1 Introduction

Expert system and its applications have risen to a great level over the last few decades with the advent of artificial intelligence. Various fields like industrial automation, robotics, bio-metric recognition and machine learning have well utilized the potential of artificial intelligence. The use of intelligent techniques such as fuzzy logic and neural network has made the automation more effective and efficient by incorporating human decision capability to expert system. Most of the automation systems usually exhibit nonlinearity and uncertainty and also suffer from disturbance and internal or external noise. So, there is a need for intelligent expert solution for smooth and efficient functioning of such systems. Conventional controllers, such as proportional–integral–derivative (PID) controller, fail to deliver efficient control for systems with nonlinear and uncertain dynamics [1]. However, till date 90% of the controllers used in industry are PID because of its many advantages such as cost-effectiveness, ease of installation and operation [2]. Therefore, researchers across the globe are more inclined to enhance the capabilities of PI/PD/PID controllers by merging artificial intelligence with basic conventional structure. Several examples of such hybrid fuzzy controllers are available in the literature, which are quite effective and robust [3,4,5,6,7,8,9,10,11,12,13].

The performance of expert fuzzy controllers is further improved by the introduction of fractional calculus as it provides extra degree of freedom and flexibility to design scheme. In these controllers, the order of integration and differentiation is generalized to non-integer values. Diverse applications of fractional order fuzzy controllers are seen in the literature. Kumar and Kumar [14] designed a fractional order (FO) fuzzy pre-compensated FOPID controller for robotic manipulator. The controller parameters are optimized using hybrid artificial bee colony-genetic algorithm (ABC-GA) for minimizing ITAE. Simulation results prove better performance of proposed controller as compared to integer order fuzzy pre-compensated PID, fuzzy PID and PID control scheme in various case studies. The authors [15] also proposed a family of interval type-2 fractional order fuzzy PID controllers, and their performance is evaluated on fractional order systems. The design variables of the controllers are optimized using ABC-GA technique. Simulation results illustrate the effectiveness of these controllers in terms of disturbance rejection, uncertainty, low variation in control signal and better tracking. Sharma et al. [16] designed a new two-layered fractional order fuzzy logic control technique for two-link robot. Performance of the designed controller is compared with single-layer fuzzy logic controller and traditional PID. Optimum parameters of the controllers are achieved using cuckoo search algorithm. Results prove that proposed control scheme outruns the other controllers in all aspects. Das et al. [17] presented different structures of fractional order fuzzy PID to control fractional order system having dead time. Non-integer operators and scaling gains of the controllers are optimally tuned using genetic algorithm, and performance is compared quantitatively for different case studies. The authors [18] also performed comparative study of fractional order fuzzy PID and its integer order variant for delayed nonlinear and open-loop unstable process. The parameters of controllers are optimized using real-coded genetic algorithm, and the performance of FO fuzzy PID is found superior over fuzzy PID. Kumar et al. [19] presented fractional order fuzzy PD for controlling active suspension system to achieve comfortable ride. The performance of designed controller is compared with its integer order counterpart for uncertain environment with different road profiles. It is found that FO-FPD outperforms FPD in every aspect. Jesus et al. [20] designed a genetic algorithm tuned fractional order fuzzy PD plus conventional integral controller. The effectiveness of controller is proved by considering different examples. The expert fractional order fuzzy controllers offer robust and effective control. Therefore, these controllers are applied in different fields such as combined cycle power plant [21], smart base-isolated structures [22], financial system [23], hybrid power plant [24], robotic system [25, 26], nuclear reactor [27], distillation column [28] and hybrid electric vehicle [29].

The degree of freedom of a controller signifies the number of closed loops that can be modified individually and plays a significant role in designing an efficient control strategy. A 1-DOF controller has only one closed loop, and hence, it is incapable of addressing multiple conflicting issues, i.e., trajectory tracking and disturbance rejection simultaneously [30]. Both conventional and hybrid fuzzy controllers are 1-DOF in nature and thus do not provide solution to such problems [2, 30, 31]. The control problems most frequently are coupled in nature and have several conflicting objectives. The recent research in this field demonstrates that 2DOF controller provides a good solution to these problems. As an illustration, Pachauri et al. [32] presented 2DOF PID-based inferential control of bioreactor. The suggested controller outperforms the PID in closed-loop control for set-point tracking, disturbance rejection and noise suppression. Ghosh et al. [33] also proved that 2DOF PID outperforms the 1DOF PID for magnetic levitation system. Richa et al. [31] compared the performance of CSA tuned 2DOF FOPID, 2DOF PID and PID for robotic manipulator, and results proved the superiority of 2DOF FOPID over other implemented controllers. Debbarma et al. [34] presented the comparative study of firefly tuned 2DOF FOPID, 2DOF PID, PID and PI controllers for AGC of power system. Simulation results claimed that 2DOF FOPID performs significantly better in all aspects over other controllers. Li et al. [35] overcome the problem of coupled robustness and dynamic response in 1DOF FOPID for fractional order system with dead time by utilizing internal model control-based 2DOF FOPID. The simple design and tuning approach of the proposed controller proves its effectiveness over 1DOF FOPID controller. Different structures of expert 2DOF fuzzy PI controller are applied to a laboratory DC drive by Precup et al. [2]. Experimental results claim that the performance of generic 2DOF fuzzy PI control structures is superior to 2DOF PI controller.

It is evident from the literature that amalgamation of fractional calculus with 2DOF PID enhances its performance. Further embedding artificial intelligence to conventional controller leads to an expert system with increased efficiency. This motivated us to design a new generic two-degree-of-freedom fractional order fuzzy proportional integral minus derivative (2DOF FOFPI-D) controller which explores the advantages of formula-based fuzzy in association with 2DOF fractional order PID structure. The structure of controller comprises of serial compensator (FOFPI) and fractional order derivative filter (FODF) to handle multiple issues simultaneously. The proposed scheme is adaptive in nature, as the gains of serial compensator are nonlinear functions of error and fractional rate of error. Further, the performance of designed controller is assessed by comparing it with 2DOF fuzzy PI-D (2DOF FPI-D) and 2DOF PI-D for coupled, nonlinear and complex two-link robotic arm in servo and regulatory modes. Fractional operators increase the flexibility of controllers at the cost of higher number of design variables, thereby making it difficult to tune. Thus, the design variables of each controller are tuned optimally using non-dominated sorting genetic algorithm-II [36] so that error index and variations in control are small. The performance of designed controllers is examined by conducting comprehensive simulation study on the basis of integral absolute error (IAE). The benefit of proposed scheme is that it incorporates simple formula-based fuzzy, wherein inputs are fuzzified using two triangular membership functions. This leads to only four if–then rules [4]. Another merit of the control scheme is that it is generic as its design does not require the model of system. Moreover, it also enjoys the benefits of 2DOF structure along with flexibility in design due to fractional operators. The key contributions of this work can be summarized as follows:

  • In this paper, a fractional order formula-based intelligent control technique, namely 2DOF FOFPI-D, is proposed. Its control law is derived analytically using linear 2DOF FOPID technique in discrete domain. Further formula-based fuzzy design methodology is incorporated in derived control law to provide expert intelligence and self-tuning control feature.

  • The proposed 2DOF FOFPI-D controller is found to be more efficient and robust in virtually created real operating environment by considering issues such as parametric uncertainty, actuator disturbance and measurement noise as compared to 2DOF fuzzy PI-D (2DOF FPI-D) and 2DOF PI-D.

  • The benefits of proposed controller make it appropriate for real-time control applications [31, 32] such as robotic arm, distillation column, bioreactor, flexible link and joint robot.

After a brief literature review in Sect. 1, the design approach of 2DOF FOFPI-D controller is presented in Sect. 2. In Sect. 3, introduction and design steps of NSGA-II algorithm are outlined. The mathematical model of robotic manipulator under consideration is given in Sect. 4. Section 5 analyzes the simulation results of 2DOF FOFPI-D and other controllers for two-link robotic arm. The discussion of results is presented in Sect. 6. Finally concluding remarks regarding significance and impact of the research work are discussed in Sect. 7.

2 Design of adaptive 2DOF FOFPI-D controller

The aim of this work is to devise an adaptive control scheme for nonlinear systems, which provides accurate control with minimum control effort while addressing several conflicting issues. The analytical formulae for proposed control scheme are derived from conventional 2DOF FOPID controller. The formula-based fuzzy design is then embedded in derived control law, which makes it intelligent and adaptive.

2.1 Mathematical formulation of 2DOF FOFPI-D control law

The general equation of control action for classical 2DOF PID controller in frequency domain is expressed as follows [30]:

$$U_{{2{\text{DOF}}\,{\text{PID}}}} = P\left( {b*R - Y} \right) + \frac{I}{s}\left( {R - Y} \right) + \frac{{K_{\text{D}} N}}{{1 + \frac{N}{s}}}\left( {c\,*\,R\, - \,Y} \right)$$
(1)

The 2DOF fractional order PID (2DOF FOPID) for non-integer order of integration and differentiation is represented as:

$$U_{{2{\text{DOF}}\,{\text{FOPID}}}} = P\left( {b*R - Y} \right) + \frac{I}{{s^{\lambda } }}\left( {R - Y} \right) + \frac{{K_{\text{D}} N}}{{1 + \frac{N}{{s^{\mu } }}}}\left( {c*R - Y} \right)$$
(2)

where P, I and KD are proportional, integral and derivative gains, respectively. R is reference signal and Y is the output of nonlinear system. b and c are weights of 2DOF controller, whereas \(\lambda\) and \(\mu\) are orders of fractional operator. N and s are the filter coefficient and complex frequency, respectively.

A large spike or kick is observed in derivative control action due to discontinuous or sudden changes in reference signal. In practical applications, the control signal drives final control element such as valve, electric actuators and fluid couplings. The large spikes in control signal may damage the element [37]. Typically derivative action for abrupt changes in reference is undesirable and is required only for system response Y. Therefore, the control law ‘\(U_{{2{\text{DOF}}{\kern 1pt} \,{\text{FOPID}}}}\)’ is modified by considering the weight ‘c’ to be zero, so that derivative action depends only on system output ‘Y’. The derivative kick is thus completely avoided, and Eq. (2) is modified as:

$$U_{{2{\text{DOF}}\,{\text{FOPI}} - {\text{D}}}} = \underbrace {{\left( {Pb + \frac{I}{{s^{\lambda } }}} \right)E\left( s \right)}}_{{U_{\text{sc}} }} - \underbrace {{\left( {\frac{{K_{\text{D}} N}}{{1 + \frac{N}{{s^{\mu } }}}} + P\left( {1 - b} \right)} \right)Y\left( s \right)}}_{{U_{\text{ff}} }}$$
(3)

where \(U_{\text{sc}} = \left( {Pb + \frac{I}{{s^{\lambda } }}} \right)E\left( s \right),\,U_{\text{ff}} = \left( {\frac{{K_{\text{D}} N}}{{1 + \frac{N}{{s^{\mu } }}}} + P\left( {1 - b} \right)} \right)Y\left( s \right){\text{ and }} E\left( s \right) = R\left( s \right) - Y\left( s \right)\) is error. The control law (Eq. 3) for two-degree-of-freedom fractional order proportional plus integral minus derivative controller comprises of a serial compensator ‘\(U_{\text{sc}}\)’ and fractional order derivative filter (FODF) ‘\(U_{\text{ff}}\)’. The serial compensator is equivalent to fractional order proportional–integral (FOPI) controller, where proportional action improves the response time of system by decreasing the time constant and integral action reduces the steady-state error [38]. Therefore, it improves transient as well as steady-state response of the system. Generally high-frequency sensor noise affects the system output, and conventional derivative term may produce unwanted large control action. This problem is solved by putting a fractional order filter with derivative term which attenuates the noise and limits the derivative action [39]. Hence, FOFD possesses predictive capability while eliminating the chattering of high-frequency noise. Further the conventional FOPI controller is replaced by fractional order fuzzy PI controller leading to a novel two-degree-of-freedom fractional order fuzzy proportional plus integral minus derivative controller. The controller is designed using analytical formulae to provide self-tuning control ability besides preserving the properties of traditional controller [12]. Analytical formulae for transforming the serial compensator to FO fuzzy PI controller are derived as:

$$U_{\text{sc}} \left( s \right) = \left\{ {Pb + \frac{I}{{s^{\lambda } }}} \right\}E\left( s \right)$$
(4)

Using backward transformation \(s = \frac{{\left( {1 - z^{ - 1} } \right)}}{T}\), where \(T > 0\) is sampling time. Further Eq. (4) is transformed into discrete form as:

$$\begin{aligned} U_{\text{sc}} \left( z \right) &= \left\{ {Pb + \frac{{IT^{\lambda } }}{{\left( {1 - z^{ - 1} } \right)^{\lambda } }}} \right\}E\left( z \right) \hfill \\ \Delta U_{\text{sc}} \left( z \right) &= \left\{ {Pb\left( {\frac{{1 - z^{ - 1} }}{T}} \right)^{\lambda } + I} \right\}E\left( z \right) \hfill \\ \end{aligned}$$
(5)

where

$$\Delta U_{\text{sc}} \left( z \right) = U_{\text{sc}} \left( z \right)\left( {\frac{{1 - z^{ - 1} }}{T}} \right)^{\lambda }$$
(6)

Expanding Eq. (5) using power series expansion [40] as

$$\Delta U_{\text{sc}} \left( z \right) = \frac{Pb}{{T^{\lambda } }}\mathop \sum \limits_{k\, = \,0}^{\infty } \left( { - \,1} \right)^{k} \left( {\begin{array}{*{20}c} \lambda \\ k \\ \end{array} } \right)z^{ - k} E\left( z \right) + IE\left( z \right)$$
(7)

Taking inverse Z-transform of Eq. (7), it yields

$$\Delta u_{\text{sc}} \left( {nT} \right) = \frac{Pb}{{T^{\lambda } }}\mathop \sum \limits_{k\, = \,0}^{\infty } \left( { - \,1} \right)^{k} \left( {\begin{array}{*{20}c} \lambda \\ k \\ \end{array} } \right)e\left( {\left( {n - k} \right)T} \right) + Ie\left( {nT} \right)$$
(8)

The first term on the right side of Eq. (8) is similar to the formulation specified by Lubich for non-integer derivative/integral of order \(\alpha\) for an arbitrary function \(x\left( {nT} \right)\) [41] and is expressed as:

$$D^{ \mp \alpha } x\left( {nT} \right) = T^{ \pm \alpha } \mathop \sum \limits_{k\, = \,0}^{\infty } \left( { - \,1} \right)^{k} \left( {\begin{array}{*{20}c} { \mp \alpha } \\ k \\ \end{array} } \right)x\left( {\left( {n - k} \right)T} \right)$$
(9)

Equating Eqs. (8) and (9), it yields

$$\Delta u_{\text{sc}} \left( {nT} \right) = PbD^{\lambda } e\left( {nT} \right) + Ie\left( {nT} \right)$$

Explicitly

$$\Delta u_{\text{sc}} \left( {nT} \right) = K_{P} e\left( {nT} \right) + K_{I} e_{\text{r}} \left( {nT} \right)$$
(10)

where \(K_{P} = I,\, K_{I} = Pb,\, e_{\text{r}} \left( {nT} \right) = D^{\lambda } e\left( {nT} \right)\,{\text{and}}\,e\left( {nT} \right) = r\left( {nT} \right) - y\left( {nT} \right).\) Similarly, solving Eq. (6) for control law gives

$$\begin{aligned} U_{{{\text{sc}}}} \left( z \right) & = T^{\lambda } \sum\limits_{{k{\kern 1pt} = {\kern 1pt} 0}}^{\infty } {\left( { - 1} \right)^{k} } \left( {\begin{array}{*{20}c} { - \lambda } \\ k \\ \end{array} } \right)z^{{ - k}} \Delta U_{{{\text{sc}}}} \left( z \right) \\ u_{{{\text{sc}}}} \left( {nT} \right) & = T^{\lambda } \sum\limits_{{k{\kern 1pt} = {\kern 1pt} 0}}^{\infty } {\left( { - 1} \right)^{k} } \left( {\begin{array}{*{20}c} { - \lambda } \\ k \\ \end{array} } \right)\Delta u_{{{\text{sc}}}} \left( {nT} \right) \\ u_{{{\text{sc}}}} \left( {nT} \right) & = D^{{ - \lambda }} \left( {\Delta u_{{{\text{sc}}}} \left( {nT} \right)} \right). \\ \end{aligned}$$
(11)

Equation (11) is transformed to fuzzy control action by multiplying right-hand side with scaling gain \(K_{u}\) as

$$u_{\text{sc}} \left( {nT} \right) = K_{u} D^{ - \lambda } \left( {\Delta u_{\text{sc}} \left( {nT} \right)} \right)$$
(12)

Thus, overall control action of 2DOF FOFPI-D is expressed as:

$$u_{{2{\text{DOF}}{\kern 1pt} {\text{FOFPI}} - {\text{D}}}} = u_{\text{sc}} - u_{\text{ff}}$$
$$u_{{2{\text{DOF}}{\kern 1pt} {\text{FOFPI}} - {\text{D}}}} = K_{u} D^{ - \lambda } \left( {\Delta u_{\text{sc}} \left( {nT} \right)} \right) - \left\{ {\frac{{ K_{D} ND^{\mu } }}{{D^{\mu } + N}} + P\left( {1 - b} \right)} \right\}y\left( {nT} \right)$$
(13)

The generalized structure of 2DOF FOFPI-D (Eqs. 10, 13) for nonlinear system is shown in Fig. 1.

Fig. 1
figure 1

Control structure of generic two-degree-of-freedom fractional order fuzzy PI-D for nonlinear system

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2.2 Framework for fuzzy control

Standard fuzzy design methodology, which includes fuzzification, control rule and defuzzification, is embedded to analytical formula derived in the last section.

2.2.1 Fuzzification

The inputs and output of serial compensator are fuzzified separately, and control rules are designed using Eq. (11). Serial compensator employs two inputs: error \(\tilde{e}\left( {nT} \right) = K_{\text{P}} e\left( {nT} \right)\) and fractional rate of error \(\tilde{e}_{\text{r}} \left( {nT} \right) = K_{\text{I}} e_{\text{r}} \left( {nT} \right)\). Two triangular membership functions, \(n\): negative and \(p\): positive, are selected for both inputs. On the other hand, output \(\Delta u_{\text{sc}} \left( {nT} \right)\) is fuzzified using four singleton membership functions, nl: negative large, ns: negative small, ps: positive small and pl: positive large. Corresponding input and output membership functions of serial compensator are shown in Fig. 2. The constant \(L\, > \,0\) is determined using NSGA-II optimization.

Fig. 2
figure 2

Membership functions of serial compensator a input and b output

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2.2.2 Control rule base

The control rule base plays a key role in designing an efficient fuzzy logic controller. These rules are framed either on the basis of operator experience or knowledge of control system dynamics. In this work, four rules for serial compensators are designed on the basis of control system knowledge. The following control rules are designed for serial compensator on the basis of input and output membership function as follows:

$$\begin{aligned} (R^{1} ):\,{\text{If}}\,\tilde{e} = \tilde{e} \cdot n\,{\text{and}}\,\tilde{e}_{\text{r}} = \tilde{e}_{\text{r}} \cdot n\,{\text{then}}\,\Delta u_{\text{sc}} = o \cdot nl \hfill \\ (R^{2} ):\,{\text{If}}\,\tilde{e} = \tilde{e} \cdot n\,{\text{and}}\,\tilde{e}_{\text{r}} = \tilde{e}_{\text{r}} \cdot p\,{\text{then}}\,\Delta u_{\text{sc}} = o \cdot ns \hfill \\ (R^{3} ):\,{\text{If}}\,\tilde{e} = \tilde{e} \cdot p\,{\text{and}}\,\tilde{e}_{\text{r}} = \tilde{e}_{\text{r}} \cdot n\,{\text{then}}\,\Delta u_{\text{sc}} = o \cdot ps \hfill \\ (R^{4} ):\,{\text{If}}\,\tilde{e} = \tilde{e} \cdot p\,{\text{and}}\,\tilde{e}_{\text{r}} = \tilde{e}_{\text{r}} \cdot p\,{\text{then}}\,\Delta u_{\text{sc}} = o \cdot pl \hfill \\ \end{aligned}$$

where \(\tilde{e} = K_{P} e = K_{P} \left( {r - y} \right)\) is error, \(\tilde{e}_{\text{r}} = K_{I} e_{\text{r}} \left( {nT} \right) = K_{I} \left( {D^{\lambda } r - D^{\lambda } y} \right)\) is the fractional rate of error and \(\Delta u_{\text{sc}}\) is fuzzy control action of serial compensator. Also, ‘\(\tilde{e} \cdot p\)’ signifies error positive, ‘\(\tilde{e}_{r} \cdot n\)’ means fractional rate of error negative and ‘\(o \cdot ns\)’ is output negative small and so on. A set of four rules decides the fuzzy control action of serial compensator, and their formulation is described as:

In rule 1 (\(R^{1}\)), \(\tilde{e} \cdot n\) (error negative) simply means that system response \(y\left( {nT} \right)\) is above the reference signal \(r\left( {nT} \right)\) and corresponding ‘\(\tilde{e}_{r} \cdot n\)’ fractional rate of error negative (\(D^{\lambda } y > D^{\lambda } r\)) implies that response is moving up at a rate faster than reference from previous instant. Explicitly from rule 1, the response is above and moving up at faster rate than reference. Therefore to bring down the response \(y\) close to reference, the fuzzy control action of serial compensator \(\Delta u_{\text{sc}}\) is set as negative large. In rule \(2 \left( {R^{2} } \right)\), the response of system is above reference but moving down at faster pace than desired trajectory. So, \(\Delta u_{\text{sc}}\) is set as negative small. Similarly, rules \(R^{3}\) and \(R^{4}\) can be explained.

2.2.3 Defuzzification

The control action of serial compensator is defuzzified by center-of-mass formula [4]. The mathematical expression for defuzzified control action \(\Delta u_{\text{sc}}\) is expressed as:

$$\Delta u_{\text{sc}} = \frac{{\mathop \sum \nolimits \left( {{\text{output}}\,{\text{corresponding}}\,{\text{to}}\,{\text{membership}}\,{\text{value}}\,{\text{of}}\,{\text{input}}\,*\,{\text{membership}}\,{\text{value}}\,{\text{of}}\,{\text{input}}} \right)}}{{\mathop \sum \nolimits {\text{membership}}\,{\text{value}}\,{\text{of}}\,{\text{input}}}}$$
(14)

The control action of serial compensator is defuzzified by decomposing inputs \(\tilde{e}\) and \(\tilde{e}_{\text{r}}\) in 20 input combination (IC) regions as shown in Fig. 3. The 2-dimensional picture is obtained by plotting membership function of \(\tilde{e}\) and \(\tilde{e}_{\text{r}}\) on horizontal and vertical axis, respectively. The rules \(R^{1}\) to \(R^{4}\) and membership functions along with IC regions are used to estimate the defuzzified control action ‘\(\Delta u_{\text{sc}}\)’. Now, let \(\tilde{e}\) and \(\tilde{e}_{\text{r}}\) fall in region IC1. It is clear from Fig. 3 that range of \(\tilde{e}\) and \(\tilde{e}_{\text{r}}\) is \(\left[ {0, L} \right]\) and \(\left[ { - \,L, 0} \right]\), respectively. Also, it is observed from Fig. 2a that value of \(\tilde{e} \cdot n < 0.5\) and \(\tilde{e}_{\text{r}} \cdot n > 0.5\). Therefore, rule \(R^{1}\) and Zadeh’s logic [42] collectively yield the following:

Fig. 3
figure 3

Input combination region for serial compensator

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IF \(\tilde{e} = \tilde{e}.n\) AND \(\tilde{e}_{\text{r}} = \tilde{e}_{\text{r}} \cdot n\) implies minimum \(\left\{ {\tilde{e} \cdot n, \,\tilde{e}_{\text{r}} \cdot n} \right\} = \tilde{e} \cdot n\)

Therefore, for rule 1

$$R^{1} \left\{ {\begin{array}{*{20}c} {{\text{corresponding}}\,{\text{input}}\,{\text{membership}}\,{\text{value}}\,{\text{is}}\,\tilde{e} \cdot n} \\ {{\text{membership}}\,{\text{value}}\,{\text{of}}\,{\text{output}}\,{\text{is}}\,o \cdot {\text{nl}} } \\ \end{array} } \right.$$

Likewise, other rules and Zadeh’s logic in IC1 lead to

$$\begin{aligned} R^{2} \left\{ {\begin{array}{*{20}c} {{\text{corresponding}}\,{\text{input}}\,{\text{membership}}\,{\text{value}}\,{\text{is}}\, \tilde{e} \cdot n} \\ {{\text{membership }}\,{\text{value}}\,{\text{of}}\,{\text{output }}\,{\text{is }}\,o \cdot ns } \\ \end{array} } \right. \hfill \\ R^{3} \left\{ {\begin{array}{*{20}c} {{\text{corresponding}}\,{\text{input}}\,{\text{membership}}\,{\text{value}}\,{\text{is}}\, \tilde{e}_{\text{r}} \cdot n} \\ {{\text{membership }}\,{\text{value}}\,{\text{of}}\,{\text{output}}\,{\text{is}}\,o \cdot ps } \\ \end{array} } \right. \hfill \\ R^{4} \left\{ {\begin{array}{*{20}c} {{\text{corresponding}}\,{\text{input}}\,{\text{membership}}\,{\text{value}}\,{\text{is}}\,\tilde{e}_{\text{r}} \cdot p} \\ {{\text{membership}}\,{\text{value}}\,{\text{of}}\,{\text{output}}\,{\text{is}}\,o \cdot pl } \\ \end{array} } \right. \hfill \\ \end{aligned}$$

It can be easily verified that above rules are true for region IC2. Therefore, in regions IC1 and IC2 the defuzzification formula gives:

$$\Delta u_{\text{sc}} = \frac{{o \cdot nl\,*\,\tilde{e} \cdot n + o \cdot ns\,*\,\tilde{e} \cdot n + o \cdot ps\,*\,\tilde{e}_{\text{r}} \cdot n + o \cdot pl\, *\,\tilde{e}_{\text{r}} \cdot p}}{{\tilde{e} \cdot n\, + \,\tilde{e} \cdot n\, + \,\tilde{e}_{\text{r}} \cdot n\, + \,\tilde{e}_{\text{r}} \cdot p}}$$
(15)

Now, mathematical representations of input membership functions are derived from the straight-line formula [4, 9], given as

$$\begin{aligned} \tilde{e} \cdot p & = \frac{{K_{P} e\left( {nT} \right) + L}}{2L}, \quad \tilde{e} \cdot n = \frac{{ - K_{P} e\left( {nT} \right) + L}}{2L} \hfill \\ \tilde{e}_{\text{r}} \cdot p & = \frac{{K_{I} e_{\text{r}} \left( {nT} \right) + L}}{2L},\quad \tilde{e}_{\text{r}} \cdot n = \frac{{ - K_{I} e_{\text{r}} \left( {nT} \right) + L}}{2L} \hfill \\ \end{aligned}$$

Now, substituting the above formulae and \(o \cdot pl = L, \,o \cdot ps = \frac{L}{2}, \,o \cdot nl = - L, \,o \cdot ns = - \frac{L}{2}\) in Eq. (15), the output \(\Delta u_{\text{sc}} \left( {nT} \right)\) in regions IC1 and IC2 is expressed as:

$$\Delta u_{\text{sc}} \left( {nT} \right) = \frac{L}{{4\left( {2L - K_{P} e\left( {nT} \right)} \right)}}\left[ {3K_{P} e\left( {nT} \right) + K_{I} e_{\text{r}} \left( {nT} \right)} \right]$$

In regions IC1 and IC2, \(\tilde{e}\left( {nT} \right) \ge 0\). Likewise, \(\Delta u_{\text{sc}}\) in regions IC5 and IC6 is expressed as:

$$\Delta u_{\text{sc}} \left( {nT} \right) = \frac{L}{{4\left( {2L + K_{P} e\left( {nT} \right)} \right)}}\left[ {3K_{P} e\left( {nT} \right) + K_{I} e_{\text{r}} \left( {nT} \right)} \right]$$

The value of \(\tilde{e}\left( {nT} \right) \le 0\) for regions IC5 and IC6. Therefore, combined formula for regions IC1, IC2, IC5 and IC6 is as follows:

$$\Delta u_{\text{sc}} \left( {nT} \right) = \frac{{L\left( {3K_{P} e\left( {nT} \right) + K_{I} e_{\text{r}} \left( {nT} \right)} \right)}}{{4\left( {2L - K_{P} \left| {e\left( {nT} \right)} \right|} \right)}}$$

Explicitly

$$\Delta u_{\text{sc}} \left( {nT} \right) = \left[ {\frac{{3LK_{P} }}{{4\left( {2L - K_{P} \left| {e\left( {nT} \right)} \right|} \right)}}} \right]e\left( {nT} \right) + \left[ {\frac{{LK_{I} }}{{4\left( {2L - K_{P} \left| {e\left( {nT} \right)} \right|} \right)}}} \right]e_{\text{r}} \left( {nT} \right)$$
(16)

The above formula of control action has structure similar to conventional FOPD controller with nonlinear gains, thereby providing self-tuning feature to the controller [12]. These variable gains play a vital role in improving the performance of controller. An increase in \(e\left( {nT} \right)\) decreases \(2L - K_{P} \left| {e\left( {nT} \right)} \right|\), which in turn increases the overall gains of ‘\(\Delta u_{\text{sc}}\)’ and vice versa. Thus, controller generates small corrective action when error diminishes but as soon as error increases, controller takes large corrective action. In the similar manner, control action of serial compensator for remaining IC regions is acquired and listed in Table 1. The brief summary of design procedure to acquire the control structure is outlined as follows:

Table 1 Formulae for \(\Delta u_{sc} \left( {nT} \right)\) in all regions of \({\text{IC}}\)
Full size table
  1. 1.

    Initially basic 2DOF fractional order PID controller is split into FOPI and FODF component in discrete domain.

  2. 2.

    Fuzzy expertise is incorporated in FOPI component (Eq. 10) by fuzzifying independent (input) variables \(K_{P} e\left( {nT} \right)\) and \(K_{I} e_{\text{r}} \left( {nT} \right)\) by simple triangular membership functions and the dependent (output) variable \(\Delta u_{\text{sc}} \left( {nT} \right)\) by singleton membership functions.

  3. 3.

    The rule base is framed to map input–output membership functions on the basis of system knowledge.

  4. 4.

    Fuzzified output \(\Delta u_{\text{sc}} \left( {nT} \right)\) is converted into crisp value using center-of-mass defuzzification algorithm. The defuzzified output is fractionally integrated and scaled by gain \(K_{u}\) to generate serial compensator output \(u_{\text{sc}} (nT)\) (Eq. 12).

  5. 5.

    The overall control action of proposed controller is obtained by algebraically adding the outputs of serial compensator and fractional order derivative filter given in Eq. (13).

Further, designed 2DOF FOFPI-D controller is transformed to its integer order equivalent (2DOF FPI-D) by considering the values of fractional order \(\lambda\) and \(\mu\) to be 1.

2.3 Implementation of fractional order operators

Over the last few decades, there has been a significant use of fractional calculus in the field of control system, filter design, system modeling and image processing [17, 25, 43, 44]. To implement fractional order operator, numerous methods are reported in the literature [45, 46]. In this work, fractional differentiator/integrator in discrete domain is designed by binomially [40] expanding backward difference transformation in operator \(s^{ \pm \alpha }\) as

$$s^{ \pm \alpha } = \left( {\frac{{1 - z^{ - 1} }}{T}} \right)^{ \pm \alpha }$$
(17)
$$s^{ \pm \alpha } = T^{ \mp \alpha } \mathop \sum \limits_{i = 0}^{\infty } \left( { - 1} \right)^{i} \frac{{\left( { \pm \,\alpha } \right)\left( { \pm \,\alpha - 1} \right)\left( { \pm \,\alpha - 2} \right) \cdots \left( { \pm \,\alpha - i + 1} \right)}}{i!}z^{ - i}$$
(18)

Denoting discrete-time differentiator and integrator operator with \(D\), thus

$$D^{ \pm \alpha } \left( {z^{ - 1} } \right) = T^{ \mp \alpha } \mathop \sum \limits_{i\, = \,0}^{\infty } \left( { - 1} \right)^{i} \left( {\begin{array}{*{20}c} { \pm \,\alpha } \\ i \\ \end{array} } \right)z^{ - i} \,{\text{where}}\, \left( {\begin{array}{*{20}c} { \pm \,\alpha } \\ i \\ \end{array} } \right) = \frac{{\left( { \pm \,\alpha } \right)\left( { \pm \,\alpha - 1} \right)\left( { \pm \,\alpha - 2} \right) \cdots \left( { \pm \,\alpha - i + 1} \right)}}{i!}$$

Therefore, differentiation and integration [41] of discrete-time arbitrary function \(x\left( {nT} \right)\), which is equivalent to well-known Grunwald–Letnikov [47], are represented as:

$$D^{ \pm \alpha } x\left( {nT} \right) = T^{ \mp \alpha } \mathop \sum \limits_{i\, = \,0}^{\infty } \left( { - \,1} \right)^{i} \left( {\begin{array}{*{20}c} { \pm \,\alpha } \\ i \\ \end{array} } \right)x\left( {\left( {n - i} \right)T} \right)$$
(19)

Realization of Eq. (19) requires computation of infinite number of coefficients and delay units. Therefore, to design practically feasible fractional order operator, principle of short memory is utilized as:

$$D^{ \pm \alpha } x\left( {nT} \right) = T^{ \mp \alpha } \mathop \sum \limits_{i\, = \,0}^{M} \left( { - \,1} \right)^{i} \left( {\begin{array}{*{20}c} { \pm \,\alpha } \\ i \\ \end{array} } \right)x\left( {\left( {n - i} \right)T} \right)$$
(20)

where \(\alpha\) is the order of operator, \(T\) is sampling time and \(M\) is number of delay elements to generate last few samples and its value is chosen as 100.

3 Multi-objective non-dominated sorting genetic algorithm-II

The metaheuristic algorithms are generally single objective in nature. Their primary aim is to discover best solution by optimizing the fitness function that lumps many objectives into one. These objectives are usually clashing and contradictory with each other, and many a times optimization of one objective, results into compromise of other objectives. Moreover, optimization by weighted sum of objectives provides a single optimum solution, and therefore, designer has no choice to trade objectives against each other according to the requirements. Another issue is to determine exact and accurate value of weights, which suits designer’s preferences, as slightest change in weights results in completely different solutions. These shortcomings may be overcome by a multi-objective algorithm, which provides a bunch of solutions instead of single best solution, thus giving the designer freedom to choose suitable final optimum solution.

NSGA-II is broadly used multi-objective optimization algorithm because of its simple population-based search approach, which is suitable for diverse fields of engineering. The Pareto-based methodology of NSGA-II has been widely utilized for contradictory multi-objective problems because of it is simple, direct and efficient non-dominant positioning technique, which results in various levels of Pareto fronts [48,49,50,51]. Further researchers have validated that NSGA-II has a superior sorting plan and consolidates elitism component as compared to NSGA-I [36]. In this work, the essential necessity is minimization of position error with least variation in control action. Therefore, objective functions [50] considered can be mathematically expressed as:

$$f_{1} = \mathop \sum \nolimits \left| {\theta_{{d_{1} }} \left( {nT} \right) - \theta_{1} \left( {nT} \right)} \right| + \mathop \sum \nolimits \left| {\theta_{{d_{2} }} \left( {nT} \right) - \theta_{2} \left( {nT} \right)} \right|$$
(21)
$$f_{2} = \mathop \sum \nolimits \left| {\tau_{1} \left( {nT + T} \right) - \tau_{1} \left( {nT} \right)} \right| + \mathop \sum \nolimits \left| {\tau_{2} \left( {nT + T} \right) - \tau_{2} \left( {nT} \right)} \right|$$
(22)

The objective functions attempt accurate position tracking with fairly small control effort. But these objectives are contradictory and at odds with each other (i.e., reducing variation in control effort results in increased tracking error and vice versa). Therefore, both objectives \(f_{1}\) and \(f_{2}\) need to be minimized simultaneously for efficient functioning of control loop. Steps for implementing NSGA-II algorithm are briefly outlined as:

  1. (a)

    Initialize consistently distributed parent population of size P on the basis of parent’s range.

  2. (b)

    Evaluate the objectives for individuals and sort the population based on non-domination.

  3. (c)

    Allot each result a rank equivalent to its non-domination level.

  4. (d)

    Use the standard binary tournament selection method.

  5. (e)

    Utilize the simulated binary crossover and polynomial mutation to make a posterity population of size P.

  6. (f)

    Merge the parent and posterity population to create broadened population of size 2P.

  7. (g)

    Sort the broadened population on the basis of non-domination.

  8. (h)

    Fill new population of size P with the entities from the sorting fronts beginning from the best.

  9. (i)

    The crowding-distance method is used to assure diversity, if a front can only partially fill the subsequent generation. The crowding-distance method keeps up variety in the population and avoids convergence to local optimum solution.

  10. (j)

    Repeat steps (b)–(i) until a stopping condition is encountered.

The parameters of 2DOF FOFPI-D, its integer order counterpart and 2DOF PI-D are optimized using NSGA-II. The designed controllers are tested on highly nonlinear two-link robotic manipulator for trajectory tracking, model uncertainty, disturbance and noise rejection.

4 Dynamics of two-link robot

Robotic manipulators are of keen interest to control engineers because of highly complex, nonlinear and coupled dynamics. The tight position tracking of end-effector for such complex system is a challenging task for the experts. Robotic manipulator is an integral part of numerous industries like process, automation, chemical, nuclear power plant, space application and medical sciences for the purpose of pick and place, precision surgeries, cutting, grinding, drilling and assembly. The mechanical model of robotic manipulator under consideration is shown in Fig. 4. It consists of two links having length \(l_{1}\) and \(l_{2}\) with their center of mass \(m_{1}\) and \(m_{2}\) lying at distal ends of links, respectively. DC motors are attached at points A and B to provide controlling torque, whereas encoders are used to estimate the angular position (\(\theta_{1} \,{\text{and}}\,\theta_{2}\)) and velocity (\(\dot{\theta }_{1} \,{\text{and}}\,\dot{\theta }_{2}\)) of the links. The descriptions of related variable of the manipulator with their nominal values are listed in Table 2. The dynamics of two-link robotic manipulator is described [52] below where parameters of link-1 and link-2 are symbolized by subscripts 1 and \(2\), respectively.

$$\begin{aligned} \tau_{1} & = m_{2} l_{2}^{2} (\ddot{\theta }_{1} + \ddot{\theta }_{2} ) + (m_{1} + m_{2} )l_{1}^{2} \ddot{\theta }_{1} + m_{2} l_{1} l_{2} \cos \theta_{2} (2\ddot{\theta }_{1} + \ddot{\theta }_{2} ) - m_{2} l_{1} l_{2} \sin \theta_{2} (\dot{\theta }_{2}^{2} ) - 2m_{2} l_{1} l_{2} \sin \theta_{2} \dot{\theta }_{1} \dot{\theta }_{2} \\ & \quad + \,(m_{1} + m_{2} )l_{1} g\cos \theta_{1} + m_{2} l_{2} g\cos (\theta_{1} + \theta_{2} ) \\ \end{aligned}$$
(23)
$$\tau_{2} = m_{2} l_{1} l_{2} \cos \theta_{2} \ddot{\theta }_{1} + m_{2} l_{2} g\cos (\theta_{1} + \theta_{2} ) + m_{2} l_{2}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) + m_{2} l_{1} l_{2} \sin \theta_{2} \dot{\theta }_{1}^{2}$$
(24)
Fig. 4
figure 4

Model of robotic manipulator

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Table 2 Description of variables with their nominal values [50]
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The control schemes derived in Sect. 2 are tested on the two-link robotic manipulator. The results obtained are discussed in the next section.

5 Results

This work presents an expert solution for nonlinear systems where multiple issues need to be addressed. A new 2DOF FOFPI-D controller is proposed for the purpose. The controller possesses the benefits of 2DOF structure along with flexibility in design and expert intelligence. The 2DOF FPI-D and 2DOF PI-D control schemes are also implemented for comparative study. The mathematical model of robotic manipulator discussed above is simulated in MATLAB to analyze the behavior of system. The simulations are performed on Intel®, CORE i3, 4 GB RAM, 1.70 GHz. personal computer using MATLAB software. Equation solver used for simulation is fourth-order Runge–Kutta with sampling time of \(\,T = 1\) ms. Controlling torques \(\tau_{1}\) and \(\tau_{2}\) are restricted within the range \(\left[ { - \,50, 50} \right]\) Nm because of actuator limitation. The closed-loop control structure for considered manipulator has dedicated controller for each link. The performance of designed controllers for two-link robotic manipulator is rigorously examined by considering issues such as trajectory tracking, model uncertainties, disturbance and noise. The control scheme for 2DOF FOFPI-D is shown in Fig. 5.

Fig. 5
figure 5

Block diagram of 2DOF FOFPI-D control scheme for 2-link robotic manipulator

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5.1 Trajectory tracking performance

Robotic manipulators are used in industrial environment to perform various tasks such as pick and place, grinding and cutting. Links of the manipulator must follow a desired trajectory to perform a particular task. In this work, a cubic polynomial function of time is considered as desired trajectory for links [50]

$$\theta_{{d_{i} }} \left( t \right) = a_{0i} + a_{1i} t + a_{2i} t^{2} + a_{3i} t^{3}$$
(25)

where \(\theta_{{d_{i} }} \left( t \right)\) is desired trajectory of link-1 and link-2 for i = 1 and 2, respectively. The boundary conditions for position of links are \(\theta_{{d_{1} }} = 1\) rad and \(\theta_{{d_{2} }} = 2\) rad at \(t = 2\) s; \(\theta_{{d_{1} }} = 0.5\) rad and \(\theta_{{d_{2} }} = 4\) rad at \(t = 4\) s. The corresponding angular velocities are \(\dot{\theta }_{{r_{i} }} = 0\) rad/s at \(t = 2\) and \(4\) s. The governing factors of NSGA-II for tuning the controllers are listed in Table 3. The search space for finding the optimum parameters of 2DOF FOFPI-D, its integer order counterpart and 2DOF PI-D is specified in Table 4. The value of filter coefficient \(N\) is considered to be ‘1000’ for all controllers. Obtained Pareto front with chosen optimum solution for designed controllers is shown in Fig. 6, and corresponding optimum parameters, fitness and integral absolute error (IAE) are recorded in Table 5. The optimum parameters of controllers remain same for entire simulation studies.

Table 3 NSGA-II parameters and their values for 2DOF FOFPI-D, 2DOF FPI-D and 2DOF PI-D controller
Full size table
Table 4 Search range of parameters for various controllers
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Fig. 6
figure 6

Pareto front for objectives \(f_{1}\) and \(f_{2}\) using NSGA-II (a) 2DOF FOFPI-D (b) 2DOF FPI-D (c) 2DOF PI-D with chosen solution marked red

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Table 5 Controller parameters, IAE and fitness functions of 2DOF FOFPI-D, 2DOF FPI-D and 2DOF PI-D controller for link-1 and link-2
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The trajectory tracking performance of 2DOF FOFPI-D and other controllers in normal operating condition is shown in Fig. 7a, and the corresponding tracking error, controller effort and path traced by end-effector in rectangular (XY) coordinates are shown in Fig. 7b–d, respectively. It is observed from Table 5 that there is a significant improvement in IAE for 2DOF FOFPI-D over other controllers. The smaller value of IAE is due to incorporation of fractional order operator in the design, which provides the extra degrees of freedom to the controller. Also, the gains of controller are nonlinear function of input signals, which provide self-tuning control capability. Thus, 2DOF FOFPI-D controller tracks the trajectory more accurately as compared to other controllers. Further robustness of controllers is also tested for parametric uncertainties, actuator disturbance due to cogging and sensor noise in feedback path.

Fig. 7
figure 7

Performance comparison of 2DOD FOFPI-D and other controllers: a reference trajectory tracking, b position error, c control effort, d path traced by end-effector in XY coordinates

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5.2 Model uncertainty

Mathematical modeling and identification of real-world robotic system are challenging issues as it is complex in nature and difficult to understand. Unavailability of exact mathematical model due to un-modeled dynamics, wear and tear and aging effect introduces uncertainty in the system. Therefore, an intelligent controller is required which is capable enough to handle such uncertainties and provide satisfactory results. In this work, uncertainty in mass and length of both links is introduced in the model to evaluate the performance of designed controllers. The uncertain dynamics of manipulator is modeled by modifying Eqs. (23) and (24).

$$\begin{aligned} \tau_{1} & = (m_{2} + \delta m_{2} )(l_{2} + \delta l_{2} )\left( {(l_{2} + \delta l_{2} )(\ddot{\theta }_{1} + \ddot{\theta }_{2} ) + g\cos (\theta_{1} + \theta_{2} )} \right) + (m_{1} + m_{2} + \delta m_{1} + \delta m_{2} )(l_{1} + \delta l_{1} )\left( {(l_{1} + \delta l_{1} )\ddot{\theta }_{1} + g\cos \theta_{1} } \right) \\ & \quad + \,(m_{2} + \delta m_{2} )(l_{1} + \delta l_{1} )(l_{2} + \delta l_{2} )\left( {\cos \theta_{2} (2\ddot{\theta }_{1} + \ddot{\theta }_{2} ) - \sin \theta_{2} (\dot{\theta }_{2}^{2} ) - 2\sin \theta_{2} \dot{\theta }_{1} \dot{\theta }_{2} } \right) \\ \end{aligned}$$
(26)
$$\begin{aligned} \tau_{2} & = (m_{2} + \delta m_{2} )(l_{1} + \delta l_{1} )(l_{2} + \delta l_{2} )\left( {\cos \theta_{2} \ddot{\theta }_{1} + \sin \theta_{2} \dot{\theta }_{1}^{2} } \right) + (m_{2} + \delta m_{2} )(l_{2} + \delta l_{2} )g\cos (\theta_{1} + \theta_{2} ) \\ & \quad + \,(m_{2} + \delta m_{2} )(l_{2} + \delta l_{2} )^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ \end{aligned}$$
(27)
$$\begin{aligned} \left| {\delta m_{1} } \right| & \le \frac{d}{100}\, * \,m_{1} \quad \left| {\delta m_{2} } \right| \le \frac{d}{100}\, * \,m_{2} \\ \left| {\delta l_{1} } \right| & \le \frac{d}{100}\, * \,l_{1 } \quad \left| {\delta l_{2} } \right| \le \frac{d}{100}\, * \,l_{2} \\ \end{aligned}$$

where \(\delta m_{i} \,{\text{and}}\,\delta l_{i}\) represents small change in mass and length, respectively, for \(i{\text{th}}\) link. \(d\) is the percentage uncertainty in system parameters.

Various cases of model uncertainty are considered by changing the value of ‘\(d\)’ from 5 to 30% with an interval of 5%. The designed controllers are employed to control the uncertain dynamics and the quantitative comparison is made on the basis of IAE value. The IAE of 2DOF FOFPI-D and other controllers for link-1 and link-2 are listed in Table 6, and the graphical representation is shown in Fig. 8. In all cases, IAE of 2DOF FOFPI-D controller for both links is smaller in contrast to other designed controllers, thereby proving its robustness against parametric uncertainty.

Table 6 IAE values of various control schemes for \(d\,\%\) uncertainty in \(m_{1} ,\,m_{2} ,\,l_{1} \,{\rm and}\,l_{2}\)
Full size table
Fig. 8
figure 8

Variation of IAE values for 2DOF FOFPI-D, 2DOF FPI-D and 2DOF PI-D with increasing % uncertainty in system parameters: a link 1 and b link 2

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5.3 Disturbance rejection

The unforeseen disturbances arising in real-world systems are major concern for engineers, as they divert the output from its actual value. In robotic systems, disturbance emerges due to cogging and eccentricity of actuator, which produce low-frequency sinusoidal torque [53]. Subsequently, there is a necessity of a proficient controller, which is capable of dismissing such disturbances so that system response precisely tracks the reference.

The competency of designed control schemes is tested by adding sinusoidal disturbance for complete time duration and is represented by \(d1\) and \(d2\) in Fig. 5. Different cases are considered by changing the amplitude of disturbance from 1 to 10 Nm, and IAE of link-1 and link-2 for disturbance dismissal is recorded in Table 7. Disturbance rejection response for \(10\text{Sin} 25t\) Nm is depicted in Fig. 9, which comprises of trajectory tracking, error, control effort and path traced by end-effector in XY coordinates. It is clear from tabular and graphical investigation that IAE values of links are considerably smaller for 2DOF FOFPI-D controller as compared to 2DOF FPI-D and 2DOF PI-D controller. Thus, disturbance rejection performance of 2DOF FOFPI-D is superior to other controllers.

Table 7 IAE values of various controllers for added sinusoidal disturbance in \(d1\) and \(d2\)
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Fig. 9
figure 9

Performance comparison of designed controllers: a trajectory tracking, b position error, c control effort, d movement of end-effector in rectangular coordinates by adding \(10\text{Sin} 25t\) Nm disturbance in \(d1\) and \(d2\)

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5.4 Noise suppression

Practically real-time systems are predominantly affected by random noise due to inherent shortcomings of sensors, which deteriorates the output. A controller must be expert enough to conquer the impact of noise created in the system. The performance of controllers is examined by adding random noise to position of links as depicted in Fig. 5. Random noise of given maximum amplitude is generated and added to the position value (\(\theta_{1}\) and \(\theta_{2}\)) in the simulation model. Different cases of random noise are considered by changing the maximum amplitude as given in Table 8. The IAE values of designed controllers for every case of noise are also recorded in Table 8. The simulation results of 2DOF FOFPI-D and other controllers for trajectory tracking, position error, controller effort and movement of end-effector in XY coordinates for added 0.01 rad amplitude noise are shown in Fig. 10. It is implied from the analysis that IAE values of 2DOF FOFPI-D controller are smaller as compared to 2DOF FPI-D and conventional 2DOF PI-D controller for all instances of noise. Hence, it is concluded that 2DOF FOFPI-D scheme suppresses the noise most efficiently in comparison with other controllers.

Table 8 IAE value of various controllers for added random noise
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Fig. 10
figure 10

Performance comparison of designed controllers: a trajectory tracking, b position error, c control effort, d tracking of end-effector in rectangular coordinates by adding \(0.01\) rad noise in links

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5.5 Simultaneous uncertainty, noise and disturbance rejection

The performance of controllers is investigated by simultaneously incorporating model uncertainties, sensor noise as well as disturbance in the system. The entire study is performed by taking 30% model uncertainty, sensor noise of \(0.01\) rad in links, disturbance of \(10\text{Sin} 25t\) Nm and desired trajectory for link-1 and link-2 as

$$\theta_{{d_{1} }} \left( t \right) = 5.3678 - 5.3678\,*\,\cos \left( {1.567t} \right)$$
(28)
$$\theta_{{d_{2} }} \left( t \right) = 3.8126\,*\,\cos \left( {2.138t} \right) - 3.8126$$
(29)

The trajectory tracking performance of designed controllers and their corresponding tracking error, torque and path traced by end-effector in rectangular coordinates are shown in Fig. 11. Further, the quantitative assessment of controllers on the basis of IAE is depicted in Fig. 12. It is found that IAE of 2DOF FPI-D and 2DOF PI-D for link-1 are approximately 1.26 and 3.68 times the IAE of 2DOF FOFPI-D controller, respectively. For link-2, they are 1.12 and 1.53 times the IAE of 2DOF FOFPI-D controller, respectively. Thus, IAE values of 2DOF FOFPI-D for both links are smallest among designed controllers. 2DOF FOFPI-D controller hence proves to be more robust and superior to its integer order counterpart and 2DOF PI-D controller.

Fig. 11
figure 11

Performance comparison of designed controllers for 30% parameter uncertainties, disturbance of \(10\text{Sin} 25t\) Nm and random noise of 0.01 rad to links: a trajectory tracking, b position error, c control effort, d movement of end-effector in rectangular coordinate

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Fig. 12
figure 12

Quantitative comparison of various controllers for link-1 and link-2 by incorporating 30% parameter uncertainties, disturbance of \(10\text{Sin} 25t\) Nm and random noise of amplitude 0.01 rad to links

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6 Discussion

In real-time robotic manipulator, the actuator may experience disturbance due to locking of the rotor at specific points. These disturbances are sinusoidal in nature and arise due to the phenomenon of cogging and eccentricity [53]. Also high-frequency noise in the feedback sensor degrades the performance of system. Inherent shortcomings, design and aging effect are the main cause of sensor noise. Another issue with real-time system is frequent change in reference and uncertain dynamics. To evaluate the capability of designed controllers in dealing with such adverse conditions, a real-time environment is virtually created in MATLAB.

The designed controllers are implemented in discrete-time domain; therefore, sampling time ‘\(T\)’ becomes an important parameter. The value of \(T\) needs to be selected in such a way that enough number of samples are obtained for analysis. If sampling time is too large, the signal information is lost and retrieved response is distorted. If sampling time is too small, large number of redundant samples will be generated which increases simulation time [32]. Thus, sampling time directly affects the system performance and it should be selected appropriately. The value of sampling time ‘\(T\)’ is taken as 0.001 s. Further total time taken by NSGA-II algorithm to generate Pareto front for 200th generation in case of 2DOF FOFPI-D is 22,167.9 s, for 2DOF FPI-D is 20,143.2 s, whereas for 2DOF PI-D is 16,884.3 s. The time taken to tune 2DOF FOFPI-D is larger because of additional design variables due to fractional order operators. This increases the computational complexity during optimization of controller parameters. The time taken to optimize also depends on the sampling time. However, the complexity in controller does not affect the time taken to obtain the controlled variable. All the designed controllers take almost same time to generate the control signal.

The control objectives for robotic manipulator are precise position tracking with minimum variation in control signal. A 2DOF PI-D controller has the capability to handle several requirements, which makes it suitable for the task at hand. The PI compensator and derivative filter components of the controller are tuned compatibly to achieve the desired objectives. Thus, IAE values are quite low for links 1 and 2, i.e., \(88.25\,*\,10^{ - 4}\) and \(33.33\,*\,10^{ - 4}\), respectively. Further the controller performance remains consistent under model uncertainties, but disturbance and noise degrade the controller performance. So to improve the performance of 2DOF PI-D controller, fuzzy expertise is embedded in PI compensator, which leads to 2DOF FPI-D. The designed controller further reduces the values of IAE as compared to 2DOF PI-D for tracking response under the influence of disturbance and noise. The controller provides good performance for introduction of parametric uncertainty with 30% as maximum change in the system parameters. This is due to the reason that formula-based fuzzy design scheme is not model specific, possesses self-tuning ability and requires only experience-based knowledge about operation of the system. The performance of controller is further enhanced by providing flexibility in design by inclusion of fractional order operator, which leads to 2DOF FOFPI-D control strategy. The results acquired for tracking control, uncertain parameters, disturbance rejection and noise suppression have lowest value of IAE, and accurate tracking is achieved. The only limitation of proposed controller is increased number of design variables due to incorporation of fractional order operators. The tuning of increased variables becomes a combinatorial problem; however, this issue may be overcome by optimization of controller parameters.

7 Concluding remarks

In this paper, a generic 2DOF FOFPI-D for control of nonlinear systems is presented. The proposed controller is a combination of serial compensator (FOFPI) and fractional order derivative filter (FODF), which is derived from traditional 2DOF FOPID. Analytical formulae for control law and implementation of fuzzy logic are derived to support the structure of controller. 2DOF FPI-D and 2DOF PI-D controllers are also designed for comparative study. A coupled, nonlinear two-link robotic manipulator is considered to evaluate the performance of designed controllers. The manipulator model replicates the real operating environment of system by considering multiple issues such as parametric uncertainty, presence of disturbance due to cogging and noise generated by sensors. The objective of the controller is to provide accurate tracking in the presence of noise and disturbance with constraints that variation in control signal is minimal. It is revealed from the results that 2DOF FOFPI-D controller efficiently handles multiple issues, due to increased degrees of freedom, fuzzy logic expertise and flexibility of fractional operator in comparison with other designed controllers. Future work may be focused toward establishing the analytical stability conditions and its real-time implementation [2, 31]. Further applicability of designed controllers to complex systems with essential nonlinearities, particularly servo system, fractional order dynamic systems, and chaotic system may also be analyzed.