ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 Published online 12 October 2007 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.096 Research Article Disturbance observer design for continuous systems with delay R. W. Jones1 * and M. T. Tham2 1 2 Mads Clausen Institute for Product Innovation, Syddansk University, Sonderborg, Denmark School of Chemical Engineering and Advanced Materials, Newcastle University, Newcastle of upon Tyne, England, UK Received 8 January 2007; Revised 16 July 2007; Accepted 13 August 2007 ABSTRACT: Disturbance observers (DOs), which are popularly used for improving the disturbance rejection capability of mechatronic servo control systems, offer several attractive features that could prove beneficial for process control systems. The tuning is simple and intuitive and they allow independent tuning of disturbance rejection characteristics, which is particularly helpful in situations in which gains need to be tuned on-line. This paper is concerned with examining DO design for continuous-time systems with delay. Two methods for incorporating time delay into traditional DO design are considered, to assess their relative performance, stability, and noise rejection characteristics. The performance of one of these time-delay-based DO approaches is then compared with proportional, integral and derivative (PID) control, designed using a maximum sensitivity approach, on a representative process model. The paper concludes by briefly examining the incorporation of a DO into a two-degrees-of-freedom (2DOF) control structure. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: disturbance observers; disturbance rejection; time-delay systems; continuous-time systems INTRODUCTION Disturbance observers (DOs), are an extremely popular way of improving the disturbance rejection performance of mechatronic systems.[1,2] The approach has similarities to the Internal Model Control (IMC) strategy of Morari and Zafiriou[3] in that both approaches use the inverse of the plant model to reject disturbances as well as to force the input–output characteristics to approximate specified dynamics. DOs have been applied primarily to motion control systems, usually providing one-degree-of-freedom control within a two-degrees-of-freedom (2DOF) control structure. With a DO designed to provide efficient disturbance rejection, the other controller is designed to provide a desirable servo response. DOs do offer several attractive features that could be beneficial for improving the disturbance rejection capability of process control systems. They allow independent tuning of disturbance rejection characteristics, which is particularly helpful in situations in which gains need to be tuned on-line. Compared to integral action, DOs also allow more flexibility via the selection of the order, relative degree, and bandwidth of any low-pass *Correspondence to: R. W. Jones, Mads Clausen Institute for Product Innovation, Syddansk University, Sonderborg, Denmark. E-mail: rjo@mci.sdu.dk 2007 Curtin University of Technology and John Wiley & Sons, Ltd. filtering of the disturbance estimate that takes place. Although the technique of appending disturbance states to a traditional state estimator is well known,[4] the DO structure does allow simple and intuitive tuning of the DO loop gains independently of the state feedback gains. In addition, it has been shown that DOs are also an effective means of suppressing the effects of nonlinearities in a class of nonlinear systems.[5] This property might also prove extremely useful in the control of process systems. The inclusion of continuous time delay into DO design for mechatronic systems was first considered in the mid-1990s but doubts over accuracy, increased calculation time requirements, and parameter sensitivity when a time-delay approximation is used led to a recommendation that the time-delay term be ignored.[6] However, the resulting model mismatch necessitates careful design of the DO and the control bandwidth. The design and implementation of discrete DOs as part of a digital 2DOF control scheme do not have the same implementation problems, as the delay is chosen as a multiple of the sample time (see for example, Ref. [7]) The use of DOs for process control has been previously considered,[8] though only for the Smith predictortype control of integrator plus time-delay systems. The control scheme used for this Smith predictor-type control was originally developed in Ref. [9], where the time-delay term was integrated within the DO design 518 R. W. JONES AND M. T. THAM procedure by mimicking the implementation used for discrete-time systems.[7] This paper introduces the use of DOs for the process control of stable time-delay processes. The disturbance rejection case only is considered, except for some comments relating to set-point response at the end of the paper. A representative first-order plus time-delay (FOPTD) process is used throughout to illustrate the different characteristics of DO design and performance. As the main emphasis is on examining DO design for continuous-time systems with time delay, two possible approaches for the implementation of time-delay within the DO are assessed. The first implementation is the approach used in Refs [8 and 9], where the time-delay term is separated from the nominal plant dynamics, and used to delay the control signal before it is used in the DO. The second implementation considers incorporating an approximation of the time-delay term directly into the inverse of the nominal model, which is used in the DO, to filter the plant output signal. This second implementation is the approach commonly used throughout model-based process control, see for example Ref. [3], and should be addressed in the context of DO design. These different implementations do have implications for the relative order required for the low-pass filter, Q(s), which is the major design parameter for the DO. The limitations placed on DO design and performance by robust stability- and noiseattenuation considerations are then addressed. The paper concludes by briefly examining the issues concerning the incorporation of a DO into a conventional PID control system. Asia-Pacific Journal of Chemical Engineering d c + u + 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Gp(s) y − Q(s)Gq (s) − + Q(s)Gn−1 (s) Figure 1. A continuous-time disturbance observer. Analysis of the DO loop is straightforward and yields the transfer functions from the signal c, and disturbance to output, respectively, as: Gcy (s) = Gp (s)Gn (s) y(s) = c(s) Gn (s) + Q(s)(Gp (s) − Gn (s)Gθ (s)) (1) Gp (s)Gn (s)(1 − Q(s)Gθ (s)) y(s) Gdy (s) = = d(s) Gn (s) + Q(s)(Gp (s) − Gn (s)Gθ (s)) (2) Equations (1) and (2) show why Q(s) is chosen as a unit gain, low-pass filter. In the absence of unmodeled dynamics, i.e. when Gp (s) = Gn (s)Gθ (s), as Q(s) → 1 at low frequencies, Gcy (s) → 1 in Eqn (1) and Gdy (s) → 0 in Eqn (2). Umeno and Hori[2] suggest the following form for the Q(s) filter: 1+ DISTURBANCE OBSERVER DESIGN Figure 1 shows the structure of the disturbance observer for a time-delay process. This implementation is that previously used in Refs [8 and 9]. The controlled output is y, while the disturbance input is d. The signal c is usually provided by an outer loop controller – in process control systems this would normally be a PI or PID controller. In this work we concentrate on the design and disturbance rejection characteristics of the DO. The DO makes use of Q(s), Gn (s), and Gθ (s) in an inner loop around the controlled plant, Gp (s), to reject disturbances. Gn (s)Gθ (s) is the nominal model of Gp (s) : Gn (s) contains the gain, lead, and lag components, while Gθ (s) is a pure time-delay term. The design initially involves choosing the structure of the Q(s) filter such that Q(s)Gn−1 (s) is proper, followed by choosing the coefficients of Q(s). The influence of measurement noise on the system is initially ignored here, partly for clarification purposes. Noise attenuation properties are discussed in the section on sensitivity functions and robustness. + Q(s) = 1+ N −m k =1 N ak (sτq )k (3) ak (sτq ) k k =1 where, for the implementation shown in Fig. 1, m represents the relative degree of Gn (s). The filter time constant is τq . A variety of low-pass filter design methods can be considered for determining the coefficients,ak , with Binomial and Butterworth designs being the most popular. DESIGN FOR CONTINUOUS-TIME SYSTEMS WITH DELAY Consider a continuous-time representation of a linear, time-invariant plant with time delay, θ , and numerator and denominator polynomials B (s) and A(s). Gp (s) = B (s) −θs e A(s) (4) Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY Two different implementations, which differ in the way the time-delay term is incorporated into the DO structure, will be examined. These are: 1. Implementation 1: The time-delay term is separated from the nominal plant dynamics, and used to delay the control signal, u, before it is used in the DO; 2. Implementation 2: This incorporates a low-order approximation of the time-delay term into the nominal model. These different implementations do have implications for the minimum relative degree required for the Q(s) filter. Motion control researchers[2] have found that the lower the relative degree of Q(s), the better the disturbance rejection performance. In the second implementation, an invertible approximation to the time-delay term has to be used. To minimize the increase in the order of the nominal model and also the relative degree required for the Q(s) filter, a (0,1) Pade approximant of the following form is used: e −θs ≈ 1 (θ s + 1) (5) Implementation 1 This is the implementation represented in Fig. 1. The time delay is separated from the other characteristics of the model for use in the DO. For a FOPTD process model, the delay-free plant model component, Gn (s), and the separated time-delay term, Gθ (s), are Gp (s) = kp e −θs , (1 + τ s) Gn (s) = Implementation 2 The commonly used approach, in model-based process control, of incorporating a low-order approximation of the time-delay term, see Eqn (5), into the FOPTD process model is now considered for the second implementation. The nominal model can be written as: Gn (s)Gθ (s) G̃n (s) = kp (1 + sτ )(1 + θ s) (9) The DO is implemented as shown in Fig. 3. Since Q(s)G̃n−1 (s) has to be proper and G̃n (s) is of second order, the Q(s) filter should have a minimum relative degree of 2. With a first-order numerator, the following Q(s) filter design, with Binomial coefficients, will be used: Q13 (s) = 3(sτq ) + 1 (sτq ) + 3(sτq )2 + 3(sτq ) + 1 3 (10) Figure 4 shows the responses of the DO with Q13 (s) using time constants: τq = 0.5τ , τ , and 1.5τ , when a kp , (1 + τ s) Gθ (s) = e −θs A FOPTD process model with a normalized timedelay value of 0.25 is representative of many process systems. For the Q(s) filter design, the time constant τq will be chosen as a function of the time constant of the process. Figure 2 shows the response of three different DOs, corresponding to choices of filter time constants of τq = 0.5τ , τ , and 1.5τ , when a unit-step change in input disturbance was introduced at t = 5.0. As expected, the smaller the value of the filter time constant, the better the disturbance rejection performance. (6) For a first-order nominal model, the relative degree of Q(s) should be at least 1. Although a Q(s) filter with a zero-order numerator could be designed, in this paper only Q(s) filters with a numerator order of at least 1 will initially be considered. In Ref. [10], simple first-order Q(s) filters are used to investigate the use of DOs for reducing interaction in decentralized control systems. From Eqn (3), and using Binomial coefficients, the Q(s) filter has the following form: Q12 (s) = 2(sτq ) + 1 (sτq )2 + 2(sτq ) + 1 (7) where the subscript ‘12’ denotes a first-order numerator and second-order denominator for the filter. For the FOPTD process model, the following parameters will be used: kp = 1.0, τ = 2.0, θ = 0.5 (8) 2007 Curtin University of Technology and John Wiley & Sons, Ltd. DO performance for a range of τq : Implementation 1. (Solid: τq = 1.0; Dashed: τq = 2.0; Dotted: τq = 3.0). This figure is available in colour online at www.apjChemEng.com. Figure 2. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj 519 520 R. W. JONES AND M. T. THAM Asia-Pacific Journal of Chemical Engineering d u c + + + Gp(s) y − S (s) = 1 − e −θs Q(s), Q(s) Figure 3. tion 2. It is straightforward to define the sensitivity, S (s), and complementary sensitivity, T (s), functions for the DO-controlled system: − + ~ Q(s)Gn−1(s) Disturbance observer: time-delay implementa- Figure 4. DO performance for a range of τq : Implementation 2 with Q13 (s) (solid: τq = 1.0; dashed: τq = 2.0; dotted: τq = 3.0). This figure is available in colour online at www.apjChemEng.com. T (s) = e −θs Q(s) (12) This direct link between the choice of Q(s) and the shaping of the sensitivity functions is another advantageous characteristic of the DO approach. The time delay term in S (s) places a limitation on the attainable disturbance rejection performance. The larger the time delay, the more severe the limitation. Figure 5 shows the sensitivity, S (s), and complementary sensitivity, T (s), functions for both implementations of the DO investigated in the previous section, with τq = 1. It can clearly be seen that Smax (s) for the Q12 (s) filter (solid line) is at a higher frequency than Smax (s) for the Q13 (s) filter (dashed line). The wider bandwidth of this filter translates into the superior disturbance rejection performance demonstrated earlier. The complementary sensitivity function provides insight into the noise attenuation at higher frequencies. It can be seen that the high-frequency roll-off of T (s) for the Q13 (s) filter (dashed line) begins at a lower frequency and is much steeper than T (s) for the Q12 (s) filter (full line), resulting in better noise attenuation. The competing requirements of the sensitivity functions brings out the classical trade-off in feedback control; namely, good tracking and disturbance rejection (S (s) small and T (s) large) must be balanced by minimizing the effect of measurement noise (S (s) large and T (s) small). Figures 6 and 7 compare the performance of both implementations of the DO, τq = 1, in the presence of zero mean measurement noise. The outputs in Fig. 6 unit step change in input disturbance was introduced at t = 5.0. Although the response trends were similar and zero-offset disturbance rejection was achieved, on comparing Figs 2 and 4 it can be seen that Implementation 2 does not provide as tight a control as Implementation 1; peak values and undershoot are generally larger and settling times longer. Though the τq can be reduced to improve the comparative performance of the DO, this approach will always provide slightly inferior performance, in the noise-free case, owing to the higher relative order of the Q(s) filter. This supports the finding from the use of DOs on mechatronic systems. SENSITIVITY FUNCTIONS AND ROBUSTNESS From Fig. 1, and assuming no model uncertainty, the open-loop transfer function for the DO system can be found to be: Gol (s) = −θs e Q(s) 1 − e −θs Q(s) (11) 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 5. Sensitivity and complementary sensitivity func- tions, S(s) and T(s), for Q12 (s) (solid) and Q13 (s) (dashed) with τq = 1. This figure is available in colour online at www.apjChemEng.com. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY multiplicative perturbation, Gp (s) = Gn (1 + (s)) (13) where (s) is due to the unmodelled dynamics. Robust stability of the DO loop is assured if the following condition[3] is satisfied: max |T (s)(s)| ≤ max |T (s)(s)| < 1. s=j ω s=j ω (14) With Q(s) = T (s), the following limitation on the design of Q(s) can be defined: |Q(s)| < Figure 6. DO performance for τq = 1; Q12 (s) (dashed) and Q13 (s) (solid). This figure is available in colour online at www.apjChemEng.com. 1 ∀s = j ω. |(s)| (15) If the time-delay term is ignored, it is easily found from Eqn (13) that:[6] (s) = e −sθ − 1 (16) The magnitude plot of 1/ provides the limit for the robust stability condition, Eqn (15). This is shown as the upper solid line in Fig. 8 for θ = 0.5. The magnitude plots for the Q12 (s) and Q13 (s) filters for τq = 1 (solid line and dashed line, respectively) are also shown. It can be seen that these designs do not cross the magnitude plot of 1/ and hence both satisfy the robust stability condition. For the Q12 (s) filter τq can be reduced to 0.8 before the stability condition is reached (dotted line), while for the Q13 (s) filter τq can be reduced to 0.58 (dashpot line). Figure 7. Control values for DOs, τq = 1; Q12 (s) (dashed) and Q13 (s) (solid). This figure is available in colour online at www.apjChemEng.com. indicate that there is little difference between the responses, though the Q13 (s) filter-based DO (solid line) appears to be slightly better than the Q12 (s) filterbased DO (dotted line). Comparing the control values in Fig. 7, the variance of the Q13 (s) filter-based DO control signal (solid line) is noticeably lower than that of the Q12 (s) filter-based DO control signal (dotted line). This is to be expected from the characteristics of the complementary sensitivity functions demonstrated in Fig. 5. The complementary sensitivity function also plays a key role in determining the robust stability of the system. If the unmodeled dynamics are treated as a 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Complementary sensitivity functions and the robust stability limit. Q12 (s) for τq = 1 (solid), Q13 (s) for τq = 1 (dashed), Q12 (s) for τq = 0.8 (dotted), Q13 (s) for τq = 0.58 (dash-pot). This figure is available in colour online at www.apjChemEng.com. Figure 8. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj 521 522 R. W. JONES AND M. T. THAM Asia-Pacific Journal of Chemical Engineering A MORE REALISTIC PROCESS EXAMPLE Table 2. IAE of disturbance rejection responses. Consider the following high-order damped process model which has been used as a representative process system in numerous publications, see Refs [11,12]: FOPTD-based DO Gp (s) = 1 2.989 SOPTD-based DO PID with Ms = 1.4 PID with Ms = 2.0 2.664 4.020 2.712 (17) (1 + s)5 Since the implementation of DOs for time-delay systems is the focus of this paper, both FOPTD and secondorder plus time-delay (SOPTD) models of Eqn (17) will be used for the DO design. In a practical situation, the high-order model dynamics would be approximated as an FOPTD or SOPTD model. Implementation 1 is used for the DO design throughout this section. These models are shown below and were identified using a relay-based identification technique.[11] Ĝp1 (s) = e −2.93s , (1 + 2.73s) Ĝp2 (s) = e −1.73s (1 + 1.89s)2 (18) The use of the FOPTD for DO design requires a Q12 (s) filter, while the use of the SOPTD for DO design requires a Q13 (s) filter. In both of these, the filter time constant was chosen as τq = 1.365 which is half the value of the time constant in the FOPTD approximate model. As before, the coefficients of the respective Q(s) filters were Binomial coefficients. To place the performance of the two DOs in some context, they will be compared with the performance of a PID controller tuned using the maximum sensitivity, Ms , the design approach of Panagopoulos et al .[12] This very powerful design approach uses the high-order process model Eqn (17) in conjunction with nonconvex optimization to design the PID parameters. The primary design goal of the approach is to provide superior disturbance rejection; hence it will provide an excellent test for the capabilities of DOs. The structure of the PID controller used is: u(s) = kc (r(s) − y(s)) + kd y(s) ki (r(s) − y(s)) −s s 1 + τf s (19) where kc is the proportional gain, ki the integral gain, kd the derivative gain, and Tf is the time constant of the first order filter on the derivative term. Table 1 shows two sets of PID controller parameters derived using the maximum sensitivity design approach on the higherorder model, Eqn (17).[12] Table 1. PID controller parameters. Ms kc ki kd Tf 1.4 2.0 0.7840 1.4700 0.2925 0.6309 0.9722 1.8375 0.0972 0.1838 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 9. Outputs: PID and DO control of a high-order process. (solid: DO using FOPTD model; dotted: DO using SOPTD model; dash-pot: PID with Ms = 1.4; dashed: PID with Ms = 2.0). This figure is available in colour online at www.apjChemEng.com. A unit step disturbance was introduced at t = 5.0, and the responses of the DOs and PID controllers are shown in Fig. 9. The DO based on the SOPTD model with the Q13 (s) filter (dotted line) provided the best disturbance rejection. This is confirmed by the Integral of Absolute Error (IAE) between controlled output and setpoint of the responses as listed in Table 2. In this example, the use of the more accurate SOPTD model overcame the disadvantage of using a Q(s) filter with a higher relative degree. Its performance was very similar to that obtained using the PID controller tuned to give a maximum sensitivity Ms = 2.0, and significantly better than the controller tuned to achieve Ms = 1.4. This is notable, as both PID controllers were designed using the exact model of the controlled process. Indeed, the performance of the DO designed using the approximated FOPTD model was only slightly worse than the PID controller tuned to a specification of Ms = 2.0. Figure 10 shows the corresponding calculated control values for each of the output responses. TWO-DEGREES-OF-FREEDOM CONTROL: PID AND DO The use of DOs has, so far, only been considered for disturbance rejection purposes. In the previous Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY for this 2DOF controller structure can be written as: y(s) = + Figure 10. Control values: PID and DO control of a highorder process. (solid: DO using FOPTD model; dotted: DO using SOPTD model; dash-pot: PID with Ms = 1.4; dashed: PID with Ms = 2.0). This figure is available in colour online at www.apjChemEng.com. section the disturbance rejection performances of both DOs demonstrate resilience to mismatch in temporal characteristics between the process and the model. It can also be seen from Eqn (2) that zero-offset regulation is independent of the accuracy of the process model and can be achieved as long as the Q(s) filter has unit gain. For set-point tracking, however, offset-free performance can be accomplished only if the model and process gains are identical. However, any offset that can be predicted from Eqn (3) can be removed by a scheme similar to that shown in Fig. 11, i.e. the DO is implemented as part of a feedback control system that contains an integral term. The integral term in the controller ensures an offset-free set-point response behavior. This control scheme is likely to be the most common way in which the DO would be used in the process industries. Although the selection of Q(s) is straightforward with a clear physical interpretation when implemented as shown in Fig. 11, the robust stability of the system now depends on the design of Gpid (s) as well as Q(s). In the absence of unmodeled dynamics, i.e. when Gp (s) = Gn (s)Gθ (s), the closed-loop transfer function r + − Gpid (s) c u + + + Gpid (s)Gn (s)Gθ (s) r(s) 1 + Gpid (s)Gn (s) Gn (s)Gθ (s)(1 − Q(s)Gθ (s)) d(s) 1 + Gpid (s)Gn (s) (20) The design of the disturbance rejection and set-point responses in Eqn (20) are obviously not independent. The control of the FOPTD plant model Eqn (8) will now be addressed using this control structure. The implemented DO uses a Q12 (s) filter, Eqn (7), with τq = 1.0. The disturbance rejection performance using only the DO is shown in Fig. 2. The feedback controller, Gpid (s), is in this case a PI controller and designed specifically to provide a good set-point response, kc = 2.0, ki = 1.0. The full line in Fig. 12 shows the corresponding set-point and disturbance rejection response of this system with a step input disturbance of −1 entering the system at t = 9. The addition of the feedback PI controller has modified the disturbance rejection response demonstrated earlier in Fig. 2. The disturbance rejection response is now slightly more aggressive with an appreciable undershoot. The performance of this interacting 2DOF control structure in the presence of plant/nominal model mismatch is also examined. With a plant gain of 0.8 and nominal model gain = 1, the response in Fig. 12 (dotted line) shows an increase in percentage overshoot for the set-point response and a longer settling time. With a plant gain of 1.2 and nominal model gain = 1, faster oscillatory modes are induced in the set-point response with a similar overshoot as in the ideal case (dashed line). In both cases of model uncertainty, the addition of a PI controller obviously compensates for any gain mismatch between the d Gp(s) y − Q(s)Gq(s) − + Q(s)Gn−1(s) Figure 11. DO in conjunction with a feedback controller. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 12. Interacting 2DOF control of the FOPTD process. (solid: no uncertainty; dotted: process gain = 0.8; dashed: process gain = 1.2). This figure is available in colour online at www.apjChemEng.com. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj 523 524 R. W. JONES AND M. T. THAM Asia-Pacific Journal of Chemical Engineering plant model and the nominal model to provide offsetfree control. For a SOPTD plant model, the controller should be chosen as a PID controller to provide offsetfree control in the presence of plant/nominal model mismatch. A simple, straightforward approach to decoupling the set-point response design and the disturbance rejection design is now implemented, see Ref. [8]. Instead of using plant output feedback in conjunction with the controller, the controller output is fed back, filtered by the nominal model transfer function, and then compared with the set-point signal. The output of the comparator, (r(s) − u(s)Gn (s)), becomes the controller input. For this 2DOF control structure, the closedloop transfer function, in the absence of unmodeled dynamics, becomes: C (s)Gn (s)Gθ (s) r(s) 1 + C (s)Gn (s) + Gn (s)Gθ (s)(1 − Q(s)Gθ (s))d(s) y(s) = and disturbance rejection response of this system with a step input disturbance of −1 entering the system at t = 9. The disturbance rejection response can be seen to be exactly the same as that obtained earlier in Fig. 2. Also, the decoupled structure allows a more aggressive setpoint response to be designed using C (s). The performance of this structure in the presence of plant/nominal model mismatch is also examined. With a plant gain of 0.8 and nominal model gain = 1, the response in Fig. 13 (dotted line) shows a decrease in percentage overshoot for the set-point response and a longer settling time. With a plant gain of 1.2 and nominal model gain = 1, a greater percentage overshoot is accompanied by a shorter settling time (dashed line). Comparing Figs 12 and 13, the response of the decoupled structure to plant/nominal model mismatch is preferable. CONCLUSIONS (21) where C (s) represents the forward path controller. With this decoupled 2DOF control structure the disturbance rejection and set-point responses can be designed independently. The DO provides the disturbance rejection, while C (s) is used to shape the set-point response. For type 0 plants, to ensure that there is no steadystate error in the face of plant/nominal model mismatch, C (s) needs to include an integral term. The control of the FOPTD plant model Eqn (8) is again addressed using the decoupled 2DOF control structure. The implemented DO uses a Q12 (s) filter, Eqn (7), with τq = 1.0. The forward path controller is a PI controller with parameters, kc = 4.0, ki = 4.0. The full line in Fig. 13 shows the corresponding set-point Two DO implementations for continuous-time systems with delay have been investigated. The approach in which the time-delay term is separated from the nominal plant dynamics and then used to delay the control signal before it is used in the DO provides the best disturbance rejection performance in a noise–free environment, though the approach in which the timedelay term is absorbed into the nominal model has superior noise attenuation characteristics. In a practical situation in which robustness and noise attenuation are important considerations, the more conservative approach, that is the approach where the time-delay term is absorbed into the nominal model, would perhaps be preferred. The choice of the Q(s) filter time constant as half the value of the process time constant, when the process was modeled as a FOPTD, was found to provide good disturbance rejection responses. The results were comparable to those obtained using PID controllers tuned for disturbance rejection using a powerful maximum sensitivity-based design approach. It was also shown that the DOs were quite robust to process–model mismatch in temporal parameters. Future work will investigate the use of error-based, as opposed to output-based, DOs for the control of process systems.[13] Current work involves investigating the use of a DO scheme to provide stiction compensation in flow valves. REFERENCES Figure 13. Decoupled 2DOF control of the FOPTD process. (solid: no uncertainty; dotted: process gain = 0.8; dashed: process gain = 1.2). This figure is available in colour online at www.apjChemEng.com. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. [1] K. Ohishi, M. Nakao, K. Ohnishi, K. Miyachi, IEEE Trans. Ind. Electron, 1987; IE–34, 44–49. [2] T. Umeno, Y. Hori, Proc. IEEE IECON’89, 1989; Vol 2, pp.313–318. [3] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs, N.J., 1989. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY [4] G.C. Goodwin, K.S. Sin, Adaptive Filtering Prediction and Control, Prentice–Hall, Englewood Cliffs, N.J., 1984. [5] S.M. Sharuz, C. Cloet, M. Tomizuka, Suppression of effects of nonlinearities in a class of nonlinear systems by disturbance observers. In Proceedings of the American Control Conference, Anchorage, USA, 2002; pp.2340–2345. [6] C.J. Kempf, S. Kobayashi, IEEE Trans. Control Syst. Technol., 1999; 7(5), 513–526. [7] A. Tesfaye, H.S. Lee, M. Tomizuka, IEEE/ASME Trans. Mechatron., 2000; 5(1), 32–38. [8] Q.-C. Zhong, J.E. Normey-Rico, IEE Proc. Control Theory Appl., 2002; 149(4), 285–290. [9] K. Hong, K. Ham, IEEE Trans. Ind. Electron., 1998; 45(2), 283–290. 2007 Curtin University of Technology and John Wiley & Sons, Ltd. [10] R.W. Jones, M.T. Tham, Proceedings of SICE-ICASE International Conference, Republic of South Korea, Busan, 2006; pp.5200–5205. [11] W.K. Ho, C.C. Hang, L.S. Cao, Automatica, 1995; 31(3), 497–502. [12] H. Panagopoulos, K.J. Astrom, T. Hagglund, IEE Proc.Control Theory, 2002; 149(1), 32–40. [13] K. Yang, Y. Choi, W.K. Chung, I.H. Suh, S.R. Oh, Robust tracking control of optical disk drive systems using error based disturbance observer and its performance measure. In Proceedings of the American Control Conference, Anchorage, USA, 2002; pp.1395–1400. Asia-Pac. J. Chem. Eng. 2007; 2: 517–525 DOI: 10.1002/apj 525

1/--страниц