Observer−Controller Design for a Class of Stable ... - ACS Publications

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Observer-Controller Design for a Class of Stable/Unstable Inverse Response Processes Carles Pedret,* Salva Alca´ntara, Ramon Vilanova, and Asier Ibeas Systems Engineering and Automatic Control Group, ETSE, UniVersitat Auto`noma de Barcelona, 08193 Bellaterra, Spain

This paper proposes the use of a two-degree-of-freedom configuration aimed at the control of inverse response processes. An observer-controller scheme based on coprime factorization results in a Smith-type predictor well suited for both stable and unstable inverse response processes. The design procedure is carried in two steps. First, the controller is designed providing an adjustable parameter to cope with the reference input tracking. Second, the observer blocks are selected for closed-loop properties, offering another tuning parameter to compromise between robust stability and disturbance rejection performance. As a result, the nominal set point response is decoupled from the disturbance response. To show the applicability of the proposed control configuration, its performance is compared by simulation with different approaches. 1. Introduction Inverse response (IR) processes are those showing an initial step response in the opposite direction to the steady-state value. It can be seen that this behavior appears because of competing effects of slow and fast dynamics.1 In the chemical process industry, this phenomenon occurs in several systems such as the drum boilers of distillation columns.2 The transfer function of these kinds of processes has one zero or an odd number of zeros in the open right half plane (RHP). Since the initial work of Iinoia and Alperter,3 it is common in the literature of process control to assume a second-order transfer function with one RHP-zero. This is the so-called second-order-process with inverse response (SOPIR) model. The reasons of using this model are that it contains the essential characteristics of inverse response processes and it can be adopted to model higher order processes.4 A process that can be described by a SOPIR model constitutes one example of a nonminimum-phase (NMP) system (a process with time delay is another example). The essential characteristic of such processes is that they do not have the smallest phase lag that is possible for processes with the same gain. This NMP characteristic introduces essential limitations in terms of achievable output performance. The difficulties associated with the control of this kind of processes become bigger when the corresponding RHP-zero approaches the origin. An important limitation due to the presence of RHP-zeros is the high-gain instability. As it is well-known from classical root-locus analysis, as the feedback gain increases toward infinity one closed-loop pole moves to the position of the open-loop zero. An additional restriction imputable to the presence of RHPzeros entails bandwidth restrictions that limit the tight control at low and at high frequencies. Finally, the presence of RHPzeros makes the condition of perfect control impossible by any stable and causal controller. The effect of the RHP-zero combined with a RHP-pole results in more stringent bandwidth limitations. The combined effect increases the minimum peak of closed-loop transfer functions such as the sensitivity transfer functions or the complementary sensitivity transfer function. The closer the RHP-pole gets to the RHP-zero, the more severe the limitations. Skogestad and Postlethwaite5 provide a compre* To whom correspondence should be addressed. E-mail: [email protected].

hensive discussion in depth. Here, some limitations imposed by RHP-zeros are briefly pointed out in order to show that it is reasonable to think about control configurations that deal explicitly with this class of processes. Two categories of control structures can be found in the literature. The first uses proportional-integral-derivative (PID) controllers within a standard unity feedback configuration with many kinds of tuning methods.4,6,7 The good results obtained with such PID approaches are due to a positive feature of the derivative action for correcting the wrong direction of the system response.1 However, performance of PID control usually degrades to keep the margin of stability. The second category uses the so-called inverse response compensators. These kinds of compensators are based on control schemes aimed at eliminating the effect of the RHP-zero from the loop transfer function. The first proposal was presented by Iinoia and Alperter.3 A more recent approach was proposed by Zhang et al.,8 where an internal model controller9 (IMC) was designed based on H∞ performance requirements. Also, Alca´ntara et al.10 presented a Smith-type predictor scheme. Therefore, it can be seen that there are few papers dealing with the control of inverse response processes, and none of them deal explicitly with the unstable plant case. This paper proposes the use of a two-degree-of-freedom (2DOF) configuration for the control of both stable and unstable inverse response processes. Certainly, despite their recognized advantages, 2DOF control configurations are not abundant in process control literature. Different 2DOF proposals exist based on the internal model control11,12 or Smith predictor schemes13,14 for particular problems such as unstable processes with time delay or integrative processes with time delay. Nevertheless, to the best knowledge of the authors, 2DOF control configurations for IR processes have not been considered before. The approach presented results in a Smith-type predictor scheme that aims to leave the nonminimum phase dynamics out of the feedback loop, facing the remaining possibly unstable dynamics in the closed loop. The design is carried out in two steps: First, he set-point tracking controller is designed providing an adjustable parameter to compromise between the speed of the set-point response and the magnitude of the undershoot. Second, the observer is built by means of a Youla parametrization offering another tuning parameter to compromise between robust stability and disturbance rejection performance. Then, the detuning does not modify the relation from the reference signal

10.1021/ie9002377 CCC: $40.75  2009 American Chemical Society Published on Web 10/28/2009

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Figure 1. Observer-controller scheme: (a) Original structure for design, (b) ideal net result in the nominal scenario, and (c) equivalent structure for implementation.

achieved in the first part of the design procedure. This is a distinguishable feature from already existing approaches. The presented approach could be regarded as a pole-placement design, in the line of the famous RST controller,15 but directly carried out at the more appealing level of transfer functions instead of polynomials, where the design of the Youla parameter is less intuitive. The paper is organized as follows: Section 2 introduces the control configuration we will work with. Section 3 presents the design procedure. In section 4, simulation examples show the applicability of the proposed approach. Concluding remarks are given in section 5. 2. The Observer-Controller Configuration The control structure used in this paper was proposed by Vidyasagar16 and used in a modified manner by some of the authors.17,18 The original control scheme is shown in Figure 1a. The controller K(s) is responsible for the stability and the set-point tracking. The two blocks, X(s) and Y(s), are used for rejecting disturbances and robustness properties. The methodology that we will use relies upon the factorization approach.16,19,20 Although the concepts are completely valid for multivariable systems, in what follows we will concentrate on the single-input single-output scenario. Arguments on transfer functions will sometimes be dropped for clarity. A useful way of representing a transfer function of a possibly unstable system is by means of a coprime factorization over RH∞. Within this framework, a coprime factorization of the transfer function P(s) is -1

P(s) ) N(s) M (s)

(1)

where N(s) and M(s) are stable coprime transfer functions. Remark 1. The stability of the coprime factors N and M implies that N contains the RHP-zeros of P, and M contains as RHP-zeros all the RHP-poles of P. Since stable transfer functions constitute an Euclidean domain, coprimeness of two transfer functions can be mathematically characterized by the existence of stable transfer functions X(s) and Y(s) satisfying the following Bezout identity: X(s) M(s) + Y(s) N(s) ) 1

(2)

To construct two stable transfer functions N and M that satisfy (1) is easy. We could divide the numerator and denominator polynomials of P by (s + F)k, where F > 0 is freely assignable and k equals the maximum of their degrees. On the other hand, to construct two stable transfer functions X and Y satisfying (2) is not so straightforward. There exists a polynomial procedure based on the Euclid’s algorithm.19 Nevertheless, the state-space procedure for computing a coprime factorization for a proper plant P is more efficient than the polynomial method.5,19,20 The feedback interconnection shown in Figure 1a is based on the reconstruction of the so-called partial state, ξ, that is, the fictitious signal that appears after the factorization of P in (1). A state feedback control interpretation of the coprime factorization (1) shall demonstrate that the state variables of any minimal realization of P are completely determined by ξ and its derivatives.21 This is the reason why ξ is called the partial state of the system. In the absence of disturbances, the observer-controller configuration can be ideally represented as in Figure 1b, and the following equations can be written: u ) Mξ - d

(3)

y ) Nξ

(4)

The two observer blocks X and Y in Figure 1a reconstruct the partial state, ξo, in order to feed it back by means of the compensator K, ξo ) Xu + Yy

(5)

By substituting (3) and (4) into (5) it follows ξo ) (XM + YN)ξ - Xd Hence, as the Bezout identity (2) holds, we have ξo ) ξ - Xd. If no disturbances entered the control system, that is, d ) 0, the partial state would be reconstructed exactly, ξo ) ξ. In the presence of disturbances, that is, d * 0, the observed partial state ξo includes a measure of the disturbances. Then, with an appropriate selection of the observer blocs X and Y, the effect that the disturbances cause to the controlled variable can be minimized in some sense. This will be dealt with in the next section.

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As can be seen in Figure 1b and noticing Remark 1, all possible RHP-zeros of P are allocated as zeros of N and left out of the feedback loop. Furthermore, all RHP-poles of P are poles of M-1. Then, the plant can be stabilized by the compensator K. The resulting ideal net serves to illustrate the operation of the observer-controller scheme in a nominal situation and to introduce the condition of internal stability. In this case, the input-output relations from the reference signal r to the output y and control signal u can be found as Tyr )

NM-1 ) NR-1 1 + KM-1

(6)

Tur )

1 ) MR-1 -1 1 + KM

(7)

RzM+K

(8)

Remark 2 Note that the possible RHP-zeros of the process are not present in the characteristic equation. Then, as N, M, X,andYarestabletransferfunctions,thenominalobserVer-controller configuration will be stable if and only if R is unimodular, that is, stable and inVertibly stable.16 This way, the set of stabilizing obserVer-controller compensators are characterized by the free, unimodular, parameter R in (8). 3. Design Procedure Having analyzed the control configuration with which we will work, we proceed with the design of their elements. In this paper, we consider the following transfer function model of processes with stable/unstable inverse response, (9)

where R, τ1, τ2 > 0 and b ) 1 is used to represent a stable model and b ) -1 is used to represent an unstable one. We assume that there does not exist pole-zero cancellation. Controller Design. In this section we show how to select the set-point tracking controller K based on an integral squared error (ISE) performance specification, that is, min |W(s)(1 Tyr(s))|22, where the weight W is chosen as 1/s for strict step changes in the reference input. The set-point tracking controller K is derived analytically and in an indirect way: first the optimum value of the unimodular parameter R is found, and then the controller K is recovered from (8): K)R-M

(10)

Let the coprime factorization for the system (9) be in the form of -Rs + 1 (µ1s + 1)(µ2s + 1)

(11)

(τ1s + b)(τ2s + 1) Kp(µ1s + 1)(µ2s + 1)

(12)

N(s) ) and M(s) )

(13)

Rs + 1 (µ1s + 1)(µ2s + 1)

(14)

and Nm(s) )

To derive the optimum value of the unimodular parameter R, it follows that |W(s)(1 - Tyr(s))| 22 ) |W(1 - NR-1)| 22 ) |W - WNaNmR-1 | 22 ) |Na(Na-1W - WNmR-1)| 22 Note that |(Na(jω)| ) 1 ∀ω. Then,

where

-Rs + 1 P(s) ) Kp (τ1s + b)(τ2s + 1)

-Rs + 1 Rs + 1

Na(s) )

In this way, N can also be factorized as N ) NaNm, that is, an all-pass portion Na and a minimum phase portion Nm. For the system at hand (9),

|W(s)(1 - Tyr(s))| 22 ) |Na-1W - WNmR-1 | 22 ) |(Na-1W)unst + (Na-1W)st WNmR-1 | 22 where the operators ( · )st and ( · )unst stand for stable and unstable transfer functions, respectively. Utilizing the orthogonality property of the H2 norm we have |W(s)(1 - Tyr(s))| 22 ) |(Na-1W)unst | 22 + |(Na-1W)st WNmR-1 | 22 Minimizing the right side we obtain the optimal R-1 as follows: Ropt-1(s) ) (WNm)-1(Na-1W)st )

(µ1s + 1)(µ2s + 1) Rs + 1 (15)

Nevertheless, Ropt-1 is not unimodular. Hence a first order lowpass filter F1(s) )

1 , λ1s + 1

λ1 > 0

(16)

is introduced. Then, R-1(s) )

(µ1s + 1)(µ2s + 1) (Rs + 1)(λ1s + 1)

(17)

where λ1 is an adjustable parameter and when it tends toward zero the optimality is recovered. Finally, it rests to recover the set-point tracking controller K with the suboptimal R-1 following (10), that is, K(s) )

k2s2 + k1s + k0 (µ1s + 1)(µ2s + 1)

(18)

where k2 ) Rλ1 - τ1τ2/Kp, k1 ) R + λ1 - (τ1 + bτ2)/Kp, and k0 ) 1 - b/Kp. It should be noted that the poles -1/µ1 and -1/µ2 introduced to obtain the coprime factors (11) and (12) vanish when the complementary sensitivity transfer function is computed. It is also important to remark that the final resulting control configuration with which we will work is that of Figure 1c. Therefore, the set-point tracking controller K in (18) is not a standard unity feedback one. As a rule of thumb, we take µ1 ) R and µ2 ) λ1. The complementary sensitivity transfer function (6) is

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Tyr ) NR-1 )

-Rs + 1 (Rs + 1)(λ1s + 1)

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(19)

Figure 2 shows the time responses for set-point tracking controller K for different values of the tuning parameter λ1. As it is known from the IMC theory,9 the optimal reference to output behavior provides the fastest step response but with a considerable undershot. With the adjustable parameter λ1, a suboptimal reference to output relation is achieved in which, increasing λ1, the undershot is reduced at expenses of a slower step response. To provide a selection criterion for the design parameter λ1, we suppose that the set point is a step signal. Then, the response of the system to this set point Yr(s) is computed as follows: -Rs + 1 1 1 Yr(s) ) Tyr ) s s (Rs + 1)(λ1s + 1)

(20)

The time domain set-point response is obtained by applying the inverse Laplace transform to (20),

yr(t) )

{

1+

R + λ1 (-1/λ1)t 2R ( ) e e -1/R t, R - λ1 R - λ1 R + 2t (-1/R)t e 1, R

λ1 * R λ1 ) R (21)

The undershoot is achieved, letting dyr(t)/dt ) 0, at tu )

{

(

)

Rλ1 R + λ1 ln , R - λ1 2λ1 0.5R,

λ1 * R

(22)

λ1 ) R

and substituting tu into (21) the undershoot is obtained yu )

{

1+

R + λ1 -tu/λ1 2R e e-tu/R, R - λ1 R - λ1 0.213,

λ1 * R λ1 ) R (23)

Figure 3 illustrates the normalized relationship between undershoot yu and λ1. It helps to strengthen the fact illustrated in Figure 2 and shows quantitatively how much the undershoot reduces as the tuning parameter λ1 increases. In the 1DOF control configuration approach presented by Zhang et al.,8 this tuning parameter should provide a suitable

Figure 3. Relationship between undershoot and λ1/R.

compromise among the nominal performance, undershoot, and robustness. The advantage of the proposed approach is that closed-loop properties such as disturbance rejection performance and robustness properties are treated in a second step. In summary, the steps for the controller design are (Step 1.1) Coprime factorization: Find a coprime factorization for the process (9) by dividing its numerator and denominator polynomials by (µ1s + 1)(µ2s + 1). As a result, we obtain the coprime factors (11) and (12). (Step 1.2) Unimodular R-1: Find the ISE-optimal value for R-1 as in (15). Add the first order filter (16) in order for R-1 to be unimodular, as in (17). (Step 1.3) K design: The final set-point tracking controller is in the form of (18). It is found with the suboptimal unimodular R-1 (17) and the coprime factor M (12). The complementary sensitivity transfer function is (19). With the adjustable parameter λ1 adjust the reference input tracking. Observer Design. The coprime transfer functions X and Y are computed following standard procedures5,19,20 with the restriction that their poles must be stable. Nevertheless, for the reason that the factors X and Y are linked with N and M by means of the Bezout identity (2), they cannot be chosen with the same freedom as the factors N and M. Then, in order to circumvent the problem caused by this intrinsic dependence, we reformulate the observer by application of a Youla-type parametrization. In this way it is possible to write the set of observers that give the partial state ξ as the set of solutions to the Bezout identity (2). If Xo and Yo are solutions of the Bezout identity and Q is a stable, but otherwise free, transfer function, the set {X, Y: X ) Xo - NQ, Y ) Yo + MQ}

(24)

characterizes all linear observers for ξ.22 With such a modification, it is easy to show that the parameter Q is canceled when one computes the transfer functions from the reference signal r. Therefore, transfer functions (6) and (7) are not modified. With the parametrization (24) the transfer functions from d to y can be found as Tyd ) P(1 - NR-1K(Yo + MQ))

Figure 2. Time responses for set-point tracking controller K by increasing the value of the tuning parameter λ1: 0.1, 1, 5, 10.

(25)

The observer and the parameter Q will be designed to consider feedback properties. However, even for the nominal system, there is no guarantee of zero steady-state error for load-

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disturbance step changes. The reason is that the design of the controller K was only made for set-point tracking purposes. To ensure zero gain at low frequencies, the gain of the parameter Q should be chosen appropriately. Recalling the sensitivity transfer function (25), we find the required gain of Q, that is, gQ, as the value that makes Tyd(0) ) 0, gQ )

Yo(0) R(0) N(0) K(0) M(0) M(0)

(26)

Now, we can write Q as Q ) gQ + Qo with Qo being a transfer function with zero gain at zero frequency. This modification entails a new solution for the components of the Bezout identity: Xˆo ) Xo - gQN

(27)

Yˆo ) Yo + gQM

(28)

and, consequently, the initial parametrization (24) is written as {X, Y: X ) Xˆo - NQo, Y ) Yˆo + MQo}

is reduced in solving the following model matching problem: ε z min |W(s) (1 - M(s) ν(s))| ∞ stable ν(s)

The trivial case of the problem is when M-1 is stable, that is, b ) 1 in (9). Then, the unique optimal is ν ) M-1 and ε ) 0. On the other hand, when M-1 is unstable, ε * 0 and a frequency characterization of the resulting error will be required. This characterization is obtained by means of the weight W in the optimization problem (33). The following form is suggested: W(s) )

Tyd

|W(s) - W(s) M(s) ν(s)| ∞ g |W(1/τ1)| Then, ε g |W(1/τ1)| and the function ν(s) )

(30)

To consider the uncertainty that always exists in practical processes, let us assume that ∆m determines the multiplicative uncertainty bound of the process, that is, ∆m(s) ) (Pr(s) - P(s))/ P(s), where Pr(s) represents the real process. This uncertainty representation can be used to collect different uncertainties encountered in practice such as plant input actuator uncertainty, plant output uncertainty, plant parameters uncertainty, among others. According to the well-known small-gain theorem9,20 the closed-loop system is robustly stable if and only if |∆mTδd|∞ < 1. Substituting (30), this condition can be rewritten as |NR KYˆo(1 + Yˆo-1MQo)| ∞ < -1

1 |∆m | ∞

(31)

It can be seen that the left-hand side of (31) consists of two terms: the first one, NR-1KYˆo, determines the initial robust stability margin, and the second, 1 + Yˆo-1MQo includes the Qo parameter. This parameter is designed to force the second term to match a desired transfer function, that is, 1 + Yˆo-1MQo ≈ F2. Since the aim in this paper is to provide a simple tuning for closed-loop properties, the following first order filter is proposed: F2(s) )

1 , λ2 s + 1

λ2 > 0

(34)

(29)

-1

) NM (1 - NR K(Yˆo + MQo)) Tδd ) NR-1K(Yˆo + MQo)

γs + 1 , γ>β βs + 1

The simplest nontrivial case is when WM has only one RHPzero, say s ) 1/τ1. If ν is stable and WMν has finite ∞-norm, then by the maximum modulus theorem19 it follows that

The transfer functions from d to y and δ can be found as -1

(33)

(32)

This filter incorporates the tuning parameter λ2 which allows us to take into account the well-known compromise between nominal disturbance rejection performance and the robust stability. That is to say, increasing λ2 tends to strengthen the robust stability of the closed loop at the expense of degrading its disturbance rejection performance. On the contrary, decreasing λ2 improves the disturbance rejection performance of the closed-loop with the loss of its robust stability in the presence of uncertainty. The design equation is QoM ) Yˆo(F2 - 1) and to solve it for Qo it is necessary to invert the transfer function M. In the unstable plant case, that is, b ) -1 in (9), it cannot be inverted directly because it will cause the instability of Qo. In such a case, we must find a stable approximate inverse of M. Let us define this stable inverse approximation as ν. Then, the problem

W(s) - W(1/τ1) W(s) M(s)

(35)

is stable, optimal, and yields ε ) |W(1/τ1)|. In this case, substituting (12) with the values stated above, µ1 ) R and µ2 ) λ1, and (35) with γ ) τ1 and β ) 0 into (35) we obtain a stable approximate inverse of M in the form of ν)

(Rs + 1)(λ1s + 1) (τ1s + 1)(τ2s + 1)

(36)

Now, the design equation for Qo is Qo ) Yˆo(F2 - 1)ν

(37)

There exist algorithms19,23 to solve the model matching problem (33) in the general case. Since the constraint (31) for tuning the adjustable parameter λ2 cannot be analytically solved, numerical simulation tests would be necessary to determine a rule-of-thumb for tuning λ2. Nevertheless, a detailed analysis for the generic process (9) is not straightforward. The reason is that, despite the proposed 2DOF control configuration provides two adjustable parameters, one for nominal tracking performance (λ1) and the other for closed-loop properties (λ2), they are not independent. As depicted in Figure 2, decreasing λ1 offers a faster step response at the expense of augmenting the undershoot. In addition, as it is well-known, the faster the step response, the lower the robust stability margin. Therefore, to tune the adjustable parameter λ2 it is recommended to fix it around the value of R in the first place. If the resultant closed-loop disturbance rejection performance or robust stability is not satisfactory, then monotonously increase or decrease the adjustable parameter λ2 until the desirable compromise between nominal disturbance rejection performance and the robust stability is satisfied. If the compromise is not satisfactory, then it will be necessary to sacrifice the speed of the set point tracking response by augmenting λ1. In summary, the steps for the observer design are as follows. (Step 2.1) Initial Observer: Find two stable transfer functions Xo and Yo satisfying the Bezout identity (2), that is, XoM + YoN ) 1. Standard procedures can be used.5,19,20 The set of linear observers is characterized by means of (24), Q being a stable transfer function. (Step 2.2) Reformulation for Tyd(0) f 0: Find the gain gQ (26) as

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the value that assures zero steady-state error for load-disturbance step changes. Reformulate the initial set of observers as in (29) with the new solutions (27), (28) and Qo being a transfer function with zero gain at zero frequency. (Step 2.3) Design for closedloop properties: Force the term 1 + Yˆo-1MQo in the robust stability condition (31) to match the filter F2 (32). Compute Qo as in (37). In the stable plant case, ν ) M-1. In the unstable plant case, use the stable inverse (36). Finally, with the parameter λ2 compromise between the disturbance rejection performance and robust stability. 4. Simulation Examples To show the applicability of the design procedure let us consider the following two examples. Example 1. Consider the inverse response process3,8 P(s) ) 3

-2s + 1 (s + 1)(2s + 1)

In the method proposed by Iinoia and Alperter,3 an inverse response compensator was used with a tuning parameter k ) 6 in addition with a feedback PI controller with parameters kc ) 1 /6 and τi ) 2. The approach proposed by Zhang et al.,8 based on H∞ control theory, provided an adjustable parameter to compromise between performance and robust stability. Assuming that a faster step response is required, the value of the tuning parameter is chosen to be λ ) 1.4. In the book of Stephanopoulos1 it is suggested that one way to control inverse response processes is by using a PID with Z&N tuning, along the same lines as Waller and Nygardas.4 Nevertheless, it is well-known that the Z&N tuning method provides set point step responses with high overshoot and very low robustness since it was conceived for 1DOF PID controllers considering disturbance rejection performance. Then, for comparative purposes, a 2DOF PID controller with the famous AMIGO tuning rule24 is chosen. The reason is that this tuning was presented as a refinement of the Z&N tuning rules and it was carried out by means of the optimization of load disturbance rejection with constraints on robustness to model uncertainties. We assume c ) 1 and then the unique prefilter parameter, b, is adjusted to a value of 0.85 to provide the same undershoot that with the method proposed by Zhang et al. To design the set-point tracking controller K in (18), we take λ1 ) 0.75 to provide the same undershoot than that reached with the 2DOF PID controller tuned with the AMIGO method and with Zhang et al. approach. Thus, the controller is obtained in the form of K(s) ) 0.667

0.625s + 1 0.75s + 1

To design the observer blocks X and Y, we follow Steps 2.1-2.3. The tuning parameter is chosen as λ2 ) 0.1. The procedure gives transfer functions X and Y of fifth and third order, respectively. By applying model reduction techniques,25 that is, a balanced realization and an optimal Hankel norm approximation, we get the following equivalent transfer functions, X(s) ) 0.167

(0.026s + 1)(1.262s + 1)(8.989s - 1) (0.1s + 1)(s + 1)(0.2s + 1)

and 1.333s + 1 0.1s + 1 A unit step change is introduced to the set-point signal at t ) 0 and a step signal of disturbance with amplitude -0.25 is Y(s) ) 0.667

Figure 4. Closed-loop responses with the nominal system for Example 1.

introduced to the process inputs at t ) 25 s. The simulations are shown in Figure 4. The set-point response performance with the proposed approach is practically the same than that with the method of Zhang et al. Despite the fact that the 2DOF PID with the AMIGO tuning rule provides the same undershoot, the set-point response presents overshoot. Compared with the other approaches, it should be noted that the proposed method provides the best disturbance response. The control signals are shown in Figure 5. The 2DOF PID and the proposed approach provide responses with similar extreme values in the disturbance rejection response. Assuming that there exist uncertainties in the parameters of the plant, let ∆m ) 25%. Within this condition, time simulations are shown in Figure 6 and Figure 7. It can be observed that the proposed method retains the control system performance better than the other approaches. Example 2. Consider the unstable inverse response process P(s) ) -4

-0.25s + 1 (s - 1)(0.5s + 1)

This kind of unstable process is studied in the book of Skogestad and Postlethwaite5 and controlled by means of an H∞ mixed sensitivity design method. Here the input weight is wu ) 1 and the performance weight is wp ) (s/Ms + ωB)/(s + ωBA) with Ms ) 2, ωB ) 2, and A ) 10-8. To design the set point tracking controller K in (18), we take λ1 ) 0.6. We obtain a controller in the form of K(s) ) -2.5

(0.41s + 1)(0.89s + 1) (0.25s + 1)(0.6s + 1)

The two observer blocs X and Y are designed by following Steps 2.1-2.3. The tuning parameter is λ2 ) 1. This results in transfer functions X and Y of seventh order. By applying a balanced realization and an optimal Hankel norm approximation, we get the following transfer functions, X(s) ) (s4 + 59.39s3 + 1157.66s2 + 1516.35s + 454.36)/(s4 + 37.92s3 + 314.59s2 + 856.76s + 411.52) and Y(s) ) (-9.25s4 - 889.43s3 - 32435.71s2 - 99768.05s 68945.77)/(s4 + 104.09s3 + 3381.73s2 + 25916.65s + 61505.85)

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Figure 5. Control signals with the nominal system for Example 1.

Figure 6. Closed-loop responses with the uncertain system for Example 1.

Figure 7. Control signals with the uncertain system for Example 1.

Figure 8. Closed-loop responses with the nominal system for Example 2.

Figure 9. Control signals with the nominal system for Example 2.

Figure 10. Closed-loop responses with the uncertain system for Example 2.

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is that the detuning in the second step does not modify the nominal relation from the reference signal achieved in the first part of the design procedure. In this way, the improvement of the closed-loop properties is deferred until the second step of the suggested design, in which the observer is built for the enhancement of the inherited observer-free controller design performed in the first step. Acknowledgment The financial support received from the Spanish CICYT programme under Grants DPI2007-64570 and DPI2007-63356 is greatly recognized. Literature Cited

Figure 11. Control signals with the uncertain system for Example 2.

A unit step reference change is introduced at t ) 0 and a step signal of disturbance with amplitude 0.5 is introduced to the process input at t ) 10 s. The time responses are shown in Figure 8. It is well-known that the effect of the RHP-pole combined with the RHP-zero produces bandwidth limitations. In addition, the combined effect increases the minimum peak of closedloop transfer functions such as the sensitivity transfer functions or the complementary sensitivity transfer function. The more the closeness of the RHP-pole and the RHP-zero, the more severe the limitations.5 The simulation shown in Figure 8 displays evidence of the difficulties of H∞ controller with the process. The weighed sensitivity requirements are not quite satisfied since the peak of the sensitivity function is 2.94 > Ms. Furthermore, the tracking response provides a large undershoot and a prominent overshoot. The proposed observer-controller design provides an improved tracking response with smaller undershoot. Furthermore, it also results in better disturbance rejection response. Control signals are shown in Figure 9. In this example, we assume parametric uncertainty and let ∆m ) 15%. Closed-loop responses within this conditions are shown in Figure 10. Also, Figure 11 shows the control signals provided by the two approaches. It can be observed that both approaches hold the robust stability. Nevertheless, the H∞ controller offers worst results. 5. Conclusions A two-degree-of-freedom control configuration is presented for the control of both stable and unstable inverse response processes. Since the first proposals3,4 there are few papers dealing with the control of inverse response processes and none of them deal explicitly with the unstable plant case. The design is performed in two-steps by tuning two parameters: the first one allows an adjustment of the speed of the set-point tracking response; the second one offers the possibility to find a trade-off between robust stability and disturbance rejection performance. A distinguishable feature

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ReceiVed for reView February 13, 2009 ReVised manuscript receiVed September 15, 2009 Accepted October 2, 2009 IE9002377