ISA-PID Controller Tuning: A combined min-max / ISE approach

Proceedings of the 26 IEEE International Conference on Control Applications Munich, Germany, October 4-6, 26 FrB11.2 ISA-PID Controller Tuning: A combined min-max / ISE approach Ramon Vilanova, Pedro Balaguer Abstract This communication addresses the problem of tuning a PID controller on the basis of a Model Reference Specification and posterior inclusion of Robustness considerations. The tuning is based upon a First Order Plus Time Delay (FOPTD) model and aims to achieve a step response specification. The industrial ISA-PID formulation is chosen. First of all the expression for the structure of optimal controllers as providers of an approximation of such a reference model is got. A tuning rule is derived where the four parameters of the ISA-PID are determined by means of two new parameters: one parameter, T M, is related to the desired closed-loop time constant and the other one, z, that characterizes the approximation problem by means of the corresponding weighting function. As it is usual designs where a weighting function is used to set up the synthesis problem there should be some guide on how to select such weight. In this communication it is shown how this can be done by taking into account an ISE criterion. The introduction of ISE-like criterions for both parameters generates the optimal controller as a PI controller and the PID controller arises when detuning is introduced in order to increase the robustness. Index Terms PID Control, Robustness, FOPDT Models. I. INTRODUCTION Proportional-Integrative-Derivative (PID) controllers are with no doubt the most extensive option that can be found on industrial control applications. Their success is mainly due to its simple structure and meaning of the corresponding three parameters. This fact makes PID control easier to understand by the control engineer than other most advanced control techniques. Because of the widespread use of PID controllers it is interesting to have simple but efficient methods for tuning the controller. In fact, since Ziegler-Nichols proposed their first tuning rules [1], an intensive research has been done. From modifications of the original tuning rules [2], [3], [4] to a variety of new techniques: analytical tuning [5], [6]; optimization methods [7], [8]; gain and phase margin optimization [7], [9], just to mention a few. Recently, tuning methods based on optimization approaches with the aim of ensuring good stability robustness have received attention in the literature [1], [11]. However these methods, although effective, use to rely on somewhat complex optimization procedures and do not provide tuning rules. Instead, the tuning of the controller is defined as the solution of the optimization problem. The authors are with the Telecomunication and System Engineering Department, ETSE, Universitat Autonoma de Barcelona, 8193 Bellaterra, Barcelona, Spain. Corresponding author Ramon.Vilanova@uab.es Financial support from CICYT DPI24-6393 is greatly appreciated The purpose of this paper is to obtain PID tuning rules based on a combination of a simple model description; First Order plus Time Delay (FOPTD); and closed loop specifications with robustness considerations. The tuning rules are given in two forms: firstly parameterized in terms of two parameters directly related to the desired reference model and approximation weighting function. To get the results as close as possible to the industrial situation, the widely used ISA structure [7] is chosen for the PID control law. The paper is organized as follows. Section 2 presents the problem formulation: process model, PID structure and the optimization problem based on a min-max formulation. Section 3 solves the min-max optimization problem and provides the controller structure and expression for the PID parameters. Starting from the optimal controller structure, Section 4 deals with the choice of the parameters that define the approximation problem in terms of Integral Square Error criteria. Simulation examples are presented in section 5 and final conclusions and considerations for further extensions are conducted in section 6. II. PROBLEM FORMULATION In this section the controller equations are presented as well as the assumed process model structure and the optimization problem that is posed in order to tune the PID controller. A. PID Controller There exists different ways to express the PID control law [12]. In this paper we concentrate on the ISA PID control law [7] [ u(s) = K p br(s) y(s)+ 1 (r(s) y(s)) (1) st i ] st d + (cr(s) y(s)) 1+sT d /N where r(s), y(s) and u(s) are the Laplace transforms of the reference, process output and control signal respectively. K p is the PID gain, T i and T d are the integral an derivative time constants, finally N is the ratio between T d and the time constant of an additional pole introduced to assure the properness of he controller. Parameters b and c are called set-point weights and constitute a simple way to obtain a 2- DOF controller. As their choice does not affect the feedback properties of the resulting controlled system, with no loss of -783-9796-7/6/$2. 26 IEEE 2956

generality here we will assume b = c =1. This way, the PID transfer function we will work with can be written as 1+s(T i + T d N K(s) =K )+s2 T it d (N+1) N p st i (1 + s T d N ) (2) B. Process Model An important category of industrial processes can be represented by a First Order Plus Dead Time (FOPDT) model as. G n (s) = Ke Ls (3) 1+Ts where K is the process gain, T the time constant and L the time delay. This kind of models are easy to find by means of a simple step response experiment to get the process reaction curve. In order to deal with the delay term is usual to use a rational approximation. Here we will work with the following simple first order Taylor expansion of the e Ls term e Ls 1 Ls. (4) Obviously, this delay approximation has to be taken into account when analyzing the control system Robustness. As it will be seen in the provided example, the uncertainty is computed with respect to the e Ls delay term expression. C. Optimization problem In order to take into account robustness considerations, the design problem must be posed accordingly. One, rather usual, approach is to use frequency dependent uncertainty descriptions and to include them into the design problem [13]. Suppose the real process G(s) is given by the nominal model (3). An uncertainty description based on a multiplicative model error, Δ m (s) is defined as Δ m (s) =. G(s) G n(s) (5) G n (s) It is well known that a controller, K(s), that stabilizes the control system on the nominal system, also stabilizes all the control systems built up arround a family F of plants such that W m (s)t (s) < 1 (6) where T (s) is the nominal Complementary Sensitivity transfer function: T (s) = K(s)G n(s) (7) 1+G n (s)k(s) and W (s) is a frequency dependent weight that defines the family of plants: F = {G(s) =G n (s)(1 + Δ m (s)) : Δ m (jw) < W m (jw) } (8) However if one uses the Internal Model Control paradigm (IMC) that can be found in [13] it turns out that T (s) has a very simple expression; T (s) =G n (s)c(s); intermsof the so called IMC controller, or Youla parameter, for stable plants. The IMC synthesis gets C(s) on a first step and recover K(s) on a second step from: r r K(s) = K(s) C(s) 1 G n (s)c(s) d (a) d u C(s) (b) u G(s) G(s) G n (s) Fig. 1. Feedback Control System: (a) Conventional Configuration, (b) Internal Model Control configuration The main feature of the IMC method is that the desired closed loop time constant is provided as a tuning parameter, commonly known as the IMC filter. Robustness is dealt through the reduction of this desired closed loop bandwidth. A detailed description of the IMC synthesis method can be found in [13] [14]. Here we will make use of the IMC formulation just to set up the min-max problem we will base the design on. This way we will directly design the C(s) transfer function as the solution to the following problem y y (9) min W (s)(m(s) C(s)G n(s)) (1) C(s) where M(s) is a Reference Model for the closed loop system response and W (s) is a weighting function. In this communication we will use the Reference Model to specify the desired closed loop time constant, T M. Therefore it will take the form: M(s) = n M (s) d M (s) = 1 1+T M s (11) With respect to the weighting function, W (s), in order to automatically include integral action and keep it as simple as possible, we will assume the following form: W (s) = n W (s) d W (s) = γ 1+zs s (12) In order to include Robustness considerations, the solution to this minimization problem must be constrained to (6). 2957

III. SOLUTION TO THE MIN-MAX OPTIMIZATION PROBLEM This section will present a solution to the optimization problem (1). Several approaches exists to solve this H problem. See [15], [16] among others. Here we will follow a particularization of the solution presented in [17] where a polynomial approach was taken. This has the advantage of providing the structure of the optimal controller. Therefore, as we will do here, the problem statement can be constrained in order to provide a solution that leads to a PID controller. The problem at hand is, in fact, a min-max approximation problem : given two transfer functions M(s), N(s) RH find C(s) RH such that the following cost function in the -norm is minimized J = E(s) = W (s)(m(s) N(s)C(s)) (13) where N(s), M(s) and W (s) are factored as N(s) = n N (s) d N (s) M(s) = n M (s) d M (s) W (s) = n W (s) d W (s) The solution to the minimization of the cost function (13) lies in optimal interpolation theory [18]. First, factorize the plant numerator n N (s) as n N (s) = n + N (s)n N (s) where the polynomial n + N (s) only has stable roots and n N (s) is the remaining part. In order to obtain a unique factorisation the polynomial n + N (s) is assumed to be monic. Let ν =deg(n N (s)) and{z 1,z 2,..., z ν } be the distinct zeros of n N (s). From equation (13) it results that the error function E(s) is subjected to the following interpolation constraints: E(z i )=M(z i ) i =1...ν (14) If z i is a zero with multiplicity ν i, then additional differential interpolation constraints should be imposed. A well established theory [18], [19], [15] that solves this problem exists and a closed form solution can be obtained from the following lemma [15]: Lemma 3.1: The optimal E o (s) which minimizes E(s) is of an all-pass form E o (s) = { ρ q(s) q(s) if ν 1 if ν = (15) where q(s) =1+q 1 s + q 2 s 2 +...+ q ν 1 s ν 1 is a strictly hurwitz polynomial and q (s) =q( s). Furthermore, the constants ρ and {q i } ν 1 i=1 are real and are uniquely determined by the interpolation constraints (14). Besides, the minimum cost of (13) is given by J o =min E(s) = E o (s) = ρ Now we will proceed with the application of this lemma in order to compute the optimal C(s) =C o (s). Note first that in our case ν =1and z 1 =1/L. Therefore the interpolation constraints give the following value for the optimal cost ρ: (L + z) ρ = W (1/L)M(1/L) = γl (L + T M ) (16) Application of the above lemma gives the following equation for the optimal parameter C o (s) then W (s)m(s) W (s)n(s)c o (s) =ρ q (s) q(s) ( ) C o (s) = (W (s)n(s)) 1 W (s)m(s) ρ q (s) q(s) d W (s)d N (s) = n W (s)n + N (s)n N (s) (17) ( nw (s)n M (s)q(s) ρq ) (s)d W (s)d M (s) d W (s)d M (s)q(s) In order for C o (s) to be a stable transfer function, n N (s) must be a factor of the numerator. That is to say, there must exist a polynomial χ(s) such that n N (s)χ(s) = n W (s)n M (s)q(s) (18) ρq (s)d W (s)d M (s) It follows that, to determine the optimal controller C o (s), the χ(s) polynomial must be known. In any case, the optimal C o (s) will obey to the following structure: C o (s) = d N (s)χ(s) n W (s)n + N (s)d M (s)q(s) (19) Moreover, as ν =1, it follows from the previous lemma that q(s) =q (s) =1.Also,d N (s) =(1+Ts), n + N (s) = K and d M (s) = (1 + T M (s)). Therefore, C o (s) further simplifies to C o (s) = 1 K (1 + Ts)χ(s) (1 + T M s)(1 + zs) (2) With respect to χ(s), it must obey to (18) so, if χ(s). = χ + χ 1 s, then: (1 Ls)(χ + χ 1 s) = (1+zs) (21) ρs γ (1 + T Ms) It is easily seen that χ =1 χ 1 = z + L ρ γ Therefore, the solution for the optimal C o (s) is C o (s) = 1 K p (1 + Ts)(1 + χ 1 s) (1 + T M s)(1 + zs) (22) (23) and the resulting optimal feedback K o (s) = C o (s)/(1 G n (s)c o (s)) becomes 2958

K o (s) = 1 K p (1 + Ts)(1 + χ 1 s) s(( ρ γ + T M)+T M ( ρ γ + z)s) = 1 (1 + Ts)(1 + χ 1 s) K p ( ρ γ + T (24) ( M) s(1 + T ρ γ +z) M ( ρ γ +TM )s) Thus, identifying (2) and (24) the following tuning relations arise K p = T i K(ρ/γ + T M ) T i = T + χ 1 T M (ρ/γ + z) (ρ/γ + T M ) T d = T M (ρ/γ + z) (ρ/γ + T M ) N (25) N = T T i ρ/γ L (ρ/γ + T M ) (ρ/γ + z) 1 Note that these relations provide all the PID parameters, including the derivative filter, N. The benefits of providing a tuning of this parameter have been reported in [2], [21]. Although the tuning relations (26) look somewhat complicated note they are directly expressed in terms of the process model (K, L and T ) and the definition of the optimization problem (γ, z and T M ). Moreover, note that γ always appear as ρ/γ. Therefore, because of (16) it results that ρ/γ is independent of γ. Without loss of generality we can assume γ =1and the previous relations simplify to: T i K p = K(ρ + T M ) (ρ + z) T i = T + χ 1 T M (ρ + T M ) (ρ + z) T d = T M (ρ + T M ) N (26) N = T T i ρ L (ρ + T M ) (ρ + z) 1 These tuning relations provide the four ISA-PID parameters parameterized in terms of the desired T M and z as determining the frequency range where the solution to (1) is to provide a better match. Next section gives z a meaning in terms of Robustness considerations. IV. z T M CONTROLLER TUNING Previous section provides, in addition of explicit tuning expressions for the K p, T i, T d and derivative filter constant N, the structure of the optimal IMC controller that leads to the ISA-PID controller. In fact, the tuning has been reformulate din terms of the z and T M parameters that define the approximation problem (1). The role of z will be to establish the frequency range where the approximation error is penalized. The choice of z will have a repercussion on the mismatch between the desired reference model and the achieved closed loop input-output relation. There are different ways of measuring this mismatch. Obviously one is the optimal value for J found above (16). However this is a worst case measure in the frequency domain. This is why we will look for the value of z that determines the approximation problem (1) in such a way that provides a minimum value for the following Integral Squared Error criterion: J ISE = [e(t)] 2 dt = [y M (t) y(t)] 2 dt (27) where y M (t) is the output of the reference model to an unit step change and y(t) the output provided by a controller that obeys to a structure given by: C o (s) = 1 (1 + Ts)χ(s) (28) K (1 + T M s)(1 + zs) If we compute E(s) =L[e(t)] = L[y M (t) y(t)] it turns out that 1 E(s) = 1+T M s (1 Ls)(1 + χ1 s) (1 + T M s)(1 + zs) = ρ 1+zs therefore the time domain solution is obtained as: (29) e(t) = ρ z e t/z (3) By introducing (3) into (27) results in: J ISE = ρ2 (31) 2z Taking the derivative of (31) with respect to z and equating to zero produces z = L. Therefore independent of T M.Asan example, figure (2) shows evaluations of the ISE functional (31) with respect to z for different values of T M and L =1. ISE.25.2.15.1 ISE cost function wrt z for different values of TM.5.5 1 1.5 2 z Fig. 2. J ISE functional (31) for different values of T M and L =1. In terms of the original optimization problem, the value of z that gives the lowest ρ is z =(z ). Note however that this will cause very low penalty ( ) fortheerrorat mid-high frequencies. Therefore not very realistic. This is 2959

the reason the value z = L can be taken as an answer to the question of how to design the weighting function. Therefore, for z L we are relaxing the matching - according to the J ISE measure - that the controller achieves. To select a value z<lor L<zcan be considered as a detuning process. What are the repercussions of such detuning? Will look at the effect that this parameter has on the achieved robustness margin. Assuming we have the set of plants defined in terms of an uncertainty description weight W m (s), the Robust Stability constraint takes the form: G n (s)c(s)w m (s) < 1 K 1 Ls 1 (1 + Ts)(1 + χ 1 s) p 1+TsK p (1 + T M s)(1 + zs) W m(s) < 1 (1 Ls)(1 + χ 1 s) (1 + T M s)(1 + zs) W m(s) < 1 (32) (1 Ljw)(1 + χ 1 jw) (1 + T M jw)(1 + zjw) < 1 W m (jw) w Now, once T M is fixed (as L is given by the process model), and recalling from (22) that χ 1 is also depending of z, the choice of z will obey to the satisfaction of a constraint of the form: (1 + χ 1 jw) 1+zjw < (1 + T M jw) 1 (1 Ljw) W m (jw) w (33) As 1/W m (s) usually has a low pass form, in order to increase the robustness of the design it would be desired that χ 1 <z. By (22) and (16) this is implied by z>t M. At this point, in order to complete the design, just the selection of T M is left to be done. This can be left as a free tuning parameter and obtain the ISA-PID parameters from (26) once it is fixed from the designer. However it seems necessary to introduce another criterion to help in the selection of T M. Up to now we have been concerned with reference step change performance. However, the choice of the desired closed-loop constant will have direct repercussion on the load-disturbance attenuation performance. Therefore it seems reasonable to introduce this kind of considerations when selecting T M. The load disturbance performance can be expressed in terms of the integrated absolute error due to a load disturbance in the form of a unit step at the process input: J IAE = e(t) dt This criterion is difficult to deal with analytically. On the other hand, the integrated error criterion J IE = e(t)dt is much more convenient. In [7] it is shown that IE = T i. Thus, the criterion IE is directly given by the integrating gain of the controller. Both criterion will be identical IE = IAE if the error is positive. Furthermore, if the system is well damped the criteria will be close. First of all we will concentrate on the situation where z = L as suggested above. Since in our case T i = T + χ 1 (ρ+z) T M (ρ+t M ),ifwesolvefor T i T M = z=l it is found that T M = L. If we put these values into (26) we obtain that T i = T, K p = T/2KL, T d = and N =. Therefore, the controller arising from a minimization problem of the form (1), defined by z and T M taking values that minimizes J ISE and J IE, respectively, is a PI controller. Moreover, this controller results identical to an IMC tuned PI controller according to [13] [22] with a closed loop constant equal to L. From this fact we give the detuning procedure we were introducing above as to change from a PI to a PID controller. Therefore, the introduction of robustness considerations makes us to change from a PI to a PID controller. Note with the choice T M = L, constraint (33) becomes (1 + χ 1 jw) 1+zjw < 1 W m (jw) w (34) V. SIMULATION EXAMPLE This section presents a simulation example that shows the application of the outlined design method. Even the example presented here is purely academical, a more complete application has been developed and can be found in [23]. The purpose is to show how, the introduction of z as a robustness parameter - detuned with respect to z = L - provides better performance over all the family set built up around the nominal design. As it has been stated, the design is based upon a FOPTD nominal process model where the time delay has been approximated by using (4). Therefore, consider the following nominal model: 1 L n s G n (s) =K n 1+T n s with K n = L n = T n =1. In addition, an uncertainty of 3% associated to each parameter is considered. The We will compare the tuning arising from the choice T M = L n and z = L n (therefore a PI controller) with that resulting from increasing z such that (32) is satisfied. In order to guide the selection of z the inverse of the multiplicative uncertainty, Δ m (jw), of the members of the plant family 1 is plotted against C(jw)G n (jw). In order to find the value of z that allows for Robust Stability the initial value of z = L =1is increased and it is found that the value of z that satisfies the robust stability constraint is z 1.4. This way, the step and load disturbance responses of the PI controller and PID controller with respect 1 It is worth to notice that the uncertainty has been computed with respect to G(s) =K e Ls. This way the effect of the delay approximation (4) is 1+Ts also taken into account. 296

Sensitivity constraint in order to better improve disturbance attenuation. This way a mixed sensitivity problem will need to be solved. Although optimization approaches based on non-convex numerical methods could be used it would be helpful if an analytical solution along the lines of the one presented could be found. Also, considerations to include the set-point weights and to design the overall ISA-PID controller are being considered. VII. ACKNOWLEDGMENTS This work is supported by the Spanish CICYT program under contract DPI24-6393. Fig. 3. Robust Stability Constraint to the plant family is shown in figure (4). It can be seen that the response of the PID controller over all the family set is closer to the nominal one that of the PI controller. Fig. 4. family Step responses of the ISA-PI and ISA-PID with respect to the plant VI. CONCLUSIONS Tuning relations for PID design have been presented. In order to get results closer to Industrial applications the discussion has concentrated on the ISA formulation. The design has been done from a min-max optimization problem stated in terms of a desired time constant for the closed-loop step response, T M. The approximation problem is also defined in terms of a weighting function characterized by a parameter z. The closed-form of the solution to the minimization problem has allowed to get an optimal way to define the problem. This is to say how to choose T M and z. These optimal values turned the controller into a PI. Starting from this PI, if more robustness is needed, deviations from the optimal situation, T M = z = L, will detune the PI controller and generate a PID controller with better robustness margins. Future work is conducted to introduce a REFERENCES [1] J. Ziegler and N. Nichols, Optimum settings for automatic controllers, Trans. ASME, pp. 759 768, 1942. [2] I. Chien, J. Hrones, and J. Reswick, On the automatic control of generalized passive systems, Trans. ASME, pp. 175 185, 1952. [3] C. Hang, K. Astrom, and W. Ho, Refinement of the ziegler nichols formula, IEE Proceedings.Part D., vol. 138, pp. 111 118, 1991. [4] K. Astrom and T. Hgglund, Revisiting the ziegler nichols step response method for pid control, Journal of Process Control, vol. 14, pp. 635 65, 24. [5] S. Hwang and H. Chang, Theoretical examination of closed-loop properties and tuning methods of single loop pi controllers, Chem. Eng. Sci, vol. 42, pp. 2395 2415, 1987. [6] S. Skogestad, Simple analytic rules for model reduction and pid controller tuning, Journal of Process Control, vol. 13, pp. 291 39, 23. [7] K. Astrom and T. Hgglund, PID Controller: Theory, Design and Tuning. Instrument of Society of America, 1995. [8] K. Astrom, H. Panagopoulos, and T. Hgglund, Design of pi controllers based on non-convex optimization, Automatica, vol. 34, pp. 585 61, 1998. [9] H. Fung, Q. Wang, and T. Lee, Pi tuning in terms of gain and phase margins, Automatica, vol. 34, pp. 1145 1149, 1998. [1] M. Ge, M. Chiu, and Q. Wang, Robust pid controller design via lmi approach, Journal of Process Control, vol. 12, pp. 3 13, 22. [11] R. Toscano, A simple pi/pid controller design method via numerical optimizatio approach, Journal of Process Control, vol. 15, pp. 81 88, 25. [12] P. Cominos and N. Munro, Pid controllers: recent tuning methods and design to specification, IEE Proceedings.Part D., vol. 149, pp. 46 53, 22. [13] M. Morari and E. Zafirou, Robust Process Control. Englewood Cliffs, NJ, Prentice-Hall, 1989. [14] Y. Lee, S. Park, M. Lee, and C. Brosilow, Pid controller tuning for desired closed-loop responses for si/so systems, AIChe J., vol. 44, pp. 16 115, 1998. [15] B. Chen, Controller synthesis of optimal sensitivity: multivariable case, IEE Proceedings.Part D., vol. 131, pp. 547 551, 1984. [16] B. Francis, A Course in H Control theory. Springer Verlag, 1987. [17] R. Vilanova and I. Serra, Model reference control in two degree of freedom control systems: adaptive min-max approach, IEE Proceedings.Part D., vol. 146, pp. 273 281, 1999. [18] D. Sarason, Generalized interpolation in, Trans. AMS, vol.127, pp. 179 23, 1967. [19] G. Zames and B. Francis, Feedback, minmax sensitivity and optimal robustness, IEEE Trans. Automat. Contr., vol. 28, pp. 585 61, 1983. [2] A. Isaakson and S. Graebe, Derivative filter is an integral part of pid design, IEE Proceedings.Part D., vol. 149, pp. 41 45, 22. [21] W. Luyben, Effect of derivative algorithm and tuning selection on the pid control of dead-time processes, Industrial Engineering Chemistry Research, vol. 4, pp. 365 3611, 21. [22] D. E. Rivera, M. Morari, and S. Skogestad, Internal model control 4. pid controller design, Ind. Eng. Chem. Res., vol. 25, pp. 252 265, 1986. [23] R. Vilanova, Three-term controller design with sensitivity considerations: application to a continuous stirred tank reactor, Submitted to UKACC, International Conference on Control, 26. 2961