control efforts for system with constrained inputs [7, 12] and time delayed systems
[13]. Lyapunov control methods have also been applied to a variety of complex mechanical
systems [14], systems with discontinuous dynamics [20], and aerospace applications [15].
Lyapunov?s stability theory is summarized concisely below.
Theorem 1 Let ?x = 0 be an equilibrium point of a system described by ?x = f(x) in a domain
D ? IRn containing the origin. Let V : D ? IR be a continuously differentiable function such
that
V(0) = 0 and V(x) > 0 in D?{0}
and
?V(x) ? 0 in D
Then ?x = 0 is a stable equilibrium point of the system ?x = f(x). Moreover, if
?V(x) < 0 in D?{0}
then ?x = 0 is an asymptotically stable equilibrium point of ?x = f(x).
3
A proof of Theorem 1 is given in [6].
Consider the system ?x = f(x,u) where x represents the states and u is a control input
and suppose it is desired to regulate the state x to a target value ?x. Lyapunov stability theory
can be applied to the design of a control u = g(x) as follows. We select a candidate Lyapunov
function,i.e., V(x) to be a scalar-valued positive definite function of the state vector x. A
common choice is the total energy of the system state, e.g. V(x) = xTx. The derivative ?V(x)
of V(x(t)) with respect to t is given by
?V(x) = (?xV)f(x,u) (2.1)
or represented as a Lie Derivative
?V(x) = LfV
Lyapunov-based control design involves the selection of a feedback control u = g(x) that will
guarantee a negative definite time derivative (2.1) in a region containing the desired equilibrium
point ?x.
Example 1 Consider the system
?x1 = x1x2 +x22
?x2 = x21 +u
(2.2)
We select the Lyapunov function V(x1,x2) = 12 parenleftbigx21 +x22parenrightbig which has the corresponding
derivative with respect to time
?V(x1,x2, ?x1, ?x2) = x1 ?x1 +x2 ?x2
where ?x2 is a function of the control input u and the states x1 and x2.
?V(x1,x2,u) = 2x21x2 +x1x22 +x2u
4
We select the control input u to guarantee ?V(x1,x2) is negative definite, hence stable using
Lyapunov?s definitions. One possible choice is
u = ?2x21 ?x1x2 ?x21x2 ?x2
resulting in negative definite function ?V(x1,x2) = ?x21x22 ?x22. This gives a closed loop stable
system using Lyapunov?s definition of stability, in the introduction of this chapter.
2.2 Input-Output Linearization
Input-output linearization is a feedback control method where the result is a direct and
simple relation between a system?s output and control input. Input-output linearization is
used for time-delayed single input single output nonlinear system?s control [11]. Singh et al
used input-output linearization to develop adaptive flight controls [17, 16] and more recently,
input-output linearization is being used to develop state observers for nonlinear systems [5, 22].
Consider a nonlinear system
?x = f(x,u) (2.3a)
y = h(x) (2.3b)
Input-Output linearization attempts to linearize the input-output relationship 2.3a at the
expense of producing nonlinear output functions. For example, consider a system described by
?x1 = x22
?x2 = ?x1 +u
y = x1
5
with an output y = x1. A change of variables
z1 = x1, z2 = x22
and the selection of a feedback control u = x1 ? v2x2 yields closed loop dynamics
?z1 = z1 (2.4a)
?z2 = v (2.4b)
y = ?z2 (2.4c)
The state transformation from x to z produces linear state equations but the output mea-
surement (2.4c) is now a nonlinear function of the state. As a result, traditional linear control
approaches cannot be applied to this revised system.
A generalized input-output linearization for single-input-single-output (SISO) systems is
described below. Consider the SISO plant
?x = f(x) +g(x)u (2.5)
y = h(x) (2.6)
with state vector x? IRn, scalar input u, and scalar plant output y. Input-output linearization
decomposes the inputuinto a nonlinear inner loop controlu(x,v) = ?(x)+?(x)v that creates a
linear dynamic relationship between the outputyand a user defined inputv[6, 19]. Performance
of the input-output linearized system is controlled by linear feedback system design methods
applied to the new input v. The new linear model state variables are the output y of (2.6) and
its derivatives diydti, i = 1,??? ,n?1. Because of this structure, input-output linearization is well
suited for tracking control problems.
6
A general form of input-output linearization follows below. Differentiate the system output
function (2.6) until the plant input u appears explicitly in the final differentiation. The general
form for the ith time derivative of the output can be expressed using Lie derivative notation
y(i) = L(i)f h+(LgL(i?1)f h)u (2.7)
The ith time derivative of the output y does not have any explicit dependence on the plant
input u for a system with relative degree r, where i<r. An explicit dependence on the input
u appears in the rth derivative of the output. That is,
LgLrfhnegationslash= 0
The input u(t) is expressed as a gain scheduled linear affine function of an external input v.
Selecting the plant input as
u(x,v) = ?(x) +?(x)v
= 1L
gLrfh
[?Lr?1f h+v] (2.8)
a linear dynamic relationship between the input variablev and the system output y is achieved.
The dynamic relationship between v and y is then described by the linear model
y(r) = d
ry
dtr = v (2.9)
The inner loop for input-output linearizing control is illustrated by the block diagram in Fig. 2.1.
The equivalent system is shown in the dashed block.
The input-output linear system (2.9) is stabilized by designing an outer loop control law
for the new input v by any linear control design method. The applicability of linear design
methods is one reason that input-output linearization is very attractive [10]. Additionally,
7
x = f(x) + g(x)u?(x) + ?(x)v h(x)
y(r)  = v
r ? n
uv
v
x y
y
Figure 2.1: Inner loop of input-output linearizing control, and equivalent model
the straightforward computation of the time derivative functions (2.7) makes input-output
linearization attractive.
There are difficulties related to input-output linearization. Canceling nonlinearities by
mathematical means may produce singularities in any control solution. More precisely,
Definition 5 A system has a singularity at x0 if an input-output linearization feedback law
u(x,v) satisfies limx?x0u(x,v)= ?. The feedback law u(x,v) is also considered singular at x0.
Example 2 The control law u(x,v) = 1x +v is singular at x = 0.
Singularities appear in the design of input-output linearization control laws (2.8) at state values
where LgLrfh is zero.
A second issue that affects input-output linearization control law design is associated with
the relative degree r of a system. The relative degree is defined below.
Definition 6 Consider the nonlinear state equations
?x = f(x) +g(x)u
y = h(x)
8
where f : D ? IRn,g : D ? IRn,and h : D ? IR are sufficiently smooth on a domain D ? IRn,
the state equations are said to have relative degree r, 1 ?r ?n, in a region D0 ?D if [6]
??i
?xg(x) = 0, i = 1,2,??? ,r?1;
??r
?x g(x) negationslash= 0
for all x?D0 where
?1(x) = h(x) and ?i+1(x) = ??i?xf(x), i = 1,2,??? ,r?1
If the relative degree r of a system is a function of the operating point x or is not well defined
over the state space, then singularities may occur in input-output linearization. That is, the
control solution u(x,v) = ?(x) +?(x)v will have a singularity at points x where the relative
degree of (2.5)-(2.6) change. Therefore, input-output linearization is not a well defined solution
for systems with poorly defined relative degree. Pseudo-linearization techniques may be able
to circumvent the usual input-output linearization techniques difficulties [4, 8, 9, 10].
If the relative degree r of a system is equal to the order of the system,i.e., n then input-
output linearization provides a linearized system from v to y. If the relative degree is less than
number of states, i.e., r <n, then the system has a set of dynamics, called internal dynamics
with an order n?r, that are not considered during input-output linearization. The system
also has zero dynamics, which are defined as the internal dynamics of the system where the
system output y is kept at zero by the input u [19, page 219]. The stability of the input-output
linearized system (2.5)-(2.6) is only guaranteed if the internal and zero dynamics are stable.
In general, the determination of the stability of the internal dynamics is very difficult due to
nonlinearities and the fact they are coupled to the other dynamics.
9
Figure 2.2: Input-output relationship
Example 3 Consider a system
?x1 = x1x2 +x22
?x2 = x21 +u
y = x1
(2.10)
We differentiate the output y until u appears explicitly in the result:
y = x1
?y = ?x1 = x1x2 +x22
?y = x2 ?x1 +x1 ?x2 + 2x2 ?x2
= x1x22 +x32 +x31 +x1u+2x2x21 +2x2u
We incorporate an auxiliary (user-defined) input v into our control law and we select the input
u as
u = ?x1x
22 ?x32 ?x31 ?2x2x21 +v
x1 +2x2
to eliminate the plant nonlinearities. This process yields the input-output linearized form
?y = v
as shown in Figure 2.2. This revised system is easily controlled by linear techniques. The final
10
x = f(x) + g(x)u
z =Az + bv
?(x) + ?(x)v T(x)
uv x z
zv
Figure 2.3: Inner loop of input-state linearizing control, and equivalent model
solution in this example can take the form
v = ?K
?
?? x1
x2
?
??
u = ?x1x
22 ?x32 ?x31 ?2x2x21 +v
x1 +2x2
where K is a set of stability state feedback gains associated with Figure 2.2.
2.3 Input-State Linearization
Recall the single input single output (SISO) plant (2.5). Input-State linearization is
achieved by the combination of a smooth, invertible state transformation z = T(x) and a
nonlinear inner loop control u = ?(x) +?(x)v, where v is a user defined input. The control u
yields a linear relationship between the new input v and the new state z, as illustrated in Fig.
2.3. The equivalent linear input-state system is shown in the dashed block. The block labeled
T(x) represents the smooth (differentiable), invertible nonlinear transformation of the state x;
the block labeled ?(x) + ?(x)v represents a nonlinear mapping from the new input v to the
11
original input u. The result of input-state linearization is a model
?z = Az+bv (2.11)
where the pair (A,b) is in controllable canonic form. An outer-loop control law for v is designed
with the controllable canonic state variable model.
Two definitions will aid in the following discussion.
Definition 7 A linearly independent set of vector fields {f1,f2,??? ,fm} is involutive if there
are scalar functions ?ijk: IRn ? IR such that
[fi,fj](x) =
msummationdisplay
k=1
?ijk(x)fk(x), ? i,j
Definition 8 An Ad field is
adfg = [f,g] = ?gf ??fg
There are two conditions that must be satisfied in order to apply input-state linearization. They
are (1) the set of vector fields
braceleftBig
g(x) adfg???adn?2f g
bracerightBig
(Lie Brackets) must satisfy
rank
braceleftBigbracketleftBig
g(x) adfg ad2fg???adn?1f g
bracketrightBigbracerightBig
= n
and must be involutive over the field of scalar nonlinear functions in x. A positive test for
involutivity exists if the Lie bracket of any two elements of a set of vector fields is spanned
by the set. Both conditions, (1) and (2), must be satisfied for the existence of a linearizing
transformation T(x).
If a linearizing transformation T(x) exists for the system ?x = f(x) +g(x)u, then in fact
an infinite number of such transformations exist. This property can be exploited to reduce the
12
complexity of choosing a transformation. Let the transformation take the form
T(x) =
?
??
??
??
??
??
??
?
T1(x)
T2(x)
...
Tn?1(x)
Tn(x)
?
??
??
??
??
??
??
?
Recall that the pair (A,b) in (2.11) is assumed to be in controllable canonic form
A =
?
??
??
??
??
??
??
?
0 1 0 ??? 0
0 0 1 ??? 0
... ...
... 0 1
0 ??? ??? 0 0
?
??
??
??
??
??
??
?
and b =
?
??
??
??
??
??
??
?
0
0
...
0
1
?
??
??
??
??
??
??
?
Thus, ?zi = zi+1 for i = 1,??? ,n?2 and ?zn will be a nonlinear function containing the input u.
The transformation T(x) from x to z is written recursively as
?z1 = T1(x) = z2
= (??xT1) ?x = (??xT1)[f(x) +g(x)u]
?z2 = (??xT2)[f(x) +g(x)u]
= (??x (??xT1)[f(x) +g(x)u])[f(x) +g(x)u]
All of the transformation elements T1???Tn are expressed recursively in terms of T1. This is a
direct result of the form ofA. Due to the structure ofb, (?xTj)g(x) = 0 for allj = 1,2,??? ,n?1
and (?xTn)g(x) negationslash= 0. Thus, the calculation of T(x) is reduced to the calculation of T1(x) in
?xTig(x) = 0, i = 1,2,??? ,n?1; ?xg(x) negationslash= 0
13
where
Ti+1(x) = ?xTif(x), i = 1,2,??? ,n?1
As with input-output linearization, input-state linearization has potential difficulties that
increase the complexity of is control design or prevent it from providing a satisfactory control
solution. The computation of the Lie brackets is extremely labor intensive for all but the
simplest of plants. Symbolic processing by computers can be attempted in principle, but the
rank conditions are often difficult to verify for all state space values. The development of
the nonlinear state transformation T(x), which requires the solution of n partial differential
equations, is very labor intensive. The input-state linearizaiton control solution may have
singularities within the operating region of the state space.
Example 4 The system
?x1 = asin(x2)
?x2 = ?x21 +u
y = x1
is in the form required of input-state linearization, Figure 2.3
f(x) =
?
?? asin(x2)
?x21
?
?? g(x) =
?
?? 0
1
?
??
We first determine the conditions under which the system is a valid input-state linearizable
system. The vector field set {g(x),adfg(x)} is examined for linear independence. The adfg(x)
component is given
adfg(x) = ?xg(x)f(x)??xf(x)g(x) =
?
?? acos(x2)
0
?
??
14
and the field matrix
{g(x),adfg(x)} =
?
??
??
?
?? 0
1
?
?? ,
?
?? acos(x2)
0
?
??
?
??
??
is only full rank when x2 negationslash= ?pi/2. Input-state linearization will only be valid for operating
regions where |x2| = pi/2.
We apply the above observations to compute recursively the state transformation. Recall
that we must solve
(?xT1)g(x) = 0
(?xT1)adfg(x) negationslash= 0
or
?T1
?x2 = 0
??T1?x
1
acos(x2) negationslash= 0
for T1(x). One possible solution is T1(x) = x21. We select this function for T1(x) and we obtain
T2(x) as
T2(x) = (?xT1(x))f(x) =
bracketleftbigg
2x1 0
bracketrightbigg
?
?? asin(x2)
?x2
?
?? = 2ax
1 sin(x2)
With the above definition of T(x) the input-state linearized model becomes
?z1 = 2x1 ?x1 = 2ax1 sin(x2) = z2
?z2 = 2ax1 cos(x2)u+2a(a?acos(x2)2 ?x31 cos(x2))
15
We choose the modified system input to cancel the nonlinearities and introduce a new control
v.
u = ?2a
2 +2a2 cos(x2)2 +2ax31 cos(x2)+v
2ax1 cos(x2)
which yields the linear input-state relationship from v to z
?z1 = z2
?z2 = v
2.4 Integrator Backstepping
The technique of integrator backstepping is applied to strict feedback systems with the
following definition.
Definition 9 We say that a system is in strict feedback form or that a system is a strict feedback
system if it can written as a cascaded set of state equations
?x1 = f1(x1)+g1(x1)x2 (2.12)
?x2 = f2(x1,x2) +g2(x1,x2)x3
?x3 = f2(x1,x2,x3) +g2(x1,x2,x3)x4
...
?xn = fn(x) +gn(x)u
A block diagram representation of (2.12) appears as a set of nested loops.
Integrator backstepping is a recursive approach to control system design. A stabilizing
control law is constructed for an inner state variable (e.g. x1), treating the state variable from
the next level (e.g. x2) as a pseudo-input. The stabilizing control law for the inner variable
is propagated to the input of the next state variable by differentiation. This process is called
backstepping. Due to this process of repeated differentiation in the design of backstepping
16
Figure 2.4: Integrator Backstepping Example Block Diagram
control laws, there is a tendency for the control laws to ?blow up? or to contain singularities
within a selected operating region of the state space.
Backstepping has advantages over the linearization techniques discussed in the previous
sections. Unlike the previous two methods, the linearization of system dynamics is not an in-
herent result or requirement of the backstepping methodology. Thus, backstepping is arguably
a more flexible design approach than input-output or input-state linearization. Nonlinearities
that are not inherently harmful to a system?s stability do not have to be mathematically can-
celed. The backstepping approach also satisfies Lyapunov stability theory at every iteration
and, in the final result, provides an explicit Lyapunov function for an entire closed loop system.
Example 5 Consider the strict feedback system
?x1 = sin(x1)x2 (2.13a)
?x2 = cos(x1)u (2.13b)
A block diagram for the system is shown in Figure 2.4. We choose the variable x2 as the input
to the subsystem (2.13a) and the Lyapunov function
V1(x1) = 12x21
The derivative of V1 with respect to time is
?V1(x1,x2) = x1 sin(x1)x2
17
Figure 2.5: Integrator Backstepping Example Block Diagram, Part 2
We therefore select the desired state x2(t) function to be
x2 = ?x1 sin(x1)
and add this to the input x2, the same input was then subtracted from the input, Figure 2.5. A
new variable ? was defined by the user.
? = x1 sin(x1) +x2
and the desired control x2 was backstepped through the integrator, Figure 2.5.
?? = d
dt (x1 sin(x1)) +cos(x1)u (2.14)
Let
V2(x1,?) = V1(x1) + 12?2
and the derivative of V2 with respect to time is
?V2(x1,?) = ?V1(x1)+???
18
The term ?V1(x1) is negative definite so we select ?? = ??. We thus solve
?? = ddt (x1 sin(x1)) +cos(x1)u
for the external input u.
u = 1cos(x
1)
bracketleftbigg
??? ddt (x1 sin(x1))
bracketrightbigg
= 1cos(x
1)
[??? ?x1 (sin(x1) +x1 cos(x1))]
= 1cos(x
1)
[?x1 sin(x1)?x2 ?sin(x1)x2 (sin(x1) +x1 cos(x1))]
Observe that the final control solution is only valid for |x1| ?pi/2.
19
Chapter 3
Ball On Beam Example
In this chapter a classical nonlinear control problem, the ball-on-beam problem, is used to
illustrate weaknesses in the four nonlinear control design techniques presented in Chapter 2 and
to demonstrate the utility of a newly proposed hierarchical control system design technique, the
subspace stabilization approach (SSA). The SSA involves the selection of a subset of system
states for an inner-loop nonlinear control law that is used to aid an outer loop robust control
law. The ball-on-beam problem involves several design challenges that are not met by the
design approaches of Chapter 2 but that are easily addressed by the SSA. In particular, the
relative degree in the ball-on-beam problems is poorly defined, which renders infeasible such
common nonlinear control methods as input-state linearization or input-output linearization.
Linear feedback control designed around a model linearization of the ball-on-beam often
performs well near nominal operating conditions. Since LQR control design is an established
technique with known robustness properties, we shall henceforth use LQR linear state feedback
design for the outer loop control law in our treatment of the SSA. The reader should notice
that this choice is made for convenience in presentation and that the SSA is not restricted to
the use of LQR control for the outer loop robust control law design.
A diagram of the ball-on-beam system is shown in Figure 3.1. One well known mathemat-
20
Figure 3.1: Ball on Beam Diagram
ical representation of this system is given by
?x1 = x2
?x2 = mx1x
24
parenleftBig
m+ Iballr2
parenrightBig + mgsin(x3)parenleftBig
m+ Iballr2
parenrightBig
?x3 = x4
?x4 = ? ?Cdx4 +gmx1 cos(x3)parenleftbigI
beam +Iball +mx21
parenrightbig
where x1 is the position p of the ball on the beam (m), x2 is the velocity v of the ball measured
along the beam (m/s), x3 is the angle ? of the beam relative to horizontal (rad), x4 is the
angular velocity of the beam (rad/s),? is a moment applied to the beam by a controlling motor
(N-m), r is the radius of the ball (m), g is a gravitational constant (m/s2), m is the mass of
the ball, Ibeam is the inertia of the beam (kgm2), Ibeam is the inertia of the beam (kgm2), and
Cd is a friction term associated with the beams pivot (N). For clarity in exposition, we simplify
these equations to be
?x1 = x2 (3.1a)
?x2 = x1x24 +sin(x3) (3.1b)
?x3 = x4 (3.1c)
?x4 = u (3.1d)
21
where u is the control input. In spite of numerous studies in the literature, a control law that
will stabilize the system (3.1a)-(3.1d) remains an active topic of research discussion [3, 4, 18].
The remainder of this chapter is organized as follows. We first review LQR control design
techniques in Section 3.1. Following this, we apply four nonlinear control design techniques
of Chapter 2 to the ball-on-beam problem in Sections 3.2?3.4, where we establish that none
of these techniques provide a satisfactory control law design. The ball-on-beam system is not
in strict feedback form, therefore integrator backstepping is not a valid design technique. In
Section 3.5 we illustrate that one issue that causes these other methods to fail is the poorly
defined relative degree of the ball-on-beam system. Following this, we illustrate our SSA method
in Section 3.6, in which we demonstrate not only a satisfactory control law, but also a relatively
simple design procedure by which this control law may be obtained. We defer a theoretical
analysis of the SSA method to Chapter 4.
3.1 Review of the Linear control
We linearize (3.1a)-(3.1d) about the origin to obtain
?x1 = x2 (3.2a)
?x2 = x3 (3.2b)
?x3 = x4 (3.2c)
?x4 = u (3.2d)
or
?x = Ax+Bu
22
where
A =
?
??
??
??
??
??
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
?
??
??
??
??
??
B =
?
??
??
??
??
??
0
0
0
1
?
??
??
??
??
??
A state-feedback law u = ?Kx is found using the above two state matrices and a linear-
quadratic regulator (LQR) design. LQR calculates the optimal gain matrix K such that the
state-feedback law minimizes a cost function J with user defined weights, i.e. Q = I4x4,R = 10.
min J =
integraldisplay
xTQx dt
subject to ?x = Ax+Bu
This state-feedback control law is able to control the nonlinear plant (Equations 3.1a - 3.1d)
for initial position values close to the origin (x(0) = 0).
3.2 Lyapunov Control Design Method
In the Lyapunov control design method the designer selects a positive definite Lyapunov
function V(x) The equations (3.1a) thru (3.1d) show that most positive definite energy-like
functions V(x) defined for all states will cause the resulting derivative ?V(x) to have the form
V(x) = f2(x1,x2,x3,x4)
where
?V(x) = (?xf2)T ?x
Due to the structure of the system (3.1a)-(3.1d) , Lyapunov-theory based control laws u will
typically include terms of the form 1/x4, which creates a singularity in the control at x4 = 0
23
(zero angular velocity). For example, consider the Lyapunov functionV(x) = 12(x21+x22+x23+x24)
whose derivative with respect to time is
?V(x) = x1 ?x1 +x2 ?x2 +x3 ?x3 +x4 ?x4
We therefore select a control input u that is decomposed as
u = u1(x) +u2(x)
where u1(x) cancels the positive nonlinear terms and u2(x) is a negative definite function.
That is, we select the control solution
u1(x) = ?x1x2x
4
? x2 sin(x3)x
4
?x3 ?x4x2x1
u2(x) = ?x32x4 ?x21x4 ?x22x4 ?x4
The control u1 has a singular point at x4 = 0. That is, the law fails at the desired operating
point. This difficulty arises in every energy-based Lyapunov design we tested.
3.3 Input-Output Linearization Method
We apply input-output linearization to the ball-on-beam example as follows. The ball-
on-beam output is y = x1 and the input is found in Equation (3.1d) as u. The output is
24
differentiated until the input explicitly appears, to obtain
y = x1
?y = ?x1 = x2
?y = ?x2 = x1x24 +sin(x3)
...y = ?x
1x24 + ?x4(2x1x4) + ?x3 cos(x3)
= x2x24 +2x1x4u+x4 cos(x3)
The function ...y is written more conveniently as ...y = f1(x)+ug1(x). Solution ...y for the control
input u gives the control solution for input-output linearization, as
u = ?f1(x)g
1(x)
(3.3)
If the denominator g1(x) of u is non-zero in the region of operation, then this control is valid
within the region of operation. If the origin is within the region of operation, such as in the ball-
on-beam system (x = 0 and g1(0) = 0), then the control contains a singularity. For the ball-on-
beam example, the origin is within the region of operation, making input-output linearization
an infeasible control design method. Approximate input-output linearization is extensively
discussed for the ball-on-beam system by Hauser et al [4]. Hauser et al neglect terms, i.e.
x1x24, in the dynamics that produce singularities within the input-output linearization control
solution.
3.4 Input-State Linearization Method
We apply input-state linearization to the ball-on-beam example as follows. As discussed
in Section 2.3, a transformation T(x) must exist for input-state linearization to be a valid
method. One of two necessary and sufficient conditions for a transformation to exist requires
25
the following set of vector fields (Lie brackets) to have full rank [19]. The vector field matrix
Vec. Field =
?
??
??
??
??
??
0 0 2x1x4 v1(x)
0 ?2x1x4 v2(x) v3(x)
0 ?1 0 0
1 0 0 0
?
??
??
??
??
??
where
v1(x) = 4x4x2 ?cos(x3)
v2(x) = ?2x4x2 +cos(x3)
v3(x) = ?2x1x34 ?2x4 sin(x3)???
?sin(x3)x4 ?2x34x1
the rank negationslash= n when x3 = 90? and x4 = 0. The ball-on-beam system does not satisfy at least one
of the necessary and sufficient vector field rank condition, so input-state linearization will not
work for the this system. Hauser et al. show that the matrix
bracketleftBig
g(x) adfg ad2fg???adn?2f g
bracketrightBig
for
the ball-on-beam equations is not involutive [4], ans so this system fails the second condition
for the existence of T(x).
3.5 Underlying Issue: Poorly Defined Relative degree
A major issue in all of these failed methods is the relation between the state x4 and the
relative degree of the system (3.1a)-(3.1d). The ball on beam system has a state dependent
relative degree of 3, as seen from the result g1(x) in Equation (3.3) and the relative degree is
not well defined at the origin (i.e. x = 0).
For the ball-on-beam equations (3.1a-3.1d), the linear control design methods produces the
only ?successful? control input u around the origin. We conclude that nonlinearities that are
26
neglected in a linearized system model must be related to the poorly defined relative degree
of the system. Examination of the system equations (3.1a)-(3.1d) and the linearized equations
(3.2a)-(3.2d) reveals that the centrifugal acceleration term x1x24 is entirely neglected in the
linearized equations. The observation is also related to results in Hauser et al [4].
3.6 Dynamic Analysis of the Ball On Beam System
The nonlinear design techniques attempt to compensate for the centrifugal acceleration
term x1x24 located in the dynamics of ?x2. Hauser et al neglect this term in their approximate
input-to-output linearization [4]. We evaluate the effects of this term in a simulation study. A
simulation was run with simple LQR gains applied to two different systems: the original system
(3.1a)-(3.1d) and a modified system with the x1x24 neglected, ?x2 = sin(x3). By neglecting the
centrifugal acceleration term x1x24 the modified system is able to return much higher initial
values x1(0) to the origin.
The position response x1 for both systems with the same parameters is seen in Figure
3.2. The centrifugal acceleration term produces an ?anti-damping? term in the dynamics ?x2.
The states associated with this centrifugal acceleration term are specifically addressed in the
following control design. Theory associated with this concept will be discussed in Chapter 4.
The state dynamics of x3 and x4 in the equations (3.1a) thru (3.1d) are separated from
the state dynamics of x1 and x2, as shown in Figure 3.3.
We chose the Lyapunov
?V(x3,x4) = x3 ?x3 +x4 ?x4 (3.4a)
?V(x3,x4) = x3x4 +x4u (3.4b)
which implies stability of x3 and x4 with
u = ?x4 ?x4x23 ?x3 (3.5)
27
0 1 2 3 4 5 6 7?1
?0.5
0
0.5
1
1.5
2
2.5
3
compare nonlinear plant with x1x42 and without
time
position, x
1
without
withwith x1x42 
without x1x42 
Figure 3.2: Comparison of a Modified System with the Original System
Figure 3.3: Block Diagram of Simplified ball-on-beam System
To verify states x3 = ? and x4 = ?? are stabilized, the system (3.1a)?(3.1a) was simulated with
an initial condition of x3(0) = pi/10. State x3 and x4 were returned to the origin, as seen in
their response shown in Figure 3.4.
28
0 1 2 3 4 5 6 7
?0.3
?0.2
?0.1
0
0.1
0.2
0.3
time,s
Angle and Anglular Velocity
x3 = angle,radians
x4 = anglular vel., rad/s
Figure 3.4: Stability of x3 and x4 confirmed
The Lyapunov designed control (3.5) and a new user defined control input v were feedback
into the original ball-on-beam system.
?x1 = x2 (3.6a)
?x2 = x1x24 +sin(x3) (3.6b)
?x3 = x4 (3.6c)
?x4 = ?x4 ?x4x23 ?x3 +v (3.6d)
The new dynamics (3.6a)?(3.6d) were linearized about the origin, and an LQR optimized set
of gains K were chosen to stabilize the origin. The following control was chosen.
unl = ?x4 ?x4x23 ?x3 ?K
?
??
??
??
??
??
x1
x2
x3
x4
?
??
??
??
??
??
29
0 10 20 30 40 50 60?20
0
20
40
60
80
100
120
Initial Condition of 100 meters, with control ?nl
time
meters and m/sec
position,state x1
velocity of ball,state x2
Figure 3.5: x1 Response with nonlinear control, unl
A plot of the position of the ball, velocity of the ball, angle of the beam and angular rate of the
beam are seen in Figure 3.5 and 3.6. Numerous initial conditions were applied to the system,
and the system was stable for initial conditions up to 150 meters (i.e. over time, all states
would return to zero x = 0).
A comparison of the LQR control input from Section 3.1 and the above control unl, is
seen in Figure 3.7. The control input unl produces a similar settling time for similar initial
conditions, and although damping in the linear control law is higher, unl maintains stability for
initial conditions far outside of the linear LQR state-feedback?s (u = ?Kx) stability range.
Hauser et al.?s [4] approximate input-output linearization mentioned in Section 3.3 is com-
pared to the above control unl in Figure 3.8. Both position responses in Figure 3.8 are simu-
lated with the same mechanical constants, i.e. ball mass and beam mass, and initial position
x1(0) = 1.92 m. The control input unl produces a slightly better settling time than Hauser?s
control input and unl maintains stability for initial conditions far outside of Hauser?s control
input stability range, x1(0) > 1.92 m.
30
0 10 20 30 40 50 60?4
?2
0
2
4
6
8
time
radians and rad/sec
Initial Condition of 100 meters, with control ?nl
angle,state x3
angular velocity,state x4
Max 
Angle = 0.4889pi  
Figure 3.6: x3 and x4 Responses with nonlinear control, unl
3.7 Concluding Remarks
The ball-on-beam example system, discussed in this chapter, has a poorly defined relative
degree. Well-known nonlinear methods fail when the relative degree is uncertain or changing.
This problem was tackled by designing a nonlinear control component that directly addresses
the state variables that affect the system?s relative degree and with the relative degree addressed,
an outer loop linear controller became significantly more effective. With these observations, a
new approach to nonlinear control design has been outlined. In the following chapter, five more
examples will be discussed and used as catalyst in outlining this new control design approach.
31
0 1 2 3 4 5 6 7 8?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
1.2
Position with ?nl and ?=?Kx
time
meters
?
?nl 
Figure 3.7: x1 Response with unl and u = ?Kx control inputs
0 2 4 6 8 10 12 14?0.5
0
0.5
1
1.5
2
2.5
Ball Position, SSA and Hauser Compared
time,s
p,position (m)
Hauser Control
SSA Control
Figure 3.8: x1 Response with unl and Hauser?s control inputs
32
Chapter 4
New Subset Stabilization Approach (SSA) Control Design
In this chapter, five example systems will supplement the ball-on-beam example in the
previous chapter. For the purposes of this dissertation, the approach used in the ball-on-beam
example will be referred to as a Subset Stabilization Approach (SSA). This approach provided a
control effort, designed through nonlinear and linear methods, that stabilized the ball-on-beam
nonlinear plant well beyond standard linear techniques. Additionally, the new SSA control was
able to provide a feasible control effort where traditional nonlinear control techniques could not.
To take a more in depth look at SSA control within the aspects of nonlinear systems, we will
examine five different nonlinear system examples. In each example, the traditional nonlinear
approaches discussed in Chapter 2 fail to provide a feasible solution, or do not provide a
stabilizing control effort. In each example system, a standard linear control technique will be
applied and compared with the new SSA.
4.1 Subset Stabilization Approach
The proposed SSA control combines the benefits of larger stability region offered by non-
linear control design with the simplicity and design ease of the linear systems approach. The
SSA was born out of two observations of many practical nonlinear problems:
? System nonlinearities are often a function of a subset of the full state vector. (Robotic
systems are a notable exception to this observation, but rigid robot dynamics are well
structured and can be controlled by all of the existing nonlinear methods.)
? In situations where existing nonlinear methods fail, a subset of the state variables con-
tribute to structural problems that result in control singularity or loss of controllability.
33
The SSA nonlinear control approach reduces the effects of variables that contribute to either
nonlinearities or structural problems. Steps in the SSA nonlinear design approach are:
1. Partition the state vector into two sets of variables
x =
?
?? z
w
?
??
where the variablesw are those contributing to the dominant nonlinearities of the system.
In some systems certain variables trigger control singularity, or impact system structure;
these variables would be included in the set of variables w. The system model can be
expressed in the form
?z = A1z+A2w+B1u+f1(z,w,u) (4.1)
?w = A3z+A4w+B2u+f2(z,w,u) (4.2)
Here Ai, i = 1,...,4 and Bj, j = 1,2 are linear matrices of appropriate dimension, and
functions fk(z,w,u), k = 1,2| represent the nonlinearities of the system.
2. Choose a candidate Lyapunov function V(w), a positive definite function of the subset of
variables w.
3. Design an inner loop control ulyp so that the time derivative ?V(w) is negative definite in
w.
4. Apply the control u = ulyp + v to the system 4.1 to stabilize the w-dynamics, and to
introduce a new input variable v for outer loop control. The partially stabilized system
is less sensitive to the nonlinear terms fk(z,w,u) associated with w.
5. Design an outer loop control v using linear system methods to stabilizes either (1) the
linearization of the uncompensated plant 4.1, or (2) a linearization of the plant with the
34
inner loop control ulyp. Past experience has been successful by applying linear quadratic
regulator (LQR) methods with either technique.
6. The composite SSA control is given by
u = ulyp +v
= ulyp +uLQR (4.3)
Local asymptotic stability is guaranteed by standard Lyapunov stability techniques and Linear
techniques around an operating point.
4.2 Five Case Studies
In this section, five example plants are studied to illustrate the challenges of standard
nonlinear control methods and the utility of the SSA control design method. A comparison of
the resulting state responses is presented for each example system.
4.2.1 System 1
Consider the system
?x1 = x2 +tanh(u)
?x2 = x1 +x22 +u (4.4)
y = x1
A Lyapunov design based on a Lyapunov function
V(x1,x2) = 12(x21 +x22) (4.5)
35
is examined first. The derivative of V is given by
?V(x1,x2,u) = 2x1x2 +x32 +x1 tanh(u) +x2u.
Notice that the expression is not an affine function in u. Therefore, an input that yields a
negative definite ?V is not determined easily and a stabilizing feedback control cannot be found
in a straightforward manner by Lyapunov?s design method and using the Lyapunov function
(4.5). Though tuning with trail and error using different Lyapunov functions could possibly
improve these results. But, for this Chapter, Lyapunov?s design method will be standardized
to include the energy like function (4.5).
Next, consider the input-output linearization approach. The system 4.4 has relative degree
r = 1, so the input u appears explicitly in the first derivative of the output y:
?y = ?x1 = x2 +tanh(u)
Applying the nonlinear input
u = arctanh(v?x2)
yields a first-order linear input-output dynamic model
?y = v
to which a linear design is easily determined and stabilizes the result. The nonlinear control
u approaches infinity, however, near the condition |v ? x2| = 1, due to the domain of the
hyperbolic arctangent, |v?x2| < 1. Therefore, the nonlinear solution is localized around the
v?x2 = 0.
For input-state linearization, a transformation must exist T(x) that will allow a linear
relationship between the input and the states. For T(x) to exist, the ad field for (4.4) must
36
be full rank. Do to structure of the system,non-affine structure, input-state linearization is
not a feasible solution for control. The system?s structure also prohibits the use of integrator
backstepping due because it is not in strict feedback form.
To design a SSA controller, the model (4.4) is first examined for nonlinear terms. The
dominant state nonlinearity is the quadratic term x22 in second state equation. The presence of
this term overwhelms any linear control law as x2 deviates from zero. Therefore, a nonlinear
control can be designed to stabilize x2, and consequently reduce the effect of the nonlinear term
x22. Heuristically, the plant with nonlinear inner loop control will behave more like a linearized
model that can be used to design a stabilizing linear control law.
Choose the Lyapunov function V(x2) = 12x22. Then
?V(x2) = x2 ?x2
= x1x2 +x32 +x2u
Choose the inner loop control
ulyp = ?x1 ?x22 ?x2
to guarantee stability of the variable x2. Apply the control ulyp +v to the original plant (4.4).
The first term is the inner loop control, and the second term v is a new input for outer loop
control. The resulting model is
?x1 = x2 +tanh(u)
?x2 = ?x2 +v
37
Where the outer loop control v is designed using LQR methods; the gains used are K =bracketleftbigg
0.3162 0.3162
bracketrightbigg
1. The SSA control is given by the following equations.
u = ulyp ?Kx.
A comparison of the state responses for LQR control and the SSA control is seen in Fig. 4.1.
This response is seen on the edge of stability for the LQR gains. If the initial condition of
Figure 4.1: Example System 1: State responses to LQR feedback and SSA control
x2(0) is increased slightly, then the LQR gains do not provide a stable feedback control law.
However, SSA control provides stability well beyond the initial condition, x2(0) ? 1.05.
1Operating point x = 0 and weights Q = I2?2,R = 10
38
4.2.2 System 2
The second case study examines the second-order plant modeled by
?x1 = x1 +x2 +x22
?x2 = x1 +x22 +u (4.6)
y = x1
Again, Lyapunov control design based on the Lyapunov function V(x) = 12(x21 +x22) yields a
undesirable control law.
u = ?x21x?12 ?2x1 ?x1x2 ?x22
The control law has a singularity at x2 = 0, so the plant input u goes to infinity as the state
approaches the equilibrium point x = 0.
This case study has a relative degree of n = 2, implying the input will explicitly appear
in the second derivative of the output equation. The input-output linearizing control designed
around System 2 yields the control law
u = ?2x1 +x2 +2x
22 +2x1x2 +2x32
2x2 +1
which has a singularity atx2 = ?0.5, and is only locally valid around the operating point x = 0.
For input-state linearization, the system is written as
?x = f(x) +g(x)u, where g(x) =
?
?? 0
1
?
?? and f(x) =
?
?? x1 +x2 +x22
x1 +x22
?
??
and the ad field
[g(x) adfg] =
?
?? 0 ?1?2x2
1 ?2x2
?
?? (4.7)
39
must have full rank for all values ofx1,2, orrank = 2: ?x ? IR2. Clearly, from (4.7), a state pair
along the line (x1,x2) = (IR,?0.5) produces a low rank ad field (rank< 2). Therefore, system
2 does not satisfy a necessary and sufficient condition for the existence of a transformation
matrix T(x) to exist. Finally, similar to System 1, System 2 does not have the correct structure
to perform integrator backstepping control design. This leads to the use of a SSA control design.
The SSA control design method will focus on the nonlinear function, x22. Following the
example design process for System 1 yields the inner loop control
ulyp = ?x1 ?x22 ?x2.
The LQR gains are K =
bracketleftbigg
4.1669 2.0714
bracketrightbigg
, and the SSA control is given by
u = ulyp +uLQR
= ?x1 ?x22 ?x2 ?Kx (4.8)
A comparison of the state responses for simple LQR control and the SSA control is seen in
Fig. 4.2. As in System 1, Figure 4.1, the system?s state response is plotted on the edge of a
Figure 4.2: Example System 2: State responses to LQR feedback and SSA control
40
stable state region (x1(0),x2(0)) = (0.125,0). If the LQR designed control were to be simulated
past the shown initial condition, then it would become unstable, and the states would go to ?.
The SSA control stabilizes System 2 beyond this initial condition and provides a faster settling
time at the current initial condition (x1(0),x2(0)).
4.2.3 System 3
Consider the second-order plant model
?x1 = sin(x2)
?x2 = u (4.9)
y = x1
which has an equilibrium point at the originx1(0),x2(0) = (0,0). Lyapunov-based design based
on the Lyapunov function V(x1,x2) = 12(x21 +x22) yields the control
u = ?x1 sin(x2)x
2
with a singularity along the state space line (x1,x2) = (IR,0) which contains the origin. This
control is not a valid solution for control around the origin.
Input-output linearization control design produces the control
u = vcos(x
2)
As with the Lyapunov control solution, this solution u has a singularity at the origin.
For input-state linearization, the ad field for the plant (4.9) is given as
bracketleftbigg
g(x) adfg
bracketrightbigg
=
?
?? 0 ?cos(x2)
1 0
?
?? (4.10)
41
The ad field (4.10) loses full rank when x2 satisfies cos(x2) = 0. As for backstepping, the plant
model is in strict feedback form, so backstepping is a feasible approach, but produces a control
with singularity under the condition 1?x21 = 0, or (x1,x2) = (1,IR).
The SSA approach is now considered as a control solution design method. The plant
nonlinearity is only a function of the variable x2, so the nonlinear aspect of the SSA control is
designed to stabilize the state x2. The Lyapunov function V(x2) = 12x22 is chosen and yields a
control
ulyp = ?x2
The linear portion of SSA control is designed by LQR methods to yield state feedback gain
K =
bracketleftbigg
0.3162 0.3162
bracketrightbigg
. The complete SSA control is totally linear in this case example:
u = ulyp +uLQR
= ?x2 ?Kx
A comparison of state responses is in Fig. 4.3. The linear control fails to compensate for
nonlinear effects of x2, as seen in a comparison of the SSA and linear responses in Figure 4.3.
4.2.4 System 4
Consider a third-order plant modeled by
?x1 = x2
?x2 = x2 +sin(x3) (4.11)
?x3 = u
y = x1
42
Figure 4.3: Example System 3: State responses to LQR feedback and SSA control
Using the same Lyapunov energy functionV(x) = 12 parenleftbigx21 +x22 +x23parenrightbig, the Lyapunov-based design
produces the control
u = ?x1x2 +x
22 +x2 sin(x3)
x3
The Lyapunov control is similar to the Lyapunov control in the previous three systems; the
control produces a singularity along the plane (x1,x2,x3) = (IR,IR,0).
System 3 has a relative degree of r = 3 and the input-output linearizing control design
yields
u = ?x2 +sin(x3)cos(x
3)
Again, the input-output linearization standard design technique produce a control law with
singularities along the plane (x1,x2,x3) = (IR,IR,0). Therefore, the equilibrium point is not
stabilized with either Lyapunov or input-output linearization design. Similar to the earlier
example systems, the ad field of the System 4 (4.11) is not full rank for specific conditions
(sin(x3) = 0 which includes the origin).
43
For this example, the SSA control design method stabilizes the variablex3, included in most
of the nonlinear aspects of System 4. An inner loop control that stabilizes x3 by Lyapunov?s
control design method is given as
ulyp = ?x3
and applying ulyp to the plant yields the new model.
?x1 = x2
?x2 = x2 +sin(x3)
?x3 = ?x3 +v
The optimized LQR gains that stabilize the response about the origin and minimize the LQR
cost function with Q,R weights are given by K =
bracketleftbigg
0.3162 4.9487 2.3162
bracketrightbigg
. The SSA
control?s final form is given by
u = ?x3 ?Kx
The responses of the three state variables are shown in Fig. 4.4 with the initial conditions
(x1,x2,x3)o = (3.75,0,0). The SSA control approach is able to compensate for nonlinearities
related tox3, as seen in Figure 4.4. The linear controlKLQR allows for must higher oscillations,
and will eventually go unstable at higher initial conditions x1(0) > 3.75.
4.2.5 System 5
The ball-on-beam example was given in the previous chapter, Chapter 3. To better under-
stand the improvements of the SSA control, a simplified version of the ball-on-beam dynamics
44
Figure 4.4: Example System 4: State responses to LQR feedback and SSA control
are modeled by the following fourth order system
?x1 = x2
?x2 = x1x24 +sin(x3)
?x3 = x4 (4.12)
?x4 = u
y = x1
As in the previous chapter, the plant output y is the ball position along the beam and the
beam angle is the state variable x3. Standard Lyapunov design based on V(x) = 12xTPx, with
P = I4?4, yields the control law
u = ?x1x2 +x1x2x
24 +x2 sin(x3) +x3x4
x4
45
with a singularity along the state space region (x1,x2,x3,x4) = (IR,IR,IR,0), including the
origin.
Input-output linearization yields the control law
u = ?x4x2 +cos(x3)x
1
and, as in the Lyapunov designed control, the control u blow up at the origin, x4 = 0. Hauser
et al. show the ball-on-beam problem cannot be controlled by input-state linearization because
the necessary conditions for state transformation are not satisfied [4], and is verified in Section
3.4. Integrator backstepping is not a feasible control design approach because the model (4.12)
is not in strict feedback form.
It is obvious, sin(x3) and the x24 are the main nonlinear terms. Therefore, in SSA control,
a nonlinear inner loop is designed to stabilize the variables x3 and x4. The Lyapunov function
V(x3,x4) = 12(x23+x24) is used to design the SSA?s inner loop control ofx3 andx4. The resulting
nonlinear inner loop control is given by
ulyp = ?x4 ?x4x23 ?x3
Applying the inner loop control law u = ulyp +v to the plant yields
?x1 = x2
?x2 = x1x24 +sin(x3)
?x3 = x4
?x4 = ?x4 ?x4x23 ?x3 +v
46
and LQR gains are designed for the outer loop control variable v, with the equilibrium point
as the origin. The final form of the SSA control is given by
u = ulyp +v
= ?x4 ?x4x23 ?x3 ?Kx
A plot of the state variable responses is shown in Fig. 4.5. and the nonlinearities compensated
Figure 4.5: Example 5 (ball-on-beam control): State responses to LQR feedback and SSA control
for with SSA control are easily identified in the LQR state variable responses. As in Chapter
3, the SSA control approach allows for a larger stability region for LQR control designed, after
an inner loop stabilization routine is applied to x3 and x4.
47
4.3 Theoretical analysis
In this section we present an analysis of the subset stabilization approach for the control of a
nonlinear system. This analysis builds upon observations made in Chapter 3 and Sections 4.2.1-
4.2.5 in which the SSA compares very favorably to other non hierarchical control approaches in
a number of examples. We here bolster these anecdotal examples with theoretical observations
that justify the use of the SSA under specific conditions. In particular, we establish that the
SSA, in a simplistic sense, acts to decrease the impact of the system nonlinearities so that
a robust control law designed about a given operating point may in fact extend its region of
convergence in closed loop operation. While any robust control law design approach could be
used, we here present our analysis in terms of the SSA as a companion to LQR control design.
While the examples presented in the previous sections establish evidence of the utility of
our SSA control design method, they do not address the question of why the SSA method occurs
at all. We now present a theoretical analysis of the SSA method and provide another example
to illustrate the method. Our analysis will use the following definitions.
Definition 10 Suppose that there exists a control input u(z,w) such that, for the correspond-
ing closed loop system
?z = A11z+A11w+f11(w)+f12(z,w) +f13(u(z,w))
?w = A11z+A11w+f21(z) +f22(z,w)+f23(u(z,w))
there is a Lyapunov function on the state subset w; that is, there exists a function V(w) > 0
such that ddtV(w(t)) < 0 for the system in closed loop with u(z,w). Then we say that the
open-loop system (4.13a)-(4.13b) satisfies the the subspace stabilizability condition.
Definition 11 Consider a system S that satisfies the subset stabilizability condition. The null
subset manifold of S is the set of points
?
?? z
w
?
??? IRn+m where w = 0.
48
Now, consider a system in the general form
?z = A11z+A12w+f11(w) +f12(z,w)+f13(u) (4.13a)
?w = A21z+A22w+f21(z) +f22(z,w) +f23(u) (4.13b)
the state vectors are z ? IRn and w ? IRm. The system is controlled by a SSA-based control
design, as shown in Figure 4.6. Let
Figure 4.6: Subset Stabilization Approach Block Diagram
ulyp(z,w) = ?ulyp(z,w) +u2(wn1,...,wnm)
be a Lyapunov designed control effort where ?ulyp(z,w) drives the system to the null subset
manifold w = 0, n is an integer, and u2 is selected so that (1) an increase in n increases the
rate of convergence of w to the null subset manifold, i.e., ?V(w,n + 1) ? ?V(w,n) for all n,
and (2) ?wu2|w=0 = 0. We interpret the action of the SSA inner loop control law ulyp in a
fashion similar to variable structure control, where state trajectories are often divided into a
reaching phase and a manifold tracking phase. In the case of SSA control, the closed loop
system may be considered to be in a reaching state when the system state is distant from the
49
null space manifold, which can be characterized either in terms of bardblwbardbl or in terms of the relative
magnitudes of the inner loop control command ulyp and the outer loop control command ulqr.
Given an appropriate choice of system states inw, once the state trajectory is ?near? to the null
subset manifold, the effects of the system nonlinearities may be greatly reduced. This leads to
two key observations regarding the success of SSA control in contrast to other control methods:
1. While effects of the nonlinearities in the underlying system may be reduced, they are not
entirely canceled. This stands in contrast to both linear robust control, which describes
linearities primarily in terms of norm bounds and the small gain theorem, and to nonlinear
control, which uses a precise model of system nonlinearities in the selection of its control
commands.
2. The use of an outer-loop robust control law is strengthened because a properly selected null
subset manifold and SSA inner loop control law effectively limits the gain of undesirable
nonlinearities that exist in the open-loop system.
As a result, we may consider the SSA closed loop system to be in a ?reaching? phase while its
states are outside of the region of convergence of the outer loop law, and consider the system
to be in a ?surface tracking phase? when its states are in the region of convergence of the outer
loop law.
We illustrate these principles and the tuning of the integer parameter n in the auxiliary
input u2 in the following example
Example 6 SSA Bounded Subset Example Consider System 2 from Section 4.2.2
?x1 = x1 +x2 +x22 (4.14a)
?x2 = x1 +x22 +u (4.14b)
A subset of the state space x? IR2 is chosen x2 ? IR1 and is bounded with a Lyapunov control
designed input ulyp. As in section 4.2.2, we use a Lyapunov function V(x2) = 12x22 to design a
50
nonlinear feedback control for the dynamics of the state x2. We define a set of parameterized
subset stabilizing control inputs as
u = ?x1 ?x22 ?x2 ?x2n+12 (4.15)
where n is an integer that is used to adjust the ?gain? of the subspace stabilization control law
u(z,w) for large values of x2.
Phase portraits based on simulation results of the subset stabilized system are shown in
Figure 4.7(a). 2 The null subset manifold x2 = 0 is easily seen as x2(t) ? 0 as t ? ?.
An increase of the exponent integer parameter n in (4.15) will increase the control effort u for
large deviations x2 from the desired null space manifold. By consequence, the impact of the
nonlinearities involving x2 are decreased once the state trajectories come near to the null subset
manifold. As seen in Figure 4.7(b) and Figure 4.7(c) the corresponding state trajectories lead
the subset stabilized phase portraits to a smaller deviation from the null subspace manifold.
The phase portraits clearly illustrate that only the state x2 is regulated by the SSA controller
(4.15). We now examine the effects of the SSA control law in closed loop with the outer loop
LQR controller (see figure 4.6. The region of convergence (ROC) refers to the a region of state
space, ROC ? IR2, where the system ?x = f(x) converges to the origin as t ? ?. An increase
in the ROC area is a valid quantitative method to compare the SSA controller in figure 4.6 and
a traditional linear LQR control solution.
The traditional linear control solution uses an LQR optimization and the weights Q and R
(4.16).
Q =
?
?? 1 0
0 1
?
?? and R = 10 (4.16)
The required linear model for the LQR optimization is based on a linearization of the nonlinear
model (4.14a)-(4.14a) evaluated at the origin. The resulting control effort for LQR linear control
2a plot of the trajectory of x1(t) with respect to the trajectory of x2(t).
51
?1.5 ?1 ?0.5 0 0.5 1 1.5
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
Phase Portrait,
 SSA with ulin = 0, ulyp = ?x1?x22?x2
x1
x 2
= initial condition [x1(0),x2(0)]
(a) u = ?x1 ?x22 ?x2
?1.5 ?1 ?0.5 0 0.5 1 1.5
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
Phase Portrait,
 SSA with ulin = 0, ulyp = ?x1?x22?x2?x23
x1
x 2
= initial condition [x1(0),x2(0)]
(b) u = ?x1 ?x22 ?x2 ?x32
?1.5 ?1 ?0.5 0 0.5 1 1.5
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
Phase Portrait,
 SSA with ulin = 0, ulyp = ?x1?x22?x2?x25
x1
x 2
= initial condition [x1(0),x2(0)]
(c) u = ?x1 ?x22 ?x2 ?x52
Figure 4.7: Subspace stabilized system phase portrait. The integer parameter n in equation (4.15) is
varied from 0 (a) to 1 (b) to 2 (c). Observe that with increased values of n the phase portrait tends to
more closely approach the null subset manifold x2 = 0.
52
is a negative linear feedback with gain K.
K =
bracketleftbigg
5.3956 3.3002
bracketrightbigg
The SSA controller was designed with the SSA control law 4.15 in closed loop with an
outer loop LQR controller designed with the same matrices (4.16) and a different linearization.
Two versions of SSA control design are compared, n = 0 and n = 2. Observe in Figure 4.8(a)
that the area of the region of convergence (blue) of the SSA-based solution is comparable to
slightly greater than the area of the region of convergence corresponding to the simple linear
LQR approach (red). Notice further in Figure 4.8(b) that the region of convergence significantly
increases with the SSA control law parameter n.
?3 ?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5?7
?6
?5
?4
?3
?2
?1
0
1
2
x1
x 2
u = ?x1?x22?2x2?K*[x1;x2]
SSA?Based
Solution
n=0
Linear
(a) uSSA = ?x1 ?x22 ?2x2 + v
?3 ?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5?7
?6
?5
?4
?3
?2
?1
0
1
2
x1
x 2
u = ?x1?x22?x2?x23?K*[x1;x2]
SSA?based
Solution
n=1
LQR
Solution
(b) uSSA = ?x1 ?x22 ?x2 ?x32 + v
Figure 4.8: Region of Convergence, Comparison of SSA-based control and LQR control efforts for System
2
53
4.4 Conclusion
Five case studies are presented in the previous chapter in which existing well-known nonlin-
ear control design methods produced either control solutions with singularities or were unable
to produce any feasible control solution. In engineering practice, standard nonlinear control
design methods may not be mathematically tractable, or may result in control laws with sin-
gularities, or the design processes cannot be completed due to system structural deficiencies.
A SSA nonlinear design proves effective in all of these specific system examples. An inner
loop nonlinear control is first designed to stabilize a subset of the plant?s state variables. Only
those variables contributing to significant plant nonlinearities are stabilized by the inner loop
control. The inner loop control law also introduces a new control variable for outer loop con-
trol. An outer loop control designed by usual linear system methods completes the SSA control
law. The SSA control design approach combines the simplicity of linear design methods with
the improved region of stability of nonlinear design methods, yet avoids the pitfalls of some
standard nonlinear control design methods.
54
Chapter 5
Conclusions
In this chapter a description of conclusions drawn for the Subset Stabilization Approach
(SSA) are discussed. The major advantages of SSA control design are re-iterated and major
design considerations are discussed. A final argument is given for using SSA control design and
in which situations SSA should be used.
5.1 Design Advantages
The Subset Stabilization Approach design approach has a direct application to a large group
of systems and has proven itself usefulness for several specific nonlinear system examples, i.e.
the ball-on-beam example in Chapter 3 and the five other example systems in Section 4.2. In
each example, the traditional nonlinear design approaches failed to provide a valid control effort
or were structurally inapplicable. The linear control method provided control efforts that were
stable within small regions around the given operating point. The SSA-based control solutions
were able to increase the region of convergence volume for the linear control effort and ,in the
ball-on-beam example, provide a seemingly globally stable control.
SSA-based control solutions were not unique and provided tuning possibility, as discussed
in Section 4.3. The system in Example 6
?x1 = x1 +x2 +x22
?x2 = x1 +x22 +u
55
was simulated with multiple control solutions
u = ?x1 ?x22 ?x2 ?x2n+12 ?K
?
?? x1
x2
?
??
where n = 0 and n = 1 were chosen. As n increased, the control input magnitude increased
for large values of x2 and the system responded faster. As with input-output linearization,
input-state linearization, Integrator backstepping, and Lyapunov control design the response
of the system may be changed by the choice of the Lyapunov function used for the design
procedure,i.e. the energy function in example 6.
5.2 SSA for Nonlinear Control
The SSA successfully provided a control solution where traditional nonlinear control meth-
ods failed and improved the region of convergence for traditional linear control method solutions.
In Section 4.3 theory for SSA was given in which a bounded subset of states for the entire state
space provided advantages for traditional linear control solutions. The SSA control design tech-
nique gives a clear method for solving nonlinear control problems where traditional nonlinear
methods fail, and improves linear design solutions for the same systems.
56
Chapter 6
Future Work
In this chapter a clear description of subsequent future work will be discussed. In previ-
ous chapters, a subset stabilization approach (SSA) control design was introduced. Questions
related to the advantages of this approach and theoretical explanations where answered for spe-
cific examples. Although SSA clearly provides advantages, i.e. the ball-on-beam problem, these
advantages are not fully explored through stability and realworld implementation. The SSA
designed control effort found in Chapter 3 should be validated with a real world experiment
and the stability requirements for a generalized SSA designed control should be analytically
examined.
6.1 Experimental Validation of the Ball-on-beam Control Effort
In chapter 3 a SSA-based control design solution was tested on Matlab Simulink model/s-
function. Dr. Uran and Dr. Jezernik in Slovenia have built a ?ball?-on-beam system and
implemented their approach for ?ball?-on-beam control. Uran and Jezernik?s Ball and Beam
like mechanism replaces the ball by a cart which slides ?frictionless? on the beam, as in Figure
6.1 [21].
The Ball-on-beam mechanism above is under actuated, like the Ball-on-beam mechanism
presented in Chapter 3. The first degree of freedom is present in the angle of a beam which is
actuated by an electric direct drive motor. The second degree of freedom is represented by the
cart sliding on the beam which is actuated by the movement of the beam. With states defined
as x1 = r, x2 = ?r, x3 = ?, and x4 = ??, the dynamics for the ball-on-beam system, in figure 6.1
57
Figure 6.1: Uran and Jezernik Ball on Beam Apparatus
are given.
parenleftbigm
vr2 +IY Y P
parenrightbig ??+2m
vr?r ??+mvgrcos(?) = M?
mv?r?mvr ??2 +mvgsin(?) = 0
The variable r is the ball position, ? is the angle of the beam, mv is a mass of the ball, IY Y P
is inertia of the beam, g is the gravity constant and M? is an input torque. In Uran and
Jezernik?s ball-on-beam like mechanism the ball(cart) position r , the angle of the beam ?, as
well as the ball(cart) position speed and the beam angle speed are measured through various
sensors. Using the predefined states x1 = r, x2 = ?r, x3 = ?, and x4 = ??. Uran and Jezernik?s
ball-on-beam dynamics equations become
parenleftbigm
vx21 +IY Y P
parenrightbig ?x
4 + 2mvx1x2x4 +mvgx1 cos(x3) = M?
mv ?x2 ?mvx1x24 +mvgsin(x3) = 0
58
and are written as a set of simple state equations using the nonlinear invertible transformation
M? = mvgx1 cos(x3) +2mvx1x2x4 +vIY Y P +vmvx21.
?x1 = x2
?x2 = x1x24 ?gsin(x3)
?x3 = x4
?x4 = v
We are communicating with Dr. Uran to implement and compare the subset stabilization
approach with Uran and Jezernik?s control solution presented in [21]. In the following sections,
the SSA control solution is formulate and structurally compared with Uran and Jezernik?s
control solution.
6.1.1 Subset Stabilization Approach to [21]
The equations presented in [21] are used for the following design. The Lyapunov design
is chosen around the angle of the beam x3 and the angular rate of the beam x4, V(x3,x4) =
1
2
parenleftbigx2
3 +x24
parenrightbig. The derivative of V(x
3,x4) with respect to time is given below.
?V = x3x4 +x4v
We chose the following control effort
ulyp = ?x3 ?x4 ?x23x4
59
and supplemented it with a user defined input w.
?x1 = x2 (6.1)
?x2 = x1x24 ?gsin(x3)
?x3 = x4
?x4 = ?x3 ?x4 ?x23x4 +w
The system is linearized using the state matrices
A =
?
??
??
??
??
??
0 1 0 0
0 0 ?g 0
0 0 0 1
0 0 ?1 ?1
?
??
??
??
??
??
,B =
?
??
??
??
??
??
0
0
0
1
?
??
??
??
??
??
and the LQR command in Matlab. The following gains were used (Q=eye(4) and R=10).
K =
bracketleftbigg
0.3162 0.7335 5.7938 2.5620
bracketrightbigg
The invertible transformation used to simplify the dynamics of ?x4 were applied to the control
effort, and the control input is shown below.
M? = mvgx1 cos(x3) +2mvx1x2x4 +vIY Y P +vmvx21
v = ?x3 ?x4 ?x23x4 +w
w = ?K
?
??
??
??
??
??
x1
x2
x3
x4
?
??
??
??
??
??
60
0 5 10 15 20 25 30 35 40?0.5
0
0.5
1
Uran and Jezernik System Simulation 
 With SSA Control Solution
time
x 1
,x 2
,x 3
,x 4
Ball Position
Ball Rate
Beam Position
Beam Angular Rate
Figure 6.2: Uran and Jezernik System Simulation with SSA control Solution
Uran and Jezernik provided the following parametric values.
Beam Length = 0.8 m
mv = 3.15 kg
IY Y P = 1.248 kgm2
Stiction of Beam = 1.3 Nm
Stiction of Cart = 4.86 N
Coulomb Friction ? 30% of stiction
A Matlab Simulink simulation provided the following state responses.
6.1.2 Comparison of State Equations and Control Solution
The state equations used for Uran and Jezernik?s paper and this dissertation, chapter 3,
are very similar. In fact, they are identical with Iball ? 0.
61
Uran/Jezernik Equations
?x1 = x2
?x2 = x1x24 ?gsin(x3)
?x3 = x4
?x4 = M??2mvx1x2x4?mvgx1 cos(x3)(m
vx21+IYYP)
Chapter 3 Equations
?x1 = x2
?x2 = Cr parenleftbigmx1x24 +mgsin(x3)parenrightbig
?x3 = x4
?x4 = ??Cdx4+gmx1 cos(x3)(mx2
1+Ibeam+Iball)
where Cr =
parenleftBig
m+ Iballr2
parenrightBig?1
Uran and Jezernik used two cascaded computed torque controllers M?,UJ given by the
following two equations.
M?,UJ = parenleftbigmvr2 +IY Y Pparenrightbig ??z +2mvr?r ??+mvgrcos(?)
??z = KpP (?z ??)?KvP ??
?z = arcsin
bracketleftbigg ?1
mvg
parenleftbigm
v?rz ?mvr ??2
parenrightbigbracketrightbigg
?rz = ?r+KvV (?rz ? ?r) +KpV (rz ?r)
The SSA control solution M?,SSA uses a cascaded Lyapunov control ulyp, LQR gain control
w = ?Kx, and an invertible transformation.
M?,SSA = parenleftbigmvr2 +IY Y Pparenrightbig ulyp +2mvr?r ??+mvgrcos(?)
ulyp = ??? ????2 ??+w
w = ?K
?
??
??
??
??
??
r
?r
?
??
?
??
??
??
??
??
K =
bracketleftbigg
0.3162 0.7335 5.7938 2.5620
bracketrightbigg
62
The two control laws are very similar in their structure and require the same state feedback
variables. The differential dynamics introduced in Uran and Jezernik?s control are a major
difference between the two control solutions.
We are working together with Dr. Uran and Dr. Jezernik to implement the SSA-based
control solution in Slovenia. After the control is implemented, a comparison will be made with
the SSA-based design approach and Uran and Jezernik?s design approach. This work should
provide great insight into the robustness of the SSA control solution for the ball-on-beam
system.
6.2 Analytical Stability
The SSA-based nonlinear design proves effective in all the examples discussed in this dis-
sertation. We presented theoretical analysis in section 4.3 and provided an example to illustrate
the SSA method. These results still fail to answer analytical questions which are posed in the
SSA-based control solution. Is there an optimal SSA-based design method which will guaran-
tee the ?reaching? phase of the control to converge to the ?tracking? phase of the control, as
discussed in section 4.3? SSA-based control acts to decrease the impact of the system non-
linearities so that a robust control law designed about a given operating point may extend its
region of convergence in closed loop operation, but is this extension of the region of convergence
guaranteed? Even with these questions unanswered, the SSA-based design approach provides
a clear method for designing control solutions.
63
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65
Appendices
66
Appendix A
Five Systems Examples,Code:
function [tspan,xL,xH] = SimAllSys1_5(sys,x1i,x2i);
% Batch Calculation for all the System?s in Chapter 3
% [] = SimAllSys1_5(); %
%sys = 2; x1i = -2.5; x2i = 2;
plotit = 0;
warning off
tspan = [0:0.01:30]?;
switch(sys)
case 1
%x1i = 1;x2i = 0;
[tL1,xL1] = ode45(@sys1L,[tspan],[x1i x2i]);
[tH1,xH1] = ode45(@sys1H,[tspan],[x1i x2i]);
if(plotit),figure(1)
for ii=1:max(size(xH1,2),size(xL1,2))
subplot(max(size(xH1,2),size(xL1,2)),1,ii)
plot(tspan(1:length(xH1)),xH1(:,ii),...
tspan(1:length(xL1)),xL1(:,ii))
if(ii==1),title(?system 1?),end
legend(?hybrid?,?linear?)
ylabel(sprintf(?x_%1d?,ii))
67
end,pause(0.1)
[qq,gg] = find(xH1<=0.02*x1i);
[hh,gg] = find(xL1<=0.02*x1i);
if(~(length(qq)<1))
fprintf(?Hybrid settling time = %1.2f sec\n?,tH1(qq(1)))
fprintf(?Linear settling time = %1.2f sec\n?,tL1(hh(1)))
end
end
xL = xL1;
xH = xH1;
case 2
%x1i = 0.124;x2i = 0;
[tL2,xL2] = ode23tb(@sys2L,[tspan],[x1i x2i]);
[tH2,xH2] = ode23tb(@sys2H2,[tspan],[x1i x2i]);
if(plotit),figure(2)
for ii=1:max(size(xH2,2),size(xL2,2))
subplot(max(size(xH2,2),size(xL2,2)),1,ii)
plot(tspan(1:length(xH2)),xH2(:,ii),...
tspan(1:length(xL2)),xL2(:,ii))
if(ii==1),title(?system 2?),end
legend(?hybrid?,?linear?)
ylabel(sprintf(?x_%1d?,ii))
end,pause(0.1)
[qq,gg] = find(xH2<=0.02*x1i);
[hh,gg] = find(xL2<=0.02*x1i);
if(~(length(qq)<1))
68
fprintf(?Hybrid settling time = %1.2f sec\n?,tH2(qq(1)))
fprintf(?Linear settling time = %1.2f sec\n?,tL2(hh(1)))
end
end
xL = xL2;
xH = xH2;
case 3
%x1i = 0.1;x2i = 0;
[tL3,xL3] = ode45(@sys3L,[tspan],[x1i x2i]);
[tH3,xH3] = ode45(@sys3H,[tspan],[x1i x2i]);
if(plotit),figure(3)
for ii=1:max(size(xH3,2),size(xL3,2))
subplot(max(size(xH3,2),size(xL3,2)),1,ii)
plot(tspan(1:length(xH3)),xH3(:,ii),...
tspan(1:length(xL3)),xL3(:,ii))
if(ii==1),title(?system 3?),end
legend(?hybrid?,?linear?)
ylabel(sprintf(?x_%1d?,ii))
end,pause(0.1)
[qq,gg] = find(xH3<=0.02*x1i);
[hh,gg] = find(xL3<=0.02*x1i);
fprintf(?Hybrid settling time = %1.2f sec\n?,tH3(qq(1)))
fprintf(?Linear settling time = %1.2f sec\n?,tL3(hh(1)))
end
xL = xL3;
xH = xH3;
case 4
69
%x1i = 5;x2i = 0;
x3i = 0;
[tL4,xL4] = ode45(@sys4L,[tspan],[x1i x2i x3i]);
[tH4,xH4] = ode45(@sys4H,[tspan],[x1i x2i x3i]);
if(plotit),figure(4)
for ii=1:max(size(xH4,2),size(xL4,2))
subplot(max(size(xH4,2),size(xL4,2)),1,ii)
plot(tspan(1:length(xH4)),xH4(:,ii),...
tspan(1:length(xL4)),xL4(:,ii))
if(ii==1),title(?system 4?),end
legend(?hybrid?,?linear?)
ylabel(sprintf(?x_%1d?,ii))
end,pause(0.1)
[qq,gg] = find(xH4<=0.02*x1i);
[hh,gg] = find(xL4<=0.02*x1i);
fprintf(?Hybrid settling time = %1.2f sec\n?,tH4(qq(1)))
fprintf(?Linear settling time = %1.2f sec\n?,tL4(hh(1)))
end
xL = xL4;
xH = xH4;
case 5
%x1i = 6;x2i = 0;
x3i = 0;x4i = 0;
[tL5,xL5] = ode45(@sys5L,[tspan],[x1i x2i x3i x4i]);
[tH5,xH5] = ode45(@sys5H,[tspan],[x1i x2i x3i x4i]);
if(plotit),figure(5)
for ii=1:max(size(xH5,2),size(xL5,2))
70
subplot(max(size(xH5,2),size(xL5,2)),1,ii)
plot(tspan(1:length(xH5)),xH5(:,ii),...
tspan(1:length(xL5)),xL5(:,ii))
if(ii==1),title(?system 5?),end
legend(?hybrid?,?linear?)
ylabel(sprintf(?x_%1d?,ii))
end,pause(0.1)
[qq,gg] = find(xH5<=0.02*x1i);
[hh,gg] = find(xL5<=0.02*x1i);
fprintf(?Hybrid settling time = %1.2f sec\n?,tH5(qq(1)))
fprintf(?Linear settling time = %1.2f sec\n?,tL5(hh(1)))
end
xL = xL5;
xH = xH5;
end
%figure(1),hold on,plot(xH2(:,1),xH2(:,2),?b?)
figure(1),hold on,plot(xL2(:,1),xL2(:,2),?g?)
%************************** SYSTEMS
function dx = sys1H(t,x)
x1 = x(1); x2 = x(2);
Kh = [0.3162 0.2538];
u = -x1 - x2^2 - x2 - Kh*[x1;x2];
dx(1,1) = x2 + tanh(u);
dx(2,1) = x1 + x2^2 + u;
71
function dx = sys1L(t,x)
x1 = x(1); x2 = x(2);
K = [1.0477 1.0477];
u = - K*[x1;x2];
dx(1,1) = x2 + tanh(u);
dx(2,1) = x1 + x2^2 + u;
%************************** 2
function dx = sys2H(t,x)
x1 = x(1); x2 = x(2);
Kh = [4.1669 2.0714];
u = -x1-x2^2-x2;%-Kh*[x1;x2]; %-x1^2*x2 extra
dx(1,1) = x1 + x2 +x2^2;
dx(2,1) = x1 + x2^2 + u;
function dx = sys2H2(t,x)
x1 = x(1); x2 = x(2);
Kh = [4.1669 2.0714];
u = -x1-x2^2-x2-x2^5;%-Kh*[x1;x2]; %-x1^2*x2 extra
dx(1,1) = x1 + x2 +x2^2;
dx(2,1) = x1 + x2^2 + u;
function dx = sys2L(t,x)
x1 = x(1); x2 = x(2);
K = [5.3956 3.3002];
u = - K*[x1;x2];
72
dx(1,1) = x1 + x2 +x2^2;
dx(2,1) = x1 + x2^2 + u;
%************************** 3
function dx = sys3H(t,x)
x1 = x(1); x2 = x(2);
Kh = [0.3162 0.3162];
u = -x2 - Kh*[x1;x2];
dx(1,1) = sin(x2);
dx(2,1) = u;
function dx = sys3L(t,x)
x1 = x(1); x2 = x(2);
K = [0.3162 0.8558];
u = - K*[x1;x2];
dx(1,1) = sin(x2);
dx(2,1) = u;
%************************** 4
function dx = sys4H(t,x)
x1 = x(1); x2 = x(2); x3 = x(3);
Kh = [0.3162 4.9487 2.3162];
u = - x3 - Kh*[x1;x2;x3];
dx(1,1) = x2;
dx(2,1) = x2 + sin(x3);
dx(3,1) = u;
73
function dx = sys4L(t,x)
x1 = x(1); x2 = x(2); x3 = x(3);
K = [0.3162 4.0279 2.8558];
u = - K*[x1;x2;x3];
dx(1,1) = x2;
dx(2,1) = x2 + sin(x3);
dx(3,1) = u;
%************************** 5
function dx = sys5H(t,x)
x1 = x(1); x2 = x(2);
x3 = x(3); x4 = x(4);
Kh = [0.3162 1.2440 1.2889 0.9178];
u = -x4 - x4*x3^2 - x3 - Kh*[x1;x2;x3;x4];
dx(1,1) = x2;
dx(2,1) = x2*x4^2 + sin(x3);
dx(3,1) = x4;
dx(4,1) = u;
function dx = sys5L(t,x)
x1 = x(1); x2 = x(2);
x3 = x(3); x4 = x(4);
K = [0.3162 1.1969 2.1070 2.0770];
u = - K*[x1;x2;x3;x4];
dx(1,1) = x2;
dx(2,1) = x2*x4^2 + sin(x3);
dx(3,1) = x4;
74
dx(4,1) = u;
75
Appendix B
Code for Five System Phase Portrait
% State Space Portrait for 5 different systems
%
%
if(~exist(?LinearVal?)&~exist(?HybridVal?)),LinearVal = 0;
HybridVal = 0;end
figurenum = 20;
x2HvecROC = [];x1HvecROC = [];
x2LvecROC = [];x1HvecROC = [];
plotit = 0;
plotPP = 1; % plot phase portrait
skipitH=0;skipitL=0;
ii = 0; jjH=1;jjL=1;
PPL = 0.1; %for system 2, keep below 0.1 pp box, others =0.5
for sys = [2];
figurenum = figurenum+1;
for x1i = [-0.6:0.1:0.2];
for x2i = [-2,2]; %-0.5:0.1:0.5
[tspan,xL,xH] = SimAllSys1_5(sys,x1i,x2i);
if(size(xH,1)<2|size(xH,2)<2),skipitH=1;xH=zeros(2,2);
end
if(size(xL,1)<2|size(xL,2)<2),skipitL=1;xL=zeros(2,2);
76
end
ii = ii+1;
x1vector(ii) = x1i;
x2vector(ii) = x2i;
v1vectorH(ii) = xH(2,1)-xH(1,1);
v2vectorH(ii) = xH(2,2)-xH(1,2);
v1vectorL(ii) = xL(2,1)-xL(1,1);
v2vectorL(ii) = xL(2,2)-xL(1,2);
if(plotPP),figure(100+figurenum),plot(x1i,x2i,?b.?)
plot(x1i,x2i,?b.?), hold on,
plot(xH(:,1),xH(:,2),?b?),title(?hybrid?),
hold on, figure(200+figurenum)
plot(x1i,x2i,?b.?), hold on,
plot(xL(:,1),xL(:,2),?r:?),
title(?linear?), hold on, pause(0.01)
end
if(abs(xH(size(xH,1),1))<=PPL & ..
abs(xH(size(xH,1),2))<=PPL & ~skipitH),
HybridVal = HybridVal + 1; end
if(abs(xL(size(xL,1),1))<=PPL & ..
abs(xL(size(xL,1),2))<=PPL & ~skipitL),
LinearVal = LinearVal + 1; end
if(abs(xH(size(xH,1),1))<=PPL & ..
abs(xH(size(xH,1),2))<=PPL & plotit & ~skipitH)
figure(figurenum)
plot(x1i,x2i,?bx?), hold on, pause(0.01)
x1HvecROC(jjH) = x1i; x2HvecROC(jjH) = x2i;
77
jjH = jjH+1;
end
if(abs(xL(size(xL,1),1))<=PPL & ..
abs(xL(size(xL,1),2))<=PPL & plotit & ~skipitL)
figure(figurenum)
plot(x1i,x2i,?ko?), hold on, pause(0.01)
x1LvecROC(jjL) = x1i; x2LvecROC(jjL) = x2i;
jjL=jjL+1;
end
end
clc
fprintf(?Hybrid Value = %f?,HybridVal)
fprintf(?\nLinear Value = %f?,LinearVal)
skipitH=0;skipitL=0;
end
end
quiver(x1vector,x2vector,v1vectorH,v2vectorH,?b?), hold on
quiver(x1vector,x2vector,v1vectorL,v2vectorL,?r?)
78
Appendix C
Ball On Beam Example Code/Diagrams
C.1 Block Diagram For Ball-on-beam Model
79
C.2 S-Function for Nonlinear Plant
function [sys,x0,str,ts] = nonlinearplant(t,x,u,flag,a,b)
% s-function [sys,x0,str,ts] = nonlinearplant(t,x,u,flag, ..
% Makes the input memory
%
%
switch flag,
case 0, [sys,x0,str,ts]=mdlInitializeSizes(a,b);
case 1, sys=mdlDerivatives(t,x,u);
case 2, sys=mdlUpdate(t,x,u);
case 3, sys=mdlOutputs(t,x,u);
case 4, sys=mdlGetTimeOfNextVarHit(t,x,u);
case 9, sys=mdlTerminate(t,x,u);
otherwise error([?Unhandled flag = ?,num2str(flag)]);
end
function [sys,x0,str,ts]=mdlInitializeSizes(a,b);
sizes = simsizes;
sizes.NumContStates = 4;
sizes.NumDiscStates = 0; %+ny; % weights of NN are the
% states plus DeltaWstates
% (nx*nx = p)
sizes.NumOutputs = 4; %+ny;
sizes.NumInputs = 1; %+ny;
sizes.DirFeedthrough = 1;
80
sizes.NumSampleTimes = 1; % at least one sample time is needed
sys = simsizes(sizes);
x0 = [a b 0 0]; % [zeros(mdem*nx+ny,1)]; % initial conditions
str = []; % str is always an empty matrix
ts = [0]; % initialize the array of sample times
return
function sys=mdlDerivatives(t,x,u);
L = 1; % Length of Beam
gg = 9.81; % gravitational constant
mass = 0.067; % mass of the ball
rr = 0.0127; % radius of the ball
Iball = 0.4*mass*rr^2; % moment of inertia of the ball
Ibeam = (1/12)*2*mass*L^2; % moment of inertia of the beam
Cd = 10; % friction coefficient at the pivot
sys(1,1) = x(2);
sys(2,1) = (mass + Iball/rr^2)^(-1) * (mass*x(1)*x(4)*x(4) + ...
mass*gg*sin(x(3)));
sys(3,1) = x(4);
sys(4,1) = (Ibeam + Iball + mass*x(1)*x(1))^(-1) * ...
(u - Cd*x(4) + mass*gg*x(1)*cos(x(3)));
return
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function sys=mdlUpdate(t,x,u);
sys = [];
return
function y=mdlOutputs(t,x,u);
% output neural network output vector and weights
C = eye(4);
y = C*x;
return
%
%==================================================
% mdlGetTimeOfNextVarHit
%===================================================
%
function sys=mdlGetTimeOfNextVarHit(t,x,u)
sampleTime = 1; % Example, set the next
sys = t + sampleTime;
% end mdlGetTimeOfNextVarHit
%
%======================================================
% mdlTerminate
% Perform any end of simulation tasks.
%======================================================
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%
function sys=mdlTerminate(t,x,u)
sys = [];
% end mdlTerminate
C.3 Feedback M File
function u = feedback(input,method)
global K
K = [1.0741 0.6168 1.6836 0.3164];
Klin = [1.3867 1.8617 16.1673 0.0230]; % Linear Gain
Khyb = [0.3162 0.7815 4.6581 2.2274]; % Hybrid Gain
x1 = input(1);
x2 = input(2);
x3 = input(3);
x4 = input(4);
g = 9.81;
c1 = (5/7);
c2 = (g)*c1;
c3 = (22/5)*0.0127^2;
m = 0.05;
k1 = (180/pi);
k2 = (180/pi);
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% Control Efforts
switch lower(method)
case {?linear?}
u = -Klin*[x1;x2;x3;x4];
case {?hybrid?}
y = -Khyb*[x1;x2;x3;x4];
v = -x4-x4*x3^2-x3 + y;
u = 10.*x4-.6573*x1*cos(x3)+.1117e-1*v+.6700e-1*v*x1^2;
case {?inputoutput?}
v = -1/2*(x4*x2+g*cos(x3))*(sign(x1))*(1/max(x1,1));
u = - (9.81)*x1*cos(x3)*m+k1*(v)*m*c3 + k2*v*m*x1^2;
case {?lyapunov?}
v = max(min(-x4*x2*c1*x1-1/x4*x2*g*c1*sin(x3)...
-x3,100),-100);
%max(min(-x2*(x1+c1*x1*x4^2+g*c1*sin(x3))/x4,100),-100);
u = - (9.81)*x1*cos(x3)*m+k1*(v)*m*c3 + k2*v*m*x1^2;
otherwise
error(?Specify a case: linear,hybrid,inputoutput, or hybridks?)
end
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