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Conversion of SISO processes with multiple time-delays to single time-delay processes
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Research Paper
Conversion of SISO processes with multiple time-delays to single time-delay processes Xiaoli Luan a , Qiang Chen a , Pedro Albertos b,∗ , Fei Liu a a b
Key laboratory of Advance Process Control for Light Industry (Ministry of Education) and the Institute of Automation, Jiangnan University, China Department of System Engineering and Control, Universidad Politécnica de Valencia, Spain
a r t i c l e
i n f o
Article history: Received 22 February 2017 Received in revised form 7 September 2017 Accepted 2 October 2017 Available online xxx Keywords: Multiple time-delays Pre-compensator design Single time-delay based control
a b s t r a c t There are many processes with multiple time-delays. These may appear in the direct path or in some feedback path. In this paper, a pre-compensator is designed in order to convert a multiple time-delays process to one with a single time-delay in the forward path. On this simpler model, any well-established approach to design the control for input/output time-delay plants can be applied. The proposed method is directly applicable to stable plants with multiple time-delays, including in the feedback path, and it can be extended to unstable plants with some constraints. Some numerical examples and case studies are provided to illustrate the application of the proposed algorithms. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Time-delays are widely found in the process industries when transporting materials or energy. These delays may also arise at the actuator inputs, at the sensor outputs as well as in the interconnection between internal/external variables, and can lead to degradation of the controlled closed-loop system behavior. Therefore control of dead-time systems has been extensively studied over the past decades. As the time-delay operator is not a rational function, the stability analysis and control design for transfer functions including time-delays is very complicated, leading to a characteristic quasipolynomial with infinite number of roots. One option to deal with dead-time processes is based on some kind of approximation of either the model response or the transfer function, like the Padé approximations [1], used in [2]. However, the major disadvantage of the approximations lies in that some model properties are lost and, for instance, the approximated model may be unstable even the original one is stable. Dead-time compensation is one of the control approaches used to simplify the design of closed-loop controlled delayed systems, and the pioneer proposal was the Smith predictor (SP) [3]. As it is well-known, in this setting, an undelayed output is predicted based on the input and output measurements,
∗ Corresponding author. E-mail addresses: [email protected] (X. Luan), [email protected] (Q. Chen), [email protected], [email protected] (P. Albertos), fl[email protected] (F. Liu).
in such a way that the delay is shifted out of the loop and the control can be designed for the rational part of the transfer function. One strong limitation of the SP is the requirement to factorize the transfer function into two terms, the delay and the rational (also called fast) part which is also required to be sTable Since the seminal work proposed by Smith, many extensions of the SP have been presented, such as the extension to unstable or integrating processes [4–6], systems with cascade structure [7–9], discrete-time stable or unstable systems [10,11], or systems with uncertainties and disturbances [12–14]. Nevertheless, the time-delay should be separated from the rational transfer function. The aforementioned techniques are suitable for single-inputsingle-output (SISO) systems with single time-delay. Some extensions have been reported for multi-input and multioutput (MIMO) plants with different time-delays in the different input/output path signals (see, for instance, [15]). However, multiple time-delays are also found in many practical SISO applications, such as industrial processes where there are different input-output paths or internal recycling [2,16,17], or when considering congestion control over a transition control protocol (TCP) network [18,19]. In some cases [15], the multi time-delay structure appears in some elements of the MIMO transfer matrix [20]. Although the control of a time-delay system is more difficult than the control of a non-delayed system, it is much easier than that of a system with multiple time-delays. Thus, different model approximation approaches were put forward to transform the systems with multiple time-delays into ones without or with a single time-delay, such as the above mentioned Padé approximation, the decentralized relay feedback estimation, constrained H2 approx-
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imation, etc. [16,21–23]. Instead of utilizing the approximating process, the state-space approach was proposed to consider the influence of multiple time-delays [24,25], by enlarging the state vector, which represents an increment in the system dimension leading to higher computational cost. To overcome these disadvantages, a predictor-observer-based control strategy for SISO disturbed systems with multiple time-delays was proposed in Ref. [26]. To simplify the controller design, the filtered Smith predictor (FSP) is extended in Ref. [27] to systems with multiple time-delays to consider the closed-loop performances for robustness, set-point tracking, disturbance rejection, and noise attenuation [28,29]. In the aforementioned references dealing with multiple timedelays systems the model approximation is necessary to achieve the desired performance by either approximating the systems with a single time-delay, or by designing the controller based on deadtime compensation [30]. This will definitely lead to inaccuracy. On the other hand, simple models are very important so they should be used in any proposed control methodology to be easily applied and tuned. In particular, this is important for complicated systems where the numerator and denominator of the transfer function contain multiple time-delays. The problem of control design for a SISO plant with multiple time-delays has not been solved in a general case. In Ref. [29], the design of low order controllers is tackled in the frequency domain, mainly dealing with the controlled plant at low frequencies and applying the generalized Nyquist criterion to the characteristic quasipolynomial. As already mentioned, the state space representation [24] has been also used to analyze the stability of this kind of plants, but the control design is not easy in this framework. In this paper, the control design is formulated in three stages:
2. Problem statement Let us represent the kind of SISO processes we are interested in as P(s), a rational function of the variables s, including some exponential terms e–s , pointing out that the dynamics are represented by differential terms and time-delays. The general case will be r1
P(s) =
ni (s).e−i s
i=1 r2
; dj (s).e
1 ≤ 2 · · · ≤ r1
and
1 ≤ 2 · · · ≤ r2 (1)
−j s
j=1
where ni (s), dj (s), are polynomials of s, showing that there are r1 delays in the numerator and r2 delays in the denominator, and i , j are different delays appearing ordered in the global transfer function. They are denoted by different variables to make easier the notation. As the plant could be considered as a complex structure composed of delayed elements connected in series, parallel and loop, the global transfer function (1) could be also expressed as a composition of elementary delayed transfer functions and, as later on explained in Section 4 and also defined in Ref. [32], it can be assumed that 1 = 0. The simplest case is when there are several subsystems and delays in series (for instance, input 1 and output 2 delays and G1 (s), G2 (s) as rational transfer functions). In this case, the total transfer function is easily reduced to a single delay: P(s) = G1 (s)G2 (s)e−(1 +2 )s = G(s)e−s
(2)
Other than that, let us consider: 1) a pre-compensator is designed to convert, under some constraints, the initial multiple time-delays process to one with a single time-delay in the forward path, without any approximation. 2) then, an already developed DTC is applied to the obtained single time-delay in order to get the rational part, that is, the undelayed output. It could be the SP or any improved solution. Otherwise, a general control design approach for single-delay processes can be applied. 3) the final control is designed for the rational transfer function following any appropriate control design technique.
a Parallel paths. There are several paths of series systems like (2), in such a way that the transfer function is
P(s) =
Gi (s)e−i s
(3)
i=1
Without loss of generality, it is assumed that 1 < 2 · · · < r . • There is a loop with both, forward and feedback delays. The full transfer function is given by P(s) =
As the last two stages are well documented and accessible in the literature (see, for instance Refs. [4] and [31]), in this paper the implementation of the first stage is presented. For that purpose, a procedure to convert a multi time-delay SISO plant into an equivalent model with a rational part and a single time-delay is presented. Then, steps 2) and 3) above can be followed to design the control. As there are pole/zero cancellations, the proposed approach is directly applicable to stable and minimum-phase (MP) systems with multiple time-delays, including in the feedback path. In order to get a stable and realizable compensator, some constraints should be fulfilled. The paper is organized as follows. In the next section the problem is defined. Then, the pre-compensator design is developed for some basic structures including elements in parallel and loop. The main result considers the general case and a procedure to apply the previous results is summarized. Some numerical examples are presented to illustrate the effectiveness of the proposed algorithms. Unstable terms can be handled under some conditions and nonminimum-phase (NMP) terms require a pre-treatment, as detailed in Section 6. Finally, some conclusions are drafted in the last section.
r
G1 (s)e−1 s 1 + G1 (s)e−1 s G2 (s)e−2 s
=
G1 (s)e−1 s 1 + Gb (s)e−b s
(4)
where, obviously, 1 ≤ b . • The process is composed of a combination of these basic structures. For instance, both, the forward and the feedback paths may have several sub-paths in parallel (as well as loops or series terms). The main goal of this work is to design a pre-compensator, C(s) such that the whole system can be represented as P(s)C(s) = G(s)e−s = L(s)
(5)
The structure and components of this pre-compensator will depend on the structure of the process, as later on shown. Thence, the number and distribution of the delays is much more general than those in the processes considered in Ref. [27] and no approximation approach is used, as proposed in Ref. [1]. Once the simplified model is obtained, a controller can be designed for the rational plant G(s) [32,33] or by any control design approach suitable for single time-delayed plants [4,34,35].
Please cite this article in press as: X. Luan, et al., Conversion of SISO processes with multiple time-delays to single time-delay processes, J. Process Control (2017), https://doi.org/10.1016/j.jprocont.2017.10.001
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Fig. 1. Type I compensator structure. Fig. 2. General type I compensator structure.
Initially, the case of stable and minimum-phase plants is considered. 3. Pre-compensator design
where Kf (s) =
The structure of the pre-compensator could be also with series, parallel and feedback components. The main requirement is that all the components should be realizable and stable. This will impose some constraints in the selection of its structure, as later on shown.
Let us assume that there are two paths in parallel. The process transfer function is P(s) = G1 (s)e−1 s + G2 (s)e−2 s ;
and Kbi (s) =
G(s)e−s G1
(6)
1 < 2
+ G2
(s)e−2 s
=
G(s)e−s G1
(s)e−1 s
/
1+
G2
(s)e−2 s
G1
(s)e−1 s
(7)
G(s)e−s G1 (s)e−1 s G(s)e−s
1+
G1 (s)e−1 s
(8)
G2 (s)e−2 s G(s)e−s
and it can be implemented by means of the structure shown in Fig. 1. The compensator (8) is given by C(s) =
Kf (s)
(9)
1+Kf (s)Kb (s)
G(s)e−s G1
(s)e−1 s
;
Kb (s) =
G2 (s)e−2 s G(s)e−s
(10)
In order to get a realizable compensator, looking at (10), the targeted transfer function delay should be 2 ≥ ≥ 1 , and (G1 ) ≤ (G) ≤ (G2 ), where (G) denotes the relative degree of the transfer function G(s). Constraints a: In order to define a realizable compensation, it should be (G1 ) ≤ (G2 ). Remark. As the initial plant (6) is stable and MP, and the targeted plant also is, the compensator (7) will be also stable and MP.
Lemma 1. A multi time-delay SISO plant with several delays in parallel, such that
n
Gi (s)e−i s ;
C(s) =
G(s)e−s n
1 ≤ 2 ≤ .. . ≤ n
Gi
n
⎞
(s)e−i s
Gi ⎜ ⎜ i=2 ⎜ == / 1+ G1 (s)e−1 s ⎜ G1 (s)e−1 s ⎝ G(s)e−s
(s)e−i s
⎟ ⎟ ⎟ ⎟ ⎠
(14)
that can be rearranged as depicted in Fig. 2 with components as shown in (12), (13). Remark. The targeted transfer function should be selected such that ≥ 1 , and (G) ≥ (G1 ). Corollary 1. If there is a non-delayed input/output path in the process, the time-delays may be cancelled by the appropriate selection of the pre-compensator. In fact, if the minimum delay is 1 = 0, the open-loop transfer function G(s) can be selected to be undelayed, as far as the constraint a is satisfied.
P(s) = 1 +
In Ref. [36] the control of the plant e−2s 1 + 1+s 1 + 4s
is approached by considering the delayed component as a disturbance, estimating it by means of a disturbance observer and then rejecting it. This involves an approximation. Now, if the suggested pre-compensator is added, assuming P(s) = 1 +
1 e−2s + = G1 (s) + G2 (s)e−2s , 1+s 1 + 4s
the pre-compensator will be G(s) = G1 (s) = 1 +
This result can be generalized by the following lemma.
P(s) =
(13)
⎛
Example.
Thatis, from(8)and(9) Kf (s) =
i = 2. . .n
G(s)e−s
i=1
This transfer function can be rewritten as C(s) =
Gi (s)e−i s
Proof. By repeating similar arrangements than in the case of two delays, the compensator elements are derived as follows. The precompensator is computed as
Thus, assuming a desired compensated transfer function (5), L(s) = G(s)e−s = P(s)C(s), the pre-compensator is easily derived as (s)e−1 s
(12)
G1 (s)e−1 s
if and only if the Constraints a, (G1 ) ≤ (Gi ); i = 2, ..., n , is fulfilled, to get realizable compensator elements (12), (13).
3.1. Parallel paths
C(s) =
G(s)e−s
(11)
1 ⇒ Kf = 1; 1+s
Kb (s) =
1+s e−2s (1 + 4s)(s + 2)
and the time-delay is cancelled. Thence, a suitable control can be designed for G(s). 3.2. Time-delays in the loop
i=1
can be converted to a single time-delay model with transfer function G(s)e−s = P(s)C(s) by adding a looped pre-compensator C(s) with the structure shown in Fig. 2
Let us assume that there is a loop in the process, with timedelays in both paths, as indicated in (4). The process block diagram and the proposed pre-compensator are depicted in Fig. 3
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Fig. 3. Type II compensator structure. Fig. 6. Combined series system.
Proof. By repeating similar arrangements than in the case of two blocks, the compensator is derived as
G(s)e−s
= Assuming that the desired transfer function of the single dead time model is given by G(s)e−s , the compensator should be as
G(s)e−s 1 + G1 (s)e−1 s G2 (s)e−2 s
C(s) = =
G1
(s)e−1 s
⎧ ⎨ K = G(s)e−s f1 G1 (s)e−1 s C(s) = Kf 1 (s) + Kf 2 (s) with ⎩ −s
(16)
G2 (s)e−2 s
as illustrated in Fig. 3. Again, in order to define a realizable control, the targeted transfer function should be such that ≥ 1 and (G) ≥ (G1 ). This result can be generalized by the following lemma. Lemma 2. A multi time-delay SISO plant with several forward and feedback delays, as shown in Fig. 4, such that G1 (s)e−1 s 1 + G1
(s)e−1 s
n
Gi
(17) (s)e−i s
i=2
can be converted into a single time-delay model with transfer function G(s)e−s = P(s)C(s) by adding a parallel pre-compensator with the structure depicted in Fig. 4 where Kf (s) =
G(s)e−s
(18)
G1 (s)e−1 s
G(s)e−s G1 (s)e−1 s
+ G(s)e−s
n
Gi (s)e−i s
(20)
i=2
Similar pre-compensator structures can be developed to simplify other complex multi time-delay processes, combining the previous compensators. There are many possibilities. In order to illustrate the procedure, let us analyze some general structures. Let us consider a series system as depicted in Fig. 5a. The blocks can be swapped to be represented as shown in Fig. 5b. Two options can be considered: i) transform the whole system into a single parallel path by introducing the first subsystem into those in parallel and then, apply the procedure outlined in Section 3.1; or ii) swap the subsystems, as depicted in Fig. 5b, and apply the procedure in Section 3.1 to the parallel path in the first block. Once this path is converted to a single time-delay it can be combined in series with the last one. Moreover, if the first subsystem in Fig. 5a is a loop system, the conversion strategy presented in Section 3.2 can be applied to it. Once transformed into a single time-delay subsystem the structure is similar to that in Fig. 5a and the above procedure can be applied. Let us now assume a more general case represented by the following transfer function P(s) =
i = 2. . .n
if the targeted transfer function G(s)e−s is properly selected.
(19)
Ga (s)e−a s + Gb (s)e−b s
(21)
1 + Gc (s)e−c s
If there is a common factor G1 (s)e−1 s in the three transfer functions, it can be rewritten as P(s) =
and Kfi (s) = Gi (s)e−i s G(s)e−s
G1 (s)e−1 s
(15)
requiring two paths in parallel to be implemented. For instance,
P(s) =
Gi
3.3. Generalization to multi time-delay structures
+ G(s)e−s G2 (s)e−2 s
Kf 2 = G(s)e
(s)e−i s
which can be implemented assuming (18) and (19) in the structure shown in Fig. 4.
G1 (s)e−1 s
G(s)e−s
n i=2
C(s) = Fig. 4. General type II compensator structure.
1 + G1
(s)e−1 s
G1 (s)e−1 s 1 + G1 (s)e−1 s G4 (s)e−4 s
[G2 (s)e−2 s + G3 (s)e−3 s ]
(22)
otherwise, it can be assumed that G1 (s)e−1 s = 1. That is, it can be considered as a feedback loop, F1 (s), in series with a parallel path, F2 (s), as shown in Fig. 6
Fig. 5. Series system with parallel delayed subsystems at: a) the output or b) the input.
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be reduced to a single time-delay by applying Lemma 2. Moreover, according to (23) it can be reduced toL1 (s) = d 1(s) . To ensure the 1
internal stability, the d1 (s) should be Hurtwitz. The realizability of the compensator’s components C1i (s) requires (d1 (s)) ≥ (dj (s));
Fig. 7. General compensator structure.
Following the steps defined in Section 3.2, a pre-compensator C1 (s) can be designed to be converted together with the first block F1 (s) into a single time-delay term and then, the remaining parallel path is treated as in Section 3.1. As a result, the compensator in Fig. 7 is obtained, being: C11 (s) =
H(s)e−s G1 (s)e−1 s
;
C12 (s) = G4 (s)e−4 s H(s)e−s
(23)
where L1 (s) = H(s)e−s = C1 (s)F1 (s) is the target for the first simplified block. Then, combining this equivalent subsystem with the F2 (s) block it yields C21 (s) =
L(s) G2
(s)e−2 s H(s)e−s
C22 (s) =
;
G3 (s)e−3 s H(s)e−s L(s)
(24)
where L(s) = C2 (s)H(s)F2 (s)e−s = C2 (s)F2 (s)L1 (s) is the target for the total converted system. Of course, the constraints for the realizability of all the elements should be fulfilled. 4. Main result Let us consider the general case of SISO plants with multiple time-delays, as described in (1). This transfer function can be obtained from any complex system composed of several elements connected in a network by applying, for instance, the Mason’s rule [33]. If this is the case, one term in the denominator will be the unity or, at least a non-delayed term, that is, ϑ1 = 0. Thus, the following theorem provides the main result. Theorem. Given a stable and minimum phase SISO plant with transfer function with multiple time-delays in the numerator and/or denominator, as described in (1), it can be converted to a single time-delay transfer function by means of a pre-compensator provided that the following assumption is fulfilled. (n1 (s)) ≤ (ni (s));
Assumption A.
i = 2, ..., r1 (25)
The pre-compensator is obtained by applying the previous lemmas. Proof. Under the assumption that ϑ1 = 0, the plant transfer function (1) can be rewritten as r1
ni (s).e−i s
r2
d1 (s) +
dj (s).e−j s
5. Examples In this part, some examples to verify the effectiveness of the proposed method are developed. 5.1. Example 1: basic model Consider the following multiple time-delay plant P(s), which is given in [25]. P(s) =
= 1+
1 d1 (s)
−0.3534 −0.5519 e−3.14s + e−2.345s 35.235s2 + 16.11s + 1 42.2025s2 + 19.1s + 1
An appropriate targeted single time-delay compensated plant model L(s) is defined. For instance, let us assumeL(s) = e−2.345s /(s + 1)2 . Applying the formulas (10) and (11) for the parallel processes, the compensator is obtained as Kf (s) = −
42.2025s2 +19.1s+1 0.5519(s + 1)
dj (s).e−j s
r1
(26)
i=1
j=2
The first component can be considered as a feedback loop with forward path d 1(s) and a parallel feedback, similar to (22), and it can
;
Kb (s) =
−0.3534(s + 1)2 e−0.795s ; 35.235s2 + 16.11s + 1
A general multiple time-delay transfer function with a long time-delay in the dominator, as reported in [26], is considered: y(s) =
(s + 2)e−3s + (s + 3)e−2s u(s) (6s + 1) + (0.5s + 1)e−5s
In order to apply the proposed methodology, the transfer function is represented as P(s) =
G2 (s)e−2 s + G3 (s)e−3 s 1 + G4 (s)e−4 s
which is similar to (22) with G1 (s)e−1 s = 1, G4 (s)e−4 s = s+2 −3s s+3 −2s G3 (s)e−3 s = 6s+1 e , andG2 (s)e−2 s = 6s+1 e . The pre-compensator is designed following the methodology for general multiple SISO processes, (23) and (24), and the detailed structure of the compensator is shown in Fig. 7, being 0.5s+1 −5s e , 6s+1
=
ni (s).e−i s
2
5.2. Example 2: general model
j=2 1 d1 (s) r2
(27)
If this is not the case, new terms can be added to d1 (s) to get d¯ 1 (s)in such a way that d¯1 (s)remains Hurtwitz and it also fulfills this condition. A new term d¯ 1 (s) − d1 (s) is added in the denominator sum (For illustration, see the example 6 in Section 6). The second term, once multiplied by L1 (s), can be treated as a parallel system, applying Lemma 1, as illustrated in (24). In this case, Assumption A is required.
C11 (s) = H(s)e−s ;
i=1
P(s) =
j = 2, ..., r2
C12 (s) = G4 (s)e−4 s H(s)e−s ; C21 (s)
L(s) G2 (s)e−2 s H(s)e−s
;
C22 (s) =
G3 (s)e−3 s H(s)e−s L(s)
Taking into account the constraints, the target for the first block can be selected as H(s)e−s = G4 (s)e−4 s , whereas the desired openloop transfer function could be selected, as L(s) =
0.5s + 1 −7s e 6s + 1
1
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leading to C11 (s) =
0.5s + 1 −5s e ; 6s + 1
C21 (s) =
6s + 1 ; s+3
C12 (s) =
C22 (s) =
0.5s + 1 2 6s + 1
e−10s ;
(s + 2) −s e (6s + 1)
(3s + 1)e−0.5s + (s + 1) e−2s y(s) = u(s) (s + 1)2 (3s + 1) (5s − 1) To make the pre-compensator stable the unstable term should not be cancelled. Thence, L(s) is defined, for example, as: L(s) =
6. Unstable and NMP plants Dealing with right hand poles/zeros, some additional limitations may appear.
e−0.5s (2s + 1) (0.5s + 1) (5s − 1)
Then, the corresponding compensator for the parallel path is computed as Kf (s) =
(s + 1)2 (2s + 1) (0.5s + 1)
Kb (s) =
(2s + 1) (0.5s + 1) −1.5s e (s + 1) (3s + 1)
6.1. Non-minimum phase model In this case, some simple transformations allow the application of the proposed approach avoiding the unstable pole/zero cancellation. Example 3. For instance, assume that the transfer function of the process with non-minimum phase zero is given by y(s) e−1.5s (s − 1) e−s + = 2 = F1 (s) + F2 (s) 2s +1 u(s) s +3s + 1
6.3. Plants with an unstable term
If the pre-compensator as defined in formula (12) is directly used, a right half plane pole will appear in Kf (s). Thus, a new equivalent model without non-minimum phase zeros is defined. A feasible way is to decompose the initial model into Gp =
−2e−s e−1.5s (s + 1) e−s + 2 + = F1 (s) + F2 (s) + F3 (s) 2 2s + 1 s +3s + 1 s +3s + 1
and the compensator elements will be computed as Kf (s) =
L(s) F2 (s) F3 (s) ; K (s) = ; Kb22 (s) = F1 (s) b21 L(s) L(s)
If the unstable pole only appears in one term in parallel, this direct approach is only applicable if the unstable term presents the shortest time-delay, as can be easily derived by considering (12) and (13), otherwise the corresponding feedback component Kbi (s) would be unstable. Example 5.
e−s 3s + 1
e−2s e−0.5s y(s) + = u(s) (s + 1) (3s + 1) (s + 1)2 (5s − 1)
L(s) =
s2 +3s + 1 (3s + 1) (s + 1)
;
Kb21 (s) =
−2 (3s + 1) ; s2 +3s + 1
Kf =
(s + 2)3
(s + 2)
Example 6.
Another option, if the term including the non-minimum-phase zero has higher time-delay y(s) e−s (s − 1) e−1.5s = = F1 (s) + F2 (s) + 2 2s + 1 u(s) s +3s + 1 is to design the compensator as follows: L(s) 3s + 1 ; Kf (s) = = 2s + 1 F2 (s)
(s + 1)2 (5s − 1)
Kb22 (s) =
2s + 1 −0.5s e 3s + 1
(3s + 1) (s − 1) −0.5s Kb (s) = F1 (s)L(s) = e s2 +3s + 1
where Kf (s) is stable and Kb (s) is non-minimum phase. 6.2. Unstable plants: common term For unstable plants, if the unstable term is common to all the paths, it can be left out of the simplification process. Example 4.
e−0.5s
the compensator is designed as
Leading to Kf (s) =
For instance, let us assume this first situation being
If the desired transfer function is assumed to be
Taking into account the constraints, L(s) can be selected as L(s) =
In this case, if the final goal is to control the compensated process, a control design approach appropriated for unstable time delayed plants should be applied.
For instance, assume that the plant is given as:
3
;
Kb =
(s + 2)3 e−1.5s (s + 1) (3s + 1)
Let us now assume the other case, that is:
y(s) e−0.5s e−2s = + s+1 s−1 u(s) It can be expressed in the general form (1) like (s − 1)e−0.5s + (s + 1)e−2s y(s) = u(s) s2 − 1 In this case, the denominator d1 (s) is not Hurtwitz. So, define, for instance, d¯ 1 (s) = s2 + 3s + 2. The process transfer function can be written as: P(s) =
y(s) = u(s) 1+
1 s2 +3s+2 1 (−3s − 3) s2 +3s+2
(s − 1)e−0.5s + (s + 1)e−2s
which is similar to (22). Identifying the different components in (23), the following selection of the transfer function will make all the compensator components stable and realizable. If H(s) = 3 3(s + 1) G1 (s) = 2 1 , then C11 (s) = 1; C12 (s) = − =− s +3s+2 s+2 s2 + 3s + 2
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And this pre-compensator C1 (s) applied to the process P(s) leads to a new process: P(s)C1 (s) =
s−1 1 −2s e−0.5s + e s+2 s2 + 3s + 2
where the approach followed in the Example 3 can be applied. 7. Conclusions In this paper, the problem of converting a process model with multiple time-delays to one with a single time-delay in the forward path has been discussed, and the pre-compensator design procedure has been outlined. The main achievement is to get a compensated system with a single time-delay thus making easier the possible design of the control. Similar to the Smith Predictor, there are some constraints in applying this procedure but it is very useful in many practical processes with multiple delays. The new proposed pre-compensator structure is able to compensate all kinds of dead times, including the complicated systems with multiple time-delays in both the numerator and denominator, as far as some constraints about the time-delays and relative order of the process components are fulfilled. Moreover, no approximation is required. A comparison with the approach used in [30] illustrates this advantage. The proposed pre-compensator structure has been illustrated with several examples considering multiple delays, multiple delayed paths as well as delayed feedback paths. As the approach is based on cancellations, some additional constraints should be taken into account when dealing with unstable and non-minimum phase plant models. The procedure can be applied for any process described in the general form (1) with the limitation described by the assumptions A in Section 4. This assumption is usually satisfied in most industrial applications, as shown in the different examples presented in the paper. The control design limitations encountered in dealing with multi time-delays SISO plants have been overcome as the design is applied to a rational transfer function once the initial plant is converted into a single time-delay plant and then the SP or any well-established DTC approach is applied. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 61473137 and 61722306, the 111 projects (Grants No. B12018 and No. B08015), and the project TIN2014-56158-C4-4-P-AR, Ministerio de Economía y Competitividad, Spain. References [1] A. Bultheel, M.V. Barel, Padé techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math. 14 (3) (1986) 401–438. [2] K.E. Kwok, M.C. Ping, P. Li, A model-based augmented PID algorithm, J. Process Control 10 (2000) 9–18. [3] O.J.M. Smith, Closed control of loops with dead time, Chem. Eng. Prog. 53 (5) (1957) 217–219. [4] J.E. Normey-Rico, E.F. Camacho, Dead-time compensators: a survey, Control Eng. Pract. 16 (4) (2008) 407–428. [5] P. Albertos, P. García, Robust control design for long time-delay systems, J. Process Control 19 (10) (2009) 1640–1648. [6] P. García, P. Albertos, T. Hagglund, Control of unstable non-minimum-phase delayed systems, J. Process Control 16 (2006) 1099–1111.
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Research Paper
Conversion of SISO processes with multiple time-delays to single time-delay processes Xiaoli Luan a , Qiang Chen a , Pedro Albertos b,∗ , Fei Liu a a b
Key laboratory of Advance Process Control for Light Industry (Ministry of Education) and the Institute of Automation, Jiangnan University, China Department of System Engineering and Control, Universidad Politécnica de Valencia, Spain
a r t i c l e
i n f o
Article history: Received 22 February 2017 Received in revised form 7 September 2017 Accepted 2 October 2017 Available online xxx Keywords: Multiple time-delays Pre-compensator design Single time-delay based control
a b s t r a c t There are many processes with multiple time-delays. These may appear in the direct path or in some feedback path. In this paper, a pre-compensator is designed in order to convert a multiple time-delays process to one with a single time-delay in the forward path. On this simpler model, any well-established approach to design the control for input/output time-delay plants can be applied. The proposed method is directly applicable to stable plants with multiple time-delays, including in the feedback path, and it can be extended to unstable plants with some constraints. Some numerical examples and case studies are provided to illustrate the application of the proposed algorithms. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Time-delays are widely found in the process industries when transporting materials or energy. These delays may also arise at the actuator inputs, at the sensor outputs as well as in the interconnection between internal/external variables, and can lead to degradation of the controlled closed-loop system behavior. Therefore control of dead-time systems has been extensively studied over the past decades. As the time-delay operator is not a rational function, the stability analysis and control design for transfer functions including time-delays is very complicated, leading to a characteristic quasipolynomial with infinite number of roots. One option to deal with dead-time processes is based on some kind of approximation of either the model response or the transfer function, like the Padé approximations [1], used in [2]. However, the major disadvantage of the approximations lies in that some model properties are lost and, for instance, the approximated model may be unstable even the original one is stable. Dead-time compensation is one of the control approaches used to simplify the design of closed-loop controlled delayed systems, and the pioneer proposal was the Smith predictor (SP) [3]. As it is well-known, in this setting, an undelayed output is predicted based on the input and output measurements,
∗ Corresponding author. E-mail addresses: [email protected] (X. Luan), [email protected] (Q. Chen), [email protected], [email protected] (P. Albertos), fl[email protected] (F. Liu).
in such a way that the delay is shifted out of the loop and the control can be designed for the rational part of the transfer function. One strong limitation of the SP is the requirement to factorize the transfer function into two terms, the delay and the rational (also called fast) part which is also required to be sTable Since the seminal work proposed by Smith, many extensions of the SP have been presented, such as the extension to unstable or integrating processes [4–6], systems with cascade structure [7–9], discrete-time stable or unstable systems [10,11], or systems with uncertainties and disturbances [12–14]. Nevertheless, the time-delay should be separated from the rational transfer function. The aforementioned techniques are suitable for single-inputsingle-output (SISO) systems with single time-delay. Some extensions have been reported for multi-input and multioutput (MIMO) plants with different time-delays in the different input/output path signals (see, for instance, [15]). However, multiple time-delays are also found in many practical SISO applications, such as industrial processes where there are different input-output paths or internal recycling [2,16,17], or when considering congestion control over a transition control protocol (TCP) network [18,19]. In some cases [15], the multi time-delay structure appears in some elements of the MIMO transfer matrix [20]. Although the control of a time-delay system is more difficult than the control of a non-delayed system, it is much easier than that of a system with multiple time-delays. Thus, different model approximation approaches were put forward to transform the systems with multiple time-delays into ones without or with a single time-delay, such as the above mentioned Padé approximation, the decentralized relay feedback estimation, constrained H2 approx-
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imation, etc. [16,21–23]. Instead of utilizing the approximating process, the state-space approach was proposed to consider the influence of multiple time-delays [24,25], by enlarging the state vector, which represents an increment in the system dimension leading to higher computational cost. To overcome these disadvantages, a predictor-observer-based control strategy for SISO disturbed systems with multiple time-delays was proposed in Ref. [26]. To simplify the controller design, the filtered Smith predictor (FSP) is extended in Ref. [27] to systems with multiple time-delays to consider the closed-loop performances for robustness, set-point tracking, disturbance rejection, and noise attenuation [28,29]. In the aforementioned references dealing with multiple timedelays systems the model approximation is necessary to achieve the desired performance by either approximating the systems with a single time-delay, or by designing the controller based on deadtime compensation [30]. This will definitely lead to inaccuracy. On the other hand, simple models are very important so they should be used in any proposed control methodology to be easily applied and tuned. In particular, this is important for complicated systems where the numerator and denominator of the transfer function contain multiple time-delays. The problem of control design for a SISO plant with multiple time-delays has not been solved in a general case. In Ref. [29], the design of low order controllers is tackled in the frequency domain, mainly dealing with the controlled plant at low frequencies and applying the generalized Nyquist criterion to the characteristic quasipolynomial. As already mentioned, the state space representation [24] has been also used to analyze the stability of this kind of plants, but the control design is not easy in this framework. In this paper, the control design is formulated in three stages:
2. Problem statement Let us represent the kind of SISO processes we are interested in as P(s), a rational function of the variables s, including some exponential terms e–s , pointing out that the dynamics are represented by differential terms and time-delays. The general case will be r1
P(s) =
ni (s).e−i s
i=1 r2
; dj (s).e
1 ≤ 2 · · · ≤ r1
and
1 ≤ 2 · · · ≤ r2 (1)
−j s
j=1
where ni (s), dj (s), are polynomials of s, showing that there are r1 delays in the numerator and r2 delays in the denominator, and i , j are different delays appearing ordered in the global transfer function. They are denoted by different variables to make easier the notation. As the plant could be considered as a complex structure composed of delayed elements connected in series, parallel and loop, the global transfer function (1) could be also expressed as a composition of elementary delayed transfer functions and, as later on explained in Section 4 and also defined in Ref. [32], it can be assumed that 1 = 0. The simplest case is when there are several subsystems and delays in series (for instance, input 1 and output 2 delays and G1 (s), G2 (s) as rational transfer functions). In this case, the total transfer function is easily reduced to a single delay: P(s) = G1 (s)G2 (s)e−(1 +2 )s = G(s)e−s
(2)
Other than that, let us consider: 1) a pre-compensator is designed to convert, under some constraints, the initial multiple time-delays process to one with a single time-delay in the forward path, without any approximation. 2) then, an already developed DTC is applied to the obtained single time-delay in order to get the rational part, that is, the undelayed output. It could be the SP or any improved solution. Otherwise, a general control design approach for single-delay processes can be applied. 3) the final control is designed for the rational transfer function following any appropriate control design technique.
a Parallel paths. There are several paths of series systems like (2), in such a way that the transfer function is
P(s) =
Gi (s)e−i s
(3)
i=1
Without loss of generality, it is assumed that 1 < 2 · · · < r . • There is a loop with both, forward and feedback delays. The full transfer function is given by P(s) =
As the last two stages are well documented and accessible in the literature (see, for instance Refs. [4] and [31]), in this paper the implementation of the first stage is presented. For that purpose, a procedure to convert a multi time-delay SISO plant into an equivalent model with a rational part and a single time-delay is presented. Then, steps 2) and 3) above can be followed to design the control. As there are pole/zero cancellations, the proposed approach is directly applicable to stable and minimum-phase (MP) systems with multiple time-delays, including in the feedback path. In order to get a stable and realizable compensator, some constraints should be fulfilled. The paper is organized as follows. In the next section the problem is defined. Then, the pre-compensator design is developed for some basic structures including elements in parallel and loop. The main result considers the general case and a procedure to apply the previous results is summarized. Some numerical examples are presented to illustrate the effectiveness of the proposed algorithms. Unstable terms can be handled under some conditions and nonminimum-phase (NMP) terms require a pre-treatment, as detailed in Section 6. Finally, some conclusions are drafted in the last section.
r
G1 (s)e−1 s 1 + G1 (s)e−1 s G2 (s)e−2 s
=
G1 (s)e−1 s 1 + Gb (s)e−b s
(4)
where, obviously, 1 ≤ b . • The process is composed of a combination of these basic structures. For instance, both, the forward and the feedback paths may have several sub-paths in parallel (as well as loops or series terms). The main goal of this work is to design a pre-compensator, C(s) such that the whole system can be represented as P(s)C(s) = G(s)e−s = L(s)
(5)
The structure and components of this pre-compensator will depend on the structure of the process, as later on shown. Thence, the number and distribution of the delays is much more general than those in the processes considered in Ref. [27] and no approximation approach is used, as proposed in Ref. [1]. Once the simplified model is obtained, a controller can be designed for the rational plant G(s) [32,33] or by any control design approach suitable for single time-delayed plants [4,34,35].
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Fig. 1. Type I compensator structure. Fig. 2. General type I compensator structure.
Initially, the case of stable and minimum-phase plants is considered. 3. Pre-compensator design
where Kf (s) =
The structure of the pre-compensator could be also with series, parallel and feedback components. The main requirement is that all the components should be realizable and stable. This will impose some constraints in the selection of its structure, as later on shown.
Let us assume that there are two paths in parallel. The process transfer function is P(s) = G1 (s)e−1 s + G2 (s)e−2 s ;
and Kbi (s) =
G(s)e−s G1
(6)
1 < 2
+ G2
(s)e−2 s
=
G(s)e−s G1
(s)e−1 s
/
1+
G2
(s)e−2 s
G1
(s)e−1 s
(7)
G(s)e−s G1 (s)e−1 s G(s)e−s
1+
G1 (s)e−1 s
(8)
G2 (s)e−2 s G(s)e−s
and it can be implemented by means of the structure shown in Fig. 1. The compensator (8) is given by C(s) =
Kf (s)
(9)
1+Kf (s)Kb (s)
G(s)e−s G1
(s)e−1 s
;
Kb (s) =
G2 (s)e−2 s G(s)e−s
(10)
In order to get a realizable compensator, looking at (10), the targeted transfer function delay should be 2 ≥ ≥ 1 , and (G1 ) ≤ (G) ≤ (G2 ), where (G) denotes the relative degree of the transfer function G(s). Constraints a: In order to define a realizable compensation, it should be (G1 ) ≤ (G2 ). Remark. As the initial plant (6) is stable and MP, and the targeted plant also is, the compensator (7) will be also stable and MP.
Lemma 1. A multi time-delay SISO plant with several delays in parallel, such that
n
Gi (s)e−i s ;
C(s) =
G(s)e−s n
1 ≤ 2 ≤ .. . ≤ n
Gi
n
⎞
(s)e−i s
Gi ⎜ ⎜ i=2 ⎜ == / 1+ G1 (s)e−1 s ⎜ G1 (s)e−1 s ⎝ G(s)e−s
(s)e−i s
⎟ ⎟ ⎟ ⎟ ⎠
(14)
that can be rearranged as depicted in Fig. 2 with components as shown in (12), (13). Remark. The targeted transfer function should be selected such that ≥ 1 , and (G) ≥ (G1 ). Corollary 1. If there is a non-delayed input/output path in the process, the time-delays may be cancelled by the appropriate selection of the pre-compensator. In fact, if the minimum delay is 1 = 0, the open-loop transfer function G(s) can be selected to be undelayed, as far as the constraint a is satisfied.
P(s) = 1 +
In Ref. [36] the control of the plant e−2s 1 + 1+s 1 + 4s
is approached by considering the delayed component as a disturbance, estimating it by means of a disturbance observer and then rejecting it. This involves an approximation. Now, if the suggested pre-compensator is added, assuming P(s) = 1 +
1 e−2s + = G1 (s) + G2 (s)e−2s , 1+s 1 + 4s
the pre-compensator will be G(s) = G1 (s) = 1 +
This result can be generalized by the following lemma.
P(s) =
(13)
⎛
Example.
Thatis, from(8)and(9) Kf (s) =
i = 2. . .n
G(s)e−s
i=1
This transfer function can be rewritten as C(s) =
Gi (s)e−i s
Proof. By repeating similar arrangements than in the case of two delays, the compensator elements are derived as follows. The precompensator is computed as
Thus, assuming a desired compensated transfer function (5), L(s) = G(s)e−s = P(s)C(s), the pre-compensator is easily derived as (s)e−1 s
(12)
G1 (s)e−1 s
if and only if the Constraints a, (G1 ) ≤ (Gi ); i = 2, ..., n , is fulfilled, to get realizable compensator elements (12), (13).
3.1. Parallel paths
C(s) =
G(s)e−s
(11)
1 ⇒ Kf = 1; 1+s
Kb (s) =
1+s e−2s (1 + 4s)(s + 2)
and the time-delay is cancelled. Thence, a suitable control can be designed for G(s). 3.2. Time-delays in the loop
i=1
can be converted to a single time-delay model with transfer function G(s)e−s = P(s)C(s) by adding a looped pre-compensator C(s) with the structure shown in Fig. 2
Let us assume that there is a loop in the process, with timedelays in both paths, as indicated in (4). The process block diagram and the proposed pre-compensator are depicted in Fig. 3
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Fig. 3. Type II compensator structure. Fig. 6. Combined series system.
Proof. By repeating similar arrangements than in the case of two blocks, the compensator is derived as
G(s)e−s
= Assuming that the desired transfer function of the single dead time model is given by G(s)e−s , the compensator should be as
G(s)e−s 1 + G1 (s)e−1 s G2 (s)e−2 s
C(s) = =
G1
(s)e−1 s
⎧ ⎨ K = G(s)e−s f1 G1 (s)e−1 s C(s) = Kf 1 (s) + Kf 2 (s) with ⎩ −s
(16)
G2 (s)e−2 s
as illustrated in Fig. 3. Again, in order to define a realizable control, the targeted transfer function should be such that ≥ 1 and (G) ≥ (G1 ). This result can be generalized by the following lemma. Lemma 2. A multi time-delay SISO plant with several forward and feedback delays, as shown in Fig. 4, such that G1 (s)e−1 s 1 + G1
(s)e−1 s
n
Gi
(17) (s)e−i s
i=2
can be converted into a single time-delay model with transfer function G(s)e−s = P(s)C(s) by adding a parallel pre-compensator with the structure depicted in Fig. 4 where Kf (s) =
G(s)e−s
(18)
G1 (s)e−1 s
G(s)e−s G1 (s)e−1 s
+ G(s)e−s
n
Gi (s)e−i s
(20)
i=2
Similar pre-compensator structures can be developed to simplify other complex multi time-delay processes, combining the previous compensators. There are many possibilities. In order to illustrate the procedure, let us analyze some general structures. Let us consider a series system as depicted in Fig. 5a. The blocks can be swapped to be represented as shown in Fig. 5b. Two options can be considered: i) transform the whole system into a single parallel path by introducing the first subsystem into those in parallel and then, apply the procedure outlined in Section 3.1; or ii) swap the subsystems, as depicted in Fig. 5b, and apply the procedure in Section 3.1 to the parallel path in the first block. Once this path is converted to a single time-delay it can be combined in series with the last one. Moreover, if the first subsystem in Fig. 5a is a loop system, the conversion strategy presented in Section 3.2 can be applied to it. Once transformed into a single time-delay subsystem the structure is similar to that in Fig. 5a and the above procedure can be applied. Let us now assume a more general case represented by the following transfer function P(s) =
i = 2. . .n
if the targeted transfer function G(s)e−s is properly selected.
(19)
Ga (s)e−a s + Gb (s)e−b s
(21)
1 + Gc (s)e−c s
If there is a common factor G1 (s)e−1 s in the three transfer functions, it can be rewritten as P(s) =
and Kfi (s) = Gi (s)e−i s G(s)e−s
G1 (s)e−1 s
(15)
requiring two paths in parallel to be implemented. For instance,
P(s) =
Gi
3.3. Generalization to multi time-delay structures
+ G(s)e−s G2 (s)e−2 s
Kf 2 = G(s)e
(s)e−i s
which can be implemented assuming (18) and (19) in the structure shown in Fig. 4.
G1 (s)e−1 s
G(s)e−s
n i=2
C(s) = Fig. 4. General type II compensator structure.
1 + G1
(s)e−1 s
G1 (s)e−1 s 1 + G1 (s)e−1 s G4 (s)e−4 s
[G2 (s)e−2 s + G3 (s)e−3 s ]
(22)
otherwise, it can be assumed that G1 (s)e−1 s = 1. That is, it can be considered as a feedback loop, F1 (s), in series with a parallel path, F2 (s), as shown in Fig. 6
Fig. 5. Series system with parallel delayed subsystems at: a) the output or b) the input.
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be reduced to a single time-delay by applying Lemma 2. Moreover, according to (23) it can be reduced toL1 (s) = d 1(s) . To ensure the 1
internal stability, the d1 (s) should be Hurtwitz. The realizability of the compensator’s components C1i (s) requires (d1 (s)) ≥ (dj (s));
Fig. 7. General compensator structure.
Following the steps defined in Section 3.2, a pre-compensator C1 (s) can be designed to be converted together with the first block F1 (s) into a single time-delay term and then, the remaining parallel path is treated as in Section 3.1. As a result, the compensator in Fig. 7 is obtained, being: C11 (s) =
H(s)e−s G1 (s)e−1 s
;
C12 (s) = G4 (s)e−4 s H(s)e−s
(23)
where L1 (s) = H(s)e−s = C1 (s)F1 (s) is the target for the first simplified block. Then, combining this equivalent subsystem with the F2 (s) block it yields C21 (s) =
L(s) G2
(s)e−2 s H(s)e−s
C22 (s) =
;
G3 (s)e−3 s H(s)e−s L(s)
(24)
where L(s) = C2 (s)H(s)F2 (s)e−s = C2 (s)F2 (s)L1 (s) is the target for the total converted system. Of course, the constraints for the realizability of all the elements should be fulfilled. 4. Main result Let us consider the general case of SISO plants with multiple time-delays, as described in (1). This transfer function can be obtained from any complex system composed of several elements connected in a network by applying, for instance, the Mason’s rule [33]. If this is the case, one term in the denominator will be the unity or, at least a non-delayed term, that is, ϑ1 = 0. Thus, the following theorem provides the main result. Theorem. Given a stable and minimum phase SISO plant with transfer function with multiple time-delays in the numerator and/or denominator, as described in (1), it can be converted to a single time-delay transfer function by means of a pre-compensator provided that the following assumption is fulfilled. (n1 (s)) ≤ (ni (s));
Assumption A.
i = 2, ..., r1 (25)
The pre-compensator is obtained by applying the previous lemmas. Proof. Under the assumption that ϑ1 = 0, the plant transfer function (1) can be rewritten as r1
ni (s).e−i s
r2
d1 (s) +
dj (s).e−j s
5. Examples In this part, some examples to verify the effectiveness of the proposed method are developed. 5.1. Example 1: basic model Consider the following multiple time-delay plant P(s), which is given in [25]. P(s) =
= 1+
1 d1 (s)
−0.3534 −0.5519 e−3.14s + e−2.345s 35.235s2 + 16.11s + 1 42.2025s2 + 19.1s + 1
An appropriate targeted single time-delay compensated plant model L(s) is defined. For instance, let us assumeL(s) = e−2.345s /(s + 1)2 . Applying the formulas (10) and (11) for the parallel processes, the compensator is obtained as Kf (s) = −
42.2025s2 +19.1s+1 0.5519(s + 1)
dj (s).e−j s
r1
(26)
i=1
j=2
The first component can be considered as a feedback loop with forward path d 1(s) and a parallel feedback, similar to (22), and it can
;
Kb (s) =
−0.3534(s + 1)2 e−0.795s ; 35.235s2 + 16.11s + 1
A general multiple time-delay transfer function with a long time-delay in the dominator, as reported in [26], is considered: y(s) =
(s + 2)e−3s + (s + 3)e−2s u(s) (6s + 1) + (0.5s + 1)e−5s
In order to apply the proposed methodology, the transfer function is represented as P(s) =
G2 (s)e−2 s + G3 (s)e−3 s 1 + G4 (s)e−4 s
which is similar to (22) with G1 (s)e−1 s = 1, G4 (s)e−4 s = s+2 −3s s+3 −2s G3 (s)e−3 s = 6s+1 e , andG2 (s)e−2 s = 6s+1 e . The pre-compensator is designed following the methodology for general multiple SISO processes, (23) and (24), and the detailed structure of the compensator is shown in Fig. 7, being 0.5s+1 −5s e , 6s+1
=
ni (s).e−i s
2
5.2. Example 2: general model
j=2 1 d1 (s) r2
(27)
If this is not the case, new terms can be added to d1 (s) to get d¯ 1 (s)in such a way that d¯1 (s)remains Hurtwitz and it also fulfills this condition. A new term d¯ 1 (s) − d1 (s) is added in the denominator sum (For illustration, see the example 6 in Section 6). The second term, once multiplied by L1 (s), can be treated as a parallel system, applying Lemma 1, as illustrated in (24). In this case, Assumption A is required.
C11 (s) = H(s)e−s ;
i=1
P(s) =
j = 2, ..., r2
C12 (s) = G4 (s)e−4 s H(s)e−s ; C21 (s)
L(s) G2 (s)e−2 s H(s)e−s
;
C22 (s) =
G3 (s)e−3 s H(s)e−s L(s)
Taking into account the constraints, the target for the first block can be selected as H(s)e−s = G4 (s)e−4 s , whereas the desired openloop transfer function could be selected, as L(s) =
0.5s + 1 −7s e 6s + 1
1
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leading to C11 (s) =
0.5s + 1 −5s e ; 6s + 1
C21 (s) =
6s + 1 ; s+3
C12 (s) =
C22 (s) =
0.5s + 1 2 6s + 1
e−10s ;
(s + 2) −s e (6s + 1)
(3s + 1)e−0.5s + (s + 1) e−2s y(s) = u(s) (s + 1)2 (3s + 1) (5s − 1) To make the pre-compensator stable the unstable term should not be cancelled. Thence, L(s) is defined, for example, as: L(s) =
6. Unstable and NMP plants Dealing with right hand poles/zeros, some additional limitations may appear.
e−0.5s (2s + 1) (0.5s + 1) (5s − 1)
Then, the corresponding compensator for the parallel path is computed as Kf (s) =
(s + 1)2 (2s + 1) (0.5s + 1)
Kb (s) =
(2s + 1) (0.5s + 1) −1.5s e (s + 1) (3s + 1)
6.1. Non-minimum phase model In this case, some simple transformations allow the application of the proposed approach avoiding the unstable pole/zero cancellation. Example 3. For instance, assume that the transfer function of the process with non-minimum phase zero is given by y(s) e−1.5s (s − 1) e−s + = 2 = F1 (s) + F2 (s) 2s +1 u(s) s +3s + 1
6.3. Plants with an unstable term
If the pre-compensator as defined in formula (12) is directly used, a right half plane pole will appear in Kf (s). Thus, a new equivalent model without non-minimum phase zeros is defined. A feasible way is to decompose the initial model into Gp =
−2e−s e−1.5s (s + 1) e−s + 2 + = F1 (s) + F2 (s) + F3 (s) 2 2s + 1 s +3s + 1 s +3s + 1
and the compensator elements will be computed as Kf (s) =
L(s) F2 (s) F3 (s) ; K (s) = ; Kb22 (s) = F1 (s) b21 L(s) L(s)
If the unstable pole only appears in one term in parallel, this direct approach is only applicable if the unstable term presents the shortest time-delay, as can be easily derived by considering (12) and (13), otherwise the corresponding feedback component Kbi (s) would be unstable. Example 5.
e−s 3s + 1
e−2s e−0.5s y(s) + = u(s) (s + 1) (3s + 1) (s + 1)2 (5s − 1)
L(s) =
s2 +3s + 1 (3s + 1) (s + 1)
;
Kb21 (s) =
−2 (3s + 1) ; s2 +3s + 1
Kf =
(s + 2)3
(s + 2)
Example 6.
Another option, if the term including the non-minimum-phase zero has higher time-delay y(s) e−s (s − 1) e−1.5s = = F1 (s) + F2 (s) + 2 2s + 1 u(s) s +3s + 1 is to design the compensator as follows: L(s) 3s + 1 ; Kf (s) = = 2s + 1 F2 (s)
(s + 1)2 (5s − 1)
Kb22 (s) =
2s + 1 −0.5s e 3s + 1
(3s + 1) (s − 1) −0.5s Kb (s) = F1 (s)L(s) = e s2 +3s + 1
where Kf (s) is stable and Kb (s) is non-minimum phase. 6.2. Unstable plants: common term For unstable plants, if the unstable term is common to all the paths, it can be left out of the simplification process. Example 4.
e−0.5s
the compensator is designed as
Leading to Kf (s) =
For instance, let us assume this first situation being
If the desired transfer function is assumed to be
Taking into account the constraints, L(s) can be selected as L(s) =
In this case, if the final goal is to control the compensated process, a control design approach appropriated for unstable time delayed plants should be applied.
For instance, assume that the plant is given as:
3
;
Kb =
(s + 2)3 e−1.5s (s + 1) (3s + 1)
Let us now assume the other case, that is:
y(s) e−0.5s e−2s = + s+1 s−1 u(s) It can be expressed in the general form (1) like (s − 1)e−0.5s + (s + 1)e−2s y(s) = u(s) s2 − 1 In this case, the denominator d1 (s) is not Hurtwitz. So, define, for instance, d¯ 1 (s) = s2 + 3s + 2. The process transfer function can be written as: P(s) =
y(s) = u(s) 1+
1 s2 +3s+2 1 (−3s − 3) s2 +3s+2
(s − 1)e−0.5s + (s + 1)e−2s
which is similar to (22). Identifying the different components in (23), the following selection of the transfer function will make all the compensator components stable and realizable. If H(s) = 3 3(s + 1) G1 (s) = 2 1 , then C11 (s) = 1; C12 (s) = − =− s +3s+2 s+2 s2 + 3s + 2
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And this pre-compensator C1 (s) applied to the process P(s) leads to a new process: P(s)C1 (s) =
s−1 1 −2s e−0.5s + e s+2 s2 + 3s + 2
where the approach followed in the Example 3 can be applied. 7. Conclusions In this paper, the problem of converting a process model with multiple time-delays to one with a single time-delay in the forward path has been discussed, and the pre-compensator design procedure has been outlined. The main achievement is to get a compensated system with a single time-delay thus making easier the possible design of the control. Similar to the Smith Predictor, there are some constraints in applying this procedure but it is very useful in many practical processes with multiple delays. The new proposed pre-compensator structure is able to compensate all kinds of dead times, including the complicated systems with multiple time-delays in both the numerator and denominator, as far as some constraints about the time-delays and relative order of the process components are fulfilled. Moreover, no approximation is required. A comparison with the approach used in [30] illustrates this advantage. The proposed pre-compensator structure has been illustrated with several examples considering multiple delays, multiple delayed paths as well as delayed feedback paths. As the approach is based on cancellations, some additional constraints should be taken into account when dealing with unstable and non-minimum phase plant models. The procedure can be applied for any process described in the general form (1) with the limitation described by the assumptions A in Section 4. This assumption is usually satisfied in most industrial applications, as shown in the different examples presented in the paper. The control design limitations encountered in dealing with multi time-delays SISO plants have been overcome as the design is applied to a rational transfer function once the initial plant is converted into a single time-delay plant and then the SP or any well-established DTC approach is applied. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 61473137 and 61722306, the 111 projects (Grants No. B12018 and No. B08015), and the project TIN2014-56158-C4-4-P-AR, Ministerio de Economía y Competitividad, Spain. References [1] A. Bultheel, M.V. Barel, Padé techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math. 14 (3) (1986) 401–438. [2] K.E. Kwok, M.C. Ping, P. Li, A model-based augmented PID algorithm, J. Process Control 10 (2000) 9–18. [3] O.J.M. Smith, Closed control of loops with dead time, Chem. Eng. Prog. 53 (5) (1957) 217–219. [4] J.E. Normey-Rico, E.F. Camacho, Dead-time compensators: a survey, Control Eng. Pract. 16 (4) (2008) 407–428. [5] P. Albertos, P. García, Robust control design for long time-delay systems, J. Process Control 19 (10) (2009) 1640–1648. [6] P. García, P. Albertos, T. Hagglund, Control of unstable non-minimum-phase delayed systems, J. Process Control 16 (2006) 1099–1111.
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