36.2.1 Fuzzy-Based PI Controller

The fuzzy-PI system implements an FL for the PI control signals e_filt and Δe_filt, the error and the delta error, respectively, the linguistic inputs to the system. The FL output is the PI control ΔUPI signal. The error signal has three fuzzy variables: positive (P), zero (Z), and negative (N). The delta error signal has three fuzzy variables: positive (P), zero (Z), and negative (N). The membership function for each fuzzy variable is a Gaussian function. Figs. 36.2 and 36.3 display the trained membership functions for the fuzzy inputs.

The linguistic control rules are established considering the dynamic behavior of the BLDC drive system and analyzing the error and its variation. The fuzzy inference system has nine rules.

Rule 1: IF speed error is negative (N) AND delta speed error is negative (N), THEN the fuzzy-PI control voltage (output of FL) is negative large (NL). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}<w_{\mathrm{meas}}$ and that the difference between them is increasing. Thus, the control action must act to change the error momentum and bring the motor rate back down to the reference rate. Therefore, a large negative change in control voltage is necessary.

Rule 2: IF speed error is negative (N) AND delta speed error is zero (Z), THEN the fuzzy-PI control voltage (output of FL) is negative medium (NM). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}<w_{\mathrm{meas}}$ and that the difference between them is steady. Thus, the control action need not change the error momentum but still bring the motor rate back down to the reference rate. Therefore, a medium negative change in control voltage is necessary.

Rule 3: IF speed error is zero (Z) AND delta speed error is negative (N), THEN the fuzzy-PI control voltage (output of FL) is negative small (NS). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}=w_{\mathrm{meas}}$ but tending toward $w_{\mathrm{r}\mathrm{e}\mathrm{f}}<w_{\mathrm{meas}}$ as the error trends more negative. Thus, the control action must act to halt the error momentum but maintain the motor rate being close to the reference rate. Therefore, a small negative change in control voltage is necessary.

Rule 4: IF speed error is negative (N) AND delta speed error is positive (P), THEN the fuzzy-PI control voltage (output of FL) is zero (ZE). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}<w_{\mathrm{meas}}$ and that the difference between them is decreasing. Thus, the control action of nothing allows the error momentum to bring the motor rate back down to the reference rate. Therefore, no change in control voltage is necessary.

Rule 5: IF speed error is zero (Z) AND delta speed error is zero (Z), THEN the fuzzy-PI control voltage (output of FL) is zero (ZE). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}=w_{\mathrm{meas}}$ and that the difference between them is steady. Thus, the control action of nothing allows the error momentum to remain negligible to leave the motor rate at the reference rate. Therefore, no change in control voltage is necessary.

Rule 6: IF speed error is positive (P) AND delta speed error is positive (P), THEN the fuzzy-PI control voltage (output of FL) is positive large (PL). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}>w_{\mathrm{meas}}$ and that the difference between them is increasing. Thus, the control action must act to change the error momentum and bring the motor rate back up to the reference rate. Therefore, a large positive change in control voltage is necessary.

Rule 7: IF speed error is positive (P) AND delta speed error is zero (Z), THEN the fuzzy-PI control voltage (output of FL) is negative medium (NM). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}>w_{\mathrm{meas}}$ and that the difference between them is steady. Thus, the control action need not change the error momentum but still bring the motor rate back down to the reference rate. Therefore, a medium positive change in control voltage is necessary.

Rule 8: IF speed error is zero (Z) AND delta speed error is positive (P), THEN the fuzzy-PI control voltage (output of FL) is positive small (PS). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}=w_{\mathrm{meas}}$ but tending toward $w_{\mathrm{r}\mathrm{e}\mathrm{f}}>w_{\mathrm{meas}}$ as the error trends more positive. Thus, the control action must act to halt the error momentum but maintain the motor rate being close to the reference rate. Therefore, a small positive change in control voltage is necessary.

Rule 9: IF speed error is positive (P) AND delta speed error is negative (N), THEN the fuzzy-PI control voltage (output of FL) is zero (ZE). This rule pertains to the condition that $w_{\mathrm{r}\mathrm{e}\mathrm{f}}>w_{\mathrm{meas}}$ and that the difference between them is decreasing. Thus, the control action of nothing allows the error momentum to bring the motor rate back down to the reference rate. Therefore, no change in control voltage is necessary.

Each rule is implemented via a weighted product of the membership variables. The fuzzy centroid method generates a crisp output from the nine-dimensional fuzzy rule vector. This scaled output corresponds to the control signal, ΔUPI. Figs. 36.4 and 36.5 illustrate the initial and trained fuzzy control output ΔUPI as a function of the error and the change of error.

36.2.2 Fuzzy-Based PD Controller

The same technique applied to the fuzzy-PI controller is applied to the fuzzy-PD controller. The controller has two inputs and one output. The fuzzy-PD system implements an FL for the PD control signal. The signals e_filt and Δe_filt, the error and the delta error, respectively, are the linguistic inputs to the system. The FL output is the PD control signal UPD. The error signal has three fuzzy variables: positive (P), zero (Z), and negative (N). The delta error signal has three fuzzy variables: positive (P), zero (Z), and negative (N). The membership function for each fuzzy variable is a Gaussian function. Figs. 36.6 and 36.7 show the trained membership functions for the PD fuzzy inputs.

Similarly, the number of fuzzy rules that are required is equal to the product of the number of fuzzy sets that make up each of the two fuzzy input variables. Therefore, a total of nine fuzzy rules are instituted. The nine fuzzy rules are composed of the same fuzzy rules generated for the fuzzy-PI controller and are not repeated here. Each rule is implemented via a weighted product of the membership variables. The fuzzy centroid method generates a crisp output from the nine-dimensional fuzzy rule vector. This scaled output corresponds to the control signal, UPD. Figs. 36.8 and 36.9 demonstrate the initial and trained fuzzy control output, UPD, as a function of the error and the change of error.

Figs. 36.8 and 36.9 provide an indication of how well the EKF learning algorithm succeeds in training the output fuzzy-PD control signal. This is an indication of how well the control process is proceeding during the experimental implementation.

36.3.1 Input Layer

The input layer passes the input value, assigned for each FNN as follows (let be the feedback error).

36.3.2 The Membership Layer

Each membership function into the fuzzy set is a Gaussian; the following evaluation occurs (note that m_ij and σ_ij are parameters):

$y_{ij}^{\mathrm{m}\mathrm{e}\mathrm{m}}=exp\frac{{\left(u_i-m_{ij}\right)}^2}{{\sigma }_{ij}^2}$ for input $i=1,2$ and member $j=1,2,3$ of that input.

For further derivations, allow the matrix y^mem to be vectorized as follows:

$\begin{array}{l}{y}_1^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{11}^{\mathrm{m}\mathrm{e}\mathrm{m}},y_2^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{12}^{\mathrm{m}\mathrm{e}\mathrm{m}},y_3^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{13}^{\mathrm{m}\mathrm{e}\mathrm{m}},y_4^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{21}^{\mathrm{m}\mathrm{e}\mathrm{m}},\\ y_5^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{22}^{\mathrm{m}\mathrm{e}\mathrm{m}},y_6^{\mathrm{m}\mathrm{e}\mathrm{m}}=y_{23}^{\mathrm{m}\mathrm{e}\mathrm{m}}\end{array}$

and allow the same vectorization for the means m matrix into a vector and sigmas σ matrix into a vector.

36.3.3 The Rule Layer

Each rule function is a product of two membership outputs. Refer to the table: each table cell is a rule layer output calculated by the product of the row and column label values.

36.3.4 The Output Layer

The weighted summation of the rules composes the output layer (note parameters W_k): $y={\Sigma }_{k=1}^9{y}_k^{\mathrm{rule}}{W}_k$ where $k=1,9$.

36.5.1 Fuzzy-Based Proportional Controller

The first step in designing the controller is to decide which state variables of the drive system can be taken as the input signals to the controller. Both the speed error, e(k), and the delayed feedback control signal, $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$, are used as the inputs to the speed controller. The output of the fuzzy-based proportional controllers is the gain, FK_P. The linguistic fuzzy variable “e(k)” has three sets: positive large (PL), zero (ZE), and negative large (NL), with each set having its own membership function. Furthermore, the linguistic fuzzy variable “$u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$” also has three sets: positive large (PL), zero (ZE), and negative large (NL), with each set having its own membership function. After specifying the fuzzy sets, it is required to determine the membership functions for these sets. Typical triangular membership functions are utilized for the e(k) and $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$. The three fuzzy sets can be symbolized by $F_i^j,i=1,2$ and $j=1,2,3\text{.}$ Their corresponding membership functions can be symbolized by $\mu F^j\left(e,u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right),j=1,2,3$_.

The next step in the design of the fuzzy subcontroller is the determination of the fuzzy IF-THEN inference rules. The number of fuzzy rules that are required is equal to the product of the number of fuzzy sets that make up each of the two fuzzy input variables. Thus, a total of nine fuzzy rules are required to relate each possible combination of the two fuzzy input variables to the output membership fuzzy sets. The fuzzy rules ensure that the output of each fuzzy controller enhances the overall speed-tracking performance. Examples of such rules are the following:

IF e(k) is positive large AND $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$ is negative large, THEN FK_P is positive medium (PM).

IF e(k) is zero AND $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$ is zero, THEN FK_P is positive small (PS).

IF e(k) is positive large AND $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$ is zero, THEN FK_P is positive very large (PVL).

In general, a typical fuzzy rule is of the form:

R^(k): IF e(k) is F₁^j, and $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$ is F₂^l, THEN f_P is C_k^l for $j=1\dots 3,l=1\dots 3,k=1\dots 9$.

The conjunction of the rule antecedents is evaluated by the fuzzy operation intersection, which is implemented by the min operator. The rule strength represents the degree of membership of the output variable for a particular rule. Defining the rule strength, ξ_i,j of a particular rule as ${\xi }_{i,j}=min\left({\mu }_{F_i},{\mu }_{F_j}\right)$ where $i\in \left[\mathit{PL},ZE,NL\right]$ is associated with the fuzzy variable, e(k), and $j\in \left[\mathit{PL},ZE,NL\right]$ is associated with the fuzzy variable, $u_{\mathrm{F}\mathrm{P}}\left(k-1\right)$. The fuzzy inference engine uses the appropriately designed knowledge base to evaluate the fuzzy rules and produce an output for each rule. Subsequently, the multiple outputs are transformed to a crisp output by the defuzzification interface. Once the aggregated fuzzy set representing the fuzzy output variable has been determined, an actual crisp control decision must be made. The process of decoding the output to produce an actual value for the controller gain FK_P is referred to as defuzzification. Thus, a fuzzy-logic controller-based center-average defuzzifier is implemented. The output of fuzzy-based proportional controller is given by $F{K}_{\mathrm{P}}=f_{\mathrm{P}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right)$, where

$f_{\mathrm{P}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right)=\frac{{\displaystyle \sum _{1=1}^9{\mu }_{{\mathrm{O}}^l}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right)\right)\right)}}{{\displaystyle \sum _{\mathrm{l}=1}^9}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right)\right)\right)}$

The linguistic fuzzy output variable “FK_P” has nine sets: negative very large (NVL), negative large (NL), negative medium (NM), and negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive large (PL), and positive very large (PVL). The two fuzzy sets, namely, negative very large (NVL) and positive very large (PVL), are added to enhance the tracking performance. After specifying the fuzzy sets, it is required to determine the membership functions for these sets μ_O^l for $l=1\dots 9$. The membership function for the fuzzy set-representing zero is a fuzzy singleton. Additionally, the other membership functions are composed of fuzzy singletons within the region defined for the fuzzy output variable. Consequently, the control signal generated by the fuzzy-proportional controller can be written as $u_{\mathrm{F}\mathrm{P}}\left(k\right)=G_{\mathrm{P}}\left\{f_{\mathrm{P}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{P}}\left(k-1\right)\right)\right\}e\left(k\right)$, where G_P is the scaling factor that is tuned experimentally using genetic optimization.

36.5.2 Fuzzy-Based Integral Controller

The same technique applied to the fuzzy-based proportional controller is applied to the fuzzy-based integral controller. The controller has two inputs and one output. The inputs to the fuzzy-based controller are (1) the angular speed error, e(k), and (2) the delayed feedback control signal, $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$. The controller output is the gain, FK_I, of the controller. Three fuzzy sets are defined for the fuzzy linguistic variable “e(k)” and three fuzzy sets for the fuzzy linguistic variable $“u_{\mathrm{F}\mathrm{I}}\left(k-1\right).”$ After specifying the fuzzy sets of the two fuzzy variables, the membership functions for these sets are derived. The membership functions are composed of the same fuzzy triangular membership functions allocated for the fuzzy-based proportional controller. Similarly, the number of fuzzy rules that are required is equal to the product of the number of fuzzy sets that make up each of the two fuzzy input variables. Therefore, a total of nine fuzzy rules are introduced. Some of these rules and their significance are explained as follows:

Rule 1: IF e(k) is positive large (PL) AND $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$ is positive large (PL), THEN FK_I is positive large (PL). This rule implies a general condition when both input signals are positive. Consequently, the output membership function assigned to this rule is a positive value to keep the controller gain within a stable range.

Rule 2: IF e(k) is zero (ZE) AND $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$ is positive large (PL), THEN FK_I is positive very small (PVS). This rule implies a general condition when the speed error is close to zero and $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$ is positive. Therefore, the output membership function assigned to this rule is a small positive value to reduce the control signal variations.

Rule 3: IF e(k) is negative large (NL) AND $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$ is positive large (PL), THEN FK_I is positive medium (PM). This rule usually deals with the case in which overshoots occur. A medium positive value is assigned in this case to ensure that the overshoot is decreased and prevent oscillations at the same time.

The rule strength represents the degree of membership of the output variable for a particular rule. Similarly, the rule strength, ξ_n,m of a particular rule is defined as

${\xi }_{n,m}=min\left({\mu }_{F_n},{\mu }_{F_m}\right)$

where $n\in \left[\mathit{PL},ZE,NL\right]$ is associated with the fuzzy variable, e(k), and $m\in \left[\mathit{PL},ZE,NL\right]$ is associated with the fuzzy variable, $u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$.

The fuzzy inference engine uses the appropriately designed knowledge base to evaluate the fuzzy rules and produce an output for each rule. Subsequently, the multiple outputs are transformed to a crisp output, by the defuzzification interface. A center-average defuzzifier method is used. Thus, the output of fuzzy-based integral controller is given by $F{K}_{\mathrm{I}}=f_{\mathrm{I}}(e\left(k\right),u_{\mathrm{F}\mathrm{I}}\left(k-1\right)$ where

$f_{\mathrm{I}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{I}}\left(k-1\right)\right)=\frac{{\displaystyle \sum _{1=1}^9{\mu }_{{\mathrm{O}}^l}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{I}}\left(k-1\right)\right)\right)\right)}}{{\displaystyle \sum _{\mathrm{l}=1}^9}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{I}}\left(k-1\right)\right)\right)\right)}$

After specifying the fuzzy sets for the fuzzy output variable, FK_I, the membership functions for these sets are derived. It is important to note that the same fuzzy singleton membership functions are used here. The control signal of the fuzzy-based integral controller is given by

$u_{\mathrm{F}\mathrm{I}}\left(k\right)=G_{\mathrm{I}}\left\{f_{\mathrm{I}}\left(e,u_{\mathrm{F}\mathrm{I}}\left(k-1\right)\right)\right\}{\displaystyle \sum _{i=0}^k}e\left(i\right)\Delta t,k=0,1,2,\dots \text{,}$

where G_I is the scaling factor that is tuned experimentally using genetic optimization.

36.5.3 Fuzzy-Based Derivative Controller

The same process applied to fuzzy-based proportional controller and fuzzy-based integral controller is applied to the fuzzy-based derivative controller. The input signals (state variables) to the controller are e(k) and $u_{\mathrm{F}\mathrm{D}}\left(k-1\right)$. The output of the controller is the gain FK_D. After specifying the fuzzy sets and the membership functions for these sets, the triangle membership functions are also used to define the degree of membership. Figs. 36.2 and 36.3 show the membership functions for the fuzzy sets. Similarly, the linguistic fuzzy output variable “FK_D” has nine sets: negative very large (NVL), negative large (NL), negative medium (NM), and negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive large (PL), and positive very large (PVL). Now, it is necessary to find the final controller output. Consequently, the output of fuzzy-based derivative controller is given by $F{K}_{\mathrm{D}}=f_{\mathrm{D}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{D}}\left(k-1\right)\right)$, where

$f_{\mathrm{D}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{D}}\left(k-1\right)\right)=\frac{{\displaystyle \sum _{\mathrm{l}=1}^9}{\mu }_{{\mathrm{O}}^l}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{D}}\left(k-1\right)\right)\right)\right)}{{\displaystyle \sum _{\mathrm{l}=1}^9}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(e\left(k\right),u_{\mathrm{F}\mathrm{D}}\left(k-1\right)\right)\right)\right)}$

The control signal of the fuzzy-based derivative controller:

$u_{\mathrm{F}\mathrm{D}}\left(k\right)=G_{\mathrm{D}}{f}_{\mathrm{D}}\left(e\left(k\right),u_{\mathrm{F}\mathrm{D}}\left(k-1\right)\right)\Delta e\left(k\right)/\Delta t$

where G_D is a scaling factor that tuned experimentally using genetic optimization.

Finally, the overall output of fuzzy PID was derived as follows:

$u_{\mathrm{FPID}}\left(k\right)=u_{\mathrm{F}\mathrm{P}}\left(k\right)+u_{\mathrm{F}\mathrm{I}}\left(k\right)+u_{\mathrm{F}\mathrm{D}}\left(k\right)$

Saturation is also incorporated in the hybrid fuzzy-PID controller structure. As such, the final output of proposed controller is

$\begin{array}{l}{u}_{\text{max}}\left(k\right),u_{\mathrm{FPID}}\left(k\right)>u_{\text{max}}\left(k\right),\\ u\left(k\right)=u_{\mathrm{FPID}}\left(k\right)u_{\text{min}}\left(k\right)\le u_{\mathrm{FPID}}\left(k\right)\le u_{\text{max}}\left(k\right)\\ u_{\text{min}}\left(k\right)u_{\mathrm{FPID}}\left(k\right)<u_{\text{min}}\left(k\right)\left(10\right)\end{array}$

where u_min and u_max are the permitted minimum and maximum inputs to the brushless drive system.

The brushless DC motor used in the section is excited by a three-phase sinusoidal supply and driven from a six-step inverter, which converts a constant voltage to a three-phase voltage with a frequency corresponding instantaneously to the rotor speed. The model of the drive system is used to design the $H\infty$ tracking controller and to simulate its dynamic behavior. However, fuzzy-logic controller does not require mathematical models and has the ability to approximate nonlinear systems. The reader is referred to [16] for details of the dynamic model of the brushless dc motor drive system. For the purpose of discussion, the dynamics of the brushless servomotor drive system can be written in state space form as follows:

$x^{\left(n\right)}=f\left(x\right)+Bu+d$

where $x={\left(x,\stackrel{.}{x},\dots ,x^{\left(n-1\right)}\right)}^T\in R^n$, d is the external disturbance, and $B\ne 0$. Since $B\ne 0$ for $x\in R^n$, the system is controllable in the whole space Rⁿ.

36.8.2 The Fuzzy H∞ Control Structure

A hybrid adaptive fuzzy controller is implemented, which employs a fuzzy-logic controller integrating an $H\infty$ tracking controller. The $H\infty$ tracking controller is capable of taking care of the robust stabilization and disturbance rejection problems, while fuzzy-logic controller is used as principle components of the adaptive fuzzy controller. Actually, it is outfitted with an adaptive control law that modifies the parameters of the controller based on a Lyapunov synthesis approach. Fig. 36.13 shows the block diagram of the proposed control structure.

36.8.3 H∞ Control Design

The controller objective is to obtain high-performance servo tracking; therefore, the control problem is to force the actual output to follow a predetermined bounded reference trajectory. The tracking performance indicates how closely the servomotor follows a desired position trajectory, velocity trajectory, or acceleration trajectory. For the purpose of discussion, we define the tracking error to be $\overline{\varepsilon }={\left[\varepsilon ,\stackrel{.}{\varepsilon }\right]}^T$, where ɛ is the rotor speed error and $\stackrel{\text{.}}{\varepsilon }$ is the acceleration error. Consequently, the motor speed error $\varepsilon ={\omega }_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\omega }_{\mathrm{act}}$ and the acceleration error $\stackrel{.}{\varepsilon }={\alpha }_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\alpha }_{\mathrm{act}}$ are used as input variables to the $H\infty$ controller. Here, ω_act is the actual speed in rev/s, ω_ref is the desired reference speed, α_ref is the desired acceleration in rev/s², and α_act is the calculated acceleration. The acceleration is calculated as the difference in the speed over two successive sampling periods. A block diagram of the $H\infty$ tracking controller is shown in Fig. 36.14. The $H\infty$ control law is defined as [17]

$u_{\infty }=-\frac{1}{r_{\mathrm{r}}}{B}^{T}P\varepsilon$

where r_r is a weighting factor and P is the solution of the simplified Riccati-like equation for $P=P^T\ge 0$

$PA+A^{T}P=-Q$

The matrices A and B are given by

$A=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -k_2\hfill & \hfill -k_1\hfill \end{array}\right]B=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$

where $k=\left[k_1,k_2\right]$ is chosen such that all roots of the polynomial $p^{\left(S\right)}=S^n+K_n{S}^{n-1}+\cdots +k_1$ are in the open left-hand plane.

36.8.4 Fuzzy Logic Control Design

The fuzzy-logic principles with center-average defuzzifier, product inference, and singleton fuzzifier are used as building blocks for the fuzzy controller. A block diagram of the fuzzy control structure is shown in Fig. 36.15. The fuzzy-logic-based controller implemented in this work has two inputs and one output. The inputs are the rotor speed error and the feedback acceleration error, with each of these inputs corresponding to a fuzzy variable.

The output is the fuzzy control decision. The fuzzy variable associated with the rotor speed error signal consists of five fuzzy sets: positive large (PL), positive medium (PM), zero (ZE), negative large (NL), and negative medium (NM). Typical example of the fuzzy membership functions used for the speed is shown in Fig. 36.16. In this way, for any given value of the rotor angle, the degree of membership μ, to which it belongs to each of these sets, can be determined, and a control decision based on this information can be obtained.

Similarly, the acceleration error can be described by a group of partially overlapping fuzzy sets as positive large (PL), positive medium (PM), zero (ZE), negative large (NL), and negative medium (NM), with each set having its own membership function. An example of typical membership functions that make up the fuzzy variable acceleration is shown in Fig. 36.17.

The five fuzzy sets can be symbolized by $F_{\mathrm{I}}^j,i=1,2$ and $j=1,2,3\dots 5$. Their membership functions can be symbolized by $\mu F_{\mathrm{I}}^j\left(x_{\mathrm{I}}\right),i=1,2$ and $j=1,2,3\dots 5$.

36.8.5 Fuzzy Logic Rules

The next step in the design of the fuzzy-logic-based controller is the determination of the fuzzy IF-THEN inference rules. The number of fuzzy rules that are required is equal to the product of the number of fuzzy sets that make up each of the two fuzzy input variables. For the fuzzy controller described here, the input variables representing the rotor speed and its acceleration consist of five fuzzy sets each. Thus, a total of 25 fuzzy rules are required. During each sampling interval that a controller is calculated, the present value of the degree of membership of the motor speed and its acceleration is calculated for each of the fuzzy sets that make up their respective fuzzy variables. This process is one of the largest computational burdens of the fuzzy-logic control algorithm if each fuzzy set is expected to have some degree of membership for the present input value of the motor speed and its acceleration. This will in addition affect the computational speed at which the fuzzy rules are processed since all the rules have to be evaluated.

Using a triangular membership function for the input variables was very desirable to reduce this computational burden and speed up the process. This ensures that only two fuzzy sets, which are overlapping, will actually have a nonzero degree of membership (Figs. 36.16 and 36.17). With this advantage, the evaluation of the degree of membership in each set is accelerated. This also speeds up the evaluation of the fuzzy rules since only those rules that correspond to nonzero values of membership function need to be evaluated. Hence, for this controller, only four fuzzy rules need to be evaluated during any sampling interval. The fuzzy inference rules have been determined experimentally using the measured responses of the brushless drive servo system. A typical fuzzy rule is of the form:

[R⁽¹⁾:] IF x₁ is F₁^j, and x₂ is F₂^j, THEN O is G₁^l for $j=1\dots 5,l=1\dots 5$.

In general,

[R^(k):] IF x₁ is F₁^j, and x₂ is F₂^j, THEN O is G_k^l for $j=1\dots 5,l=1\dots 5,k=1\dots 25$.

The conjunction of the rule antecedents is evaluated by the fuzzy operation intersection, which is implemented by the min operator. The rule strength represents the degree of membership of the output variable for a particular rule. Defining the rule strength, ξ_i,j, of a particular rule as

${\xi }_{i,j}=min\left({\mu }_{F_i^{\mathrm{speed}}},{\mu }_{F_j^{\mathrm{accel}}}\right)$

where $i\in \left[\mathit{PL},\mathit{PM},ZE,NM,NL\right]$ is associated with the fuzzy variable, speed, and $j\in \left[\mathit{PL},\mathit{PM},ZE,NM,NL\right]$ is associated with the fuzzy variable, acceleration. From above, it is clear that for all the rules where at least one of the degrees of membership in the corresponding fuzzy sets is zero, the output of the min operator will be zero, and hence, these rules do not have to be analyzed. The fuzzy inference engine uses the appropriately designed knowledge base to evaluate the fuzzy rules and produce an output for each rule. Subsequently, the multiple outputs are transformed to a crisp output by the defuzzification interface.

36.8.6 Defuzzification

Once the aggregated fuzzy set representing the fuzzy output variable has been determined, an actual crisp control decision must be made. The process of decoding the output to produce an actual value for the control signal is referred to as defuzzification. Thus, a fuzzy-logic controller-based center-average defuzzifier is implemented. The control output can be written in general form:

$O\left(x\right)={\displaystyle \sum _{\mathrm{l}=1}^{25}}{\mu }_{{\mathrm{O}}^l}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(x\right)\right)\right)/{\displaystyle \sum _{\mathrm{l}=1}^{25}}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(x\right)\right)\right)$

which can be written in the form

$O\left(x\right)={\theta }^T\xi \left(x\right)={\xi }^T\left(x\right)\theta$

where

${\xi }^l\left(x\right)={\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(x\right)\right)/{\displaystyle \sum _{\mathrm{l}=1}^{25}}\left({\displaystyle \underset{i=1,2}{min}}\left({\mu }_{{\mathrm{F}}_i^l}\left(x\right)\right)\right)$

$\theta =\left({\mu }_{{\mathrm{O}}^1},\dots {\mu }_{{\mathrm{O}}^{25}}\text{,}\right)$ is a parameter vector representing the degree of membership of the output, which is tuned online using an adaptation law.

36.8.7 On-Line Adaptation

By using an adaptive law, the fuzzy parameters, θ, can be tuned online based on the Lyapunov synthesis approach. The feedback control law can be expressed as follows [17]:

$u={\xi }^T\left(x\right)\theta -u_{\infty }/g$

where θ is a vector denoting the parameter estimate of fuzzy-logic controller. The parameter θ is adjusted using the following adaptive law:

$\stackrel{.}{\theta }=\gamma \xi \left(x\right)g{B}^{T}P\varepsilon$

where γ is an adaptation gain and g is a constant that depends on machine parameters. A block diagram of the hybrid adaptive fuzzy controller is shown in Fig. 36.18.

The DC-DC converters are generally divided into two groups: hard-switching converters and soft-switching converters. In hard-switching converters, the power switches cut off the load current within the turn-on and turn-off times under the hard-switching conditions. The output voltage is controlled by adjusting the on-time of the power switch, which in turn adjusts the width of a voltage pulse at the output. This is known as PWM control. In soft-switching converters, resonant components are used to create oscillatory voltage or current waveforms so that the zero-voltage switching or zero-current switching conditions could be created for the power switches. For many years, control design for converters has been carried out through analog circuits, which limited them to mostly PI controller structure. The PI controllers generally give overshoot in output voltage and high initial current when rise time of response is reduced. Feed-forward types of controllers have also been designed by sensing the input voltage to improve line regulation in applications with a wide range of input voltages and load currents. However, direct sensing of the input voltage through a feed-forward loop may induce large-signal disturbances that could upset the normal duty cycle of the converter. Using human linguistic terms and common sense, several fuzzy-logic controllers have been developed and implemented for the DC-DC converters [18–20]. These controllers have shown promise in dealing with nonlinear systems and achieving voltage regulation in buck converters. Fuzzy-logic control uses humanlike linguistic terms in the form of IF-THEN rules to capture the nonlinear system dynamics. Once in place, the fuzzy rules will not be able to adapt themselves to adequately capture the dynamics and external disturbances of the converter. Although achieving many practical successes, fuzzy control has not been viewed as a rigorous science due to a lack of formal analysis and synthesis techniques. As a result of this, a lot of work has been done to develop adaptive fuzzy controllers and automate the modeling process as much as possible. Recently, the resurgence of interest in ANN has injected a new driving force into fuzzy literature. One of the major features of ANNs is their learning capability. A learning algorithm is usually used to update the nonlinear parameters of the network architecture. A drawback in using an ANN for control is that there is so much freedom in structural implementation choice that it is often difficult to decide how complex a structure is actually necessary for the desired control. Besides, the implementation is not at all intuitive, and the inner workings of the network are very much invisible to the designer. The integration of neural network architectures with fuzzy inference system has resulted in a very powerful strategy known as adaptive network-based fuzzy inference system. Some researchers suggest that neural networks and fuzzy control are in fact special instances of adaptive networks [21,22]. The suggested fuzzy inference system has the advantage that it can not only take linguistic information from human experts but also adapt itself using numerical data to achieve better performance.

36.10.1 DC-DC Switch-Mode Power Converter

The DC-DC switch-mode power converter consists of six distinct circuit networks as shown in Fig. 36.21, working together to form a complete DC-to-DC dual-output switch-mode power regulator [23]. The six distinct circuit networks are (1) the bias voltage regulator network, (2) the 100 KHZ clock, (3) the pulse-width modulator (PWM), (4) the power switch driver interface, (5) the feedback error amplifier, and (6) the integrated-magnetic (IM) power stage. In order to implement the proposed adaptive fuzzy-neural-network controller, the converter system shown in Fig. 36.21 was modified by removing the bias voltage regulator network, the 100 KHz clock, and the pulse-width modulator (PWM), which is the core of the control system. The power-stage concept is based on that of a “dual-output forward” configuration operating in a continuous mode of energy storage. When the power MOSFET switch Q1 is turned ON, energy is transferred from the input power source (V_IN) to the two secondary sides of the transformer. The voltage potential across the terminals of C1 will be that of the reflected line voltage, namely, $N_{\mathrm{S}1}\times V_{\mathrm{IN}}/N_{\mathrm{P}}$. The ON time is a fraction of the clock period, and it has to be short and long enough to bring the output to a level above the desired voltage (5 and 15 V). Diodes D1 and D4 conduct allowing the energy transfer to be used to supply 5 and 15 V load power, respectively, and to replenish the energy lost in the “output” inductors during the previous OFF period of Q1. Diodes D2 and D3 do not conduct during this time, since they are reverse-biased.

When Q1 turns OFF, the 5 and 15 V load power is sustained by the energies of the two “inductors” (L1 and L2). The output stored energy, located in two inductors, will then start discharging causing the voltages to drop. Diodes D2 and D3 conduct during this time with diodes D1 and D4 forced off. As soon as the voltages go below the desired values, another impulse will be applied on Q1 (Q1 ON), to bring the voltage back upward. Capacitors C2 (0.001 μf) and C3 (100 μf) serve to prevent the dynamic ripple currents in the “inductors” from appearing at the 5 and 15 V load terminals, respectively.

36.11.1 Membership Functions and Rules Generation

The chief problem of the construction of the fuzzy controller is related to the choice of the regulator parameters. Certainly, there is no systematic procedure for the design of a fuzzy controller (membership functions and rules generation), and this section is devoted to the determination of the fuzzy sets and the linguistic rules. The number of the linguistic sets of the proposed controller has been determined using laboratory experimentation and the responses of the DC-DC converter that have been compared with different numbers of the fuzzy sets. The actual responses have been compared with three, four, five, and seven fuzzy sets. It has been found that the use of more than five linguistic sets does not improve the tracking accuracy of the DC-DC converters but in fact increases the computation time. Consequently, four linguistic sets, positive large (PL), positive small (PS), negative large (NL), and negative small (NS), have been chosen for the input variable voltage error, e(k), and three linguistic sets, positive medium (PM), zero (ZE), and negative medium (NM), have been chosen for that of the change in voltage error, Δe(k). Additionally, five linguistic sets, positive large (PL), positive small (PS), zero, negative large (NL), and negative small (NL), have been chosen for the output variable of the fuzzy controller structure. The fuzzy controller utilizes triangular membership functions on the controller input. The triangular membership function is chosen owing to its simplicity. For the change in voltage error, Δe(k), the initial values of the premise parameters (the corner coordinates of the triangle) are chosen so that the membership functions are equally spaced along the operating range of each input variable. The scaled input and output membership functions sets are shown in Figs. 36.25 through 36.27. The actual error measured from the feedback network ranges from 0.8 to about 6.0 V, while the control signal ranges from 16% to 50% duty cycle. The scaling factors ${\xi }_1=0.167$ and ${\xi }_2=0.01$ were determined using experimental tests in such a way that the normalized inputs e(k) and Δe(k) are well adapted to the universe of discourse $\left[-1,1\right]$ for any operating point. The linguistic control rules are established considering the dynamic behavior of the DC-DC converter and analyzing the error and its variation. To obtain control decision, the max-min inference method is used. It is based on the minimum function to describe the “AND” operator present in each control rule and the maximum function to describe the “OR” operator.

The output of the fuzzy controller structure is crisp, and thus a combined output fuzzy set must be defuzzified. To express the qualitative action in a quantitative action, the sum-product composition method was used. It calculates the crisp output as the weighted average of the centroids of all output membership functions. The output of the controller can be expressed as

$O_o=\frac{\left({\displaystyle \sum _m}{O}_m^{\ast }{a}_{Cm}\times b_{Cm}\right)}{{\displaystyle \sum _m}\left(O_m^{\ast }{b}_{Cm}\right)}$

where a_Cm and b_Cm are the centers and widths of the output fuzzy sets, respectively, for $m=1,\dots ,5$. The values of the b_Cm s were chosen to be unity. This scaled output corresponds to the control signal (percent duty cycle) to be applied to maintain the output voltage at a constant value. Fig. 36.28 represents the normalized output O_o of the proposed fuzzy controller structure as a function of the error e(k) and the change of error Δe(k). It clearly illustrates the nonlinear characteristics of the proposed fuzzy controller structure.

36.12.1 Topology of the Neural-Fuzzy Controller (NFC)

The proposed ANFIS is a multilayer neural network-based fuzzy-logic controller. The network architecture is built, such that the designer recognizes the internal nodes as they relate to the components of fuzzy controller. The system has a total of five layers. The architecture of the network is shown in Fig. 36.30. The scaled input and output membership functions sets are shown in Figs. 36.3–36.5. The actual error measured from the feedback network ranges from 0.8 to about 6.0 V, while the control signal ranges from 16% to 50% duty cycle.

36.12.1.2 Layer 2: Input Membership Layer

Each node in this layer acts as a linguistic label of one of the input variables in layer 1, that is, the membership value specifying the degree to which an input value belongs to a fuzzy set is determined in this layer. The triangular membership function is chosen owing to its simplicity. For the change in voltage error, Δe(k), the initial values of the premise parameters (the corner coordinates of the triangle) are chosen so that the membership functions are equally spaced along the operating range of each input variable. Due to the nonlinearity in the converter, the triangular membership for the error, e(k), was slightly modified. The weights between the input and membership level are assumed to be unity. The output of neuron j in the second layer can be obtained as follows:

when $\left(X_i\ge a_j\right)\mathrm{and}\left(X_i\le b_j\right)$

$O_j^2=f_j^2\left({\mathrm{n}\mathrm{e}\mathrm{t}}_j^2\right)=\left(X_i-a_j\right)/\left(b_j-a_j\right)$

else when $\left(X_i\ge b_j\right)\mathrm{and}\left(X_i\le c_j\right)$

$O_j^2=f_j^2\left({\mathrm{n}\mathrm{e}\mathrm{t}}_j^2\right)=\left(X_i-c_j\right)/\left(b_j-c_j\right)$

where a_j,b_j, and c_j are the corners of the jth membership function in layer 2 shown in Fig. 36.31 and X_i is the ith input variable to the node of layer 2, which could be either the value of the error or the change in error.

36.12.2 Learning Algorithm

The only weights that are trained are those between layer 3 and layer 4 of Fig. 36.30. The back-propagation network is used to train the weights of this layer. The weights of the neural network were trained offline before they were used in the experimental setup. The learning algorithm used can be described in the following steps:

Step 1: Calculate the error for the change in the control signal (duty cycle) based on the type of the converter.

$E_{\mathrm{o}}=T_{\mathrm{o}}-O_{\mathrm{o}}^5$

where E_o,T_o, and O_o⁵ are the output error, the target control signal, and the actual control signal, respectively.

Step 2: Calculate the error gradient.

$\begin{array}{l}{\delta }_m=E_o\times f^{\prime }\left(O_1^4,\dots ,O_m^4\right)\\ \Rightarrow {\delta }_m=\left(T_o-O_o^5\right)\times f^{\prime }\left(O_1^4,\dots ,O_m^4\right)\end{array}$

where

$f\left(O_1^4,\dots ,O_m^4\right)=\frac{{\displaystyle \sum _{j=1}^m}{O}_j^4\times a_{Cj}}{{\displaystyle \sum _{j=1}^m}{O}_j^4}$

and the partial derivative of f with respect to O_m⁴ is given by

${f_{O_m^4}}^{\prime }\left(O_1^4,\dots ,O_m^4\right)=\frac{{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne m\end{array}}^{m-1}}{O}_j^4\times \left(a_{Cm}-a_{Cj}\right)}{{\left({\displaystyle \sum _{j=1}^m}{O}_j^4\right)}^2}$

where a_Ci for $i=1,\dots ,5$ are the centers of the output fuzzy sets and O_j⁴ is the firing strength from node j in layer 4.

Step 3: Calculate the weight correction

$\Delta w_{\mathit{km}}=\eta \times {\delta }_m\times O_k^3$

to increase the learning rate, the Sejnowski-Rosenberg updating mechanism was used, which takes into account the effect of past weight changes on the current direction of the movement in the weight space. This is given by

$\Delta w_{\mathit{km}}\left(t\right)=\eta \times \left(1-\alpha \right)\times {\delta }_m\times O_m^3+\alpha \times \Delta w_{\mathit{km}}\left(t-1\right)$

where α is a smoothing coefficient in the range of 0–1.0 and η is the learning rate.

Step 4: Update the weights.

$w_{\mathit{km}}\left(t+1\right)=w_{\mathit{km}}\left(t\right)+\Delta w_{\mathit{km}}\left(t\right)$

where t is the iteration number
The weights linking the rule layer (layer 3) and the output membership layer (layer 4) are trained to capture the system dynamics and therefore minimize the ripples around the operating point. The training results of the link weights are shown in Figs. 36.32 and 36.33.

In the initialization of the weights, wt5 and wt6 in Fig. 36.32 were set to a very small number of 0.001 with the initial assumption that those weights are almost redundant before the training begins. However, the neurofuzzy system was capable of correcting those assumptions. Weight wt5 finally converges to a positive value that indicates an excitatory rule weight, while wt6 converges to a negative number that is an inhibitory rule weight. Some of the weights change sign during the training process. Weight wt1 was initially set at a positive value but converges to a negative value that makes that rule weight inhibitory.