ay inThe classical Proportional-Integral-Derivative (PID) controller is still the most common control algorithm used in many real applications due to its simplicity. The key idea of designing a PID controller is to determine the three gains (i.e. proportional gain Kp, integral gain Ki and derivative gain Kd) of the controller. A good performance of the controller can be obtained, if all the model parameters and operating conditions are exactly known, if not, the adaptation of the PID gains must be considered. The APID control scheme is a mean to continuously adjust the controller parameters to maintain consistent optimal plant performance and guarantee the stability of the overall system.
In (Ghanadan, Reza and Blankenship, 1991), a systematic method to design an APID controller for nonlinear systems of second order dynamics has been developed. The controller in (Baek, Seung-Min and Kuc, 1997) consists of a feedback input from the APID controller as well as a feedforward input from the learning controller. It is proved that all the error signals are bounded and the tracking error converges globally and asymptotically as the learning proceeds.
A self-tuning PID control system is proposed in (Ohnishi, Yamamoto and Shah, 2000 ) for multivariable systems with unknown parameters and time-delays. The controlled object is equipped with an internal model in order to compensate the time-delay as well as unstable zeros. An APID learning control scheme is presented in (Kuc, Tae-Yong and Han , 2000 ) for control of uncertain robot manipulator performing periodic tasks. It is composed of a fixed/adaptive PID feedback control scheme and a feedforward input learning scheme implementing the control strategy: linear PID feedback stabilization and feedforward learning for nonlinear compensation and tracking. In (Grassi, Tsakalis, Dash, Gaikwadand Stein, 2000 ), the tuning of PID parameters with a loop-shaping objective are considered. A different update law that approximates the constrained minimization of the norm of the error system is designed. The work in (Badreddine and Feng, 2001 ) proposes and analyzes a direct (APID) control scheme for off-line and on-line tuning of PID parameters. The adaptive backstepping-based PID presented in (Benaskeur and Desbiens, 2002) is improved in (Ranger and Desbiens , 2003) by including the integral action through the definition of the first backstepping error, by placeing the regulation poles to any stable locations and by adding the switching sigma modification.
On the other hand, the neural networks have been recently used in control system design due to their powerful learning and adaptive abilities. Specifically, some PID control architectures, by using the neural networks, have been proposed as in (Chang, Hwang and Hsieh, 2003). The main idea is that output nodes of the fully connected neural networks are regarded as the gains Kp, Ki and Kd of PID controller, respectively. These gains adjusted on-line based on certain adaptation laws so as to achieve control objective.
In (Chang and Yan, 2005), using the sliding mode, a robust APID control tuning is newly proposed to deal with the control problem for a class of uncertain systems with external disturbance. Three gains of PID controller are considered as on-line adjustable parameters. A self-tuning PID design methodology utilizing just-in-time-learning (JITL) modelling technique is developed for nonlinear control systems in (Kansha, Jia and Chiu, 2008). PID controller parameters are updated based on the JITL information and a self-tuning algorithm derived from the Lyapunov method.
In (Tamura and Ohmori, 2007), the APID control for output tracking problem by constructing Lyapunov’s function based on PID control properties in a class of MIMO system and the reference model is designed. The controlled MIMO system with m-input and m-output n-state is considered and the APID control with constant and variable gain matrices is proposed. However the tuning laws of PID parameter matrices are derived from satisfying the Lyapunov’s stability theorem under some assumptions that the controlled system’s zero-dynamics is asymptotically stable and some properties i.e. (CB = 0 and CAB > 0) or (CB > 0) and some rank conditions concerning system and reference model matrices.
However, the almost strict positive realness condition (ASPR) (i.e. and minimum phase of the system) is shown not to be satisfied in most controlled plants. To improve this situation, the so-called parallel feedforward compensator (PFC) is introduced in (Iwai, Mizumoto and Nakashima, 2006).
In (Iwai, Mizumoto, Liu, Shah and Jiang, 2006), a APID controller for SISO plant is proposed. PFC is constructed to achieve the ASRPness of the controlled system. Thus the design always gives stable PID control system.. Few years later, that controller is modified to improve the windup phenomenon in (Mizumoto, Harada, Fujimoto and Iwai, 2011) and (Minami, Mizumoto and Iwai, 2010). Following the same methodology, ( Mizumoto, Ikeda, Hirahata and Iwai, 2010) deals with the design of discrete time APID controller with PFC.
The most well known tuning rules are still based on SISO plant because of the complexity and difficulty to handle the region of stability concerning PID parameters in MIMO system. There have been several methods of tuning MIMO PID controllers. Although most of these methods are simple extension of the SISO case, the determination of many PID controller parameters of multivariable systems is very complicated. In (Iwai, Mizumoto and Nakashima, 2006), a very simple design of multivariable PFC is proposed. It is determined by approximating the controlled plant model so that the stability of MIMO PID control system can easily be guaranteed as far as we can obtain an approximate mathematical model of the MIMO plant (Iwai, Mizumoto and Nakashima, 2006).
In this paper, the APID controller for output tracking problem of MIMO systems in (Tamura and Ohmori, 2007) is improved by adding an external system called parallel feedforward compensator PFC following (Iwai, Mizumoto, Liu, Shah and Jiang, 2006),(Mizumoto, Harada, Fujimoto and Iwai, 2011) methodology. Although (Iwai, Mizumoto, Liu, Shah and Jiang, 2006) used a PFC for SISO systems only, yet it is shown in this paper that this method can ensure that the condition and minimum phase of the MIMO system is satisfied yielding the ASPR condition which in turn guarantees stability of the APID control for MIMO systems. Thus the MIMO APID controller can be applied to stable and square MIMO systems (Iwai, Mizumoto and Nakashima, 2006) for tracking problems. Simulation results in tracking the reference input signal demonstrate the effectiveness of that APID controller.
A challenging problem in designing a PID controller is to find its appropriate gain values (i.e., proportional gain kP, integral gain kI, and derivative gain kD) (Aström and Hägglund, 1995). Moreover, in case where some of the system parameters or operating conditions are uncertain, unknown, or varying during operation, a conventional PID controller would not change its gains to cope with the system changes. Therefore a tuning method is needed. Various PID controller tuning techniques have been reported in the literature. It is classified into two groups, offline tuning methods as Zeigler-Nichols method and online tuning methods or adaptive PID. APID can tune the PID gains to force the system to follow a desired performance even with the existence of some changes in system characteristics [Mansour, 2011].
Adaptive control has been commonly used during the past decades specially the model reference adaptive control (MRAC). Its objective is to adapt the parameters of the control system to force the actual process to behave like some given ideal model which is demonstrated in [Haykin, 2002; Åström and Wittenmark, 1995]. There are two main categories of adaptive control. (1) Indirect. It starts with controlled system identification and then uses those estimated parameters to design the controller as presented in [Tian, Su, and Chu, 2000; Ramos et al.,2009 ; Xiao, Li and Liu,2012]. (2) Direct. This is more practical than indirect method. It uses a parameter estimation method to get the controller parameters directly the same as introduced in [Prabhu and Bhaskaran, 2013; Prakash and Anita, 2010].
An adaptive PID controller is presented in [Liu et al., 2009] using least square method which is an offline parameter estimation method. On the other hand, an optimal self-tuning PID controller is introduced in [Tian, Su, and Chu, 2000] using RLS to estimate the model from its dynamic data. RLS is a recursive algorithm for online parameter estimation that is frequently used because it has a fast rate of convergence. In [Wakasa, Tanaka and Nishimura , 2012] an online type of controller parameter tuning method is presented by utilizing RLS algorithm. It develops the standard offline fictitious reference iterative tuning FRIT method to be used as a modified estimation error for RLS algorithm. Also the controllers in [Wakasa, Tanaka and Nishimura , 2012; Silveira, Coelho and Gomes , 2012; Ionescu and Keyse , 2005] present online tuning based on input and output data of the system. In the case of unstable systems, few researchers study the behavior of the adaptive PID techniques on unstable systems and examine its ability to stabilize them as verified in [Badreddine and Feng , 2001; Arora, Y. Hote and Rastogi , 2011; Paz-Ramos et al. , 2004; O'Dwyer , 2009].
The problem of finding proper gain values (i.e. proportional gain KP, integral gain KI and derivative gain KD) while designing a PID controller is very challenging [Aström and Hägglund, 1995]. The problem get even more complicated in case some of the system parameters or operating conditions are uncertain, unknown or varying during operation, a conventional PID controller wouldn’t change its gains to cope with the system changes. Therefore, the need of tuning method exists. Different PID Controller tuning techniques have been investigated in the literature. It classified into two categories, off-line tuning methods as Zeigler-Nichols method and online tuning methods or adaptive PID (APID). The basic idea of APID is to tune the PID gains to force the system to follow a desired performance even with the existence of some changes in system characteristics [Mansour, 2011].
Model Reference Adaptive Control (MRAC) has been commonly used during the past decades. MRAC objective is to adapt the parameters of the control system to force the actual process to behave like some given ideal model which is demonstrated in [Haykin , 2002; Åström, 1995]. Adaptive control methods have two main categories. 1) Indirect: it starts with controlled system identification then use those estimated parameters to design the controller as presented in [Tian, Su and Chu, 2000; Ramos et al.,2009; S. Xiao, Li and Liu, 2012]. 2) Direct: this is more practical than indirect method. It uses a parameter estimation method to get the controller parameters directly the same as introduced in [Prabhu and MuraliBhaskaran , 2013; Prakash and Anita , 2010].
In [Liu et al., 2009], least square method is used as an offline parameter estimation method in APID controller designing. On the other hand, an optimal self tuning PID controller is presented in [Tian, Su and Chu, 2000] using RLS to estimate the model parameters. Recursive Least Square (RLS) is a recursive algorithm used for estimating parameters online and it is frequently used because of its fast convergence rate. An online type of controller parameter tuning method is introduced in [Wakasa, 2012] by developing a standard offline fictitious reference iterative tuning FRIT method and using it as a modified estimation error for the RLS algorithm. Also the controllers in [Wakasa, 2012; Silveira, Coelho and Gomes , 2012; Ionescu, Keyser ,2005] present online tuning based on input and output data of the system. Few researchers study the behavior of the adaptive PID techniques in the case of unstable systems and examine its ability to stabilize them as verified in [Badreddine and Feng, 2001; Arora, Hote and Rastogi , 2011; Paz-Ramos , 2004; O'Dwyer , 2009].
The heuristic adaptive approach in [Fahmy , Badr and Rahman, 2014] uses RLS algorithm as adaptation mechanism to tune the PID gains automatically online to force the actual process to behave like the desired reference model. Although no stability analysis is given, yet simulation results show its ability to stabilize a simple unstable system even when some variations in system parameters occur.
While designing a PID controller, it is a tricky problem to find its appropriate gain values (i.e. proportional gain KP, integral gain KI and derivative gain KD) [Aström and Hägglund, 1995; Mansour, 2011]. Moreover, in case which some of the system parameters or operating conditions are uncertain, unknown or varying during operation, a fixed parameter PID controller wouldn’t tune its gains to cope with the system changes. A new strategy called adaptive PID is needed which automatically detects the changes in the controlled system and readjusts its gains thereby, improving the response of the PID controller.
Regarding Adaptive control techniques, major advances have occurred in both understanding and designing during the past decades especially in Model Reference Adaptive Control (MRAC). Its objective is to adapt the parameters of the control system to force the actual process to behave like some given desired model as demonstrated in [Haykin , 2002; Aström and Wittenmark, 1995]. There are two main categories of adaptive control; indirect and direct. Indirect adaptive control starts with controlled system identification then use those estimated parameters to design the controller as presented in [Ramos et al.,2009; Tian, Su and Chu, 2000; Xiao, Li and Liu, 2012]. Since direct adaptive control uses a parameter estimation method to get the controller parameters directly the same as introduced in [Prabhu and MuraliBhaskaran , 2013; Prakash and Anita , 2010] therefore it is more practical than indirect method.
Combining the advantages of both techniques, PID with its simplicity and MRAC with its adaptation mechanism, the idea of adaptive PID (APID) seems very promising. In [Liu et al., 2009], APID is designed by means of least square method in which an offline parameter estimation method is used. On the other hand, an optimal self tuning PID controller is introduced in [Tian, Su and Chu, 2000] using RLS to estimate the model from its dynamic data. RLS is a recursive algorithm for online parameter estimation that is frequently used because it has a fast rate of convergence. In [Wakasa, 2012] an online type of controller parameter tuning method is presented by utilizing RLS algorithm. It develops the standard offline fictitious reference iterative tuning FRIT method to be used as a modified estimation error for RLS algorithm. Also the controllers in [Wakasa, 2012; Ionescu, Keyser ,2005; Silveira, Coelho and Gomes , 2012] present online tuning based on input and output data of the system.
The design of Multivariable control system is highly applicable in industry and academia since it can handle more real processes as in [Chang, Hwang and Hsieh, 2003; Yu, Chang and Yu, 2007]. Few literature work propose multivariable controllers with different techniques and methodologies such as generalized predictive control (GPC) algorithm in [Uduehi, Ordys and Grimble, 2002] and predictive PID as in [Saeed, Uddin and Katebi, 2010]. Most of the previously proposed controllers are applied under the assumptions or conditions on the system characteristics [Mansour, 2011]. Several of them are restricted to stable, minimum phase [Costa, Hsu and Imai , 2003] and ASPR (Almost Restrict Positive Realness) systems [Tamura and Ohmori, 2006; Tamura and Ohmori, 2007]. Thus, there is a great demand to study the general MIMO systems with no restrictions. The controller in [Fahmy et al., 2013] overcomes a part of the problem that appears in [Tamura and Ohmori, 2007] where a multivariable adaptive PID controller is developed by adding a Parallel feed forward compensator designed in [Iwai, Mizumoto, Nakashim, 2006] to generalize the application of APID controllers but only for stable square MIMO systems. Few researchers study the behavior of the adaptive PID techniques on unstable systems and examine its ability to stabilize them as shown in [Arora, Hote and Rastogi , 2011; Badreddine and Feng, 2001; O'Dwyer , 2009; Paz-Ramos et al., 2004].
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