Unit 8: Real-World Applications: Control, Decision Making & AI

1. Architecture of a Fuzzy Inference System (FIS)

A Fuzzy Inference System (FIS) is a mathematical framework that maps crisp numerical inputs to crisp numerical outputs using fuzzy reasoning and expert knowledge. The standard architecture comprises four essential components:

1. Fuzzification Interface: Transforms crisp sensor measurements into degrees of membership across linguistic terms.

2. Rule Base (Knowledge Base): A collection of fuzzy IF-THEN rules reflecting human domain expertise.

3. Inference Engine: Applies fuzzy logic operators (t-norms, implications) to aggregate rule activations.

4. Defuzzification Interface: Synthesizes the aggregated fuzzy output into a single crisp control action.

2. Mamdani Fuzzy Inference (1975)

Introduced by Ebrahim Mamdani to control a steam engine, this is the most widespread and intuitive FIS model.

• Rule Structure: Both antecedent and consequent are fuzzy linguistic propositions:

where $A_i^k$ and $B^k$ are fuzzy sets with continuous membership functions.

• Firing Strength: $w_k = \min(\mu_{A_1^k}(x_1), \mu_{A_2^k}(x_2))$.
• Implication: The consequent fuzzy set is clipped (Mamdani min implication: $\mu_{B^{*k}}(y) = \min(w_k, \mu_{B^k}(y))$) or scaled (Larsen product implication).
• Aggregation: $\mu_{\text{out}}(y) = \max_k \mu_{B^{*k}}(y)$.
• Output: Requires explicit continuous integration (e.g. Centroid) to defuzzify.

3. Takagi-Sugeno-Kang (TSK) Fuzzy Inference (1985)

Introduced by Tomohiro Takagi and Michio Sugeno to provide computationally efficient control and mathematical stability proofs.

• Rule Structure: The antecedent is fuzzy, but the consequent is a crisp functional polynomial of the input variables:

(If $p_1 = p_2 = 0$, it is a Zero-Order Sugeno FIS where consequents are constant scalars).

• Defuzzification: Replaced by a simple, exact weighted average:

Zero computational integration required! The TSK model is smooth, differentiable, and forms the mathematical foundation of Adaptive Neuro-Fuzzy Inference Systems (ANFIS).

1. Fuzzification and Rule Evaluation

Given crisp input vector $\mathbf{x}^ = (x_1^, \dots, x_n^*)$:

1. For each rule $R_k$, compute the membership of each input: $\mu_{A_i^k}(x_i^*)$.
2. Combine antecedent conditions using a t-norm (typically minimum or product) to find the rule firing strength:

2. Comparative Defuzzification Formulas

Let $\mu_{\text{agg}}(y) = \max_{k=1}^R \mu_{B^{*k}}(y)$ be the aggregated fuzzy output on universe $Y$.

1. Centroid Method (Center of Gravity / Area):

• Smooth and continuous response under varying inputs. Highly sensitive to all active rules.

2. Bisector of Area (BOA):

The vertical line $y^*_{\text{BOA}}$ that divides the total area into two equal halves:

3. Middle of Maxima (MOM):

• Fast, but can jump discontinuously when rule activations switch.

1. The Inverted Pendulum Benchmark

The inverted pendulum on a moving cart is the quintessential non-linear benchmark for control theory. The dynamical state is characterized by two state variables:

1. Error $e(t) = \theta(t)$: Angular deviation of the pole from vertical.

2. Change of Error $\Delta e(t) = \dot{\theta}(t)$: Angular velocity.

2. Fuzzy Rule Matrix for Pendulum Balancing

Each input is partitioned into 5 linguistic sets:

• Negative Big (NB), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Big (PB).

The control action is the horizontal force $F$ applied to the cart:

| $\Delta e \setminus e$ | NB | NS | ZE | PS | PB | |:---:|:---:|:---:|:---:|:---:|:---:| | NB | NB | NB | NS | NS | ZE | | NS | NB | NS | NS | ZE | PS | | ZE | NS | NS | ZE | PS | PS | | PS | NS | ZE | PS | PS | PB | | PB | ZE | PS | PS | PB | PB |

• Physical Intuition: If angle is Positive Big (leaning right) and velocity is Positive Big (falling rapidly right), apply Positive Big force to accelerate the cart rightwards and catch the falling pole!
• If angle is Positive Small but velocity is Negative Small (already swinging back towards center), apply Zero force to prevent overshooting.

1. Decision Making under Linguistic Uncertainty

In classical Multi-Criteria Decision Making (MCDM), human decision makers are forced to assign exact numerical ratings (e.g. "Alternative A is exactly 3.5 times better than B"). Fuzzy MCDM replaces crisp pairwise ratings with Triangular Fuzzy Numbers (TFNs) to capture subjective doubt.

2. Fuzzy Analytic Hierarchy Process (Fuzzy AHP)

Introduced by Buckley (1985) and Chang (1996):

1. Construct the fuzzy pairwise comparison matrix $\tilde{A} = (\tilde{a}_{ij})_{n \times n}$ where $\tilde{a}_{ij} = (l_{ij}, m_{ij}, u_{ij})$.
2. The reciprocal condition satisfies $\tilde{a}_{ji} = \tilde{a}_{ij}^{-1} = (1/u_{ij}, 1/m_{ij}, 1/l_{ij})$.
3. Compute the fuzzy synthetic extent values:

1. Calculate the degree of possibility $V(\tilde{S}_i \ge \tilde{S}_k)$ and normalize to obtain crisp priority weights $w_i$.

3. Fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)

1. Alternatives are evaluated against criteria using linguistic variables converted to fuzzy decision matrices.
2. Determine the Fuzzy Positive Ideal Solution (FPIS) $A^*$ and Fuzzy Negative Ideal Solution (FNIS) $A^-$.
3. Compute vertex-based distances $d_i^*$ and $d_i^-$ for each alternative.
4. Calculate the closeness coefficient:

1. Rank alternatives in descending order of $CC_i$.

1. Fuzzy c-Means Clustering (Bezdek, 1981)

In classical k-means clustering, each data point $\mathbf{x}_k$ belongs strictly to a single cluster. In Fuzzy c-Means (FCM), every point has a degree of membership $u_{ik} \in [0, 1]$ across all $c$ clusters such that $\sum_{i=1}^c u_{ik} = 1$.

Definition 8.1 (FCM Objective Function):

where:

• $N$ is the number of data points, $c$ is the number of clusters.
• $m > 1$ is the fuzziness exponent (typically $m = 2$).
• $\mathbf{v}_i$ is the center of cluster $i$.

Alternating Optimization Algorithm:

1. Update cluster centers:

1. Update membership matrix:

Iterate until convergence $\|\Delta U\| < \epsilon$.

2. Adaptive Neuro-Fuzzy Inference Systems (ANFIS, Jang 1993)

ANFIS is a multilayer feedforward network that integrates the linguistic interpretability of fuzzy logic with the self-learning capability of neural networks.

• Layer 1 (Fuzzification): Adaptive nodes computing membership parameters (e.g. Gaussian bell centers and widths).
• Layer 2 (Rule Nodes): Fixed T-norm product nodes: $w_k = \mu_{A^k}(x) \mu_{B^k}(y)$.
• Layer 3 (Normalization): Fixed ratio nodes: $\bar{w}_k = \frac{w_k}{\sum w_j}$.
• Layer 4 (Consequent Layer): Adaptive linear nodes: $\bar{w}_k f_k = \bar{w}_k (p_k x + q_k y + r_k)$.
• Layer 5 (Summation): Fixed output node: $y = \sum \bar{w}_k f_k$.

Trained using hybrid learning: Recursive Least Squares (RLS) for linear consequent parameters, and Backpropagation Gradient Descent for non-linear antecedent membership parameters.