Znd. Eng. Chem. Res. 1994,33,3196-3201
3196
Multirate Output Control for SISO Non-Minimum Phase Systems Zalman J. Palmor,. Yoram Halevi, and Zahi Rom Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
The application of multirate output control (MROC) to SISO non-minimum phase (NMP)systems exhibiting severe inverse response is considered in this paper. It is shown that the MROC improves considerably the control performance of such systems, and the reasons for this are clarified. A detailed comparison between the MROC and linear time-invariant (LTI) control structures (both continuous and discrete) like the PID, IMC, and observerlstate feedback is carried out on a wide range of simple processes. The main advantage in the application of the MROC to NMP systems is that a drastic improvement in disturbance attenuation can be achieved while maintaining similar tracking capabilities. Sufficiently small sampling intervals yield a n almost perfect disturbance rejection but degrades the robustness properties. This trade-off is discussed in the paper, and it is shown that a n appropriate design of the MROC leads to good robustness properties with excellent performance incomparable to LTI control schemes. For the design of the MROC, a tuning algorithm for optimal pole placement, based upon undershoot and settling time, is developed for two poles and one zero systems.
I. Introduction
2. Preliminaries
In recent years several new types of multirate digital controllers were developed (Araki and Hagiwara, 1986; Mita et al. 1987; Hagiwara and Araki, 1988; Hagiwara et al., 1990; and others). These controllers have been shown to offer considerably more design freedom than conventional linear time-invariant (LTI)controllers. "he multirate output controller (MROC),proposed by Hagiwara and Araki (19881, is a relatively new type of controller which detects the plant output at N uniformly spaced times and changes the plant input once during one frame period, T . This control structure achieves equivalent state feedback without observers, is simple t o realize, and unlike other types of multirate controllers, does not introduce inputs with high-frequency contents. Although various advantages of the MROC were clarified through theoretical studies, to the best of the authors' knowledge no attempt t o investigate the practical aspects of those advantages has been done so far. In this paper we identify a class of processes, namely non-minimum phase (NMP) systems exhibiting inverse responses to step inputs, in which the MROC is found to be superior to LTI control structures, both continuous and discrete. NMP systems are known to present control difficulties (Holt and Morari, 1985) and are quite common in loops in chemical systems such as boilers and distillation columns. It is shown that an appropriate design of the MROC can lead t o considerable improvements in disturbance rejection for such systems as compared to LTI control structures while maintaining good robustness properties. To this end, we focus in this paper on one zero and two poles stable and unstable systems. A brief review of the MROC as well as of some properties of NMP systems are given in section 2. A simple method for optimal pole placement needed for the design of the MROC is developed in section 3. Section 4 contains a detailed study of the disturbance rejection capabilities of the MROC relative to LTI control structures for a wide range of simple processes. A summary is provided in section 5.
The purpose of this section is to introduce briefly the basic relations of the MROC and the associated notation. "he reader is referred to Hagiwara and Araki (1988)for details. Suppose that a SISO LTI continuoustime plant is given by
* To whom correspondence should be addressed. Tel.: 9724-292086. fax: 972-4-3245 33. E-mail:
[email protected].
$ = e ATIN
R ( t ) = A&)
+ bu(t)
y(t)=C d t )
(1)
where x E Rn is the state, u is the plant input, and y is the plant output. As stated in the introduction, the MROC operates such that the control signal, u(t), is changed once during a frame period T and is applied to the plant through a ZOH. u ( t ) is therefore given by u ( t ) = u(kT)
kT 5 t
(k
+ 1)T
(2)
The output y ( t ) is detected a t N uniformly spaced times during the frame period T . That is, the output is sampled every TIN units of time. It can be shown that the basic formula of the MROC sampling mechanism is given by
&((K where
E
+ 1)T) = jr(kT) - &(kT)
R N x n& , E R N x l ,and jr
and where rl-N
=-
c$
E
(3)
R N x lare given by
e-AUbdv (U
0888-5885l94I2633-3196$04.5QlQ 0 1994 American Chemical Society
= 0, 1, ..., N - 1 ) ( 5 )
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3197 0
'
U t
0.5
P U t
o i
-0.5
Figure 1. Closed-loop configuration with MROC.
-1
-1.5
-2
I
I
0
2
4
6
10
8
Time I
1
Figure 2. Frequency domain representation of the closed loop with MROC.
The control law of the MROC takes the following general form:
u((k
+ 1)T)= Mu(kl3 + H&(kT)
(6)
where M is a scalar, H E R l x N and , &(kT) E RNxlis the multirate error vector (see Figure 1)defined in a similar fashion t o fr(kT) in (4) with e ( t ) = r(t) - y(t). Theorem (Hagiwara and Araki,1988). Suppose that (A, c ) is an observable pair and that N L n. Then, for almost every frame period T , the MROC control law ( 6 ) can be made equivalent to any state feedback control law:
u(k2') = -Fx(kT)
(k L 1)
Figure 3. Effect of the zero location on the step response of a NMP system.
is a finite impulse response (FIR) filter and that D(Z) is an integrator if M = 1. P(s) is the plant transfer function corresponding to (l),d represents an input disturbance, and r is the reference. The relation between the transformed output and the transformed reference can be derived using the mutlirate Z transform and is given by y(z,) =
+ z[c(zN)zN(~)jD(z)(l- 2-l)
1
4:)
z=z#
(12)
(7)
Remarks: (a) If N = n, then H and M are uniquely determined as follows:
H = FC-l
M = FC-' G
(8)
+
(b) If N is selected to be n 1and the plant has no zero a t the origin, then M can be chosen arbitrarily and H is uniquely determined by
H = [CG1-l[FM1
(9)
+
(c) If N > n 1, then more degrees of freedom for selecting the parameters, M and H,of the MROC, are available. According to remark b above, if N is selected to be n 1 and M is chosen to be unity, then in addition to realizing an arbitrary state feedback the MROC contains a discrete integrator which is essential for removing offsets. In addition, Hagiwara and Araki (1988) showed that the poles of the closed loop system with a MROC are those of the equivalent state feedback plus one pole at the origin. That is, a delay of one frame period. In order to gain a better insight into the properties of MROC, its frequency domain representation is depicted in Figure 2. The MROC consists of two components: C(ZN)and D ( z ) which are as follows:
+
(10) 3. Optimum Pole Placement
D(z) = z/(z - M>
(11)
where z and Z N are the Z and the ZN transforms variables respectively. z is associated with the frame period, T , whereas Z N with the period TIN. The hi's are the components of H . It is worthwhile noting that C(ZN)
In the previous section we have seen that the MROC can place the poles of the system in an arbitrary fashion like any state feedback controllers, but it cannnot shift or remove RHP zeros. Since the location of the poles relative to the RHP zeros determines the characteristics of the response, it is evident that a method for deter-
3198 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 180 160
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mining the best (in a sense to be defined in the sequel) pole locations is required. We will focus on one zero two poles (1Z2P) plants and develop a method for optimum pole placement. The transfer function of the plant is given by
We next pose the following problem: Given z (the location of the zero), what should be the values of p1 and p2 such that the response of P(s)to a step input is the best? It was noted in the previous section that by moving the poles, both the US and the ST vary such that when the first increases the latter decreases and vice versa. We therefore formulate the following criterion:
+ @US
as
I I
1s
2
25
3
35
4
4s
s
POLEZERO RATIO Pn Figure 5. Optimal pole/zero ratio vs ST.
POLEZERO RATIO P/Z
Figure 4. Optimal pole/zero ratio vs US.
J = ST
01 0
(14)
where e is a weighting factor and seek for p1 and p2 such that J is minimized. J is not a standard performance criterion, but it characterizes very well the problem we are dealing with and involves quantities that are physically meaningful. Notice that by including both ST and US in the criterion, it is qualitatively equivalent to defining a "reference trajectory" with small undershoot and settling time and minimizing a norm (e.g., Lz) of the error. The minimization of J can be solved analytically only in part and requires some numerics even for the 1Z2P system in (13). The details can be found in Rom (1991). However, several conclusions can be stated. First, for all practical purposes and for all e's, J attains a minimum for p1 = p2 A p . This conclusion holds particularly for ST's based upon 99%. Second, for the J defined in (14) and for the above ST's the optimal poles are real (and equal). On the basis of the results of the minimization of J , two design diagrams are constructed. Figure 4 enables one to determine the optimal poles' location for a given z and a desired US. Similarly, one can determine optimum p for a given z and a desired ST via Figure 5. Note, that the ST in Figure 5 is normalized to z . These figures will be used for the design of the MROC in the next section. 4. Disturbance Rejection Capabilities of MROC
In this section we investigate the disturbance rejection capabilities of the MROC for NMP systems. We do it for a wide range of 1Z2P plants. The reason for using relatively simple NMP systems is 2-fold. First,
such systems are quite common in the chemical and the process industries and also can be found in simple mechanical systems (see Rom, 19911, and secondly even for such simple systems the MROC is found to perform remarkably better than various LTI controllers. The design of the MROC will be applied to three cases of 1Z2P systems: a stable system, a system containing an integrator, and an unstable system. To assess the performance of the MROC, it is compared, where applicable, with conventional controllers (like PI, PID) to internal model control (IMC; Morari and Zafirious, 1989) and to state feedback plus observer. The motivation for using the MROC for input disturbance rejection is quite clear. LTI dynamic controllers (like the IMC, for example) always increase the order of the closed loop. In cases where stable poles of the process are cancelled by the controller, they remain poles of the closed loop, and depending on the particular input they may or may not be excited. Typically the cancelled poles disappear from the reference output relation but not from the transference from the input disturbance (see Figure 2) to the output. The same applies to the state feedback observer, where the observer dynamics is not excited by the reference but is excited by input disturbances. A s the MROC achieves state feedback with no observer, it does not suffer from the previous deficiencies, and its potential improvement lies therefore in its disturbance rejection properties. An important parameter in the design of the MROC is the frame period, T. Er and Anderson (1991) showed that too small and too large (in case of unstable processes) T 's may lead to close to singular and 6 (eq (4)) and consequently to excessively large gains indicating high sensitivity. They constructed some quantitative guidelines for choosing T. It is shown in the sequel that similar conclusions can be derived by direct examination of robustness properties and that suitable frame periods can be determined in a straightforward manner using stability margins. Case 1: 1Z2P Stable N M P System. The transfer function of the process is given by 0.1 - s
(s
+ l)(s + 2)
(15)
In Figure 6 the responses of the system with PI, PID, and IMC controllers to unit step change in set point are depicted. The PI and PID are tuned via Ziegler-Nichols rules. Although the response with the PI has a 50% US, its settling time is clearly unacceptable. The PID (which includes a filter on the derivative) has a much
Ind. Eng. Chem. Res., Vol. 33,No. 12,1994 3199 1
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i !
!
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15
20
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25
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Timr
Figure 6. Responses to a unit set point change of case 1with PI, PID, and IMC controllers. 1
0.5
'.
t *------
-1 0 -1
I
0
5
10
15
20
25
30
35
I
40
Timr
Figure 7. Set-point responses of process (15) with IMC controller and MROC with T = 0.3.
better ST but an excessive undershoot. Although one may try to optimize the tuning, it is clear that the conventional PI or PID controllers are not suited to deal effectively with NMP processes with severe inverse responses. The IMC controller, though tuned quite conservatively, is considerably better. We now turn to the design of the MROC for the system (15). We do it for two different frame periods: T = 0.3 and T = 0.06. Since the order of the process is n = 2 and since we wish the MROC to contain an integrator (Le., M = 11,N = 3 is chosen. We design a continuous IMC controller and the MROC such that both have similar responses to setpoint changes (with particularly the same US). The IMC controller for the process (15) is completely determined except for A, the time constant of the IMC filter, which is taken t o be 1.0. The resulting US,as shown in Figure 7, is 66%. For z = 0.1 and a US of 66%, we find from Figure 4 that the two poles should be placed at p = 0.234. The corresponding discrete poles are given by q 2= e-pT
(16)
Then the state feedback gain vector, F,is determined for the two frame periods. Having the F'! and the corresponding T's we calculate the matrix C and the vector 2: via (4) and determine H using (9). This completes the design of the two MROC's (for the two frame periods). The responses of the closed loop with the IMC controller and the MROC (with T = 0.3) to set point change are shown in Figure 7. It is seen that both have the same US but that the MROC has a slightly better ST. The corresponding responses to unit change in input disturbance are depicted in Figure 8 together with the associated control efforts. The responses of the two MROC's are considerably better in any respect though the IMC controller is a continuous one whereas
- MROC :T=O.3 -._-." : T-0.06
'.
-_ 5
10
15
20
25
I
Figure 8. Responses to unit step change in input disturbance in case 1: (a) output, (b) control effort. Table 1. Stability Margins of the Control Systems in Case 1 gain phase delay margin (gm) margin (pm) margin (dm) IMC (continuous) 1.85 53.8" 14.7 MROC (T= 0.3) 3.25 88.1" 13.1 MROC (T= 0.06) 2.17 71.3" 0.08
the MROC's are discrete (!I. Also, it is interesting to watch the control efforts: while the IMC controller is "confused", giving initially a correction in the "wrong" direction, the MROC's respond in the right direction. Another conclusion that can be drawn from Figure 8 is that the smaller the frame period, the better the response. Indeed, with T = 0.06, an almost perfect attenuation of the disturbance is achieved. However, as noted previously, decreasing T by too much leads to deterioration in robustness properties. To see this we calculated the gain, phase and delay margins for the three controllers. They are summarized in Table 1. It is clearly seen that while the slower MROC (corresponding to T = 0.3) has better robustness properties (as measured by the margins above) than the IMC, the fast MROC (with T = 0.06) is completely nonrobust as indicated by its extremely small delay margin. It means that an additional delay of a little more than one frame period destabilizes the system. Thus, it can be concluded that a MROC with a relatively large frame period, determined such that robustness properties equal to or better than a properly designed continuous IMC controller are guaranteed, rejects considerably better input disturbances. Next, a state feedback plus observer controllers are designed and compared to the robust MROC (with T = 0.3). For a fair comparison an integrator is added to the state controller, and the gains of the state feedback are determined such that the responses to set point changes are similar to those of the robust MROC. Two observers were designed: one which led to design with stability margins similar to those of the MROC, and the
3200 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 0.2
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second a dead beat observer (DBO)which gives a much less robust system. In Figure 9 the responses are depicted. Despite the fact that the SF observer controllers operate with a sampling interval of 0.1, the response of the robust MROC is significantly better. Case 2: 1Z2P N M P System with an Integrator. 1Z2P transfer functions with an integrator are commonly found in stream drum level control. Satisfactory control is difficult t o achieve in such systems particularly when the time constant associated with the RHP zero is larger than the one corresponding t o the stable pole as in the following transfer function: (17)
Following the same steps as in case 1, we first design an IMC controller with a filter with A = 1. Then an MROC which gives the same US (52% in this case) to reference step change is designed. The MROC includes an integrator ( M = 1 in (ll)),and N is set to its minimum value which is 3. T, the frame period is chosen based upon robustness characteristics and an appropriate value which gives satisfactory and wide stability margins was found to be T = 0.15. The responses of the continuous IMC controller and the robust MROC to step change in input disturbance are compared in Figure loa. The corresponding control effort is shown in Figure lob. The superiority of the MROC is evident. Case 3: 1Z2P Unstable NMP System. The process t o be considered is as follows:
4d(s) =
-MROC
- - - IMC
-1.5 0
5
10
1s
20
25
30
TLnc
Figure 10. Responses to unit step change in input disturbance, case 2, with continuous IMC and robust MFtOC: (a) output, (b) control effort.
I
.3
0
I
I 2
4
6
8
10
I2
rimr
1 - 2s
(s
+ 2)(s - 1)
For this case a two degree of freedom IMC controller was designed as the one degree of freedom IMC designs were completely unsatisfactory. The feedback and feedforward IMC controllers Qd and qr are given by (S
1
b /
+
P(s)=
20
rtmc
Figure 9. Responses to unit step change in input disturbance in case 1with state feedback observers and MROC.
1 - 5s P(s)= s(s + 1)
I
"
+ 2)(1 - 7s)(l - S) (1+ 2s)(s + 1) (s
= (2s
+ 2x1 - s)
+ l)(h+ 1)
For definitions of Qd and qr see Morari and Zafririou (19891, and for details of the design, see Rom (1991). With A = 0.5 a 47%US is obtained for a unit change of set point. Then we designed a MROC which gives similar responses to step changes in set point. The MROC is designed with M = 1 and N = 3. T is
-MROC
0
2
4
6
8
10
12
rime
Figure 11. Responses to a step input disturbance, case 3: (a) output, (b)control effort.
determined to be 0.2 via robustness measures in a similar fashion as in the previous cases. The performances of the controllers are shown in Figure 11. The advantage of the MROC is once again evident. 5. summary
The paper identifies an area in which the MROC is found to be superior t o LTI control structures, namely
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3201 in rejecting input disturbances in NMP systems with severe inverse responses to step inputs. The reasons for that capability were clarified and shown t o be a direct consequence of the ability of the MROC to achieve state feedback without directly measuring the state. The paper focused on a wide range of 1Z2P NMP systems. It was shown that even for such relatively simple systems, the performance of the MROC is incomparable to both continuous and discrete LTI controllers. Two issues related to the practical design of the MROC, one the optimal pole placement and the other the choice of a frame period, were treated. A method for optimally tuning the MROC and a way to select the period through direct frequency domain robustness measures were presented. The paper demonstrates that MROCs which are both robust and advantageous relative to LTI controllers can be constructed.
Er, M. J.; Anderson, B. D. 0. Practical Issues in Multirate Output Controllers. Znt. J . Control 1991,53,1005-1020. Hagiwara, T.; Araki, M. Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output. ZEEE Trans. Autom. Control 1988,33,812-819. Hagiwara, T.; Fujimura, T.; Araki, M. Generalized MultirateOutput Controllers. Znt. J . Control 1990,52,597-612. Holt, B. R.; Morari, M. Design of Resilient Processing Plants VI. The Effect of Right-Half-Plane Zeros on Dynamic Resilience.
Chem. Eng. Sci. 1986,40,59-74. Mita, T. ;Pang, B. C.; Liu, K. Z. Design of Optimal Strongly Stable Digital Control Systems and Application to Output Feedback Control of Mechanical Systems. Znt. J . Control 1987,45,2071-
2082. Morari, M.; Zdiriou, E. Robust Process Control; Prentice-Hall: New York, 1989. Rom, Z. Control of Non-minimum phase systems. MSc. thesis, Mechanical Eng., Technion, Israel Institute of Technology, 1991.
Received for review February 1, 1994 Accepted July 28, 1994@
Literature Cited Araki, M.; Hagiwara, T. Pole Assignment by Multirate SampledData Output Feedback. Znt. J . Control 1986,44,1661-1673.
@
Abstract published in Advance ACS Abstracts, October 1,
1994.
