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Properties and control of the quadruple-tank process with multivariable dead-times
Journal of Process Control 20 (2010) 18–28
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Properties and control of the quadruple-tank process with multivariable dead-times D. Shneiderman, Z.J. Palmor * Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 5 April 2009 Received in revised form 28 July 2009 Accepted 27 October 2009
Keywords: Multiple delays Multi-input multi-output process Right-half plane zero Control education
a b s t r a c t The well known laboratory quadruple-tank process (QTP) has been introduced in the laboratories of many schools around the world as it is ideally suited to illustrate concepts in multivariable control. In this paper the QTP is extended to include independent multivariable dead times (DTs) and their effects on the properties and control of the QTP are studied. DTs are very common in many various processes and make the control of the QTP more interesting and challenging. The addition of DTs may introduce infinite, finite or not any number of non-minimum-phase (NMP) zeros. As shown in the paper it depends on a particular combination of the multivariable DTs. The conditions for each case are stated and the location and behavior of the zeros closest to the imaginary axis due to the DTs are specified. Other properties of the QTP with DTs as the output real NMP zero directions, the decentralized integral controllability of the process and time-domain bounds on closed loop performance are derived and discussed. Also, a novel laboratory QTP with DTs is described and used to demonstrate the main results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The well known multivariable laboratory process, called the quadruple-tank process (QTP), consists of four interconnected liquid tanks, two pumps and two valves, [1], and is shown schematically in Fig. 1. The inputs are the voltages to the two pumps ðv 1 ; v 2 Þ and the outputs are the liquid levels in the lower tanks ðh1 ; h2 Þ. The linearized dynamics of the process exhibits a multivariable zero that can be moved from one side of the complex plane to the other one by changing the valves’ positions c1 ; c2 . This process was found to be ideally suited to illustrate many concepts in multivariable control, particularly performance limitation due to the non-minimum-phase zero and its output direction. It is not surprising, therefore, that the laboratory QTP was incorporated in control laboratories in many schools around the world. See [2–4,20] for example. Multivariable processes containing dead-times (DTs), or delays, between inputs and outputs are very common in practice and the presence of DTs further limits achievable control performance. Hence adding multivariable DTs to the QTP makes its control even more challenging. It is therefore of interest to investigate the effect of multivariable DTs on the properties and control of the QTP. In particular, the number of the non-minimum-phase (NMP) zeros and the location of the dominant ones (dominant zeros are the NMP zeros closest to the imaginary axis), the output directions of * Corresponding author. Tel.: +972 4 829 2086; fax: +972 4 8295172. E-mail addresses: [email protected] (D. Shneiderman), palmor@technion. ac.il (Z.J. Palmor). 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.10.010
the real NMP zeros, the process’ decentralized integral controllability (DIC) and closed loop interaction bounds associated with NMP zeros. These issues are dealt with in this paper and it is shown that the presence of DTs may or may not produce NMP zeros. Surprisingly, it depends on a particular combination of the DTs. A novel laboratory QTP, called the quadruple-tank process with dead-times (QTPwDT), that enables the introduction of independent DTs between each input and output while retaining the basic physics of the original QTP, was constructed in the control laboratory of the ME faculty at the Technion, [5–7]. It represents a modification of the original QTP. The QTPwDT was constructed primarily to investigate properties and capabilities of a novel auto-tuner for decentralized dead-time compensators, [5,8] and is utilized to demonstrate the performance degradation due to NMP zeros. The linearized model of the QTPwDT is used for the investigation in this paper. The rest of the paper is organized as follows. In Section 2 a nonlinear model of the QTPwDT is linearized and the corresponding transfer matrix derived. The poles of the QTPwDT as well as the zero polynomial from which the NMP zeros can be evaluated, are obtained via the transfer matrix. Section 3 discusses the effect of multivariable DTs on the properties and control of the QTP. The number of the NMP zeros originating from the DTs as well as the location of the dominant ones and their behavior as function of the DTs are discussed and characterized. Results on the output directions of the real NMP zeros as well as on the DIC of the QTPwDT and time-domain bounds on the achievable performance associated with the NMP zeros are also presented in this section. Section 4 contains short description of the laboratory set-up of
D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
19
Seemingly one might be tempted to think that independent delays may be achieved if DTs were added to the inputs and the outputs of the original QTP. It can be shown, however, that such structure doesn’t result in independent delays between each input and output. The QTPwDT with independent delays between each input and output is depicted schematically in Fig. 2. The independent DTs are accomplished by introducing two additional pumps instead of the two valves in the original QTP. Hence, there are four pumps in the QTPwDT, but their voltage inputs are dependent as explained next. The control input voltages v 1 and v 2 are divided between the pumps to yield mi ði ¼ 1; . . . ; 4Þ, the input voltage to pump i, and delayed at time t by the DTs ai ði ¼ 1; . . . ; 4Þ as follows:
m1 ¼ c1 v 1 ðt a1 Þ m4 ¼ ð1 c1 Þ v 1 ðt a4 Þ m2 ¼ c2 v 2 ðt a2 Þ m3 ¼ ð1 c2 Þ v 2 ðt a3 Þ
Fig. 1. The quadruple-tank process.
the QTPwDT and presents both simulation and experimental results demonstrating some of the results of this paper. Concluding remarks are given in Section 5. Proofs of the main lemmas are deferred to Appendix. 2. The QTP with dead-times In this section we utilize a simple non-linear mathematical model of the QTPwDT and derive its transfer matrix from which the poles of the QTPwDT as well as the zero polynomial, whose roots are the zeros of the process, are obtained.
where c1 ; c2 2 ð0; 1Þ are constants. Clearly, at steady state and m1 =m4 ¼ c1 =ð1 c1 Þ; m2 þ m3 ¼ m2 and m1 þ m4 ¼ m1 m2 =m3 ¼ c2 =ð1 c2 Þ. This new configuration enables one to apply any c1 ; c2 2 ð0; 1Þ and four independent DTs, ai ði ¼ 1; . . . ; 4Þ, between inputs and outputs. When all the ai s are set to 0 the QTPwDT functions essentially as the original QTP. Like in the QTP the target is to control the liquid levels ðh1 ; h2 Þ in the lower two tanks with voltage inputs v 1 and v 2 . Suppose that about its operating point each pump i produces a flowrate given by ki mi ki2 , where ki ; ki2 are constants associated with pump i, then mass balances and Bernoulli’s law yield the following simple non-linear equations:
8 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dh1 > ¼ A11 a1 2gh1 þ a3 2gh3 þ ðc1 k1 v 1 ðt a1 Þ k12 Þ > dt > > > > pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > > < dhdt2 ¼ A1 a2 2gh2 þ a4 2gh4 þ ðc2 k2 v 2 ðt a2 Þ k22 Þ 2 pffiffiffiffiffiffiffiffiffiffiffi dh3 > 1 > ð Þ ¼ a c Þk v ðt a Þ k 2gh þ ð1 > 3 3 2 3 32 3 2 A3 > dt > > > dh pffiffiffiffiffiffiffiffiffiffiffi > : 4 ¼ 1 a4 2gh þ ðð1 c Þk4 v 1 ðt a4 Þ k42 Þ 4 1 A4 dt
Fig. 2. The quadruple-tank process with dead-times.
ð1Þ
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
where Ai is the cross-section of tank i; ai the cross-section of the outlet hole in tank i; hi the water level in tank i; and g the acceleration of gravity. It is seen that the model (1) of the QTPwDT is similar to the QTP model derived in [1] except for the four independent DTs in the inputs. Under steady-state conditions, the non-linear model in (1) reduces essentially to that of the QTP. Hence similarly to the QTP [1], there exists a unique constant input ðv 01 ; v 02 Þ giving the 0 0 steady-state levels ðh1 ; h2 Þ if and only if the matrix
c 1 k1
ð1 c2 Þk3
ð1 c1 Þk4
c2 k2
is non-singular, i.e., if and only if c1 c2 ð1 c1 Þ ð1 c2 Þ k3 k4 =ðk1 k2 Þ – 0. Introduce the deviation variables D 0 ^xi ¼ hi hi ; D
^j ¼ ^xj ; y D ^j ¼ u
fact allows us to study the effect of multivariable DTs on the properties of the QTP. From the linearized state-space realization in (2) as well as from the corresponding transfer matrix in (3) it is evident that the QTPwDT is stable as it’s eigenvalues given by
si ¼ 1=T i ;
detðPðsÞÞ ¼
j ¼ 1; 2 j ¼ 1; 2
0
where hi ði ¼ 1; . . . ; 4Þ; v 0j ðj ¼ 1; 2Þ are equilibrium values of hi and v j ^j ðj ¼ 1; 2Þ are the outputs around their equilibrespectively and y rium values. Then the linearized state-space equations around the equilibrium are
8 4 > < d^xðtÞ ¼ A ^xðtÞ þ P B u i ^ ðt ai Þ dt i¼1 > : ^ðtÞ ¼ C ^xðtÞ y
ð2Þ
ð4Þ
are real and located in the open left half plane. Consequently, according to [9–11], the RHP transmission zeros, which are the focus in this paper, may be obtained from the determinant of (3):
i ¼ 1; . . . ; 4
v j v 0j ;
i ¼ 1; . . . ; 4
c1 c2 T 1 T 2 k1 k2 =ðA1 A2 Þ
Q4 ð1 þ s T i Þ i¼1 ð1 þ s T 3 Þ ð1 þ s T 4 Þ esða1 þa2 Þ k3 k4 ð1 c1 Þ ð1 c2 Þ sða3 þa4 Þ e k1 k2 c1 c2
ð5Þ
Hence the RHP zeros of the QTPwDT are determined via the numerator of (5): k3 k4 ð1 c1 Þð1 c2 Þ sða1 þa2 a3 a4 Þ esða1 þa2 Þ ð1þsT 3 Þð1þsT 4 Þ e ¼0 k1 k2 c1 c2
ð6Þ that is, the RHP zeros of the QTPwDT are the RHP roots of the term in the brackets in (6). 3. Properties of the QTP with dead-times
where
2
1=T 1
0
A3 =ðA1 T 3 Þ
0
1=T 2
0
0
0
1=T 3
6 6 A¼6 4
0 2 c1 k1 =A1 6 0 6 B1 ¼ 6 4 0 2
0 0
0 3 0 07 7 7; 05
0 2
C¼
A4 =ðA2 T 4 Þ 7 7 7; 5 0
0
0
1=T 4 3
6 0 c k =A 7 27 6 2 2 B2 ¼ 6 7; 40 5 0
0 0
0
3
7 60 0 7 6 B3 ¼ 6 7; 4 0 ð1 c2 Þk3 =A3 5
Non-minimum-phase zeros, those particularly closest to the imaginary axis, restrict the closed loop performance. The effect of multivariable DTs on the properties of the QTP is discussed in this section. Specifically, the number of the non-minimum-phase (NMP) zeros and the location of the dominant ones (that is, the NMP zeros closest to the imaginary axis), the output directions of the real NMP zeros, the process’ decentralized integral controllability (DIC) and its relative gain array (RGA) as well as performance limitations imposed by the NMP zeros are also discussed in this section.
3
0
2
6 6 B4 ¼ 6 4
0 0 1 0 0 0
0
3
0
0
0
07 7 7; 05
0 ð1 c1 Þk4 =A4
3.1. Effect of delays on the number and location of the NMP zeros In order to investigate the effect of multivariable DTs on the number and location of the NMP zeros we define two parameters:
0
D
b ¼ a1 þ a2 a3 a4
0 1 0 0
ð7Þ
D
K ¼ k3 k4 =ðk1 k2 Þ
and the T i ði ¼ 1; . . . ; 4Þ in A are
Ai Ti ¼ ai
sffiffiffiffiffiffiffiffiffiffiffiffi 0 2 hi ; g
The dimensionless parameter K depends upon the pumps constants ki ði ¼ 1; . . . ; 4Þ and for a given setup is constant. It is 1 if each pair of pumps (1&4 and 2&3) consists of identical pumps. b is the difference between the DTs on the diagonal and those on the off-diagonal of the QTPwDT transfer matrix (3). Using (7) it follows from (6) that the zeros of the QTPwDT are the roots of
i ¼ 1; . . . ; 4
^ to y ^; PðsÞ, is given by Then the transfer matrix from u
^ðsÞ ¼ PðsÞ u ^ ðsÞ ¼ C ðs I AÞ1 y
4 X
! sak
Bk e
^ ðsÞ u
k¼1
where
PðsÞ ¼
P11 ðsÞ P12 ðsÞ P21 ðsÞ P22 ðsÞ
2
ðc1 T 1 k1 =A1 Þesa1 1þsT 1
¼ 4 ðð1c
sa4 1 ÞT 2 k4 =A2 Þe ð1þsT 4 Þð1þsT 2 Þ
ð1 þ s T 3 Þ ð1 þ s T 4 Þ K
3 sa3
ðð1c2 ÞT 1 k3 =A1 Þe ð1þsT 3 Þð1þsT 1 Þ
ðc2 T 2 k2 =A2 Þesa2 1þsT 2
5 ð3Þ
The transfer matrix of the QTPwDT in (3) contains independent DTs between each input and output. Upon setting the DTs to zero, (3) reduces to the transfer matrix of the QTP, [1]. Hence, the transfer matrix of the QTPwDT keeps the basic physics of the original QTP. This
ð1 c1 Þ ð1 c2 Þ
c1 c2
esb ¼ 0
ð8Þ
While in the QTP the zeros depend on the valves’ positions c1 and c2 , the zeros of the QTPwDT depend also on the DTs. However (8) reveals that the latter dependence is associated with a particular combination of the DTs namely on b, and not on the individual DTs, ai ði ¼ 1; . . . ; 4Þ, as might have been expected. In the following two sections the effect of b and c1 ; c2 on the NMP zeros, in particular, on the dominant NMP zeros, which dominate the performance limitation, are investigated.
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
3.1.1. The case b 6 0 Introduce the function: D
GðsÞ ¼
K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ sb e ð1 þ s T 3 Þ ð1 þ s T 4 Þ
ð9Þ
where c1 ; c2 2 ð0; 1Þ and T 3 ; T 4 > 0. Clearly GðsÞ is stable. With definition (9), (8) can be rewritten as follows:
1 þ GðsÞ ¼ 0
(iii) The line jGð0Þj ¼ K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ ¼ 1 divides the c1 c2 plane into minimum-phase and non-minimumphase regions for different K values as shown in Fig. 4. (iv) Note that b ¼ 0 means either no DTs, that is, ai ¼ 0ði ¼ 1; . . . ; 4Þ, or that a1 þ a2 ¼ a3 þ a4 . Although the two cases are completely different both are identical as far as the number of NMP zeros is considered.
ð10Þ
As the structure of (10) is similar to the structure of a closed loop characteristic equation, the number of the finite NMP zeros in the case b 6 0 can be determined by the Nyquist stability theorem, [12]. The magnitude jGðjwÞj and the phase argðGðjwÞÞ of GðsÞ are:
K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ ffi jGðjwÞj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ ðw T 3 Þ2 Þ ð1 þ ðw T 4 Þ2 Þ
ð11Þ
argðGðjwÞÞ ¼ p þ b w tan1 ðw T 3 Þ tan1 ðw T 4 Þ
ð12Þ
The magnitude (11) is a monotonic strictly decreasing function of w and its maximum value is:
jGð0Þj ¼ jGðjwÞkw¼0 ¼ K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ
ð13Þ
The typical polar plot of GðsÞ in (9) and a circle with radius jGð0Þj are depicted in Fig. 3. The next lemma follows directly from the Nyquist stability theorem. Lemma 1. Let jGð0Þj be defined in (13). Let b in (7) be non-positive. Then: (i) if jGð0Þj < 1, the QTPwDT has no NMP zeros independent of the DTs; (ii) if jGð0Þj > 1, the QTPwDT has at least one NMP zero (one NMP zero only when b ¼ 0) and the number of the NMP zeros increases when b decreases (as specified in Lemma 2). Remark 1
As stated in Lemma 1(ii) the number of the NMP zeros increases as b decreases. The next lemma clarifies that dependence. For jGð0Þj > 1 let bcrðkÞ ðk ¼ 1; 2; . . .Þ and wcr be defined as follows:
bcrðkÞ ¼ 2kp þ tan1 ðwcr T 3 Þ þ tan1 ðwcr T 4 Þ =wcr ;
wcr ¼
k ¼ 1; 2; . . .
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r 2 u u T 23 þ T 24 þ T 23 T 24 þ 4T 23 T 24 jGð0Þj2 t 2T 23 T 24
ð14Þ
ð15Þ
Note that (15) is the positive solution w ¼ wcr of jGðjwÞj ¼ 1, with jGðjwÞj defined in (11) and (14) is the solution of argðGðjwÞÞjw¼wcr ¼ ð2k þ 1Þp for k ¼ 1; 2; . . ., where argðGðjwÞÞ is defined in (12). Noting that jGðjwÞj and argðGðjwÞÞ are both monotonically decreasing functions of w and recalling the typical polar plot in Fig. 3, it follows that: (a) for 0 P b > bcrð1Þ the Nyquist plot of GðsÞ in (9) has one clockwise encirclement of 1 þ j0; (b) for b ¼ bcrðkÞ the Nyquist plot of GðsÞ traverses the 1 þ j0 with phase ð2k þ 1Þp radians at frequency wcr ; (c) for bcrðkÞ > b > bcrðkþ1Þ the Nyquist plot of GðsÞ has 2k þ 1 clockwise encirclement of 1 þ j0. With the above we have the following lemma. Lemma 2. Let jGð0Þj > 1 and b 6 0. Then:
(i) It is quite surprising that despite of the presence of DTs between inputs and outputs what determines whether there are or not finite NMP zeros are just the c1 ; c2 like in the QTP (without DTs)! (ii) Note that for K ¼ 1 the conditions jGð0Þj < 1 and jGð0Þj > 1 reduce to 1 < c1 þ c2 < 2 and 0 < c1 þ c2 < 1, respectively, which are the conditions for minimum-phase and non-minimum-phase of the QTP [1].
(i) for 0 P b > bcrð1Þ the QTPwDT has one NMP real zero; (ii) for bcrðkÞ > b > bcrðkþ1Þ the QTPwDT has k pairs of NMP complex conjugate zeros and one NMP real zero; (iii) for b ¼ bcrðkÞ ðk ¼ 1; 2; . . .Þ one of the k pairs is located on the imaginary axis at jwcr . Lemmas 1 and 2 are concerned with the number of finite NMP zeros. We next consider the locations of those zeros. When b ¼ 0 and jGð0Þj > 1, (8) has two real roots one of which is located in the right-half plane exactly as in the original QTP. Let us denote this NMP real zero by a0 . According to Lemma 2 there always exists one NMP real zero for b 6 0 and jGð0Þj > 1. We denote this real zero by a. The next lemma specifies the location of the NMP zeros. Lemma 3. Let jGð0Þj > 1 and b 6 0. Then: (i) the NMP real zero, a, decreases from a0 as b decreases. When b ! 1 then a ! 0þ ; (ii) for bcrðkÞ > b > bcrðkþ1Þ the k pairs of the complex conjugate NMP zeros, fðzi ; zi Þ; i ¼ 1; . . . ; kg, lie in the following strip 0 < Refzi g < aði ¼ 1; . . . ; kÞ. The proofs of Lemmas 2 and 3 are presented in Appendix.
Fig. 3. Typical polar plot of GðsÞ.
Remark 2. Lemmas 2 and 3 have a clear physical interpretation. It is well known that control deteriorates with NMP zeros and more so as these zeros get closer to the imaginary axis. jGð0Þj > 1 means that larger portions of the flows go to the upper tanks rendering the control of the outputs (the heights of the lower tanks) more dif-
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
Fig. 4. Minimum-phase and non-minimum-phase regions in c1 c2 plane for different K in the case b 6 0.
ficult. Control becomes even harder when the input delays to the upper tanks increase which is the case when b decreases. In this circumstance, according to Lemma 3, more and more NMP zeros lie closer to the imaginary axis as the strip in which they are situated moves towards the imaginary axis.
We recall that when b ¼ 0 and jGð0Þj > 1, (8) has two real roots one of which is a0 (NMP zero) as defined just before Lemma 3. For 0 < b < T 3 þ T 4 and jGð0Þj > 1 let aextr and bextr be the real positive solutions of the following two equations:
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < a ¼ ðbT 3 2T 3 T 4 þbT 4 Þ2 4bT 3 T 4 ðbT 3 T 4 Þ bT 3 2T 3 T 4 þbT 4 2bT 3 T 4
2bT 3 T 4
3.1.2. The case b > 0 Unlike the case b 6 0, in this case it turns out that there is an infinite number of NMP zeros as summarized in the following lemma.
Define
Lemma 4. Let b > 0. Then the QTPwDT has infinite number of NMP zeros with arbitrarily large positive real part.
crðrÞ ¼ 2rp þ tan1 ðwcr T 3 Þ þ tan1 ðwcr T 4 Þ =wcr ; b
D
:
eab ð1þaT 3 Þð1þaT 4 Þ
1 ¼ jGð0Þj
r ¼ 0; 1; . . .
ð16Þ
ð17Þ
Proof of Lemma 4. Introduce the new variable z ¼ s b with b > 0. Then (8) can be rewritten as an exponential polynomial in z and the Lemma follows at once from Pontrjagin’s theorem ([13], Theorem 13.1).
where wcr was defined in (15). It can be shown that
Remark 3. Note that while for b 6 0 the existence of NMP zeros depends on jGð0Þj only, for b > 0 NMP zeros are always present independent of jGð0Þj.
The next lemma provides information on the location of the dominant NMP zeros as b varies from 0 to 1.
While the number of the NMP zeros is infinite independent of jGð0Þj the latter does affect their locations as indicated in Lemmas 5 and 6. The proofs of Lemmas 5 and 6 are given in Appendix. Lemma 5. Let jGð0Þj < 1 and b > 0. Then the dominant NMP zero is real, say b, unique and approaches the imaginary axis as b increases. The other NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > b for all b > 0. Remark 4. Lemma 5 has the following physical interpretation. jGð0Þj < 1 means that larger portions of the flows go to the lower tanks rendering the control of the outputs (the heights of the lower tanks) easier. However, the control becomes harder when the input delays to the lower tanks increase which is the case when b increases. In this circumstance, according to Lemma 5 the real NMP zero moves closer to the imaginary axis and many more zeros, located to the right of that zero, move with it.
crð0Þ 0 < bextr < b
ð18Þ
Lemma 6. Let jGð0Þj > 1 and b > 0. Then: (i) for 0 6 b 6 bextr the dominant NMP zero is real and increases from a0 when b ¼ 0 to aextr when b ¼ bextr . A second real NMP zero, say a2 , moves on the real axis from þ1 when b ¼ 0 to aextr when b ¼ bextr . The rest of the NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > a2 for all b in the range; (ii) for bextr < b < bextr þ db , where db ! 0þ the multiple real NMP zero, aextr , breaks away from the real axis as b increases and this pair of complex conjugate NMP zeros is dominant; crðrÞ þ eðr ¼ 0; 1; . . .Þ, crðrÞ e to b (iii) when b increases from b where e is a positive constant and e 2p=wcr , there is a dominant pair of complex conjugate NMP zeros which moves from the right-half to the left half complex plane crðrÞ ðr ¼ 0; 1; . . .Þ crosses the imaginary axis at and for b ¼ b jwcr .
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
Fig. 5. Dominant NMP zero locus ðb > 0Þ.
Remark 5. For the purpose of illustrating Lemma 6 we show in Fig. 5 a typical NMP zero locus as function of bð0 < b < 1Þ. Of the infinite branches that exist only four dominant branches are shown. Two branches start on the real axis, one (solid) at a0 and moves to the right and the second one (dashed) at þ1 and moves to the left as b increases. For b ¼ bextr the two branches coincide at aextr and when b increases further the two branches leave the real axis and split to two complex conjugate ones which approach the crð0Þ at jw . The 3rd branch imaginary axis and cross it when b ¼ b cr (dotted) and its counterpart the 4th branch (not shown) originate with their real parts at þ1 and move to the left. However, for all crð0Þ Þ their real part is always larger than the corresponding b 2 ð0; b real parts of the first two branches. That is, branches 1&2 contain the dominant zeros until they cross the imaginary axis when crð0Þ . Then the 3rd & the 4th branches become dominant. These b¼b branches approach the imaginary axis and cross it at jwcr when crð1Þ ; b crð2Þ Þ another pair of crð1Þ . When b increases further in ðb b¼b branches moves towards the imaginary axis and crosses it at crð2Þ and so forth. jwcr for b ¼ b Remark 6. It can be easily deduced from the previous remark that the pair of complex conjugate NMP zeros corresponding to b in crð0Þ is dominant and approaches the imaginary axis bextr < b < b crð0Þ . as b approaches b
Define for the transfer matrix PðsÞ in (3) the output direction of a NMP real zero z as a vector w 2 R2 of unit length such as wT PðzÞ ¼ 0:
w1 w2
T
2 c1 T 1 k1 6 4
A1
ð1c2 ÞT 1 k3 za 3 e A1
eza1
1þzT 1 ð1c1 ÞT 2 k4 za 4 e A2
3
ð1þzT 3 Þð1þzT 1 Þ 7 c2 T 2 k2 za 2 e
ð1þzT 4 Þð1þzT 2 Þ
A2
5¼
T 0 0
ð19Þ
1þzT 2
It follows from (19) that w1 ; w2 –0, so the zero z is never associated with only one output. Solving (19) for c2 yields:
w1 ð1 c1 Þ ezða1 a4 Þ ð1 þ z T 1 ÞT 2 k4 A1 ¼ w2 c1 ð1 þ z T 4 Þð1 þ z T 2 ÞT 1 k1 A2 With no delays or when a1 ¼ a4 the above reduces to that derived in [1] for the QTP. Thus, while in the latter case zero direction is determined solely by the value of c1 , in the QTPwDT it also depends on the difference a1 a4 as indicated next: if c1 1, then z is mostly associated with the first output ðw1 w2 Þ; if c1 1, then z is mostly associated with the second output ðw1 w2 Þ; if a1 a4 , then z is mostly associated with the first output; if a1 a4 , then z is mostly associated with the second output.
3.2. NMP real zeros and their output directions
3.3. DIC of the QTPwDT
Each zero of a multivariable system has both location and direction. There are output and input zero directions. From a practical point of view, the output zero direction is usually of more interest than the input one, because output zero direction gives information about which output may be difficult to control, [12]. In this section the output directions of the NMP real zeros for QTPwDT transfer matrix are found.
A desirable property of a decentralized control system is that the closed loop system should remain stable as subsystem controllers are brought in and out of service. This property is called decentralized integral controllability (DIC), [14]. The relative gain array (RGA) [12] is a steady-state measure of interactions for decentralized control. The RGA is defined as D RGA ¼ Pð0Þ PT ð0Þ, where the asterisk denotes the element-by-
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
element matrix multiplication (Schur product). Although the RGA is associated with steady-state it is a useful tool in the process industry to decide on input–output pairing for decentralized controllers, [12,14]. For a 2 2 transfer matrix to be DIC it is a necessary and sufficient that the (1,1) element of the RGA be positive. The QTPwDT transfer matrix (3) has two inputs and two outputs, consequently we can determine the necessary and sufficient condition for QTPwDT to be DIC. The RGA of the QTPwDT transfer matrix is given by
RGA ¼
k
1k 1 D k ¼ 1 jGð0Þj
1k k
1 ðw ð^r eÞ ejw2 jÞ ezðts1 aÞ 1 1
^21 the interaction, e the settling level, t s1 where yu1 is the undershoot, y D ¼ . the settling time, a maxfminfa1 ; a3 g; minfa2 ; a4 gg and ts1 > a ¼ 0. Remark 7. The bound in (23) reduces to the one in [15] for a That is when in each row there is at least one element with no delay.
ð20Þ ^21 P w1 yu1 þ jw2 jy
where jGð0Þj was defined in (13), and the necessary and sufficient condition for QTPwDT to be DIC is that k in (20) be positive. Therefore the DIC property of the QTPwDT depends on jGð0Þj only, as stated in the next lemma. Lemma 7. Let jGð0Þj be defined in (13). Then the QTPwDT is DIC iff jGð0Þj < 1. Since the RGA is a steady-state property DTs do not affect it. Then assuming K ¼ 1 and for the same c1 and c2 , the RGA for both the QTP in [1] and the QTPwDT are identical. Indeed k in (20) (assuming K ¼ 1) reduces to that derived in [1] for the QTP. The RGA analysis, [1,5,7], suggests that for jGð0Þj > 1 another input–output pairing for decentralized integral control should be chosen. Let
e ¼ P 12 ðsÞ P11 ðsÞ PðsÞ P22 ðsÞ P21 ðsÞ
w1^r ezðts1 aÞ 1
ð24Þ
Hence, like the case with no DTs, the above bound suggests that the undershoot is expected to be large for RHP zero close to the origin and/or for small interactions. Hence the trade off between undershoot in the setpoint response in the first loop and the interaction to the second loop and vice versa. Also, if one has two similar systems, both having the same zero and w1 , one system without DTs and the other one with, then both will have the same bound (the right hand side in (24)) if the settling time of the system with DTs . This is another way to look on the performance deteis longer by a rioration due to the DTs.
4. Experimental study
^ 1 and u ^2 be the QTPwDT transfer matrix PðsÞ in (3) with the inputs u e permuted. With this pairing the (1,1) element of the RGA for PðsÞ is given by 1 k and is positive. Therefore the new pairing, ðv 2 ; h1 Þ and ðv 1 ; h2 Þ, guarantees that the QTPwDT is DIC. 3.4. Effect of RHP zeros on closed – loop performance RHP zeros are known to limit the achievable performance of linear multivariable systems [12,19]. Since DTs may introduce, as shown above, many RHP zeros, their effect on the achievable performance of the QTPwDT with feedback control is of central interest. Johansson in [15] derived time-domain lower bounds on interactions in linear MIMO systems due to RHP zeros. Those bounds are extended in the following to include processes with DTs and will be presented for 2 2 systems, though their extension to any mxm system with DTs is quite straightforward. Hence the notations in [15, Definition 2] will be used throughout this section. Let the QTPwDT be controlled by a feedback controller such that the closed loop is stable. It is well known [12] that the transmission zeros of the closed loop are the transmission zeros of the open loop, particularly so the RHP transmission zeros of the QTPwDT and each such RHP zero, say z, satisfies the following
In this section the laboratory QTPwDT will be described briefly and simulations as well as experimental results, demonstrating the performance degradation in the control of the QTPwDT with NMP zeros and illustrating some of the results of this paper will be presented. The laboratory setup illustrated schematically in Fig. 2, is shown in Fig. 6. The parameters of the setup are given in Table 1. The acceleration of gravity is g ¼ 981 ½cm=s2 . Four similar gear pumps are employed, each with a capacity of 2:3½‘= min and voltage range of 3–12 [V]. The height of each tank is 23 ½cm . With those parameters K and b, defined in (7), are K ¼ 1:004; b ¼ 5 ½s . As b (see (7)) is negative in the setup considered here, NMP cases may be obtained according to Lemma 1 if jGð0Þj > 1 whereas when jGð0Þj < 1 the process is minimum-phase (MP). Two cases with different c1 and c2 are considered, one MP and the second one NMP. The c1 and c2 values and the corresponding linearized models of the QTPwDT according to (3) are as follows:
ð21Þ
where TðsÞ is the complementary sensitivity given by
yðsÞ ¼ TðsÞrðsÞ ¼ PðsÞCðsÞðI þ PðsÞCðsÞÞ1 rðsÞ
ð23Þ
Remark 8. For a small settling level e the bound in (23) reduces to
;
wT PðzÞ ¼ wT TðzÞ ¼ 0
^21 P w1 yu1 þ jw2 jy
ð22Þ
with CðsÞ the transfer matrix of the controller and rðsÞ and yðsÞ the Laplace transforms of the setpoint r and output y respectively. The next lemma extends Theorem 3 in [15] to 2 2 cases with DTs : Lemma 8. Given the stable closed loop system in (22) with zero initial T conditions at t ¼ 0 and let rðtÞ ¼ ½ ^r 0 for t > 0. Assume that PðsÞ has a real RHP zero z > 0 with zero direction w 2 R2 and w1 > 0. Then the setpoint response satisfies
Fig. 6. The experimental setup for the QTPwDT.
D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
25
Table 1 Parameter values of the laboratory QTPwDT. i
Ai ðcm2 Þ
ai ðcm2 Þ
ki ðcm3 =ðs VÞÞ
ki2 ðcm3 =sÞ
ai ðsÞ
1 2 3 4
12 20 12 20
0.26 0.26 0.21 0.21
3.808 3.831 3.904 3.752
2.712 2.006 3.069 3.499
5 6 7 9
MP case ðc1 ¼ 0:65; c2 ¼ 0:60Þ
2 P ðsÞ ¼ 4
1:585es5 1þ7:685s
es7 ð1þ4:778sÞð1þ7:685sÞ
0:778es9 ð1þ3:705sÞð1þ11:85sÞ
1:362es6 1þ11:85s
3 5
ð25Þ
NMP case ðc1 ¼ 0:40; c2 ¼ 0:35Þ
2 Pþ ðsÞ ¼ 4
0:834es5 1þ6:57s
1:39es7 ð1þ10:231sÞð1þ6:57sÞ
1:271es9 ð1þ14:05sÞð1þ11:29sÞ
0:757es6 1þ11:29s
3 5
ð26Þ
The corresponding (1,1) elements in the RGA in (20) are k ¼ 1:56 and k ¼ 0:56 for the MP and NMP cases respectively. Consequently accordingly to Lemma 7 the QTPwDT is DIC in the MP case and is not DIC in the NMP case. To make the NMP case a DIC one we follow the pairing change discussed in Section 3.3 which leads to:
2 e þ ðsÞ ¼ 4 P
1:39es7 ð1þ10:231sÞð1þ6:57sÞ
0:834es5 1þ6:57s
0:757es6 1þ11:29s
1:271es9 ð1þ14:05sÞð1þ11:29sÞ
3 5
ð27Þ
Two different controllers were designed for each of the two cases. A decentralized PI controller tuned via the BLT method described in [16], and a decentralized dead-time compensator (DTC) with the structure suggested by Jerome and Ray [17]. The primary controller of the DTC was tuned by a novel method, called EVS (eigenvalue sensitivity), see [18] for details. For completeness, the EVS method is briefly described. The method utilizes characteristic loci, [12], and consists of two stages. In the first stage interactions are ignored and the components of the decentralized primary controller are tuned in a single-input single-output fashion using predetermined gain and DT margins. In the second stage the primary controller’s gains are reduced until the minimum distance to 1 þ j0 of the character-
Fig. 7. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 1 in the minimum phase QTPwDT (simulation).
Fig. 8. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 1 for the minimum phase QTPwDT (experiment).
istic loci of the sensitivity matrix of the closed loop with interactions is equal or larger to that obtained in the first stage. In Figs. 7 and 8 responses of the QTPwDT, corresponding to P ðsÞ, to a unit change in the first setpoint with the feedback controllers described above are depicted. Fig. 7 shows simulations with the non linear process while experimental results are given e þ ðsÞ, the in Fig 8. Responses of the QTPwDT, corresponding to P NMP case, to a unit change in the second setpoint controlled by the decentralized PI and by the DTC tuned via the EVS method are depicted in Fig. 9 (simulation) and Fig. 10 (experimental results). The agreement between the simulations and the experiments is remarkable. One can see that faster and tighter control is obtained by the DTC in both cases. The deterioration in the control quality due to RHP zeros is evident: the settling time in the NMP case is many times larger than in the MP case for both the decentralized DTC and the decentralized PI. Applying Lemma 2 to the NMP case indicates that there is a single NMP real zero as b ¼ 5 ½s > bcrð1Þ ¼ 40:03 ½s . The location of that zero is z ¼ 0:042 and the poles of P þ ðsÞ are f0:1522;
Fig. 9. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 2 for the non-minimum-phase QTPwDT (simulation).
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the DTs. Time-domain bounds on control performance imposed by the NMP zeros indicating the deterioration in performance due to the NMP zeros are derived. Experimental results of the laboratory QTPwDT demonstrating both the control difficulties due to the RHP zeros as well as some of the results were presented.
Acknowledgement The support of the Israel Electric Company for this work is gratefully acknowledged. Appendix A The purpose of the appendix is to present proofs for Lemmas 2, 3, 5, 6 and 8. Detailed proofs can be found in [5]. A.1. Proof of Lemma 2 Fig. 10. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 2 in the non-minimum-phase QTPwDT (experiment).
0:0886; 0:0977; 0:0712g. Consequently this zero is dominating. Hence the significant deterioration in the performance (see Remark 2). The zero z ¼ 0:042 of P þ ðsÞ has zero direction wT ¼ ½ w1 w2 ¼ ½ 0:575 0:818 . With settling level e 0 Lemma 8 for a unit step in r 2 ðr ¼ ½ r1 r2 T ¼ ½ 0 1 T Þ gives
w2 yu2
^12 P þ jw1 jy
w2 ezðts2 a Þ 1
¼ 6 ½s to which reduces for t s2 ¼ 66 ½s and a
^12 P 0:072: 0:818yu2 þ 0:575y As seen in Fig. 10 undershoot is very small, yu2 0, hence we get a lower bound on the interaction
(i) Follows directly from (a). (ii) According to (c) there are 2k þ 1 zeros. Since jGðsÞjs¼r j for real r is a monotonically decreasing function there is only one real solution of jGðsÞjs¼r j ¼ 1. Hence the rest are k pairs of complex conjugate zeros. (iii) Follows from (b). A.2. Proof of Lemma 3 (i) It follows from Lemma 2 that for all b 6 0 there is one real NMP zero which is the solution of jGðsÞjs¼r j ¼ 1. Clearly this zero, denoted a, decreases with b and the Lemma follows. (ii) A Nyquist path shifted to the right by r P 0 as shown in Fig. A.1, is used. For r > a; jGðsÞjs¼r j < 1. Hence no encirclements of 1 þ j0 by the locus of GðsÞ corresponding to the path in Fig. A.1 are obtained. Consequently none of the k pairs of complex conjugate NMP zeros is enclosed by the modified path in Fig. A.1 and Lemma follows.
^12 P 0:125 ½cm y which agrees well with both the simulation and experimental results. 5. Conclusion A new laboratory quadruple-tank process with multivariable DTs, called here the QTPwDT, was constructed, described and used to study how the DTs affect the properties and the control of this interesting process. It is well known that DTs deteriorate the achievable control performance partly because they may give rise to NMP zeros. The conditions for the appearance of NMP zeros, their number, location and the behavior of the dominant ones are analyzed in the paper. Interestingly enough, it depends on b, a particular combination of the DTs, and on jGð0Þj, the magnitude of the steady-state gain of an artificial transfer matrix defined in the paper. Not in all circumstances the existence of DTs brings about NMP zeros. Such is the case when b is negative and jGð0Þj < 1. However when jGð0Þj > 1 a finite number of NMP zeros appear and they increase in number and become more dominant as b decreases. It was further shown that completely different situation occurs when b is positive. In those cases infinite number of NMP zeros exist no matter what value jGð0Þj assumes. However the latter do influence the location, dominance and behavior of the zeros as specified in the paper. Physical interpretations for the main results are given and it is also shown that the DIC property of the QTPwDT is independent of
A.3. Proofs of Lemmas 5 and 6 Preliminaries – Since b > 0; GðsÞ in (9) is non-causal. Hence in order to examine the properties and locations of the infinite number of NMP zeros in this case we introduce the modified Nyquist path which encloses the left half plane possibly penetrating into the right-half plane up to a finite r P 0 as depicted in Fig. A.2. For any r P 0 and any b > 0, the magnitude jGðjw þ rÞj and the phase argðGðjw þ rÞÞ of GðsÞ as w varies in 0 6 w 6 1 are: jGð0Þjerb ð1þrT Þð1þrT Þ
3 4 jGðjwþ rÞj ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 wT 3 4 1þ 1þwT 1þ 1þrT 3 rT 4
argðGðjwþ rÞÞ ¼ p bwþtan1
ðA:1Þ
wT 3 wT 4 þtan1 1þ r T 3 1þ r T 4 ðA:2Þ
respectively with jGð0Þj given in (13). The magnitude (A.1) is a monotonically strictly decreasing function of w for r P 0; b > 0 and therefore its maximum value is:
D GðrÞj ¼ jGðjw þ rÞjw¼0 ¼
jGð0Þj erb ð1 þ r T 3 Þ ð1 þ r T 4 Þ
ðA:3Þ
Clearly GðsÞ maps the infinite half circle in Fig. A.2 to the origin and recall that GðsÞ in (9) has two negative poles.
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
conclude that there exists r ¼ b þ e, where e ! 0þ , for which the locus of GðsÞ corresponding to the path in Fig. A.2 has one clockwise encirclement of the 1 þ j0. Hence, according to the Nyquist stability theorem, the QTPwDT has two zeros within the modified Nyquist path in Fig. A.2 for 0 6 r < b and three zeros for b < r < b þ e and it follows that b is the dominant NMP real zero. Moreover b is the unique real zero because r ¼ b is the unique positive solution of jGðrÞj ¼ 1. Since according to Lemma 4 the QTPwDT has infinite number of NMP zeros with arbitrarily large positive real part we conclude that the rest of the NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > b for all b > 0. Finally as jGð0Þj < 1 and b > 0 it is clear from both the derivative of (A.3) with respect to r and the proof of the existence and the uniqueness of b above that as b increases b moves from þ1 (for b ! 0þ Þ to 0þ (for b ! þ1Þ. This completes the proof.
Fig. A.1. Modified Nyquist path in [s] plane ðb 6 0Þ.
Fig. A.2. Modified Nyquist path in [s] plane ðb > 0Þ.
A.4. Proof of Lemma 5 If jGð0Þj < 1 the locus of GðsÞ corresponding to the path in Fig. A.2 for r ¼ 0 has no encirclements of the 1 þ j0 and the Nyquist stability theorem implies that (10) has two zeros in the left half complex plane. Next we apply the modified Nyquist path in Fig. A.2 with increasing values of r. This results in encirclements of 1 þ j0 if and only if jGðrÞj > 1. For jGð0Þj < 1; b > 0 and T 3 ; T 4 > 0, denote the positive solution of jGðrÞj ¼ 1 by b. This solution always exists and is unique as discussed next. For b > ðT 3 þ T 4 Þ the existence and the uniqueness of b follow from the fact that the derivative of (A.3) with respect to r is positive for all r P 0 and that jGð0Þj < 1. For b ¼ T 3 þ T 4 the derivative above is zero at r ¼ 0 and positive for all r > 0. Hence for b P ðT 3 þ T 4 Þ (A.3) is monotonically increasing function for r > 0. For 0 < b < T 3 þ T 4 , the derivative of (A.3) with respect to r is negative for 0 6 r < a, zero for r ¼ a and positive for r > a where a was defined in (16). Consequently (A.3) is less than one for 0 6 r < a and monotonically increases in r > a. Therefore the existence and the uniqueness of b for b > 0 follow. Thus, the locus of GðsÞ corresponding to the path in Fig. A.2 has no encirclements of the 1 þ j0 for 0 6 r < b and at least one clockwise encirclement for r > b as according to Lemma 4 the number of the zeros of (10) enclosed by the closed path in Fig. A.2 can not decrease as r increases. Since the magnitude (A.1) is a monotonically strictly decreasing function of w for any r P 0 we
A.5. Proof of Lemma 6 crðrÞ ðr ¼ 0; 1; . . .Þ in (17) are all the b valProof of Lemma 6(iii). b ues for which the polar plot of GðsÞ traverses the 1 þ j0 point at crðrÞ there exists a single pair of complex w ¼ wcr . Hence for b ¼ b zeros, jwcr , on the imaginary axis. Single due to the fact that the magnitude in (A.1) is monotonically decreasing. Next we draw the locus of GðsÞ corresponding to the closed path in Fig. A.2 with r ¼ 0. The number of encirclements of 1 þ j0 for bcrðrÞ þ e, where e 2p=wcr , is larger by two relative to that for bcrðrÞ e which implies that the above pair moves from the right-half to the left half crðrÞ . crðrÞ e; b plane. Clearly that pair is dominant for b 2 ðb Proof of Lemma 6(i). Whether or not the locus of GðsÞ corresponding to the closed path in Fig. A.2 encircles the 1 þ j0 depends on the behavior of the positive solutions of jGðrÞj ¼ 1 for any b > 0. From the properties of jGðrÞj in (A.3) discussed in the previous proof it is apparent that in this case, where jGð0Þj > 1; jGðrÞj ¼ 1 may have, depending on b, either zero or two positive solutions. To find the value of b for which two identical solutions exist we solve minr;b jGðrÞj ¼ 1 which reduces to the following two equations: djGðrÞj=dr ¼ 0, and jGðrÞj ¼ 1, and (16) is obtained, the positive solutions of which were denoted aextr and bextr . Consequently for b ¼ bextr there are two identical solutions (r ¼ aextr Þ, for b < bextr two different positive solutions and no solutions exist for b > bextr . Denote by a1 and a2 ð0 < a1 < a2 Þ the two solutions for b < bextr . It is apparent from the inspection of jGðrÞj ¼ 1, where jGðrÞj defined in (A.3), that as b increases from 0 to bextr a1 increases from r ¼ a0 to r ¼ aextr while a2 decreases from r ¼ þ1 to r ¼ aextr . a1 and a2 are the NMP real zeros of (10) since they satisfy jGðrÞj ¼ 1. jGðrÞj > 1 for 0 < r < a1 and r > a2 , while jGðrÞj < 1 for a1 < r < a2 . Therefore when r ¼ a1 the number of the clockwise encirclement of the 1 þ j0 by the locus of GðsÞ corresponding to the path in Fig. A.2 increases by one and consequently a1 is dominant. Proof of Lemma 6 (ii). From the previous section it is clear that one has jGðrÞk r ¼ a ¼ 1. Hence a small positive perturbation in ext
b ¼ bext w¼0 b, say b1 ¼ bextr þ db , where db ! 0þ , leads to an increase in the amplitude, that is jGðrÞk r ¼ a > 1. Consequently and due to ext
b ¼ b1 w¼0 the fact that the amplitude in (A.1) is monotonically decreasing function of w for fixed r and b, it is clear that there exists a frequency w0 > 0 for which jGðrÞk r ¼ a ¼ 1 which implies a pair ext
b ¼ b1 w ¼ w0 of zeros given approximately by aext j w0 . Hence the break away from the real axis of the zero locus at r ¼ aextr . This results in an
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increase by two in the number of clockwise encirclements of 1 þ j0 by the locus of GðsÞ corresponding to the closed path in Fig. A.2, as r is increased from slightly below to slightly above aextr . Hence the dominance of the pair of zeros follows. This completes the proof. A.6. Proof of Lemma 8 Using (21) and (22) and the Laplace transform of the output y gives
wT yðs ¼ zÞ ¼
2 X
wk
Z
1
ezt yk ðtÞ dt ¼ 0;
0
k¼1
which is equivalent to
Z
t s1
ezt
2 X
0
wk yk ðtÞ dt ¼
Z
k¼1
1
ezt
2 X
t s1
wk yk ðtÞ dt
ðA:4Þ
k¼1
Since the response yk can not start before the smallest DT in the kth row has elapsed, the left-hand side of (A.4) can be written as
Z
t s1
ezt
0
¼
2 X
wk yk ðtÞ dt
k¼1
Z
fa1 ;a3 gt s1
min
ezt w1 y1 ðtÞ dt
Z
fa2 ;a4 gt s1
min
ezt w2 y2 ðtÞ dt:
leading to:
Z
fa1 ;a3 gts1
ezt w1 y1 ðtÞ dt
min
¼
Z
fa2 ;a4 gt s1
ezt w2 y2 ðtÞ dt
min
Z
1
ezt
t s1
2 X
wk yk ðtÞ dt
ðA:5Þ
k¼1
The left-hand side of (A.5) satisfies
Z
fa1 ;a3 gts1
min
Z
6 a
¼
t s1
ezt w1 y1 ðtÞ dt
Z
fa2 ;a4 gt s1
min
ezt w2 y2 ðtÞ dt
^21 ezt dt w1 yu1 þ jw2 j y
eza ezts1 ^21 : w1 yu1 þ jw2 j y z
Substituting the above into (A.5) and following the same steps as in the proof of [15, Theorem 3] leads to (23) at once.
References [1] K.J. Johansson, The quadruple-tank process: a multivariable laboratory process with an adjustable zero, IEEE Trans. Contr. Syst. Tech. 8 (3) (2000) 456–465. [2] I. Alvarado, D. Limon, W. Garcia-Gabin, T. Alamo, E.F. Camacho, An educational plant based on the quadruple-tank process, in: Proceedings of the Seventh IFAC Symposium on Advances in Control Education, Madrid, Spain, vol. 7, Part 1, June 2006. [3] E. Rusli, S. Ang, R.D. Braatz, A quadruple-tank process control experiment, Chem. Eng. Educ. 38 (2004) 174–181. [4] R. Vadigepalli, E.P. Gatzke, F.J. Doyle, Robust control of a multivariable experimental four-tank system, Ind. Eng. Chem. Res. 40 (8) (2001) 1916–1927. [5] D. Shneiderman, Properties of the Quadruple-Tank Process with Multiple Dead Times and its Control via Auto-Tuned Decentralized Dead-Time Compensator, M.Sc. Thesis, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel, 2005. [6] Z.J. Palmor, D. Shneiderman, Properties of a novel quadruple tank process with dead times, in: Proceedings of the 30th Israeli Conference on Mechanical Engineering, Tel-Aviv, Israel, May 2005, pp. 176–177. [7] Z.J. Palmor, D. Shneiderman, Effect of multivariable dead times on the properties of the quadruple-tank process, in: Proceedings of the Ninth Biennial ASME Conference on Engineering Systems Design and Analysis, Haifa, Israel, ESDA2008-59033, July 2008. [8] Z.J. Palmor, D. Shneiderman, U. Burshtein, N. Albersheim, An auto-tuner for decentralized dead-time compensators and its application to the quadruple tank process with dead times, in: Proceedings of the 30th Israeli Conference on Mechanical Engineering, Tel-Aviv, Israel, May 2005, pp. 174–175. [9] N.F. Jerome, W.H. Ray, Model-predictive control of linear multivariable systems having time delays and right-half-plane zeros, Chem. Eng. Sci. 47 (4) (1992) 763–785. [10] L. Pandolfi, On the zeros of transfer functions of delayed systems, Syst. Control Lett. 1 (3) (1981) 204–210. [11] L. Pandolfi, The transmission zeros of systems with delays, Int. J. Control 36 (6) (1982) 959–976. [12] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control, Analysis and Design, second ed., John Wiley & Sons Ltd., 2005. [13] R. Bellman, K. Cooke, Differential-Difference Equation, Academic Press, New York, 1963. [14] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs, NJ, 1989. [15] K.J. Johansson, Interaction bounds in multivariable control systems, Automatica 38 (2002) 1045–1051. [16] W.L. Luyben, Simple method for tuning SISO controllers in multivariable systems, Ind. Eng. Chem. Process Des. Dev. 25 (1986) 654–660. [17] N.F. Jerome, W.H. Ray, High performance multivariable control strategies for systems having time delays, AIChE J. 32 (6) (1986) 914–931. [18] U. Burshtein, Design and Auto-Tuning of Decentralized Multivariable Dead-Time Compensators, M.Sc. Thesis, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel, 2002. [19] M.M. Seron, J.H. Braslavsky, G.C. Goodwin, Fundamental Limitations in Filtering and Control, Springer-Verlag, 1997. [20] G.C. Goodwin, S.F. Graebe, M.E. Salgado, Control System Design, Prentice-Hall, NJ, 2001.
Contents lists available at ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
Properties and control of the quadruple-tank process with multivariable dead-times D. Shneiderman, Z.J. Palmor * Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 5 April 2009 Received in revised form 28 July 2009 Accepted 27 October 2009
Keywords: Multiple delays Multi-input multi-output process Right-half plane zero Control education
a b s t r a c t The well known laboratory quadruple-tank process (QTP) has been introduced in the laboratories of many schools around the world as it is ideally suited to illustrate concepts in multivariable control. In this paper the QTP is extended to include independent multivariable dead times (DTs) and their effects on the properties and control of the QTP are studied. DTs are very common in many various processes and make the control of the QTP more interesting and challenging. The addition of DTs may introduce infinite, finite or not any number of non-minimum-phase (NMP) zeros. As shown in the paper it depends on a particular combination of the multivariable DTs. The conditions for each case are stated and the location and behavior of the zeros closest to the imaginary axis due to the DTs are specified. Other properties of the QTP with DTs as the output real NMP zero directions, the decentralized integral controllability of the process and time-domain bounds on closed loop performance are derived and discussed. Also, a novel laboratory QTP with DTs is described and used to demonstrate the main results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The well known multivariable laboratory process, called the quadruple-tank process (QTP), consists of four interconnected liquid tanks, two pumps and two valves, [1], and is shown schematically in Fig. 1. The inputs are the voltages to the two pumps ðv 1 ; v 2 Þ and the outputs are the liquid levels in the lower tanks ðh1 ; h2 Þ. The linearized dynamics of the process exhibits a multivariable zero that can be moved from one side of the complex plane to the other one by changing the valves’ positions c1 ; c2 . This process was found to be ideally suited to illustrate many concepts in multivariable control, particularly performance limitation due to the non-minimum-phase zero and its output direction. It is not surprising, therefore, that the laboratory QTP was incorporated in control laboratories in many schools around the world. See [2–4,20] for example. Multivariable processes containing dead-times (DTs), or delays, between inputs and outputs are very common in practice and the presence of DTs further limits achievable control performance. Hence adding multivariable DTs to the QTP makes its control even more challenging. It is therefore of interest to investigate the effect of multivariable DTs on the properties and control of the QTP. In particular, the number of the non-minimum-phase (NMP) zeros and the location of the dominant ones (dominant zeros are the NMP zeros closest to the imaginary axis), the output directions of * Corresponding author. Tel.: +972 4 829 2086; fax: +972 4 8295172. E-mail addresses: [email protected] (D. Shneiderman), palmor@technion. ac.il (Z.J. Palmor). 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.10.010
the real NMP zeros, the process’ decentralized integral controllability (DIC) and closed loop interaction bounds associated with NMP zeros. These issues are dealt with in this paper and it is shown that the presence of DTs may or may not produce NMP zeros. Surprisingly, it depends on a particular combination of the DTs. A novel laboratory QTP, called the quadruple-tank process with dead-times (QTPwDT), that enables the introduction of independent DTs between each input and output while retaining the basic physics of the original QTP, was constructed in the control laboratory of the ME faculty at the Technion, [5–7]. It represents a modification of the original QTP. The QTPwDT was constructed primarily to investigate properties and capabilities of a novel auto-tuner for decentralized dead-time compensators, [5,8] and is utilized to demonstrate the performance degradation due to NMP zeros. The linearized model of the QTPwDT is used for the investigation in this paper. The rest of the paper is organized as follows. In Section 2 a nonlinear model of the QTPwDT is linearized and the corresponding transfer matrix derived. The poles of the QTPwDT as well as the zero polynomial from which the NMP zeros can be evaluated, are obtained via the transfer matrix. Section 3 discusses the effect of multivariable DTs on the properties and control of the QTP. The number of the NMP zeros originating from the DTs as well as the location of the dominant ones and their behavior as function of the DTs are discussed and characterized. Results on the output directions of the real NMP zeros as well as on the DIC of the QTPwDT and time-domain bounds on the achievable performance associated with the NMP zeros are also presented in this section. Section 4 contains short description of the laboratory set-up of
D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
19
Seemingly one might be tempted to think that independent delays may be achieved if DTs were added to the inputs and the outputs of the original QTP. It can be shown, however, that such structure doesn’t result in independent delays between each input and output. The QTPwDT with independent delays between each input and output is depicted schematically in Fig. 2. The independent DTs are accomplished by introducing two additional pumps instead of the two valves in the original QTP. Hence, there are four pumps in the QTPwDT, but their voltage inputs are dependent as explained next. The control input voltages v 1 and v 2 are divided between the pumps to yield mi ði ¼ 1; . . . ; 4Þ, the input voltage to pump i, and delayed at time t by the DTs ai ði ¼ 1; . . . ; 4Þ as follows:
m1 ¼ c1 v 1 ðt a1 Þ m4 ¼ ð1 c1 Þ v 1 ðt a4 Þ m2 ¼ c2 v 2 ðt a2 Þ m3 ¼ ð1 c2 Þ v 2 ðt a3 Þ
Fig. 1. The quadruple-tank process.
the QTPwDT and presents both simulation and experimental results demonstrating some of the results of this paper. Concluding remarks are given in Section 5. Proofs of the main lemmas are deferred to Appendix. 2. The QTP with dead-times In this section we utilize a simple non-linear mathematical model of the QTPwDT and derive its transfer matrix from which the poles of the QTPwDT as well as the zero polynomial, whose roots are the zeros of the process, are obtained.
where c1 ; c2 2 ð0; 1Þ are constants. Clearly, at steady state and m1 =m4 ¼ c1 =ð1 c1 Þ; m2 þ m3 ¼ m2 and m1 þ m4 ¼ m1 m2 =m3 ¼ c2 =ð1 c2 Þ. This new configuration enables one to apply any c1 ; c2 2 ð0; 1Þ and four independent DTs, ai ði ¼ 1; . . . ; 4Þ, between inputs and outputs. When all the ai s are set to 0 the QTPwDT functions essentially as the original QTP. Like in the QTP the target is to control the liquid levels ðh1 ; h2 Þ in the lower two tanks with voltage inputs v 1 and v 2 . Suppose that about its operating point each pump i produces a flowrate given by ki mi ki2 , where ki ; ki2 are constants associated with pump i, then mass balances and Bernoulli’s law yield the following simple non-linear equations:
8 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dh1 > ¼ A11 a1 2gh1 þ a3 2gh3 þ ðc1 k1 v 1 ðt a1 Þ k12 Þ > dt > > > > pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > > < dhdt2 ¼ A1 a2 2gh2 þ a4 2gh4 þ ðc2 k2 v 2 ðt a2 Þ k22 Þ 2 pffiffiffiffiffiffiffiffiffiffiffi dh3 > 1 > ð Þ ¼ a c Þk v ðt a Þ k 2gh þ ð1 > 3 3 2 3 32 3 2 A3 > dt > > > dh pffiffiffiffiffiffiffiffiffiffiffi > : 4 ¼ 1 a4 2gh þ ðð1 c Þk4 v 1 ðt a4 Þ k42 Þ 4 1 A4 dt
Fig. 2. The quadruple-tank process with dead-times.
ð1Þ
20
D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
where Ai is the cross-section of tank i; ai the cross-section of the outlet hole in tank i; hi the water level in tank i; and g the acceleration of gravity. It is seen that the model (1) of the QTPwDT is similar to the QTP model derived in [1] except for the four independent DTs in the inputs. Under steady-state conditions, the non-linear model in (1) reduces essentially to that of the QTP. Hence similarly to the QTP [1], there exists a unique constant input ðv 01 ; v 02 Þ giving the 0 0 steady-state levels ðh1 ; h2 Þ if and only if the matrix
c 1 k1
ð1 c2 Þk3
ð1 c1 Þk4
c2 k2
is non-singular, i.e., if and only if c1 c2 ð1 c1 Þ ð1 c2 Þ k3 k4 =ðk1 k2 Þ – 0. Introduce the deviation variables D 0 ^xi ¼ hi hi ; D
^j ¼ ^xj ; y D ^j ¼ u
fact allows us to study the effect of multivariable DTs on the properties of the QTP. From the linearized state-space realization in (2) as well as from the corresponding transfer matrix in (3) it is evident that the QTPwDT is stable as it’s eigenvalues given by
si ¼ 1=T i ;
detðPðsÞÞ ¼
j ¼ 1; 2 j ¼ 1; 2
0
where hi ði ¼ 1; . . . ; 4Þ; v 0j ðj ¼ 1; 2Þ are equilibrium values of hi and v j ^j ðj ¼ 1; 2Þ are the outputs around their equilibrespectively and y rium values. Then the linearized state-space equations around the equilibrium are
8 4 > < d^xðtÞ ¼ A ^xðtÞ þ P B u i ^ ðt ai Þ dt i¼1 > : ^ðtÞ ¼ C ^xðtÞ y
ð2Þ
ð4Þ
are real and located in the open left half plane. Consequently, according to [9–11], the RHP transmission zeros, which are the focus in this paper, may be obtained from the determinant of (3):
i ¼ 1; . . . ; 4
v j v 0j ;
i ¼ 1; . . . ; 4
c1 c2 T 1 T 2 k1 k2 =ðA1 A2 Þ
Q4 ð1 þ s T i Þ i¼1 ð1 þ s T 3 Þ ð1 þ s T 4 Þ esða1 þa2 Þ k3 k4 ð1 c1 Þ ð1 c2 Þ sða3 þa4 Þ e k1 k2 c1 c2
ð5Þ
Hence the RHP zeros of the QTPwDT are determined via the numerator of (5): k3 k4 ð1 c1 Þð1 c2 Þ sða1 þa2 a3 a4 Þ esða1 þa2 Þ ð1þsT 3 Þð1þsT 4 Þ e ¼0 k1 k2 c1 c2
ð6Þ that is, the RHP zeros of the QTPwDT are the RHP roots of the term in the brackets in (6). 3. Properties of the QTP with dead-times
where
2
1=T 1
0
A3 =ðA1 T 3 Þ
0
1=T 2
0
0
0
1=T 3
6 6 A¼6 4
0 2 c1 k1 =A1 6 0 6 B1 ¼ 6 4 0 2
0 0
0 3 0 07 7 7; 05
0 2
C¼
A4 =ðA2 T 4 Þ 7 7 7; 5 0
0
0
1=T 4 3
6 0 c k =A 7 27 6 2 2 B2 ¼ 6 7; 40 5 0
0 0
0
3
7 60 0 7 6 B3 ¼ 6 7; 4 0 ð1 c2 Þk3 =A3 5
Non-minimum-phase zeros, those particularly closest to the imaginary axis, restrict the closed loop performance. The effect of multivariable DTs on the properties of the QTP is discussed in this section. Specifically, the number of the non-minimum-phase (NMP) zeros and the location of the dominant ones (that is, the NMP zeros closest to the imaginary axis), the output directions of the real NMP zeros, the process’ decentralized integral controllability (DIC) and its relative gain array (RGA) as well as performance limitations imposed by the NMP zeros are also discussed in this section.
3
0
2
6 6 B4 ¼ 6 4
0 0 1 0 0 0
0
3
0
0
0
07 7 7; 05
0 ð1 c1 Þk4 =A4
3.1. Effect of delays on the number and location of the NMP zeros In order to investigate the effect of multivariable DTs on the number and location of the NMP zeros we define two parameters:
0
D
b ¼ a1 þ a2 a3 a4
0 1 0 0
ð7Þ
D
K ¼ k3 k4 =ðk1 k2 Þ
and the T i ði ¼ 1; . . . ; 4Þ in A are
Ai Ti ¼ ai
sffiffiffiffiffiffiffiffiffiffiffiffi 0 2 hi ; g
The dimensionless parameter K depends upon the pumps constants ki ði ¼ 1; . . . ; 4Þ and for a given setup is constant. It is 1 if each pair of pumps (1&4 and 2&3) consists of identical pumps. b is the difference between the DTs on the diagonal and those on the off-diagonal of the QTPwDT transfer matrix (3). Using (7) it follows from (6) that the zeros of the QTPwDT are the roots of
i ¼ 1; . . . ; 4
^ to y ^; PðsÞ, is given by Then the transfer matrix from u
^ðsÞ ¼ PðsÞ u ^ ðsÞ ¼ C ðs I AÞ1 y
4 X
! sak
Bk e
^ ðsÞ u
k¼1
where
PðsÞ ¼
P11 ðsÞ P12 ðsÞ P21 ðsÞ P22 ðsÞ
2
ðc1 T 1 k1 =A1 Þesa1 1þsT 1
¼ 4 ðð1c
sa4 1 ÞT 2 k4 =A2 Þe ð1þsT 4 Þð1þsT 2 Þ
ð1 þ s T 3 Þ ð1 þ s T 4 Þ K
3 sa3
ðð1c2 ÞT 1 k3 =A1 Þe ð1þsT 3 Þð1þsT 1 Þ
ðc2 T 2 k2 =A2 Þesa2 1þsT 2
5 ð3Þ
The transfer matrix of the QTPwDT in (3) contains independent DTs between each input and output. Upon setting the DTs to zero, (3) reduces to the transfer matrix of the QTP, [1]. Hence, the transfer matrix of the QTPwDT keeps the basic physics of the original QTP. This
ð1 c1 Þ ð1 c2 Þ
c1 c2
esb ¼ 0
ð8Þ
While in the QTP the zeros depend on the valves’ positions c1 and c2 , the zeros of the QTPwDT depend also on the DTs. However (8) reveals that the latter dependence is associated with a particular combination of the DTs namely on b, and not on the individual DTs, ai ði ¼ 1; . . . ; 4Þ, as might have been expected. In the following two sections the effect of b and c1 ; c2 on the NMP zeros, in particular, on the dominant NMP zeros, which dominate the performance limitation, are investigated.
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
3.1.1. The case b 6 0 Introduce the function: D
GðsÞ ¼
K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ sb e ð1 þ s T 3 Þ ð1 þ s T 4 Þ
ð9Þ
where c1 ; c2 2 ð0; 1Þ and T 3 ; T 4 > 0. Clearly GðsÞ is stable. With definition (9), (8) can be rewritten as follows:
1 þ GðsÞ ¼ 0
(iii) The line jGð0Þj ¼ K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ ¼ 1 divides the c1 c2 plane into minimum-phase and non-minimumphase regions for different K values as shown in Fig. 4. (iv) Note that b ¼ 0 means either no DTs, that is, ai ¼ 0ði ¼ 1; . . . ; 4Þ, or that a1 þ a2 ¼ a3 þ a4 . Although the two cases are completely different both are identical as far as the number of NMP zeros is considered.
ð10Þ
As the structure of (10) is similar to the structure of a closed loop characteristic equation, the number of the finite NMP zeros in the case b 6 0 can be determined by the Nyquist stability theorem, [12]. The magnitude jGðjwÞj and the phase argðGðjwÞÞ of GðsÞ are:
K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ ffi jGðjwÞj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ ðw T 3 Þ2 Þ ð1 þ ðw T 4 Þ2 Þ
ð11Þ
argðGðjwÞÞ ¼ p þ b w tan1 ðw T 3 Þ tan1 ðw T 4 Þ
ð12Þ
The magnitude (11) is a monotonic strictly decreasing function of w and its maximum value is:
jGð0Þj ¼ jGðjwÞkw¼0 ¼ K ð1 c1 Þ ð1 c2 Þ=ðc1 c2 Þ
ð13Þ
The typical polar plot of GðsÞ in (9) and a circle with radius jGð0Þj are depicted in Fig. 3. The next lemma follows directly from the Nyquist stability theorem. Lemma 1. Let jGð0Þj be defined in (13). Let b in (7) be non-positive. Then: (i) if jGð0Þj < 1, the QTPwDT has no NMP zeros independent of the DTs; (ii) if jGð0Þj > 1, the QTPwDT has at least one NMP zero (one NMP zero only when b ¼ 0) and the number of the NMP zeros increases when b decreases (as specified in Lemma 2). Remark 1
As stated in Lemma 1(ii) the number of the NMP zeros increases as b decreases. The next lemma clarifies that dependence. For jGð0Þj > 1 let bcrðkÞ ðk ¼ 1; 2; . . .Þ and wcr be defined as follows:
bcrðkÞ ¼ 2kp þ tan1 ðwcr T 3 Þ þ tan1 ðwcr T 4 Þ =wcr ;
wcr ¼
k ¼ 1; 2; . . .
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r 2 u u T 23 þ T 24 þ T 23 T 24 þ 4T 23 T 24 jGð0Þj2 t 2T 23 T 24
ð14Þ
ð15Þ
Note that (15) is the positive solution w ¼ wcr of jGðjwÞj ¼ 1, with jGðjwÞj defined in (11) and (14) is the solution of argðGðjwÞÞjw¼wcr ¼ ð2k þ 1Þp for k ¼ 1; 2; . . ., where argðGðjwÞÞ is defined in (12). Noting that jGðjwÞj and argðGðjwÞÞ are both monotonically decreasing functions of w and recalling the typical polar plot in Fig. 3, it follows that: (a) for 0 P b > bcrð1Þ the Nyquist plot of GðsÞ in (9) has one clockwise encirclement of 1 þ j0; (b) for b ¼ bcrðkÞ the Nyquist plot of GðsÞ traverses the 1 þ j0 with phase ð2k þ 1Þp radians at frequency wcr ; (c) for bcrðkÞ > b > bcrðkþ1Þ the Nyquist plot of GðsÞ has 2k þ 1 clockwise encirclement of 1 þ j0. With the above we have the following lemma. Lemma 2. Let jGð0Þj > 1 and b 6 0. Then:
(i) It is quite surprising that despite of the presence of DTs between inputs and outputs what determines whether there are or not finite NMP zeros are just the c1 ; c2 like in the QTP (without DTs)! (ii) Note that for K ¼ 1 the conditions jGð0Þj < 1 and jGð0Þj > 1 reduce to 1 < c1 þ c2 < 2 and 0 < c1 þ c2 < 1, respectively, which are the conditions for minimum-phase and non-minimum-phase of the QTP [1].
(i) for 0 P b > bcrð1Þ the QTPwDT has one NMP real zero; (ii) for bcrðkÞ > b > bcrðkþ1Þ the QTPwDT has k pairs of NMP complex conjugate zeros and one NMP real zero; (iii) for b ¼ bcrðkÞ ðk ¼ 1; 2; . . .Þ one of the k pairs is located on the imaginary axis at jwcr . Lemmas 1 and 2 are concerned with the number of finite NMP zeros. We next consider the locations of those zeros. When b ¼ 0 and jGð0Þj > 1, (8) has two real roots one of which is located in the right-half plane exactly as in the original QTP. Let us denote this NMP real zero by a0 . According to Lemma 2 there always exists one NMP real zero for b 6 0 and jGð0Þj > 1. We denote this real zero by a. The next lemma specifies the location of the NMP zeros. Lemma 3. Let jGð0Þj > 1 and b 6 0. Then: (i) the NMP real zero, a, decreases from a0 as b decreases. When b ! 1 then a ! 0þ ; (ii) for bcrðkÞ > b > bcrðkþ1Þ the k pairs of the complex conjugate NMP zeros, fðzi ; zi Þ; i ¼ 1; . . . ; kg, lie in the following strip 0 < Refzi g < aði ¼ 1; . . . ; kÞ. The proofs of Lemmas 2 and 3 are presented in Appendix.
Fig. 3. Typical polar plot of GðsÞ.
Remark 2. Lemmas 2 and 3 have a clear physical interpretation. It is well known that control deteriorates with NMP zeros and more so as these zeros get closer to the imaginary axis. jGð0Þj > 1 means that larger portions of the flows go to the upper tanks rendering the control of the outputs (the heights of the lower tanks) more dif-
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
Fig. 4. Minimum-phase and non-minimum-phase regions in c1 c2 plane for different K in the case b 6 0.
ficult. Control becomes even harder when the input delays to the upper tanks increase which is the case when b decreases. In this circumstance, according to Lemma 3, more and more NMP zeros lie closer to the imaginary axis as the strip in which they are situated moves towards the imaginary axis.
We recall that when b ¼ 0 and jGð0Þj > 1, (8) has two real roots one of which is a0 (NMP zero) as defined just before Lemma 3. For 0 < b < T 3 þ T 4 and jGð0Þj > 1 let aextr and bextr be the real positive solutions of the following two equations:
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < a ¼ ðbT 3 2T 3 T 4 þbT 4 Þ2 4bT 3 T 4 ðbT 3 T 4 Þ bT 3 2T 3 T 4 þbT 4 2bT 3 T 4
2bT 3 T 4
3.1.2. The case b > 0 Unlike the case b 6 0, in this case it turns out that there is an infinite number of NMP zeros as summarized in the following lemma.
Define
Lemma 4. Let b > 0. Then the QTPwDT has infinite number of NMP zeros with arbitrarily large positive real part.
crðrÞ ¼ 2rp þ tan1 ðwcr T 3 Þ þ tan1 ðwcr T 4 Þ =wcr ; b
D
:
eab ð1þaT 3 Þð1þaT 4 Þ
1 ¼ jGð0Þj
r ¼ 0; 1; . . .
ð16Þ
ð17Þ
Proof of Lemma 4. Introduce the new variable z ¼ s b with b > 0. Then (8) can be rewritten as an exponential polynomial in z and the Lemma follows at once from Pontrjagin’s theorem ([13], Theorem 13.1).
where wcr was defined in (15). It can be shown that
Remark 3. Note that while for b 6 0 the existence of NMP zeros depends on jGð0Þj only, for b > 0 NMP zeros are always present independent of jGð0Þj.
The next lemma provides information on the location of the dominant NMP zeros as b varies from 0 to 1.
While the number of the NMP zeros is infinite independent of jGð0Þj the latter does affect their locations as indicated in Lemmas 5 and 6. The proofs of Lemmas 5 and 6 are given in Appendix. Lemma 5. Let jGð0Þj < 1 and b > 0. Then the dominant NMP zero is real, say b, unique and approaches the imaginary axis as b increases. The other NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > b for all b > 0. Remark 4. Lemma 5 has the following physical interpretation. jGð0Þj < 1 means that larger portions of the flows go to the lower tanks rendering the control of the outputs (the heights of the lower tanks) easier. However, the control becomes harder when the input delays to the lower tanks increase which is the case when b increases. In this circumstance, according to Lemma 5 the real NMP zero moves closer to the imaginary axis and many more zeros, located to the right of that zero, move with it.
crð0Þ 0 < bextr < b
ð18Þ
Lemma 6. Let jGð0Þj > 1 and b > 0. Then: (i) for 0 6 b 6 bextr the dominant NMP zero is real and increases from a0 when b ¼ 0 to aextr when b ¼ bextr . A second real NMP zero, say a2 , moves on the real axis from þ1 when b ¼ 0 to aextr when b ¼ bextr . The rest of the NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > a2 for all b in the range; (ii) for bextr < b < bextr þ db , where db ! 0þ the multiple real NMP zero, aextr , breaks away from the real axis as b increases and this pair of complex conjugate NMP zeros is dominant; crðrÞ þ eðr ¼ 0; 1; . . .Þ, crðrÞ e to b (iii) when b increases from b where e is a positive constant and e 2p=wcr , there is a dominant pair of complex conjugate NMP zeros which moves from the right-half to the left half complex plane crðrÞ ðr ¼ 0; 1; . . .Þ crosses the imaginary axis at and for b ¼ b jwcr .
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
Fig. 5. Dominant NMP zero locus ðb > 0Þ.
Remark 5. For the purpose of illustrating Lemma 6 we show in Fig. 5 a typical NMP zero locus as function of bð0 < b < 1Þ. Of the infinite branches that exist only four dominant branches are shown. Two branches start on the real axis, one (solid) at a0 and moves to the right and the second one (dashed) at þ1 and moves to the left as b increases. For b ¼ bextr the two branches coincide at aextr and when b increases further the two branches leave the real axis and split to two complex conjugate ones which approach the crð0Þ at jw . The 3rd branch imaginary axis and cross it when b ¼ b cr (dotted) and its counterpart the 4th branch (not shown) originate with their real parts at þ1 and move to the left. However, for all crð0Þ Þ their real part is always larger than the corresponding b 2 ð0; b real parts of the first two branches. That is, branches 1&2 contain the dominant zeros until they cross the imaginary axis when crð0Þ . Then the 3rd & the 4th branches become dominant. These b¼b branches approach the imaginary axis and cross it at jwcr when crð1Þ ; b crð2Þ Þ another pair of crð1Þ . When b increases further in ðb b¼b branches moves towards the imaginary axis and crosses it at crð2Þ and so forth. jwcr for b ¼ b Remark 6. It can be easily deduced from the previous remark that the pair of complex conjugate NMP zeros corresponding to b in crð0Þ is dominant and approaches the imaginary axis bextr < b < b crð0Þ . as b approaches b
Define for the transfer matrix PðsÞ in (3) the output direction of a NMP real zero z as a vector w 2 R2 of unit length such as wT PðzÞ ¼ 0:
w1 w2
T
2 c1 T 1 k1 6 4
A1
ð1c2 ÞT 1 k3 za 3 e A1
eza1
1þzT 1 ð1c1 ÞT 2 k4 za 4 e A2
3
ð1þzT 3 Þð1þzT 1 Þ 7 c2 T 2 k2 za 2 e
ð1þzT 4 Þð1þzT 2 Þ
A2
5¼
T 0 0
ð19Þ
1þzT 2
It follows from (19) that w1 ; w2 –0, so the zero z is never associated with only one output. Solving (19) for c2 yields:
w1 ð1 c1 Þ ezða1 a4 Þ ð1 þ z T 1 ÞT 2 k4 A1 ¼ w2 c1 ð1 þ z T 4 Þð1 þ z T 2 ÞT 1 k1 A2 With no delays or when a1 ¼ a4 the above reduces to that derived in [1] for the QTP. Thus, while in the latter case zero direction is determined solely by the value of c1 , in the QTPwDT it also depends on the difference a1 a4 as indicated next: if c1 1, then z is mostly associated with the first output ðw1 w2 Þ; if c1 1, then z is mostly associated with the second output ðw1 w2 Þ; if a1 a4 , then z is mostly associated with the first output; if a1 a4 , then z is mostly associated with the second output.
3.2. NMP real zeros and their output directions
3.3. DIC of the QTPwDT
Each zero of a multivariable system has both location and direction. There are output and input zero directions. From a practical point of view, the output zero direction is usually of more interest than the input one, because output zero direction gives information about which output may be difficult to control, [12]. In this section the output directions of the NMP real zeros for QTPwDT transfer matrix are found.
A desirable property of a decentralized control system is that the closed loop system should remain stable as subsystem controllers are brought in and out of service. This property is called decentralized integral controllability (DIC), [14]. The relative gain array (RGA) [12] is a steady-state measure of interactions for decentralized control. The RGA is defined as D RGA ¼ Pð0Þ PT ð0Þ, where the asterisk denotes the element-by-
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
element matrix multiplication (Schur product). Although the RGA is associated with steady-state it is a useful tool in the process industry to decide on input–output pairing for decentralized controllers, [12,14]. For a 2 2 transfer matrix to be DIC it is a necessary and sufficient that the (1,1) element of the RGA be positive. The QTPwDT transfer matrix (3) has two inputs and two outputs, consequently we can determine the necessary and sufficient condition for QTPwDT to be DIC. The RGA of the QTPwDT transfer matrix is given by
RGA ¼
k
1k 1 D k ¼ 1 jGð0Þj
1k k
1 ðw ð^r eÞ ejw2 jÞ ezðts1 aÞ 1 1
^21 the interaction, e the settling level, t s1 where yu1 is the undershoot, y D ¼ . the settling time, a maxfminfa1 ; a3 g; minfa2 ; a4 gg and ts1 > a ¼ 0. Remark 7. The bound in (23) reduces to the one in [15] for a That is when in each row there is at least one element with no delay.
ð20Þ ^21 P w1 yu1 þ jw2 jy
where jGð0Þj was defined in (13), and the necessary and sufficient condition for QTPwDT to be DIC is that k in (20) be positive. Therefore the DIC property of the QTPwDT depends on jGð0Þj only, as stated in the next lemma. Lemma 7. Let jGð0Þj be defined in (13). Then the QTPwDT is DIC iff jGð0Þj < 1. Since the RGA is a steady-state property DTs do not affect it. Then assuming K ¼ 1 and for the same c1 and c2 , the RGA for both the QTP in [1] and the QTPwDT are identical. Indeed k in (20) (assuming K ¼ 1) reduces to that derived in [1] for the QTP. The RGA analysis, [1,5,7], suggests that for jGð0Þj > 1 another input–output pairing for decentralized integral control should be chosen. Let
e ¼ P 12 ðsÞ P11 ðsÞ PðsÞ P22 ðsÞ P21 ðsÞ
w1^r ezðts1 aÞ 1
ð24Þ
Hence, like the case with no DTs, the above bound suggests that the undershoot is expected to be large for RHP zero close to the origin and/or for small interactions. Hence the trade off between undershoot in the setpoint response in the first loop and the interaction to the second loop and vice versa. Also, if one has two similar systems, both having the same zero and w1 , one system without DTs and the other one with, then both will have the same bound (the right hand side in (24)) if the settling time of the system with DTs . This is another way to look on the performance deteis longer by a rioration due to the DTs.
4. Experimental study
^ 1 and u ^2 be the QTPwDT transfer matrix PðsÞ in (3) with the inputs u e permuted. With this pairing the (1,1) element of the RGA for PðsÞ is given by 1 k and is positive. Therefore the new pairing, ðv 2 ; h1 Þ and ðv 1 ; h2 Þ, guarantees that the QTPwDT is DIC. 3.4. Effect of RHP zeros on closed – loop performance RHP zeros are known to limit the achievable performance of linear multivariable systems [12,19]. Since DTs may introduce, as shown above, many RHP zeros, their effect on the achievable performance of the QTPwDT with feedback control is of central interest. Johansson in [15] derived time-domain lower bounds on interactions in linear MIMO systems due to RHP zeros. Those bounds are extended in the following to include processes with DTs and will be presented for 2 2 systems, though their extension to any mxm system with DTs is quite straightforward. Hence the notations in [15, Definition 2] will be used throughout this section. Let the QTPwDT be controlled by a feedback controller such that the closed loop is stable. It is well known [12] that the transmission zeros of the closed loop are the transmission zeros of the open loop, particularly so the RHP transmission zeros of the QTPwDT and each such RHP zero, say z, satisfies the following
In this section the laboratory QTPwDT will be described briefly and simulations as well as experimental results, demonstrating the performance degradation in the control of the QTPwDT with NMP zeros and illustrating some of the results of this paper will be presented. The laboratory setup illustrated schematically in Fig. 2, is shown in Fig. 6. The parameters of the setup are given in Table 1. The acceleration of gravity is g ¼ 981 ½cm=s2 . Four similar gear pumps are employed, each with a capacity of 2:3½‘= min and voltage range of 3–12 [V]. The height of each tank is 23 ½cm . With those parameters K and b, defined in (7), are K ¼ 1:004; b ¼ 5 ½s . As b (see (7)) is negative in the setup considered here, NMP cases may be obtained according to Lemma 1 if jGð0Þj > 1 whereas when jGð0Þj < 1 the process is minimum-phase (MP). Two cases with different c1 and c2 are considered, one MP and the second one NMP. The c1 and c2 values and the corresponding linearized models of the QTPwDT according to (3) are as follows:
ð21Þ
where TðsÞ is the complementary sensitivity given by
yðsÞ ¼ TðsÞrðsÞ ¼ PðsÞCðsÞðI þ PðsÞCðsÞÞ1 rðsÞ
ð23Þ
Remark 8. For a small settling level e the bound in (23) reduces to
;
wT PðzÞ ¼ wT TðzÞ ¼ 0
^21 P w1 yu1 þ jw2 jy
ð22Þ
with CðsÞ the transfer matrix of the controller and rðsÞ and yðsÞ the Laplace transforms of the setpoint r and output y respectively. The next lemma extends Theorem 3 in [15] to 2 2 cases with DTs : Lemma 8. Given the stable closed loop system in (22) with zero initial T conditions at t ¼ 0 and let rðtÞ ¼ ½ ^r 0 for t > 0. Assume that PðsÞ has a real RHP zero z > 0 with zero direction w 2 R2 and w1 > 0. Then the setpoint response satisfies
Fig. 6. The experimental setup for the QTPwDT.
D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
25
Table 1 Parameter values of the laboratory QTPwDT. i
Ai ðcm2 Þ
ai ðcm2 Þ
ki ðcm3 =ðs VÞÞ
ki2 ðcm3 =sÞ
ai ðsÞ
1 2 3 4
12 20 12 20
0.26 0.26 0.21 0.21
3.808 3.831 3.904 3.752
2.712 2.006 3.069 3.499
5 6 7 9
MP case ðc1 ¼ 0:65; c2 ¼ 0:60Þ
2 P ðsÞ ¼ 4
1:585es5 1þ7:685s
es7 ð1þ4:778sÞð1þ7:685sÞ
0:778es9 ð1þ3:705sÞð1þ11:85sÞ
1:362es6 1þ11:85s
3 5
ð25Þ
NMP case ðc1 ¼ 0:40; c2 ¼ 0:35Þ
2 Pþ ðsÞ ¼ 4
0:834es5 1þ6:57s
1:39es7 ð1þ10:231sÞð1þ6:57sÞ
1:271es9 ð1þ14:05sÞð1þ11:29sÞ
0:757es6 1þ11:29s
3 5
ð26Þ
The corresponding (1,1) elements in the RGA in (20) are k ¼ 1:56 and k ¼ 0:56 for the MP and NMP cases respectively. Consequently accordingly to Lemma 7 the QTPwDT is DIC in the MP case and is not DIC in the NMP case. To make the NMP case a DIC one we follow the pairing change discussed in Section 3.3 which leads to:
2 e þ ðsÞ ¼ 4 P
1:39es7 ð1þ10:231sÞð1þ6:57sÞ
0:834es5 1þ6:57s
0:757es6 1þ11:29s
1:271es9 ð1þ14:05sÞð1þ11:29sÞ
3 5
ð27Þ
Two different controllers were designed for each of the two cases. A decentralized PI controller tuned via the BLT method described in [16], and a decentralized dead-time compensator (DTC) with the structure suggested by Jerome and Ray [17]. The primary controller of the DTC was tuned by a novel method, called EVS (eigenvalue sensitivity), see [18] for details. For completeness, the EVS method is briefly described. The method utilizes characteristic loci, [12], and consists of two stages. In the first stage interactions are ignored and the components of the decentralized primary controller are tuned in a single-input single-output fashion using predetermined gain and DT margins. In the second stage the primary controller’s gains are reduced until the minimum distance to 1 þ j0 of the character-
Fig. 7. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 1 in the minimum phase QTPwDT (simulation).
Fig. 8. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 1 for the minimum phase QTPwDT (experiment).
istic loci of the sensitivity matrix of the closed loop with interactions is equal or larger to that obtained in the first stage. In Figs. 7 and 8 responses of the QTPwDT, corresponding to P ðsÞ, to a unit change in the first setpoint with the feedback controllers described above are depicted. Fig. 7 shows simulations with the non linear process while experimental results are given e þ ðsÞ, the in Fig 8. Responses of the QTPwDT, corresponding to P NMP case, to a unit change in the second setpoint controlled by the decentralized PI and by the DTC tuned via the EVS method are depicted in Fig. 9 (simulation) and Fig. 10 (experimental results). The agreement between the simulations and the experiments is remarkable. One can see that faster and tighter control is obtained by the DTC in both cases. The deterioration in the control quality due to RHP zeros is evident: the settling time in the NMP case is many times larger than in the MP case for both the decentralized DTC and the decentralized PI. Applying Lemma 2 to the NMP case indicates that there is a single NMP real zero as b ¼ 5 ½s > bcrð1Þ ¼ 40:03 ½s . The location of that zero is z ¼ 0:042 and the poles of P þ ðsÞ are f0:1522;
Fig. 9. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 2 for the non-minimum-phase QTPwDT (simulation).
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
the DTs. Time-domain bounds on control performance imposed by the NMP zeros indicating the deterioration in performance due to the NMP zeros are derived. Experimental results of the laboratory QTPwDT demonstrating both the control difficulties due to the RHP zeros as well as some of the results were presented.
Acknowledgement The support of the Israel Electric Company for this work is gratefully acknowledged. Appendix A The purpose of the appendix is to present proofs for Lemmas 2, 3, 5, 6 and 8. Detailed proofs can be found in [5]. A.1. Proof of Lemma 2 Fig. 10. Responses of the water levels ðh1 ; h2 Þ and control signals ðu1 ; u2 Þ to step in the reference signal to Tank 2 in the non-minimum-phase QTPwDT (experiment).
0:0886; 0:0977; 0:0712g. Consequently this zero is dominating. Hence the significant deterioration in the performance (see Remark 2). The zero z ¼ 0:042 of P þ ðsÞ has zero direction wT ¼ ½ w1 w2 ¼ ½ 0:575 0:818 . With settling level e 0 Lemma 8 for a unit step in r 2 ðr ¼ ½ r1 r2 T ¼ ½ 0 1 T Þ gives
w2 yu2
^12 P þ jw1 jy
w2 ezðts2 a Þ 1
¼ 6 ½s to which reduces for t s2 ¼ 66 ½s and a
^12 P 0:072: 0:818yu2 þ 0:575y As seen in Fig. 10 undershoot is very small, yu2 0, hence we get a lower bound on the interaction
(i) Follows directly from (a). (ii) According to (c) there are 2k þ 1 zeros. Since jGðsÞjs¼r j for real r is a monotonically decreasing function there is only one real solution of jGðsÞjs¼r j ¼ 1. Hence the rest are k pairs of complex conjugate zeros. (iii) Follows from (b). A.2. Proof of Lemma 3 (i) It follows from Lemma 2 that for all b 6 0 there is one real NMP zero which is the solution of jGðsÞjs¼r j ¼ 1. Clearly this zero, denoted a, decreases with b and the Lemma follows. (ii) A Nyquist path shifted to the right by r P 0 as shown in Fig. A.1, is used. For r > a; jGðsÞjs¼r j < 1. Hence no encirclements of 1 þ j0 by the locus of GðsÞ corresponding to the path in Fig. A.1 are obtained. Consequently none of the k pairs of complex conjugate NMP zeros is enclosed by the modified path in Fig. A.1 and Lemma follows.
^12 P 0:125 ½cm y which agrees well with both the simulation and experimental results. 5. Conclusion A new laboratory quadruple-tank process with multivariable DTs, called here the QTPwDT, was constructed, described and used to study how the DTs affect the properties and the control of this interesting process. It is well known that DTs deteriorate the achievable control performance partly because they may give rise to NMP zeros. The conditions for the appearance of NMP zeros, their number, location and the behavior of the dominant ones are analyzed in the paper. Interestingly enough, it depends on b, a particular combination of the DTs, and on jGð0Þj, the magnitude of the steady-state gain of an artificial transfer matrix defined in the paper. Not in all circumstances the existence of DTs brings about NMP zeros. Such is the case when b is negative and jGð0Þj < 1. However when jGð0Þj > 1 a finite number of NMP zeros appear and they increase in number and become more dominant as b decreases. It was further shown that completely different situation occurs when b is positive. In those cases infinite number of NMP zeros exist no matter what value jGð0Þj assumes. However the latter do influence the location, dominance and behavior of the zeros as specified in the paper. Physical interpretations for the main results are given and it is also shown that the DIC property of the QTPwDT is independent of
A.3. Proofs of Lemmas 5 and 6 Preliminaries – Since b > 0; GðsÞ in (9) is non-causal. Hence in order to examine the properties and locations of the infinite number of NMP zeros in this case we introduce the modified Nyquist path which encloses the left half plane possibly penetrating into the right-half plane up to a finite r P 0 as depicted in Fig. A.2. For any r P 0 and any b > 0, the magnitude jGðjw þ rÞj and the phase argðGðjw þ rÞÞ of GðsÞ as w varies in 0 6 w 6 1 are: jGð0Þjerb ð1þrT Þð1þrT Þ
3 4 jGðjwþ rÞj ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 wT 3 4 1þ 1þwT 1þ 1þrT 3 rT 4
argðGðjwþ rÞÞ ¼ p bwþtan1
ðA:1Þ
wT 3 wT 4 þtan1 1þ r T 3 1þ r T 4 ðA:2Þ
respectively with jGð0Þj given in (13). The magnitude (A.1) is a monotonically strictly decreasing function of w for r P 0; b > 0 and therefore its maximum value is:
D GðrÞj ¼ jGðjw þ rÞjw¼0 ¼
jGð0Þj erb ð1 þ r T 3 Þ ð1 þ r T 4 Þ
ðA:3Þ
Clearly GðsÞ maps the infinite half circle in Fig. A.2 to the origin and recall that GðsÞ in (9) has two negative poles.
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
conclude that there exists r ¼ b þ e, where e ! 0þ , for which the locus of GðsÞ corresponding to the path in Fig. A.2 has one clockwise encirclement of the 1 þ j0. Hence, according to the Nyquist stability theorem, the QTPwDT has two zeros within the modified Nyquist path in Fig. A.2 for 0 6 r < b and three zeros for b < r < b þ e and it follows that b is the dominant NMP real zero. Moreover b is the unique real zero because r ¼ b is the unique positive solution of jGðrÞj ¼ 1. Since according to Lemma 4 the QTPwDT has infinite number of NMP zeros with arbitrarily large positive real part we conclude that the rest of the NMP zeros, say zi ði ¼ 1; 2; . . .Þ, satisfy Reðzi Þ > b for all b > 0. Finally as jGð0Þj < 1 and b > 0 it is clear from both the derivative of (A.3) with respect to r and the proof of the existence and the uniqueness of b above that as b increases b moves from þ1 (for b ! 0þ Þ to 0þ (for b ! þ1Þ. This completes the proof.
Fig. A.1. Modified Nyquist path in [s] plane ðb 6 0Þ.
Fig. A.2. Modified Nyquist path in [s] plane ðb > 0Þ.
A.4. Proof of Lemma 5 If jGð0Þj < 1 the locus of GðsÞ corresponding to the path in Fig. A.2 for r ¼ 0 has no encirclements of the 1 þ j0 and the Nyquist stability theorem implies that (10) has two zeros in the left half complex plane. Next we apply the modified Nyquist path in Fig. A.2 with increasing values of r. This results in encirclements of 1 þ j0 if and only if jGðrÞj > 1. For jGð0Þj < 1; b > 0 and T 3 ; T 4 > 0, denote the positive solution of jGðrÞj ¼ 1 by b. This solution always exists and is unique as discussed next. For b > ðT 3 þ T 4 Þ the existence and the uniqueness of b follow from the fact that the derivative of (A.3) with respect to r is positive for all r P 0 and that jGð0Þj < 1. For b ¼ T 3 þ T 4 the derivative above is zero at r ¼ 0 and positive for all r > 0. Hence for b P ðT 3 þ T 4 Þ (A.3) is monotonically increasing function for r > 0. For 0 < b < T 3 þ T 4 , the derivative of (A.3) with respect to r is negative for 0 6 r < a, zero for r ¼ a and positive for r > a where a was defined in (16). Consequently (A.3) is less than one for 0 6 r < a and monotonically increases in r > a. Therefore the existence and the uniqueness of b for b > 0 follow. Thus, the locus of GðsÞ corresponding to the path in Fig. A.2 has no encirclements of the 1 þ j0 for 0 6 r < b and at least one clockwise encirclement for r > b as according to Lemma 4 the number of the zeros of (10) enclosed by the closed path in Fig. A.2 can not decrease as r increases. Since the magnitude (A.1) is a monotonically strictly decreasing function of w for any r P 0 we
A.5. Proof of Lemma 6 crðrÞ ðr ¼ 0; 1; . . .Þ in (17) are all the b valProof of Lemma 6(iii). b ues for which the polar plot of GðsÞ traverses the 1 þ j0 point at crðrÞ there exists a single pair of complex w ¼ wcr . Hence for b ¼ b zeros, jwcr , on the imaginary axis. Single due to the fact that the magnitude in (A.1) is monotonically decreasing. Next we draw the locus of GðsÞ corresponding to the closed path in Fig. A.2 with r ¼ 0. The number of encirclements of 1 þ j0 for bcrðrÞ þ e, where e 2p=wcr , is larger by two relative to that for bcrðrÞ e which implies that the above pair moves from the right-half to the left half crðrÞ . crðrÞ e; b plane. Clearly that pair is dominant for b 2 ðb Proof of Lemma 6(i). Whether or not the locus of GðsÞ corresponding to the closed path in Fig. A.2 encircles the 1 þ j0 depends on the behavior of the positive solutions of jGðrÞj ¼ 1 for any b > 0. From the properties of jGðrÞj in (A.3) discussed in the previous proof it is apparent that in this case, where jGð0Þj > 1; jGðrÞj ¼ 1 may have, depending on b, either zero or two positive solutions. To find the value of b for which two identical solutions exist we solve minr;b jGðrÞj ¼ 1 which reduces to the following two equations: djGðrÞj=dr ¼ 0, and jGðrÞj ¼ 1, and (16) is obtained, the positive solutions of which were denoted aextr and bextr . Consequently for b ¼ bextr there are two identical solutions (r ¼ aextr Þ, for b < bextr two different positive solutions and no solutions exist for b > bextr . Denote by a1 and a2 ð0 < a1 < a2 Þ the two solutions for b < bextr . It is apparent from the inspection of jGðrÞj ¼ 1, where jGðrÞj defined in (A.3), that as b increases from 0 to bextr a1 increases from r ¼ a0 to r ¼ aextr while a2 decreases from r ¼ þ1 to r ¼ aextr . a1 and a2 are the NMP real zeros of (10) since they satisfy jGðrÞj ¼ 1. jGðrÞj > 1 for 0 < r < a1 and r > a2 , while jGðrÞj < 1 for a1 < r < a2 . Therefore when r ¼ a1 the number of the clockwise encirclement of the 1 þ j0 by the locus of GðsÞ corresponding to the path in Fig. A.2 increases by one and consequently a1 is dominant. Proof of Lemma 6 (ii). From the previous section it is clear that one has jGðrÞk r ¼ a ¼ 1. Hence a small positive perturbation in ext
b ¼ bext w¼0 b, say b1 ¼ bextr þ db , where db ! 0þ , leads to an increase in the amplitude, that is jGðrÞk r ¼ a > 1. Consequently and due to ext
b ¼ b1 w¼0 the fact that the amplitude in (A.1) is monotonically decreasing function of w for fixed r and b, it is clear that there exists a frequency w0 > 0 for which jGðrÞk r ¼ a ¼ 1 which implies a pair ext
b ¼ b1 w ¼ w0 of zeros given approximately by aext j w0 . Hence the break away from the real axis of the zero locus at r ¼ aextr . This results in an
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D. Shneiderman, Z.J. Palmor / Journal of Process Control 20 (2010) 18–28
increase by two in the number of clockwise encirclements of 1 þ j0 by the locus of GðsÞ corresponding to the closed path in Fig. A.2, as r is increased from slightly below to slightly above aextr . Hence the dominance of the pair of zeros follows. This completes the proof. A.6. Proof of Lemma 8 Using (21) and (22) and the Laplace transform of the output y gives
wT yðs ¼ zÞ ¼
2 X
wk
Z
1
ezt yk ðtÞ dt ¼ 0;
0
k¼1
which is equivalent to
Z
t s1
ezt
2 X
0
wk yk ðtÞ dt ¼
Z
k¼1
1
ezt
2 X
t s1
wk yk ðtÞ dt
ðA:4Þ
k¼1
Since the response yk can not start before the smallest DT in the kth row has elapsed, the left-hand side of (A.4) can be written as
Z
t s1
ezt
0
¼
2 X
wk yk ðtÞ dt
k¼1
Z
fa1 ;a3 gt s1
min
ezt w1 y1 ðtÞ dt
Z
fa2 ;a4 gt s1
min
ezt w2 y2 ðtÞ dt:
leading to:
Z
fa1 ;a3 gts1
ezt w1 y1 ðtÞ dt
min
¼
Z
fa2 ;a4 gt s1
ezt w2 y2 ðtÞ dt
min
Z
1
ezt
t s1
2 X
wk yk ðtÞ dt
ðA:5Þ
k¼1
The left-hand side of (A.5) satisfies
Z
fa1 ;a3 gts1
min
Z
6 a
¼
t s1
ezt w1 y1 ðtÞ dt
Z
fa2 ;a4 gt s1
min
ezt w2 y2 ðtÞ dt
^21 ezt dt w1 yu1 þ jw2 j y
eza ezts1 ^21 : w1 yu1 þ jw2 j y z
Substituting the above into (A.5) and following the same steps as in the proof of [15, Theorem 3] leads to (23) at once.
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