European Journal of Control (2003)9:118±132 # 2003 EUCA
Loop Performance Assessment for Decentralized Control of Stable Linear Systems Juan I. Yuz and Graham C. Goodwin School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia
Modern control system design methods, including robust control, typically lead to controllers of high complexity. However, in practice, there is usually strong motivation to implement controllers having simpler forms. One issue is clearly the degree of the controllers. However, of equal, if not greater, importance is the issue of the architecture (or connectivity) of the control law. Indeed, industry often prefers simpler, decentralized, architectures. In this context, the current paper presents a methodology for obtaining a benchmark for the performance of decentralized controllers. This benchmark has several potential areas of application including determining input±output pairing, quantifying the degradation in performance when restricted complexity architectures are used, and as a tool to assess existing decentralized control loops. Keywords: Benchmarks, Performance Assessment
Decentralized
Control,
1. Introduction Modern control design methods, including model based design, robust control and model predictive control, usually lead to controllers having high Correspondence and offprint requests to: J.I. Yuz, and G.C. Goodwin, School of Electrical Engineering & Computer Science, The University of Newcastle, NSW 2308, Australia. Tel.: 61-2-4921-7072; Fax: 61-2-4960-1712. E-mail:
[email protected],
[email protected]
complexity. These solutions, though strongly supported by the academic community, are often viewed with caution by industry. Thus, the majority of control systems found in practice, comprise relatively simple decentralized architectures, typically based on single loop PID controllers. There are many reasons why industry prefers to use controllers of restricted complexity. These include familiarity, ease of design, intuitive appeal, maintainability, perceived robustness to sensor/ actuator failure and cabling issues. However, use of these simple architectures raises the issue of the performance loss inherent in limiting the architecture of the control law. It is thus desirable to develop benchmarks for the performance of both centralized and decentralized controllers, so that a realistic assessment of the relative performance can be made. The topic of performance assessment for unrestricted control loops has been the subject of substantial prior work. The most commonly used benchmarks for loop performance include: (i) minimum variance control, and (ii) integral (or summed, in the discrete time case) square output error due to a step output disturbance. Not surprisingly, these two performance benchmarks are closely related and approach each other if, for example, the process disturbances have
Received 8 May 2002; Accepted in revised form 12 November 2002. Recommended by H. Hjalmarsson and I.D. Landau.
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Performance Assessment for Decentralized Control of Stable Linear Systems
relatively low bandwidth compared to that of the plant. Prior work on loop performance assessment in the unrestricted controller case may be found in, for example, [1,11±13,21±25,27,28,31±33,35,46,49±51]. Surveys of available results are given in [39] and [34], and the book [29] gives an excellent treatment of this field. The topic of loop performance assessment is naturally closely connected to the problem of fundamental performance limitations. Elucidation of fundamental limitations on the performance of feedback control loops has been a topic of interest since the seminal work of Bode during the 1940s. The tools for analyzing limits of performance include logarithmic sensitivity integrals, limiting quadratic optimal control and entropy measures. Early work focused on linear feedback systems see, for example, [2,3,7± 9,14,16,37,40,42,47,48]. There has also been some recent work on performance limitations for nonlinear feedback systems, see for example [30,43,44]. Our focus here will be on the quantification of the relative performance of centralized and decentralized controllers. Note that much of the existing work on control performance monitoring does not use an explicit process model, whereas the method proposed here requires that a model be available. Decentralized controllers are widely used in industry. There has been a lot of work regarding the structure of this kind of controllers, particularly concerning the input/output pairing problem, considering different approaches ± see, for example [4,5,10,15, 38,52], which is a good review on this issue. However, there exists few systematic design methods for these restricted complexity control laws. One difficulty is that enforcing a decentralized controller into the free parameter of the class of all stabilizing controllers for a given plant renders the associated optimization problem non-convex [45]. Nonetheless, some recent theoretical advances have been made. For example, for discrete-time systems, [45] has formulated the decentralized design problem as an l1 decentralized model matching problem. Also, [41] has proposed a form of time varying control that is optimal for a quadratic cost function. A weighted cost function is used in [18] which convexifies the problem. Another class of design strategies for decentralized control uses sequential design for each loop (see, for example, [6,20,26,36]). Our goal in this paper is to give a methodology for obtaining a benchmark for the performance of decentralized control laws. We also develop some additional insights into the centralized case as a prelude to the decentralized case.
The benchmark developed here has several potential uses, e.g.: (i) assisting in the determination of appropriate input±output pairing, (ii) evaluating the inherent degradation in loop performance, resulting from the use of a decentralized architecture, and (iii) as a tool in assessing the performance of an existing decentralized control system. An overview of the remainder of the paper is as follows: In Section 2, we briefly review a measure of control loop performance for the centralized case. Section 3 introduces the decentralized control strategy to be used in the sequel. Section 4 presents a method to obtain a redesigned decentralized controller leading to thedecentralizedperformancebenchmark.Examplesof the proposed methodology are presented in Section 5. Section 6 presents conclusions.
2. A Performance Benchmark for Centralized Control To provide a framework for our subsequent treatment of the decentralized problem, we first review the unrestricted controller case. For simplicity, we limit our treatment to stable, discrete-time, square, linear multivariable systems. Thus, consider a stable m m discrete-time transfer function G(z) where z is the Z-transform variable. It is well known [17] that the class of all stabilizing proper control laws for G(z) can be parametrized as: C
z I Q
zG
z 1 Q
z Q
zI G
zQ
z 1 ,
1
where Q(z) is any proper stable transfer function. The associated sensitivity and complementary sensitivity function are given respectively by T
z G
zQ
z,
2
S
z I
3
G
zQ
z:
An interesting observation (which to the best of our knowledge, has not been previously explicitly stated in the literature) is the following. Lemma 1. In the case of open-loop stable plants, with unrestricted control architecture, the columns of the sensitivity and the complementary sensitivity matrices can be independently optimized. Proof. The result follows immediately from (2) and (3) since the ith column of T(z) and S(z) depend only on
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J.I. Yuz and G.C. Goodwin
the i-th column of Q(z) which can therefore be designed independently of other columns. & Remark 1. Lemma 1 applies only a column by column basis. Indeed, there is usually a trade-off involved in optimizing the individual elements within a given column of S(z) or T(z). Thus, in general, one needs to specify the relative importance of each element in a given column. Example 1. Consider the following simple 2 2 discrete time system: 2 z 1 z :
4 G
z z 2 "z 1 We will ®rst study the centralized designs for this problem. One convenient way to reduce the design problem to its simplest form is to introduce the interactor matrix [19]. Using the left interactor matrix, (z), we have that:
z G
z G
z
5
where G
z is biproper. For the system given in equation (4): 1 z 0 0 z 1 )
z
6
z "z2 z2 "z 1 z 2 Now, G
z for our example, is minimum phase. Thus, we can choose: 1 Q
z G
z Q
z
7
k0
To ensure existence of these two integrals, i.e., the convergence to zero of the steady state error, we require that: S11
1 1
a0
a1 0:
S21
1 "a0 "a1 b0 b1 0:
14
15
Substituting into (12) and (13) we see that: J11 1 j1
a0 j 2 :
J21 j"a0 j2 j" b0 j2 :
16
17
Clearly J21 is optimized by taking b0 ". However, we then see that there is a clear trade-off when choosing a0 between minimizing J11 and J21. Specifically we obtain the following alternative solutions: (i) J11 is optimized by the choice: a0 1 ) J11 J21 1 "2 :
18
(ii) J21 is optimized by the choice:
yielding: S
z I
z 1 Q
z:
8
Now, let: q
z q12
z : Q
z 11 q21
z q22
z
9
Then, the columns of S(z) in (8) can be written as:
1 S11
z S1
z S21
z 0 S2
z
We next consider a unit step disturbance on the first output. The resultant sum of squares of the closedloop error response on outputs one and two are given respectively by: 2 Z 1 X 1 S11
e j! y1
k2 d!
12 J11 2 1 e j! k0 2 Z 1 X 1 S21
e j! 2 y2
k
13 J21 1 e j! d! 2
S12
z 0 S22
z 1
q11
z : q21
z
z 1 "z 1
z
z 1 "z 1
10 q12
z 0 : z 2 q22
z
0
2
11 We consider the ®rst column of S(z). It turns out that it suf®ces to take q11
z a0 a1 z 1 and q21
z b0 b1 z 1 .
a0 0 ) J11 J21 2:
19
(iii) The sum of the outputs squared, i.e., J11 J21, is optimized by the choice: a0
1 "2 ) J J 1 : 11 21 1 "2 1 "2
20
Using a similar argument, with q21
z c0 c1 z 1 and q22
z d0 d1 z 1 , we consider: 2 Z 1 X 1 S12
ej! J12 y1
k2 d!:
21 2 1 e j! k0 2 Z 1 X 1 S22
ej! 2 y2
k d!:
22 J22 2 1 e j! k0 We then have the following: (i) J22 is optimized by the choice: c0
1 1 ) J12 J22 1 2 " "
23
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Performance Assessment for Decentralized Control of Stable Linear Systems
(ii) J12 is optimized by the choice: c0 0 ) J12 J22 2
The various trade-offs illustrated in Fig. 1 are shown in a different fashion in Fig. 2. This ®gure shows the trade-off between optimizing different elements of sensitivity. For example, bringing J22 to its minimum value of 1 is associated with increasing J12 from 0 to 4.
24
(iii) The sum of the outputs squared, i.e., J12 J22, is optimized by the choice: " 1 ) J12 J22 1
25 c0 1 "2 1 "2
3. Decentralized Control
We see from (23) that, when " is small, there is a major performance penalty involved in bringing J22 to its absolute minimum value of 1 since this implies that J12 becomes 1/"2. Figure 1 shows the individual sensitivity cost functions obtaining by optimizing the following weighted combinations: J1
1 1 J11
1 J2
2
1
1 J21 ,
We next turn to the case where the architecture of the controller is restricted to a decentralized form. So as to end up with a diagonal (decentralized) control law, we choose a nominal diagonal model comprising the diagonal elements of G(z). Thus, we write:
26
2 J12 2 J22
G
z Gd
z G"
z I G
zGd
z
27
where Gd(z) is diagonal, G"(z) represents the additive modelling error, and G(z) is the (left) relative modelling error. For a stable open loop system, all proper stabilizing control laws for Gd(z) can be parametrized as [17]:
for 1 2 [0, 1], 2 2 [0, 1] and for the particular choice " 0.5. Note, in particular, that with 1 2 0.5 the optimal costs, in accordance with (20) and (25), are: J11 1:04,
J21 0:16,
J12 0:16,
Cd
z I
28
4
4
3
3 J12
J11
J22 1:64:
2
1
30
2
0
0.2
0.4
λ1
0.6
0.8
0
1
2
2
1.5
1.5
1
0.5
0
Qd
zGd
z 1 Qd
z,
1
J22
J21
0
0
0.2
0.4
0
0.2
0.4
λ2
0.6
0.8
1
0.6
0.8
1
1
0.5
0
0.2
0.4
λ1
0.6
0.8
29
1
0
Fig. 1. Individual sensitivity cost depending on s.
λ2
122 2
2
1.5
1.5
J22
J11
J.I. Yuz and G.C. Goodwin
1
0.5
0
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0
0
1
2 J12
J21
3
4
Fig. 2. Trade-off between the costs.
where Qd(z) is any diagonal proper stable transfer function. We next turn to the question of the achievable performance using a decentralized control law. Two initial ideas might be: (i) ignore the off-diagonal terms in G(z) and simply evaluate the corresponding SISO benchmarks by optimizing the diagonal elements of the sensitivity matrix, or (ii) design the control law, Qd(z), based on Gd(z) but then evaluate the true MIMO sensitivities via [17]: S I
Gd Qd I G Gd Qd
Sd I G" Qd 1 :
1
31
One might reasonably anticipate that the benchmark provided by suggestion (i) may be too optimistic since MIMO interactions are totally ignored, whereas the benchmark provided by suggestion (ii) may be too pessimistic since, although MIMO interactions are considered in evaluating the true sensitivities, the design is optimized with respect to the nominal diagonal plant only. This suggests that a mid-course is needed. This topic is taken up in the next section.
4. Redesign of the Decentralized Controller Let Qd
z denote the value of the transfer function Qd(z) optimized using only the diagonal transfer function Gd(z). We have argued in Section 3 that the true MIMO sensitivity based on using Qd
z may give a pessimistic view of the achievable performance. Thus it seems desirable to optimize (some function of) this true MIMO sensitivity given in equation (31). Unfortunately, this achieved sensitivity is a non-linear function of Qd(z) and this makes direct optimization of Qd(z) a formidable task. We therefore make an appropriate approximation of the true MIMO
sensitivity. (We have actually found that the form of this approximation needs to be carefully chosen to end up with a realistic design.) As a prelude to the subsequent development, we write the redesigned matrix Qd(z) as a relative displacement from Qd
z, i.e. we express it as: Qd
z Qd
zI Q
z,
32
where Q(z) is a stable proper transfer function. Remark 2. If we restrict both, Qd(z) and Qd
z, to give MIMO integral action for the plant, then Q(z) must be zero at d.c. In addition, we assume that Q(z) is small for all frequencies of interest. Note that these facts and the use of the relative form (32) are pivotal in the subsequent development. Based on the above, we have the following result. Lemma 2. Given a diagonal controller parametrized as in (30), with Qd(z) expressed as a relative displacement from Qd
z as in Eq. (32), then the true MIMO sensitivity (31) can be expressed as: S
z S1
z S2
z,
33
where S1 Sd
Td Q
Sd S G Td Q S
34
with Sd I
Gd Qd ,
35
Td Gd Qd , S I G Td
36 1
37
and where S2(z) contains terms of order (Q)2 and higher.
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Performance Assessment for Decentralized Control of Stable Linear Systems
Proof. Using Eqs (36), (35), (32) and (31) we have that: S
z I Sd
1
Gd Qd
I Q I G Gd Qd
I Q Td Q I I G Td 1 G Td Q
I G Td
1
Expanding the central term in the last expression as a Taylor's series in terms of Q(z), we obtain (33). & Remark 3. Note that S1(z) in (34) is linear in Q(z). This property will be used to simplify the associated design problem. Remark 4. Of course, the utility of using S1(z) for design depends on S(z) being close, in some sense, to S1(z). We will see in Section 5, that S(z) and S1(z) are indeed close, at least for the cases considered there. We will next proceed to optimize an appropriate function of S1(z). We first make the following observation. Remark 5. Whereas the columns of the unrestricted sensitivity can be optimized separately, this is not true in the decentralized case (Compare the proof of Lemma 1 with the form of S1(z) given in (34)). Thus, in general, we will need to consider all elements of S1(z) simultaneously. By analogy to the unrestricted case, we propose the following performance benchmark: Z m X m S1
e j! 2 1 X ij
39 J min d! e j! Q
z2 2 i 1 j 1 1 where is the class of stable proper diagonal transfer functions. Remark 6. The cost function in (39) represents the sum of the square error at every output due to a unit step disturbance applied to each output of the plant. Remark 7. Since S1(z) is linear in Q(z), the optimization given in (39) can be carried out by a number of equivalent methods, including Spectral Factorization and Linear Quadratic Regulator theory. To illustrate, we will utilize the latter technique. We can write the m2 elements of S1(z) as an m2 1 vector: S121
z
S1mm
zT
40
Now, for both Qd
z and Qd(z) to give integral action, we require that Q(z) be zero at dc Hence, we write:
z
1 Q
z Q
z 1
z 22 q11 q
z vecQ
z q
z
T qmm
z :
1
38
V
z vecS1
z S111
z
z in vector We also express the diagonal matrix Q form as:
41
42 Since S1(z) is linear in Q(z), it is also linear in
z, and thus we can write: Q V
z S1
z vec M
z q
z N
z 1 z 1 1 z 1
43 for some m2 m matrix transfer function M(z) and an m2 1 vector transfer function N(z). Let u(k) be the inverse Z-transform of the vector q
z and introduce minimal state space models for M(z) and N(z) as follows: N
z CN
zI M
z CM
zI
AN 1 BN ,
44
AM 1 BM DM :
45
Then, via Parseval's theorem, the cost function given in (39) can be written as the following cheap regulator problem: 1 X y
kT y
k,
46 J k0
where x
k 1 Ax
k Bu
k,
with x
0 xo
47
y
k Cx
k Du
k 0 0 AN , B , A BM 0 AM
48 BN , xo 0
49 C CN
CM ,
D DM :
50
The optimal controller for this in®nite horizon LQR problem can be expressed as: u
k
K x
k
51
where K is the optimal feedback gain. Then, combining (47) and (51) we have that: q
z
K zI
A BK 1 xo
52
This determines q
z. We can then recover Q(z) via equation (42) and (41). Of course, the optimal value of Q(z) is guaranteed to be stable by the properties of the optimal LQR solution. We denote it by Q
z.
124
When the basic methodology proposed above is applied (as will be done in Section 5) there are several minor technical issues that need to be considered to solve the associated optimal controller design problem1. Integral action should be carefully introduced in the sensitivity equations (43), to avoid the presence of fake uncontrollable modes due to the step response. Also, the standard LQR solutions lead to a strictly proper controller. However, a biproper controller can be obtained by rewriting Eq. (43) as: 1 ~ q
z N
z
53 vec S
z 1 M
z~ 1 z ~ zM
z. In this where q~
z z 1 q
z and M
z way, we obtain a strictly proper solution q~
z, so & q
z is biproper. Remark 8. Since Q(z) is stable, this guarantees stability of the nominal (diagonal) plant. However, in view of the approximations involved, our procedure does not necessarily guarantee stability of the true MIMO loop. Nonetheless, our experience with the procedure is that stability is rarely a problem save in the presence of severe multivariable interactions. Moreover, adding a stability constraint would increase the benchmark cost, thus our result can be always considered as a lower bound on the achievable performance.
5. Examples In this section, we show how the proposed approach can be applied to various MIMO plants. First, we consider the same system as in Example 1 on page 4. Example 2. Consider the pure time delay plant, given in (4) on page 4. Since we plan to use a decentralized architecture, we first compute the relative gain array (RGA) [4]. Taking " 0.5, we have that: 2 3 " 1 6" 1 " 17 1 2 6 7 RGA
G
z 4 2 1 1 " 5 " 1 " 1
54 This suggests that we should to swap the input variables. This yields the following model: 1 z 2 z :
55 G
z "z 1 z 2 Full controller As a first step, we follow the same procedure as in Example 1 to obtain an optimal centralized controller. 1
For example, using MATLAB.
J.I. Yuz and G.C. Goodwin
Considering a first order controller given by: a o a1 z 1 bo b1 z 1 : Q
z c o c 1 z 1 do d1 z 1
56
The parameters can be chosen to minimize the cost functions (26)±(27). Of course, in the centralized case, the ordering of the inputs plays no role. Hence, the trade-off in optimal values for the individual elements is exactly as previously shown in Figs 1 and 2. When the different terms in (26) and (27) are equally weighted, i.e., 1 2 0.5, then the minimum value for the cost functions is achieved with the controller parametrized by
0:5 4 6 1 2 12 1 z 5 5 z
57 Q
z Q
z 5 5 1 2 The optimal responses to step output disturbances are as shown in Fig. 3. The corresponding individual components of the cost are: J11 1:04, J12 0:16, J22 1:64
J21 0:16,
58
and then J J11 J12 J21 J22 3. Nominal diagonal optimal controller We next consider the diagonal terms of the plant (55). In this case, we obtain the following simplified decoupled nominal system: 1 0 z
59 Gd
z 0 z 2 The relative modelling error, from Eq. (29), is: 0 1 1 G
z
G
z Gd
zGd
z " 0
60 Again, we will take " 0.5. The nominal optimal diagonal controller for the diagonal plant (59), is given by equation (30), with 1 0 I
61 Qd
z 0 1 The resulting optimal nominal sensitivities are: 1 z 0 ,
62 Td
z Gd
z 0 z 2 1 z 1 0 : Sd
z I Gd
z 0 1 z 2
63
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Performance Assessment for Decentralized Control of Stable Linear Systems d1 y1
1
d2 y1
1
0.5
0.5
0
0
–0.5
–0.5 0
2
4
6
8
10 d1 y2
1
0
2
4
6
8 d2 y2
1
0.5
0.5
0
0
–0.5
10
–0.5 0
2
4
6
8
10
0
2
4
6
8
10
Fig. 3. Step disturbance responses for the closed-loop with centralized optimal controller.
Figure 4 shows the response of the nominal system when there is a unit step disturbance on the ®rst output (left part) and when there is a step disturbance on the second output (right part). As a consequence of the simpli®ed diagonal model, there is no interaction in the nominal control loop. Hence, for this nominal diagonal system, the minimum value of the cost functions previously de®ned are: J1 J11 J21 1 0 1 J J1 J2 1 2 3 J2 J12 J22 0 2 2
64 We recall, however, that these results apply only to the simpli®ed diagonal model of the plant (59). If we use the nominal controller (61) on the true MIMO plant we expect that the achieved performance will be quite different. This is con®rmed in Fig. 5. The cost function given by (64) now grows to J 2 4 6. Redesigned diagonal controller The diagonal controller given by Qd
z can be redesigned using the procedure described in Section 4. In particular, we use the method explained in Remark 7 to obtain the increment Q(z) for the decentralized control law (32). Figure 6 shows the response of the actual system to step disturbances in each of the outputs for the redesigned diagonal controller. Table 1 summarizes the various optimal values of the cost function for this first example. Notice that the true MIMO cost with Q(z) 0 is 6 which is rather pessimistic, whereas the nominal
diagonal design cost is 3 which is too optimistic. The cost with the redesigned controller is 4.553. This represents an increase of 50% over the cost with a centralized controller. This may indicate that decentralized control is not a good option for this plant. Finally, note that the true achieved cost is within 6% of the predicted cost based on the approximation S1(z) given in (34). This strongly supports the use of this approximation, at least for the previous example. Example 3. We next consider an alternative plant:
z G
z "z
1 1
"z z
2 2
,
65
where 0 " 1 is a parameter which re¯ects the coupling of the system. In fact, the Relative Gain Array (RGA) [4] for this system is given by: 2 61 RGA
G
z 6 4
1 "2
"2 1 "2
3 "2 1 "2 7 7 1 5 1 "2
66
which con®rms that when " is small the plant is diagonal dominant. However, when " approaches 1, the appropriate pairing between the input and output variables becomes unclear. We do not repeat the centralized design since this follows similar steps as for Example 1.
126
J.I. Yuz and G.C. Goodwin
d1 y1
1
d2 y1
1
0.5
0.5
0
0
–0.5
–0.5 0
5
10
15
20 d1 y2
1
0
5
10
15 d2 y2
1
0.5
0.5
0
0
–0.5
20
–0.5 0
5
10
15
20
0
5
10
15
20
Fig. 4. Response of the nominal system to step output disturbances.
d1 y1
1
d2 y1
1
0.5
0.5
0
0
–0.5
–0.5 0
5
10
15
20 d1 y2
1
0
0.5
0
0
–0.5
–0.5 5
10
15
10
15
20
20 d2 y2
1
0.5
0
5
0
5
10
Fig. 5. Response of the actual system to step output disturbances.
15
20
127
Performance Assessment for Decentralized Control of Stable Linear Systems
d1 y1
1
d2 y1
1
0.5
0.5
0
0
–0.5
–0.5 0
5
10
15
20 d1 y2
1
0
5
10
15 d2 y2
1
0.5
0.5
0
0
–0.5
20
–0.5 0
5
10
15
20
0
5
10
15
20
Fig. 6. Response of the actual system with the redesigned controller to step output disturbances.
Table 1. Summary of the values for the cost functions in Example 2. J2
J
1.2 1 2 1.668 1.770
1.8 2 4 2.885 3.055
3 3 6 4.553 4.825
The corresponding diagonal nominal model is given by 1 0 z
67 Gd
z 0 z 2 This is the same nominal model that appears in Example 2, so the optimal nominal controller and the nominal sensitivities are as found previously in Eqs (61), (62) and (63), respectively. Also the nominal cost function values are the same. However, the relative modeling error is now given by 0 " 1 : G
z
G
z Gd
z Gd
z " 0
68
J(ε)
Centralized design Diagonal model True model and Q(z) 0 Predicted cost for Q(z) optimal True cost for Q(z) optimal
J1
15
Centralized design Nominal controller (QΔ=0) Predicted with Q*Δ True with Q*Δ
10 5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 7. Plot of cost functions versus the parameter ".
We consider a range of values of " 2 [0, 0.9]. Notice that the optimal cost is J 3 for the nominal diagonal system independent of ". Table 2 summarizes the various optimal values for the different cost functions. The various costs are also plotted as a function of " in Fig. 7. Notice that the predicted and true costs are in very close agreement up to " 0.8. This is further veri®ed in Fig. 8, which shows the predicted and achieved output of the closed loop system, when a unit step disturbance is injected in each channel, and the controller is given by the optimal Q(z) (with " 0.7) obtained using (34). Notice that decentralized control gives only a modest increase compared with centralized control for " up to about 0.6.
128
J.I. Yuz and G.C. Goodwin
Table 2. Summary of values of the cost function for different " in Example 3.
" 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Centralized
JQ(z) 0
Predicted JQ
z
True JQ
z
J1 J2 J 123 1.01 1.99 3 1.04 1.96 3 1.08 1.92 3 1.14 1.86 3 1.20 1.80 3 1.27 1.73 3 1.33 1.67 3 1.39 1.61 3 1.45 1.55 3
J1 J2 J 123 1.02 2.01 3.03 1.08 2.04 3.12 1.19 2.11 3.30 1.36 2.22 3.58 1.60 2.40 4.00 1.98 2.71 4.69 2.61 3.28 5.89 3.86 4.47 8.33 7.62 8.17 15.79
J1 J2 J 123 1.02 2.01 3.03 1.08 2.04 3.12 1.17 2.09 3.26 1.30 2.16 3.46 1.46 2.26 3.72 1.67 2.40 4.07 1.95 2.62 4.57 2.37 2.98 5.35 3.18 3.74 6.92
J1 J2 J 123 1.02 2.01 3.03 1.08 2.04 3.12 1.17 2.09 3.26 1.30 2.16 3.46 1.46 2.26 3.72 1.69 2.43 4.12 2.04 2.72 4.76 2.82 3.47 6.29 unstable !
1
1
0.5
0.5
0
0 d1 True y1 Predicted y1
–0.5 –1 0
5
10
15
–1 0
1
1
0.5
0.5
0
0 d1 True y2 Predicted y2
–0.5 –1 0
5
10
15
d2 True y1 Predicted y1
–0.5
20
5
10
15
20
d2 True y2 Predicted y2
–0.5 –1 20
0
5
10
15
20
Fig. 8. Predicted and actual disturbance response of the (redesigned) closed loop for " 0.7.
Example 4. Distillation column. Finally, we present a more realistic, pilot scale, process. We use a 2 input±2 output model of an ethanol±water distillation column found in [17], where all other variables of the system are supposed already controlled. The inputs u1 and u2 are related to the reflux flow and reboiler steam flow, respectively. The output y1 is the concentration of ethanol in the final product at the top of the column and y2 is the bottom plate temperature. The system is shown schematically in Fig. 9. A simplified discrete-time model (assuming sampling time T 1[s]) for the linearized continuous-time
system ([17] p. 171), is: 2 3 0:1z 4 0:05 z 2 6 1 0:85z 1 7 1 0:85 z 1 7: G
z 6 1 2 4 0:05z 10 5 0:12
1 0:9z z 1 0:85z 1
1 0:75z 1
1 0:95 z 1
69 First, we design a centralized controller following the method in Remark 7. As established in Lemma 1, every column of the sensitivity S(z) can be optimized independently.
129
Performance Assessment for Decentralized Control of Stable Linear Systems
For this particular case, to explicitly force integral action on the control loop, we write the controller as: Q
z G
1
1
z 1 Q
z
1
70
z. and we ®nd the optimal Q
z Q Figure 10 shows the responses obtained for the centralized control loop when unit step output Condenser Heat exchanger Column shell
Reflux (liquid)
Vapour flow
Cooling water
disturbances are applied in each channel. The values of the optimal cost functions for each sensitivity matrix column are J1 3.691 and J2 2.309. Thus the total cost is J 6. The RGA associated with the distillation column model (69) is RGA
G
z
0:210 : 1:210
1:210 0:210
71
This suggests that the pairing (u1, y1), (u2, y2) is sensible. Then the corresponding diagonal nominal model is: 2
0:1z 4 6 1 0:85z Gd
z 6 4 0
Liquid flow Distillate (liquid)
Feed (liquid)
3 0
1
1
1
2
0:12
1 0:9z z 0:75z 1
1 0:95z 1
7 7: 5
72
Boilup (vapour)
The optimal nominal diagonal controller Qd(z) is Reboiler Heat exchanger
2
Steam
6 Qd
z 6 4
1
0:85z 0:1 0
Fig. 9. Schematic of distillation column.
d1 y1
0
1
Botton (liquid)
1
3
1
7 7: 0:75z
1 0:95z 1 5 0:12
1 0:9z 1
73 1
d2 y1
1
0.5
0.5
0
0
–0.5
–0.5 0
5
10
15
20 d1 y2
1
0
5
10
15 d2 y2
1
0.5
0.5
0
0
–0.5
20
–0.5 0
5
10
15
20
0
5
10
15
Fig. 10. Step output disturbance responses for the centralized control loop.
20
130
J.I. Yuz and G.C. Goodwin
1
1
0.5
0.5
0
0 d1 y1 with Q*Δ
–0.5
y1 with QΔ=0
–1 0
10
20
30
40
0 1
0.5
0.5
0
0 d1 y2 with Q*Δ
0
10
20
30
10
20
40
y2 with QΔ=0
–1 40
30
d2 y2 with Q*Δ
–0.5
y2 with QΔ=0
–1
y1 with QΔ=0
–1
1
–0.5
d2 y1 with Q*Δ
–0.5
0
10
20
30
40
Fig. 11. Comparison of responses to step output disturbances for the distillation column.
The corresponding nominal sensitivity matrices are: 4 z 1 z 4 0 0 , Sd
z : Td
z 0 z 2 0 1 z 2
74 The nominal optimal cost (J J1 J2) based on the diagonal model for the plant is J 4 2 6. However, when Qd
z is used to control the (full) model of the distillation column (69), the achieved sensitivity is given by (31) (with Q(z) 0) and the associated cost is then J 4.728 2.912 7.640. Next, we redesign the diagonal controller as discussed is Section 4. For this example, this predicts a relatively small charge in the cost to 7.201. The improved performance is illustrated in Figure 11, which compares the responses of the system to step output disturbances in the first and second channel. The true cost function value in this case is 7.224, which represents an increment of 20% compared with that achieved with the centralized design. The predicted and the true cost with the optimal Q
z Q
z are shown in the Table 3, together with the values of the cost previously obtained. We see that the incremental cost of using decentralized control for this problem is quite small indicating that there is a little motivation to use a centralized controller. This supports, at least for this
Table 3. Summary of values of the cost functions for Example 4.
Centralized design Diagonal model True model and Q(z) 0 Predicted cost for Q(z) optimal True cost for Q(z) optimal
J1
J2
J
3.691 4 4.728 4.456 4.475
2.309 2 2.912 2.745 2.749
6 6 7.640 7.201 7.224
example, a decision that is typical taken in practice, i.e., to use a decentralized control structure for this kind of system.
6. Conclusions This paper has developed a benchmark for the performance of decentralized controllers applied to stable square discrete transfer functions. This benchmark is based on a particular approximation to the achieved sensitivity matrix under decentralized control. Simulation results confirm that the particular approximation employed here is a very good predictor of the true measured MIMO sensitivity. It is believed that the methodology developed in the paper has important practical implications in
Performance Assessment for Decentralized Control of Stable Linear Systems
assessing the relative performance of centralized and decentralized controllers. The results obtained in the examples are actually a little surprising since they show that the best performance achieved by decentralized control is frequently not significantly worse than that achieved with centralized control (at least in the examples studied here). This gives support to the wide spread use of decentralized controllers in practical systems.
References 1. AÊstroÈm KJ. Assessment of achievable performance of simple feedback loops. Int J Adaptive Control Signal Process, 1991; 5(1): 3±19 2. AÊstroÈm KJ. Fundamental limitations of control system performance. Preprint, 1995 3. Bode HW. Network Analysis and Feedback Amplifier Design. Van Nostrand, New York, 1945 4. Bristol EH. On a new measure of interaction for multivariable process control. Automatica 1966; 123±134 5. Bristol EH. Recent results on interaction in multivariable process control. 71st AIChE Conf. 1978. 6. Bryant GF, Yeung LF. New sequential procedures for multivariale systems based on Gauss Jordan factorisation. IEE Proc. ± Control Theory Appl. 1994; 141(5): 427±436 7. Jie Chen. Logarithmic integrals, interpolation bounds and performance limitations in MIMO feedback systems. IEEE Transactions on Automatic Control 2000; 45(6): 1098±1115 8. Jie Chen, Nett C. Sensitivity integrals for multivariable discrete time systems. Automatica 1995; 31(8): 1113±1124 9. Jie Chen, Li Qiu, Toker O. Limitations in maximal tracking accuracy. IEEE Transactions on Automatic Control 2000; 45(2): 326±331 10. Conley A, Salgado ME. Gramian based interaction measure. In: Proceedings. of the 39th IEEE CDC, vol 5. 2000, pp 5020±5022 11. Desborough L, Harris TJ. Performance assessment measures for univariate feedback control. Canad J Chem Eng 1992; 70: 1186±1197 12. Desborough L, Harris TJ. Performance assessment measures for univariate feedforward/feedback control. Canad J Chem Eng 1993; 71: 605 13. DeVries WR, Wu SM. Evaluation of process control effectiveness and diagnosis of variation in paper basis weight via multivariate time-series analysis. IEEE Trans Autom Control 1978; 23: 702±708 14. Freudenberg JS, Looze DP. Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Trans Autom Control 1985; AC±30(6): 555±565 15. Glover K. All optimal hankel norm approximations of linear multivariable systems and their l1-error bound. Int J Control 1984; 39(6): 1115±1193 16. GoÂmez G, Goodwin GC. Vectorial sensitivity constraints for linear multivariable systems. Automatica 1996; 32(4): 499±518
131
17. Goodwin GC, Graebe SF, Salgado ME. Control System Design Prentice-Hall, New Jersey 2001 18. Goodwin GC, SeroÂn MM, Salgado ME. H2-design of decentralized controllers. In: Proceedings of the American Control Conference. 1999 19. Goodwin GC, Sin K. Adaptive Filtering Prediction and Control. Prentice Hall, New Jersey, 1984 20. GuÈcluÈ AN, OÈzguÈler B. Diagonal stabilisation of linear multivariable systems. Int J Control 1986; 43(3): 965±980 21. Gustafsson F, Graebe S. Closed-loop performance monitoring in the presence of system changes and disturbances. Automatica 1998; 34(11): 1311±1326 22. HaÈgglund T. A control-loop performance monitor. Control Eng Practice 1995; 3: 1543±1551 23. Harris T. Assessment of control loop performance. Canadian Journal of Chem Eng 1989; 67: 856±861 24. Harris T, Boudreau F, MacGregor J. Performance assessment of multivariable feedback controllers. Automatica 1996; 32: 1505±1518 25. Hinde RF, Cooper DJ. Using pattern recognition in controller adaptation and performance evaluation. In: American Control Conference, San Francisco. June 2±4, 1993, pp 74±78 26. Hovd M, Skogestad S. Sequential design of decentralized controllers. Automatica 1994; 30(10): 1601±1607 27. Huang B, Shah S, Kwok K. Good, bad or optimal? performance assessment of MIMO processes. Automatica 1997; 33(6): 1175±1183 28. Huang B, Shah SL. Feedback control performance assessment of non-minimum phase MIMO systems. In: AIChE, Annual Meeting, Los Angeles. November 16±21, 1997 29. Huang B, Shah SL. Performance Assessment of Control Loops: Theory and Applications. Springer, London. 1999. ISBN 1-85233-639-0. 30. Iglesias PA. An analogue of bode's integral for stable nonlinear systems: relations to entropy. In: Proceedings of the 40th IEEE CDC, Orlando, Florida, USA, 2001; 4: pp. 3419±3420 31. Kendra S, Cinar A. Controller performance assessment by frequency domain techniques. J Process Control 1997; 7(3): 181±194 32. Ko B, Edgar TF. Assessment of achievable PI control performance for linear processes with dead time. In: American Control Conference, Philadelphia, PA. June 1998. 33. Kozub D, Garcia C. Monitoring and diagnosis of automated controllers in the chemical process industries. In: AIChE, Annual Meeting, St. Louis, MI 1993 34. Kozub DJ. Controller performance monitoring and diagnosis: experiences and challenges. In: J.C. Kantor, C.E, Garcia, B.C. Carnahan (eds), Fifth International Conference on Chemical Process Control, Tahoe, CA 1996, pp 83±96 AIChE and CACHE 35. Lynch C, Dumont G. Control loop performance monitoring. IEEE Trans Control Systems Technol 1996, 4(2): 185±192 36. Mayne DQ. The design of linear multivariable systems. Automatica 1973; 201±207 37. Middleton RH. Trade-offs in linear control systems design. Automatica 1991; 27(2): 281±292
132 38. Niederlinski A. A heuristic approach to the design of linear multivariable interacting control systems. Automatica 1971; 7: 691±701 39. Qin SJ. Control performance monitoring ± a review and assessment. Comput Chem Eng 1998; 23: 173±186 40. Li Qiu, Davidson EJ. Performance limitations of non-minimum phase systems in the servomechanism problem. Automatica 1993; 29(2): 337±349 41. Savkin AV, Petersen IR. Optimal stabilisation of linear systems via decentralized output feedback. IEEE Trans Autom Control 1998; 43(2): 292±294 42. SeroÂn MM, Braslavsky JH, Goodwin GC. Fundamental Limitations in Filtering and Control. Springer, London 1997 43. SeroÂn MM, Goodwin GC. Sensitivity limitations in nonlinear feedback control. Systems Control Lett. 1996; 27: 249±254 44. Shamma J. Performance limitations in sensitivity reduction for nonlinear plants. Systems Control Lett 1991; 17: 43±47 45. Sourlas DD, Manousiouthakis V. Best achievable decentralized performance. IEEE Trans Autom Control 1995; 40(11): 1858±1871
J.I. Yuz and G.C. Goodwin
46. Stanfelj N, Marlin T, MacGregor J. Monitoring and diagnosing process control performance; the singleloop case. Ind Eng Chem Res 1993; 32: 301±314 47. Sule VR, Athani VV. Directional sensitivity trade-offs in multivariable feedback systems. Automatica 1991; 27(5): 869±872 48. Sung HK, Hara S. Properties of sensitivity and complementary sensitivity functions in single-input single-output digital control systems. Int J Control 1998; 48(6): 2429±2439 49. Swanda A, Seborg D. Evaluating the performance of PID-type feedback control loops using normalized settling time. In: ADCHEM 97, Banff, Canada, June 9±11 1997 50. Thornhill N, Oettinger M, Fedenczuk P. Refinery-wide control loop performance assessment. J Process Control 1999; 9: 109±124 51. Tyler M, Morari M. Performance monitoring of control systems using likelihood methods. Automatica 1996; 32: 1145±1162 52. van de Wal M, de Jager B. A review of methods for input/output selection. Automatica 2001; 37(4): 487±510