Simple Analytical minâmax Model Matching Approach to Robust

690

Ind. Eng. Chem. Res. 2010, 49, 690–700

Simple Analytical min-max Model Matching Approach to Robust Proportional-Integrative-Derivative Tuning with Smooth Set-Point Response S. Alcańtara,*,† C. Pedret,† R. Vilanova,† and W. D. Zhang*,‡ Telecommunication and Systems Engineering Department, ETSE, UniVersitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain, and Department of Automation, Shanghai Jiaotong UniVersity, Shanghai 200030, People’s Republic of China

This communication addresses the set-point robust proportional-integrative-derivative (PID) tuning for stable first order plus time delay systems from a general min-max model matching formulation. As opposed to some recent optimization-based numerical procedures, the derivation is carried out analytically, and it is based on a Smith-type inverse response configuration. Within the considered context, several choices result in a standard PID. This work investigates the simplest one, leading to a PID controller solely depending on a single design parameter. This contrasts with other analytical approaches resulting in more involved tuning. Attending to common performance/robustness indicators, the free parameter is finally fixed to provide an automatic tuning solely dependent on the model information. Toward this transition, dimensional analysis proves fruitful and allows us to establish that the proposed tuning rule is very robust for lead-dominant plants in the presence of parametric uncertainty. Lastly, simulation examples show that the suggested compensator yields very good results. 1. Introduction Despite the modern control theory state of the art, proportionalintegrative-derivative (PID) controllers continue to be the most common option in the realm of control applications, with an absolute dominance within the process control industry nowadays.1,2 This is explained because of their simplicity both in implementation and in understanding. As a matter of fact, in most of the situations a PID controller can perform reasonably well and is indeed all that is required. Recent advances in optimal methods have boosted the control solutions based on optimization procedures. In particular, a plethora of optimization-based PID designs have been reported in the literature during the last years (see, for example ref 3-5) However, many of them, although effective, rely on somewhat complex numerical optimization procedures and/or fail to provide tuning rules. At the present time, it constitutes a more recent trend to derive analytical PID solutions based on different performance criteria which finally yield the desired tuning rules2,6,7 while ensuring good stability margins. Following this line, the present communication addresses the analytical derivation, within a min-max model matching context, of the simplest robust PID tuning rules for smooth setpoint response. To achieve results as close as possible to the industrial situation, the widely used ISA PID structure1 is chosen for the control law. The analytical min-max model matching approach to PID design was already conducted in ref 6, but the proposed solution involved two tuning parameters. In this work, a simpler solution solely depending on one parameter and with very similar associated performance is obtained. For design purposes, a first order plus time delay (FOPTD) model is employed since it has been shown to be representative of an important category of industrial processes.8 The suggested design extends the authors’ works,9,10 and it is based on approximating the FOPTD model by a second order * To whom correspondence should be addressed. E-mail: [email protected], [email protected]. † Universitat Auto`noma de Barcelona. ‡ Shanghai Jiaotong University.

plus inverse response (SOPIR) one. The Smith-type inverse response (IR) control configuration presented in ref 9 is then applied to the latter and the analytical procedure in ref 10 finally yields the solution to the proposed model matching problem. This solution, as already stated, depends on a single tuning parameter which provides a simple way of adjusting the robustness/performance trade-off. Afterward, both nominal and robust stability analysis are provided in terms of this design parameter, which is ultimately fixed so as to provide an automatic tuning solely dependent on the model information. This is accomplished by means of using dimensional analysis,11 which proves to be a very useful tool, in conjunction with standard robustness and performance indicators in a genuine way. Representative simulation examples validate the proposed automatic tuning, which is compared with other ones found in the literature. The article is organized as follows. Section 2 is devoted to the problem statement. In section 3 the general min-max model matching problem is solved for a particular setup leading to a PID compensator with a single tuning parameter. Section 4 addresses the stability of the derived controller. In section 5 the tuning parameter is conveniently fixed, thus providing automatic tuning. Simulation examples to show the applicability of the proposed method are provided in section 6 while concluding remarks end the paper in section 7. 2. Problem Statement In this section the control framework and the model matching problem on which the controller derivation is based are introduced. The latter obeys to a min-max optimization problem that captures the performance objective. 2.1. The Control Framework. The conventional one-degreeof-freedom (1-DOF) scenario is assumed throughout this work for the control configuration. The customary unity feedback controller is depicted in Figure 1. As it is well-known, closed-loop performance and robustness are typically evaluated in terms of the sensitivity S(s) and the complementary sensitivity T(s) transfer functions,12 respectively:

10.1021/ie9010194  2010 American Chemical Society Published on Web 12/04/2009

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

S(s) z

1 1 + L(s)

T(s) z 1 - S(s) )

691

(1)

L(s) 1 + L(s)

(2)

where L(s) z P(s) K(s) is the loop transfer function. As it has already been stated, the model Po(s) for the real process P(s) is given by 1 e-sh (3) τs + 1 For design purposes, it is convenient to approximate the delay term in 3 so as to achieve a purely rational process model. By using the first order Pade´ expansion e-sh ≈ (-(h/2)s + 1)/ ((h/2)s + 1), 3 can be approximated as follows: Po(s) ) Kg

h - s+1 2 Po(s) ≈ Kg (4) h (τs + 1) s + 1 2 Regarding the control law, the following ISA PID form1 is chosen:

(

[

u(s) ) Kp r(s) - y(s) +

)

1 (r(s) - y(s)) + sTi sTd (r(s) - y(s)) 1 + sTd /N

]

(5)

where r(s), y(s), and u(s) are the Laplace transforms of the reference, process output, and control signal, respectively. Kp is the PID gain, whereas Ti and Td are its integral and derivative time constants. Finally, N is the ratio between Td and the time constant of an additional pole introduced to ensure the properness of the controller. This way, the following transfer function for the controller K(s) is assumed:

(

)

Td Td + s2Ti (N + 1) N N (6) K(s) ) Kp Td sTi 1 + s N 2.2. The Model Matching Problem. The controller design will be based on a desired input-to-output response. Mathematically, the following min-max optimization problem is posed to capture the performance objective: 1 + s Ti +

min

stabilizing K(s)

(

Figure 2. Inverse response control configuration.

)

|W(s)(Mref(s) - T(s))| ∞

(7)

where Mref(s) is a desired reference model for the closed-loop system response, W(s) is a weighting function, and T(s) is the complementary sensitivity function, which is also by definition the transfer function from the input to the output of the closedloop system. In section 3 the control problem eq 7 will be solved for a suitable particular case yielding a regulator K(s) in the form of eq 6. As has already been commented, eq 7 will be finally solved by approximating the model Po(s) as in eq 4. As a result, we

Figure 3. Inverse response control configuration ideal net result.

are indeed considering a SOPIR model. The Smith-type inverse response control configuration depicted in Figure 2, introduced in the context SOPIR systems,9,10 will be adopted here to simplify the analytical solution derivation of eq 7. The terms N(s), M(s) are such that the nominal model Po(s) for P(s) can be factored in the following way: Po(s) ) N(s)M-1(s)

(8)

where N(s) is a proper, all-pass and nonminimum-phase factor, whereas M-1(s) is a strictly proper minimum-phase factor. For the approximate model in eq 4, the following selection is performed: -(h/2)s + 1 1 N(s) ) (9) τs + 1 (h/2)s + 1 In the case of no plant/model mismatch and no disturbances, the proposed configuration can be conceptually represented as in Figure 3. Now, by defining M-1(s) ) Kg

R(s) z

Ksp(s) M(s) + Ksp(s)

(10)

the input to output relations (dropping the s argument for clarity) are given by eq 11. (Note that R(s), or simply R, has nothing to do with the reference signal, for which the time and Laplace transform pair is r(t) T r(s). Furthermore, if arguments are dropped, r may be used indistinctly for both the time and the frequency domains.)

() (

u -RN -RM RM ) y P(1 - RN) 1 - RN RN

)(

di do r

)

(11)

For internal stability we require R(s) to be stable. It is worth noting that the sensitivity and complementary sensitivity transfer functions simplify to S(s) ) 1 - R(s)N(s)

Figure 1. Feedback control sheme.

(12)

T(s) ) R(s)N(s) (13) In this new setting, the control design can be posed in terms of R(s). Once this is done, an equivalent unity feedback controller can be recovered easily from eq 10 and Figure 2:

692


R(s)M(s) (14) 1 - R(s)N(s) The resulting K(s) will be realizable if the closed-loop system is well-posed in the sense that all possible transfer functions-those in eq 11-are defined and proper. This happens if and only if both R(s) N(s) and R(s) M(s) are proper. See ref 13 for a discussion regarding the concept of good implementation of a control law. K(s) )

3. Analytical Solution to the Model Matching Problem We are now concerned with finding a simple solution to the problem in 7, which is aimed at minimizing the following functional: J ) |W(s)(Mref(s) - T(s))| ∞

(15)

Several methods could be followed in order to solve this H∞ general problem. See, for example,14,15 However, our interest focuses on simple instances of the problem in eq 7 leading to a controller in the form of eq 6. Following this rationale, the above min-max problem was solved in ref 6 for the following particular setup: W(s) ) (1 + zs)/s, Mref(s) ) 1/(1 + TMs). Additionally, a first order Taylor approximation for the delay in eq 3 was taken into account. The resulting controller, depite being a commercial PID, depends on two tuning parameters: z, TM. The role of TM is clear: it specifies the desired speed of response whereas z allows for adjustment of the robustness of the control system. In what follows, a different setup is suggested, resulting in a single-parameter PID control law, which provides very similar performance but with more intuitive and easier tuning. The suggested setting for minimizing eq 15 is W(s) ) 1/s, Mref(s) ) 1. In terms of the control configuration presented in section 2.2 (see Figure 2) the general problem of eq 7 concerning eq 15 simplifies to 1 min | (1 - R(s)N(s))| ∞ s

(16)

R(s)∈RH∞

The weight W(s) ) 1/s is the simplest one ensuring integral action in the design, whereas the selected reference model Mref(s) ) 1 specifies the ideal input-to-output relation. Needless to say, this is not achievable in practice: if a very quick response is desired, this would be at the expense of a large overshoot in the output transient. As it will be seen, controlling the overshoot will be an easy task once the optimum controller has been derived. Problem 16 is indeed a sensitivity one: according to the Tydo(s) relation in eq 11, eq 16 minimizes the effect of a step disturbance do(s) entering at the output of the plant. It is well-known that this is equivalent to minimizing the effect of a step reference r(s) in the tracking error e(s) ) r(s) - y(s). These additional considerations make problem 16 completely meaningful. Regarding the process model, the approximation in eq 4 is used. Note that the simple setting W(s) ) 1/s, Mrefs ) 1 now allows use of a better approximation for the nominal model. Let us assume now that (1 - R(s) N(s))/s is stable in eq 16, that is, analytic in the complex RHP (this will be the case if R(s) ∈ RH∞ and (1 - RN)|s)0 ) 0). Then, by the well-known maximum modulus principle,16 the maximum modulus of (1 R(s) N(s))/s within the RHP is attained over the jω-axis. From eqs 9 and 10, we can check that |(1/s)(1 - R(s) N(s))|s)2/h ) h/2, consequently: h 1 | (1 - R(s)N(s))| ∞ g s 2

(17)

The minimum value solution leads to h R(s) ) s + 1 2

(18)

For internal stability and realization issues it is necessary to extend the optimum R(s) in 18 with a filter. The following suboptimum R(s) is proposed (the optimum solution is recovered as λ f 0): h s+1 2 R(s) ) (19) (λs + 1)2 An equivalent unity feedback controller can be recovered from eq 14 and cast into the commercial form 6 providing the following tuning rule: χ(λ) Kg(4λ + h) χ(λ) Ti ) 2 Td 2λ2 ) N 4λ + h τh(4λ + h) N+1) 2λ2χ(λ) Kp )

(20)

where χ(λ) )

4λ(2τ + h) + (τ + h)2 - τ2 - 4λ2 4λ + h

(21)

When designing a control system within the standard 1-DOF scenario, all the control requirements, namely: nominal/robust stability (robustness), good command following (performance), good disturbance rejection (performance), small magnitude of control signal, mitigation of measurement (high-frequency) noise on the plant output, etc., have to be met by means of a frequency-dependent robustness/performance trade-off adjustment.12 This is clear from the equality S(s) + T(s) ) 1 and the fact that the closed-loop relations depend on S(s) and T(s). For instance, from eqs 11-13: y(s) ) T(s)r(s) + S(s)do(s) + P(s)S(s)di(s)

(22)

u(s) ) K(s)S(s)r(s) - K(s)S(s)do(s) - T(s)di(s)

(23)

and

In our case, both the reference and the disturbances will be assumed step signals, and the design will be aimed at providing smooth set-point response. It will be seen in the next sections that the λ parameter precisely provides a simple means for adjusting the shapes of the S(s) and T(s) functions, responsible for the final design features. Indeed, the proposed approach allows design of the controller utilizing the robustness/ performance compromise, which requires the tuning of a single parameter, λ, and then provides the map to the four-dimensional ISA PID parameters space. In view of eq 20, it is clear that designing the ISA PID compensator directly would be more complicated and less intuitive, involving four parameters (and more complicated expressions) instead of one. As it will be shown, the requirement of smooth set-point will automatically yield a robust design with low levels of control usage. In addition, acceptable disturbance rejection will be obtained generally. Consequently, for the problem at hand, a single design parameter is really all that is necessary. In particular, it has been observed that the alternative setting presented in ref 6 for minimizing eq 15, involving two tuning parameters, does not


693

provide any real advantage in practice, one of the parameters now being redundant. Remark 1. Although the focus in this work is on FOPTD models, the above analytical procedure is readily applicable to other important low-order, stable models in process control. (The design, as it has been presented here, is not applicable to unstable plants since it would result in an unstable pole/zero cancellation between the plant P(s) and the controller K(s).) • Delayed integrating processes (DIP), with ε being sufficiently small: h - s+1 Kg e-sh e-sh 2 ≈ Kg ≈ Kg s s+ε ε 1 h s+1 s+1 ε 2 • First order delayed integrating processes (FODIP), with ε being sufficiently small:

(

)(

)

Kg -hs + 1 e-hs ≈ s(τs + 1) ε 1 s + 1 (τs + 1) ε • Second order processes with time delay (SOPTD): Kg

(

)

Figure 4. Dual locus diagram.

e-hs -hs + 1 ≈ Kg Kg (τ1s + 1)(τ2s + 1) (τ1s + 1)(τ2s + 1) • Second order processes with inverse response (SOPIR): Kg

-Rs + 1 (τ1s + 1)(τ2s + 1)

The dual locus diagram technique is now applied: very briefly stated, the closed-loop system is stable if the locus of L1(s) reaches the intersection point earlier than L2(s). The loci of L1(s) and L2(s) have been displayed in Figure 4 for positive frequencies of the imaginary axis only. The intersection frequency can be determined by solving the equation

|

This case was considered in ref 10 by the authors. If τ1 and τ2 constitute a complex conjugate pair, the proposed technique proves also useful for plants with underdamped or oscillatory response.

-

h s 2

)

h s+1 2

|

)1

(26)

from which the positive frequency of interest can be seen to be

4. Stability Analysis This section addresses how the λ parameter influences both the nominal and the robust stability of the proposed controller. The main objective is to prepare the groundwork for assisting in the selection of the tuning parameter based on robustness considerations. This task is finally accomplished in section 5, where a free-of-λ tuning rule is proposed. 4.1. Nominal Stability. Since we have considered the approximation eq 4 for the adopted FOPTD model, the basic requirement of nominal stability is not guaranteed for a FOPTD plant even when all its parameters are perfectly known. The nominal stability issue is dealt with here by means of the dual locus technique along the lines of ref 17. This technique, based upon the argument principle,16 can be regarded as a modified version of the well-known Nyquist criterion.12 Taking the unity feedback controller from eq 6 and eq 20 together with the model 3 results in the following characteristic equation: h s+1 2

1 + L(s) ) 1 +

h λ s + 2λ + s 2 which can be rewritten in the form 2 2

(

)

e-sh ) 0

L1(s) - L2(s) ) 0

(

λ2s2 + 2λ + h s+1 2

h s 2

)

(24)

(25)

ωc )

√-4 - 2µ + √(-4 - 2µ)2 + 4 √2λ

L2(s) ) e-sh

(27)

where µ ) h/λ. The phase angles of L1(s) and L2(s) at ωc are, respectively, 1 2+ µ 2 1 - arctan µλωc φ1 ) arctan λωc 2

(28)

φ2 ) -hωc

(29)

and

The stability condition is satisfied only when the phase angle of L1(s) is larger (in absolute value) than that of L2(s) at ωc, that is, if φ1 - φ2 < 0. From the fact that µ ) h/λ and eq 28 it can be seen that the function φ1 - φ2 is ultimately only a function of µ. In Figure 5 the function φ1 - φ2 is plotted against the µ parameter. It can be finally concluded that the resulting closedloop system, in the nominal case, is stable, provided that the PID controller tuning parameter λ ) h/µ is chosen such that 1 h ≈ 0.074h (30) 13.5135 4.2. Robust Stability. To account for model uncertainty we will assume that the dynamic behavior of a plant is described not only by a single linear time invariant (LTI) model but by a whole family, usually referred to as the uncertainty set, built λ>

by making the following assignments: L1(s) ) -

(

λ2s2 + 2λ +

694


|T(jω)|
0.074 the closed-loop is stable for a perfectly known FOPTD system. The necessary minimum λsthus, providing the fastest responsesyielding robust stability can be determined graphically by plotting the magnitudes of eq 36 and eq 35 for a given parametric uncertainty pattern, increasing λ until eq 37 is satisfied. This method could be followed by the control system designer in order to conveniently adjust the robustness/ performance trade-off. To make this procedure completely automatic, the following section proposes a way to fix λ, providing thus an autotuning of the proposed controller. 5. Automatic PID Tuning Derivation

Figure 5. φ1 - φ2 vs µ ) h/λ.

around the nominal one. More precisely, we will consider the possible plants at hand belonging to the following set: F ) {P(s) ) Po(s)(1 + ∆m(s))}

(31)

where Po(s) ∈ F denotes the nominal model of eq 3, ∆m(s) is the relative (multiplicative) model error ∆m(s) z

P(s) - Po(s) Po(s)

(32)

satisfying |∆m(jω)| e |Wm(jω)|. Wm(s) is a frequency weight bounding the modeling error. It is well-known12,18 that a controller K(s) that stabilizes the nominal plant Po(s), also stabilizes all the plants in eq 31 provided that |Wm(s)T(s)| ∞ < 1

(33)

where T(s) is the nominal complementary sensitivity function eq 2. Now, we will assume nominal closed loop stability with respect to the model Po(s) and will consider uncertainty in the plant parameters. Condition 33 evaluates robust stability in terms of the complementary sensitivity function T(s). We need to compute the relative model error between the model and the real plant. We will compute this relative error with respect to the model Po(s) given by eq 3, whereas the real plant is considered to be P(s) ) Kg(1 + rk)

1 e-h(1+rh)s (1 + rτ)τs + 1

(34)

for rk, rτ, rh in the interval (-1, +1). It can be seen that if we denote by δk, δτ, δh the maximum (positive) values of rk, rτ, rh, respectively, then the worst case relative error ∆m of eq 32, that is, the most difficult plant to stabilize, is given by ) (1 + δk) ∆*(s) m

τs + 1 e-shδh - 1 (1 - δτ)τs + 1

(35)

According to the control configuration employed in section 2.2: h - s+1 2 T(s) ) R(s)N(s) ) (λs + 1)2

This section is aimed at conveniently fixing the value of λ in the tuning relations of eq 20, giving rise to a tuning rule solely dependent on the model. So far, the particular model matching problem in eq 16 has been solved. Its solution, given in eq 20, has been found to depend on the FOPTD model in addition to an extra tuning parameter, λ. The lower the value of λ is, the lower is the value of the functional 15. However, an excessively low value for λ providing very fast responses is not desirable since it is bound to produce large overshoots in the step response. This is not taken into account by the adopted performance criterion. Besides, from the stability analysis of section 4, in order to make the closed loop robust, λ has to provide the necessary detuning and cannot be so small in practice. In accordance with this, the following summarizes the requirements to be met. • Performance. A sufficiently fast, free-of-oVershoot nominal set-point response. This performance specification obeys the fact that in many processes such as chemical or mechanical systems an excessive overshoot is not acceptable. Consequently, λ has to produce a small value for the functional eq 15 while ensuring smooth set-point response. • Robustness. As the controller is obtained from the model, it has to be chosen in such a way that the closed-loop is not too sensitive to variations in process dynamics. Making direct use of the robust stability condition 37 with eq 27 and eq 36 is not easy and would be restricted to parametric uncertainty. Instead, a more general and simpler robustness measure will be used. In spite of this, condition 37 will be used at a later stage to assess robustness in the face of parametric uncertainty. Sensitivity to modeling errors can alternatively be captured by the peak of the sensitivity function:

(36)

In our case, it is easy to see that the robust stability condition 33 holds if and only if

Ms z |S(jω)| ∞ z max ω

|

1 1 + L(jω)

|

(38)

which indicates the inverse of the shortest distance from the Nyquist plot to the critical point. Having Ms < 2 is a traditional robustness indicator.12 It is evident that both the overshoot and the sensitivity peak will depend on the loop function L(s). On the other hand, as T(s) ) L(s)/(1 + L(s)) ) (-(h/2)s + 1)/(λs + 1)2 is a function of just h and λ, L(s) depends only on h, λ as well. Consequently, if we define q1 ) “overshoot”, q2 ) “sensitivity peak” it is clear that there exist functional relations f1, f2 such that qi ) fi(h, λ), i ) 1, 2. In these functional relations we have two variables and only one independent unit (time). By applying the Buckingham pi theorem from dimensional analysis (see for instance, refs 5 and 11) it is possible to describe the same relationships by using only one dimensionless parameter. In particular,


Figure 6. Output overshoot (yov) and sensitivity peak (Ms) against λ/h. Table 1. Proposed ISA PID Tuning Kp

Ti

Td/N

N+1

0.4Ti/(Kgh)

τ + 0.1h

0.4h

1.25(τ/Ti)

relations qi ) fi(h, λ) can be expressed more compactly as πi ) φi(λ/h) where πi contains the quantity of interest qi, proving that both the overshoot and the sensitivity peak depend only on λ/h. This dependence can be seen in Figure 6, from which the zero overshoot requirement is met for λ > 0.6h. However, at λ/h ) 0.6 the sensitivity peak curve slope is still significant. For the sake of an improvement in robustness, some extra nominal performance in terms of closed-loop bandwidth is sacrificed by choosing λ ) h, the point on which Ms ≈ 1.1134 and the sensitivity curve has a slope of almost zero. This indicates that it is not worth slowing down the nominal response further. With the choice λ ) h, the tuning rule of eq 20 becomes that of Table 1. Condition 37 can be used now to give an idea of the achieved robustness with λ ) h in terms of parametric uncertainty. First, note that for λ ) h eq 36 becomes h - s+1 2 (39) T(s) ) (hs + 1)2 in eq 35 Let us consider the worst case uncertainty ∆*(s) m with δk ) δτ ) δh ) δ ) 0.65sthat is, assume 65% of simultaneous parametric uncertaintysand T(s) in eq 40 and define the variable q3 ) “frequency distance between bode plots It is clear that q3 ) f(h, τ). By invoking of T(s) and 1/∆*(s). m again the Buckingham pi theorem, the same functional relationship can be expressed in the more compact form π3 ) φ(h/τ), where π3 contains the quantity of interest q3. One can try out different values for h/τ until the magnitude bode plots of 1/∆*(s) m and T(s) almost intersect. This has been found to happen for h/τ ) 0.63, see Figure 7. Consequently, it can be claimed that 65% of parametric uncertainty is allowed for any FOPTD system for which h/τ ) 0.63. The described procedure can be repeated for different values of δ. This experiment yields the bounds shown in Table 2. It is worth noting that the proposed tuning rule is very robust for lead-dominant systems, tolerating almost 100% of uncertainty in the plant parameters. Proceeding likewise, the worst case overshoots for different values of δ have been plotted in Figure 8. The different robustness/performance tests performed

695

Figure 7. Robust stability condition for the nominal plant e-0.63s/(s + 1), assuming 65% of parametric uncertainty in Kg, τ, h. Table 2. Permissible Simultaneous Parametric Uncertainty in Kg, τ, h h/τ )

0.1

0.25

0.5

0.63

1

5

10

δ × 100

38%

45%

59%

65%

84%

97%

98%

so far may be taken as a rough guide to the limits of the achievable PID performance for robust smooth set-point. If lower overshoots and faster responses are required for the control of an overshoot-sensitive system, a multiloop controller may be a more suitable solution.19 However, considerably better results would come with the expense of additional complexity and more agressiveness of the controller. 5.1. Control Effort Constraints. As it will be seen in section 6, moderate control usage is required by the proposed controller in Table 1. However, as the control efforts are important in practical applications, we provide here quantitative guidelines for selecting λ according to saturation limits and slew rate constraints. If we consider the nominal plant approximation in 4, we have from eq 11 that the control signal associated with a set-point change is given by

Figure 8. Worst case overshoots for different levels of parametric uncertainty.

696


u(t) ) L

-1

{

}

1 R(s)M(s) ) L s

-1

{

1 Kg

}

h (τs + 1) s + 1 2 1 s (λs + 1)2 (40)

(

)

which represent, respectively, the highest rates of change in the increasing and decreasing directions. For the proposed tuning rule λ ) h the above two expressions simplify to |u˙+(t)| ∞ )

-1

where L denotes the inverse Laplace transform. By calculating 40 the maximum value for u(t) can be found to be

(

1 τh - 2τλ - hλ + 2λ2 |u(t)| ∞ ) 1× Kg 2λ2 2τh - 2τλ - hλ exp (-2λ + h)(τ - λ)

(

))

1 if τ e λ (41) Kg From this point on, it is assumed that λ/h > 0.5 (indeed, it was shown in Figure 6 that λ/h g 0.6 for zero nominal overshoot). From 41 it can be seen that |Kgu(t)|∞ only depends on h/τ and λ/h. Thus, once a given plant has been modeled according to a FOPTD description, a particular h/τ relation is obtained. With this in mind, |Kgu(t)|∞ only depends on λ/h and λ can be tuned so as to obtain acceptable peaks on the control signal. On the other hand, if λ/h is fixed first, then |Kgu(t)|∞ becomes a function of h/τ. This function can be used to give an idea of the maximum control effort required along different lead-lag ratios. In our case, we fixed λ/h ) 1. Substituting λ ) h in eq 41 leads to

|u(t)| ∞ )

{

(

(

) (

))

if h < τ if h g τ (42)

which can be easily expressed as a function of Kg and h/τ. Slew rate constraints can be similarly tackled. From eq 40 the derivative of the control signal is

u˙(t) ) L

-1

{R(s)M(s)} ) L

{

-1

1 Kg

h (τs + 1) s + 1 2 (λs + 1)2

(

|u˙-(t)| ∞ )

if τ > λ

|u(t)| ∞ )

1 1 -τ + h h 1exp Kg 2 h -τ + h 1 Kg

and

)

}

u˙(t) ) u˙+(t) + u˙-(t)

u˙(t) if u˙(t) > 0 0 if u˙(t) e 0

(44)

and

-u˙(t) if u˙(t) < 0 0 if u˙(t) g 0 the following expressions can be easily obtained u˙-(t) z

|u˙+(t)| ∞ ) -

(

|u˙-(t)| ∞ ) 0 if τ e λ

if h < τ

(49)

if h g τ

Table 3. Processes within the Experimental Setup Together with their FOPTD Approximationsa real process

3τh + 2λ2 - 4τλ - 2hλ (-2λ + h)(τ - λ)

1.2e-1.2s 0.8s + 1

P2(s) )

1 (1 + s)(1 + 0.1s)(1 + 0.01s)(1 + 0.001s)

e-0.073s 1.073s + 1

P3(s) )

e-s (1 + 0.05s)2

e-s 0.093s + 1

P4(s) )

1 (1 + s)4

(46)

P5 ≡

)

if τ > λ (47)

FOPTD model

P1(s) )

(45)

1 2τh - 2τλ - hλ 2 Kgλ3

1 2λ2 - 2τλ - hλ + τh × 2 Kgλ3 exp -

)

In this section we will evaluate the proposed simple automatic tuning rule of Table 1 through simulations. The objective is to cover a representative set of examples so as to properly obtain conclusions regarding the performance and robustness of the suggested method. Table 3 collects the information of the experimental setup. First, four linear processes are considered including the lag-dominant, lead-dominant, and balanced lag and delay cases. The first one consists of a FOPTD plant for which there is only parametric uncertainty, whereas the other three systems are linear processes with relatively complex dynamics which are modeled as FOPTD plants for control purposes. These three last examples are taken from ref 2. Additionally, a fifth nonlinear system is taken into account. This last process represents the isothermal series/parallel Van de Vusse reaction20,21 taking place in an isothermal continuous stirred tank reactor (CSTR). The corresponding approximate FOPTD model has been derived assuming the system in a stationary point. For the sake of comparison, other approaches to PID design considering FOPTD models are examined. Since a complete comparison is not possible because of the large number of existing tuning rules (see ref 8) we will concentrate on two existing methods also conceived in the spirit of simplicity: (i) S-IMC tuning rule (leading to a PI). A really simple and effective tuning proposed in ref 7. (ii) AMIGO tuning rule

and |u˙-(t)| ∞ )

(

6. Simulation Examples

where

{ {

1h - τ τ exp 2 K h2 τ-h g 0

(48)

(43)

By calculating eq 43 and considering the following decomposition (a similar decomposition was not used for u(t) in eq 41 because u(t) g 0 for t g 0):

u˙+(t) z

{

1 2Kgh

{

x˙ ) f(x, u) x2 y )

e-s s+1

e-1.42s 2.9s + 1 0.0126e-0.0085s 0.01s + 1

a P1-4 are linear processes. Regarding P5, f(x, u) ) (f1(x, u), f2(x, u)) ) (- 50x1 - 10x12 + (10 - x1)u, 50x1 - 100x2 - x2u). The assumed working point is (x*, u*) ) (3, 1.117, 34.2805).

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 Table 4. Results of Performance/Robustness Evaluation for the Set of Plants

697

5a {P}i)1

performance set-point plant

tuning

robustness Ms

P1

proposed SIMC AMIGO

1.74 2.12 1.85

P2

proposed SIMC AMIGO

P3

IAE

disturbance

TV

yov

IAE

TV

|u(t)|∞

2.15 2.4 1.7

1.1 1.64 6.13

3.3 20.95 10.96

1.25 1.235 0.91

0.55 0.79 0.72

0.51 0.6 0.56

1.33 1.56 1.31

0.21 0.24 0.24

11.03 16.11 11.56

6.36 23.42 23.55

0.09 0.04 0.027

0.57 0.75 0.75

0.53 0.62 0.62

proposed SIMC AMIGO

1.42 1.6 1.46

2.5 2.18 1.94

1 1.09 1.46

0 4.27 0

1.25 1.09 0.97

0.5 0.54 0.56

0.5 0.52 0.5

P4

proposed SIMC AMIGO

1.66 2 1.62

3.76 4.08 3.23

1.78 2.68 2.02

5.1 18.77 15.54

1.77 1.54 1.26

0.61 0.82 0.67

0.53 0.6 0.58

P5

proposed SIMC AMIGO

1.4 1.57 1.38

0.0026 0.0026 0.0022

0.0005 0.0047 0.0041

2.3 2.5 3

34.46 34.48 34.73

11.1 15.6 51.4

0.4 1.6 1.1

a As the system P5 is nonlinear, the robustness indicator Ms has been computed with respect to its linearization on the working point, which turns out to be (-1.117s + 188.8)/(s2 + 278.6s + (1.937 × 104)).

Figure 9. P1(s) time responses for set-point change and load disturbance for the proposed (solid), S-IMC (dotted), AMIGO (dashed)tuning rules.

(leading to a PID). A rule along the lines of the classical Ziegler-Nichols method (see ref 2). To evaluate the robustness and the performance obtained with the different methods at hand, the following standard measures will be used: • Robustness. The peak of the sensitivity function, Ms (eq 38), is the inverse of the minimum distance from the Nyquist plot to the critial point and constitutes a quite standard robustness indicator.12 • Output performance. The integrated absolute error (IAE) of the error e ) r - y will be computed. IAE )

∫

∞

0

|e(t)| dt

• Input performance. To evaluate the manipulated input usage, the total variation (TV) of the control signal u(t) will be computed. TV )

∫

∞

0

|u˙(t)| dt

To provide a more global and complete comparison framework, the performance measures above will be calculated for both a set-point change and load disturbance. In addition, the percent overshoot of the output y(t), denoted as yov, will be taken into account for set-point output performance. Similarly, the peak of the control signal u(t), that is, |u(t)|∞, will be indicated for load disturbance performance. Table 4 summarizes the results obtained. It follows from Figures 9-13 that the proposed tuning rule generates quite smooth responses requiring inexpensive control. Table 4 shows that the required control usage is lower than that associated with the other two methods. In particular, for plants P1 and P5 the proposed tuning control usage is far below that of the AMIGO tuning. It can also be seen that the proposed method provides, generally, both the minimum output overshoots and the control signal peaks. With respect to set-point evaluation, the AMIGO tuning rule gives better IAEs. However, if one inspects Figures 9-13 it is clear that the set-point

698


Figure 10. P2(s) time responses for set-point change and load disturbance for the proposed (solid), S-IMC (dotted), AMIGO (dashed) tuning rules.


responses of the proposed method are qualitatiVely better than those for the AMIGO tuning, which exhibit significantly larger overshoots. Regarding disturbance rejection, the proposed method provides an inferior performance with respect to the S-IMC and AMIGO proposals. This is a quite expected result since the proposed tuning rule was derived for smooth set-point. Nevertheless, disregarding the lag-dominant plant P2, the disturbance rejection responses are not significantly inferior, and in the case of the S-IMC, they are indeed quite comparable. This is explained in part due to the fact that the optimization problem in eq 16 is, as a matter of fact, a sensitivity optimization problem. A deeper analysis of the serVo-regulator trade-off tuning within a generalized version of the presented framework is being currently conducted and will be the topic of a future work. Lastly, Table 4 shows that for the five considered systems the robustness indicator for the proposed method is always very close to that associated with the best method. This robustness

is in accordance with the smoothness of the corresponding control and output signals. 7. Conclusions Starting by considering a general model matching problem in the supremum norm, we have presented tuning relations for PID design modeling the plant as a FOPTD system. For the derivation of the suggested tuning, the general model matching problem has been customized so as to arrive at a PID control solution. The concrete model matching problem finally solved leads to a single tuning parameter. Since the controller is derived by means of a purely rational approximation, the nominal stability region for this parameter has been determined. Finally, an automatic robust tuning rule has been derived for smooth set-point changes. Along the way, dimensional analysis has proved to be a useful tool and the limits of the suggested solution have been explored. Representative simulation examples have


699



revealed the good robustness and performance of the proposed approach in practice. The primary goal of the presented method has been to obtain a good set-point response. However, good disturbance responses are also desirable. Because responses to set-point and load disturbance are usually conflicting, future work is conducted to show how the trade-off between the servo and regulator modes can be tackled within the presented framework (in fact, it can be seen that the optimization problem 16 has a normindependent solution structure), providing a solid base for some conclusions of the previous work,22 relying on a numerical analysis. Along the same lines, the presented approach can be used to simplify the servo-regulator trade-off methodology in ref 23. If instead of achieving a compromise, one desires the optimum performance for the two different controller modes, then a more complex controller structure has to be used.5,24 The task of designing commercial 2-DOF PID compensators analyti-

cally in a similar way as it has been done in this paper is currently underway. Acknowledgment The financial support received from the Spanish CICYT programme under Grants DPI2007-64570 and DPI2007-63356 and from the National Science Foundation of China under Grant 60874005 is greatly appreciated. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Astrom, K.; Hagglund, T. PID Controller: Theory, Design and Tuning; Instrument Society of America: 1995. (2) Astrom, K.; Hagglund, T. Revisiting the Ziegler-Nichols Step Response Method for PID Control. J. Process Control 2004, 14, 635–650. (3) Ge, M.; Chiu, M.; Wang, Q. Robust PID Controller Design via LMI Approach. J. Process Control 2002, 12, 3–13.

700


(4) Toscano, R. A Simple PI/PID Controller Design Method via Numerical Optimization Approach. J. Process Control 2005, 15, 81–88. (5) Tavakoli, S.; Griffin, I.; Fleming, P. Robust PI Control Design: A Genetic Algorithm Approach. Int. J. Soft Comput. 2007, 2, 401–407. (6) Vilanova, R. IMC Based Robust PID Design: Tuning Guidelines and Automatic Tuning. J. Process Control 2008, 18, 61–70. (7) Skogestad, S. Simple Analytic Rules for Model Reduction and PID Controller Tuning. J. Process Control 2003, 13, 291–309. (8) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules, 2nd ed.; Imperial College Press: London; 2006. (9) Alcańtara, S.; Pedret, C.; Vilanova, R.; Balaguer, P.; Ibeas, A. Control Configuration for Inverse Response Processes. Proc. 16th Mediterranean Conf. Control Autom. (MED08) 2008, 582–586. (10) Alcańtara, S.; Pedret, C.; Vilanova, R.; Zhang, W. Analytical H∞ Design for a Smith-Type Inverse Response Compensator. Proc. Am. Control Conf. (ACC09) 2009, 1604–1609. (11) Buckingham, E. On Physically Similar Systems: Illustrations of the Use of Dimensional Equations. Phys. ReV. 1914, 4, 345–376. (12) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Control; Wiley: New York, 1997. (13) Kwok, W.; Davison, D. Implementation of Stabilizing Control Laws—How Many Controller Blocks Are Needed for a Universally Good Implementation? Control Syst. Mag., IEEE 2007, 27, 55–60. (14) Francis, B. A. A Course in H∞ Control Theory; Lecture Notes in Control and Information Sciences; Springer-Verlag: New York, 1987. (15) Vilanova, R.; Serra, I. Model Reference Control in Two-Degreeof-Freedom Control Systems: Adaptive Min-Max Approach. IEE Proc., Part D 1999, 146, 273–281.

(16) Churchill, R.; Brown, J. Complex Variable and Applications; McGraw Hill: New York, 1986. (17) Zhong, Q.-C. Control of Integral Processes with Dead-Time. Part 3: Deadbeat Disturbance Response. Automatica 2003, 39, 1309–1312. (18) Morari, M.; Zafirou, E. Robust Process Control; Prentice-Hall International: New York, 1989. (19) Tran, T.; Ha, Q.; Nguyen, H. Robust Non-Overshoot Time Responses Using Cascade Sliding Mode-PID Control. J. AdV. Comput. Intell. Intell. Inf. 2007, 11, 1224–1231. (20) de Vusse, J. G. V. Plug-Flow Type Reactor versus Tank Reactor. Chem. Eng. Sci. 1964, 19, 964. (21) Kravaris, C.; Daoutidis, P. Nonlinear State Feedback Control of Second Order Nonminimum-Phase Nonlinear Systems. Comput. Chem. Eng. 1990, 14, 439–449. (22) Arrieta, O.; Vilanova, R. Servo/Regulation Tradeoff Tuning of PID Controllers with a Robustness Consideration. Proc. 46th Conf. Decis. Control 2007, 1838–1843. (23) Vilanova, R.; Arrieta, O. Combined SensitiVity/Complementary SensitiVity min-max Approach for Load Disturbance/Setpoint Tradeoff Design; Recent Advances in Industrial Engineering and Operations Research; Springer-Verlag: New York, 2007. (24) Zlosnikas, V.; Baskys, A. PID Controller with Enhanced Disturbance Rejection. Electron. Electr. Eng. 2008, 5, 65–68.

ReceiVed for reView June 23, 2009 ReVised manuscript receiVed November 3, 2009 Accepted November 10, 2009 IE9010194

Simple Analytical minâmax Model Matching Approach to Robust

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