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\begin{document}



%\addtolength{\hoffset}{-0.5cm}
%\newtheorem{lem}{Lemma}%[section]
%\newtheorem{thm}{Theorem}%[section]
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%\maketitle

%\begin{center}
%\date{\today}
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%\begin{frontmatter}

\title{Synergetic Regulator Synthesis for a Class of Nonlinear Systems  }

\author{Nusawardhana\thanks{ Cummins, Inc., 1460 National Road, Columbus, IN 47201}\;\;\; Jianming Lian\thanks{ Center for Advanced Power Systems, Florida State University, 2000 Levy Street, Tallahassee, FL 32310}\;\;\; Mohamed Zribi\thanks{Electrical Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait}\;\;\; Stanis{\l}aw H. \.{Z}ak\thanks{ School of Electrical and Computer Engineering,
Purdue University, 465 Northwestern Avenue, West Lafayette, IN 47907-2035}
}

%\author{Nusawardhana$^{\dagger}$,\thanks{$^{\dagger}$ Cummins, Inc., 1460 National Road, Columbus, IN 47201} Jianming Lian$^{\dagger}$, Mohamed Zribi$^{\ddagger}$\thanks{$^{\ddagger}$ Department of Mathematical Sciences, San Diego State University, San Diego, CA 92182.}, Stanislaw H. \.{Z}ak$^{\dagger}$}

\maketitle
%\author{Nusawardhana}
%\address{Cummins, Inc., \\1460 National Road, Columbus, IN 47201}
%%    \corauth[cor]{Corresponding author.}
%    \ead{nusa.nusawardhana@cummins.com}
%
%\author{Jianming Lian}
%\address{Center for Advanced Power Systems,\\ Florida State University,
%2000 Levy Street, Tallahassee, FL 32310}
%    \ead{jianming.lian@gmail.com}
%
%\author{Mohamed Zribi}
%\address{Electrical Engineering Department, Kuwait University\\P.O. Box 5969, Safat 13060, Kuwait}
%    \ead{mzribi10@yahoo.com}
%
%\author{Stanis{\l}aw H. \.{Z}ak\corauthref{cor}}
%\address{School of Electrical and Computer Engineering,
%  Purdue University, \\465 Northwestern Avenue, West Lafayette, IN 47907-2035}
%  \corauth[cor]{Corresponding author.}
%    \ead{zak@purdue.edu}

\begin{abstract}
The integral action method combined with the synergetic control design approach are employed to synthesize regulators for a class of nonlinear
dynamical systems.  The integral action is commonly used in the
design of controllers to track constant or almost constant
reference signals. The proposed method offers a closed-form
analytic solution. An attractive feature of the
proposed approach is that the control law is obtained by
solving the first-order differential equation. The stability
analysis of the closed-loop system driven by the proposed regulator is performed and illustrated with two simulation examples: the regulator design for a wing-rock suppression problem and a two-link manipulator.
\end{abstract}


{\bf Keywords:}
%\begin{keyword}
Synergetic control, invariant manifold, output regulation, integral
action
%\end{keyword}
%\end{frontmatter}


\section{Introduction}

%In this paper, we combine \emph{synergetic control} approach with \emph{sliding mode control} to synthesize a regulator for a class of dynamical systems.
An attractive feature of synergetic control is that solving
the first-order differential equation  provides a closed-form solution to the regulator problem.
For an elementary introduction to the subject of synergetic
control and its applications to power electronics, we recommend
Refs.~\cite{santi03,santi04,nwzakasme06,nwzakcrossleygnc07}. In this paper,
we combine a synergetic control approach with an integral action method in the regulator design. The
integral action is commonly used in the design of controllers
to track constant or almost constant reference
signals~\cite{behcetcorless02}.

The paper is organized as follows. In the following section, we
present elements of synergetic control approach that we employ
in this paper. Then, we apply the synergetic control approach to
the synthesis of a regulator using the integral
action.  Stability analysis of the closed-loop system is
performed, and an algorithm for the construction of a stable
invariant manifold, an essential element of  synergetic
regulator design, is formulated.  Finally, we test the proposed algorithm for the regulator construction to design regulators for a nonlinear wing-rock
suppression problem and a two-link robotic manipulator.


\section{Synergetic Control Method}
Consider a class of nonlinear dynamical systems modeled as
\begin{equation}\label{eq:nonlinear}
\dot{\bfx} = \b{f}( \bfx) + \bfb ( \bfx ) \bfu,
\end{equation}
where $\bfx$ is a vector of state variables of dimension $n$,
$\bfu$ is the input vector of dimension $m$, $\b{f}(\bfx)$ is a
state-dependent function of dimension $n$, $\bfb(\bfx)$ is a
state-dependent input function matrix of dimension $n\times m$
and it has full column rank everywhere.


Let $\bfsi = \b{s}(\bfx):\mathbb{R}^n\to \mathbb{R}^m$ be a
vector function of dimension $m$.


The objective of the control problem, discussed in this paper,
is to find a control policy that stabilizes
the system~(\ref{eq:nonlinear}) by
steering the system~(\ref{eq:nonlinear}) trajectories to a
manifold,
\begin{equation}
\cal{M} = \{\bfx : \bfsi = \mathbf{0}\} \label{eq:manifold}
\end{equation}
and force the trajectories to stay on the manifold
thereafter.

We further require that, (i) the system~(\ref{eq:nonlinear})
confined to the manifold $\cal{M} = \{\bfx : \bfsi = \mathbf{0}\}$
is asymptotically stable to the origin, (ii) $\bfsi=\b s(\b x)$ is
constructed so that the square $m\times m$ matrix $\bfs_{\bfx}
(\bfx) \bfb (\bfx)$ is invertible, where $\bfs_{\bfx}$ is the
Jacobian matrix of $\bfs(\bfx)$ with respect to $\bfx$. It is
shown in Refs.~\cite{nwzakasme06}, \cite{nwzakcrossleygnc07} that
when these two conditions are satisfied, the solution to the above
control problem is obtained by solving the associated
first-order ordinary differential equation
\begin{equation}\label{eq:odesoc}
\bfT \bfsid + \bfsi = \mathbf{0},
\end{equation}
where $\bfT\in \mathbb{R}^{m\times m}$ is a symmetric positive
definite matrix. The control policy is the solution to
the first-order differential equation~(\ref{eq:odesoc}), that is,
\begin{eqnarray}
\bfu &=& - (\bfT
\bfs_{\bfx}(\bfx)\bfb(\bfx))^{-1}\left(\bfT \bfs_{\bfx}(\bfx)
\b{f}(\bfx) +  \bfs ( \bfx)\right)\nonumber\\
&=& - (\bfs_{\bfx}(\bfx)\bfb(\bfx))^{-1}\bfs_{\bfx}(\bfx)
\b{f}(\bfx) - (\bfs_{\bfx}(\bfx)\bfb(\bfx))^{-1}\bfT^{-1}\bfs (
\bfx).
\label{eq:lawsoc}
\end{eqnarray}

The use of the control law~(\ref{eq:lawsoc}) in the nonlinear
dynamical systems~(\ref{eq:nonlinear}) makes the invariant
manifold $\cal{M} = \{\bfx:\bfsi=\mathbf{0}\}$ attractive. The
control law~(\ref{eq:lawsoc}) drives closed-loop trajectories
of~(\ref{eq:nonlinear}) from any initial condition towards the
invariant manifold $\cal{M}$. As stated above, $\bfsi$ is constructed in such a way that
all trajectories of~(\ref{eq:nonlinear}) confined on $\cal{M}$
converge to the origin. Therefore, the stability of the
closed-loop trajectories is achieved in two modes: (i) closed-loop
trajectories converging to the manifold $\cal{M}
=\{\bfx:\bfsi=\mathbf{0}\}$, and (ii) closed-loop trajectories
moving along the manifold $\cal{M} =\{\bfx:\bfsi=\mathbf{0}\}$
towards the origin.

The synergetic control design method employs the following two-step design procedure:
\begin{enumerate}
    \item Construct an asymptotically stable invariant manifold $\cal{M} =\{\bfx:\bfsi=\mathbf{0}\}$,
    (i.e. all trajectories of the system confined to $\cal{M}$ converge asymptotically to the origin).
    \item Construct the controller ${\bfu}$ using~(\ref{eq:lawsoc}).
\end{enumerate}


The synergetic control approach shares similarity with that of the
 variable structure sliding mode control~(SMC).
This relation was discussed in Refs.~\cite{santi03},
\cite{santi04}, \cite{nwzakasme06}, \cite{nwzakcrossleygnc07}.
It was observed in Refs.~\cite{nwzakasme06},
\cite{nwzakcrossleygnc07} that the first term of the control
policy~(\ref{eq:lawsoc}) is the so-called ``{equivalent
control},'' while the second term may be viewed as an
``{approximation to the discontinuous control law}'' of the
SMC. Additionally, the method of constructing a stable
invariant manifold proposed in \cite{Zak03} will be directly applied to the
construction of the invariant manifold in this paper.

In the development of the proposed regulator, we
adopt the integral tracking method  used in the sliding mode
tracking controller design described in
Ref.~\cite{edwardsspurgeon98}. The resulting controller drives
the closed-loop dynamics so that the integral of the tracking
error converges to zero. This is essentially a regulation
problem of the ``integral error'' for which regulation methods
using synergetic control~\cite{nwzakasme06, nwzakcrossleygnc07}
can be applied.

\section{System Description}
The class of
nonlinear systems that we consider is modeled by
\begin{eqnarray}
\dot{ \bfx } &=& \b{f}( \bfx) + \bfb ( \bfx ) \bfu\nonumber\\
&=& \bfa(\bfx) \bfx + \hat{\bfb} \tilde{\bfu}.
\label{eq:nonlinear-controllable}
\end{eqnarray}
We assume that the system~(\ref{eq:nonlinear-controllable})  is
transformable into the following ``regular form''~\cite{utkin92},
\cite{edwardsspurgeon98},
\begin{equation}\label{eq:regular}
\left[\begin{array}{c} \dot{\bfx}_1 \\
\dot{\bfx}_2 \end{array}\right] = \left[\begin{array}{cc}
\bfa_{11} &
\bfa_{12} \\
\bfa_{21}(\bfx) & \bfa_{22}(\bfx) \end{array}\right]
\left[\begin{array}{c} \bfx_1
\\ \bfx_2 \end{array}\right] + \left[\begin{array}{c}
\b{O} \\ \bfb_u \end{array}\right] \tilde{\bfu},
\end{equation}
where the state vector $\bfx_1$ is in $\mathbb{R}^{n-m}$, the
state vector $\bfx_2$ is in $\mathbb{R}^m$, and the control
vector $\bfu$ is in $\mathbb{R}^m$. The matrix $\b{O} \in
\mathbb{R}^{(n-m)\times m}$ is a matrix with zero entries. The
matrix $\bfb_u \in \mathbb{R}^{m \times m}$ is invertible and
the matrices $\bfa_{ij}$, $i=1,2$, $j=1,2$ have appropriate
dimensions. Note that $\bfa_{1j}$'s, $j=1,2$, are constant
matrices, while $\bfa_{2j}$'s, $j=1,2$, are state-dependent
matrices. For a constructive algorithm that can be used to
transform~(\ref{eq:nonlinear-controllable}) into the regular
form given by~(\ref{eq:regular}), we refer to
Ref.~\cite[p.~66]{edwardsspurgeon98}.

An example of physical systems that can be modeled using the above non-linear regular form are natural Lagrangian systems. A simple example of a natural Lagrangian system whose model can be represented inn the regular form is a planar two-link robot. Indeed, let $\b x_1=\left[\theta_1\; \theta_2\right]^\top$ be the vector of joint angles, $\b\tau=[\tau_1\;\tau_2]^\top$ be the vector of torques applied to the joints. A detailed modeling of such a robot can be found
in~\cite[p.~396]{slotineli91}, and its dynamics are given by
\begin{equation}\label{eqn:tlkoriginal}
\b H(\b x_1)\ddot{\b x}_1+\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 + \b g(\b x_1)=\b\tau,
\end{equation}
where $\b H(\b x_1)$ is the manipulator inertia matrix and $\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1$ is the vector of centripetal and Coriolis torques, and $\b g(\b x_1)=\left[\begin{array}{cc}
g_1(\b x_1) & g_2(\b x_1)
\end{array}\right]^{\top}$ is the vector of gravitational torques. Because the inertia matrix $\b H(\b x_1)$ is positive definite over the whole workspace, we know that $\b H^{-1}(\b x_1)$ exists and is also positive definite. Let
\[
\b G(\b x_1)=\left[\begin{array}{cc}
\frac{g_1(\b x_1)}{x_1} & 0\\
0 & \frac{g_2(\b x_1)}{x_2}
\end{array}\right].
\]
Using the above we can represent the term $\b g(\b x_1)$ as
\begin{equation}
\b g(\b x_1)=\b G(\b x_1)\b x_1.
\label{eqn:gterm}
\end{equation}
Taking into account~\eqref{eqn:gterm}, we can rewrite~\eqref{eqn:tlkoriginal} as
\begin{equation}\label{eqn:tlk}
\ddot{\b x}_1=-\b H^{-1}(\b x_1)\b
C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 - \b H^{-1}(\b x_1)\b G(\b x_1)\b x_1 + \b H^{-1}(\b x_1)\b\tau.
\end{equation}
Let $\b x_2=\dot{\b x}_1$ and $\tilde{\b u}=\b H^{-1}(\b x_1)\b\tau$.
Then, the dynamics of the planar two-link robot modeled by~\eqref{eqn:tlk} can be represented in the form
of~\eqref{eq:regular}, that is,
\begin{equation}\label{eq:two_link_regular}
\left[\begin{array}{c} \dot{\bfx}_1 \\
\dot{\bfx}_2 \end{array}\right] = \left[\begin{array}{cc}
\b O &
\b I_2 \\
- \b H^{-1}(\b x_1)\b G(\b x_1) & -\b H^{-1}(\b x_1)\b
C(\b x_1,\b x_2) \end{array}\right]
\left[\begin{array}{c} \bfx_1
\\ \bfx_2 \end{array}\right] + \left[\begin{array}{c}
\b{O} \\ \b I_2 \end{array}\right] \tilde{\b u}.
\end{equation}


\section{Regulator Design}
Suppose that the output $\bfy\in \mathbb{R}^p$ of system~(\ref{eq:regular}) has the form,
\begin{equation}
\bfy = \left[\begin{array}{cc} \bfc_1 & \bfc_2
\end{array}\right]\left[\begin{array}{c}
\bfx_1 \\ \bfx_2 \end{array}\right].
\end{equation}
Our objective is to design a regulator so that the output $\b y$
tracks a constant reference vector $\bfr \in
\mathbb{R}^p$. We now introduce the vector variable $\bfx_r$,
defined by the following differential equation,
\begin{equation}\label{eq:errordynamic}
\dot{\bfx}_r = \bfr - \bfy.
\end{equation}
The controller should be designed in such a way so that
\[
\lim_{t\to \infty} \dot{\b x}_r(t)=\b 0.
\]


To proceed, we define the augmented state variable
vector,
\begin{equation}\label{eq:augmentedvector} \tilde{\bfx}
= \left[\begin{array}{c} \bfxr \\ \bfx
\end{array}\right] = \left[\begin{array}{c}
\bfxr \\ \hline \\ \underbrace{\left \{ \begin{array}{c} \bfx_1 \\
\bfx_2 \end{array} \right \}}_{\bfx}  \end{array}\right]\in \mathbb{R}^N,
\end{equation}
where $N=n+p$. The dynamics of the overall augmented system are governed by the
state equation,
\begin{equation}\label{eq:augmenteddynamics}
 \left[\begin{array}{c} \dot{\bfx}_r \\
\dot{\bfx}_1 \\
\dot{\bfx}_2 \end{array}\right] =
\underbrace{\left[\begin{array}{ccc} \b{O} &
-\bfc_1 & -\bfc_2 \\
\b{O} & \bfa_{11} & \bfa_{12} \\
\b{O} & \bfa_{21}(\bfx) & \bfa_{22}(\bfx)
\end{array}\right]}_{\tilde{\bfa}} \left[\begin{array}{c} \bfxr \\
\bfx_1 \\ \bfx_2
\end{array}\right] + \underbrace{\left[\begin{array}{c} \b{I} \\ \b{O} \\ \b{O}
\end{array}\right]}_{\bfb_r} \bfr +
\underbrace{\left[\begin{array}{c} \b{O} \\ \b{O} \\ \bfb_u
\end{array}\right]}_{\tilde{\bfb}} \bfu.
\end{equation}
Note that the matrices in the augmented state
equation~(\ref{eq:augmenteddynamics}) are labeled as
$\tilde{\bfa}$, $\bfb_r$, and $\tilde{\bfb}$, respectively.
Hence, the augmented system model~(\ref{eq:augmenteddynamics})
can be represented as
\begin{equation}\label{eq:compact_augmented}
\dot{\tilde{\bfx}} = \tilde{\bfa} \tilde{\bfx} + \bfb_r \bfr +
\tilde{\bfb} \bfu.
\end{equation}
We make the following assumption:

{\bf Assumption 1}: The pair
\[
\left(\left[\begin{array}{cc}
\b O & -\b C_1\\
\b O & \b A_{11}
\end{array}\right], \quad \left[\begin{array}{c}
-\b C_2\b B_u\\
\b A_{12}\b B_u
\end{array}\right]\right)
\]
is controllable.


An essential element of the synergetic controller design is the construction of
an invariant manifold $\bfsi=\b 0$. Let the manifold function $\bfsi$ be defined as
\begin{equation}\label{eq:manifoldfunction}
\bfsi = \bfsi(\tilde{\bfx})= \bfsi(\bfxr, \bfx_1, \bfx_2) = -
\bfS_r \bfxr + \bfS_{1} \bfx_1 + \bfS_{2} \bfx_2 =
\left[\begin{array}{ccc} -\bfS_r & \bfS_{1} & \bfS_{2}
\end{array}\right] \tilde{\bfx} = \tilde{\bfS}\tilde{\bfx}.
\end{equation}
The negative sign in front of $\b S_r$ is for the purpose of simplified notation.
Proper construction of $\bfsi$ means that the system modeled
by~(\ref{eq:augmenteddynamics}) restricted to the manifold
$\cal{M}=\{\tilde{\bfx}:\bfsi=\mathbf{0}\}$ is asymptotically stable,
that is, the  trajectories of the system restricted to the
manifold $\cal{M}$ converge asymptotically toward the origin. A
construction of such $\tilde{\bfS}$ is provided in the
section. We assume that
\begin{equation}\label{eq:detSB}
 \det\left( \tilde{\bfS}\tilde{\bfb}
\right) = \det\left(\b S_2 \b B_u  \right) \ne 0.
\end{equation}
Using~\eqref{eq:odesoc}, we perform manipulations to obtain,
\begin{eqnarray}
\mathbf{0}&=&\bfT \dot{\bfsi} + \bfsi \nonumber \\
&=& \bfT\tilde{\bfS} \dot{\tilde{\bfx}} +
\tilde{\bfS} \tilde{\bfx} \nonumber \nonumber \\
&=& \bfT\tilde{\bfS} (\tilde{\bfa} \tilde{\bfx} + \bfb_r \bfr +
\tilde{\bfb}
\tilde{\bfu}) + \tilde{\bfS} \tilde{\bfx} \nonumber \nonumber \\
&=& \bfT\tilde{\bfS}\tilde{\bfa} \tilde{\bfx} + \bfT \tilde{\bfS}
\bfb_r \bfr + \bfT \tilde{\bfS} \tilde{\bfb}\tilde{\bfu} + \tilde{\bfS}
\tilde{\bfx}. \label{eq:raw4control}
\end{eqnarray}
%\begin{equation}
%\bfu = -(\bfS_{2}\bfb_u)^{-1} \left\{\tilde{\bfS}\tilde{\bfa} \tilde{\bfx}
%-\bfS_r \bfr + T^{-1} \tilde{\bfS} \tilde{\bfx}\right\}
%\end{equation}
It follows from~(\ref{eq:augmenteddynamics}) that $\tilde{\bfS}\bfb_r = -\bfS_r$.
Hence, we can re-write~(\ref{eq:raw4control}) as
\begin{equation}\label{eq:clean4control}
 \mathbf{0} =\bfT\tilde{\bfS}\tilde{\bfa} \tilde{\bfx} - \bfT \tilde{\bfS}_r \bfr
 + \bfT \tilde{\bfS} \tilde{\bfb}\tilde{\bfu} + \tilde{\bfS}
\tilde{\bfx}.
\end{equation}
Solving~(\ref{eq:clean4control}) for $\bfu$ yields
\begin{equation}\label{eq:soclawtracking}
\fbox{$\displaystyle \tilde{\bfu} = -(\tilde{\bfS}\tilde{\bfb})^{-1}(\tilde{\bfS} \tilde{\bfa}
\tilde{\bfx} + \bfT^{-1} \tilde{\bfS} \tilde{\bfx} - \bfS_r \bfr)$}
\end{equation}


Substituting~(\ref{eq:soclawtracking}) into~(\ref{eq:augmenteddynamics}),
we obtain closed-loop system dynamics model,
\begin{equation}\label{eq:closedlooptracking}
\dot{\tilde{\bfx}} = \left(\b{I} - \tilde{\bfb}
(\tilde{\bfS}\tilde{\bfb})^{-1} \tilde{\bfS}\right)\tilde{\bfa}
\tilde{\bfx} - \tilde{\bfb} (\tilde{\bfS}\tilde{\bfb})^{-1}
\bfT^{-1}\tilde{\bfS} \tilde{\bfx} +
\tilde{\bfb}(\tilde{\bfS}\tilde{\bfb})^{-1}\bfS_r \bfr+\bfb_r
\b{r}.
\end{equation}

In the following section, we show that the control law given by~\eqref{eq:soclawtracking} forces the above closed-loop system to track the reference signal $\b r$.


%\section{Constructing the Invariant Manifold}
\section{Analysis of the Closed-Loop System Behavior}
To proceed with our analysis, we need the following lemma, which can be viewed as a corollary of Theorem~6.2 in~\cite[p.~333]{Zak03}; see also~\cite{zakhui93a}.
\begin{lem}\label{lem_pre_construct}
Consider the system model given by~\eqref{eq:augmenteddynamics}, where
$\mb{rank}\;(\tilde{\b B})=m$, Assumption~1 is satisfied, and a
given set of $N-m$ complex numbers,
$\{\lambda_1,\ldots,\lambda_{N-m}\}$ is such that any $\lambda_i$
whose imaginary part is nonzero appears in the above set in a
conjugate pair. Then, there exist full-rank matrices $\b W\in
\mathbb{R}^{N\times (N-m)}$ and $\b W^g\in \mathbb{R}^{(N-m)\times
N}$ such that
\[
\b W^g\b W=\b I_{N-m},\quad \b W^g\tilde{\b B}=\b O,
\]
and the eigenvalues of $\b W^g\b A\b W$ are
\[
\mb{eig}(\b W^g\tilde{\b A}\b W)=\{\lambda_1,\ldots, \lambda_{N-m}\}.
\]
\end{lem}
{\bf Proof: }  We begin our proof by
defining an $N\times (N-m)$ real matrix of the form,
\begin{equation}\label{eq:v11}
\bfV = \left[\begin{array}{c}
\b I_{N-m}\\
\hline  \b O
\end{array}\right].
\end{equation}
Therefore,
\[
\left[\begin{array}{cc} \bfV & \tilde{\bfb}\end{array}\right]^{-1}
= \left[\begin{array}{cc}\b I_{N-m} & \bfO \\ \bfO &
\bfb_u\end{array}\right]^{-1} = \left[\begin{array}{cc}
\b I_{N-m} &\bfO \\ \bfO & \bfb_u^{-1} \end{array} \right] =
\left[\begin{array}{c} \bfV^g \\ \tilde{\b B}^g \end{array}\right].
\]
Hence,
\[
\left[\begin{array}{c} \bfV^g \\ \tilde{\bfb}^g \end{array}\right]\left[\begin{array}{cc} \bfV & \tilde{\bfb}\end{array}\right]=\left[\begin{array}{cc}
\b V^g\b V & \b V^g\tilde{\bfb}\\
\tilde{\bfb}^g\b V & \tilde{\bfb}^g\tilde{\b B}
\end{array}\right]=\left[\begin{array}{cc}
\b I_{N-m} & \b O\\
\b O & \b I_m
\end{array}\right]=
\b I_N.
\]
Note that
\begin{equation}
\b V^g\b V=\b I_{N-m}\;\;\mbox{ and }\;\; \b V^g\tilde{\b B}=\b O.
\label{eq:provgandb}
\end{equation}
By Assumptions~1, the pair
\[
\left(\bfV^g \tilde{\bfa} \bfV,   \bfV^g
\tilde{\bfa} \tilde{\bfb}\right)=\left(\left[\begin{array}{cc}
\b O & -\b C_1\\
\b O & \b A_{11}
\end{array}\right], \quad \left[\begin{array}{c}
-\b C_2\b B_u\\
\b A_{12}\b B_u
\end{array}\right]\right)
\]
is controllable. Therefore, for any set of complex numbers,
\[
\left\{\lambda_1,\lambda_2,\ldots,\lambda_{N-m}  \right\}
\]
symmetric with respect to the real axis, we can always find a
matrix $\b F$ such that
\[
\{\lambda_1,\lambda_2,\ldots,\lambda_{N-m}\} =
\mbox{eigenvalues} \left(\bfV^g \tilde{\bfa} \bfV - \bfV^g
\tilde{\bfa} \tilde{\bfb} \b{F}\right).
\]
The matrix $\b{F}$ can be determined using any available method
for pole placement. We proceed by performing the
following manipulations,
\begin{equation}
\bfV^g \tilde{\bfa} \bfV - \bfV^g \tilde{\bfa} \tilde{\bfb}
\b{F}= \bfV^g\tilde{\bfa}\left(\bfV-\tilde{\bfb}\b{F}\right) =
\b{W}^g \tilde{\b{A}} \b{W}, \label{eq:fromvtow}
\end{equation}
where $\bfW^g=\bfV^g$ and $\bfW = \bfV-\tilde{\bfb}\b{F}$. Note
that by~(\ref{eq:provgandb}),
\begin{equation}
\b W^g\b W = \b V^g\left(\bfV-\tilde{\bfb}\b{F}  \right) = \b
V^g\b V = \b I_{N-m}. \label{eq:wgwisinmm}
\end{equation}
We also have, $\b V^g\tilde{\b B}=\b W^g\tilde{\b B}=\b O$, which completes the proof of the lemma.\\
\hspace*{5.5in} $\Box$
%\end{proof}

\begin{thm}\label{thm_tracking}
Suppose that the assumptions of Lemma~\ref{lem_pre_construct} are satisfied. Then there exists the invariant manifold matrix $\tilde{\b S}\in\mathbb{R}^{m\times N}$
such that $\tilde{\b S}\tilde{\b B}=\b I_m$ and the closed-loop system~\eqref{eq:closedlooptracking} tracks the reference signal $\b r$, that is, $\b y(\infty)=\b r$.
\end{thm}
{\bf Proof: }  Consider the matrix, $\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]$. Note that
\[
\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]=\left[\begin{array}{cc}
\b I_{N-m} & \b O\\
\b F & \b I_m
\end{array}\right]\left[\begin{array}{c}
\b V^g\\
\tilde{\bfb}^g
\end{array}\right].
\]
Thus the matrix $\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]$ is invertible and its inverse has the form
\[
\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^g\\
\tilde{\bfb}^g+\b F\b V^g
\end{array}\right].
\]
Let
\[
\tilde{\b S}=\tilde{\bfb}^g+\b F\b V^g=\left[\begin{array}{cc}
\b F & \b B_u^{-1}
\end{array}\right].
\]
Note that by~\eqref{eq:provgandb},
\[
\tilde{\b S}\tilde{\bfb}=\left(\tilde{\bfb}^g+\b F\b V^g\right) \tilde{\bfb}=\b I_m.
\]
Consider the following state-space transformation,
\[
\b z=\left[\begin{array}{c}
\b z_1\\
\b z_2
\end{array}\right]=\left[\begin{array}{c}
\b W^g\\
\tilde{\b S}
\end{array}\right]\tilde{\b x}=\left[\begin{array}{c}
\b V^g\\
\tilde{\b S}
\end{array}\right]\tilde{\b x}.
\]
Note that $\b z_2=\tilde{\b S}\tilde{\b x}=\b{\sigma}$. We also
have
\begin{equation}\label{eq:transformmatrix}
\left[\begin{array}{c} \bfW^g \\ \tilde{\bfS}
\end{array}\right]\left[\begin{array}{cc} \bfW & \tilde{\bfb}
\end{array}\right] = \left[\begin{array}{cc}
\bfW^g\bfW & \bfW^g \tilde{\bfb}\\
\tilde{\bfS}\bfW & \tilde{\bfS}\tilde{\bfb}\end{array}\right] =
\left[\begin{array}{cc} \b{I}_{N-m} & \b{O} \\
\b{O} & \b{I}_m\end{array}\right],
\end{equation}
where $\b{I}_{N-m}$ is the identity matrix of dimension
$(N-m)\times (N-m)$. Therefore, the matrices in~(\ref{eq:transformmatrix}) satisfy the
following relations,
\begin{equation}\label{eq:Wproperties}
\left.
\begin{array}{cc}
\begin{array}{rcl}
\bfW^g \bfW &=& \b{I}_{N-m},\\
\quad \tilde{\bfS}\bfW &=& \b{O} \in \mathbb{R}^{m\times (N-m)},\\
\end{array} &
\begin{array}{rcl}
\bfW^g \tilde{\bfb} &=& \b{O} \in \mathbb{R}^{(N-m)\times m},\\
\quad \tilde{\bfS}\tilde{\bfb}&=&\b{I}_{m}.
\end{array}
\end{array}\right\}
\end{equation}
Using matrix relations shown
in~(\ref{eq:Wproperties}), we represent the closed-loop dynamics
in the $\bfz$ coordinates as
\begin{eqnarray}
\left[\begin{array}{c} \dot{\bfz}_1 \\
\dot{\bfz}_2 \end{array}\right] &=& \left[\begin{array}{c} \bfW^g \\
\tilde{\bfS} \end{array}\right] (\tilde{\bfa} -
\tilde{\bfb}(\tilde{\bfS}\tilde{\bfb})^{-1}\tilde{\bfS}\tilde{\bfa}
- \tilde{\bfb}(\tilde{\bfS}\tilde{\bfb})^{-1}
\bfT^{-1}\tilde{\bfS}) \left[\begin{array}{cc} \bfW & \tilde{\bfb}
\end{array}\right] \left[\begin{array}{c} \bfz_1 \\ \bfz_2
\end{array}\right] \nonumber \\ && \quad
- \left[\begin{array}{c} \bfW^g \\ \tilde{\bfS} \end{array}\right]
\tilde{\bfb}(\tilde{\bfS}\tilde{\bfb})^{-1} \tilde{\bfS}\bfb_r
\bfr + \left[\begin{array}{c} \bfW^g \\ \tilde{\bfS}
\end{array}\right] \bfb_r \b{r}
\nonumber  \\
&=& \left[\begin{array}{cc} \bfW^g \tilde{\bfa} \bfW & \bfW^g
\tilde{\bfa}
\tilde{\bfb} \\
\b{O} & -\bfT^{-1} \end{array}\right] \left[\begin{array}{c}\bfz_1 \\
\bfz_2 \end{array}\right] + \left[\begin{array}{c} \bfW^g \\
\tilde{\bfS} \end{array}\right]\left(-
\tilde{\bfb}(\tilde{\bfS}\tilde{\bfb})^{-1} \tilde{\bfS} \bfb_r
+\bfb_r\right)\bfr \nonumber \\
&=&\left[\begin{array}{cc} \bfW^g \tilde{\bfa} \bfW & \bfW^g
\tilde{\bfa}
\tilde{\bfb} \\
\b{O} & -\bfT^{-1} \end{array}\right] \left[\begin{array}{c}\bfz_1 \\
\bfz_2 \end{array}\right] + \left[\begin{array}{c} \b{O} + \b{W}^g \b{B}_r \\
-\tilde{\b S} \bfb_r + \tilde{\b S} \bfb_r \end{array}\right]\b{r}    \nonumber \\
&=& \left[\begin{array}{cc} \bfW^g \tilde{\bfa} \bfW & \bfW^g
\tilde{\bfa}
\tilde{\bfb} \\
\b{O} & -\bfT^{-1} \end{array}\right] \left[\begin{array}{c}\bfz_1 \\
\bfz_2 \end{array}\right] + \left[\begin{array}{c} \b{W}^g \bfb_r
\\ \b{O} \end{array}\right] \b{r}.
\label{eq:closedloop_z}
\end{eqnarray}
Observe that $\b{W}^g\tilde{\bfb}(\tilde{\b S} \tilde{\b
B})^{-1}\tilde{\b S} \bfb_r = \b O$ because ${\bfW^g
\tilde{\bfb}=\b{O}}$. Since $\bfT$ is positive definite, the eigenvalues of $-\bfT^{-1}$
are all in the open left-hand complex plane. This implies
that the dynamics of $\bfz_2=\b{\sigma}$ are asymptotically
stable, that is, the invariant manifold, $\{\b{\sigma}=\b 0\}$,
is asymptotically attractive. By~(\ref{eq:fromvtow}), the
matrix $\bfW$ can be chosen so that the eigenvalues of the
constant matrix $\bfW^g \tilde{\bfa} \bfW$ are in the open
left-hand complex plane. In
the steady-state the overall system trajectories converge to
the point on the manifold, $\{\b z_2=\b{\sigma}=\b 0\}$, given
by
\begin{equation}
\b{0} = \b W^g \tilde{\b A} \b W \b z_1 + \b W^g \bfb_r \b r.
\label{eq:steady-state-z1-r}
\end{equation}
We partition $\b F$ as
\begin{equation}\label{eq:assume-F}
\b F = \left[\begin{array}{cc} \b F_1 & \b F_2 \end{array}\right],
\end{equation}
where $\b F_1 \in \mathbb{R}^{m \times p}$ and $\b F_2 \in
\mathbb{R}^{m \times (n-m)}$. We have,
\begin{equation}\label{eq:w-from-assume}
\b W = \b V - \tilde{\b B} \b F = \left[\begin{array}{cc}
    \b I_{p} & \b{O} \\
    \b{O} & \b I_{n-m} \\
    -\b B_u \b F_1 & - \b B_u \b F_1 \end{array}\right] \quad \mbox{and} \quad
    \b W^g \bfb_r = \left[\begin{array}{l}\b I_p \\ \b{O}_{(n-m) \times p} \end{array}\right].
\end{equation}
Recall that $\b W^g=\b V^g$. Using the above and the fact that
\[
\b z_1=\b W^g\tilde{\b x}=\left[\begin{array}{c}
\b x_r\\
\b x_1
\end{array}\right],
\]
we express~(\ref{eq:steady-state-z1-r}) in the form,
\begin{eqnarray*}
\b{0} &=& \b V^g \left[\begin{array}{ccc} \b O & - \b C_1 & - \b
C_2 \\
\b O & \b A_{11} & \b A_{12} \\
\b O & \b A_{21}(\b x) & \b A_{22}(\b x) \end{array}\right] \b W
\b z_1 + \left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r \\
& = & \left[\begin{array}{cc}
    \b C_2 \b B_u \b F_1 & -\b C_1 + \b C_2 \b B_u \b F_2 \\
    -\b A_{12} \b B_u \b F_1 & \b A_{11} - \b A_{12} \b B_u \b F_2
    \end{array}\right] \b W^g \tilde{\b x}+ \left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r \\
& = & \left[\begin{array}{cc}
\b C_2 \b B_u \b F_1 & -\b C_1 + \b C_2 \b B_u \b F_2 \\
    -\b A_{12} \b B_u \b F_1 & \b A_{11} - \b A_{12} \b B_u \b F_2
    \end{array}\right] \left[\begin{array}{c}\b x_r \\ \b x_1
    \end{array}\right] +\left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r,
\end{eqnarray*}
or, equivalently,
\begin{eqnarray}
\b{0} &=& \b C_2 \b B_u \b F_1 \b x_r -\left(\b C_1 - \b C_2 \b
B_u \b
F_2 \right)\b x_1 + \b r, \label{eq:first-row} \\
\b{0} &=& -\b A_{12} \b B_u \b F_1 \b x_r + \left( \b A_{11} - \b
A_{12} \b B_u \b F_2 \right) \b x_1. \label{eq:second-row}
\end{eqnarray}
%From the matrix equality~(\ref{eq:Wproperties}),
%%\[\left[\begin{array}{c}\b W^g \\ \tilde{\b S} \end{array}\right]
%%\left[\begin{array}{cc} \b W & \tilde{\b B} \end{array}\right],
%%\]
%we get
%\begin{equation}
%\left. \begin{array}{ccc} \b B_u \b F_1 = - \b B_u \b S_r & \mbox{
%and } & \b B_u \b F_2 = \b B_u \b S_1. \end{array}\right.
%\label{eq:B-relations}
%\end{equation}
In the steady-state, $\b z_2 = \b{0}$, that is,
\[
\tilde{\b S}\tilde{\b x}= \left[\begin{array}{ccc} -\bfS_r & \bfS_{1} & \bfS_{2}
\end{array}\right]\left[\begin{array}{c}
\b x_r\\
\b x_1\\
\b x_2
\end{array}\right]=\b 0.
\]
Hence,
\begin{equation}
\b S_r \b x_r = \b S_1 \b x_1 + \b S_2 \b x_2.
\label{eq:S-steady-state}
\end{equation}
Substituting $\b S_2=\b B_u^{-1}$ and~(\ref{eq:S-steady-state})
into~(\ref{eq:first-row}), we obtain
\begin{eqnarray*}
\b{0} &=& -\b C_2 \b B_u \b S_r \b x_r - \b C_1 \b x_1 + \b C_2 \b
B_u \b S_1 \b x_1 + \b r \\
&=&\b C_2 \b B_u(- \b S_1 \b x_1 - \b S_2 \b x_2)-\b C_1 \b x_1 +
\b C_2 \b
B_u \b S_1 \b x_1 + \b r \\
&=& - \b C_2 \b x_2 - \b C_1 \b x_1 + \b r \\
&=& -\b y + \b r,
\end{eqnarray*}
that is, $\b y(\infty)=\b r$, which completes the proof of the theorem.\\
\hspace*{5.5in} $\Box$


We summarize the above considerations in the form of an algorithm for constructing a matrix $\tilde{\bfS}$ of the invariant manifold.
\begin{center}
{\bf Algorithm for the Construction of an Invariant Manifold}
\end{center}
\begin{enumerate}
    \item Find a matrix $\bfV$ such that $\left[\begin{array}{ccc} \bfV  &
    \vline & \tilde{\bfb} \end{array}\right]$ is invertible.
    \item The first $(N-m)$ rows of $\left[\begin{array}{ccc} \bfV &
    \vline & \tilde{\bfb} \end{array}\right]^{-1}$ form the matrix $\bfV^g$, that is,
    \[
    \left[\begin{array}{ccc}
    \bfV & \vline &  \tilde{\bfb} \end{array}\right]^{-1} =
    \left[\begin{array}{c}
    \bfV^g \\ \hline  \tilde{\bfb}^g \end{array}\right].
    \]
    \item \label{step:F} Check if the pair $ \left(\bfV^g \tilde{\bfa} \bfV, \bfV^g
\tilde{\bfa} \tilde{\bfb}   \right) $ is controllable. If yes,
construct a gain matrix $\b{F}$ such that the eigenvalues of
the
    matrix
    \[
    %\label{eq:V-statefeedback}
    \bfV^g \tilde{\bfa} \bfV - \bfV^g \tilde{\bfa} \tilde{\bfb} \b{F}
    \]
    are in desired locations.
%    \item \label{step:G} Determine the matrix $\bfW$ from the relation
%    \begin{equation}\label{eq:WfromV}
%    \bfW = \bfV - \tilde{\bfb} \b{F}.
%    \end{equation}
    \item Construct the matrix  $\tilde{\bfS}$ using the relation,
    \[
\tilde{\b S}=\tilde{\bfb}^g+\b F\b V^g=\left[\begin{array}{cc}
\b F & \b B_u^{-1}
\end{array}\right].
\]
    \end{enumerate}
\begin{center}
-----------------------------------------------------------------
\end{center}

%\section{Stability Analysis of the Closed-Loop System Model}
%
%Our analysis is for the constant reference signal $\b r(t)=\b r
%\in \mathbb{R}^p$.
%
%
%
%%\subsection{Closed-Loop System Stability Analysis}
%In our stability analysis of the above closed-loop
%system, we employ the following state variable transformation,
%\begin{equation}\label{eq:xtoz}
%\tilde {\bfx} =  \left [\begin{array}{cc} \bfW &
%\tilde{\bfb}\end{array}\right]\bfz.
%\end{equation}
%Observe that
%\[
%\left [\begin{array}{cc} \bfW &
%\tilde{\bfb}\end{array}\right]=\left [\begin{array}{cc} \bfV-\tilde{\bfb}\b{F} &
%\tilde{\bfb}\end{array}\right]=\left[\begin{array}{cc}
%\b V & \tilde{\b B}
%\end{array}\right]\left[\begin{array}{cc}
%\b I_{N-m} & \b O\\
%-\b F & \b I_m
%\end{array}\right],
%\]
%where $\det\left[\begin{array}{cc}
%\b V & \tilde{\b B}
%\end{array}\right]\ne 0$.
%Hence, $ \det\left[\begin{array}{cc} \bfW & \tilde{\bfb}
%\end{array}\right]\ne 0$.
%The inverse state variable transformation from $\tilde{\bfx}$
%to $\bfz$ is
%\begin{equation}\label{eq:ztox}
%\bfz= \left[\begin{array}{c}
%\b z_1\\
%\b z_2
%\end{array}\right]=
%\left[\begin{array}{cc} \bfW & \tilde{\bfb}
%\end{array}\right]^{-1}\tilde{\bfx} =
%\left[\begin{array}{c} \bfW^g \\ \tilde{\bfS}
%\end{array}\right] \tilde{\bfx}.
%\end{equation}
%
%
%The above arguments can be summarized in the form of the
%following invariant manifold design algorithm. We mention that
%this is essentially the same algorithm that was proposed for
%the design of a sliding surface in the context of sliding mode
%controller construction. For further discussion of the above
%algorithm, we refer to~\cite[Section~IV]{zakhui93a},
%\cite[pp.~42--43]{zakhui93b}, \cite[Section~6.7]{Zak03}. The
%output of the algorithm is the matrix $\tilde{\bfS}$ defined
%in~(\ref{eq:manifoldfunction}).
%
%
%
%
%We next show that, in steady-state, the condition~(\ref{eq:steady-state-z1-r})
%and the condition for perfect tracking,
%\begin{equation}
%\b{0} = \b r - \b y, \label{eq:perfect-tracking}
%\end{equation}
%are equivalent. In our proof we employ the arguments presented at the beginning
%of this section, except now we use a special form for the matrix $\b V$ and
%we explicitly write down the transformed matrices of the augmented tracking system.
%
%Recall now that $N=n+p$. Without loss of generality,
%see~(\ref{eq:v11}), let
%\begin{equation}\label{eq:assume-v-vg}
%\b V = \left[\begin{array}{c} \b{V}_{11} \\
%\hline \b{V}_{21}
%\end{array}\right] =
%\left[\begin{array}{ll} \b I_{p} & \b{O}_{p\times (n-m)} \\
% \b{O}_{(n-m)\times p} & \b I_{n-m} \\
%\hline \b{O}_{m\times p} & \b{O}_{m\times(n-m)}
%\end{array}\right]\: \mbox{ and }\: \b V^g =
%\left[\begin{array}{lll} \b I_p & \b O_{p
%\times (n-m)} & \b O_{p \times m} \\
%\b O_{(n-m) \times p} & \b I_{n-m} & \b O_{(n-m) \times m}
%\end{array}\right].
%\end{equation}
%In addition,
%which is~(\ref{eq:perfect-tracking}). Thus, (\ref{eq:steady-state-z1-r}) and (\ref{eq:perfect-tracking}) are equivalent.


\section{Examples}
\subsection{Roll-trajectory following in wing-rock suppression system}
To illustrate the  use of the integral synergetic control for trajectory tracking,
we consider the following nonlinear wing-rock dynamics~\cite{passino05},
\begin{equation}\label{eq:wingrock}
\ddot{\phi} = \alpha_1 {\phi} + \alpha_2 \dot{\phi} + \alpha_3
\dot{\phi}^2 + \alpha_4 \phi^2 \dot{\phi} + \alpha_5 \dot{\phi}^2
\phi + b u,
\end{equation}
where $\phi$ is the aircraft roll-angle, in degrees, and $u$ is
the control input. The parameters for this system are:
$\alpha_1=-0.0148927$,  $\alpha_2=0.0415424$,
$\alpha_3=0.01668756$, $\alpha_4=-0.06578382$,
$\alpha_5=0.08578836$, and $b=0.100$. The control objective is
to suppress sustained limit-cycle oscillations induced by the
wing-rock. We construct a synergetic controller so that the
angle $\phi$ follows a piecewise constant reference signal
profile. Let $x_1=\phi$, $x_2=\dot{\phi}$, and let
%\begin{equation}\label{eq:reference-wr}
$\dot{x}_r = r - x_1$,
%\end{equation}
where $r=\phi_r$ is the reference, $y=x_1$ is the output.
Using the above notation, we combine the plant model and
the reference dynamics to obtain the following augmented model,
\begin{equation}\label{eq:wr-augmented}
\left[\begin{array}{c} \dot{x}_r \\ \dot{x}_1 \\
\dot{x}_2 \end{array}\right] =
\underbrace{\left[\begin{array}{ccc} 0 & -1 & 0
\\ 0 & 0 & 1 \\ 0 & \alpha_1 + \alpha_5 x_2^2 & \alpha_2 +
\alpha_3 x_2 + \alpha_4 x_1^2 \end{array}\right]}_{\tilde{\bfa}}
\underbrace{\left[\begin{array}{c} x_r \\ x_1 \\ x_2
\end{array}\right]}_{\tilde{\bfx}} + \left[\begin{array}{c} 1 \\ 0 \\ 0
\end{array}\right] r + \underbrace{\left[\begin{array}{c} 0 \\ 0
\\ b
\end{array}\right]}_{\tilde{\bfb}} u.
\end{equation}
It is easy to verify that
\[
\dot{\tilde{\bfx}} = \left[\begin{array}{c} r-x_1 \\
x_2 \\
\alpha_1x_1 + \alpha_5 x_2^2x_1+\alpha_2 x_2+ \alpha_3 x_2^2 +
\alpha_4 x_1^2x_2 + b u \end{array}\right].
\]
We then  employ the algorithm
described above to construct an attractive invariant manifold.
We begin by selecting
\[
\bfV = \left[\begin{array}{cc}1 & 0 \\ 0 & 1 \\ 0 & 0
\end{array}\right] \quad \mbox{and} \quad \b{F} =
\left[\begin{array}{cc} -0.75 & 2.00 \end{array}\right]
\]
to place the eigenvalues of $\bfV^g \tilde{\bfa} \bfV - \bfV^g
\tilde{\bfa} \tilde{\bfb} \b{F}$ at $-0.5$ and $-1.5$. Performing
manipulations described in the algorithm, we obtain the manifold
\begin{equation}\label{eq:manifoldwr}
\bfsi = \left[\begin{array}{ccc} -\frac{1}{2b} & \frac{3}{2b} &
\frac{1}{b} \end{array}\right]\tilde{\bfx}=\tilde{\bfS}
\tilde{\bfx}=\b 0.
\end{equation}
The synergetic control law has the form,
\begin{equation}\label{eq:u4wr}
\bfu(\tilde{\bfx}) =
-(\tilde{\bfS}\tilde{\bfb})^{-1}\left(\tilde{\bfS}\tilde{\bfa}
\tilde{\bfx} + \tilde{\bfS}\bfb_r r + \bfT^{-1}
\tilde{\bfS}\right)\tilde{\bfx}.
\end{equation}
The resulting closed loop dynamics are described by
\[
\left[\begin{array}{c} \dot{\bfz}_1 \\
\dot{z}_2 \end{array}\right] = \left[\begin{array}{cccc} 0 & -1 &
\vline & 0
\\ 0.5 & -1.5 & \vline& b \\
\hline 0 & 0 & \vline &-\bfT^{-1} \end{array}\right]
\left[\begin{array}{c} \bfz_1 \\ z_2 \end{array}\right] +
\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] r,
\]
where $\bfz_1 = \left[\begin{array}{cc} x_r & x_1
\end{array}\right]^\top $, $\bfz_2 = \bfsi$. Note that the eigenvalues of the closed
loop dynamics are $-0.5$, $-1.5$, and $-\bfT^{-1}$, which
guarantee that the  tracking error converges to zero, that is,
at steady-state, $x_1=r$. In this example, we use $\b{T} = 1$.
To see this, at steady-state, we have
\[
\left[\begin{array}{c} 0 \\ 0 \end{array}\right] =
\left[\begin{array}{cc} 0 & -1 \\ 0.5 & -1.5 \end{array}\right]
\left[\begin{array}{c} x_r \\ x_1 \end{array} \right] +
\left[\begin{array}{c} 1 \\ 0 \end{array}\right] r.
\]
Solving the above matrix equation for $x_r$ and $x_1$, we
obtain $x_r = 3r$ and $x_1 = r$. Therefore, at steady-state
$x_1 = r$.


We present simulation results illustrating the performance of
integral synergetic controller in Figures~\ref{fig:time_history}
and~\ref{fig:phase_portrait}. We show in Figure~\ref{fig:time_history}
trajectories of~(\ref{eq:wr-augmented}) driven by the control
strategy~(\ref{eq:u4wr}) where solid lines represent $x_1(t)$,
dotted lines represent $x_2(t)$, and dashed-lines represent
$\phi_r(t)$. We can see that as the transient become negligible,
$x_1(t)$ closely tracks $\phi_r(t)$.
%\begin{center}
\begin{figure}
\epsfxsize=3.50in \epsfclipon
\centerline{\epsffile{zoom-time-tracking_legend.eps}} \caption{The
roll angle $\phi(t)$, denoted $x_1(t)$, tracking reference angle
$\phi_r(t)$: several plots for different initial conditions.}
\label{fig:time_history}
\end{figure}
%\end{center}

%\begin{center}
\begin{figure}
\epsfxsize=3.25in \epsfclipon
%\centerline{\epsffile{phase_potrait_integral.eps}}
\centerline{\begin{tabular}{cc}
\epsffile{outline-phase-portrait-tracking-wr.eps} &
\epsffile{zoom-phase-tracking-wr_legend.eps}\end{tabular}}\caption{Phase
trajectories of the closed-loop system driven by the integral
synergetic controller originating from several initial conditions:
(a) general view, (b) magnified view.} \label{fig:phase_portrait}
\end{figure}
%\end{center}
In Figure~\ref{fig:phase_portrait}, we show phase portraits of
nonlinear wing-rock dynamics. In Figure~\ref{fig:phase_portrait}(a),
dotted lines indicate sustained limit-cycle oscillations of the
uncompensated closed-loop system. Thick solid lines denote the
closed-loop phase trajectories emanating from different initial
conditions of the closed-loop system driven by the synergetic
controller. In Figure~\ref{fig:phase_portrait}(b), we show a magnified
portion of the phase portrait in the neighborhood of the origin
from Figure~\ref{fig:phase_portrait}(a). The solid line indicates transitions from several
constant reference values, denoted by the sequence of symbols:
$\triangle$,~$\diamondsuit$,~$\nabla$,~$\Box$ (see these values
from Fig.~\ref{fig:time_history}) towards the ultimate
destination: the origin, denoted by the symbol $\bigodot$.




\subsection{Two-link robotic manipulator}
We now synthesize a regulator for the two-link robotic manipulator depicted in Figure~\ref{fig:dbl_pend}.
\begin{figure}
\centering
\includegraphics[scale=0.7]{DBL_PEND.eps}
\caption{A two-link robotic manipulator.}
\label{fig:dbl_pend}
\end{figure}
We first concern ourselves with constructing a mathematical model.  The Lagrangian has the form,
\begin{eqnarray*}
L&=&\frac{1}{2}m_1l_1^2\omega_1^2+\frac{1}{2}m_2\left( l_1^2\omega_1^2+l_2^2\omega_2^2+2l_1l_2\cos\left( \theta_2 - \theta_1\right) \omega_1\omega_2 \right)\\
&{}&\quad +m_1gl_1\cos(\theta_1)+m_2g\left(l_1\cos(\theta_1)+l_2\cos(\theta_2)\right),
\end{eqnarray*}
where $\omega_i=\dot{\theta}_i$, $i=1,2$, are the angular velocities of the links, and $m_i$ and $l_i$ are the link masses and their lengths, respectively. We assume that the input torques, $\tau_1$ and $\tau_2$ are applied at each joint. The parameter values are given in Table~\ref{tab:dbl_pendpar}.
\begin{table}[h]
\caption{Parameter numerical values of the two-link robotic manipulator.}
\begin{center}
\begin{tabular}{ll}\hline
$m_1=m_2$ & 1 kg\\
$l_1=l_2$ & 0.5 m\\
%$I_1=I_2$& 1/48 kg$\cdot$m$^2$\\
$g$& 9.81 m/sec$^2$ \\ \hline
\end{tabular}
\end{center}
\label{tab:dbl_pendpar}
\end{table}
We next write the Lagrange equations of motion for the system,
\begin{eqnarray}
&{}&\left[\begin{array}{cc}
(m_1+m_2)l_1^2 & m_2l_1l_2\cos(\theta_2 - \theta_1)\\
m_2l_1l_2\cos(\theta_2 - \theta_1) & m_2l_2^2
\end{array}\right]\left[\begin{array}{c}
\ddot{\theta}_1\\
\ddot{\theta}_2
\end{array}\right]\nonumber\\
&{}&\qquad\qquad = \left[\begin{array}{c}
\tau_1 - \tau_2 + m_2l_1l_2\dot{\theta}_2^2 \sin(\theta_2 - \theta_1) - (m_1 + m_2) g l_1 \sin(\theta_1)\\
\tau_2-m_2l_1l_2\dot{\theta}_1^2\sin(\theta_2 - \theta_1) -m_2gl_2\sin(\theta_2)
\end{array}\right].
\label{eq:matftwlags}
\end{eqnarray}
Note that the manipulator inertia matrix, $\b H$, on the left-hand side of the above equation is symmetric and positive definite, and its determinant is
\[
\det \b H = (m_1+m_2)m_2l_1^2l_2^2 - m_2^2l_1^2l_2^2 \cos^2(\theta_2 - \theta_1) > 0.
\]

We represent the above model in state-space format, where the state variables are defined as, $x_1=\theta_1$, $x_2=\theta_2$, $x_3=\dot{\theta}_1$, and $x_4= \dot{\theta}_2$. First, using the above notation, we represent~\eqref{eq:matftwlags} as
\[
\b H(\b x_1)\ddot{\b x}_1= -\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 - \b g(\b x_1)+\b B_u\b\tau,
\]
where
\[
\b B_u=\left[\begin{array}{cc}
1 & -1\\
0 & 1
\end{array}\right],
\]
and
\[
\b g(\b x_1)=\b G(\b x_1)\b x_1=\left[\begin{array}{cc}
-(m_1+m_2)gl_1\frac{\sin(x_1)}{x_1} & 0\\
0 & -m_2gl_2\frac{\sin(x_2)}{x_2}
\end{array}\right]\b x_1.
\]
With the above notation in place we represent as in~\eqref{eq:two_link_regular}, where
\[
\tilde{\b u}=\b H^{-1}\b B_u\b{\tau}.
\]
We select
\[
\b T=0.05\b I_2,
\]
and
\[
\{\lambda_1, \lambda_2,\lambda_3,\lambda_4\}=\{-3, -3.5, -4, -4.5\}.
\]
Applying the design algorithm, we obtain
\[
\tilde{\b S} =\left[\begin{array}{cccccc}
-18. &  0  &  8.5 &  0 & 1 & 0\\
0 & -10.5 & 0 & 6.5 & 0 & 1
\end{array}\right].
\]
With the above data, we construct the synergetic regulator~\eqref{eq:soclawtracking}. In our simulations, we select the initial conditions to be,
\[
\b x_r(0)=\b 0\quad\mb{and}\quad \b x(0)^{\top}=\left[\begin{array}{cccc}
-3/4\pi & \pi & 0 & 0
\end{array}\right].
\]
The components of the reference signal vector $\b r$ are plotted in Figures~\ref{fig:theta1} and~\ref{fig:theta2}, where we show plots of link angles $\theta_1$ and $\theta_2$ versus time.
\begin{figure}
\centering
\includegraphics[scale=0.45]{theta1.eps}
\caption{A plot of $\theta_1$ and its reference versus time.}
\label{fig:theta1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.45]{theta2.eps}
\caption{A plot of $\theta_2$ and its reference versus time.}
\label{fig:theta2}
\end{figure}
In Figures~\ref{fig:tip_x} and~\ref{fig:tip_y}, we show plots of the time history of the $x$ and $y$ coordinates, respectively, of the end effector of the robotic manipulator.
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_x.eps}
\caption{A plot of the $x$-coordinate of the end effector versus time.}
\label{fig:tip_x}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_y.eps}
\caption{A plot of the $y$-coordinate of the end effector versus time.}
\label{fig:tip_y}
\end{figure}
In Figure~\ref{fig:tip_xy}, we plot the path of the end effector in the $(x,y)$ plane.
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_xy.eps}
\caption{A plot of the path of the end effector in  the $(x,y)$-plane.}
\label{fig:tip_xy}
\end{figure}
As can be seen from the Figures the reference signal is tracked as predicted by the theory.

\section{Conclusions}
We presented a regulator design algorithm for nonlinear
dynamical systems using integral synergetic control. The
solution to the above nonlinear regulator problem is derived for a
class of nonlinear systems in the``regular form'' with
nonlinearities satisfying the matching condition using a synergetic control approach.

The proposed  synergetic regulator design methodology incorporates
an additional state vector representing the integral of the
tracking error. The construction of an
asymptotically stable invariant manifold uses the augmented state
vector in order to drive the integral of the tracking error to zero.
Using the augmented state vector, we convert the initial
tracking problem into the equivalent regulation problem for which we can readily
apply the synergetic control design methodology.

This design methodology consists of two steps: the manifold
design and the control law design. An algorithm is presented for the
construction of an asymptotically stable invariant manifold in such a
way that all trajectories confined to the manifold converge to the
origin. Once a manifold is constructed, the synergetic control strategy can be
easily obtained by solving the first-order differential equation
of regulated variables. Finally, the
proposed design procedure is illustrated on
a nonlinear wing-rock suppression problem and a two-link robotic manipulator.
Simulations show that
integral synergetic regulator yields an excellent tracking performance.

%\subsection*{Acknowledgement}
%We thank the reviewers for their constructive comments on
%improving the manuscript.

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\[
 \left[ \begin {array}{ccc} 1&0&0\\\noalign{\medskip}0&1&0\\\noalign{\medskip}-{\it Bu}\,{\it F1}-{\it Bu}\,{\it Sr}&-{\it Bu}\,{
\it F2}+{\it Bu}\,{\it S1}&{\it Bu}\,{\it S2}\end {array} \right]
\]
\[
\left[ \begin {array}{ccc}
1&0&0\\\noalign{\medskip}0&1&0\\\noalign{\medskip}-{\it Bu}\,{\it
F1}&-{\it Bu}\,{\it F2}&{\it Bu}
\end {array} \right]
\]

\[
\left[ \begin {array}{ccc}
1&0&0\\\noalign{\medskip}0&1&0\\\noalign{\medskip}-{\it Sr}&{\it
S1}&{\it S2}\end {array} \right]
\]

\[
 \left[ \begin {array}{cc} 1&0\\\noalign{\medskip}0&1\\\noalign{\medskip}-{\it Bu}\,{\it F1}&-{\it Bu}\,{\it F2}\end {array}
 \right]
%
\]

\[
\left[ \begin {array}{cc} {\it C2}\,{\it Bu}\,{\it F1}&-{\it
C1}+{\it C2}\,{\it Bu}\,{\it F2}\\\noalign{\medskip}-{\it
A12}\,{\it Bu}\,{\it F1}&{\it A11}-{\it A12}\,{\it Bu}\,{\it
F2}\end {array} \right]
\]

\author{
Nusawardhana%
    \thanks{School of Aeronautics and Astronautics, 315 N. Grant Street, Purdue University, West Lafayette, IN 47907-2023 }
  \ , S.H. \.{Z}ak \thanks{School of Electrical and Computer Engineering,
  Electrical Engineering Building, 465 Northwestern Avenue, Purdue University, West Lafayette, IN 47907-2035}
  \ , and W.A. Crossley \thanks{School of Aeronautics and Astronautics, 315 N. Grant Street,  Purdue University, West Lafayette, IN 47907-2023}\\
  {\normalsize\itshape
   Purdue University, West Lafayette, IN 47907}\\
  %\and
  %Third C. Author%
  % \thanks{Job Title, Department, Address, and AIAA Member Grade.}\\
  %{\normalsize\itshape
  %Business or Academic Affiliation, City, Province, Zipcode, Country}
 }
