Disturbance rejection performance analyses of closed loop control ...
ISA Transactions 55 (2015) 63–71
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio Baris Baykant Alagoz n, Furkan Nur Deniz, Cemal Keles, Nusret Tan Inonu University, Electrical–Electronics Engineering, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 15 November 2013 Received in revised form 14 July 2014 Accepted 19 September 2014 Available online 11 October 2014 This paper was recommended for publication by Prof. Y. Chen
This study investigates disturbance rejection capacity of closed loop control systems by means of reference to disturbance ratio (RDR). The RDR analysis calculates the ratio of reference signal energy to disturbance signal energy at the system output and provides a quantitative evaluation of disturbance rejection performance of control systems on the bases of communication channel limitations. Essentially, RDR provides a straightforward analytical method for the comparison and improvement of implicit disturbance rejection capacity of closed loop control systems. Theoretical analyses demonstrate us that RDR of the negative feedback closed loop control systems are determined by energy spectral density of controller transfer function. In this manner, authors derived design criteria for specifications of disturbance rejection performances of PID and fractional order PID (FOPID) controller structures. RDR spectra are calculated for investigation of frequency dependence of disturbance rejection capacity and spectral RDR analyses are carried out for PID and FOPID controllers. For the validation of theoretical results, simulation examples are presented. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Disturbance rejection control Disturbance suppression PID controllers Fractional order PID controllers Reference to disturbance ratio (RDR)
1. Introduction In real applications, control systems are exposed to environmental disturbances, mainly, in intermittent character and therefore these unpredictable disturbances may severely deteriorate the control performance. Robust control performance is possible for control systems in real world applications, when they exhibit adequate degree of disturbance rejection. Disturbance rejection control (DRC) aims a controller design reducing the negative effects of disturbances on the control performance, and it became one of the major concerns in the design of feedback control systems [1]. DRC design strategies provide a step towards disturbance tolerant control systems. Several works were presented to deal with undesired influence of disturbance signals on the system output in the literature. These works can be taken into account in two folds: The one is explicit approaches that are employing additional functions and blocks designed for disturbance rejection proposes. The second is implicit approaches that are based on utilization of structural disturbance rejection capacity of control systems. Explicit methods contain additional functional blocks such as filters [2–4], disturbance and state observers [1,5], disturbance estimator [6], and a class of n
Corresponding author. E-mail addresses: [email protected] (B.B. Alagoz), [email protected] (F.N. Deniz), [email protected] (C. Keles), [email protected] (N. Tan). http://dx.doi.org/10.1016/j.isatra.2014.09.013 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
robust adaptive state feedback controllers [7]. Since additional function and blocks developed for disturbance rejection may complicate structure of control systems, explicit methods can lead to increased computational complexity in controller design and operations. Implicit approaches rely on inherent disturbance rejection capacity of the conventional control structures without need for any additional blocks. Implicit approaches were addressed in many studies in the literature. Szita et al. suggested a frequency domain design methodology to obtain acceptable disturbance rejection according to a pre-specified reference model for time delay systems [9]. Koussiouris et al. presented frequency-domain conditions for disturbance rejection and decoupling with stability or pole placement [10]. In many works, disturbance rejection controller design problem were taken into account as the minimization of sensitivity function amplitude [11–13]. Besides, the disturbance rejection control based on high gain control was proposed the suppression of unknown disturbances, which are not usually accessible for measurement [1,8]. High-gain control was addressed in many works [14–17]. High-gain feedback design problems were solved for almost disturbance decoupling to find out a sequence of dynamic compensators for linear systems [14,16,17]. It was reported that one of the major problems with disturbance decoupling is that it could be achieved approximately with an arbitrary degree of accuracy [14]. Although high-gain control can reduce effects of unknown disturbance signal on system output, they may also encounter drawbacks of control
64
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
saturation and high peaks in practice and therefore, they need precisely tuning and well-constrained designs for the real world applications. Yu et al. present the performance assessment of PI control loops for rejecting the input load disturbances by using a DS-d IAE-based performance index and a survey on stochastic and deterministic disturbance performance indices [18]. There is a need for the deterministic method for assessment of actual disturbance rejection capacity of closed loop control systems, which considers the tradeoff between reference and disturbance transmission toward the system output. The current study aims to derive straightforward analytical expressions showing spectral dependence of disturbance rejection capacity of closed loop control systems. This paper is devoted in investigation of implicit disturbance rejection capacity of closed loop control systems on the bases of communication channel limitation. We analyzed reference to disturbance ratio (RDR) of closed loop control systems in a similar manner the signal to noise ratio (SNR) defined for communication channels. RDR measure is used to calculate quantitative assessment of reference input dominancy on disturbance at the system output. A control system shows a satisfactory disturbance rejection performance in the case of RDR c 1. If RDR r 1, it can be stated that the control system does not exhibit any disturbance rejection skill. RDR spectra are obtained for spectral investigation of disturbance rejection control performance in this study. We assume unmeasured input disturbance, where unknown additive disturbance signals come into the control system from the plant input. For the improvement of disturbance rejection performance, the proposed method utilizes limitations of control loop communication channel modeling for predominance of reference against the disturbances at the system output. Concepts of communication channel modeling in control loops were discussed in details [19]. In recent years, Rojas presented theoretical works imposing a communication channel in the control loop and addressed problem of feedback stabilization subject to a constraint on the channel signal-to-noise ratio (SNR) [20,21]. Rojas et al. demonstrated that communication channels impose limitations to feedback control [22]. Signal-to-noise ratio performance limitations were investigated for disturbance rejection and stability of the closed loop control system in [21]. For the analytical formulation of RDR method, we consider a negative feedback closed loop control system as the superposition of two low-pass communication channels. One is from reference input to the system output and the other is from disturbance input to the output. RDR spectrum is derived for closed loop control system and detailed analyses are carried out for PID controller family. Frequency dependence of disturbance rejection is investigated for classical PID and FOPID controller structures. Moreover, design criteria based on RDR performance specification are introduced for disturbance rejection PID and FOPID controller design problems. Matlab/Simulink closed loop control system simulations validate our theoretical findings. Disturbance rejection performances are compared for PID and FOPID controller designs. Motivation of this study comes from the requirement for straightforward and practical formulation of disturbance rejection capacity of closed loop PID and FOPID control systems. This makes disturbance rejection performance of control systems measurable and comparable. Moreover elaborating impacts of design coefficients on disturbance rejection performance of closed loop PID controller family is very useful for controller practice. RDR specification in control design problems helps to improve robust performance of PID and FOPID controllers in real control application. The paper is organized as follows: the following section was devoted for reference and disturbance channel modeling of a negative feedback control system and a general formula to express the ratio of reference signal to disturbance signal at the output is
derived. In the further section, PID and FOPID controller design constraints for desired disturbance rejection are derived and illustrative examples are presented.
2. Methodology 2.1. Reference to disturbance ratio Consider a closed loop control system as the superposition of two low-pass communication channels. The first channel is from the reference signal to output of control system and referred as to Reference Channel Control (RCC) and the other is from input of plant to the output of control system and referred as to Disturbance Channel Control (DCC). Fig. 1 shows a closed loop control system model with additive disturbance signal. Assuming a linear time invariant system, the system in Fig. 1 can be written as a superposition of two systems, namely RCC and DCC models, as illustrated in Fig. 2. Fig. 2(a) shows RCC system obtained in the case of a zero disturbance ðd ¼ 0Þ, and Fig. 2(b) shows DCC system in Fig. 1, when the reference signal is zero ðr ¼ 0Þ. Lets denote the transfer function of RCC system given in Fig. 2 by P r ðsÞ. The P r ðsÞ can be written as, P r ðsÞ ¼
Q ðsÞjd ¼ 0 CðsÞGðsÞ ¼ 1 þ CðsÞGðsÞ rðsÞ
ð1Þ
Lets denote the transfer function of DCC system given in Fig. 2 by P d ðsÞ. The P d ðsÞ can be written as, P d ðsÞ ¼
Q ðsÞjr ¼ 0 GðsÞ ¼ 1 þ CðsÞGðsÞ dðsÞ
ð2Þ
One can express output of the closed loop control system by superposing the RCC system output Q r ðsÞ ¼ Q ðsÞjd ¼ 0 and the DCC system output Q d ðsÞ ¼ Q ðsÞjr ¼ 0 as, Q ðsÞ ¼ Q r ðsÞ þ Q d ðsÞ
ð3Þ
Ogata also expressed Q ðsÞ in an open form with respect to controller, plant and feedback transfer functions in order to figure out the necessary conditions for disturbance suppression at the control system output [8]. Ogata suggested the condition of jCðsÞGðsÞj c 1 to improve disturbance rejection performance of closed loop control systems with unity feedback. Because, in the case of high open loop gain (jCðsÞGðsÞj c 1), the transfer function from disturbance to system output P d ðsÞ ¼ Q ðsÞ=dðsÞ goes to almost zero. However, suppression of reference signal by the control system should be also taken into account for a realistic evaluation of disturbance rejection performance in control point of view. We further deepen this perfective for the measure of disturbance
d
e r
C (s )
G (s )
u
Q
Fig. 1. A closed loop control system with additive input disturbance model.
e
r
C (s )
u
G (s )
Q
d C (s )
u
G (s )
Q
Fig. 2. (a) RCC system model obtained for zero disturbance ðd ¼ 0Þ; (b) DCC system model obtained for zero reference signal ðr ¼ 0Þ of closed loop control system with additive disturbance model.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
rejection capacity by a fair assessment between reference and disturbance transmissions to the system output. Spectral power of reference signal at the output of closed loop control system can be expressed as jQ r ðjwÞj2 ¼ jP r ðjwÞj2 jrðjwÞj2
ð4Þ
Spectral power of disturbance signal at the output of closed loop control system can be expressed as: jQ d ðjwÞj2 ¼ jP d ðjwÞj2 jdðjwÞj2
ð5Þ
In analogy with Signal to Noise Ratio (SNR) of communication channels, Reference to Disturbance Ratio (RDR) can be defined as, RDR ¼
2
2
2
jQ r ðjwÞj jP r ðjwÞj jrðjwÞj ¼ jQ d ðjwÞj2 jP d ðjwÞj2 jdðjwÞj2
ð6Þ
In order to evaluate the disturbance attenuation effect of the closed loop control system comparable with the power of reference signal, input of RCC and DCC systems have equal powers, jrðjwÞj2 ¼ jdðjwÞj2 , and thus RDR of the closed loop system can be obtained as, RDR ¼
jP r ðjwÞj2 jP d ðjwÞj2
ð7Þ
When transfer functions of RCC and DCC systems are used in Eq. (7), one obtains an elegant expression of RDR for a closed loop system as: RDR ¼ jCðjwÞj2
ð8Þ
Eq.(8) clearly demonstrates that disturbance suppression strength of a closed loop system strongly depends on energy spectral density of the controller transfer function. This is a significant result stating that disturbance rejection capacity of a negative feedback closed loop control system is solely determined by controller structures. Hence a disturbance tolerant closed loop control system design problem can be reduced to controller design problem. It might be convenient to express the RDR in decibel (dB) as, RDRdB ¼ 20 log jCðjwÞj:
ð9Þ
2.2. Relation of RDR spectrum with sensitivity function In the control literature, sensitivity function is considered to improve disturbance rejection performance of closed loop control systems [11,12,13]. To ensure a good output disturbance rejection, a sensitivity function constraint was given for a desired frequency range ( 8 w r ws rad/s) [11] as follows, 1 SðjwÞ ¼ rB ð10Þ 1 þ CðjwÞGðjwÞdB Although, minimization of sensitivity function increases disturbance rejection performance of a closed loop control system, it does not provide a relevant measurement for disturbance rejection capacity. Sensitivity function, defined as S ¼ eðsÞ=rðsÞ, is a transfer function from the reference input to tracking error [12] and it does not consider power of disturbance on system output, directly. Hence, RDR analysis is convenient for assessment of disturbance rejection capacity. High RDR value is possible by increase of controller amplitude jCðjwÞj, and this decreases sensitivity function. Relation between RDR and sensitivity can be written as, j1 SðjwÞj2 RDR ¼ jCðjwÞj2 ¼ jSðjwÞj2 jGðjwÞj2
ð11Þ
Very low amplitude of sensitivity function results in a high RDR performance. On the other hand, by considering Eq. (7), one can
65
see that RDR also appears as the ratio of energy spectral density of complementary sensitivity function ðjCðjwÞGðjwÞ=ð1 þ CðjwÞ GðjwÞÞj2 Þ and energy spectral density of load disturbance sensitivity function ðjGðjwÞ=ð1 þCðjwÞGðjwÞÞj2 Þ. Essentially, a high RDR performance is obtainable by increasing transmission capacity from reference input to the system output and decreasing transmission capacity from disturbance input to the system output.
3. RDR analysis for PID controller family In order to investigate effects of gain coefficients (kp ,ki ,kd ) on RDR performance of closed loop control system, RDR of classical PID controller structure is derived. To see effects of order coefficients (λ,μ) on RDR performance of closed loop control system, RDR of fractional order PID controller structure is investigated in this section. 3.1. RDR performance of PID controller Transfer function of PID controller structure is commonly written as, CðsÞ ¼ kp þ
ki þ kd s; s
ð12Þ
where, (kp ,ki ,kd ) are PID controller coefficients used for tuning the controller performance. Accordingly, the amplitude response of a PID controller can be written as, 1=2 ðki kd w2 Þ2 þ ðkp wÞ2 jCðjwÞj ¼ : ð13Þ jwj Then, by considering Eq. (13), RDR of the closed loop PID control system is obtained as, 2 ki 2 RDR ¼ kd w þ kp : ð14Þ w Eq. (14) indicates that RDR of closed loop PID control system depends on angular frequency w as the following: Solely integrator component, tuned by ki , increases RDR performance at low frequencies. The derivative component, tuned by kd , improves RDR performance at high frequencies, and the gain component, tuned by kp , contributes to RDR performance regardless of angular frequency. The following RDR response characteristics can be observed depending on conditions of PID coefficients (kp ,ki ,kd ): (i) When kp a 0, ki a0 and kd a 0, RDR performance of PID controllers is expressed by Eq. (14). Its RDR spectrum exhibits a pffiffiffiffiffiffiffiffiffiffiffi minima at the frequency wm ¼ ki =kd . Minimum RDR performance of the PID controller is determined by the proportional 2 gain as kp at the frequency wm . Fig. 3(a) illustrates the RDR spectrum of PID controller for (kp ¼ 1,ki ¼ 1,kd ¼ 1). In this figure, minima of RDR performance appears at w ¼ 1 as the value of one due to kp ¼ 1. Around the minima, RDR performance increases depending on ki for lower frequencies and kd for higher frequencies. (ii) When kp a0, ki a0 and kd ¼ 0, PID controller structure implements a Proportional-Integrator (PI) controller and RDR of PI controller can be written as, 2 ki 2 RDR ¼ þ kp : ð15Þ w this equation implies that as the angular frequency increases, RDR performance of a PI controller decreases. For very low frequencies or near to DC, RDR can reach to enormously high values. For DC signals, RDR performance of PI controller theoretically goes to infinity. So, it exhibits very strong disturbance attenuation for very low frequencies. Increasing kp improves the disturbance
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
30
30
25
25
20
20
RDRdB
RDRdB
66
15
15
10
10
5
5
0 0
0.5
1
1.5 w
2
2.5
0
3
0
0.5
1
1.5 w
2
2.5
3
10 0.5 RDRdB
RDRdB
8 6 4
-0.5
2 0
0
0
0.5
1
1.5 w
2
2.5
3
-1
0.5
1
1.5 w
2
2.5
3
Fig. 3. (a) RDR spectra of PID controller for (kp ¼ 1,ki ¼ 1,kd ¼ 1); (b) RDR spectrum of PI controller for (kp ¼ 1, ki ¼ 1); (c) RDR spectrum of PD controller for (kp ¼ 1, kd ¼ 1); (d) RDR performance of P controller for ðkp ¼ 1Þ.
rejection for low frequency disturbances [23]. Fig. 3(b) illustrates RDR spectrum of PI controller for (kp ¼ 1,ki ¼ 1). (iii) When kp a0, ki ¼ 0 and kd a 0, PID controller structure implements a Proportional-Derivative (PD) controller and RDR of PD controller can be written as, 2
RDR ¼ ðkd wÞ2 þ kp :
ð16Þ
this equation implies that as the angular frequency increases, RDR of a PD controller increases according to second-order polynomial characteristics. For the very low frequencies or near to DC, RDR 2 approximates to the term of kp . In other words, the lowest RDR performance of PD controller can be adjusted by kp for the low frequency region. Fig. 3(c) illustrates the RDR spectrum of PD controller for (kp ¼ 1, kd ¼ 1). (iv) When kp a0, ki ¼ 0 and kd ¼ 0, PID controller structure implements a proportional controller (P) and RDR of Proportional (P) controller can be expressed simply as, 2
RDR ¼ kp
ð17Þ
this RDR performance of P controller is independent of the angular frequency as illustrated in Fig. 3(d). The minimal and frequency-independent value of RDR, which is adjusted by kp , spreads uniformly in the whole RDR spectrum. Utilization of a RDR performance as a design criteria in PID tuning problems can be very beneficial for the enhancement of disturbance rejection skills of closed loop PID control systems. PID tuning algorithms or methods should consider RDR performance design constraint in parameter optimization process to benefit from inherent disturbance tolerance property of closed loop control structures. A general RDR design constraint ensuring a given minimal RDR performance (α) can be defined as, αr
ki kd w w
2
2
þ kp
ð18Þ
It can be clearly observed from Eq. (18) that the proportional gain kp is a fundamental term for PID, PD, PI and P controllers to determine minimal RDR performance. In control practice, reference signal and disturbance signal mostly contain low frequency components. PI can be more advantageous to obtain higher RDR spectrum compared to PID controllers due to the term of ððki =wÞ kd wÞ. Fig. 4 shows RDR spectra of PI versus PID controller for equal common coefficients. In this figure, the ki coefficient of PID controller set to π 2 kd in order to move the minima frequency ðwm Þ towards the angular frequency of π in the spectrum. As seen, PI controller provides a better RDR compared to RDR spectrum of PID controller at high frequency region. It is concluded that due to the derivative term in PID, PID control system can be more sensitive to high frequency disturbance signals. In practice, high frequency disturbance may originate from environmental or system noise. Fig. 5 shows ki and kd dependence of RDR for w ¼ 0:1, w ¼ 1, w ¼ 2 and w ¼ 3 frequencies. α 4 20 dB lines in figures are used to indicate (ki ,kd ) search regions for tuning algorithms. Intersection of stability ranges of coefficients and α 4 20 dB region promises a disturbance tolerant, stable PID control system design. As a result, use of RDR design constraint enhances effectiveness of heuristic optimization algorithms due to narrowing their search regions. Increasing kp results in expanding α 420 dB regions and kp provides improvement of RDR independent of angular frequency for the whole spectrum. Fig. 5 also demonstrates that at low frequencies ðw ¼ 0:1Þ, RDR strongly depends on value of ki parameter, but at the high frequencies ðw ¼ 3Þ, RDR becomes very dependent of kd values.
3.2. Disturbance rejection capacity of fractional order PID controllers Transfer function of FOPID controllers are commonly written as, C f o ðsÞ ¼ kp þ
ki þ kd sμ ; sλ
ð19Þ
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
where, (kp ,ki ,kd ) are gain coefficients and (λ,μ) are fractional order coefficients of fractional order PID controller. By using sα ¼ jα wα and jα ¼ cos ððπ=2ÞαÞ þ j sin ððπ=2ÞαÞ, the amplitude response of FOPID controllers can be derived as, π π 2 jC f o ðjwÞj ¼ kp þ ki cos λ w λ þ kd cos μ wμ 2 2 π π 2 1=2 ð20Þ þ kd sin μ wμ ki sin λ w λ 2 2 then, by considering Eq. (8), RDR of the closed loop FOPID control system is written as, π π 2 RDRf o ¼ kp þ ki cos λ w λ þ kd cos μ wμ 2 2 2 π π μ þ kd sin μ w ki sin λ w λ : ð21Þ 2 2 Effects of gain coefficients (ki ,kd ) on RDR performance of FOPID strongly depends on (λ,μ) fractional order coefficients due to the factors cos ððπ=2ÞλÞ, cos ððπ=2ÞμÞ, sin ððπ=2ÞλÞ and sin ððπ=2ÞμÞ. Besides, (λ,μ) coefficients appear on the power of angular frequency as w λ and wμ . This complicates simplified analyses as given in previous section for PID controller. However, for λ¼ 1 and
PID PI
35
RDRdB
30 25
67
μ¼1, FOPID works as classic PID structure and the RDR formulation of FOPID simplifies to Eq. (14). For ki 4 0 and kd 4 0, a local minimum of RDRfo spectrum characteristic can be found by solving 0 ¼ ðkd sin ððπ=2ÞμÞwμ ki sin ððπ=2ÞλÞ w λ Þ2 equation as, 1=ðλ þ μÞ ki sin ððπ=2ÞλÞ ð22Þ wm ¼ kd sin ððπ=2ÞμÞ Fig. 6 reveals (λ,μ) dependence of RDR performance for FOPID controller at various frequencies. It is clearly seen in Fig. 6(a) and (d) that RDR performance shows dependence on λ at low frequency (w ¼ 0:1 rad/s) and μ at high frequency (w ¼ 3 rad/s). Fig. 6(b) shows that RDR performance is severely deteriorated at w ¼ 1 rad/s. The figure indicates very low RDR performance around λ ¼ 1:5 and μ ¼ 1:5 at w ¼ 1 rad/s for ki ¼ 1, kd ¼ 1 and kp ¼ 1. Fig. 7 shows λ and μ dependence of RDR for FOPID controllers. Fig. 8 shows RDR spectrum for various λ and μ. Fractional orders λ and μ are very effective on RDR spectrum due to the terms of w λ and wμ . Decrease of λ and μ values more flattens RDR spectrum. As values of λ and μ increase, RDR spectrum exhibits more dependence on the angular frequency. A general RDR design constraint ensuring a given minimal RDR performance (α) for FOPID controller can be defined as, π π 2 α r kp þ ki cos λ w λ þkd cos μ wμ 2 2 π π 2 þ kd sin μ wμ ki sin λ w λ ð23Þ 2 2
20 15
4. Simulation examples
10
This section shows disturbance tolerant PID controller examples to validate theoretical findings. Matlab/Simulink simulations were used in these illustrative examples:
5 0.5
1
1.5 w
2
2.5
3
Fig. 4. A comparison of RDR spectra for PID controllers (kp ¼ 1, ki ¼ 9:8, and kd ¼ 1) and PI controllers (kp ¼ 1, ki ¼ 9:8).
Example 1. This example compares disturbance rejection performance of two PID controller designs for a stable, second order plant function given as GðsÞ ¼ 1=ðs2 þ 4s þ 3Þ.
30
5
25
10
20
15
15
20
10
20
25
5
0
30
80
10
60
15
kd
kd
5
40
20 25 30
5
10
15
20
25
30
5
10
ki
15
20
25
30
0
ki
35 30
5
kd
20
15
15
20
10
25 30
5 5
10
15 ki
20
25
30
kd
25
10
35
5
30
10
25
15
20 15
20
10
25 30
5 5
10
15
20
ki
Fig. 5. RDR versus ki and kd drawing for kp ¼ 1 and various angular frequencies.
25
30
68
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
35 0.5
-10
25
1
0
0.5
30
1
20
-20
15
1.5
1.5
-30
10 2
0.5
1
1.5
2
2
0.5
1
1.5
2
18
10 9
0.5
16
0.5
14
8 1
12
1
7
10
6
1.5
1.5
8
5 2
0.5
1
1.5
2
-40
6
2
4
0.5
1
1.5
2
Fig. 6. RDR versus λ and μ for ki ¼ 1, kd ¼ 1 and kp ¼ 1 at various angular frequencies.
30
40
20
w=3
10
w=2
0 -10
w = 0 .1
20
w = 0 .1 RDRdB
RDRdB
30
w=3 10
w= 2 0
w =1 0
w =1
0.5
1
1.5
λ
-10
2
0
0.5
1
1.5
μ
2
Fig. 7. λ and μ dependence of RDR for ki ¼ 1, kd ¼ 1 and kp ¼ 1.
50
30 20
30
RDRdB
RDRdB
40
20
0.6 0.8
10 0 -10
0.6 0.8
10 0
1.4 0.5
1.2 1
1.0
1.5 w
-10 2
2.5
3
1.0 1.2
1.4 0.5
1
1.5 w
2
2.5
3
Fig. 8. RDR spectrum for ki ¼ 1 kd ¼ 1 kp ¼ 1 various λ ¼ {0.6,0.8,1.0,1.2,1.4} in (a) and μ¼ {0.6,0.8,1.0,1.2,1.4} in (b).
Fig. 9 shows RDR spectra for two PID designs; one has a low disturbance rejection performance with the low gain PID coefficients (kp ¼ 1, ki ¼ 1, kd ¼ 1) and the other presents a high disturbance rejection performance with the high gain PID
coefficients (kp ¼ 30, ki ¼ 5, kd ¼ 10). RDR values calculated from simulations confirm analytical results calculated by using Eq. (14). Fig. 10 (a) shows step response of RCC in the case of a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1 for a zero disturbance.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
It settles properly. Fig. 10(b) shows step response of DCC for a zero reference input. A step at disturbance input results in a peak with the amplitude of 0.2 at the output in Fig. 10(b). To enhance disturbance tolerance of this control system, the PID coefficients were tuned as kp ¼ 30, ki ¼ 5 and kd ¼ 10 in order to improve the RDR spectrum as illustrated in Fig. 9. This provides a faster step response for RCC as in Fig. 10(c) and considerably reduces peak amplitude obtained from DCC at the output in Fig. 10(d). Fig. 10 reveals impacts of rise time on disturbance rejection performance of control systems. Faster closed loop control systems are more disturbances tolerant so that they can respond fast enough against to the disturbance signal and thus better holds their outputs at the setpoint. Fig. 11 reveals improvements in disturbance rejection effect for sinusoidal input signals. Fig. 11(a) shows response of PID control system (kp ¼ 1,ki ¼ 1,kd ¼ 1) for a sinusoidal reference signal (amplitude of one and the frequency of 0.1 rad/s) and a sinusoidal disturbance signal (amplitude of one and the frequency of 0.5 rad/s). The disturbance signal caused large ripples at output. When PID
:k p=1 k i=1 k d=1 :k p=30 k i=5 k d=10
RDRdB
40
:Simulation
30 20 10 0
0.5
1
1.5 w
2
2.5
3
Fig. 9. A comparison of RDR spectra for PID controllers for a low (kp ¼ 1, ki ¼ 1, and kd ¼ 1) and for a high (kp ¼ 30, ki ¼ 5, kd ¼ 10) disturbance rejection performance.
control system was configured to kp ¼ 30, ki ¼ 5, and kd ¼ 10 to improve RDR performance as shown in Fig. 9, ripples caused from the disturbance remarkably diminished and the output can well follow the reference signal. Fig. 12 (a) and (b) illustrate step response of DCC for PD and PI controller respectively. Since PD controller exhibits better RDR at high frequencies, effects of disturbance on the output contain mainly low frequency and DC components as in Fig. 12(a). However, a PI control system presents higher RDR at low frequencies or DC region, PI control system exhibits rather high frequency components at the output as illustrated in Fig. 12(b). These results confirm the theoretical analysis presented in previous sections.
Example 2. This example compares RDR performance of two PID controllers and a FOPID controller design and demonstrates disturbance response of these controllers for an unstable, time delay plant model, given as GðsÞ ¼ ð1=ðs 1ÞÞe 0:5s [9]. System instability is a major factor limiting disturbance rejection capacity of control systems so that unstable control systems are practically useless. The time delay of plant system considerably limits the stability interval of controller coefficients and accordingly RDR performance of controllers. Since GðsÞ plant is an unstable time delayed system, it considerably limits design ranges of controller coefficients in this example. We applied set and trial methods to find out a PID and a FOPID designs with satisfying RDR performances and compared it with a disturbance rejection PID design (kp ¼ 2:168, ki ¼ 0:693, kd ¼ 0:358) proposed by Szita et al. [9]. Fig. 13 shows RDR spectra and disturbance response of Szita et al.'s PID, the proposed PID (kp ¼ 2:5, ki ¼ 0:8, kd ¼ 0:4) and FOPID (kp ¼ 0:5, ki ¼ 1:6, kd ¼ 1:5, λ ¼ 0:8, μ ¼ 0:9) controller designs. FOPID is implemented by Carlson approximation [24]. As a result of imperfection of the approximate FOPID implementation, RDR performance of FOPID can be 3–4 dB less than theoretical calculations.
1.2
1
1
0.8
0.8
Q
Q
1.2
0.6
Q(t) r(t) d(t)
0.4 0.2
0.6 0.4
0.2
Q
50
69
0.1
5
0.2
10
15
20
t
0
0 0
20
40
60
20
80
40
60
80
100
t
t
1.2
1.2 1
1
0.8
0.8
0.6
0.6
0.02
0.4
0.01
0.03
Q
Q
Q(t) r(t) d(t)
0
Q(t) r(t) d(t)
0.4 0.2
0.2
40
60 t
80
0 10
0
0 20
Q(t) r(t) d(t)
100
20
30
t
20
40
60
80
100
t
Fig. 10. Step response of RCC in (a) and DCC in (b) for a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1. Step response of RCC in (c) and DCC in (d) for a PID controller with kp ¼ 30, ki ¼ 5, kd ¼ 10.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
1.5
1.5
1
1
0.5
0.5
0
0
Q
Q
70
-0.5 -1 -1.5
-0.5
Q(t) r(t) d(t) 0
20
40
60
80
Q(t) r(t) d(t)
-1 -1.5
100
0
20
40
60
80
100
t
t
Fig. 11. (a) Sinus response of a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1. (b) Sinus response of a PID controller with kp ¼ 30, ki ¼ 5, kd ¼ 10, which is more disturbance tolerant.
0.05
0.03
0.02
Q
Q
Q(t) r(t) d(t)
0.04
0.03
Q(t) r(t) d(t)
0.01
0.02 0.01 0
0 5
10
15
20
25
5
10
15
t
20
25
t
Fig. 12. (a) Effect of PD controller (kp ¼ 30, ki ¼ 0, kd ¼ 10) on disturbance rejection (b) Effect of PI controller (kp ¼ 30, ki ¼ 5, kd ¼ 0) on disturbance rejection.
0.8
20
Q
RDRdB
30
d(t) Szita et al. PID FOPID
1
Szita et al. PID FOPID
0.6 0.4
10
0.2
0 0
0.5
1
1.5 w
2
2.5
3
0
5
6
7
8
t
Fig. 13. (a) A comparison of RDR spectra for Szita et al.'s PID (kp ¼ 2:168, ki ¼ 0:693, kd ¼ 0:358) and proposed PID (kp ¼ 2:5, ki ¼ 0:8, kd ¼ 0:4) and FOPID (kp ¼ 0:5, ki ¼ 1:6, kd ¼ 1:5,λ ¼ 0:8,μ ¼ 0:9) controller designs. (b) A comparison of step disturbance response of controllers.
Due to the fact that environmental disturbance is in random nature; it is convenient to illustrate disturbance rejection performance of controllers against a noisy disturbance signal generated with normally distributed random numbers. Fig. 14 demonstrates noise suppression performance of controllers at the output for PID designs. It is clearly seen that control systems attenuate and smooth the band limited white noise with the power of 0.001, noticeably. It is observed that there are slight differences between controller responses for noisy disturbance. However, the FOPID controller design can give slightly better response for noisy disturbance because FOPID controller presents higher RDR values at the high frequency region compared to other controllers as seen in Fig. 13(b). Unpredictable disturbance naturally appears in real control systems and robust performance of control system is very dependent of disturbance rejection performance.
It is also noteworthy that disturbance rejection capacity of different controller designs can be compared by means of RDR spectrum analyses, and this makes possible to select a better controller designs with a desirable RDR spectrum via RDR spectrum comparisons.
5. Conclusions This study presented RDR analysis method for evaluation of disturbance rejection capacity of closed control systems. This method is based on utilization of communication channel limitations to improve disturbance rejection performance of control systems. RDR spectrum well describes the spectral dependence of disturbance rejection capacity. Eq. (8) shows that RDR spectra
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
provided by larger kp , enhance overall disturbance rejection performance of PID controller family.
1.5 r(t) d(t) Szita et al. PID FOPID
1
Q
0.5
71
This study elaborates disturbance rejection control for PID controller family. Findings of this paper contribute to enhancement of robust performance of PID and FOPID controllers in control engineering practice.
0 -0.5
References
-1 -1.5
0
20
40
60
80
100
t
Fig. 14. Effects of white noise disturbance with the power of 0.001 on controller performance.
of closed loop control systems are determined solely by energy spectral density of controller function and therefore, RDR performance of a control system can be also adjusted by only tuning controller parameters. Disturbance tolerant PID and FOPID controller design constraints were derived for tuning problems. The theoretical findings were validated by simulation results. Some important remarks can be summarized as follows: – It is shown that energy spectral density of the controller transfer function determines RDR of closed loop systems. – RDR analysis provides quantitative evaluation of disturbance rejection capacity of controllers and yields straightforward analytical solutions that are very useful for controller tuning problems. – Frequency dependence of RDR spectrum is characterized by fractional orders λ and μ due to the terms of w λ and wμ . In the case of λ ¼ 1 and μ ¼ 1, RDR of PID controller exhibits frequency dependence characterized by the term ðki =w kd wÞ in Eq. (14). PD controller structures are effective at high frequency region according to the term of kd w, PI controller structures are effective at low frequency region according to the term of 2 ki =w. p PID controller exhibits a minima with the value of kp at ffiffiffiffiffiffiffiffiffiffiffi wm ¼ ki =kd . – The coefficient kp is frequency independent fundamental component of RDR spectrum. It can be effectively used to adjust value of minimal RDR performance of PID and FOPID control systems regardless of angular frequency. – (λ,μ) fractional order coefficients of FOPID scales gain coefficients (kp ,ki ,kd ) by sinusoidal functions cos ððπ=2ÞλÞ, cos ððπ=2ÞμÞ, sin ððπ=2ÞλÞ and sin ððπ=2ÞμÞ. – Large time-delays or low-gain controllers reduce disturbance rejection performance due to delaying controller action to deal with disturbances. Faster PID and FOPID controls, particularly
[1] Chang JL. Robust output feedback disturbance rejection control by simultaneously estimating state and disturbance. J Control Sci Eng 2011;2011:1–13. [2] Vukic Z, Kuljaca O. Lect PID Controllers 2002:23. [3] Vrancic D, Strmčnik S, Kocijan J, de Moura Oliveirac PB. Improving disturbance rejection of PID controllers by means of the magnitude optimum method. ISA Trans 2010;49(1):47–56. [4] Vandeursen JME, Peperstraete JA. Model-based and PID controllers for disturbance rejection in processes with time delay: a comparison. ISA Trans 1996;35(3):225–36. [5] Busawon KK, Kabore P. Disturbance attenuation using proportional integral observers. Int J Control 2001;74(6):618–27. [6] Shamsuzzoha M, Lee M. Enhanced disturbance rejection for open-loop unstable process with time delay. ISA Trans 2009;48(2):237–44. [7] Jin XZ, Yang GH. Robust adaptive fault-tolerant compensation control with actuator failures and bounded disturbances. Acta Autom Sin 2009;35:305–9. [8] Ogata K. Modern Control Engineering. New Jersey: Prentice Hall; 2010. [9] Szita G, Sanathanan CK. Robust design for disturbance rejection in time delay systems. J Franklin Inst 1997;334(4):611–29. [10] Koussiouris TG, Tzierakis KG. Frequency-domain conditions for disturbance rejection and decoupling with stability or pole placement. Automatica 1996;32(2):229–34. [11] Monje CA, Vinagre BM, Feliu V, Chen YQ. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng Pract 2008;16: 798–812. [12] Doyle JC, Francis B, Tannenbaum A. Feedback Control Theory. New York: Macmillan Publishing; 1992. [13] Aström KJ, Richard MM. Feedback Systems: An Introduction for Scientists and Engineers. New Jersey: Princeton University Press; 2008. [14] Willems JC. Almost invariant subspaces: an approach to high-gain feedback design – Part I: Almost controlled invariant subspaces. IEEE Trans Autom Control 1981;26(1):235–52. [15] Young KD. Disturbance decoupling by high gain feedback. IEEE Trans Autom Control 1982;27(4):970–1. [16] Trentelman HL, Willems JC. Guaranteed roll-off in a class of high-gain feedback design problems. Syst Control Lett 1983;3(6):361–9. [17] Willems JC. Almost invariant subspaces: an approach to high gain feedback design – part II: almost conditionally invariant subspaces. IEEE Trans Autom Control 1982;27(5):1071–85. [18] Yu Z, Wang J. Assessment of proportional ıntegral control loop performance for ınput load disturbance rejection. Indus Eng Chem Res 2012;51:11744–52. [19] Elia N. When bode meets Shannon: control-oriented feedback communication schemes. IEEE Trans Autom Control 2004;49(9):1477–88. [20] Rojas AJ. Signal-to-noise ratio fundamental limitations in the discrete-time domain. Syst Control Lett 2012;61(1):55–61. [21] Rojas AJ. Signal-to-noise ratio performance limitations for input disturbance rejection in output feedback control. Syst Control Lett 2009;58(5):353–8. [22] Rojas AJ, Middleton RH, Freudenberg JS, Braslavsky JH. Input disturbance rejection in channel signal-to-noise ratio constrained feedback control. In: Proceedings of the 2008 American Control Conference, Seattle, Washington, USA; 2008. p. 3100–5. [23] Aström KJ, Hagglund T. Advanced PID Control. Chapter 4: Controller Design. The ISA Society; 2006. [24] Carlson GE, Halijak CA. Approximation of fractional capacitors (1/s)\widehat1/n by a regular Newton process. IEEE Trans Circuit Theory 1964;7:210–3.
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio Baris Baykant Alagoz n, Furkan Nur Deniz, Cemal Keles, Nusret Tan Inonu University, Electrical–Electronics Engineering, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 15 November 2013 Received in revised form 14 July 2014 Accepted 19 September 2014 Available online 11 October 2014 This paper was recommended for publication by Prof. Y. Chen
This study investigates disturbance rejection capacity of closed loop control systems by means of reference to disturbance ratio (RDR). The RDR analysis calculates the ratio of reference signal energy to disturbance signal energy at the system output and provides a quantitative evaluation of disturbance rejection performance of control systems on the bases of communication channel limitations. Essentially, RDR provides a straightforward analytical method for the comparison and improvement of implicit disturbance rejection capacity of closed loop control systems. Theoretical analyses demonstrate us that RDR of the negative feedback closed loop control systems are determined by energy spectral density of controller transfer function. In this manner, authors derived design criteria for specifications of disturbance rejection performances of PID and fractional order PID (FOPID) controller structures. RDR spectra are calculated for investigation of frequency dependence of disturbance rejection capacity and spectral RDR analyses are carried out for PID and FOPID controllers. For the validation of theoretical results, simulation examples are presented. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Disturbance rejection control Disturbance suppression PID controllers Fractional order PID controllers Reference to disturbance ratio (RDR)
1. Introduction In real applications, control systems are exposed to environmental disturbances, mainly, in intermittent character and therefore these unpredictable disturbances may severely deteriorate the control performance. Robust control performance is possible for control systems in real world applications, when they exhibit adequate degree of disturbance rejection. Disturbance rejection control (DRC) aims a controller design reducing the negative effects of disturbances on the control performance, and it became one of the major concerns in the design of feedback control systems [1]. DRC design strategies provide a step towards disturbance tolerant control systems. Several works were presented to deal with undesired influence of disturbance signals on the system output in the literature. These works can be taken into account in two folds: The one is explicit approaches that are employing additional functions and blocks designed for disturbance rejection proposes. The second is implicit approaches that are based on utilization of structural disturbance rejection capacity of control systems. Explicit methods contain additional functional blocks such as filters [2–4], disturbance and state observers [1,5], disturbance estimator [6], and a class of n
Corresponding author. E-mail addresses: [email protected] (B.B. Alagoz), [email protected] (F.N. Deniz), [email protected] (C. Keles), [email protected] (N. Tan). http://dx.doi.org/10.1016/j.isatra.2014.09.013 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
robust adaptive state feedback controllers [7]. Since additional function and blocks developed for disturbance rejection may complicate structure of control systems, explicit methods can lead to increased computational complexity in controller design and operations. Implicit approaches rely on inherent disturbance rejection capacity of the conventional control structures without need for any additional blocks. Implicit approaches were addressed in many studies in the literature. Szita et al. suggested a frequency domain design methodology to obtain acceptable disturbance rejection according to a pre-specified reference model for time delay systems [9]. Koussiouris et al. presented frequency-domain conditions for disturbance rejection and decoupling with stability or pole placement [10]. In many works, disturbance rejection controller design problem were taken into account as the minimization of sensitivity function amplitude [11–13]. Besides, the disturbance rejection control based on high gain control was proposed the suppression of unknown disturbances, which are not usually accessible for measurement [1,8]. High-gain control was addressed in many works [14–17]. High-gain feedback design problems were solved for almost disturbance decoupling to find out a sequence of dynamic compensators for linear systems [14,16,17]. It was reported that one of the major problems with disturbance decoupling is that it could be achieved approximately with an arbitrary degree of accuracy [14]. Although high-gain control can reduce effects of unknown disturbance signal on system output, they may also encounter drawbacks of control
64
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
saturation and high peaks in practice and therefore, they need precisely tuning and well-constrained designs for the real world applications. Yu et al. present the performance assessment of PI control loops for rejecting the input load disturbances by using a DS-d IAE-based performance index and a survey on stochastic and deterministic disturbance performance indices [18]. There is a need for the deterministic method for assessment of actual disturbance rejection capacity of closed loop control systems, which considers the tradeoff between reference and disturbance transmission toward the system output. The current study aims to derive straightforward analytical expressions showing spectral dependence of disturbance rejection capacity of closed loop control systems. This paper is devoted in investigation of implicit disturbance rejection capacity of closed loop control systems on the bases of communication channel limitation. We analyzed reference to disturbance ratio (RDR) of closed loop control systems in a similar manner the signal to noise ratio (SNR) defined for communication channels. RDR measure is used to calculate quantitative assessment of reference input dominancy on disturbance at the system output. A control system shows a satisfactory disturbance rejection performance in the case of RDR c 1. If RDR r 1, it can be stated that the control system does not exhibit any disturbance rejection skill. RDR spectra are obtained for spectral investigation of disturbance rejection control performance in this study. We assume unmeasured input disturbance, where unknown additive disturbance signals come into the control system from the plant input. For the improvement of disturbance rejection performance, the proposed method utilizes limitations of control loop communication channel modeling for predominance of reference against the disturbances at the system output. Concepts of communication channel modeling in control loops were discussed in details [19]. In recent years, Rojas presented theoretical works imposing a communication channel in the control loop and addressed problem of feedback stabilization subject to a constraint on the channel signal-to-noise ratio (SNR) [20,21]. Rojas et al. demonstrated that communication channels impose limitations to feedback control [22]. Signal-to-noise ratio performance limitations were investigated for disturbance rejection and stability of the closed loop control system in [21]. For the analytical formulation of RDR method, we consider a negative feedback closed loop control system as the superposition of two low-pass communication channels. One is from reference input to the system output and the other is from disturbance input to the output. RDR spectrum is derived for closed loop control system and detailed analyses are carried out for PID controller family. Frequency dependence of disturbance rejection is investigated for classical PID and FOPID controller structures. Moreover, design criteria based on RDR performance specification are introduced for disturbance rejection PID and FOPID controller design problems. Matlab/Simulink closed loop control system simulations validate our theoretical findings. Disturbance rejection performances are compared for PID and FOPID controller designs. Motivation of this study comes from the requirement for straightforward and practical formulation of disturbance rejection capacity of closed loop PID and FOPID control systems. This makes disturbance rejection performance of control systems measurable and comparable. Moreover elaborating impacts of design coefficients on disturbance rejection performance of closed loop PID controller family is very useful for controller practice. RDR specification in control design problems helps to improve robust performance of PID and FOPID controllers in real control application. The paper is organized as follows: the following section was devoted for reference and disturbance channel modeling of a negative feedback control system and a general formula to express the ratio of reference signal to disturbance signal at the output is
derived. In the further section, PID and FOPID controller design constraints for desired disturbance rejection are derived and illustrative examples are presented.
2. Methodology 2.1. Reference to disturbance ratio Consider a closed loop control system as the superposition of two low-pass communication channels. The first channel is from the reference signal to output of control system and referred as to Reference Channel Control (RCC) and the other is from input of plant to the output of control system and referred as to Disturbance Channel Control (DCC). Fig. 1 shows a closed loop control system model with additive disturbance signal. Assuming a linear time invariant system, the system in Fig. 1 can be written as a superposition of two systems, namely RCC and DCC models, as illustrated in Fig. 2. Fig. 2(a) shows RCC system obtained in the case of a zero disturbance ðd ¼ 0Þ, and Fig. 2(b) shows DCC system in Fig. 1, when the reference signal is zero ðr ¼ 0Þ. Lets denote the transfer function of RCC system given in Fig. 2 by P r ðsÞ. The P r ðsÞ can be written as, P r ðsÞ ¼
Q ðsÞjd ¼ 0 CðsÞGðsÞ ¼ 1 þ CðsÞGðsÞ rðsÞ
ð1Þ
Lets denote the transfer function of DCC system given in Fig. 2 by P d ðsÞ. The P d ðsÞ can be written as, P d ðsÞ ¼
Q ðsÞjr ¼ 0 GðsÞ ¼ 1 þ CðsÞGðsÞ dðsÞ
ð2Þ
One can express output of the closed loop control system by superposing the RCC system output Q r ðsÞ ¼ Q ðsÞjd ¼ 0 and the DCC system output Q d ðsÞ ¼ Q ðsÞjr ¼ 0 as, Q ðsÞ ¼ Q r ðsÞ þ Q d ðsÞ
ð3Þ
Ogata also expressed Q ðsÞ in an open form with respect to controller, plant and feedback transfer functions in order to figure out the necessary conditions for disturbance suppression at the control system output [8]. Ogata suggested the condition of jCðsÞGðsÞj c 1 to improve disturbance rejection performance of closed loop control systems with unity feedback. Because, in the case of high open loop gain (jCðsÞGðsÞj c 1), the transfer function from disturbance to system output P d ðsÞ ¼ Q ðsÞ=dðsÞ goes to almost zero. However, suppression of reference signal by the control system should be also taken into account for a realistic evaluation of disturbance rejection performance in control point of view. We further deepen this perfective for the measure of disturbance
d
e r
C (s )
G (s )
u
Q
Fig. 1. A closed loop control system with additive input disturbance model.
e
r
C (s )
u
G (s )
Q
d C (s )
u
G (s )
Q
Fig. 2. (a) RCC system model obtained for zero disturbance ðd ¼ 0Þ; (b) DCC system model obtained for zero reference signal ðr ¼ 0Þ of closed loop control system with additive disturbance model.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
rejection capacity by a fair assessment between reference and disturbance transmissions to the system output. Spectral power of reference signal at the output of closed loop control system can be expressed as jQ r ðjwÞj2 ¼ jP r ðjwÞj2 jrðjwÞj2
ð4Þ
Spectral power of disturbance signal at the output of closed loop control system can be expressed as: jQ d ðjwÞj2 ¼ jP d ðjwÞj2 jdðjwÞj2
ð5Þ
In analogy with Signal to Noise Ratio (SNR) of communication channels, Reference to Disturbance Ratio (RDR) can be defined as, RDR ¼
2
2
2
jQ r ðjwÞj jP r ðjwÞj jrðjwÞj ¼ jQ d ðjwÞj2 jP d ðjwÞj2 jdðjwÞj2
ð6Þ
In order to evaluate the disturbance attenuation effect of the closed loop control system comparable with the power of reference signal, input of RCC and DCC systems have equal powers, jrðjwÞj2 ¼ jdðjwÞj2 , and thus RDR of the closed loop system can be obtained as, RDR ¼
jP r ðjwÞj2 jP d ðjwÞj2
ð7Þ
When transfer functions of RCC and DCC systems are used in Eq. (7), one obtains an elegant expression of RDR for a closed loop system as: RDR ¼ jCðjwÞj2
ð8Þ
Eq.(8) clearly demonstrates that disturbance suppression strength of a closed loop system strongly depends on energy spectral density of the controller transfer function. This is a significant result stating that disturbance rejection capacity of a negative feedback closed loop control system is solely determined by controller structures. Hence a disturbance tolerant closed loop control system design problem can be reduced to controller design problem. It might be convenient to express the RDR in decibel (dB) as, RDRdB ¼ 20 log jCðjwÞj:
ð9Þ
2.2. Relation of RDR spectrum with sensitivity function In the control literature, sensitivity function is considered to improve disturbance rejection performance of closed loop control systems [11,12,13]. To ensure a good output disturbance rejection, a sensitivity function constraint was given for a desired frequency range ( 8 w r ws rad/s) [11] as follows, 1 SðjwÞ ¼ rB ð10Þ 1 þ CðjwÞGðjwÞdB Although, minimization of sensitivity function increases disturbance rejection performance of a closed loop control system, it does not provide a relevant measurement for disturbance rejection capacity. Sensitivity function, defined as S ¼ eðsÞ=rðsÞ, is a transfer function from the reference input to tracking error [12] and it does not consider power of disturbance on system output, directly. Hence, RDR analysis is convenient for assessment of disturbance rejection capacity. High RDR value is possible by increase of controller amplitude jCðjwÞj, and this decreases sensitivity function. Relation between RDR and sensitivity can be written as, j1 SðjwÞj2 RDR ¼ jCðjwÞj2 ¼ jSðjwÞj2 jGðjwÞj2
ð11Þ
Very low amplitude of sensitivity function results in a high RDR performance. On the other hand, by considering Eq. (7), one can
65
see that RDR also appears as the ratio of energy spectral density of complementary sensitivity function ðjCðjwÞGðjwÞ=ð1 þ CðjwÞ GðjwÞÞj2 Þ and energy spectral density of load disturbance sensitivity function ðjGðjwÞ=ð1 þCðjwÞGðjwÞÞj2 Þ. Essentially, a high RDR performance is obtainable by increasing transmission capacity from reference input to the system output and decreasing transmission capacity from disturbance input to the system output.
3. RDR analysis for PID controller family In order to investigate effects of gain coefficients (kp ,ki ,kd ) on RDR performance of closed loop control system, RDR of classical PID controller structure is derived. To see effects of order coefficients (λ,μ) on RDR performance of closed loop control system, RDR of fractional order PID controller structure is investigated in this section. 3.1. RDR performance of PID controller Transfer function of PID controller structure is commonly written as, CðsÞ ¼ kp þ
ki þ kd s; s
ð12Þ
where, (kp ,ki ,kd ) are PID controller coefficients used for tuning the controller performance. Accordingly, the amplitude response of a PID controller can be written as, 1=2 ðki kd w2 Þ2 þ ðkp wÞ2 jCðjwÞj ¼ : ð13Þ jwj Then, by considering Eq. (13), RDR of the closed loop PID control system is obtained as, 2 ki 2 RDR ¼ kd w þ kp : ð14Þ w Eq. (14) indicates that RDR of closed loop PID control system depends on angular frequency w as the following: Solely integrator component, tuned by ki , increases RDR performance at low frequencies. The derivative component, tuned by kd , improves RDR performance at high frequencies, and the gain component, tuned by kp , contributes to RDR performance regardless of angular frequency. The following RDR response characteristics can be observed depending on conditions of PID coefficients (kp ,ki ,kd ): (i) When kp a 0, ki a0 and kd a 0, RDR performance of PID controllers is expressed by Eq. (14). Its RDR spectrum exhibits a pffiffiffiffiffiffiffiffiffiffiffi minima at the frequency wm ¼ ki =kd . Minimum RDR performance of the PID controller is determined by the proportional 2 gain as kp at the frequency wm . Fig. 3(a) illustrates the RDR spectrum of PID controller for (kp ¼ 1,ki ¼ 1,kd ¼ 1). In this figure, minima of RDR performance appears at w ¼ 1 as the value of one due to kp ¼ 1. Around the minima, RDR performance increases depending on ki for lower frequencies and kd for higher frequencies. (ii) When kp a0, ki a0 and kd ¼ 0, PID controller structure implements a Proportional-Integrator (PI) controller and RDR of PI controller can be written as, 2 ki 2 RDR ¼ þ kp : ð15Þ w this equation implies that as the angular frequency increases, RDR performance of a PI controller decreases. For very low frequencies or near to DC, RDR can reach to enormously high values. For DC signals, RDR performance of PI controller theoretically goes to infinity. So, it exhibits very strong disturbance attenuation for very low frequencies. Increasing kp improves the disturbance
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
30
30
25
25
20
20
RDRdB
RDRdB
66
15
15
10
10
5
5
0 0
0.5
1
1.5 w
2
2.5
0
3
0
0.5
1
1.5 w
2
2.5
3
10 0.5 RDRdB
RDRdB
8 6 4
-0.5
2 0
0
0
0.5
1
1.5 w
2
2.5
3
-1
0.5
1
1.5 w
2
2.5
3
Fig. 3. (a) RDR spectra of PID controller for (kp ¼ 1,ki ¼ 1,kd ¼ 1); (b) RDR spectrum of PI controller for (kp ¼ 1, ki ¼ 1); (c) RDR spectrum of PD controller for (kp ¼ 1, kd ¼ 1); (d) RDR performance of P controller for ðkp ¼ 1Þ.
rejection for low frequency disturbances [23]. Fig. 3(b) illustrates RDR spectrum of PI controller for (kp ¼ 1,ki ¼ 1). (iii) When kp a0, ki ¼ 0 and kd a 0, PID controller structure implements a Proportional-Derivative (PD) controller and RDR of PD controller can be written as, 2
RDR ¼ ðkd wÞ2 þ kp :
ð16Þ
this equation implies that as the angular frequency increases, RDR of a PD controller increases according to second-order polynomial characteristics. For the very low frequencies or near to DC, RDR 2 approximates to the term of kp . In other words, the lowest RDR performance of PD controller can be adjusted by kp for the low frequency region. Fig. 3(c) illustrates the RDR spectrum of PD controller for (kp ¼ 1, kd ¼ 1). (iv) When kp a0, ki ¼ 0 and kd ¼ 0, PID controller structure implements a proportional controller (P) and RDR of Proportional (P) controller can be expressed simply as, 2
RDR ¼ kp
ð17Þ
this RDR performance of P controller is independent of the angular frequency as illustrated in Fig. 3(d). The minimal and frequency-independent value of RDR, which is adjusted by kp , spreads uniformly in the whole RDR spectrum. Utilization of a RDR performance as a design criteria in PID tuning problems can be very beneficial for the enhancement of disturbance rejection skills of closed loop PID control systems. PID tuning algorithms or methods should consider RDR performance design constraint in parameter optimization process to benefit from inherent disturbance tolerance property of closed loop control structures. A general RDR design constraint ensuring a given minimal RDR performance (α) can be defined as, αr
ki kd w w
2
2
þ kp
ð18Þ
It can be clearly observed from Eq. (18) that the proportional gain kp is a fundamental term for PID, PD, PI and P controllers to determine minimal RDR performance. In control practice, reference signal and disturbance signal mostly contain low frequency components. PI can be more advantageous to obtain higher RDR spectrum compared to PID controllers due to the term of ððki =wÞ kd wÞ. Fig. 4 shows RDR spectra of PI versus PID controller for equal common coefficients. In this figure, the ki coefficient of PID controller set to π 2 kd in order to move the minima frequency ðwm Þ towards the angular frequency of π in the spectrum. As seen, PI controller provides a better RDR compared to RDR spectrum of PID controller at high frequency region. It is concluded that due to the derivative term in PID, PID control system can be more sensitive to high frequency disturbance signals. In practice, high frequency disturbance may originate from environmental or system noise. Fig. 5 shows ki and kd dependence of RDR for w ¼ 0:1, w ¼ 1, w ¼ 2 and w ¼ 3 frequencies. α 4 20 dB lines in figures are used to indicate (ki ,kd ) search regions for tuning algorithms. Intersection of stability ranges of coefficients and α 4 20 dB region promises a disturbance tolerant, stable PID control system design. As a result, use of RDR design constraint enhances effectiveness of heuristic optimization algorithms due to narrowing their search regions. Increasing kp results in expanding α 420 dB regions and kp provides improvement of RDR independent of angular frequency for the whole spectrum. Fig. 5 also demonstrates that at low frequencies ðw ¼ 0:1Þ, RDR strongly depends on value of ki parameter, but at the high frequencies ðw ¼ 3Þ, RDR becomes very dependent of kd values.
3.2. Disturbance rejection capacity of fractional order PID controllers Transfer function of FOPID controllers are commonly written as, C f o ðsÞ ¼ kp þ
ki þ kd sμ ; sλ
ð19Þ
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
where, (kp ,ki ,kd ) are gain coefficients and (λ,μ) are fractional order coefficients of fractional order PID controller. By using sα ¼ jα wα and jα ¼ cos ððπ=2ÞαÞ þ j sin ððπ=2ÞαÞ, the amplitude response of FOPID controllers can be derived as, π π 2 jC f o ðjwÞj ¼ kp þ ki cos λ w λ þ kd cos μ wμ 2 2 π π 2 1=2 ð20Þ þ kd sin μ wμ ki sin λ w λ 2 2 then, by considering Eq. (8), RDR of the closed loop FOPID control system is written as, π π 2 RDRf o ¼ kp þ ki cos λ w λ þ kd cos μ wμ 2 2 2 π π μ þ kd sin μ w ki sin λ w λ : ð21Þ 2 2 Effects of gain coefficients (ki ,kd ) on RDR performance of FOPID strongly depends on (λ,μ) fractional order coefficients due to the factors cos ððπ=2ÞλÞ, cos ððπ=2ÞμÞ, sin ððπ=2ÞλÞ and sin ððπ=2ÞμÞ. Besides, (λ,μ) coefficients appear on the power of angular frequency as w λ and wμ . This complicates simplified analyses as given in previous section for PID controller. However, for λ¼ 1 and
PID PI
35
RDRdB
30 25
67
μ¼1, FOPID works as classic PID structure and the RDR formulation of FOPID simplifies to Eq. (14). For ki 4 0 and kd 4 0, a local minimum of RDRfo spectrum characteristic can be found by solving 0 ¼ ðkd sin ððπ=2ÞμÞwμ ki sin ððπ=2ÞλÞ w λ Þ2 equation as, 1=ðλ þ μÞ ki sin ððπ=2ÞλÞ ð22Þ wm ¼ kd sin ððπ=2ÞμÞ Fig. 6 reveals (λ,μ) dependence of RDR performance for FOPID controller at various frequencies. It is clearly seen in Fig. 6(a) and (d) that RDR performance shows dependence on λ at low frequency (w ¼ 0:1 rad/s) and μ at high frequency (w ¼ 3 rad/s). Fig. 6(b) shows that RDR performance is severely deteriorated at w ¼ 1 rad/s. The figure indicates very low RDR performance around λ ¼ 1:5 and μ ¼ 1:5 at w ¼ 1 rad/s for ki ¼ 1, kd ¼ 1 and kp ¼ 1. Fig. 7 shows λ and μ dependence of RDR for FOPID controllers. Fig. 8 shows RDR spectrum for various λ and μ. Fractional orders λ and μ are very effective on RDR spectrum due to the terms of w λ and wμ . Decrease of λ and μ values more flattens RDR spectrum. As values of λ and μ increase, RDR spectrum exhibits more dependence on the angular frequency. A general RDR design constraint ensuring a given minimal RDR performance (α) for FOPID controller can be defined as, π π 2 α r kp þ ki cos λ w λ þkd cos μ wμ 2 2 π π 2 þ kd sin μ wμ ki sin λ w λ ð23Þ 2 2
20 15
4. Simulation examples
10
This section shows disturbance tolerant PID controller examples to validate theoretical findings. Matlab/Simulink simulations were used in these illustrative examples:
5 0.5
1
1.5 w
2
2.5
3
Fig. 4. A comparison of RDR spectra for PID controllers (kp ¼ 1, ki ¼ 9:8, and kd ¼ 1) and PI controllers (kp ¼ 1, ki ¼ 9:8).
Example 1. This example compares disturbance rejection performance of two PID controller designs for a stable, second order plant function given as GðsÞ ¼ 1=ðs2 þ 4s þ 3Þ.
30
5
25
10
20
15
15
20
10
20
25
5
0
30
80
10
60
15
kd
kd
5
40
20 25 30
5
10
15
20
25
30
5
10
ki
15
20
25
30
0
ki
35 30
5
kd
20
15
15
20
10
25 30
5 5
10
15 ki
20
25
30
kd
25
10
35
5
30
10
25
15
20 15
20
10
25 30
5 5
10
15
20
ki
Fig. 5. RDR versus ki and kd drawing for kp ¼ 1 and various angular frequencies.
25
30
68
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
35 0.5
-10
25
1
0
0.5
30
1
20
-20
15
1.5
1.5
-30
10 2
0.5
1
1.5
2
2
0.5
1
1.5
2
18
10 9
0.5
16
0.5
14
8 1
12
1
7
10
6
1.5
1.5
8
5 2
0.5
1
1.5
2
-40
6
2
4
0.5
1
1.5
2
Fig. 6. RDR versus λ and μ for ki ¼ 1, kd ¼ 1 and kp ¼ 1 at various angular frequencies.
30
40
20
w=3
10
w=2
0 -10
w = 0 .1
20
w = 0 .1 RDRdB
RDRdB
30
w=3 10
w= 2 0
w =1 0
w =1
0.5
1
1.5
λ
-10
2
0
0.5
1
1.5
μ
2
Fig. 7. λ and μ dependence of RDR for ki ¼ 1, kd ¼ 1 and kp ¼ 1.
50
30 20
30
RDRdB
RDRdB
40
20
0.6 0.8
10 0 -10
0.6 0.8
10 0
1.4 0.5
1.2 1
1.0
1.5 w
-10 2
2.5
3
1.0 1.2
1.4 0.5
1
1.5 w
2
2.5
3
Fig. 8. RDR spectrum for ki ¼ 1 kd ¼ 1 kp ¼ 1 various λ ¼ {0.6,0.8,1.0,1.2,1.4} in (a) and μ¼ {0.6,0.8,1.0,1.2,1.4} in (b).
Fig. 9 shows RDR spectra for two PID designs; one has a low disturbance rejection performance with the low gain PID coefficients (kp ¼ 1, ki ¼ 1, kd ¼ 1) and the other presents a high disturbance rejection performance with the high gain PID
coefficients (kp ¼ 30, ki ¼ 5, kd ¼ 10). RDR values calculated from simulations confirm analytical results calculated by using Eq. (14). Fig. 10 (a) shows step response of RCC in the case of a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1 for a zero disturbance.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
It settles properly. Fig. 10(b) shows step response of DCC for a zero reference input. A step at disturbance input results in a peak with the amplitude of 0.2 at the output in Fig. 10(b). To enhance disturbance tolerance of this control system, the PID coefficients were tuned as kp ¼ 30, ki ¼ 5 and kd ¼ 10 in order to improve the RDR spectrum as illustrated in Fig. 9. This provides a faster step response for RCC as in Fig. 10(c) and considerably reduces peak amplitude obtained from DCC at the output in Fig. 10(d). Fig. 10 reveals impacts of rise time on disturbance rejection performance of control systems. Faster closed loop control systems are more disturbances tolerant so that they can respond fast enough against to the disturbance signal and thus better holds their outputs at the setpoint. Fig. 11 reveals improvements in disturbance rejection effect for sinusoidal input signals. Fig. 11(a) shows response of PID control system (kp ¼ 1,ki ¼ 1,kd ¼ 1) for a sinusoidal reference signal (amplitude of one and the frequency of 0.1 rad/s) and a sinusoidal disturbance signal (amplitude of one and the frequency of 0.5 rad/s). The disturbance signal caused large ripples at output. When PID
:k p=1 k i=1 k d=1 :k p=30 k i=5 k d=10
RDRdB
40
:Simulation
30 20 10 0
0.5
1
1.5 w
2
2.5
3
Fig. 9. A comparison of RDR spectra for PID controllers for a low (kp ¼ 1, ki ¼ 1, and kd ¼ 1) and for a high (kp ¼ 30, ki ¼ 5, kd ¼ 10) disturbance rejection performance.
control system was configured to kp ¼ 30, ki ¼ 5, and kd ¼ 10 to improve RDR performance as shown in Fig. 9, ripples caused from the disturbance remarkably diminished and the output can well follow the reference signal. Fig. 12 (a) and (b) illustrate step response of DCC for PD and PI controller respectively. Since PD controller exhibits better RDR at high frequencies, effects of disturbance on the output contain mainly low frequency and DC components as in Fig. 12(a). However, a PI control system presents higher RDR at low frequencies or DC region, PI control system exhibits rather high frequency components at the output as illustrated in Fig. 12(b). These results confirm the theoretical analysis presented in previous sections.
Example 2. This example compares RDR performance of two PID controllers and a FOPID controller design and demonstrates disturbance response of these controllers for an unstable, time delay plant model, given as GðsÞ ¼ ð1=ðs 1ÞÞe 0:5s [9]. System instability is a major factor limiting disturbance rejection capacity of control systems so that unstable control systems are practically useless. The time delay of plant system considerably limits the stability interval of controller coefficients and accordingly RDR performance of controllers. Since GðsÞ plant is an unstable time delayed system, it considerably limits design ranges of controller coefficients in this example. We applied set and trial methods to find out a PID and a FOPID designs with satisfying RDR performances and compared it with a disturbance rejection PID design (kp ¼ 2:168, ki ¼ 0:693, kd ¼ 0:358) proposed by Szita et al. [9]. Fig. 13 shows RDR spectra and disturbance response of Szita et al.'s PID, the proposed PID (kp ¼ 2:5, ki ¼ 0:8, kd ¼ 0:4) and FOPID (kp ¼ 0:5, ki ¼ 1:6, kd ¼ 1:5, λ ¼ 0:8, μ ¼ 0:9) controller designs. FOPID is implemented by Carlson approximation [24]. As a result of imperfection of the approximate FOPID implementation, RDR performance of FOPID can be 3–4 dB less than theoretical calculations.
1.2
1
1
0.8
0.8
Q
Q
1.2
0.6
Q(t) r(t) d(t)
0.4 0.2
0.6 0.4
0.2
Q
50
69
0.1
5
0.2
10
15
20
t
0
0 0
20
40
60
20
80
40
60
80
100
t
t
1.2
1.2 1
1
0.8
0.8
0.6
0.6
0.02
0.4
0.01
0.03
Q
Q
Q(t) r(t) d(t)
0
Q(t) r(t) d(t)
0.4 0.2
0.2
40
60 t
80
0 10
0
0 20
Q(t) r(t) d(t)
100
20
30
t
20
40
60
80
100
t
Fig. 10. Step response of RCC in (a) and DCC in (b) for a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1. Step response of RCC in (c) and DCC in (d) for a PID controller with kp ¼ 30, ki ¼ 5, kd ¼ 10.
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
1.5
1.5
1
1
0.5
0.5
0
0
Q
Q
70
-0.5 -1 -1.5
-0.5
Q(t) r(t) d(t) 0
20
40
60
80
Q(t) r(t) d(t)
-1 -1.5
100
0
20
40
60
80
100
t
t
Fig. 11. (a) Sinus response of a PID controller with kp ¼ 1, ki ¼ 1, kd ¼ 1. (b) Sinus response of a PID controller with kp ¼ 30, ki ¼ 5, kd ¼ 10, which is more disturbance tolerant.
0.05
0.03
0.02
Q
Q
Q(t) r(t) d(t)
0.04
0.03
Q(t) r(t) d(t)
0.01
0.02 0.01 0
0 5
10
15
20
25
5
10
15
t
20
25
t
Fig. 12. (a) Effect of PD controller (kp ¼ 30, ki ¼ 0, kd ¼ 10) on disturbance rejection (b) Effect of PI controller (kp ¼ 30, ki ¼ 5, kd ¼ 0) on disturbance rejection.
0.8
20
Q
RDRdB
30
d(t) Szita et al. PID FOPID
1
Szita et al. PID FOPID
0.6 0.4
10
0.2
0 0
0.5
1
1.5 w
2
2.5
3
0
5
6
7
8
t
Fig. 13. (a) A comparison of RDR spectra for Szita et al.'s PID (kp ¼ 2:168, ki ¼ 0:693, kd ¼ 0:358) and proposed PID (kp ¼ 2:5, ki ¼ 0:8, kd ¼ 0:4) and FOPID (kp ¼ 0:5, ki ¼ 1:6, kd ¼ 1:5,λ ¼ 0:8,μ ¼ 0:9) controller designs. (b) A comparison of step disturbance response of controllers.
Due to the fact that environmental disturbance is in random nature; it is convenient to illustrate disturbance rejection performance of controllers against a noisy disturbance signal generated with normally distributed random numbers. Fig. 14 demonstrates noise suppression performance of controllers at the output for PID designs. It is clearly seen that control systems attenuate and smooth the band limited white noise with the power of 0.001, noticeably. It is observed that there are slight differences between controller responses for noisy disturbance. However, the FOPID controller design can give slightly better response for noisy disturbance because FOPID controller presents higher RDR values at the high frequency region compared to other controllers as seen in Fig. 13(b). Unpredictable disturbance naturally appears in real control systems and robust performance of control system is very dependent of disturbance rejection performance.
It is also noteworthy that disturbance rejection capacity of different controller designs can be compared by means of RDR spectrum analyses, and this makes possible to select a better controller designs with a desirable RDR spectrum via RDR spectrum comparisons.
5. Conclusions This study presented RDR analysis method for evaluation of disturbance rejection capacity of closed control systems. This method is based on utilization of communication channel limitations to improve disturbance rejection performance of control systems. RDR spectrum well describes the spectral dependence of disturbance rejection capacity. Eq. (8) shows that RDR spectra
B.B. Alagoz et al. / ISA Transactions 55 (2015) 63–71
provided by larger kp , enhance overall disturbance rejection performance of PID controller family.
1.5 r(t) d(t) Szita et al. PID FOPID
1
Q
0.5
71
This study elaborates disturbance rejection control for PID controller family. Findings of this paper contribute to enhancement of robust performance of PID and FOPID controllers in control engineering practice.
0 -0.5
References
-1 -1.5
0
20
40
60
80
100
t
Fig. 14. Effects of white noise disturbance with the power of 0.001 on controller performance.
of closed loop control systems are determined solely by energy spectral density of controller function and therefore, RDR performance of a control system can be also adjusted by only tuning controller parameters. Disturbance tolerant PID and FOPID controller design constraints were derived for tuning problems. The theoretical findings were validated by simulation results. Some important remarks can be summarized as follows: – It is shown that energy spectral density of the controller transfer function determines RDR of closed loop systems. – RDR analysis provides quantitative evaluation of disturbance rejection capacity of controllers and yields straightforward analytical solutions that are very useful for controller tuning problems. – Frequency dependence of RDR spectrum is characterized by fractional orders λ and μ due to the terms of w λ and wμ . In the case of λ ¼ 1 and μ ¼ 1, RDR of PID controller exhibits frequency dependence characterized by the term ðki =w kd wÞ in Eq. (14). PD controller structures are effective at high frequency region according to the term of kd w, PI controller structures are effective at low frequency region according to the term of 2 ki =w. p PID controller exhibits a minima with the value of kp at ffiffiffiffiffiffiffiffiffiffiffi wm ¼ ki =kd . – The coefficient kp is frequency independent fundamental component of RDR spectrum. It can be effectively used to adjust value of minimal RDR performance of PID and FOPID control systems regardless of angular frequency. – (λ,μ) fractional order coefficients of FOPID scales gain coefficients (kp ,ki ,kd ) by sinusoidal functions cos ððπ=2ÞλÞ, cos ððπ=2ÞμÞ, sin ððπ=2ÞλÞ and sin ððπ=2ÞμÞ. – Large time-delays or low-gain controllers reduce disturbance rejection performance due to delaying controller action to deal with disturbances. Faster PID and FOPID controls, particularly
[1] Chang JL. Robust output feedback disturbance rejection control by simultaneously estimating state and disturbance. J Control Sci Eng 2011;2011:1–13. [2] Vukic Z, Kuljaca O. Lect PID Controllers 2002:23. [3] Vrancic D, Strmčnik S, Kocijan J, de Moura Oliveirac PB. Improving disturbance rejection of PID controllers by means of the magnitude optimum method. ISA Trans 2010;49(1):47–56. [4] Vandeursen JME, Peperstraete JA. Model-based and PID controllers for disturbance rejection in processes with time delay: a comparison. ISA Trans 1996;35(3):225–36. [5] Busawon KK, Kabore P. Disturbance attenuation using proportional integral observers. Int J Control 2001;74(6):618–27. [6] Shamsuzzoha M, Lee M. Enhanced disturbance rejection for open-loop unstable process with time delay. ISA Trans 2009;48(2):237–44. [7] Jin XZ, Yang GH. Robust adaptive fault-tolerant compensation control with actuator failures and bounded disturbances. Acta Autom Sin 2009;35:305–9. [8] Ogata K. Modern Control Engineering. New Jersey: Prentice Hall; 2010. [9] Szita G, Sanathanan CK. Robust design for disturbance rejection in time delay systems. J Franklin Inst 1997;334(4):611–29. [10] Koussiouris TG, Tzierakis KG. Frequency-domain conditions for disturbance rejection and decoupling with stability or pole placement. Automatica 1996;32(2):229–34. [11] Monje CA, Vinagre BM, Feliu V, Chen YQ. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng Pract 2008;16: 798–812. [12] Doyle JC, Francis B, Tannenbaum A. Feedback Control Theory. New York: Macmillan Publishing; 1992. [13] Aström KJ, Richard MM. Feedback Systems: An Introduction for Scientists and Engineers. New Jersey: Princeton University Press; 2008. [14] Willems JC. Almost invariant subspaces: an approach to high-gain feedback design – Part I: Almost controlled invariant subspaces. IEEE Trans Autom Control 1981;26(1):235–52. [15] Young KD. Disturbance decoupling by high gain feedback. IEEE Trans Autom Control 1982;27(4):970–1. [16] Trentelman HL, Willems JC. Guaranteed roll-off in a class of high-gain feedback design problems. Syst Control Lett 1983;3(6):361–9. [17] Willems JC. Almost invariant subspaces: an approach to high gain feedback design – part II: almost conditionally invariant subspaces. IEEE Trans Autom Control 1982;27(5):1071–85. [18] Yu Z, Wang J. Assessment of proportional ıntegral control loop performance for ınput load disturbance rejection. Indus Eng Chem Res 2012;51:11744–52. [19] Elia N. When bode meets Shannon: control-oriented feedback communication schemes. IEEE Trans Autom Control 2004;49(9):1477–88. [20] Rojas AJ. Signal-to-noise ratio fundamental limitations in the discrete-time domain. Syst Control Lett 2012;61(1):55–61. [21] Rojas AJ. Signal-to-noise ratio performance limitations for input disturbance rejection in output feedback control. Syst Control Lett 2009;58(5):353–8. [22] Rojas AJ, Middleton RH, Freudenberg JS, Braslavsky JH. Input disturbance rejection in channel signal-to-noise ratio constrained feedback control. In: Proceedings of the 2008 American Control Conference, Seattle, Washington, USA; 2008. p. 3100–5. [23] Aström KJ, Hagglund T. Advanced PID Control. Chapter 4: Controller Design. The ISA Society; 2006. [24] Carlson GE, Halijak CA. Approximation of fractional capacitors (1/s)\widehat1/n by a regular Newton process. IEEE Trans Circuit Theory 1964;7:210–3.
