Sage Journals: Discover world-class research

Abstract

Design of centralised disturbance rejection controllers for an highly interacting MIMO quadruple tank process is tedious. Recently, centralised disturbance rejection Fractional Order LQI (FOLQI) controller is designed for such system to meet the desired specifications with better performance than various disturbance rejection controllers that are available in the literature. In this paper, an optimisation problem is formulated to obtain the optimal parameters of FOLQI controller providing minimum control effort by applying various widely used heuristic methods like Cuckoo Search (CS), Accelerated Particle Swarm Optimisation (APSO) and FireFly (FF) algorithms. A detailed simulation study is carried out to compare the performance of the FOLQI and Integer Order LQI (IOLQI) controllers obtained by these heuristic algorithms under disturbance and parameter uncertainty conditions. From the simulation study it is inferred that (i) FOLQI controller provides better time domain specifications % M_p, t_s and J in comparison to IOLQI controllers and (ii) FF tuned FOLQI and IOLQI controllers provide better robustness characteristics compared to CS and APSO tuned controllers.

Keywords

Quadruple tank process minimum and non-minimum phase FOLQI cuckoo search accelerated particle swarm optimisation firefly

Introduction

The control system design plays a major role for an highly interacting non-linear Multi Input Multi Output (MIMO) system of various engineering domains like (i) in mechanical: fuzzy PID controller provides better set point tracking of twin-rotor system compared to conventional PID controller,¹ (ii) in electrical: optimal Linear Quadratic Gaussian (LQG) controller for AC/HVDC system to provide accurate state estimation and better reference signal tracking,² (iii) in signal processing: discrete time state space methodology for digital modelling and design of optimal PID controller using Linear Quadratic Regulator (LQR) for system with multiple time delay,³ (iv) in aerospace: Hammerstein model based Model Predictive Control (MPC) is proposed for UAV to provide better real time target tracking under external disturbances and robustness characteristics, compared to conventional PID controller⁴ and (v) in process control: developing decentralised PI controller for quadruple tank process.⁵ In specific, controller design for MIMO system with non-minimum phase operating mode is a challenging task as it impose limitations for control bandwidth of linear feedback design.^5–7 The complexity due to interaction effect is analysed by various methods along with decoupling algorithms proposed in Liu et al.⁸ and Chen et al.⁹ providing classification of different algorithms based on their characteristics/application domains and decisions about selecting centralised or decentralised type of controllers. Many researchers presented QTP as a laboratory bench mark system which exhibits the characteristics of an highly interacting MIMO system operating in both minimum and non-minimum phase operating conditions.^5,10,11

For QTP, different conventional control schemes are presented by various researchers and some of them are: (i) control methodologies like decentralised PI, multivariable Internal Model Controller (IMC) and $μ$ -analysis based $H_{\infty}$ controllers are designed. These controllers performance are analysed by robustness measures for stability and performance under tracking/disturbance rejection conditions. For the above cases the IMC and $H_{\infty}$ controllers provide similar and better performances than decentralised PI controller,⁷ (ii) implementation of decentralised PI controller by analysing the severity of the interaction using relative gain array principle and the performance comparison of the step response plots show that non-minimum phase condition has nearly 10 times lower band width than minimum phase operating condition,¹² (iii) tuning of decentralised PI controller using direct Nyquist array to shape the Gershgorin bands meeting the required frequency domain specifications which outperforms PI controller tuned using ZN method,¹³ (iv) LMI and inverse dynamics approach based tuning of robust decentralised PID controller provide better step response tracking performance for minimum and non-minimum phase systems respectively¹⁴ and (v) design of robust optimal decentralised PI controller meeting the desired gain margin, phase margin and overshoot constraints outperforms conventional decentralised PI controller.¹⁵

On the other hand, few authors developed advanced control schemes for QTP and some of them are: (i) a nonlinear zero dynamic attack based on the Byrness-Isidori normal form representation along with Lyapunov analysis is presented for QTP working in non-minimum phase which ensures stealthy to the proposed attack until some of the tank became overflow or empty in the presence of uncertainty and small parameter variations condition,¹⁶ (ii) the robust $H_{\infty}$ observer based controller is designed for Takagi-Sugeno fuzzy systems with time varying delays subjected to parameter uncertainties and external disturbances. Lyapunov Krasovskii function ensures the asymptotic stability of the proposed controller,¹⁷ (iii) disturbance observer based integral backstepping controller is designed for two tank system with external disturbances and found better performance in terms of avoiding chattering, steady state error and disturbance suppression characteristics compared to sliding mode controller,¹⁸ (iv) pair of disturbance/uncertainty rejection control law are designed and compared for system with disturbance and parameter uncertainty conditions for effective set point tracking problem,¹⁹ (v) arbitrary single-variable synthesis based optimal controller design with minimum ITSE/ITAE under disturbance conditions providing better performance than conventional controllers,²⁰ (vi) design of adaptive inverse evolutionary neural controller outperforming conventional PID controller in terms of error criteria,²¹ (vii) design of linear active disturbance rejection controller provides better set point tracking and disturbance rejection condition than conventional active disturbance rejection controller²² and (viii) design of robust sliding mode adaptive controller outperforms continuous time adaptive pole placement controller in terms of set point tracking, disturbance rejection and parameter uncertainty.²³

Recently, Fractional Order (FO) calculus is employed in modelling and controller design for industrial applications. Few authors designed FO controllers for different class of systems and are as follows:

(i) FO-PID controller for magnetic levitation system providing better performance in terms of IAE, ISE, ITAE and control effort compared to integer order PID controller,²⁴ (ii) FO-PI/FO-PD controllers providing better suppression of limit cycle over conventional PID controller for servo plant with separable non-linearity,^25–27 (iii) unified controller parameter expressions of FO controllers are derived for universal plant having complex coefficient plus fractional complex order derivative with dead time to meet desired frequency domain specifications,²⁸ (iv) FO adaptive controller gives better performance in terms of control effort than classical controllers for variable time delay process system,²⁹ (v) FO-PI along with conventional feedforward controller design using classical frequency domain approach for level control system provide better performance than conventional controllers along with feedforward controller in terms of settling time, overshoot and level tracking,³⁰ (vi) FO-Sliding Mode Controller (SMC) provides better performance characteristics like finite time convergence and reduced chattering effect in the presence of parameter uncertainties compared to conventional SMC³¹ and (vii) dual mode adaptive FO-PI with feedforward controller resulting better performance than decentralised PI, Quantitative Feedback Therory (QFT) and SMC controller in terms of settling time, overshoot and ISE.¹⁰

On the other hand, controller parameters are tuned using various meta-heuristic algorithms by formulating objective function with constraints. Some of them are: (i) building energy optimisation using grey wolf and butterfly optimisation algorithms reducing the annual energy consumption of an office building in Seattle weather conditions compared to PSO in terms of number of building simulations required and convergence rate,^32,33 (ii) a novel hybrid forecasting model based on long short-term memory neural network and empirical wavelet transform decomposition along with CS algorithm are developed for digital currency time series to minimise the negative situation by increasing the forecasting achievement. These algorithms are tested by estimating digital currencies such as BTC, XRP, DASH and LTC for its improved performances,³⁴ (iii) a novel hybrid wind speed forecasting model is developed based on long short-term memory neural network decomposition method and grey-wolf optimiser. The proposed combined model has the capability to capture the non-linear characteristics of wind speed time series by achieving accurate forecasting performance than single forecasting models,³⁵ (iv) PSO and Harris Hawks Optimisation (HHO) algorithms based PID controller parameters design for attitude and altitude control of the quadrator in different geometrical paths are studied. It is observed that HHO based controller provide better response in terms of simplicity, flexibility and ability to search randomly by avoiding local optima,³⁶ (v) hybrid HHO-GWO based optimal path planning and tracking algorithms are employed to carry out the payload hold-release mission of UAV by avoiding obstacles which outperforms controller tuned using PSO and GWO,³⁷ (vi) various meta-heuristic approaches to tune optimal PID controller and their application/limitations are discussed in Joseph et al.,³⁸ (vii) PID controller parameters obtained using improved CS algorithm gives better results in terms of peak time, overshoot and settling time compared to conventional CS and Particle Swarm Optimisation (PSO),³⁹ (viii) PID controller design for automatic voltage regulator system using CS gives better performance than PSO and Artificial Bee Colony (ABC) algorithms in terms of overshoot, settling time and steady state error,⁴⁰ (ix) combination of PSO and CS algorithms used to design PID controller for quadrotor system providing better efficiency in comparison to conventional CS and classical reference model in terms of ISE error,⁴¹ (x) tuning of fuzzy controller’s membership function using FF algorithm for autonomous mobile robot for better actuator function⁴² and (xii) LQI controller with optimal Q and R matrices tuned using GA⁴³ and bat algorithms⁴⁴ provide better result in comparison to conventional LQI controllers in both transient and steady state responses.

For QTP, few authors have presented various optimal controllers and are as follows: (i) PID controller tuned using GA provides better performance compared to PI/PID controllers tuned using IMC, PSO and Bacterial Foraging Optimisation (BFO) algorithms,⁴⁵ (ii) robust optimal decentralised PI controller based on non-linear optimisation provides improved bandwidth for specified stability margins and robustness against parameter uncertainty,⁴⁶ (iii) the robust $H_{\infty}$ observer based controller providing better stability performance under parameter uncertainty and disturbances,¹⁷ (iv) linear quadratic regulator controller tuning under disturbance gives better performance than other optimal controllers,⁴⁷ (v) linear quadratic Gaussian controller augmented with integrator provides better disturbance rejection capability than $H_{\infty}$ , loop-shaping, feedback linearisation and model predictive control for non-minimum phase operating condition¹¹ and (vi) design of centralised fractional order LQI controller tuned using sequential quadratic programming optimisation approach under load disturbance and parameter uncertainty condition provides better responses in terms of (a) settling time than IOLQI controller, (b) steady state error than LADR controller and (c) control effort than both IOLQI and LADR controllers.⁴⁸ It is observed that this method provides limited robustness characteristics in the presence of load disturbance and parameter uncertainty. This motivate us to optimise FOLQI controller parameters using heuristic methods for QTP under minimum and non-minimum phase operating modes in the presence of large uncertainty/disturbance conditions. The contributions of the paper are summarised as follows:

An optimal tuning of FOLQI controller parameters using CS, APSO and FF algorithms is employed for QTP under disturbance conditions.

These heuristic algorithms provide an optimal FOLQI controller parameters by solving the inequality constrained optimisation problem to minimise the control effort along with time domain constraints such as overshoot, settling time and steady state error.

Simulations are carried out for minimum and non-minimum phase operating modes of QTP under disturbance and parameter uncertainty conditions. The performance of FOLQI controller obtained by using various heuristic algorithms are compared and also to show the superiority of FOLQI controller, the results are compared with optimally tuned IOLQI controller using heuristic algorithms.

This paper is organised as follows: Section 2 show the physical construction, parameter descriptions and mathematical model of QTP. Section 3 explains the basics of LQI. The proposed optimisation problem with time domain constraints is shown in Section 4. Section 5 presents CS, APSO and FF heuristic algorithms used in this paper. Section 6 explains briefly about the simulation procedure and discuss the results in detail. Finally, Section 7 concludes the paper along with future research directions.

Description of QTP

Physical construction

The physical construction of the QTP is shown in Figure 1 which consists of four tanks with orifice, reservoir, two pumps, two directional control valves and four level sensors. Pump $P_{1}$ directs the inlet flow to tank $T_{1}$ and $T_{4}$ through directional control valve $V_{1}$ while pump $P_{2}$ directs the inlet flow to Tank $T_{2}$ and $T_{3}$ through directional control valve $V_{2}$ . The flow from the pumps $P_{1}$ and $P_{2}$ are controlled by their corresponding input voltages $u_{1}$ and $u_{2}$ respectively. Each tank is provided with an orifice which leads the flow to (i) reservoir from tank $T_{1}$ and $T_{2}$ , (ii) tank $T_{1}$ from tank $T_{3}$ and (iii) tank $T_{2}$ from tank $T_{4}$ . The level of the liquid in the tanks are measured by Level Transmitters (LT). The symbols of QTP and its representation are shown in Table 1.

Figure 1.

MIMO process configuration of QTP.

Table 1.

Symbols of QTP and its representation.

Symbols	Representation
$A_{i},$ $i = 1$ –4	Cross sectional area of tank $T_{i}$ $(c m^{2})$
$a_{i},$ $i = 1$ –4	Cross sectional area of orifice $O_{i}$ $(c m^{2})$
$x_{i},$ $i = 1$ –4	Level of water in tank $T_{i}$ $(cm)$
$L T_{i},$ $i = 1$ –4	Level Transmitter in tank $T_{i}$
$u_{i},$ $i = 1, 2$	Pump $P_{i}$ input voltage $(volts)$
$p_{i},$ $i = 1, 2$	Pump constant for $P_{i}$
$σ_{i},$ $i = 1, 2$	Flow parameter between two tanks
$g$	Gravitational constant $(cm / se c^{2})$

Mathematical modelling

The non-linear differential equations governing the mass balance for QTP¹⁸ are described as:

{\overset{\cdot}{x}}_{1} = - \frac{a_{1}}{A_{1}} \sqrt{2 g x_{1}} + \frac{a_{3}}{A_{1}} \sqrt{2 g x_{3}} + σ_{1} \frac{p_{1}}{A 1} u_{1}

(1)

{\overset{\cdot}{x}}_{2} = - \frac{a_{2}}{A_{2}} \sqrt{2 g x_{2}} + \frac{a_{4}}{A_{2}} \sqrt{2 g x_{4}} + σ_{2} \frac{p_{2}}{A 2} u_{2}

(2)

{\overset{\cdot}{x}}_{3} = - \frac{a_{3}}{A_{3}} \sqrt{2 g x_{3}} + (1 - σ_{2}) \frac{p_{2}}{A 3} u_{2}

(3)

{\overset{\cdot}{x}}_{4} = - \frac{a_{4}}{A_{4}} \sqrt{2 g x_{4}} + (1 - σ_{1}) \frac{p_{1}}{A 4} u_{1}

(4)

The equilibrium inputs $u_{1 o}$ and $u_{2 o}$ for (1)–(4) of QTP provide the following equilibrium points:

x_{1 o} = (\frac{1}{2 g}) {(\frac{(1 - σ_{2}) p_{1} u_{2 o} + σ_{1} p_{1} u_{1 o}}{a_{1}})}^{2}

(5)

x_{2 o} = (\frac{1}{2 g}) {(\frac{(1 - σ_{1}) p_{1} u_{1 o} + σ_{2} p_{2} u_{2 o}}{a_{2}})}^{2}

(6)

x_{3 o} = (\frac{1}{2 g}) {(\frac{(1 - σ_{2}) p_{2} u_{2 o}}{a_{3}})}^{2}

(7)

x_{4 o} = (\frac{1}{2 g}) {(\frac{(1 - σ_{1}) p_{1} u_{1 o}}{a 4})}^{2}

(8)

Taylor series expansion is used to linearise the differential equations (1)–(4) around the equilibrium points (5)–(8). The linearised state space representation of QTP is given as:

\begin{matrix} [\begin{matrix} Δ {\overset{\cdot}{x}}_{1} \\ Δ {\overset{\cdot}{x}}_{2} \\ Δ {\overset{\cdot}{x}}_{3} \\ Δ {\overset{\cdot}{x}}_{4} \end{matrix}] = [\begin{matrix} - \frac{1}{T 1} & 0 & \frac{A_{3}}{A_{1} . T_{3}} & 0 \\ 0 & - \frac{1}{T_{2}} & 0 & \frac{A_{4}}{A_{2} . T_{4}} \\ 0 & 0 & - \frac{1}{T_{3}} & 0 \\ 0 & 0 & 0 & - \frac{1}{T_{4}} \end{matrix}] [\begin{matrix} Δ x_{1} \\ Δ x_{2} \\ Δ x_{3} \\ Δ x_{4} \end{matrix}] \\ \begin{matrix} + \end{matrix} [\begin{matrix} \frac{σ_{1} . p_{1}}{A_{1}} & 0 \\ 0 & \frac{σ_{2} . p_{2}}{A_{2}} \\ 0 & \frac{(1 - σ_{2}) . p_{2}}{A_{3}} \\ \frac{(1 - σ_{1}) . p_{1}}{A_{4}} & 0 \end{matrix}] [\begin{matrix} Δ u_{1} \\ Δ u_{2} \end{matrix}] \end{matrix}

(9)

\begin{matrix} Δ y \end{matrix} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] [\begin{matrix} Δ x_{1} \\ Δ x_{2} \\ Δ x_{3} \\ Δ x_{4} \end{matrix}]

(10)

where, $T_{i} = \frac{A_{i}}{a_{i}} * \sqrt{\frac{2 x_{io}}{g}},$ and $i = 1$ to 4.

Introduction to LQI

A preliminaries about LQI³² is presented in this section. In general, an optimal LQI controller is tuned by applying weightages on either or both control signal $u (t)$ and states $x (t)$ for the given state space equation¹¹ by minimising the cost function.¹²

State space equation:

\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + Bu (t) \\ y (t) = Cx (t) + Du (t) \end{matrix}}

(11)

Cost function:

\int_{0}^{\infty} (x^{T} (t) Qx (t) + u^{T} (t) Ru (t)) dt

(12)

where, the matrices $Q$ and $R$ are positive definite and symmetric. Under the assumption of (11) is completely observable and controllable, the optimal control law is given as

u (t) = \frac{K_{i}}{s} e (t) - Kx (t)

(13)

where, $K = R^{- 1} B^{T} P$ and $K_{i}$ is the integral gain. The matrix $P$ is obtained by solving the Riccati equation $A^{T} P + PA - PB R^{- 1} B^{T} P + Q = 0$ .

Optimisation problem

The closed loop schematic representation of QTP with FOLQI controller tuned under continuous load disturbance is shown in Figure 2. The FOLQI controller parameters for QTP at given operating conditions are tuned by formulating an optimisation problem to meet the desired specifications with minimum control effort under continuous load disturbance $d (t)$ . The proposed problem is solved by recently developed CS, APSO and FF heuristic algorithms.

Figure 2.

Closed loop configuration of QTP with FOLQI controller.

Cost function:

\begin{matrix} \underset{(Controller parameters, Q, R)}{Minimize} J = \int_{0}^{T} Δ u^{2} (t) dt \end{matrix}

(14)

Subject to inequality constraints:

Overshoot / undershoot $(% M_{p})$

Settling time $(t_{s})$

Steady state error $(e_{ss})$

where, $Δ u (t)$ is the required change in controller output and it is expressed as

\begin{matrix} [Δ u (t)]_{m * 1} = - [K]_{m * n} [Δ y (t)]_{n * 1} \\ + {[\begin{matrix} \frac{1}{s^{α_{1}}} & 0 & . & . & 0 \\ 0 & \frac{1}{s^{α_{2}}} & . & . & 0 \\ . & . & . & . & 0 \\ . & . & . & . & . \\ 0 & 0 & 0 & 0 & \frac{1}{s^{α_{m}}} \end{matrix}]}_{m * m} [K_{i}]_{m * m} [Δ e (t)]_{m * 1} \end{matrix}

(15)

where, $n$ and $m$ are the number of states and number of control inputs of the system respectively.

Heuristic algorithms

In this section, nature inspired meta-heuristic based algorithms such as CS, APSO and FF used for solving the proposed constraint optimisation problem given in Section 4 are discussed.

CS algorithm

CS algorithm introduced by Yang and Slowik⁴⁹ in 2009 is based on obligate brood parasitic habitual of cuckoo birds and levy flight habitual of some birds/fruit flies. CS algorithm mimics the egg laying behaviour of cuckoo bird in other host birds nest. The egg which successfully gets hatched is considered to be best and travels to next generation. The technique that cuckoo birds use to search for nest and spawn is mathematically represented by levy flight mechanism. The host bird finds the egg and if it is not his own, it throws out the egg or moves to some other nest by leaving the cuckoo egg which fails to results in hatching. The three stages of CS algorithm are as follows:

Cuckoo lays egg only in one randomly selected nest at an instant.

Best nest with high quality of egg is considered to be success for moving to the next generation. The best egg/solution in the current generation is replaced by the best egg/solution in the next generation.

The number of host nest (n) is fixed and the probability of finding alien egg $(p_{a})$ by host bird is assumed to be $0.25$ .

The generated solution of the levy flight formulation is given as:

x_{i} (t + 1) = x_{i} (t) + μ \oplus + Levy (λ)

(16)

where, ( $μ$ , ⊕ and $λ$ ) represents step size, entry wise multiplication and Levy distribution parameter respectively.

The pseudocode of CS algorithm is given as follows:

Algorithm 1: - CS algorithm
Require: (i) Objective function: $J$ , (ii) Inequality constraints: $% M_{p}$ , $t_{s}$ and $e_{ss}$ , (iii) $Ma x_{iterations}$ (T) and (iv) $Probability factor (P_{a})$ 1: Initial population generation of n host nests $x_{i}, (i = 1, 2, . . ., n)$ 2: while t ≤ $T$ do 3: Get a cuckoo randomly by Levy flights 4: Evaluate its objective function $f (x_{i} (t))$ along with inequality constraints 5: Choose a nest among n (say, j) randomly 6: if $F_{i}$ > $F_{j}$ then 7: replace j by the new solution 8: end if loop 9: A fraction (pa) of worse nests are abandoned and new ones are built 10: Keep the best solutions 11: Rank the solutions and find the current best 12: t → t + 1 13: end while loop 14: Post process results with best solution

Algorithm 1: - CS algorithm

Require: (i) Objective function:

J

, (ii) Inequality constraints:

% M_{p}

t_{s}

and

e_{ss}

, (iii)

Ma x_{iterations}

(T) and (iv)

Probability factor (P_{a})

1: Initial population generation of n host nests

x_{i}, (i = 1, 2, . . ., n)

2: while t ≤

T

do
3: Get a cuckoo randomly by Levy flights
4: Evaluate its objective function

f (x_{i} (t))

along with inequality constraints
5: Choose a nest among n (say, j) randomly
6: if

F_{i}

F_{j}

then
7: replace j by the new solution
8: end if loop
9: A fraction (pa) of worse nests are abandoned and new ones are built
10: Keep the best solutions
11: Rank the solutions and find the current best
12: t → t + 1
13: end while loop
14: Post process results with best solution

APSO algorithm

PSO algorithm is proposed by Kennedy and Eberhart in 1995.⁵⁰ This algorithm is inspired by the swarm behaviour of fishes and birds that follow while in search of food. Each bird is referred as particle in search space and it changes its flying characteristics following the particle which is nearer to its prey. APSO differs its function from PSO by avoiding calculation of individual best by introducing randomness in the initial guess which results in increased accuracy and fast convergence.⁵¹

In this algorithm, velocity vector is computed during each iteration and the corresponding position is updated. The updated velocity and position vectors $v_{i} (t + 1)$ and $x_{i} (t + 1)$ are given as:

v_{i (t + 1)} = v_{i (t)} + λ (g^{*} - x_{i} (t)) + μ \in_{t}

(17)

x_{i (t + 1)} = x_{i (t)} + v_{i (t + 1)}

(18)

where, $λ$ and $μ$ are the acceleration constants, $g^{*}$ is the global best value, $x_{i}$ represents position of $i^{th}$ particle at time ’t’ and $\in_{t}$ is drawn from N[0,1].

The pseudocode of APSO algorithm is given as follows:

Algorithm 2: - APSO algorithm
Require: (i) Objective function: $J$ , (ii) Inequality constraints: $% M_{p}$ , $t_{s}$ and $e_{ss}$ and (iii) $Ma x_{iterations}$ (T) 1: Generate an initial population of N particles $x_{i} (i = 1, 2 . . ., N)$ 2: while (t ≤ T) do 3: for i = 1 to N do 4: Update velocity vector 5: Update position vector 6: Evaluate objective functions at new locations $x_{i}^{t + 1}$ along with constraints 7: Find current best 8: end for loop 9: Find global best 10: t → t + 1 11: end while loop 12: Post process results with best solution

Algorithm 2: - APSO algorithm

Require: (i) Objective function:

J

, (ii) Inequality constraints:

% M_{p}

t_{s}

and

e_{ss}

and (iii)

Ma x_{iterations}

(T)
1: Generate an initial population of N particles

x_{i} (i = 1, 2 . . ., N)

2: while (t ≤ T) do
3: for i = 1 to N do
4: Update velocity vector
5: Update position vector
6: Evaluate objective functions at new locations

x_{i}^{t + 1}

along with constraints
7: Find current best
8: end for loop
9: Find global best
10: t → t + 1
11: end while loop
12: Post process results with best solution

FF algorithm

FF algorithm is proposed by Yang and Slowik in 2008.⁵² This algorithm is inspired by flashing behaviour of fireflies belonging to a particular species. The behavioural assumptions of fireflies are as follows:

Fireflies are unisex and attracted towards each other based on the intensity of light it produces.

Less intensity fireflies are attracted towards high intensity fireflies.

Intensity of the light increases when fireflies comes closer.

If both the fireflies have same brightness, they move randomly without attracting each other.

The relative movement occurring between less bright firefly and more bright firefly is represented as follows:

x_{i (t + 1)} = x_{i (t)} + β_{0} e^{- γ r_{ij}^{2}} (x_{j} (t) - x_{i} (t)) + μ_{t} \in_{i} (t)

(19)

where, $β_{0}$ is the brightness of the firefly, $γ$ indicates light absorption co-efficient, r represents Euclidean distance between fireflies and it is given as: $r_{ij}$ = $\sqrt{{(x_{i} - x_{j})}^{2}}$ , $μ (t)$ is the random parameter and $\in_{i} (t)$ is a vector of random numbers which belongs to Gaussian or random distribution at time t.

The pseudocode of FF algorithm is given as follows:

Algorithm 3: - FF algorithm
Require: (i) Objective function: $J$ , (ii) Inequality constraints: $% M_{p}$ , $t_{s}$ and $e_{ss}$ and (iii) $Ma x_{iterations}$ (T) 1: State Generate initial fireflies population $X_{i} (i = 1, 2, . . ., n)$ 2: Determine $I_{i}$ at $X_{i}$ using f $(X_{i})$ 3: while t < $Ma x_{iteratios}$ 4: for i = 1:n all n fireflies 5: for j = 1:n all n fireflies 6: if $(I_{j} > I_{i})$ , Move firefly i towards j in d-dimension 7: end if 8: Evaluate new solutions and update light intensity 9: end for j loop 10: end for i loop 11: Rank the fireflies and find the best 12: end while loop 13: post process results and visualisation

Algorithm 3: - FF algorithm

Require: (i) Objective function:

J

, (ii) Inequality constraints:

% M_{p}

t_{s}

and

e_{ss}

and (iii)

Ma x_{iterations}

(T)
1: State Generate initial fireflies population

X_{i} (i = 1, 2, . . ., n)

2: Determine

I_{i}

X_{i}

using f

(X_{i})

3: while t <

Ma x_{iteratios}

4: for i = 1:n all n fireflies
5: for j = 1:n all n fireflies
6: if

(I_{j} > I_{i})

, Move firefly i towards j in d-dimension
7: end if
8: Evaluate new solutions and update light intensity
9: end for j loop
10: end for i loop
11: Rank the fireflies and find the best
12: end while loop
13: post process results and visualisation

Results and discussion

In this section, the heuristic algorithms CS, APSO and FF are used to tune the FOLQI controller parameters and the results are compared with IOLQI controller. To realise the FOLQI controller, the fractionality in the integrator is approximated to an integer order using oustaloup approximation of order five for the range of frequencies [0.001, 1000] $rad / \sec$ .⁵³ The closed loop simulation is performed in the presence of continuous load disturbance $d (t) = 1 cm$ introduced at $t = 50^{th} \sec$ . The parameters and equilibrium points of QTP for both minimum and non-minimum phase operating conditions used in simulation are shown in Table 2.

Table 2.

Parameters and equilibrium points of QTP.

Parameters	Non-minimum phase	Minimum phase
$A_{i},$ $i = 1$ –4	176.71	176.71
$a_{i},$ $i = 1, 2$	2.54	2.54
$a_{i},$ $i = 3, 4$	1.43	1.43
$h_{i} eq,$ $i = 1$ –4	20.20, 13.33, 22.94, 15.29	16.47, 16.71, 8.69, 8.09
$p_{i},$ $i = 1, 2$	186.86	180.29
$σ_{i},$ $i = 1, 2$	0.45, 0.35	0.60, 0.60
g	981	981

The FOLQI controller parameters are tuned to suppress the continuous load disturbance with minimum controller effort in addition to meet the desired constraints: less than 10% in $% M_{p}$ , less than $70$ $\sec$ in $t_{s}$ (tolerance band = ± $3 %$ ) and less than 1% in $e_{ss}$ . The bounds for the controller parameters are: $Q_{ii}$ ( $i = 1$ to $4$ ) ∈ $[0.01, 2.5]$ , $R_{ii}$ ( $i = 1$ and $2$ ) ∈ $[0.1, 5]$ , $K_{i 1}$ ∈ $[0.1, 10]$ , $K_{i 2}$ ∈ $[0.1, 10]$ , $α_{1}$ ∈ $[0.1, 2]$ and $α_{2}$ ∈ $[0.1, 2]$ .

For optimisation, the parameters of CS, APSO and FF algorithms presented in Table 3 are chosen and $10^{- 6}$ is selected as convergence tolerance. Similarly, the presented heuristic algorithms are also used to tune the parameters of the existing IOLQI controller in order to compare its performance with the FOLQI controller. Table 4 show the converged values of FOLQI and IOLQI controllers satisfying the required time domain specifications with minimum control effort for both minimum and non-minimum phase operating conditions.

Table 3.

Heuristic algorithms and its parameters.

Heuristic algorithm	Parameters
CS	$n = 25$
	$p_{a} = 0.25$
	$T = 100$
APSO	$n = 25$
APSO	$T = 100$
FF	$n = 100$
	$T = 100$
	$μ = 0.3$
	$β = 0.3$
	$γ = 1$

Table 4.

Converged controller parameters of IOLQI and FOLQI controllers.

Heuristic algorithm	Controller/Operating condition	$Q_{11}$	$Q_{2} 2$	$Q_{3} 3$	$Q_{4} 4$	$R_{1} 1$	$R_{2} 2$	$K_{i} 1$	$K_{i} 2$	$α_{1}$	$α_{2}$
CS	IOLQI minimum	2.4409	0.9294	0.0743	1.5406	2.8543	1.6239	4.9467	0.5440	-	-
	IOLQI non-minimum	2.4266	1.1754	0.1141	2.3565	4.9639	3.8190	1.0003	1.0205	-	-
	FOLQI minimum	0.4431	0.3027	0.4434	1.5722	4.2253	2.5994	1.7411	7.1716	0.9814	0.8437
	FOLQI non-minimum	1.0788	1.0036	0.1676	1.7933	3.6276	3.7317	6.9248	2.6781	0.8020	0.8700
APSO	IOLQI minimum	0.5514	0.9915	1.5566	1.0882	2.8910	4.2427	0.8557	4.8845	-	-
	IOLQI non-minimum	1.9712	1.3166	1.6154	1.6593	2.4662	4.6690	5.5342	0.1686	-	-
	FOLQI minimum	0.4670	0.5505	0.3838	1.0798	4.3735	4.0577	7.1449	1.7352	0.8212	0.9198
	FOLQI non-minimum	2.1809	2.2659	0.7991	1.9047	3.2796	4.1326	8.5300	2.8451	0.8229	0.9219
FF	IOLQI minimum	1.2571	0.8219	1.8114	1.5922	3.0720	2.9747	2.0416	0.6452	-	-
	IOLQI non-minimum	1.2201	0.6425	1.4486	1.8297	1.0615	1.7583	2.9393	5.3533	-	-
	FOLQI minimum	0.8815	0.4995	0.6352	1.5440	2.4191	1.8231	8.3252	5.8941	0.9099	0.9834
	FOLQI non-minimum	1.0342	0.8677	0.1859	0.6687	4.0277	4.7713	0.2204	1.2102	0.9994	0.8071

The convergence of objective function $J$ using CS, APSO and FF for IOLQI and FOLQI controllers with minimum and non-minimum phase operating conditions of QTP are shown in Figure 3. The convergence epochs and statistical parameters (mean and Standard Deviation (SD)) for CS, APSO and FF of IOLQI and FOLQI controllers are shown in Table 5. It is noted that FF algorithm provides (i) faster convergence epochs for FOLQI minimum, IOLQI minimum and non-minimum conditions compared to CS and APSO algorithms, (ii) less statistical parameter – Mean compared to CS and APSO algorithms and (iii) better statistical parameter – SD for minimum phase systems. Hence, it is believed that FF algorithm provides better performance than CS and APSO algorithms.

Figure 3.

Convergence plot using CS, APSO and FF algorithms.

Table 5.

Convergence epochs and statistical index of CS, APSO and FF algorithms.

Methods	Controllers	Convergence epochs	Statistical index
Methods	Controllers	Convergence epochs	Mean	SD
CS	IOLQI: minimum phase	74	1403.34	118.10
	IOLQI: non-minimum phase	66	1334.19	40.07
	FOLQI: minimum phase	70	1241.73	91.88
	FOLQI: non-minimum phase	47	1222.34	55.06
APSO	IOLQI: minimum phase	72	1442.49	161.13
	IOLQI: non-minimum phase	67	1490.91	61.44
	FOLQI: minimum phase	74	1239.97	98.09
	FOLQI: non-minimum phase	61	1250.20	78.12
FF	IOLQI: minimum phase	65	1288.23	102.02
	IOLQI: non-minimum phase	60	1541.80	62.71
	FOLQI: minimum phase	54	1237.66	30.62
	FOLQI: non-minimum phase	60	1213.70	88.50

In general, the FF algorithm has both advantage and disadvantages compared to CS and APSO. FF algorithm takes long convergence time and has the characteristics to lock in local minima region. But proper selection of ranges for controller parameters will give the advantage of converging to best optimum point compared to CS and APSO.⁵² We have made many trail and error to select the best controller parameter range which led FF algorithm to explore the best optimal controller parameter.

Stability analysis

The stability analysis of QTP with the obtained IOLQI/FOLQI controllers tuned using CS, APSO and FF algorithms are studied using frequency response method. Tables 6 –8 show that, gain crossover frequencies and phase margins for all the control loops are found to be $\infty$ . The obtained results concludes that both IOLQI/FOLQI controllers are stable with enough stability margins.

Table 6.

Stability analysis of controllers tuned using CS algorithm.

Heuristic algorithm	Controller	Operating mode	Loop interaction		Gain crossover frequency (rad/sec)	Phase margin (°)
Heuristic algorithm	Controller	Operating mode	Input	Output	Gain crossover frequency (rad/sec)	Phase margin (°)
CS	IOLQI	Minimum	1	1	1.740	2.09
			1	2	1.770	2.07
			2	1	0.576	6.34
			2	2	0.587	6.28
		Non-minimum	1	1	0.617	2.71
			1	2	0.611	2.74
			2	1	0.685	2.91
			2	2	0.617	2.71
	FOLQI	Minimum	1	1	1.030	4.17
			1	2	1.050	4.15
			2	1	2.230	15.3
			2	2	2.270	15.3
		Non-minimum	1	1	1.900	18.9
			1	2	1.690	18.7
			2	1	1.120	13.5
			2	2	0.997	13.2

Table 7.

Stability analysis of controllers tuned using APSO algorithm.

Heuristic algorithm	Controller	Operating mode	Loop interaction		Gain crossover frequency (rad/s)	Phase margin (°)
Heuristic algorithm	Controller	Operating mode	Input	Output	Gain crossover frequency (rad/s)	Phase margin (°)
APSO	IOLQI	Minimum	1	1	0.724	3.77
			1	2	0.737	3.73
			2	1	1.730	1.57
			2	2	1.760	1.55
		Non-minimum	1	1	1.590	1.63
			1	2	1.430	1.51
			2	1	0.277	9.44
			2	2	0.253	10.30
	FOLQI	Minimum	1	1	2.250	17.30
			1	2	2.290	17.30
			2	1	0.164	9.67
			2	2	0.160	9.64
		Non-minimum	1	1	2.110	17.20
			1	2	1.880	17.10
			2	1	1.150	9.24
			2	2	1.030	9.09

Table 8.

Stability analysis of controllers tuned using FF algorithm.

Heuristic algorithm	Controller	Operating mode	Loop interaction		Gain crossover frequency (rad/s)	Phase margin (°)
Heuristic algorithm	Controller	Operating mode	Input	Output	Gain crossover frequency (rad/s)	Phase margin (°)
FF	IOLQI	Minimum	1	1	1.120	2.72
			1	2	1.140	2.70
			2	1	0.628	4.85
			2	2	0.640	4.82
		Non-minimum	1	1	1.160	3.56
			1	2	1.040	3.35
			2	1	1.570	2.64
			2	2	1.410	2.46
	FOLQI	Minimum	1	1	2.350	9.54
			1	2	2.390	9.48
			2	1	1.910	3.21
			2	2	1.940	3.14
		Non-minimum	1	1	0.319	4.77
			1	2	0.289	5.55
			2	1	0.723	19.40
			2	2	0.643	19.60

Case 1: Under disturbance condition

The simulation is performed for QTP with FOLQI and existing IOLQI controllers obtained using CS, APSO and FF algorithms for a duration of $350 \sec$ . A continuous load disturbance at the outputs $x_{1} (t)$ and $x_{2} (t)$ of magnitude $1 cm$ is introduced at $50^{th} \sec$ as shown in Figure 4. The time domain performance indices $% M_{p}$ , $t_{s}$ , $e_{ss}$ and $J$ of the IOLQI and FOLQI controllers are measured for minimum and non-minimum phase operating modes of QTP and are given in Table 9. The output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) and controller responses ( $u_{1} (t)$ and $u_{2} (t)$ ) for QTP with IOLQI and FOLQI controllers are shown in Figures 5 –12.

Figure 4.

Load disturbance signal.

Table 9.

Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance conditions.

Controller/Operating condition	Heuristic algorithm	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
IOLQI minimum phase	CS	$x_{1}$	5.7977	24.1030	0.0014	1.827	3.6639
	CS	$x_{2}$	6.1123	24.1750	0.0077	1.836	3.6639
	APSO	$x_{1}$	5.5180	28.3980	0.0425	1.845	3.7028
	APSO	$x_{2}$	5.8153	29.9640	0.0105	1.858	3.7028
	FF	$x_{1}$	5.6629	26.9330	0.0148	1.792	3.5972
	FF	$x_{2}$	5.9180	29.2720	0.0334	1.805	3.5972
FOLQI minimum phase	CS	$x_{1}$	4.3728	2.7920	0.0069	1.765	3.5371
	CS	$x_{2}$	4.6702	2.7200	0.0040	1.772	3.5371
	APSO	$x_{1}$	4.3744	1.7480	0.0141	1.761	3.5342
	APSO	$x_{2}$	4.6647	1.7800	0.0317	1.773	3.5342
	FF	$x_{1}$	5.2375	4.9460	0.1809	1.766	3.5719
	FF	$x_{2}$	5.5283	4.9670	0.1615	1.806	3.5719
IOLQI non-minimum phase	CS	$x_{1}$	6.2989	51.5870	0.0322	1.720	3.6708
	CS	$x_{2}$	8.1258	62.0230	0.0355	1.951	3.6708
	APSO	$x_{1}$	4.9419	26.5330	0.0035	1.784	3.8153
	APSO	$x_{2}$	8.3099	55.8790	0.1366	2.032	3.8153
	FF	$x_{1}$	6.5241	46.7880	0.0622	1.804	3.8519
	FF	$x_{2}$	7.5291	48.4790	0.0503	2.048	3.8519
FOLQI non-minimum phase	CS	$x_{1}$	4.2540	2.2600	0.1369	1.627	3.5416
	CS	$x_{2}$	3.4549	23.2500	0.3901	1.914	3.5416
	APSO	$x_{1}$	4.4437	2.2310	0.0264	1.650	3.5494
	APSO	$x_{2}$	3.7544	21.5190	0.1069	1.900	3.5494
	FF	$x_{1}$	5.6173	13.8690	0.7047	1.700	3.5129
	FF	$x_{2}$	4.6486	9.5570	0.6279	1.813	3.5129

Figure 5.

Output $x_{1} (t)$ and controller $u_{1} (t)$ responses for minimum phase condition of QTP with IOLQI controller.

Figure 6.

Output $x_{2} (t)$ and controller $u_{2} (t)$ responses for minimum phase condition of QTP with IOLQI controller.

Figure 7.

Output $x_{1} (t)$ and controller $u_{1} (t)$ responses for non-minimum phase condition of QTP with IOLQI controller.

Figure 8.

Output $x_{2} (t)$ and controller $u_{2} (t)$ responses for non-minimum phase condition of QTP with IOLQI controller.

Figure 9.

Output $x_{1} (t)$ and controller $u_{1} (t)$ responses for minimum phase condition of QTP with FOLQI controller.

Figure 10.

Output $x_{2} (t)$ and controller $u_{2} (t)$ responses for minimum phase condition of QTP with FOLQI controller.

Figure 11.

Output $x_{1} (t)$ and controller $u_{1} (t)$ responses for non-minimum phase condition of QTP with FOLQI controller.

Figure 12.

Output $x_{2} (t)$ and controller $u_{2} (t)$ responses for non-minimum phase condition of QTP with FOLQI controller.

From the table, it is observed that FOLQI controllers provide better time characteristics $% M_{p}$ , $t_{s}$ and $J$ in comparison to IOLQI controllers. However, it is noted that IOLQI controllers provide better $e_{ss}$ than FOLQI controllers tuned using CS, APSO and FF algorithms.

Remarks 1: The performance measures listed in Table 9 indicate that the controllers tuned using heuristic methods are not providing consistency with respect to minimum and non-minimum phase conditions. However the responses obtained using IOLQI and FOLQI controllers are meeting the given closed loop constraints.

Remarks 2: The simulation results indicate that for all cases, the control effort utilised for controlling $x_{2}$ is higher than control effort required for controlling $x_{1}$

Case 2: Under disturbance and parameter uncertainty

The robustness of the FOLQI controllers are analysed by introducing parameter uncertainty in $A_{i}$ and $a_{i}$ of ± $30$ % along with continuous load disturbance $d (t)$ at $50^{th}$ $\sec$ . The performance of FOLQI controllers are compared with its integer order counterparts. The time domain characteristics $% M_{p}$ , $t_{s}$ , $e_{ss}$ and $J$ of IOLQI and FOLQI controllers are computed for minimum and non-minimum phase of QTP and are given in Tables 10 –12. The outputs $x_{1} (t)$ and $x_{2} (t)$ for QTP with IOLQI and FOLQI controllers obtained using (i) CS algorithm are shown in Figures 13 –16, (ii) APSO algorithm are shown in Figures 17 –20 and (iii) FF algorithm are shown in Figures 21 –24.

Table 10.

Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using CS algorithm.

Controller/Operating condition	Parameter uncertainty (%)	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
CS-IOLQI minimum phase	−30%	$x_{1}$	5.8089	20.6680	0.0049	0.903	1.8122
	−30%	$x_{2}$	6.1224	20.7270	0.0425	909.2	1.8122
	Nominal	$x_{1}$	5.7977	24.1030	0.0014	1.827	3.6639
	Nominal	$x_{2}$	6.1123	24.1750	0.0077	1.836	3.6639
	+30%	$x_{1}$	6.0560	29.0490	0.0008	3.072	6.1559
	+30%	$x_{2}$	6.3768	30.8900	0.0132	3.084	6.1559
CS-FOLQI minimum phase	−30%	$x_{1}$	4.3322	2.5380	0.0119	0.865	1.735
	−30%	$x_{2}$	4.6280	2.4840	0.0083	0.870	1.735
	Nominal	$x_{1}$	4.3728	2.7920	0.0069	1.765	3.5371
	Nominal	$x_{2}$	4.6702	2.7200	0.0040	1.772	3.5371
	+30%	$x_{1}$	4.5716	3.5200	0.0168	2.982	5.9742
	+30%	$x_{2}$	4.8736	3.4840	0.0113	2.992	5.9742
CS-IOLQI non-minimum phase	−30%	$x_{1}$	6.4819	43.6580	0.3433	0.8305	1.8160
	−30%	$x_{2}$	8.3504	52.2830	0.4847	0.9855	1.8160
	Nominal	$x_{1}$	6.2989	51.5870	0.0322	1.720	3.6708
	Nominal	$x_{2}$	8.1258	62.0230	0.0355	1.951	3.6708
	+30%	$x_{1}$	6.6453	67.0240	0.2686	2.949	6.1970
	+30%	$x_{2}$	8.1421	79.1180	0.3158	3.248	6.1970
CS-FOLQI non-minimum phase	−30%	$x_{1}$	4.1406	2.1340	0.4431	0.7718	1.7361
	−30%	$x_{2}$	3.2853	27.1110	1.1946	0.9643	1.7361
	Nominal	$x_{1}$	4.2540	2.2600	0.1369	1.627	3.5416
	Nominal	$x_{2}$	3.4549	23.2500	0.3901	1.914	3.5416
	+30%	$x_{1}$	4.9472	2.4790	0.0416	2.802	5.9831
	+30%	$x_{2}$	7.4995	17.7240	0.0761	3.182	5.9831

Table 11.

Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using APSO algorithm.

Controller/Operating condition	Parameter uncertainty (%)	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
APSO-IOLQI minimum phase	−30%	$x_{1}$	5.7473	29.9520	0.0944	0.9212	1.8513
	−30%	$x_{2}$	6.0479	29.8620	0.0145	0.9302	1.8513
	Nominal	$x_{1}$	5.5180	28.3980	0.0425	1.845	3.7028
	Nominal	$x_{2}$	5.8153	29.9640	0.0105	1.858	3.7028
	+30%	$x_{1}$	5.6510	32.0650	0.0170	3.087	6.1898
	+30%	$x_{2}$	5.9519	33.9000	0.0048	3.103	6.1898
APSO-FOLQI minimum phase	−30%	$x_{1}$	4.2845	1.6550	0.0295	0.8623	1.7333
	−30%	$x_{2}$	4.5714	1.6820	0.0715	0.8709	1.7333
	Nominal	$x_{1}$	4.3744	1.7480	0.0141	1.761	3.5342
	Nominal	$x_{2}$	4.6647	1.7800	0.0317	1.773	3.5342
	+30%	$x_{1}$	4.6015	1.8930	0.0065	2.977	5.9695
	+30%	$x_{2}$	4.8976	1.9390	0.0091	2.993	5.9695
APSO-IOLQI non-minimum phase	−30%	$x_{1}$	4.8412	22.8300	0.0173	0.8599	1.8835
	−30%	$x_{2}$	9.2901	47.4860	0.9350	1.024	1.8835
	Nominal	$x_{1}$	4.9419	26.5330	0.0035	1.784	3.8153
	Nominal	$x_{2}$	8.3099	55.8790	0.1366	2.032	3.8153
	+30%	$x_{1}$	5.1532	38.5570	0.0232	3.040	6.3955
	+30%	$x_{2}$	8.0681	76.7930	0.3393	3.355	6.3955
APSO-FOLQI non-minimum phase	−30%	$x_{1}$	4.3313	2.1080	0.2745	0.7881	1.7407
	−30%	$x_{2}$	3.5948	25.2610	0.7400	0.9526	1.7407
	Nominal	$x_{1}$	4.4437	2.2310	0.0264	1.650	3.5497
	Nominal	$x_{2}$	3.7544	21.5190	0.1069	1.900	3.5497
	+30%	$x_{1}$	4.5975	2.4820	0.1237	2.829	5.9944
	+30%	$x_{2}$	3.9477	13.9610	0.2747	3.165	5.9944

Table 12.

Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using FF algorithm.

Controller/Operating condition	Parameter uncertainty (%)	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
FF-IOLQI minimum phase	−30%	$x_{1}$	5.8759	26.7950	0.0258	0.8852	1.7790
	−30%	$x_{2}$	6.1253	28.7900	0.0749	0.8938	1.7790
	Nominal	$x_{1}$	5.6629	26.9330	0.0148	1.792	3.5972
	Nominal	$x_{2}$	5.9180	29.2720	0.0334	1.805	3.5972
	+30%	$x_{1}$	6.0770	30.7940	0.0003	3.026	6.0686
	+30%	$x_{2}$	6.3546	33.4520	0.0192	3.043	6.0686
FF-FOLQI minimum phase	−30%	$x_{1}$	5.2602	4.2330	0.2694	0.8644	1.7563
	−30%	$x_{2}$	5.5461	4.2860	0.2415	0.8919	1.7563
	Nominal	$x_{1}$	5.2375	4.9460	0.1809	1.766	3.5719
	Nominal	$x_{2}$	5.5283	4.9670	0.1615	1.806	3.5719
	+30%	$x_{1}$	5.3317	6.4830	0.1315	2.984	6.0210
	+30%	$x_{2}$	5.6275	6.4840	0.1165	3.037	6.0210
FF-IOLQI non-minimum phase	−30%	$x_{1}$	6.4270	36.0870	0.3769	0.8716	1.8973
	−30%	$x_{2}$	7.5023	37.4690	0.3136	1.026	1.8973
	Nominal	$x_{1}$	6.5241	46.7880	0.0622	1.804	3.8519
	Nominal	$x_{2}$	7.5291	48.4790	0.0503	2.048	3.8519
	+30%	$x_{1}$	6.7010	67.7480	0.1609	3.068	6.4556
	+30%	$x_{2}$	7.6919	69.7580	0.1291	3.388	6.4556
FF-FOLQI non-minimum phase	−30%	$x_{1}$	5.3932	12.0800	0.4283	0.8233	1.7244
	−30%	$x_{2}$	4.1344	8.4150	0.4153	0.901	1.7244
	Nominal	$x_{1}$	5.6173	13.8690	0.7047	1.700	3.5129
	Nominal	$x_{2}$	4.6486	9.5570	0.6279	1.813	3.5129
	+30%	$x_{1}$	6.3565	15.6810	0.8626	2.894	5.9369
	+30%	$x_{2}$	5.6796	10.7130	0.7417	3.043	5.9369

Figure 13.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with IOLQI controller using CS algorithm.

Figure 14.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with IOLQI controller using CS algorithm.

Figure 15.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with FOLQI controller using CS algorithm.

Figure 16.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with FOLQI controller using CS algorithm.

Figure 17.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with IOLQI controller using APSO algorithm.

Figure 18.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with IOLQI controller using APSO algorithm.

Figure 19.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with FOLQI controller using APSO algorithm.

Figure 20.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with FOLQI controller using APSO algorithm.

Figure 21.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with IOLQI controller using FF algorithm.

Figure 22.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with IOLQI controller using FF algorithm.

Figure 23.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for minimum phase operating condition of QTP with FOLQI controller using FF algorithm.

Figure 24.

Output responses ( $x_{1} (t)$ and $x_{2} (t)$ ) for non-minimum phase operating condition of QTP with FOLQI controller using FF algorithm.

From the table, it is observed that (i) FOLQI controllers provide better time characteristics $% M_{p}$ , $t_{s}$ and $J$ in comparison to IOLQI controllers, (ii) IOLQI controllers provide better $e_{ss}$ in comparison to FOLQI controllers, (iii) CS algorithm tuned IOLQI and FOLQI controllers fails to meet the required $t_{s}$ and $e_{ss}$ respectively for non-minimum phase QTP, (iv) APSO tuned IOLQI controller fails to meet the required $t_{s}$ for non-minimum phase QTP and (v) FF tuned IOLQI and FOLQI controllers meet all the required specifications for both minimum and non-minimum phase of QTP. Based on the above observation, it is concluded that FF tuned FOLQI controller provides better time domain characteristics $% M_{p}$ , $t_{s}$ , $e_{ss}$ and $J$ compared to APSO and CS tuned FOLQI controllers.

Case 3: Tracking performances under nominal/disturbance condition

The equilibrium points of QTP with both IOLQI/FOLQI controllers are increased to the magnitude of $1$ $cm$ at the time instant of $50^{th}$ $\sec$ under nominal condition. The controller parameters tuned using CS, APSO and FF algorithms are employed for the simulation of QTP under nominal/output disturbance conditions for a duration of $350 \sec$ . The time performance characteristics like $% M_{p}$ , $t_{s}$ , $e_{ss}$ and $J$ of IOLQI and FOLQI controllers tuned using CS, APSO and FF algorithms are computed for minimum and non-minimum phase of QTP and are given in Tables 13 and 14.

Table 13.

Tracking performance characteristics for IOLQI and FOLQI controllers of QTP under nominal conditions.

Controller/Operating condition	Heuristic algorithm	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
IOLQI minimum phase	CS	$x_{1}$	5.7839	22.5640	0.0181	2.072	4.1539
	CS	$x_{2}$	5.7074	26.0420	0.1711	2.082
	APSO	$x_{1}$	5.7283	30.0950	0.2061	2.087	4.1901
	APSO	$x_{2}$	5.6521	28.3980	0.0320	2.103
	FF	$x_{1}$	5.7688	24.7960	0.0526	2.035	4.0843
	FF	$x_{2}$	5.6906	27.2780	0.1807	2.050	4.0843
FOLQI minimum phase	CS	$x_{1}$	5.7268	1.7590	0.1078	2.004	4.0170
	CS	$x_{2}$	5.6510	4.1176	0.0587	2.013	4.0170
	APSO	$x_{1}$	5.7268	1.7220	0.0716	2.001	4.0170
	APSO	$x_{2}$	5.6510	1.7650	0.1592	2.016	4.0170
	FF	$x_{1}$	5.7271	4.9550	0.2981	2.003	4.0530
	FF	$x_{2}$	5.6510	4.0390	0.2661	2.050	4.0530
IOLQI non-minimum phase	CS	$x_{1}$	5.0668	113.8760	2.1401	1.944	4.1659
	CS	$x_{2}$	6.0659	—–	3.0222	2.222	4.1659
	APSO	$x_{1}$	4.8259	24.7550	0.1090	1.971	4.2263
	APSO	$x_{2}$	4.9334	—–	5.1436	2.256	4.2263
	FF	$x_{1}$	4.9896	174.3880	2.9595	2.046	4.3787
	FF	$x_{2}$	6.0689	34.9970	2.4065	2.333	4.3787
FOLQI non-minimum phase	CS	$x_{1}$	4.7247	2.1820	1.6160	1.831	4.0006
	CS	$x_{2}$	6.9816	—–	4.1661	2.169	4.0006
	APSO	$x_{1}$	4.7381	2.1780	1.5262	1.857	4.0082
	APSO	$x_{2}$	6.9899	—–	3.8133	2.152	4.0082
	FF	$x_{1}$	4.7444	5.4306	2.0702	1.949	4.0356
	FF	$x_{2}$	6.9922	5.4340	1.7261	2.087	4.0356

Table 14.

Tracking performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance conditions.

Controller/Operating condition	Heuristic algorithm	Output (cm)	Overshoot (%)	Settling time (s)	Steady state error (e_ss)	Controller effort (×10³)	Effective controller effort (J × 10³)
IOLQI minimum phase	CS	$x_{1}$	5.7268	0.001	0.0098	1.872	3.7549
	CS	$x_{2}$	5.6510	0.001	0.0890	1.882	3.7549
	APSO	$x_{1}$	5.7268	0.001	0.1225	1.867	3.7477
	APSO	$x_{2}$	5.6510	0.001	0.0213	1.881	3.7477
	FF	$x_{1}$	5.7268	0.001	0.0338	1.869	3.7528
	FF	$x_{2}$	5.6510	0.001	0.1054	1.883	3.7528
FOLQI minimum phase	CS	$x_{1}$	5.7268	0.001	0.0506	1.871	3.7494
	CS	$x_{2}$	5.6510	0.001	0.0275	1.879	3.7494
	APSO	$x_{1}$	5.7268	0.001	0.0424	1.869	3.7512
	APSO	$x_{2}$	5.6510	0.001	0.0944	1.882	3.7512
	FF	$x_{1}$	5.7268	0.001	0.2341	1.853	3.7493
	FF	$x_{2}$	5.6510	0.001	0.2090	1.897	3.7493
IOLQI non-minimum phase	CS	$x_{1}$	4.7191	0.001	1.0561	1.750	3.7545
	CS	$x_{2}$	6.9811	0.001	1.4954	2.004	3.7545
	APSO	$x_{1}$	4.7191	0.001	0.0539	1.727	3.7277
	APSO	$x_{2}$	6.9811	18.565	2.6364	2.001	3.7277
	FF	$x_{1}$	4.7191	0.001	0.1757	1.756	3.7899
	FF	$x_{2}$	6.9811	0.001	0.5085	2.034	3.7899
FOLQI non-minimum phase	CS	$x_{1}$	4.7191	0.001	0.8728	1.711	3.7353
	CS	$x_{2}$	6.9811	0.001	2.2621	2.025	3.7353
	APSO	$x_{1}$	4.7191	0.001	0.7756	1.731	3.7369
	APSO	$x_{2}$	6.9811	0.001	1.9542	2.006	3.7369
	FF	$x_{1}$	4.7191	0.001	0.6981	1.815	3.7556
	FF	$x_{2}$	6.9811	0.001	0.5704	1.940	3.7556

From the table, it is observed that (i) under nominal condition of minimum phase operating mode shown in Table 13, for both IOLQI and FOLQI controllers tuned using CS, APSO and FF algorithms meet the required time specifications whereas in non-minimum phase of operating mode only FOLQI controller tuned using FF algorithms meet the required time specifications (ii) under output disturbance condition, for all the mode of operating condition shown in Table 14, both IOLQI and FOLQI controllers meet the required time specifications.

Remarks 3: The performance measures listed in Table 13 indicate that the controllers tuned using CS and APSO algorithms are not meeting required time specifications to non-minimum phase conditions. However the responses obtained using IOLQI and FOLQI controllers tuned using CS, APSO and FF algorithms are meeting the given closed loop constraints for minimum phase operating mode. From the above inferences, it is concluded that FF algorithm provide better tracking performances.

Remarks 4: Under disturbance condition, since both the disturbance and equilibrium set point is increased uniformly for the magnitude of $1 cm$ , the time specifications are met. The performance may vary if the equilibrium point for tracking is given in negative direction.

Conclusions

In this paper, various heuristic algorithms such as CS, APSO and FF were used to tune the FOLQI controllers for QTP under continuous load disturbance condition. The constrained optimisation problem was formulated and solved using these presented heuristic algorithms. Simulations were performed for minimum and non-minimum phase operating modes of QTP under disturbance and parameter uncertainty conditions. To show the superiority of FOLQI controller, the results were compared with optimally tuned IOLQI controller using heuristic algorithms. From the simulation, it is observed that (i) FOLQI controller outperforms IOLQI in terms of $% M_{p}$ , $t_{s}$ and $J$ , whereas IOLQI controller outperforms FOLQI in terms of $e_{ss}$ and (ii) FF tuned FOLQI and IOLQI controllers provide better robustness characteristics compared to CS and APSO tuned FOLQI and IOLQI controllers.

The future directions of this research can be extended to (i) validate the performance of heuristic algorithms on other practical systems (ii) explore on any promising heuristic algorithms for tuning optimal FOLQI controller parameters.

Footnotes

Ethical declarations

Hereby, the authors solely assure that for this manuscript “Heuristic algorithms based optimal tuning of FOLQI controller for quadruple tank process under disturbance conditions” the following ethics are fulfilled:

(1) This manuscript is the authors own work, which has not been published or currently being considered elsewhere for publication.

(2) The authors own contribution in research and analysis of the problem is presented in a truthful manner.

(3) All citation sources are disclosed properly.

(4) The contribution of all authors have been actively involved in this work.

(5) All the software used while preparing this manuscript is originally licenced by the institution “Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala, India”.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Narayanasamy Selvaganesan

References

Juang

Liu

Lin

. A hybrid intelligent controller for a twin rotor mimo system and its hardware implementation. ISA Trans 2011; 50(4): 609–619.

Rakhshani

Naveh

IMH

Mehrjerdi

, et al. An optimized LQG servo controller design using LQI tracker for VSP-based AC/DC interconnected systems. Int J Electr Power Energy Syst 2021; 129: 106752.

Chang

Shieh

Liu

, et al. Digital modeling and PID controller design for mimo analog systems with multiple delays in states, inputs and outputs. Circuits Syst Signal Process 2009; 28(1): 111–145.

Altan

Hacıoğlu

. Model predictive control of three-axis gimbal system mounted on UAV for real-time target tracking under external disturbances. Mech Syst Signal Process 2020; 138: 106548.

Johansson

. The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Trans Control Syst Technol 2000; 8(3): 456–465.

Tofighi

Merrikh-Bayat

. A benchmark system to investigate the non-minimum phase behaviour of multi-input multi-output systems. J Contr Decis 2018; 5(3): 300–317.

Vadigepalli

Gatzke

Doyle

. Robust control of a multivariable experimental four-tank system. Ind Eng Chem Res 2001; 40(8): 1916–1927.

Liu

Tian

Xue

, et al. A review of industrial mimo decoupling control. Int J Control Autom Syst 2019; 17(5): 1246–1254.

Chen

Zhang

. A review of mutual coupling in mimo systems. IEEE Access 2018; 6: 24706–24719.

10.

Roy

. Dual mode adaptive fractional order pi controller with feedforward controller based on variable parameter model for quadruple tank process. ISA Trans 2016; 63: 365–376.

11.

Grebeck

A comparison of controllers for the quadruple tank system. Technical Report, Lund Institute of Technology (LTH), Lund, Sweden, 1998.

12.

Rosinová

Kozáková

. Decentralized robust control of mimo systems: Quadruple tank case study. IFAC Proc 2012; 45(11): 72–77.

13.

Husek

. Decentralized pi controller design based on phase margin specifications. IEEE Trans Control Syst Technol 2014; 22(1): 346–351.

14.

Rosinová

Markech

. Robust control of quadruple tank process. ICIC Express Lett 2008; 2(3): 231–238.

15.

Maghade

Patre

. Decentralized pi/pid controllers based on gain and phase margin specifications for tito processes. ISA Trans 2012; 51(4): 550–558.

16.

Park

Lee

Shim

. On stealthiness of zero-dynamics attacks against uncertain nonlinear systems: A case study with quadruple-tank process. In: International symposium on mathematical theory of networks and systems (ISMTNS), 2018, pp.10–17.

17.

Naami

Ouahi

Rabhi

, et al. Design of robust control for uncertain fuzzy quadruple-tank systems with time-varying delays. Granul Comput 2022; 7(4): 951–964.

18.

Meng

, et al. Disturbance observer-based integral backstepping control for a two-tank liquid level system subject to external disturbances. Math Probl Eng 2020; 2020: 1–22.

19.

Huang

Canuto

Novara

. The four-tank control problem: comparison of two disturbance rejection control solutions. ISA Trans 2017; 71: 252–271.

20.

Navratil

Pekar

Matusu

. Control of a multivariable system using optimal control pairs: a quadruple-tank process. IEEE Access 2020; 8: 2537–2563.

21.

Son

. Level control of quadruple tank system based on adaptive inverse evolutionary neural controller. Int J Control Autom Syst 2020; 18(9): 2386–2397.

22.

Meng

Zhang

, et al. Liquid level control of four-tank system based on active disturbance rejection technology. Measurement 2021; 175: 1–10.

23.

Osman

Kara

Arıcı

. Robust adaptive control of a quadruple tank process with sliding mode and pole placement control strategies. IETE J Res 2023; 69: 2412–2425.

24.

Gole

Barve

Kesarkar

, et al. Investigation of fractional control performance for magnetic levitation experimental set-up. In: 2012 International conference on emerging trends in science, engineering and technology (INCOSET), Tiruchirappalli, India, 13–14 December 2012, pp.500–504. New York, NY: IEEE.

25.

Kesarkar

Selvaganesan

. Superiority of fractional order controllers in limit cycle suppression. Int J Autom Control 2013; 7(3): 166–182.

26.

Kesarkar

Selvaganesan

Priyadarshan

. Novel controller design for plants with relay nonlinearity to reduce amplitude of sustained oscillations: illustration with a fractional controller. ISA Trans 2015; 57: 295–300.

27.

Perumal

Selvaganesan

. Input dependent nyquist plot for limit cycle prediction and its suppression using fractional order controllers. Trans Inst Meas Contr 2019; 41(13): 3847–3860.

28.

Sathishkumar

Selvaganesan

. Fractional controller tuning expressions for a universal plant structure. IEEE Control Syst Lett 2018; 2(3): 345–350.

29.

Copot

Ghita

Ionescu

. Simple alternatives to PID-type control for processes with variable time-delay. Processes 2019; 7(3): 146.

30.

Roy

Krishna Roy

. Fractional order pi control applied to level control in coupled two tank mimo system with experimental validation. Control Eng Pract 2016; 48: 119–135.

31.

Sutha

Lakshmi

Sankaranarayanan

. Fractional-order sliding mode controller design for a modified quadruple tank process via multi-level switching. Comput Electr Eng 2015; 45: 10–21.

32.

Ghalambaz

Jalilzadeh Yengejeh

Davami

. Building energy optimization using grey wolf optimizer (GWO). Case Stud Therm Eng 2021; 27: 101250.

33.

Ghalambaz

Jalilzadeh

Davami

. Building energy optimization using butterfly optimization algorithm. Therm Sci 2022; 26(5 Part A): 3975–3986.

34.

Altan

Karasu

Bekiros

. Digital currency forecasting with chaotic meta-heuristic bio-inspired signal processing techniques. Chaos Solitons Fractals 2019; 126: 325–336.

35.

Altan

Karasu

Zio

. A new hybrid model for wind speed forecasting combining long short-term memory neural network, decomposition methods and grey wolf optimizer. Appl Soft Comput 2021; 100: 106996.

36.

Altan

. Performance of metaheuristic optimization algorithms based on swarm intelligence in attitude and altitude control of unmanned aerial vehicle for path following. In: 2020 4th international symposium on multidisciplinary studies and innovative technologies (ISMSIT), Istanbul, Turkey, 22–24 October 2020, pp.1–6. New York, NY: IEEE.

37.

Belge

Altan

Hacıoğlu

. Metaheuristic optimization-based path planning and tracking of quadcopter for payload hold-release mission. Electronics 2022; 11: 1208.

38.

Joseph

Dada

Abidemi

, et al. Metaheuristic algorithms for PID controller parameters tuning: review, approaches and open problems. Heliyon 2022; 8: e09399.

39.

Jin

Jiang

, et al. Novel improved cuckoo search for PID controller design. Trans Inst Meas Contr 2015; 37(6): 721–731.

40.

Bingul

Karahan

. A novel performance criterion approach to optimum design of PID controller using cuckoo search algorithm for avr system. J Franklin Inst 2018; 355(13): 5534–5559.

41.

El Gmili

Mjahed

El Kari

, et al. Particle swarm optimization and cuckoo search-based approaches for quadrotor control and trajectory tracking. Appl Sci 2019; 9(8): 1719.

42.

Lagunes

Castillo

Soria

. Optimization of membership function parameters for fuzzy controllers of an autonomous mobile robot using the firefly algorithm. Cham: Springer International Publishing, 2018. pp.199–206.

43.

Souza

de Mesquita

Reis

LLN

, et al. Optimal LQI and PID synthesis for speed control of switched reluctance motor using metaheuristic techniques. Int J Control Autom Syst 2021; 19(1): 221–229.

44.

Ahmadi

Mohammadi-Ivatloo

Anvari-Moghaddam

, et al. Optimal robust LQI controller design for z-source inverters. Appl Sci 2020; 10(20): 7260.

45.

Al-Awad

. Optimal control of quadruple tank system using genetic algorithm. Int J Comput Digit Syst 2019; 8(1): 51–59.

46.

Mahapatro

Subudhi

Ghosh

. Design of a robust optimal decentralized PI controller based on nonlinear constraint optimization for level regulation: an experimental study. IEEECAA J Autom Sin 2019; 7(1): 1–13.

47.

Altabey

. Model optimal control of the four tank system. Int J Syst Sci Appl Math 2016; 1(4): 30–41.

48.

Mohankumar

Selvaganesan

Jayakumar

, et al. Centralised fractional order LQI controller design for quadruple tank process — an optimisation approach. Results Control Optim 2023; 10: 100202.

49.

Yang

Slowik

. Cuckoo search algorithm. In: Slowik

(ed.) Swarm intelligence algorithms. Boca Raton, FL: CRC Press, 2020, pp.109–120.

50.

Kennedy

Eberhart

. Particle swarm optimization. In: Proceedings of ICNN’95-international conference on neural networks, vol. 4, 1995, pp.1942–1948. New York, NY: 6.

51.

Yang

Deb

Fong

. Accelerated particle swarm optimization and support vector machine for business optimization and applications. In: S

Fong

(ed.) Networked Digital Technologies. Berlin, Heidelberg: Springer, 2011, pp.53–66.

52.

Yang

Slowik

. Firefly algorithm. In: Slowik

(ed.) Swarm intelligence algorithms. Boca Raton, FL: CRC Press, 2020, pp.163–174.

53.

Oustaloup

Levron

Mathieu

, et al. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans Circuits Syst I Fundam Theory Appl 2000; 47(1): 25–39.