Sliding Mode Based Self-Tuning PID Controller for Second Order Systems

In this paper, a sliding mode based self-tuning PID controller is proposed for second order systems. While developing the controller, it is assumed that the system model has a part which contains nonlinear terms similar to PID structure. The controller and update rules for PID parameters are obtained from Lyapunov stability analysis. Numerical simulations are conducted on a Twin-Rotor Multi-Input Multi-Output System (TRMS) model to show the performance of the proposed controller.


Introduction
PID control is the most preferred control technique in industrial applications due to its simple structure and convenience in implementation [1].However, the effectiveness of the PID controller is based on the accurate selection of its parameters.Despite the good performance results in linear systems, the selection of the parameters might be very difficult and time wasting with the rise of nonlinearities of the system.To deal with this problem many approaches of self-tuning PID controllers have been presented till today.These approaches can be divided into two main categories: i) model based approaches and ii) rule-based approaches.In model based approaches, the tuning mechanism is based on the knowledge of the system model [2].In rule based approaches, the tuning is based on some optimization or estimation rules without model knowledge, which basically mimics an experienced operator's behavior [2].A good survey can be found in [2] on this topic.
In the literature, many studies can be found on selftuning PID controller and its applications.In [3], An et.al. presented a self-tuning method for PID controllers based on the theory of adaptive interaction for the quadrotor system.In [4], a self-tuning PID control scheme based on support vector machine (SVM) and particle swarm optimization (PSO) were presented.Jiang and Jiang proposed a fuzzy based self-tuning PID controller for temperature control [5].Zheng et.al. used fuzzy module to tune PID controller parameters according to the error and change in error [6].In [7] and [8], genetic algorithm was utilized to tune the PID parameters.Na presented a study on water level control of a nuclear steam generator with PID controller of which parameters were tuned by model predictive control (MPC) [9].In [1], least squares support vector machine identifier was utilized to tune parameters of PID controller.Fan et.al. used neural network to tune PID controller for position tracking of a pneumatic artificial muscle [10].Gundogdu and Komurgaz presented a self-tuning algorithm for PID controller based on adaptive interaction approach [11].In [12], Howell and Best used continuous action reinforcement learning automata (CARLA) method to tune the PID controller parameters while controlling engine idle-speed.In [13], Shih and Tseng designed a self-tuning PID controller by using integral of timeweighted absolute error (ITAE) optimal control principle and the pole-placement approach to control position of a servo-cylinder.Dong and Mo presented model reference adaptive PID controller for motor control system with backlash [14].In [15], Chamsai et.al. presented an adaptive PID controller combined with sliding mode controller for uncertain nonlinear systems.Chang and Yan proposed an adaptive PID controller based on sliding mode controller for uncertain chaotic systems [16].Kuo et.al. presented an adaptive sliding mode controller with PID tuning method for a class of uncertain systems [17].
In this paper, a sliding mode based self-tuning PID controller is proposed for uncertain second order systems.Different from the literature, it is assumed that the model contain nonlinear terms similar to PID structure.The controller and update rules for PID parameters are obtained from Lyapunov stability analysis.Numerical simulations are conducted on a Twin-Rotor Multi-Input Multi-Output System (TRMS) model to test the performance of controller and parameter update rule.
The rest of the paper is organized as follows; the system model is presented in Section 2. Control and parameter update rule design are presented in Section 3. Numerical simulation results are given in Section 4.

Proceedings of the International Symposium of Mechanism and Machine Science, 2017
AzC IFToMM -Azerbaijan Technical University 11-14 September 2017, Baku, Azerbaijan

94
Finally concluding remarks are presented in Section 5.

System Model
The following second order system is considered in this paper, ) where where g(•):R 2 → R is unknown function, kp, kd and ki are unknown system parameters.
Assumption 2: It is assumed that the system parameters, kp, kd and ki are in known bounded regions.
Assumption 3: It is assumed that x(t) is available and continuous.

Control and Parameter Update Rule Design
The objective of the controller is to utilize that ) ( 1 t x track a desired trajectory while updating PID parameters.To achieve this objective, the error system is designed as follows, where xd1 and xd2 are desired trajectories.To construct sliding mode controller, the filtered error signal is designed as The derivative of ( 7), which will be utilized later, is The control input is designed as where uR is sliding part of the controller and By substituting the ( 9) and ( 10) in ( 8), it is obtained as where The Lyapunov function in ( 13) is utilized to construct update rules for PID gains and design uR The derivative of ( 13) is obtained as From ( 14), the update rules of p k ˆ, d k ˆ and i k ˆ are selected as in (15) to eliminate the terms with gain errors.
After substitution of ( 15) in ( 14), V  is obtained as The input signal uR should be designed to make V  negative.To achieve this purpose, uR will be investigated by separating into three terms as u1, is designed as to eliminate first two terms in ( 16) as ) sgn( 2 To eliminate the term sg in (16), the condition in assumption 1 can be utilized.From (4) the following inequality can be obtained By using (19), u2 is designed as follows (20) By substituting ( 18) and ( 20) in ( 16), V  is obtained as, where An upper bound for L can be defined as where p k , d k and i k are upper bounds of kp, kd and ki, respectively.
Hence, the following inequality can be written Remark 1: In (23), Lm may go to infinity for 0 1  d x since integral term.But it should be kept in mind that the main interested term is |s|Lm.So if it can be proven that s(t) converge to 0, fast enough, then, it can be assumed that the term |s|Lm stays bounded.So Lm can be accepted as bounded.In the rest of the paper, it will be proven that s(t) converge to 0 with a tunable rate.From (24), So, u3 can be obtained as By substituting ( 9) and ( 15) in (28), it is obtained as, If k is selected as

Numerical Simulations
The performance of the control law in (9) and update rule in (15) were evaluated by conducting numerical simulation by using the dynamic model of a 2-DOF helicopter which is known as TRMS.During the simulation, the parameter values of input signal were selected as λ=diag(30,30), k=diag (

Conclusions
In this paper, a sliding mode based self-tuning PID controller was designed for second order systems.While designing the controller, it was assumed that the system model contain nonlinear terms similar to PID structure.The controller and update rule for PID parameters were obtained from Lyapunov stability analysis.The effectiveness of the controller and update rule were evaluated by conducting numerical simulation and achieved satisfactory results.
i k ˆ are estimates of kp, kd ve ki , respectively.