Design of a Fractional-order Self-tuning Regulator using Optimization Algorithms

Deepyaman Maiti, Mithun Chakraborty, Ayan Acharya, and Amit Konar

Department of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata, India

Abstract — The self-tuning regulators form an important

sub-class of adaptive controllers. This paper introduces a novel scheme for designing a fractional order self-tuning regulator. Original designs for all the sub-modules of the self-tuning regulator are proposed. The particle swarm optimization algorithm is utilized for online identification of the parameters of the dynamic fractional order process while the subsequent tuning of the controller parameters is performed by differential evolution. Results show that the proposed self-tuning regulator is both precise and robust.

Index Terms — Controller tuner, differential evolution, fractional order self-tuning regulator, parameter identifier, particle swarm optimization.

I. INTRODUCTION

Development of a control system involves many tasks such as modeling, design of a control law, implementation and validation. The self-tuning regulator (STR) attempts to automate several of these tasks. This is illustrated in Fig. 1, which shows the block diagram of a process with an STR. It is assumed that the structure of the process is specified.

Parameters of the process are estimated on-line by the block labeled ‘parameter identifier’. The block labeled ‘controller tuner’ contains computations that are required to perform the design of a controller with a specified method and a few design parameters that can be chosen externally. The design problem is called the underlying design problem for systems with known parameters. The block labeled ‘controller’ is an implementation of the controller whose parameters are obtained from the ‘controller tuner’ block [1].

In this paper we have presented a scheme for the design of a fractional order STR. This means that the physical process that needs to be controlled as well as the controller that will control it are both of fractional order. The real world objects or processes that we want to estimate are generally of fractional order [2].

A typical example of a non-integer (fractional) order system is the voltage-current relation of a semi-infinite lossy RC line or diffusion of heat into a semi-infinite solid, where heat flow q(t) is equal to the half-derivative

of temperature T(t): ).t(qdt

)t(Td5.0

5.0=

So far, however, the usual practice when dealing with a fractional order process has been to use an integer order approximation. Disregarding the fractional order of the

system was caused mainly by the non-existence of simple mathematical tools for the description of such systems. Since major advances have been made in this area recently, it is possible to consider also the real order of the dynamical systems. Such models are more adequate for the description of dynamical systems with distributed parameters than integer-order models with concentrated parameters.

It is necessary to understand the theory of fractional calculus in order to realize the significance of fractional order integration and derivation. Fractional calculus is the branch of calculus that generalizes the derivative or integral of a function to non-integer order, allowing calculations such as deriving a function to 1/2 order. For instance sα indicates derivation to the order α. Knowledge in the subject of fractional calculus is essential to design fractional order controllers. Of the several definitions of fractional derivatives, the Grunwald-Letnikov and Riemann-Liouville definitions are the most used [3]. These definitions are required for the realization of discrete control algorithms.

The design of an STR can be divided into sub-tasks such as designing the modules for parameter estimation [4] and controller design [5]. For a chosen structure of a fractional order system, we have designed the ‘parameter identifier’ and ‘controller tuner’ blocks of Fig. 1. As a matter of fact, the controller we have considered is of a fractional order PID type, which differs from the usual integral order PID controller by the property that the orders of integration and derivation are positive real rather than restricted to only positive integers (conventionally unity). Fractional order PID controllers are much superior to their integer order counterparts, especially when controlling fractional order processes [6].

For designing the ‘parameter identifier’ block, we have made use of a stochastic optimization strategy from the family of evolutionary computation, namely particle swarm optimization (PSO) [7] – [8]. The ‘controller tuner’ block utilizes differential evolution (DE) [9] – [10].

Verification of the precision of our design is performed by operating the STR on a system with known parameters, and also by simulation. The robustness of our design is also displayed.

Proceedings of 11th International Conference on Computer and Information Technology (ICCIT 2008)25-27 December, 2008, Khulna, Bangladesh

1-4244-2136-7/08/$20.00 ©2008 IEEE 470

Fig. 1. Block diagram of a self-tuning regulator

II. STOCHASTIC OPTIMIZATION ALGORITHMS

A. The Optimization Problem

The optimization problem consists in determining the global optimum (in our case, minimum) of a continuous real-valued function of n independent variables x1, x2, x3,

…, xn, mathematically represented as ⎟⎟⎠

⎞⎜⎜⎝

⎛→Xf , where

)x,.....x,x,x(X n321=→

is called the parameter vector. Then the task of any optimization algorithm reduces to searching the n-dimensional hyperspace to locate a

particular point with position-vector DX→

such that

⎟⎟⎠

⎞⎜⎜⎝

⎛→DXf is the global optimum of ⎟⎟

⎠

⎞⎜⎜⎝

⎛→Xf .

B. Particle Swarm Optimization

The PSO algorithm [7] - [8] attempts to mimic the natural process of group communication of individual knowledge, which occurs when a social swarm elements flock, migrate, forage, etc. in order to achieve some optimum property such as configuration or location.

The ‘swarm’ is initialized with a population of random solutions. Each particle in the swarm is a different possible set of the unknown parameters to be optimized. Representing a point in the solution space, each particle adjusts its flying toward a potential area according to its own flying experience and shares social information among particles. The particles swarm toward the best fitting solution encountered in previous iterations with the intent of encountering better solutions through the course of the process and eventually converging on a single minimum error solution.

Let the swarm consist of N particles moving around in a D-dimensional search space. Each particle is initialized with a random position and a random velocity. Each

particle modifies its flying based on its own and companions’ experience at every iteration. The ith particle is denoted by Xi, where Xi = (xi1, xi2, …, xiD). Its best previous solution (pbest) is represented as Pi = (pi1, pi2, …, piD). Current velocity (position changing rate) is described by Vi, where Vi = (vi1, vi2, …, viD). Finally, the best solution achieved so far by the whole swarm (gbest) is represented as Pg = (pg1, pg2, …, pgD). The fitness function evaluates the performance of particles to determine whether the best fitting solution is achieved. The particles are manipulated according to the following equations:

id id 1 1 id id

2 2 gd id

v (t 1) ωv (t) c . .(p (t) x (t)) c . .(p (t) x (t))

ϕϕ

+ = + − +−

)1t(v)t(x)1t(x ididid ++=+ . (The equations are presented for the dth dimension of the

position and velocity of the ith particle.) Here, c1 and c2 are two positive constants, called

cognitive learning rate and social learning rate respectively, ϕ1 and ϕ2 are two random functions in the range [0,1], ω is the time-decreasing inertia factor which balances the global wide-range exploitation and the nearby exploration abilities of the swarm.

C. Differential Evolution

DE [9] - [10] belongs to the class of evolutionary algorithms guided by the principles of Darwinian Evolution and Natural Genetics where each time-varying parameter vector (candidate solution) in the population is called a chromosome and each time-step represents a generation. The first step of the algorithm, as usual, is:

Initialization: This step is identical to the random initialization of position vectors in PSO.

Each iteration consists of the following three steps:

Mutation: For each chromosome )t(Xi→

belonging to the

current generation, three other chromosomes )t(X p→

,

)t(X q→

and )t(X r→

are randomly selected from the same

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generation (i, p, q and r are distinct); the scaled difference

of )t(X q→

and )t(X r→

is added to )t(X p→

to generate a

donor vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−×+=+

→→→→)t(X)t(XF)t(X)1t(V rqpi

where F is a constant scalar in (0,1). We take F = 0.8.

Recombination: A trial offspring vector )1t(T i +→

is

created for each current-generation parent vector )t(Xi→

by first choosing a constant CR (0<CR<1) called the crossover constant and then setting the jth component

)1t(T j,i +→

of )1t(T i +→

according to:

( ) if (0,1)( 1)

( ) otherwise,

jV t rand CR,i, j

T ti, j X ti, j

<+ =

⎧⎪⎨⎪⎩

where randj (0, 1) is a random number selected from the interval (0, 1), j = 1, 2,..., n. We take CR = 0.96.

Selection: This step is be mathematically expressed as: ( 1) if ( ( 1)) ( ( ))

( 1) ( ) otherwise,

1 2

T t f T t f X t ,i i iX ti X tii , , .......,m.

+ + <+ =

∀ =

⎧⎪⎨⎪⎩

Thus, the next-generation population is generated, keeping the population-size m always unchanged.

III. DETAILED DESIGN SCHEME OF THE PARAMETER

IDENTIFIER BLOCK

We have considered a fractional process whose transfer

function is of the form 321 asasa

1++ βα

. It should be

noted that without loss of generality, we may presume the dc gain to be unity so that the dc gain and its possible fluctuations are included in the coefficients a1, a2 and a3. This fractional order system has five varying parameters, namely three coefficients a1, a2 and a3, and two fractional powers α and β. (Of course any other structure can be chosen due to the generalness of our design method. This particular structure merely serves as an example.)

A. Design Principles of the Identifier

Let R(s), C(s), E(s) and U(s) denote respectively the reference input, the output, the error signal (input to the controller) and the control signal (input to the process) respectively.

The controller we have considered (as already mentioned) is a fractional order PID controller, whose transfer function is of the form δλ− ++ sTsTK dip .

dip T,T,K are the proportional, integral and derivative constants respectively, δλ, are the orders of integration and derivation. We can have by simple block diagram algebra,

321dip

dip

asasasTsTK

sTsTK

)s(R)s(C

+++++

++=

βαδλ−

δλ−,

⎥⎦

⎤⎢⎣

⎡−=−=⇒

)s(R)s(C1)s(R)s(C)s(R)s(E .

For reference input R(s) = s1 (unit step),

( )( )( )321dip

321dip

asasasTsTKs

asasasTsTK)s(U

+++++

++++=

βαδλ−

βαδλ−,

( )321dip

dip

asasasTsTKs

sTsTK)s(C

+++++

++=⇒

βαδλ−

δλ−.

We sample the output values c(t) in discrete time domain and store them. The PSO algorithm will search the solution space to come up with a process model which will replicate the observed values of c(t) for the same signal U(s). At the moment of parameter identification, the controller has retained its old values of δλ,,T,T,K dip (fixed the last time the controller was tuned), but the process parameters 321 a,,a,,a βα have changed. So the optimization problem is five-dimensional since we know the controller parameters from its last tuning.

Let, for a process model, the output response is p(t). We

will define a parameter ∑=

−=f

it

2)]t(p)t(c[F , which gives

a measure of the deviation of the output of the trial process model from the output of the actual process. F is the fitness function that the PSO algorithm will try to minimize. At F = 0, the unknown parameters are optimized. The position vector of the best particle, i.e. the optimized value of }a,,a,,a{ 321 βα is the identified parameter set. The process model for the optimized solution set should provide output identical to c(t).

There are five unknown parameters to be optimized. So the present problem of system identification can be solved by a straightforward application of the PSO algorithm for optimization on a five-dimensional solution space, each particle having a five-dimensional position and velocity vector. This approach gives very accurate results, even when the input data is corrupted to a significant extent. We studied this scheme under both ideal and non-ideal (random error component added to readings) conditions.

B. Illustration of the Identifier

Synthetic data for c(t) is created using a1 = 0.8, α = 2.2, a2 = 0.5, β = 0.9, a3 = 1. The controller parameters are:

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pK =410, iT =650, dT =50, λ =0.25, δ =1.25. Sampling frequency is 20 samples per second. Error in taking readings of c(t) is simulated by adding a random number in the range [ 05.0− , 05.0 ] to each reading.

The PSO parameters used are: the inertia factor ω decreases linearly from 0.9 to 0.4; the cognitive learning rate c1=1.4, the social learning rate c2=1.4. Number of particles in the population is 50. The PSO algorithm is run for 200 iterations, and this is kept as the stop criterion. The search ranges are: 1a , 2a , 3a : 0 to 2.0 (for all three parameters), α: 2.0 to 2.4 and β: 0.7 to 1.1. The velocity ranges are: 21 a,a , 3a : –0.5 to 0.5 (for all three parameters), α, β: -0.1 to 0.1 (for both parameters).

Fig. 2. Output waveform c(t) used for identification

Fig. 3. Variation of best fitness with number of iterations

TABLE I.

IDENTIFICATION AND PERCENTAGE ERRORS Parameter Estimate Percentage Error

1a 0.8000 0.0002 α 2.2000 0.0000

2a 0.5000 0.0035

β 0.9000 0.0042

3a 1.0000 0.0030

Table 1 gives results of the identification when the readings are accurate (error component is not added). After 200 iterations, the best fitness is 2.5946 X 10-14.

Even when the random error component is added, the identification is very accurate.

C. Analysis

The challenge in fractional order system identification is that the fractional powers are not restricted to assume only discrete integral values, but are distributed in a continuous interval. However, the use of the stochastic optimization strategy makes an accurate identification possible.

So the parameter identifier block can estimate the process parameters with a very high degree of accuracy. Now the controller tuner module should use these values of the identified parameters to tune the controller parameters on-line.

IV. DETAILED DESIGN SCHEME OF THE CONTROLLER

TUNER BLOCK

A. Design Principles: Formulation of the Objective Function and its Optimization

Our approach is based on the root locus method (dominant poles method) of designing integral PID controllers [11]. As in the traditional root locus method, the peak overshoot Mp and rise time trise (or, in other words, requirements of stability and damping levels) are specified. From these specifications, we find out the damping ratio ζ and the undamped natural frequency ωn, making use of (1), (2):

22p

p

)}M{ln(

)Mln(

π+−=ζ (1)

2rise

21

n1t

1tan

ζ−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ζζ−

−π

=ω

−

(2)

Using these computed values of ζ and ωn, we then determine the desired positions of the dominant poles p1,2 of the closed loop system:

jba1jp 2nn2,1 ±−=ζ−ω±ζω−= (3)

where na ζω= , 2n 1b ζ−ω= , a, b > 0.

Let Gp(s) be the transfer function of the process we want to control, Gc(s) the transfer function of the controller to be designed and H(s) the transfer function of the feedback-path. Then, the closed loop transfer function of the

controlled system is: )s(H)s(G1

)s(G)s(T

+= ,

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C PG(s) = G (s).G (s) being the forward path transfer

function: δλ− ++= sTsTK)s(G dipc (4)

Assuming unity feedback, we have H(s)=1. In this case, the characteristic equation becomes 0)s(G1 =+ ,

0)s(G).s(G1 pc =+⇒ 0)s(Q)s(P

).s(G1 c =+⇒

0)s(P).sG)s(Q c =+⇒ (5) where P(s) and Q(s) are respectively the numerator and denominator polynomials of G(s), P(s) and Q(s) have no common factor.

As p1,2 must be poles of the closed loop system, each of them must be a root of the characteristic equation and hence must satisfy equation (5). Thus, putting

jbaps 1 ±−== in equation (5), we obtain 0)p(P).p(Gp(Q 11c)1 =+

( ) 0)jba(P.)jba(T)jba(TK)jba(Q dip =+−+−++−+++−⇒ δλ−

(6) Equation (6) is a complex equation in five unknowns,

namely Kp, Ti, Td, λ, δ and our problem of designing a controller that makes the closed loop dominant poles of the system coincide with 2,1p now reduces to determining the set of values of {Kp, Ti, Td, λ, δ} for which (6) holds good. But as the number of unknowns exceeds the number of equations, there exists an infinite number of solution sets and the equation cannot be unambiguously solved by traditional methods.

This necessitates the application of stochastic global search techniques, which, in turn, requires the formulation of a suitable objective function or cost function.

Let, R = real part of the L.H.S. of equation (6), I = imaginary part of the L.H.S. of equation (6),

P = RItan 1− .

We define ),,T,T,K(f dip δλ = ⏐R⏐ + ⏐I⏐ + ⏐P⏐ as our objective function. Clearly, f ≥ 0, in general, and f = 0 if and only if R = 0 and I = 0 and P = 0, i.e. if and only if equation (6) is satisfied.

So, we now employ DE to scour the five-dimensional search space and home in on the optimal solution set

p i d{K T T λ δ }* * * * *, , , , for which f = f min = 0. It is germane to mention here that the formulae (1), (2)

and (3) hold strictly only for a second order system with its complex conjugate pole-pair at 2,1p . However, for a higher-order system with a dominant pole-pair (real part of this pole-pair is much smaller than those of other poles), these formulae are widely used in control engineering applications with a fair degree of accuracy [11]. Nevertheless, after designing our controller with the

help of these formulae, we perform a simulation to obtain the unit step response of the closed-loop control system as a check.

B. Illustration of the Tuner

The process (control objective) has the transfer function

1s5.0s8.0

1)s(G9.02.2p

++= . We want to design a

controller such that the closed loop system has a peak overshoot Mp ≤ 10% and rise-time trise ≤ 0.3 seconds.

Using the formulae (1) and (2), we obtain ζ = 0.5912 and ωn = 9.107 s-1. Thus the dominant poles for the closed loop controlled system should lie at

)345.7j384.5(p1 +−= and )345.7j384.5(p 2 −−= . As per convention, we use unity feedback. The

controller transfer function is given by (4). Putting )345.7j384.5(ps 1 +−== in the characteristic equation,

01)345.7j384.5(5.0)345.7j384.5(8.0

)345.7j384.5(T)345.7j384.5(TK1

9.02.2dip =

++−++−

+−++−++

δλ−

(7) After separating the real and imaginary parts, we have:

R ( 13.4235) cos(2.203 )9.107

(9.107) cos(2.203 )

ip

d

TK

T

λ

δ

λ

δ

= + + + (8)

I sin(2.203 )9.107

(9.107) sin(2.203 ) 98.9237

i

d

T

T

λ

δ

λ

δ

= − +

−

(9)

P = RItan 1− . (10)

Thus, (8), (9) and (10) give us our objective function ),,T,T,K(f dip δλ = ⏐R⏐ + ⏐I⏐ + ⏐P⏐.

We set the limits on the components of the position-vectors of the chromosomes (i.e. the controller parameters) as follows: as a practical consideration, we assume 1000T,T,K0 dip ≤≤ , 2,0 ≤δλ≤ .

The stop criterion used is a tolerance value f = 0.0001. After the DE algorithm is run, the position vector of the

best particle gives the parameters of the optimum PID controller. The transfer function of the identified controller is 37.179.1 s27.46s55.19780.962 ++ − .

Although we have constructed the objective function f by making use of the dominant pole jbap1 +−= in the second quadrant, we would arrive at the same f if we had started with the third-quadrant dominant pole

jbap 2 −−= . This is because, in the latter case, the imaginary part of the reduced characteristic equation (6) would just be the negative of what we have obtained in (7) so that f, which involves absolute values only, would

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remain unaltered. This is true not only for the particular problem in question but in general as well.

Fig. 4 shows the unit step response of the uncontrolled system as well as the unit step response of the system controlled by the designed STR. The peak overshoot is 5.2%, and the rise time is 0.054 seconds.

Fig. 4. Unit step responses of the control objective

C. Analysis of the Tuner Designing Scheme

The controller meets the rise-time and peak overshoot requirements more than satisfactorily. The rise-time performance especially far exceeds the design requirement, which means that the controller tuner is actually over-performing and we can have a significant safety margin. We also considered using PSO in the ‘controller tuner’ block as well. However we found that DE offers significantly better performance in this case.

An important point is to be noted in this context. For the final system to exhibit the desired performance, it is necessary that the evaluated pole-values 2,1p truly correspond to the dominant poles of the closed-loop controlled system. However, the optimization of our cost function ensures only that 2,1p are poles of the system but does not guarantee that these are the dominant poles. In other words, equation (7) embodies a necessary, but not sufficient, condition for the dominance of 2,1p . Obviously, there may be a number of possible combinations of values of the parameters

],,T,T,K[ dip δλ for which the resulting characteristic

equation is satisfied by 2,1p but, in each such case, the

characteristic equation will have other roots (closed-loop system poles), too, and these poles will also play an important role in determining the overall system response. This is a possible explanation of the observation that the performance of the final designed system with fractional controller is actually better than desired.

V. COMMENTS AND CONCLUSIONS

An elegant and effective design scheme for a fractional order STR is proposed. A simple computer program in C or MATLAB using sampled data acquisition and other interfaces with the physical process can actually implement the complete STR. Of course, the same design can easily be employed to implement an integer order STR as well. Apart from the example shown, we also considered several other similar problems and solved them by our proposed method. These have not been included in this paper owing to paucity of space. We also intend to apply other well-known algorithms such as Genetic Algorithm, Bacterial Foraging Optimization and their many variants to the same optimization problems and compare their performance with the presented scheme.

REFERENCES

[1] K. J. Astrom, and B. Wittenmark, Adaptive Control, Second Edition, Pearson Education, Inc., 1995.

[2] P. J. Torvik, and R. L. Bagley, “On the appearance of the fractional derivative in the behaviour of real materials,” Trans. of the ASME, vol. 51, June 1984, pp. 294-298.

[3] K. B. Oldham, and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

[4] J. P. Norton, An Introduction to Identification, Academic Press, London, 1986.

[5] I. Petras, “The fractional order controllers: Methods for their synthesis and application,” Journal of Electrical Engineering, vol. 50, no. 9-10, pp. 284-288, 1999.

[6] S. Milos, and C. Martin, “The fractional-order PID controller outperforms the classical one,” 7th Intl. Scientific-Technical Conf. – Process Control 2006, Czech Republic.

[7] J. Kennedy, and R. C. Eberhart, “Particle swarm optimisation,” Proc. of the IEEE Intl. Conf. on Neural Networks, pp.1942-1948, 1995.

[8] Y. Shi, and R. C. Eberhart, “A modified particle swarm optimiser,” Proc. of the IEEE Congress on Evolutionary Computation, pp. 69-73, 1998.

[9] R. Storn, and K. Price, “Differential evolution – A simple and efficient heuristic for global continuous spaces,” Journal of Global Optimization, 11(4) (1997) 341–359.

[10] S. Das, A. Konar, and U. K. Chakraborty, “Two improved differential evolution schemes for faster global search,” ACM-SIGEVO Proc. of Genetic and Evolutionary Computation Conf. (GECCO-2005), 2005.

[11] I. J. Nagrath, and M. Gopal, Control Systems Engineering, Fifth Edition, New Age International Publishers, 2007.

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