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Structural modification of systems using discretization and generalized sampled-data hold functions
Automatica 42 (2006) 1935 – 1941 www.elsevier.com/locate/automatica
Brief paper
Structural modification of systems using discretization and generalized sampled-data hold functions夡 Amir G. Aghdam a,∗ , Edward J. Davison b , Rafael Becerril-Arreola c a Electrical and Computer Engineering Department, Concordia University, Montréal, Que., Canada H3G 1M8 b Electrical and Computer Engineering Department, University of Toronto, Toronto, Ont., Canada M5S 3G4 c Electrical and Computer Engineering Department, Concordia University, Montréal, Que., Canada H3G 1M8
Received 16 August 2004; received in revised form 10 May 2006; accepted 5 June 2006 Available online 30 August 2006
Abstract This paper investigates the decentralized control of LTI continuous-time plants using generalized sampled-data hold functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the discretetime equivalent model to form a hierarchical system model of the plant. This is a new application of discretization and has, as its motivation, the design of decentralized controllers using centralized methods. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Decentralized control; Large-scale system; Generalized sampled-data hold functions
1. Introduction Sampled-data system applications are widely found in industry, mostly in order to take advantage of the application of computers as digital controllers. This is usually accomplished by using a simple zero-order hold (ZOH). The idea of using generalized sampled-data hold functions (GSHF) instead of a simple ZOH (or first-order hold) in control systems, was first introduced by Chammas and Leondes (1979) and Kabamba (1987). Kabamba investigated several advantages of using GSHFs in control systems and pointed out that, by using them, one can obtain many of the advantages of state feedback controllers, without the requirement of using state estimation procedures (Kabamba, 1987); in particular he showed that GSHFs can 夡 This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant RGPIN-262127-03 and in part by the natural Sciences and Engineering Research Council of Canada under Grant no. A4396. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised from by Associate Editor Toshiharu Sugie under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 514 848 2424; fax: +1 514 848 2802. E-mail addresses: [email protected] (A.G. Aghdam), [email protected] (E.J. Davison), [email protected] (R. Becerril-Arreola).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.06.005
significantly improve the performance of the closed-loop system. In addition, it is known that discrete-time controllers can have a significant effect on improving the overall performance of certain classes of decentralized control systems. In a discretized model, each transfer function in the resulting transfer function matrix is a function of the system parameters, and also the sampling period and type of hold function (ZOH, first-order hold, . . .). The question arises: using GSHFs, to what extent can one modify these transfer functions so “something desirable” happens? In this paper it will be shown how one can use generalized sampled-data hold functions to convert a continuous-time plant to a discrete-time equivalent system with a certain desired structure, in particular a hierarchical structure, which can directly be used to simplify the design of decentralized control systems for the plant. In D’Andrea and Dullerud (2003), a state-space approach is proposed to control systems with a highly structured interconnection topology and it is shown that design of controllers by discarding interconnection variables can cause instability. However, for a system with a hierarchical structure, or more precisely, a system with an acyclic graph structure, internal stability of each individual subsystem is a necessary and sufficient condition for stability of the overall system. Systems with hierarchical graph structures are investigated in the literature extensively and have many
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A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
g11
g31
1 g21
g21 g11
1
g12
g32 2
g23
3
g22
g33
g31
2 g32
g33
g22
3
g13 Fig. 1. Digraph of a continuous-time LTI system with three control agents.
applications in formation flight, underwater vehicles, automated highway, robotics, satellite constellation, just to name a few (Fax & Murry, 2004; Stankovic, Stanojevic, & Šiljak, 2000). Modifying a system’s structure to a hierarchical one can also have applications in decentralized adaptive control (Gavel & Šiljak, 1989). This paper is organized as follows. Digraphs are introduced in Section 2 to clarify some of the points that will be addressed in the main results. Section 3 then presents necessary and sufficient conditions for the existence of GSHF’s that make the structure of the plant hierarchical. Section 4 presents an example of control design for a plant that satisfy the conditions introduced in Section 3. Finally, Section 5 briefly discusses the importance of the results obtained. 2. Digraphs and system structure Consider the following strictly proper continuous-time decentralized LTI system with m control agents: x(t) ˙ = Ax(t) +
m
bj uj (t),
(1a)
j = 1, . . . , m
(1b)
j =1
yj (t) = cj x(t),
where x(t) ∈ Rn is the state vector, and uj (t) ∈ Rqj , and ¯ . . . , m} are the control vector and outyj (t) ∈ Rrj , j ∈ m={1, put vector of agent #j , respectively, and A, bj , and cj are matrices of appropriate dimensions. The transfer function matrix relating the inputs and outputs of this system can be written as ⎡ Y (s) ⎤ ⎡ g (s) . . . g (s) ⎤ ⎡ U (s) ⎤ 11 1m 1 1 ⎥⎢ . ⎥ ⎢ .. ⎥ ⎢ . . ⎢ ⎥ .. ⎦ ⎣ .. ⎦ , ⎣ . ⎦ = ⎣ .. Ym (s)
gm1 (s) . . .
gmm (s)
Um (s)
¯ represents the directed where gij (s)ci (sI −A)−1 bj , i, j ∈ m, arc from node j to node i in the digraph of this system. For example, the digraph of a system with three input–output agents is depicted in Fig. 1. In the control of decentralized systems, the structure of the interconnections generally has a significant role to play and, in some decentralized control design procedures, it is often assumed that there exists a bound on the interconnections between different nodes. For example, in decentralized adaptive control methods, such an assumption is often made or, alternatively, certain structural constraints on the interconnections are often assumed in order to assure the stability of the overall system
Fig. 2. Digraph of a continuous-time hierarchical LTI system with three control agents.
(Gavel & Šiljak, 1989). In the special case for systems which have a hierarchical structure, one can directly apply centralized control methods to each subsystem (represented by each node in the digraph of the system), with no assumptions required to be made on the interconnections, since it is guaranteed that signals coming from a higher level subsystems to a lower level subsystems will always be bounded, once the higher level subsystems are stabilized. For example, assume that the transfer functions g12 (s), g13 (s), and g23 (s) in Fig. 1 are all equal to zero; then the corresponding system will have the hierarchical structure shown in Fig. 2. Assuming that the system has no unstable decentralized fixed modes (DFM) (Wang & Davison, 1973), one can design a centralized stabilizing controller for each subsystem separately. In this case, subsystem #1, which has the highest level in the hierarchical structure, will be internally stable. Since there is no input signal coming into this subsystem other than its own bounded control input, this implies that all output signals coming out of this subsystem will be stable. The output signals of subsystem #1 includes the interconnection signal going to subsystem #2 and subsystem #3 through g21 (s) and g31 (s), respectively. Now subsystem #2, which has the second highest level, will also be internally stable and this implies that all output signals of this subsystem will also be bounded because all incoming signals, including the control input and the interconnection signal from subsystem #1, are bounded. Thus, all interconnection signals coming from the higher level subsystems to subsystem #3 will now be bounded and, since this subsystem will also be internally stable, one can conclude, using a similar argument, that the input and output signals of subsystem #3 are also bounded. Thence, the complete system will be internally stable. This implies that the decentralized controller design problem for a hierarchical system can be carried out in a centralized way for each subsystem. Note that the transfer function matrix of the hierarchical system of Fig. 2 has a lower-triangular form. In general case, it can be shown that a system with an acyclic digraph (i.e., a system whose digraph has no directed cycle) can be converted to a hierarchical system by adding some redundant edges and renumbering the vertices properly (Callier, Chan, & Desoer, 1978). Our goal now is to determine if and how one can modify the structure of a system by using generalized sampling. 3. Main results Consider the continuous-time LTI system represented by (1). It is desired to discretize the system by applying GSHFs fj (t),
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
j ∈ m, ¯ with a sampling period T, to each control agent. This problem can be formulated as follows. Let
1937
fj (t + T ) = fj (t)
where ∈ sp(A), and g(A) is an invertible matrix for nonpathological sampling periods (Chen & Francis, 1995). This implies that for almost all sampling periods, the unobservable subspace of ([ci1 . . . cij −1 ] , A) is equivalent to the unobservable subspace of its discrete-time equivalent pair ([ci1 . . . cij −1 ] , Ad ). As a result
for k=0, 1, . . . . As discussed in (Kabamba, 1987), the discretetime equivalent model is thence described by
[ci1 . . . cij −1 ] (zI − Ad )−1 ij = 0,
uj (t) = fj (t)u˜ j [k],
j ∈ m, ¯ t ∈ [kT , (k + 1)T ),
x[k + 1] = Ad x[k] +
or equivalently:
m
bdj u˜ j [k],
cdik (zI − Ad )−1 bdij = 0, j = 2, . . . , m,
j =1
yj [k] = cdj x[k],
j = 1, . . . , m,
(2)
where Ad = eAT , bdj =
T
(3a) eA(T−) bj fj () d,
j = 1, . . . , m,
(3b)
0
cdj = cj ,
j = 1, . . . , m.
(3c)
It is desired now to determine if one can choose a set of sampled-data hold functions fj (t), j ∈ m, ¯ and a sampling period T, so that certain elements of the transfer function matrix corresponding to the discrete-time equivalent model become equal to zero. Definition 1 (Chen and Francis, 1995). Consider the system (1). The sampling period T is pathological (with respect to A), if A has two eigenvalues with equal real parts but their imaginary parts differ by an integer multiple of 2/T . Note that there is no pathological sampling period for the system if all of the eigenvalues of A are real. Theorem 1. Consider the system (1). For every nonpathological sampling period T, there exists a sampled-data hold function fp (t) for each agent p ∈ m, ¯ so that the discrete-time equivalent model has a hierarchical structure, iff there exist distinct integers i1 , . . . , im ∈ m, ¯ so that the intersection of the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A) has a nonzero element for any j ∈ {2, . . . , m}. Proof (Proof of sufficiency). Assume that ij is a nonzero vector belonging to the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A). It follows from (Kabamba, 1987), that there exists a sampled-data hold function fij (t) for any sampling period T, so that in the discretetime equivalent model described by (2) and (3), bdij = ij . On the other hand, it is known that
Ad − eT I [ ci1
...
cij −1 ]
=
g(A)
0
0
I
A − I [ ci1
...
cij −1 ]
,
k = 1, . . . , j − 1.
This means that one can find the sampled-data hold functions f1 (t), . . . , fm (t), and rearrange the elements of the input and output vectors (i.e., by choosing the integers i1 , . . . , im properly), so that the transfer function matrix corresponding to the discrete-time equivalent model has a lower-triangular form, which corresponds to a hierarchical structure. Proof of necessity: Assume that the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A) have no nonzero vector in common. For nonpathological sampling periods the unobservable subspace of a continuoustime system and its discrete-time equivalent model are the same. Thus, there are no sampled-data hold functions and no input–output arrangement such that the upper-diagonal terms in the transfer function matrix of the discrete-time equivalent model (2) can be set equal to zero. The following theorem provides sufficient conditions for a system to have a hierarchical discrete-time equivalent. Theorem 2. Consider the system (1). For any sampling period T, there exists a sampled-data hold function fp (t) for each agent p ∈ m, ¯ so that the discrete-time equivalent model (2), (3) has a hierarchical structure, if there exist integers l1 , l2 ∈ m, ¯ l1 = l2 , so that ] , A) is unobservable; (a) the pair ([c1 . . . cl2 −1 cl2 +1 . . . cm and (b) the pair (A, bj ) is controllable for every j ∈ m, ¯ j = l1 ((A, bl1 ) may or may not be controllable).
Proof. Assume that conditions (a) and (b) of Theorem 2 both hold. Let i1 = l1 , im = l2 , and choose i2 , . . . , im−1 so that {i1 , i2 , . . . , im }= m. ¯ Now, since the pair ([ci1 . . . cim−1 ] , A) is unobservable, this implies that the pair ([ci1 . . . cim−1 ] , Ad ) will also be unobservable for any arbitrary sampling period T (Chen & Francis, 1995). This, in turn, means that the pair ([ci1 . . . cij −1 ] , Ad ), j =2, . . . , m is also unobservable, which implies that the columns of the matrix [ci1 . . . cij −1 ] (zI − Ad )−1 are linearly dependent. Therefore, there exists a nonzero vector ij so that [ci1 . . . cij −1 ] (zI − Ad )−1 ij = 0,
j = 2, . . . , m.
(4)
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A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
On the other hand, since the pair (A, bij ), j = 2, . . . , m is controllable, it follows from (Kabamba, 1987) that for any sampling period T, there exists a sampled-data hold function fij for each agent j (j =2, . . . , m), so that bdij =ij . Such (nonunique) sampled-data hold function may be obtained by using the following equation: fij (t) = bij e(T −t)A WT−1 ,ij ij ,
(5)
where WT ,ij =
T 0
etA bij bij etA dt.
This results in cdis (zI − Ad )
−1
bdij = 0,
j, s ∈ m, ¯ j > s.
Thence, using any arbitrary nonzero sampled-data hold function (such as a simple ZOH) for fi1 , along with the set of sampleddata hold functions given in (5), the corresponding transfer matrix for the discrete-time equivalent model will have a lowertriangular form. Remark 1. It is to be noted that the results of Theorem 1 are valid for all nonpathological sampling periods while those of Theorem 2 are valid for all sampling periods. The choice of sampling period is important, however, in terms of output performance. A method is proposed in Aghdam (2006) which considers the sampling period as one of the design parameters to minimize a prespecified performance index. For the systems that do not meet the conditions of Theorems 1 and 2 but have flexible sensor and actuator structures, one can make a modification in the input and output configuration to satisfy the desired conditions. Remark 2. Theorems 1 and 2 provide conditions under which the digraph of a system can be modified to a hierarchical structure by using sampling. However, they do not guarantee that the transfer function matrix of the discretized model will not have undesirable zeros. For example, it is not desirable to have a transfer function matrix with all elements of one row equal to zero. It can be easily verified that there will be at least one nonzero entry in each row of the transfer function matrix of the discretized model, iff cj , ∀j ∈ m, ¯ is in the controllability subspace of the discretized model. A nonpathological sampling period and controllability of the continuoustime system are sufficient conditions for avoiding such undesirable structures. It is desired now to obtain conditions under which the resulting discretized model does not have any DFMs (Wang & Davison, 1973). Theorem 3. Given (1), assume that the pair (A, bj ) is controllable for all j ∈ m. ¯ Then eT ∈ sp(Ad ) is not a DFM of the discrete-time equivalent model (2), with respect to the block
diagonal gain matrix K = diag(K1 , . . . , Km ), Kj ∈ Rsj ×rj , if the following three conditions all hold: (i) the sampling period T is nonpathological; T (ii) 0 e−t fj (t) dt = 0; ] , A) is observable. (iii) the pair ([c1 . . . cm Proof. eT ∈ sp(Ad ) is a DFM of (2) if and only if there exists a permutation of m ¯ denoted by distinct integers i1 , . . . , im , such that the rank of the matrix: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Ad − eT I
bdi1
bdi2
...
cil+1
0
0
...
cil+2
0
0
...
.. .
.. .
.. .
..
cim
0
0
...
.
bdil
⎤
0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ .. ⎥ . ⎦ 0
is less than n for some l ∈ m ¯ (Anderson & Moore, 1981). Assume now that conditions (i)–(iii) are all satisfied. Conditions (i) and (ii) imply that the resulting discretized system will not lose controllability and observability (corresponding to any controllable or observable pairs in the continuous-time model) (Middleton & Freudenberg, 1995). Thus, the pair (Ad , bdj ) is ] , A ) is controllable for all j ∈ m, ¯ and the pair ([c1 . . . cm d observable. This implies that the rank condition given above for the existence of a DFM does not hold for any l ∈ m. ¯ Remark 3. The results of Theorems 1 and 2 are interesting in the sense that they introduce a completely new application for sampling in control. The results can be very useful in the design of decentralized controllers, when the structure of the original system does not permit one to directly apply centralized controller design methods to decentralized systems. In this case, if the conditions given in Theorem 1 or Theorem 2 are met, one can find a set of GSHFs to modify the structure of the system in the discrete-time equivalent model (2), (3), so that centralized digital control methods can now be directly applied to each interconnected subsystem. Remark 4. When conditions of Theorem 1 are not satisfied, the question arises: is it possible to reduce the magnitude of certain interconnections by applying sampled-data hold functions? The motivation to this question is that systems with weak interconnections often have the property that decentralized control can still effectively be applied, using centralized control design methods for each subsystem. This problem has been investigated in Becerril, Aghdam, and Davison (2005), where the strength of interconnections has been measured in a closed form using the Hankel-norm. It is to be noted that the main objective of the present work is stability of the interconnected system. An extensive decentralized control performance degradation analysis in presence of interconnections is presented in Krtolica and Šiljak (1980). The problem of high-performance decentralized control design for interconnected systems remains an open area for research.
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
g21 g11
1
gd (z)
g22
2
11
1939
1
gd (z) 21
2
gd (z) 22
g12 Fig. 4. Digraph of the discrete-time equivalent mass-spring system of Example 1.
Fig. 3. Digraph of the mass-spring system of Example 1.
and gd22 are given by
4. Numerical example Example 1. Consider a 2-input 2-output mass-spring system with the following state-space matrices: ⎡
0
⎢ ⎢ −0.2 ⎢ ⎢ ⎢0 A=⎢ ⎢ ⎢1 ⎢ ⎢ ⎣0 1
0
0
0
−0.02
0.1
0.01
0.1
0
0
1
0
0.1
−1
−0.1
0
0
0
0
0
⎥ 0.01 ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ ⎥ ⎥ 1 ⎦
0.1
0
0
−1
−0.1
b1 = [0 0 0 1 0 0] ,
0
⎤
1
b2 = [0 0 0 0 0 1] ,
c1 = [0 0 0 1 0 0],
c2 = [0 0 0 0 1 1].
(6a)
(6b) (6c)
The digraph of this system is depicted in Fig. 3, where g11 , g12 , g21 , and g22 are given by
−0.163(z4 − 6.295z3 + 7.274z2 − 4.752z + 1.175) , z5 − 2.450z4 + 3.045z3 − 2.449z2 + 1.187z − 0.333 1.398(z5 − 1.19z4 + 1.706z3 − 1.137z2 + 0.528z + 0.023 , z6 − 3.45z5 +5.494z4 −5.494z3 +3.637z2 −1.52z+0.333 0.1 , z−1 respectively. In addition, it can be easily verified that the conditions of Theorem 3 for the given model, GSHF, and sampling period hold true, which means that the discrete-time equivalent model has no DFMs. This implies that one can design a decentralized controller for this system by applying a digital controller for each subsystem independently and using centralized methods. For instance, consider the controllers: v11 [k + 1] = v11 [k] + 5(ym1 [k] − yref,1 ),
(7a)
v12 [k + 1] = 0.6876v12 [k] + ym1 [k],
(7b)
u˜ 1 [k] = − 0.3591ym1 [k] − 0.005629v11 [k] − 0.04669v12 [k],
(7c)
(s 4 + 0.12s 3 + 1.201s 2 + 0.02s + 0.1)/den(s),
and
0.001(s 2 + 20s + 100)/den(s),
v21 [k + 1] = v21 [k] + 5(ym2 [k] − yref,2 ),
(8a)
0.001(s + 21s + 120s + 100)/den(s),
v22 [k + 1] = −0.8004v22 [k] + ym2 [k],
(8b)
(s + 1.12s + 1.321s + 1.221s + 0.12s + 0.1)/den(s),
u˜ 2 [k] = − 9.393ym2 [k] − 0.6006v21 [k]
3
5
2
4
3
2
respectively, and den(s)=s(s 4 +0.22s 3 +2.212s 2 +0.24s+1.2). Here, (c1 , A) is unobservable but (c2 , A) is not, while (A, b1 ) and (A, b2 ) are both controllable. Thus, the conditions of Theorem 2 hold for l1 =1 and l2 =2. In other words, one can use generalized sampling to construct a hierarchical discrete-time model, with the first subsystem at a higher level. Accordingly, let us assume that i1 = 1 and i2 = 2 (which means that l1 = 1 and l2 = 2 in Theorem 2), and let us use Eq. (3b) to obtain a sampled-data hold function for control agent #2, so that the resultant discretized model will be hierarchical with subsystem #1 at the higher level. In this case, it can be shown that, for all values of the sampling period T, the vector [1 0 1 0 1 0] is a basis for the null-space of c1 (zI − Ad )−1 (in fact, the null-space is one dimensional). Solve now the minimum energy problem for (3b) using the controllability gramian as described in Kabamba (1987), with T = 5 s. Then, on applying the resultant sampled-data hold function to control agent #2 and a simple ZOH to control agent #1, the hierarchical model of Fig. 4 is obtained, where gd11 , gd21 ,
− 1.086 × 10−4 v22 [k],
(8c)
which have been designed for subsystem #1 and subsystem #2, respectively, independently of each other, using centralized methods to solve the robust servomechanism problem. Here yref,1 and yref,2 in (7) and (8) denote constant reference signals for control agent #1 and control agent #2, respectively. The eigenvalues of the resultant closed-loop discrete-time system corresponding to subsystem #1 and subsystem #2 are given by: sp1 = {0.9246, 0.2935 ± 0.7898i, 0.7906 ± 0.08309i, 0.5514 ± 0.5753i} and sp2 = {−0.8004, 0.5303 ± 0.2824i}, respectively. The eigenvalues of the overall closed-loop discrete-time system are thus given by: sp = sp1 ∪ sp2 . Assume now that x[0] = [1 1 1 1 1 1] and v11 [0] = v12 [0] = v21 [0] = v22 [0] = 0. Figs. 5(a) and (b) give the outputs of the resultant closed-loop system, for a unit step reference input in control agent #1 and a zero reference input in control agent #2, using the local controllers (7) and (8). The control signals, corresponding to agents #1 and #2, are given in Figs. 5(c) and (d), respectively.
1940
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
1.5
6 4 ym2
ym1
1
0.5
0
0
0
100
(a)
200
-2
300
(sec)
0
300
200
300
-10 -20
-0.2
u2
u1
200 (sec)
0
0
-30 -40
-0.4
(c)
100
(b)
0.2
-0.6
2
-50 0
100
200
-60
300
(sec)
(d)
0
100 (sec)
Fig. 5. Closed-loop simulations for Example 1: (a) samples of the output response in agent #1; (b) samples of the output response in agent #2; (c) samples of the control signal in agent #1; (d) samples of the control signal in agent #2.
5. Conclusions This paper introduces a new application for GSHFs, which can be used to simplify the structure of the resulting discretized model. The conditions under which such a discretized model can have a hierarchical digraph are discussed and a method for the synthesis of the sampled-data hold functions corresponding to each control agent is presented. The motivation for simplifying the plant structure to become hierarchical is that the controller design, particularly for decentralized control, can be greatly simplified. For example, one can design decentralized controllers for the hierarchical discrete-time model by applying centralized methods directly to each control agent. Simulation results are given to show how such a discretization procedure can result in a simple hierarchical digraph that simplifies the decentralized controller design in solving the robust servomechanism problem.
References Aghdam, A. G. (2006). Decentralized control design using piecewise constant hold functions. In Proceedings of the 2006 American control conference (pp. 207–212). Minneapolis, MN, USA, June 2006. Anderson, B. O. D., & Moore, J. B. (1981). Time-varying feedback laws for decentralized control. IEEE Transactions on Automatic Control, AC-26(5), 1133–1138. Becerril, R. A., Aghdam, A. G., & Davison, E. J. (2005). Minimizing interactions of subsystems in large-scale interconnected systems using generalized sampling. 2005 IFAC world congress, Paper Code Tu-E02Tp/13, Prague, Czech Republic. Callier, F. M., Chan, W. S., & Desoer, C. A. (1978). Input–output stability of interconnected systems using decompositions: An improved formulation. IEEE Transactions on Automatic Control, AC-23(2), 150–163.
Chammas, A. B., & Leondes, C. T. (1979). On the finite time control of linear systems by piecewise constant output feedback. International Journal of Control, 30(2), 227–234. Chen, T., & Francis, B. A. (1995). Optimal sampled-data control systems. Berlin: Springer. D’Andrea, R., & Dullerud, G. E. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions on Automatic Control, AC-48, 1478–1495. Fax, J. A., & Murry, R. M. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, AC-49(9), 1465–1476. Gavel, D. T., & Šiljak, D. D. (1989). Decentralized adaptive control: Structural conditions for stability. IEEE Transactions on Automatic Control, AC34(4), 413–426. Kabamba, P. T. (1987). Control of linear systems using generalized sampleddata hold functions. IEEE Transactions on Automatic Control, AC-32(9), 772–783. Krtolica, R., & Šiljak, D. D. (1980). Suboptimality of decentralized stochastic control and estimation. IEEE Transactions on Automatic Control, AC25(1), 76–83. Middleton, R. H., & Freudenberg, J. S. (1995). Non-pathological sampling for generalized sampled-data hold functions. Automatica, 31(2), 315–319. Stankovic, S. S., Stanojevic, M. J., & Šiljak, D. D. (2000). Decentralized overlapping control of a platoon of vehicles. IEEE Transactions on Control System Technology, AC-8(5), 816–832. Wang, S. H., & Davison, E. J. (1973). On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control, AC-18, 473–478. Amir G. Aghdam received the B.A.Sc. degree from Isfahan University of Technology, Isfahan, Iran and the M.A.Sc. degree from Sharif University of Technology, Tehran, Iran, both in Electrical Engineering. He received his Ph.D. degree from the University of Toronto in 2000 and worked as a development engineer at Voyan Technology, Santa Clara, California from 2000 to 2001. He was a postdoctoral researcher at the systems control group of the University of Toronto for six months and then joined the
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941 Department of Electrical and Computer Engineering at Concordia University as an assistant professor in June 2002. Dr. Aghdam is a member of Professional Engineers of Ontario and a senior member of the IEEE. He served as Chair of the IEEE Montreal Section and Chair of the Control Systems Chapter of the IEEE Montreal Section in 2005 and 2006. Dr. Aghdam is an associate editor of the IEEE Systems Journal.
Edward J. Davison received the A.R.C.T. degree in piano in 1958, the B.A.Sc. degree in Engineering-Physics and the M.A. degree in Applied Mathematics from the University of Toronto in 1960, 1961 respectively, and the Ph.D. degree and Sc.D. degree from Cambridge University in 1964 and 1977, respectively. He was appointed as University Professor of Electrical and Computer Engineering at the University of Toronto in 2001, and as University Professor Emeritus in 2004. He was inducted into the University of Toronto’s Engineering Alumni Hall of Distinction in 2003. Dr. Davison has received several awards including the National Research Council of Canada’s E.W.R. Steacie Memorial Fellowship 1974–1977, the Canada Council Killam Research Fellowship 1979–1980, 1981–1983, the Athlone Fellowship (Cambridge University) 1961–1963, IEEE Transactions on Automatic Control Outstanding Paper Awards, and a Current Contents Classic Paper Citation. He was elected a Fellow of the Canadian Academy of Engineering in 2005, a Fellow of the International Federation of Control (IFAC) in 2005, a Fellow of the Institute of Electrical and Electronic Engineers in 1977, a Fellow of the Royal Society of Canada in 1977, an Honorary Professor of Beijing Institute of Aeronautics and Astronautics in 1986, and has been
1941
a designated Consulting Engineer of the Province of Ontario since 1979. In 1998, he was elected a member of the Academy of Nonlinear Sciences (ANS), Moscow, Russia, in 1996 he received the Outstanding Member Service Award from IFAC, in 1984, he received the IEEE Centennial Medal and was elected a Distinguished Member of the IEEE Control Systems Society. He has served on numerous positions in the IEEE Control Systems Society including President in 1983 and Consulting Editor of the IEEE Transactions on Automatic Control in 1985. He was Chairman of the IFAC Theory Committee in 1988-1990, Vice-Chairman of the IFAC Technical Board in 1990–1993, and a member of the IFAC Council in 1991–1996. He has served on numerous Editorial Boards of various journals. In 1993, he was awarded the triennial Quazza Medal from the International Federation of Automatic Control, in 1997, he received the IEEE Control System Society’s Hendrick W. Bode Lecture Prize, and in 2003, he received the Canada Council Killam Prize in Engineering. Rafael Becerril Arreola was born in Zitacuaro, Mexico, in 1978. He received a degree in Electronic Systems Engineering from the ITESM Campus Toluca, Mexico, in December 2000. Working on the nonlinear control of linear machines for magnetic levitation, he obtained his M.A.Sc degree from the University of Toronto, Canada, in 2003. Before starting a career in industry, he joined the Department of Electrical and Computer Engineering at Concordia University as Research Associate, working on linear time-varying decentralized control for large-scale systems and the application of nonlinear control strategies to on-ramp metering in highways.
Brief paper
Structural modification of systems using discretization and generalized sampled-data hold functions夡 Amir G. Aghdam a,∗ , Edward J. Davison b , Rafael Becerril-Arreola c a Electrical and Computer Engineering Department, Concordia University, Montréal, Que., Canada H3G 1M8 b Electrical and Computer Engineering Department, University of Toronto, Toronto, Ont., Canada M5S 3G4 c Electrical and Computer Engineering Department, Concordia University, Montréal, Que., Canada H3G 1M8
Received 16 August 2004; received in revised form 10 May 2006; accepted 5 June 2006 Available online 30 August 2006
Abstract This paper investigates the decentralized control of LTI continuous-time plants using generalized sampled-data hold functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the discretetime equivalent model to form a hierarchical system model of the plant. This is a new application of discretization and has, as its motivation, the design of decentralized controllers using centralized methods. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Decentralized control; Large-scale system; Generalized sampled-data hold functions
1. Introduction Sampled-data system applications are widely found in industry, mostly in order to take advantage of the application of computers as digital controllers. This is usually accomplished by using a simple zero-order hold (ZOH). The idea of using generalized sampled-data hold functions (GSHF) instead of a simple ZOH (or first-order hold) in control systems, was first introduced by Chammas and Leondes (1979) and Kabamba (1987). Kabamba investigated several advantages of using GSHFs in control systems and pointed out that, by using them, one can obtain many of the advantages of state feedback controllers, without the requirement of using state estimation procedures (Kabamba, 1987); in particular he showed that GSHFs can 夡 This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant RGPIN-262127-03 and in part by the natural Sciences and Engineering Research Council of Canada under Grant no. A4396. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised from by Associate Editor Toshiharu Sugie under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 514 848 2424; fax: +1 514 848 2802. E-mail addresses: [email protected] (A.G. Aghdam), [email protected] (E.J. Davison), [email protected] (R. Becerril-Arreola).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.06.005
significantly improve the performance of the closed-loop system. In addition, it is known that discrete-time controllers can have a significant effect on improving the overall performance of certain classes of decentralized control systems. In a discretized model, each transfer function in the resulting transfer function matrix is a function of the system parameters, and also the sampling period and type of hold function (ZOH, first-order hold, . . .). The question arises: using GSHFs, to what extent can one modify these transfer functions so “something desirable” happens? In this paper it will be shown how one can use generalized sampled-data hold functions to convert a continuous-time plant to a discrete-time equivalent system with a certain desired structure, in particular a hierarchical structure, which can directly be used to simplify the design of decentralized control systems for the plant. In D’Andrea and Dullerud (2003), a state-space approach is proposed to control systems with a highly structured interconnection topology and it is shown that design of controllers by discarding interconnection variables can cause instability. However, for a system with a hierarchical structure, or more precisely, a system with an acyclic graph structure, internal stability of each individual subsystem is a necessary and sufficient condition for stability of the overall system. Systems with hierarchical graph structures are investigated in the literature extensively and have many
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A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
g11
g31
1 g21
g21 g11
1
g12
g32 2
g23
3
g22
g33
g31
2 g32
g33
g22
3
g13 Fig. 1. Digraph of a continuous-time LTI system with three control agents.
applications in formation flight, underwater vehicles, automated highway, robotics, satellite constellation, just to name a few (Fax & Murry, 2004; Stankovic, Stanojevic, & Šiljak, 2000). Modifying a system’s structure to a hierarchical one can also have applications in decentralized adaptive control (Gavel & Šiljak, 1989). This paper is organized as follows. Digraphs are introduced in Section 2 to clarify some of the points that will be addressed in the main results. Section 3 then presents necessary and sufficient conditions for the existence of GSHF’s that make the structure of the plant hierarchical. Section 4 presents an example of control design for a plant that satisfy the conditions introduced in Section 3. Finally, Section 5 briefly discusses the importance of the results obtained. 2. Digraphs and system structure Consider the following strictly proper continuous-time decentralized LTI system with m control agents: x(t) ˙ = Ax(t) +
m
bj uj (t),
(1a)
j = 1, . . . , m
(1b)
j =1
yj (t) = cj x(t),
where x(t) ∈ Rn is the state vector, and uj (t) ∈ Rqj , and ¯ . . . , m} are the control vector and outyj (t) ∈ Rrj , j ∈ m={1, put vector of agent #j , respectively, and A, bj , and cj are matrices of appropriate dimensions. The transfer function matrix relating the inputs and outputs of this system can be written as ⎡ Y (s) ⎤ ⎡ g (s) . . . g (s) ⎤ ⎡ U (s) ⎤ 11 1m 1 1 ⎥⎢ . ⎥ ⎢ .. ⎥ ⎢ . . ⎢ ⎥ .. ⎦ ⎣ .. ⎦ , ⎣ . ⎦ = ⎣ .. Ym (s)
gm1 (s) . . .
gmm (s)
Um (s)
¯ represents the directed where gij (s)ci (sI −A)−1 bj , i, j ∈ m, arc from node j to node i in the digraph of this system. For example, the digraph of a system with three input–output agents is depicted in Fig. 1. In the control of decentralized systems, the structure of the interconnections generally has a significant role to play and, in some decentralized control design procedures, it is often assumed that there exists a bound on the interconnections between different nodes. For example, in decentralized adaptive control methods, such an assumption is often made or, alternatively, certain structural constraints on the interconnections are often assumed in order to assure the stability of the overall system
Fig. 2. Digraph of a continuous-time hierarchical LTI system with three control agents.
(Gavel & Šiljak, 1989). In the special case for systems which have a hierarchical structure, one can directly apply centralized control methods to each subsystem (represented by each node in the digraph of the system), with no assumptions required to be made on the interconnections, since it is guaranteed that signals coming from a higher level subsystems to a lower level subsystems will always be bounded, once the higher level subsystems are stabilized. For example, assume that the transfer functions g12 (s), g13 (s), and g23 (s) in Fig. 1 are all equal to zero; then the corresponding system will have the hierarchical structure shown in Fig. 2. Assuming that the system has no unstable decentralized fixed modes (DFM) (Wang & Davison, 1973), one can design a centralized stabilizing controller for each subsystem separately. In this case, subsystem #1, which has the highest level in the hierarchical structure, will be internally stable. Since there is no input signal coming into this subsystem other than its own bounded control input, this implies that all output signals coming out of this subsystem will be stable. The output signals of subsystem #1 includes the interconnection signal going to subsystem #2 and subsystem #3 through g21 (s) and g31 (s), respectively. Now subsystem #2, which has the second highest level, will also be internally stable and this implies that all output signals of this subsystem will also be bounded because all incoming signals, including the control input and the interconnection signal from subsystem #1, are bounded. Thus, all interconnection signals coming from the higher level subsystems to subsystem #3 will now be bounded and, since this subsystem will also be internally stable, one can conclude, using a similar argument, that the input and output signals of subsystem #3 are also bounded. Thence, the complete system will be internally stable. This implies that the decentralized controller design problem for a hierarchical system can be carried out in a centralized way for each subsystem. Note that the transfer function matrix of the hierarchical system of Fig. 2 has a lower-triangular form. In general case, it can be shown that a system with an acyclic digraph (i.e., a system whose digraph has no directed cycle) can be converted to a hierarchical system by adding some redundant edges and renumbering the vertices properly (Callier, Chan, & Desoer, 1978). Our goal now is to determine if and how one can modify the structure of a system by using generalized sampling. 3. Main results Consider the continuous-time LTI system represented by (1). It is desired to discretize the system by applying GSHFs fj (t),
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
j ∈ m, ¯ with a sampling period T, to each control agent. This problem can be formulated as follows. Let
1937
fj (t + T ) = fj (t)
where ∈ sp(A), and g(A) is an invertible matrix for nonpathological sampling periods (Chen & Francis, 1995). This implies that for almost all sampling periods, the unobservable subspace of ([ci1 . . . cij −1 ] , A) is equivalent to the unobservable subspace of its discrete-time equivalent pair ([ci1 . . . cij −1 ] , Ad ). As a result
for k=0, 1, . . . . As discussed in (Kabamba, 1987), the discretetime equivalent model is thence described by
[ci1 . . . cij −1 ] (zI − Ad )−1 ij = 0,
uj (t) = fj (t)u˜ j [k],
j ∈ m, ¯ t ∈ [kT , (k + 1)T ),
x[k + 1] = Ad x[k] +
or equivalently:
m
bdj u˜ j [k],
cdik (zI − Ad )−1 bdij = 0, j = 2, . . . , m,
j =1
yj [k] = cdj x[k],
j = 1, . . . , m,
(2)
where Ad = eAT , bdj =
T
(3a) eA(T−) bj fj () d,
j = 1, . . . , m,
(3b)
0
cdj = cj ,
j = 1, . . . , m.
(3c)
It is desired now to determine if one can choose a set of sampled-data hold functions fj (t), j ∈ m, ¯ and a sampling period T, so that certain elements of the transfer function matrix corresponding to the discrete-time equivalent model become equal to zero. Definition 1 (Chen and Francis, 1995). Consider the system (1). The sampling period T is pathological (with respect to A), if A has two eigenvalues with equal real parts but their imaginary parts differ by an integer multiple of 2/T . Note that there is no pathological sampling period for the system if all of the eigenvalues of A are real. Theorem 1. Consider the system (1). For every nonpathological sampling period T, there exists a sampled-data hold function fp (t) for each agent p ∈ m, ¯ so that the discrete-time equivalent model has a hierarchical structure, iff there exist distinct integers i1 , . . . , im ∈ m, ¯ so that the intersection of the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A) has a nonzero element for any j ∈ {2, . . . , m}. Proof (Proof of sufficiency). Assume that ij is a nonzero vector belonging to the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A). It follows from (Kabamba, 1987), that there exists a sampled-data hold function fij (t) for any sampling period T, so that in the discretetime equivalent model described by (2) and (3), bdij = ij . On the other hand, it is known that
Ad − eT I [ ci1
...
cij −1 ]
=
g(A)
0
0
I
A − I [ ci1
...
cij −1 ]
,
k = 1, . . . , j − 1.
This means that one can find the sampled-data hold functions f1 (t), . . . , fm (t), and rearrange the elements of the input and output vectors (i.e., by choosing the integers i1 , . . . , im properly), so that the transfer function matrix corresponding to the discrete-time equivalent model has a lower-triangular form, which corresponds to a hierarchical structure. Proof of necessity: Assume that the controllable subspace of (A, bij ) and the unobservable subspace of ([ci1 . . . cij −1 ] , A) have no nonzero vector in common. For nonpathological sampling periods the unobservable subspace of a continuoustime system and its discrete-time equivalent model are the same. Thus, there are no sampled-data hold functions and no input–output arrangement such that the upper-diagonal terms in the transfer function matrix of the discrete-time equivalent model (2) can be set equal to zero. The following theorem provides sufficient conditions for a system to have a hierarchical discrete-time equivalent. Theorem 2. Consider the system (1). For any sampling period T, there exists a sampled-data hold function fp (t) for each agent p ∈ m, ¯ so that the discrete-time equivalent model (2), (3) has a hierarchical structure, if there exist integers l1 , l2 ∈ m, ¯ l1 = l2 , so that ] , A) is unobservable; (a) the pair ([c1 . . . cl2 −1 cl2 +1 . . . cm and (b) the pair (A, bj ) is controllable for every j ∈ m, ¯ j = l1 ((A, bl1 ) may or may not be controllable).
Proof. Assume that conditions (a) and (b) of Theorem 2 both hold. Let i1 = l1 , im = l2 , and choose i2 , . . . , im−1 so that {i1 , i2 , . . . , im }= m. ¯ Now, since the pair ([ci1 . . . cim−1 ] , A) is unobservable, this implies that the pair ([ci1 . . . cim−1 ] , Ad ) will also be unobservable for any arbitrary sampling period T (Chen & Francis, 1995). This, in turn, means that the pair ([ci1 . . . cij −1 ] , Ad ), j =2, . . . , m is also unobservable, which implies that the columns of the matrix [ci1 . . . cij −1 ] (zI − Ad )−1 are linearly dependent. Therefore, there exists a nonzero vector ij so that [ci1 . . . cij −1 ] (zI − Ad )−1 ij = 0,
j = 2, . . . , m.
(4)
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A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
On the other hand, since the pair (A, bij ), j = 2, . . . , m is controllable, it follows from (Kabamba, 1987) that for any sampling period T, there exists a sampled-data hold function fij for each agent j (j =2, . . . , m), so that bdij =ij . Such (nonunique) sampled-data hold function may be obtained by using the following equation: fij (t) = bij e(T −t)A WT−1 ,ij ij ,
(5)
where WT ,ij =
T 0
etA bij bij etA dt.
This results in cdis (zI − Ad )
−1
bdij = 0,
j, s ∈ m, ¯ j > s.
Thence, using any arbitrary nonzero sampled-data hold function (such as a simple ZOH) for fi1 , along with the set of sampleddata hold functions given in (5), the corresponding transfer matrix for the discrete-time equivalent model will have a lowertriangular form. Remark 1. It is to be noted that the results of Theorem 1 are valid for all nonpathological sampling periods while those of Theorem 2 are valid for all sampling periods. The choice of sampling period is important, however, in terms of output performance. A method is proposed in Aghdam (2006) which considers the sampling period as one of the design parameters to minimize a prespecified performance index. For the systems that do not meet the conditions of Theorems 1 and 2 but have flexible sensor and actuator structures, one can make a modification in the input and output configuration to satisfy the desired conditions. Remark 2. Theorems 1 and 2 provide conditions under which the digraph of a system can be modified to a hierarchical structure by using sampling. However, they do not guarantee that the transfer function matrix of the discretized model will not have undesirable zeros. For example, it is not desirable to have a transfer function matrix with all elements of one row equal to zero. It can be easily verified that there will be at least one nonzero entry in each row of the transfer function matrix of the discretized model, iff cj , ∀j ∈ m, ¯ is in the controllability subspace of the discretized model. A nonpathological sampling period and controllability of the continuoustime system are sufficient conditions for avoiding such undesirable structures. It is desired now to obtain conditions under which the resulting discretized model does not have any DFMs (Wang & Davison, 1973). Theorem 3. Given (1), assume that the pair (A, bj ) is controllable for all j ∈ m. ¯ Then eT ∈ sp(Ad ) is not a DFM of the discrete-time equivalent model (2), with respect to the block
diagonal gain matrix K = diag(K1 , . . . , Km ), Kj ∈ Rsj ×rj , if the following three conditions all hold: (i) the sampling period T is nonpathological; T (ii) 0 e−t fj (t) dt = 0; ] , A) is observable. (iii) the pair ([c1 . . . cm Proof. eT ∈ sp(Ad ) is a DFM of (2) if and only if there exists a permutation of m ¯ denoted by distinct integers i1 , . . . , im , such that the rank of the matrix: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Ad − eT I
bdi1
bdi2
...
cil+1
0
0
...
cil+2
0
0
...
.. .
.. .
.. .
..
cim
0
0
...
.
bdil
⎤
0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ .. ⎥ . ⎦ 0
is less than n for some l ∈ m ¯ (Anderson & Moore, 1981). Assume now that conditions (i)–(iii) are all satisfied. Conditions (i) and (ii) imply that the resulting discretized system will not lose controllability and observability (corresponding to any controllable or observable pairs in the continuous-time model) (Middleton & Freudenberg, 1995). Thus, the pair (Ad , bdj ) is ] , A ) is controllable for all j ∈ m, ¯ and the pair ([c1 . . . cm d observable. This implies that the rank condition given above for the existence of a DFM does not hold for any l ∈ m. ¯ Remark 3. The results of Theorems 1 and 2 are interesting in the sense that they introduce a completely new application for sampling in control. The results can be very useful in the design of decentralized controllers, when the structure of the original system does not permit one to directly apply centralized controller design methods to decentralized systems. In this case, if the conditions given in Theorem 1 or Theorem 2 are met, one can find a set of GSHFs to modify the structure of the system in the discrete-time equivalent model (2), (3), so that centralized digital control methods can now be directly applied to each interconnected subsystem. Remark 4. When conditions of Theorem 1 are not satisfied, the question arises: is it possible to reduce the magnitude of certain interconnections by applying sampled-data hold functions? The motivation to this question is that systems with weak interconnections often have the property that decentralized control can still effectively be applied, using centralized control design methods for each subsystem. This problem has been investigated in Becerril, Aghdam, and Davison (2005), where the strength of interconnections has been measured in a closed form using the Hankel-norm. It is to be noted that the main objective of the present work is stability of the interconnected system. An extensive decentralized control performance degradation analysis in presence of interconnections is presented in Krtolica and Šiljak (1980). The problem of high-performance decentralized control design for interconnected systems remains an open area for research.
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
g21 g11
1
gd (z)
g22
2
11
1939
1
gd (z) 21
2
gd (z) 22
g12 Fig. 4. Digraph of the discrete-time equivalent mass-spring system of Example 1.
Fig. 3. Digraph of the mass-spring system of Example 1.
and gd22 are given by
4. Numerical example Example 1. Consider a 2-input 2-output mass-spring system with the following state-space matrices: ⎡
0
⎢ ⎢ −0.2 ⎢ ⎢ ⎢0 A=⎢ ⎢ ⎢1 ⎢ ⎢ ⎣0 1
0
0
0
−0.02
0.1
0.01
0.1
0
0
1
0
0.1
−1
−0.1
0
0
0
0
0
⎥ 0.01 ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ ⎥ ⎥ 1 ⎦
0.1
0
0
−1
−0.1
b1 = [0 0 0 1 0 0] ,
0
⎤
1
b2 = [0 0 0 0 0 1] ,
c1 = [0 0 0 1 0 0],
c2 = [0 0 0 0 1 1].
(6a)
(6b) (6c)
The digraph of this system is depicted in Fig. 3, where g11 , g12 , g21 , and g22 are given by
−0.163(z4 − 6.295z3 + 7.274z2 − 4.752z + 1.175) , z5 − 2.450z4 + 3.045z3 − 2.449z2 + 1.187z − 0.333 1.398(z5 − 1.19z4 + 1.706z3 − 1.137z2 + 0.528z + 0.023 , z6 − 3.45z5 +5.494z4 −5.494z3 +3.637z2 −1.52z+0.333 0.1 , z−1 respectively. In addition, it can be easily verified that the conditions of Theorem 3 for the given model, GSHF, and sampling period hold true, which means that the discrete-time equivalent model has no DFMs. This implies that one can design a decentralized controller for this system by applying a digital controller for each subsystem independently and using centralized methods. For instance, consider the controllers: v11 [k + 1] = v11 [k] + 5(ym1 [k] − yref,1 ),
(7a)
v12 [k + 1] = 0.6876v12 [k] + ym1 [k],
(7b)
u˜ 1 [k] = − 0.3591ym1 [k] − 0.005629v11 [k] − 0.04669v12 [k],
(7c)
(s 4 + 0.12s 3 + 1.201s 2 + 0.02s + 0.1)/den(s),
and
0.001(s 2 + 20s + 100)/den(s),
v21 [k + 1] = v21 [k] + 5(ym2 [k] − yref,2 ),
(8a)
0.001(s + 21s + 120s + 100)/den(s),
v22 [k + 1] = −0.8004v22 [k] + ym2 [k],
(8b)
(s + 1.12s + 1.321s + 1.221s + 0.12s + 0.1)/den(s),
u˜ 2 [k] = − 9.393ym2 [k] − 0.6006v21 [k]
3
5
2
4
3
2
respectively, and den(s)=s(s 4 +0.22s 3 +2.212s 2 +0.24s+1.2). Here, (c1 , A) is unobservable but (c2 , A) is not, while (A, b1 ) and (A, b2 ) are both controllable. Thus, the conditions of Theorem 2 hold for l1 =1 and l2 =2. In other words, one can use generalized sampling to construct a hierarchical discrete-time model, with the first subsystem at a higher level. Accordingly, let us assume that i1 = 1 and i2 = 2 (which means that l1 = 1 and l2 = 2 in Theorem 2), and let us use Eq. (3b) to obtain a sampled-data hold function for control agent #2, so that the resultant discretized model will be hierarchical with subsystem #1 at the higher level. In this case, it can be shown that, for all values of the sampling period T, the vector [1 0 1 0 1 0] is a basis for the null-space of c1 (zI − Ad )−1 (in fact, the null-space is one dimensional). Solve now the minimum energy problem for (3b) using the controllability gramian as described in Kabamba (1987), with T = 5 s. Then, on applying the resultant sampled-data hold function to control agent #2 and a simple ZOH to control agent #1, the hierarchical model of Fig. 4 is obtained, where gd11 , gd21 ,
− 1.086 × 10−4 v22 [k],
(8c)
which have been designed for subsystem #1 and subsystem #2, respectively, independently of each other, using centralized methods to solve the robust servomechanism problem. Here yref,1 and yref,2 in (7) and (8) denote constant reference signals for control agent #1 and control agent #2, respectively. The eigenvalues of the resultant closed-loop discrete-time system corresponding to subsystem #1 and subsystem #2 are given by: sp1 = {0.9246, 0.2935 ± 0.7898i, 0.7906 ± 0.08309i, 0.5514 ± 0.5753i} and sp2 = {−0.8004, 0.5303 ± 0.2824i}, respectively. The eigenvalues of the overall closed-loop discrete-time system are thus given by: sp = sp1 ∪ sp2 . Assume now that x[0] = [1 1 1 1 1 1] and v11 [0] = v12 [0] = v21 [0] = v22 [0] = 0. Figs. 5(a) and (b) give the outputs of the resultant closed-loop system, for a unit step reference input in control agent #1 and a zero reference input in control agent #2, using the local controllers (7) and (8). The control signals, corresponding to agents #1 and #2, are given in Figs. 5(c) and (d), respectively.
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A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941
1.5
6 4 ym2
ym1
1
0.5
0
0
0
100
(a)
200
-2
300
(sec)
0
300
200
300
-10 -20
-0.2
u2
u1
200 (sec)
0
0
-30 -40
-0.4
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Fig. 5. Closed-loop simulations for Example 1: (a) samples of the output response in agent #1; (b) samples of the output response in agent #2; (c) samples of the control signal in agent #1; (d) samples of the control signal in agent #2.
5. Conclusions This paper introduces a new application for GSHFs, which can be used to simplify the structure of the resulting discretized model. The conditions under which such a discretized model can have a hierarchical digraph are discussed and a method for the synthesis of the sampled-data hold functions corresponding to each control agent is presented. The motivation for simplifying the plant structure to become hierarchical is that the controller design, particularly for decentralized control, can be greatly simplified. For example, one can design decentralized controllers for the hierarchical discrete-time model by applying centralized methods directly to each control agent. Simulation results are given to show how such a discretization procedure can result in a simple hierarchical digraph that simplifies the decentralized controller design in solving the robust servomechanism problem.
References Aghdam, A. G. (2006). Decentralized control design using piecewise constant hold functions. In Proceedings of the 2006 American control conference (pp. 207–212). Minneapolis, MN, USA, June 2006. Anderson, B. O. D., & Moore, J. B. (1981). Time-varying feedback laws for decentralized control. IEEE Transactions on Automatic Control, AC-26(5), 1133–1138. Becerril, R. A., Aghdam, A. G., & Davison, E. J. (2005). Minimizing interactions of subsystems in large-scale interconnected systems using generalized sampling. 2005 IFAC world congress, Paper Code Tu-E02Tp/13, Prague, Czech Republic. Callier, F. M., Chan, W. S., & Desoer, C. A. (1978). Input–output stability of interconnected systems using decompositions: An improved formulation. IEEE Transactions on Automatic Control, AC-23(2), 150–163.
Chammas, A. B., & Leondes, C. T. (1979). On the finite time control of linear systems by piecewise constant output feedback. International Journal of Control, 30(2), 227–234. Chen, T., & Francis, B. A. (1995). Optimal sampled-data control systems. Berlin: Springer. D’Andrea, R., & Dullerud, G. E. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions on Automatic Control, AC-48, 1478–1495. Fax, J. A., & Murry, R. M. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, AC-49(9), 1465–1476. Gavel, D. T., & Šiljak, D. D. (1989). Decentralized adaptive control: Structural conditions for stability. IEEE Transactions on Automatic Control, AC34(4), 413–426. Kabamba, P. T. (1987). Control of linear systems using generalized sampleddata hold functions. IEEE Transactions on Automatic Control, AC-32(9), 772–783. Krtolica, R., & Šiljak, D. D. (1980). Suboptimality of decentralized stochastic control and estimation. IEEE Transactions on Automatic Control, AC25(1), 76–83. Middleton, R. H., & Freudenberg, J. S. (1995). Non-pathological sampling for generalized sampled-data hold functions. Automatica, 31(2), 315–319. Stankovic, S. S., Stanojevic, M. J., & Šiljak, D. D. (2000). Decentralized overlapping control of a platoon of vehicles. IEEE Transactions on Control System Technology, AC-8(5), 816–832. Wang, S. H., & Davison, E. J. (1973). On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control, AC-18, 473–478. Amir G. Aghdam received the B.A.Sc. degree from Isfahan University of Technology, Isfahan, Iran and the M.A.Sc. degree from Sharif University of Technology, Tehran, Iran, both in Electrical Engineering. He received his Ph.D. degree from the University of Toronto in 2000 and worked as a development engineer at Voyan Technology, Santa Clara, California from 2000 to 2001. He was a postdoctoral researcher at the systems control group of the University of Toronto for six months and then joined the
A.G. Aghdam et al. / Automatica 42 (2006) 1935 – 1941 Department of Electrical and Computer Engineering at Concordia University as an assistant professor in June 2002. Dr. Aghdam is a member of Professional Engineers of Ontario and a senior member of the IEEE. He served as Chair of the IEEE Montreal Section and Chair of the Control Systems Chapter of the IEEE Montreal Section in 2005 and 2006. Dr. Aghdam is an associate editor of the IEEE Systems Journal.
Edward J. Davison received the A.R.C.T. degree in piano in 1958, the B.A.Sc. degree in Engineering-Physics and the M.A. degree in Applied Mathematics from the University of Toronto in 1960, 1961 respectively, and the Ph.D. degree and Sc.D. degree from Cambridge University in 1964 and 1977, respectively. He was appointed as University Professor of Electrical and Computer Engineering at the University of Toronto in 2001, and as University Professor Emeritus in 2004. He was inducted into the University of Toronto’s Engineering Alumni Hall of Distinction in 2003. Dr. Davison has received several awards including the National Research Council of Canada’s E.W.R. Steacie Memorial Fellowship 1974–1977, the Canada Council Killam Research Fellowship 1979–1980, 1981–1983, the Athlone Fellowship (Cambridge University) 1961–1963, IEEE Transactions on Automatic Control Outstanding Paper Awards, and a Current Contents Classic Paper Citation. He was elected a Fellow of the Canadian Academy of Engineering in 2005, a Fellow of the International Federation of Control (IFAC) in 2005, a Fellow of the Institute of Electrical and Electronic Engineers in 1977, a Fellow of the Royal Society of Canada in 1977, an Honorary Professor of Beijing Institute of Aeronautics and Astronautics in 1986, and has been
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a designated Consulting Engineer of the Province of Ontario since 1979. In 1998, he was elected a member of the Academy of Nonlinear Sciences (ANS), Moscow, Russia, in 1996 he received the Outstanding Member Service Award from IFAC, in 1984, he received the IEEE Centennial Medal and was elected a Distinguished Member of the IEEE Control Systems Society. He has served on numerous positions in the IEEE Control Systems Society including President in 1983 and Consulting Editor of the IEEE Transactions on Automatic Control in 1985. He was Chairman of the IFAC Theory Committee in 1988-1990, Vice-Chairman of the IFAC Technical Board in 1990–1993, and a member of the IFAC Council in 1991–1996. He has served on numerous Editorial Boards of various journals. In 1993, he was awarded the triennial Quazza Medal from the International Federation of Automatic Control, in 1997, he received the IEEE Control System Society’s Hendrick W. Bode Lecture Prize, and in 2003, he received the Canada Council Killam Prize in Engineering. Rafael Becerril Arreola was born in Zitacuaro, Mexico, in 1978. He received a degree in Electronic Systems Engineering from the ITESM Campus Toluca, Mexico, in December 2000. Working on the nonlinear control of linear machines for magnetic levitation, he obtained his M.A.Sc degree from the University of Toronto, Canada, in 2003. Before starting a career in industry, he joined the Department of Electrical and Computer Engineering at Concordia University as Research Associate, working on linear time-varying decentralized control for large-scale systems and the application of nonlinear control strategies to on-ramp metering in highways.