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Robust controller design based on simplified triangular model∗
PII: S0005–1098(97)00162–3
Automatica, Vol. 34, No. 3, pp. 319—325, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
Brief Paper
Robust Controller Design Based on Simplified Triangular Model* GUO-HUA WU,- YU-GENG XI‡ and ZHONG-JUN ZHANG‡ Key Words—Model approximation; model-based control; predictive control; robust control; stability analysis.
In practice, if the controller design is mainly for regulation rather than optimization, modest design based on simplified model is often accepted. In this case, the complex plant may be approximated by a highly simplified model which should be easy to be identified. The controller design based on the simplified model would be simpler than that based on the complete model. The closed-loop stability can also be analyzed in a simple way. Astrom (1980) designed a robust controller by using a strongly simplified model for the plant with monotone step response. This model has a simple dead-time step form with only two parameters, the gain and the sampling period. The robust stability of the resulted closed-loop system is guaranteed under simple design conditions. In a straightforward way, Lu and Kumar (1984) proposed another approximation model with N parameters, i.e. the staircase step response model. It has been shown that for a linear time-invariant plant and a suitable sampling period, one can choose a set of parameters in control law so that the asymptotic stability can be ensured. In this paper a simplified triangular model with only three parameters is proposed as a crude approximation for the impulse response of a stable plant. The triangular model is superior to the Astrom’s one in the sense of higher precision of modeling while keeping less computational effort. Two kinds of robust controllers are developed based on this approximation model. The robust dead-beat controller extends the result of Astrom (1980) to more general stable plants without restriction of monotone step response. By selecting the parameters properly, the robust stability of the closed-loop system can be ensured. For robust predictive controller with IMC structure, the robust stability and zero tracking error of the resulted closed-loop system are guaranteed under certain model-plant mismatches and the upper bounds of the model-plant mismatch can be given. Simulation results demonstrate that the robust controllers are capable of tracking the desired output exactly, even in the piecewise-constant disturbances’ environment.
Abstract—In this paper a simplified triangular model is suggested for approximating the impulse response of a stable system. The triangular model has only three parameters and is easy to identify. Both dead-beat controller and extended horizon predictive controller are redesigned based on this triangular model. Under simple conditions the robust stability and zero tracking error of the resulted closed-loop system are guaranteed. Simulations show that two kinds of robust controllers are effective and work well even with unknown piecewise-constant disturbances. ( 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION
Model-based control has found wide acceptance in the process control fields. To achieve high control performance the model used for controller design is often required to approximate the real plant as close as possible. Richalet et al. (1978) established model algorithmic control (MAC) based on finite impulse response (FIR) model of a plant. Garcia and Morari (1982) studied the stability and robustness of the closed-loop system and put forward internal model control (IMC) which is inherent in all model-based control schemes. The impulse response model is used by many industrialists because it is physically intuitive and easy to obtain. However, with much more model parameters, it is more difficult to identify on-line than the conventional state-space or parameterized model. On the other hand, based on the parameterized model, generalized predictive control (Clarke et al., 1987) or extended horizon predictive control (Ydstie, 1984) was developed in the field of self-tuning control. Scattolini and Bittanti (1990) provided some simple criteria for the design of the prediction horizon in these control schemes so as to guarantee the closed-loop stability. * Received 31 December 1996; received in revised form 19 February 1997. This paper was not presented at any meeting. This paper was recommended for publication by Associate Editor P. J. Gawthrop under the direction of Editor C. C. Hang. Corresponding author Prof. Yugeng Xi. Tel. #86 21 62812806; Fax #86 21 62820892, E-mail [email protected]. - Computer Center, Air China, Beijing Capital International Airport, Beijing 100621, People’s Republic of China. ‡ Institute of Automation, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. 319
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2. PLANT DESCRIPTION AND TRIANGULAR MODEL
Consider a stable plant described by the following infinite impulse response model: = y(t)" + h(i)u(t!i), (1) i/1 where u(t) and y(t) are the plant input and output, respectively, h(i) the unit impulse response of the plant which satisfies lim h(i)"0. The step rei?= sponse coefficients of the plant are given by j s( j )" + h(i). (2) i/1 Without loss of generality, assume that s(R)'0. Equation (1) can be approximated by the following impulse response model with truncation length P to any desired degree of accuracy:
where
= y(t)" + h(i)u(t!i)"G(z~1)u(t) i/1 P + + h(i)u(t!i)"G (z~1)u(t), p i/1
(3)
G (z~1)"h(1)z~1#2#h(P)z~P. p The above nominal model description is widely used in model predictive control algorithms. However, since there are much more parameters to be determined than those in a parameterized model, it is not suitable for on-line estimation. In the following, a triangular model is introduced to approximate the impulse response of the plant, see Fig. 1. The corresponding step response is compared with the simplified step response model suggested by Astrom (1980) in Fig. 2. It is obvious that the triangular model is superior to the Astrom’s one because higher precision of modeling can be achieved. The coefficients of impulse response of the triangular model will be given by:
G
H 0(i4n, i, n H h (i)" (N!i), n(i4N, m N!n 0,
(4)
i'N,
where H is the peak value of the triangular model, n is the corresponding sampled point and N is the model truncation length. Generally N4P is chosen. The triangular model can be described by where
y (t)"Hw(t!1)"G (z~1)u(t), . .
G (z~1)"h (1)z~1#2#h (N)z~N . . .
Fig. 1. Impulse response model (solid) and triangular model (dash).
(5)
Fig. 2. Step response of Astrom’s model and the triangular model.
and extended input w(t!1) 1 n w(t!1)" + iu(t!i) n i/1 N 1 + (N!i)u(t!i). (6) # N!n i/n`1 Denote the model-plant mismatch *h(i)"h(i)! h (i) due to the approximation and truncation . error, one can rewrite (1) into y(t)"y (t)#g(t), (7) . where g(t)"+= *h(i)u(t!i). i/1 The plant is then described by the triangular impulse response model together with an error term including the approximation and the truncation error. It should be noted that the triangular response model is characterized by three parameters, a peak value of the triangular model H, its sampling point n and the model truncation length N. The model y (t)"Hu(t!1) suggested by As. trom (1980) is a special case of the triangular model with n"1 and N"2. 3. ROBUST CONTROLLER DESIGN AND STABILITY ANALYSIS
3.1. Robust dead-beat controller design Similar to Astrom (1980), a dead-beat controller with integral action based on the triangular model
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Case 2. When h(i)(H/(N!1) for i"1, 2 , N!1,
(5) is designed as follows: 1 w(t)"w(t!1)# [y*(t#1)!y(t)]. H
(8)
It follows the control law n u(t)"u(t!1)# [y*(t#1)!y(t)] H n n # + i[u(t!i)!u(t#1!i)]# N!n i/2 N ] + (N!i)[u(t!i)!u(t#1!i)]. (9) i/n`1 With (1), (5) and (8) the closed-loop system has the form N y*(t#1)"h (1)u(t)# + [h(i)#h (i#1) . . i/1 = !h (i)]u(t!i)# + h(i)u(t!i) . i/N`1 " ¹(z~1)u(t), (10) where the characteristic polynomial ¹(z~1)" += t z~i has the coefficients i/1 i H/n, i"0,
G
h(i)#H/n, 14i4n!1, (11) t" i h(i)!H/(N!n), n4i4N!1, h(i),
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i5N.
The stability of the closed-loop system will depend on the coefficients of ¹(z~1) and may be tested by usual stability criteria if the controller design is based on the triangular model (5). In order to compare the triangular model with the simplified model suggested by Astrom (1980), one may consider the special case n"1 by choosing proper sampling period ¹. At first, the following preliminary lemma is introduced.
C
D
N~1 1 N~1 = + h(i)' + h(i)# + D h(i) D . (13b) 2 i/1 i/1 i/N Proof. According to (11), the coefficients of the characteristic polynomial for n"1 are given as follows:
G
H,
i"0,
t " h(i)!H/(N!1), 14i4N!1, (14) i h(i), i5N. Case 1. When h(i)'H/(N!1) for i"1, 2 , N!1, it follows from (13a)
K
K
K
K
N~1 H = inf D ¹(z~1)D5H! + h(i)! ! + D h(i)D N!1 D z D51 i/1 i/N N~1 = "2H! + h(i)! + D h(i) D'0. i/1 i/N Case 2. When h(i)4H/(N!1) for i"1, 2 , N!1, a straightforward calculation in view to (13b) gives N~1 H = inf D ¹(z~1)D5H! + h(i)! ! + D h(i)D N!1 D z D51 i/1 i/N N~1 = " + h(i)! + D h(i) D'0. i/1 i/N According to Lemma 1 the closed-loop system is asymptotically stable if the sufficient conditions (13) are satisfied.
then the related closed-loop system will be asymptotically stable (Desoer and Vidyasagar, 1975).
Remark 1. Theorem 1 gives sufficient conditions for designing dead-beat controller based on (5) to ensure the closed-loop stability. With one controller the closed-loop system will be always stable for various plants with different step responses if the conditions in Theorem 1 are satisfied. It is robust against the different behavior of the plants. No restrictions are placed on the plant dynamics such as monotone step response. It is suitable to arbitrary stable plants whose h(i) may be zero (not all zero) and negative. In the following, more simple conditions will be given for a plant with monotone step response.
¹heorem 1. For a stable time-invariant linear system, the closed-loop system with the dead-beat controller (9) is always asymptotically stable if n"1 and the parameters H and N are chosen so that the following sufficient conditions are satisfied. Case 1. When h(i)5H/(N!1) for i"1, 2 , N!1,
Corollary 1. For a stable time-invariant linear system with monotone step response, the closed-loop system with the dead-beat controller (9) is always asymptotically stable if the parameters H and N are chosen so that the sufficient conditions given below are satisfied. Case 1. When h(i)5H/(N!1) for i"1, 2 , N!1,
¸emma 1. If the characteristic polynomial ¹(z~1)" += t z~i has the property that i/1 i (12) inf D ¹(z~1) D'0, D z D51
C
D
1 N~1 = H' + h(i)# + D h(i) D . 2 i/1 i/N
(13a)
s(R) H' . 2
(15a)
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Case 2. When h(i)(H/(N!1) for i"1, 2 , N!1, N~1 s(R) + h(i)' . 2 i/1
(15b)
u(t)
Corollary 2. For a stable time-invariant linear system with monotone step response, the closed-loop system with the dead-beat controller (9) with n"1 and N"2 is always asymptotically stable if the parameter H and the sampling period are chosen so that the conditions s(R) h(1)' 2 s(R) H' 2
(16a)
(16b)
2n u(t)" y*(t#k)!y(t)#y (t) . Hk(k#1)
(21)
C
n + iu(t#k!i) i/k`1 N H + (N!i)u(t#k!i) . ! (22) N!n i/n`1 Equations (21) and (22) have the form of constant future predictive controller (Astrom and Wittenmark, 1989) or extended horizon predictive controller (Ydstie, 1984). If k"1, it is just the deadbeat controller presented above. H ! n
are satisfied. Remark 2. Condition (16) in Corollary 2 is the same as in Astrom (1980). So Theorem 1 is more general than the theorem in Astrom (1980). It also suits the plants without monotone step response. With flexible choice of N, there is no restriction on the sampling period and the model prediction may be more accurate. 3.2. Robust predictive controller design In this section we prefer to take k-step ahead prediction rather than one-step prediction as in dead-beat design. From equation (7), one has yL (t#k)"y (t#k)#e(t#k/t), (17) . where yL denotes the estimated value of the output, k is the prediction horizon, e(t#k/t) denotes the error prediction at t#k based on the information available at t which includes the model-plant mismatch and disturbances. It was suggested to take the form by Richalet et al. (1978) and Rouhani and Mehra (1982): e(t#k/t)"y(t)!y (t). (18) . The control proposed is to determine the manipulated variable under the assumption u(t#i)"u(t) for i"1, 2 , k!1 to minimize a given cost function min J"DD yL (t#k)!y*(t#k) DD2,
¹heorem 2. For the triangular model based predictive controller (21) or (22), if the parameters k, n, N and H as well as model-plant mismatch *h(i) satisfy the following conditions: Case 1. When n((N/2)!1,
S
1 N(N!n) 1 k'N! ! # 2 2 4
(23a)
and = 2N#4Nk!2k2!2k!Nn!N2 + D *h(i) D( H. 2(N!n) i/1 (23b) Case 2. When n5(N/2)!1,
S
nN 1 1 # ! (k4n 2 4 2
and
1 y*(t#k)!y(t)#y (t) . + k h (i) i/1 . N ! + h (i)u(t#k!i) . (20) . i/k`1
D
D
(19)
where y*(t#k) is known as the reference output. According to (17) and (19), the optimal manipulated variable can be solved as follows:
C
2(N!n) " H[(n#1)(N!n)#(k!n)(2N!k!n!1)] ][y*(t#k)!y(t)#y (t) . H N ! + (N!i)u(t#k!i)]. N!n i/k`1 Case 2. If k4n.
and
u(t)"
The control law will be discussed for two cases with respect to the relationships between k and n. Case 1. If k'n,
S : 1 S : 2
(24a)
= 2k2#2k!Nn + D *h(i) D( H. (24b) 2n i/1 The resulting closed-loop system is robustly stable, and lim [y(t)!y*(t)]"0. t?=
Proof. (S ) One can derive the closed-loop rela1 tionships between the system output y(t), the model output y (t), the reference input y*(t#k) and the .
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Case 1. When n((N/2)!1, it is easy to proof N!1!J1 N(N!n)#1'n. From (23a) we 2 2 4 have k'n. The characteristic equation of the closed-loop system may be obtained from (21) with (1) and (5):
C
N#2Nk!k2!k!Nn H 2(N!n)
Fig. 3. The block diagram of the control system.
control signal u(t) using the following equations: y*(t#k)"y(t)!y (t) . k N # + h (i)# + h (i)zk~i u(t), . . i/1 i/k`1 (25)
C
D
= y(t)"G(z~1)u(t)" + h(i)u(t!i), (26) i/1 N y (t)"G (z~1)u(t)" + h (i)u(t!i), (27) . . . i/1 where = G(z~1)" + h(i)z~i, i/1 N G (z~1)" + h (i)z~i. . . i/1 The block diagram of the closed-loop system with IMC structure is shown in Fig. 3 and the characteristic equation of the closed-loop system may be obtained from (25). 1 u(t) y*(t#k)"y(t)!y (t)# . G (z~1) # "¹(z~1)u(t)
C
where
k N " + h (i)# + h (i)zk~i . . i/1 i/k`1 = # + (h(i)!h (i))z~i u(t), . i/1
D
H # N!n
D
= # + (h(i)!h (i))z~i u(t)"y*(t#k). (29) . i/1 The characteristic polynomial ¹ (z~1) has the form N#2Nk!k2!k!Nn ¹ (z~1)" H 2(N!n) H # N!n
1 G (z~1)" . # +k h (i)#+N h (i)zk~i . . i/1 i/k`1 According to the theorem of Garcia and Morari (1982) the robust stability of the closed-loop system with IMC structure depends on the location of the roots of the characteristic polynomial ¹(z~1) if the polynomials G (z~1), G (z~1) and G(z~1) are all # . stable. In this paper in order to ensure that the resulting closed-loop system is robustly stable one must guarantee that the polynomial ¹(z~1) is stable as well as the predictive controller is also stable. In accordance with the different relationships between k and n, two cases will be discussed.
N + (N!i)zk~i i/k`1
= # + (h(i)!h (i))z~i. . i/1 According to the condition (23b), one has
(30)
N#2Nk!k2!k!Nn H inf D¹(z~1)D5 H! 2(N!n) N!n D z D51 N = ] + (N!i)! + Dh(i)!h (i)D . i/k`1 i/1 2N#4Nk!2k2!2k!Nn!N2 H " 2(N!n) = ! + D *h(i)D'0. i/1 It gives by equation (23a)
K
(31)
K
1 G (z~1) DzD51 # k N 5 + h (i)! + h (i) . . i/1 i/k`1 2N#4Nk!2k2!2k!Nn!N2 H'0. " 2(N!n) (32) inf
(28)
N + (N!i)zk~i i/k`1
Case 2. When n5(N/2)!1, it follows from (24a) k4n. The characteristic equation of the closed-loop system may be obtained from (22) with (1) and (5).
C
k(k#1) H n + izk~i H# 2n n i/k`1 H N # + (N!i)zk~i N!n i/n`1 = # + (h(i)!h (i))z~i u(t)"y*(t#k). . i/1
D
(33)
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The characteristic polynomial ¹(z~1) is H n k(k#1) + izk~i H# ¹(z~1)" n 2n i/k`1 H N # + (N!i)zk~i N!n i/n`1 = # + (h(i)!h (i))z~i. . i/1 One has by (24b)
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4. ILLUSTRATIVE EXAMPLES
k(k#1) H n~k inf D¹(z~1)D5 H! + (k#i) 2n n D z D51 i/1 H N~k ! + (N!k!i) N!n i/n~k`1 = ! + Dh(i)!h (i)D . i/1 2k2#2k!Nn = " H! + D *h(i) D'0 2n i/1 . (35) It gives by (24a) inf DzD51
K
K
1 k N 5 + h (i)! + h (i) . . G (z~1) # i/1 i/k`1 2k2#2k!Nn " H'0. 2n
ling period must be large enough to meet the stability conditions. In Lu and Kumar (1984), although there is no explicit restriction for the sampling period, the parameters of the staircase model are much more so that it is difficult to choose these parameters from practical point of view. The sufficient conditions for the closed-loop stability are also more complex than that of Theorem 2.
(36)
According to Lemma 1 and the theorem in Garcia and Morari (1982), the closed-loop system and the controller are robustly stable in both cases. (S ) Consider the zero tracking property of the 2 closed-loop system when the stability conditions are satisfied. Equation (20) gives k N y*(t)" + h (i)u(t!k)# + h (i)u(t!i) . . i/1 i/k`1 #y(t!k)!y (t!k). . So one can obtain y(t)!y*(t)"[y(t)!y(t!k)] k # + h (i) [u(t!k!i)!u(t!k)] . i/1 N # + h (i)[u(t!k!i)!u(t!i)]. . i/k`1 Since the closed-loop system is asymptotically stable, it is straightforward to show lim [y(t)!y*(t)]"0. tPR
Remark 3. In Theorem 2 no restriction is placed on the sampling period ¹ and it can be selected arbitrarily small. However, in Astrom (1980), the samp-
In order to demonstrate the effectiveness of the proposed controllers, two simulation examples are given in the following. Example 1. Consider a stable plant with monotone step response given in Astrom (1980) whose transfer function is 1 G(s)" . (s#1)6 For this plant if the robust dead-beat controller is adopted one may take the sampling period ¹"5, and set the parameters n"1, N"3 in the triangular model. It can be found that H"1 will satisfy the stability condition (15b) in Corollary 1. At the same time, with respect to Astrom’s controller, one has to choose the sampling period ¹57.5 to satisfy the stability conditions. As comparison the simulation result of Astrom’s controller with ¹"8, H"1.5 is also given. For the robust predictive controller, one may take sampling period ¹"1 and choose the parameters n"5, N"13 in the triangular model. According to Theorem 2 one can set the parameters H"0.25, k"7 to meet the stability condition. The same controller but designed by the true model with the prediction horizon k"7 is also shown for comparison. All simulation results to a square wave set-points of amplitude 10 and the period 50 are shown in Fig. 4. The simulations show that two kinds of robust controllers presented here may achieve good control performance with low cost of modeling and design. Example 2. Consider a second-order plant with non-monotone step response given in Lu and Kumar (1984); the system transfer function is 2 G(s)" . s2#2s#2 In the simulation a square wave of disturbances varying between #1 and !1 with a period of 50 is acted on the system output. One may choose the sampling period ¹"0.5. For the robust dead-beat controller, according to Theorem 1 the parameters
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designed by the true model. For robust predictive controller, with respect to the case 1 of Theorem 2 the parameters n"2, N"7, H"0.6, k"4 are selected. While k"4 is chosen for the robust predictive controller designed by the true model. The results are compared in Fig. 5(b). Simulation results show that both robust controllers are effective and work well even in the piecewise-constant disturbances’ environment. It should be mentioned that the Astrom’s controller cannot be used in this example while the controller by Lu and Kumar requires more parameters (N"9) for modeling and design. Fig. 4. Output response of Example 1. (——) dead-beat controller based on triangular model (¹"5, n"1, N"3, H"1); (— ) —) dead-beat controller based on Astrom’s model (¹"8, n"1, N"2, H"1.5); ( ) ) ) ) ) predictive controller based on triangular model (¹"1, n"5, N"13, H"0.25, k"7); (— — —) predictive controller based on true model (¹"1, k"7).
5. CONCLUSIONS
In this paper a novel approximate model, triangular impulse response model, is proposed to model a stable plant. Two kinds of robust controllers, dead-beat controller and extended horizon predictive controller, are designed based on the triangular model. Some simple criteria for the design of the controller parameters are provided. The robust stability and convergence of the resulted closed-loop system have been established. Simulation results show that these control schemes are simple and efficient and may achieve good control performances. Acknowledgements—The authors sincerely thank the reviewers for their helpful comments and suggestions. This work was supported by the Chinese Postdoctoral Science Foundation and the National Natural Science Foundation of China.
REFERENCES
Fig. 5. Output response of Example 2: (a) dead-beat controller (——) triangular model based (n"1, N"5, H"2.6); () ) ) ) ) )) true model based; (b) predictive controller (——) triangular model based (n"2, N"7, k"4, H"0.6); () ) ) ) ) )) true model based (k"4).
n"1, N"5, H"2.6 are chosen to satisfy condition (13b). The responses of the closed-loop systems are shown in Fig. 5(a) and compared with one
Astrom, K. J. (1980). A robust sampled regulator for stable systems with monotone step response. Automatica, 16, 313—315. Astrom, K. J. and B. Wittenmark (1989). Adaptive Control. Addison-Wesley, Reading, MA. Clarke, D. W., C. Mohtadi and P. S. Tuffs (1987). Generalized predictive control—Part 1. Basic algorithm. Automatica, 23, 137—148. Desoer, C. A. and M. Vidyasagar (1975). Feedback System: Input—Output Properties. Academic Press, New York. Garcia, C. E. and M. Morari (1982). Internal model control: 1. A unifying reviews and some new results. I&EC Process Des. Dev., 21, 308—323. Lu, W. S. and K. S. Kumar (1984). A staircase model for unknown multi-variable systems and design of regulators. Automatica, 20, 109—112. Richalet, J., A. Rault, J. Testud and J. Papon (1978). Model predictive heuristic control: applications to industrial process. Automatica, 14, 413—428. Rouhani, R. and R. K. Mehra (1982). Model algorithmic control (MAC): basic theoretical properties. Automatica, 18, 401—414. Scattolini, R. and S. Bittanti (1990). On the choice of the horizon in the long-range predictive control—some simple criteria, Automatica, 26, 915—917. Ydstie, B. E. (1984). Extended horizon adaptive control. In 9th ¼orld Congress of the IFAC, Budapest, Hungary.
Automatica, Vol. 34, No. 3, pp. 319—325, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
Brief Paper
Robust Controller Design Based on Simplified Triangular Model* GUO-HUA WU,- YU-GENG XI‡ and ZHONG-JUN ZHANG‡ Key Words—Model approximation; model-based control; predictive control; robust control; stability analysis.
In practice, if the controller design is mainly for regulation rather than optimization, modest design based on simplified model is often accepted. In this case, the complex plant may be approximated by a highly simplified model which should be easy to be identified. The controller design based on the simplified model would be simpler than that based on the complete model. The closed-loop stability can also be analyzed in a simple way. Astrom (1980) designed a robust controller by using a strongly simplified model for the plant with monotone step response. This model has a simple dead-time step form with only two parameters, the gain and the sampling period. The robust stability of the resulted closed-loop system is guaranteed under simple design conditions. In a straightforward way, Lu and Kumar (1984) proposed another approximation model with N parameters, i.e. the staircase step response model. It has been shown that for a linear time-invariant plant and a suitable sampling period, one can choose a set of parameters in control law so that the asymptotic stability can be ensured. In this paper a simplified triangular model with only three parameters is proposed as a crude approximation for the impulse response of a stable plant. The triangular model is superior to the Astrom’s one in the sense of higher precision of modeling while keeping less computational effort. Two kinds of robust controllers are developed based on this approximation model. The robust dead-beat controller extends the result of Astrom (1980) to more general stable plants without restriction of monotone step response. By selecting the parameters properly, the robust stability of the closed-loop system can be ensured. For robust predictive controller with IMC structure, the robust stability and zero tracking error of the resulted closed-loop system are guaranteed under certain model-plant mismatches and the upper bounds of the model-plant mismatch can be given. Simulation results demonstrate that the robust controllers are capable of tracking the desired output exactly, even in the piecewise-constant disturbances’ environment.
Abstract—In this paper a simplified triangular model is suggested for approximating the impulse response of a stable system. The triangular model has only three parameters and is easy to identify. Both dead-beat controller and extended horizon predictive controller are redesigned based on this triangular model. Under simple conditions the robust stability and zero tracking error of the resulted closed-loop system are guaranteed. Simulations show that two kinds of robust controllers are effective and work well even with unknown piecewise-constant disturbances. ( 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION
Model-based control has found wide acceptance in the process control fields. To achieve high control performance the model used for controller design is often required to approximate the real plant as close as possible. Richalet et al. (1978) established model algorithmic control (MAC) based on finite impulse response (FIR) model of a plant. Garcia and Morari (1982) studied the stability and robustness of the closed-loop system and put forward internal model control (IMC) which is inherent in all model-based control schemes. The impulse response model is used by many industrialists because it is physically intuitive and easy to obtain. However, with much more model parameters, it is more difficult to identify on-line than the conventional state-space or parameterized model. On the other hand, based on the parameterized model, generalized predictive control (Clarke et al., 1987) or extended horizon predictive control (Ydstie, 1984) was developed in the field of self-tuning control. Scattolini and Bittanti (1990) provided some simple criteria for the design of the prediction horizon in these control schemes so as to guarantee the closed-loop stability. * Received 31 December 1996; received in revised form 19 February 1997. This paper was not presented at any meeting. This paper was recommended for publication by Associate Editor P. J. Gawthrop under the direction of Editor C. C. Hang. Corresponding author Prof. Yugeng Xi. Tel. #86 21 62812806; Fax #86 21 62820892, E-mail [email protected]. - Computer Center, Air China, Beijing Capital International Airport, Beijing 100621, People’s Republic of China. ‡ Institute of Automation, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. 319
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2. PLANT DESCRIPTION AND TRIANGULAR MODEL
Consider a stable plant described by the following infinite impulse response model: = y(t)" + h(i)u(t!i), (1) i/1 where u(t) and y(t) are the plant input and output, respectively, h(i) the unit impulse response of the plant which satisfies lim h(i)"0. The step rei?= sponse coefficients of the plant are given by j s( j )" + h(i). (2) i/1 Without loss of generality, assume that s(R)'0. Equation (1) can be approximated by the following impulse response model with truncation length P to any desired degree of accuracy:
where
= y(t)" + h(i)u(t!i)"G(z~1)u(t) i/1 P + + h(i)u(t!i)"G (z~1)u(t), p i/1
(3)
G (z~1)"h(1)z~1#2#h(P)z~P. p The above nominal model description is widely used in model predictive control algorithms. However, since there are much more parameters to be determined than those in a parameterized model, it is not suitable for on-line estimation. In the following, a triangular model is introduced to approximate the impulse response of the plant, see Fig. 1. The corresponding step response is compared with the simplified step response model suggested by Astrom (1980) in Fig. 2. It is obvious that the triangular model is superior to the Astrom’s one because higher precision of modeling can be achieved. The coefficients of impulse response of the triangular model will be given by:
G
H 0(i4n, i, n H h (i)" (N!i), n(i4N, m N!n 0,
(4)
i'N,
where H is the peak value of the triangular model, n is the corresponding sampled point and N is the model truncation length. Generally N4P is chosen. The triangular model can be described by where
y (t)"Hw(t!1)"G (z~1)u(t), . .
G (z~1)"h (1)z~1#2#h (N)z~N . . .
Fig. 1. Impulse response model (solid) and triangular model (dash).
(5)
Fig. 2. Step response of Astrom’s model and the triangular model.
and extended input w(t!1) 1 n w(t!1)" + iu(t!i) n i/1 N 1 + (N!i)u(t!i). (6) # N!n i/n`1 Denote the model-plant mismatch *h(i)"h(i)! h (i) due to the approximation and truncation . error, one can rewrite (1) into y(t)"y (t)#g(t), (7) . where g(t)"+= *h(i)u(t!i). i/1 The plant is then described by the triangular impulse response model together with an error term including the approximation and the truncation error. It should be noted that the triangular response model is characterized by three parameters, a peak value of the triangular model H, its sampling point n and the model truncation length N. The model y (t)"Hu(t!1) suggested by As. trom (1980) is a special case of the triangular model with n"1 and N"2. 3. ROBUST CONTROLLER DESIGN AND STABILITY ANALYSIS
3.1. Robust dead-beat controller design Similar to Astrom (1980), a dead-beat controller with integral action based on the triangular model
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Case 2. When h(i)(H/(N!1) for i"1, 2 , N!1,
(5) is designed as follows: 1 w(t)"w(t!1)# [y*(t#1)!y(t)]. H
(8)
It follows the control law n u(t)"u(t!1)# [y*(t#1)!y(t)] H n n # + i[u(t!i)!u(t#1!i)]# N!n i/2 N ] + (N!i)[u(t!i)!u(t#1!i)]. (9) i/n`1 With (1), (5) and (8) the closed-loop system has the form N y*(t#1)"h (1)u(t)# + [h(i)#h (i#1) . . i/1 = !h (i)]u(t!i)# + h(i)u(t!i) . i/N`1 " ¹(z~1)u(t), (10) where the characteristic polynomial ¹(z~1)" += t z~i has the coefficients i/1 i H/n, i"0,
G
h(i)#H/n, 14i4n!1, (11) t" i h(i)!H/(N!n), n4i4N!1, h(i),
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i5N.
The stability of the closed-loop system will depend on the coefficients of ¹(z~1) and may be tested by usual stability criteria if the controller design is based on the triangular model (5). In order to compare the triangular model with the simplified model suggested by Astrom (1980), one may consider the special case n"1 by choosing proper sampling period ¹. At first, the following preliminary lemma is introduced.
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D
N~1 1 N~1 = + h(i)' + h(i)# + D h(i) D . (13b) 2 i/1 i/1 i/N Proof. According to (11), the coefficients of the characteristic polynomial for n"1 are given as follows:
G
H,
i"0,
t " h(i)!H/(N!1), 14i4N!1, (14) i h(i), i5N. Case 1. When h(i)'H/(N!1) for i"1, 2 , N!1, it follows from (13a)
K
K
K
K
N~1 H = inf D ¹(z~1)D5H! + h(i)! ! + D h(i)D N!1 D z D51 i/1 i/N N~1 = "2H! + h(i)! + D h(i) D'0. i/1 i/N Case 2. When h(i)4H/(N!1) for i"1, 2 , N!1, a straightforward calculation in view to (13b) gives N~1 H = inf D ¹(z~1)D5H! + h(i)! ! + D h(i)D N!1 D z D51 i/1 i/N N~1 = " + h(i)! + D h(i) D'0. i/1 i/N According to Lemma 1 the closed-loop system is asymptotically stable if the sufficient conditions (13) are satisfied.
then the related closed-loop system will be asymptotically stable (Desoer and Vidyasagar, 1975).
Remark 1. Theorem 1 gives sufficient conditions for designing dead-beat controller based on (5) to ensure the closed-loop stability. With one controller the closed-loop system will be always stable for various plants with different step responses if the conditions in Theorem 1 are satisfied. It is robust against the different behavior of the plants. No restrictions are placed on the plant dynamics such as monotone step response. It is suitable to arbitrary stable plants whose h(i) may be zero (not all zero) and negative. In the following, more simple conditions will be given for a plant with monotone step response.
¹heorem 1. For a stable time-invariant linear system, the closed-loop system with the dead-beat controller (9) is always asymptotically stable if n"1 and the parameters H and N are chosen so that the following sufficient conditions are satisfied. Case 1. When h(i)5H/(N!1) for i"1, 2 , N!1,
Corollary 1. For a stable time-invariant linear system with monotone step response, the closed-loop system with the dead-beat controller (9) is always asymptotically stable if the parameters H and N are chosen so that the sufficient conditions given below are satisfied. Case 1. When h(i)5H/(N!1) for i"1, 2 , N!1,
¸emma 1. If the characteristic polynomial ¹(z~1)" += t z~i has the property that i/1 i (12) inf D ¹(z~1) D'0, D z D51
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D
1 N~1 = H' + h(i)# + D h(i) D . 2 i/1 i/N
(13a)
s(R) H' . 2
(15a)
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Case 2. When h(i)(H/(N!1) for i"1, 2 , N!1, N~1 s(R) + h(i)' . 2 i/1
(15b)
u(t)
Corollary 2. For a stable time-invariant linear system with monotone step response, the closed-loop system with the dead-beat controller (9) with n"1 and N"2 is always asymptotically stable if the parameter H and the sampling period are chosen so that the conditions s(R) h(1)' 2 s(R) H' 2
(16a)
(16b)
2n u(t)" y*(t#k)!y(t)#y (t) . Hk(k#1)
(21)
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n + iu(t#k!i) i/k`1 N H + (N!i)u(t#k!i) . ! (22) N!n i/n`1 Equations (21) and (22) have the form of constant future predictive controller (Astrom and Wittenmark, 1989) or extended horizon predictive controller (Ydstie, 1984). If k"1, it is just the deadbeat controller presented above. H ! n
are satisfied. Remark 2. Condition (16) in Corollary 2 is the same as in Astrom (1980). So Theorem 1 is more general than the theorem in Astrom (1980). It also suits the plants without monotone step response. With flexible choice of N, there is no restriction on the sampling period and the model prediction may be more accurate. 3.2. Robust predictive controller design In this section we prefer to take k-step ahead prediction rather than one-step prediction as in dead-beat design. From equation (7), one has yL (t#k)"y (t#k)#e(t#k/t), (17) . where yL denotes the estimated value of the output, k is the prediction horizon, e(t#k/t) denotes the error prediction at t#k based on the information available at t which includes the model-plant mismatch and disturbances. It was suggested to take the form by Richalet et al. (1978) and Rouhani and Mehra (1982): e(t#k/t)"y(t)!y (t). (18) . The control proposed is to determine the manipulated variable under the assumption u(t#i)"u(t) for i"1, 2 , k!1 to minimize a given cost function min J"DD yL (t#k)!y*(t#k) DD2,
¹heorem 2. For the triangular model based predictive controller (21) or (22), if the parameters k, n, N and H as well as model-plant mismatch *h(i) satisfy the following conditions: Case 1. When n((N/2)!1,
S
1 N(N!n) 1 k'N! ! # 2 2 4
(23a)
and = 2N#4Nk!2k2!2k!Nn!N2 + D *h(i) D( H. 2(N!n) i/1 (23b) Case 2. When n5(N/2)!1,
S
nN 1 1 # ! (k4n 2 4 2
and
1 y*(t#k)!y(t)#y (t) . + k h (i) i/1 . N ! + h (i)u(t#k!i) . (20) . i/k`1
D
D
(19)
where y*(t#k) is known as the reference output. According to (17) and (19), the optimal manipulated variable can be solved as follows:
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2(N!n) " H[(n#1)(N!n)#(k!n)(2N!k!n!1)] ][y*(t#k)!y(t)#y (t) . H N ! + (N!i)u(t#k!i)]. N!n i/k`1 Case 2. If k4n.
and
u(t)"
The control law will be discussed for two cases with respect to the relationships between k and n. Case 1. If k'n,
S : 1 S : 2
(24a)
= 2k2#2k!Nn + D *h(i) D( H. (24b) 2n i/1 The resulting closed-loop system is robustly stable, and lim [y(t)!y*(t)]"0. t?=
Proof. (S ) One can derive the closed-loop rela1 tionships between the system output y(t), the model output y (t), the reference input y*(t#k) and the .
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Case 1. When n((N/2)!1, it is easy to proof N!1!J1 N(N!n)#1'n. From (23a) we 2 2 4 have k'n. The characteristic equation of the closed-loop system may be obtained from (21) with (1) and (5):
C
N#2Nk!k2!k!Nn H 2(N!n)
Fig. 3. The block diagram of the control system.
control signal u(t) using the following equations: y*(t#k)"y(t)!y (t) . k N # + h (i)# + h (i)zk~i u(t), . . i/1 i/k`1 (25)
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D
= y(t)"G(z~1)u(t)" + h(i)u(t!i), (26) i/1 N y (t)"G (z~1)u(t)" + h (i)u(t!i), (27) . . . i/1 where = G(z~1)" + h(i)z~i, i/1 N G (z~1)" + h (i)z~i. . . i/1 The block diagram of the closed-loop system with IMC structure is shown in Fig. 3 and the characteristic equation of the closed-loop system may be obtained from (25). 1 u(t) y*(t#k)"y(t)!y (t)# . G (z~1) # "¹(z~1)u(t)
C
where
k N " + h (i)# + h (i)zk~i . . i/1 i/k`1 = # + (h(i)!h (i))z~i u(t), . i/1
D
H # N!n
D
= # + (h(i)!h (i))z~i u(t)"y*(t#k). (29) . i/1 The characteristic polynomial ¹ (z~1) has the form N#2Nk!k2!k!Nn ¹ (z~1)" H 2(N!n) H # N!n
1 G (z~1)" . # +k h (i)#+N h (i)zk~i . . i/1 i/k`1 According to the theorem of Garcia and Morari (1982) the robust stability of the closed-loop system with IMC structure depends on the location of the roots of the characteristic polynomial ¹(z~1) if the polynomials G (z~1), G (z~1) and G(z~1) are all # . stable. In this paper in order to ensure that the resulting closed-loop system is robustly stable one must guarantee that the polynomial ¹(z~1) is stable as well as the predictive controller is also stable. In accordance with the different relationships between k and n, two cases will be discussed.
N + (N!i)zk~i i/k`1
= # + (h(i)!h (i))z~i. . i/1 According to the condition (23b), one has
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N#2Nk!k2!k!Nn H inf D¹(z~1)D5 H! 2(N!n) N!n D z D51 N = ] + (N!i)! + Dh(i)!h (i)D . i/k`1 i/1 2N#4Nk!2k2!2k!Nn!N2 H " 2(N!n) = ! + D *h(i)D'0. i/1 It gives by equation (23a)
K
(31)
K
1 G (z~1) DzD51 # k N 5 + h (i)! + h (i) . . i/1 i/k`1 2N#4Nk!2k2!2k!Nn!N2 H'0. " 2(N!n) (32) inf
(28)
N + (N!i)zk~i i/k`1
Case 2. When n5(N/2)!1, it follows from (24a) k4n. The characteristic equation of the closed-loop system may be obtained from (22) with (1) and (5).
C
k(k#1) H n + izk~i H# 2n n i/k`1 H N # + (N!i)zk~i N!n i/n`1 = # + (h(i)!h (i))z~i u(t)"y*(t#k). . i/1
D
(33)
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The characteristic polynomial ¹(z~1) is H n k(k#1) + izk~i H# ¹(z~1)" n 2n i/k`1 H N # + (N!i)zk~i N!n i/n`1 = # + (h(i)!h (i))z~i. . i/1 One has by (24b)
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4. ILLUSTRATIVE EXAMPLES
k(k#1) H n~k inf D¹(z~1)D5 H! + (k#i) 2n n D z D51 i/1 H N~k ! + (N!k!i) N!n i/n~k`1 = ! + Dh(i)!h (i)D . i/1 2k2#2k!Nn = " H! + D *h(i) D'0 2n i/1 . (35) It gives by (24a) inf DzD51
K
K
1 k N 5 + h (i)! + h (i) . . G (z~1) # i/1 i/k`1 2k2#2k!Nn " H'0. 2n
ling period must be large enough to meet the stability conditions. In Lu and Kumar (1984), although there is no explicit restriction for the sampling period, the parameters of the staircase model are much more so that it is difficult to choose these parameters from practical point of view. The sufficient conditions for the closed-loop stability are also more complex than that of Theorem 2.
(36)
According to Lemma 1 and the theorem in Garcia and Morari (1982), the closed-loop system and the controller are robustly stable in both cases. (S ) Consider the zero tracking property of the 2 closed-loop system when the stability conditions are satisfied. Equation (20) gives k N y*(t)" + h (i)u(t!k)# + h (i)u(t!i) . . i/1 i/k`1 #y(t!k)!y (t!k). . So one can obtain y(t)!y*(t)"[y(t)!y(t!k)] k # + h (i) [u(t!k!i)!u(t!k)] . i/1 N # + h (i)[u(t!k!i)!u(t!i)]. . i/k`1 Since the closed-loop system is asymptotically stable, it is straightforward to show lim [y(t)!y*(t)]"0. tPR
Remark 3. In Theorem 2 no restriction is placed on the sampling period ¹ and it can be selected arbitrarily small. However, in Astrom (1980), the samp-
In order to demonstrate the effectiveness of the proposed controllers, two simulation examples are given in the following. Example 1. Consider a stable plant with monotone step response given in Astrom (1980) whose transfer function is 1 G(s)" . (s#1)6 For this plant if the robust dead-beat controller is adopted one may take the sampling period ¹"5, and set the parameters n"1, N"3 in the triangular model. It can be found that H"1 will satisfy the stability condition (15b) in Corollary 1. At the same time, with respect to Astrom’s controller, one has to choose the sampling period ¹57.5 to satisfy the stability conditions. As comparison the simulation result of Astrom’s controller with ¹"8, H"1.5 is also given. For the robust predictive controller, one may take sampling period ¹"1 and choose the parameters n"5, N"13 in the triangular model. According to Theorem 2 one can set the parameters H"0.25, k"7 to meet the stability condition. The same controller but designed by the true model with the prediction horizon k"7 is also shown for comparison. All simulation results to a square wave set-points of amplitude 10 and the period 50 are shown in Fig. 4. The simulations show that two kinds of robust controllers presented here may achieve good control performance with low cost of modeling and design. Example 2. Consider a second-order plant with non-monotone step response given in Lu and Kumar (1984); the system transfer function is 2 G(s)" . s2#2s#2 In the simulation a square wave of disturbances varying between #1 and !1 with a period of 50 is acted on the system output. One may choose the sampling period ¹"0.5. For the robust dead-beat controller, according to Theorem 1 the parameters
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designed by the true model. For robust predictive controller, with respect to the case 1 of Theorem 2 the parameters n"2, N"7, H"0.6, k"4 are selected. While k"4 is chosen for the robust predictive controller designed by the true model. The results are compared in Fig. 5(b). Simulation results show that both robust controllers are effective and work well even in the piecewise-constant disturbances’ environment. It should be mentioned that the Astrom’s controller cannot be used in this example while the controller by Lu and Kumar requires more parameters (N"9) for modeling and design. Fig. 4. Output response of Example 1. (——) dead-beat controller based on triangular model (¹"5, n"1, N"3, H"1); (— ) —) dead-beat controller based on Astrom’s model (¹"8, n"1, N"2, H"1.5); ( ) ) ) ) ) predictive controller based on triangular model (¹"1, n"5, N"13, H"0.25, k"7); (— — —) predictive controller based on true model (¹"1, k"7).
5. CONCLUSIONS
In this paper a novel approximate model, triangular impulse response model, is proposed to model a stable plant. Two kinds of robust controllers, dead-beat controller and extended horizon predictive controller, are designed based on the triangular model. Some simple criteria for the design of the controller parameters are provided. The robust stability and convergence of the resulted closed-loop system have been established. Simulation results show that these control schemes are simple and efficient and may achieve good control performances. Acknowledgements—The authors sincerely thank the reviewers for their helpful comments and suggestions. This work was supported by the Chinese Postdoctoral Science Foundation and the National Natural Science Foundation of China.
REFERENCES
Fig. 5. Output response of Example 2: (a) dead-beat controller (——) triangular model based (n"1, N"5, H"2.6); () ) ) ) ) )) true model based; (b) predictive controller (——) triangular model based (n"2, N"7, k"4, H"0.6); () ) ) ) ) )) true model based (k"4).
n"1, N"5, H"2.6 are chosen to satisfy condition (13b). The responses of the closed-loop systems are shown in Fig. 5(a) and compared with one
Astrom, K. J. (1980). A robust sampled regulator for stable systems with monotone step response. Automatica, 16, 313—315. Astrom, K. J. and B. Wittenmark (1989). Adaptive Control. Addison-Wesley, Reading, MA. Clarke, D. W., C. Mohtadi and P. S. Tuffs (1987). Generalized predictive control—Part 1. Basic algorithm. Automatica, 23, 137—148. Desoer, C. A. and M. Vidyasagar (1975). Feedback System: Input—Output Properties. Academic Press, New York. Garcia, C. E. and M. Morari (1982). Internal model control: 1. A unifying reviews and some new results. I&EC Process Des. Dev., 21, 308—323. Lu, W. S. and K. S. Kumar (1984). A staircase model for unknown multi-variable systems and design of regulators. Automatica, 20, 109—112. Richalet, J., A. Rault, J. Testud and J. Papon (1978). Model predictive heuristic control: applications to industrial process. Automatica, 14, 413—428. Rouhani, R. and R. K. Mehra (1982). Model algorithmic control (MAC): basic theoretical properties. Automatica, 18, 401—414. Scattolini, R. and S. Bittanti (1990). On the choice of the horizon in the long-range predictive control—some simple criteria, Automatica, 26, 915—917. Ydstie, B. E. (1984). Extended horizon adaptive control. In 9th ¼orld Congress of the IFAC, Budapest, Hungary.