Stabilizing of Subspaces Based on DPGA and Chaos Genetic Algorithm for Optimizing State Feedback Controller

Abstract

The main purpose of the paper is to optimize state feedback parameters using intelligent method, GA, Hermite-Biehler, and chaos algorithm. GA is implemented for local search but it has some deficiencies such as trapping into a local minimum and slow convergence, so the combination of Hermite-Biehler and chaos algorithm has been added to GA to avoid its deficiencies. Dividing search space is usually done by distributed population genetic algorithm (DPGA). Moreover, using generalized Hermite-Biehler Theorem can find the domain of parameters. In order to speed up the convergence at the first step, Hermite-Biehler method finds some intervals for controller, in the next step the GA will be added, and, finally, chaos disturbance will help the algorithm to reach a global minimum. Therefore, the proposed method can optimize the parameters of the state feedback controller.

1. Introduction

One of the first applications of state-space approach in state feedback is to shift the system eigenvalues. According to Kalman et al. [1], J. Bertram in 1959 was the first person who discovered that if the realization of the system is in the controllable mode, characteristic equation can be used with any equation of state feedback and the poles can be located in the appropriate places [2]. Three years later in 1962, Rosenbrock used state feedback for shifting system eigenvalues to obtain better response. As a fact, design of state feedback is used for stabilizing and improving the performance of systems [3].

In this paper, genetic algorithm with distributed parameters is used to obtain the parameter of state feedback controller for unstable systems with time delay. To do so, the theory of Hermite-Biehler is used in order to calculate stable areas of controller in DPGA. As mentioned above, this theory can be applied to the stable systems. As a result, in the first step, FOPDT, which is an unstable system, must be stabilized with proportional or other types of controllers as an inner loop. In the next step, the stable state feedback parameters are found by genetic algorithm to optimize the main system [4].

In DPGA, the search space is divided into several subspaces. Genetic algorithm is done in each of these subspaces to achieve the best solution. Then by comparing the best answers of subspaces, the best subspaces is selected for next division. Therefore, the performance of algorithm is improved by decreasing the search space [4]. Then in the new division the space is divided into several subspaces that each one does not cross others.

The paper is organized as follows. A brief review of genetic algorithm is described in Section 2 and combination of DPGA and chaos GA is presented in Section 3. An experiment that is designed by combination of DPGA and chaos algorithm is proposed in Section 4 and finally we draw conclusion of this method in Section 5.

2. A Brief Review of Genetic Algorithm

A genetic algorithm is a method to search an optimal solution and genetic algorithm is a kind of evolutionary computing, which is part of artificial intelligence. This algorithm is not random search algorithm because of special properties. The main idea of this method is inspired by biological processes of living organisms. The nature of successive generations, according to the law of “adaptation for survival” selects creatures. This law, first, was raised by Darwin who suggested that genetic algorithm can be used to solve real world problems [5]. The first idea of evolutionary computation, in 1960, is proposed by Rechenberg and this theory was later used by other researchers and has been developed. Since the basic principle of genetic algorithms was presented by Holland and colleagues at the University of Michigan in 1962, then in 1975, their mathematical foundation was published in a book in the Netherlands with the name Adaptation in natural and artificial systems [6]. In 1992, Koza used genetic algorithm to perform specific tasks in the program and called this method evolutionary programming [7]. Its goal is to evaluate the performance of algorithm in order to find the answer to any problem. In recent years, genetic algorithms have been usually used in optimization problems. Important property of this algorithm is its robustness, so that a flexible balance exists between performance and the characteristics required for survival in the environment.

3. Combination of DPGA and Chaos GA

3.1. Hermite-Biehler Theorem

In this section we consider 𝛿(𝑠)=𝛿₀+𝛿₁𝑠+⋯𝛿_𝑒𝑠^𝑛 as a real polynomial of transfer function of degree 𝑛and define 𝛿(𝑠)as 𝛿(𝑠)=𝛿_𝑒(𝑠²)+𝑠𝛿_𝑜(𝑠²), where𝛿_𝑒(𝑠²),𝛿_𝑜(𝑠²) are the components of 𝛿(𝑠)made up of even and odd powers of𝑠 for𝜔∈𝑅, 𝛿(𝑗𝜔)=𝑝(𝜔)+𝑗𝑞(𝜔) where𝑝(𝜔)=𝛿_𝑒(−𝜔²),𝑞(𝜔)=𝜔𝛿_𝑜(−𝜔²)[8].

Let 𝜔_𝑒1,𝜔_𝑒2,… and 𝜔_𝑜1,𝜔_𝑜2,…denote the nonnegative real zeros of 𝛿_𝑒(−𝜔²) and𝛿_𝑜(−𝜔²) are arranged in ascending order. Then 𝛿(𝑠) is Hurwitz stable if and only if all zeros of𝑝(𝜔), 𝑞(𝜔)are real and distinct and0<𝜔_𝑒1<𝜔_𝑜1<𝜔_𝑒2<𝜔_𝑜2<⋯. Let 0=𝜔₀<𝜔₁<⋯<𝜔_𝑚−1 be the real, nonnegative, distinct finite zeros of 𝑞(𝜔) with odd multiplicities. Also define 𝜔_∞=∞, then 𝜎(𝛿(𝑠=𝑝𝜔))sgn₀−2sgn𝑝𝜔₁+⋯+(−1)^𝑚−12sgn𝑝𝜔_𝑚−1+(−1)^𝑚2sgn𝑝𝜔_𝑚𝑝𝜔⋅𝛾if𝑛iseven,sgn₀−2sgn𝑝𝜔₁+2sgn𝑝𝜔₂+⋯+(−1)^𝑚−12sgn𝑝𝜔_𝑚−1⋅𝛾if𝑛isodd, (3.1) where 𝛾=(−1)^𝑙−1sgn[𝑞(𝑘_𝑝,∞)] and 𝜎(𝛿(𝑠))= number of open left half plane zeros of 𝛿(𝑠)−number of open right half plane zeros of 𝛿(𝑠) [9].

3.2. Chaos Algorithm

Chaos algorithm is used in nonlinear systems. This algorithm is named due to chaotic disturbance that avoid being trapped in local minimum.

Integrating with chaos algorithm, genetic algorithm uses its dynamic characteristic as well as its random property for fast convergence and finding optimal solution. In the following, we discuss chaos in detail.

We use the following logical equation for the production of initial variables: 𝑋_𝑛+1=𝜇𝑋_𝑛1−𝑋_𝑛𝑛=0,1,…,𝑁, (3.2) where 𝜇=4,(𝑋₀≠0.25,0.5,0.75),0<𝑋₀<1.

Chaos variables are very sensitive to initial values; little change in initial values causes completely changes in chaotic variables [10].

Chaos algorithm has characteristics of high efficiency such as regularity, ergodicity, and stochastic properties and uses chaotic variables for optimal search. Using chaos algorithm in GA forms new method which helps to avoid trapping into local minimum and results in a fast convergences to a global optimal solution.

3.3. Stable Region for Unstable FOPDT

FODPT is described by the transfer function𝐺(𝑠)=(𝑘𝑒^−𝜃𝑠)/(𝑇𝑠−1)that is usually normalized by the delay time (𝐿=𝜃/𝑇) and when it is applied to the system; 𝐺(𝑠)is as below: 𝐺(𝑠)=𝑒^−𝐿𝑠.𝑠−1 (3.3) Controller with closed-loop characteristic equation will be as follows: 𝛿(𝑠)=𝐷(𝑠)+𝑘_𝑝𝑁(𝑠). (3.4) There are many classical methods to determine the value of 𝑘_𝑝 that 𝛿(𝑠) is Hurwitz such as technique of root locus, Nyquist stability criterion, and the Routh-Hurwitz criterion. Since the first two methods are graphical methods and fail to provide analytical characteristic of values of stable𝑘_𝑝, the Routh-Hurwitz criterion is chosen according to its ability in analytical solution. However, using this method for the stabilization process of the FOPDT does not always have a desired value of𝑘_𝑝. To obtain stable range of𝑘_𝑝, a structure with two controllers in [11] has been adopted.

In Figure 1, 𝐶₁(𝑠) is a controller that is shown in (3.5). The purpose of controlling system is stabilization and the main feedback controller is applied to the overall system which is shown in Figure 1. 𝐶₁(𝑠)=𝑘_𝑝1𝑘𝐿≤1_𝑝11+𝑇_𝑑1𝑠1<𝐿<2. (3.5)

Figure 1 

Controller of inner loop.

3.4. Stable Domain of the State Feedback Controller by DPGA

The controller design is in the following form: 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)(𝑡)(𝑡)=𝐴𝑥+𝐵𝑢=𝐶𝑥,𝑢=−𝐾𝑥+𝑟, (3.6) where 𝑟 is a reference input and 𝐾=𝑘₁,𝑘₂,…,𝑘_𝑛, (3.7) where 𝐾is state feedback controller with variable range of the overall transfer function. So the closed-loop characteristic polynomial can be obtained as follows: 𝜍(𝑠)=𝜍_𝑒(𝑠)+𝑠𝜍_𝑜(𝑠). (3.8)Equation (3.8) is the transfer function of the system and 𝜍_𝑒 and 𝜍_𝑜 contained the controller parameters and the closed-loop characteristic polynomial, and 𝜍(𝑠,𝐾)is Hurwitz. By considering Hermite-Biehler Theorem, 𝜍_𝑒(𝑤) and 𝜍_𝑜(𝑤)can be obtained to have a  𝑘_𝑖 in the odd powers and these 𝑘_𝑖 will not depend on 𝑘_𝑗 in the even powers.

3.5. Procedures of DPGA and Chaos GA

Step 1. Size of population and length of chromosome are selected and the objective function is chosen as follows [12]: 𝐽_IAE=^𝑡₀|𝑒(𝑡)|𝑑𝑡,𝐽_ITAE=^𝑡₀𝑡|𝑒(𝑡)|𝑑𝑡,𝐽=𝐽_IAE+𝐽_ITAE, (3.9) where 𝑒(𝑡) is the closed loop error and the fitness function of each gene is calculated as follows: 1𝐹=.1+𝐽 (3.10)

Step 2. Assume𝑘_𝑗=0, stable region of 𝑘_𝑖, is obtained based on Routh-Hurwitz criterion [4].

Step 3. The number of search subspace (𝑁_𝑧) is chosen and the scan space is (𝑘_𝑖max−𝑘_𝑖min)/𝑁_𝑧(for𝑖=1,…,𝑟).

Step 4. In each subspace, 𝑘_𝑖𝑠are selected and for every 𝑘_𝑖 suitable domain of 𝑘_𝑗 are obtained (for𝑗=1,…,𝑙) and the biggest region of  𝑘_𝑗 are chosen as the search subspace.

Strings 𝜄_𝑡 are chosen in a way that (3.11) is satisfied as follows: 𝑛−𝜎𝑁^′=𝜄₀−2𝜄₁+(−1)^𝑚−12𝜄_𝑚−1+(−1)^𝑚𝜄_𝑚𝛾for𝑚^′𝜄+𝑛even₀−2𝜄₁+(−1)^𝑚−12𝜄_𝑚−1𝛾for𝑚^′+𝑛odd, (3.11) and the ranges of 𝑘_𝑗𝑠are determined by solving the following inequalities (for𝑡=0,1,…,𝑚): 𝜍_𝑒𝑤_𝑡>0if𝜄_𝑡𝜍=1,_𝑒𝑤_𝑡<0if𝜄_𝑡=−1. (3.12)

Step 5. Chaos algorithm is applied in the following steps.

Step 5.1. Generation of initial population according to (3.2).

Step 5.2. Continuing the steps as standard genetic algorithm.

Step 5.3. Selection of top 10 to 15 percent of the population.

Step 5.4. Repetition of Steps 2 and 3 to achieve desired condition.

Step 6. The parameters of genetic algorithm are generated for every subspace such as selection, crossover, and mutation.

Step 7. If the end condition is met, the best solution among subspaces is selected.

4. Simulation Results

The following examples illustrate that combination of DPGA and chaos algorithm has a good response. For this algorithm, the number of bits for any variable is 8, and number of iterations equals to 50 and𝑁_𝑧=3.

The transfer functions of the plants are given as follows: 𝐺₁(𝑠)=𝑒^−3.5𝑠,𝐺3𝑠−1₂(𝑠)=4𝑠+1(𝑠+1)³. (4.1) The results of inner loop of 𝐺₁(𝑠) are shown in Table 1 and the ranges of state feedback controller are represented in Tables 3 and 5. The search subspaces of DPGA and chaos GA based on Hermite-Biehler Theorem are shown in Tables 2 and 4. The transfer functions of systems using Integrated Time Absolute Error (ITAE) as performance index have small overshoots, short rise times, and fast transition process according to ITAE [12]. Therefore, the fitness function is chosen to have a good result. Figures 2 and 4 show the change of fitness values. Figures 3 and 5 show that combination of DPGA and chaos algorithm for feedback controller has a better result than standard GA and chaos.

Table 1 

The parameters of inner loop of 𝐺₁(𝑠).

Table 2 

The search subspaces of 𝐺₁(𝑠) with Hermite-Biehler Theorem.

Table 3 

The parameters of controller for 𝐺₁(𝑠).

Table 4 

The search subspaces of 𝐺₂(𝑠) with Hermite-Biehler Theorem.

Table 5 

The parameters of controller for 𝐺₂(𝑠).

Figure 2 

Convergence in DPGA and chaos GA.

Figure 3 

Comparison of three methods for 𝐺1(𝑠).

Figure 4 

Convergence in DPGA and chaos GA.

Figure 5 

Comparison of three methods for 𝐺2(𝑠).

5. Conclusion

A distributed population genetic algorithm with the chaos algorithm has been presented to optimize state feedback controller and this algorithm is used for that stable and unstable FODPT processes. In the first step unstable FODPT is stabilized by the inner loop and then the feedback controller is designed. A chaos genetic algorithm takes the advantage of chaotic variables to make the search of optimal values more effective and faster. In other words, chaos genetic algorithm integrates powerful global searching capability of the GA with that of powerful local searching capability of the chaos. The results show that the combination of DPGA and chaos algorithm has a good performance such as speed of convergence.