Heuristic algorithms based optimal tuning of FOLQI controller for quadruple tank process under disturbance conditions

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 % Mp, ts 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.


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 controller 4 and (v) in process control: developing decentralised PI controller for quadruple tank process. 56][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 nonminimum phase operating conditions. 5,10,11or 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 m-analysis based H ' 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 ' 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 respectively 14 and (v) design of robust optimal decentralised PI controller meeting the desired gain margin, phase margin and overshoot constraints outperforms conventional decentralised PI controller. 15n 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 ' 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 controller 22 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. 23ecently, 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][26][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 SMC 31 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 holdrelease 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 function 42 and (xii) LQI controller with optimal Q and R matrices tuned using GA 43 and bat algorithms 44 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 ' 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 ' , loop-shaping, feedback linearisation and model predictive control for nonminimum phase operating condition 11 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. 48It 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: 1.An optimal tuning of FOLQI controller parameters using CS, APSO and FF algorithms is employed for QTP under disturbance conditions.2. 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.3. 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.

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.

Mathematical modelling
The non-linear differential equations governing the mass balance for QTP 18 are described as: The equilibrium inputs u 1o and u 2o for ( 1)-( 4) of QTP provide the following equilibrium points: 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: where, , and i = 1 to 4.

Introduction to LQI
A preliminaries about LQI 32 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 equation 11 by minimising the cost function. 12tate space equation: Cost function: Table 1.Symbols of QTP and its representation.

Symbols Representation
Flow parameter between two tanks g Gravitational constant (cm=sec 2 ) 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 where, K = R À1 B T P and K i is the integral gain.The matrix P is obtained by solving the Riccati equation

Optimisation problem
The Subject to inequality constraints: 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 Slowik 49 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:  The generated solution of the levy flight formulation is given as: where, (m, È and l) represents step size, entry wise multiplication and Levy distribution parameter respectively.The pseudocode of CS algorithm is given as follows: PSO algorithm is proposed by Kennedy and Eberhart in 1995. 50This 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. 51n 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: where, l and m are the acceleration constants, g Ã is the global best value, x i represents position of i th particle at time 't' and 2 t is drawn from N[0,1].
The pseudocode of APSO algorithm is given as follows: FF algorithm FF algorithm is proposed by Yang and Slowik in 2008. 52This algorithm is inspired by flashing behaviour of fireflies belonging to a particular species.The behavioural assumptions of fireflies are as follows: 1. Fireflies are unisex and attracted towards each other based on the intensity of light it produces.2. Less intensity fireflies are attracted towards high intensity fireflies.3. Intensity of the light increases when fireflies comes closer.4. 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: where, b 0 is the brightness of the firefly, g indicates light absorption co-efficient, r represents Euclidean distance between fireflies and it is given as:

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. 53The 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.
For optimisation, the parameters of CS, APSO and FF algorithms presented in Table 3 are chosen and 10 À6 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.
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.
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. 52We 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 '.The obtained results concludes that both IOLQI/FOLQI controllers are stable with enough stability margins.

Case 1: Under disturbance condition
The simulation is performed for QTP with FOLQI and existing IOLQI controllers obtained using CS, APSO    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 6 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  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

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.
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 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 nonminimum 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.

Figure 2 .
Figure 2. Closed loop configuration of QTP with FOLQI controller.
, (ii) APSO algorithm are shown in Figures 17-20 and (iii) FF algorithm are shown in Figures 21-24 .

Figure 5 .
Figure 5. Output x 1 (t) and controller u 1 (t) responses for minimum phase condition of QTP with IOLQI controller.

Figure 6 .
Figure 6.Output x 2 (t) and controller u 2 (t) responses for minimum phase condition of QTP with IOLQI controller.

Figure 7 .
Figure 7. Output x 1 (t) and controller u 1 (t) responses for non-minimum phase condition of QTP with IOLQI controller.

Figure 8 .
Figure 8.Output x 2 (t) and controller u 2 (t) responses for non-minimum phase condition of QTP with IOLQI controller.

Figure 9 .
Figure 9. Output x 1 (t) and controller u 1 (t) responses for minimum phase condition of QTP with FOLQI controller.

Figure 10 .
Figure 10.Output x 2 (t) and controller u 2 (t) responses for minimum phase condition of QTP with FOLQI controller.

Figure 11 .
Figure 11.Output x 1 (t) and controller u 1 (t) responses for non-minimum phase condition of QTP with FOLQI controller.

Figure 12 .
Figure 12.Output x 2 (t) and controller u 2 (t) responses for non-minimum phase condition of QTP with FOLQI controller.

1 .
Cuckoo lays egg only in one randomly selected nest at an instant.2. Best nest with high quality of egg is considered to be success for moving to the next generation.
Require: (i) Objective function: J, (ii) Inequality constraints: %M p , t s and e ss and (iii) Max iterations (T)1: Generate an initial population of N particles x i (i = 1, 2:::, N) is the random parameter and 2 i t ð Þ 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 1: -CS algorithm Require: (i) Objective function: J, (ii) Inequality constraints: %M p , t s and e ss , (iii) Max 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 4 T do 3: Get a cuckoo randomly by Levy flights 4: Evaluate its objective function f (x i (t)) along with

Table 2 .
Algorithm 3: -FF algorithmRequire: (i) Objective function: J, (ii) Inequality constraints: %M p , t s and e ss and (iii) Max 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 ) Parameters and equilibrium points of QTP.

Table 3 .
Heuristic algorithms and its parameters.

Table 4 .
Converged controller parameters of IOLQI and FOLQI controllers.

Table 5 .
Convergence epochs and statistical index of CS, APSO and FF algorithms.

Table 6 .
Stability analysis of controllers tuned using CS algorithm.

Table 7 .
Stability analysis of controllers tuned using APSO algorithm.

Table 8 .
Stability analysis of controllers tuned using FF algorithm.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 Figure4.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 Table9.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 inFigures 5-12. and

Table 9 .
Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance conditions.

Table 10 .
Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using CS algorithm.

Table 11 .
Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using APSO algorithm.

Table 12 .
Performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance and parameter uncertainty conditions using FF algorithm.

Table 12 .
(continued)Figure15. Output responses (x 1 (t) and x 2 (t)) for minimum phase operating condition of QTP with FOLQI controller using CS algorithm.Figure18.Output responses (x 1 (t) and 2 (t)) 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 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.
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.Figure16.Output responses (x 1 (t) and x 2 (t)) for non-minimum phase operating condition of QTP with FOLQI controller using CS algorithm.Figure17.Output responses (x 1 (t) and x 2 (t)) for minimum phase operating condition of QTP with IOLQI controller using APSO algorithm.

Table 13 .
Tracking performance characteristics for IOLQI and FOLQI controllers of QTP under nominal conditions.

Table 14 .
Tracking performance characteristics for IOLQI and FOLQI controllers of QTP under disturbance conditions.