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\begin{document}
\title{On Fuzzy Estimators}
\author{Stanislaw H. \.{Z}ak \thanks{School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907}}
\maketitle
\begin{abstract}
A fuzzy estimator is presented and a stability analysis of the closed-loop system driven by the combined fuzzy estimator-controller compensator is performed.
\end{abstract}
\section{Fuzzy Estimator}
One of the essential elements of the control problem is a model of a dynamical system to be controlled. In many situations a mathematical model of the system is not available or is incomplete. However, there may be experts who can provide a linguistic description, in terms of IF-THEN rules, of the system. Combining the available mathematical description of the system with its linguistic description results in a fuzzy system model. An example of a fuzzy system model is the so called Takagi-Sugeno~(T-S), also known as the Takagi-Sugeno-Kang~(TSK), fuzzy model. One can construct a TSK model if local description of the dynamical system to be controlled is available in terms of linear models
\[
\dot{\b x}(t)=\b A_i\b x(t)+\b B_i\b u(t),\quad i=1,2,\ldots,r,
\]
where the state vector $\b x(t) \in \mb{\Bb R}^n$, the control input $\b u(t) \in \mb{\Bb R}^m$, and the matrices $\b A_i$ and $\b B_i$ are of appropriate dimensions.
The above information is then fused with the available IF-THEN rules, where the $i$-th rule can have the form:
\[
\mb{Rule i }:\mb{ IF } z_1(t)\mb{ is } F_1^i\mb{ AND } \cdots\mb{ AND } z_q(t)\mb{ is }F_q^i,
\]
\[
\mb{ THEN }\left\{\begin{array}{lcl}
\dot{\b x}(t)&=&\b A_i\b x(t)+\b B_i\b u(t)\\
\b y&=&\b C_i\b x(t),
\end{array}\right.
\]
where $F_j^i$, $j=1,2,\ldots,q$, is the $j$th fuzzy set of the $i$th rule, and $z_1(t),\ldots,z_q(t)$ are measurable system variables. Let $\mu_j^i(x_j)$ be the membership function of the fuzzy set $A_j^i$, and let
\[
w^i = w^i(\b z)=\prod_{j=1}^n\mu_j^i(z_j),
\]
where $r$ is the number of fuzzy rules. Then, given a pair $(\b x(t),\b u(t))$, the resulting fuzzy system model is inferred as the weighted average of the local models, and has the form
\begin{eqnarray}
\dot{\b x} &=& \frac{\sum_{i=1}^r w^i\left( \b A_i\b x+\b B_i\b u\right)}{\sum_{i=1}^r w^i}\nonumber\\
&=&\sum_{i=1}^r \alpha_i\left( \b A_i\b x+\b B_i\b u\right)\nonumber\\
&=&\left(\sum_{i=1}^r \alpha_i \b A_i\right)\b x + \left(\sum_{i=1}^r \alpha_i\b B_i\right)\b u \nonumber\\
&=& \b A(\b \alpha)\b x+\b B(\b \alpha)\b u,
\label{eq:fuzzy_model}
\end{eqnarray}
where for $i=1,2,\ldots,r$,
\[
\alpha_i=\frac{w^i}{\sum_{i=1}^r w^i}.
\]
Similarly,
\begin{eqnarray}
\b y &=& \frac{\sum_{i=1}^r w^i  \b C_i\b x}{\sum_{i=1}^r w^i}\nonumber\\
&=&\left(\sum_{i=1}^r \alpha_i \b C_i\right)\b x \nonumber\\
&=& \b C(\b \alpha)\b x.
\label{eq:outofpla}
\end{eqnarray}
Note that
\begin{equation}
\label{eq:alphas}
\alpha_i\ge 0,\quad \sum_{i=1}^r \alpha_i=1\quad \mb{and}\quad \b \alpha =\left[\begin{array}{cccc}
\alpha_1&\alpha_2&\cdots&\alpha_r
\end{array}\right]^T
\in [0,1]^r.
\end{equation}
A fuzzy state estimator for the system modeled by~(\ref{eq:fuzzy_model}) and~(\ref{eq:outofpla}) can be represented as
\begin{equation}
\dot{\tilde{\b x}}=\b A(\b{\alpha})\tilde{\b x}+\b B(\b{\alpha})\b u+\b L(\b{\alpha})\left( \b y-\tilde{\b y}\right),
\label{eq:staestpoly}
\end{equation}
where
\[
\b L(\b{\alpha})=\sum_{i=1}^r \alpha_i\b L_i.
\]
The state estimator~(\ref{eq:staestpoly}) can be represented equivalently as
\begin{equation}
\fbox{$ \displaystyle \dot{\tilde{\b x}}=\sum_{i=1}^{r}\alpha_i \b A_i \tilde{\b x}+\sum_{i=1}^{r}\alpha_i \b B_i \b u + \sum_{i=1}^{r}\alpha_i \b L_i \left( \b y-\tilde{\b y}\right) $}
\label{eq:eqrepofste}
\end{equation}
We note that the state estimator~(\ref{eq:eqrepofste}) can be viewed as a result of implementing the following $r$ fuzzy rules:
\[
\mb{Rule i }:\mb{ IF } z_1(t)\mb{ is } F_1^i\mb{ AND } \cdots\mb{ AND } z_q(t)\mb{ is }F_q^i,
\]
\[
\mb{ THEN }\left\{\begin{array}{lcl}
\dot{\tilde{\b x}}&=&\b A_i\tilde{\b x}+\b B_i\b u+\b L_i\left( \b y-\tilde{\b y}\right)\\
\tilde{\b y}&=&\b C_i\tilde{\b x}.
\end{array}\right.
\]
We use the state estimate in the controller implementation to get
\begin{equation}
\b u=-\sum_{j=1}^r \b K_j\tilde{\b x}+\b v.
\label{eq:statesincont}
\end{equation}
Substituting the above in to the plant model yields
\begin{equation}
\dot{\b x}=\sum_{i=1}^r \alpha_i \b A_i \b x -\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \b B_i\b K_j \tilde{\b x} + \sum_{i=1}^r \alpha_i \b B_i \b v.
\label{eq:plancontr}
\end{equation}
Substituting~(\ref{eq:outofpla}) and~(\ref{eq:statesincont}) into the estimator equation~(\ref{eq:eqrepofste}), we obtain
\begin{eqnarray}
\dot{\tilde{\b x}}&=&\sum_{i=1}^{r}\alpha_i \b A_i \tilde{\b x}-\sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \b B_i \b K_j\tilde{\b x} + \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \b L_i \b C_j\left( \b x-\tilde{\b x}\right)+\sum_{i=1}^r \alpha_i \b B_i \b v\nonumber \\
&=& \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \left( \b A_i-\b B_i \b K_j\right)\tilde{\b x}+ \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \b L_i \b C_j\left( \b x-\tilde{\b x}\right)+\sum_{i=1}^r \alpha_i \b B_i \b v.
\label{eq:estiwfecoandc}
\end{eqnarray}
Equations~(\ref{eq:plancontr}) and~(\ref{eq:estiwfecoandc}) constitute the closed-loop system driven by the combined fuzzy controller-estimator compensator. We represent the above two equations modeling the closed-loop system in matrix form as
\begin{eqnarray}
\left[\begin{array}{c}
\dot{\b x}\\
\dot{\tilde{\b x}}
\end{array}\right] &=& \left[\begin{array}{cc}
\sum_{i=1}^r \alpha_i \b A_i &  -\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \b B_i\b K_j\\
\sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \b L_i \b C_j & \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \left( \b A_i-\b B_i \b K_j- \b L_i \b C_j \right)
\end{array}\right]\left[\begin{array}{c}
\b x\\
\tilde{\b x}
\end{array}\right]\nonumber\\
&{}& \quad+\left[\begin{array}{c}
\sum_{i=1}^r \alpha_i \b B_i\\
\sum_{i=1}^r \alpha_i \b B_i
\end{array}\right]\b v \label{eq:staeclo}\\
\b y &=& \left[\begin{array}{cc}
\sum_{i=1}^r \alpha_i \b C_i & \b O
\end{array}\right]\left[\begin{array}{c}
{\b x}\\
{\tilde{\b x}}
\end{array}\right].
\label{eq:mafooffuzcls}
\end{eqnarray}
\section{The Comparison Method for Linear Systems}
Let $\b A$ be a constant $n\times n$ matrix and $e^{\b A t}$ denote the matrix exponential. We will now give a necessary and sufficient condition for the matrix exponential to have all its elements non-negative for $t\ge 0$. This result appears in~\cite[page~171]{bellman60}.
\begin{lemma}
A necessary and sufficient condition that all elements of $e^{\b A t}$ be non-negative for $t\ge 0$ is that
\begin{equation}
a_{ij}\ge 0 \quad i\ne j.
\label{eq:nonnegnasc}
\end{equation}
\end{lemma}
{\bf Proof}\quad We follow the arguments of Bellman~\cite[page~172]{bellman60}, see also~\cite{beckenbach65}. We first prove the necessary condition. Since
\[
e^{\b At}=\b I_n+\b At +\frac{1}{2}\b A^2 t^2+\cdots
\]
it is clear that~(\ref{eq:nonnegnasc}) is necessary for the result to hold for small $t$, whch establishes the necessity condition

To prove the sufficiency, let $c_1$ be a scalar so that all the elements of $\b A+c_1\b I_n$ are non-negative. Then, obviously, all the elements of $e^{\b A+c_1\b I_n}$ are non-negative. Also, the elements of $e^{-c_1\b I_n t}$ are non-negative since exponentials are always non-negative. The matrices $\b A+c_1\b I_n$ and $-c_1\b I_n$ commute. Hence, we can write
\begin{eqnarray*}
e^{\b A t} &=& e^{\left(\b A+c_1\b I_n \right)t-c_1\b I_n t}\\
&=& e^{\left(\b A+c_1\b I_n \right)t}e^{-c_1\b I_n t},
\end{eqnarray*}
which establishes the desired non-negativity condition and the sufficiency condition.\\
\hspace*{5.5in} $\Box$

Using the above Lemma, we can prove the following lemma due to Bellman~\cite{bellman62}. We use the following notation: $\b x\le \b y$ implies component-by-component majorization.
\begin{lemma}
\label{lemma:veccomp}
If $a_{ij}\ge 0$, $i\ne j$, then
\begin{equation}
\frac{d\b x}{dt}\le \b A\b x, \quad \b x(0)=\b c,
\label{eq:derofxleax}
\end{equation}
implies $\b x\le \b y$, where
\begin{equation}
\frac{d\b y}{dt}= \b A\b y, \quad \b y(0)=\b c.
\label{eq:licomeqfl}
\end{equation}
\end{lemma}
{\bf Proof}\quad We can represent~(\ref{eq:derofxleax}) as
\begin{equation}
\frac{d \b x}{dt}=\b A\b x-\b f(t),\quad \b x(0)=\b c,
\label{eq:alrepofdifin}
\end{equation}
where $\b f(t)$ is a vector-valued function whose components are non-negative for all $t\ge 0$. The solution of~(\ref{eq:alrepofdifin}) can be represented as
\begin{eqnarray}
\b x&=&e^{\b A t}\b c - \int_{0}^t e^{\b A(t-s)}\b f(s)ds\nonumber \\
&=& \b y - \int_{0}^t e^{\b A(t-s)}\b f(s)ds.
\label{eq:alsoofperde}
\end{eqnarray}
The components of the integrand of the second term in~(\ref{eq:alsoofperde}) are all non-negative for $t\ge s$. Hence,
\[
\int_{0}^t e^{\b A(t-s)}\b f(s)ds \ge \b 0,\quad\mb{for}\quad t\ge 0,
\]
which implies
\[
\b x\le \b y.
\]
The proof is complete.\\
\hspace*{5.5in} $\Box$

Equation~(\ref{eq:licomeqfl}) is referred as the {\em equation of comparison\/} for~(\ref{eq:derofxleax}).

\section{Stability Analysis of the Closed-Loop System}
We now perform a stability analysis of the closed-loop system given by~(\ref{eq:staeclo}) and~(\ref{eq:mafooffuzcls}). We first apply a state variable transformation to bring the system into a form convenient for further analysis. The transformation that we will use has the form,
\begin{equation}
\left[\begin{array}{cc}
\b I_n & \b O\\
-\b I_n & \b I_n
\end{array}\right]\left[\begin{array}{c}
\b x\\
\tilde{\b x}
\end{array}\right].
\label{eq:bigtranstocls}
\end{equation}
The closed-loop system model in the new coordinates is
\begin{eqnarray}
\left[\begin{array}{c}
\dot{\b x}\\
\dot{\tilde{\b x}}-\b x
\end{array}\right] &=& \left[\begin{array}{cc}
 \sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \left(\b A_i-\b B_i\b K_j\right)&  -\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \b B_i\b K_j\\
\b O & \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \left( \b A_i- \b L_i \b C_j \right)
\end{array}\right]\left[\begin{array}{c}
\b x\\
\tilde{\b x}-\b x
\end{array}\right]\nonumber\\
&{}& \quad+\left[\begin{array}{c}
\sum_{i=1}^r \alpha_i \b B_i\\
\b O
\end{array}\right]\b v \label{eq:innecofipart}\\
\b y &=& \left[\begin{array}{cc}
\sum_{i=1}^r \alpha_i \b C_i & \b O
\end{array}\right]\left[\begin{array}{c}
{\b x}\\
{\tilde{\b x}}
\end{array}\right].
\label{eq:fuzclsinncoo}
\end{eqnarray}
Let $\b e=\tilde{\b x}-\b x$. We will now present a sufficient condition for the unforced closed-loop system to be globally uniformly asymptotically stable, that is, we will give a condition for uniform asymptotic stability in the large of the system
\begin{equation}
\left[\begin{array}{c}
\dot{\b x}\\
\dot{\b e}
\end{array}\right] = \left[\begin{array}{cc}
 \sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \left(\b A_i-\b B_i\b K_j\right)&  -\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \b B_i\b K_j\\
\b O & \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \left( \b A_i- \b L_i \b C_j \right)
\end{array}\right]\left[\begin{array}{c}
\b x\\
\b e
\end{array}\right]
\label{eq:unfcllosy}
\end{equation}
This condition was proposed by Ma, Sun and He~\cite{ma98}.
\begin{thm}
\label{thm:staboffuzobs}
If there exist two scalar functions $V(\b x):\mb{\Bb R}^n \to \mb{\Bb R}$ and $\tilde{V}(\b e):\mb{\Bb R}^n \to \mb{\Bb R}$ and positive real numbers $\gamma_i$, $\tilde{\gamma}_i$, $i=1,\ldots,4$ such that
\begin{equation}
\gamma_1\|\b x\|^2 \le V(\b x) \le \gamma_2\|\b x\|^2, \quad \tilde{\gamma}_1\|\b e\|^2 \le \tilde{V}(\b e) \le \tilde{\gamma}_2\|\b e\|^2,
\label{eq:cononvanvt}
\end{equation}
\begin{equation}
\left.\begin{array}{l}
\frac{\partial V(\b x)}{\partial \b x}\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \left(\b A_i-\b B_i \b K_j \right)\b x \le -\gamma_3 \|\b x\|^2\\ \frac{\partial \tilde{V}(\b e)}{\partial \b e}\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \left(\b A_i-\b L_i \b C_j \right)\b e \le -\tilde{\gamma}_3 \|\b e\|^2,
\end{array}\right\}
\label{eq:conontimder}
\end{equation}
and
\begin{equation}
\left\Arrowvert \frac{\partial V(\b x)}{\partial \b x}\right\Arrowvert \le \gamma_4 \|\b x\|, \quad \left\Arrowvert \frac{\partial \tilde{V}(\b e)}{\partial \b e}\right\Arrowvert \le \tilde{\gamma}_4 \|\b e\|,
\label{cononthegraofvtv}
\end{equation}
then the system~(\ref{eq:unfcllosy}) is globally uniformly asymptotically stable.
\end{thm}
{\bf Proof}\quad We first evaluate the time derivative of $V$ on the trajectories of the system~(\ref{eq:unfcllosy}). We obtain
\begin{eqnarray}
\dot{V}(\b x) &=& \frac{\partial V(\b x)}{\partial \b x}\dot{\b x}\nonumber\\
&=&\frac{\partial V(\b x)}{\partial \b x}\sum_{i=1}^r\sum_{j=1}^r \alpha_i \alpha_j\left(\b A_i-\b B_i\b K_j\right) \b x - \frac{\partial V(\b x)}{\partial \b x}\sum_{i=1}^r \sum_{j=1}^r \alpha_i \alpha_j \b B_i\b K_j {\b e}\nonumber\\
&\le & -\gamma_3\|\b x\|^2 + \left\Arrowvert \frac{\partial V(\b x)}{\partial \b x}\right\Arrowvert \sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \| {\b e}\|\nonumber\\
&\le & -\frac{\gamma_3}{2}\|\b x\|^2 -\frac{\gamma_3}{2}\|\b x\|^2 + \gamma_4 \sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \| {\b e}\|\|\b x\|.
\end{eqnarray}
To proceed, we observe that
\begin{equation}
\frac{\gamma_3}{2}\|\b x\|^2 - \gamma_4 \sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \| {\b e}\|\|\b x\|+\frac{\gamma_4^2}{2 \gamma_3} \left(\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \right)^2 \|{\b e}\|^2\ge 0.
\label{eq:adandsubtoggeo}
\end{equation}
Indeed, we can represent the left-hand side of the above as
\begin{equation}
\left[\begin{array}{cc}
\|\b x\| & \|\b e\|
\end{array}\right]\left[\begin{array}{cc}
\frac{\gamma_3}{2} & -\frac{\gamma_4}{2}\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \\
-\frac{\gamma_4}{2}\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert & \frac{\gamma_4^2}{2 \gamma_3}\left(\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert\right)^2
\end{array}\right]\left[\begin{array}{c}
\|\b x\|\\
\|\b e\|
\end{array}\right]
\label{eq:inforofsqma}
\end{equation}
It is easy to check that the above quadratic form is positive semi-definite. We use this fact and~(\ref{eq:cononvanvt}) to represent~(\ref{eq:adandsubtoggeo}) as
\begin{eqnarray}
\dot{V}(\b x) &\le & -\frac{\gamma_3}{2}\|\b x\|^2 + \frac{\gamma_4^2}{2 \gamma_3}\left(\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \right)^2 \|\b e\|^2\nonumber\\
&\le & -\frac{\gamma_3}{2 \gamma_2} V(\b x) + \frac{\gamma_4^2}{2 \gamma_3\tilde{\gamma}_1}\left(\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \right)^2 \tilde{V}(\b e).
\label{eq:evaltdovoncecc}
\end{eqnarray}
Let $a=\left(\sum_{i=1}^r \sum_{j=1}^r \left\Arrowvert \b B_i\b K_j\right\Arrowvert \right)^2$. Then, we represent~(\ref{eq:evaltdovoncecc}) as
\begin{equation}
\dot{V}(\b x) \le  -\frac{\gamma_3}{2 \gamma_2} V(\b x)+\frac{a\gamma_4^2}{2 \gamma_3\tilde{\gamma}_1}\tilde{V}(\b e).
\label{eq:fifoofvdo}
\end{equation}
We now evaluate the time derivative of $\tilde{V}(\b e)$ on the trajectories of~(\ref{eq:unfcllosy}) to obtain
\begin{eqnarray}
\dot{\tilde{V}}(\b e) &=& \frac{\partial \tilde{V}(\b e)}{\partial \b e}\dot{\b e}\nonumber\\
&\le & \frac{\partial \tilde{V}(\b e)}{\partial \b e} \sum_{i=1}^{r}\sum_{j=1}^r \alpha_i \alpha_j \left( \b A_i- \b L_i \b C_j \right)\b e\nonumber\\
&\le & -\tilde{\gamma}_3\|\b e\|^2\nonumber\\
&\le & -\frac{\tilde{\gamma}_3}{\tilde{\gamma}_2}\tilde{V}(\b e).
\label{eq:tderoftilv}
\end{eqnarray}
We combine~(\ref{eq:fifoofvdo}) and~(\ref{eq:tderoftilv}) into a single matrix differential inequality of the form,
\begin{eqnarray}
\left[\begin{array}{c}
\dot{V}(\b x)\\
\dot{\tilde{V}}(\b e)
\end{array}\right]&\le & \left[\begin{array}{cc}
-\frac{\gamma_3}{2 \gamma_2} & \frac{a\gamma_4^2}{2 \gamma_3\tilde{\gamma}_1}\\
0 & -\frac{\tilde{\gamma}_3}{\tilde{\gamma}_2}
\end{array}\right]\left[\begin{array}{c}
{V}(\b x)\\
{\tilde{V}}(\b e)
\end{array}\right]\nonumber\\
&=& \b A \left[\begin{array}{c}
{V}(\b x)\\
{\tilde{V}}(\b e)
\end{array}\right].
\label{eq:comofvandtilv}
\end{eqnarray}
We then refer to Lemma~\ref{lemma:veccomp} to conclude that the system~(\ref{eq:unfcllosy}) is globally uniformly asymptotically stable. Indeed, since the equation of comparison for~(\ref{eq:comofvandtilv}) is globally uniformly asymptotically stable, we have to have
\[
V(\b x)\to 0\quad \mb{and}\quad \tilde{V}(\b e) \to 0\quad \mb{for}\quad t\to 0.
\]
Since $V$ and $\tilde{V}$ are both positve definite decrescent and radially ubounded functions, the above can hold provided $\b x\to \b 0$ and $\b e\to \b 0$ uniformly with respect to $t$ and irrespective of the initial conditions $\b x(0)$ and $\b e(0)$. Therefore, the closed-loop system is globally uniformly asymptotically stable.\\
\hspace*{5.5in} $\Box$

\section{Example---Four Wheel Steering Control System Design}
The objective of a four wheel steering (4WS) system is to enhance a vehicle's handling performance. A common 4WS system operates by steering the rear wheels as a function of the front steer angle. It is well known among 4WS engineers that the magnitude and direction of rear steering, in-phase or out-of-phase, is a function of the vehicle's longitudinal speed. For instance, at low speeds the rear wheels are steered out-of-phase with respect to the front wheels to reduce the vehicle turning radius. For highway speeds, the rear wheels are steered in-phase with the front wheels to improve the vehicle's steerability. With in-phase steering, a 4WS vehicle translates more easily and with less yaw rotation than a corresponding two wheel steer vehicle. For the case of evasive or emergency maneuvers, such as steering to avoid an object on a highway, a combination of both out-of-phase and in-phase rear wheel steering would be recommended. This is because out-of-phase rear wheel steering excites the vehicle's yaw rate that causes the vehicle to rotate about its yaw axis. The in-phase rear wheel steering, on the other hand, dampens the yaw rate and laterally stabilizes the vehicle. Our goal in this paper is to give a method for a 4WS control system design. A block diagram of the 4WS control system is shown in Figure~\ref{fig_fwsblk}.
\begin{figure}
\vspace{3.5in}
\hspace{1in}\special{bmp:c:/tony/fwsblk.bmp x=4.67in y=3.5in}
\caption{Four wheel steer (4WS) control system block diagram.}
\label{fig_fwsblk}
\end{figure}

One of the essential elements of the control design process is a model of the dynamical system to be controlled. A detailed model that captures the vital dynamic characteristics of a vehicle is highly nonlinear and too complicated for controller design purposes. Following Friedland~\cite{friedland96}, we refer to such a model as the truth model, and we use it in simulations to evaluate the design. For purposes of the controller design, we use a simpler model that captures the important features of the plant. We refer to such a model as the design model. In this example, we use a fuzzy dynamical model of a vehicle to design a 4WS control system, and we test the controller performance on a nonlinear truth model. 


The type of fuzzy dynamical model we use is referred to as in the literature as the Takagi-Sugeno-Kang (TSK) or just the Takagi-Sugeno (TS) fuzzy model. 



Vehicles, in particular automobiles, are highly nonlinear dynamical systems, where the major nonlinearity comes from the tires. The amount of lateral force a tire develops is a nonlinear function of tire slip. Tire slip is defined as the difference between the tire's actual travel direction and the tire's desired direction, which is the steer angle. The ratio of the tire lateral force to tire slip is called the cornering stiffness. A tire's cornering stiffness is constant for small values of tire slip. A typical curve of tire lateral force versus tire slip is shown in Figure~\ref{fig:dugoff}. 
\begin{figure}
\vspace{3.5in}
\hspace{1in}\special{bmp:c:/tony/dugoff1.bmp x=4.66in y=3.5in}
\caption{A typical curve of tire lateral force versus tire slip.}
\label{fig:dugoff}
\end{figure}
Cornering stiffness changes with the tire slip angle. Vehicle linear models assume that the tire cornering stiffness as a constant. Yet in reality, the lateral force as a function of tire slip is nonlinear, that is, the cornering stiffness varies for different values of tire slip. In this example, we derive local linear models that describe dynamical behavior of a vehicle about chosen operating points. Specifically, we construct local linear models corresponding to different cornering stiffnesses. We then combine the obtained local models using fuzzy rules to arrive at a fuzzy dynamical model that we treat as a design model. Then, we use the design model to construct a four wheel steering controller. The controller's performance is tested on the design model and on the truth model.


\subsection{Vehicle Fuzzy Model}
We now develop a fuzzy model of a four wheel steering vehicle. In the model development, we specify two local linear models and the fuzzy rule base associated with the model. We begin with the development of a linear model of a four wheel steering vehicle.
\subsubsection{A Linear Model of a Four Wheel Steer Vehicle}
\label{sec:4ws}
In our derivation, we will aid ourselves with the free body diagram shown in Figure~\ref{fig:free}. We assume that the left and right front tire slip angles are the same. Also, we assume that the left and right rear tire slip angles are the same. Another assumption is that the front and rear tire steer angles are small, and so we can use small angle approximations when deriving a linear model. Further, it is assumed that the longitudinal velocity, denoted $v_x$, is constant and there is no braking. We neglect roll and pitch dynamics. 
\begin{figure}
\vspace{3.5in}
\hspace{1.25in}\special{bmp:c:/tony/bike3b.bmp x=4.67in y=3.5in}
\caption{Four wheel steer vehicle free body diagram.}
\label{fig:free}
\end{figure}
The local coordinate frame, that is attached to the vehicle, is denoted using lower case letters, $x$ and $y$. The global, fixed, coordinate frame is denoted using capital letters, $X$ and $Y$. Our derivation follows that of Wong~\cite{wong93}. 
The total lateral force denoted $F_y$ is 
\begin{equation}
F_y = 2C_f\alpha_f + 2 C_r\alpha_r,
\label{eq:forcey}
\end{equation}
where $C_f$ and $C_r$ are the front and rear tire cornering stiffness coefficients, respectively. The front and rear tire slip angles are denoted $\alpha_f$ and $\alpha_r$, respectively. The front tire slip angle expression is modeled as
\begin{equation}
\alpha_f = \delta_f - \frac{\dot{y} + l_1\dot{\theta}}{v_x},
\label{eq:alphaf}
\end{equation}
where $\delta_f$ is the front steer angle and $l_1$ is the distance from the vehicle's center-of-gravity to its front axle. The rear tire slip angle is modeled by 
\begin{equation}
\alpha_r = \delta_r + \frac{l_2\dot{\theta}-\dot{y}}{v_x},
\label{eq:alphar}
\end{equation}
where $\delta_r$ denotes the rear steer angle and $l_2$ is the distance from the vehicle's center-of-gravity to its rear axle. The body side slip angle, denoted by $\beta$, is modeled as
\begin{equation}
\beta = \tan^{-1}\left(\frac{\dot{y}}{v_x}\right).
\label{eq:beta}
\end{equation}
Let $m_{tot}$ denote the vehicle total mass seen at the vehicle's center-of-gravity. Then, using Newton's second law, we get 
\begin{equation}
m_{tot}a_y= F_y,
\label{eq:fy}
\end{equation}
where $F_y$ is the total lateral force acting on the vehicle and $a_y$ denotes the lateral acceleration given by
\begin{equation}
a_y = \ddot{y} + v_x \dot{\theta}.
\label{eq:ay}
\end{equation}
The total torque acting on the vehicle, denoted $\Gamma$, is
\begin{equation}
\Gamma = 2aC_f\alpha_f-2bC_r\alpha_r.
\label{eq:gamma}
\end{equation}
Upon substituting~(\ref{eq:alphaf}) and~(\ref{eq:alphar}) into~(\ref{eq:gamma}), we obtain
\begin{equation}
\Gamma = 2 l_1 C_f\left(\delta_f - \frac{\dot{y} + l_1 \dot{\theta}}{v_x} \right)-2l_2C_r\left(\delta_r - \frac{l_2\dot{\theta}-\dot{y}}{v_x}\right)
\label{eq:gamma2}
\end{equation}
Let $I_{cg}$ denote the vehicle moment of inertia about the normal to the $x-y$ plane. Then, by Newton's second law,
\begin{equation}
I_{cg}\ddot{\theta}= \Gamma.
\label{eq:icg}
\end{equation}
Let $x_1=\dot{y}$ and $x_2=\dot{\theta}$. We combine the above equations and represent them in matrix form to get
\begin{eqnarray}
\label{eq:state}
\left[\begin{array}{c}
\dot{x}_1\\
\dot{x}_2
\end{array}\right] &=& \left[\begin{array}{c}
-U \dot{\theta} +\frac{1}{m_{tot}}F_y\\
\frac{1}{I_z}\Gamma
\end{array}\right] \nonumber\\
&=& \left[\begin{array}{cc}
-2\frac{{C}_f+{C}_r}{m_{tot}U} & -U-2\frac{a{C}_f-b{C}_r}{m_{tot}U}\\
-2\frac{a{C}_f-b{C}_r}{I_zU} &-2\frac{a^2{C}_f+b^2{C}_r}{I_zU}
\end{array}\right]\left[\begin{array}{c}
x_1\\
x_2
\end{array}\right]\nonumber\\
&{}&\quad
+\left[\begin{array}{c}
2\frac{{C}_f}{m_{tot}}\\
2\frac{a{C}_f}{I_z}
\end{array}\right]\delta_f
+
\left[\begin{array}{c}
2\frac{{C}_r}{m_{tot}}\\
-2\frac{b{C}_r}{I_z}
\end{array}\right]\delta_r.
\end{eqnarray}

\subsection{Vehicle Local Models}
In the previous section, we mentioned that the tires are the dominant source of nonlinearity in a vehicle model. So in specifying our fuzzy model, we take into account this phenomena. As can be seen in Figure~\ref{fig:dugoff}, the tire cornering stiffness is large for tire slip values less than 0.05 rad. However, for slip values greater than 0.05 rad, the cornering stiffness is significantly smaller. We account for this fact in our fuzzy model construction in that we consider two local models. We refer to one model as the big slip model and the other as the small slip model. Both models are derived from~(\ref{eq:state}). However, they have different values for their front and rear cornering stiffness coefficients, $C_f$ and $C_r$. 

In the small slip local model, we let $C_f=50000$~N/rad and $C_r=50000$~N/rad.  Using these values and the data found in Table~\ref{tab:4wsveh}, we get
\begin{equation}
\dot{\b x} = \b A_1 \b x + \b B_{1f} \delta_f + \b B_{1r} \delta_r,
\label{eq:smallslip}
\end{equation}
where
\begin{equation}
\b A_1 =
\left[\begin{array}{cc}
-7.31301& -18.4488\\
0.0306& -6.4007\\
\end{array}\right],\; 
\b B_{1f}=\left[\begin{array}{c}
66.8896\\
48.1200
\end{array}\right],\;\mb{and}\;
\b B_{1r}=\left[\begin{array}{c}
66.8896\\
-48.6800
\end{array}\right].
\label{eq:local1}
\end{equation}
\begin{table}
\caption{Vehicle data for linear local models.}
\label{tab:4wsveh}
\begin{center}
\begin{tabular}{cc}\hline
$g$ & $9.81\,\mb{m}/\mb{s}^2$\\
$a$ & $1.203\,\mb{m}$\\
$b$ & $1.217\,\mb{m}$\\
$m_{tot}$ & $1495.0\,\mb{kg}$\\
$I_z$ & $2500\,\mb{kgm}^2$\\
$U$ & $18.3\,\mb{m}/\mb{s}$\\ \hline
\end{tabular}
\end{center}
\end{table}
For the big slip model, we let $C_f=30000$~N/rad and $C_r=30000$~N/rad in~(\ref{eq:state}) to get
\begin{equation}
\dot{\b x} = \b A_2 \b x + \b B_{2f} \delta_f + \b B_{2r} \delta_r,
\label{eq:bigslip}
\end{equation}
where
\begin{equation}
\b A_2 =
\left[\begin{array}{cc}
-4.3862& -18.2693\\
0.0184& -3.8404\\
\end{array}\right],\; 
\b B_{2f}=\left[\begin{array}{c}
40.1338\\
28.8720
\end{array}\right],\;\mb{and}\;
\b B_{2r}=\left[\begin{array}{c}
40.1338\\
-29.2080
\end{array}\right].
\label{eq:local2}
\end{equation}

\subsection{Fuzzy Rule Base}
In this subsection, we specify fuzzy sets and a fuzzy rule base for our fuzzy model of a vehicle. We use one linguistic variable for front wheel slip, and the linguistic values associated with this variable are: BIG and SMALL. Fuzzy sets describing these linguistic values are shown in Figure~\ref{fig:mem1}. 

Using our insight and experience in vehicle modeling, we propose the following two fuzzy rules. 

\begin{center}

Rule 1:\quad IF  $|\alpha_{f}|$ is SMALL\\
\quad\quad\quad THEN $\dot{\b x} = \b A_{1} \b x + \b B_{1f}\delta_f + \b B_{1r}\delta_r$\\

Rule 2: IF  $|\alpha_{f}|$ is BIG \\
\quad\quad\quad THEN $\dot{\b x} = \b A_{2} \b x + \b B_{2f}\delta_f + \b B_{2r}\delta_r$,\\ 
\end{center}
where
\[
\b x^T = \left[\begin{array}{cc}
\dot{y}&\dot{\theta}
\end{array}\right].
\]
We now fuse the above IF-THEN rules with linear local models given by equations~(\ref{eq:smallslip})--(\ref{eq:local2}) to obtain a vehicle fuzzy model. Let the membership function for the front tire slip fuzzy set be denoted by $\mu_j(\alpha_f)$. The fuzzy model's structure is a function of the constituent local models associated with each rule. The strength of each rule is referred to as its firing weight. The effective weight for each rule is normalized, and this normalized weight is 
\[
w_j = w_j(\alpha_f)= \frac{\mu_j(\alpha_f)}{\sum_{i=1}^{m}\mu_i(\alpha_f)}. 
\]
where $m$ is the number of rules. We can now express the fuzzy model in general as
\[
\dot{\b x} = \sum_{i=1}^{m} w_i \left(\b A_i \b x + \b B_{if} \delta_f+ \b B_{ir}\delta_r\right),
\]
and explicitly, for our system, as
\begin{equation}
\dot{\b x} = \left( w_1 \b A_1 + w_2 \b A_2 \right)\b x 
+ \left( w_{1} \b B_{1f} + w_{2} \b B_{2f}\right)\delta_f 
+ \left( w_{1} \b B_{1r} + w_{2} \b B_{2r}\right)\delta_r.
\label{eq:ourfuzzy}
\end{equation}
\begin{figure}
\vspace{3.5in}
\hspace{0.85in}
\special{bmp:c:/tony/mem1.bmp x=4.66in y=3in}
\caption{Front wheel slip membership functions.}
\label{fig:mem1}
\end{figure}

\section{Fuzzy Model Validation}
In this section, we compare the dynamic response of the TSK model with both the truth model and a vehicle linear model. Specifically,  we compare the lateral displacement, lateral acceleration, body side slip, yaw rate, front tire slip and lateral acceleration responses for the TSK, truth and linear models. The equations describing the truth model are given in Appendix~A. The numerical parameters we use in our simulations are given in Table~\ref{tab_param} found in Appendix~B. In the simulations, we apply the same step steering wheel input, $\delta_f$, shown in Figure~\ref{fig:tstep}, to the TSK model, truth model, and linear model and we compare the responses. In all the cases, we set $\delta_r=0$. The steering wheel input, $\delta_f$, represents the input to a vehicle during a lane change maneuver. The linear model used in our simulations is the small slip model given by~(\ref{eq:smallslip}). The results of our simulations are shown in Figures~\ref{fig:tsk1}--\ref{fig:tsk6}. The plots for the truth, fuzzy and linear vehicle models are represented by dashed, solid and dotted lines, respectively.
\begin{figure}
\vspace{3.5in}
\hspace{.5in}\special{bmp:c:/tony/tskstep.bmp x=4.66in y=3.5in}
\caption{Simulation steering input, $\delta_f$.}
\label{fig:tstep}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/tsk1.bmp x=4.66in y=3.5in}
\caption{Lateral displacement responses of the linear, truth, and TSK models.}
\label{fig:tsk1}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tsk2.bmp x=4.66in y=3.5in}
\caption{Yaw rate responses of the linear, truth, and TSK models.}
\label{fig:tsk2}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tsk3.bmp x=4.66in y=3.5in}
\caption{Body side slip angle responses of the linear, truth, and TSK models.}
\label{fig:tsk3}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tsk4.bmp x=4.66in y=3.5in}
\caption{Yaw angle responses of the linear, truth, and TSK models.}
\label{fig:tsk4}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tsk5.bmp x=4.66in y=3.5in}
\caption{Front tire slip angle responses of the linear, truth, and TSK models.}
\label{fig:tsk5}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tsk6.bmp x=4.66in y=3.5in}
\caption{Lateral acceleration responses of the linear, truth, and TSK models.}
\label{fig:tsk6}
\end{figure}
It can be observed that the fuzzy (TSK) model response is closer to the truth model in comparison with the linear model. The fuzzy model accounts for the tire nonlinearity, and with this quantity included in the TSK model, we obtain the TSK model response that closely follows the response of the truth model. This is significant because the fuzzy model's complexity is close to that of the linear model, however its performance has the integrity of the truth model. This fuzzy model performs better than the linear model.  Recall that the first stage of our control system design is to obtain a viable design model. In our controller synthesis, we will use the fuzzy (TSK) model as our design model. The fuzzy controller design entails developing local controllers for local linear models. In the next section, we first construct optimal controllers for the local models and then use these local controllers in the proposed fuzzy controller design. 


\section{Four Wheel Steering Controller Design}
\label{sec:opt}
In this section, we describe our method of 4WS control system design using a vehicle fuzzy model as a design model. Recall that the purpose of a 4WS control is to enhance a vehicle's handling by steering the rear wheels in conjunction with the front wheels. To accomplish this goal, we design a fuzzy controller for the rear wheel steer input. The front steering input is the primary input to the vehicle, while the role of the rear steering input is to enhance the handling performance of the vehicle for arbitrary front steering input. In our rear steering controller design, we set the front steer input, $\delta_f$, equal to zero. We begin by constructing feedback controllers for each local model. In particular, we construct linear quadratic regulator feedback controllers. Several other investigators have used the LQR method in the design of 4WS control systems. In particular, Higuchi and Saitoh~\cite{higuchi93} developed a 4WS control system using optimal control theory to minimize the side slip angle, $\beta$. In their implementation, the front steering wheel angle, $\delta_f$, is fed forward and the yaw rate, $\dot{\theta}$, and side slip angle, $\beta$, are fed back. Young and Kim~\cite{young95} developed another optimal 4WS control system using optimal control theory with dynamic state feedback. In their work, they proposed using a vehicle speed to determine the rear-to-front steering ratio as a function of vehicle speed. Smith and Benton~\cite{smith96} developed a nonlinear optimal 4WS controller for emergency steering maneuvers. They used an optimization routine based on Powell's method to find the optimal performance index to minimize the vehicle travel distance upon making a emergency steering maneuver. Shiosuka, Nagamatsu, and Yoshida~\cite{shiotsuka93} developed an adaptive 4WS controller using neural networks. In their design, the control system is of the model-following type. Three neural networks are used in their controller design. One neural network is trained to learn the tire cornering stiffness as a function of the tire slip angle, road coefficient of adhesion, and vehicle speed. A second neural network is used to schedule the feedforward control gain determined by the optimal regulator. The third neural network is used schedule the feedback control gain determined by the optimal regulator. 

A common control objective in the aforementioned optimal control designs involves compensating for tire nonlinearity.  Tire nonlinearity is the primary factor we are concerned with in our fuzzy modeling as well as in our fuzzy controller design. In our design approach, we combine fuzzy modeling and optimal control theory. We use the LQR design to specify our control objectives analytically.  We desire to develop a 4WS control system that minimizes the vehicle's side slip angle, $\beta$. We construct a quadratic performance index that takes into account the above design requirements. In the proposed controller implementation, we use both the yaw rate, $dot{\theta}$, and lateral velocity, $dot{y}$, to control the rear wheel steer, $\delta_r$. This leads us to minimizing the cost function
\[
 J = \int_{0}^{\infty}\left(\b x^{T}\b Q \b x + R \delta_r^2\right)dt
\] 
subject to 
\[
\dot{\b x}=\b A_i \b x + \b B_{ir} \delta_r,\quad i=1,2,
\]
where
\[
\b x^T = \left[\begin{array}{cc}
\dot{y}&\dot{\theta}
\end{array}\right],
\]
and $\b Q$ and $R$ are weight parameters that a designer has to select. The optimal rear steering control feedback, for each local model, takes the form
\[
\delta_r = -\frac{1}{R}\b B_{ir}^T \b P_i \b x
\]
where $\b P_i$, $i=1,2$, are symmetric and positive semidefinite matrices that satisfy the following algebraic Riccati equations
\[
-\b P_i\b A_i -\b A_i^{T}\b P_i + \b P_i\b B_{ir}R^{-1}\b B_{ir}^T \b P_i -\b Q= \b 0, \quad i=1,2.
\label{eq:riccati1}
\]
We choose the weight parameters, $\b Q$ and $R$ to minimize side slip angle. 
By trial and error, we settled on
\[\b Q = \left[\begin{array}{cc}
50&0\\
0&0
\end{array}\right],
\] 
and
\[
R = 1.0.
\]
Solving the corresponding Riccati equations for both the big slip and small slip linear models yields the following two optimal gain vectors. The first optimal gain vector, for the big slip model, is
\[\b K_1 = \left[\begin{array}{cc} 
6.9645& -0.2629 
\end{array}
\right].
\]
The second  optimal gain vector, for the small slip model, is
\[\b K_2=  \left[\begin{array}{cc} 
6.9655& -0.4353 
\end{array}
\right].
\]
The resulting fuzzy rear steering controller is 
\begin{equation}
\label{eq:fbk}
\delta_r =-\left( w_1 \b K_1 + w_2 \b K_2 \right)\b x.
\end{equation}
In the following section, we discuss the stability of the closed-loop system consisting of the design model given by~(\ref{eq:ourfuzzy}), with $\delta_f=0$, and the above fuzzy controller.

\section{Stability Analysis of Fuzzy Control System}
In this section, we analyze the stabilizing properties of the fuzzy controller given by~(\ref{eq:fbk}). Although, we have constructed stabilizing controllers for each local models, this does not guarantee that the closed-loop fuzzy feedback control system is stable. The feedback control system stability theory for fuzzy systems proposed by Tanaka and Sugeno~\cite{tanaka92} and further developed by Wang, Tanaka, and Griffin~\cite{wang96} enables us to check the stability of the closed-loop system. A sufficiency condition for the stability of the closed-loop system, given by Wang et al.~\cite{wang96}, requires that there exists a common symmetric positive definite matrix $\b P$ such that for
\begin{equation}
\b H_{ij}=\b A_i-\b B_i \b K_j
\end{equation}
we have
\begin{equation}
\label{eq:stab}
\b H_{ij}^{T}\b P  +  \b P \b H_{ij}  < 0, \quad i,j=1,2,
\end{equation}
that is, matrices given by~(\ref{eq:stab}) are negative definite. For our fuzzy system model
\begin{eqnarray*}
\b H_{11}&=&\b A_1-\b B_1 \b K_1\\
&=&\left[
\begin{array}{cc}
 -473.1655 &  -0.6632\\
 339.0641 & -19.1989
\end{array}\right],
\end{eqnarray*}
\begin{eqnarray*}
\b H_{12}&=&\b A_1-\b B_1 \b K_2\\
&=&\left[
\begin{array}{cc}
-264.9299  & -0.6584 \\
  191.5374 & -14.3662
\end{array}\right],
\end{eqnarray*}
\begin{eqnarray*}
\b H_{21}&=&\b A_2-\b B_2 \b K_1\\
&=&\left[
\begin{array}{cc}
-475.8051  & 10.6902 \\
  340.5286 & -27.4615
\end{array}\right],
\end{eqnarray*}
and
\begin{eqnarray*}
\b H_{22}&=&\b A_2-\b B_2 \b K_2\\
&=&\left[
\begin{array}{cc}
  -263.8030 &   -7.0163 \\
  190.7174 & -9.7391 
\end{array}\right].
\end{eqnarray*}
A common $\b{P}$ satisfying the stability condition given by~(\ref{eq:stab}) is 
\[
\b P=\left[
\begin{array}{cc}
  0.0019 &   0.0012 \\
    0.0012  &  0.0504
\end{array}\right].
\]
In conclusion, the closed-loop system consisting of fuzzy model~(\ref{eq:ourfuzzy}) driven by fuzzy controller~(\ref{eq:fbk}), for $\delta_f=0$, is globally uniformly asymptotically stable. So far, we have proven the stabilizability property of our fuzzy controller on the vehicle fuzzy model. However, the vehicle fuzzy model only serves as a design model. For the controller to be useful, it must pass a test on the simulation model of the system. Recall that the simulation model is also called the truth model. In the next section, we show simulation results of our testing of the fuzzy controller on the vehicle fuzzy model and on the truth model.

\section{Simulations of 4WS Fuzzy Controller}
\subsection{Fuzzy Controller with Fuzzy Model and Truth Model}
In this subsection, we discuss the results of the simulation experiment where we compare the fuzzy controller performance on the vehicle fuzzy model and on the truth model. The driver's input, $\delta_f$, applied to both models is the same. It is the signal shown in Figure~\ref{fig:tstep} multiplied by a factor of 2. It can be seen in Figures~\ref{fig:tcom1a}--\ref{fig:tcom6a} that the fuzzy controller performs well on the truth model. Obviously, the two systems do not yield identical results. This difference does not indicate that the proposed controller's performance is undesirable. In the next subsection, we compare the performance of the  fuzzy controller connected to the truth model with the performance of the truth model. In both cases, $\delta_r=0$ and $\delta_f$ is the same as in the previous experiment. 
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom1a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system lateral displacement responses for the vehicle truth and fuzzy models.}
\label{fig:tcom1a}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom2a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system yaw rate responses for the vehicle truth and fuzzy models.}
\label{fig:tcom2a}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom3a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system body side slip angle responses for the vehicle truth and fuzzy models.}
\label{fig:tcom3a}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom4a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system yaw angle responses for the vehicle truth and fuzzy models.}
\label{fig:tcom4a}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom5a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system front tire slip angle responses for the vehicle truth and fuzzy models.}
\label{fig:tcom5a}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{0.5in}
\special{bmp:c:/tony/tcom6a.bmp x=4.66in y=3.5in}
\caption{Closed-loop system lateral acceleration responses for the vehicle truth and fuzzy models.}
\label{fig:tcom6a}
\end{figure}

\subsection{4WS versus Two Wheel Steering (2WS)}
In this subsection, we compare the performance of the proposed fuzzy controller with that of a front wheel steered vehicle truth model. Note that a front wheel steered vehicle truth model, called the 2WS system, is the truth model where we set $\delta_r=0$. Again, we use the same input to both models as before. In Figures~\ref{fig:fcom1}--\ref{fig:fcom6}, we present the lateral displacement, yaw rate, body side slip, front tire slip, lateral acceleration and yaw angle responses.
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom1.bmp x=4.66in y=3.5in}
\caption{Lateral displacement responses for the 2WS and 4WS systems.}
\label{fig:fcom1}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom2.bmp x=4.66in y=3.5in}
\caption{Yaw rate responses for the 2WS and 4WS systems.}
\label{fig:fcom2}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom3.bmp x=4.66in y=3.5in}
\caption{Body side slip angle responses for the 2WS and 4WS systems.}
\label{fig:fcom3}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom6.bmp x=4.66in y=3.5in}
\caption{Yaw angle responses for the 2WS and 4WS systems.}
\label{fig:fcom4}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom4.bmp x=4.66in y=3.5in}
\caption{Front tire slip angle responses for the 2WS and 4WS systems.}
\label{fig:fcom5}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hspace{1.0in}
\special{bmp:c:/tony/fcom5.bmp x=4.66in y=3.5in}
\caption{Lateral acceleration responses for the 2WS and 4WS systems.}
\label{fig:fcom6}
\end{figure}
Notice that in Figure~\ref{fig:fcom1} the lateral displacements do not differ much. However, the orientation responses of the vehicles shown in Figure~\ref{fig:fcom6} differ significantly. The body side slip angle is noticeably reduced by the 4WS controller as compared with that of the 2WS vehicle. The yaw rate and yaw angle responses, whose plots are found in Figures~\ref{fig:fcom2} and~\ref{fig:fcom6}, respectively, indicate that the fuzzy controller makes the vehicle dynamics response to  resemble that of a linear model. One can say that the fuzzy controller tends to linearize the vehicle dynamics in the presence of the tire nonlinearity.

In this example, we showed how to use the TSK fuzzy model as a design model to construct a  four wheel steering control system.  An important aspect of this approach is that there exists a theory that allows stability analysis of the fuzzy control system to be conducted. This is significant because automotive engineers have been skeptical about using fuzzy logic controllers, in chassis control applications, since there was no way to theoretically show their stability or robustness.

The constituents of the fuzzy model are local linear models glued together with a linguistic description of the vehicle. The linear models are simple, and thereby are tractable within the linear quadratic regulator (LQR) design framework. Although simple, these local linear models combined with the linguistic description of the vehicle well mimic the dynamic performance of the more complicated nonlinear vehicle model. We combined the LQR control theory and fuzzy modeling to construct a novel vehicle steering controller. 
The controller globally stabilizes the design model, and results in a very good performance when it is connected to the nonlinear simulation vehicle model that we referred to as the truth model. 


\begin{figure}
\vskip3.5in
\hskip.75in\special{eps:c:/tony/lateral.eps y=4in}
\caption{Lateral displacement.}
\label{fig:lateral}
\end{figure}



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