TRUONG NGUYEN LUAN VU MULTI LOOP PID CONTROLLER ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY PRESS HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY[.]
FUNDAMENTALS OF MATHEMATICAL
Understanding vector, matrix, and norm theories is essential foundational knowledge for mastering this subject Recognizing the differences between various norms of vectors and matrices enables accurate performance measurement A solid grasp of these concepts is crucial for comprehending the material in this book and applying them effectively in different contexts.
Let A be square n x n matrix, which can be expressed as:
The transpose of a matrix A is a n x n matrix and defined as:
The determinant of a matrix is a crucial concept in linear algebra, particularly for solving systems of linear equations It is defined exclusively for square matrices and plays a key role in understanding matrix invertibility and solving for unknowns The determinant can be calculated using minors of the matrix, which simplifies the process of evaluating complex matrices This mathematical operation is essential for analyzing the properties of matrices and optimizing problem-solving in various applications.
A A (1.3) where Cij denotes the cofactor of aij
A square matrix with a determinant of zero is called a singular matrix, indicating that its rows or columns are linearly dependent The concept of minors helps define the matrix's rank as the order of the largest non-zero minor, which measures the number of linearly independent rows or columns In the case of square matrices, a rank deficiency signifies singularity, emphasizing the matrix's dependence structure.
By definition, the inverse of a non-singular matrix A, denoted A -1 , satisfies A -1 A = AA -1 = I, and is defined as:
Eigenvalues appear in the solution of linear system of equations and are often referred to the solution of the roots of a characteristic equation Consider a matrix operation on a nonzero vector, xx
The eigenvalues i are the n solutions to n’th-order characteristic equation
The eigenvalue decomposition of a matrix can be expressed as
The eigenvalue matrix containing n eigenvalues of A in the diagonal can be given by:
The matrix X is the eigenvector matrix whose columns correspond to the eigenvector xi associated with the eigenvalue i
- The eigenvectors are usually normalized to have unit length, i.e
- The largest of the absolute values of the eigenvalues of A is the spectral radius of A
- The sum of the eigenvalues of A is equal to the sum of the diagonal elements of A
- The product of the eigenvalues of A is equal to the determinant of A
- A and A T have the same eigenvalues (different eigenvectors)
Gershgorin’s theorem states that the eigenvalues of an n x n matrix A are located within the union of n circles in the complex plane Each circle is centered at aii, the diagonal entries of A, with a radius equal to the sum of the absolute values of the off-diagonal elements in the corresponding row, given by ri = ∑_{j ≠ i} |a_{ij}| Additionally, these eigenvalues also reside within a union of n circles centered at aii, but with radii defined by alternative sums, such as r'i = ∑_{j ≠ i} |a_{ji}|, providing multiple bounds for the spectral location of matrix eigenvalues.
- The eigenvalues of a Hermitian matrix are real
- A Hermitian matrix is positive definitely if and only if all its eigenvalues are positive
Let A be an n x n complex (constant) matrix, then the positive square roots of the eigenvalues A H A (where A H means the complex conjugate transpose of A) are called the singular values of A
The maximum and minimum singular values of A are denoted by
The singular values bound the magnitude of the eigenvalues:
The RGA of a complex non-singular n x n matrix A, denoted RGA(A) or A is defined as:
RGA A A A A T (1.17) where the operation denotes element-by-element multiplication (Hadamard or Schur product)
2 The sum of elements in each row (or each column) of RGA is 1
The concept of norm is used to measure the size of a vector, matrix, signal, or system It is denoted by , that satisfies the following properties:
Vector 1-norm This is sometimes referred to the “taxi-cab” norm
Vector 2-norm The most common norm is Euclidean norm
Vector 2-norm The -norm gives the largest magnitude in the vector max i i a a (1.22)
Consider an m x n matrixA = a ij , i1, 2, , j 1, 2,n ,m A norm on a matrix A is a matrix norm if it satisfies the multiplicative property (also called the consistency condition)
Sum matrix norm This is the sum of the element magnitudes:
The Euclidean norm (or Frobenious matrix norm) for a matrix is analogous to the vector 2-norm,
A A A A (1.25) where the trace (tr) operator is the sum of the diagonal elements of the matrix
Max element norm This is the largest element magnitude, max max, ij i j a
1.7 VECTORS, MATRICES, AND NORMS IN MATLAB
Matlab is a powerful mathematics analysis software, widely recognized as an essential tool for control engineering Over the years, it has established itself as the industry-standard platform for control system design, enabling engineers to develop and optimize complex control solutions efficiently.
Table 1.1 Matlab commands are frequently used in the control engineering
A sum(sum(abs(A))) max max, ij i j a
Matlab offers a gentle learning curve, allowing users to quickly experiment and adapt independently This article focuses on tutorials utilizing Matlab and its object-oriented functions within the Control System Toolbox The goal is to highlight useful Matlab features, demonstrate how complex control problems can be efficiently solved with Matlab, and show how it can be applied to various multi-loop control challenges discussed in this book.
TRADITIONAL MULTI-LOOP PID TUNING
TRADITIONAL MULTI-LOOP PID TUNING
The Continuous Cycling method, introduced in 1942, is a well-known approach for determining PID controller parameters This method is particularly effective for first-order plus time delay models, allowing for precise tuning of PID settings Using this technique, PID parameters can be calculated based on specific equations, such as the formula involving a coefficient like 0.6, to optimize controller performance and ensure stable system behavior.
The second one is called Process Reaction Curve formula proposed in 1943 Based on this method, the PID controller can be obtained from the following equation:
2.1.1 Advantages and Disadvantages of Z-N method
The features of Z-N method are shown in Table 2.1:
Table 2.1 Analysis of the advantages and disadvantages of Z-N method
1 It is easy to understand and apply in the industry
It does not work for plants whose root loci do not cross the imaginable axis for any values of gain
2 It is one of the standard techniques which many authors consider and extend
The results of closed-loop system are often more oscillated than desirable
Fig.2.1 Set-point response for WB column 2.1.2 Case Study
Example 2.1 Wood and Berry (WB) distillation column
Wood and Berry [3] introduced the following transfer function model of a pilot-scale distillation column, which consists of an eight-tray plus reboiler separating methanol and water,
Considering to Eq.2.1, the single-loop Z-N controller and a multi- loop PID controller designed by McAvoy [2] can be obtained as:
Figure 2.1 demonstrates that single-loop tuning, where both loops are automatic, is not feasible for MIMO systems due to the potential risks associated with the closed-loop response, highlighting the challenges in applying traditional tuning methods to complex multi-input, multi-output systems.
2.2 BIGGEST LOG MODULUS TUNING METHOD (BLT) [4]
The biggest log modulus tuning (BLT) provides such a standard tuning methodology proposed by Luyben in 1986
Fig 2.2 Block diagram for the multi-loop control system
The key feature of this method is the maximum closed-loop log modulus, which indicates the height of the resonant peak in the log modulus Bode plot of the closed-loop servo transfer function In a multi-loop feedback control system, such as the one shown in Fig 2.2, the closed-loop log modulus provides essential insights into system stability and response characteristics The closed-loop servo transfer function can be derived from the system configuration, allowing for accurate analysis of resonance behavior and control performance Understanding the maximum closed-loop log modulus is crucial for optimizing control system stability and achieving desired dynamic responses.
G G I G G (2.6) where G(s) is a matrix of open-loop process transfer that is the typical form
The feedback controller matrix G c(s) is diagonal since we use N- SISO controllers:
Figure 2.3 illustrates the BLT tuning procedure, beginning with the calculation based on Ziegler-Nichols settings for each individual loop The process involves identifying the ultimate gain and ultimate frequency from the diagonal transfer function Gii(s), which are derived from the process model Controller gains (Kc) are then calculated by dividing the Ziegler-Nichols gain by a factor F, where a higher F results in a more stable but slower system response To optimize performance, controller parameters are adjusted to produce a maximum peak of +2N dB in the closed-loop servo log magnitude curve, where N represents the process order Proper selection of gain and reset times ensures the overall system stability while achieving acceptable load responses.
The number of encirclements of the (-1, 0) point by W(iω) as ω varies from 0 to infinity indicates the number of zeros of the closed-loop characteristic equation in the right half of the s-plane Designing a feedback controller to achieve a maximum resonant peak in the closed-loop log modulus plot enhances system stability and resonance characteristics Understanding the relationship between encirclements and pole-zero locations is essential for effective control system design.
Fig.2.3 The block diagram for BLT tuning method
The peak in plot Lcm over the entire frequency range is the biggest log modulus L max cm , the F factor is varied until L max cm is equal to 2N, where
N is order of the system For N = 1, in the SISO case, we get the familiar
+2 dB maximum closed-loop log modulus criterion For a 2x2 system, a + 4 dB value of L max cm is used; for 3x3, +6 dB; and so forth
IfL cm max 2N, increase F log 1
The ultimate gain K ui and ultimate frequency ui
2.2.1 Advantages and Disadvantages of BLT method
The features of BLT are shown in Table 2.2:
Table 2.2 Analysis of the advantages and disadvantages of BLT method
1 It is such a standard tuning methodology so that it is easy to understand and used by control engineers
The ability of the detuned loop to reject disturbances is reduced, leading to poorer transients
2 The most useful frequency- domain specification is the maximum closed loop log modulus
The BLT procedure was usually applied with PI controllers The method can be extended to include derivative action However, it is complicated in use
The frequency response curves for the closed-loop servo transfer function can be efficiently determined graphically using a Nichols chart Additionally, these curves can be effortlessly generated with the aid of digital computers, streamlining the analysis process.
The unbalanced responses between loops are seen
4 It is easy to design multi-loop
Ziegler-Nichol methods and detuning the factor F
It sometimes provides a poor stability coupled with robustness and unsatisfied performance for multivariable control systems
5 It is better than the phase margin criterion in case of large deadtime process (The
ZN settings give very large phase margins for large deadtime)
It gives poor results in calculated the Integral-Absolute-Error (IAE)
6 - Detuning with independent parameters may be considered with the added advantage of reducing interactions by different amounts of the other loop
Example 2.1 Wood and Berry (WB) distillation column
Accordingly to the procedure of BLT method, we can easily find the detuning factor and tuning parameter at the Fig 2.4 as below:
Fig.2.4 The scalar W(jω) = −1 + det[I + G(jω)Gc(jω)] function for WB column with BLT tuning
Figure 2.5 expresses the closed-loop responses of BLT that compare with empirical method and shows that BLT method has a stable performance
Fig.2.5 Closed-loop time responses of each loop for Example 1 (WB)
(Solid line: BLT, Dash dotted line: Empirical)
Example 2.2 Ward and Wood column (WW) [3]
Considering the block diagram in Fig 3, it provided the controller for WW column as following:
Figure 2.6 demonstrates how well the closed-loop responses in set- point change and disturbances rejection in both loop
Fig.2.6 Closed-loop time responses of each loop for example 2 (WW)
(Solid line: BLT-PI, Dash dotted line: Empirical-PI)
2.3.1 Sequential Loop Closing with Relay Auto-Tuning
The MIMO sequential relay tuning procedure involves a systematic process to optimize controller parameters First, a relay feedback test is performed between y1 and u1 with loop 2 on manual, enabling the setting of a Ziegler-Nichols PI/PID controller using the ultimate gain and frequency Next, the controllers for loop 2 are designed via a second relay feedback test between y2 and u2, while loop 1 operates in automatic mode The process continues by conducting a relay feedback experiment between y1 and u1 with loop 2 in automatic mode to update the tuning constants for loop 1 This iterative procedure is repeated until the control settings converge, ensuring optimal combined dynamics for the MIMO system.
Tuning of this loop closed becomes similar to tuning the faster loop first
In accordance with set-point weighting, the normal PI control law becomes:
(2.16) where β is generally between 0 and 1
The closed-loop transfer function relating the set-point, R(s), and the plant outputs, Y(s), is given by:
2.3.3 Advantages and Disadvantages of SAT method
The features of SAT method are shown in Table 2.3:
Table 2.3 Analysis of the advantages and disadvantages of SAT method
1 It makes problem become simple
It is not necessary to identify all individual transfer for purpose controllers tuning
2 It identifies processed information around the important frequency, the ultimate frequency (the frequency where the phase angle is -π )
Loops designed earlier are usually faster than later loops
3 It is a closed-loop test so the process will not drift away from the nominal operating point
For the high dead time and nonlinear process, it sometimes obtains high overshoot and disturbance
4 For process with a long time constant, it is a more time- efficient method than conventional step or pulse testing The experimental time roughly equals to 2~4 times of the ultimate period
It is sometimes has poor performance together with robustness
5 It operates in an efficient manner
The IAE values is significantly large
6 It is a more accurate approach -
Fig 2.7 Sequential tuning procedure for 2x2 systems y 1
Example 2.3 Wood and Berry (WB) distillation column
The design of PID controller for the WB by following the relay feedback approach which is showed in Fig.2.8:
Fig.2.8 Closed-loop time responses of each loop for Example 3 (WB)
(Solid line: SAT, Dash dotted line: BLT)
Figure 2.8 shows the comparison between SAT method and BLT method and we can also say that SAT method makes control system more balance and be a good performance than BLT
2.4 GAIN AND PHASE MARGIN (GPM) METHOD [6]
Gain and phase margins, illustrated in Fig 4.1, are essential frequency-domain specifications that indicate system stability robustness These margins measure how close the open-loop transfer function Gc(iω)Gp(iω) is to the critical point (-1, 0) on the polar plot By assessing gain and phase margins, engineers can determine the stability boundaries of the control system, ensuring reliable performance under disturbances and parameter variations Accurate consideration of these margins is vital for designing stable and resilient control systems.
The phase margin (PM) is the angle between the negative real axis and the line from the origin to the intersection point of the unit circle and the GpGc curve It is a crucial stability parameter in control systems, indicating how much phase shift can occur before the system becomes unstable A larger phase margin signifies a more robust system, capable of tolerating greater phase variations without loss of stability Understanding phase margin helps in designing control systems that maintain desired performance under various operating conditions.
A larger phase margin indicates a more stable closed-loop system, while a negative phase margin signifies instability Typically, a phase margin of around 45° is considered ideal for system stability Additionally, the gain margin plays a crucial role in assessing system robustness and stability.
Gain Margin (GM) refers to the distance between the origin point and the intersection point of the negative real axis and the GpGc curve, indicating system stability A higher gain margin signifies a more stable control system, with a GM of approximately 2 commonly considered optimal Monitoring and maintaining an adequate gain margin is essential for ensuring robust system performance and stability.
By following to Ho, W.K., Hang, C C and Cao, L S., (1995), the gain margin Am and phase margin φ m of qii(s) are shown in Fig 4.2 for multivariable feedback system which can be expressed as:
(2.23) where, B is the intersection point of Gershgorin circle and unit circle at the gain crossover frequency ω g ; C is the intersection point of
Gershgorin circle and the negative real axis at the phase crossover frequency ω p ; qij is open-loop transfer stable system; gij(s) is the process transfer function
Fig 2.9 The Nyquist plot of GMGC plane
The PI controller can be designed for each loop using the gain and phase margin tuning formulas for SISO system
Fig 2.10 A typical Nyquist diagram with the Gershgorin circle at the pi ii ci m ii k A K
Ii pi ii pi ii
Note that θ is the delay time in process
2.4.1 Advantages and Disadvantages of GPM method
The features of GPM tuning method can be analyzed as shown in Table 4.1:
Table 2.4 Analysis the Advantages and Disadvantages of GPM method
1 It can be solved by numerical methods or Bode plots
It is not fit for adaptive control and auto-tuning
2 It is fit for implementing self- tuning control
It is extended from SISO PI/PID controller (Ho et al.,1995) to MIMO PI/PID controller so that sometimes it happens the unbalance in each controller
3 It can be used to design a multi-loop controller which satisfies desired system robustness and stability
It does not require the shaping of the Gershgorin band through trial and error graphically
4 The interaction can be controlled based on Gerhgorin bands
Example 2.4 Wood and Berry (WB) distillation column
In accordance with Eqs 2.26 and 2.27, the controller can be obtained by:
Fig.2.11 Closed-loop time responses of each loop for Example 1 (WB)
(Solid line: GMP-PI, dash dotted line: BLT) The output responses for both GM and BLT design methods can be seen from Fig 11
Following the same procedure above, the controller can be obtained by:
Fig 2.12 Closed-loop time responses of each loop for example 2 (WW)
(Solid line: GPM-PI, Dash dotted line: BLT-PI)
The output responses for both GM and BLT design methods can be seen from Fig 12
D Chen and D.E Seborg (2002) introduced a multi-loop PID tuning method based on Gershgorin bands, which is essential for calculating ultimate gains and ultimate frequencies The ultimate point for the lth Gershgorin band is identified as the crossing point of G(s) on the negative real axis, possessing the maximum magnitude, with the ultimate gain (K cul) and the lth ultimate frequency (ω cul ) defining this point This ultimate point, represented as point A in Fig 2.13, is crucial for tuning PID controllers and is derived from the corresponding ultimate gain and frequency functions.
min , ul K cl K cl K cl
Fig 2.13 The ultimate point of lth Gershgorin band
The procedures to design controller for MIMO system:
Step 1 Calculate the ultimate gain and the ultimate frequency for each control loop
Step 2 Set the parameter controller by using Ziegler and Nichols
Table 2.5.: Modified Ziegler-Nichols tuning rules
Table 2.6: PI controller setting for the Wood and Berry column (using the multi-loop Gershgorin bands design method)
2.5.1 Advantages and Disadvantages of GB method
The features of GB tuning method can be analyzed as shown in Table 2.7:
Table 2.7 Analysis the Advantages and Disadvantages of GB method
1 It is no limited to second- order plus dead time process or any other specific forms
It depends on the detuning factor f and each value of f is obtained new controller parameters
2 It uses Nyquist array analysis to tune multi-loop PI/PID controler so it is easy to understand
Sometimes the closed-loop performance with f = 1 is similar to BLT performance It means that the MIMO controller may be unbalanced
3 A pre-compensator is required to achieve the diagonal dominance
4 It provides a good set-point tracking and disturbance rejection for moderately interacting systems
According to (2.32), (2.33) and (2.34), the PI controllers can be obtained by:
The comparative study between GB and BLT design methods is showed in Fig 2.14
Fig.2.14 Closed-loop time responses of each loop for example 2 (WW)
(Solid line: GB-PI, Dash dotted line: BLT-PI)
[1] Ziegler, J.G., & Nichols N.B (1942) Optimum settings for automatic controllers Trans ASME, 64(11), 759-768
[2] McAvoy, T J (1983) Interaction Analysis Instrument Society of America, Research Triangle Park, USA
[3] Wood, R K., & Berry, M W (1973) Terminal composition control of a binary distillation column Chemical Engineering Science, 28(9), 1707-1717
[4] Luyben, W L (1986) Simple method for tuning SISO controllers in multivariable systems Industrial & Engineering Chemistry Process Design and Development, 25(3), 654-660
[5] Loh, A P (1993) Autotuning of multivariable proportional- integral controllers using relay feedback Ind Eng Chem Res., 32, 1002-1007
[6] Ho, W.K., Hang, C C., & Cao, L S (1995) Tuning of PID controllers based on gain and phase margins specifications Automatica, 31 (3), 497-502
[7] Chen, D., & Seborg, D E (2002) Multi-loop PI/PID controller design based on Gershgorin bands IEE Proceedings-Control Theory and Applications, 149(1), 68-73.
MULTI-LOOP PID CONTROLLER DESIGN FOR
MULTI-LOOP PID CONTROLLER DESIGN FOR ENHANCED DISTURBANCE REJECTION
The multi-loop PI/PID controller has been extensively studied over several decades, with the IMC approach emerging in the 1980s and 1990s as a prominent tuning method for multi-loop PID controllers, initially introduced by Garcia and Morari Notable design strategies for multi-loop IMC controllers, considering control loop interactions, were proposed by Rivera and Morari, Economou and Morari, and Basualdo and Marchetti The BLT method, developed by Luyben, remains a widely used and effective approach for multi-loop PI control systems During the 1990s, advancements included Loh et al.’s auto-tuning procedure to enhance closed-loop frequency responses in MIMO systems and Jung et al.’s decentralized lambda tuning (DLT) method aimed at improving stability and robustness More recently, the generalized IMC-PID approach has been developed by Lee et al for multi-loop PID control systems, further advancing the field.
This article discusses the limitations of traditional multi-loop tuning methods, which are primarily designed for set-point tracking in SISO systems, as highlighted by Lee et al [9] Despite its importance in industry, disturbance rejection remains underrepresented in existing approaches, often leading to sluggish responses in multivariable processes While multi-loop PI/PID controllers are commonly employed due to their simplicity and widespread use, most design methods, including IMC and direct synthesis techniques, focus on creating controllers optimized for set-point tracking rather than effective disturbance rejection Consequently, these PID controllers tend to perform poorly when managing disturbances, resulting in less responsive control in practical applications.
Figure 3.1 A standard multi-loop control system
We propose a novel design method based on the generalized IMC-PID approach combined with Ms Criterion to optimize multi-loop PI controllers This method aims to improve disturbance rejection and set-point tracking performance, ensuring more robust and accurate control system behavior.
The performance and robustness of a multi-loop IMC control system primarily depend on the closed-loop time constant (), which can be optimally determined using Ms Criteria This method effectively compensates for disturbances by targeting the dominant poles in the process transfer function's diagonal elements Extending the generalized IMC-PID approach to multi-loop PI controllers offers significant advantages, including enhanced disturbance rejection—since the feedback signal cancels out disturbance influences—and the flexibility to implement complex nonlinear control algorithms within the IMC structure without stability concerns Overall, the proposed control strategy improves the robustness of MIMO processes by effectively managing disturbances and maintaining system stability.
Ms Criterion is also suggested by adjusting the design parameter () for the tradeoff between robust stability and performance
3.2 GENERALIZED IMC-PID METHOD BASED ON DISTURBANCE REJECTION FOR MULTI-LOOP PI/PID CONTROLLER DESIGN
Consider the multi-loop feedback control system in Figure 3.1, the closed-loop transfer function can be expressed as
+ r f r y where G(s) is the n x n open-loop stable processes, G c (s) is multi- loop controller, and y (s), r f (s) and d (s) are the controlled variable vector, the set-point vector, and disturbance vector, respectively
The closed-loop set-point transfer function for the multivariable feedback control system
The article discusses the closed-loop transfer function, denoted as H(s), which is defined by the relationship H(s) = G(s) * G_c(s) Here, G(s) represents the process transfer function, which is open-loop stable, while G_c(s) is a multi-loop controller with diagonal elements only The variables y(s) and r(s) correspond to the controlled variable and the set-point, respectively, emphasizing the control system's structure and stability considerations.
Deriving from the design strategy of multi-loop IMC controllers [1-
4] and putting the IMC filter that reject dominant pole in each element of the process transfer function, the desired closed-loop response R i of the ith loop typically chosen by
In the discussed control system design, Gii+ is chosen as the all-pass form, representing the non-minimum phase component, with λi serving as an adjustable parameter to optimize performance and ensure system stability The parameter ni is selected sufficiently large to make the IMC controller effective, while βij is configured to cancel the dominant poles within each process element, typically targeting one or two poles Ensuring that Gii+ at zero frequency equals one (Gii+(0) = 1) is crucial for achieving accurate set-point tracking of the controlled variable.
The desired closed-loop response matrix R(s) can be articulated as
Note that i is analogous to the closed-loop time constant and thus it determines the speed of the closed-loop response The multi-loop controller ~ ( ) c s
G with integral term can be expressed in a Maclaurin series as ci 1 c0 c1 c2 2 3
G = G + G + G + (3.5) where G c0 , G c1 and G c2 keep up a correspondence with the integral, proportional, and derivative terms of the multi-loop PID controllers, respectively
The multivariable process, G(s), can be written in a Maclaurin series as
The closed-loop transfer function, H(s), can be expressed in a Maclaurin series by substituting Eqs (3.5) and (3.6) into Eq (3.2)
R can also be expressed in a Maclaurin series as
As discussed in Chapter 2, the impact of proportional and derivative terms (i.e., 1 ~ 2
At high frequencies, G) dominates and should be designed considering process characteristics specific to high-frequency performance Conversely, at low frequencies, the integral term's design becomes crucial, ensuring system stability and accuracy across the entire frequency spectrum.
G c , the interaction effect between the control loops cannot be neglected Therefore, the analytical tuning rule for the integral time constant of the multi-loop PID controller can be derived
3.3 DESIGN OF MULTI-LOOP PROPORTIONAL GAIN
The influence of G c0 and G c1 on the PID formula is significant at high frequencies, while their effect is negligible at low frequencies Therefore, designing a multi-loop PI controller should focus on the process characteristics observed at high frequencies to ensure optimal performance.
Given that G(jw)G c (jw) 1 at high frequencies, H(s) can be approximated to
Dropping the off-diagonal terms in G(s), the ideal multi-loop controller G c (s)can be obtained as
The ideal controller G c (s)of the ith loop can be designed as
-1 m j ii- ij i j=1 ci m ii i n j i ii+ ij j=1
(3.11) where Q i(s) is IMC controller given by
Because Gii+(0) is 1 , Eq (3.11) can be rewritten in a Maclaurin series with an integral term as
Following Equation (3.12), it is evident that the first three s-terms are the most critical components of the controller, while the remaining terms are negligible The proportional gain and derivative time of the multi-loop PI controller can be effectively determined based on these key terms.
3.4 DESIGN OF MULTI-LOOP INTEGRAL TIME CONSTANT
At low frequencies, the integral term Gc0 dominates, making the interaction characteristics in this frequency range particularly significant By analyzing the off-diagonal elements of the first-order s-terms in Equations (3.7) and (3.8), the integral time constant can be accurately determined, providing valuable insights into system behavior at low frequencies.
The lead term (βi+1) in Equation (3.3) can result in excessive overshoot in the set-point response, reducing control accuracy Implementing a two-degree-of-freedom structure effectively addresses this issue by designing a set-point filter qi, enhancing the system's stability and response quality.
Tuning formulae based on Eq (3.13) and Eq (3.14) offer a significant advantage in solving the optimization problem for determining PID parameters By expressing all PID parameters through a single design parameter, λ, the approach simplifies the process and reduces the search space for optimization This streamlined method enhances the efficiency and effectiveness of tuning PID controllers for various processes.
3.5 MS CRITERION FOR ENHANCE THE ROBUSTNESS OF THE MULTI-LOOP CONTROL SYSTEM
Ms tuning is a frequency-domain method that focuses on the resonant peak Ms, which is directly related to the sensitivity function's resonant frequency The value of Ms indicates the relative stability and robustness of a closed-loop control system, with a smaller Ms suggesting a more stable and resilient system Since 1996, Ms tuning has been recognized as an effective approach for optimizing control system performance.
Skogestad and Postlethwaite [10] employed Ms as a tool for measuring system robustness
In 1998, Astrum et al highlighted that the ideal setting for the maximum sensitivity (Ms) in SISO systems ranges from 1.2 to 2., ensuring system robustness Ms tuning serves as a crucial method to define the closed-loop time constant, facilitating the determination of optimal controller parameters for enhanced system performance.
The sensitivity function in the multi-loop control system can be represented by
The sensitivity frequency response can be found by setting s = j in term of and as follows
The sensitivity function can be expressed by the matrix form as
The maximum sensitivity Ms is obtained as the maximum value of the sensitivity function over frequencies
The peak magnitude of the sensitivity function can be expressed by the matrix form as
The proposed Ms tuning method enhances the performance and robustness of closed-loop frequency responses in multi-loop control systems by optimizing a single parameter, λ This approach ensures that the multi-loop control system meets stability bounds, with all PID parameters expressible through a unified design parameter, λi The optimization process in the frequency domain aims to minimize λ and ω0i while satisfying stability constraints, resulting in a more reliable and efficient control system.
DESIGN OF MULTI-LOOP PID CONTROLLERS
DESIGN OF MULTI-LOOP PID
GENERALIZED IMC-PID METHOD WITH MP
Multi-loop PID controllers continue to be widely used in the process industries despite advancements in multivariable controller synthesis for MIMO (multiple input, multiple output) systems Their popularity is primarily due to their simplicity, ease of implementation, and proven effectiveness in managing complex processes Multi-loop PID controllers offer reliable control performance, cost-effectiveness, and familiarity for engineers, making them a preferred choice in various industrial applications.
Multi-loop PID controllers can be easily implemented
They are easy to understand the control structure
They require fewer parameters to tune than full multivariable controllers
Loop failure tolerance of control system can be assured during the design stage
Therefore, many controller tuning methods have been suggested to tune the multi-loop control system
This chapter introduces a straightforward design method for tuning multi-loop PID controllers that accounts for loop interactions, enhancing overall system performance Based on the generalized IMC-PID approach, the integral component of PID controllers dominates at low frequencies, while the proportional and derivative terms are more influential at high frequencies The proposed PID controller aims to improve the closed-loop performance of multi-loop control systems by effectively balancing these frequency-dependent control actions.
To improve the stability and performance of PID control systems, the resonant peak (Mp)—the maximum magnitude of the closed-loop frequency response—is used as a key design criterion to balance robustness and performance The weighted sum of Mp serves as an effective objective function to optimize overall system performance, accounting for both diagonal and off-diagonal elements in multivariable processes Numerous examples demonstrate the application of this approach across various complex control systems, ensuring enhanced stability and robust control performance.
Figure 4-1 Block diagram for the multi-loop control system
4.2 GENERALIZED IMC-PID APPROACH FOR MULTI-LOOP PID CONTROLLER DESIGN
Figure 4-1 shows the standard of multi-loop feedback control system, of which G(s) denotes the process transfer function matrix ~ ( ) c s
G represents the multi-loop controller with diagonal elements only; y(s) and r(s) are the controlled variable and the set-point vectors, respectively
The closed-loop response to the set-point change is obtained by
( s H s r s I G s G c s 1 G s G c s r s y (4.1) where H(s) is the closed-loop transfer function matrix
The desired closed-loop response of the multi-loop system is given in term of the diagonal matrix as follows:
In accordance with the design strategy of the IMC controller [2-4], the desired closed-loop response R i of the ith loop is chosen by y ( ) ( ) r ( ) ( ) (λ 1) i i ii i n i i s G s s R s s
(4.3) where G ii+ is the non-minimum part of G ii , which is included any time delays and right-half plane zeros and chosen to be the all pass form
G ii+ must have a steady-state gain of one to ensure proper system behavior The order n i, a positive integer typically chosen as n i = 1, should be sufficiently large to make the IMC controller realizable The desired closed-loop time constant, λ i, which is adjustable, influences the speed of the response — smaller λ i results in faster response Adjusting λ i allows for balancing robust stability and performance, enabling the control system to meet specific stability criteria.
G with integral term can be expressed in a Maclaurin series as
G (4.4) where G c 0 ,G c 1 and G c 2 correspond to the integral, proportional and derivative terms of the multi-loop PID controller, respectively
It is obvious from Eq (4.4) that the impact of proportional and derivative terms (i.e., 1 ~ 2
G ) becomes more predominant at high frequencies, and thus they should be designed based on the process characteristics at high frequencies On the other hand, the integral term
G c is dominating at low frequencies, and thus it should be designed based on the process characteristics at low frequencies
In multi-loop systems, the characteristics of closed-loop interactions vary across different frequency ranges By leveraging these frequency-dependent properties, the analytical design of multi-loop PID controllers can be significantly simplified This approach ensures that the interaction effects are fully considered, leading to more effective and robust control system performance.
At high frequencies, the magnitude of open-loop gain becomes to be ( ) ( )G jG c j 1 so that H(s) can be approximated to
The design of G and G ~ c1 can focus solely on the main diagonal elements within the multivariable process G(s), enabling the direct application of the single-loop IMC-PID method for tuning the proportional and derivative terms of a multi-loop PID controller This approach simplifies the controller design process, ensuring the implementation of an ideal multi-loop feedback system that achieves the desired closed-loop response.
Therefore, the ideal controller of the ith loop can be designed by
(4.7) where G ii- is the minimum part of G ii
Since G ii+ (0)=1, Eq (4.7) can be rewritten in a Maclaurin series with an integral term as
The standard PID control algorithm is given by
By comparing Equation (4.8) with Equation (4.9), we derive analytical tuning rules for the proportional gain and the derivative time constant of the multi-loop PID controller These tuning rules specify the relationships between controller parameters, including the proportional gain \( K_{i} \) and derivative time constant \( D_{i} \), to optimize control performance Implementing these formulas can enhance the stability and responsiveness of multi-loop PID control systems, making them essential for accurate process regulation.
Now, we consider the design of the integral term ~ 0
G c At low frequencies, the interaction effect between the control loops cannot be neglected The expansion of G(s) in a Maclaurin series gives
Substituting Eq (4.4) and Eq (4.11) into Eq (4.1) and rearranging it, H(s) can be rewritten as
The desired closed-loop response R ~ can also be extended in Maclaurin series as
Comparing the diagonal element of H(s) in Eq (4.12) and R(s)in
Eq (4.13) for the first-order term, we get the analytical tuning rule for the integral time constant of the multi-loop PID controller as follows: ii ci i i ii Ii
Using the tuning formulas from Eq (4.10) and Eq (4.14), we can easily derive analytical multi-loop PID tuning rules for various transfer function models of multivariable processes All PID parameters can be expressed through a single design parameter, λᵢ, significantly reducing the optimization search space This approach simplifies the tuning process, enhances efficiency, and improves control performance for complex multivariable systems.
4.3 WEIGHTED SUM MP CRITERION FOR MULTI-LOOP
Effective control system tuning aims to achieve optimal closed-loop performance in the time domain, but analyzing time response analytically can be challenging due to the lack of unified design methods Conversely, the frequency domain offers numerous indices that are applicable beyond specific process models and provides a more convenient framework for assessing system robustness against uncertainties Additionally, frequency-domain characteristics can be leveraged to estimate key time-domain properties, facilitating comprehensive control system analysis and design.
The resonant peak (Mp) in a SISO system represents the maximum magnitude of the closed-loop frequency response, specifically the complementary sensitivity function Mp is a key indicator of a control system's performance and robustness, providing valuable insights into how the system responds to various frequencies A lower Mp typically signifies better robustness and reduced resonance, making it an essential parameter in control system design Overall, managing Mp is crucial for ensuring optimal system stability and performance in practical applications.
Mp should lie between 1.1 and 1.4 in the SISO systems [5]
In the MIMO system, the complementary sensitivity function (closed-loop transfer function) can be expressed in term of the matrix form
H (4.15) where H ij represents the closed-loop transfer function of the ith loop to the set-point change in the jth loop
Overall control performance depends on both the diagonal and off-diagonal elements of the closed-loop transfer function matrix H Relying solely on the Mp specification for diagonal elements, similar to the complementary sensitivity in SISO systems, does not guarantee optimal performance To achieve a well-balanced control response, Hii should be close to unity across a wide frequency range, while H ij should remain near zero throughout the same range To incorporate these considerations into controller design, a weighted sum of individual Mp values is used as the objective function for optimizing PID parameters, ensuring a comprehensive approach to system robustness and stability.
This article discusses a control system optimization where the goal is to ensure the diagonal elements of the matrix Mp satisfy the inequality Mp ii ≥ Mp low, guaranteeing a minimum required speed The matrix Mp is defined by the maximum value of H ij (jωλ, ) over non-negative frequencies, representing the system's frequency response characteristics An adjusting factor w, ranging from 0 to 1, is introduced to balance the performance weighting between diagonal and off-diagonal closed-loop responses, allowing for tailored optimization of system stability and response performance This approach helps achieve a robust control design by maintaining the lower bound of the diagonal elements while considering overall system performance trade-offs.
Figure 2-2 Optimum solution map of the WB column
The optimal values of Mp low and w in a multi-loop system depend on the priority of specific closed-loop transfer functions and can serve as key tuning variables However, within certain ranges, their impact on performance is minimal, allowing these parameters to be fixed at constant values for simplicity Our simulation results indicate that, for balanced closed-loop performance, the ideal w value lies between 0.5 and 0.75, while Mp low should be between 1.1 and 1.4 As demonstrated in Figure 2-2, the integral absolute error (IAE) surface is relatively flat within these recommended ranges, signifying robustness, but becomes sharply steep outside them, emphasizing the importance of proper parameter selection for optimal system performance. -**Sponsor**Need help rewriting your article to be SEO-friendly and coherent? It can be tough making sure each paragraph is impactful! With [Article Generation](https://pollinations.ai/redirect-nexad/ygM9n8g0?user_id=983577), you can instantly create 2,000-word SEO-optimized articles Forget struggling with rewrites – save over $2,500 a month compared to hiring a writer and get your content ranking higher! It's like having a content team without the hassle!
The closed-loop performance of a multi-loop control system can be effectively evaluated using key performance indices, including the Integral Absolute Error (IAE), the Integral of the Square of Error (ISE), and the Integral Time-Weighted Absolute Error (ITAE) These metrics collectively provide a comprehensive measure of system accuracy and responsiveness, aligning with best practices in control system analysis and optimization Incorporating these indices into performance assessment helps ensure precise, reliable, and efficient multi-loop control system operation.
In this chapter, the IAE is considered in the case of a step set-point change, which is defined as
IAE (4.17) where R and Y are the input and output signals, respectively T is a finite time, which is chosen for the integral approach the steady-state value
To illustrate the superior performance of the proposed method, the closed-loop responses by the proposed controller were compared with those by several well-known tuning methods as follows:
(1) The biggest log modulus tuning (BLT) method of Luyben [6]
(2) The decentralized lambda tuning (DLT) method of Jung et al [7]
(3) The sequential auto tuning (SAT) method of Loh et al [8]
Several standard multivariable processes were also considered to study for ensuring the effectiveness of proposed method
Example 4.1: Wood and Berry (WB) distillation column [6]
The WB column model in Eq (4.18) is a diagonally dominant process widely used in the process industry, serving as a benchmark for various multi-loop PI/PID controller design methods such as BLT, DLT, and SAT The proposed control strategy was compared with these existing methods through simulation studies involving sequential step changes in set-points across individual loops In the simulations, the process model order dictated setting ni in Eq (4.3) to 1 for all loops, with optimal λi values identified as 0.029 and 3.227 for loops 1 and 2, respectively All control parameters are detailed in Table 4.1, which also presents the Integral Absolute Error (IAE) values of the closed-loop responses; these were used to evaluate performance Additionally, values of Mp low (1.2) and w (0.75) were selected for all examples in the chapter Figure 4-3 illustrates the closed-loop responses achieved by various design methods when applying sequential unit set-point changes at the first and second loops, demonstrating the effectiveness of the proposed approach.
Figure 4-3 Closed-loop time responses for the WB column
IMC-PID CONTROLLER FOR IDEAL
IMC-PID CONTROLLER FOR IDEAL
Decentralized (multi-loop) and centralized control schemes are commonly used to manage complex process interactions Multi-loop PI/PID controllers are effective and simple options for controlling multivariable processes with modest interactions that are closely decoupled, offering fault tolerance and adequate performance However, these controllers often struggle with significant interactions, making centralized (fully cross-coupled) PID controllers more suitable in such scenarios Centralized control can be implemented through a pure centralized strategy or a decoupling network combined with multi-loop controllers, with decoupling networks being highly attractive in both academic research and industrial applications Numerous decoupling schemes, primarily focused on two-input, two-output (TITO) systems with dynamic decoupling, have been developed; however, many real-world processes in industry involve more than two inputs and outputs, presenting ongoing challenges and opportunities for control system design.
The proposed method’s effectiveness was demonstrated through several examples of interacting multivariable processes Simulation results showed that the proposed method consistently performed better than other existing methods
Consider a multivariable process with the following transfer matrix:
G(i,j)(s) can be expressed as G(i,j)(s) = G(i,j)0(s) e^{-Lij s}, where G(i,j)0(s) represents strictly proper, stable scalar rational functions, and Lij are nonnegative time delays associated with each transfer function The delay-free component of the process is denoted by G0 = [gij0].
The multivariable Smith predictor control scheme is illustrated in Fig 5.1, featuring the process G(s), its model ˆG, and the modified model ˆG₀ with all delays removed The primary controller C(s) is designed to optimize process control When the model perfectly matches the actual process (i.e., ˆG = G and ˆG G₀ = 0), the closed-loop transfer function from the reference input R to the output Y is significantly simplified, enhancing control accuracy and system stability This control strategy effectively compensates for process delays, improving multivariable process regulation.
The analysis shows that (I + G₀C) contains no delays, which is also reflected in the multivariable controller C, allowing the primary controller C to be designed based on the delay-free part G₀—highlighting the control scheme's main advantage However, unlike the SISO case, even if C is designed to achieve desired performance with H₀ = G₀C[I + G₀C]⁻¹, the actual system performance cannot be fully guaranteed This limitation is evident from the closed-loop transfer function.
System performance can be significantly degraded due to the presence of GG - 10 When delays are identical across all elements in each row of the transfer matrix, the finite poles and zeros in GG - 10 are canceled out, resulting in GG - 10 = diag{e^{-L s_{ii}}} Under these conditions, the system output effectively becomes a delayed version of the original output H₀ However, this ideal property generally does not hold in most practical scenarios, affecting overall system performance.
Fig.5.2 Decoupling Smith control scheme
To enhance the performance of the multivariable Smith predictor control system, a decoupling scheme is introduced, featuring a decoupler D for the plant G, with Q representing the decoupled process GD and Q0 identical to Q but without delays Decoupling GD results in Q and Q0 becoming diagonal matrices, simplifying the multivariable Smith predictor design This approach reduces the complex multivariable control problem to multiple single-loop Smith predictors, enabling the use of various straightforward design methods for improved control system performance.
5.2.1 Ideal Decoupling Smith Control Scheme
The decoupled process in the ideal decoupling system is expected to have a diagonal transfer function matrix in the following form:
For GD to be decoupled, it is clear that the (i, j) th element of an ideal decoupler can be expressed generally as:
5.2.2 Simplified Decoupling Design for Typical Processes
This section analytically develops an ideal decoupling for 2 2
In accordance with the decoupling requirement in (8), four equations can be established as:
The decoupler elements are then given by the solution:
The aforementioned procedure can also be simply applied to derive analytical forms of decoupling elements for other high-dimensional multivariable processes with multiple time delays
As the order of the process increases, the transfer functions of the decoupler elements become too complex for straightforward use in decoupling system design Therefore, it is essential to approximate these high-order functions with reduced-order models to simplify analysis and implementation Any reduction technique can be employed to effectively fit the transfer functions into lower-order representations, facilitating more practical and efficient control system design.
Fig.5.3 Block diagram of feedback control strategies (a) Classical feedback control (b) Internal model control
5.3 DESIGN OF IMC-PID CONTROLLER
5.3.1 IMC-PID Approach for PID Controller Design
The multivariable Smith predictor is designed based on the multiple single-loop Smith predictor control designs Let the primary controller be:
The diagonal element c₍ᵢᵢ₎ is designed in relation to the delay-free component q₍ᵢᵢ₀₎ of q₍ᵢᵢ₎ to ensure the closed-loop system—comprising c₍ᵢᵢ₎(s) and q₍ᵢᵢ₀₎—achieves the desired performance.
The PID controller can be designed using the Internal Model Control (IMC) approach for processes with inherent delays, as illustrated in Fig 3 Key components include the process transfer function (Gp), the process model (G̃p), disturbance (q), feedback controller (Gc), IMC controller (Gd), and set-point filter (fR) In this control scheme, the set-point input (r), disturbance (d), and controlled variable (y) are interconnected through the IMC structure The relationship between the controlled variable and the set-point is established based on the IMC control framework, ensuring accurate and robust process regulation.
For the nominal case (i.e., G P G P ), the set-point and disturbance rejection responses are simplified to:
In the classical feedback control structure, the set-point and disturbance responses are represented by: c P R c P
According to the IMC parameterization (Morari and Zafiriou [27]), the process model G P is decomposed into two parts:
In control system modeling, the parameter G is defined as G = P P̃, where P_M and P_A represent the portions of the model that are inverted and not inverted by the controller, respectively It is important to note that P_A is typically a non-minimum phase component, often containing a time delay term and/or a right-half-plane zero, which significantly impacts system stability and controller design.
Then, the IMC controller can be described as:
The numerator m i=1 i α s 1 i causes an excessive overshoot in the servo response, which can be eliminated by introducing a set-point filter to compensate the overshoot in the servo response
Therefore, the ideal feedback controller for achieving the desired loop response can be easily obtained by
The controller derived from equation (5.18) is physically realizable but does not conform to the standard PID-type controller structure To ensure compatibility with conventional control systems, it is essential to convert this controller into a suitable PID form This conversion is achieved using advanced approximation techniques, specifically a novel application of the Maclaurin series Unlike previous methods, this approach focuses on closely approximating the ideal feedback controller with a PID controller, enhancing system performance and simplicity.
5.3.2.IMC-PID Tuning Rules for Typical Process without Time Delay Models
The most commonly used approximate model for chemical processes is the FOP model as given below:
(5.19) where K and τdenote the process gain and the time constant, respectively The optimum IMC filter structure in this case is found as:
Hence, the ideal feedback controller is obtained by:
The value of αis evaluated as: λ 2 τ 1 1 α τ (5.22) The analytical tuning rules of the PID controller can be obtained by:
For the other process models, one can be easily applied the similar above-mentioned procedure
In order to have a fair comparison, the IAE criterion is considered here for the set-point tracking
To evaluate the magnitude of the manipulated input usage, the total up and down movement of the control signal is considered as
TV is a good measure of the smoothness of controller output and should be small [28]
The robustness of a control system is crucial in controller design due to the inherent uncertainties in real plant dynamics, which can lead to degraded performance or instability This study introduces a well-known robust stability method [28] to provide a fair comparison with other existing controller design techniques, emphasizing the importance of ensuring reliable and stable control under uncertain conditions.
Robust stability analysis is essential when considering output multiplication uncertainty in complex systems For a multi-delay process subject to output multiplicative uncertainty denoted by Δ₀, the upper bound of the system’s robust stability can be precisely determined, ensuring reliable performance despite uncertainties This approach provides a critical assessment framework for maintaining stability under various uncertainties in multi-delay control systems.
Table 5.1 Controller parameters for the WB column
Tuning method Loop K ci Ii i
Table 5.2 Resulting performance indices for the WB column
To ensure a fair comparison, the robust stability degree will be maintained consistently across all the design methods evaluated In the simulation study, the proposed multi-loop PI controller is tuned by adjusting the closed-loop time constant, λ_i, to keep the γ value of the control system equal to or higher than that of the other methods This approach allows for an accurate assessment of the controller's performance while adhering to stability criteria.
This section examines a pilot-scale distillation column with eight trays and a reboiler designed for the separation of methanol and water [29] The system's dynamic behavior is characterized by its open-loop transfer function matrix, providing essential insights into its control and stability.
The ideal decoupler network is designed based on (5.8):
The decoupled process in the ideal decoupling system can be obtained by:
Considering (5.22-5.25), the controller parameters of PID controller can be calculated and listed in Table 5.1