Design of PID controllers for dead-time systems using

380

Int. J. Automation and Control, Vol. 4, No. 4, 2010

Design of PID controllers for dead-time systems using simulated annealing algorithms G. Kanthaswamy* Department of Electrical and Electronics Engineering, PSG College of Technology, Coimbatore 641004, India E-mail: [email protected] *Corresponding author

Jovitha Jerome Department of Instrumentation and Control Engineering, PSG College of Technology, Coimbatore 641004, India E-mail: [email protected] Abstract: The method of simulated annealing (SA)-based optimisation provides a non-oscillatory bounded solution closed-loop response for a large class of optimisation problems. The idea is to find a proportional-integralderivative (PID) controller setting based on minimisation of cost function that makes the response from the set point to plant output as close as possible. In this paper, the performance of a PID controller designed using the SA, directed search SA based on simplex simulated annealing, heuristic pattern search-based simulated annealing (HPSSA) algorithms are exhaustively discussed and the results are found promising compared to the earlier nonheuristic methods proposed by Kookos and George Syrcos; Luyben; Visioli; and Chidambaram. Keywords: PID control; proportional-integral-derivative control; SA; simulated annealing; SSA; simplex simulated annealing; DSSA; directed search simulated annealing; HPSSA; heuristic pattern search-based simulated annealing; time-delay systems; ISE; integral square error. Reference to this paper should be made as follows: Kanthaswamy, G. and Jerome, J. (2010) ‘Design of PID controllers for dead-time systems using simulated annealing algorithms’, Int. J. Automation and Control, Vol. 4, No. 4, pp.380–397. Biographical notes: G. Kanthaswamy is a PhD Student at the Department of Electrical and Electronics Engineering, PSG College of Technology, Coimbatore, India. He holds a Master’s Degree in Control and Instrumentation Engineering from Anna University, Chennai, India. His domain of interest is mathematical modelling and optimal control design. Jovitha Jerome is the Professor and the Head in the Department of Instrumentation and Control Engineering, PSG College of Technology, Coimbatore, India. She obtained her PhD from the Asian Institute of Technology, Thailand. Her domain of interest is power system stability and control. She has been teaching and conducting research since 1981 and has

Copyright © 2010 Inderscience Enterprises Ltd.

Design of PID controllers for dead-time systems

381

widely published research papers in international journals. She is also a Reviewer for numerous journals. She is an Active Member of various technical societies.

1

Introduction

Heuristics are typically used to solve complex (large, non-linear, non-convex (i.e. containing many local minima) multivariate combinational) optimisation problems that are difficult to solve to optimality. Heuristics are useful in dealing with local optima without getting trapped in them while searching for the global optimum as given by Osman and Kelly (1996). Simulated annealing (SA) is considered one of the powerful methods in metaheuristic optimisation as given by Kirkpatrick (1984), Kirkpatrick et al. (1983) and Press and Teukolsky (1991). It draws from analogy between the cooling of a material (search for minimum energy state) and the solving of an optimisation problem. The analogy being that if a liquid material cools and anneals too quickly, then the material will solidify into a suboptimal configuration. If the liquid material cools slowly, the crystals within the material will solidify optimally into a state of minimum energy (i.e. ground state). This ground state corresponds to the minimum of the cost function in an optimisation problem. Metaheuristic methods are accepted as good solvers in the design of controllers as given by Martí (2002). However, these methods suffer from the high computational cost due to their slow convergence, especially when applied to complex problems. The main reason for this slow convergence is that these methods explore the global search space by creating random movements without using much local information about promising search direction (Osman and Kelly, 1996). In contrast, local search methods have faster convergence due to their use of local information to determine the most promising search direction by creating logical movements. However, local search methods can be easily entrapped in local minima.

min

J

K P , K I , K D

s.t.

(1)

for K P , K I , K D  S where J is a real-valued function defined on the search space, S  R n . J denotes the cost function to minimise namely the integral square error (ISE) or integral time absolute error (ITAE) or integral absolute error (IAE). Usually, the search space S is defined as PID  R n : PID m,!,o  ª¬lowerm ,!,o , upperm,!,o º¼ , m, n, o 1, 2,3,! , n where

^

`

PID [ K P , K I , K D ] . The feasible region is defined by the constraints denoted by f Ž S . The solution of an unconstrained optimisation controller design problem given by (1) shall be addressed by several approaches. One approach shall combine meta-heuristic methods with local search methods to design more efficient methods with relatively faster convergence than pure meta-heuristic methods. Moreover, these hybrid methods are not easily entrapped in local minima because they still maintain the merits of the meta-heuristic methods. Direct search methods, as local search methods, have received

382

G. Kanthaswamy and J. Jerome

much attention in these combinations. In addition, Kvasnicka and Pospichal (1997) proposed a hybrid model of a controlled random search method, which is a generalisation of the Nelder–Mead (1965) method and SA. In this paper, proportional-integral-derivative (PID) controllers are designed using the general SA algorithm and some improved forms of SA algorithms. The methods of improvement employ to hybridise SA, as a meta-heuristic method, and direct search methods, as local search methods. Simple direct search (SDS) method being a local search method is hybridised with standard SA to design a new method called simplex simulated annealing (SSA) with faster convergence. Finally, direct search simulated annealing (DSSA) method is obtained by modifying the SSA with acceleration of cooling schedule in SSA and applying Kelley’s modification of the Nelder–Mead method on best solutions proposed by Kelley (1999). The DSSA algorithm is found to be very robust and effective. Other methods of hybridising SA with simplex methods include the following. Press and Teukolsky (1991) add a positive logarithmically distributed variable, proportional to the control annealing temperature T, to the function associated with every vertex of the simplex. Likewise, they subtract a similar random variable from the function value at every new replacement point. Then, their method may accept a new simplex whose actual function values at its vertices are not better than those at the previous simplex. Subsequently, this method was studied by Cardoso et al. (1996, 1997). The other main results were presented by Kvasnicka and Pospichal (1997). Their method depends on the use of the SA acceptance in a controlled random search method. More precisely, the controlled random search uses a simplex method on randomly selected simplex sets from the population. So, to avoid being entrapped in local minima, they applied SA acceptance on the updating procedure. The common idea underlying these hybrid approaches and the approach given in this paper is to use simplex method to generate new logical movements while applying SA. In this paper, an approach given by Hedar and Fukushima (2002) to fix some disadvantages of SA like its slowness and its wandering near the global minimum in the final stage of search is used for design of PID controllers. Pattern search (PS)-based SA methods constitute a subclass of direct search methods with exploratory moves from the current solution to trial points made along pattern directions with a certain step size. If these exploratory moves give no improvement, then the step size is decreased to refine the search as given by Hedar and Fukushima (2004). In this paper, the heuristic pattern search based simulated annealing (HPSSA) based on approximate descent direction (ADD), a derivative-free heuristic method is used to produce an ADD at the current solution as given by Hedar and Fukushima (2004). In the HPS method, the ADD method is recalled to obtain an ADD, v at the current iterate. If no improvement is obtained along the vector v , then v is used to prune the set of PS directions to generate other exploratory moves. Finally, SA is hybridised with HPS to construct a global search method, called the simulated annealing heuristic pattern search (SAHPS) method. The SAHPS tries to get better movements through the SA acceptance procedure or by using the HPS procedure. This paper is organised as follows. Section 2 describes the general PID controller and the optimisation problem formulation. The general approach of simulated annealing for designing PID controllers is dealt in Section 3, the design of PID controller using DSSA in Section 4 and the description for the controller design using SAHPS in Section 5. The simulation results for class of time-delay systems are compared in Section 6. Section 7 deals with the comparison of conclusion and future scope of the discussed work.

Design of PID controllers for dead-time systems

2

383

PID controller design

The PID controller is given by Chidambaram (1998) as follows

G (s)

KP 

KI  KDs S

(2)

where KP

kc

KI

kc Ti

KD

kcTd

Under the case of unconstrained optimisation of PID controller, the cost function is chosen as ISE index given by Equation (3) J

min

¦

e2 (3)

K P , KI , K D

for K P , K I , K D  S Under the case of constrained optimisation of PID controller, the cost function shall be chosen as ISE index given by Equation (4) min

J

KP , K I , KD

s.t. (4)

g (e)  1 h(e)  1.5 for K P , K I , K D  S Nomenclature

KP

Proportional controller gain

KI

Integral controller gain

KD

Derivative controller gain

kc

Controller gain

J e

ISE

g (e)

Process output

h (e)

Error band

Error Signal

384

3

G. Kanthaswamy and J. Jerome

Simulated annealing

Annealing makes the particles of the solid material to reach a minimum energy state. This is due to the fact that when the solid body is heated to the very high temperature, the particles of the solid body are allowed to move freely and when it is cooled slowly, the particles are able to arrange themselves so that their energy is made minimum. The mathematical equivalent of the thermodynamic annealing as described above is called SA. The energy of the particle in thermodynamic annealing process can be compared with the cost function to be minimised in an optimisation problem. The particles of the solid can be compared with the independent variables used in the minimisation function. Initially, the values assigned to the variables are randomly selected from the wide range of values. The cost function corresponding to the selected values are treated as the energy of the current state. Selecting particular values from the wide range of possible values can be compared with the particles flowing in the solid body when it is kept in high temperature. The next energy state of the particles is obtained when the solid body is slowly cooled. This is equivalent to randomly selecting the next set of the values. When the solid body is slowly cooled, the particles of the body try to reach a lower energy state. But as the temperature is high, random flow of the particles continues to take place and hence there may be chance for the particles to reach higher energy state during this transition. The SA process can be implemented using the Boltzmann Probability distribution of an energy level E (t 0) at temperature T described by Equation (5). p ( E ) D exp(  E / KT ) with the Boltzmann constant K and

D

§ 1 · ¨© ¸ KT ¹

(5)

In the same fashion, the values are randomly selected so that cost of the currently selected random values is minimum compared with the previous cost function value. At the same time, the values corresponding to the higher cost function compared with the previous cost function are also selected with some probability. The probability depends upon the current simulated temperature ‘T’. If the temperature is large, probability of selecting the values corresponding to higher energy levels are more. A system at high temperature has uniform probability of being at any energy state; but at low temperature, it has small probability of being at high energy state. This process of selecting the values randomly is repeated for the finite number of iteration. The values obtained after the finite number of iteration can be assumed as the values with lowest energy state (i.e.) lowest cost function. Figure 1 shows the flow chart for implementation of the PID controller scheme using the general SA algorithm. Variables [ P, I , D] are the gains of the PID controller used. The [ P, I , D] ranges are initialised to meet the range of the design vector.

Design of PID controllers for dead-time systems Figure 1

Flow chart for SA-based PID controller design

385

386

4

G. Kanthaswamy and J. Jerome

Directed search simulated annealing

SSA is hybridised using direct search method to achieve faster convergence. The modification obeys the scheme given by Hedar and Fukushima (2002): 1

accelerate the cooling schedule in SSA using small value of reduction factor for temperature, T

2

store the best solutions found by the accelerated SSA in a list called ‘best list’ as mentioned earlier and apply another local search method starting from each element of the best list to further improve these best solutions.

DSSA ( S , H , f , TMIN , M ) 0.

Select an initial simplex S with vertices [ K P , K I , K D ] : set the parameters of the cooling schedule: the initial temperature T , TMIN and M : set the size of the best list.

Choose a sufficiently small number H > 0: 1.

Order: order and relabel the vertices of S so that f ( K P1 , K I1 , K D1 )  f ( K P2 , K I2 , K D )  f ( K Pi1 , K I , K D ) holds. 2

i 1

i 1

2.

Best list: store the m best points in the best list.

3.

If f ( K Pi1 , K I , K D )  f ( K P , K I , K D )  H or T  TMIN ; then go to step 5. i 1

i 1

Otherwise, go to step 4. 4.

Repeat the following steps 4.1–4.4 M times.

4.1 Let k

1

4.2 Reflect: compute the k reflected points {[ K Pi , K Ii , K Di ]r }nn 1k  2 by ªKP , KI , KD º i i ¼ ¬ i

r

centroid ª¬ K Pi , K I i , K Di º¼



 U * centroid ª¬ K Pi K Ii K Di º¼  ª¬ K Pi , K I i , K Di º¼



(6)

where i

(7)

n  1, n,! , n  k  2

centroid > K P , K I , K D @

1 n  k 1

n  k 1

¦ ª¬ K i 1

Pi , K Ii

, K Di º¼

(8)

Choose U  (0.9,1.1) and find f ([ K Pi , K I i , K Di ]r ) and store the value f *  KP , KI , KD

r min n 1 i  n  k  2 f § ª¬ K Pi , K I i , K Di º¼ · © ¹

4.2.1 If f * ( K P , K I , K D )  f ( K P1 , K I1 , K D1 ), then go to step 4.2.3.

(9)

Design of PID controllers for dead-time systems

387

4.2.2 Compute p

^



exp  f *  K P , K I , K D  f K P1 , K I1 , K D1

/ T `

(10)

and choose U randomly from the interval (0: 1): If p < U; then go to step 4.2.3. Otherwise, let k k  1 and go to step 4.4. 4.2.3 Set ªKP , KI , KD º i i ¼ ¬ i

r

ªKP , KI , KD º , i i i ¼ ¬ i

n  1, n,! , n  k  2

(11)

Go to step 4.3. 4.3 Sort: Sort the vertices of S so that (1) holds and update the best list. 4.4 If k  n ; then go to step 4.2. 5.

Reduce the temperature T and go to step 2.

6.

From each point in the best list, construct a smaller simplex. Then, apply Kelley’s modification of the Nelder–Mead method on each of these simplices.

5

Hybrid pattern search simulated annealing

The SA approach is combined with the HPS to form the hybrid method SAHPS, which is expected to have a higher ability to detect global minima. At each major iteration of the SAHPS method, SA acceptance trials are repeated m1 times. Each time, a trial point is generated by using an exploring point to guide the SA search along a promising direction and to avoid making a blind random search. Specifically, an exploring point zk close to the current iterate xk is generated and an SA trial is generated along the direction sin( f ( xk )  f ( zk ))( zk  xk ), with a certain step size. If more than mac out of m1 trials are accepted, then we immediately proceed to the next major iteration of SAHPS. Otherwise, within the same major iteration, we repeat the HPS iterations m2 times. In the early stage of the search, diversification is needed more than intensification. However, the converse is needed in the final stage of the search. Since the HPS represents the intensification part of the SAHPS, it is better to initialise the value of m2 at a moderate value and increase it while the search is going on. In the end of the search, we complete the algorithm by applying a fast local search method to refine the best point obtained by the search so far. The final step of the algorithm uses Kelley’s modification (1999) of the Nelder–Mead method (1965). The detailed description of the algorithm is clearly given in Hedar and Fukushima (2004) and hence not explained in this paper.

6

Case studies

The response for unit step set point change was simulated for the case studies having different time-delay/lag ratios. The computation was performed using a Core 2 duo processor based, 2 GHz/3 Gigabyte computer with computational time in the ranges same

388

G. Kanthaswamy and J. Jerome

as that of Kookos and Syrcos (2005), having values in the order of few seconds to estimate the optimal values of the controller gains. The controller configuration used in most of the simulation was Smith PID configuration given in Chidambaram (1998) except for some cases.

6.1 Control of integrating processes with time delay 6.1.1 Example 1 Integrating processes with time delay (IPTD) are most commonly modelled among industrial processes. For example, processes involving level exhibit the integrating behaviour. They are found in distillation columns, boiler controls and several industrial applications. The transfer function of an integrating process with time delay is given by Equation (12) as: kp

g ( s)

s

e  ds

(12)

The PID controller tuning values for this model have been given by a number of researchers namely Kookos and Syrcos (2005), Tyreus and Luyben (1992), Luyben (1996a,b), Chidambaram (1998) and Chidambaram and Padma Sree (2003). Consider the industrial example of level control in a distillation column having the open loop transfer function as in Equation (13) 0.0506 6 s e s

g ( s)

(13)

with parameters k p = 0.0506 and d = 6 sec. The SA, DSSA and HPSSA-based search optimisation were carried out for minimisation of the ISE and the tuning parameters achieved by the above techniques using Smith PID, given in Chidambaram (1998) configuration is compared with other methods as given in Table 1. The results of the simulation tabulated shows that the SA, DSSA and HPSSA-based PID scheme yield lower value of ISE compared to other methods. Table 1

Comparison of the tuning parameters and ISE for unit step set point response of IPDT system – example 1

Method

KP

Luyben (1992)

2.563

0.0455

Visioli (2001)

4.5

0.5033

15.93

14.1482

Chidambaram (1998) and Chidambaram and Padma Sree (2003)

4.066

0.1504

10.979

9.1920

SA-based Smith PID

3.6763

0.4970

2.1120

2.7798

SA-based PID

2.0969

0

0.7473

9.1565

DSSA-based PID HPSSA-based Smith PID

KI

3.2134

0.0035

100.5919

107.2938

KD

9.130

11.1965 0

ISE , M

8.2763

7.3073 0.2904

Design of PID controllers for dead-time systems Figure 2

Unit step set point response of the closed-loop IPDT system – Example 1 using smith PID

Figure 3

Unit step set point response of the closed-loop IPDT system – Example 1 using PID

389

Figures 2 and 3 compares the closed-loop performance of the SA, DSSA and Smith PID/PID controller and the tunings proposed by Tyreus and Luyben (1992), Visioli (2001) and that of Chidambaram (1998) and Chidambaram and Padma Sree (2003).

390

G. Kanthaswamy and J. Jerome

The unit step set point response for the proposed controller configuration results in a more or less dead beat response compared to the other schemes. The comparison of the tuning factors and the value of the performance index, ISE is tabulated in Table 1.

6.1.2 Example 2 Consider the second example of level control process in a distillation column given by Chen and Fruehauf and referred by Kookos and Syrcos (2005) has the open loop transfer function as given in Equation (14) g ( s)

0.2 7.4 s e s

(14)

with parameters k p = 0.2 andd = 7.4 sec. Figures 4 and 5 shows the closed-loop performance of the SA, DSSA and HPSSAbased controller and the tunings proposed by Kookos and Syrcos (2005) and Luyben (1992). The obtained tunings were used to compute the performance index given by Equation (3). The SA, DSSA and HPSSA-based search optimisation were carried out for minimisation of the ISE and the tuning parameters achieved by the same for Smith PID/PID method are compared with other methods as given in Table 2. The system response for unit step set point and unit step set point change was simulated for a time of t = 150 sec. The comparison of the tuning factors and the values of the performance indices ITAE and ISE obtained are tabulated in Table 2. Figure 4

Unit step set point response of the closed-loop IPDT system – Example 2 using Smith PID

Design of PID controllers for dead-time systems Figure 5

Table 2

391

Unit step set point response of the closed-loop IPDT system – Example 2 using PID

Comparison of the tuning parameters and ISE for unit step set point response of the closed-loop IPDT system – example 2

Method

KP

KI

KD

ISE , M

Luyben (1992)

0.33

0.0051

0

21.7991

Kookos and Syrcos (2005)

0.553

0.0178

0

17.0434

SA-based Smith PID

3.1993

0.6935

0.8144

0.8144

DSSA-based PID

0.6457

0.0007539

2.8136

9.0255

0

0.2902

HPSSA-based Smith PID

25.4451

27.1368

6.2 Systems with inverse response and time delay Consider an example of a time delay process with inverse response and positive zero, which was investigated by Kookos and Syrcos (2005), having the transfer function as given in Equation (15). The delay component when approximated using numerical methods such as Taylor series, Pade’s method gives infinite number of zeroes. For example, the first-order pade approximant of the delay component gives rise to addition of one zero and one pole to the system transfer function; while the second-order gives addition of two zeros and two poles and similarly adds consequent poles and zeros for Nth order expansion making the system dynamics complicated. In such methods of approximations, the expansion could lead to infinite number of zeros which is very difficult to account for controller design as system order increases. The proposed SA, HPSSA, DSSA-based PID/Smith PID used no approximation technique to arrive at the equivalent of the delay component and gives close optimal solution using optimisation approach.

392

G. Kanthaswamy and J. Jerome 1  1.6s

g ( s)

(1  s )

2

e 1.6 s

(15)

The process has time delay of 1.6 sec Figure 6 compares the closed-loop performance of the SA, DSSA and HPSSA-based PID controller and the tunings proposed by Kookos and Syrcos (2005). The SA, DSSA and HPSSA-based scheme shows improved unit step set point as well as set point response. The simulations show that the DSSA-based PID and HPSSA PID achieved same settings and comparatively similar response to that of Kookos. The comparison of the tuning factors and the value of the performance index, ISE are tabulated in Table 3. Figure 6

Table 3

Unit step set point response of the system with inverse response and time delay – Case study 6.2

Comparison of the tuning parameters and ISE for unit step set point response of the system with inverse response and time delay

Method

KP

KI

KD

ISE , M

11.824

3.0560

Kookos and Syrcos (2005)

6.905

1.707

SA-based Smith PID

4.0368

0.3513

4.4795

3.5851

SA-based PID

3.2218

0.1432

4.7610

5.9133

DSSA-based Smith PID DSSA-based PID HPSSA-based Smith PID HPSSA-based PID

28.1077

0

32.6316

0.7251

6.2415

0.6490

14.7161

2.2611

24.7686

2.1709

32.2883

0.7464

6.2416

0.6489

14.7157

2.2611

Design of PID controllers for dead-time systems

393

6.3 Control of an unstable process Chemical reactors and complex systems have unstable steady state. A number of researchers have proposed methods for tuning unstable processes with time delay as given by Kookos and Syrcos (2005). Consider the following open-loop system as given by Equation (16) with initial tunings proposed by Kookos and Syrcos (2005).

e 0.5 s (5s  1)(2s  1)(0.5s  1)

g ( s)

(16)

The comparison of the tuning factors and the value of the performance index, ISE are tabulated in Table 4. Figures 7 and 8 compares the closed-loop performance of the SA, DSSA and HPSSA-based scheme and the tunings proposed by Kookos and Syrcos (2005). The SA, DSSA and HPSSA-based controller shows a comparatively better response than that of Kookos and Syrcos (2005) with reduced error index. Table 4

Comparison of the tuning parameters and ISE for unit step set point response of unstable system with time delay KP

Method

KI

Kookos and Syrcos (2005)

6.905

1.707

SA-based Smith PID

4.0368

0.3513

SA-based PID DSSA-based Smith PID DSSA-based PID HPSSA-based Smith PID HPSSA-based PID Figure 7

3.2218 28.1077

0.1432

KD

ISE , M

11.824

3.0560

4.4795

3.5851

4.7610

5.9133

0

32.6316

0.7251

6.2415

0.6490

14.7161

2.2611

24.7686

2.1709

32.2883

0.7464

6.2416

0.6489

14.7157

2.2611

Unit step set point response of unstable process with time delay – Case study 6.3 using smith PID

394

G. Kanthaswamy and J. Jerome

Figure 8

Unit step set point response of unstable process with time delay – Case study 6.3 using PID

6.4 Control of integrating process with time delay and inverse response Consider the general form of transfer function of an integrating process with time delay and inverse response given in Equation (17) having a right half plane zero at 1/ W z , delay d and a pole at zero. g ( s)

kp

1  W z s  ds e s (W P s  1)

(17)

Kookos and Syrcos (2005) and Luyben (1996a,b, 2001) have SA, DSSA and HPSSAbased controller PID controller setting for a integrating process with time delay and inverse (IPTD & IV) and consider the general form of transfer function of an system having transfer function of form given in Equation (18) g ( s)

0.457

1  0.418s 0.1s e s (1.06 s  1)

(18)

Figures 9 and 10 compare the closed-loop performance of the SA, DSSA and HPSSAbased controller and the tunings proposed by Kookos and Syrcos (2005). The responses of proposed DSSA-based Smith PID and proposed HPSSA-based Smith PID both yielded same settings for the Smith PID controller and hence the responses of both the techniques are identical as shown in Table 5. The SA, DSSA and HPSSA-based controller shows a comparatively similar response to that of Kookos with closeness in performance.

Design of PID controllers for dead-time systems Figure 9

Figure 10

Unit step set point response of integrating process with inverse response and time delay – Case study 6.4 using Smith PID

Unit step set point response of integrating process with inverse response and time delay – Case study 6.4 using PID

395

396

G. Kanthaswamy and J. Jerome

Table 5

Comparison of the tuning parameters and ISE for unit step set point response of integrating process with inverse response and time delay KP

KI

Kookos and Syrcos (2005)

1.117

0.1767

0

2.9751

Luyben (1992)

0.854

0.03713

0

2.4801

Method

KD

ISE , M

SA-based Smith PID

2.9718

0.0025

2.1263

1.0964

DSSA-based PID

2.1220

0

1.3116

1.5018

HPSSA-based PID

2.1220

0

1.3116

1.5018

7

Conclusion

In this paper, a novel Meta-heuristic-based PID controller for a class of time-delay systems is designed using SA, DSSA and HPSSA algorithms. The usefulness of the designed control scheme is compared with different case studies involving delays. The results show that the PID/Smith PID designed using SA, HPSSA and DSSA offers good sensitivity for closed-loop performance with reduced performance indices and the results are comparatively similar and more easily achieved for most of the cases than the techniques proposed by Kookos and Syrcos (2005), Luyben (1992, 1996a,b), Visioli (2001) and Chidambaram (1998, 2003). Even more encouraging was the fact that, the settling time and ISE for the PID controller coefficients selected by SA, DSSA and HPSSA are seemingly better than the choices made by other schemes for majority of the case studies considered. Most of the results show that both the methods of DSSA and HPSSA yield same controller gain settings and is found to have more or less similar convergence. The design scheme proposed in this paper can be further improved in performance by hybridisation and improvement of the convergence of the SA algorithm to yield bounded and close optimal solution.

References Cardoso, M.F., Salcedo, R.L. and de Azevedo, S.F. (1996) ‘The simplex-simulated annealing approach to continuous non-linear optimization’, Computer and Chemical Engineering, Vol. 20, pp.1065–1080. Cardoso, M.F., Salcedo, R.L., de Azevedo, S.F. and Barbosa, D. (1997) ‘A simulated annealing approach to the solution of MINLP problems’, Computer and Chemical Engineering, Vol. 21, pp.1349–1364. Chidambaram, M. (1998) Applied Process Control, New Delhi: Allied Publishers Ltd. Chidambaram, M. and Padma Sree, R. (2003) ‘A simple method of tuning PID controllers for integrator/dead-time processes’, Computational Chemical Engineering, Vol. 27, pp.211–215. Hedar, A. and Fukushima, M. (2002) ‘Hybrid simulated annealing and direct search method for nonlinear unconstrained global optimization’, Optimization Methods and Software, Vol. 17, pp.891–912. Hedar, A. and Fukushima, M. (2004) ‘Heuristic pattern search and its hybridization with simulated annealing for nonlinear global optimization’, Optimization Methods and Software, Vol. 19, pp.291–308.

Design of PID controllers for dead-time systems

397

Kelley, C.T. (1999) ‘Detection and remediation of stagnation in the Nelder–Mead algorithm using a sufficient decrease condition’, SIAM Journal of Optimization, Vol. 10, pp.43–55. Kirkpatrick, S. (1984) ‘Optimization by simulated annealing: quantitative studies’, Journal of Statistical Physics, Vol. 34, Nos. 5/6, p.975. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P (1983) ‘Optimization by simulated annealing’, Science, Vol. 220, No. 4598, pp.671–680. Kookos, I. and Syrcos, G. (2005) ‘PID controller tuning using mathematical programming’, Chemical Engineering and Processing, Vol. 44, pp.41–49. Kvasnicka, V. and Pospichal, J. (1997) ‘A hybrid of simplex method and simulated annealing’, Chemometrics and Intelligent Laboratory Systems, Vol. 39, pp.161–173. Luyben, W.L. (1996a) ‘Design of proportional integral and derivative controllers for integrating dead-time processes’, Indian Engineering Chemical Research, Vol. 35, p.3480. Luyben, W.L. (1996b) ‘Identification and tuning of integrating processes with dead time and inverse response’, Indian Engineering Chemical Research, Vol. 42, pp.3030–3035. Luyben, W.L. (2001) ‘Effect of derivative algorithm and tuning selection on the PID control of dead-time processes’, Indian Engineering Chemical Research, Vol. 40, pp.3605–3611. Martí, R. (2002) ‘Multi-start methods’, In F. Glover and G. Kochenberger (Eds.), Handbook of MetaHeuristics, Boston, MA: Kluwer Academic Publishers, pp.355–368. Nelder, J.A. and Mead, R. (1965) ‘A simplex method for function minimization’, The Computer Journal, Vol. 7, pp.308–313. Osman, I.H. and Kelly, J.P. (1996) Meta-Heuristics: Theory and Applications, Boston, MA: Kluwer Academic Publishers. Press, W.H. and Teukolsky, S.A. (1991) ‘Simulated annealing optimization over continuous spaces’, Computer of Physics, Vol. 5, pp.426–429. Tyreus, B. and Luyben, W.L. (1992) ‘Tuning PI controllers for integrator/dead time processes’, Indian Engineering Chemical Research, Vol. 31, pp.2625–2628. Visioli, A. (2001) ‘Optimal tuning of PID controllers for integral and unstable processes’, IEE Proceedings of Control Theory Application, Vol. 148, p.180.