KEYWORDS
Mixed 
Optimal Control; Particle Swarm Optimization Algorithm; Structure-Specified Controller
1. INTRODUCTION
Recently, mixed 
optimal control problems have received a great deal of attention from the viewpoint of theoretical design because it is an advanced technique for designing robust and optimal controllers for systems associated with sources of uncertainties. It was firstly proposed by Bernstein [1], and has been further developed by many researchers [2,3]. Although GA is a useful tool for solving optimization problems and has been applied successfully in many control systems, it still has limitations due to its stochastic searching characteristic and complex computation that make it slow convergence to global optimum. PSO is a powerful method for solving complex and ill-defined optimization problems because of its oriented searching and simple computation search [4]. Many researchers have become increasingly interested in the use of PSO as a means to design various classes of control systems. Kao [5] used PSO to design a self-tuning PID controller for a slidercrank mechanism. Chang [6] used PSO to design a PID controller for chaotic synchronization. In this paper, we propose a method to design the structure-specified mixed 
controllers by using PSO algorithm. The aims of this design are to obtain both robust stability and good performance, for instances, small tracking error, less control energy, etc. In the method, model uncertainty of the system is represented by multiplicative uncertainty, and the system is assumed to be affected by external unit step disturbances and then the structure-specified controller is defined. Finally, PSO is used to search for parameters of an admissible structure-specified controller that minimizes the integral of squared error (H2 norm) subjected to robust stability constraints (H∞ norm) against model uncertainty and external disturbances. The paper is organized as follows. Section 2, 3 and 4 explain a systematic procedure of the proposed controllers design algorithm. Simulation and results are presented in Section 5. Finally, Section 6 concludes the paper.
2. STRUCTURE-SPECIFIED MIXED 
CONTROL
Consider a system with 
inputs and 
outputs controlled system as shown in Figure 1, where 
is nominal plant model, 
is plant perturbation, 
is controller, 
is reference input, 
is trackingerror, 
is external disturbance, and 
is output of the system [7]. The plant perturbation 
is assumed upper bounded by a known stable weighting function matrix 
(1)
where 
denotes the maximum singular value of a matrix
.
It is proved that if a controller 
is designed so that:
1) The nominal closed-loop system (
and
) is asymptotically stable.
2) The robust stability performance against plant perturbation satisfies the following inequality
(2)
3) The disturbance attenuation performance satisfies the following inequality
(3)
Then, the closed-loop system is also asymptotically stable with 
and
, where 
is a stable weighting function matrix specified by the designers. 
and 
are the sensitivity functions, and the complementary sensitivity of the system, respectively [8]
(4)
(5)
In many control systems, not only the robust stability against plant perturbation and external disturbancesbut also small tracking error is also important. The problem of minimizing the tracking error of a system can be defined as minimizing the cost function, called the integral of the squared error (ISE)
(6)
where 
is the system error between input and output which can be obtain from inverse Laplace transformation of 
with 
and 
(7)
The handling of constraints in Equation (2), and Equation (3) is to recast the constraints as objectives tobeminimized and, consequently, a weighted sum approach is conveniently used with suitable weightings 
and
, which can be calculated by the designer. Therefore, the objective function of the investigated problem of designing mixed 
optimal controllers will be as follows
(8)
In this paper, the value of 
and 
is chosen equal to 1, and a suitable structure specified PID controller will be chosen depending on the number of the inputs and the number of the outputs.
3. PSO ALGORITHM
Particle swarm optimization is a heuristic global optimization method and also an optimization algorithm, which is based on swarm intelligence. It comes from the research on the bird and fish flock movement behavior. In an initial moment, all the particles are positioned randomly in the searching space, in which the solution must be. The movement of each particle is the result of a vector sum of three distinct terms; the first contribution is related to the inertia of the particle (a particle’s personal component), the second is related to the best position occupied by the particle (a personal component-memory) and the third is relative to the best position found by the group (group contribution-cooperation). Let the search space be 
-dimensional, and the particle i is represented by an 
-dimensional position vector
Figure 1. Control system with plant perturbation and external disturbance.
. The velocity is represented by an 
-dimensional velocity vector
. The fitness of particles is evaluated by the objective function of the optimization problem. The best previously function of the optimization problem. The best previously visited position of particle 
is noted as its individual best position,
. The position of the best individual of the whole swarm is noted as the global best position,
. At each step of searching process, the velocity of particle and its new position are updated according to the following two equations [9]
(9)
(10)
where w, called inertia weight, controls the impact of previous velocity of the particle.
, 
are random variables in the range of [0, 1].
, 
are positive constant parameters called acceleration coefficients. The value of each component in v is limited to the range 
to control excessive roaming of particles outside the search space.
4. PSO-BASED STRUCTURE-SPECIFIED MIXED 
CONTROL
A procedure for designing PSO-based structure-specified mixed 
controllers for the problem defined in Section 3.1 is presented below.
Step 1: Define a structure-specified controller of the form
(11)
and specify the upper bound of plant uncertainty, 
, weighting function for disturbance rejection,
.
Step 2: Set particle 
to
, the number of parameters of the controller in Equation (11) is the dimension of particle,
.
Step 3: Initialize a random swarm of 
particles as 
when the swarm size is set to
.
Step 4: For each generation, evaluate objective function of each particle using the objective function expressed in (6), and also evaluate the constraints (2) and (3). The cost function can then be calculated as following:
• If 
has right half-plane poles, then set 
• If 
then set 
else 
Determine the individual best, and the global best.
Step 5: Update the velocity of particle and its new position using (9) and (10).
Step 6: When the maximum number of iterations is arrived, stop the algorithm. Otherwise, go to Step 4.
5. SIMULATION RESULTS
In this section, an PID and PI examples are given to illustrate the proposed design procedures and a comparison study with GA algorithm is carried out to illustrate the effectiveness. Consider a highly coupled distillation column model [8]
(12)
The bound 
of the plant uncertainties 
is
(13)
To attenuate disturbance, the stable weighting function
is chosen as
(14)
5.1. PI Problem
Since there are 2-inputs and 2-outputs, the structuredspecified PI controller will be
A typical controller 
obtained from the POSbased method with 10 runs
With the following value of performance indices 
, and
. The step response, the disturbance response, and Step response with the following value uncertainty for the resultant system using PSO is shown in Figures 2(a)-(d).
5.2. PID Problem
Since there are 2-inputs and 2-outputs the structuredspecified PI controller will be
A typical controller 
obtained from the PSObased method with 10 runs (equation at the end of the page).
The step response, the disturbance response, and step response with the following value uncertainty of the resultant system using PSO is shown in Figures 3(a)-(d). Table 1 shows the statistical results from the 10 runs of problems 1 and 2. Figure 4 shows the convergence of the best trial from both PSO-based and GA-based algorithms. It can be seen that PSO-based algorithm is convergent at about 30 generations whereas GA-based algorithm takes about 45 generations.
6. CONCLUSION
This paper proposed a PSO-based algorithm for designing structure-specified mixed 
MIMO controllers with robust stability and disturbance attenuation. The high performance and validity of the proposed method are demonstrated by a MIMO system with PI and
Table 1. Performance of the pso-based controller from 10 runs.
Figure 4. Convergence of the cost for PSO-based and GA-based optimization algorithms.
PID controllers. It shown empirically that the performance of the proposed method is more superior than that of existing GA due to the presence of very good speed and fitness with a few numbers of iteration and less complexity to reach the optima simulation and experimental results show the robustness and efficiency of the proposed controller. The PSO-based method can be most widely used for designing high-performance optimal controllers.