Stabilizing PID Controllers for a Class of Time Delay Systems

It is well known that delay exists in many practical systems such as industrial systems, biological systems, engineering systems, population dynamic models, etc... Time delay may appear as part of the internal dynamics or may result because of the actuators and sensors used. Communication delays between different parts of a system are also sources of this phenomenon. Existence of time delays may cause oscillations and even instability; moreover it complicates the analysis and control of these systems (Zhong, 2006; Normey-Rico & Camacho, 2007). For all these reasons there is an extensive literature on stability analysis, stabilization and control of time delay processes, which continue to be active areas of research.

through a long cable (Zhong, 2006).It can be used to represent the dynamic behavior of temperature control in a mix process (Normey-Rico & Camacho, 2007).One of the main contributions of this work is the determination of admissible values of one of the controller's parameters.This is done using two approaches, the first approach is based on the use of an extension of the Hermite-Biehler theorem and the second approach is characterized by the application of a necessary condition which is a consequence of Kharitonov's lemma.The Ddecomposition method is then applied to find stabilizing regions in the parameter space of the remaining two parameters.By sweeping over the admissible ranges of the first parameter, the complete set of stabilizing PID controllers for this class of systems is obtained.It can be easily shown that the proposed method is applicable to first order controllers, being phase lead or phase lag controllers, by carrying out the appropriate modifications.
Stabilization being the most basic requirement in most controller design problems, determining the set of all stabilizing PID controllers for this class of systems is a first step in searching, among such controllers, those that satisfy further performance criteria, such as those imposed on the unit step response of the closed loop system.Once this first step is done, genetic algorithms are used to minimize several performance measures of the closed loop system.Genetic algorithms (GAs) (Chen et al., 2009;Chen & Chang, 2006;Jan et al., 2008 ;Goldberg, 1989;Haupt & Haupt, 2004;Melanie, 1998) are stochastic optimization methods.Unlike classical optimization methods, they are not gradient based which makes GAs suitable to minimize performance measures such as maximum percent overshoot, rise time, settling time and integral square error.Moreover, genetic algorithms explore the entire admissible space to search the optimal solution.This is another reason for using GAs after finding the complete set of stabilizing PID controllers which form the set of admissible solutions.In this work, the ranges of proportional, derivative and integral gains are not set arbitrary but they are set within the stabilizing regions determined, so that we are searching among the stabilizing values of the PID controller.
The chapter is organized as follows.In Section 2, a constant gain stabilizing algorithm is used to determine the admissible ranges of one of the controller's parameters.Next, the Ddecomposition method is introduced and a necessary condition is used to compute the admissible ranges of proportional and derivative gains.In Section 3, the stabilizing regions in the space of the controller's parameters are determined and the genetic algorithm method is used to minimize several performance measures of the step response of the closed loop system.In Section 4, illustrative examples are given.Finally, Section 6 contains some concluding remarks.

Determining controller's stabilizing parameters
In this work, we consider determining the stabilizing regions in the parameter space of a PID controller, 2 () Applied to an n-th order all poles system with delay, 0 () () k values analytically is solved for the case of a first order plant with dead time (Silva et al., 2002).Therefore, instead of determining the exact range of p k ( or d k ) a necessary condition will be used to get an estimate of the stabilizing range.

 
 not all zero and 1 l  their greatest common divisor is unique and it is denoted by is the difference between the number of its

 
It is possible to state a necessary and sufficient condition for the Hurwitz stability of () s  in terms of its even-odd components (,) ab .Stability is characterized by the interlacing property of the real, negative, and distinct roots of the even and odd parts.This result is known as the Hermite-Biehler theorem.Below is a generalization of the Hermite-Biehler theorem applicable to not necessarily Hurwitz stable polynomials.Let us define the signum function (Saadaoui & Ozguler, 2003)   The following result, determines the number of real negative roots of a real polynomial.
Lemma 2. (Saadaoui & Ozguler, 2005   . We now describe a slight extension of the constant stabilizing gain algorithm of (Saadaoui & Ozguler,2003).Given a plant where is the set of all real  such that (, ) s   has signature equal to r.Let (,) hg and (,) f e be the even-odd components of 0 () qs and 0 () , p s respectively, so that With this setting, we have and if they exist, let the real negative roots with odd multiplicities of () , with 0 :0 v  and 1 : The following algorithm determines whether 00 (,) r p q  is empty or not and outputs its elements when it is not empty (Saadaoui & Ozguler, 2005) : Algorithm 1.
2. Choose all the sequences that satisfy


is equal to the union of intervals max min (,)  for each sequence of signums I that satisfy step 3.
The algorithm above is easily specialized to determine all stabilizing proportional controllers () Cs   for the plant 0 () Gs.This is achieved by replacing r in step 2 of the algorithm by n , the degree of (, ) s   .

Remark 1. By
Step 2 of Algorithm 1, a necessary condition for the existence of a 00 (,) negative roots with odd multiplicities.When solving a constant stabilization problem, this

Determining the admissible ranges of p k using padé approximation
In Gu kFu  has all its roots real, negative and distinct is equivalent to stabilizing the new constructed polynomial 1 () s  .This new constructed problem can be solved using Algorithm 1 as only one parameter appears.

An alternative method for computing the admissible ranges of p k
In this part an alternative method is used to get the admissible values of p k without the need of using Padé approximation.Moreover this method allows the calculation of the admissible ranges of the parameters ( , ) , with main term 0 0 r h  , and 1 0 The characteristic function of the closed loop system formed by the PID controller and the time delay system (2) is given by, Since the term Ls e  has no finite roots, the quasi-polynomial 1 () s  and 1 () () Ls ss e   have the same roots, therefore stability of () s  is equivalent to stability of 1 () s  , see (Kharitonov et al., 2005).In the sequel, the quasi-polynomial, 2 () () ( ) will be used to study stability of the closed loop system.Now, using Lemma 3, if Note that only two parameters ( , ) The D-decomposition method (Hohenbicher & Ackermann, 2003;Gryazina & Polyak, 2006) is used to determine the stabilizing regions in the ( , ) p d kk plane.The D-decomposition method is based on the fact that roots of the quasi-polynomial (9) change continuously when the coefficients are changed continuously.Hence, a stable quasi-polynomial can become unstable if and only if at least one of its roots crosses the imaginary axis.Using this fact, the (,) p d kk plane can be partitioned into regions having the same number of roots of (9) in the left half plane.Stability can be checked by choosing a point inside a region and applying classical methods such as the Nyquist criterion.
Evaluating the characteristic function at the imaginary axis is equivalent to replacing s by ,0 j  , in (9) we get, where, Applying the D-decomposition method implies equating the real and imaginary parts of (10) to zero.Two cases must be considered: By sweeping over values of 0   , the ( , ) kk plane can be partitioned into root invariant regions.Using ( 13), ( 14) and ( 15) stabilizing regions in the ( , ) p d kk plane can be determined.

PID controller design
In this section, first the set of all stabilizing PID controllers are determined, this forms the set of admissible solutions for optimization.Next, GAs are used to optimize these PID parameters.

Stabilizing regions
The admissible ( , ) Note that the matrix at the left-hand of ( 17) is singular.The singular frequencies (Hohenbicher & Ackermann, 2003) are determined as the solutions of equation ( 18 solving this system we get, For 0,   the above two equations partition the (,) p i kkplane into root-invariant regions.
Hence stabilizing regions, if any, can be found.It is interesting to note here that depending on which parameter we fix first, either p k or d k , the obtained stabilizing region is different.
This can be explained by the fact that in the first case the matrix is always singular and in the second case it is always non-singular.

PID controller design using GAs
Determining the total set of stabilizing PID controllers is a first step in the design process.
Once this set of stabilizing PID controllers is found, it is natural to search within this set, controllers that meet extra performance specifications.Four performance measures will be considered: , where e(t) is the error at time t.
Optimization is done using genetic algorithms.The first step in this design procedure, which consists of determining the total set of stabilizing PID controllers, is very important.It enhances the application of the genetic algorithm by fixing the search space, unlike other works on optimizing PID controllers using GAs (Chen et al., 2009;Chen & Chang, 2006;Jan et al., 2008) where the ranges of (,,) p i d kkk are set arbitrary.Moreover, it improves the optimization time and increases the chances of obtaining the global optimum.Optimization of the PID controllers is done by minimizing each of the four cost functions: maximum percent overshoot, settling time, rise time and integral square error.GAs tries to find the global optimum by evaluating many points of the solution space in each generation.There are mainly three operations: reproduction, crossover and mutation, used to form new generations until a termination condition is reached (Goldberg, 1989;Haupt & Haupt, 2004;Melanie, 1998).In what follows, we briefly describe the parameters and tools used in the implementation of the genetic algorithm: 1. Since stability is an essential property for any control system, the search space consists of the stabilizing values of the PID controller which are determined in section 3.1.Population size is chosen to be 150 individuals per population.In the beginning the initial population is randomly chosen within the stabilizing values of the PID controller.These individuals are candidate solutions to the problem.The number of generations is chosen as 100.2. Evaluation of a generation is done by calculating the cost function for each individual.
Since minimization problems are considered, the fitness function will be the inverse of the cost function.3. Genetic operators: reproduction, crossover and mutation are used to form new generations.Individuals are chosen for reproduction according to their fitness value.Fittest ones have larger probability of selection.The Roulette Wheel selection method is used.4. In order to improve next generations, crossover which is a process of combining parts of the parent individuals to produce new offsprings is done.Crossover value used is 0.8.
To avoid local minimum and explore new parts of the search space, mutation process is applied.It consists of randomly modifying individuals in the generation.
1. Mutation probability is always set to a small value so that the search algorithm is not turned to a random search algorithm.The value chosen in our case is 0.01.2. Steps 2-5 are repeated until a termination condition is reached.The maximum generation termination condition is adopted.

Examples
In this section three illustrative examples are given.

Example 1:
Consider stabilizing the second order integrating system, by a PID controller.This transfer function represents a system composed of master-slave parts.The slave part is a model of a mobile robot which can move in one direction (Seuret et al., 2006).This robot is controlled through a communication network which introduces delays in the control loop.See (Seuret et al., 2006) for more details about this system.First, the admissible values of (, ) p d kk are calculated, see Fig. 1.The first approach proposed consists of fixing p k and using ( 16), ( 18) and ( 19) to determine stabilizing regions in the plane of the remaining two parameters.By sweeping over the admissible values of p k , the complete stabilizing regions of (,,) p i d kkk values are found as shown in Fig. 2. Using this stabilizing region of (,) p i kkvalues, we apply the genetic algorithm described in section 3.2, to minimize each of the performance indices: maximum percent overshoot (Opt1), settling time (Opt2), rise time (Opt3) and integral square error (opt4).Although a simple controller, PI controller, is used for an integrating system with delay, the results obtained are satisfactory as shown by the step response of the closed loop system in Fig. 5.

Example 2:
Consider stabilizing the first order system given in (Silva et al., 2005)  Table 1 summarizes the different results obtained using our method and classical methods.For each performance criteria we can see that our method gives better results.However it should be stressed here that minimizing one performance criteria can deteriorate other performance indices, for example we minimize overshoot at the expense of a larger settling time as shown in the results below.Optimizing the integral square error (opt4) gives better results as an overall performance.Another advantage of our method is to answer the question: what is the best performance we can get with such a controller?This is possible as optimization is done over all the admissible stabilizing values of the controller.applying the procedure developed in this work, we get (the four cases given in (Wang, 2007a) will be investigated):


Case 1: For 1 0.2 T  and 2 0. Although a necessary condition is used to determine the admissible ranges of p k , in the four cases above the intervals obtained are the same as the ones given in (Wang, 2007a).

Conclusion
In this work, a new method is given for calculating all stabilizing values of a PID controller which is applied to a special class of time delay systems.The proposed approach is based on determining first the admissible ranges of one of the controller's parameters.This step is solved using two different approaches.Next, the D-decomposition method is applied to obtain the stabilizing regions in the controller's parameter space.The genetic algorithm optimization method is then applied to find among those stabilizing controllers those that satisfy further performance specifications.Four time domain measures, which are maximum Let us first fix the notation used in this work.Let R denote the set of real numbers and C denote the set of complex numbers and let R  , 0 R and R  denote the points in the open left-half, jw -axis and the open right-half of the complex plane, respectively.The derivative of a polynomial or a quasi-polynomial  is denoted by '  .The set H of stable polynomials or quasi-polynomials are, the multiplicity of u = 0 as a root of b(u), and (0 ) (0) m b denotes the value at u = 0 of the m0-th derivative of b(u).
gain.The following result will be used in determining the admissible values of p k (alternatively d k ).Lemma 3: (Kharitonov et al., 2005) Consider the quasi-polynomial,

Fig. 5 .
Fig. 5. Step responses of the closed loop system.

Fig. 6 .
Fig. 6.Step responses of the closed loop system for example 2.Determining the set of stabilizing controllers and using GAs to optimize these stabilizing values, gives the step responses shown in Fig 7.

Fig. 7 .
Fig. 7.  Step responses of the closed loop system for example 2.
with a Padé approximation of order 3.
In fact, the problem of determining the exact range of stabilizing p C  roots and C  roots.
this part, our aim is to find all admissible values of p k .Replacing the time delay by a where H , G and F are given by (4).Note that ,

Table 1 .
Results obtained by different methods.