Optimal fuzzy control of an inverted pendulum

Abstract

In this paper, three controllers, including OFCHA (optimal fuzzy control using hedge algebras – HAs), FCHA (fuzzy control using HAs) and CFC (conventional fuzzy control) are designed. Attention is paid to the stability in the vertical position of a damped-elastic-jointed inverted pendulum subjected to a time-periodic follower force. The different values of the angular coefficient of the follower force are considered. Simulation results are exposed to illustrate the effect of OFCHA in comparison with FCHA and CFC.

Keywords

Genetic algorithm hedge algebras inverted pendulum optimal fuzzy control periodic follower force

1. Introduction

Fuzzy set theory introduced by Zadeh (1965) has provided a mathematical tool useful for modeling uncertain (imprecise) and vague data and has been presented in many real situations. As typical unstable nonlinear models, pendulum systems are often used as a benchmark for verifying the effect of a new control method because of the simplicity of the structure. Recently, many researches on the stabilization of pendulum systems have been done.

Yi and Yubazaki (2000) presented a fuzzy controller for the stabilization control of inverted pendulum systems based on Single Input Rule Modules (SIRMs) dynamically connected to a fuzzy inference model. The fuzzy controller has four input items, each with a SIRM and a dynamic importance degree. Yi et al. (2002) presented a fuzzy controller with six input items and one output item for stabilizing a parallel-type double-inverted pendulum system based on SIRMs dynamically connected to a fuzzy inference model. Becerikli and Celikb (2007) presented a fuzzy controller for an inverted pendulum system in two stages: investigation of fuzzy control system modeling methods and solution of the ‘Inverted Pendulum Problem’ by using Java programming with Applets for internet-based control education. Tao et al. (2008) proposed a fuzzy hierarchical swing-up and sliding position controller for the swing-up and position controls of an inverted pendulum–cart system that includes a fuzzy switching controller, a fuzzy swing-up controller and a twin-fuzzy-sliding-position controller. Li and Xu (2009) investigated the adaptive fuzzy logic control of dynamic balance and motion for wheeled inverted pendulums with parametric and functional uncertainties. The proposed adaptive fuzzy logic control, based on the physical properties of wheeled inverted pendulums, makes use of a fuzzy logic engine and a systematic online adaptation mechanism to approximate the unknown dynamics. Sinha and Butcher (1997) used a technique that employs both Picard iteration and expansion in shifted Chebyshev polynomials to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. A double-inverted pendulum subjected to a periodic follower force was considered as an example. Galán et al. (2005) studied a chain of identical pendulums coupled with damped elastic joints subject to vertical sinusoidal forcing of its base. Particular attention was paid to the stability of the upright equilibrium configuration with a view to understanding recent experimental results on the stabilization of an unstable stiff column under parametric excitation. Ma and Butcher (2005) presented a stability analysis for elastic columns under the influence of periodically varying follower forces whose orientation is retarded, that is, dependent on the position of the system at a previous time. Mailybaev and Seyranian (2009) studied the stability of a linear vibrational system with multiple degrees of freedom subjected to parametric excitation. A double-inverted pendulum subjected to periodic excitation of support was considered as an example.

Hedge algebras (HAs) were introduced and have been investigated since 1987 (Ho and Wechler, 1990, 1992; Ho and Nam, 2002; Ho, 2007; Ho and Long, 2007; Ho et al., 2008). The authors of HAs discovered that linguistic values can formulate an algebraic structure (Ho and Wechler, 1990, 1992) and that the algebraic structure is a Complete Hedge Algebras Structure (Ho, 2007; Ho and Long, 2007) with a main property being that the semantic order of linguistic values is always guaranteed. It is even a rich enough algebraic structure (Ho and Nam, 2002) to describe completely reasoning processes. HAs can be considered as a mathematical order-based structure of term-domains, the ordering relation of which is induced by the meaning of linguistic terms in these domains. It is shown that each term-domain has its own order relation induced by the meaning of terms, called the semantically ordering relation. Many interesting semantic properties of terms can be formulated in terms of this relation and some of these can be taken to form an axioms system of HAs. These algebras form an algebraic foundation to study a type of fuzzy logic, called linguistic-valued logic, and provide a good mathematical tool to define and investigate the concept of fuzziness of vague terms and the quantification problem and some approximate reasoning methods. In Ho et al. (2008), HA theory began applying to fuzzy control and it provided very much better results than fuzzy control alone. The studied object in Ho et al. (2008), which is a single-undeformable pendulum without external loads, where its state equations are solved by the Euler method with a sample time of one second, is too simple to evaluate completely its control effect.

That reason suggests to us, in this paper, applying HAs and the GA (genetic algorithm) in active fuzzy control of a structure, which is a damped-elastic-jointed inverted pendulum subjected to a time-periodic follower force with three controllers (conventional fuzzy control (CFC), fuzzy control using hedge algebras (FCHA) and optimal fuzzy control using hedge algebras (OFCHA)), in order to compare their control effect.

2. The damped-elastic-jointed inverted pendulum

The diagram of the system is shown in Figure 1. An inverted pendulum of length l has a concentrated mass m at its end. A spring (of constant stiffness k) and a damper (of constant damping factor c) connect the inverted pendulum to a fixed base to avoid a large rotation angle φ. A periodic follower force (F = P₁ + P₂cos ωt) acts on the mass m and makes an angle γφ with the vertical axis. A control moment u, applied at the base of the inverted pendulum, brings the angle φ to zero position.

Figure 1.

Configuration of the inverted pendulum system.

The kinetic energy T, the potential energy Π and the dissipative function Φ of the system can be expressed in the following forms (Galán et al., 2005):

T = \frac{1}{2} m ({\overset{\cdot}{x}}_{m}^{2} + {\overset{\cdot}{y}}_{m}^{2}) = \frac{1}{2} m {\overset{\cdot}{ϕ}}^{2} l^{2}

(1)

Π = {mgy}_{m} + \frac{1}{2} k ϕ^{2} = mgl \cos ϕ + \frac{1}{2} k ϕ^{2}

(2)

Φ = \frac{1}{2} c {\overset{\cdot}{ϕ}}^{2}

(3)

where

x_{m} = l \sin ϕ

y_{m} = l \cos ϕ

and g is the gravity acceleration.

Using the Lagrange equations of the second type:

\frac{d}{d t} (\frac{\partial T}{\partial \overset{\cdot}{ϕ}}) - \frac{\partial T}{\partial ϕ} = - \frac{\partial Π}{\partial ϕ} - \frac{\partial Φ}{\partial \overset{\cdot}{ϕ}} + M

(4)

The differential equation of motion for this system can then be written in the form:

{ml}^{2} \overset{··}{ϕ} + c \overset{\cdot}{ϕ} + k ϕ - mgl \sin ϕ = u + Fl \sin (ϕ - γ ϕ)

(5)

Denoting $[x_{1}, x_{2}]^{T} = [ϕ, \overset{\cdot}{ϕ}]^{T}$ , Equation (5) can be rewritten into state-space form as

\begin{matrix} {\overset{\cdot}{x}}_{1} = \overset{\cdot}{ϕ} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \overset{··}{ϕ} = - \frac{k}{{ml}^{2}} x_{1} + \frac{g}{l} \sin x_{1} \\ + \frac{Fl \sin (x_{1} - γ x_{1})}{{ml}^{2}} - \frac{c}{{ml}^{2}} x_{2} + \frac{u}{{ml}^{2}} \end{matrix}

(6)

Equations (6) could be solved by using the fourth-order Runge–Kutta method (Chapra and Canale, 2006).

3. Fuzzy control using hedge algebras and optimal fuzzy control using hedge algebras

In this section, the basic content of FCHA and OFCHA are described based on Ho and Wechler (1990, 1992), Ho and Nam (2002), Ho (2007), Ho and Long (2007) and Ho et al. (2008).

3.1. Algebraic structure of term-domains

To easily understand the algebraic approach to fuzzy control problems, it is necessary to recall some concepts of HAs. The algebraic approach to term-domains of linguistic variables is to discover the algebraic structure of term-domains induced by the natural term meaning. For instance, by the term meaning we can observe that extremely small < very small < small < approximately small < little small < big < very big < extremely big … So, we have a new viewpoint: term-domains can be modeled by a poset (partially ordered set), a semantics-based order structure.

Consider TRUTH as a linguistic variable 𝒳 and let X be its terms-set. Assume that its linguistic hedges used to express the truth are Extremely, Very, Approximately, Little, which for short are denoted correspondingly by E, V, A and L, and its primary terms are false and true. Then, X = {true, V true, E true, EA true, A true, LA true, L true, L false, false, A false, V false, E false…} ∪ {0, W , 1} is a term-domain of 𝒳, where 0, W and 1 are specific constants called absolutely false, neutral and absolutely true, respectively. According to Zadeh (1965), X can be produced by a context-free grammar and it may be finite or infinite. In general, its elements are of the form h_n,…h₁x, where h₁,…,h_n ∈ H.

Now, we discover an inherent semantics-based order structure of each term-domain. This structure can be axiomatized, based on the semantic properties of term-domains that can be expressed in terms of the relation ≤ of 𝒳, called the semantics-based ordering relation.

A term-domain X can be ordered based on the following observations.

- Each primary term has a sign that expresses a semantic tendency. For instance, true has a tendency of ‘going up’, called positive one, and it is denoted by c⁺, while false has a tendency of ‘going down’, called negative one, denoted by c⁻. In general, we always have c⁺≥ c⁻, semantically.

- Each hedge has also a sign. It is positive if it increases the semantic tendency of the primary terms and negative if it decreases this tendency. For instance, V is positive with regard to both primary terms, while L has a reverse effect and hence it is negative. Denote by H⁻ the set of all negative hedges and by H⁺ the set of all positive ones of 𝒳. If both hedges h and k belong to H⁺ or H⁻ together, then it may happen that one of these hedges may modify the terms more strongly than the other. For example, we have L, A ∈ H⁻ and L ≥ A, since Lfalse ≥ Afalse ≥ false. Note that I ≤ A and I ≤ V, where I is the identity on X. So, H⁺ and H⁻ become posets under the relation ≤, where the notation ≤ causes no confusion because H⁺, H⁻ and X are assumed to be disjoint.

- Further, we observe that each hedge may increase or decrease the semantic intensity of any other one. For instance, we have false ≤ very approximately false ≤ approximately false, which shows that very decreases the intensity of approximately. If k decreases the semantic intensity of h, it is said to be negative with regard to h. Conversely, if k increases the semantic intensity of h, it is positive with regard to h. This is called the PN-property of hedges.

- An important semantic property of hedges is the so-called heredity of hedges, which stems from the fact that for every h, the term hx inherits the meaning of x. This property may also be formulated in term of ≤: if the meaning of hx and kx can be expressed by hx ≤ kx, then we have h’hx ≤ k’kx, which says that h’ and k’ preserve the relative meaning of hx and kx and hence they cannot change the semantic-based order relationship between hx and kx. So, as a consequence, the inequality H(hx) ≤ H(kx) follows from hx ≤ kx, where H(y) = {h_n…h₁y: ∀h₁,_…, h_n ∈ H, ∀n ∈ N }.

Assume that there is a semantics-based ordering relation ≤ defined on X. So, X can be considered formally as an abstract algebra AX = (X, G, C, H, ≤), where G = {c⁻, c⁺}, C = {0, W , 1}, H = H⁺ ∪ H⁻ is the set of unary operations (H⁺ is the set of all positive hedges and H⁻ is the set of all negative ones), X\C = H(G) is the set generated from G - and ≤ is a partially ordering relation on X. We have:

h > x if kx < x, for all h ∈ H⁺, k ∈ H⁻;

if h < k then (hx > x ⇒ kx > x) and (x > hx ⇒ hx > kx);

if ∃x (x < hx < khx or x > hx > khx) then {(y < hy ⇒ y < hy < ky) and (hy < y ⇒ khy < hy < y)} for all y;

if ∃x (x < khx < hx or x > khx > hx) then {(y < hy ⇒ y < khy < hy) and (hy < y ⇒ hy < khy < y)} for all y.

3.2. Fuzziness measure of vague terms and hedges of term-domains

We can consider the structure of HAs as a qualitative model of the relative meaning of terms in a term-domain of a linguistic variable. However, in applications we often have to deal with quantitative quantities and, therefore, there is a natural demand to investigate the quantification problem of term meanings of HAs. In order to quantify HAs, first we study the notion of fuzziness and the fuzziness measure of linguistic terms. Then, we define the notion of semantically quantifying mappings (SQMs) and show a close relation between the fuzziness measure of terms and the SQMs.

Fuzziness of vague terms is a concept that is very difficult and there are many different approaches to define this concept in the fuzzy sets framework, in which the fuzziness of a vague term is, of course, defined by taking full advantage of the shape of the fuzzy set that represents its meaning. In addition, fuzzy sets are very subjective and, hence, these definitions depend merely on the specific designed fuzzy sets, while fuzziness is a more general characteristic of the own linguistic terms.

HAs provide an appropriate formal basis to define this concept, since they model immediately the semantics of term-domains. Putting in mind that elements of H(x) still express a certain meaning stemming from x, we can interpret the set H(x) as a model of the fuzziness of the term x. Hence, the ‘size’ of the set H(x) is used to measure the fuzziness degree of x. Let the reference domain of a linguistic variable 𝒳 be a real interval [a, b]. Since, by a linear transformation, the domain [a, b] can be transformed into [0, 1], we can assume for normalization that the reference domains of all physical linguistic variables are the common unit interval [0, 1], called the semantic domain of linguistic variables. In fuzzy control, it is natural to require that fuzzy sets be transformed into [0, 1]. It allows taking a mapping f from X into [0, 1] into consideration, which preserves the ordering relation on X. Then, the ‘size’ of set H(x), for x ∈ X, can be measured by the diameter of f(H(x)) ⊆ [0, 1], which will be considered as a measure of fuzziness of x.

An fm: X → [0, 1] is said to be a fuzziness measure of terms in X if:

(fm1) fm(c⁻) + fm(c⁺) = 1 and $\sum_{h \in H} fm (hu) = fm (u)$ , ∀u ∈ X; in this case fm is called complete;

(fm2) for the constants 0, W and 1, fm(0) = fm( W ) = fm(1) = 0;

(fm3) ∀x, y ∈ X, ∀h ∈ H, $\frac{fm (hx)}{fm (x)} = \frac{fm (hy)}{fm (y)}$ , that is, this proportion does not depend on specific elements and, hence, it is called the fuzziness measure of the hedge h and denoted by µ(h).

The condition (fm1) means that the primary terms and hedges under consideration are complete for modeling the semantics of the whole real interval of a physical variable. That is, except the primary terms and hedges under consideration, there are no more primary terms and hedges. The condition (fm2) is intuitively evident. The condition (fm3) seems also to be natural in the sense that applying a hedge h to different vague concepts, the relative modification effect of h is the same, that is, this proportion does not depend on terms they apply to.

We assume that H⁻ = {h_–1,…,h_–q}, where h_–1 < h_–2 <…< h_–q, H⁺= {h₁,…, h_p}, where h₁<…<h_p. For each fuzziness measure fm on X the following statements hold:

fm(hx) = µ(h)fm(x), for every x ∈ X;

fm(c⁻) + fm(c⁺) = 1;

$\sum_{- q \leq i \leq p, i \neq 0} fm (h_{i} c) = fm (c)$ , c ∈{c⁻, c⁺};

$\sum_{- q \leq i \leq p, i \neq 0} fm (h_{i} x) = fm (x)$ ;

$\sum_{- q \leq i \leq - 1} μ (h_{i}) = α$ and $\sum_{1 \leq i \leq p} μ (h_{i}) = β$ , where α, β > 0 and α + β = 1.

Given the fuzziness measure of a primary term and hedges regarded as the parameters of the quantification method, we can calculate easily the fuzziness measure of any terms of a HA, using this proposition. With such parameters, the algebraic approach to the semantics of term-domains seems to be also very flexible to adapt the application.

3.3. Quantification of term-domains of a linguistic variable

In this section, some results on the relationship between the fuzziness measure of vague terms and SQMs are presented. For a given SQM, a reasonable concept of fuzziness measure and, conversely, for a given fuzziness measure on a HA, we can define the corresponding SQM.

A mapping φ: 𝒳 → [0, 1] is said to be a SQM of 𝒜𝒳, provided that the following conditions hold:

φ is a one-to-one mapping from X into [0,1] and preserves the order on X , that is, for all x, y ∈ X , x < y ⇒ φ(x) < φ(y) and φ(0) = 0, φ(1) = 1, where 0, 1 ∈ C;

continuity: ∀x ∈ X , φ( Φ x) = infimum φ(H(x)) and φ( Σ x) = supremum φ(H(x)).

Note that, for convenience in presenting the quantitative semantics formalism in studying the meaning of vague terms, we have assumed that the common reference domain of the linguistic variables is the interval [0, 1], called the semantic domain of the linguistic variables. In applications, we need to use the values in the reference domains, for example, the interval [a, b] of the linguistic variables and, therefore, we have to transform the interval [a, b] into [0, 1] and vice versa. The transformation of the interval [a, b] into [0, 1] is called a semantization and its converse transformation from [0, 1] into [a, b] is called a desemantization.

In the algebraic approach, it is interesting that we can define the sign of terms in every term-domain, that is Sign(x) ∈ {−1, 0, 1}, for all x ∈ X. A function Sign: X → {−1, 0, 1} is a mapping that is defined recursively as follows, for h, h'∈ H and c ∈ {c⁻, c⁺}:

Sign(c⁻) = − 1, Sign(c⁺) = +1;

Sign(hc) = − Sign(c), if h is negative with regard to c;

Sign(hc) = + Sign(c), if h is positive with regard to c;

Sign(h'hx) = − Sign(hx), if h’hx ≠ hx and h' is negative with regard to h;

Sign(h'hx) = + Sign(hx), if h’hx ≠ hx and h' is positive with regard to h;

Sign(h'hx) = 0 if h’hx = hx.

The meaning of the function Sign will be given in the following statement: for any h and x, if Sign(hx) = +1 then hx > x, if Sign(hx) = −1 then hx < x and if Sign(hx) = 0 then hx = x.

There exists a natural and close connection between a fuzziness measure and a SQM. To deal with this, we define a mapping from X into [0, 1] associated with a given fuzziness measure fm: Let fm be a fuzziness measure on X. A mapping φ: X → [0, 1], which is induced by fm on X, is defined as follows (x = h_m … h₂h₁c^+/):

ν( W ) = θ = fm(c⁻), ν(c⁻) = θ − αfm(c⁻) = βfm(c⁻), φ(c⁺) = θ + αfm(c⁺);

fm(x) = fm(h_m … h₂h₁c^+/−) = µ(h_m) … µ(h₂)µ(h₁)fm(c^+/−);

if j ≤ p then

\begin{matrix} ν (h_{j} x) = ν (x) + Sign (h_{j} x) [\sum_{i \neq j}^{p} fm (h_{j} x) \\ - 0.5 (1 - Sign (h_{j} x) Sign (h_{i} h_{j} x) (β - α)) fm (h_{j} x)]; \end{matrix}

if j > p then

\begin{matrix} ν (h_{j} x) = ν (x) + Sign (h_{j} x) [\sum_{i \neq p + 1}^{j} fm (h_{j} x) \\ - 0.5 (1 - Sign (h_{j} x) Sign (h_{i} h_{j} x) (β - α)) fm (h_{j} x)] . \end{matrix}

3.4. Optimal FCHA

The main aim of a control problem is to regulate the controlled object into its stable state or to minimize a goal function defined on the whole controlled process. In order to find the optimal parameters for designing a fuzzy controller, we have to examine an optimal problem for the whole controlled process.

(1) Determine the control problem. Suppose that we have a fuzzy control problem of an application for which we can determine the following factors:

- a Fuzzy Associative Memory (FAM) table with linguistic descriptions, which represents an expert knowledge of the application domain for defining the state and control action values of the controlled object in every control cycle;

- establish a discrete control model, denoted by CM, which defines the computation relationship between numeric-valued states of the controlled object and numeric-valued control actions in each control cycle;

- a criterion for evaluating the effectiveness of the control method: goal function g(x₁, … , x_m), where x₁, x₂, … , x_m and u can be computed by formulas defined by the established CM.

(2) Construct an optimal control algorithm based on SQMs. This step consists of the following tasks:

- considering SQMs of the linguistic variables of the states of the controlled object and the control action as design variables and determining their intervals;

- constructing a fitness algorithm to evaluate the fitness level of the parameters for the GA based on the calculation of the state values of the controlled object and the control action.

- optimizing the parameters using the GA.

4. Fuzzy control systems of the inverted pendulum

The inverted pendulum fuzzy control system is based on the closed-loop fuzzy system shown in Figure 2.

Figure 2.

The inverted pendulum fuzzy control system.

To control the system, we assume the universe of discourse for the two state variables to be –1 rad ≤ x₁ ≤ 1 rad and –2 rad/s ≤ x₂ ≤ 2 rad/s. The universe of discourse for the control moment is –50 Nm ≤ u ≤ 50 Nm.

4.1. Conventional fuzzy control of the inverted pendulum

In this section, the CFC of the inverted pendulum is designed (Ross, 2004) with nine control rules.

4.1.1. Constructing the membership functions

The membership functions x₁, x₂ and u on their universes, that are, for the values negative big (NB), negative (N), zero (Z), positive (P), positive big (PB) are constructed as shown in Figures 3–5, respectively.

Figure 3.

Input x₁, partitioned.

Figure 4.

Input x₂, partitioned.

Figure 5.

Output u, partitioned.

4.1.2. Constructing the rule base

After the stage of constructing membership functions, we construct the FAM table for the rule bases, which is shown in Table 1. Then we determine the fuzziﬁcation method with a Singleton Fuzziﬁer, an inference system with the Mamdani Method, and the defuzziﬁcation method with a centroid for this fuzzy control system, as shown in Figure 6.

Figure 6.

Methods of the conventional fuzzy control system.

Table 1.

Fuzzy Associative Memory table for the inverted pendulum fuzzy control system rule base

x ₁	x ₂
x ₁	N	Z	P
N	NB	N	Z
Z	N	Z	P
P	Z	P	PB

4.2. Fuzzy control using hedge algebras of the inverted pendulum

In this section, the fuzzy controller using HAs is designed. The original FAM table for this problem is defined by a knowledge domain given in Table 1. Now, we need to determine HAs for the linguistic variables occurring in Table 1 (nine control rules). They are linguistic variables x_1s and x_2s for the state variables x₁ and x₂ of the controlled object, and u_s for the control moment u. The HAs are defined as follows:

the set of primary terms: C = {0, small, W , large, 1} = {0, c⁻, W , c⁺, 1}; W = 0.5;

the set of hedges: H = {H⁻, H⁺} = {Little, Very} = {h_–1, h₁}. Hence, p = 1; q = 1.

However, as above, the terms occurring in Table 1 are not compatible with the semantic domain [0, 1], so we have to transform them into the new terms defined by the transformation, given in Table 2. Hence, the given FAM table (Table 1) is transformed into the Semantic Associative Memory (SAM) table (Table 3).

Table 2.

Terms transformation

NB	N	Z	P	PB
Very small	Small	W	Large	Very large

Table 3.

Semantic Associative Memory table

x _1s	x _2s
x _1s	Small	W	Large
Small	Very small	Small	W
W	Small	W	Large
Large	W	Large	Very large

Then, fuzziness measures of the hedges H (µ) and the primary terms C (fm) are determined as follows:

μ (Little) = μ (h_{- 1}) = α = 0.5; μ (Very) = (h_{1}) = β = 0.5

fm (small) = W = 0.5; fm (l \arg e) = 1 - fm (small) = 0.5

The SQMs ν are determined and are shown in Table 4.

Table 4.

Parameters of semantically quantifying mappings for the inverted pendulum control problem

Very small	Small	W	Large	Very large
0.125	0.25	0.5	0.75	0.875

In the same way, we have the SAM table with SQMs, as shown in Table 5.

Table 5.

Semantic Associative Memory table with semantically quantifying mappings

x _1s	x _2s
x _1s	Small: 0.25	W : 0.5	Large: 0.75
Small: 0.25	Very small: 0.125	Small: 0.25	W : 0.5
W : 0.5	Small: 0.25	W : 0.5	Large: 0.75
Large: 0.75	W : 0.5	Large: 0.75	Very large: 0.875

The semantizations for each linguistic variable are defined by the transformations given in Figures 7–9. The new terminology ‘semantization’ was defined and accepted in Ho et al. (2008).

Figure 7.

Transformation: x₁ to x_1s.

Figure 8.

Transformation: x₂ to x_2s.

Figure 9.

Transformation: u to u_s.

The Quantifying Semantic Curve (QSC) is linearly established through the points corresponding to the control rules shown in Table 5, as shown in Figure 10.

Figure 10.

Quantifying semantic curve.

4.3. Optimal fuzzy control using hedge algebras of the inverted pendulum

The values of linguistic variables shown in Table 5 are now considered as design variables; their intervals are determined so as to ensure the semantic order of SQMs, as shown in Table 6.

Table 6.

The interval parameters of semantically quantifying mappings

[Very small]	[Small]	[W]	[Large]	[Very large]
0–0.1875	0.1876–0.375	0.3751–0.625	0.6251–0.8125	0.8126–1

The goal function G is defined as follows:

G = \sum_{i = 0}^{n} [x_{1}^{2} (i) + x_{2}^{2} (i)] = min

(7)

where n is the number of control cycles. The parameters of the optimal algorithm using the GA are defined as follows (Chipperﬁeld et al., 1994): number of individuals per subpopulations: NIND = 10; number of generations: MAXGEN = 1000; recombination probability: GGAP = 0.8; number of variables: NVAR = 5; and fidelity of solution: PRECI = 10.

5. Results and discussion

In order to study the changing rule of x₁, x₂ and u using CFC, FCHA and OFCHA, the following numerical values are used for simulations: m = 2 kg, k = 2 Nm, l = 1.5 m, P_l = 10 N, P₂ = 1 N, c = 0.005 Nms, ω = π/2 and g = 9.81 m/s², the sample time Δt = 0.01 s. The initial conditions are x₁(0) = 0.6 rad and x₂(0) = 1 rad/s.

Three studied cases with different values of angular coefficient of the follower force γ (external factor) are listed in Table 7.

Table 7.

Three studied cases

Notation	Case 1	Case 2	Case 3
γ	0.1	0.5	0.9

The error signal e, which expresses the state of the pendulum, is defined as follows:

e = \sqrt{ϕ^{2} + {\overset{\cdot}{ϕ}}^{2}} = \sqrt{x_{1}^{2} + x_{2}^{2}}

(8)

The values of x₁, x₂, u and e obtained from OFCHA, FCHA and CFC corresponding to the studied cases are shown in following sections.

5.1. Results corresponding to Case 1 (γ = 0.1)

The optimal QSC of Case 1 is shown in Figure 11. The values of x₁, x₂, u and e obtained from OFCHA, FCHA and CFC corresponding to Case 1 are shown in Figures 12–15.

Figure 11.

The optimal Quantifying Semantic Curve (QSC) of Case 1.

Figure 12.

The angle x₁, rad (Case 1: γ = 0.1).

Figure 13.

The angular velocity x₂, rad/s (Case 1: γ = 0.1).

Figure 14.

The control moment u, Nm (Case 1: γ = 0.1).

Figure 15.

The error e (Case 1: γ = 0.1).

5.2. Results corresponding to Case 2 (γ = 0.5)

The optimal QSC of Case 2 is shown in Figure 16. The values of x₁, x₂, u and e obtained from OFCHA, FCHA and CFC corresponding to Case 2 are shown in Figures 17–20.

Figure 16.

The optimal Quantifying Semantic Curve (QSC) of Case 2.

Figure 17.

The angle x₁, rad (Case 2: γ = 0.5).

Figure 18.

The angular velocity x₂, rad/s (Case 2: γ = 0.5).

Figure 19.

The control moment u, Nm (Case 2: γ = 0.5).

Figure 20.

The error e (Case 2: γ = 0.5).

5.3. Results corresponding to Case 3 (γ = 0.9)

The optimal QSC of Case 3 is shown in Figure 21. The values of x₁, x₂, u and e obtained from OFCHA, FCHA and CFC corresponding to Case 3 are shown in Figures 22–25.

Figure 21.

The optimal Quantifying Semantic Curve (QSC) of Case 3.

Figure 22.

The angle x₁, rad (Case 3: γ = 0.9).

Figure 23.

The angular velocity x₂, rad/s (Case 3: γ = 0.9).

Figure 24.

The control moment u, Nm (Case 3: γ = 0.9).

Figure 25.

The error e (Case 3: γ = 0.9).

5.4. Discussion

From the all studied cases, the results obtained from OFCHA are better than those of FCHA and CFC; the system is stabilized, although a small vibration appears during the stabilization process.

In Case 1 (γ = 0.1, Figures 12–15), the complete stabilization (e < 0.001) times are 3.1, 4 and 4.3 s corresponding to OFCHA, FCHA and CFC, respectively. The maximum control moments are about 34, 32 and 17 Nm corresponding to OFCHA, FCHA and CFC, respectively.

In Case 2 (γ = 0.5, Figures 17–20), the complete stabilization times are 3.9, 4.2 and 4.5 s corresponding to OFCHA, FCHA and CFC, respectively. The maximum control moments are about 33, 32 and 17 Nm corresponding to OFCHA, FCHA and CFC, respectively.

In Case 3 (γ = 0.9, Figures 22–25), the complete stabilization times are 3.5, 3.6 and 4.8 s corresponding to OFCHA, FCHA and CFC, respectively. The maximum control moments are about 33, 32 and 17 Nm corresponding to OFCHA, FCHA and CFC, respectively.

The control moments of CFC in all studied cases are relatively rough.

The results indicate that the angular coefficient γ of the follower force has little influence on the control performance.

6. Conclusions

In the present work, a new fuzzy controller based on HAs is applied for the stabilization control of a damped-elastic-jointed inverted pendulum with periodic follower force. The main results are summarized as follows.

The contributions of the paper are that HAs and the GA are successfully applied in optimal fuzzy control for stabilizing a damped-elastic-jointed inverted pendulum subjected to a time-periodic follower force. It is clear that the OFCHA and FCHA are simpler, more effective, offer higher control quality and are more understandable in comparison with the method based on CFC.

The algebraic approach to term-domains of linguistic variables is quite different from the fuzzy sets one in the representation of the meaning of linguistic terms and the methodology of solving the fuzzy multiple conditional reasoning problems. It allows one to linearly establish the QSC through the points corresponding to the control rules. The semantic order is always guaranteed.

In fuzzy logic, many important concepts, such as fuzzy set, T-norm, S-norm, intersection, union, complement, composition, etc., are used in approximate reasoning. This is an advantage for the process of flexible reasoning, but there are too many factors, such as shape and number of membership function, defuzzification method, etc., influencing the precision of the reasoning process and it is difficult to optimize. Those are subjective factors that cause error in determining the values of control process. Meanwhile, approximate reasoning based on HAs, from the beginning, does not use the fuzzy set concept and its precision is obviously not influenced by this concept. Therefore, the method based on HAs does not need to determine the shape and number of membership functions, neither does it need to solve the defuzzification problem. Moreover, in calculation, while there is a large number of membership functions, the number of calculations based on fuzzy control increases quickly, while the number of calculations based on HAs does not increase much with very simple calculations. With these above advantages, it is definitely possible to use HA theory for many different controlling problems.

Footnotes

Acknowledgements

The authors would like to thank reviewers for their useful comments.

Funding

This work was supported by the Vietnam National Foundation for Science & Technology Development (NAFOSTED).

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