Genetic Algorithm Applied to State Feedback Control

1

Genetic Algorithm Applied to State Feedback Control Design P. S. Oliveira, L. S. Barros and L. G. de Q. Silveira Júnior

Abstract--Control design to MIMO systems requires the state space representation. In the state space, the control strategy is the states feedback and the most used techniques are the LQR and the LQG. In despite of the good results obtained from these methods, the control design is not a straightforward task due to the trial and error method involved in the definition of weight matrices. In such cases, may be hard tuning the controller parameters in order to obtain the optimal behavior of the system. In this work, it proposes a states feedback technique in which there are no trial and error processes involved and the control design is carried out to fulfill specifications, for maximum overshoot and accommodation time. The proposed technique is based on the modification of the states transition matrix and uses genetic algorithms. The obtained results show that is possible to design controllers which fulfill design specifications. Index Terms— Genetic Algorithm, States Feedback Control Design, States Transition Matrix Modification.

I. INTRODUCTION

C

URRENTLY, requirements for control systems are more strict due to the need to perform more complex and higher precision tasks. In order to satisfy the increasing demands for the control design it is necessary to use modern control theory, which applies to complex multiple-input and multiple-output systems (MIMO), linear or nonlinear, [1]. The control design for MIMO systems requires the system state space representation and it uses state feedback as control strategy. The control systems performance depends on the parameters calculated through the use of optimization techniques. The most used techniques for state feedback control design are the Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian Regulator (LQG), [2]. In despite of the good results obtained from these methods, their startup realizations are not straightforward tasks due to the trial and error method involved in the definition of weight matrices. In such cases, may be hard tuning the controller parameters in order to obtain the optimal behavior of the system. The authors are grateful to Capes (Improvement Coordination of Upper Level) for the finantial support provided through the PRÓ-ENGENHARIA Program. P. S. Oliveira and L. S. Barros are with the Federal University of SemiArid Region, Mossoro-RN, Brazil (e-mails: [email protected] and [email protected]). L. G de Q. Silveira Junior is with the Communications Engineering Department, Federal University of Rio Grande do Norte, Natal-RN, Brazil (email: [email protected]).

In this work, it proposes a genetic algorithm (GA) whose the evolution occurs in order to attend the design specifications. Hence, there are no weight matrices to be chosen and it avoids the trial and error method in the control parameters search. The design specifications are the maximum overshoot and the accommodation time. The obtained results provided by the proposed GA are compared to the obtained results provided by the LQR technique. The proposed GA presented results as good as the LQR results, and the control parameters tuning is simpler for the proposed technique. II. STATES FEEDBACK CONTROL DESIGN A engineering systems trend is increasing its complexity mostly due to the need for performing complex and higher precision tasks. Due to the performance stringent requirements of control systems, the modern control theory and design of complex control systems have been developing since 1960. The classical control theory is applicable only for linear, time invariant, single input and output systems and works in the frequency domain. The modern control theory is applied to multiple inputs and outputs systems, linear or nonlinear, and works in the time domain. The modern control theory is based on the state concept. State is the smaller set of variables at = which in connection with the input for ≥ determines fully the behavior of the system for any time, ≥ . State variables are the variables , , … , which together determine the state of the system in a given time, . State vector is the set of n state variables which describe fully the behavior of the system. State-space is the n-dimensional space whose the coordinate axes are , , … , . State-space equations involve three types of variables present in the modeling of dynamic systems: input variables, output variables and state variables. The state equations are represented as follows, [2]. = + = +

Where, is the states vector, order ; is the input vector; order ; is the output vector, order and , , and are the state-space model matrices.

(1) (2)

2

A. State Equation Solution The Laplace Transform approach for the state equation solution for homogeneous case is presented below. = (3) Considering the Laplace Transform of equation (3), it obtains: () − (0) = () (4) Where, X(s) = ℒ[x] and (0) is the initial state. Solving equation (4) for X(s) () =

(0) = ( − )( (0) −

(5)

The Inverse Laplace Transform of this last equation provides the solution: () = * (0) +,

This approach can be extended to the vector case:

(6)

() = ()

(7)

() − (0) = ()

(8)

Considering the Laplace Transform in both sides of equation (7), it obtains:

Therefore,

(0 − ) () = (0)

(9)

() = (0 − )( (0)

(10)

() = ℒ ( [(0 − )( ](0)

(11)

Where I is the identity matrix. Pre-multiplying both sides of (9) by (0 − )( ,

The Inverse Laplace Transform of () provides the solution (). So, Which results in

() = * 2, (0)

B. State Transition Matrix

(12)

We can write the solution of homogeneous state equation as:

() = Φ() (0)

(13)

Φ() = * 2,

(14)

Where Φ() is an × matrix, which is given by The unique solution of:

Φ () = Φ(t),

To verify this, note that:

Φ(0) = I

(0) = Φ(0)(0) = (0)

() = Φ () (0) = Φ(0)(0) = ()

(15)

(16)

(17)

We confirm, therefore, that Equation (6) is the solution of Equation (3). From equations (11), (12) and (13), it obtains:

Note that

Φ() = * 2, = ℒ ( [(0 − )( ]

(18)

Φ( () = * (2, = Φ(−)

(19)

From Equation (7), we noted that the solution in Equation (6) is simply a transformation of initial conditions. Therefore, the matrix Φ() is called state transition matrix. It contains all information about the free response of the system defined by Equation (1), [1]. If the eigenvalues 7 , 7 , … , 7 of the matrix are distinct, then Φ() contains n exponentials * 89, , * 8:, , … , * 8; , . In a particular case, if is diagonal, then Φ() = * 2,

* 89 , > = = = = = < 0

* 8: ,

.

.

0

.

B A A A A * 8; , @

(20)

III. GENETIC ALGORITHM APPLIED TO CONTROL DESIGN A. Control Strategy In the control strategy in state-space the pole placement can be performed in order to achieve a satisfactory dynamic response. In sophisticated systems this task can require computerassisted calculation, but we are especially interested in obtain a procedure for controller parameters tuning which gives dynamic response with a minimum overshoot and short accommodation time, both previously defined by the designer. However, it is recognized that searching an optimum (in the sense of overshoot and transient period minimization) is not a simple task. Let us consider the general linear state space model described by the equations (1) and (2), and let the states feedback law be = −C (21) Thus, the Equation (1) becomes = ( − C)

(22)

Φ() = * (2(DE)F

(23)

As shown in (6) and (12), the general solution of the state equation is given as function of the state-transition matrix, Φ(), which for the fedback system in (22) is Now, the time behavior of the state variables depends on the feedback gain matrix C. For an order system, it has H () H () Φ() = G ⋮ H ()

H () H () ⋮ H ()

… H () … H () K ⋱ ⋮ … H ()

where all elements of Φ are functions of the elements of C. For the system having inputs:

L L C=G ⋮ LM

L L ⋮ LM

2. = P

… L … L K ⋱ ⋮ … LM

0 −1

3

0 0.9 Q; = P Q; = [ 0 1 −0.03

1 ]; = [ 0 ]

In this work, a Genetic Algorithm (GA) is used in order to obtain elements of C which minimizes the maximum overshoot and the accommodation time response of the fedback system. To achieve this goal, the elements of Φ are taken as simultaneous cost functions. B. Genetic Algorithm Proposed

IV. NUMERICAL RESULTS Digital simulations has carried out ,using the MATLAB, [8], in order to investigate and compare the performance improvements produced by the proposed GA and the LQR method. This section presents such performance rformance results obtained for two systems: −1 1. = P 6.5

−1 0 Q; = P Q; = [ 0 0 1

Fig. 1. Steps of Genetic Algorithm Proposed.

For the system 1,, the elements of C obtained by the proposed GA and by the LQR method are shown in Table 1. A. System 1 Results

Table 1. System 1 control design data.

1 ]; = [ 0 ]

Controller parameters C = [−6.228714 228714 8.673320] C = [−3.4198 − 8.0259]

GA LQR

For the GA, the specifications used were S = 5% and NO = 2 . Since the LQR method consists in a trial and error design process, the search for its gain C was done until the eigenvalues of the matrix − C be equal to t the eingenvalues obtained by the GA. The eigenvalues are shown in Table 2. Table 2. System 1 eigenvalues. eigenvalues

Eigenvalues 7 = −3.4255 4255 and 7 = −6.2478 7 = −2.9570 9570 and 7 = −6.0689

GA LQR

Figures 2, 3, 4 and 5 illustrate the comparison between GA and LQR for the cost functions. Besides, the result for the original system, without control, is shown. 1 Without Control Linear Quadratic Regulator Genetic Altorithm

0.8 0.6 0.4

Φ11(t)

It shouldd be noted that the goal in controller design techniques is the same employed by minimizing a cost function which models the system performance on combinatorial optimization problem, [3].. Genetic algorithms, when applied to solve such problems, can provide optimum solutions at shorter time than required to evaluate valuate all possible solutions [4]. The first step in any application where GA is used consist consists in define a chromosome representation in order to describe each individual in the population of interest,, [5]. Specifically, each individual in the population (or chromosome) is formed by a sequence of genes defined in a certain alphabet (real or binary values, besides others) [6].. In this work only real realvalued representation of each gene is considered. Before starting the GA, the user specifies value values for the accommodation time, NO , and maximum overshoot overshoot, S, which are the scores used to valuate a solution. Afterwards Afterwards, our GA starts with an initial population,, usually a set of randomly selected possible solutions to the controller setting, and through application of operators like crossover and mutation a new intermediate generation is obtained. Applying these same operators in present population, a new generation is formed, and then this process continues until a convergence criterion is reached (where the better solution will have the higher fitness value), or the desired number of generations is accomplished [7]. As shown in (24), the he state transition matrix generates expressions for an order system. Usin Using these cost functions, it is possible evaluate the accommodation time and the maximum overshoot for each expression, which allow to compare these values es with the specified by user. If they are lower than or equal to specification,, then the fitness function assigns one point to potential solution, otherwise none. Thus, an individual may have up to four points for each criterion in a system of order 2. Figure 1 shows the entire processing flow of the proposed method.

0.2 0 -0.2 -0.4 -0.6 -0.8

0

1

2

3

4

5 6 Time(sec)

7

Fig. 2. Conparison results for Φ .

8

9

10

4

Where (0) and (0) are the initial conditions for the state variables. So, the time response for the state variables depends on the H functions behavior. The state variables time behaviors are shown in figures 6 and 7.

0.2 Without Control Linear Quadratic Regulator Genetic Altorithm

0.15 0.1 0.05

1

Φ12(t)

0

x 1 - Without Control

-0.05

-0.2 -0.25

0

1

2

3

4

5 6 Time(sec)

7

8

9

10


2

State variable - x1

-0.15

-0.3

x 1 - Linear Quadratic Regulator

0.5

-0.1

0

-0.5

-1

-1.5

Without Control Linear Quadratic Regulator Genetic Altorithm

1.5

x 1 - Genetic Algorithm

-2

0

2

4

6 Time(sec)

8

10

12

Fig. 6. System 1 free response for x1 from the initial condition x1(0)=1 and x2(0)=5.

1

0.5

Φ21(t)

6 x 2 - Without Control

0

5

x 2 - Linear Quadratic Regulator x 2 - Genetic Algorithm

4

-0.5

-1.5

0

1

2

3

4

5 6 Time(sec)

7

8

9

10


1

1 0

-2 -3

0.6

0

2

4

6 Time(sec)

8

10

12


0.4

Φ22(t)

2

-1


0.8

State variable - x2

3 -1

0.2

From these figures, it can observe that the performance obtained with the proposed technique is the same obtained from conventional LQR method. However, GA design process is done from design specifications.

0 -0.2 -0.4 -0.6 -0.8

For the system 2, the elements of C obtained by the proposed GA and by the LQR method are shown in Table 3. B. System 2 Results

0

1

2

3

4

5 6 Time(sec)

7

8


9

10

These results show that the performance of the proposed GA is as good as the LQR and both of them present satisfactory oscillations damping. Since for this system we have () = H (). (0) + H (). (0) () = H (). (0) + H (). (0)

Table 3. System 2 control design data.

GA LQR

Controller parameters C = [8.623016 10.824402] C = [9.0499 10.7538]

For the GA, the specifications used were S = 10% and NO = 5 . The eigenvalues are shown in Table 4.

5 Table 4. System 2 eigenvalues.

Eigenvalues 7 = −0.8672 and 7 = −9.9872 7 = −0.9167 and 7 = −9.8671

Φ11(t)

Figures 8, 9, 10 and 11 illustrate the comparison between GA and LQR for the cost functions. Besides, the result for the original system, without control, is shown.

1 0.8 0.6 0.4 0.2

Φ22(t)

GA LQR

0

1

-0.2

0.8

-0.4

0.6

-0.6

0.4

-0.8

0.2

-1

Without Control Linear Quadratic Regulator Genetic Altorithm 0

1

-0.2

-0.6 Without Control Linear Quadratic Regulator Genetic Altorithm

-0.8

0

1

2

3

4

5 6 Time(sec)

7

8

9

10

4

5 6 Time(sec)

7

8

9

10

4


1

x 1 - Without Control 3


2 State variable - x1


0.8 0.6 0.4 0.2

Φ12(t)

3

Again, the behavior of the cost functions is improved for both control techniques. The effects of the control action on the state variables time behavior is shown in figures 12 and 13.

-0.4

-1

2


0

0

1 0 -1 -2

-0.2

-3

-0.4 -4 -0.6 -0.8 -1

0

1

2

3

4

5 6 Time(sec)

7

8

9

10

2

4

6

8

10 12 Time(sec)

14

16

18

20



1

0

4 x 2 - Without Control 3


State variable - x2

2 0.5

Φ21(t)

0

1 0 -1 -2

-0.5

-3 -1

-4

Without Control Linear Quadratic Regulator Genetic Altorithm -1.5

0

1

2

3

4

5 6 Time(sec)

7


8

9

10

0

2

4

6

8

10 12 Time(sec)

14

16

18

20


6

From these figures, it can observe that the performance obtained with the proposed technique is the same obtained from conventional LQR method. V. CONCLUSION In this paper it presented a method which uses the genetic algorithm technique applied to the state feedback control design. The control strategy consists in the modification of the state transition matrix for the fedback system. This method has been proposed to outline the difficult at the control design process when the LQR method is used. This difficult consist consists in the definition on of weight matrices, since th these matrices do not present a well defined relation with the time response of the controlled system. Both oth techniques have been applied to two systems and their performances compared, among them, and to the system without control. In according ccording to the obtained results, it noted that the proposed method presented performance equivalent to the LQR, however, owever, it presented the guarantee of the response to satisfy itss specifications for maximum overshoot and accommodation time time, since the proposed method has the design process based on the these specifications of the system time response. The designer can also change these specifications in order to change requirements to adjust the time which the system returns to its initial state and the maximum overshoot tolerated, in cases of disturbances. It is important to emphasize that for faster and damper time response, the energy required by the controller is larger. VI. ACKNOWLEDGEMENT The authors are grateful Coordination of Upper Level).

to Capes (Improvement

VII. REFERENCES [1]

K. Ogata, “Engenharia de Controle Moderno – 4ª EDIÇÃO”, Editora: Prentice-Hall, ISBN-9788587918239, 2003.

[2]

R. C. Dorf and R. H. Bishop, “Sistemas Sistemas de Controle Modernos – 11ª Edição”, Editora: LTC, ISBN-9788521617143,, 200 2009. R. Subbu, K. Goebel, and D. K. Frederick, “Evolutionary design and optimization of aircraft engine controllers,” IEEE Trans. Syst., Man, Cybern. C, Appl. Rev.,vol. 35, no. 4, pp. 554–565, 565, Nov. 2005.

[3]

[4] [5]

S. N. Sivanandam, S. N. Deepa. Introduction to Genetic Algorithms. Springer, 2008. Adaptation in Natural and Artificial Systems Systems”, MI: Univ. J. H. Holland, “Adaptation of Michigan Press, 1975.

[6]

G. Wang, M. Zhang, X. Xu, C. Jiang, "Optimization of Controller Parameters Based oh The Improved Genetic Algorithms," Proceedings of the 6th World Congress on Intelligence Control and Automation, June 21-23, 2006, Dalian, China.

[7]

R. I Haupt, S. E. Haupt. Practical Genetic Algorithms. John Wiley and Sons, segunda edição, 2004.

[8]

MATLAB – Version 7.6, The Language of Technical Computing Computing. The Mathworks, 2008.

VIII. BIOGRAPHIES IOGRAPHIE Phelipe Sena Oliveira iveira was born in Utinga, Utinga Brazil. He completed eted his graduate degree de in 2008 in Computer Science cience at the State University of Southwest Bahia. Currently, Currently he is the master degree student of the partnership program between the Federal University of Semi Arid Region and the State University of Rio Grande do Norte. Norte His interest areas involve artificial intelligence, specifically evolutionary algorithms.

Luciano Sales Barros was born in Campina Grande, Brazil. He concluded his graduate degree in Electrical Engineering ngineering in Federal University of Paraiba, iba, 2000. He Concluded his M.Sc. and Ph.D. degrees in Electrical lectrical Engineering in Federal University of Campina Grande, 2002 and 2006, respectively. Currently, he is with the Federal University of Semi-Arid Semi Region, where is professor of Energy Engineering. ngineering. His interest areas are control theory, power systems analysis and control and wind energy. He received his BEng, MSc, and PhD degrees in Electrical Engineering from the Federal University of Campina Grande. He is currently professor in the Communications Engineering Department at the Federal University of Rio Grande do Norte, Brazil, where his efforts are concentrated in the areas of advanced schemes for Wireless Communications Systems and Digital Signal Processing. Dr Silveira has published many papers in international conferences ces and journals. He is a member of the Brazilian Society of Telecommunications (SBrT), Brazilian Society of Automation (SBA) and co-founder founder of IQUANTA.