Settling time assignment with pid - Nguyen Doan Phuoc

Vietnam Journal of Science and Technology 55 (3) (2017) 357-367 DOI: 10.15625/2525-2518/55/3/8808 SETTLING TIME ASSIGNMENT WITH PID Nguyen Doan Phuoc, Nguyen Hoai Nam Hanoi University of Science and Technology, 1 Dai Co Viet Street, Hanoi *Email: nam.nguyenhoai@hust.edu.vn Received: 24 October 2016; Accepted for publication: 21 February 2017 ABSTRACT In this paper, the settling time of closed-loop systems controlled with PID, which is established by using magnitude and symmetric optimum methods, is analyzed. These methods have some limitations such as limited frequency bands, existence of overshoot and larger settling time. Thus, our aim is to improve these performances of the closedloop system for a certain class of transfer functions. The proposed method will be tested on a DC motor by simulation. It can be concluded that the method provides a larger range of frequency, smaller settling time and no overshoot. Keywords: PID controller, magnitude optimum criterion, symmetric optimum criterion, first- order system, first-order plus integral system. 1. INTRODUCTION Many PID tuning methods have been developed since 1940s. These methods can be classified into groups: practical approach [1 - 4]; optimal approach [5 - 8]; gain and phase margin specification based approach [9 - 13] and others [14 - 18]. However, there have not been yet any method that concurrently dealing with both the settling time and the overshoot of the closed-loop system. It is known that when plants, having transfer functions of first-order, second-order and third-order systems the magnitude optimum method based parameter determination , p Ik T and DT [5 - 8] for the PID controller seems to be one of the most effective methods, validated in reality. However, there has been no fundament until now to conclude that the magnitude optimum method is the only one to provide PID controller parameters to make the closed-loop system stable and the settling time short in the sense that the closed-loop system with transfer function ( ) ( )( ) 1 ( ) ( )= +c R s G s G s R s G s has Nguyen Doan Phuoc, Nguyen Hoai Nam 358 { } ( ) 1R+Ω = ∈ ≈cG jω ω (1) in small range of frequency, which is large enough. A method, proposed in this work, has an approach that the closed-loop system, obtained as follows, / 1( ) 1 = + cG s T s (2) has as small /T as possible (when /T is smaller, Ω is larger), which will demonstrate the conclusion. 2. PID CONTROLLER DESIGN 2.1. Control of First-Order Systems Firstly, let’s assume that the plant has a form of first-order system and a PI controller is used. Then, the transfer function of the plant is represented as follows ( ) 1 = + k G s Ts and the PI controller is ( )11( ) 1 + = + =    p I p I I k T s R s k T s T s Thus, the open-loop transfer function is ( ) ( ) 1( ) ( ) ( ) 1 + = = + p I o I k k T s G s R s G s T s Ts (3) and the closed-loop transfer function is 2 ( ) 1( ) 1 ( ) 11 1 + = = +   + + +     o I c o I I p p G s T s G s G s TT s T s k k k k (4) Since the first-order system (2) with unit gain is always stable, the steady state error is zero and the settling time is smaller if the time constant is shorter, in this case the two parameters pk and IT are determined such that the closed-loop transfer function (4) has the expected form as (2). In order to do that, pk and IT are chosen to cancel out a zero by a pole. The zero is 1= − Is T . This is equivalent to 2 11 1 0   + + + =     I I p p TT s T s k k k k as 1 = − I s T or ( )0 1= − + +I p I pT T k k T k k Settling time assignment with PID 359 Thus .=IT T Next, the closed-loop system becomes / 2 1 1( ) 111 1 + = =   + + + +     I c I I p p T s G s T sTT s T s k k k k with the selected parameters as above, if the time constant /T satisfies the following equation: ( )( ) ( )( ) ( ) / / 2 2 2 11 1 1 1 1 1 1 1   + + = + + = + + +     + = + + I I I p p p p p TT T s T s T s Ts s T s k k k k T k kT s s k k k k ⇔ ( ) ( )2/ / 2 2 11 1++ + + = + +p p p T k kT T T s T Ts s s k k k k ⇔ ( )/ 1+ = − = p p p T k k T T T k k k k This implies ( )2 1 1( ) 1 1 1 + = =  + + + +  I c p I I p p T s G s Tk k TTs T k k s s k k Hence, in the case of =IT T the bigger pk is chosen, the smaller settling time of closed- loop system will be. Furthermore, the closed-loop system is stable and has no overshoot. Obviously, in comparison to the performance of the magnitude optimum criterion method, the proposed PI controller provides a more flexible band of frequency (Ω). When the parameter pk is bigger, the frequency range Ω becomes wider: lim ( ) 1, →∞ = ∀ p c k G s s Therefore, the desired settling time 5%T of closed-loop system can be assigned by choosing 5% ln 20 =p T k kT which is clearly obtained from its unit step response /( ) 1 −= − t Th t e or /5%0.95 1 −= − T Te 2.2. Control of Second-Order Systems When the plant has the form of a second-order system, we use a PID controller. The plant model Nguyen Doan Phuoc, Nguyen Hoai Nam 360 and the controller are represented, respectively, as follows − Plant: ( )( )1 2( ) 1 1= + + k G s Ts T s − PID controller: ( )( )1 11( ) 1 + + = + + =    p A B p D I I k T s T s R s k T s T s T s where + =A B IT T T and =A B I DT T TT , it means that whenever AT and BT are known we can obtain IT and DT . The open-loop system is ( )( ) ( )( ) ( ) ( ) ( ) ( ) / 1 2 1 2 1 1 1 1( ) ( ) ( ) 1 1 1 1 + + + + = = = + + + + p A B p A B o I A k k T s T s k k T s T s G s R s G s T s Ts T s T s Ts T s (5) where / = p A p I k T k T Thus, if it is chosen that 2=BT T then the transfer function (5) has the same form as (3) shown in the first case ( ) ( ) / 1 1( ) 1 + = + p A o A k k T s G s T s Ts where the parameters IT and pk are replaced by the parameters AT and / pk , respectively. So, with the set of parameters in the case 1: 1=AT T and / pk arbitrary, the closed-loop transfer function becomes a first-order system with unity gain and the time constant / 1 1 /= = = I I p A pp T TT T T kk T kkk k and therefore, the bigger pk is chosen, the smaller its time constant /T will be. Thus, we have 1 2 1 2 1 2 and = + = + = = + A B I A B D I T T TT T T T T T T T T T In summary, the larger pk is optionally selected the smaller the settling time associated with /T will be. The assignment of desired settling time 5%T for the closed-loop system can be easily realized with ( )1 2 5% ln 20+ =p T T k kT In addition, there is no overshoot in the closed-loop system. 2.3. Extended Method to Integral First-Order Systems Settling time assignment with PID 361 For the basic control theory of linear systems, it is well known that the symmetric optimum method is applied to find the parameters , p Ik T and DT of the PID controller 1( ) 1 = + +    p D I R s k T s T s (6) for integral first-order systems and integral second-order systems, in order to make the closed- loop system stable and keep the phase margin larger. However, the disadvantage of the method is big overshoot. Although the overshoot can be decreased with the usage of an input filter, this causes an additional cost. The proposed method for determination of PID parameters later will guarantee that the closed-loop system has no overshoot and arbitrary small settling time. It is similar to first-order or second-order systems, the idea of this approach is also to find parameters , p Ik T and DT of the PID controller for the integral first-order plant ( )( ) 1= + k G s s Ts (7) such that the closed-loop system has the following form: ( )( )1 2 1( ) 1 1 + = + + t c m m Ts G s T s T s (8) Consider a closed-loop system, which includes an integral first-order system (7) and a PID controller (6) ( )( )1 11( ) 1 + + = + + =    p A B p D I I k T s T s R s k T s T s T s with + =A B IT T T and =A B I DT T TT . The open-loop system is as follows ( )( ) ( )2 1 1( ) ( ) ( ) 1 + + = = + p A B o I kk T s T s G s R s G s T s Ts Thus, if =BT T then = +I AT T T and the closed-loop system is ( ) ( ) ( )2 1( )( ) 1 ( ) 1 + = = + + + + p Ah c h A p B kk T sG s G s G s T T s kk T s So, the closed-loop system will be the same as the expected form (8) if there the following equation must be satisfied ( ) ( ) ( ) ( )( )2 1 2 1 1 , 1 11 + + = ∀ + ++ + + p A t m mA p A kk T s Ts s T s T sT T s kk T s ⇔ ( ) ( )( ) ( ) ( ) ( )21 21 1 1 1 1 ,  + + + = + + + + ∀ p A m m A p A tkk T s T s T s T T s kk T s Ts s Nguyen Doan Phuoc, Nguyen Hoai Nam 362 Controlled Object PID Input filter Figure 1. Control with PID and input filter. ⇔ ( ) ( ) 1 2 1 2 1 2 1 2 + + + = +   + = + +  +  =  A A m m m m A t p A t A m m A t m m A p T T T T T T T T T kk T T T T T T T T T T T kk ⇔ ( ) 1 2 1 2 1 2  + =   = +  + =  A m m p t m m A t m m A p T T T T kk T T T T T T T T T kk ⇔ 1 2 1 2 + =   = = + A m m p A t m m T T T T kk T T T T This implies that ( ) 1 2 1 2 1 2 1 2 1 2 1 2  + + + = =   = + = + +  + = =  + + A m m p m m m m I A m m m mA B D I m m T T T T T k kT T kT T T T T T T T T T TT T T T T T T (9) It can be seen that with the above equations, 1 mT and 2mT are freely chosen. The expected transfer function (8) becomes ( ) ( )( ) 1 2 1 2 1( ) 1 1 + + = + + m m c m m T T s G s T s T s Thus, with an addition of an input filter as illustrated in Fig.1: ( )1 2 1( ) 1 = + +m m M s T T s to obtain the closed-loop system: ( )( )1 2 1( ) ( ) 1 1 = + +c m m G s M s T s T s we always can choose 1mT and 2mT such that this closed-loop system has a settling time 5%T as desired. Obviously, there are two possible combinations of time constants 1mT and 2.mT The first case is 1 2= =m m mT T T and the second one is 1 2≠m mT T . Now some steps, about how to choose these time constants such that the desired settling time can be achieved, will be shown. Firstly, Settling time assignment with PID 363 let’s consider the case one. With the usage of input filter as shown in Fig. 1, the closed-loop transfer function is as follows ( )2 1( ) 1 = + c m G s T s So, the unit step response is / /( ) 1 − −= − −m mt T t T m t h t e e T Thus, the desired settling time 5%T will be the solution to the following equation / / 0.05 0− −+ − =m mt T t T m t e e T Let 5%= mx T T . Then ( ) (1 ) 0.05 0−= + − =xf x e x (10) As 0, (0) 0.95= =x f and when = ∞x it is obtained ( ) 0.05.∞ = −f In addition, / ( ) 0, 0.−= − xf x e x x That means there must be a unique solution to the equation (10). The solution is approximately equal to 4.7439. Hence, 5% 1 2= 4.7439 = =m m m T T T T (11) Then, the PID parameters are determined from Eq. (9). For the second case, the closed-loop system with input filter is ( )( )1 2 1( ) 1 1 = + +c m m G s T s T s Without loss of generality, let 1 2m s mT k T= where 0 1.sk< < Doing the same procedure as the first case, the following equation is obtained 1 2/ / 1 2 1 2 0.05 0 − − − − = − m mt T t T m m m m T e T e T T (12) Obviously, the expected settling time will be the solution to the equation (12). Let 5% 2= mx T T , then substitute it and 1mT into the equation (12) to have / ( , ) 0.05 0 1 − − − = − = − sx k x s s s k e e f x k k (13) It can be easily verified that (0, ) 0.95, ( , ) 0.05= ∞ = −s sf k f k and / ( , ) 0<sf x k . This implies the equation (13) must have only one solution with respect to .sk The Tab. 1 shows solutions for some values of sk . Table 1. Solution with respect to sk . sk 0.01 0.3 0.5 0.7 0.99 *x 3.005 3.352 3.6761 4.067 4.7202 Nguyen Doan Phuoc, Nguyen Hoai Nam 364 Thus, 5% 2 *=m T T x and 5%1 * .m s T T k x = (14) Hence, given a desired settling time, first we have to choose sk , then time constants 1mT and 2mT are determined by using Eq. 14, and finally the PID parameters are calculated from Eq. 9. It can be concluded that as 1→sk then * 4.7439→x . Thus, the second case becomes the first case. 3. SIMULATION RESULTS 3.1. DC Motor Speed Control Given a speed transfer function of a DC motor as follows [19] ( ) 1 = + k G s Ts where 100=k , 1.9568=T , the input is the current applied to the DC motor and the output is the speed of the rotor. The I controller parameter is determined by using the magnitude optimum method is 2 391.4= =IT kT , whereas the proposed method provides values for the PI controller as 1.9568= =IT T and 0.02=pk ( 0.1=pk for the second case) which can be arbitrarily chosen such that the settling time is smaller than any expected value. Simulation results for two cases are shown in Fig. 2 and Fig. 3. For the first case, the reference is a step function 0.5=r rad/s, but for the second case, the reference is a sine function 0.5sin(0.3 )=r t . It can be seen that the proposed method gives much better performances in terms of settling time and overshoot. Especially, when the reference signal is a sine function, the magnitude optimum method failed to keep the stability and other performances of the close-loop system. To compare performances obtained by using different methods, the IAEs (integral absolute error) at a set point change and IAEd at process disturbance change are used [4]. IAEs is calculated as follow 0 5 10 15 20 250 0.1 0.2 0.3 0.4 0.5 seconds ra d/ s The speed of rotor Magnitude optimum method Proposed method Setpoint signal Figure 2. Responses to a step function. 0 5 10 15 20 25 30-0.5 0 0.5 seconds ra d/ s The speed of rotor Magnitude optimum Proposed method Reference signal Figure 3. Responses to a sine function. Settling time assignment with PID 365 IAE ( ) f i t s t e t dt∫= (15) where e is the error between the plant output and the set point, it is the initial time, which is the time of the step change, ft is an appropriate final time. For the IAEd, the formula is the same as Eq. (15) but the initial time will be the time of a disturbance change. The IAEs and IAEd for the two cases are shown in Tab. 2. A step disturbance with amplitude of 0.01 is applied to the input of the plant at the time of 30 seconds to measure the IAEd. In conclusion, the proposed method provides better performances in term of disturbance compensation and set point change for both cases. Table 2. IAE and IAEd for the two cases. Methods Case 1 Case 2 IAE IAEd IAE IAEd Magnitude Optimum 2.23 4.265 9.254 15.046 Proposed Method 0.4892 0.9788 0.5405 0.7585 3.2. DC Rotor Position Control In this section, the PID controllers are designed to control the angular position of the rotor [19] which has transfer function as below ( )( ) 1= + k G s s Ts where the values of parameters are the same as in the previous section. The PI controller parameters are chosen by using the symmetric optimum method as follows 9, 9 17.61= = =Ia T a and 1 0.0017= =pk kT a but our method gives ( ) 1 2 1 2 1 2 1 2 1 2 0.1575, 2.8, 0.5893 + + = = = + + = + = = + + m m p I m m m m m m D m m T T T k T T T T kT T T T T T T T T where the desired settling time 5% 2=T seconds and 1 2 0.4216.= =m mT T For the second case, if 0.5=sk and 5% 2=T , then 2 0.5441=mT and 1 0.2720.=mT Thus, 37.4724=pk , =2.7729IT and 0.5759.=DT For zero overshoot, an input filter ( ) M s is used to cancel the zero of the close-loop system out. The two proposed PID controllers provide similar performances as shown in Fig. 4. The results show that the proposed method gives better performances, such as smaller settling time as desired. When parameters 1mT and 2mT increase the settling time is larger but the Nguyen Doan Phuoc, Nguyen Hoai Nam 366 overshoot is smaller and vice versa. The IAEs and IAEd are shown in Tab. 3, where a step disturbance with amplitude of 0.01 is applied to the input of the plant at the time of 70 seconds to measure the IAEd. Again, the IAEs and IAEd proved that the proposed method gives much better performance than the symmetric optimum method. 0 10 20 30 40 50 60 700 1 2 3 4 5 6 seconds ra d The angular position of rotor Symmetric optimum Proposed method Setpoint signal Figure 4. Responses to a step function. Table 3. IAEs and IAEd. Methods Performance IAEs IAEd Symmetric Optimum 24.66 104.94 Proposed Method 0.7858 0.2080 4. CONCLUSIONS AND FUTURE WORKS In the work, a PID design method with desired settling time is proposed for plants with transfer functions of the following types: first-order system, second-order system and integral first-order system. The method provides better performances than both the magnitude optimum method and the symmetric optimum method in the sense of overshoot and settling time. 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