hybrid predictive control design based on particle swarm optimization and

7/31/2019 Hybrid Predictive Control Design Based on Particle Swarm Optimization And

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Hybrid Predictive Control Design based on Particle Swarm Optimization and

Genetic Algorithm

Yaser Mohammad Nezhad1 and Mehdi Shahbazian2Department of Instrumentation and Automation

Petroleum University of Technology

Ahwaz, Iran

[email protected] [email protected]

AbstractThis paper discusses a model predictive control

approach to hybrid systems with continuous and discrete

inputs. The algorithm, which takes into account a model of a

hybrid system, described as Hybrid Automaton. However, toavoid computational complexity and computation time, the

nonlinear optimization problem is solved by evolutionary

algorithms (EA) such as Genetic Algorithms (GA) and Particle

Swarm Optimization (PSO). We have applied both GA and

PSO algorithms for nonlinear optimization in Hybrid

Predictive Control (HPC) for the start-up of a Continuous

Stirred-Tank Reactor (CSTR). The simulation results show the

good performance of approaches and their capability to use in

online application.

Keywords-component; Hybrid Systems; Mixed Integer

Programming; Particle Swarm Optimization; Genetic Algorithm;

I. INTRODUCTIONIn general, processes that comprises continuous as well

as on/off variables and logic decision of operation can beformulated as hybrid systems where continuous variablesare represented by real variables and the logic and on/offvariables can be converted into binary variables in a set ofinequalities. Hybrid dynamic models have been identied asa suitable means to account for dynamic-dependenttransitions between dierent models and for thecombination of continuous as well as discrete (switching)inputs, see e. g. [1].

Dierent approaches to the optimization of hybridsystems have been published in recent years, ranging from

rather generic formulations to tailor-made methods forcertain subtypes of hybrid systems, see e.g. [2]-[6].

One branch of methods follows the idea of transformingthe hybrid dynamics into a set of algebraic (in-) equalitiesthat serve as constraints for a mixed-integer program [7].

In order to appropriately control processes that containdiscrete and/or continuous variables (hybrid systems), hybrid

predictive control techniques has been developed. This leads,in the context of Model Predictive Control (MPC) [8], to amixed integer optimization problem that must be solved atevery sampling time [9]. The philosophy of MPC techniquesis based on the on-line use of an explicit process model, to

predict and thereby optimize the effect of future control

actions (manipulated variable changes) on the controlledvariables (process outputs) over a finite horizon.

The essential drawback with mixed integerquadratic/linear programs (MIQP/MILP) is its computationalcomplexity, which increases the time needed to finding thesolution due to the presence of logical optimization variables[10]. So, very often, the computation times are not as short asdesired, preventing its real-time implementation for manyindustrial processes. To overcome this drawback theoptimization problem can be solved efciently, by usingevolutionary algorithms. Therefore, two efcient andoptimization evolutionary algorithms, GA and PSO have

been used in this paper. PSO is similar to GA in a sense butemploys different strategies and computational effort. GAmimics the natural biological evolution and has been popular.PSO is a relatively recent heuristic search method whose

mechanism is inspired by the swarming social behavior ofspecies.

In this paper, a hybrid predictive control strategy hasbeen used to provide a safe start-up operation of thecontinuous stirred tank reactor. HPC consists of optimizingthe process behavior to obtain optimal future control actions

by using the model of the system with mixed inputs.Therefore, it is nontrivial to design a controller which willcontrol the process. The optimization problem has beensolved with the EA such as PSO and GA.

II. THE OPTIMAL CONTROL PROBLEM

The task under investigation is to determine the

continuous and discrete inputs of a hybrid system such thatthe process is driven from an initial state into a target regionwith minimal costs. The model considered in thiscontribution is formulated as a hybrid automaton accordingto [9]. Such a model is suitable to represent the transition

procedure, as it includes continuous and discrete inputs, andit can express the state-dependent switching betweendierent continuous dynamics. A short summary of hybridautomaton is given as follows:

A. Hybrid AutomatonThe hybrid automaton model may be described briey as

follows. The discrete part of the dynamics is modeled by

___________________________________978-1-61284-840-2/11/$26.00 2011 IEEE


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means of a graph whose vertices are called discrete states or

modes, and whose edges are transitions. The continuous

state takes values in a vector space X. For each mode there

is a set of trajectories, which represents the continuousdynamics of the system. Interaction between the discrete

dynamics and continuous dynamics takes place through

invariants and transition relations. Each mode has an

invariant associated to it, which describes the conditions that

the continuous state has to satisfy at this mode. Each

transition has an associated transition relation, which

describes the conditions on the continuous state under which

that particular transition may take place and effect that the

transition will have on the continuous state. The notion of a

transition relation is split up into two components, namely a

guard which specifies the subset of the state space where a

certain transition is enabled, and a reset map which is a (set-

valued) function that specifies how new continuous statesare related to previous continuous states for a particular

transition. Schematic representation of a hybrid automaton

is illustrated in Fig 1.

A hybrid automaton is a collection

( , , , , , , , )Q X f Init Inv E G R= .

1, 2{ ,..., }NQ q q q= is a finite set of discrete states,

nX is a set of continuous states,

1 1, ... ,

n

c c c cm cmU u u u u + + = \ is the

continuous input space,

{ }1 2, ,...,d d d dnU u u u= is the discrete input space

with a nite number nd of discrete inputsn

\

:f Q X X is a vector field,

Init Q X is a set of initial states,

: ( )Inv Q P X describe the invariants,

E Q Q is a set of edges,

: ( )G E P X is a guard condition,

: ( )R E P X X is a reset map.

Note that P(X) is a power set ofX , i.e., the collectionof all subset ofX . Hence, note that the guard condition G

gives for each mode a subset of0 . Let 0 1 2{ , , ,...}T t t t =

be an ordered set of time points

0

kt

\

. The hybrid statevariable of the system is given by ( , ) ( )q x Q X . The

initial hybrid state 0 0( , )q x of trajectories of hybrid

automaton lies in the initial set Init. From this hybrid statethe continuous state x evolves according to the differential

equation 0 , )(x xf q= with 0(0)x x= and the discrete state

q remains constant 0( )q t q= . The continuous evolution

can go on as long asx stays in 0( )Inv q . If at some point the

continuous state x reaches the guard 0 1( , )G q q , it says that

the transition is enabled. The discrete state may then change

to 1q , and the continuous from the current value x to the

new value x + with 0 1( , ) ( , )x x R q q +

. After this transition,

the continuous evolution resumes and the whole process arerepeated. In addition, a set of forbidden hybrid

Figure 1. Schematic representation of a hybrid automaton with 3 discrete

states.

state sets can be introduced as { ,..., }, ( , )1 j jF F F F q xn j F F = = .

Using F to explicitly exclude states that must not beencountered (e.g. for safety reasons). The general optimalcontrol task is finding the input trajectories

1 2( , ,...)uc c cu u = and 1 2( , ,...)ud d d u u = that transfer the

HA from initial hybrid state 0 0( , )q x into a target operating

region ( , )tar tar q x such that no forbidden region is

encountered, and a given cost function is minimized. We

assume that { }0 1, ,..., fT t t t = is nite and dene a subset

{ }0 1, ,...,s sT t t t T = of time points at which the inputs

can be changed.

B. Structure of Hybrid Model Predictive ControlThe philosophy of model predictive control can be stated

as follows: A model of the process is used to predict itsfuture behavior as a function of the present and future controlactions. The determination of the control action is an

optimization problem that is solved within a nite horizon h ,i.e., for a given number of time steps ahead. For each time

step k a sequence of optimal input vectors is acquired,which minimizes the given cost function while consideringthe constraints of the inputs, outputs and system states.However, only the optimal control signals corresponding tothe present time step are applied to the process during thecurrent time step, and the whole procedure is repeated in thenext time step. The model is updated with new informationof the evolution of the system, and a new optimization at

time step 1k + is performed. The hybrid predictive controlstrategy is a generalization of MPC, where the predictionmodel contains both discrete and continuous inputs. The


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following optimization problem is solved over a limited

prediction horizon h :

,min ( , , )c d

u uc d

k h u ut +

(1)

),...,,( 11 ++= hkck

c

k

ccuuuu and

1 1( , ,..., )k k k hd d dud u u u+ +

= are continuous and discrete

inputs respectively, which cu and du are piecewise

constant on [ , ]1t tk k+ . is based on the transition time

and the distance to the target region.

As mentioned earlier, due to the presence of discretevariables accompany with continuous ones the optimization

problems which now lie in the class of MIQP or MILP arecomputationally involving NP-complete problems, which

means that, in the worst case, the solution time growsexponentially with the problem size. To overcome thisdrawback two efficient optimization evolutionary algorithms,PSO and GA have been used in this paper.

III. EVOLUTIONARY ALGORITHMS

A. Particle Swarm OptimizationPSO [11] is a robust stochastic optimization algorithmwhich is dened by the behavior of a swarm of particles in amultidimensional search space looking for the best solution.Particles represent a population of candidate solutions. Each

particle is characterized by a position (i is the index of the

particle) and a velocity (both are dimensional, where d isthe dimension of the solution vector). These particles movewithin the search space with a certain law and find theglobal best result after some iteration. The particles have atness associated with the solution quality which derivedfrom the objective function as in (1). Therefore the tness isdetermined by evaluating the tness function dened as:

1f =

(2)

First the particles are initialized with a random positionand random velocity. At each iteration each particle update

its velocity vector, based on its momentum and the

influence of its best solution has been achieved so far ( idp )

and the best solution of all particles ( dp ), then computes a

new point to examine:

)()( 22111 t

id

t

gd

t

id

t

id

t

id

t

id zprczprcwvv ++=+

(3)

1 1t t t

id id id z z v+ +

= + (4)

Where w is inertia weight. 1c and 2c are cognitive

acceleration and social acceleration, respectively. 1r and 2r

are random numbers uniformly distributed in the range [0,1].

The algorithm for searching best solution stops, if either thevalue of the objective function reaches certain tolerance orthe maximum number of iteration has been reached.

The above description of PSO was restricted for solvingcontinuous problems. In our application, the PSO algorithmhas been adapted in order to add discrete variables in the

solution. To meet the need, the discrete version of PSO has

been used, which developed by [12]. It is designed forbinary optimization; to use the discrete problem a binary

encoding is required. Discrete PSO is composed of the

binary variable (0 or 1). So the velocity must be transformedinto the change of probability, which is the chance of the

binary variable taking the value one. In this version of PSO

the updating of velocity is same as continuous PSO, but foradjusting the new position the probability function of the

particle velocity was used as follows:

)exp(11)(

t

id

t

idv

vS+

= (5)

- =

+

notif

vSifz

t

idt

id

0

)(11

(6)

Where ( )tidS v and are probability function of the

particle velocity and random number respectively.

B. Genetic AlgorithmGenetic Algorithms (GA) were rst introduced in [13]

and popularized by [14]. GA provides an ecient means tosolve nonlinear optimization problems, GA maintains a setof candidate solutions called population and repeatedlymodies them. At each step, the GA selects individuals fromthe current population to be parents and uses them to producethe children for the next generation. Candidate solutions areusually represented as strings of xed length, calledchromosomes. A tness or objective function is used toreect the goodness of each member of population as in (2).Given a random initial population GA operates in cyclescalled generations, as follows:

Each member of the population is evaluated usinga tness function.

The population undergoes reproduction in anumber of iterations. One or more parents arechosen stochastically, but strings with higher

tness values have higher probability of

contributing an offspring.

Genetic operators, such as crossover and mutationare applied to parents to produce offspring.

The offspring are inserted into the population, andthe process is repeated.


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parameters have been used: w=0.95 and c1=c2=2. For bothalgorithms a population size of 20 individuals and an upperlimit of 30maxn = for the maximum number of generations

has been considered. The simulation tests are implemented ina digital computer having a core 2 Duo CPU 2 GHz with3GB of RAM using MATLAB 7.6 software package.

The sampling time is 100s, and the maximum time needed tocalculate the solution in one sampling time is obtained 5.9sand 7.1s for PSO and GA respectively, So is much shorterthan the sampling time. This means that the bothevolutionary algorithms-based control strategies are suitablefor real-time control and implementation. The simulationresults show that the optimal transition time are 107s and125s for PSO and GA, respectively. The target region isobtained after 21 sampling times for PSO and 25 for GA.These results show that PSO is faster than GA, although GAhas better performance as regards with mean tness value of

1.35 in comparison with the 1.013 of

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

ConcentrationofA

[kg/m3]

GA

PSO

Target Region

Figure 3. Concentration component A ( AC )

0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [s]

Volume[m3]

GA

PSO

Target Region

Figure 4. Volume ( RV )

0 200 400 600 800 1000 1200 1400 1600 1800300

305

310

315

320

325

330

335

340

345

350

Time [s]

Temprature[k]

GA

PSO

Target Region

Figure 5. Temperature ( RT )

the PSO. The state trajectories as the optimization results,using PSO and GA has been shown in Figures 3 to 5. Fig. 3shows the curve of concentration component A ( AC ), while

Fig. 4 and Fig. 5 display evolution of volume andtemperature states, respectively.

V. CONCLUSION

In this paper, a nonlinear model predictive controlstrategy is used for the control of a hybrid system with HAmodeling and mixed inputs. Since evolutionary algorithmshave not been widely applied in hybrid systems, weinvestigate their performance in these systems. In the

proposed HPC scheme, two evolutionary algorithms (PSOand GA) which simplify the solution of the optimization

problem have been used to solve the hybrid optimizationproblem. The comparative analysis showed a similarity inthe convergence between PSO and GA, Although PSO couldconverge faster than GA. Simulation results showed that the


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proposed control strategies using PSO and GA are suitablefor real-time control applications.

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[3] Broucke, M., M.D. Di Benedetto, S. Di Gennaro and A. Sangiovanni-Vincentelli (2000). Theory of optimal control using bisimulations. In:Hybrid Systems: Comp. and Control. Vol. 1790 of LNCS. Springer.

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[8] Camacho, E. F., and Bordons, C. (2004). Model predictive control(2nd edition). Springer Verlag.

[9] Bemporad, A., and Morari, M. (1999). Control of systems integratinglogic, dynamics, and constraints.Automatica, 35: 407427.

[10] Olarua, S., Dumur, D., Thomasb, J., and Zainea, M. (2008).Predictive control for hybrid systems. Implications of polyhedral pre-computations.Nonlinear Analysis: Hybrid Systems, 2:510-531.

[11] Kennedy, J., and Eberhart, R. (1995). Particle Swarm Optimization.Proceedings of the IEEE International Conference on NeuralNetworks, Perth, Australia, pp. 1942-1945.

[12] Kennedy, J., Eberhart, R. (1997). A discrete binary version of theparticle swarm algorithm. Systems, Man, and Cybernetics. IEEEInternational Conference on Computational Cybernetics andSimulation, 5: 4104-4108.

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hybrid predictive control design based on particle swarm optimization and

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