RHP Right-Haft PlaneSymbols Gs Process transfer function matrix Ds Decoupler matrix Qs Decoupled matrix Cs, G c s orG~ s c Multi-loop controller matrix G ii+ Non-minimum part of proc
CLASSICAL DESIGN METHODS OF
Most industrial processes are inherently multivariable, with interactions among process variables that make designing independent-loop controllers challenging Decoupling control offers a more effective way to reduce these interactions and enhance overall control performance In addition, unlike traditional multi-loop control systems, decoupling control does not require the process transfer matrix to exhibit diagonal dominance.
There are three classical decoupling control methods used in industrial process control: ideal decoupling, simplified decoupling, and inverted decoupling The literature extensively covers the analysis, design, and tuning of decoupling control systems, highlighting how these approaches aim to mitigate coupling between process variables Key topics include ensuring properness and realizability, maintaining causality, and preserving stability across a range of operating conditions This overview underscores the ongoing development and application of decoupling techniques to improve controller performance in complex multivariable processes.
Figure 1.1 General structure of a decoupled system
The general structure of a decoupling system is shown in Fig 1.1, wherein D(s) denotes the transfer function matrix of decoupling network,
G(s) represents the multivariable process, and C(s) is the transfer function matrix of decentralized controller The variables y, r, and u are denoted the output, input, and manipulated variables, respectively
After using the input-output pairing technique (i.e., relative gain array), the relative interaction between different input and output pairings should be the first problem to be considered in the multiple input, multiple output (MIMO) decoupling control, the design of an appropriate decoupler network is then the next important step Finally, the effective multi-loop control approach can be designed in terms of a centralized controller with a series of decouplers or a series of single input, single output (SISO) decentralized controllers.
Consider a general two inputs, two outputs (TITO) process model, which is shown in Fig 1.2, the controller, decoupler and process model can be given as follows:
Decoupling at the input of the TITO process G(s) requires the design of a transfer matrix D(s), such that G(s)D(s) is a diagonal transfer matrix T(s):
Figure 1.2 Decoupling control system of a TITO process
Hence, the decoupling network matrix can be obtained by:
The elements G 11 (s), G 12 (s), G 21 (s), and G 22 (s) that are the transfer function of process model are supposed to be known Q 11 (s) and Q 22 (s) are unknown elements that represent the desired dynamics of the decoupled system.
The ideal decoupling scheme provides great convenience for the controller design, but it is rarely used in practice because of its complicated decoupling elements.
A first decoupling control design consists in selecting the transfer functions Q 11 (s) and Q 22 (s) The decoupling transfer matrix D(s) is then deduced from Eq (1.6) The diagonal controller elements C 1 (s) and C 2 (s) are independently and respectively tuned based on Q 11 (s) and Q 22 (s) A logical choice for Q(s) is:
Therefore, the product matrix can be obtained as Eq (1.5).
Then, controllers can be tuned in the same way under ideal decoupling The controller does not need to be retuned even if different loops are set in different modes.
While ideal decoupling offers theoretical advantages, its practical realization is hindered by the complexity of the decoupling matrix D(s), which contains sums of transfer functions In addition, its sensitivity to model errors and the impact of system dimension pose additional challenges in applications Consequently, ideal decoupling is rarely used in practice.
The simplified decoupling scheme has a simple decoupler, but its decoupled process is intricate The controller cannot be directly designed without introducing more approximations.
The simplified decoupling is widely used in the literature A common expression of a simplified decoupled system is illustrated in Fig 1.3.
The resulting transfer function matrix Q(s) is then:
Resolving the decoupler realization problem is possible, but the diagonal transfer matrix Q(s) remains highly complex, which complicates controller tuning Researchers typically employ approximation techniques to map the diagonal elements to standard forms in order to facilitate tuning However, applying the simplified decoupling scheme to high-dimensional processes continues to face the same challenges as ideal decoupling.
The inverted decoupling structure, also named feed-forward decoupling control, overcomes the drawbacks of simplified decoupling and achieves the purpose of ideal decoupling for multivariable processes As a widely used decoupling algorithm, it can derive a decoupled process model that matches the ideal decoupler for two-input two-output (TITO) systems Figure 1.4 depicts the inverted decoupling topology and its feed-forward arrangement, illustrating how this approach enables effective decoupling in practical control applications.
For simplicity, the decoupler networks can be given as:
Inverted decoupling has the same decoupled transfer function as ideal decoupling and the same convenient realization as simplified decoupling; however, as the process dimension increases, selecting the optimal system-structure configuration becomes more complex, and each method presents its own advantages and drawbacks In practice, the most suitable choice is driven by the process dynamics and the requirements of the control design.
1.5 ANALYSIS OF THE ADVANTAGES AND DISADVANTAGES
The summation of the advantages and disadvantages of each decoupling method is shown in Table 1 as follows:
Table 1 Advantages and disadvantages of each decoupling method
AN EXTENDED METHOD OF SIMPLIFIED
AN EXTENDED METHOD OF SIMPLIFIED DECOUPLING CONTROL
Figure 2.1 shows a decoupling control system for an n×n stable process In this block diagram, G̃C denotes a multi-loop controller and D denotes the decoupling matrix The processes G̃ and Q̃ are the multivariable and decoupled apparent processes, respectively.
Decoupling aims to determine a decoupling matrix D so that GD = Q̃ is diagonal, transforming a multivariable process into decoupled, independent channels This diagonal condition enables the design of a multi-loop controller as a set of n independent SISO controllers, each tuned on the decoupled apparent process.
Figure 2.1 Block diagram of a decoupling control system
From Eq (2.1), the two relations below follow:
1 1 1 1 1 1 i i i,i- i- ,i ii ii i,i+ i+ ,i in ni ii g d +…+ g d + g d + g d +…+ g d = q (2.2)
(2.3) where g ij and d ij denote the (i, j) th elements of G and D, respectively The i th element of Q is found by rearranging Eq (2.2):
( 1 1 2 2 , -1 -1, , 1 1, ) ir ic ii ii ii i i i i i i i i i i i i in ni ii ii q g d= + g d g d+ +…+g d +g d + + +…+g d =g d g d+
(2.4) where g i r denotes the i th row vector of G dropping the element g ii , i.e., r 1 2 , -1 , 1 i =g g i i …g g i i i i + …g in g , and d i c is the i th column vector of D, dropping the element d ii , i.e., d i c =d d 1 i 2 i …d d i i -1, i i + 1, …d ni T
+ G d g (2.5) where G i denotes a matrix, dropping the i th column and row in G, and g i c the i th column vector of G, dropping the element g ii
From matrix theory, for any square matrix G we have G adj(G) = det(G) I and adj(G) G = det(G) I, where det(G) is the determinant of G and I is the identity matrix If C is defined as the transpose of the cofactor matrix associated with the entries of G, then C equals adj(G), linking the adjugate to the cofactors that determine det(G).
From Eq (2.9), the i th diagonal element of G can be written as
( 1 1 2 2 , -1 , -1 , 1 , 1 ) r c g 1 ii i i i i i i i i i i i i in in ii i i ii ii g C g C g C g C g C
G g C (2.11) c 1 2 , -1 , 1 where C i =C C i i …C C i i i i + …C in T Note that C i c corresponds to the i th column vector of adj G, dropping the elementC ii
Substituting Eq (2.13) into Eq (2.6) gives d i c as
Accordingly, the (j, i) th element of a decoupler can be expressed generally as
Substituting Eq (2.11) and Eq (2.14) into Eq (2.4) yields ii ii ii q d
Furthermore, each diagonal element of the DRGA matrix (Witcher and McAvoy, 1977; Bristol, 1979; Skogestad and Poslethwaite, 1996; Truong and Lee, 2010a) can be calculated as:
G (2.17) where Λ ii denotes the i th diagonal element of the DRGA, and ⊗ denotes element-by-element multiplication (the Hadamard or Schur product)
Therefore, from Eq (2.15) and Eq (2.16), the elements of the decoupled apparent process can be expressed as ii ii ii ii q = d Λg (2.18)
To design a simplified decoupling for a stable, square MIMO process with n inputs and outputs, the diagonal elements of the decoupling matrix, d_ii, are commonly set to unity This convention yields simple, general forms for both the decoupler and the decoupled apparent processes, enabling straightforward analysis and control design By fixing d_ii = 1, the decoupler emphasizes off-diagonal coupling terms, while the decoupled apparent processes behave as nearly independent channels, preserving stability and allowing decoupled control across all channels.
Accurate input–output (IO) pairing is critical in decoupler design because it can significantly affect the resulting decoupler network, the decoupled apparent process, and thus the decoupler’s performance and realizability In this work, direct IO pairings are adopted as the baseline For other IO configurations, the same design procedure can be applied by rearranging the IO pairing variables along the diagonal and running the design with the reordered diagonal.
2.2 SIMPLIFIED DECOUPLING DESIGN FOR TYPICAL PROCESSES
This section analytically develops simplified decoupling for
2 2, 3 3, and 4 4× × × processes using Eq (2.19) and Eq (2.20).
The following decoupler matrix results from Eq (2.19):
The cofactor of G is easily given by:
C = G (2.23) and the decoupler elements follow:
The decoupled apparent processes are obtained from Eq (2.20):
These results are in exact agreement with those from the most reported approaches for the simplified decoupling of TITO processes.
The transfer function of a 3 3× process is given by
The decoupler matrix is easily obtained from Eq (2.19):
Hence, the decoupler elements can be analytically found (see Table 2.1). Equation (2.20) allows the decoupled apparent processes for the first, second, and third loops to be analytically constituted as:
From Eqs (2.27) and (2.29), it follows that the decoupler elements and the decoupled apparent processes become more complex as the process size increases, highlighting a key limitation of simplified decoupling.
The transfer function of a 4 4× process is given by:
The decoupler matrix can be found using the above procedure:
Table 2.1 lists the analytical forms of the decoupling elements, and the same procedure can be readily applied to derive analytical forms for decoupling elements in other high-dimensional multivariable processes with multiple time delays.
With increasing process order, the transfer functions of the decoupler elements become too complex to be directly used in decoupling system design Consequently, these dynamics must be approximated with reduced-order models Any appropriate model-reduction technique can be applied to fit the decoupler into lower-order representations, enabling practical design while preserving essential behavior.
Table 2.1 Analytical forms of simplified decoupler elements for some typical processes Process Decoupler elements
2.3 REDUCTION TECHNIQUE FOR DECOUPLER REALIZABILITY
Designing decouplers for dynamic decoupling structures must respect realizability by ensuring stability, propriety, and causality In simplified decoupling configurations, setting the diagonal elements to unity makes the decoupler design more manageable, but the off-diagonal elements still must satisfy the same three foundational constraints Therefore, while the diagonal terms are fixed, the challenge remains to shape the off-diagonal relationships so that the overall decoupler remains stable, proper, and causal, delivering effective decoupling performance in real-world systems.
(i) Causality requires only present and past input values to compute the output Accordingly, decoupling elements must not involve non-causal time delay
Decoupling elements are considered proper only when the denominator term’s order is equal to or greater than the numerator term’s order Stability is achieved if and only if there are no right-half-plane (RHP) poles in the system.
Decoupler realizability for TITO processes can be readily checked by the direct calculations of G12/G11 and G21/G22 For processes that involve time delay and non-minimum phase zeros, the resulting decouplers may introduce elements with an exponential prediction term e^{θ s} and/or right-half-plane (RHP) poles Wade (1997) proposed an essential approach for inverted decoupling of TITO processes to address this, but that technique is not applicable to the simplified decoupling of high-dimensional multivariable processes with multiple time delays, because the decoupling elements contain sums of transfer functions whose dynamics are complicated by differing signs and time delays even when the underlying elements have simple dynamics To overcome this, any suitable model reduction technique can be used to fit the complex decoupling dynamics into low-order models, including, but not limited to, least squares, polynomial approximation, Laguerre expansion, and the Gaussian frequency-domain approach In this paper, the coefficient matching (CM) method proposed by Truong and Lee (2010a) is selected and extended to obtain reduced-order decoupler elements that satisfy the realizability requirements, and the decoupler design procedure is illustrated.
CM is briefly outlined here.
Decoupler elements given by Eq (19) can be expanded in Maclaurin series in s as eff ji ( ) ji 1 ji ji 2 ji 3 ( )4 ji ji ji b c d d s a s s s O s a a a
Static gain, pure delay, first-order lead/lag, and first-order lead/lag with time delay are commonly used representative dynamics for decoupler elements in decoupling structures because of their simplicity and reasonable performance In practical modeling and design, the first-order lead/lag dynamics often closely approximates the decoupler element’s actual dynamics, providing a simple yet effective approximation.
Expanding the reduced dynamics in Eq (34) as a Maclaurin series in s gives:
{ } r_eff 2 3 r ra rb ra rb rb
Equation 2.35 expresses ds as a function of a gain Kr, two time constants τ_ra and τ_rb, and higher‑order terms, with Kr, τ_ra, and τ_rb to be determined so that d_i i_eff is approximated as closely as possible over the relevant control frequency ranges To achieve this close approximation, Kr, τ_ra, and τ_rb must be chosen to minimize the error across the frequencies of interest By comparing the first, second, and third terms in Equation 2.35 with those in Equation 2.32, explicit expressions for Kr, τ_ra, and τ_rb are obtained.
K =a (2.36a) rb ii ii c τ = −b (2.36b) ra ii ii ii ii b c a b τ = − (2.36c)
Table 2.2 lists the associations between the polynomial coefficients and the decoupler element parameters for the remaining dynamic models, providing the essential links needed to determine the decoupler parameters θ_r, τ_ra, and τ_rb By establishing these relationships, the calculation of the decoupler element parameters is streamlined, enabling straightforward computation from the model coefficients.
For the resulting decoupler element to be realizable, the values of θ r , τra , and τ rb must be real and positive This is demonstrated by various case studies that consider Table 2.2
Case A addresses non-realizable elements caused by non-causal time delays, right-half-plane (RHP) poles, and improper decoupler elements; in such cases, design the decoupler elements as steady-state decoupled process gains to force the non-realizable dynamics into realizability.
Case b: If there are non-realizable elements due to non-causal time delay, it is recommended to design the decoupler elements using the first- order lead/lag model.
Case c: If there are non-realizable elements due to RHP poles and/or improper decoupler elements, it is recommended to design the decoupler element using the pure delay model.
Case d: If the decoupler elements are realizable, it is recommended to design them using the first-order lead/lag with time delay model.
AN EXTENDED METHOD OF INVERTED DECOUPLING
AN EXTENDED METHOD OF INVERTED
Considering the design method given by J Garrido et al (2011) [1], the design of the inverted decoupler network for an n × n process can be obtained by the following steps:
Figure 3.1 Matrix representation of inverted decoupling
Figure 3.2 Inverted decoupling control system of a TITO process
As shown in Fig 3.1, the decoupler D(s) is decomposed into two matrices: Dd(s) in the direct path (between controller outputs c and process inputs u) and Do(s) in the feedback loop (between process inputs u and controller outputs c) The Dd(s) matrix must contain only n non-zero elements, ensuring a single direct connection for each process input By contrast, there is no such restriction on the Do(s) matrix Furthermore, because the signal flow direction in Do(s) is opposite to that of the direct path, its interpretation involves reversed flow considerations.
Dd(s), the corresponding elements of Do(s) that must equal zero are the transpose non-zero elements of Dd(s).
This expression enables straightforward calculation of the elements of the inverted decoupling Its main advantage is its computational simplicity regardless of the system size, because Q(s) is chosen as a diagonal matrix and the subtraction (Dd(s))^{-1} − Do(s) forms a transfer matrix in which each position requires computing only one element.
There are two possible configurations to choose for the Dd matrix: diagonal elements (configuration 1-2) or off-diagonal elements Do
Table 3.1 Cases of 2×2 inverted decoupling with two unitary elements
Case Decoupler elements Decoupled process
Figure 3.3 Control schemes of the four cases of 2x2
Setting two elements to their unity value yields four possible configurations, identical to the four cases described for configurations 1–2 In this setup, the only pair of decoupler elements that differs from unity is specified by Equation (3.9).
For 3×3 processes, the procedure is the same However, in this case, the number of configurations is increased, and then the configuration 1-2-3 is a favorite choice in applications.
1 g g g do do q q q g g g do do q q q g g g do do q q dd dd d q d
Consider the case in which the elements of the Dd matrix are equal to unity In this case and using configuration 1-2-3, the following expressions are obtained:
3.4 GENERAL EXPRESSIONS OF INVERTED DECOUPLING
Assuming that the configuration is p p 1 − 2 − − − − p i p n − 1 − p n , the elements of the Dd and Do matrix are obtained as follows:
When the non-zero elements of the Dd matrix in (3.14) are fixed to unity, the expressions of the decoupler elements and apparent processes q i (s) are given by
Realizability of the decoupler requires that every element be proper, causal, and stable To prevent non-causal time delays in decoupler elements, the time delays must satisfy a defined relationship; this relation is derived from the realizability conditions and guarantees that the decoupler operates with physically realizable timing.
For the row i with the smallest time delay, the element dd_ki of Dd must be nonzero Decoupler elements must be proper, meaning their relative degrees rij must be greater than or equal to zero.
Let g_ik be the transfer function of row i that has the smallest right-half-plane (RHP) zero multiplicity η_ik The element d_ki must be non-zero This RHP zero must appear in the q_i apparent process with a multiplicity η_qi that fulfils the required condition.
Whenever two or more Dd elements must be selected in the same column to satisfy the prior conditions for all rows, there are no realizable configurations To overcome this, insert an additional diagonal block N(s) between the plant G(s) and the inverted decoupler D(s); this modification changes the process dynamics and forces the previously non-realizable elements into realizability Consequently, inverted decoupling can be applied to the updated process.
N(s) is a diagonal block that carries the required extra dynamics If there are no realizability problems in row i, the diagonal element N(i,i) equals unity When non-realizability arises from a non-causal time delay, an additional delay term e^{−θ_i s} is inserted into the corresponding diagonal entry of N(s) to restore realizability If the non-realizability comes from a right-half-plane (RHP) zero z that has become an unstable pole, the corresponding diagonal element of N(s) is formed as s z s z*.
(3.22) where z* is the complex conjugate of z If it comes from a properness problem, a simple stable pole with the adequate multiplicity can be inserted as follows:
Generally, it is preferable to add the minimum extra dynamics After evaluating the necessary additional dynamics for each configuration, we select the configuration with the fewest RHP zeros or time delays in N(s) The task of determining the minimal N(s) for a given configuration can be formulated as a linear programming problem, providing a systematic optimization framework to minimize dynamic complexity.
Consider a process with the following transfer function matrix:
In this case, to obtain realizability, it is necessary to add an extra time delay associated with the first input The diagonal block N(s) is given by
N Then the new apparent process is obtained:
Adopting the case 1 in Table 3.1, the elements of decoupler matrix are derived as follows:
After determining the decoupler matrix, the process becomes decoupled and diagonal, enabling independent tuning of the decentralized controllers for each q_i(s) In this work, a standard PID tuning approach is applied to ensure both performance and robustness of the design, with gain-margin specifications guiding the PID parameter tuning according to the rules in [2] The resulting control parameters for the two loops are presented in Table 3.2.
Table 3.2 The control parameters of example 1
Figure 3.4 The step responses of the control system and corresponding control signals
1 Garrido, J.; Vazquez, F.; Morilla, F., An extended approach of inverted decoupling, Journal of Process Control, 2011, 21, pp 55-68.
A O’Dwyer (2001) develops a PI and PID controller tuning design for processes with delay that maintains constant gain and phase margins for all delay values The approach is presented in the Proceedings of the Irish Signals and Systems Conference, 2001, pages 96–100.
DESIGN OF SIMPLIFIED DECOUPLING
Multivariable processes with multiple time delays are common in industrial systems, where interactions between process variables make feedback controller design challenging To address this, most control engineers choose between decentralized and centralized control strategies Decentralized control typically uses traditional single-loop PI/PID controllers for multivariable systems because of their simplicity, effectiveness, and ease of implementation, though they perform well only for modestly interactive processes For strongly interacting channels, centralized control is recommended, either via a decoupling network combined with a diagonal matrix controller or through a pure centralized approach This work focuses on centralized control with a decoupling network to convert a multi-input multi-output (MIMO) system into multiple independent SISO loops, enabling the use of conventional design methods to govern the entire system.
Simplified decoupling is the preferred approach because of its straightforward decoupling network, unit-diagonal elements, and robustness When solving decoupling problems, complex transfer functions must be approximated to standard forms while addressing realizability to avoid improper (non-causal) transfer functions A broad set of approximation techniques and reduced-order models is available, including the Prediction Error Method (PEM), least-squares algorithms, and linear least-squares approximation.
[7] or Gaussian approach in the frequency domain, and coefficient matching
CM method [2,5] suffers from heavy computations as the order of MIMO systems increases To overcome this limitation, this work adopts the particle swarm optimization (PSO) algorithm to approximate the complex decoupling elements with well-known industrial process models, specifically first-order plus time delay (FOPTD), second-order plus time delay (SOPTD), and SOPTD with a negative zero (SOPTDNZ) A minor modification is proposed to handle improper transfer functions that may arise from the PSO-based approximation.
However, time delays still exist in the diagonal elements of the apparent decoupled matrix It is obvious that processes with significant dead times are hard to analyze and design based on standard feedback controllers and also make sluggish responses in outputs Smith predictor (SP) scheme, known as dead time compensator (DTC), is one of the appropriate methods to deal with this situation by removing the delay terms from the closed- loop transfer functions [13] However, the original scheme is only applied for SISO systems with their controllers to be tuned for a tradeoff between robustness and performance To overcome this drawback, many variations of the SP were proposed, and the most frequently used structure is a first- order lag filter (FOLF) on the feedback loop, also called filter Smith predictor (FSP) [14, 15] From this method, the DTC structure can handle integrating and unstable processes [16, 17] In addition, other scholars also extended this scheme into the MIMO case as well [18-23]
In order to expand SP to MIMO systems, two approaches normally can be used The first one is a FSP structure which is applied directly into the whole multivariable systems [18-19, 21-22] The second one is a combination with decoupling techniques which simplify the transfer function matrix of processes, and then, address a SP scheme for each single loop as that of the SISO systems [20] In the present work, the latter one is adopted by using the simplified decoupling technique and the original
To simplify, the delay terms are removed from the diagonal elements of the decoupled apparent matrix, creating a delay-free portion that can be used directly for feedback controller design When needed, a first-order lag filter may be placed behind the PI/PID controllers to improve tracking performance and robustness.
Internal model control (IMC) is well known for its effectiveness in designing PI/PID controllers In this work, we present a generalized controller form that includes PI/PID controllers and a FOLF in series The proposed method relies on the direct synthesis approach, where the primary PI/PID controllers are designed to meet desired closed-loop transfer functions Owing to the SP structure, the design proceeds by addressing only the delay-free parts of the diagonal elements of the decoupled apparent matrix to realize multi-loop PI/PID control Consequently, explicit analytical tuning rules are derived for several common industrial processes To assess robustness, the M-Δ structure is analyzed in the presence of diagonal multiplicative output uncertainty.
4.2.1 Simplified decoupling Smith predictor structure
Consider an n-input and n-output open-loop multivariable process with the general transfer function matrix for stable, square, and multi- delays which is represented as the following matrix:
(4.1) where g s ij ( )=g ij 0 ( ) s e − θ ij s , ,i j=1,2, ,… n , of which g ij 0 ( ) s denotes the physically proper, stable, and delay-fee transfer function θ ij represents the time delay.
Figure 4.1 shows the multivariable Smith predictor control scheme, featuring the actual process Gs(s) and its model Gs_hat(s); the delay-free version of the model, Gs_hat^0(s), represents the same process with all delays removed The primary controller Gs_c(s) is implemented in a form designed to work with the Smith predictor, enabling effective delay compensation and interaction handling in the multivariable system.
The closed-loop transfer function matrix from set points to outputs becomes:
When the model is perfect, Gˆ ( ) = ( )s G s , equation (4.3) is rewritten:
Figure 4.1 Block diagram of multivariable Smith predictor control
Time-delay terms have been removed from the denominators of the closed-loop system transfer matrix, and, for a perfect model (Ĝ(s) = G(s)), there are also no delay terms in the transfer-function matrix from disturbances to outputs As a result, the primary controller Gc(s) can be designed for the delay-free part Ĝ0(s), which is the main attractiveness of this control scheme Eq (4.4) can therefore be rewritten as follows:
T G G T (4.5) where T 0 ( ) s = G ˆ 0 ( ) ( ) s G c s ( I G + ˆ 0 ( ) ( ) s G c s ) − 1 is the closed-loop transfer matrix corresponding to the designed controller of the delay-free part
It can be seen that the actual system performance cannot be guaranteed because of the delay term in Eq (4.5), regardless of how good the performance of T0(s) is In the special case where all delay terms of each row of the transfer-function matrix Ĝ(s) are identical, Ĝ(s) can be rewritten as Ĝ(s) = diag(e^{−θ_i s}) Ĝ(0)(s) Consequently, T(s) = diag(e^{−θ_i s}) T0(s), and the system outputs are the delayed outputs of T0(s); to some extent, the system performance and robustness still meet the requirements.
To overcome the loss of the desired property in multivariable Smith predictor control and enhance performance, a decoupling-Smith predictor (DSP) structure is proposed (see Fig 4.2) In this framework, D(s) is the decoupling matrix for the system transfer-function matrix G(s), and Q(s) denotes the decoupled process, defined by Q(s) = G(s) D(s) A delay-free version, Q0(s), is defined identically to Q(s) except that all delays are removed It is evident that Q(s) and Q0(s) share the same nominal dynamics aside from the delays, which facilitates improved decoupling and predictor performance.
Using decoupling techniques, the Q matrix becomes diagonal, which makes controller design for multivariable systems with multiple time delays tractable by reducing the problem to several independent single-loop designs based on Smith predictors This simplification enables the use of various simple design methods to implement effective control systems.
Figure 4.2 Decoupling Smith predictor control system
Decoupling techniques can be implemented using ideal, simplified, or inverted decoupling methods [4] In this study, the simplified decoupling approach is selected due to its robustness and straightforward decoupling network, where the diagonal elements of the decoupling matrix, D_s, are set to unity [4].
5] Truong and Lee proposed an extended method of simplified decoupling for general forms of a MIMO case where the decoupled apparent processes are given as follows [5]:
=C = ≠ (4.6) ii ii ii q = g Λ (4.7) where C=(adj )G T , adjGis the transpose of its cofactor matrix; [ ( 1) ] T ii − ii Λ = G⊗ G , ⊗ is the Hadamard or Schur product, element-by- element multiplication
Using Eq (4.6), the decoupler elements for some typical processes are obtained and summarized in Table 4.1
Table 4.1 The matrix elements of the simplifier decoupler for 2 2× and
Table 4.2 presents the diagonal elements of the decoupled process, each of which remains intricate and not readily usable for analysis and design problems The coefficient matching (CM) method, based on a Maclaurin series expansion and proposed by Truong, aims to reduce the decoupled transfer functions to standard forms and address decoupler realizability; however, its main drawback is the heavy algebraic computation that grows with the order of MIMO systems To overcome these challenges, this work introduces a novel approach for simplified decoupling using a modified PSO algorithm [27–29].
Table 4.2 The diagonal elements of the decoupled matrix for 2 2× and 3 3× processes Process Diagonal elements of decoupled process
4.2.2 The modified PSO algorithm for model reduction
Particle Swarm Optimization (PSO) is a swarm intelligence technique inspired by how groups of animals search for rich food sources In PSO, each potential solution is represented as a particle in a search space with a position that encodes parameter values and a velocity that drives its movement At every iteration, each particle is updated using its personal best position (Pbest) and the swarm's global best position (Gbest)—the best solution found by any particle up to that point The classic PSO update equations adjust a particle’s velocity by combining a cognitive component that pulls it toward Pbest and a social component that pulls it toward Gbest, followed by updating the particle’s position with the new velocity.
( 1) ( ) ( ) ( ) i i Pbest i Gbest i v k+ =ωv k c+ ω x −x k +cω x −x k (4.8) max min max k