Note on "Optimal PID Controller Design with Adjustable maximum sensitivity"
Harshit Kumar1∗, Ankur Sharma1, Aniket Jain1, Yogesh V. Hote1, Shivam Jain1 1 Electrical Engineering Department, Indian Institute of Technology Roorkee, Roorkee, India
* E-mail: harshit211997@gmail.com
ISSN 1751-8644 doi: 0000000000 www.ietdl.org
Abstract: In this short note, it is shown that the Lemma 1 and theorem 1 in paper [1] are not correct. Further, the results obtained in paper [1] are not generalized in nature. As the foundation of this paper is based on Lemma 1 and theorem 1, the correct analysis of the existing concepts will be helpful to readers. The claims about the inaccuracy of [1] are justified through mathematical analysis, numerical examples, and supporting graphs.
1 Incorrectness in Lemma 1
1.1 Lemma 1 as given in [1]
C1(s) and C2(s) are two controllers that are designed for the pro- cesses G(s − a) and G(s) respectively. If M1 and M2 are the maximum sensitivity of the processes with loop transfer function C1(s)G(s) and C2(s)G(s) respectively, then M1 and M2 will satisfy the following relation:
M1≤M2 ∀a≥0, (1) M1>M2 ∀a<0, (2)
1.2 Proposed Justification
Lemma 1 as stated above, is not valid for all values of a. However, it may hold only for some particular range of a. This is justified below:
1. First of all, there is an error in equations (3) and (4) in [1]. The correct equations are given below [2].
Now, dividing both sides of (5) by 2 and taking advantage of the monotonicity of sine function in [− π , π ], we get
22
sin(PMC1(s)G(s)/2) ≥ sin(PMC2(s)G(s)/2) ∀a ≥ 0, (8)
Multiplying both sides of Equation (8) by 2, we obtain
2sin(PMC1(s)G(s)/2) ≥ 2sin(PMC2(s)G(s)/2) Taking inverse of (9), we get
∀a ≥ 0, (9)
1 ≤ 1 ∀a≥0,
2sin(P MC1 (s)G(s) /2) 2sin(P MC2 (s)G(s) /2) Using (7), it can be deduced that
(10)
Ms≥ GM GM − 1
Ms≥ 1 2sin(P M/2)
2. Equation(15),(16)and(4)in[1]aregivenbelow
(3) (4)
M1 < 1 (11) 2sin(P MC1 (s)G(s) /2)
M2 < 1 (12) 2sin(P MC2 (s)G(s) /2)
Hence, from (10), (11) and (12), there are two possible cases for M2 CaseI:
1 <M2≤ 1 (13) 2sin(P MC1 (s)G(s) /2) 2sin(P MC2 (s)G(s /2)
Case II :
M2 <
1 (14) 2sin(P MC1 (s)G(s) /2)
PMC1(s)G(s) ≥ PMC2(s)G(s) PMC1(s)G(s) < PMC2(s)G(s)
1
Ms < 2sin((PM)/2)
∀a ≥ 0, (5)
∀a < 0,
(6) (7)
For case I, Equation (1) holds. However, for case II i.e., from (11) and (14), Equation (1) does not necessarily hold. Similar analysis canbedonefor∀a<0.
Also, even if we consider the correct equation (4), instead of (7), we can arrive at the same conclusion as shown below.
Rewriting equations (11) and (12).
where, P MY (s) denotes the phase margin of transfer function Y (s). Using equations (5), (6) and (7), the authors of paper [1] have obtained the results of Lemma 1 as given in (1) and (2).
The authors in [1] have mentioned that these results are gener- alised in nature for any process G(s) and for all the values of a. According to the analysis shown below, it is clear that results do not hold for all values of a.
M1 ≥
1 (15) 2sin(P MC1 (s)G(s) /2)
1
M2 ≥ 2sin(PMC2(s)G(s)/2) (16)
Hence from (10), (15) and (16), there are two possible cases for
Consider the case when P M ∈ [−π, π]. Then, [− π , π ].
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P M/2 ∈
M1
1
Case I :
1
2sin(P MC1 (s)G(s) /2)
Case II :
≤ M1 <
1 (17) 2sin(P MC2 (s)G(s /2)
M1 ≥
1 (18) 2sin(P MC2 (s)G(s) /2)
For case I, Equation (1) holds. But for case II i.e., from (16) and (18), Equation (1) does not necessarily hold. Similar analysis can be done for ∀a < 0.
3. We will now give an example to support the above mentioned proof.
Rewriting equations (9) and (10) of paper [1]:
P MC1 (s)G(s−a) = φ = φm1 + φ1s (19)
PMC2(s)G(s) = φ = φm2 + φ2s (20) Let us consider a plant G(s) with transfer function given below:
G(s) = 100 (21) s(s+2)
The phase margin of G(s) is calculated as φm2 = 11.4209◦. Now, let us consider a controller C2(s) whose maximum phase shift is 48.5791◦ so as to provide an optimal phase shift of 60◦(satisfying (19)).
The transfer function of C2(s) obtained using the concept of lead lag compensator design [3], is given below:
C2(s) = 0.2672s + 1 (22) 0.03819s + 1
Fig. 1: Maximum sensitivity M1 and M2 vs a
Using the notion of lead lag compensator design, transfer function
of controller C1(s) is given as:
C1(s) = 0.0456s + 1 (26)
and G(s)C2(s) as M2:
M1 = 1.4284 M2 = 1.5509
Here, we have taken a < 0 and obtained M1 < M2. However, according to (1), M1 > M2.
Fig. 1. illustrates the relation between maximum sensitivity vs. a for the transfer function given in equation(21). It can be seen, for values near a = 8, that M1 becomes greater than M2, , contradicting equation(1). Similarly, near a = −8, M2 becomes greater than M1, thus, contradicting equation(2).
Hence, it can be concluded from above analysis that Lemma 1 of [1] does not hold.
Now, consider following two cases for a: Case I : a = 8
Now, G(s − a) will become: G(s − 8) =
100 (s−8)(s−6)
(23)
Phase margin of G(s − a) is evaluated as φm1 = −88.2953◦. Let us take another controller C1(s) whose maximum phase shift is φ1s = 148.2953◦ so as to provide an optimal phase margin of 60◦(satisfying (20)).
The transfer function of C1(s) obtained using the concept of lead lag compensator design [3], is given below:
C1 (s) = 0.2512s + 1 (24) 0.07811s + 1
We find maximum sensitivity of G(s)C1(s) as M1 and G(s)C2(s) as M2 :
M1 = 2.0456 M2 = 1.5509
Here, we have taken a > 0 and obtained M1 > M2. However, according to (1), M1 ≤ M2.
Case II : a = −8 Similarly,
G(s + 8) = 100 (25) (s+8)(s+14)
Phase margin of G(s − a) is computed as φm1 = 127.1822◦. Let us take another controller C1(s) whose maximum phase shift is φ1s = −67.1822◦ so as to provide an optimal phase margin of 60◦(satisfying (20)).
2
2
2.1
Incorrectness in theorem 1
Theorem 1 as given in [1]
1.12s + 1
Now we will also find maximum sensitivity of G(s)C1(s) as M1
For PID controller structure C(s) tuned by minimising the ISE, ITSE, IETSE, ITETSE functions for any process G(s) which gives different controller designs C11(s),C12(s), C21(s) and C22(s), respectively, which provides the maximum sensitivity M11(s), M12(s), M21(s) and M22(s), respectively.
M21 =M11 andM22 =M12 M21 >M11 andM22 >M12 M21 <M11 andM22 <M12
∀a=0, ∀a<0, ∀a>0,
and ∆M = f(a), where, ’a’ is the constant in the exponen- tial weighted error function and ∆M is the change in maximum sensitivity.
This theorem does not hold for all values of a and any process G(s).
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and (21) of [1]. On a closer observation of Fig. 3, it can be seen that the value of maximum sensitivity at a = −3 (the case when we are minimising IETSE), i.e., 1.1547 is clearly less than the maximum sensitivity corresponding to the minimisation of ISE at a = 0 (the case when we are minimising ISE), i.e., 1.4666.
This implies that the value of maximum sensitivity obtained via minimisation of IETSE is lesser than that computed by minimisation of ISE. Clearly, this is a contradiction of Theorem 1.
Fig. 2: Effect of a on maximum sensitivity for K = 100
Fig. 4: Effect of a on maximum sensitivity for a ∈ [0, 3]
Fig. 3: Effect of a on maximum sensitivity for K = 1 2.2 Justification
In this subsection, via simulation results, we will show the fallacy of Theorem 1.
2.2.1 Numerical Example 1: Consider the same numerical example as taken in [1] in section 6.1. The error function of the process is
E(s) = s + Kk (27) s2 + Kks + K
Fig. 2 depicts the plot of maximum sensitivity (corresponding to the controller obtained by minimisation of IETSE) versus constant a as given in [1]. Using this figure plotted in the range a ∈ [−5, 5] and for K = 100, the authors in [1] justified the validity of Theorem 1.
As Theorem 1 does not impose any restriction on the transfer function of plant, we could as well choose K = 1 and the theorem should hold. To investigate the applicability of Theorem 1, we select K = 1 and the resultant plot of maximum sensitivity versus a can be seen in Fig. 3. The value of maximum sensitivity at a = 0 cor- responds to the value of sensitivity, when the controller is obtained via the minimisation of ISE. This can be seen from Equation (19)
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2.3
Fig. 5: Effect of a on maximum sensitivity for a ∈ [−1, 0]
Numerical Example 2
Consider an unstable system given in [1] as follows:
e−0.2s
G(s)= s−1 (28)
Fig. 4 depicts the plot of maximum sensitivity (corresponding to con- troller design by IETSE minimisation) versus a for a ∈ [0, 3]. The
3
same plot is shown in Fig. 4(c) in [1]. Using this figure,the authors of [1] justified the validity of Theorem 1.
To further investigate the claim made in [1], a is varied in the range [−1, 0] and the values of maximum sensitivity are observed. The corresponding plot is shown in Fig. 5. On a closer observation of Fig. 5, it can be seen that the value of maximum sensitivity at a = −0.9 (the case when we are minimising IETSE), i.e., 3.5109 is clearly less than the maximum sensitivity corresponding to the min- imisation of ISE at a = 0 (the case when we are minimising ISE), i.e., 4.5427.
This implies that the value of maximum sensitivity obtained via minimisation of IETSE is lesser than that computed by minimisation of ISE. Clearly, this is a contradiction of Theorem 1.
Conclusion
It can be concluded from the aforementioned mathematical analysis, numerical examples and plots, that the Lemma 1 and Theorem 1 as proposed by the authors in [1] does not hold for all values of a, rendering it ineffective and fallacious.
References
[1] Verma, Bharat; Padhy, Prabin K.: ’Optimal PID controller design with adjustable maximum sensitivity’, IET Control Theory & Applications, 2018, 12, (8), p. 1156-1165
[2] Johnson, M.A., Moradi, M.H.: ’PID control new identification and design methods’(Springer-Verlag London, 2005)
[3] Gopal, M.: ’Control Systems : Principles and Design’(Tata McGraw-Hill Education, 2002)