Approximation Properties and Continuous Differentiability of a Class of ...

Approximation Properties and Continuous Differentiability of a Class of SISO Fuzzy Systems Barnabas Bede1,2

Matthew P. Peterson

Department of Mathematics DigiPen Institute of Technology 9931 Willows Rd. NE Redmond, WA, 98052, USA 2 Óbuda Univeristy Bécsi út 96/B, H-134 Budapest, Hungary Email:1 [email protected] 2 [email protected]

DigiPen Institute of Technology 9931 Willows Rd. NE Redmond, WA, 98052, USA Email:[email protected]

1

Abstract—The goal of the present paper is to investigate approximation and smoothness properties of Larsen type single input single output (SISO) fuzzy systems, that is, fuzzy logic systems using the maximum as aggregation for the individual rule outputs, product (Goguen) t-norm as the conjunctive operator and center of gravity defuzzification. We prove that the function providing the output of the above considered Larsen type fuzzy system is capable of approximating any continuous function. Also, it is continuously differentiable under very relaxed conditions as e.g. continuous differentiability of the antecendents except at their core, and continuous differentiability of the consequences of fuzzy rules except at their core and the enpoints of their support. We show practical examples regarding the approximation and smoothness of the Larsen type operators, showing also by an example that the conditions on the antecedent part cannot be in general weakened further without loosing continuous differentiability. The present paper provides a theoretical background for claims in literature stating that the output of a fuzzy control system of this type is smooth.

are smooth, so will be the output of the TS fuzzy system. The case of a fuzzy system of Larsen type is not immediate and it is interesting, because of the usage of the maximum operator which is known to destroy differentiability.

I. I NTRODUCTION

The fuzzy set 𝐴𝑖 can be seen as a model for the expression 𝑥 is about 𝑥𝑖 , 𝑖 = 1, ..., 𝑛. The consequence part 𝐵𝑖 will be considered integrable and it is constructed such that 𝐵𝑖 (𝑦𝑖 ) = 1 and it has the support

The study of approximation capability of fuzzy systems started from B. Kosko’s paper [7]. Most of the papers available in literature use the Takagi-Sugeno approach [13], [11], [2] or they are using sum as the aggregation method for the fuzzy rules [9], [14], [6]. In this situation smoothness properties have been investigated in e.g., [3]. Also, min→ aggregation was considered [4], [5], however, smoothness properties were not investigated for this case. Our goal in the present paper is to investigate the approximation problem of an unknown bounded function 𝑓 : [𝑎, 𝑏] → [𝛼, 𝛽] by a SISO Fuzzy System of Larsen type, i.e., having maximum as aggregation operator of the individual rule outputs, product playing the role of conjunction, and the defuzzification being the most widespread method, i.e., the center of gravity. This problem of approximation capability was also considered in detail in the recent monograph [1]. In the present paper we also consider the problem of continuous differentiability of Larsen type SISO fuzzy systems. In many works it is mentioned that Mamdani type fuzzy controllers, [8] have a smooth output. Also, smoothness for Takagi Sugeno systems is relatively easily addressed [3]. If the antecedents

II. A PPROXIMATION PROPERTIES OF L ARSEN FUZZY SYSTEMS

Let us construct now the framework of our problem. Let 𝑎 = 𝑥0 < 𝑥1 < ... < 𝑥𝑛 < 𝑥𝑛+1 = 𝑏 be a partition of the input domain. Let 𝑓 (𝑥𝑖 ) = 𝑦𝑖 , 𝑖 = 0, ..., 𝑛 + 1. We consider fuzzy systems with the support of each fuzzy set in the antecedent part, satisfying 𝐴𝑖 (𝑥𝑖 ) = 1 and let us suppose that there exist 𝜀 > 0, 𝑟 ∈ ℕ, 𝑟 < 𝑛, such that the 𝜀-level sets satisfy (𝐴𝑖 )𝜀 ⊆ [𝑥𝑖−𝑟 , 𝑥𝑖+𝑟 ], 𝑖 = 1, ..., 𝑛.

(𝐵𝑖 )𝜀 ⊆ [min{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }, max{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }] for 𝑖 = 1, ..., 𝑛. The fuzzy set 𝐵𝑖 can be seen as a model for the expression 𝑦 is about 𝑦𝑖 , 𝑖 = 1, ..., 𝑛. The fuzzy rule base If 𝑥 is 𝐴𝑖 then 𝑦 is 𝐵𝑖 , 𝑖 = 1, ..., 𝑛, helps us construct the Larsen SISO fuzzy system ´ 𝛽 ⋁𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑦 ⋅ 𝑑𝑦 . 𝐹 (𝑓, 𝑥) = 𝛼´ 𝛽 ⋁ 𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑑𝑦 𝛼 The following theorem is a generalization of Theorem 7.16 that was proposed recently in [1]. In the above cited Theorem the conditions imposed on the antecedents and consequences are stronger, namely, they restrict the supports of the antecedents and consequences to be small.

We will provide error estimates in terms of the modulus of continuity defined as

where 𝑧𝑗−𝑟 = min{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 },

𝜔(𝑓, 𝛿) = sup{∣𝑓 (𝑥) − 𝑓 (𝑦)∣ : ∣𝑥 − 𝑦∣ ≤ 𝛿}. Theorem 1: Any continuous function 𝑓 : [𝑎, 𝑏] → [𝛼, 𝛽] can be approximated by the Larsen fuzzy system ´ 𝛽 ⋁𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑦 ⋅ 𝑑𝑦 𝐹 (𝑓, 𝑥) = 𝛼´ 𝛽 ⋁ 𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑑𝑦 𝛼

𝑧𝑗+𝑟 = max{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }. Taking now into account that (𝐵𝑖 )𝜀 ⊆ [min{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }, max{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }] we obtain

with any membership functions for the antecedents and consequences 𝐴𝑖 , 𝐵𝑖 , 𝑖 = 1, ..., 𝑛 such that there exist 𝜀 > 0, 𝑟 ∈ ℕ, 𝑟 < 𝑛, such that (i) 𝐴𝑖 continuous, 𝐴𝑖 (𝑥𝑖 ) = 1

≤

+

≤

𝑦𝑖 = 𝑓 (𝑥𝑖 ), 𝑖 = 1, ..., 𝑛. Moreover the following error estimate holds true ∥𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∥ ≤ 2𝑟𝜔 (𝑓, 𝛿) + 𝜀2 (𝛽 − 𝛼)2 𝑀

𝛼

.

𝑖=1

∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣ ´ 𝛽 ⋁   [ 𝑛 (𝐴 (𝑥) ⋅ 𝐵 (𝑦))] ⋅ (𝑦 − 𝑓 (𝑥))𝑑𝑦  𝑖  𝛼 𝑖=1 𝑖  =  ´ 𝛽 ⋁𝑛   (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝑖=1

´ 𝛽 ⋁𝑛 ∣[ 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦))] ⋅ (𝑦 − 𝑓 (𝑥))∣ 𝑑𝑦 ≤ 𝛼 . ´ 𝛽 ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝛼 By Lemma 7.10 in [1] we have ∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣ ´ 𝛽 ⋁𝑛 [ (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦))] ⋅ ∣𝑦 − 𝑓 (𝑥)∣ 𝑑𝑦 ≤ 𝛼 𝑖=1 . ´ 𝛽 ⋁𝑛 (𝐴 (𝑥) ⋅ 𝐵 (𝑦)) ⋅ 𝑑𝑦 𝑖 𝑖 𝑖=1 𝛼 Also, any fixed value of 𝑥 ∈ [𝑎, 𝑏] belongs to the 𝜀-level set of at most 2𝑟 antecedents. Suppose that 𝑥 ∈ (𝐴𝑗−𝑟 )𝜀 ∪ ... ∪ (𝐴𝑗+𝑟 )𝜀 . Then we have ´ 𝑧𝑗+𝑟 ⋁𝑛

∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣

𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ ∣𝑦 − 𝑓 (𝑥)∣ 𝑑𝑦 ´ 𝛽 ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝛼 ´ ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ ∣𝑦 − 𝑓 (𝑥)∣ 𝑑𝑦 [𝛼,𝑧𝑗−𝑟 ]∪[𝑧𝑗+𝑟 ,𝛼] + , ´ 𝛽 ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝛼

≤

𝑧𝑗−𝑟

𝜀2 (𝛽 − 𝛼)2

𝑖=1 (𝐴𝑖 (𝑥)

⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦

Finally we obtain

with

)−1

Proof: We have

𝛼

⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦

∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣ ≤ 2𝑟𝜔(𝑓, 𝛿) + 𝜀2 (𝛽 − 𝛼)2 𝑀

𝑖=1,...,𝑛

𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑑𝑦

𝑖=1 (𝐴𝑖 (𝑥)

∣𝑦 − 𝑓 (𝑥)∣ 𝑑𝑦

𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝜔(𝑓, 2𝑟𝛿)𝑑𝑦 ´ 𝛽 ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝛼

𝛼

𝛼

𝛿 = max {𝑥𝑖 − 𝑥𝑖−1 }

[

[𝛼,𝑧𝑗−𝑟 ]∪[𝑧𝑗+𝑟 ,𝛽]

´ 𝛽 ⋁𝑛

+ ´ 𝛽 ⋁𝑛

with

𝑀=

´

𝛼

(𝐵𝑖 )𝜀 ⊆ [min{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }, max{𝑦𝑖−𝑟 , ..., 𝑦𝑖+𝑟 }],

𝑛 𝛽 ⋁

𝜀2

´ 𝛽 ⋁𝑛

(ii) 𝐵𝑖 integrable, 𝐵𝑖 (𝑦𝑖 ) = 1,

(ˆ

𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝜔(𝑓, 𝑧𝑗+𝑟 − 𝑧𝑗−𝑟 )𝑑𝑦 ´ 𝛽 ⋁𝑛 𝑖=1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦 𝛼

𝑧𝑗−𝑟

(𝐴𝑖 )𝜀 ⊆ [𝑥𝑖−𝑟 , 𝑥𝑖+𝑟 ], 𝑖 = 1, ..., 𝑛;

and

∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣

´ 𝑧𝑗+𝑟 ⋁𝑛

(ˆ 𝑀=

𝑛 𝛽 ⋁

𝛼 𝑖=1

)−1 (𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)) ⋅ 𝑑𝑦

.

Corollary 2: A Larsen SISO fuzzy system 𝐹 (𝑓, 𝑥) given as above, can approximate any continuous function 𝑓 (𝑥) with arbitrary accuracy. Proof: If 𝛿 → 0 and 𝜀 → 0 then it is easy to see that 𝐹 (𝑓, 𝑥) → 𝑓 (𝑥), and the result is immediate. Let us remark that results in [1] do not cover the important case of Gaussian antecedents and consequences. This case can be covered now easily. Corollary 3: A Larsen SISO fuzzy system 𝐹 (𝑓, 𝑥) with Gaussian antecedents and consequences can approximate any continuous function 𝑓 (𝑥) with arbitrary accuracy. Proof: The proof is immediate from the previous Theorem, since for Gaussian membership functions we can find a set of parameter values 𝑟, 𝜀, 𝑀 such that we have ∣𝐹 (𝑓, 𝑥) − 𝑓 (𝑥)∣ arbitrarily small. III. C ONTINUOUS DIFFERENTIABILITY OF L ARSEN FUZZY SYSTEMS

The fact that a fuzzy system provides a smooth output is very intuitive and this was mentioned in several works as e.g. in [10]. In fact this is widely known as being one of the main advantages of fuzzy controllers of Mamdani types or Larsen types. Yet a proof of this property can not be found in the literature as to the best of the authors knowledge. In the present paper we address this problem and we prove that the Larsen type SISO fuzzy system described above, under very

relaxed conditions is continuously differentiable. The proof is elementary but non-trivial. Theorem 4: Let 𝑓 : [𝑎, 𝑏] → ℝ be a monotone function and let 𝑦𝑖 = 𝑓 (𝑥𝑖 ), 𝑖 = 1, ..., 𝑛.. We consider the Larsen type SISO fuzzy system ´ 𝛽 ⋁𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑦 ⋅ 𝑑𝑦 𝐹 (𝑓, 𝑥) = 𝛼´ 𝛽 ⋁ 𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑑𝑦 𝛼 with any membership functions for the antecedents and consequences 𝐴𝑖 , 𝐵𝑖 , 𝑖 = 1, ..., 𝑛 satisfying (i) 𝐴𝑖 monotone increasing and differentiable on (−∞, 𝑥𝑖 ) and monotone decreasing and differentiable on (𝑥𝑖 , ∞), with the closure of its support being (𝐴𝑖 )0 = [𝑥𝑖−1 , 𝑥𝑖+1 ], 𝑖 = 1, ..., 𝑛; (ii) 𝐵𝑖 strictly increasing and differentiable on [min{𝑦𝑖−1 , 𝑦𝑖 , 𝑦𝑖+1 }, 𝑦𝑖 ) and strictly decreasing and differentiable on (𝑦𝑖 , max{𝑦𝑖−1 , 𝑦𝑖 𝑦𝑖+1 }],

ˆ +

𝛽

𝑐(𝑥)

ˆ =

𝑐(𝑥)

𝐴′1 (𝑥) ⋅ 𝐵1 (𝑦) ⋅ 𝑦𝑑𝑦 +

𝛼

and 𝑔 ′ (𝑥) = ˆ +

𝐴′2 (𝑥) ⋅ 𝐵2 (𝑦) ⋅ 𝑦𝑑𝑦 − 𝑐′ (𝑥) ⋅ 𝐴2 (𝑥) ⋅ 𝐵2 (𝑐(𝑥)) ⋅ 𝑐(𝑥)

ˆ

𝛼

𝛽

𝑐(𝑥)

ˆ

=

𝑐(𝑥)

ˆ

𝛽

𝑐(𝑥)

𝐴′2 (𝑥) ⋅ 𝐵2 (𝑦) ⋅ 𝑦𝑑𝑦

𝐴′1 (𝑥)⋅𝐵1 (𝑦)𝑑𝑦 +𝑐′ (𝑥)⋅𝐴1 (𝑥)⋅𝐵1 (𝑐(𝑥))⋅𝑐(𝑥)

𝐴′2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦 − 𝑐′ (𝑥) ⋅ 𝐴2 (𝑥) ⋅ 𝐵2 (𝑐(𝑥)) ⋅ 𝑐(𝑥) 𝑐(𝑥)

𝛼

𝐴′1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 +

ˆ

𝛽

𝑐(𝑥)

𝐴′2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦.

As a conclusion the function 𝐹 (𝑓, 𝑥) s differentiable and its derivative is 𝐹 ′ (𝑓, 𝑥) (𝐵𝑖 )0 = [min{𝑦𝑖−1 , 𝑦𝑖 , 𝑦𝑖+1 }, max{𝑦𝑖−1 , 𝑦𝑖 , 𝑦𝑖+1 }], 𝑖 = 1, ..., 𝑛; ) (ˆ ˆ 𝛽 𝑐(𝑥) Then the Larsen type system given above is continuous and ′ ′ 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦) ⋅ 𝑦𝑑𝑦 + 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦) ⋅ 𝑦𝑑𝑦 = continuously differentiable function (class 𝐶 1 ) on [𝑎, 𝑏]. 𝛼 𝑐(𝑥) Proof: Let us remark that for the SISO fuzzy system ) (´ ´𝛽 𝑐(𝑥) considered in this theorem, i.e., 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑑𝑦 + 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑑𝑦 1 1 2 2 𝛼 𝑐(𝑥) ⋅ (´ ´ 𝛽 ⋁𝑛 )2 ´ 𝑐(𝑥) 𝛽 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑦 ⋅ 𝑑𝑦 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 + 𝑐(𝑥) 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦 𝐹 (𝑓, 𝑥) = 𝛼´ 𝛽 ⋁ 𝛼 𝑛 [ 𝑖=1 𝐴𝑖 (𝑥) ⋅ 𝐵𝑖 (𝑦)] ⋅ 𝑑𝑦 (ˆ ) 𝛼 ˆ 𝛽 𝑐(𝑥) ′ ′ only two rules are fired at a time, i.e., for any input value − 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 + 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦 𝛼 𝑐(𝑥) 𝑥 ∈ [𝑎, 𝑏] there exists 𝑗 ∈ {1, 2, ..., 𝑛 − 1} such that 𝑥 ∈ (´ ) ´𝛽 (𝐴𝑗 )0 ∩(𝐴𝑗+1 )0 . Without restricting the generality we suppose 𝑐(𝑥) 𝐴 (𝑥) ⋅ 𝐵 (𝑦) ⋅ 𝑦𝑑𝑦 + 𝐴 (𝑥) ⋅ 𝐵 (𝑦) ⋅ 𝑦𝑑𝑦 1 1 2 2 𝛼 𝑐(𝑥) that 𝑗 = 1. ⋅ (´ )2 . ´𝛽 First let us assume that 𝑥 ∈ (𝑥1 , 𝑥2 ). Then we have 𝑐(𝑥) 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 + 𝑐(𝑥) 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦 𝛼 ´𝛽 (𝐴 (𝑥) ⋅ 𝐵 (𝑦) ∨ 𝐴 (𝑥) ⋅ 𝐵 (𝑦)) ⋅ 𝑦𝑑𝑦 1 1 2 2 𝐹 (𝑓, 𝑥) = 𝛼´ 𝛽 For the case 𝑥 ∈ (𝑥𝑗 , 𝑥𝑗+1 ) the proof is exactly the same (𝐴1 (𝑥) ⋅ 𝐵1 (𝑦) ∨ 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦))𝑑𝑦 as above. 𝛼 Now let us consider the extreme case 𝑥 = 𝑥1 . Then since We have to prove that this function is differentiable with its derivative being continuous. Let us remark first that the 𝐴2 (𝑥) is differentiable at 𝑥1 and since (𝐴2 )0 = [𝑥1 , 𝑥3 ] we have 𝐴′2 (𝑥1 ) = 0. Also we observe that 𝑐(𝑥) is right consequences 𝐵1 and 𝐵2 overlap on (𝑥1 , 𝑥2 ). We define 𝑐(𝑥) to be the unique solution in 𝑦 of the equation continuous at 𝑥1 . We obtain the derivative to the right of 𝑥1 as (´ ) (´ ) 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦) = 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦). 𝛽 ′ 𝛽 𝐴 (𝑥) ⋅ 𝐵 (𝑦) ⋅ 𝑦𝑑𝑦 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑑𝑦 1 1 1 1 𝛼 𝛼 We can easily see that 𝐹𝑟′ (𝑓, 𝑥) = (´ )2 𝛽 ( )−1 ( ) 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 𝛼 𝐵2 𝐴1 (𝑥) 𝑐(𝑥) = 𝐵1 𝐴2 (𝑥) (´ ) (´ ) 𝛽 𝛽 ′ 𝐴 (𝑥) ⋅ 𝐵 (𝑦) ⋅ 𝑦𝑑𝑦 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑑𝑦 1 1 1 1 𝛼 𝛼 is differentiable and we can rewrite − = 0. (´ )2 ´𝛽 ´ 𝑐(𝑥) 𝛽 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑦𝑑𝑦 + 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑦𝑑𝑦) 𝐴 (𝑥) ⋅ 𝐵 (𝑦)𝑑𝑦 1 1 2 1 1 𝛼 𝑐(𝑥) 2 𝛼 . 𝐹 (𝑓, 𝑥) = ´ 𝑐(𝑥) ´𝛽 𝐴1 (𝑥) ⋅ 𝐵1 (𝑦)𝑑𝑦 + 𝑐(𝑥) 𝐴2 (𝑥) ⋅ 𝐵2 (𝑦)𝑑𝑦) Similar reasoning can be performed at the left of 𝑥1 and 𝛼 we get 𝐹𝑟′ (𝑓, 𝑥) = 0. The choice of 𝑥1 did not restrict the We denote that numerator as 𝑓 (𝑥) and the denominator as generality of our proof, and we obtain similarly continuous 𝑔(𝑥). Then we observe that derivative at each 𝑥𝑗 , 𝑗 = 1, ..., 𝑛 and one-sided differentiabilˆ 𝑐(𝑥) ity can be obtained at the endpoints 𝑥0 and 𝑥𝑛+1 . Finally we 𝐴′1 (𝑥)⋅𝐵1 (𝑦)⋅𝑦𝑑𝑦+𝑐′ (𝑥)⋅𝐴1 (𝑥)⋅𝐵1 (𝑐(𝑥))⋅𝑐(𝑥) 𝑓 ′ (𝑥) = obtain that 𝐹 (𝑓, 𝑥) is continuously differentiable. 𝛼

Figure 1.

Figure 2.

Gaussian antecedents of a SISO fuzzy system

Gaussian consequences of a SISO fuzzy system

Figure 3. Larsen SISO fuzzy system associated to 𝑓 (𝑥) = 𝑥2 , Gaussian membership functions

Figure 4.

Smooth splines as antecedents of a SISO fuzzy system

IV. E XAMPLES Remark 5: It is an interesting remark that the fuzzy system constructed above does not have have the consequences 𝐵𝑖 differentiable, nor the target function 𝑓 does not need to be differentiable and moreover it does not even need it to be continuous, yet the output will be differentiable. This indeed proves the hypothesys that we have provided at the begining of the paper that fuzzy systems and as a consequence fuzzy controllers have a smoothing effect in practical problems. Further let us remark that the requirement that 𝑓 is monotone is restrictive in the proof of the previous theorem mention and it is necessary to have only two overlapping antecedents each time.

We exemplify our theoretical findings through a few examples. Surely a visual confirmation of smoothness properties is not necessarry, however it is useful to understand properties and limitations of Larsen type fuzzy systems. First we consider 𝑓 : [0, 1] → [0, 1], 𝑓 (𝑥) = 𝑥2 and a Larsen fuzzy system with Gaussian Antecedents Fig. 1, Gaussian Consequences Fig. 2. This exemplifies Theorem 1 (see Fig. 3) To exemplify smooth monotonic spline antecedents and consequences (we use here the monotonic splines described in [12]). The antecedents used for our construction are shown

Figure 5.

Smooth splines as consequences of a SISO fuzzy system

Figure 6. Larsen SISO fuzzy system associated to 𝑓 (𝑥) = 𝑥2 smooth spline membership

in Fig. 4, the consequences are shown in Fig. 5 and the results are to be found in Fig. 7. Let us observe that differentiability of the antecedents 𝐴𝑖 , 𝑖 = 1, ..., 𝑛 is required at the endpoints of the support 𝑥𝑖−1 , 𝑥𝑖+1 but differentiability at the core 𝑥𝑖 is not required. To examplify that we consider monotone splines that are not differentiable at the core, both as antecedents and consequences for a Larsen fuzzy system. We can also see in Fig. 10 a counterexample. If we consider triangular Antecedents and Consequences we do not obtain continuously differentiable output. Let us observe that in Theorem 4 besides monotony and boundedness, nothing is required on the target function 𝑓 (𝑥).

Figure 7.

Figure 8.

Antecedents that are not differentiable at the core

Consequences that are not differentiable at the core

As a conclusion we obtain continuously differentiable output even if the target function is discontinuous. We consider { 2 𝑥 if 𝑥 < 0.5 𝑓 (𝑥) = 𝑥 otherwise together with the spline antecedents in Fig. 7. The Larsen SISO fuzzy system in this case is illustrated in Fig. 11 V. C ONCLUSION AND FURTHER RESEARCH Approximation and Smoothness of Larsen fuzzy systems is investigated and our findings are that the output of a Larsen fuzzy system is able to approximate any continuous function with arbitrary accuracy and it is continuously differentiable, under very relaxed conditions. For future research we plan to

Figure 9. Larsen SISO fuzzy system associated to 𝑓 (𝑥) = 𝑥2 sharp splines

Figure 10. Larsen SISO fuzzy system associated to 𝑓 (𝑥) = 𝑥2 with triangular Antecedents and Consequences.

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Figure 11. Larsen SISO fuzzy system associated to a discontinuous function.

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