Advances in Mathematical Physics
Volume 2017 (2017), Article ID 1938513, 11 pages
https://doi.org/10.1155/2017/1938513

Research Article

PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

Wichai Witayakiattilerd

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Paholyothin Road, Klong Luang, Rangsit, Pathumthani 12121, Thailand

Correspondence should be addressed to Wichai Witayakiattilerd; wichai@mathstat.sci.tu.ac.th

Received 27 July 2017; Revised 8 October 2017; Accepted 17 October 2017; Published 12 November 2017

Academic Editor: Alkesh Punjabi

Copyright © 2017 Wichai Witayakiattilerd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result.

1. Introduction

In this document, we tackle a combined problem of impulsive condition, singular perturbation and PID controller. They are introduced separately as follows.

Impulsive differential equations play an important role in many areas of science such as physics, engineering, biology, and medicine. They can represent some natural phenomena better than ordinary differential equations, phenomena such as controlling chaos and bifurcation in engineering systems, epidemic model with impulsive birth, and modelling and control of complex dynamic systems ([1–16] and reference therein). Some of them are reviewed here. In 1989, Lakshmikantham et al. wrote an important book in this field and provided a theory named the theory of impulsive differential equations which have been cited by many researchers. In 2007, Jankowski investigated the existence of solutions and quasisolutions of advanced impulsive differential equations by applying Schauder’s fixed-point theorem. In 2009, Witayakiattilerd and Chonwerayuth proved the regularity of piecewise continuous almost periodic solutions for nonlinear impulsive systems by using contraction principle on Banach space. In 2012, Abdel-Rady et al. proved the existence and uniqueness of the solution of first-order impulsive differential equation and applied the results to second-order impulsive differential equation. In 2015, Li et al. derived the existence and global attractiveness of unique periodic solutions by using Lyapunov’s second method and the contraction mapping principle. In 2016, Li and Wu studied the stability of nonlinear differential systems with state-dependent delayed impulses. In 2017, Li and Cao presented a new impulsive delay inequality that involves unbounded and nondifferentiable time-varying delay.

Some phenomena represented by an impulsive system may be perturbed by small parameters, in particular singularity perturbation. An impulsive differential equation with singularity perturbation is quite complicated and is difficult to find an exact solution, so there are not many researches in this field. Most of the researches were done by a few groups of mathematicians before year 2000 (see [3, 6, 9, 17–19]). For example, in 1992, Kulev studied the uniform asymptotic stability of impulsive perturbed systems of differential equations. In 1996, Bainov et al. investigated the uniform asymptotic stability of impulsive perturbed systems of differential equations. In 2007, Zhu et al. proved the exponential stability of singularly perturbed impulsive delay differential equations.

A proportional-integral-derivative controller (PID controller) is, today, found in several areas where control has to be exerted. It is the most common form of feedback and is an important component of a distributed control system. It is always combined with logic, sequential functions, selectors, and simple function blocks for building complicated automation systems used for energy production, transportation, and manufacturing. Many sophisticated control strategies such as model predictive control are also organized hierarchically. For more details, please see ([20–23] and reference therein).

The Abstract Cauchy Problem (ACP) of the singular perturbation of nonlinear functional impulsive differential equations on a Banach space of the problem is written as follows:

where ,  ,   and are given continuous operators in ,   and denote the jump of states and at time with the magnitude of jumps and , , respectively. If and are globally Lipschitz and uniformly bounded in , then will be of order faster than . Consequently, we call the slow variable and the fast variable of the system. In this case, we assume that the system is controlled by PID controller:

where and is reference variable called set point at time . The constants ,  , and are nonnegative, called proportional gain, integral gain, and derivative gain, respectively.

The complexity of the system makes it difficult to find its exact solution. This paper presents the approximation method for solving the system. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable.

The paper is organized as follows: in Section 2, some basic concepts and notations which are useful for reading the next section are introduced; in Section 3, an approximated system is introduced and an approximate solution is presented; in Section 4, the existence of solution of optimal control problem is solved; an example is presented in Section 5.

2. Preliminaries

In this section, we state the definitions and theories that are used in this study. Throughout the paper, denotes a Banach space with norm and denotes another Banach space with norm . A linear transformation from into is bounded on a domain of ,   if there exists a constant such that for all . The linear space of all bounded linear operators from into is denoted by and denote by . In the proof of the existence and the uniqueness of solution, we use fixed-point theorem on Banach spaces or contraction mapping principle. An operator is strictly contraction if there exists a constant with such that for all . The contraction mapping principle states that if is a strict contraction on Banach space . Then the equation has a unique solution in ; that is, has a unique fixed-point . Furthermore, if is a closed subset of the Banach space and the strict contraction-operator maps into itself, then the equation has a unique solution . Let be a one-parameter family of bounded linear operators from into such that is uniformly continuous, that is,

Definition 1. A one-parameter family of bounded linear operators from into is a semigroup of bounded linear operators on if

(1), ( is the identity operator on );(2) for every (the semigroup property).

Define a linear operator by for all where . Then is called the infinitesimal generator of a semigroup . Definition 1 implies that for a semigroup has a unique infinitesimal generator. If is uniformly continuous, its infinitesimal generator is a bounded operator. On the other hand, every bounded linear operator is the infinitesimal generator of a uniformly continuous semigroup and this semigroup is unique. For more detail, please see [24].

Definition 2. A one-parameter family of bounded linear operators from into is a group of bounded linear operators on if

(1), ( is the identity operator on );(2) for every (the semigroup property);(3).

Theorem 3. Let be an infinitesimal generator of the -semigroup . Then

(1)for all , ;(2)for all and ;(3)for all and ;(4)for all ,  .

3. Main Results

In this section, we investigate the Abstract Cauchy Problem (ACP) of the singular perturbation of nonlinear functional impulsive differential equations on a Banach , System (1). A suggested form of the system (1) is as follows:

with

where and is a set point at time . ,  , and are constants.

3.1. Approximation Method

Suppose that is arbitrary initial value. Then we can choose an initial such that the fast equation becomes stationary, that is, . This implies that

Accordingly, for small enough , it seems reasonable to substitute (3) with a differential algebraic equation:

where and .

Suppose that solves the following system:

Therefore, by substituting into (6), we have

where . In other words, we can approximate the slow equation by the inhomogeneous ACP (8).

First, we solve semilinear ACP:

Let be an admissible control set.

Definition 4. Let . A mild solution on of the semilinear ACP (9) with respect to control is a continuous function such that satisfies the integral equation:

where is the -semigroup generated by the operator .

We start with the following classical result which assures the existence and uniqueness of mild solutions of (9) for Lipschitz continuous function and .

Theorem 5. Let . Let ,   be continuous in on and uniformly Lipschitz continuous (with constant and ) on . If is infinitesimal generator of a -semigroup on , then for every the semilinear ACP (9) has a unique mild solution with respect to control . Moreover, the mapping is Lipchitz continuous from into .

Proof. For a given . Define a mapping by

Denoting by the supremum norm of as an element of , it follows from definition of and the uniformly Lipschitz continuous of and that

where is a bound of on . Using (11) and (12) and induction on , we have

This implies that

For sufficiently large ,   and by a well-known extension to the contraction mapping principle, there is a unique such that

Therefore, the semilinear ACP (9) has a unique mild solution . The Lipchitz continuity of the mapping is consequences of the following argument. Let and be the mild solutions of (9) with the initial values and , respectively. Then it follows from the definition of mild solution and the uniformly Lipschitz continuous of and that

for some constant . By using Gronwall’s lemma, it can be implied that

which yields the Lipchitz continuity of the mapping .

Corollary 6. Under the conditions of Theorem 5, for every , the integral equation

has a unique solution .

Proof. The proof is similar to the proof of Theorem 5 by defining .

3.2. Regularity of the System without Impulses

Definition 7. Let . A classical solution of the inhomogeneous ACP (9) with respect to control is a function such that

(1) is continuously differentiable on ;(2) for all ;(3) satisfies the semilinear ACP (9).

Theorem 8 (regularity). Let and be the infinitesimal generator of a -group on a Banach space . If and are continuously differentiable on and , respectively, then the mild solution of (9) with is a classical solution of the semilinear ACP (9) with respect to control .

Proof. Let be the solution of (9) with . Set

The continuous differentiability of and implies the continuity differentiability of . Define a function by

It follows from the assumptions that and that the function is continuous in on and uniformly Lipschitz in and Corollary 6 assures that the integral equation

has a unique solution . Moreover, from our assumption, we obtain

This implies that

where denotes a little- notation and as uniformly on . We will show that is differentiable on by regarding the convergence of as in norm . It follows from the definition of , (20), (21), and (23), that

Since , and as ,

for some nonnegative function as . Therefore, we have

where . By applying Gronwall’s lemma to (26), we have

Thus as . This implies that is differentiable on and its derivative is which is an element in . Consequently, is continuously differentiable on . Next, we will show that satisfies the semilinear ACP (9). Since is differentiable on and is -group,

Hence satisfies the semilinear ACP (9). Finally, we show that for all . It follows from (28) and that

Using the assumption of and , and the continuity of , we obtain the result that is continuous. This implies that for all .

3.3. System with Impulses

Next, we prove the existence of solution of the impulsive system (8). Throughout this section, we denote as all type -piecewise continuous function from to such with discontinuous at .

Definition 9. Let . A mild solution on of the semilinear impulsive ACP (8) with respect to control is a function such that satisfies the integral equation

where is the -semigroup generated by the operator .

Definition 10. Let . A classical solution of the inhomogeneous ACP (8) with respect to control is a function such that

(1) is continuously differentiable on , , and for all ;(2) for all ;(3) satisfies the semilinear ACP (8).

Theorem 11. Let . Let be the infinitesimal generator of a -group on a Banach space . If is continuously differentiable on , and for all and is continuously differentiable on ,  , and for all , then the mild solution of (8) with for all , is a classical solution of the semilinear ACP (8) with respect to control .

Proof. Let us consider the following system:

Theorems 5 and 8 imply that System (31) has a unique mild solution such that

and the mild solution is a classical solution on . So

Next, we consider the following system:

By applying Theorems 5 and 8, we have

That is,

is a classical solution on . Continuing this process, we have that

is a classical solution on .

4. Optimal Control Problem

In this section, we study an optimal control problem of the approximated system (8) such that is controlled by a controller variable . We denote each solution with respect to a control by and call the ordered pair a pairwise control pair, sometimes written shortly as . Define a cost functional by integrating an error function, that is,

where is the error function. We want to investigate an optimization control problem, Problem (P).

Problem (P). Problem (P) is to find the pairwise control pair such that for all .

Theorem 12. Suppose that the error function satisfies the following conditions:

(1) is Borel measurable.(2) is sequentially lower semicontinuous on .(3) is convex on .(4)There are constants such that for all and .

Under the assumptions of Theorem 11, Problem (P) constrained by system (8) has at least one solution; that is, there exists a pairwise control pair such that for all .

Proof. Let . If , the theorem is already true. Assume that . By condition (), there are constants such that for all and . Then

for some and for all . The minimum of implies that there is a sequence of the minimum point, say , of the cost functional such that and . Thus, there is and such that for all . This implies that . Consequently, for all . Hence is a bounded sequence in with respect to the norm . Since is a reflexive Banach space and is unit ball of , there exists a subsequence of such that for some . Let be a mild solution with respect to a control and let be a sequence of mild solution corresponding to the sequence of control , that is,

By the continuous differentiability of and , there is a constant such that

By Gronwall’s lemma, there is an such that

Since ,  . By using conditions () and (),

Thus, , that is, for all .

5. Example

In this section, an example is presented to clarify the main results. Consider a perturbed differential system with control in , for all :

where and . Assume that a set point function of the example is . Let and . Then system (44) becomes

Since is a solution of with as , we can approximate (45) with

where . Then, (46) satisfy the condition of Theorem 11. Therefore, an approximate mild solution of system (45) is in the following form:

where . One can apply Picard iteration to (47). For example, we set . Then the iteration of approximation mild solution is

where with ,   and ,  ,  .

If we set an initial function and , then, for ,

We can continue the process until this condition is met “if then , for some small given ,   where ”. In fact, the value of is called deviation concerning with manipulated variable (the manipulated variable is the variable that causes a change in the controlled variable) that we want to eliminate. Define a cost functional by

Problem (P) is to find the pairwise control pair such that for all . We use Theorem 12 to confirm that Problem (P) constrained by system (45) has at least one solution.

6. Conclusion

In this study, we consider the PID control system of the impulsive differential equation with singular perturbation. Because of the complexity of the system, it is difficult to find its exact solution. So the approximation method is introduced for solving the system. The idea of the method is to eliminate the fast variable in the system under some assumptions. We found an approximated mild solution of the system from the approximation method. The approximated mild solution of the system is written in the following form:

where is the infinitesimal generator of a -semigroup . We showed that the mild solution is also the classical solution of the approximated system. Moreover, we proved the theorem confirming that Problem (P) constrained by the PID control system has at least one solution. In this article, we do not mention how to design controllers, but it is well known that there are many different ways to design PID control (please see [21–23, 25] and references therein). The result form this paper can be used as a prerequisite to PID controller design. For example, the -semigroup property of the operator will be a necessary condition for evaluating the controller parameters , , and that play important role in PID controller design process.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author acknowledges all the support from Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University.

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