8684
Ind. Eng. Chem. Res. 2008, 47, 8684–8692
PROCESS DESIGN AND CONTROL Synthesis of PID Tuning Rule Using the Desired Closed-Loop Response Rames C. Panda* Department of Chemical Engineering, CLRI (CSIR), Adyar, Chennai - 600020, India
Proportional-integral-derivative (PID) controllers are widely used in industry, because of their simple structure and ease in implementation. A tuning method based on the IMC-PID rule is synthesized for stable singleinput-single-output systems. The controller transfer function is expanded, in the vicinity of zero in the complex plane, using a Laurent series, to derive analytical expressions for controller parameters. A tuning parameter is chosen to determine the stability of the response/closed-loop system. PID parameters are obtained for different types of processes, and related robustness issues are also discussed. 1. Introduction Classical proportional-integral-derivative (PID) controllers are widely used in process industries, despite the continued advancement in control technology. According to Astrom and Hagglund,1 more than 95% of the control loops are of the PID type. These types of controllers are popular because of their ease in operation, robust behavior, and easy maintenance. Generally, PID controllers have four different structures. The ideal PID with the following structure has limited use, because of problems in implementation, and it is used mainly for academic interests:
(
) KC 1 + GPID0 C
)
1 + τDs τIs
(1)
Primarily, the series structure (eq 2) and the parallel structure (eq 3) are used in process industries:
(
) KC 1 + GPID1 C
(
GPID2 ) KC 1 + C
)(
τDs + 1 1 τIs RτDs + 1
) )
(2)
τDs 1 + τIs RτDs + 1
(3)
Other than these three structures, there exists another form that is often used in IMC designs that have a filter in series with the ideal structure:
(
GPID4 ) KC 1 + C
)(
1 1 + τDs τIs τfs + 1
)
(4)
The accuracy and performance of these controllers are greatly dependent on the method of tuning controller parameters, namely, KC, τI, and τD. A great deal of work that has provided tuning laws based on frequency and time domain approaches have been reported in the literature. Tuning formulas that have been reported by Ziegler and Nichols,2 Cohen and Coon,3 Astrom and Hagglund,1 and Tyreus and Luyben4 were based on process reaction curves. Tuning rules based on error minimization criteria were proposed by Rivera et al.,5 Zhuang and Artherton,6 and many other researchers (for example, Dwyer7). Smith,8 Chien and Fruehauf,9 Chen and Seborg,10 Morari and Zafiriou,11 and Skogestad12 used * To whom correspondence should be addressed. Tel.: +91(44)2491 6706. Fax: +91(44)2491 1589. E-mail address:
[email protected].
a direct synthesis approach to obtain controller constants in IMC-PID form. Huang et al.,13 Lee et al.,14 and Panda et al.15 have recently discussed tuning rules based on the desired closedloop response. In the latter two references, an expression for the true controller (GCTrue), which contains a time delay term (e-DPs) is simplified using a power series to approximate its value. This true controller is finally rearranged in a suitable manner such that an ideal PID controller is obtained. The main advantage of this type of tuning technique lies in the fact that the desired closed-loop response contains a tuning parameter that is often called the closed-loop time constant, λ (which is generally inherited in the analytical expressions of controller coefficients); this term can be tuned by the user to select a tradeoff between better performance and robustness. Thus, the PID parameters are simplified to a single tuneable parameter λ, which is related to the proportional gain, mainly to adjust the speed of response, giving advantage in accommodating actuator saturation. With processes that have zero time delay, the PID formula results in a pole zero cancellation to a proportional-integral (PI) structure. In addition to the aforementioned conventional tuning, PID controllers can also be autotuned,16 using the system’s ultimate properties in the frequency domain. The PID controller transfer function, GC(s), is a function of s. Because the closed-loop system must be stable, a Taylor series is used to expand GC(s) about s ) 0. When the first three terms of an expanded Taylor series of a function are considered to be important (for example, as in the case of GC(s), where the first three terms represent the proportional (P), integral (I), and derivative (D) actions), the Taylor series of that function around zero is called a Maclaurin series. However, for functions that are less well-behaved (undefined) or nondifferentiable or singular at a point, a Laurent series can be used. A Laurent series (of a function) is a two-sided infinite power series (including positive and negative terms) whose terms can be used to approximate some functions. A close look into the literature for IMC design (see Figure 1) with a time delay process shows that Rivera et al.5 used a Pade approximation of the exponential term that appears in the denominator of GCTrue), whereas, in 2002, Chen and Seborg10 used a Taylor series and, somewhat earlier, Lee and co-workers14,17 obtained a superior performance using a Maclaurin series on GCTrue (by expanding it around the origin of the s-plane) and presented analytical expressions for the PID controller param-
10.1021/ie800258c CCC: $40.75 2008 American Chemical Society Published on Web 10/15/2008
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8685
Figure 1. Schematic block diagrams of (a) PID and (b) IMC control schemes.
eters (abbreviated here as IMC-Mac). They derived PID parameters for stable and unstable low-order models.17 However, for first-order-plus-dead-time (FOPDT) systems with high DP/τ processes, their tuning rule gives negative values of τD. Chien and Fruehauf9 suggested that many chemical processes can be modeled as integrator-plus-dead-time (IPDT) systems for suitable feedback control design. Hence, looking at IPDT types of processes, the method reported by Lee et al.17 faced some problems in deriving analytical expressions for the controller parameters from GCTrue, because the s term in the denominator disappears at s ) 0. Therefore, they approximated IPDT processes to a FODUP model, possessing an unstable pole near zero, and they limited their discussions to unstable processes. Rice and Cooper18 presented an extensive comparative study on the closed-loop performance of IPDT processes using PID controllers in which parameters were obtained using different PID synthesis procedures, and they showed that IMC design can be equally as good as that of other methods. Based on the aforementioned facts, it can be stated that the motivation and objective behind this article is to determine a correct approximation function that can give a PID controller equation almost the same as GCTrue (using the desired closed-loop response technique or control signature). This approach illustrates IMC PID parameter synthesis using a Laurent series. In fact, the Laurent series has an advantage over other approximating power series because it generalizes a Maclaurin series that, again, generalizes a Taylor series. Moreover, it can be specifically applied to solving singularity problems, i.e, to synthesize PID parameters for the IPDT type of systems. For the latter case (for the IPDT system), the controller was synthesized by Lee et al.17 using a Maclaurin series, thereby avoiding singularity problems by approximating the IPDT process by adding an extra pole nearer to the origin (whose dynamics can be neglected) and the resulting transfer function was used to derive the PID parameters. In the present method, there is no need to add an extra pole, because the Laurent series is able to handle singularity problems. Because the Laplace variable s is a complex variable, GC(s) (which is a function of s) can be expanded near the vicinity of s ) 0 using the Laurent series. In a global sense, the Taylor and Maclaurin series are special cases (when positive coefficients are considered) of the Laurent series. Thus the remaining portion of this paper is organized as follows: In section 2, the mathematical preliminaries behind the development of the true controller using the Laurent series is presented. Analytical expressions of PID controller parameters are derived in section 3; this section contains PID parameters for a family of standard transfer functions. Results are discussed using different process examples in section 4. Conclusions are drawn at the end of the paper.
2. Mathematical Development A controller is designed to have the closed-loop system be stable, i.e, in the low-frequency region (controller works nearer to the ultimate frequency, ωu), and, hence, the controller function should be linearized. The ideal or true controller can be approximated by a power series in a complex or the s-plane, by expanding it near the vicinity of zero. Although it can be done via a Taylor or Maclaurin series (where terms with s0, s1, and s2 appear), we choose the Laurent series (because the controller is given by eq 1, where terms with s0, s-1, and s1 appear, the series no longer belongs to the Taylor or Maclaurin type, but becomes Laurent type, with other coefficients as zeros) because it addresses complex coefficients that are important especially to investigate the behavior of functions near singularity. Generally, this series can be used to express holomorphic functions and, when used to expand the function f(s) near s ) 0, it reduces to a Taylor series or a Maclaurin series (cases when the first three terms are important). However, the Maclaurin series generalizes the Taylor series and become subsets of the Laurent series, which can be globally used in the application of convergent power series. Singularity problems of functions (i.e., GP ) e-DPs/S, as in the case of IPDT systems) can be handled by the Laurent series and not directly by the other two types of series mentioned here. Let f(s) be an analytic function (analytic at z0 and in its neighborhood) in a unit circle |z - z0| < r, where z be any point inside a circle with z0 as its center. With s0 and z0 as poles, we consider C to be a positively oriented circle of radius r such that |z - z0| < r < C. For s ∈ C, we then get 1 1 ) s - z s - z0
[ ]
∞ (z - z0)j 1 ) j z - z0 j)0 (s - z0) 1s - s0
∑
(5)
Because z - z0/(s - s0) < 1, this series results in an uniform convergence. Thus, ∞
ds ) ∑ ∫ ∫ sf(s) -z j)0 C
C
(
)
f(s) ds (z - z0)j (s - z0)j+1
or f(z) )
)
1 2πj
ds ∫ sf(s) -z
∞
( ∫(
C
1 ∫ ∑ 2πj j)0 ∞
1 ∑ 2πj j)1
)
C
f(s) ds (z - z0)j + (s - z0)j+1
C
f(s) 1 ds (s - z0)-j+1 (z - z0)j
)
(6)
8686 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 ∞
)
∑
∞
+
GP y e-DPs ) ) R (λs + 1) (λs + 1)
bj
∑ (z - z )
aj(z - z0)j +
j
j)0
j)1
0
which can be equated to a complementary sensitivity function as
where 0 < |z - z0| < r and aj )
1 2πj
∫ (s -f(s)z )
j+1
C
ds
j ) 0, 1, 2,...)
(for
(8a)
GTrue y C GP ) R 1 + GTrueG C P
0
and bj )
1 2πj
∫ (s -f(s) z)
-j+1
C
ds
(for
j ) 1, 2,...)
(8b)
The sum of the limits of these two series can be written as
GTrue C )
∞
∑ c (z - z )
j
j
(9)
0
j)-∞
GIMC C 1 - (y ⁄ R)dGIMC C
1 2πj
∫ (s -f(s)s )
j+1
C
ds
(for
j ) 0, (1, (2,...)
0
b1 ) Res f(z) ) s ) z0
{
}
dm-1 1 lim [(z - z0)mf(z)] (m - 1)! zfz0 dzm-1
(10)
n! f (z0) ) 2πj
∫
GP(s) )
KPe-DPs τPs + 1
1 GP-
where GP- )
KP τPs + 1
and GP+ ) e-Ds Let us consider a desired closed-loop response as
φ(s) (1 + βs)f(s) ) s(1 + βs) s(1 + βs)
(17)
or, using eq 3, as GTrue C ) where β ) RτD This true controller can be expanded near the vicinity of s ) 0, using the Laurent series, as GTrue C (s) )
1 s(βs + 1)
[
∞
∑ c (s)
j
j
j)-∞
[
]
]
1 φ′′(0)s2 + ... ... + φ(0) + φ′(0)s + s(βs + 1) 2!
(18) Comparing the coefficients of the s terms of eqs 18 and 3, we get
(12)
where KP is the process gain, τP is the process time constant, and DP is the time delay. This transfer function has two parts: one invertible). The IMC controller can be expressed as GIMC C )
(16)
(11)
where f n represents the nth derivative of function f(z). Now the true controller can be derived as follows. We consider a process with transfer function
(15.1)
f(s) s
GTrue C )
)
f(s) ds C (s - z0)n+1
(λs + 1) - e-Ds
In fact, the standard form of a PID controller can be given as
and aj can be determined using n
(15)
(λs + 1) - GP+
1 ⁄ GP-
GTrue C )
Because it is difficult to determine the coefficients aj and bj (eqs 8a and 8b), some people generally prefer to express f(s) in Taylor series form, where it is expanded in a power series. As far as the objective of this paper is concerned, we are only interested in coefficients b1, a0, and a1 (or, in other words, c-1, c0, and c1). The terms in the Laurent series (eq 9) with j g 0 are called the analytic part and those terms with j > 0 are called the principal part. If the series has a finite number of terms in the principal part, it is said to have a pole of order m at z0 with c-m as the lowest nonzero coefficient. The residue of a pole is the value of the coefficient c-1. The b1 coefficient can be determined using the residue theorem, as for f(s) having an mthorder pole,
)
1 ⁄ GP-
The right-hand side of this equation can be written or rearranged to
where cj )
(14)
Thus, the true controller can be expressed as follows (considering only the invertible part of GP):
0
f(z) )
(13)
KC ) a0 ) φ′(0) ) f′(0) + βf(0)
(19a)
KC ) b1 ) φ(0) ) f(0) τI
(19b)
KCτD ) a1 )
φ′′(0) f′′(0) + 2βf′(0) ) 2! 2
(19c)
where GC(s) )
φ(s) s(βs + 1)
(20a)
φ(s) ) (βs + 1)f(s) (20b) Therefore, eqs 19a-19c give formulas for KC, τI, and τD. 3. PID Tuning Parameters The method described in the earlier section is applied to some standard transfer functions, and the comprehensive results are presented in Table 1. Most of the chemical processes can be modeled as first-order-plus-dead-time (FOPDT) processes.
Table 1. Analytical Expressions for PID Controller Parameters for Standard Transfer Functions
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8687
8688 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 2. Closed-loop response of example processes using a PID controller (dotted line represents the present tuning rule and solid line represents (a) Lee et al.14 data for example 1 (Ex-1) and example 2 (Ex-2) and (b) IMC data from Skogestad12 for example 3 (Ex-3)). Table 2. λ Selection Rule
PI PID
2βτP -
FOPDT
SOPDT
λ ) max(1.7D, 0.2τP) λ ) max(0.25D, 0.2τP)
λ ) max(0.25D, 0.2τP) λ ) max(0.25D, 0.2τP)
Applying eqs 19a-19c, we get exclusive expressions for the controller parameters for a FOPDT system: KC )
τI KP(λ + DP)
τI ) (τP + β) +
DP2 2(λ + DP)
(21)
(22)
τD )
DP2τI DP3 + 3(λ + DP) (λ + DP) +β 2τI
(23)
These obtained expressions are similar to that of Lee et al.,14 except for the excess term β (i.e., if β becomes 0, then we get back expressions for the controller parameters given by Lee et al.14). There are few advantages of having the present form of analytical expressions of controller parameters: (i) the presence of the β term increases the speed of response (Kc), (ii) the possibility of τD becoming negative can be avoided even for processes with a high DP/τP ratio (there is a possibility of τD becoming negative in the 1998 work of Lee et al.14). Similarly, for the second-order-plus-dead-time (SOPDT) systems, the controller parameters are shown in Table 1.
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8689
Figure 3. Closed-loop response ((a) setpoint change and (b) load disturbance) of a higher-order process (example 4, Ex-4) with the present controller scheme (the solid line represents data from Lee et al.,17 the dash-dotted line represents data from Skogestad,12 and the dotted line represents data obtained using the present method). Table 3. Closed-Loop Controller Parameters and Performance IMC-MACa
Present PID Tuning Kc
τI
τD
IAER
IAEL
λ
Kc
τI
τD
IAER
IAEL
λ
1.587 1.931 0.355 72
1.0713 14.785 30.63 3.61
0.065 3.171 1.80 1.139
2.119 3.848 26.26 0.858
0.687 7.656 95 0.05
0.425 1.414 12.58 0.2
1.55 1.908 0.614 40
1.046 14.608 32.56 2.86
0.042 3.028 3.28 1.19
2.12 3.94 17.64 1.186
0.6872 7.76 53 0.071
0.425 1.414 12.58 0.2
example Ex1 Ex2 Ex3 Ex4 a
Data taken from Lee et al.14
Table 4. Controller Performance and PID Parameter Values Computed by Different Methods of Tuning for the Process GP ) 0.0506 e-6s/s
c
method
KC
present tuning method IMCa SIMCb ZNc
1.3788 2.99 1.219 3.0
τI 60.25 26.4 64.8 12
λ
IAER
10.2 10.2 10.2
25.45 23.49 27.36 70.26
a Data taken from Rice and Cooper.18 b Data taken from Skogestad.12 Data taken from Ziegler and Nichols.2
It can be seen that the controller parameters are functions of the process model parameters and two other parameters (λ and β). The problem is now to provide proper guide rules for selecting λ and β. 3.1. Selection of λ. We consider the following selection rule for the calculation of the equivalent desired closed-loop time constant, λ, as suggested by Luyben.19 For FOPDT Systems :
λ ) max(0.2τP, 1.7DP)
For SOPDT Systems :
λ ) max(0.2τP, 0.25DP)
For IPDT Systems :
λ ) DP√10 (from Rice and Cooper18)
3.2. Selection of β. Naturally, this parameter is equal to RτD, where R is a constant (R ) 0.1) and τD is the derivative time. Thus, the presence of the τD term in φ(s) (i.e., the appearance of β in φ(s), φ′(s), and φ′′(s)) makes it difficult to get exclusive solutions of KC, τI, and τD. To make the closed-loop response faster (and stable), in many literature reports,19 τD is approximated as τD ) τP/2. We consider selecting β based on the following rule: β ) R(0.25)max(τP, DP)
(24)
This selection is based on a conservative selection of τD ) τP/2 with an adjustable constant factor of 0.25 (as determined using Ziegler-Nichols (ZN) tuning criteria2 (τD/τI ) 0.25) and Chien and Freuhauf9 (τI ≈ τP)). Therefore, with the help of these two
parameters (λ and β) and eqs 19a-19c, the PID parameters can be evaluated. 4. Results and Discussions The present controller has an IMC-PID structure. The controller parameters are implemented in PID2 form (eq 3). The performance of the proposed controller is evaluated by simulation with the following examples: Example Example Example Example
1 2 3 4
exp(-0.25s)/(s + 1) 2.0 exp(-s)/[(10s + 1)(5s + 1)] 0.2 exp(-7.4s)/s (s2 + 2s + 0.25)/(s4 + 6.5s3 + 15s2 + 14s + 4)
4.1. Performance. The closed-loop performance of the system is given by its complementary sensitivity function, T(s) )
GP(s)GC(s) 1 + GP(s)GC(s)
(25)
Example 1: FOPDT Process. A first-order-plus-time-delay process with KP ) 1, τP ) 1, and DP ) 0.25 is chosen for simulation study. The controller is designed with a first-order filter. Closed-loop responses with the present controller are obtained (see Figure 2, Ex-1) and IAEs are calculated. The results are shown in Table 3. The figure shows that the closedloop response with present tuning (IAER ) 2.119 and IAEL ) 0.687) is faster and its settling time is less, compared to that of IMC-Mac (IAER ) 2.12 and IAEL ) 0.6872). The present tuning scheme yields higher values of KC and τD (but slightly lower τI, because of the presence of the β term in the analytical expression of τI), which makes the response faster, compared to that of IMC-Mac. Moreover, with increasing time lag (keeping the process time constant), the performance is seen to be improved with the present tuning rule. Note that, with increases in the λ value, the tuning parameter τI approaches τP and as the KC and τD values almost vanish, the controller reduces to an integral controller. Example 2: SOPDT Process. A SOPDT overdamped process with KP ) 2, τP1 ) 10, τP2 ) 5, and DP ) 1 is considered. A
8690 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 4. Variation of (a) cutoff frequency (ωc) versus the normalized closed-loop time constant (λ/D), (b) gain margin (gm) versus (λ/D), (c) phase margin (pm) and gain margin (gm) versus (ωc), and (d) sensitivity versus frequency (ω) and closed-loop time constant (λ/D).
second-order filter is assumed and, with the help of the specific desired closed-loop response, the controller is designed and used for simulation. The controller parameters and IAE values are shown in Table 3. The closed-loop response for this example is shown in Figure 2 (Ex-2). As evident from Table 3, with the present tuning (KC ) 1.931, τI ) 14.785, τD ) 3.171), the setpoint and load-change performances are 3.848 and 7.656, respectively, whereas, with IMC-MAC, the set-point and loadchange performances become 3.94 and 7.76, respectively. When an underdamped system is selected with KP ) 1, τP ) 3, ξ ) 0.4, and DP ) 1, the performance of the PID controller (λ ) 1.25) is seen to be almost similar to the present results (IAER ) 12.83, IAEL ) 16.83), that of Lee et al.14 (IAER ) 12.23, IAEL ) 16.23), and that with SMIC (IAER ) 14.61, IAEL ) 20.43), and the responses are slightly oscillatory. A close look at the tuning formula (SOPDT) reveals that they involve a 2λ term, whereas in the case of FOPDT, it involves only λ. Example 3: IPDT Process. According to Luyben20 and Chien and Fruehauf,9 many chemical processes can be modeled as integrator-plus-dead-time (IPDT) types of systems. Here, an IPDT process with a transfer function GP ) 0.2 e-7.4s/s (from Chien et al.21) is considered for controller design and simulation. The PID controller parameter values and closed-loop performance results are calculated using the present tuning scheme and the Lee et al.17 data separately and are provided in Table 3. Values of λ and β are chosen to be 12.58 and 0.0463 (with proposed tuning), respectively, during simulation. The closedloop response is shown in Figure 2 (Ex-3), from which it can be observed that the present method produces a faster response, compared to the results of Lee et al.17 The response with present tuning is rather smoother and can be made faster by further retuning. The performance obtained using the method of Skogestad12 yields a PI controller with KC ) 0.3378 and τI )
59.2. The λ value was calculated using λ ) max(1.7 × dp, 0.2 × τ) ) 12.58, and the performance was calculated to be IAER ) 28.24 and IAEL ) 67.93. With increases in the value of λ, a PID controller reduces to a PI controller and then to an integral one. A widely used transfer function GP ) 0.0506 e-6s/s for the tank level control process is also tested with the present method, as well as with other available methods of PI tuning (IMC by Rice and Cooper,18 SIMC by Skogestad,12 and the method reported by Ziegler and Nichols2). The controller parameters and performance values are provided in Table 4. We can see from the result that the use of Ziegler-Nichols (ZN) tuning, which brings the process to the brink of stability with ultimate gain and period, yields the worst performance. Example 4: Higher-Order Process. A higher-order system with a dominant lead time constant is used with a transfer function of s2 + 2s + 0.25 (26) s4 + 6.5s + 15s2 + 14s + 4 This yields an open-loop step response with large overshoot because of a strong lead term. Please note that, a zero (βs + 1) is already included in the numerator of GC and, therefore, a first-order filter is sufficient for realization of the controller (GC). However, in the case of IMC-MAC, a second-order filter must be used. The actual process transfer function is reduced to a SOPDT system using Skogestad’s rule12 as GP )
0.0670(7.46s + 1)e-0.25s (26.1) (2s + 1)(0.75s + 1) Based on this reduced model (eq 26.1) (neglecting the numerator zero), a PID controller (SIMC) is synthesized (KC GP )
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8691
) 59.7, τI ) 2, and τD ) 0.75 with λ ) 0.2). The designed controller is used with the actual process for obtaining closedloop results (Figure 3), which shows faster responses (with the present control scheme) and, thus, better performance, compared to IMC-MAC. With the present control scheme, the closedloop response shows an overshoot of ∼40% and a settling time of ∼15 s. The corresponding performance values are shown in Table 3. With the SIMC controller, the performance value for the set-point change case (IAER ) 1.5833) is higher, compared to the other two control schemes, but shows a better performance with the load-change case as IAEL ) 0.0335. 4.2. Robustness. Under process uncertainty (where lm represents multiplicative uncertainty), stability will be guaranteed if |lm(jω)T(jω)| < 1. This becomes
| | -Djω
e 1 < λjω + 1 ||lm(jω)||
Naturally, the aforementioned relation converges to a specific value. A controller in IMC structure is followed and accompanied by a filter transfer function. Thus, the true controller (GCTrue, or φ(s)) can be analyzed for robustness purposes. To do this, let us consider a process GP with a nominal transfer function model GPm, given as GPm )
KPe-DPs τPs
(27)
Therefore, the loop transfer function becomes e-DPs (28) λs + 1 - e-DPs where GCTrue ) GCIMC/(1 - GPGC) can be seen in eq 15. After introducing normalized process parameters (θ ) τP/DP and µ ) λ/DP) and substituting s ) jω, eq 28 becomes L(s) ) GP(s)GTrue C (s) )
cos ω - j sin ω e-jω ) ) (jµω + 1 - e-jω) jµω + 1 - cos ω + j sin ω (A cos ω - B sin ω) - j(A sin ω + B cos ω) (29) A2 + B2 where A ) (1 - cos ω) and B ) (µω + sin ω). Therefore, the amplitude ratio and phase can be determined from eq 29. At the normalized cutoff frequency, |L(jωc)| ) 1 gives us L(jω) )
µ2ωc2 + 2µωc sin(ωc) - 2cos(ωc) + 1 ) 0
(30)
Equation 30 can be solved for ωc with different normalized closed-loop time constant (µ) values between µmin ) 0 and µmax ) 1. The phase angle can be calculated from eq 31:
( 180π )
φm ) [π + arg(L(jωc))]
(31)
where
The robustness or sensitivity is computed from eq 32 S)
|
1 1 + L(jω)
|
(32)
and is shown in Figure 4d. When a disturbance of amount W(s) ) GP(s) - GPm(s) enters the process, the proposed controller is sufficiently robust to encounter the effect of disturbance and retain closed-loop stability. To be a robust controller,
|G
|
True C W(s) ∞ < 1
(33)
That leads one to obtain a sufficient condition as
|
|
K(jµω + 1 - e-jω) ∀ω (34) jθω + 1 Equation 34 reveals that the robustness increases as µ increases. If A is permitted to be the state space matrix of the process GP, the eigenvalues can be determined by solving |λI - A| ) 0. W(jω)
