Control of processes with dead time and input constraints using control signal shaping

Control of processes with dead time and input constraints using control signal shaping Q.-C. Zhong and C.-C. Hang Abstract: Using the idea of shaping the control signal, the authors generalise the time-delayfilter-based deadbeat control for processes with dead time to a two-level control so that actuator saturation is avoided. The controller mimics experienced manual operation to provide a two-level control signal. At the first stage (level), the controller outputs a value very close to the actuator saturation bound to provide the largest acceleration and then the controller outputs a smaller value to maintain the steady-state output at the same level as the set point. The system quickly settles in a finite time, which is explicitly determined by the saturation bound and is independent of the controller and (almost) of the sampling period. The disturbance response can be freely tuned according to the desired phase or gain margin. Three examples are given to show the effectiveness of the proposed controller. 1 Introduction \Regulation is often the main task of many of the controllers used in industry, however, in many cases it is also very important to obtain a fast response to set-point changes. These two control problems can be decoupled using twodegree-of-freedom (2-DOF) techniques; see [1 3] for delayfree systems and [4 6] for delay systems, where a pre-filter is frequently used as the second DOF to weight the set-point change so that the set-point response is desirable. Pre-filters that have particular properties have been studied in the literature. A variable set-point weighting scheme was proposed in [7], where adaptive techniques were used to adjust the set-point weighting. A deadbeat set-point response was obtained by using a time-delay filter in [8], where the reference signal for the closed-loop control system is converted to a pulse-step signal from the original step signal. An optimal control strategy, called pulse-step control, was proposed in [9] to obtain a settling time close to the time at which the impulse response reaches its maximum (when there is almost no input constraint). This is a feedforward control law used with a proportional-integral-derivative (PID) feedback controller for regulation. The major drawback of the approach is that a set of nonlinear equations has to be solved to obtain the optimal switching times, which complicates the design. A minor drawback of the proposed controller is that it is only available for systems having a relative degree higher than two, which means that it is not directly applicable for most of the chemical processes that are modelled as a first-order plus dead-time (FOPDT) model. While expecting a fast set-point response, it is also important for the response to have little or no overshoot. q IEE, 2004 IEE Proceedings online no. 20040553 doi: 10.1049/ip-cta:20040553 Paper first received 27th August 2003 and in revised form 22nd March 2004 Q.-C. Zhong is with the School of Electronics, University of Glamorgan, Pontypridd, CF37 1DL, UK and is also with the College of Electrical and Information Engineering, Hunan University, Changsha 410082, China C.-C. Hang is with the Department of Electrical & Computer Engineering, National University of Singapore, Singapore 119260, Singapore An interesting time-domain method was proposed in [10] to design an optimal PID controller. Minimising the switch time weighted squared error criterion the step response achieves a negligible overshoot. The cost paid for this is a relatively slow response because, for a conventional PID control system without using special techniques, little or no overshoot simply means that the system is not fast enough; see Section 2.1 for further details. Another important issue when attempting to obtain a fast response is that the input constraint has to be taken into account, at least to some extent. The approach proposed in [8] offers a deadbeat set-point response, which is very fast. However, there can be problems in practice when using this control strategy. First of all, the first pulse in the converted set point has a relatively large amplitude. This might cause the actuator to become saturated and the system performance therefore becomes degraded. Secondly, the set-point response was designed to reach the desired output in one sampling period, in addition to the inherent dead time. This might push the controller too hard and, intuitively, it might be difficult to obtain such a fast response. We intend to add more freedom to the controller and explicitly consider the actuator saturation in the design by shaping the control signal. The previously mentioned two drawbacks are avoided and hence the proposed control strategy is more practical. The response is still deadbeat and the deadbeat time is dependent on the saturation bound of the controller. A prominent property of the controller is that it provides a two-level control signal, which mimics experienced manual operation: the control outputs a value close to the saturation bound to provide the largest acceleration for the process at first and then the control outputs a smaller value to keep the system output at the set point. Since the controller provides the largest acceleration at the first stage, the step response is the fastest among those achievable. The control action at the second stage guarantees that this fastest response has no overshoot. 2 Controller design Since a large number of chemical processes can be modelled by the following FOPDT model: IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 473

Fig. 1 The control structure GðsÞ ¼ Ke ts Ts þ 1 where K is the static gain, t is the dead time and T is the apparent time constant, the controller will be designed for this model. However, the proposed technique can be easily applied to multi-lag processes, as will be shown by an example in Section 3. The control structure under consideration is shown in Fig. 1, where r 0 is the converted set point and e is the error signal (which is not r y here). The controller consists of a compensator C(z) and a feedforward controller F(z). Taking into account the sample-hold effect, the generalised plant is: GðzÞ ¼K 1 a z a z l where l ¼ t=t s is a positive integer and a ¼ e T s=t : Assume that the compensator C(z) is: CðzÞ ¼ ð1 az 1 ÞNðzÞ DðzÞ ð1þ where the order of the polynomials N(z) and D(z) inz 1 is n and m, respectively, then the closed-loop transfer function of the feedback control system is: Kð1 aþnðzþz ðlþ1þ T yr ðzþ ¼FðzÞ ð2þ DðzÞþKð1 aþnðzþz ðlþ1þ and the transfer function from the set point r to the control signal u is: ð1 az 1 ÞNðzÞ T ur ðzþ ¼FðzÞ ð3þ DðzÞþKð1 aþnðzþz ðlþ1þ Note that it is not necessary to place a zero z ¼ a in the controller (1). However, this does not affect the explanation of our key idea and, as can be seen later, this does simplify the controller design and is helpful for system stability and performance. 2.1 Shaping the control signal Figure 2a shows a typical control signal in a unity feedback control systems, as shown in Fig. 1 with FðzÞ ¼1; involving a PI controller. The control signal consists of three stages: (i) due to the integrating effect of the controller, the control signal increases in stage I until the actuator saturates; (ii) the integrator winds up and the actuator saturates; and (iii) the control signal settles down. If the proportional gain is very large, stage I may disappear. In order to obtain a fast setpoint response the proportional gain is expected to be large, however, it cannot be too large otherwise the actuator saturates very quickly. This means that the potential of the controller is often not fully used to speed up the system response, as can be seen from the shaded area in Fig. 2a. The integrator windup in stage II requires that the error signal has an opposite sign for a long period to drag the integrator back to normal. This causes a large overshoot and a long settling time. The oscillatory stage III is not desirable either, which causes a long settling time. There are some 474 Fig. 2 Control signals in different shapes a A typical control signal associated with a PI controller b The desired control signal when the set-point change is the bound r of all step changes techniques available to avoid windup of an integrator [11]. We intend to propose a different technique to avoid the actuator saturation and take full use of the controller potential to speed up the system response while reducing the overshoot. In order to obtain a fast transient response, the desired control signal (with respect to a step set-point change) is shown in Fig. 2b, where the set-point change, r; is assumed to be the bound or typical amplitude of allowable step changes. This is in order to mimic the experienced manual operation: the controller outputs a value very close to the saturation bound u to drive the plant as hard as possible at the early stage and then the controller outputs a smaller value to keep the system output at the set point. This results in a two-level control signal. Assume that the moment at which the change occurs is ðn þ 1ÞT s ; then this kind of signal can be approximately denoted as: 1 a nþ1 z n 1 Kð1 a nþ1 Þ r This corresponds to the following desired transfer function from r to u: TurðzÞ d ¼ 1 anþ1 z n 1 Kð1 a nþ1 ð4þ Þ The amplitude of the first level of the control signal varies when the order n of N(z) changes. For a given control bound u and a given bound r of the set-point changes, the following IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

condition has to be satisfied to keep the controller output below and close to the saturation bound u : r Kð1 a nþ1 Þ u ð5þ which requires n T ln K u T s K u r 1 ð6þ The smaller the saturation bound u; the larger the value of n; the larger the ratio T=T s ; the larger the value of n. It is worth noting that n does not depend on the dead time of the plant. The idea of shaping is not new: the input-shaping technique [12, 13] is an open-loop control approach using delay elements to eliminate any residual vibration of flexible structures; set-point pre-filtering (and the more complicated reference trajectory) has been widely used model predictive control [14]; and a reference model is often used to provide the ideal response for the compensator to track. These techniques shape the set point of a control system. We intend to shape the control signal in a feedback control system. 2.2 Design of F(z) and N(z) The objective of the controller design is to obtain a control signal in the two-level form shown in Fig. 2b by choosing F(z) and C(z), or equivalently, F(z), D(z) and N(z). Simply, F(z) can be chosen to cancel the closed-loop poles provided that the closed-loop system is stable, i.e.: FðzÞ ¼ DðzÞ Kð1 aþ þ NðzÞz ðlþ1þ ð7þ As a result, the two transfer functions in (2) and (3) become: T yr ðzþ ¼NðzÞz ðlþ1þ ð8þ and 1 az 1 T ur ðzþ ¼NðzÞ ð9þ Kð1 aþ Since the output y is expected to start just after the inherent dead time, N(0) cannot be zero. As a result, D(0) cannot be zero in order to guarantee the causality of the controller. According to (9) and (4), the desired N(z) can be derived as: NðzÞ ¼ 1 anþ1 z n 1 1 az 1 1 a P ni¼0 1 a nþ1 ¼ a i z i P ni¼0 a i ð10þ Obviously, N(z) satisfies the constraint Nð1Þ ¼1 to guarantee the zero steady-state error for the set-point response, according to (8). The set-point response reaches the steady state at: ðl þ n þ 1ÞT s t þ T ln K u K u r where the is due to the approximate selection of n in (6). For a given plant, this time is independent of the controller and the sampling period and solely depends on the saturation bound u: In other words, this is an inherent characteristic of the system and there is no way to make the response any faster. There is no braking control (a large negative action, which is common in the time-optimal control strategy) in the control signal. This is because: (i) there is no need for such a brake here because the response reaches the steady state in a finite time and there is no overshoot; (ii) the benefit of a large negative action is very small when u is not very large [9], which is the common case in practice; and (iii) the control strategy is more sensitive when there is a large negative control action [15]. The function of D(z), N(z) and F(z) is now clear: D(z) is designed to guarantee the stability of the closed-loop system; F(z) is designed to obtain the desired set-point response and N(z) is designed to shape the control signal as well as the set-point response. 2.3 Design of D(z): Stability of the closed-loop system There are many options to design the compensator, in particular D(z), because the function of D(z) is to guarantee the system stability. One simple option is to choose D(z) to be an integrator: DðzÞ ¼ 1 z 1 K I where K I is an integral coefficient such that the steady-state error with respect to step disturbances is zero. However, this normally brings a sluggish or oscillatory disturbance response. A better design is to choose: 1 z 1 DðzÞ ¼ NðzÞ ð11þ K I to obtain a better stability margin and a faster disturbance response. Another advantage by doing so is that the stability analysis and the parameter tuning can be considerably simplified. This choice of D(z) offers the following PI controller: CðzÞ ¼ ð1 az 1 ÞNðzÞ DðzÞ ¼ K I 1 az 1 1 z 1 ð12þ which is, by chance, almost the same as the one proposed in [8]. The corresponding open-loop transfer function is: LðzÞ ¼CðzÞGðzÞ ¼ K IKð1 aþ ðz 1Þz l ð13þ A typical root locus of this system with respect to K I is shown in Fig. 3. Since L(z) has no zero, all the ðl þ 1Þ loci approach asymptotically to ðl þ 1Þ straight lines, which start at z ¼ 1=ðl þ 1Þ with angles: Fig. 3 Typical root locus of the closed-loop system IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 475

y ¼ 2k þ 1 l þ 1 180 ðk ¼ 0; 1; 2; ; lþ The ðl þ 1Þ loci can be categorised into two groups: one is the ðl 1Þ loci starting at the origin and then approaching 1; the other is the two loci starting at z ¼ 0 and z ¼ 1; which meet together and then break to approach 1 (if there is no zero z ¼ a in the controller, then the locus starting at the origin in the second group starts at z ¼ a; which is on the right-hand side of z ¼ 0). Since the pole z ¼ a of the plant is cancelled by the zero z ¼ a in the controller, the two loci closest to the right-half real axis are pushed towards the lefthalf plane. This means that a larger gain margin or a faster disturbance response can be obtained. This is why such a zero is placed in the controller (1). As can be seen from the root locus in Fig. 3, there always exists a critical gain K Ic such that only one real pole or a pair of complex poles arrive(s) at the unity circle and the others remain inside the unity circle. For any gain 0 < K I < K Ic ; the closed-loop system is always stable. Since the open-loop transfer function (13) is very similar to that obtained in [8], the stability lemma obtained there still holds and is cited below. Lemma 1: [8] The closed-loop system designed above is stable if: 2 0 < K I < Kð1 aþ sin p 4l þ 2 In order to obtain a phase margin of f m ; the integral coefficient K I can be chosen as: 2 K I ¼ Kð1 aþ sin p 2f m 4l þ 2 In order to obtain a gain margin of g m ; the integral coefficient K I can be chosen as: 2 p K I ¼ sin Kð1 aþg m 4l þ 2 Proof: see [8]. When n ¼ 0; we have NðzÞ ¼1: This offers: T yr ¼ z ðlþ1þ 1 az 1 and T ur ¼ Kð1 aþ These responses are the same as those obtained in [8]. According to (6) or (5), the saturation bound should not be less than 1=ðKð1 aþþ: Otherwise, the approach proposed in [8] makes the actuator saturated and hence it cannot be used. 2.4 An alternative implementation The control system designed above can be implemented in an alternative structure shown in Fig. 4, where the filter F u ðzþ is: F u ðzþ ¼ 1 anþ1 z n 1 Kð1 a nþ1 Þ and G m ðzþ is the model of the nominal plant, i.e.: G m ðzþ ¼K 1 a z ðlþ1þ 1 1 az The desired set-point response is y m ðzþ ¼G m ðzþf u ðzþrðzþ and, for rðzþ ¼1; it is y m ðzþ ¼NðzÞz ðlþ1þ : This control structure, which has appeared in [9], can be regarded as an 476 A Fig. 4 An alternative implementation open-loop controller F u ðzþ combined with a closed loop controller C(z). F u ðzþ is used to supply a desired control profile u o while the desired set-point response y m is supplied through an ideal model of the plant G m ðzþ: C(z) is designed to govern the disturbance response, the stability and the robustness. The control signal u consists of two terms: u ¼ u o þ u c where u o is the desired (open-loop) control signal and u c is the contribution of the (closed-loop) compensator C due to disturbances and model uncertainties. When there is no disturbance or model uncertainty, u c is always zero and the system works in open loop. The compensator C works only when it is needed: when there are model uncertainties and =or disturbances. This structure well combines open-loop control with closed-loop control. It can also be regarded as an extension of the input-shaping technique [12, 13, 16] to deal with model uncertainties and disturbances. Indeed, the s-domain equivalent of the pre-filter F u ðzþ is: ^F u ðsþ ¼ 1 e ðnþ1þðtsþ1þt s=t Kð1 a nþ1 Þ This is a typical input shaper [12, 13, 16]. Moreover, the nominal transfer function from r to y (in the s-domain): T yr ðsþ ¼^F u ðsþgðsþ ¼ 1 aðnþ1þðtsþ1þ ð1 a nþ1 ÞðTs þ 1Þ e ts is a pseudo-differential polynomial [17, 18] and hence the nominal response y to a step change r is deadbeat [19, 20]. There are some advantages in the use of this structure. It is clear that the desired control signal can be designed in an open-loop way if the plant is stable. The extension is the injection of this desired control signal into the model G m of the process and thus obtaining the error between the model output y m and the process output y for error feedback. Another advantage of this implementation is that the compensator C does not affect the shape of the control signal, which is not explicit in the case discussed before (where N(z) is a part of the controller). This means that the controller may not be limited to a PI controller as designed before. In other words, the proposed technique can be regarded as a bolt-on any standard well-tuned PID controllers [11, 21]. The compensator C can then be designed to meet the requirement of robustness and disturbance rejection. This issue has been extensively studied in the literature. The sampling periods for the feedforward controller F u ðzþ and the feedback loop can be different to give more freedom to the design of the compensator. Using this structure, it is also possible to extend the proposed idea to general high-order plants. IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

3 Examples Three examples will be studied in this Section. The first example is an FOPDT having a long dead time; the second one is an FOPDT having a short dead time; the third one is a multi-lag process which can be modelled as an FOPDT model. In the first example, attention will be focused on showing the relationship between the converted set point, the error signal, the control signal and the set-point response. In the second example, attention will be focused on the comparison of the disturbance response with respect to well-tuned PI controllers for both nominal and uncertain cases. In the last example, attention will be focused on the effectiveness of the proposed method for multi-lag processes, i.e. robustness of the controller. 3.1 A process with a long dead time Consider the following FOPDT model having a long dead time: Fig. 6 The converted set point (m ¼ 5; l ¼ 20; T s ¼ 0:25) GðsÞ ¼ e 5s s þ 1 which was studied in [8]. Here, we choose the sampling period as T s ¼ 0:25 s: The parameters of the generalised plant are a ¼ 0:7788 and l ¼ 20: Assume that the upper bound of the controller output is u ¼ 1:45 and the typical amplitude (or bound) of step changes is r ¼ 1; then n 3:68: We choose n ¼ 4: As a result: NðzÞ ¼0:31 þ 0:2414z 1 þ 0:1881z 2 þ 0:1464z 3 þ 0:1141z 4 The parameter K I is chosen as K I ¼ 0:173 to obtain a phase margin of 45 : The corresponding distribution of the closedloop poles is shown in Fig. 5: all the 21 poles are inside the unity circle. The converted set point r 0 for r ¼ 1 is shown in Fig. 6 and the error signal is shown in Fig. 7. The set point has been changed into four Sections: the first Section is a staircase signal larger than the original value due to the effect of D(z) in F(z); the second Section is zero (this piece disappears if m>l); the third Section is another staircase signal due to the effect of N(z)inF(z); the last Section is the original set point. It is worth noting that the large magnitude in the first Section, which is simply a value in the controller, does not cause the actuator to be saturated. The error signal Fig. 7 The error signal e ¼ r 0 y is very clean and it remains zero after m steps, even when the system output is increasing from zero to the steady state. The control signal, as shown in Fig. 8, has two levels and stays below the upper bound. The system output reaches the steady state after n þ 1 ¼ 5 sampling steps, in addition to the inherent dead time of the plant, as shown in Fig. 8. 3.2 A process with a short dead time Consider the following FOPDT model having a short dead time: GðsÞ ¼ e 0:5s s þ 1 Fig. 5 The distribution of the closed-loop poles Fig. 8 Example 1: the output y and the control signal u IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 477

Fig. 9 Example 2: the output y and the control signal u which was studied in [8, 22]. Here, we choose the sampling period as T s ¼ 0:25 s: The parameters of the generalised plant are a ¼ e 0:25 and l ¼ 2: Assume that the upper bound of the controller output is u ¼ 1:45 and the typical amplitude (or bound) of step changes is r ¼ 1; then n 3:68: We choose n ¼ 4: As a result: NðzÞ ¼0:31 þ 0:2414z 1 þ 0:1881z 2 þ 0:1464z 3 þ 0:1141z 4 As mentioned before, N(z) does not depend on the dead time of the plant. The N(z) obtained here is the same as the one obtained in the previous example. K I is chosen as 1.101 to obtain a phase margin of 55 : The set-point response and the corresponding control signal are shown in Fig. 9. The control signal changes from a larger value to a smaller value at the m ¼ n þ 1 ¼ 5th step and the system output reaches the steady state at the l þ n þ 1 ¼ 7th step: There is no oscillation in the control signal and no overshoot in the system response. The simulation results are compared with two discreteversion PI controllers, of which the continuous ones K p ð1 þ 1=T i sþ are tuned by Zigler-Nichols method (noted as Z-N in Figures) and the Ho-Hang-Cao method [22] (noted as H-H- C in Figures). The parameters tuned by the Z-N method are K p ¼ 1:8 and T i ¼ 1:5 s and the parameters tuned by the Ho-Hang-Cao method are K p ¼ 1:05 and T i ¼ 1:0 s to obtain a gain margin of 3 db and a phase margin of 60 : All the controller outputs in the simulations are bounded by u ¼ 1:45: A step disturbance d ¼ 0:2 acts at t ¼ 7 s: The control signals, the system outputs and the error signals in the nominal case are shown in Fig. 10. The responding speeds with respect to the set-point changes are very close to each other, however, the response obtained by the proposed controller has no overshoot whereas the other two have about a 20% overshoot. The settling time obtained by the proposed controller is much shorter than the other two. The excellent set-point response does not cause a degradation of the disturbance response. In fact, the disturbance response is still the best (slightly better than that obtained by H-H-C). As for the control signal, there are only two levels in the control signal (in the part corresponding to the set-point change) of the proposed controller, but there are many levels in the other two. The proposed controller does not saturate but both the other two do saturate. A prominent property can be seen from the error signal shown in Fig. 10c: the error signal e (see Fig. 1) remains at zero after the effect of D(z), i.e. after m steps, until there is a load disturbance. In other words, the proposed controller does not have error 478 Fig. 10 a The system outputs b The control signals c The error signals Example 2: responses in the nominal case accumulation if there is no disturbance (or uncertainty, of course), but the other two controllers do have error accumulation. When there exists uncertainty in the plant, the proposed controller still behaves much better than the other two, as can be seen from Fig. 11, where the responses in three cases having different uncertainties are shown. This indicates that the sampling period T s may be chosen such that t is not an exact multiple of T s : 3.3 A multi-lag process Consider the multi-lag process: 1 GðsÞ ¼ ðs þ 1Þ 5 IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

Fig. 12 Example 3: responses subjected to a single set-point change a The system outputs b The control signals NðzÞ¼0:1484 þ 0:1329z 1 þ 0:1191z 2 þ 0:1067z 3 þ 0:0956z 4 þ 0:0856z 5 þ 0:0767z 6 þ 0:0687z 7 þ 0:0616z 8 þ 0:0552z 9 þ 0:0495z 10 Fig. 11 Example 2: responses in three uncertain cases a Case 1: K increased by 20% b Case 2: T increased by 20% c Case 3: t increased by 20% studied in [22]. Here, we design the controller using the following model: GðsÞ ¼ e 3s 2:73s þ 1 ð14þ which is slightly different from that given in [22]. The sampling period is chosen as T s ¼ 0:3 s and the parameters of the generalised plant are a ¼ 0:8959 and l ¼ 10: Assume that u ¼ 1:45 and r ¼ 1 again, then n 9:65: We choose n ¼ 10: As a result: The integral gain K I is tuned as K I ¼ 0:4791 to obtain a phase margin of 60 : The simulation results are shown in Fig. 12 with a comparison to those (noted by H-H-C in the Figures) obtained by the discrete version of the PI controller 0:49ð1 þ 1=ð2:74sÞÞ tuned in [22] (to obtain a phase margin of 60 and a gain margin of 3 db). Since the proposed controller is designed according to the approximate model (14), the set-point is no longer deadbeat. However, it is still much faster than the one obtained by the PI controller whereas the disturbance response is slightly better. The overshoot is also smaller. The control signals are quite different at the beginning: the proposed control signal outputs a value very close to the actuator saturation bound from the beginning, but the PI controller has to integrate the error signal. The response to a series of set-point changes r ¼ 0:4 att ¼ 0 with an additional step change of 0.6 at t ¼ 20 is shown in Fig. 13. It is possible to design a nonlinear or adaptive pre-filter F u ðzþ in Fig. 4 so that the desired control signal changes according to different set-point changes and the current contribution u c of the compensator C. This is left for future research. Some complicated research on this general topic can be found in [23, 24]. IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 479

processes. When there are uncertainties in the model, e.g. when the dead time is not an exact multiple of the period or the plant is a multi-lag process, the system response will be worse than the nominal response, but it is still better than that obtained by a conventional well-tuned PID controller. 5 Acknowledgments The authors would like to thank the review panel for the helpful suggestions. This work was partially supported by the EPSRC under Portfolio Partnership grant GR/S61256/01. 6 References Fig. 13 Example 3: responses subjected to a series of set-point changes a The system outputs b The control signals 4 Conclusions Instead of shaping the set point, the idea of shaping the control signal has been proposed to offer a two-level control for processes with dead time and input constraints. The control signal (responding to the set-point change) consists of two levels: a large-value level and then a small-value level. The system output reaches the steady state in a finite time, which is determined by the actuator saturation bound and the plant parameters and is independent of the control parameters, even the sampling period. Hence, the sampling period can be freely chosen to obtain a satisfactory disturbance response. The disturbance response is governed by a PI controller, which can be tuned to guarantee a specified gain or phase margin. As a matter of fact, the proposed idea can be regarded as a bolt-on to any welldesigned control systems. In other words, the robustness and the disturbance rejection of the system can be guaranteed by the design of the compensator. Three examples have shown that the two-level control indeed offers a very fast set-point response with no or only a small overshoot. The technique is shown with respect to the FOPDT model, however, as has been shown by an example, it is also applicable to multi-lag 1 Horowitz, I.M.: Synthesis of feedback systems (Academic Press, New York, NY, 1969) 2 Youla, D.C., and Bongiorno, J.J., Jr.: A feedback theory of two-degreeof-freedom optimal Wiener-Hopf design, IEEE Trans. Autom. Control, 1985, 30, (7), pp. 652 665 3 Morari, M., and Zafiriou, E.: Robust process control (Prentice-Hall, Englewood Cliffs, NJ, 1989) 4 Zhong, Q.-C., and Li, H.X.: Two-degree-of-freedom PID-type controller incorporating the Smith principle for processes with deadtime, Ind. Eng. Chem. Res., 2002, 41, (10), pp. 2448 2454 5 Mirkin, L., and Zhong, Q.-C.: 2DOF controller parametrization for systems with a single I/O delay, IEEE Trans. Autom. Control, 2003, 48, (11), pp. 1999 2004 6 Mirkin, L., and Zhong, Q.-C.: Coprime parametrization of 2DOF controller to obtain sub-ideal disturbance response for processes with dead time. Proc. 40th IEEE Conf. on Decision & control, Orlando, FL, USA, December 2001, pp. 2253 2258 7 Hang, C.C., and Cao, L.S.: Improvement of transient response by means of variable set point weighting, IEEE Trans. Ind. Electron., 1996, 43, (4), pp. 477 484 8 Zhong, Q.-C., Xie, J.Y., and Jia, Q.: Time delay filter-based deadbeat control of process with dead time, Ind. Eng. Chem. Res., 2000, 39, (6), pp. 2024 2028 9 Wallén, A., and Åström, K.J.: Pulse-step control. Proc. 15th IFAC World Congress, Barcelona, Spain, 2002 10 Dan-Isa, A., and Atherton, D.P.: Time-domain method for the design of optimal linear controllers, IEE Proc., Control Theory Appl., 1997, 144, (4), pp. 287 292 11 Åström, K.J., and Hägglund, T.: PID Controllers: Theory, Design, and Tuning (Instrument Society of America, Research Triangle Park, NC, 1995, 2nd edn.) 12 Bodson, M.: Adaptive algorithm for the tuning of two input shaping methods, Automatica, 1998, 34, pp. 771 776 13 Pao, L.Y.: Multi-input shaping design for vibration reduction, Automatica, 1999, 35, (1), pp. 81 89 14 Maciejowski, J.M.: Predictive control with constants (Addison- Wesley, Harlow, UK, 2002) 15 Åström, K.J., and Furuta, K.: Swinging up a pendulum by energy control, Automatica, 2000, 36, (2), pp. 287 295 16 Zhong, Q.-C.: Time Delay Control & Its Applications. PhD Thesis, Shanghai Jiao Tong University, Shanghai, China, Dec. 1999 17 Dym, H., Georgiou, T., and Smith, M.C.: Explicit formulas for optimally robust controllers for delay systems, IEEE Trans. Autom. Control, 1995, 40, (4), pp. 656 669 18 Medvedev, A.: Disturbance attenuation in finite-spectrum-assignment, Automatica, 1997, 33, (6), pp. 1163 1168 19 Nobuyama, E., Shin, S., and Kitamori, T.: Deadbeat control of continous-time systems: MIMO case. Proc. 35th IEEE Conf. on Decision & Control, Kobe, Japan, 1996, pp. 2110 2113 20 Zhong, Q.-C.: Control of integral processes with dead-time. Part 3: deadbeat disturbance response, IEEE Trans. Autom. Control, 2003, 48, (1), pp. 153 159 21 Cominos, P., and Munro, N.: PID controllers: recent tuning methods and design to specification, IEE Proc., Control Theory Appl., 2002, 149, (1), pp. 46 53 22 Ho, W.K., Hang, C.C., and Cao, L.S.: Tuning of PID controllers based on gain and phase margin specifications, Automatica, 1995, 31, (3), pp. 497 502 23 Bemporad, A., Casavola, A., and Mosca, E.: Nonlinear control of constrained linear systems via predictive reference management, IEEE Trans. Autom. Control, 1997, 42, (3), pp. 340 349 24 Casavola, A., Mosca, E., and Angeli, D.: Robust command governors for constrained linear systems, IEEE Trans. Autom. Control, 2000, 45, (11), pp. 2071 2077 480 IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004