Decentralized supervisory based switching control for uncertain multivariable plants with variable input output pairing

Decentralized supervisory based switching control for uncertain multivariable plants with variable input output pairing

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  ISA Transactions 51(2012) 132–140 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Decentralized supervisory based switching control for uncertain multivariable plants with variable input–output pairing Omid Namaki-Shoushtari∗ , Ali Khaki-Sedigh1 Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, P.O. Box 16315-1355, Tehran 1431714191, Iran a r t i c l e i n f o Article history: Received 4 April 2011 Received in revised form 9 August 2011 Accepted 28 August 2011 Available online 13 October 2011 Keywords: Uncertain multivariable control system Supervisory switching control Decentralized control Changing Control Configuration QFT Bumpless transfer a b s t r a c t In this paper, the design of decentralized switching control for uncertain multivariable plants is considered. In the proposed strategy, the uncertainty region is divided into smaller regions with a nominal model and specific control structure. The underlying design is based on the quantitative feedback theory (QFT). It is assumed that a MIMO-QFT controller exists for robust stability and performance of the individual uncertain sets. The proposed control structure is made up by these local decentralized controllers, which commute among themselves in accordance with the decision of a high level decision maker called the supervisor. The supervisor makes the decision by comparing the local models’ behaviors with the one of the plant and selects the controller corresponding to the best fitted model. A hysteresis switching logic is used to slow down the switching to guarantee the overall closed loop stability. It is shown that this strategy provides a stable and robust adaptive controller to deal with complex multivariable plants with input–output pairing changes during the plant operation, which can facilitate the development of a reconfigurable decentralized control. Also, the multirealization technique is used to implement a family of controllers to achieve bumpless transfer. Simulation results are employed to show the effectiveness of the proposed method. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction In spite of the indisputable performance advantage of central- ized multivariable controllers, many complex multivariable plants employ decentralized controllers [1]. The main advantages that have led to the widespread use of decentralized controllers are the ease of implementation and maintenance, the design and tun- ing simplicity, and fault tolerance. To effectively design such con- trollers, input–output pairing prior to the commencement of the design is a key step for the achievement of the desired closed loop stability and performance. There are different approaches to in- put–output pairing and RGA is the first and the most widely used analytical tool for this problem [1–3]. In the face of large plant parameter variations, unknown or un- certain multivariable plants, the input–output structure or the so called control configuration of the plant may endure fundamental changes, which will severely degrade the decentralized controller performance and can easily lead to closed loop instability [3,4]. The quadruple-tank process [5], the Wood and Berry distillation col- umn [6], and the Shell heavy oil fractionator [7] are examples of such plants. The well-known input–output pairing techniques are ∗ Corresponding author. Tel.: +98 21 8406 2161; fax: +98 21 8846 2066. E-mail addresses: onamakis@dena.kntu.ac.ir, onamakis@ieee.org (O. Namaki-Shoushtari), sedigh@kntu.ac.ir (A. Khaki-Sedigh). 1 Tel.: +98 21 8406 2317; fax: +98 21 8846 2066. unable to analyze the effect of uncertainty on input–output pair- ing and only recently, pairing methods are proposed for uncertain multivariable plants [3,8]. These methods can only provide a suit- able input–output pairing in the presence of model uncertainties and are unable to tackle the problem of major structural changes in the plant and are also unable to provide an online input–output se- lection based on the present state of the plant [4]. A reconfigurable structure for the design of the decentralized controller based on the nonlinear adaptive control strategies is proposed in [4]. The conventional adaptive control has some inherent limita- tions which have been well recognized in the literature. Most notably, if unknown parameters enter the process model in com- plicated ways, it may be very difficult to construct a continuously parameterized family of candidate controllers [9]. To overcome the above mentioned difficulties with the classical approach to adapta- tion and control, researchers have focused on the switching control systems during the past decade [9–12]. Finding a single controller which can deal with the entire range of parameter variations may be impossible for a highly uncertain process. To design a satisfactory control plant in the presence of large modeling uncertainties, noise, and disturbances, a hierarchical control structure could be used. The control architecture consists of a bank of candidate controllers supervised by a logic-based switching [9]. In each fixed predetermined region of uncertainty, the local controller can achieve the desired performance. Switching is made between the local controllers to support all ranges of uncertainties. 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.08.008
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  O. Namaki-Shoushtari, A.Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 133 The overall control architecture consists of a bank of controllers (multi-controllers), and a supervisor. At each time instant, a high level decision maker, the so called supervisor, determines which controller should be placed in the feedback loop. In other words, when estimates of the plant are changed, a new controller may be selected, which is similar to the idea of adaptive control. But unlike the traditional adaptive control strategies, this adaptation takes place in a discrete fashion. One of the main advantages of the supervisory control is its modularity. Designing multi-estimators, multi-controllers, and switching logic can be done in a mutually independent manner. This feature enables the designer, to use ‘‘off-the-shelf’’ robust control laws [9]. The objective of this paper is to provide a practical solution to the problem of decentralized control of uncertain multivariable plants with variable control configuration. Decentralized MIMO-QFT is used as the underlying linear robust design strategy, because it is a powerful design methodology that provides a transparent tradeoff between different, often conflicting design specifications, and suggests a controller with minimum cost of feedback that satisfies the set of performance specifications in spite of the plant uncertainty [13]. However, as with any other linear robust design methodology, it fails in the face of variable control configuration. By combining and switching robust designs, the designed controller optimizes the time response of the plant by fast adaptation of the controller parameters during the transient response based on the error amplitude and ensures closed loop stability [14]. The problem of robust adaptive control via combining QFT and switching supervisory control is introduced in [15] for single- input–single-output (SISO) plants; the entire region of uncertainty is partitioned into smaller regions. For each region, a QFT controller is employed to attain robust stability and performance despite uncertainties and disturbances. A supervisory architecture orchestrates controller selection. This selection is based on the values of monitoring signals. In this paper, the combined switching QFT control is extended for multi-input–multi-output (MIMO) plants. The key idea is to divide the original uncertainty set into smaller subsets with certain control configuration as in [16]. Then, for each uncertainty subset, a decentralized MIMO QFT control is worked out to satisfy the robust stability and tracking requirements. Finally, a switching supervisory structure is used to recognize to which uncertainty subset, the ‘‘real’’ uncertain plant is matched with. Then, the supervisor places the corresponding controller in the feedback loop. This paper is organized as follows. In Section 2, switching supervisory control is briefly reviewed. Fundamentals of MIMO- QFT are given in Section 3. Our proposed design structure is described in Section 4. Simulation results and conclusions are made in Sections 5 and 6, respectively. 2. Switching supervisory control In this section, the supervisory control framework for adap- tively controlling plants with large uncertainty is briefly re- viewed [11,12]. The basic idea is to divide the uncertainty set, which is compact, into a finite number of subsets with nominal val- ues and then employ a family of controllers, one for each nominal value. Switching among the controllers is orchestrated by a super- visor in such a way that closed-loop stability is assured, as is shown in Fig. 1. The benefits gained by this approach include (i) simplicity and modularity in design: controller design amounts to controller design for linear time-invariant plants with smaller uncertainty, for which various computationally efficient controller design tools are available; (ii) ability to handle larger classes of plants than is possible with other approaches (see [9] for more discussion). Fig. 1. The supervisory control framework. Consider an uncertain linear plant Mp parameterized by a parameter p, and denote by p∗ the true but unknown parameter. The corresponding realization is: Mp∗ :  ˙x = Ap∗ x + Bp∗ u y = Cp∗ x, (1) where x ∈ Rnx is the state, u ∈ Rnu is the input, and y ∈ Rny is the output. The parameter p∗ ∈ Rnp belongs to a known finite set P := {p1, . . . , pm}, where m is the cardinality of P. It is assumed that (Ap, Bp) is stabilizable and (Ap, Cp) is detectable for every p ∈ P. The architecture of supervisory adaptive control comprises: (1) a family of controllers, and (2) a decision maker (supervisory unit). The decision maker, consisting of a multi-estimator, a monitoring signal generator and a switching logic, produces a switching signal that indicates at every time the active controller. • Multi-estimator: A multi-estimator is a collection of models, one for each fixed parameter p ∈ P. The multi-estimator takes in the input u and produces a bank of outputs yp, p ∈ P. The multi-estimator should have the following matching property: there is ˆp ∈ P such that |yˆp(t) − y(t)| ≤ cee−λe(t−t0) |yˆp(t0) − y(t0)| (2) for all t ≥ t0, for all u, and for some ce ≥ 0 and λe > 0. One such multi-estimator for (1) is constructed as follows with the state xE = (ˆxT 1 , . . . , ˆxT m)T where xE ∈ Rm×nx and, the dynamics ˙ˆxp = Ap ˆxp + Bpu + Lp(yp − y), yp = Cp ˆxp, (3) for all p ∈ P, and the property (2) is satisfied with ˆp = p∗ . The matrices Lp in (3) are such that Ap + LpCp are Hurwitz ∀p ∈ P. (since (3) with p = p∗ and (1) imply that (d/dt)(xp∗ − x) = (Ap + LpCp) (xp∗ − x) and y = Cp∗ x). • Multi-controller: A family of candidate controllers {Gp} is designed such that the closed loop plant meets the desired robust stability and performance specifications for every p ∈ P . Then the family of controllers is Gp, p ∈ P. (4) • Monitoring signals: Monitoring signals µp, p ∈ P are norms of the output estimation errors, ep = yp − y. Here, the monitoring signals are generated as µp := ε0 + ∫ t 0 e−λ(t−s) γ ‖yp(s) − y(s)‖2 ds (5) for some γ , ε0 > 0 and λ ∈ (0, λ0), where ‖.‖ denotes the vector norm. The numbers γ , ε0, and λ are design parameters, and λ0 is related to the eigenvalues of the closed-loop plant (for
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  134 O. Namaki-Shoushtari,A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 Fig. 2. The two-degrees of freedom structure of QFT. details on λ0, see [11]). Note that the interactions between the outputs of the multivariable plant are taken into account in the estimation error vector. • Switching logic: The switching signal, which nominates the active controller at each time instant, is produced by the scale independent hysteresis switching logic [17]: σ(t) :=    argmin q∈P µq(t) if ∃q ∈ P such that (1 + h)µq(t) ≤ µσ(t−)(t) σ(t− ) else, (6) where h > 0 is called a hysteresis constant and is a design parameter that conveniently prevents excessive switching. The control signal applied to the plant is u(t) = uσ (t). (See Fig. 1.) 3. Decentralized quantitative feedback design The decentralized MIMO-QFT design technique provides a design procedure to synthesize a fixed diagonal controller transfer function matrix G(s) and pre-filter F(s) to satisfy specifications on the closed-loop plant shown in Fig. 2, where P(s) is the MIMO uncertain plant. In solving an n × n multivariable design problem, the synthesis problem is converted into n equivalent single-loop multi-input–single-output (MISO) problems, where parameter uncertainties, external disturbances and performance tolerances are derived from the original problem, and the coupling effects between MISO subsystems are treated as disturbance inputs. The objective of the design is to achieve set point tracking, while minimizing the outputs due to the disturbance inputs (cross- coupling effects) [13]. It is desired that the closed-loop system is stable and: |tij(jω)| ≤ bij(ω), i ̸= j, 0 ≤ aii(ω) ≤ |tii(jω)| ≤ bii(ω), (7) where T(s) = [tij(s)] is the control ratio matrix, (note that the closed-loop plant control ratio tij = yi/rj relates the ith output to the jth input.) A(s) = [aij(s)] and B(s) = [bij(s)] are the desired lower and upper tracking bounds for the MIMO plant. 4. Switching supervisory QFT control Combining switching and QFT for SISO plants was first introduced in [14] to prioritize some specifications over others according to the state of the plant at each time. Switching is used to select the appropriate controller which is determined based on the error amplitude. However, the proposed method in [14] requires that the underlying controllers must have the same poles and this constraint, limits controllers type and therefore the performance of the plant. A more general approach for SISO plants is introduced in [15] to overcome this limitation in which the quantitative feedback design was used in the context of switching supervisory adaptive control to achieve robust stability and performance for highly uncertain SISO plants. In the following, the decentralized switching-QFT method is given for MIMO plants based on the SISO version of [15]. It is shown that the proposed method could be implemented for complex uncertain plants with I/O pairing Fig. 3. The supervisory based switching QFT control architecture. or control configuration changes during the multivariable plant operation. In the proposed method, the idea is to divide the original uncertainty set into smaller subsets with certain I/O pairing. Then, for each uncertainty subset a decentralized MIMO QFT control is worked out to satisfy the robust stability and tracking requirements. Finally, a switching supervisory structure as shown in Fig. 3 is used to match the proper uncertainty subset to the ‘‘real’’ uncertain plant. Then, the supervisor places the corresponding controller in the feedback loop. 4.1. Problem formulation Regarding highly uncertain MIMO plants, two distinct cases may occur: • A single decentralized QFT controller exists for the entire uncertainty set. However, the closed loop performance may not be improved further as desired. • A single decentralized QFT controller does not exist to ensure closed loop robust stability and performance. The second case is more severe, especially in cases where parameter uncertainty can lead to input–output pairing change which is detrimental to the decentralized control of multivariable plants. The goal is to design a stable closed-loop control system which can track a predetermined set point in spite of large plant uncertainty and/or disturbances. 4.2. Class of admissible plants The plant is assumed to be modeled by a stabilizable and detectable MIMO linear plant with control input u and measured output y, perturbed by a bounded disturbance input d. It is assumed that the plant transfer function belongs to a known class of admissible transfer function matrices of the form: M :=  p∈P Mp where p is a parameter taking values in some index set. Mp is also a family of transfer functions ‘‘centered’’ around some known nominal process model transfer function νp [18]. Allowable unmodeled dynamics around the nominal process model transfer functions νp could be specified as: Mp :=  νp(1 + δm) + δa : ‖δm‖∞,λ ≤ ε, ‖δa‖∞,λ ≤ ε  , where ε > 0 and λ ≥ 0 are two arbitrary numbers. Here, ‖.‖∞,λ denotes eλt -weighted H∞ norm of a transfer function matrix:
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  O. Namaki-Shoushtari, A.Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 135 ‖v‖∞,λ := supRe[s]≥0 ¯σ(v(s −λ)), ( ¯σ(.): the peak of the maximum singular value) [18]. Throughout the paper, we will take P to be a compact subset of a finite-dimensional normed linear vector space. By this definition, the entire region of plant uncertainty is partitioned into a set of smaller regions. Each smaller region is presented by a parameter value p, and Mp is a model of the plant in that small region. Considering all permissible uncertainties and disturbances in each smaller region, based on the appropriate I/O pairing, a decentralized MIMO-QFT controller is designed to achieve robust stability and robust set point tracking specifications. 4.3. Multi-estimator and multi-controller A state-shared multi-estimator of the following form could be utilized here. ˙xE = AE xE + LE y + BE u, yp = CpxE , ep = yp − y, (8) where p ∈ P, and AE is an asymptotically stable matrix. This type of structure is quite common in adaptive control [19]. One option for AE , LE and BE is as follows (using the multi-estimators described in (3)): xE = (ˆxT 1 , . . . , ˆxT m)T , AE = diag[Ap + LpCp], LE = block diag[−Lp], BE = block diag[Bp], p ∈ P. For practical reasons, the bank of local controllers can be efficiently implemented (multirealized) by means of a state-shared parameter dependent feedback system. The method provided can implement bumpless transfer, which is an effective way to improve poor transient response of the switched systems [20,21]. 4.4. Bumpless transfer using multirealization techniques A major problem in the practical implementation of switching based control strategies is the undesirable transient behavior of the plant outputs during switching times. The key response property that can guarantee a smooth transition during the switching times is the bumpless transfer [20]. That is, control input u must be continuous even in the time instants at which the supervisor changes the controller. By bumpless transfer, the control signal is smooth and there is no need to apply sudden changes to system inputs by actuators, while unexpected changes in the control signal can yield undesirable transient responses in switching times. On the other hand, in a multicontroller architecture, only one of the controllers at any instant of time is placed into the feedback loop. Hence, at each time instant it is only necessary to generate one control signal. This can be employed to simplify the multicontroller architecture by generating all control signals by a single controller. This concept is commonly identified as state sharing [20]. State sharing can also be used in the multiestimation part of switching multiple model control methods [19]. The state-shared implementation for a family of multivariable LTI controllers with strictly proper transfer function matrices using the right matrix function description (MFD) with denominator matrix in the Popov form is presented in [21], in the form of {A0 + B0Ki, B0, Ci} or its dual realization {A0 + LiC0, Bi, C0} (i = 1, 2, . . . , m). The multirealization form {A0 + LiC0, Bi, C0} is preferred for the implementation of multicontroller structures to achieve bumpless transfer. However, it is more convenient to investigate the dual form {A0 + B0Ki, B0, Ci} because of the results on invariant description of linear multivariable plants [22]. All controller transfer function matrices can be written in the form of a right MFD in which all the denominator matrices are in the Popov form. Based on the results in [21], a modified and simpler algorithm is presented in [23], which uses the Hermite form polynomial matrix to achieve a multirealization in the form of {A0 +Ai, B0, Ci}. In this multirealization form, A0 is a stable matrix and (A0, B0) is controllable. Also, to achieve bumpless transfer, the duality property can be employed to obtain a multirealization in the form {A0 +Ai, Bi, C0}. In the following, the method based on the Hermite form polynomial matrix to achieve a multirealization in the form of {A0 + Ai, B0, Ci} is briefly reviewed. A given n×n nonsingular polynomial matrix can be transformed to a unique row Hermite form DH (s) by elementary column operations, where DH (s) is an upper triangular polynomial matrix with each diagonal element monic and of higher degree than any other element in the same row, if a diagonal element is 1, all other entries in that row will be zero [24]. With a unique denominator DH (s) in the row Hermit form polynomial matrix, a unique and canonical right MFD for H(s) = NH (s)D−1 H (s) is obtained and is called the row Hermite form MFD. In general, we can always write a given polynomial matrix D(s) in the following form: D(s) = SL(s)Dhr + ΨL(s)Dlr where SL(s) = diag{sli , ı = 1, 2, . . . , n}, and li is the degree of the ith row of D(s). Moreover, Dhr is the highest-row-degree coefficient matrix of D(s) and the term ΨL(s)Dlr accounts for the lower-row-degree terms of D(s), with Dlr a matrix of coefficients and ΨL(s) = block diag{[sli−1 , sli−2 , . . . , 1], i = 1, 2, . . . , n} [24]. Theorem 1 ([23]). Consider a polynomial matrix DH (s) which is in row Hermite form. Let li denote the degree of the ith row of DH (s) and µi is the controllability index corresponding to the ith column of a matrix B (µi independent vectors bi, Abi, . . . , Aµi−1 bi), where (A, B) is a controllable pair that corresponds to a state realization of D−1 H (s). Then, µi = li. Theorem 2 ([23]). Consider a set of ni-input no-output strictly proper systems Hi(s) (i = 1, 2, . . . , m). There exist a set of realiza- tions in the canonical controllability form {Ai, B0, Ci} where Ai is an n×n matrix and {Ai, B0, Ci} are controllable realizations of Hi(s) (i = 1, 2, . . . , m) if and only if there exists a right row Hermite form MFD for all Hi(s), with Hi(s) = NHi (s)D−1 Hi (s), and deg{DHi (s)} = n for i = 1, 2, . . . , m, provided that lik = ljk, i, j = 1, 2, . . . , m and k = 1, 2, . . . , ni where lik is the degree of the kth row of the DHi (s). Based on Theorem 2, the following algorithm is given to obtain the multirealization in the form of {A0 + Ai, B0, Ci}. The Multirealization algorithm: Step 1. Write irreducible row Hermite form MFD descriptions for all the MIMO systems Hi(s) = NHi (s)D−1 Hi (s). Step 2. Let lmaxj = maxi lij, where lij denotes the degree of the jth column in DHi (s). Then, for all DHi (s) matrices, multiply the jth column by (s + a)lmaxj − lij for arbitrary chosen a > 0. Step 3. Applying changes in the previous steps, it may be possible that DHi (s) is not in the row Hermite form. Convert all DHi (s) to the row Hermite form and apply appropriate changes on NHi (s) to have Hi(s) unchanged (Hi(s) = ˜NHi (s)˜D−1 Hi (s)). Step 4. Using the controllable canonical form realization for systems Hi(s) = ˜NHi (s)˜D−1 Hi (s), (i = 1, 2, . . . , m), the multirealization in the {A0 + Ai, B0, Ci} form is obtained. In the following section, the proposed algorithm is used for practical implement of the bank of controllers. Instead of switching between different controller transfer function matrices, a multirealization of these controllers is employed. In the first
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  136 O. Namaki-Shoushtari,A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 Table 1 Plant conditions used in Example 1. No. γ11 γ22 γ12 γ21 δ11 δ22 δ12 δ21 1 1 2 0.5 1 1 2 2 3 2 1 2 0.5 1 0.5 1 1 2 3 1 2 0.5 1 0.2 0.4 0.5 1 4 1 2 4 5 2 3 1 2 5 1 2 4 5 1 2 0.5 1 6 1 2 4 5 0.5 1 0.2 0.4 7 10 8 2 4 1 2 2 3 8 10 8 2 4 0.5 1 1 2 9 10 8 2 4 0.2 0.4 0.5 1 10 5 8 16 20 2 3 1 2 11 5 8 16 20 1 2 0.5 1 12 5 8 16 20 0.5 1 0.2 0.4 step, the multirealization algorithm is applied to the transpose of transfer function matrices, HT i (s). Then, using the duality theorem, the multirealization in the {A0 +Ai, Bi, C0} form is obtained. Finally, the controller with the bumpless transfer property is implemented using this multirealization technique. 5. Simulation results Example 1. In this section, a numerical example is used to illustrate the proposed design method. Consider the following 2×2 uncertain plant: M(s) =     γ11 1 + sδ11 γ12 1 + sδ12 γ21 1 + sδ21 γ22 1 + sδ22     . The corresponding twelve different parameter cases are given in Table 1. The complexity of this example lies in the fact that the appropriate input–output pair changes as the parameters vary and a single robust decentralized controller (regardless of the underlying design methodology) cannot effectively control the plant. Closed loop specifications (for all plants) (1)—The tracking specifications, Tlij ≤ |T(jω)|ij ≤ Tuij (i, j = 1, 2.) are basically non-interacting, and are enforced to ω ≤ ωh = 10 rad/s. On-diagonal: Tuii(ω) =     25 s2 + 6s + 25     s=jω and Tlii(ω) =     4 s2 + 4.4s + 4     s=jω . Off-diagonal: Tuij(ω) = 0.1, and Tlij = 0. (2)—Stability margin: |1(1 + Li)| ≤ 3.5 dB for all plants. Where Li corresponds to the ith loop gain and this would indicate a gain margin and phase margin of 9.6 dB and 39 deg respectively [13]. The relative gain array (RGA) was introduced by Bristol [2] as a measure of interaction in multivariable control plants and is a practical tool for control configuration selection in many multivariable plants. The RGA is defined as Λ = M(0)∗ M−T (0), where the asterisk denotes the Schur product and −T is the inverse transpose. The elements of the RGA for the uncertain process M(s) is as follows: Λ = [ λ 1 − λ 1 − λ λ ] , λ = γ11γ22 γ11γ22 − γ12γ21 . Fig. 4. Variations of the plant parameters. Fig. 5. Switching signal, note that the change in RGA matrix is recognized by supervisor. For cases 1–3, λ = 1.33; therefore, pairing along the diagonal is proposed. In cases 4–6, the structural change in the multivariable plant clearly leads to λ = −0.11 and the off diagonal input–output pairing is proposed. If a fixed structure decentralized controller is employed, the parameter variation leads to closed-loop instability. However, in the case of a switching supervisory control design methodology shown in Fig. 3, the change in the RGA matrix resulting from the plant parameter variations, shown in Fig. 4, is recognized by the supervisor as shown in Fig. 5. Hence, when the change has occurred, a new input–output pairing is chosen. Fig. 6 shows that the decentralized switching-QFT controller can easily handle the new control configuration. The corresponding control effort is shown in Fig. 7. For cases 7–9, λ = 1.11 where diagonal pairing is the appropri- ate input–output pair. Finally, in cases 10–12, λ = −0.14 which leads to the off diagonal input–output pairing. In order to design the multiple-model based decentralized switching-QFT architecture, the region of uncertainty is divided into the following smaller regions: Cases 1, 2 and 3 Cases 4, 5 and 6 Cases 7, 8 and 9 Cases 10, 11 and 12.
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  O. Namaki-Shoushtari, A.Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 137 Fig. 6. Plant outputs in the proposed switching structure. (The family of controllers are implemented using multirealization technique to achieve bumpless transfer.) Fig. 7. Control inputs in the proposed switching structure. (The family of controllers are implemented using multirealization technique to achieve bumpless transfer.) Tank 3 Pump 1 1ν Pump 1 2ν 1 y 2 y Tank 1 Tank 4 Tank 2 Fig. 8. Schematic diagram of the quadruple-tank process. Now, for each region, a decentralized MIMO-QFT design is employed to attain robust stability and performance in spite of uncertainties and disturbances. The decentralized controllers and pre-filters are designed as follows: G1(s) =     185(s + 1.1) s(s + 13.5) 0 0 10(s + 1.4) s(s + 5.3)     , F1(s) =    6 s + 6 0 0 5.8 s + 5.8    G2(s) =     0 57.9(s + 2.9) s(s + 21.7) 38.4(s + 1.7) s(s + 72.2) 0     , F2(s) =    0 2.7 s + 2.7 6.5 s + 6.5 0    G3(s) =     23.1(s + 1) s(s + 16.3) 0 0 18.6(s + 5.9) s(s + 84.8)     , F3(s) =    2.1 s + 2.1 0 0 1.9 s + 1.9    G4(s) =     0 95.6(s + 2.5) s(s + 83.3) 24.7(s + 1.2) s(s + 99.8) 0     , F4(s) =    0 1.7 s + 1.74 1.8 s + 1.8 0    . Finally, a supervisory architecture determines the active decentralized controller which should be placed in the feedback loop. This selection is based on the values of the monitoring signal. A schematic diagram of the overall multiple-model based switching structure is depicted in Fig. 3. Example 2 (Quadruple-Tank Process). The quadruple-tank process shown in Fig. 8 is considered to illustrate the effectiveness of the proposed design procedure. The tank consists of four interconnected water tanks and two pumps. The input signals are the voltages v1 and v2 applied to the two pumps and the outputs are y1 and y2 representing the water level in tanks 1 and 2, respectively. There are two valves that facilitate flows to the tanks. One of the features of the linear dynamic model for the process is the variable appropriate input–output pairing depending on the user adjustable valve settings. This provides an interesting challenge for linear decentralized design paradigms [5]. The control objective is the control of two lower tanks levels, h1 and h2, using the two pumps. Using the mass balances and Bernoulli’s law, the equations describing the plant are dh1 dt = − a1 A1  2gh1 + a3 A1  2gh3 + γ1k1 A1 v1 dh2 dt = − a2 A2  2gh2 + a4 A2  2gh4 + γ2k2 A2 v2
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  138 O. Namaki-Shoushtari,A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 Fig. 9. Results of switching PI control for the MP setting. The plots show simulations with the nonlinear physical model. dh3 dt = − a3 A3  2gh3 + (1 − γ2)k2 A3 v2 dh4 dt = − a4 A4  2gh4 + (1 − γ1)k1 A4 v1 where hi, Ai and ai for i = 1, . . . , 4 denote the level, cross-section and outlet hole cross sections of the ith tank, respectively. Also, g denotes the gravity acceleration. In the above equations, kivi is the corresponding flow to the pump voltage, vi. Moreover, γ1, γ2 ∈ (0,1) indicate the division ratio of flow using the valves. And, the measured level signals are y1 = kc h1 and y2 = kc h2, where kc is a measurement constant. The linearized dynamics at a given stationary operating point is determined from the state space dx dt =            − 1 T1 0 A3 A1T3 0 0 − 1 T2 0 A4 A2T4 0 0 − 1 T3 0 0 0 0 − 1 T4            x +            γ1k1 A1 0 0 γ2k2 A2 0 (1 − γ2)k2 A3 (1 − γ1)k1 A4 0            u y = [ kc 0 0 0 0 kc 0 0 ] with the variables xi := hi − ho i and ui := vi − vo i and the time constants are given by Ti = Ai ai  2 ho i g , i = 1, . . . , 4. At a particular ho i , the stationary control signal is obtained by solving the following equations a1 A1  2gho 1 = γ1k1 A1 vo 1 + (1 − γ2)k2 A1 vo 2 a2 A2  2gho 2 = (1 − γ1)k1 A2 vo 1 + γ2k2 A2 vo 2. Finally, the transfer function matrix corresponding to the stationary operating point is M(s) =     γ1T1k1kc A1(1 + sT1) (1 − γ2)T1k2kc A1(1 + sT1)(1 + sT3) (1 − γ1)T2k1kc A2(1 + sT2)(1 + sT4) γ2T2k2kc A2(1 + sT2)     . The corresponding RGA is Λ = [ λ 1 − λ 1 − λ λ ] where λ = γ1γ2 γ1+γ2−1 . If λ > 0 or 1 < γ1 +γ2 < 2 then (y1 −u1/y2 − u2) is the appropriate input–output pair for decentralized control. However, for 0 < γ1 + γ2 < 1, the closed-loop performance will be better if y1 and y2 are permuted in the control structure and the decentralized control is considered with pairings (y1, u2) and (y2, u1). It is also shown that the system is minimum phase (MP) for 1 < γ1 + γ2 < 2, and non-minimum phase (NMP) for 0 < γ1 + γ2 < 1 that is inherently more difficult to control. Note that, the aforementioned permutation, however, does not change the location of the right half-plane zero, and experiments have shown that the settling times are still much larger than the minimum- phase setting [5]. The chosen operating points and the parameter values corresponding to the MP and NMP settings are given in Table 2. In this section, decentralized PI control is applied to the nonlinear process model. A decentralized PI controller as a
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  O. Namaki-Shoushtari, A.Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 139 Fig. 10. Results of switching PI control for the NMP setting. The plots show simulations with the nonlinear physical model. Table 2 Nominal operating conditions and parameter values of the quadruple-tank process. Symbol State/parameters Value (MP setting) Value (NMP setting) ho 1, ho 2, ho 3, ho 4 Nominal levels (cm) 12.4, 12.7, 1.8, 1.4 12.6, 13.0, 4.8, 4.9 vo 1, vo 2 Nominal pomp voltages (V) 3.00, 3.00 3.15, 3.15 A1, A2, A3, A4 Areas of the tanks (cm2 ) 28, 32, 28, 32 28, 32, 28, 32 a1, a2, a3, a4 Areas of the drain in the tank (cm2 ) 0.071, 0.057, 0.071, 0.057 0.071, 0.057, 0.071, 0.057 γ1 Ratio of flow in tank 1 to flow in tank 4 0.70 0.43 γ2 Ratio of flow in tank 2 to flow in tank 3 0.60 0.34 k1, k2 Pump constants (cm3 /Vs) 3.33, 3.35 3.14, 3.29 T1, T2, T3, T4 Time constants in the linearized model (s) 62, 90, 23, 30 63, 91, 39, 56 kc Measurement constant (V/cm) 0.50 0.50 G Gravitation constant (cm/s2 ) 981 981 combination of two SISO PI controllers is considered. PI controllers of the form gl(s) = Kl  1 + 1 τils  , l = 1, 2 are tuned based on the linear physical model which is accurate and agree with the response of the real process. Then, PI controllers are incorporated into the proposed switching control structure to deal with the variable input–output pairing of the multivariable process. For the MP setting, it is easy to find controller parameters which yield good performance. The controller settings are adopted from [5] as (K1, τi1) = (3.0, 30) and (K2, τi2) = (2.7, 40). Due to the fact that the non-minimum phase process is normally harder to control than the minimum phase one, it is difficult to find the controller parameters which give good closed-loop performance for the NMP setting. The controller parameters (K1, τi1) = (0.94, 187.3) and (K2, τi2) = (0.99, 197.3) which are borrowed from [25] stabilize the process and give reasonable performance. Thus, the decentralized controllers are as follows: G1(s) =     3.0  1 + 1 30s  0 0 2.7  1 + 1 40s      , G2(s) =     0 0.94  1 + 1 187.3s  0.99  1 + 1 197.3s  0     . If a diagonal decentralized controller is employed, the appro- priate input–output pairing corresponding to NMP setting leads to much slower responses than in the minimum-phase case. The settling time for the step responses, which is an important char- acteristic of the system, is approximately ten times longer for the nonminimum-phase setting [5]. However, in the case of a switch- ing supervisory control design, the appropriate input–output pair is recognized by the supervisor. Figs. 9 and 10 show that the decen- tralized switching-PI controller can easily handle the two MP and NMP settings. To proceed with simulation, assume that the process is working in MP settings. In addition, the hysteresis constant h in the switching law (6) is set to be h = 0.1. Suppose that the second candidate controller is connected into the loop initially, that is, σ(0) = 2. Fig. 9 depicts the closed-loop input–output trajectories. These exhibit satisfactory closed-loop performance. The switching signal identifies the ‘‘right’’ controllers via 1 switch. Furthermore, we set the first controller initially, i.e., σ(0) = 1, and carry out the simulation again for the NMP settings. Simulation results are given in Fig. 10. The ‘‘right’’ controller corresponding to the appropriate
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  140 O. Namaki-Shoushtari,A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 input–output pair is identified by the supervisor and the closed loop performance is good enough. 6. Conclusion This paper presents the application of multivariable switching multiple model based adaptive control for complex multivariable plants. Large plant uncertainties lead to different input–output pairings in multivariable plants. The proposed control structure consists of a bank of candidate controllers and a supervisor. The underlying design strategy is decentralized MIMO QFT. Each of the candidate controllers is designed in order to achieve the demanded performance in a subregion of the plant uncertainty. The supervisor consists of a multi-estimator, a performance signal generator and a hysteresis switching logic scheme. The supervisor selects the active controller corresponding to the local model which best fits the plant input–output data. Simulation results are presented to show the effectiveness of the proposed methodology in the face of varying control configuration in complex multivariable plants. References [1] Skogestad S, Postlethwaite I. Multivariable feedback control analysis and design. 2nd ed. John Wiley & Sons; 2005. [2] Bristol E. On a new measure of interaction for multivariable process control. IEEE Trans Autom Control 1966;11:133–4. [3] Khaki-Sedigh A, Moaveni B. Control configuration selection for multivariable plants. LNCIS, vol. 391. Berlin (Heidelberg): Springer-Verlag; 2009. [4] Moaveni B, Khaki-Sedigh A. Reconfigurable controller design for linear multivariable systems. Int J Model Ident Control 2007;2:138–46. [5] Johansson KH. The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Trans Control Syst Technol 2000;8(3): 456–465. [6] Wood RK, Berry MW. Terminal composition control of a binary distillation column. Chem Eng Sci 1973;28:1707–17. [7] Chen D, Seborg DE. Relative gain array analysis for uncertain process models. AIChE J 2002;48(2):302–10. [8] Moaveni B, Khaki-Sedigh A. Input–output pairing analysis for uncertain multivariable processes. J Process Control 2008;18:527–32. [9] Hespanha JP, Liberzon D, Morse AS. Overcoming the limitations of adaptive control by means of logic-based switching. Systems Control Lett 2003;49(1): 49–65. [10] Böling JM, Seborg DE, Hespanha JP. Multi-model adaptive control of a simulated pH neutralization process. Control Eng Pract 2007;15:663–72. [11] Liberzon D. Switching in systems and control. Boston: Birkhäuser; 2003. [12] Sun Z, Ge SS. Stability theory of switched dynamical systems. London: Springer-Verlag; 2011. [13] Houpis CH, Rasmussen SJ, Garcia-Sanz M. Quantitative feedback theory: fundamentals and applications. 2nd ed. Florida: A CRC Press Book, Taylor & Francis; 2006. [14] Garcia-Sanz M, Elso J. Beyond the linear limitations by combining switching & QFT: application to wind turbines pitch control systems. International Journal of Robust and Nonlinear Control 2009;19(1):40–58. [Special issue: Wind turbines: new challenges and advanced control solutions]. [15] Namaki-Shoushtari O, Khaki Sedigh A. Design of supervisory based switching QFT controllers for improved closed loop performance. In: Proceedings of the 18th Iranian conference on electrical engineering. ICEE 2010. 2010. p. 599–604. [16] Namaki-Shoushtari O, Khaki Sedigh A. Design of decentralized supervisory based switching QFT controller for uncertain multivariable plants. In: Proceedings of the 48th IEEE conference on decision and control. 2009. p. 934–9. [17] Hespanha JP, Liberzon D, Morse AS. Bounds on the number of switchings with scale independent hysteresis: applications to supervisory control. In: Proceedings of the 39th IEEE conf. on decision and contr. vol. 4. 2000. p. 3622–7. [18] Hespanha JP. Tutorial on supervisory control. Orlando, FL: Lecture Notes for the Workshop Control Using Logic and Switching for the 40th IEEE CDC; 2001. [19] Morse AS. Supervisory control of families of linear set-point controllers—part 1: exact matching. IEEE Trans Autom Control 1996;41(10):1413–31. [20] Morse AS. Control using logic-based switching. In: Isidori A, editor. Trends in control: an European perspective. London: Springer-Verlag; 1995. p. 69–113. [21] Su S, Anderson B, Brinsmead T. Minimal multirealization of MIMO linear systems. IEEE Trans Autom Control 2006;51(4):690–5. [22] Popov VM. Invariant description of linear, time-invariant controllable systems. SIAM J Control 1972;10:252–64. [23] Sadeghi H. State sharing in multivariable systems and its application in switching control. MSc. thesis, K.N. Toosi University of Technology; Spring 2010. [24] Kailath T. Linear systems. New Jersey: Prentice-Hall; 1980. [25] Vadigepalli R, Gatzke EP, Doyle III FJ. Robust control of a multivariable experimental four-tank system. Ind Eng Chem Res 2001;40(8):1916–27.