Lyapunov approach based design of a gain adaptive interval type-2 fuzzy controller for servo systems

Abstract

This paper focuses on the development of a stable Mamdani type-2 fuzzy logic based controller for automatic control of servo systems. The stability analysis of the fuzzy controller has been done by employing the concept of Lyapunov. The Lyapunov approach results in the derivation of an original stability analysis that can be used for designing the rule base of our proposed online gain adaptive Interval Type-2 Fuzzy Proportional Derivative controller (IT2-GFPD) for servo systems with assured stability. In this approach a quadratic positive definite Lyapunov function is used and sufficient stability conditions are satisfied by the adaptive type-2 fuzzy logic control system. Illustrative simulation studies with linear and nonlinear models as well as experimental results on a real-time servo system validate the stability and robustness of the developed intelligent IT2-GFPD. A comparative performance study of IT2-GFPD with other controllers in presence of noise and disturbance also proves the superiority of the proposed controller.

Keywords

Type-2 fuzzy control Lyapunov stability self-tuning servo position control and real time experimentation

1 Introduction

Fuzzy logic controllers (FLCs) are being successfully used for control schemes where there are difficulties in deriving mathematical models or having performance limitations with conventional linear control. Generally, fuzzy controller works with error (e) and change of error (Δe) as input variables. The output control action, u for proportional derivative (PD-type) and Δu the incremental control action for proportional integral (PI-type) are used as the fuzzy output variables. Here our designed fuzzy controller is an interval Type-2 fuzzy PD-type [1], so, the control output is represented by u, the rule consequent. In recent times, Type-2 fuzzy controllers are becoming relatively popular [2 –8] for providing improved responses for processes with measurement uncertainty, model uncertainty as well as strong measurement noise. In such cases, Type-2 fuzzy controllers are found to be more competent for maintaining the desired process output compared to Type-1 or conventional fuzzy controllers. A good review on various successful industrial applications of Type-2 fuzzy controllers is reported in [8].

In this paper, we have worked with stability of fuzzy control which has been one of the major issues since the time of the Mamdani’s pioneer work [9, 10]. A close look at the review [11] and the comprehensive works [12 –19] on the stability of Fuzzy model-based control system justifies that stability has been a challenging work in fuzzy control. But Zadeh has often explained that as fuzzy control has been accepted as a task-oriented control, hence, unlike conventional set point-oriented control, it does not require a rigorous mathematical analysis of stability [15]. Sugeno has also pointed out that the reliability to control hardware is considered more important than the stability of control [19].

The success of fuzzy control [20] shows that we do not require a stability theory for it. But for surety, the works on stability theory will certainly give us a wider view of future development of fuzzy control. Fuzzy controller themselves are basically nonlinear systems and stability analysis cannot be done by traditional stability analysis like the frequency analysis method (Bode and Nyquist criteria) or the s-plane methods (Nyquist criteria and Routh-Hurwitz). Therefore, many researcher have worked on it and improved the stability analysis of fuzzy control [11 –17]. For the stability analysis, majority of the researchers [11 –14] have used Lyapunov stability method. In this paper, we have also used the Lyapunov stability [14] to build a stable Mamdani Type-2 Fuzzy Logic controller for a very commonly used system [21, 22], the servo positioning mechanism, in this case, a servo system SRV02 by Quanser, Canada, [23]. In general, DC servo motors are usually employed in position control applications and such processes are of integrating in nature with small dead time. Here, by establishing the stability of each fuzzy subsystem the stability of the closed-loop system is guaranteed. The stability method allows to modify the controller rule base so that the controlled servo system becomes stable in the operating range fixed by the manipulated variable constraints.

In a conventional fuzzy controller, there is little scope for performance enhancement during plant operation as the number of control rules, related membership functions (MFs) and their corresponding scaling factors (SFs) remain unchanged [24]. Moreover, like non-fuzzy controllers, standard tuning guidelines for fuzzy controllers are also not available till date. But, it is found that satisfactory close-loop response may be achieved by choosing appropriate input-output SFs [25 –32] and a proper fuzzy rule base [33]. To obtain improved performance for systems with nonlinearity and dead time, instead of having fixed tuning for fuzzy controllers, parameters (preferably SFs) need to be tuned online to provide necessary variation in control action to counteract the changes in process behavior [28 –32].

However, for proper tuning of an FLC, the order of their (SFs) significance is quite important. Initially, we tune the SFs with global consequences and later on SFs with local consequences are chosen. Out of the various tunable parameters, SFs have the utmost priority due to their global effect on the control performance [34]. From the early days of fuzzy control, researchers have studied [25] the outcome of modifying the SFs based on the performance characteristics of FLCs. In this regard, a number of online tuning schemes for FLCs are reported in [26 –32].

In this study, to enhance the performance of a conventional interval type-2 fuzzy PD controller (IT2-FPD), an online adaptation scheme of its input scaling factors (SFs) [26] is proposed. Both the input SFs for error (G_e) and change of error (G_Δe) are varied online depending on the current process error (e) and change of error (Δe). Further this controller’s rule base is also designed by the Lyapunov stability analysis approach for servo system. Hence, the resulting IT2-FPD with online gain adaptation mechanism and Lyapunov stability proved rule-base is termed as gain adaptive fuzzy PD controller (IT2-GFPD).

This paper is organized as follows: In Section 2, we have presented the introductory explanation of type-2 FLC’s. In Section 3, we explained the proposed controller, IT2-GFPD and its online gain adaptation scheme. The next Section 4, gives an illustration of the Lyapunov stability analysis approach to build a general rule base of IT2-GFPD for servo system. In Section 5, validation of the proposed controller has been presented with simulation and practical results and the concluding remarks are made in Section 6.

2 Type-2 fuzzy Logic control system structure

Type-2 fuzzy logic control system [2] consists of a process and a Type-2 fuzzy logic controller (T2-FLC) as shown in Fig. 1. Here, let XCRⁿ be a universe of discourse. The accepted single input non-linear dynamical systems modeled by the state-space equations can be represented as $\dot{x} = f (x) + b (x) u (t)$ (1)

Fig. 1

Block diagram of an IT2-FLC.

Where x ∈ X, x = [x₁x₂ . . x_n] ^T is the state vector, n is the number of input variables, $\dot{x} = {[{\dot{x}}_{1} {\dot{x}}_{2} . . {\dot{x}}_{n}]}^{T}$ , is the derivative of x with respect to the time t, f ( x ) = [f₁ (x) f₂ (x) … f_n (x)] ^T and b ( x ) = [b₁ (x) b₂ (x) … b_n (x)] ^T are the functions describing the dynamics of the process. u (t) is the control signal fed to the process, obtained by centre-of-set (COS) type-reduction [3] scheme for Mamdani type-2 FLCs.

Assuming each of x is partitioned by N number of membership functions, Type-2 fuzzy sets that are labeled by ${\overset{ˇ}{X}}_{1}^{q}$ (q = 1,2, ... ,N), ${\overset{ˇ}{X}}_{2}^{q}$ (q = 1,2, ... ,N),... ${\overset{ˇ}{X}}_{n}^{q}$ (q = 1,2, ... ,N) describe the linguistic terms (LTs) or MFs. Uncertainty in the primary MF of type-2 fuzzy set, $\overset{ˇ}{X}$ , consists of a bounded region called the Footprint of Uncertainty (FOU). This concept of FOU models the uncertainties in the shape and position of type-1 fuzzy sets and introduces lower and upper membership functions in it [2, 3].

Our objective is to design the rule-base of a Mamdani type-2 fuzzy controller that will position the servo-system at its desired position. The system has two-degree of freedom, process error e (k) and change of error Δe (k) ; and the k^th sampling instant is defined as:

$e (k) = y (k) - r,$ (2a) $Δ e (k) = e (k) - e (k - 1)$ (2b) where r is the desired output or set value, and y (k) is the process output at k^th sampling instant during close-loop operation. Here, the two (n= 2) input variables e, Δe and output variable u are partitioned by N number of Type-2 fuzzy sets (MFs), labeled by ${\overset{ˇ}{X}}_{1}^{q} (q = 1, 2, \dots, N)$ , ${\overset{ˇ}{X}}_{2}^{q} (q = 1, 2, \dots, N)$ and U^q (q = 1, 2, … . N) respectively. Assuming that for each possible combination of N input MFs there is a rule for these two inputs ${\overset{ˇ}{X}}_{1}, {\overset{ˇ}{X}}_{2}$ , the rule-base comprises of P rules. Then the p^th IF – THEN fuzzy (control) rule based on the Mamdani-FLC is of the form (3):

$\begin{matrix} R^{p} : if e is {\overset{ˇ}{X}}_{1}^{p} and Δ e is check X_{2}^{p} then u is U^{p} \\ (p = 1, 2, \dots, P) \end{matrix}$ (3) where, U^p is the distinct Type-2 fuzzy set defined in the interval $[u_{-}^{p}, {\bar{u}}^{p}]$ . Membership grades of input variables e, Δe on each ${\overset{ˇ}{X}}_{1}^{p}, {\overset{ˇ}{X}}_{2}^{p}$ are calculated as $[\underline{μ} {\overset{ˇ}{X}}_{1}^{p} (e), {\bar{μ}}_{{\overset{ˇ}{X}}_{1}^{p}} (e)]$ and $[\underline{μ} {\overset{ˇ}{X}}_{2}^{p} (Δ e), {\bar{μ}}_{{\overset{ˇ}{X}}_{2}^{p}} (Δ e)]$ respectively for p = 1, 2 . . P. Here, $\underline{μ} {\overset{ˇ}{X}}_{1}^{p}$ (e), and ${\bar{μ}}_{{\overset{ˇ}{X}}_{1}^{p}} (e)$ are the lower and upper membership grades of the interval Type-2 fuzzy set for input variable ${\overset{ˇ}{X}}_{1}^{p}$ . Similarly, $\underline{μ} {\overset{ˇ}{X}}_{2}^{p} (Δ e)$ and ${\bar{μ}}_{{\overset{ˇ}{X}}_{2}^{p}} (Δ e)$ are the lower and upper membership grades of input variable ${\overset{ˇ}{X}}_{2}^{p}$ . Next, we compute the firing interval of the p^th rule as given by

$\begin{matrix} F^{p} (e, Δ e) = [\underline{f^{p}} (e, Δ e), \bar{f^{p}} (e, Δ e)], \\ where p = 1, \dots P . \end{matrix}$ (4)

$\begin{matrix} = [\underline{μ} {\overset{ˇ}{X}}_{1}^{p} (e) \times \underline{μ} {\overset{ˇ}{X}}_{2}^{p} (Δ e), {\bar{μ}}_{{\overset{ˇ}{X}}_{1}^{p}} (e) \times {\bar{μ}}_{{\overset{ˇ}{X}}_{2}^{p}} (Δ e)], \\ p = 1, 2 \dots P . \end{matrix}$ (5)

With F^p (e, Δe) and the corresponding rule consequents, we perform the COS type reduction as given by $U_{COS (e, Δ e)} = \underset{\begin{matrix} f^{p} \in F^{p} (e, Δ e) \\ u^{p} \in U^{p} \end{matrix}}{\cup} \frac{\sum_{p = 1}^{P} f^{p} u^{p}}{\sum_{p = 1}^{P} f^{p}} = [u_{l}, u_{r}],$ (6) $\begin{matrix} u_{l} = min_{m \in [1, P - 1]} \frac{\sum_{p = 1}^{m} u_{-}^{p} {\bar{f}}^{p} + \sum_{p = m + 1}^{P} u_{-}^{p} f_{-}^{p}}{\sum_{p = 1}^{m} {\bar{f}}^{p} + \sum_{p = m + 1}^{P} f_{-}^{p}} \\ = \frac{\sum_{p = 1}^{L} u_{-}^{p} {\bar{f}}^{p} + \sum_{p = L + 1}^{P} u_{-}^{p} f_{-}^{p}}{\sum_{p = 1}^{L} {\bar{f}}^{p} + \sum_{p = L + 1}^{P} f_{-}^{p}} \end{matrix}$ (7a) $\begin{matrix} u_{r} = max_{m \in [1, P - 1]} \frac{\sum_{p = 1}^{m} {\bar{u}}^{p} f_{-}^{p} + \sum_{p = m + 1}^{P} {\bar{u}}^{p} {\bar{f}}^{p}}{\sum_{p = 1}^{m} f_{-}^{p} + \sum_{p = m + 1}^{P} {\bar{f}}^{p}} \\ = \frac{\sum_{p = 1}^{R} {\bar{u}}^{p} f_{-}^{p} + \sum_{p = R + 1}^{P} {\bar{u}}^{p} {\bar{f}}^{p}}{\sum_{p = 1}^{R} f_{-}^{p} + \sum_{p = R + 1}^{P} {\bar{f}}^{p}} \end{matrix}$ (7b) where the switching points L and R are determined by Karnik and Mendel (KM) algorithm [3] such that $u_{-}^{L} ⩽ u_{l} ⩽ u_{-}^{L + 1} and {\bar{u}}^{R} ⩽ u_{r} ⩽ {\bar{u}}^{R + 1}$ . Finally, the defuzzified output is given by $U = \frac{u_{l} + u_{r}}{2} .$ (8)

Here, the input and output variables are partitioned by 5 membership functions (MF) shown in Fig. 2. The 25 rules have been established following Lyapunov stability, as discussed in Section 4.

Fig. 2

MFs for e, Δe and u with 5 Type-2 fuzzy sets.

3 Gain adaptation scheme

Block diagram of the proposed IT2-GFPD is shown in Fig. 3. Normalized error (e_N) and change of error (Δe_N) are defined by Equations (9a) and (9b) respectively. The effective input scaling factors $G_{e}^{A}$ and $G_{Δ e}^{A}$ for e and Δe [26] are continuously modified by relations (9c) and (9d). Here, β is a nonlinear parameter which causes the required variations in the input scaling factors (SFs) from their nominal settings (G_e and G_Δe) so that an enhanced process response can be achieved.

Fig. 3

Block diagram of the proposed IT2-GFPD controller.

$e_{N} (k) = G_{e}^{A} e (k)$ (9a)

$Δ e_{N} (k) = G_{Δ e}^{A} Δ e (k)$ (9b) $G_{e}^{A} = G_{e} (k_{2} β)$ (9c) $G_{Δ e}^{A} = G_{Δ e} (1 + k_{3} β)$ (9d)

Here, k₂ and k₃ are two positive constants which determine the extent of variation of the input SFs with the alteration of β. The value of β at k^th sampling instant is determined by the mono-polar sigmoid function as given by the following relation: $β (k) = \frac{1}{(1 + e^{\frac{- α (k)}{k_{1}}})}$ (10a) where, $α (k) = e_{N} (k - 1) \times Δ e_{N} (k - 1) .$ (10b)

In relation (10a), k₁ is a positive constant, which determines the nature of variation of β with α. The expression for α is given by relation (10b), which is the product of e_N and Δe_N of the last sampling instant. Figure 4 shows the variation of β with α for different values of k₁. From Fig. 4 we find that the value of k₁ influences β in the following manners: (a) it governs the degree of nonlinearity of β which determines the sensitivity of $G_{e}^{A}$ and $G_{Δ e}^{A}$ at a specific operating point, (b) it determines the extent to which β varies for every possible values of α in the range [– 1, 1]. So, β effectively determines the nature of continuous variations of $G_{e}^{A}$ and $G_{Δ e}^{A}$ during close-loop operation. Thus, the required nonlinear variation of $G_{e}^{A}$ and $G_{Δ e}^{A}$ at different operating points can be provided by selecting an appropriate value of k₁ in order to achieve the desired variation in control action. In this study, we consider, k₁ = 0.2 which provides the required nonlinearity of β (shown in Fig. 4) for having desired gain variation during both the simulation and hardware experiments.

Fig. 4

Variation of BETA (β) with ALPHA (α) for different values of k₁.

3.1 Tuning strategy

The proposed online SFs adjustment for IT2-GFPD are suggested based on the concept of an operator’s control policy: when the process output is close to the desired value and approaching towards it then the control action is required to be weak to avoid unwanted oscillations, on the contrary, when the process output is moving away from the desired value, control action is required to be strong. As per the basic knowledge of close-loop dynamics, we can expect that such dynamic behavior in control action will give an improved servo response as well as improved load regulation without resulting large oscillation. Thus, the proposed gain adaptive mechanism is capable to provide the desired variation for both the proportional and derivative gains around the set point towards improving the transient and steady state responses. Note that during steady state condition when both error e = 0 and change of error Δe = 0, normalized inputs to the fuzzifier i.e. e_N and Δe_N will also be zero as they are defined by the product of non-zero scaling factors ( $G_{e}^{A}$ and $G_{Δ e}^{A}$ ) and zero valued error (e = 0) and change of error (Δe = 0). So, the control action of the proposed IT2-GFPD will be zero during steady state condition. Hence, we find that our proposed online gain adaptive scheme attempts to provide the appropriate control action at each sampling instant for improved transient response during close-loop operation. Moreover, a rule base has also been designed for IT2-GFPD based on the Lyapunov stability approach.

4 Stability analysis approach

According to our literature survey [11 –16] application of the Lyapunov function based stability analysis proposed by Wong et al. [16], for systems controlled by Takagi-Sugeno type Type-1 FLCs is well acknowledged. Similar Lyapunov based method for the stability analysis of systems controlled by Mamdani type Type-1 FLCs is developed by Precup et al. [14]. This method is extended to the stability analysis of systems controlled by Type-2 FLCs considering the upper membership functions only, since the lower membership range is included in the upper one [13]. The idea behind all these above Lyapunov function based stability analysis methods is to break down the problem of analyzing the stability of the whole fuzzy system into analyzing each fuzzy sub-system individually. It requires the verification of the Lyapunov criterion for any sub-system consisting of a fuzzy rule and the state-space model of the system to be controlled. The derivative of the crisp Lyapunov function is computed for each rule taking into account the MFs used in the rule. The derivative is to be negative for all the crisp values within the ranges of the MFs in order that the sub-system formed by the rule becomes stable [13]. If each sub-system is proved to be stable then the whole fuzzy system becomes stable. Similar type of work on Fuzzy Lyapunov synthesis method for the design of stable type-2 Fuzzy controllers has also been found in [12 , 17].

Here, the fuzzy subsystem consists of a practical servo process, SRV02 by Quanser, Canada [23] and a Type-2 Mamdani FLC. The schematic diagram of SRV02 is given in Fig. 5. The mathematical expression of the system has been derived theoretically by combining the Electrical and Mechanical equations combining the effect of load shaft speed in terms of the applied motor voltage V_m (t). $J_{eq} \frac{{dw}_{l} (t)}{dt} + B_{eq, v} w_{l} (t) = A_{m} V_{m} (t)$ (11)

Fig. 5

SRV02 DC motor armature circuit and gear train.

Where $B_{eq, v} = B_{eq} + \frac{n_{g} K_{g}^{2} n_{m} k_{t}}{R_{m}}$ , and $A_{m} = \frac{n_{g} K_{g} n_{m} k_{t}}{R_{m}}$ , w_l is the speed of the load shaft, τ_m is the applied motor torque, K_g is the gear ratio, n_g is the gearbox efficiency, k_t is the current-torque constant, and n_m is the motor efficiency. The final derived Equation (11) represents a first order system. Here, according to Quanser, Canada made SRV02 [23], the motor armature resistance, R_m = 2.6Ω, n_g = 0.9, K_g = 70, n_m = 0.69, k_t = 7.68 × 10^-3N - m/A, high-gear equivalent viscous damping coefficient B_eq = 0.015Nm (rad/s), and high-gear equivalent moment of inertia without external load J_eq = 2.087 × 10^-3Kg - m².

Then, J_eq is evaluated assuming disc load is attached to the high gear. Next, the value of B_eq,v and A_m can be theoretically calculated.

Performing the Bump test in the laboratory we have found the value of open loop steady state gain, K = 1.53rad/s/V, and time constant, τ = 0.0254s.

Hence, the open loop transfer function for position control system with SRV02 [23] is given by $G (s) = \frac{1.53}{s (0.0254 s + 1)} .$ (12)

We can represent the state-space equation given in Equation 1 for the servo process model (Equation (12)) as, $\dot{x} = f (x) + b (x) u$ (13) where x ɛ X , x = [x₁x₂] ^T, derivative of x with respect to t is $\dot{x} = [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}]$ , f ( x ) and b ( x ) describe the dynamics of the process and u is the control signal fed to the process.

The stability analysis approach by Precup et al. [14] requires the verification of the Lyapunov criterion for any subsystem constituted by a rule and the subsystem to be controlled. Derivative of a common Lyapunov function is calculated for each rule, with reference to the membership functions of the inputs used in the rule. For each of the subsystems related to a particular rule it is necessary to prove that the derivative is always negative between the extremes of the membership functions range. Thus, the overall system will also be stable satisfying the following theorem:

Theorem 1: If P is a positive definite matrix then:

V ( x ) = x ^TP x → ∞ as ∥ x ∥ → ∞ , V ( 0 ) =0

$\dot{V} ⩽ 0, \forall x \in$ for all fuzzy subsystems,

Also the set ${x \in X | \dot{V} (x) = 0}$ does not contain any other trajectory of the system except the trivial trajectory x (t) =0 for t ⩾ 0, then the fuzzy control system with self-tuning scheme defined in Section 3 and the process described by Equation (12) is globally asymptotically stable in the origin.

Here, the mathematical model of the servo motor derived in Equation (12) can be written in the form of differential equation: $m \ddot{x} + \ddot{x} = Ku$ (14)

Where m = 0.0254 and K = 1.53; mandKare the open loop time constant (τ) and steady state gain respectively, x is the angular displacement and u is the input to the servo motor.

Now, we will define a set of state variables sufficient to describe the system in Equation (13) comprising of x and $\dot{x} (\frac{dx (t)}{dt})$ as {z₁, z₂} with $z = [\begin{matrix} z_{1} \\ z_{2} \end{matrix}] \in X = [- 1, 1] \times [- 1, 1], z_{1} (t) = x, z_{2} (t) = \dot{x} .$ (15)

Therefore, the state-space equation for the servo motor given by Equation (13) is represented as $\dot{z} = f (z) + b (z) u$ (16) $f (z) = [\begin{matrix} z_{2} \\ \frac{1}{m} (- z_{2}) \end{matrix}], b (z) = [\begin{matrix} 0 \\ K / m \end{matrix}] .$ (17)

In this study, without loss of generality, we may assume that z₁ (t) = e (t) and $z_{2} (t) = \dot{e} (t),$ corresponding to error (e) and change of error (Δe) respectively. To prove the servo system is stable by the stability analysis Theorem 1 [13], the following Lyapunov function candidate in (18) is considered: $V (z) = z^{T} P z = \frac{1}{2} (\frac{4 {Kz}_{1}^{2}}{m} + z_{2}^{2})$ (18)

where $P = (\begin{matrix} K / 2 m & 0 \\ 0 & 1 / 2 \end{matrix}) .$ (19)

Hence, from Equation (18), V is positive and for $\begin{matrix} z \end{matrix} \to \infty$ , V (z)→ ∞. Therefore V ( 0 ) > 0, ∀ z ≠ 0 and differentiating V, we have $\begin{matrix} \dot{V} (z) = {\dot{z}}^{T} Pz + z^{T} P \dot{z} \\ = \frac{1}{m} ({Kz}_{1} z_{2} - z_{2}^{2} + {Kz}_{2} u) \end{matrix}$ (20)

Hence, it is required that $\frac{1}{m} ({Kz}_{1} z_{2} - z_{2}^{2} + {Kz}_{2} u) < 0$ (21) in the neighborhood of (0, 0) ^T.

Therefore, we need to design the rule-base of a Mamdani type-2 fuzzy controller that will be able to do smooth position-control of a servo-system and satisfy Equation (20). The fuzzy control rule base of IT2-GFPD is consisting of 25 rules and it is designed as shown in Table 1. Five Type-2 fuzzy sets - NB (negative big), NS (negative small), ZE (zero), PS (positive small), and PB (positive big) defined over the common normalized domain (– 1, 1) are used for defining the input (e, Δe) and output (u) variables for designing the FLC as shown in Fig. 2.

Table 1

Rule base of the IT2-GFPD controller

Δe/e	NB	NS	ZE	PS	PB
NB	PB	PB	PB	PS	ZE
	R1	R2	R3	R4	R5
NS	PB	PB	PS	ZE	NS
	R6	R7	R8	R9	R10
ZE	PB	PS	ZE	NS	NB
	R11	R12	R13	R14	R15
PS	PS	ZE	NS	NB	NB
	R16	R17	R18	R19	R20
PB	ZE	NS	NB	NB	NB
	R21	R22	R23	R24	R25

The system stability can be proved by considering following five cases. Therefore, in each group of case 1 to case 5 , z₁, z₂ and u represent their MFs within the crisp ranges they are defined. However, to compute the value of the crisp Lyapunov function, $\dot{V} (z)$ defined by Equation (20) for a particular fuzzy sub-system all the crisp values of z₁ and z₂ within the ranges (between the extremes) of their corresponding MFs used in that rule are considered [13]. Hence, while computing $\dot{V}$ (z) for any of the 25 fuzzy sub-systems corresponding to total 25 rules (Table 1) z₁ and z₂ are considered as crisp variables (of course locally) within the ranges on which their respective MFs are defined. In that sense, for the stability proof of any fuzzy sub-system using the crisp Lyapunov function $\dot{V}$ (z) in Equation (20), z₁ and z₂ should be locally crisp variables.

case 1: if z₁, z₂ are of same signs and z_2, u are of opposite signs, then here z₂ (- z₂ + Ku) 〈 0, Kz₁z₂ 〉 0, and|z₂ (z₂ + Ku) | > Kz₁z₂, this is for rules R1, R2, R6, R7, R19, R20, R24 and R25.

case 2: When z₁, z₂ are of opposite sign and u is also opposite sign of z₂ or ZE, then $| \begin{matrix} z_{2} \end{matrix} | ⩾ | \begin{matrix} u \end{matrix} |$ and $| \begin{matrix} z_{2} \end{matrix} | ⩾ | \begin{matrix} z_{1} \end{matrix} |$ , Kz₁z₂ < 0, Kz₂u < 0, this is in R3, R4, R5, R8, R9, R10, R17, R21, R22, R23.

case 3: Again when z₁, z₂ are of opposite sign and u is of same sign of z₂, then $| \begin{matrix} z_{2} \end{matrix} | > | \begin{matrix} u \end{matrix} |$ , also ${Kz}_{1} z_{2} < 0, z_{2}^{2} < 0$ , and $| {Kz}_{1} z_{2} + z_{2}^{2} | > {Kz}_{1} z_{2}$ , followed in rules R10, R16.

case 4: When z₂ = ZE, $- z_{2}^{2} < 0$ , and as z₁ and u are of opposite sign, and (z₁ - u) ⩽0 that is observed in rules R11, R12, R14 and R15.

case 5 : The condition z₁, z₂, u are all ZE, in R13, $\dot{V}$ will be negative or equal to zero, when $z_{2}^{2} ⩾ (z_{1} z_{2} + {Kz}_{2} u)$ . According to the proposed self-tuning fuzzy controller, the updating scaling factors in Equation (9b) and (9c) show $G_{e}^{A} < G_{Δ e}^{A}$ and the value of output scaling factor $G_{u} ⪡ G_{Δ e}^{A}$ . So, the value of $| \begin{matrix} z_{2} \end{matrix} | ⩾ | \begin{matrix} z_{1} \end{matrix} |$ and also $| \begin{matrix} z_{2} \end{matrix} | ⩾ | \begin{matrix} u \end{matrix} |$ .

Thus, $\dot{V}$ is expected to be negative.

Therefore, taking Equation (20) into account for all the above cases where all possible fuzzy rules have been considered, $\dot{V} (z) ⩽ 0$ . Hence, the Condition 2 of Theorem 1 is satisfied, i.e., $\dot{V}$ is negative semi-definite.

Next, it will be proved that the Condition 3 in Theorem 1 is also satisfied. Assuming that there is a trajectory with z₂ (t) = 0, z₁ (t) ≠ 0, which results ${\dot{z}}_{2} = \frac{K}{m} u \neq 0,$ (22) that means z₂ (t) cannot be constant. So, z (t) = 0 , is the only possible trajectory for which $\dot{V} (z) = 0 .$ Therefore, the system contains no other trajectory except the trivial trajectory z (t) = 0 for t ⩾ 0 for the set ${z \in X | \dot{V} (z) = 0}$ . Hence, the Condition 3 of Theorem 1 is also satisfied. The designed rule base in Table 1 which is satisfying Theorem 1 has been clearly shown in Fig. 6. The control surface corresponding to Table 1 is also shown in Fig. 7. Thus, we can infer according to Theorem 1, fuzzy control system with gain adaptive Type-2 fuzzy PD (IT2-GFPD) controller defined in Section 3 is globally asymptotically stable at the origin.

Fig. 6

Lyapunov derivative surface plot.

Fig. 7

Control surface of IT2-GFPD.

Observe that, in the proposed IT2-GFPD, input scaling factors are adapted online and they are multiplied with the input variables error (e) and change of error (Δe) to normalize them (Equations 9a and 9b). These normalized values of error (e_N) and change of error (Δe_N) usually remain within the universe of discourse [+ 1, –1]. However, during close-loop operation, if the value of e_N or Δe_N exceeds above +1 or less than – 1 then the value is saturated to +1 or – 1 respectively, according to Fig. 2. Since, every combination of e_N and Δe_N fires some rules within the designed fuzzy rule-base of Table 1 satisfying Lyapunov stability Theorem 1. Therefore, the adaptive scaling factor seems to have little impact on the stability of our designed controller (IT2-GFPD) if the parameters (k₁, k₂, and k₃) of the adaptive scaling factors ( $G_{e}^{A}$ and $G_{Δ e}^{A}$ defined in Equation 9c and 9d, respectively) are chosen judiciously. This fact will be substantiated in the next section providing responses of the servo model with±50% perturbations in k₁, k₂, and k₃ from their nominal values:

5 Results

To verify the effectiveness of the proposed IT2-GFPD, we have tested its performance on close-loop servo position control application through simulation study as well as real time experimentation. Here, it is to mention that same set of input-output SFs, G_e = 1.0 ; G_Δe = 3.0, G_u = 3.0 for all the IT2-FPD type controllers are taken for both the servo process model and the real time servo system SRV02 [23]. The value of tuning parameter k₂ and k₃ present in gain updating relations 9(c) and 9(d) are heuristically chosen as k₂ = k₃ = 5 through trial and error for both the simulation study and hardware based experimental verification. As per previous discussions in Section 3, it is observed that the desired nonlinear variation in β is obtained with k₁ = 0.2. To make comprehensive quantitative assessment among all the reported controllers several performance indices are evaluated. Percentage overshoot (% OS) and settling time (t_s) are calculated to evaluate the controller performance during transient response whereas integral absolute error (IAE) and integral time absolute error (ITAE) are calculated for steady state performance evaluation. For clear comparison among the controllers transient responses with different legends are provided during simulation as well as real time verification. The nature of MFs for Type-2 fuzzy PD controllers (IT2-FPD) as shown in Fig. 2 is considered to be identical to Type-1 and with 20% footprint of uncertainty (FOU). Control surfaces of IT2-FPD for 25 rules are given in Fig. 7 which is found to be highly non-linear. Presence of FOU in Type-2 fuzzy sets [35 –37] includes an extra mathematical dimension to deal system uncertainties and measurement noise in a more efficient manner. For simplicity we consider secondary MF’s for Type-2 fuzzy sets as rectangular in nature. Performance of the proposed IT2-GFPD is compared with conventional PD, PID, type-1 fuzzy PID (FPID), type-2 fuzzy PD controller, Z-slicing [31] type-2 fuzzy, recently reported model based interval type-2 fuzzy PID with an online self-tuning mechanism (OST-IT2-FPID [32] and peak observer based interval type-2 fuzzy PID controller PO-IT2-FPID [35] under both set point change and load disturbance.

Table 2
Simulation results of (23) with step input and load disturbance

θ _d Controller %OS t_s (s) IAE ITAE

θ_d = 0.1 IT2-GFPD 0.9 3.1 1.33 9.58

IT2-FPD 17.3 4.1 1.84 21.7

IT2-ZFPD 17.3 4.8 1.79 18.6

OST-IT2-FPID 25.6 8.8 3.84 56.2

PO-IT2-FPID 25.2 29.9 4.60 54.6

PD 11.8 7.5 2.117 25.29

PID 76.0 21.9 5.92 87.7

FPID 16.1 28.6 11.34 211.3

θ_d = 0.15 IT2-GFPD 0.10 0.1 1.40 12.2

IT2-FPD 31.2 6.6 2.23 27.9

IT2-ZFPD 27.2 5.6 2.11 25.9

OST-IT2-FPID 32.3 10.3 4.20 68.1

PO-IT2-FPID 30.3 11.4 3.544 43.2

PD 13.3 7.5 2.144 26.04

PID 78.5 24.3 6.35 97.1

FPID 16.1 28.0 11.34 213.0

θ _d	Controller	%OS	t_s (s)	IAE	ITAE
θ_d = 0.1	IT2-GFPD	0.9	3.1	1.33	9.58
	IT2-FPD	17.3	4.1	1.84	21.7
	IT2-ZFPD	17.3	4.8	1.79	18.6
	OST-IT2-FPID	25.6	8.8	3.84	56.2
	PO-IT2-FPID	25.2	29.9	4.60	54.6
	PD	11.8	7.5	2.117	25.29
	PID	76.0	21.9	5.92	87.7
	FPID	16.1	28.6	11.34	211.3
θ_d = 0.15	IT2-GFPD	0.10	0.1	1.40	12.2
	IT2-FPD	31.2	6.6	2.23	27.9
	IT2-ZFPD	27.2	5.6	2.11	25.9
	OST-IT2-FPID	32.3	10.3	4.20	68.1
	PO-IT2-FPID	30.3	11.4	3.544	43.2
	PD	13.3	7.5	2.144	26.04
	PID	78.5	24.3	6.35	97.1
	FPID	16.1	28.0	11.34	213.0

5.1 Simulation study

Related servo system is given by a first-order integrating plus dead-time (FOIPD) model: $G_{p} (s) = \frac{{Ke}^{- θ_{d} s}}{s (τ s + 1)}$ (23)

During simulation study, for the above process model, we consider K = 1 ; τ = 1 ; and dead-time θ_d = 0.1 . Robustness and noise sensitivity of the proposed IT2-GFPD along with other reported controllers are also tested (keeping same set of controller settings) in presence of wide model uncertainty with considerable (50%) perturbation in process dead-time. Responses of the PD, PID, FPID, IT2-GFPD, IT2-FPD, IT2-ZFPD [31], OST-IT2-FPID [32] and PO-IT2-FPID [35] for the nominal model are depicted in Fig. 8 and for the perturbed model (50%) in Fig. 9. Performance indices for both the nominal and perturbed (50%) models are given in Table 2. In case of the proposed controller the overshoot almost disappears and an improved performance compared to the other controllers is also observed. The performance study with measurement noise (noise power = 0.001) is depicted in Fig. 10 and related performance indices are given in Table 3. Noise immunity of IT2-GFPD is found to be much superior compared to OST-IT2-FPID [32] and PO-IT2-FPID [35]. Responses with changing reference levels have also been illustrated in Figs. 11a and 11b, and the corresponding performance indices are given in Table 4. Here also, IT2-GFPD shows better performance than others. Figure 11c shows the responses of (23) with±50% perturbations in k₁, k₂, and k₃ from their nominal values, which reveals that the adaptive SFs have little impact on the stability of our proposed IT2-GFPD as explained earlier in Section 4.

Fig. 8a

Responses of (23) with θ_d = 0.1 for IT2-FPD, IT2-GFPD, IT2-ZFPD controllers.

Fig. 8b

Responses of (23) with θ_d = 0.1 for OST-IT2-FPID and PO-IT2-FPID controllers.

Fig. 8c

Responses of (23) with θ_d = 0.1 for FPID, PID and PD controllers.

Fig. 9a

Responses of (23) withθ_d = 0.15 for IT2-FPD, IT2-GFPD, IT2-ZFPD controllers.

Fig. 9b

Responses of (23) with θ_d = 0.15 for OST-IT2-FPID and PO-IT2-FPID controller.

Fig. 9c

Responses of (23) with θ_d = 0.15 for FPID, PID and PD controllers.

Fig. 10a

Responses of (23) for IT2-GFPD, IT2-FPD and IT2-ZFPD in presence of measurement noise.

Fig. 10b

Responses of (23) for PO-IT2-FPID and OST-IT2-FPID in presence of measurement noise.

Fig. 10c

Responses of (23) for FPID and PID in presence of measurement noise.

Fig. 11a

Responses of (23) for PD, FPD, FPID, IT2-FPD, IT2-GFPD controllers with changing reference.

Fig. 11b

Responses of (23) for PO-IT2-FPID, OST-IT2-FPID controllers with changing reference.

Fig. 11c

Responses of (23) for IT2-GFPD with±50% perturbations in k_1, k₂ and k₃ from their nominal values.

Table 3

Simulation results of (23) with noise

θ _d	Controller	IAE	ITAE
leθ_d = 0.1	IT2-GFPD	1.071	2.691
	IT2-FPD	1.345	3.177
	IT2-ZFPD	1.387	3.172
	OST-IT2-FPID	2.371	6.126
	PO-IT2-FPID	3.740	22.170
	PID	3.917	17.250
	FPID	6.580	34.940

Table 4

Simulation results of (23) for changing reference

Controller	IAE	ITAE
PD	5.644	433.1
FPID	31.840	2717.0
FPD	6.699	542.0
IT2-FPD	5.508	448.5
IT2-GFPD	5.364	426.2
PO-IT2-FPID	19.930	1711.0
OST-IT2-FPID	11.830	934.4

Non-linear model:

In our simulation study, we also consider the following non-linear model, $\frac{d^{2} y}{{dt}^{2}} + 0.4 y \frac{dy}{dt} = u (t - θ_{d})$ (24)

Table 5

Simulation results of (24) with step input and load disturbance

θ _d	Controller	%OS	t_s (s)	IAE	ITAE
θ_d = 0.1	IT2-GFPD	2.6	3.3	1.268	6.883
	IT2-FPD	33.4	6.1	2.017	24.55
	IT2-ZFPD	38.8	7.3	2.495	32.09
	OST-IT2-FPID	39.1	16.1	5.158	95.44
	PO-IT2-FPID	32.3	20.3	3.585	55.54
	PID	43.9	20.3	4.443	50.75
θ_d = 0.15	IT2-GFPD	3.0	3.6	1.268	6.88
	IT2-FPD	33.4	6.1	2.017	24.55
	IT2-ZFPD	38.8	7.3	2.495	32.09
	OST-IT2-FPID	44.6	17.3	5.861	102.54
	PO-IT2-FPID	35.8	18.2	3.841	53.70
	PID	43.3	26.1	4.395	50.55

Robustness of our proposed IT2-GFPD is tested on the above non-linear model even with 50 % perturbation in its dead-time (θ_d) using the same set of controller settings. Responses of PID, IT2-GFPD, IT2-FPD, IT2-ZFPD [31], OST-IT2-FPID [32] and PO-IT2-FPID [35] for the non-linear system in (24) with θ_d = 0.1 are depicted in Figs. 12a, and 12b, and the performance comparison is provided in Table 5. Performance study is also observed with changing references. Responses in Figs. 13a and 13b, and the performance indices in Table 6 reveal that the proposed controller is superior compared to PD, FPID, type-1 fuzzy PD (FPD), IT2-FPD, IT2-ZFPD [31], OST-IT2-FPID [32], and PO-IT2-FPID [35].

Fig. 12a

Responses of (24) for IT2-FPD, IT2-GFPD, IT2-ZFPD, PID controllers.

Fig. 12b

Responses of (24) with OST-IT2-FPID and PO-IT2-FPID controller.

Fig. 13a

Responses of (24) for PD, FPD, FPID, IT2-FPD, IT2-GFPD controllers with changing reference.

Fig. 13b

Responses of (24) for PO-IT2-FPID, OST-IT2-FPID controllers with changing reference.

Table 6

Simulation results of (24) for changing reference

Controller	IAE	ITAE
PD	10.65	975.9
FPID	26.82	2436
FPD	7.162	579.7
IT2-FPD	5.898	478.8
IT2-GFPD	5.934	506.1
PO-IT2-FPID	15.43	1350
OST-IT2-FPID	11.25	961.1

5.2 Real-time experimentation

Effectiveness of the proposed IT2-GFPD is also experimentally verified on a DC servo position control system, SRV02 by Quanser, Canada [23] shown in Fig. 14. SRV02 incorporates high efficiency low inductance Faulhaber Coreless DC Motor. With the help of QUARC and MATLAB-Simulink in the PC and the data acquisition device, Quanser Q2-USB, reported controllers are implemented on SRV02 [23] for their performance evaluation. Observe that, SRV02 servo position control system is identified as a first-order integrating plus dead-time (FOIPD) model as derived in Section 4.

Fig. 14a

Servo system, SRV02.

Fig. 14b

Experimental setup with SRV02.

Set point tracking and load regulation responses of different controllers are plotted in Fig. 15 and related performance indices are given in Table 7. Performance of SRV02 is also studied for PID, FPID, IT2-GFPD, IT2-FPD, IT2-ZFPD [31], PO-IT2-FPID [35], OST-IT2-FPID [32], FPID and PID in presence of measurement noise (noise power 0.00001). The responses of SRV02 under noise are shown in Fig. 16 and the corresponding performance indices are given in Table 8. Based on the responses and performance indices it can clearly be justified that the proposed IT2-GFPD is capable to provide enhanced close-loop response under model uncertainty and in presence of measurement noise compared to conventional Type-2 PD (IT2-FPD), PO-IT2-FPID [35] and OST-IT2-FPID [32], FPID, and PID controllers.

Fig. 15a

Responses of SRV02 for IT2-FPD, IT2-GFPD, and IT2-ZFPD controllers.

Fig. 15b

Responses of SRV02 for PO-IT2-FPID and OST-IT2-FPID controllers.

Fig. 15c

Responses of SRV02 for FPID and PID controllers.

Fig. 16a

Responses of SRV02 with noise for IT2-GFPD, IT2-FPD, and IT2-ZFPD controllers.

Fig. 16b

Responses of SRV02 with noise for PO-IT2-FPID and OST-IT2-FPID controllers.

Fig. 16c

Responses of SRV02 with noise for PID and FPID controllers.

Table 7

Performance comparison for SRV02

Controller	%OS	t_s (s)	IAE	ITAE
IT2-GFPD	4.1	0.2	0.027	0.01
IT2-FPD	8.1	0.2	0.028	0.02
IT2-ZFPD	5.9	0.2	0.021	0.009
OST-IT2-FPID	22.8	—	0.866	3.783
PO-IT2-FPID	22.2	—	0.977	4.77
PID	5.06	0.37	0.044	0.019
FPID	20.93	—	0.130	0.066

Table 8

Results of Quanser SRV02 under noise power 10^–5

Controller	IAE	ITAE
IT2-GFPD	0.5004	2.273
IT2-FPD	0.5827	2.878
IT2-ZFPD	0.5886	2.909
OST-IT2-FPID	1.553	6.415
PO-IT2-FPID	1.285	5.073
PID	0.617	2.503
FPID	1.022	3.361

Observe that, the DC servo system is modeled as a linear system, neglecting the nonlinearity effect caused by backlash and nonlinear friction, which are difficult to model. But for high performance servo system, these nonlinear factors cannot be neglected. Hence, desired performance cannot be achieved under conventional (fuzzy or non-fuzzy) controllers with fixed settings. For such reasons adaptive controllers and particularly adaptive fuzzy controllers are mostly suggested in the literature. Furthermore, Interval Type-2 fuzzy controllers (IT2-FLCs) are found to exhibit superior performances than Type-1 fuzzy controllers and other conventional controllers in case of model uncertainty and noisy environment. As a result, the IT2-FLC with online adaptive scaling factors has made our proposed (IT2-GFPD) controller to perform superior than PID, FPID, IT2-FPD, PO-IT2-FPID controllers as demonstrated above.

6 Conclusion

In this study, an online gain adaptive interval type-2 fuzzy PD controller (IT2-GFPD) has been proposed. The rule-base of our IT2-GFPD has been designed based on the Lyapunov stability analysis approach. Effectiveness of the proposed controller has been tested through simulation experiments on both linear and non-linear integrating plus dead-time systems as well as real-time implementation on a practical DC servo system (SRV02). Performance of the IT2-GFPD has been compared with conventional (PD and PID) and fuzzy (type-1 and type-2) controllers with respect to various performance indices. Superior performance of our IT2-GFPD has been observed than other reported controllers. The designed IT2-GFPD controller has been found to be quite robust with model uncertainties and in presence of measurement noise compared to conventional, Z-slicing type-2 fuzzy controller, type-2 peak observer based controller and online self-tuning type-2 fuzzy PID controller reported in the literature.

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