Output Regulation of Large-Scale Hydraulic ... - Semantic Scholar
JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007
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Output Regulation of Large-Scale Hydraulic Networks Claudio De Persis, Tom Nørgaard Jensen, Romeo Ortega, and Rafał Wisniewski
Abstract The problem of output regulation for a class of hydraulic networks found in district heating systems is addressed. The results show that global asymptotic and semi-global exponential output regulation is achievable using a set of decentralized proportional-integral controllers. The fact that the result is global and independent of the number of end-users has the consequence that structural changes such as enduser addition and removal can be made in the network while maintaining the stability properties of the system. Furthermore, the decentralized nature of the control architecture eases the implementation of structural changes in the network.
Index Terms Nonlinear systems, Robust control, Decentralized control, Output regulation, Hydraulic networks
I. I NTRODUCTION The work presented here concerns the output regulation of a class of large-scale hydraulic networks found in district heating systems. The case study considers a new paradigm for the design of district heating systems, which has been proposed to reduce the diameter of the pipes This work is supported by The Danish Research Council for Technology and Production Sciences within the Plug and Play Process Control Research Program. C. De Persis is with ITM, Faculty of Mathematics and Natural Sciences, University of Groningen, 9747 AG Groningen, The Netherlands and Department of Computer, Control and Management Engineering A. Ruberti, Sapienza Universit`a di Roma, Via Ariosto 25, 00185 Roma, Italy. (e-mail:[email protected]). T.N. Jensen and R. Wisniewski are with Department of Electronic Systems, Aalborg University, Fredrik Bajers Vej 7C, 9220 Aalborg Denmark. (e-mail:{tnj,raf}@es.aau.dk). R. Ortega is with Laboratoire des Signaux et Syst`emes, CNRS-SUPELEC, 91192, Gif-sur-Yvette, France. (email:[email protected]). July 9, 2012
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in the network. By reducing the pipe diameter, it is possible to reduce the heat dispersion from the pipes and thereby reduce the energy losses in the system [2]. On the other hand, the reduced diameters induce increased pressure losses throughout the network which must be compensated by multiple pumps. Studies held that the multi-pump architecture is the technology which can compensate for the increased pressure losses while still achieving a substantial energy saving ([1]). The multi-pump architecture raises the question of how the pumps should be operated to control the network in appropriate way. The new paradigm also gives rise to a flexible network structure in which end-users can be added to or removed from the network. The case study is part of the ongoing research program Plug & Play Process Control [3] which considers automatic reconfiguration of the control system if components such as sensors, actuators or subsystems are added to or removed from a system. To fulfill the control objective, which is to keep the pressure across the so-called end-user valves at a constant reference, a set of proportional-integral control actions is proposed. The control actions are decentralized in the sense that the individual controllers use only locally available information, which is the pressure measurement at the end-user. The results show that it is possible to achieve global asymptotic and semi-global exponential output regulation using this control architecture. These results represent an interesting addition to [5], where semi-global practical stability was achieved by positive proportional controllers, and we believe they are instrumental for further developments which are briefly discussed in conclusions of the paper. In Section II, the model of the system is introduced along with the output regulation problem. The main results of the paper are presented in Section III. In Section IV, experimental results and numerical simulations of the closed loop system are presented. Finally, conclusions are given in Section V along with possible future research directions. Nomenclature: For a vector x ∈ Rn , xi denotes the ith element of x. Let M(n, m; R) denote the set of n × m matrices with real entries, and M(n; R) = M(n, n; R). For a square matrix A, A > 0 means that A is positive definite. For a square matrix A, A = diag(xi ) means that A has xi as entries on the main diagonal and zero elsewhere. For a matrix A, Aij will be used to denote the entry in the ith row and jth column of A. A matrix A is said to be Hurwitz if all eigenvalues of A have strictly negative real part. Throughout the paper, C 1 denotes a continuously differentiable map. A continuous map is said to be proper if the inverse image of July 9, 2012
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a compact set is compact. A function f : R → R is called monotonically increasing if it is order preserving, i.e., for all x and y such that x < y then f (x) < f (y). For a map f : Rn → Rm , let Df (·) denote the Jacobian matrix of f (·). II. S YSTEM M ODEL The system under consideration is the hydraulic network of a district heating system. The model has been derived in detail in [4]. To make the paper self-contained, the essentials of the model are briefly recalled below. The hydraulic network is the interconnection of several two-terminal components, namely valves, pipes and pumps. Each component is described by a constitutive equation given by ∆hk = Jk q˙k + λk (qk ) + µk (qk ) − ∆hp,k ,
(1)
where Jk > 0 is a parameter which is non-zero for pipe components and zero for other components; the function µk is non-zero for valve components and zero for other components; the function λk is non-zero for pipe components and zero for other components; the input ∆hp,k is non-zero for pump components and zero for other components. The functions λk , µk are C 1 , monotone increasing, zero at qk = 0 and proper. The precise expression of λk , µk is unknown. A graph G is associated with the hydraulic network, where the nodes are the terminals of the components and the m edges are the components. The graph satisfies the following: Assumption 2.1: [5] G is a connected graph. Let T be a spanning tree of G, that is a connected sub-graph of G which contains all the nodes of G but no cycles. The edges of G which are not in T are the chords of G. Denote by n the number of chords of G. The hydraulic networks considered in this paper have the following property: Assumption 2.2: [5] Each chord in G corresponds to a pipe. Moreover, the pipe is in series with a valve and a pump (see Fig. 1). By definition of T , adding an edge to T which is not in G, a cycle is obtained. These cycles are called the fundamental cycles of G. The hydraulic networks considered in this paper additionally satisfy the following: Assumption 2.3: [5] There exists one and only one component called the heat source. It corresponds to a valve of the network, and it lies in all the fundamental cycles. July 9, 2012
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Valve Pump
Pipe
qi
Remaining network
Fig. 1.
The series connection associated with each end-user [4].
The pipes, valves and pumps defined in Assumption 2.2 are referred to as end-user pipes, valves and pumps ([4]). Moreover, a series connection comprising an end-user pipe, valve and pump is referred to as an end-user. The flows through the end-user pipes form a set of independent flow variables, in the sense that they can be set independently without violating Kirchhoff’s law. The m edges of the graph G and the corresponding components are labeled via the integers 1, 2, . . . , m. Following [4], the edges corresponding to the end-user pipes are labeled via the integers 1, 2, . . . , n < m. The vector q ∈ Rn of all independent flows is taken as the state variable of the system. Denoted by B ∈ M(n, m; R) the fundamental loop matrix associated to the graph ([7], [4]), the m-dimensional vector x of all flows (dependent and independent) through the components in the system is given by x = B T q. The dynamic model of the hydraulic network is characterized in the following: Proposition 2.1: [5] Any hydraulic network satisfying Assumptions 2.1 and 2.2 admits the representation J q˙ = f (B T q) + u yi (qi ) = µi (qi ) , i = 1, . . . , n,
(2) (3)
where q ∈ Rn is the vector of independent flows; u ∈ Rn is a vector of independent inputs consisting of a linear combination of the delivered pump pressures; yi is the measured pressure drop across the ith end-user valve; J ∈ M(n; R), J > 0; f (·) is a C 1 vector field; µi (·) is the fundamental law of the ith end-user valve. The vector field f (·) can be written as f (x) = −B(λ(x) + µ(x)), ∀x ∈ Rm ,
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(4)
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where λ(x) = [λ1 (x1 ), . . . , λm (xm )]T ; µ(x) = [µ1 (x1 ), . . . , µm (xm )]T and λi (·) is non-zero if and only if i is a pipe and µi (·) is non-zero if and only if i is a valve. The matrix J in (2) is given by J = BJ B T
(5)
where J = diag(J1 , . . . , Jm). A. Output Regulation Problem It is desired to regulate the pressure yi across the ith end-user valve to a given reference value ri with the use of a feedback controller having available only local information. The vector r = (r1 , . . . , rn ) of reference values takes values in a known compact set R: R = {r ∈ Rn | 0 < rm ≤ ri ≤ rM }.
(6)
For the purpose of asymptotic output regulation, a set of decentralized proportional-integral controllers is the focus of the work presented here. The controllers considered will be of the form ξ˙i = −Ki (yi (qi ) − ri )
(7)
ui = ξi − Ni (yi (qi ) − ri )
(8)
where Ki , Ni > 0 and i = 1, 2, . . . , n. III. S TABILITY
PROPERTIES OF THE CLOSED LOOP SYSTEM
In this section, the stability properties of the closed loop system will be examined. First, it is shown that global asymptotic output regulation can be proved. Secondly, semi-global exponential stability is shown. A. Global Asymptotic Stability In this subsection, global asymptotic stability of the desired equilibrium point of the closed loop system will be shown. To this end, first define the proportional gain matrix N = diag(Ni ). The following lemma will be instrumental in deriving the closed loop stability properties of the system.
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Lemma 3.1: Let the matrix G(q) ∈ M(n, R) be given by G(q) ≡ NDy(q) − Df (B T q),
(9)
then G(q) ≥ 0. Proof: Recall that x = B T q and f (x) = −B (λ(x) + µ(x))
(10)
where λ(x) = (λ1 (x1 ), . . . , λm (xm ))T and µ(x) = (µ1 (x1 ), . . . , µm (xm ))T . Then −Df (B T q) is given by −Df (B T q) = −
∂x ∂ f (x) ∂x ∂q
(11)
= BΛ(x)B T .
(12)
where Λ(x) = diag( ∂x∂ i (λi (xi ) + µi (xi ))). By Assumption 2.2, each chord in the graph G described by the network corresponds to a pipe in series with a user valve. Therefore, bearing in mind that we have rearranged the numbering of the components such that the first n components are the pipes in the chords of G, (12) can be rewritten as −Df (B T q) =
In F
Λ1 (x)
0
0 Λ2 (x)
In F
T
= Λ1 (x) + F Λ2 (x)F T ,
(13) (14)
where Λ1 (x) = diag( ∂x∂ i (λi (xi ))) for i = 1, . . . , n, Λ2 (x) = diag( ∂x∂ i (λi (xi ) + µi (xi ))) for i = n + 1, . . . , m, and F is a matrix with entries in {0, 1} such that B = (In F ) ([5]). Since λi (·) are monotonically increasing functions the matrix Λ1 (x) is positive semi definite for all x. Furthermore, since µi (·) are monotonically increasing functions, the matrix Λ2 (x) is positive semi definite for all x (recall that λi (·) is non-zero only for pipe components and µi (·) is non-zero only for valve components). Then it follows that −Df (B T q) is positive semi definite. The matrix Dy(q) is given by Dy(q) = diag(
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∂ yi (qi )). ∂qi
(15)
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Recall that yi (qi ) = µi (qi ), and µi (·) is a monotonically increasing function. As a consequence Dy(q) is positive semi definite. Since N is diagonal and positive definite and Dy(q) is a positive semi definite diagonal matrix, it follows that NDy(q) is positive semi definite. From the derivations above, it is concluded that NDy(q)−Df (B T q) is a positive semi definite matrix. Since the functions µi (·) are monotonically increasing and proper, they admit inverses µ−1 i (·). Now, let qi∗ ≡ µ−1 i (ri ),
(16)
that is: qi∗ is the flow through the ith end-user valve which produce the reference output. Furthermore, define q˜ ≡ q − q ∗ ,
(17)
y˜i (˜ qi ) ≡ µi (˜ qi + qi∗ ) − ri .
(18)
and the variable
The main result of the subsection will rely on the following Lemma, which has been derived in [5]: Lemma 3.2: The function y˜i (˜ qi ) is monotonically increasing and zero at q˜i = 0, and moreover y˜i (˜ qi )˜ qi > 0, ∀ q˜i 6= 0 .
(19)
Now, the main result of this subsection can be stated. Proposition 3.1: The point (˜ q T , q˙T ) = 0 is a globally asymptotically stable equilibrium point of the closed loop system given by (2), (3), (7) and (8). Proof: The closed loop system defined by (2), (3), (7) and (8) is J q˙ = f (B T q) + ξ − N y˜(˜ q) ξ˙ = −K y˜(˜ q)
(20) (21)
From (20) the following can be derived J q¨ = Df (B T q)q˙ + ξ˙ − NDy(q)q˙
(22)
J q¨ = −G(q)q˙ − K y˜(˜ q ).
(23)
which can be rewritten as
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In the above J > 0, G(q) ≥ 0 by Lemma 3.1 and K y˜(˜ q ) can be written as ∇W (˜ q ), with W (˜ q) given as W (˜ q) =
n X i=1
Ki
Z
q˜i
y˜i (s)ds,
(24)
0
and W (˜ q) > 0 by Lemma 3.2. Therefore, the structure of (23) is similar to that of a mechanical system in the standard Lagrangian form [9]. This motivates the choice of the Lyapunov function candidate V : R2n → R, which can be seen as an equivalent of the total energy function for a mechanical system: Z q˜i n X 1 Ki V (˜ q , q) ˙ = ˙ y˜i (s)ds + q˙T J q, 2 0 i=1
(25)
which is positive definite and radially unbounded. The first term in the definition of the function V (˜ q, q) ˙ in (25) can be interpreted as the equivalent of the potential energy function of a mechanical system described by (23); whereas, the second term can be interpreted as the kinetic energy function. Note, however, that the physical dimensions of the two terms mismatch in the case of the hydraulic network, and as such the Lyapunov function candidate does not have a direct physical interpretation. The time derivative of V (˜ q , q) ˙ is d V (˜ q , q) ˙ = q˙ T J q¨ + q˙T K y˜(˜ q) dt
(26)
= −q˙T G(q)q˙ ≤ 0
(27)
where the last inequality comes from Lemma 3.1. Using (9) and (14) the following bound is obtained n
X ∂ d V (˜ q , q) ˙ ≤− λi (qi )q˙i2 . dt ∂qi i=1 Recalling that
∂ λ (q ) ∂qi i i
> 0 for every qi 6= 0 and
∂ λ (q ) ∂qi i i
(28)
= 0 if and only if qi = 0, it is
concluded by the LaSalle invariance principle, that the closed loop system converges to the set Q given by Q = {(q, q) ˙ ∈ R2n | qi = 0 ∨ q˙i = 0}.
(29)
Since, qi = 0 implies q˙i = 0 which in turn implies q¨i = 0, from (23) it follows that y˜i (˜ qi ) = 0 is the largest invariant set of the closed loop system fulfilling
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d V dt
(˜ q , q) ˙ = 0. From this it follows
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that K y˜(˜ q) → 0 as t → ∞. Since y˜i (·) is monotonically increasing and zero in zero, it is concluded that q˜ → 0 as t → ∞. From (16) and (18) it can be seen that y˜i (˜ qi ) → 0 is equivalent to µi (qi ) → ri . That is, Proposition 3.1 shows that the desired equilibrium point of the closed loop system is globally asymptotically stable. Moreover, since the result holds for arbitrary values of control gains Ni > 0 and Ki > 0, asymptotic output regulation is guaranteed for arbitrary choice of gains. Since Proposition 3.1 is independent of the number n of end-users in the system, it follows that end-users can be added to or removed from the system while maintaining asymptotic output regulation. B. Semi-global Exponential Stability In this section, semi-global exponential output regulation of the closed loop system is shown. This result rests on the assumption that
∂ (µi (xi ))|xi=qi∗ ∂xi
6= 0. Requiring the variation of the
pressure drop with respect to the flow at the end-user valves to be non-zero at the working point is a realistic condition which is often satisfied in practice. Some preliminaries are instrumental in the analysis below. If q = q ∗ , the input c(q ∗ ) such that the output y equals the reference r is given by f (B T q ∗ ) + c(q ∗ ) = 0. The solution ξ ∗ to (7)-(8) which produces the input c(q ∗ ) when y˜(˜ q ) = 0 is ξ ∗ = −f (B T q ∗ ). The ability by (7)(8) to reproduce c(q ∗ ) constitutes the so-called internal model property. Perform the change of coordinates ξ˜ ≡ ξ + f (B T q ∗ ),
(30)
J q˜˙ =f˜(˜ q ) + ξ˜ − N y˜(˜ q)
(31)
so as to obtain
˙ ξ˜ = − K y˜(˜ q ),
(32)
where f˜(˜ q ) = f (B T (˜ q + q ∗ )) − f (B T q ∗ ). The variable q˜ measures the distance of the state q from the point where the regulation error vanishes to zero; whereas, the variable ξ˜ measures the distance of the controller state ξ from the variable ξ ∗ which provides the value of the control when the regulation error is zero.
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It is evident from (32) that the addition of the internal model introduces a sub-system which is not asymptotically stable. This is a well-known obstacle towards the solution of the regulation problem. A way to overcome this obstacle is to rely on the canonical parametrization of the internal model introduced in [10]. In what follows, we mimic the analysis in [10], although the special structure of the system allows us to proceed in a more straightforward way. Let F˜ ∈ M(n, R) be a Hurwitz matrix and define further the new coordinate ([10]) χ = ξ˜ + F˜ J q˜ which yields J q˜˙ =f˜(˜ q) + χ − F˜ J q˜ − N y˜(˜ q)
(33)
χ˙ = − K y˜(˜ q ) + F˜ f˜(˜ q ) + F˜ χ − F˜ 2 J q˜ − F˜ N y˜(˜ q ).
(34)
Lemma 3.3: [11] Let f : Rn → R be a C 1 function in a convex neighborhood U of 0 in Rn , with f (0) = 0. Then f (x1 , . . . , xn ) =
n X
xi gi (x1 , . . . , xn )
(35)
i=1
for some suitable C 1 functions gi : Rn → R defined in U, with gi (0) = By Lemma 3.3 the map f˜(˜ q) can be written as ˆ q )˜ f˜(˜ q ) = φ(˜ q
∂ f (0). ∂xi
(36)
ˆ q ) a matrix of continuously differentiable functions. In order to make (34) independent with φ(˜ of K, we set K = −F˜ N. Observe that to preserve the decentralized nature of the controller, K must be diagonal. This in turn requires F˜ to be diagonal as well. Also observe that F˜ Hurwitz and N positive definite guarantee that K is positive definite as well. The closed-loop system can be written more simply as χ˙ =F˜ χ − φ(a) (˜ q)˜ q
(37)
J q˜˙ =φ(b) (˜ q )˜ q + χ − N y˜(˜ q)
(38)
ˆ q ) + F˜ 2 J φ(a) (˜ q ) = −F˜ φ(˜
(39)
ˆ q ) − F˜ J. φ(b) (˜ q ) = φ(˜
(40)
where
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Again by Lemma 3.3, the relation between q˜i and y˜i can be written as y˜i (˜ qi ) = gi (˜ qi )˜ qi . Observe that, as
∂ (˜ y (˜ q ))|q˜i =0 ∂ q˜i i i
=
∂ (µi (qi ))|qi=qi∗ ∂qi
(41)
6= 0, and y˜i (˜ qi ) = 0 if and only if q˜i = 0,
for every q˜i the function gi (·) on the right-hand side is positive. Moreover, if q˜i range over a compact set, then by continuity of gi (·) there exists m > 0 such that gi (˜ qi ) ≥ m > 0 ,
i = 1, 2, . . . , n.
(42)
Then the main result of this subsection can be stated. Proposition 3.2: Given system (2), (3), a vector r of reference values satisfying (6), and a compact set of initial conditions Q ⊂ Rn × Rn , there exist diagonal positive definite matrices N and K in (7), (8), such that every trajectory (q(t), ξ(t)) of the closed-loop system (2), (3), (7), (8) with initial condition in Q is bounded and satisfies limt→+∞ yi (t) = ri for i = 1, 2, . . . , n. Proof: Let P > 0 be such that F˜ T P + P F˜ = −2I and consider the Lyapunov function candidate V : R2n → R given by 1 1 V (χ, q˜) = χT P χ + q˜T J q˜ 2 2 which is positive definite and radially unbounded. Compute the time derivative along the trajectories of the system, to obtain d V (χ, q˜) =χT P F˜ χ − χT P φ(a) (˜ q)˜ q+ dt + q˜T φ(b) (˜ q)˜ q + q˜T χ − q˜T N y˜(˜ q) .
(43)
Let S be a level set of V (χ, q˜) containing the set of initial conditions of the system. Bearing in mind (42), the time derivative of V (χ, q˜) can be written in compact form as d V (χ, q˜) ≤ − |χ|2 + |χ|||P φ(a)(˜ q )|||˜ q|+ dt + |˜ q |||φ(b) (˜ q )|||˜ q| + |˜ q ||χ| − m˜ q T N q˜ .
(44)
Let Θ be a positive constant such that max{||P φ(a) (˜ q )||, ||φ(b)(˜ q )|| | q˜ ∈ S} ≤ Θ, which exists by continuity of φ(a) (·) and φ(b) (·). Then |χ|2 d V (χ, q˜) ≤ − + (Θ2 + Θ + 1)|˜ q|2 − m˜ q T N q˜ . dt 2
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˜ , with Ni ’s the diagonal entries of N and N ˜ to design, so that Let N1 = . . . = Nn = m−1 N |χ|2 d ˜ q |2 . V (χ, q˜) ≤ − + (Θ2 + Θ + 1 − N)|˜ dt 2 ˜ ∗ = Θ2 + Θ + 1 + 1 σM (J) , where σM (·) denotes the maximum eigenvalue of a (symmetric) Set N 2 σM (P ) ˜ ˜∗ matrix. Then, for all N ≥ N 1 d V (χ, q˜) ≤ − V (χ, q˜). dt σM (P )
(45)
This shows that the trajectories (χ(t), q˜(t)) of the system are bounded and converge to the origin. Bearing in mind the definition of (χ, q˜), it also shows that (q(t), ξ(t)) are bounded and q(t) → q ∗ , ξ(t) → −f (B T q ∗ ) as t → ∞. By continuity of µ(·), it is concluded that µ(q(t)) → µ(q ∗ ) = r as t → ∞, i.e. the thesis. Remark 3.1: In view of the quadratic nature of the Lyapunov function V (χ, q˜) the proof actually shows exponential convergence of (˜ q , χ) to the origin. Bearing in mind (41) and the boundedness of the state (˜ q, χ), also the regulation error y˜i (˜ qi ) = µi (qi ) − ri converges exponentially to zero. A number of comments are in order. –
The Lyapunov function used in the proof is given by the sum of the Lyapunov functions of the two unforced subsystems. The proof amounts to dominate the cross-terms in the time derivative of the Lyapunov functions via appropriate tuning of the parameter N.
–
The result shows that a sufficiently large N eventually semi-globally exponentially stabilizes the system. In the proof, a lower bound on N above which stability is achieved is provided. However, as it depends on the physical parameters of the system which are uncertain or completely unknown, this bound is of limited use in the actual gain design. In practice, the tuning of the gain N is carried out by a trial and error procedure (see Section IV).
–
Combining Propositions 3.1 and 3.2, one can observe that choosing K = −F˜ N the solution will converge to the prescribed set-point from any initial condition and that within a certain neighborhood of the set-point (which can be enlarged at will by increasing the gains N) the convergence will be exponential.
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IV. E XPERIMENTAL
AND
13
N UMERICAL R ESULTS
The closed loop system has been tested using experiments on a special laboratory setup and numerical simulations. The setup is the same one used and described in [5], where additional details can be found. It emulates a small scale district heating system. However, due to physical constraints imposed by the size of the setup, the dynamics are 5-10 times faster than what would be expected in a real system. A picture of the setup is shown in Fig. 2.
Fig. 2.
The laboratory setup used in experiments and simulations.
The laboratory setup consists of four end-users, and the diagram of the corresponding hydraulic network is shown in Fig. 4. Referring to the figure, the end-users in the system are comprised of the connections {c9 , c10 , c11 }, {c27 , c28 , c29 }, {c23 , c24 , c25 } and {c19 , c20 , c21 }, with the electronically controlled valves c10 , c28 , c24 , c20 modelling the pressure drop across the primary side of the heat exchangers at the end-users, and the valve c17 modelling the pressure drop across the secondary side of the heat exchanger at the heat source. It is straightforward to verify that the network satisfies Assumptions 2.1-2.3. Despite the setup have been designed to emulate the four end-user scenario, currently valves c4 and c14 are closed, the measured outputs are limited to signals y3 = dp3 and y4 = dp4 , which are the differential pressure measurements across the valves c24 and c20 respectively. The input is limited to Pumps 4, 5 and 6 (components c23 , c19 and c1 ). This scenario corresponds to the two end-user case. The experiments have been conducted for this scenario.
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A. Experimental Results The control signals are given as u3 =∆hp,1 + ∆hp,23
(46)
u4 =∆hp,1 + ∆hp,19 . The control signal ui is calculated by the proportional-integral control based on the measurement dpi and then distributed to the pumps in the system based on the following rule ∆hp,1 =0.7 min(u3, u4 ) ∆hp,23 =u3 − ∆hp,1
(47)
∆hp,19 =u4 − ∆hp,1 . In view of Propositions 3.1 and 3.2, any choice of K, N satisfying K = −F˜ N for some Hurwitz matrix F˜ guarantees global asymptotic and local exponential convergence. The gains used in the experiments are N = 2I2 and K = 1.5I2 . Observe that K = −F˜ N, with F˜ = −0.75I2 . These gains have been chosen via a trial and error procedure to guarantee a good response in terms of overshoot and settling time. As it is evident from Fig. 3, asymptotic output regulation is achieved for the closed loop system. Proposition 3.2 predicts that increasing the gain N would enlarge the region of exponential convergence of the solution. However, the fast response illustrated in Fig. 3 supports the idea that the set of initial conditions is already included in the region of exponential convergence with the choice N = 2I2 and K = 1.5I2 . B. Numerical Results The purpose of the simulation is to verify that the closed-loop system is able to handle the addition/removal of end-users while keeping the system on-line and still achieve asymptotic stability of the desired equilibrium point. This is a feature guaranteed by the decentralized nature of the controllers, and the fact that Proposition 3.1 is independent of the number of end-users in the system. Therefore, a scenario, where the end-user connections consisting of {c9 , c10 , c11 } and {c27 , c28 , c29 } in Fig. 4 are removed from and later re-inserted into the system, has been simulated. The functions λi (·) used in the simulations are given by λi (x) = pi |x|x, July 9, 2012
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PI test loop 4: Ref step 0.2→ 0.3→ 0.2, N4=2, K4=1.5 0.35
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Plot of the results obtained in the experiments on the laboratory setup, see Fig. 4. At the time the experiments were
performed, only end-user connections 3 and 4 were available for experiments. The top plots shows the pressure measurements dp3 and dp4 obtained at end-users 3 and 4. The dashed lines indicates the two reference values used to obtain a step response of the closed loop system. The bottom plots shows the corresponding control inputs u3 and u4 .
and the functions µi (·) are given by µi (x) = vi |x|x.
(49)
The parameters used in the simulations are: J11 = 0.8488, J12 = J21 = 0.3183, J22 = J44 = 0.7427, J13 = J23 = J32 = J31 = 0.2122, J33 = 0.6366, J14 = J24 = J34 = J43 = J42 = J41 = 0.1061, p2 = p3 = p6 = p7 = p12 = p13 = p15 = p16 = 0.0141, p8 = p11 = p22 = p25 = p26 = p29 = 0.4503, p18 = p21 = 0.6755, v10 = v28 = v24 = v20 = 0.005, v17 = 0.0013 and controller gain matrices N = 2I4 , K = 1.5I4 . To simulate the removal of the end-users 1 and 2 the valve parameters v10 and v28 has been changed to large values (0.005 + 5.25 · 102 ), thus reducing the flows through valves c10 and c28 to almost zero. Furthermore, some logic is introduced to the controllers to disable control signals u1 and u2 and reset their respective integrators while end-users 1 and 2 are disconnected from the system.
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Pump
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C 20
Pipeline
16
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Pressure diff. sensor
Pump 5: UPE 25−60
dp4
Pump 4: UPE 25−60
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dp3
Pump 1: UPE 25−60
dp2
Expansion tank
C 19 19 l
C 18 C1 Heat source
C2
Pump 6: Pipe, D: 20 mm, UPE 25−60 Length: 6 m
C 17
C 16 Pipe, D: 20 mm, Length: 6 m
Fig. 4.
Pipe, D: 10 mm, Length: 9 m
Pipe, D: 10 mm, Length: 9 m
C 21
C 23 C 22
C3
Pipe, D: 10 mm, Length: 6 m
C9 Pipe, D: 10 mm, Length: 6 m
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C 25 C4
C5
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C 27
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dp1
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C8
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C 11
C7 Pipe, D: 20 mm, Length: 6 m
C 13
C 12
Pipe, D: 20 mm, Length: 6 m
Pipe, D: 20 mm, Length: 6 m
The hydraulic network diagram for the laboratory setup in Fig. 2.
Furthermore, the control signals have been distributed to the pumps according to the rule ∆hp,1 =0.7 min(u1 , u2, u3 , u4 ) ∆hp,5 =0.7 min(u1 , u2) − ∆hp,1 ∆hp,9 =u1 − ∆hp,1 − ∆hp,5
(50)
∆hp,27 =u2 − ∆hp,1 − ∆hp,5 ∆hp,23 =u3 − ∆hp,1 ∆hp,19 =u4 − ∆hp,1 when all four end-users are present and to the rule (47) when only end-user 3 and 4 are present. The results of the simulations are shown in Fig. 5. In Fig. 5 it is seen that the closed loop system achieves asymptotic output regulation when all four end-users are present in the system as well as when only two of the end-users are present in the system. The qualitative difference between Fig. 3 and Fig. 5 (no overshoot and oscillations in simulations) is due to the fact that the the dynamics of the experimental setup are much faster than in a reallife hydraulic network. Since the actuators installed in the laboratory equipment are the same as in a real system (and hence much slower than the dynamics of the laboratory setup), there is a large delay in the control input which results in the difference between experimental results and numerical simulations.
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V. C ONCLUSION The problem of output regulation for large-scale hydraulic networks comprising a district heating system was addressed. The results show that global asymptotic and semi-global exponential output regulation is achievable using a set of decentralized proportional-integral control actions. Furthermore, as the former result is global and independent on the number of end-users in the system, it is concluded that the property of asymptotic output regulation is maintained if end-users are arbitrarily added to or removed from the system. The results were supported by experiments for a laboratory setup with two end-users and by numerical simulations of a four end-user system. The incorporation of positive constraints on the control signals is seen as a natural extension of the results presented here. Since the centrifugal pumps used in the network are only capable of delivering positive pressures, the explicit incorporation of this constraint in the stability analysis of the closed loop system will be necessary. This increases considerably the difficulty of the control problem and requires a deep independent study to be carried out in the future. ACKNOWLEDGMENT The authors would like to thank Carsten Kallesøe from Grundfos Management A/S for providing his helpful insights on the topic of the paper. R EFERENCES [1] F. Bruus, B. Bøhm, N.K. Vejen, J. Rasmussen, N. Bidstrup, K.P. Christensen, and H. Kristjansson (2004). EFP-2001 Supply of district heating to areas with low heat demand. Danish Energy Authority, Journal No. 1373/01-0035 (In Danish). [2] C.S. Kallesøe, “Simulation of a district heating system with a new network structure”, Technical Report, Grundfos Management A/S, 2007. [3] J. Stoustrup, “Plug & play control: Control technology towards new challenges,” European Journal of Control, vol. 15, no. 3-4, pp. 311-330, 2009. [4] C. De Persis, and C.S. Kallesøe, “Pressure regulation in nonlinear hydraulic networks by positive controls”, Proc. 10th European Control Conference, Budapest, Hungary, pp. 4102-4107, August 2009. [5] C. De Persis, and C.S. Kallesøe, “Pressure regulation in nonlinear hydraulic networks by positive and quantized control”, IEEE Transactions on Control Systems Technology, 19(6), 1371–1382, 2011. [6] C. De Persis, and C.S. Kallesøe, “Quantized controllers distributed over a network: An industrial case study”, Proc. 17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece, pp. 616-621, June 2009. [7] C. A. Desoer and E. S. Khu, Basic Circuit Theory, New York: Mc- Graw-Hill, 1969. [8] J.A. Roberson, and C.T. Crowe, Engineering fluid mechanics, fifth ed., Houghton Mifflin Company, 1993.
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[9] R. Ortega, A. Lor´ıa, P.J. Nicklasson, and H. Sira-Ram´ırez, Passivity-based Control of Euler-Lagrange Systems, SpringerVerlag, 1998. [10] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal nonlinear output regulation with adaptive internal model”, IEEE Transactions on Automatic Control, vol. 46, pp. 1178-1194, 2001. [11] J. Milnor, Morse Theory, Princeton University Press, 1973.
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Fig. 5. Result of a numerical simulation of the four end-user system in Fig. 4. The figure shows control inputs u1 , u2 , u3 , u4 , the controlled variable dp1 , dp2 , dp3 , dp4 , and the flow through valves c10 , c28 , c24 , c20 obtained with the proportional-integral feedback control. At time 100 s, the end-user connections consisting of {c9 , c10 , c11 } and {c27 , c28 , c29 } are removed from the system. At time 200 s the end-user connections are re-inserted into the system. The dashed line at 0.2 bar in the two middle plots indicates the reference value.
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Output Regulation of Large-Scale Hydraulic Networks Claudio De Persis, Tom Nørgaard Jensen, Romeo Ortega, and Rafał Wisniewski
Abstract The problem of output regulation for a class of hydraulic networks found in district heating systems is addressed. The results show that global asymptotic and semi-global exponential output regulation is achievable using a set of decentralized proportional-integral controllers. The fact that the result is global and independent of the number of end-users has the consequence that structural changes such as enduser addition and removal can be made in the network while maintaining the stability properties of the system. Furthermore, the decentralized nature of the control architecture eases the implementation of structural changes in the network.
Index Terms Nonlinear systems, Robust control, Decentralized control, Output regulation, Hydraulic networks
I. I NTRODUCTION The work presented here concerns the output regulation of a class of large-scale hydraulic networks found in district heating systems. The case study considers a new paradigm for the design of district heating systems, which has been proposed to reduce the diameter of the pipes This work is supported by The Danish Research Council for Technology and Production Sciences within the Plug and Play Process Control Research Program. C. De Persis is with ITM, Faculty of Mathematics and Natural Sciences, University of Groningen, 9747 AG Groningen, The Netherlands and Department of Computer, Control and Management Engineering A. Ruberti, Sapienza Universit`a di Roma, Via Ariosto 25, 00185 Roma, Italy. (e-mail:[email protected]). T.N. Jensen and R. Wisniewski are with Department of Electronic Systems, Aalborg University, Fredrik Bajers Vej 7C, 9220 Aalborg Denmark. (e-mail:{tnj,raf}@es.aau.dk). R. Ortega is with Laboratoire des Signaux et Syst`emes, CNRS-SUPELEC, 91192, Gif-sur-Yvette, France. (email:[email protected]). July 9, 2012
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in the network. By reducing the pipe diameter, it is possible to reduce the heat dispersion from the pipes and thereby reduce the energy losses in the system [2]. On the other hand, the reduced diameters induce increased pressure losses throughout the network which must be compensated by multiple pumps. Studies held that the multi-pump architecture is the technology which can compensate for the increased pressure losses while still achieving a substantial energy saving ([1]). The multi-pump architecture raises the question of how the pumps should be operated to control the network in appropriate way. The new paradigm also gives rise to a flexible network structure in which end-users can be added to or removed from the network. The case study is part of the ongoing research program Plug & Play Process Control [3] which considers automatic reconfiguration of the control system if components such as sensors, actuators or subsystems are added to or removed from a system. To fulfill the control objective, which is to keep the pressure across the so-called end-user valves at a constant reference, a set of proportional-integral control actions is proposed. The control actions are decentralized in the sense that the individual controllers use only locally available information, which is the pressure measurement at the end-user. The results show that it is possible to achieve global asymptotic and semi-global exponential output regulation using this control architecture. These results represent an interesting addition to [5], where semi-global practical stability was achieved by positive proportional controllers, and we believe they are instrumental for further developments which are briefly discussed in conclusions of the paper. In Section II, the model of the system is introduced along with the output regulation problem. The main results of the paper are presented in Section III. In Section IV, experimental results and numerical simulations of the closed loop system are presented. Finally, conclusions are given in Section V along with possible future research directions. Nomenclature: For a vector x ∈ Rn , xi denotes the ith element of x. Let M(n, m; R) denote the set of n × m matrices with real entries, and M(n; R) = M(n, n; R). For a square matrix A, A > 0 means that A is positive definite. For a square matrix A, A = diag(xi ) means that A has xi as entries on the main diagonal and zero elsewhere. For a matrix A, Aij will be used to denote the entry in the ith row and jth column of A. A matrix A is said to be Hurwitz if all eigenvalues of A have strictly negative real part. Throughout the paper, C 1 denotes a continuously differentiable map. A continuous map is said to be proper if the inverse image of July 9, 2012
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a compact set is compact. A function f : R → R is called monotonically increasing if it is order preserving, i.e., for all x and y such that x < y then f (x) < f (y). For a map f : Rn → Rm , let Df (·) denote the Jacobian matrix of f (·). II. S YSTEM M ODEL The system under consideration is the hydraulic network of a district heating system. The model has been derived in detail in [4]. To make the paper self-contained, the essentials of the model are briefly recalled below. The hydraulic network is the interconnection of several two-terminal components, namely valves, pipes and pumps. Each component is described by a constitutive equation given by ∆hk = Jk q˙k + λk (qk ) + µk (qk ) − ∆hp,k ,
(1)
where Jk > 0 is a parameter which is non-zero for pipe components and zero for other components; the function µk is non-zero for valve components and zero for other components; the function λk is non-zero for pipe components and zero for other components; the input ∆hp,k is non-zero for pump components and zero for other components. The functions λk , µk are C 1 , monotone increasing, zero at qk = 0 and proper. The precise expression of λk , µk is unknown. A graph G is associated with the hydraulic network, where the nodes are the terminals of the components and the m edges are the components. The graph satisfies the following: Assumption 2.1: [5] G is a connected graph. Let T be a spanning tree of G, that is a connected sub-graph of G which contains all the nodes of G but no cycles. The edges of G which are not in T are the chords of G. Denote by n the number of chords of G. The hydraulic networks considered in this paper have the following property: Assumption 2.2: [5] Each chord in G corresponds to a pipe. Moreover, the pipe is in series with a valve and a pump (see Fig. 1). By definition of T , adding an edge to T which is not in G, a cycle is obtained. These cycles are called the fundamental cycles of G. The hydraulic networks considered in this paper additionally satisfy the following: Assumption 2.3: [5] There exists one and only one component called the heat source. It corresponds to a valve of the network, and it lies in all the fundamental cycles. July 9, 2012
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Valve Pump
Pipe
qi
Remaining network
Fig. 1.
The series connection associated with each end-user [4].
The pipes, valves and pumps defined in Assumption 2.2 are referred to as end-user pipes, valves and pumps ([4]). Moreover, a series connection comprising an end-user pipe, valve and pump is referred to as an end-user. The flows through the end-user pipes form a set of independent flow variables, in the sense that they can be set independently without violating Kirchhoff’s law. The m edges of the graph G and the corresponding components are labeled via the integers 1, 2, . . . , m. Following [4], the edges corresponding to the end-user pipes are labeled via the integers 1, 2, . . . , n < m. The vector q ∈ Rn of all independent flows is taken as the state variable of the system. Denoted by B ∈ M(n, m; R) the fundamental loop matrix associated to the graph ([7], [4]), the m-dimensional vector x of all flows (dependent and independent) through the components in the system is given by x = B T q. The dynamic model of the hydraulic network is characterized in the following: Proposition 2.1: [5] Any hydraulic network satisfying Assumptions 2.1 and 2.2 admits the representation J q˙ = f (B T q) + u yi (qi ) = µi (qi ) , i = 1, . . . , n,
(2) (3)
where q ∈ Rn is the vector of independent flows; u ∈ Rn is a vector of independent inputs consisting of a linear combination of the delivered pump pressures; yi is the measured pressure drop across the ith end-user valve; J ∈ M(n; R), J > 0; f (·) is a C 1 vector field; µi (·) is the fundamental law of the ith end-user valve. The vector field f (·) can be written as f (x) = −B(λ(x) + µ(x)), ∀x ∈ Rm ,
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where λ(x) = [λ1 (x1 ), . . . , λm (xm )]T ; µ(x) = [µ1 (x1 ), . . . , µm (xm )]T and λi (·) is non-zero if and only if i is a pipe and µi (·) is non-zero if and only if i is a valve. The matrix J in (2) is given by J = BJ B T
(5)
where J = diag(J1 , . . . , Jm). A. Output Regulation Problem It is desired to regulate the pressure yi across the ith end-user valve to a given reference value ri with the use of a feedback controller having available only local information. The vector r = (r1 , . . . , rn ) of reference values takes values in a known compact set R: R = {r ∈ Rn | 0 < rm ≤ ri ≤ rM }.
(6)
For the purpose of asymptotic output regulation, a set of decentralized proportional-integral controllers is the focus of the work presented here. The controllers considered will be of the form ξ˙i = −Ki (yi (qi ) − ri )
(7)
ui = ξi − Ni (yi (qi ) − ri )
(8)
where Ki , Ni > 0 and i = 1, 2, . . . , n. III. S TABILITY
PROPERTIES OF THE CLOSED LOOP SYSTEM
In this section, the stability properties of the closed loop system will be examined. First, it is shown that global asymptotic output regulation can be proved. Secondly, semi-global exponential stability is shown. A. Global Asymptotic Stability In this subsection, global asymptotic stability of the desired equilibrium point of the closed loop system will be shown. To this end, first define the proportional gain matrix N = diag(Ni ). The following lemma will be instrumental in deriving the closed loop stability properties of the system.
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Lemma 3.1: Let the matrix G(q) ∈ M(n, R) be given by G(q) ≡ NDy(q) − Df (B T q),
(9)
then G(q) ≥ 0. Proof: Recall that x = B T q and f (x) = −B (λ(x) + µ(x))
(10)
where λ(x) = (λ1 (x1 ), . . . , λm (xm ))T and µ(x) = (µ1 (x1 ), . . . , µm (xm ))T . Then −Df (B T q) is given by −Df (B T q) = −
∂x ∂ f (x) ∂x ∂q
(11)
= BΛ(x)B T .
(12)
where Λ(x) = diag( ∂x∂ i (λi (xi ) + µi (xi ))). By Assumption 2.2, each chord in the graph G described by the network corresponds to a pipe in series with a user valve. Therefore, bearing in mind that we have rearranged the numbering of the components such that the first n components are the pipes in the chords of G, (12) can be rewritten as −Df (B T q) =
In F
Λ1 (x)
0
0 Λ2 (x)
In F
T
= Λ1 (x) + F Λ2 (x)F T ,
(13) (14)
where Λ1 (x) = diag( ∂x∂ i (λi (xi ))) for i = 1, . . . , n, Λ2 (x) = diag( ∂x∂ i (λi (xi ) + µi (xi ))) for i = n + 1, . . . , m, and F is a matrix with entries in {0, 1} such that B = (In F ) ([5]). Since λi (·) are monotonically increasing functions the matrix Λ1 (x) is positive semi definite for all x. Furthermore, since µi (·) are monotonically increasing functions, the matrix Λ2 (x) is positive semi definite for all x (recall that λi (·) is non-zero only for pipe components and µi (·) is non-zero only for valve components). Then it follows that −Df (B T q) is positive semi definite. The matrix Dy(q) is given by Dy(q) = diag(
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(15)
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Recall that yi (qi ) = µi (qi ), and µi (·) is a monotonically increasing function. As a consequence Dy(q) is positive semi definite. Since N is diagonal and positive definite and Dy(q) is a positive semi definite diagonal matrix, it follows that NDy(q) is positive semi definite. From the derivations above, it is concluded that NDy(q)−Df (B T q) is a positive semi definite matrix. Since the functions µi (·) are monotonically increasing and proper, they admit inverses µ−1 i (·). Now, let qi∗ ≡ µ−1 i (ri ),
(16)
that is: qi∗ is the flow through the ith end-user valve which produce the reference output. Furthermore, define q˜ ≡ q − q ∗ ,
(17)
y˜i (˜ qi ) ≡ µi (˜ qi + qi∗ ) − ri .
(18)
and the variable
The main result of the subsection will rely on the following Lemma, which has been derived in [5]: Lemma 3.2: The function y˜i (˜ qi ) is monotonically increasing and zero at q˜i = 0, and moreover y˜i (˜ qi )˜ qi > 0, ∀ q˜i 6= 0 .
(19)
Now, the main result of this subsection can be stated. Proposition 3.1: The point (˜ q T , q˙T ) = 0 is a globally asymptotically stable equilibrium point of the closed loop system given by (2), (3), (7) and (8). Proof: The closed loop system defined by (2), (3), (7) and (8) is J q˙ = f (B T q) + ξ − N y˜(˜ q) ξ˙ = −K y˜(˜ q)
(20) (21)
From (20) the following can be derived J q¨ = Df (B T q)q˙ + ξ˙ − NDy(q)q˙
(22)
J q¨ = −G(q)q˙ − K y˜(˜ q ).
(23)
which can be rewritten as
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In the above J > 0, G(q) ≥ 0 by Lemma 3.1 and K y˜(˜ q ) can be written as ∇W (˜ q ), with W (˜ q) given as W (˜ q) =
n X i=1
Ki
Z
q˜i
y˜i (s)ds,
(24)
0
and W (˜ q) > 0 by Lemma 3.2. Therefore, the structure of (23) is similar to that of a mechanical system in the standard Lagrangian form [9]. This motivates the choice of the Lyapunov function candidate V : R2n → R, which can be seen as an equivalent of the total energy function for a mechanical system: Z q˜i n X 1 Ki V (˜ q , q) ˙ = ˙ y˜i (s)ds + q˙T J q, 2 0 i=1
(25)
which is positive definite and radially unbounded. The first term in the definition of the function V (˜ q, q) ˙ in (25) can be interpreted as the equivalent of the potential energy function of a mechanical system described by (23); whereas, the second term can be interpreted as the kinetic energy function. Note, however, that the physical dimensions of the two terms mismatch in the case of the hydraulic network, and as such the Lyapunov function candidate does not have a direct physical interpretation. The time derivative of V (˜ q , q) ˙ is d V (˜ q , q) ˙ = q˙ T J q¨ + q˙T K y˜(˜ q) dt
(26)
= −q˙T G(q)q˙ ≤ 0
(27)
where the last inequality comes from Lemma 3.1. Using (9) and (14) the following bound is obtained n
X ∂ d V (˜ q , q) ˙ ≤− λi (qi )q˙i2 . dt ∂qi i=1 Recalling that
∂ λ (q ) ∂qi i i
> 0 for every qi 6= 0 and
∂ λ (q ) ∂qi i i
(28)
= 0 if and only if qi = 0, it is
concluded by the LaSalle invariance principle, that the closed loop system converges to the set Q given by Q = {(q, q) ˙ ∈ R2n | qi = 0 ∨ q˙i = 0}.
(29)
Since, qi = 0 implies q˙i = 0 which in turn implies q¨i = 0, from (23) it follows that y˜i (˜ qi ) = 0 is the largest invariant set of the closed loop system fulfilling
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d V dt
(˜ q , q) ˙ = 0. From this it follows
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that K y˜(˜ q) → 0 as t → ∞. Since y˜i (·) is monotonically increasing and zero in zero, it is concluded that q˜ → 0 as t → ∞. From (16) and (18) it can be seen that y˜i (˜ qi ) → 0 is equivalent to µi (qi ) → ri . That is, Proposition 3.1 shows that the desired equilibrium point of the closed loop system is globally asymptotically stable. Moreover, since the result holds for arbitrary values of control gains Ni > 0 and Ki > 0, asymptotic output regulation is guaranteed for arbitrary choice of gains. Since Proposition 3.1 is independent of the number n of end-users in the system, it follows that end-users can be added to or removed from the system while maintaining asymptotic output regulation. B. Semi-global Exponential Stability In this section, semi-global exponential output regulation of the closed loop system is shown. This result rests on the assumption that
∂ (µi (xi ))|xi=qi∗ ∂xi
6= 0. Requiring the variation of the
pressure drop with respect to the flow at the end-user valves to be non-zero at the working point is a realistic condition which is often satisfied in practice. Some preliminaries are instrumental in the analysis below. If q = q ∗ , the input c(q ∗ ) such that the output y equals the reference r is given by f (B T q ∗ ) + c(q ∗ ) = 0. The solution ξ ∗ to (7)-(8) which produces the input c(q ∗ ) when y˜(˜ q ) = 0 is ξ ∗ = −f (B T q ∗ ). The ability by (7)(8) to reproduce c(q ∗ ) constitutes the so-called internal model property. Perform the change of coordinates ξ˜ ≡ ξ + f (B T q ∗ ),
(30)
J q˜˙ =f˜(˜ q ) + ξ˜ − N y˜(˜ q)
(31)
so as to obtain
˙ ξ˜ = − K y˜(˜ q ),
(32)
where f˜(˜ q ) = f (B T (˜ q + q ∗ )) − f (B T q ∗ ). The variable q˜ measures the distance of the state q from the point where the regulation error vanishes to zero; whereas, the variable ξ˜ measures the distance of the controller state ξ from the variable ξ ∗ which provides the value of the control when the regulation error is zero.
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It is evident from (32) that the addition of the internal model introduces a sub-system which is not asymptotically stable. This is a well-known obstacle towards the solution of the regulation problem. A way to overcome this obstacle is to rely on the canonical parametrization of the internal model introduced in [10]. In what follows, we mimic the analysis in [10], although the special structure of the system allows us to proceed in a more straightforward way. Let F˜ ∈ M(n, R) be a Hurwitz matrix and define further the new coordinate ([10]) χ = ξ˜ + F˜ J q˜ which yields J q˜˙ =f˜(˜ q) + χ − F˜ J q˜ − N y˜(˜ q)
(33)
χ˙ = − K y˜(˜ q ) + F˜ f˜(˜ q ) + F˜ χ − F˜ 2 J q˜ − F˜ N y˜(˜ q ).
(34)
Lemma 3.3: [11] Let f : Rn → R be a C 1 function in a convex neighborhood U of 0 in Rn , with f (0) = 0. Then f (x1 , . . . , xn ) =
n X
xi gi (x1 , . . . , xn )
(35)
i=1
for some suitable C 1 functions gi : Rn → R defined in U, with gi (0) = By Lemma 3.3 the map f˜(˜ q) can be written as ˆ q )˜ f˜(˜ q ) = φ(˜ q
∂ f (0). ∂xi
(36)
ˆ q ) a matrix of continuously differentiable functions. In order to make (34) independent with φ(˜ of K, we set K = −F˜ N. Observe that to preserve the decentralized nature of the controller, K must be diagonal. This in turn requires F˜ to be diagonal as well. Also observe that F˜ Hurwitz and N positive definite guarantee that K is positive definite as well. The closed-loop system can be written more simply as χ˙ =F˜ χ − φ(a) (˜ q)˜ q
(37)
J q˜˙ =φ(b) (˜ q )˜ q + χ − N y˜(˜ q)
(38)
ˆ q ) + F˜ 2 J φ(a) (˜ q ) = −F˜ φ(˜
(39)
ˆ q ) − F˜ J. φ(b) (˜ q ) = φ(˜
(40)
where
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Again by Lemma 3.3, the relation between q˜i and y˜i can be written as y˜i (˜ qi ) = gi (˜ qi )˜ qi . Observe that, as
∂ (˜ y (˜ q ))|q˜i =0 ∂ q˜i i i
=
∂ (µi (qi ))|qi=qi∗ ∂qi
(41)
6= 0, and y˜i (˜ qi ) = 0 if and only if q˜i = 0,
for every q˜i the function gi (·) on the right-hand side is positive. Moreover, if q˜i range over a compact set, then by continuity of gi (·) there exists m > 0 such that gi (˜ qi ) ≥ m > 0 ,
i = 1, 2, . . . , n.
(42)
Then the main result of this subsection can be stated. Proposition 3.2: Given system (2), (3), a vector r of reference values satisfying (6), and a compact set of initial conditions Q ⊂ Rn × Rn , there exist diagonal positive definite matrices N and K in (7), (8), such that every trajectory (q(t), ξ(t)) of the closed-loop system (2), (3), (7), (8) with initial condition in Q is bounded and satisfies limt→+∞ yi (t) = ri for i = 1, 2, . . . , n. Proof: Let P > 0 be such that F˜ T P + P F˜ = −2I and consider the Lyapunov function candidate V : R2n → R given by 1 1 V (χ, q˜) = χT P χ + q˜T J q˜ 2 2 which is positive definite and radially unbounded. Compute the time derivative along the trajectories of the system, to obtain d V (χ, q˜) =χT P F˜ χ − χT P φ(a) (˜ q)˜ q+ dt + q˜T φ(b) (˜ q)˜ q + q˜T χ − q˜T N y˜(˜ q) .
(43)
Let S be a level set of V (χ, q˜) containing the set of initial conditions of the system. Bearing in mind (42), the time derivative of V (χ, q˜) can be written in compact form as d V (χ, q˜) ≤ − |χ|2 + |χ|||P φ(a)(˜ q )|||˜ q|+ dt + |˜ q |||φ(b) (˜ q )|||˜ q| + |˜ q ||χ| − m˜ q T N q˜ .
(44)
Let Θ be a positive constant such that max{||P φ(a) (˜ q )||, ||φ(b)(˜ q )|| | q˜ ∈ S} ≤ Θ, which exists by continuity of φ(a) (·) and φ(b) (·). Then |χ|2 d V (χ, q˜) ≤ − + (Θ2 + Θ + 1)|˜ q|2 − m˜ q T N q˜ . dt 2
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˜ , with Ni ’s the diagonal entries of N and N ˜ to design, so that Let N1 = . . . = Nn = m−1 N |χ|2 d ˜ q |2 . V (χ, q˜) ≤ − + (Θ2 + Θ + 1 − N)|˜ dt 2 ˜ ∗ = Θ2 + Θ + 1 + 1 σM (J) , where σM (·) denotes the maximum eigenvalue of a (symmetric) Set N 2 σM (P ) ˜ ˜∗ matrix. Then, for all N ≥ N 1 d V (χ, q˜) ≤ − V (χ, q˜). dt σM (P )
(45)
This shows that the trajectories (χ(t), q˜(t)) of the system are bounded and converge to the origin. Bearing in mind the definition of (χ, q˜), it also shows that (q(t), ξ(t)) are bounded and q(t) → q ∗ , ξ(t) → −f (B T q ∗ ) as t → ∞. By continuity of µ(·), it is concluded that µ(q(t)) → µ(q ∗ ) = r as t → ∞, i.e. the thesis. Remark 3.1: In view of the quadratic nature of the Lyapunov function V (χ, q˜) the proof actually shows exponential convergence of (˜ q , χ) to the origin. Bearing in mind (41) and the boundedness of the state (˜ q, χ), also the regulation error y˜i (˜ qi ) = µi (qi ) − ri converges exponentially to zero. A number of comments are in order. –
The Lyapunov function used in the proof is given by the sum of the Lyapunov functions of the two unforced subsystems. The proof amounts to dominate the cross-terms in the time derivative of the Lyapunov functions via appropriate tuning of the parameter N.
–
The result shows that a sufficiently large N eventually semi-globally exponentially stabilizes the system. In the proof, a lower bound on N above which stability is achieved is provided. However, as it depends on the physical parameters of the system which are uncertain or completely unknown, this bound is of limited use in the actual gain design. In practice, the tuning of the gain N is carried out by a trial and error procedure (see Section IV).
–
Combining Propositions 3.1 and 3.2, one can observe that choosing K = −F˜ N the solution will converge to the prescribed set-point from any initial condition and that within a certain neighborhood of the set-point (which can be enlarged at will by increasing the gains N) the convergence will be exponential.
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IV. E XPERIMENTAL
AND
13
N UMERICAL R ESULTS
The closed loop system has been tested using experiments on a special laboratory setup and numerical simulations. The setup is the same one used and described in [5], where additional details can be found. It emulates a small scale district heating system. However, due to physical constraints imposed by the size of the setup, the dynamics are 5-10 times faster than what would be expected in a real system. A picture of the setup is shown in Fig. 2.
Fig. 2.
The laboratory setup used in experiments and simulations.
The laboratory setup consists of four end-users, and the diagram of the corresponding hydraulic network is shown in Fig. 4. Referring to the figure, the end-users in the system are comprised of the connections {c9 , c10 , c11 }, {c27 , c28 , c29 }, {c23 , c24 , c25 } and {c19 , c20 , c21 }, with the electronically controlled valves c10 , c28 , c24 , c20 modelling the pressure drop across the primary side of the heat exchangers at the end-users, and the valve c17 modelling the pressure drop across the secondary side of the heat exchanger at the heat source. It is straightforward to verify that the network satisfies Assumptions 2.1-2.3. Despite the setup have been designed to emulate the four end-user scenario, currently valves c4 and c14 are closed, the measured outputs are limited to signals y3 = dp3 and y4 = dp4 , which are the differential pressure measurements across the valves c24 and c20 respectively. The input is limited to Pumps 4, 5 and 6 (components c23 , c19 and c1 ). This scenario corresponds to the two end-user case. The experiments have been conducted for this scenario.
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A. Experimental Results The control signals are given as u3 =∆hp,1 + ∆hp,23
(46)
u4 =∆hp,1 + ∆hp,19 . The control signal ui is calculated by the proportional-integral control based on the measurement dpi and then distributed to the pumps in the system based on the following rule ∆hp,1 =0.7 min(u3, u4 ) ∆hp,23 =u3 − ∆hp,1
(47)
∆hp,19 =u4 − ∆hp,1 . In view of Propositions 3.1 and 3.2, any choice of K, N satisfying K = −F˜ N for some Hurwitz matrix F˜ guarantees global asymptotic and local exponential convergence. The gains used in the experiments are N = 2I2 and K = 1.5I2 . Observe that K = −F˜ N, with F˜ = −0.75I2 . These gains have been chosen via a trial and error procedure to guarantee a good response in terms of overshoot and settling time. As it is evident from Fig. 3, asymptotic output regulation is achieved for the closed loop system. Proposition 3.2 predicts that increasing the gain N would enlarge the region of exponential convergence of the solution. However, the fast response illustrated in Fig. 3 supports the idea that the set of initial conditions is already included in the region of exponential convergence with the choice N = 2I2 and K = 1.5I2 . B. Numerical Results The purpose of the simulation is to verify that the closed-loop system is able to handle the addition/removal of end-users while keeping the system on-line and still achieve asymptotic stability of the desired equilibrium point. This is a feature guaranteed by the decentralized nature of the controllers, and the fact that Proposition 3.1 is independent of the number of end-users in the system. Therefore, a scenario, where the end-user connections consisting of {c9 , c10 , c11 } and {c27 , c28 , c29 } in Fig. 4 are removed from and later re-inserted into the system, has been simulated. The functions λi (·) used in the simulations are given by λi (x) = pi |x|x, July 9, 2012
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PI test loop 4: Ref step 0.2→ 0.3→ 0.2, N4=2, K4=1.5 0.35
0.3
0.3
dp (bar)
0.25
4
dp3 (bar)
PI test loop 3: Ref step 0.2→ 0.3→ 0.2, N3=2, K3=1.5 0.35
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Fig. 3.
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50
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time (s)
u (bar)
u3 (bar)
time (s)
40
time (s)
Plot of the results obtained in the experiments on the laboratory setup, see Fig. 4. At the time the experiments were
performed, only end-user connections 3 and 4 were available for experiments. The top plots shows the pressure measurements dp3 and dp4 obtained at end-users 3 and 4. The dashed lines indicates the two reference values used to obtain a step response of the closed loop system. The bottom plots shows the corresponding control inputs u3 and u4 .
and the functions µi (·) are given by µi (x) = vi |x|x.
(49)
The parameters used in the simulations are: J11 = 0.8488, J12 = J21 = 0.3183, J22 = J44 = 0.7427, J13 = J23 = J32 = J31 = 0.2122, J33 = 0.6366, J14 = J24 = J34 = J43 = J42 = J41 = 0.1061, p2 = p3 = p6 = p7 = p12 = p13 = p15 = p16 = 0.0141, p8 = p11 = p22 = p25 = p26 = p29 = 0.4503, p18 = p21 = 0.6755, v10 = v28 = v24 = v20 = 0.005, v17 = 0.0013 and controller gain matrices N = 2I4 , K = 1.5I4 . To simulate the removal of the end-users 1 and 2 the valve parameters v10 and v28 has been changed to large values (0.005 + 5.25 · 102 ), thus reducing the flows through valves c10 and c28 to almost zero. Furthermore, some logic is introduced to the controllers to disable control signals u1 and u2 and reset their respective integrators while end-users 1 and 2 are disconnected from the system.
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Pump
End−user 4
End−user 3
End−user 2
Elec. control valve
Elec. control valve
Elec. control valve
C 20
Pipeline
16
C 24
End−user 1 Elec. control valve
C 10
C 28
Valve dp
Pressure diff. sensor
Pump 5: UPE 25−60
dp4
Pump 4: UPE 25−60
Pump 2: UPE 25−60
dp3
Pump 1: UPE 25−60
dp2
Expansion tank
C 19 19 l
C 18 C1 Heat source
C2
Pump 6: Pipe, D: 20 mm, UPE 25−60 Length: 6 m
C 17
C 16 Pipe, D: 20 mm, Length: 6 m
Fig. 4.
Pipe, D: 10 mm, Length: 9 m
Pipe, D: 10 mm, Length: 9 m
C 21
C 23 C 22
C3
Pipe, D: 10 mm, Length: 6 m
C9 Pipe, D: 10 mm, Length: 6 m
C 26
C 25 C4
C5
C6
Pump 3: Pipe, D: 20 mm, UPE 25−60 Length: 6 m
Pipe, D: 20 mm, Length: 6 m
C 15
C 27
Pipe, D: 10 mm, Length: 6 m
C 14
Pipe, D: 20 mm, Length: 6 m
dp1
Pipe, D: 10 mm, Length: 6 m
C 29
Pipe, D: 10 mm, Length: 6 m
C8
Pipe, D: 10 mm, Length: 6 m
C 11
C7 Pipe, D: 20 mm, Length: 6 m
C 13
C 12
Pipe, D: 20 mm, Length: 6 m
Pipe, D: 20 mm, Length: 6 m
The hydraulic network diagram for the laboratory setup in Fig. 2.
Furthermore, the control signals have been distributed to the pumps according to the rule ∆hp,1 =0.7 min(u1 , u2, u3 , u4 ) ∆hp,5 =0.7 min(u1 , u2) − ∆hp,1 ∆hp,9 =u1 − ∆hp,1 − ∆hp,5
(50)
∆hp,27 =u2 − ∆hp,1 − ∆hp,5 ∆hp,23 =u3 − ∆hp,1 ∆hp,19 =u4 − ∆hp,1 when all four end-users are present and to the rule (47) when only end-user 3 and 4 are present. The results of the simulations are shown in Fig. 5. In Fig. 5 it is seen that the closed loop system achieves asymptotic output regulation when all four end-users are present in the system as well as when only two of the end-users are present in the system. The qualitative difference between Fig. 3 and Fig. 5 (no overshoot and oscillations in simulations) is due to the fact that the the dynamics of the experimental setup are much faster than in a reallife hydraulic network. Since the actuators installed in the laboratory equipment are the same as in a real system (and hence much slower than the dynamics of the laboratory setup), there is a large delay in the control input which results in the difference between experimental results and numerical simulations.
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V. C ONCLUSION The problem of output regulation for large-scale hydraulic networks comprising a district heating system was addressed. The results show that global asymptotic and semi-global exponential output regulation is achievable using a set of decentralized proportional-integral control actions. Furthermore, as the former result is global and independent on the number of end-users in the system, it is concluded that the property of asymptotic output regulation is maintained if end-users are arbitrarily added to or removed from the system. The results were supported by experiments for a laboratory setup with two end-users and by numerical simulations of a four end-user system. The incorporation of positive constraints on the control signals is seen as a natural extension of the results presented here. Since the centrifugal pumps used in the network are only capable of delivering positive pressures, the explicit incorporation of this constraint in the stability analysis of the closed loop system will be necessary. This increases considerably the difficulty of the control problem and requires a deep independent study to be carried out in the future. ACKNOWLEDGMENT The authors would like to thank Carsten Kallesøe from Grundfos Management A/S for providing his helpful insights on the topic of the paper. R EFERENCES [1] F. Bruus, B. Bøhm, N.K. Vejen, J. Rasmussen, N. Bidstrup, K.P. Christensen, and H. Kristjansson (2004). EFP-2001 Supply of district heating to areas with low heat demand. Danish Energy Authority, Journal No. 1373/01-0035 (In Danish). [2] C.S. Kallesøe, “Simulation of a district heating system with a new network structure”, Technical Report, Grundfos Management A/S, 2007. [3] J. Stoustrup, “Plug & play control: Control technology towards new challenges,” European Journal of Control, vol. 15, no. 3-4, pp. 311-330, 2009. [4] C. De Persis, and C.S. Kallesøe, “Pressure regulation in nonlinear hydraulic networks by positive controls”, Proc. 10th European Control Conference, Budapest, Hungary, pp. 4102-4107, August 2009. [5] C. De Persis, and C.S. Kallesøe, “Pressure regulation in nonlinear hydraulic networks by positive and quantized control”, IEEE Transactions on Control Systems Technology, 19(6), 1371–1382, 2011. [6] C. De Persis, and C.S. Kallesøe, “Quantized controllers distributed over a network: An industrial case study”, Proc. 17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece, pp. 616-621, June 2009. [7] C. A. Desoer and E. S. Khu, Basic Circuit Theory, New York: Mc- Graw-Hill, 1969. [8] J.A. Roberson, and C.T. Crowe, Engineering fluid mechanics, fifth ed., Houghton Mifflin Company, 1993.
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[9] R. Ortega, A. Lor´ıa, P.J. Nicklasson, and H. Sira-Ram´ırez, Passivity-based Control of Euler-Lagrange Systems, SpringerVerlag, 1998. [10] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal nonlinear output regulation with adaptive internal model”, IEEE Transactions on Automatic Control, vol. 46, pp. 1178-1194, 2001. [11] J. Milnor, Morse Theory, Princeton University Press, 1973.
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0.9
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Fig. 5. Result of a numerical simulation of the four end-user system in Fig. 4. The figure shows control inputs u1 , u2 , u3 , u4 , the controlled variable dp1 , dp2 , dp3 , dp4 , and the flow through valves c10 , c28 , c24 , c20 obtained with the proportional-integral feedback control. At time 100 s, the end-user connections consisting of {c9 , c10 , c11 } and {c27 , c28 , c29 } are removed from the system. At time 200 s the end-user connections are re-inserted into the system. The dashed line at 0.2 bar in the two middle plots indicates the reference value.
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