Stabilization of Inverted Cart-Pendulum System Using PID Controller: A Frequency-Domain Approach

This paper focuses on the angular stabilization of inverted cart-pendulum system using PID controller. The tuning of PID controller is formulated as a nonlinear optimization problem, in which the objective function is composed of five design conditions in frequency domain. Particle swarm optimization technique has been used for optimizing PID parameters. Also a PID controller has been designed based on same specifications, and a comparative study has been carried out. All the responses have been calculated using FOMCON toolbox of Matlab/Simulink.


Introduction
In recent years, Fractional-order PID (PI D ) controller has become a good generalization of standard PID controller using fractional calculus. Compared to PID controller, the tuning of PI D is more complex and remains a challenging problem. As the PID controller is one of the most commonly used controllers in industries, and in recent years fractional calculus [1][2][3][4][5] has been used to present the more generalized form of PID controller, that is, PI D controller.
It is well known that inverted cart-pendulum [6] is an example of underactuated, nonminimum phase and highly unstable system. Because of this, it is very difficult to design a controller for a system like inverted pendulum. The design becomes more difficult because of the physical constraints on track length, applied voltage, and the pendulum angle.
Podlubny has proposed a generalization of the PID controller as PI D controller. He also demonstrated that the fractional order PID controller has better response than classical PID controller [1,7]. Also, many valuable studies have been done for fractional order controllers and their implementations [8][9][10][11][12][13][14]. Tuning of the PI D controller using the frequency-domain approaches is studied in many papers. For example, [15] propose a method based on optimization strategies. Tuning of H∞ controllers for fractional singleinput single-output (SISO) system was suggested in [12]. In [16], tuning method for PI D based on Ziegler-Nichols and Astrom-Hagglund tuning rules is presented.
In this study, a PI D controller has been designed for an unstable system (inverted cart-pendulum system) using frequency-domain approach, and particle swarm optimization (PSO) has been used for obtaining the PI D parameters ( , , , , ).
This work is organized as follows: Section 2 covers inverted cart-pendulum system, Section 3 covers FOPID controller, Section 4 presents particle swarm optimization, Section 5 presents FOPID controller design, Section 6 presents simulation results, and the conclusion is provided in Section 7.

Inverted Cart-Pendulum System
The inverted pendulum system [6] has two degrees of freedom of motions as shown in Figure 1. The first is linear motion of the cart in the -axis and is given bÿ where = 2 ( + cos 2 ) + ( + ). The second is rotation of the pendulum about theplane and is described bÿ Now linearising (1) and (2) for small angle of from the equilibrium point and neglecting the friction coefficient (which is very small compared to other parameters), the following equations are obtained: Now taking the Laplace Transform of (3) and (4), the following transfer function is obtained: From Table 1 putting all values in (5), Now the DC motor used to convert control voltage to force is represented by only a gain block of gain = 15 for simplicity. Then 1 ( ) become

PI D Controller
The differential equation of the PI D controller is described by [16,17] ( ) = ( ) + − ( ) + ( ) . (2) X-position of cart from the reference (m) ±0.3 (3) -angle of the pendulum with respect to vertical (rad) ≤0.1 m-mass of the pendulum (kg) 0.23 (6) L-length of the pendulum (m) 0.4 (7) J-moment of inertia of pendulum (kg⋅m 2 ) 0.099 (8) b-coefficient of cart friction (Ns/m) 0.055 (9) g-acceleration owing to gravity (m/s 2 ) 9 . After the introduction of this definition, it is easily seen that classical types of PID controllers such as integral order PID, PI, or PD become special cases of the most general fractional order PID controller. In other words, the PI D controller expands the integer-order PID controller from point to plane, as shown in Figure 2, thereby adding flexibility to controller design and allowing us to control our real world processes more accurately but only at the cost of increased design complexity.
Taking Laplace Transform of (8), the controller expression in -domain is obtained as

Particle Swarm Optimization Method
The PSO method [18] is a wide category of swarm intelligence methods for solving the optimization problems. This technique inspired by bird flocks, fish schools, and animal herds constitute representative examples of natural systems where aggregated behaviors are met, producing impressive, collision-free, and synchronized moves. The PSO is population-based stochastic optimization which was developed by Kennedy and Eberhart in 1995. The major advantage of the PSO over other stochastic optimization methods is its simplicity. It is a population-based search algorithm. The basic idea of the PSO is the mathematical modeling and simulation of the solution searching activities in multidimension search space where optimal solution exists. In a PSO system, particles fly around in a multi-dimensional search space. During flight, each particle adjusts its position according to its own experience and the experience of the neighboring particles. Update velocity and position of particle are depended upon inertia, cognitive, and social factors. Each particle has a memory, and hence it is capable of remembering the best previous position in the search space ever visited by it. The position corresponding to the best fitness is known as best , and the overall best out of all the particles in the population is called best . The modified velocity and position of each particle can be calculated using the current velocity and position as follows. Velocity update equation is given by Position update equation is given by where = number of iterations, = velocity of particle, = inertia weight factor = max − [( max − min )/ ] * iteration number, 1 , 2 = cognitive and social acceleration factors respectively, rand() = random numbers uniformly distributed in the range (0, 1), = position of particle, and = th particle. As shown in Figure 3, the PSO algorithm consists of just three steps, which are repeated until some stopping condition is met.
(1) Evaluate the fitness value of each particle.
(2) Update individual and global best fitness values and positions. (3) Update velocity and position of each particle.
The first two steps are fairly trivial. Fitness evaluation is conducted by supplying the candidate solution to the

PI D Controller Design
The concept of frequency domain design of PI D controllers was first proposed at [8]. If ( ) is the transfer function of the process, then the objective is to find out a controller ( ), so that the open loop system ( ) ( ) meets the following design specifications.

Phase Margin Specification. Consider
Arg ( ( gc ) ( gc )) = − + , where is the desired phase margin and gc is the desired gain crossover frequency.

Robustness against System's Gain Variation.
Consider

Complementary Sensitivity Specification. Consider
where is the specified magnitude of the complementary sensitivity function or noise attenuation for frequencies for all ≥ (rad/s). Evaluate g best p best(i) = p best(i−1) p best(i) = p best(i) Fitness = fitness p best iter = 1 iter = iter + 1 g best(iter) = g best(iter) g best(iter) = g best(iter−1) fitness p best(iter) < fitness p best(iter−1) iter = iter max

Sensitivity Specification. Consider
( ) where is the specified magnitude of the sensitivity function or load disturbance rejection for frequencies for all ≤ (rad/s). Now five specifications (12)- (16) have been solved using PSO to obtain the PI D controller parameters, and also a conventional PID controller has been designed using the same specifications.

Simulation Results
In Figure 4, the block diagram of whole system is given. The angle of the rod is controlled via the PI D controller where  the aim is to hold the rod at the upright position. Some parameters for PSO program are given in Table 2.
The values of 1 , 2 , 3 , 4 , and 5 are given in Table 4, and the overall value of objective function is less in case of PI D controller as compared to PID controller. In (23), the values of all weights are considered as 1 = 2 = 3 = 4 = 5 = 1.
Here design specifications have been considered as gc = 20 rad/sec, = 60 ∘ , = 0.01 rad/sec, = 100 rad/sec, and = = −20 dB. Using the objective function of (23) and after running the PSO program with parameters given in Table 2, the PI D parameters are given in Table 3, and by taking = = 1, PID controller has been designed using the same conditions as used for PI D controller. The PID parameters are given in Table 3. These are the values for particular case and may vary for running the PSO many times, but the results are almost same for all cases.
The initial angle has been taken as 0.1 rad. The simulation results are shown in Figures 5, 6, and 7 for cart angle, control voltage, and error, respectively, using both PI D and PID controllers. The Bode diagrams of open loop system are    given in Figures 8 and 9 for both PI D and PID cases. All results are obtained using FOMCON toolbox of MAT-LAB/SIMULINK [19]. As it can be noticed in Figures 5 and 7 that cart angle has less deviation around its upright position for PI D controller and settles to upright position slightly faster than PID controller. In Figure 6, PI D controller takes less control effort than PID controller. In Figures 8 and 9, the phase margin condition is almost satisfied for both PI D controller and PID controller.

Conclusion
In this work, an attempt has been made to test the effectiveness of PI D controller for a highly unstable system like inverted cart pendulum. Two different controllers (PI D and PID controllers) using frequency domain approach based on the same specifications have been designed. It is shown in Figures 5-7 that PI D controller performs better than conventional PID controller. In Bode diagrams as shown in Figures 8 and 9, it is clear that gain margin is better for PI D controller than that for conventional PID controller.
It is also seen that the use of particle swarm optimization technique for calculating controller parameters is very simple and provides good convergence towards optimal values.
In this work, only cart angle has been stabilized at its upright position, but on the other hand, cart position stabilization along with cart angle using PI D controller and real time implementation might be the subject of future work.