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sensor scheduling framework for distributed parameter processes
Computers and Chemical Engineering 29 (2005) 867–876
A new integrated output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes Michael A. Demetrioua,∗ , Nikolaos Kazantzisb a
Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA
b
Available online 11 November 2004
Abstract The present study proposes a new methodological framework that allows the comprehensive integration of an output feedback controller synthesis method with a collocated actuator and sensor scheduling policy. The proposed actuator scheduling policy is algorithmically realized through a simple closed-loop performance requirement that forces the actuator to move close to a location, where the maximum deviation of the process state from the reference design steady state occurs in the presence of spatiotemporally varying disturbances. Using a collocated set of sensors and actuators, simple local output feedback controllers are designed that are capable of achieving process regulation at the design steady state conditions. Within the proposed framework, only information emanating from the sensor placed at the location where the actuator has moved to is used for control purposes and in accordance to the aforementioned scheduling policy. Consequently, in the proposed approach, the use of minimal information and computational requirements for generating the requisite local control actions is ensured, while attaining the overall process control objectives. Practical implementation aspects of the proposed method are discussed, and its performance is evaluated in an illustrative case study through simulations. © 2004 Elsevier Ltd. All rights reserved. Keywords: Distributed parameter systems; Actuator scheduling; Actuator and sensor collocation; Supervisory control
1. Introduction Increasingly stringent performance requirements imposed on the design of control and supervision systems for transport processes have spawned considerable research effort over the recent years (Christofides, 2001). Technologically important transport processes and the associated phenomena dominate almost all emerging fields such as nanotechnology, materials processing and biotechnology, and as a result, they offer ample motivation for the study of a wide array of interesting operational problems associated with the control, diagnostics and intelligent supervision of spa∗
Corresponding author. E-mail addresses: [email protected] (M.A. Demetriou), [email protected] (N. Kazantzis). 0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.09.017
tially extended processes, as well as production systems (Christofides, 2001; Garcia-Osorio & Ydstie, 2004). One notable and quite important problem pertains to the optimal placement and scheduling activities of sensors and actuators such that a set of prespecified performance criteria and requirements are met (Alonso, Kevrekidis, Banga, & Frouzakis, 2004; Antoniades & Christofides, 2000, 2001; Kumar & Seinfeld, 1978; van de Wal & de Jager, 2001). It should be pointed out, that traditionally the optimal sensor and actuator placement problem was addressed on the basis of open-loop considerations with the emphasis being placed on enhanced controllability and observability requirements (Christofides, 2001; Curtain & Zwart, 1995; Kumar & Seinfeld, 1978). However, a group of researchers recognized the need for the development of a framework, within which the controller synthesis and the actuator/sensor placement problem can be
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addressed in a unified manner (Antoniades & Christofides, 2000, 2002; Kumar & Seinfeld, 1978; Demetriou, 1999; van de Wal & de Jager, 2001), with very interesting results reported in (Antoniades & Christofides, 2001) on an output feedback control strategy implemented through an optimally placed network of actuators and sensors. Furthermore, in recent studies and within the aforementioned methodological framework, robustness issues with respect to unmodeled process dynamics as well as time-varying disturbances were also addressed (Antoniades & Christofides, 2002; Armaou & Christofides, 2001; Christofides, 1998; Christofides & Baker, 1999). Even though noticeably enhanced performance was achieved within the above methodological framework, the latter resulted in actuating and sensing devices permanently located at fixed positions along the spatial domain of the process or the host structure of interest. Recently, a moving actuator and scheduling policy has been proposed, that offers the operational advantage of successfully compensating for the effect of spatiotemporally varying disturbances, since actuators moving closer to the disturbance can exercise higher control authority, and thus, ensure enhanced closed-loop performance (Demetriou & Kazantzis, 2004a,b). In the above approaches, the controller synthesis problem was conceptually “decoupled” and separately addressed from the actuator placement and scheduling/activation one, under the assumption that the local state feedback controllers designed had access to measurements of the full process state vector (Demetriou & Kazantzis, 2004a,b). The present research study aims at the development of a systematic framework that allows the attainment of the following objectives: (i) the comprehensive supervisory integration of an output feedback controller synthesis method with a collocated actuator and sensor scheduling policy, that rely only on the availability of local process measurements and actuation functions through an optimally placed network of sensors and actuators; (ii) the use of minimal information and computational requirements for generating the requisite local control actions, while attaining the overall process control objective which is the successful process regulation (or equivalently the fixed set-point tracking) in the presence of spatiotemporally varying disturbances. In particular, within the proposed framework, it is assumed that the process under consideration has either multiple actuators and the desirable arrangement is to activate only one of them during a given time interval while all remaining ones are kept dormant, or a single actuator capable of moving at a priori selected positions (in order to reduce input/energy requirements, simplify the local controller design, as well as cope more effectively with spatiotemporally varying disturbances (Demetriou & Kazantzis, 2004a,b)). The proposed actuator activation and scheduling policy is algorithmically realized through a simple closed-loop performance require-
ment that forces the actuator to move close to a location where it is most needed, namely to a location where the maximum deviation/excursion of the process state from the reference design steady state occurs in the presence of spatiotemporal disturbances. Please notice, that the above maximum state deviation criterion serves as a concrete quantitative measure that is parameterized by the actuator location. Therefore, as time progresses, one can make explicit use of the above quantitative criterion, and identify the location at which the actuator should move to and exercise its control authority over a certain time interval/window. As a result, an actuator scheduling procedure emerges that can be algorithmically implemented over a sequence of time-windows, within which the actuator remains active at a location identified through the maximum state deviation decision-making criterion mentioned above. Using a collocated network of sensors and actuators, simple local output feedback controllers are designed that are capable of achieving process regulation at the nominal design steady state conditions. It should be pointed out, that even though all the available sensors which are spatially distributed provide measurement signals and valuable information for process monitoring purposes, only information emanating from the sensor placed at the location where the actuator has moved to (and become active) is used for control purposes, in accordance to the aforementioned scheduling policy. Please notice, that in such a supervisory and decentralized manner, a process control strategy is proposed that is endowed with the characteristics of a hybrid (switching) system due to the underlying simultaneous evolution of the process state in the continuous-time domain and the discrete-event nature of the proposed actuator activation and scheduling policy. Furthermore, the proposed framework introduces time-scale multiplicity associated with the latent process dynamics and the discrete-event dynamics that corresponds to the supervisory maximum state deviation decision-making and the proposed actuator scheduling policy. Finally, in light of the above remarks, it should be emphasized, that the proposed approach introduces only minimal computational requirements for a computer-based implementation of its process control strategy. This is certainly a desirable design feature for decentralized supervisory process control systems, that ought to satisfactorily cope with process dynamic components that evolve in both the continuous, as well as the discrete-time domain (Garcia-Osorio & Ydstie, 2004). The present paper is organized as follows: In Section 2 a few mathematical preliminaries are presented, as well as the problem formulation. Section 3 contains the paper’s main results associated with the proposed output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes. In Section 4 implementation aspects of the proposed approach and method are discussed, and its performance is evaluated through simulation studies performed in an illustrative example that is presented in Section 5. Finally, a few concluding remarks are provided in Section 6.
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2. Mathematical preliminaries—problem formulation The transport processes under consideration are modelled through the following 1D controlled diffusion parabolic PDE ∂ ∂ ∂ x(t, ξ) = κ(ξ) x(t, ξ) + b(ξ)u(t) + d(ξ)w(t), ∂t ∂ξ ∂ξ (2.1) accompanied by Dirichlet boundary conditions x(t, 0) = 0 = x(t, ), and initial conditions x(0, ξ) ∈ L2 (0, ), where x(t, ξ) denotes the state, ξ ∈ Ω = [0, ] ⊂ R is the spatial coordinate, t ∈ [t0 , ∞[ is the time variable, u(t) denotes the control signal, b(ξ) denotes the spatial distribution of the actuating device, w(t) the unknown exogenous input signal (disturbances) and d(ξ) the spatial distribution of the disturbance. The thermal diffusivity is denoted by κ(ξ) > 0, ξ ∈ Ω (Friedman 1964). The state space in this case is X = L2 (0, ) endowed with the standard L2 inner product and norm which are denoted by ·, · X and · X , respectively (Robinson, 2001). The control objective is to choose the signal u ∈ L2 ([t0 , ∞)[; R) so that regulation of the state x(t, ξ) to zero (or to some equilibrium state xe (ξ)) is achieved while a certain cost functional, which penalizes both the regulation error as well as the total energy or control effort, is minimized. It is assumed that the spatial distribution of the actuating devices is spanned over a portion of the process spatial domain Ω centered at a location ξa and is given by 1 if ξ − ≤ ξ ≤ ξ + a a b(ξ) = χ[ξa −,ξa +] (ξ) 2 . = 0 otherwise It should be noted that the above approximation for b(ξ) avoids any regularity problems that may arise due to the unbounded nature of a pointwise (in space) actuator distribution (i.e. a spatial delta function) and also guarantees √ that b ∈ L2 (0, ) with norm 1/ 2. Furthermore, one must ensure that the location ξa of the actuator is such that approximate controllability of (2.1) is attained (Curtain & Zwart, 1995). While it is mathematically convenient to consider b(ξ) = δ(ξ − ξa ), with δ(ξ) denoting the spatial delta function, for numerical implementation purposes one uses δ(ξ − ξa ) ≈ χ[ξa −,ξa +] (ξ), where χ[ξa −,ξa +] (ξ) denotes the characteristic function over the interval [ξa − , ξa + ]. The above system may be placed in an abstract setting written as an evolution system (Dautray & Lions, 2000; Lions & Magenes, 1972) x˙ (t) = Ax(t) + Bu(t) + Dw(t), (2.2) in the state space X. Under the above representation, the system’s second order (strongly) elliptic operator is given by (Dautray & Lions, 2000) d d Aφ = κ(ξ) φ, dξ dξ
φ ∈ Dom(A),
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with domain Dom(A) = H 2 (0, ) ∩ H01 (0, ) = {ψ ∈ 2 L (0, ) | ψ, ψ are abs. continuous , ψ ∈ L2 (0, ) and ψ (0) = 0 = ψ( )} and the input operator by Bu(t) = b(ξ)u(t),
B ∈ L(R, X).
The disturbance operator D is given via Dw(t) = d(ξ)w(t),
D ∈ L(R, X).
One way to address the control objective is to employ a single actuator, preferably placed in some “optimal” location, and subsequently design a robust controller (e.g. H∞ ) that would provide adequate levels of robustness with respect to certain classes of disturbances (Keulen, 1993). As mentioned in the introductory Section 1, in order to improve the ability of the system to compensate for spatiotemporal disturbance variations, i.e. considering Dφ = d(t, ξ)φ, a moving actuator scheme is proposed, in which the actuator is capable of moving at preselected positions in the spatial domain, reside for a certain time interval, and then move to another location according to the rules of an intelligent guidance and scheduling scheme that is based on closed-loop performance criteria. A meaningful criterion for a process regulation problem in the presence of spatiotemporally varying disturbances would be to force the actuator to move to a location at which the process state exhibits the maximum deviation/excursion from the nominal design steady state conditions (equilibrium). Furthermore, due to hardware considerations, the actuator may only move to a finite number of locations, and thus the guidance and scheduling policy simplifies to that of moving the actuating device at these preselected candidate positions (please see Fig. 1). Therefore, the primary process control objectives are: (i) to find the finite number of optimal (with respect to certain optimality measure) positions that the actuator can reside at during a given time interval, (ii) the actuator guidance and scheduling policy, and (iii) the control signal that must be supplied to the actuating device by the controller while residing at a given location. The attainment of the above objectives represents the focus of the next section.
3. Main results 3.1. Actuator placement A meaningful way to address the optimality of the finite locations that the actuating device will be residing at during a given time interval [tk , tk + t[, would be to enhance the controllability properties of the system. Please notice, that the input operator B(·) is parameterized by the spatial locations θ ∈ Θ in order to emphasize the dependence of the input operator on the actuator location. The set of candidate actuator
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locations Θ is defined via Θ = {θ ∈ Ω : (A, B(θ)) is approx. controllable}. Remark 1. The fact that the operator A is a Riesz-spectral operator with simple eigenvalues {λn , n ≥ 1} and corresponding eigenvectors {ϕn , n ≥ 1} admitting the represen tation Az = ∞ n=1 λn zψn ϕn , with {ψn , n ≥ 1} being the eigenvectors of A∗ , along with the definition of the input operator Bu = bu, b ∈ X, implies that the test of approximate controllability reduces to the requirement: bϕn = 0 for all n, (Curtain & Zwart, 1995). In the event that the thermal diffusivity is spatially invariant, one obtains: ϕn = √ (2/ ) sin(nπξ/ ), and thus the controllability test reduces to: sin(nπξa / ) = 0, n ≥ 1. In this case, the set of candidate locations is given by
nπθ Θ = θ ∈ Ω : sin
= 0, n ≥ 1 .
Using purely open-loop (enhanced controllability) measures, one may choose the “optimal” actuator locations θ from the above set Θ that maximize the bound α = α(θ) in the following inequality: Wc (θ)φ, φ X ≥ α φ 2X ,
φ ∈ X,
(3.1)
where Wc (θ) denotes the θ-parameterized controllability Gramian defined as follows Wc (θ)φ1 , A∗ φ2 X + A∗ φ1 , Wc (θ)φ2 X = −B∗ (θ)φ1 , B∗ (θ)φ2 X ,
θ ∈ Θ,
(3.2)
for φ1 , φ2 ∈ D(A∗ ). Equivalently stated, the “optimal” actuator locations are given by θ opt = arg sup α(θ), θ∈Θ
where α(θ) is given via (3.1) and (3.2). Please notice, that while there is a continuum of locations that provide controllability, one can only consider a finite number m of these candidate positions. Thus, one must consider an m-dimensional subset Θm = {θ1 , . . . , θm } of Θ, with each location θi being one of the m largest values of α(θ), i.e. choose Θm to contain the first m maxima of Θ. 3.2. Actuator guidance and scheduling policy Once the m actuator locations are found, a guidance and scheduling policy must be developed that would indicate to the moving actuating device where to move to and stay for the duration of a time interval: [tk , tk + t[. As mentioned earlier, it would be meaningful to move the actuator in a location that is most needed; namely to a location where the maximum deviation of the process state from its equilibrium xe (θ)(≡0) occurs. Indeed, the measure maxθ∈Ω |x(t, θ) − xe (θ)| would provide the location in the spatial domain associated with the
Fig. 1. Depiction of moving actuator scheme.
above largest state deviation criterion. Furthermore, since it is desired to place the actuating device at points that render the system controllable (i.e. in Θ) as well, one then restricts the search within the set of the controllable points. Thus, one may consider the following criterion: θ opt = arg max |x(t, θ) − xe (θ)|, θ∈Θ
t ∈ [tk , tk + t[,
(3.3)
Alternatively, one may consider a quadratic performance criterion: θ opt = arg maxθ∈Θ |x(t, θ) − xe (θ)|2 . Moreover, since the actuator can only reside at the a priori selected locations θ ∈ Θm , one considers θ opt = arg maxm |x(t, θ) − xe (θ)|2 . θ∈Θ
(3.4)
While the above choice conforms with intuitively appealing performance requirements, it can still lead to a situation in which the maximum deviation is exactly halfway between two adjacent locations, as depicted in Fig. 2a. Another possibility is to have two identical maxima occurring at two different locations θi and θj , as seen in Fig. 2b. To differentiate between two different actuator positions, one may then include the effects of the control effort associated with each of the candidate positions to arrive at
θ opt = arg maxm |x(t, θ) − xe (θ)|2 + |u(t, θ)|2 , (3.5) θ∈Θ
where u(t, θ) is the control signal supplied to the actuating device at a position θ ∈ Θm . While full state measurements
Fig. 2. Placement scenarios; (a) maximum of |x(t, ξ)| is between two neighboring elements θi and θj , (b) two equal maxima of |x(t, ξ)| located at θi and θj .
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Fig. 3. Spatial distribution of actuator placement measures; (a) using the controllability Gramian: AW + WAT = −B(ξ)BT (ξ), (b) using the H∞ norm of I(s − (A − B(ξ)C(ξ)))−1 D.
might be desirable, and in some cases feasible,1 in most applications one seldom has access to the entire state x(t, ξ). Instead, state measurements are available at discrete points in the domain (e.g. using thermocouples for temperature measurements) or at the boundary. For the system under consideration, it is assumed that one has m available (pointwise) measurements given by y1 (t) x(t, ξ1 ) C(ξ1 )x(t) . .. .. = y(t, ξ) = . . .. = ym (t) x(t, ξm ) C(ξm )x(t) x(t, ξ)δ(ξ − ξ1 ) dξ 0 .. , = (3.6) . x(t, ξ)δ(ξ − ξm ) dξ 0
where δ(ξ − ξi ) denotes the spatial delta function. Mutatis mutandis, it is straightforward to consider sensors with finite support and avoid regularity problems associated with the above delta functions in a spirit similar to the actuators case and the spatial distribution function b(ξ). Furthermore, if one chooses the sensor locations to coincide with 1
Indeed, in the case of thermal manufacturing processes (Demetriou, Paskaleva, & Vayena, 2003), an infrared pyrometry camera may be used. This infrared camera provides a non-contact method for measuring the entire temperature field.
the candidate actuator positions, i.e. ξi = θi , i = 1, . . . , m, or C(θi ) = B∗ (θi ), then one has collocated inputs and outputs (actuators/sensors). Please notice, that as will be seen in the sequel, the choice of a collocated system of actuators and sensors exhibits methodological advantages and considerably simplifies the controller design problem. 3.3. Controller design Using the fact that m measurements are available for control, one may only utilize the output that corresponds to the active actuator location within a given time interval, and therefore, move the actuating device at a location θi (resulting in a specific B(θi )) and use a simple static output feedback control law: u(t) = −yi (t). The resulting closed-loop system dynamics with the actuator at a location θi is governed by x˙ (t) = Ax(t) − B(θi )yi (t)
= (A − B(θi )B∗ (θi ))x(t),
,
θi ∈ Θ m ,
(3.7)
since u(t, θi ) = −yi (t) = −B∗ (θi )x(t). While it can be shown that Acl (θ) A − B(θi )B∗ (θi ), is the generator of an exponentially stable C0 semigroup for any θi ∈ Θm (Pazy, 1983), one needs to impose the additional condition Pφ1 , Acl (θi ), φ2 X + A∗cl (θi )φ1 , Pφ2 X = −Qφ1 , φ2 X ,
φ1 , φ2 ∈ Dom(A),
(3.8)
for all θi ∈ Θm to ensure closed-loop stability. Indeed, the above condition provides a common Lyapunov function for
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Fig. 4. Spatial distribution of four representative disturbance functions.
all actuator locations V (x(t)) = x(t), Px(t) X ,
(3.9)
and the closed-loop dynamics (3.7), and therefore, guarantees asymptotic stability of the closed-loop system regardless of the actuator position used. As a result, any actuator location that is chosen from the set Θm will result in a stable closedloop system. Please notice, that the above controller synthesis problem is considerably simplified using a collocated network of actuators and sensors, and the implementation of the proposed control strategy is characterized by computational efficiency and transparency in meeting the primary design objectives. It should be pointed out, that the above assumption of collocation may be in principle relaxed at the expense of a more complex, computationally demanding, less transparent and harder to coordinate process control and actuator activation policies. Also notice, that the Lyapunov function V (x(t)) when evaluated at the initial time corresponds to the optimal value of the quadratic functional ∞ J= x(τ), Qx(τ) X dτ, (3.10) 0
i.e. one easily obtains: J = V (x(0)) = x(0), Px(0) X . Therefore, an interesting connection between closed-loop stability and optimality (performance) can be established within the proposed framework. Remark 2. It should be noted that even though all m sensors will be providing information on the system’s performance, only the information emanating from the ith sensor will be fed back to the ith actuator which is active (according to the
rules of the scheduling policy delineated previously) via the control signal: u(t, θi ) = −yi (t). In light of the above considerations, the proposed actuator guidance/scheduling and control policy attains the following form: θ opt = arg maxθ∈Θm (|y(t, θ) − ye (θ)|2 + |u(t, θ)|2 ) , uopt (t) = −y(t, θ opt ) = −C(θ opt )x(t) t ∈ [tk , tk + t[.
(3.11)
4. Implementation aspects While the collocated static output feedback control scheme can be implemented with no additional computational load, for obtaining the candidate actuator locations one must first compute the optimal actuator locations comprising the set Θm . To do so, one first approximates the infinite dimensional system (2.1) using an exponential stabilizabilitypreserving approximation scheme (Atwell, Borggaard, & King, 2001; Christofides, 2001; Shvartsman & Kevrekidis, 1998). Thus one arrives at a finite-dimensional system mathematically represented by a system of linear ODEs in state space x˙ n (t) = Axn (t) + B(θ)u(t) + Dw(t),
(4.1)
where xn is the finite-dimensional state vector and A, B, D constant matrices of appropriate dimensionality. Employing
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Fig. 5. Evolution of the L2 norm.
a standard Galerkin scheme, the state x(t, ξ) is approximated via
x(t, ξ) =
n i=1
M −1 B0 (θ) with [M]ij = φin (ξ)φjn (ξ) dξ, 0
xin (t)φin (ξ),
=−
κ(ξ)
0
[K]ij
n dφin (ξ) dφj (ξ) dξ, dξ dξ
where {φjn ∈ H01 (0, )} are the standard B-splines (Prenter et al., 1975; Schultz, 1973) on the spatial interval [0, ] with 2 respect to the uniform mesh {0, 1 n , n , . . . , }, that is, for j = 0, 1, 2, . . . ,
[B0 (θ)]i =
1 − nξ − j , ξ ∈ (j − 1) , (j + 1) , n n φjn (ξ) = 0, otherwise, on [0, ].
Remark 3. Under the stated assumptions and the stabilizability-preserving approximation scheme employed, one can readily deduce that the eigenspectrum of the closedloop system’s characteristic matrix Acl = A − B(θ)C(θ) is assignable through the proposed output feedback control law and the assumed collocated network of actuators/sensors. Hence, for the finite-dimensional system closed-loop stability is ensured regardless of the actuator position, since a common Lyapunov function: V (x) = xT Px can be straightforwardly computed, with P being the solution to the Lyapunov matrix equation: PAcl + Acl TP = −Q, and Q being the matrix representation of the operator Q with respect to the aforementioned function sequence {φjn ∈ H01 (0, )}. Furthermore, for
Let V n = span {φjn }n−1 j=1 be the appropriate function sequence for the finite dimensional subspaces of H01 (0, ) and consider the system of Eq. (4.1) resulting from the weak form of the approximated PDE with test functions: φjn (ξ) ∈ V n . The matrix A is the matrix representation of A and B(θ) is the vector representation of the θ-parameterized input operator B(θ). Specifically, one has A = M −1 K and B(θ) =
0
1 = 2
χ[θ−,θ+] (ξ)φin (ξ) dξ
θ+
θ−
φin (ξ) dξ ≈ φin (θ)
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Fig. 6. Spatial distribution (a) at t = tf /2 and (b) at t = tf ; fixed actuator (dotted) and moving actuator (solid).
parabolic PDE systems with a time-scale separation property characterizing the eigenspectrum of the differential operator, one could follow the technical arguments introduced in (Christofides, 2001; Christofides & Daoutidis, 1997), and by considering only the low-dimensional dominant dynamics, to show that the proposed pole-placing regulator is adequate for the infinite-dimensional system as well. Remark 4. In addition to the controllability criteria imposed on the set of m admissible candidate locations θ ∈ Θ, one may impose additional conditions that provide a certain robustness with respect to disturbances. In this case, one may choose the locations θ ∈ Θm ⊂ Θ that minimize the L2 (rms) gain γ in the following polytopic LMIs
ATi Σ + ΣAi + CzT Cz
ΣD
DT Σ
−γ 2 I
≤ 0,
where Ai A − B(θi )BT (θi ), and D, Cz denote the matrix distributions of the disturbance and controlled output respectively in the closed-loop system representation x˙ n (t) = Ai xn (t) + Dw(t)
zn (t) = Cz xn (t).
To maximize the effects of the disturbance on all the states, one may choose Cz ≡ In and in order to account for the “worst” spatial distribution of the disturbances, one may choose D ≡ 1n , where 1n denotes the n-dimensional disturbance vector which is derived from setting d(ξ) = 1 in the
finite dimensional expansion for (2.1). This “worst” spatial distribution accounts for a disturbance which excites all the modes, thus having a distribution equal to the algebraic sum of all the eigenfunctions, i.e.
d(ξ) ≡ 1 ≈
n i=1
φin (ξ).
In relation to the original PDE (2.1), the above translates into having the disturbance entering at every point in the spatial domain, i.e. d(ξ) = 1. Furthermore, the solution to the above LMIs allows one to avoid placing the actuators at or near nodal points of the disturbance distribution function, thereby ensuring that the disturbances will be affected by the locations of the actuating device. Finally, invoking the Bounded Real Lemma guarantees that the linear system is nonexpansive with (Boyd, Ghaoui, Feron, & Balakrishnan, 1994):
T 0
|y(t, θi )|2 dt ≤ γ
0
T
|w(t)|2 dt,
∀θi ∈ Θm .
Remark 5. The above LMI-based method for generating the set Θm is based on closed-loop criteria as opposed to openloop/controllability criteria presented earlier. By parameterizing the collocated static output feedback controller with the locations θ ∈ Θ, one is able to enhance the performance of the controller and increase its robustness with respect to the worst case spatial distribution of disturbances. In summary,
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Fig. 7. Actuator guidance sequence.
one considers the closed loop system x˙ n (t) = (A − B(θ)C(θ)) xn (t) + Dw(t) zn (t) = In xn (t)
,
θ ∈ Θ,
and minimizes the H∞ norm of the transfer function G(s, θ) = In (sIn − (A − B(θ)C(θ)))−1 D, i.e. minimizes the largest singular value of G(s, θ) over all θ ∈ Θ.
5. Illustrative example A controlled diffusion process modelled through the parabolic PDE (2.1) is considered. Closed loop simulations with a diffusivity κ = 0.01(1 + 0.1 sin(2πξ/ )) were performed using n = 50 linear elements (splines). The two measures for choosing the candidate actuator locations, namely using the controllability Gramian and the H∞ norm of the closed-loop transfer function with a disturbance d(ξ) = 1 are depicted in Fig. 3. Fig. 3(a) has a maximum at the midpoint, in accordance with earlier numerical studies (Demetriou & Borggaard, 2003), and with a disturbance distribution of d(ξ) ≡ 1, while Fig. 3(b) shows a minimum again at the midpoint ξ = /2. While these two measures (open-loop and closed-loop) provide the same optimal location with no apparent differentiation of the spatial robustness measure, further examination of the latter provides further insight into the effects of spatial distributions on the actuator locations, a case not feasible when only controllability measures are employed. Fig. 4 depicts the H∞ norm of the closed-loop transfer function with four different spatial disturbance distributions: d1 (ξ) = χ[0.2 ,0.3 ] (ξ) d2 (ξ) = χ[0.4 ,0.6 ] (ξ), d3 (ξ) =
χ[0.8 ,0.9 ] (ξ), and d4 (ξ) = χ[0, ] (ξ). This suggests that when one has knowledge of the spatial distribution of disturbances, the optimal location is no longer situated at the mid point. For the remainder of the numerical studies, we consider nine equidistant points in [0, ] at which the actuator can reside at. The closed-loop system was simulated in the time interval [0, 6] with a fixed actuator placed at ξ = 0.5 and the proposed moving actuator scheme was applied. As an initial condition, we considered x(0, ξ) = 1000x3 ( − x3 ), 0 ≤ ξ ≤ = 1. To emphasize the effects of the moving actuator, a spatiotemporally varying disturbance (the term d(ξ)w(t)) was included and given by the following expression: d(ξ)w(t) = 10−7 χ[0.2,0.3] (ξ) sin(πt) + χ[0.4,0.6] (ξ) sin π
π
× πt − + χ[0.8,0.9] (ξ) sin πt + . 2 2 Fig. 5 depicts the evolution of the L2 (0, ) norm for the openloop case (dashed), for the case of a permanently mounted (fixed) actuator at ξ = 0.5 (dotted) and the case of a moving actuator (solid). It is easily observed that when a moving actuator is used, the closed-loop performance is enhanced, via a faster convergence of the norm to zero. The spatial distribution at two different time instances (t = 3 and t = 6) is depicted in Fig. 6, where it is once again observed that a moving actuator can enhance closed-loop performance. Finally, the actuator moving sequence is given in Fig. 7.
6. Concluding remarks A new framework that allows the integration of an output feedback controller synthesis method with a collocated
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actuator and sensor scheduling policy was presented. The proposed method enforces the actuator to move close to a location where the maximum deviation of the process state from the reference design steady state occurs in the presence of spatiotemporally varying disturbances, thus providing a quantitative criterion for the development of an actuator scheduling algorithm. Furthermore, the use of a collocated set of sensors and actuators, enables the design of simple local output feedback controllers that are capable of achieving process regulation at the design steady state conditions. It was shown, that since only information emanating from the sensor placed at the location where the actuator has moved to (in accordance to the aforementioned scheduling policy) is used for control purposes, minimal computational requirements for attaining the overall control objectives are needed.
References Alonso, A. A., Kevrekidis, I. G., Banga, J. R., & Frouzakis, C. E. (2004). Optimal sensor location and reduced order observer design for distributed process systems. Computers and Chemical Engineering, 28, 27. Antoniades, C., & Christofides, P. D. (2000). Computation of optimal actuator locations for nonlinear controllers in transport-reaction processes. Computers and Chemical Engineering, 24. Antoniades, C., & Christofides, P. D. (2001). Integrating nonlinear output feedback control and optimal actuator/sensor placement for transportreaction processes. Chemical Engineering Science, 56, 4517. Antoniades, C., & Christofides, P. D. (2002). Integrated optimal actuator/sensor placement and robust control of uncertain transport-reaction processes. Computers and Chemical Engineering, 26, 187. Armaou, A., & Christofides, P. D. (2001). Robust control of parabolic PDE systems with time-dependent spatial domains. Automatica, 37, 61. Atwell, J. A., Borggaard, J. T., & King, B. B. (2001). Reduced order controllers for Burgers’ equation with a nonlinear observer. Journal of Applied Mathematics and Computer Science 11(6), 1311–1330. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. SIAM studies in applied mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics. Christofides, P. D. (2001). Nonlinear and robust control of PDE systems: Methods and applications to transport-reaction processes. Boston: Birkhauser. Christofides, P. D., & Daoutidis, P. (1997). Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. Journal of Mathematical Analysis and Applications, 216, 398. Christofides, P. D. (1998). Robust control of parabolic PDE systems. Chemical Engineering Science, 53, 2949.
Christofides, P. D., & Baker, J. (1999). Robust output feedback control of quasi-linear parabolic PDE systems. Systems & Control Letters, 36, 307. Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinite dimensional linear systems theory, texts in applied mathematics (vol. 21). Berlin: Springer-Verlag. Dautray, R., Lions, J. L. (2000). Mathematical analysis and numerical methods for science and technology: Functional and variational methods (vol. 2). New York, Berlin: Springer Verlag, Heidelberg. Demetriou, M. A. (1999). Numerical investigation on optimal actuator/sensor location of parabolic pde’s. In Proceedings of the 1999 American control conference, San Diego, CA, USA, June 2–4 (vol. 3, pp. 1722–1726). Demetriou, M. A., & Borggaard, J. (2003). Optimization of an integrated actuator placement and robust control scheme for distributed parameter processes subject to worst case spatial disturbance distribution. In Proceedings of the 2003 American control conference, Denver, Colorado, June 4–6. Demetriou, M. A., Paskaleva, A., Vayena, O., & Doumanidis, H. (2003). Scanning actuator guidance scheme in an 1-D thermal manufacturing process. IEEE Transactions on Control Systems Technology, 11, 757. Demetriou, M. A., & Kazantzis, N. (2004). A new actuator activation policy for performance enhancement of controlled diffusion processes. Automatica, 40, 415. Demetriou, M. A., & Kazantzis, N. (2004). Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling. International Journal of Robust Nonlinear Control, 14, 181. Friedman, A. (1964). Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall. Garcia-Osorio, V., & Ydstie, B. E., (2004). Distributed, asynchronous and hybrid simulation of process networks using recording controllers. Internatonal Journal Robust and Nonlinear Control, 14, 227. Keulen, B. V. (1993). H∞ -control for distributed parameter systems: A statespace approach. Boston, Basel, Berlin: Birkh¨auser. Kumar, S., & Seinfeld, J. H. (1978). Optimal location of measurements for distributed parameter estimation. IEEE Trans. Autom. Contr., 23, 690. Lions, J. L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problems, I. New York: Springer-Verlag. Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag. Prenter, P. M. (1975). Splines, variational methods. New York: Wiley. Robinson, J. C. (2001). Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge, United Kingdom: Cambridge University Press. Schultz, M. H. (1973). Spline analysis. Englewood Cliffs, N.J: Prentice Hall. Shvartsman, S. Y., & Kevrekidis, I. G. (1998). Nonlinear model reduction for control of distributed systems: A computer-assisted study. AIChE Journal, 44, 1579. van de Wal, M., & de Jager, B. (2001). A review of methods for input/output selection. Automatica, 37, 487.
A new integrated output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes Michael A. Demetrioua,∗ , Nikolaos Kazantzisb a
Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA
b
Available online 11 November 2004
Abstract The present study proposes a new methodological framework that allows the comprehensive integration of an output feedback controller synthesis method with a collocated actuator and sensor scheduling policy. The proposed actuator scheduling policy is algorithmically realized through a simple closed-loop performance requirement that forces the actuator to move close to a location, where the maximum deviation of the process state from the reference design steady state occurs in the presence of spatiotemporally varying disturbances. Using a collocated set of sensors and actuators, simple local output feedback controllers are designed that are capable of achieving process regulation at the design steady state conditions. Within the proposed framework, only information emanating from the sensor placed at the location where the actuator has moved to is used for control purposes and in accordance to the aforementioned scheduling policy. Consequently, in the proposed approach, the use of minimal information and computational requirements for generating the requisite local control actions is ensured, while attaining the overall process control objectives. Practical implementation aspects of the proposed method are discussed, and its performance is evaluated in an illustrative case study through simulations. © 2004 Elsevier Ltd. All rights reserved. Keywords: Distributed parameter systems; Actuator scheduling; Actuator and sensor collocation; Supervisory control
1. Introduction Increasingly stringent performance requirements imposed on the design of control and supervision systems for transport processes have spawned considerable research effort over the recent years (Christofides, 2001). Technologically important transport processes and the associated phenomena dominate almost all emerging fields such as nanotechnology, materials processing and biotechnology, and as a result, they offer ample motivation for the study of a wide array of interesting operational problems associated with the control, diagnostics and intelligent supervision of spa∗
Corresponding author. E-mail addresses: [email protected] (M.A. Demetriou), [email protected] (N. Kazantzis). 0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.09.017
tially extended processes, as well as production systems (Christofides, 2001; Garcia-Osorio & Ydstie, 2004). One notable and quite important problem pertains to the optimal placement and scheduling activities of sensors and actuators such that a set of prespecified performance criteria and requirements are met (Alonso, Kevrekidis, Banga, & Frouzakis, 2004; Antoniades & Christofides, 2000, 2001; Kumar & Seinfeld, 1978; van de Wal & de Jager, 2001). It should be pointed out, that traditionally the optimal sensor and actuator placement problem was addressed on the basis of open-loop considerations with the emphasis being placed on enhanced controllability and observability requirements (Christofides, 2001; Curtain & Zwart, 1995; Kumar & Seinfeld, 1978). However, a group of researchers recognized the need for the development of a framework, within which the controller synthesis and the actuator/sensor placement problem can be
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addressed in a unified manner (Antoniades & Christofides, 2000, 2002; Kumar & Seinfeld, 1978; Demetriou, 1999; van de Wal & de Jager, 2001), with very interesting results reported in (Antoniades & Christofides, 2001) on an output feedback control strategy implemented through an optimally placed network of actuators and sensors. Furthermore, in recent studies and within the aforementioned methodological framework, robustness issues with respect to unmodeled process dynamics as well as time-varying disturbances were also addressed (Antoniades & Christofides, 2002; Armaou & Christofides, 2001; Christofides, 1998; Christofides & Baker, 1999). Even though noticeably enhanced performance was achieved within the above methodological framework, the latter resulted in actuating and sensing devices permanently located at fixed positions along the spatial domain of the process or the host structure of interest. Recently, a moving actuator and scheduling policy has been proposed, that offers the operational advantage of successfully compensating for the effect of spatiotemporally varying disturbances, since actuators moving closer to the disturbance can exercise higher control authority, and thus, ensure enhanced closed-loop performance (Demetriou & Kazantzis, 2004a,b). In the above approaches, the controller synthesis problem was conceptually “decoupled” and separately addressed from the actuator placement and scheduling/activation one, under the assumption that the local state feedback controllers designed had access to measurements of the full process state vector (Demetriou & Kazantzis, 2004a,b). The present research study aims at the development of a systematic framework that allows the attainment of the following objectives: (i) the comprehensive supervisory integration of an output feedback controller synthesis method with a collocated actuator and sensor scheduling policy, that rely only on the availability of local process measurements and actuation functions through an optimally placed network of sensors and actuators; (ii) the use of minimal information and computational requirements for generating the requisite local control actions, while attaining the overall process control objective which is the successful process regulation (or equivalently the fixed set-point tracking) in the presence of spatiotemporally varying disturbances. In particular, within the proposed framework, it is assumed that the process under consideration has either multiple actuators and the desirable arrangement is to activate only one of them during a given time interval while all remaining ones are kept dormant, or a single actuator capable of moving at a priori selected positions (in order to reduce input/energy requirements, simplify the local controller design, as well as cope more effectively with spatiotemporally varying disturbances (Demetriou & Kazantzis, 2004a,b)). The proposed actuator activation and scheduling policy is algorithmically realized through a simple closed-loop performance require-
ment that forces the actuator to move close to a location where it is most needed, namely to a location where the maximum deviation/excursion of the process state from the reference design steady state occurs in the presence of spatiotemporal disturbances. Please notice, that the above maximum state deviation criterion serves as a concrete quantitative measure that is parameterized by the actuator location. Therefore, as time progresses, one can make explicit use of the above quantitative criterion, and identify the location at which the actuator should move to and exercise its control authority over a certain time interval/window. As a result, an actuator scheduling procedure emerges that can be algorithmically implemented over a sequence of time-windows, within which the actuator remains active at a location identified through the maximum state deviation decision-making criterion mentioned above. Using a collocated network of sensors and actuators, simple local output feedback controllers are designed that are capable of achieving process regulation at the nominal design steady state conditions. It should be pointed out, that even though all the available sensors which are spatially distributed provide measurement signals and valuable information for process monitoring purposes, only information emanating from the sensor placed at the location where the actuator has moved to (and become active) is used for control purposes, in accordance to the aforementioned scheduling policy. Please notice, that in such a supervisory and decentralized manner, a process control strategy is proposed that is endowed with the characteristics of a hybrid (switching) system due to the underlying simultaneous evolution of the process state in the continuous-time domain and the discrete-event nature of the proposed actuator activation and scheduling policy. Furthermore, the proposed framework introduces time-scale multiplicity associated with the latent process dynamics and the discrete-event dynamics that corresponds to the supervisory maximum state deviation decision-making and the proposed actuator scheduling policy. Finally, in light of the above remarks, it should be emphasized, that the proposed approach introduces only minimal computational requirements for a computer-based implementation of its process control strategy. This is certainly a desirable design feature for decentralized supervisory process control systems, that ought to satisfactorily cope with process dynamic components that evolve in both the continuous, as well as the discrete-time domain (Garcia-Osorio & Ydstie, 2004). The present paper is organized as follows: In Section 2 a few mathematical preliminaries are presented, as well as the problem formulation. Section 3 contains the paper’s main results associated with the proposed output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes. In Section 4 implementation aspects of the proposed approach and method are discussed, and its performance is evaluated through simulation studies performed in an illustrative example that is presented in Section 5. Finally, a few concluding remarks are provided in Section 6.
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2. Mathematical preliminaries—problem formulation The transport processes under consideration are modelled through the following 1D controlled diffusion parabolic PDE ∂ ∂ ∂ x(t, ξ) = κ(ξ) x(t, ξ) + b(ξ)u(t) + d(ξ)w(t), ∂t ∂ξ ∂ξ (2.1) accompanied by Dirichlet boundary conditions x(t, 0) = 0 = x(t, ), and initial conditions x(0, ξ) ∈ L2 (0, ), where x(t, ξ) denotes the state, ξ ∈ Ω = [0, ] ⊂ R is the spatial coordinate, t ∈ [t0 , ∞[ is the time variable, u(t) denotes the control signal, b(ξ) denotes the spatial distribution of the actuating device, w(t) the unknown exogenous input signal (disturbances) and d(ξ) the spatial distribution of the disturbance. The thermal diffusivity is denoted by κ(ξ) > 0, ξ ∈ Ω (Friedman 1964). The state space in this case is X = L2 (0, ) endowed with the standard L2 inner product and norm which are denoted by ·, · X and · X , respectively (Robinson, 2001). The control objective is to choose the signal u ∈ L2 ([t0 , ∞)[; R) so that regulation of the state x(t, ξ) to zero (or to some equilibrium state xe (ξ)) is achieved while a certain cost functional, which penalizes both the regulation error as well as the total energy or control effort, is minimized. It is assumed that the spatial distribution of the actuating devices is spanned over a portion of the process spatial domain Ω centered at a location ξa and is given by 1 if ξ − ≤ ξ ≤ ξ + a a b(ξ) = χ[ξa −,ξa +] (ξ) 2 . = 0 otherwise It should be noted that the above approximation for b(ξ) avoids any regularity problems that may arise due to the unbounded nature of a pointwise (in space) actuator distribution (i.e. a spatial delta function) and also guarantees √ that b ∈ L2 (0, ) with norm 1/ 2. Furthermore, one must ensure that the location ξa of the actuator is such that approximate controllability of (2.1) is attained (Curtain & Zwart, 1995). While it is mathematically convenient to consider b(ξ) = δ(ξ − ξa ), with δ(ξ) denoting the spatial delta function, for numerical implementation purposes one uses δ(ξ − ξa ) ≈ χ[ξa −,ξa +] (ξ), where χ[ξa −,ξa +] (ξ) denotes the characteristic function over the interval [ξa − , ξa + ]. The above system may be placed in an abstract setting written as an evolution system (Dautray & Lions, 2000; Lions & Magenes, 1972) x˙ (t) = Ax(t) + Bu(t) + Dw(t), (2.2) in the state space X. Under the above representation, the system’s second order (strongly) elliptic operator is given by (Dautray & Lions, 2000) d d Aφ = κ(ξ) φ, dξ dξ
φ ∈ Dom(A),
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with domain Dom(A) = H 2 (0, ) ∩ H01 (0, ) = {ψ ∈ 2 L (0, ) | ψ, ψ are abs. continuous , ψ ∈ L2 (0, ) and ψ (0) = 0 = ψ( )} and the input operator by Bu(t) = b(ξ)u(t),
B ∈ L(R, X).
The disturbance operator D is given via Dw(t) = d(ξ)w(t),
D ∈ L(R, X).
One way to address the control objective is to employ a single actuator, preferably placed in some “optimal” location, and subsequently design a robust controller (e.g. H∞ ) that would provide adequate levels of robustness with respect to certain classes of disturbances (Keulen, 1993). As mentioned in the introductory Section 1, in order to improve the ability of the system to compensate for spatiotemporal disturbance variations, i.e. considering Dφ = d(t, ξ)φ, a moving actuator scheme is proposed, in which the actuator is capable of moving at preselected positions in the spatial domain, reside for a certain time interval, and then move to another location according to the rules of an intelligent guidance and scheduling scheme that is based on closed-loop performance criteria. A meaningful criterion for a process regulation problem in the presence of spatiotemporally varying disturbances would be to force the actuator to move to a location at which the process state exhibits the maximum deviation/excursion from the nominal design steady state conditions (equilibrium). Furthermore, due to hardware considerations, the actuator may only move to a finite number of locations, and thus the guidance and scheduling policy simplifies to that of moving the actuating device at these preselected candidate positions (please see Fig. 1). Therefore, the primary process control objectives are: (i) to find the finite number of optimal (with respect to certain optimality measure) positions that the actuator can reside at during a given time interval, (ii) the actuator guidance and scheduling policy, and (iii) the control signal that must be supplied to the actuating device by the controller while residing at a given location. The attainment of the above objectives represents the focus of the next section.
3. Main results 3.1. Actuator placement A meaningful way to address the optimality of the finite locations that the actuating device will be residing at during a given time interval [tk , tk + t[, would be to enhance the controllability properties of the system. Please notice, that the input operator B(·) is parameterized by the spatial locations θ ∈ Θ in order to emphasize the dependence of the input operator on the actuator location. The set of candidate actuator
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locations Θ is defined via Θ = {θ ∈ Ω : (A, B(θ)) is approx. controllable}. Remark 1. The fact that the operator A is a Riesz-spectral operator with simple eigenvalues {λn , n ≥ 1} and corresponding eigenvectors {ϕn , n ≥ 1} admitting the represen tation Az = ∞ n=1 λn zψn ϕn , with {ψn , n ≥ 1} being the eigenvectors of A∗ , along with the definition of the input operator Bu = bu, b ∈ X, implies that the test of approximate controllability reduces to the requirement: bϕn = 0 for all n, (Curtain & Zwart, 1995). In the event that the thermal diffusivity is spatially invariant, one obtains: ϕn = √ (2/ ) sin(nπξ/ ), and thus the controllability test reduces to: sin(nπξa / ) = 0, n ≥ 1. In this case, the set of candidate locations is given by
nπθ Θ = θ ∈ Ω : sin
= 0, n ≥ 1 .
Using purely open-loop (enhanced controllability) measures, one may choose the “optimal” actuator locations θ from the above set Θ that maximize the bound α = α(θ) in the following inequality: Wc (θ)φ, φ X ≥ α φ 2X ,
φ ∈ X,
(3.1)
where Wc (θ) denotes the θ-parameterized controllability Gramian defined as follows Wc (θ)φ1 , A∗ φ2 X + A∗ φ1 , Wc (θ)φ2 X = −B∗ (θ)φ1 , B∗ (θ)φ2 X ,
θ ∈ Θ,
(3.2)
for φ1 , φ2 ∈ D(A∗ ). Equivalently stated, the “optimal” actuator locations are given by θ opt = arg sup α(θ), θ∈Θ
where α(θ) is given via (3.1) and (3.2). Please notice, that while there is a continuum of locations that provide controllability, one can only consider a finite number m of these candidate positions. Thus, one must consider an m-dimensional subset Θm = {θ1 , . . . , θm } of Θ, with each location θi being one of the m largest values of α(θ), i.e. choose Θm to contain the first m maxima of Θ. 3.2. Actuator guidance and scheduling policy Once the m actuator locations are found, a guidance and scheduling policy must be developed that would indicate to the moving actuating device where to move to and stay for the duration of a time interval: [tk , tk + t[. As mentioned earlier, it would be meaningful to move the actuator in a location that is most needed; namely to a location where the maximum deviation of the process state from its equilibrium xe (θ)(≡0) occurs. Indeed, the measure maxθ∈Ω |x(t, θ) − xe (θ)| would provide the location in the spatial domain associated with the
Fig. 1. Depiction of moving actuator scheme.
above largest state deviation criterion. Furthermore, since it is desired to place the actuating device at points that render the system controllable (i.e. in Θ) as well, one then restricts the search within the set of the controllable points. Thus, one may consider the following criterion: θ opt = arg max |x(t, θ) − xe (θ)|, θ∈Θ
t ∈ [tk , tk + t[,
(3.3)
Alternatively, one may consider a quadratic performance criterion: θ opt = arg maxθ∈Θ |x(t, θ) − xe (θ)|2 . Moreover, since the actuator can only reside at the a priori selected locations θ ∈ Θm , one considers θ opt = arg maxm |x(t, θ) − xe (θ)|2 . θ∈Θ
(3.4)
While the above choice conforms with intuitively appealing performance requirements, it can still lead to a situation in which the maximum deviation is exactly halfway between two adjacent locations, as depicted in Fig. 2a. Another possibility is to have two identical maxima occurring at two different locations θi and θj , as seen in Fig. 2b. To differentiate between two different actuator positions, one may then include the effects of the control effort associated with each of the candidate positions to arrive at
θ opt = arg maxm |x(t, θ) − xe (θ)|2 + |u(t, θ)|2 , (3.5) θ∈Θ
where u(t, θ) is the control signal supplied to the actuating device at a position θ ∈ Θm . While full state measurements
Fig. 2. Placement scenarios; (a) maximum of |x(t, ξ)| is between two neighboring elements θi and θj , (b) two equal maxima of |x(t, ξ)| located at θi and θj .
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Fig. 3. Spatial distribution of actuator placement measures; (a) using the controllability Gramian: AW + WAT = −B(ξ)BT (ξ), (b) using the H∞ norm of I(s − (A − B(ξ)C(ξ)))−1 D.
might be desirable, and in some cases feasible,1 in most applications one seldom has access to the entire state x(t, ξ). Instead, state measurements are available at discrete points in the domain (e.g. using thermocouples for temperature measurements) or at the boundary. For the system under consideration, it is assumed that one has m available (pointwise) measurements given by y1 (t) x(t, ξ1 ) C(ξ1 )x(t) . .. .. = y(t, ξ) = . . .. = ym (t) x(t, ξm ) C(ξm )x(t) x(t, ξ)δ(ξ − ξ1 ) dξ 0 .. , = (3.6) . x(t, ξ)δ(ξ − ξm ) dξ 0
where δ(ξ − ξi ) denotes the spatial delta function. Mutatis mutandis, it is straightforward to consider sensors with finite support and avoid regularity problems associated with the above delta functions in a spirit similar to the actuators case and the spatial distribution function b(ξ). Furthermore, if one chooses the sensor locations to coincide with 1
Indeed, in the case of thermal manufacturing processes (Demetriou, Paskaleva, & Vayena, 2003), an infrared pyrometry camera may be used. This infrared camera provides a non-contact method for measuring the entire temperature field.
the candidate actuator positions, i.e. ξi = θi , i = 1, . . . , m, or C(θi ) = B∗ (θi ), then one has collocated inputs and outputs (actuators/sensors). Please notice, that as will be seen in the sequel, the choice of a collocated system of actuators and sensors exhibits methodological advantages and considerably simplifies the controller design problem. 3.3. Controller design Using the fact that m measurements are available for control, one may only utilize the output that corresponds to the active actuator location within a given time interval, and therefore, move the actuating device at a location θi (resulting in a specific B(θi )) and use a simple static output feedback control law: u(t) = −yi (t). The resulting closed-loop system dynamics with the actuator at a location θi is governed by x˙ (t) = Ax(t) − B(θi )yi (t)
= (A − B(θi )B∗ (θi ))x(t),
,
θi ∈ Θ m ,
(3.7)
since u(t, θi ) = −yi (t) = −B∗ (θi )x(t). While it can be shown that Acl (θ) A − B(θi )B∗ (θi ), is the generator of an exponentially stable C0 semigroup for any θi ∈ Θm (Pazy, 1983), one needs to impose the additional condition Pφ1 , Acl (θi ), φ2 X + A∗cl (θi )φ1 , Pφ2 X = −Qφ1 , φ2 X ,
φ1 , φ2 ∈ Dom(A),
(3.8)
for all θi ∈ Θm to ensure closed-loop stability. Indeed, the above condition provides a common Lyapunov function for
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Fig. 4. Spatial distribution of four representative disturbance functions.
all actuator locations V (x(t)) = x(t), Px(t) X ,
(3.9)
and the closed-loop dynamics (3.7), and therefore, guarantees asymptotic stability of the closed-loop system regardless of the actuator position used. As a result, any actuator location that is chosen from the set Θm will result in a stable closedloop system. Please notice, that the above controller synthesis problem is considerably simplified using a collocated network of actuators and sensors, and the implementation of the proposed control strategy is characterized by computational efficiency and transparency in meeting the primary design objectives. It should be pointed out, that the above assumption of collocation may be in principle relaxed at the expense of a more complex, computationally demanding, less transparent and harder to coordinate process control and actuator activation policies. Also notice, that the Lyapunov function V (x(t)) when evaluated at the initial time corresponds to the optimal value of the quadratic functional ∞ J= x(τ), Qx(τ) X dτ, (3.10) 0
i.e. one easily obtains: J = V (x(0)) = x(0), Px(0) X . Therefore, an interesting connection between closed-loop stability and optimality (performance) can be established within the proposed framework. Remark 2. It should be noted that even though all m sensors will be providing information on the system’s performance, only the information emanating from the ith sensor will be fed back to the ith actuator which is active (according to the
rules of the scheduling policy delineated previously) via the control signal: u(t, θi ) = −yi (t). In light of the above considerations, the proposed actuator guidance/scheduling and control policy attains the following form: θ opt = arg maxθ∈Θm (|y(t, θ) − ye (θ)|2 + |u(t, θ)|2 ) , uopt (t) = −y(t, θ opt ) = −C(θ opt )x(t) t ∈ [tk , tk + t[.
(3.11)
4. Implementation aspects While the collocated static output feedback control scheme can be implemented with no additional computational load, for obtaining the candidate actuator locations one must first compute the optimal actuator locations comprising the set Θm . To do so, one first approximates the infinite dimensional system (2.1) using an exponential stabilizabilitypreserving approximation scheme (Atwell, Borggaard, & King, 2001; Christofides, 2001; Shvartsman & Kevrekidis, 1998). Thus one arrives at a finite-dimensional system mathematically represented by a system of linear ODEs in state space x˙ n (t) = Axn (t) + B(θ)u(t) + Dw(t),
(4.1)
where xn is the finite-dimensional state vector and A, B, D constant matrices of appropriate dimensionality. Employing
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873
Fig. 5. Evolution of the L2 norm.
a standard Galerkin scheme, the state x(t, ξ) is approximated via
x(t, ξ) =
n i=1
M −1 B0 (θ) with [M]ij = φin (ξ)φjn (ξ) dξ, 0
xin (t)φin (ξ),
=−
κ(ξ)
0
[K]ij
n dφin (ξ) dφj (ξ) dξ, dξ dξ
where {φjn ∈ H01 (0, )} are the standard B-splines (Prenter et al., 1975; Schultz, 1973) on the spatial interval [0, ] with 2 respect to the uniform mesh {0, 1 n , n , . . . , }, that is, for j = 0, 1, 2, . . . ,
[B0 (θ)]i =
1 − nξ − j , ξ ∈ (j − 1) , (j + 1) , n n φjn (ξ) = 0, otherwise, on [0, ].
Remark 3. Under the stated assumptions and the stabilizability-preserving approximation scheme employed, one can readily deduce that the eigenspectrum of the closedloop system’s characteristic matrix Acl = A − B(θ)C(θ) is assignable through the proposed output feedback control law and the assumed collocated network of actuators/sensors. Hence, for the finite-dimensional system closed-loop stability is ensured regardless of the actuator position, since a common Lyapunov function: V (x) = xT Px can be straightforwardly computed, with P being the solution to the Lyapunov matrix equation: PAcl + Acl TP = −Q, and Q being the matrix representation of the operator Q with respect to the aforementioned function sequence {φjn ∈ H01 (0, )}. Furthermore, for
Let V n = span {φjn }n−1 j=1 be the appropriate function sequence for the finite dimensional subspaces of H01 (0, ) and consider the system of Eq. (4.1) resulting from the weak form of the approximated PDE with test functions: φjn (ξ) ∈ V n . The matrix A is the matrix representation of A and B(θ) is the vector representation of the θ-parameterized input operator B(θ). Specifically, one has A = M −1 K and B(θ) =
0
1 = 2
χ[θ−,θ+] (ξ)φin (ξ) dξ
θ+
θ−
φin (ξ) dξ ≈ φin (θ)
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Fig. 6. Spatial distribution (a) at t = tf /2 and (b) at t = tf ; fixed actuator (dotted) and moving actuator (solid).
parabolic PDE systems with a time-scale separation property characterizing the eigenspectrum of the differential operator, one could follow the technical arguments introduced in (Christofides, 2001; Christofides & Daoutidis, 1997), and by considering only the low-dimensional dominant dynamics, to show that the proposed pole-placing regulator is adequate for the infinite-dimensional system as well. Remark 4. In addition to the controllability criteria imposed on the set of m admissible candidate locations θ ∈ Θ, one may impose additional conditions that provide a certain robustness with respect to disturbances. In this case, one may choose the locations θ ∈ Θm ⊂ Θ that minimize the L2 (rms) gain γ in the following polytopic LMIs
ATi Σ + ΣAi + CzT Cz
ΣD
DT Σ
−γ 2 I
≤ 0,
where Ai A − B(θi )BT (θi ), and D, Cz denote the matrix distributions of the disturbance and controlled output respectively in the closed-loop system representation x˙ n (t) = Ai xn (t) + Dw(t)
zn (t) = Cz xn (t).
To maximize the effects of the disturbance on all the states, one may choose Cz ≡ In and in order to account for the “worst” spatial distribution of the disturbances, one may choose D ≡ 1n , where 1n denotes the n-dimensional disturbance vector which is derived from setting d(ξ) = 1 in the
finite dimensional expansion for (2.1). This “worst” spatial distribution accounts for a disturbance which excites all the modes, thus having a distribution equal to the algebraic sum of all the eigenfunctions, i.e.
d(ξ) ≡ 1 ≈
n i=1
φin (ξ).
In relation to the original PDE (2.1), the above translates into having the disturbance entering at every point in the spatial domain, i.e. d(ξ) = 1. Furthermore, the solution to the above LMIs allows one to avoid placing the actuators at or near nodal points of the disturbance distribution function, thereby ensuring that the disturbances will be affected by the locations of the actuating device. Finally, invoking the Bounded Real Lemma guarantees that the linear system is nonexpansive with (Boyd, Ghaoui, Feron, & Balakrishnan, 1994):
T 0
|y(t, θi )|2 dt ≤ γ
0
T
|w(t)|2 dt,
∀θi ∈ Θm .
Remark 5. The above LMI-based method for generating the set Θm is based on closed-loop criteria as opposed to openloop/controllability criteria presented earlier. By parameterizing the collocated static output feedback controller with the locations θ ∈ Θ, one is able to enhance the performance of the controller and increase its robustness with respect to the worst case spatial distribution of disturbances. In summary,
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Fig. 7. Actuator guidance sequence.
one considers the closed loop system x˙ n (t) = (A − B(θ)C(θ)) xn (t) + Dw(t) zn (t) = In xn (t)
,
θ ∈ Θ,
and minimizes the H∞ norm of the transfer function G(s, θ) = In (sIn − (A − B(θ)C(θ)))−1 D, i.e. minimizes the largest singular value of G(s, θ) over all θ ∈ Θ.
5. Illustrative example A controlled diffusion process modelled through the parabolic PDE (2.1) is considered. Closed loop simulations with a diffusivity κ = 0.01(1 + 0.1 sin(2πξ/ )) were performed using n = 50 linear elements (splines). The two measures for choosing the candidate actuator locations, namely using the controllability Gramian and the H∞ norm of the closed-loop transfer function with a disturbance d(ξ) = 1 are depicted in Fig. 3. Fig. 3(a) has a maximum at the midpoint, in accordance with earlier numerical studies (Demetriou & Borggaard, 2003), and with a disturbance distribution of d(ξ) ≡ 1, while Fig. 3(b) shows a minimum again at the midpoint ξ = /2. While these two measures (open-loop and closed-loop) provide the same optimal location with no apparent differentiation of the spatial robustness measure, further examination of the latter provides further insight into the effects of spatial distributions on the actuator locations, a case not feasible when only controllability measures are employed. Fig. 4 depicts the H∞ norm of the closed-loop transfer function with four different spatial disturbance distributions: d1 (ξ) = χ[0.2 ,0.3 ] (ξ) d2 (ξ) = χ[0.4 ,0.6 ] (ξ), d3 (ξ) =
χ[0.8 ,0.9 ] (ξ), and d4 (ξ) = χ[0, ] (ξ). This suggests that when one has knowledge of the spatial distribution of disturbances, the optimal location is no longer situated at the mid point. For the remainder of the numerical studies, we consider nine equidistant points in [0, ] at which the actuator can reside at. The closed-loop system was simulated in the time interval [0, 6] with a fixed actuator placed at ξ = 0.5 and the proposed moving actuator scheme was applied. As an initial condition, we considered x(0, ξ) = 1000x3 ( − x3 ), 0 ≤ ξ ≤ = 1. To emphasize the effects of the moving actuator, a spatiotemporally varying disturbance (the term d(ξ)w(t)) was included and given by the following expression: d(ξ)w(t) = 10−7 χ[0.2,0.3] (ξ) sin(πt) + χ[0.4,0.6] (ξ) sin π
π
× πt − + χ[0.8,0.9] (ξ) sin πt + . 2 2 Fig. 5 depicts the evolution of the L2 (0, ) norm for the openloop case (dashed), for the case of a permanently mounted (fixed) actuator at ξ = 0.5 (dotted) and the case of a moving actuator (solid). It is easily observed that when a moving actuator is used, the closed-loop performance is enhanced, via a faster convergence of the norm to zero. The spatial distribution at two different time instances (t = 3 and t = 6) is depicted in Fig. 6, where it is once again observed that a moving actuator can enhance closed-loop performance. Finally, the actuator moving sequence is given in Fig. 7.
6. Concluding remarks A new framework that allows the integration of an output feedback controller synthesis method with a collocated
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actuator and sensor scheduling policy was presented. The proposed method enforces the actuator to move close to a location where the maximum deviation of the process state from the reference design steady state occurs in the presence of spatiotemporally varying disturbances, thus providing a quantitative criterion for the development of an actuator scheduling algorithm. Furthermore, the use of a collocated set of sensors and actuators, enables the design of simple local output feedback controllers that are capable of achieving process regulation at the design steady state conditions. It was shown, that since only information emanating from the sensor placed at the location where the actuator has moved to (in accordance to the aforementioned scheduling policy) is used for control purposes, minimal computational requirements for attaining the overall control objectives are needed.
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