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\begin{document}
\title{Adaptive Fuzzy Control}
\author{Stanislaw H. \.{Z}ak \thanks{School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907}}
\maketitle
\begin{abstract}
Adaptive fuzzy tracking controller for a class of dynamical systems is analyzed. It is shown that the closed-loop system driven by this fuzzy adaptive controller stable and the adaptation parameters are bounded.
\end{abstract}
\section{Introduction}


We consider a class of nonlinear dynamical systems modeled by the differential equation
\begin{equation}
x^{(n)}=f\left(x,\dot{x},\ldots,x^{(n-1)}\right)+g\left(x,\dot{x},\ldots,x^{(n-1)}\right)u,
\label{eq:nthodemod}
\end{equation}
where $f=f\left(x,\dot{x},\ldots,x^{(n-1)}\right)$ and $g=g\left(x,\dot{x},\ldots,x^{(n-1)}\right)$ are unknown to us. We define
\[
\b x=\left[\begin{array}{cccc}
x & \dot{x} & \cdots & x^{(n-1)}
\end{array}\right]^T\quad \mb{and}\quad \b x_d=\left[\begin{array}{cccc}
x_d & \dot{x}_d & \cdots & x^{(n-1)}_d
\end{array}\right]^T.
\]
Let
\[
\b e=\b x-\b x_d=\left[\begin{array}{cccc}
x-x_d & \dot{x}-\dot{x}_d & \cdots & x^{(n-1)}-x^{(n-1)}_d
\end{array}\right]^T=\left[\begin{array}{cccc}
e & \dot{e} & \cdots & e^{(n-1)}
\end{array}\right]^T.
\]
We wish to construct a controller $u$ such that
\[
\lim_{t \to \infty} \b e(t) = \b 0.
\]
Let
\[
\b k^T=\left[\begin{array}{cccc}
k_0 & k_1 & \cdots & k_{n-1}
\end{array}\right]^T
\]
be such that
\begin{equation}
e^{(n)}+k_{n-1}e^{(n-1)}+\cdots +k_1 \dot{e} + k_0 e =0,
\label{eq:trackerr}
\end{equation}
represents the desired dynamical behavior of the tracking error. If $f$ and $g$ were known exactly to us, then the controller of the form
\begin{equation}
u^{*}=\frac{1}{g}\left(-f+x_d^{(n)}-\b k^T \b e \right)
\label{eq:idealcon}
\end{equation}
would force the tracking error, $e$, to behave as described by~(\ref{eq:trackerr}). However, $f$ and $g$ are unknown to us. We will use an adaptive fuzzy logic controller to approximate, as closely as possible, the performance of~(\ref{eq:idealcon}). Recall from Chapter~\ref{chp:fuzzysys}, that a fuzzy logic controller with $q$ inputs, $v_1, v_2,\ldots,v_q$, and the center average defuzzifier can be represented as
\begin{equation}
u=\frac{\sum_{l=1}^M\prod_{j=1}^q\mu_{A_j^l}(v_j)\theta_l}{\sum_{l=1}^M\prod_{j=1}^q\mu_{A_j^l}(v_j)},
\label{eq:fuz_lo_co}
\end{equation}
where $M$ is the number of fuzzy rules used to construct the controller, $A_j^l$ is the $j$-th fuzzy set corresponding to the $l$-th fuzzy rule, and $c_l$ can be taken as a center of the $l$-th fuzzy set corresponding to the controller's output. To proceed, we represent~(\ref{eq:fuz_lo_co}) as
\begin{equation}
u=\sum_{l=1}^M\left(\frac{\prod_{j=1}^q\mu_{A_j^l}(v_j)}{\sum_{l=1}^M\prod_{j=1}^q\mu_{A_j^l}(v_j)} \right) \theta_l.
\label{eq:alt_rp_flc}
\end{equation}
Let
\[
\b{\theta}=\left[\begin{array}{cccc}
\theta_1 & \theta_2 & \cdots & \theta_M
\end{array}\right]^T
\]
and
\[
\b{\xi}=\b{\xi}(v_1,v_2,\ldots,v_M)=\left[\begin{array}{ccc}
\frac{\prod_{j=1}^q\mu_{A_j^1}(v_j)}{\sum_{l=1}^M\prod_{j=1}^q\mu_{A_j^l}(v_j)} & \cdots & \frac{\prod_{j=1}^q\mu_{A_j^M}(v_j)}{\sum_{l=1}^M\prod_{j=1}^q\mu_{A_j^l}(v_j)}
\end{array}\right]^T.
\]
Then, we can represent~(\ref{eq:fuz_lo_co}) as
\begin{equation}
u=\b{\theta}^T\b{\xi}.
\label{eq:flc_as_xitheat}
\end{equation}
We assume that there exists a vector $\b{\theta}^{*}$ such that
\begin{equation}
u^{*}=\frac{1}{g}\left(-f+x_d^{(n)}-\b k^T \b e \right)
=\b{\theta}^{* T}\b{\xi}.
\label{eq:ideal_ada}
\end{equation}
We will now show that if the above assumption is satisfied then the control law
\begin{equation}
u=\b{\theta}^{T}\b{\xi},
\label{eq:real_adaflc}
\end{equation}
where $\b{\theta}$ is being appropriately adapted, will force the tracking error, $\b e$, to converge to zero. Furthermore, the adaptation parameters remain bounded. To see this, consider $e^{(n)}$. We have
\begin{eqnarray*}
e^{(n)} &=& x^{(n)}-x_d^{(n)}\\
&=& f+gu-x_d^{(n)}.
\end{eqnarray*}
Adding and subtracting $g u^{*}$ to the right hand side of the above equation and substituting into it $g u^{*}$ obtained from~(\ref{eq:idealcon}) gives
\begin{eqnarray}
e^{(n)} &=& f+g\left( u-u^{*}\right) + gu^{*}- x_d^{(n)}\nonumber\\
&=& f+g\left( u-u^{*}\right) -f+x_d^{(n)}-\b k^T \b e  - x_d^{(n)}\nonumber\\
&=& \b k^T \b e + g\left( u-u^{*}\right).
\label{eq:enmkegums}
\end{eqnarray}
Let
\begin{equation}
\b{\phi}=\b{\theta}-\b{\theta^{*}}.
\label{eq:nottmtsta}
\end{equation}
Substituting~(\ref{eq:ideal_ada}) and (\ref{eq:real_adaflc}) into~(\ref{eq:enmkegums}) and taking into account~(\ref{eq:nottmtsta}) yields
\begin{equation}
\fbox{$\displaystyle e^{(n)}=-k_{n-1}e^{(n-1)}-\cdots +k_1 \dot{e} - k_0 e + g\b{\phi}^T\b{\xi} $}
\label{eq:exforeror}
\end{equation}
We now represent the above equation in state space format. Let
\[
\b A=\left[\begin{array}{ccccccc}
0 & 1 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & 0 & \cdots & 0 & 0\\
\vdots& {}& {}& {}& \ddots&{} &\vdots\\
0 & 0 & 0 & 0 & \cdots & 0 & 1\\
-k_0 & -k_1 & -k_2 & -k_4 & \cdots & -k_{n-2} & -k_{n-1}
\end{array}\right]\quad\mb{and}\quad \b b=\left[\begin{array}{c}
0\\
0\\
\vdots\\
0\\
1
\end{array}\right].
\]
Then~(\ref{eq:exforeror}) can be represented in an equivalent way as
\begin{equation}
\dot{\b e}=\b A\b e+\b b \left(g \b{\phi}^T\b{\xi} \right).
\label{eq:statspfore}
\end{equation}
\begin{thm}
Consider a dynamical system model given by~(\ref{eq:statspfore}). Suppose that $g$ is a positive constant, the components $\phi_i=\phi_i(t)$ of the vector-valued function $\b{\phi}=\b{\phi}(t)\in \mb{\Bb R}^M$ are adjustable parameters according to the adaptation law
\begin{equation}
\dot{\b{\phi}}=\dot{\b{\theta}}=-\b{\Gamma}\tilde{e}\b{\xi},
\label{eq:adaplawex}
\end{equation}
where $\b{\Gamma}=\b{\Gamma}^T>0$ is a design parameter matrix. The vector-valued function $\b{\xi}=\b{\xi}(t)\in \mb{\Bb R}^M$ is bounded for all $t$. The matrix $\b A$ is asymptotically stable by construction. The real-valued function, $\tilde{e}$, satisfies the equation
\begin{equation}
\tilde{e}=\b b^T\b P\b e,
\label{eq:conforc}
\end{equation}
where $\b P=\b P^T>0$ is the solution of the matrix Lyapunov equation
\begin{equation}
\b A^T\b P+\b P\b A=-2\b Q
\label{eq:layafordaptlaw}
\end{equation}
for some $\b Q=\b Q^T>0$. Then, the erorr $\b e$ and $\b{\phi}$ are both bounded and
\[
\lim_{t\to \infty}\b e(t)=\b 0.
\]
\end{thm}
{\bf Proof:} We will use the Lyapunov function candidate,
\begin{equation}
V(\b e, \b{\phi})=\frac{1}{2}\b e^T\b P\b e+\frac{1}{2}g\b{\phi}^T\b{\Gamma}^{-1}\b{\phi}
\label{eq:layfuncanforada}
\end{equation}
and Barbalat's lemma to prove the theorem. The time derivative of the Lyapunov function candidate $V=V(\b e, \b{\phi})$ evaluated on the trajectories of the system
\[
\left.
\begin{array}{lll}
\dot{\b e} & = & \b A\b e +\b b(g\b{\phi}^T\b{\xi})\\
\dot{\b{\phi}} &=& -\b{\Gamma} \tilde{e} \b{\xi}
\end{array}
\right\}
\]
is
\begin{eqnarray*}
\dot{V} &=& \b e^T\b P\dot{\b e}+g\b{\phi}^T\b{\Gamma}^{-1}\dot{\b{\phi}}\\
&=& \b e^T\b P\left(\b A\b e + \b b\left(g\b{\phi}^T\b{\xi}\right)\right)-g\b{\phi}^T e \b{\Gamma}^{-1}\b{\Gamma}\b{\xi}\\
&=& -\b e^T\b Q\b e + g\left(\tilde{e}\b{\phi}^T-\tilde{e}\b{\phi}^T \right)\b{\xi}\\
&=& -\b e^T\b Q\b e,
\end{eqnarray*}
where we used the fact that $\tilde{e}=\b b^T\b P\b e=\b e^T \b P\b b$. Thus, $\dot{V}$ is negative semi-definite in the $\left[\begin{array}{c}
\b e\\
\b{\phi}
\end{array}\right]$-space, which implies that $\b e$ and $\b{\phi}$ are bounded.

To show the convergence of $\b e$, we use Barbalat's lemma. For this, we first show that $\dot{V}$ is uniformly continuous. Indeed, since
\[
\ddot{V}=-2\b e^T\b Q\dot{\b e}
\]
is bounded, because $\b e$ and $\dot{\b e}$ are bounded, $\dot{V}$ is uniformly continuous. Hence, $\dot{V}\to 0$ as $t\to \infty$. Since $\dot{V}=-\b e^T\b Q\b e$ and $\b Q=\b Q^T>0$, we have to have $\b e(t)\to \b 0$ as $t\to 0$.\\
\hspace*{5.5in} $\Box$

If the lower bounds on the parameters of $\b{\theta}^{*}$ are known and it is desired that $\theta_i$s will not go below these bounds, then we modify the adaptation law to achieve this goal. Let $\theta_{i,low}$ denote the lower bound of $\theta^*_i$. Fischle and Schr\"{o}der~\cite{fischle99} propose the following modification;
\begin{equation}
\dot{\theta}=-\b{\Gamma}\tilde{e}\b L\b{\xi},
\label{eq:modadalaw}
\end{equation}
where the matrix $\b L$ is a diagonal matrix whose diagaonal elements are defined as
\[
l_{ii}=\left\{\begin{array}{cl}
0 & \mb{if}\;\theta_i=\theta_{i,low}\;\mb{and}\; \tilde{e}\xi_i>0\\
1 & \mb{else}.
\end{array}
\right.
\]



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