Synthesis of Multivariable PID Controllers via InterCommunicative Decentralized Multi-Scale Control for TITO Processes Jobrun Nandong1, 2 1
Department of Chemical Engineering; 2 Curtin Sarawak Research Institute Curtin University Sarawak 98009 Miri, Sarawak, MALAYSIA
[email protected]
Abstract—A new method of centralized PI/PID control system synthesis for TITO processes using the non-conventional InterCommunicative Decentralized Multi-Scale Control (ICD-MSC) scheme is presented. Explicit mathematical tuning equations are constructed which are used to systematically convert the intercommunicative multi-scale controllers into an industrial implementable form of centralized PID controllers. An illustrative example shows the effectiveness of the proposed synthesis method which is easy to use and understand. Keywords—multivariable PID control; decentralized PID control; MIMO control design; multi-scale control
I. INTRODUCTION Over the last few decades, various forms of advanced control techniques have been developed, such as the linearquadratic control [1], generalized model predictive control [2] and model predictive control (MPC) [3]. In spite of all the advances in control theories, the PID control is still the most widely used controller in industries [4]. In process industry, most of the systems of interest are multi-input and multi-output (MIMO) in nature where multi-loop (decentralized) PID controllers are often adopted to control such systems. The main challenge for designing effective decentralized PID controllers for a MIMO process arises from the presence of process interactions, which fairly often impose a limitation on control performance. It has been recognized that even for single-input and single-output (SISO) processes, the tuning of a PID controller is often difficult without a systematic procedure [5]. The PID controller tuning for a MIMO process is even more challenging because the number of tuning parameters increases with the system dimension, i.e., an n × n MIMO system will leads to 2n, 3n and 4n numbers of tuning parameters for multi-loop PI, PID and PID with filter controllers, respectively. In the fully centralized PID control system, there are 3n 2 number of controller parameters that need to be tuned. Hence, it would be much more difficult to design the multivariable PID control system than the decentralized one.
The existing methods for decentralized PID controller tuning can be broadly categorized as in [6]: (1) the detuning, (2) sequential loop closing, (3) iterative or trial-and-error, (4)
simultaneous equation solving or optimization, and (5) independent methods. The detuning method based on the BLT [7] is probably among the most commonly used in process industry because of its simplicity. But the BLT method often leads to under satisfactory performance, i.e., can be either too sluggish or too oscillatory. Another popular design approach is based on the sequential tuning where the PID controllers are designed sequentially one after another [8]. The main issue in this approach is that one has to select the right order of the loop closing sequence as the control performance is strongly influenced by this order. In addition to the sequential and detuning approaches, several researchers have developed PID tuning methods based on the independent approach, e.g., see [9], 10]. In process control, one common approach to improve the performance of decentralized PID control is to augment the control system with decouplers, which are used to mitigate the adverse effect of process interactions. There are several forms of decoupling methods, where the basic techniques can be divided into 3 categories [11]: (a) the ideal, (b) simplified, and (c) inverted decoupling methods. One of the major problems in using the decoupling techniques is that the decouplers might not be physically realizable, especially those that are based on the ideal decoupling method. There are several methods of decentralized PID control design which include the decoupling controllers, e.g., see [12], [13], [14]. Note that, Nandong and Zang [15] have recently proposed the Inter-Communicative Decentralized Multi-scale Control (ICD-MSC) scheme as an alternative to traditional decoupling approach for overcoming the performance limitation imposed by the process interactions. The ICD-MSC is basically founded on the more basic Multi-scale Control (MSC) scheme for SISO nonminimum-phase processes; see [16], [17]. Additionally, the same basic MSC scheme has recently been adopted in the construction of several PID tuning relations; see [18], [19]. In the present work, the ICD-MSC scheme [15] and the procedure for constructing some PID tuning relations in [18] are combined together in order to synthesize a multivariable PI/PID control system for the two-input and two-output (TITO) processes. It is worth noting that the ICD-MSC scheme in [15]
represents a generalized approach for designing the multi-loop multi-scale controllers together with the inter-communicative controllers to overcome process interactions. The resulting controllers might not be equivalent to the conventional PID controllers. In this regard, the main contribution in this work is to devise a simple way by which, one can readily convert these ICD-MSC controllers into the standard multivariable PI/PID controllers, hence permitting an industrial implementation of the ICD-MSC scheme. The rest of this paper is organized as follows. Section 2 presents the construction of mathematical tuning relations based on the ICD-MSC scheme for TITO processes. Section 3 provides the proposed design procedure for multivariable PI/PID control system. An illustrative example is given in Section 4. Finally, Section 5 highlights some concluding remarks and future research directions.
R2
The details about the ICD-MSC scheme can be found in [15] while the construction procedure of PID controller tuning formulas can be found in [18]. The basic idea of the ICD-MSC scheme is to reduce the effect of MIMO process interactions via the inner-layer of a given multi-scale controller. The multiscale controller has two loops – a fast inner feedback loop and a relatively slow outermost feedback loop. The effect of the process interactions is treated as an input disturbance, which is to be removed by the fast feedback inner loop. In contrast to this idea, in the conventional decoupling techniques, the loop interactions are mitigated via the conventional feedforward cancellation approach. In the ICD-MSC, the loop interactions are removed or reduced by the fast feedback control action of the given multi-scale controller – a feedback approach. Fig. 1 shows the block diagram of an Input-Input structure of ICD-MSC scheme. Other ICD-MSC structures are called the Input-Output and Output-Output structures [15]. In Fig. 1, gii denotes the diagonal transfer function, gij the off-diagonal transfer function, Wii the multi-scale predictor, I ij the intercommunicative controller, K ii ,1 the inner sub- controller, K ii ,0 the outermost sub-controller and Fri the setpoint pre-filter; Ri and Yi denote the setpoint and controlled signals, respectively. In this work, the Input-Input structure of ICD-MSC scheme (Fig. 1) is adopted to synthesize a centralized PI/PID control system for TITO processes. Let us consider a TITO process given as follows
(1)
Here, we assume that the diagonal transfer functions can be represented as the first-order plus deadtime (FOPDT) model:
Fr1
+
K11,0
E1
E2 Fr2
K22,0
+
-
C1
+
K22,1
+
g11
I12
g12
I21
g21 U2
g11
+
-
+
+
+
C2
-
U1
K11,1
+
+
+ +
Y1
Y2
W21
Fig. 1. Block diagram of the Input-Input Structure of the ICD-MSC scheme.
R1
II. CONSTRUCTION OF TUNING RELATIONS
⎡ g11 ( s ) g12 ( s ) ⎤ P( s) = ⎢ ⎥ ⎣⎢ g 21 ( s ) g 22 ( s )⎦⎥
W11 R1
R2
Fr1
+
E1
K11,0
C1
U1
+
h11 h12
g12
h21
g21 +
E2 Fr1
+
K22,0 -
g11
+
h22
C2
+
g11
U2
+
+
+ +
Y1
Y2
Fig. 2. Simplified block diagram of the Input-Input Structure of the ICD-MSC scheme.
R1
Fr1
+
E1
U1
+
Gc11
g11
+
Gc12 Fr1
E2 +
+
-
Y1
g21 +
Gc22
+
g12
Gc21
R2
+
g11
U2
+ +
Y2
Fig. 3. Equivalent multivariable control block diagram of the InputInput Structure of the ICD-MSC scheme.
g ii ( s ) = K pii exp( −θ ii s ) (τ ii s + 1)
(2)
First, the dead-time in (2) is approximated using the 1/1 Padé formula. Then, the approximated model is decomposed into a sum of an outermost mode ( mii ,0 ) and an inner-layer mode ( mii ,1 ) as given by g ii ( s ) = k ii ,0 (τ ii s + 1) + k ii ,1 (α ii s + 1)
mii ,0
mii ,1
(3)
where α ii = 0.5θii and it is assumed that α ii < τ ii . In (3), the gains for the outermost and innermost modes are as
k ii ,0 = K pii (τ ii + 0.5θ ii ) ( τ ii − 0.5θ ii ) k ii ,1 = K pii θ ii (0.5θ ii − τ ii )
(4)
o cii ,1
(α ii s + 1) (τ cii,1s + 1)
(14)
where βij denotes the IC controller gain.
(5) o hij ( s ) = β ijo k cjj ,1 (α ii s + 1) (τ cii ,1 s + 1)
o h ij = β ijo k cjj ,1
(16)
In (15) or (16), the overall inter-communicative controller gain ( βijo ) is given in the form of βijo = βij (1 + k cii ,1k ii ,1 )
(7)
In (7), the overall sub-controller gain and the closed-loop time constant are respectively given by (8) and (9):
(15)
Meanwhile, if the IC controller form in (14) is used, a static form of (11) is obtained as follows
(6)
Let the multi-scale predictor Wii = mii ,1 and the inner subcontroller Kii,1 = kcii ,1 (a Proportional controller is chosen for the innermost mode). Hence, (6) can be simplified to (7): hii ( s ) = k
β ij (τ cii ,1s + 1)(τ cjj,1 s + 1) (α ii s + 1)(α jj s + 1)
After substituting (13) into (12), and then into (11)
Note that, Fig. 1 can be simplified to Fig. 2. Then, based on Fig. 2 the inner-loop transfer function corresponding to each diagonal element of (1) is written as in (6): hii ( s ) = U i ( s ) Ci ( s ) = K ii,1 (1 + K ii,1Wii ( s ) )
I ij ( s ) =
(17)
Now, let us define an MSC parameter (see reference [18]), which is a ratio of the open-loop time constant to that of the closed-loop of the inner mode given as follows
o k cii ,1 = k cii ,1 (1 + k cii ,1 k ii ,1 )
(8)
λii,1 = α ii τ cii ,1 = 1 + k cii ,1k ii,1
τ cii,1 = α ii (1 + k cii,1k ii ,1 )
(9)
Upon rearrangement of (18), the sub-controller gain can be calculated in term of the MSC ratio, i.e.:
(18)
Again referring to Fig. 2, the off-diagonal inner-loop transfer function from C2 to U1 is given as follows
k cii ,1 = (λii ,1 − 1) k ii ,1
hij ( s ) = h jj ( s ) I ij ( s ) (1 + K ii,1Wii ( s ) )
It follows from (19) that the overall sub-controller gain in (8) can now be expressed in term of λii ,1 :
(10)
where I ij denotes Inter-Communicative (IC) controller. Next, (10) can readily be simplified to (11) below:
hij ( s ) =
o I ijo ( s )k cjj ,1 (α ii s + 1)(α jj s + 1)
(τ cii ,1 s + 1)(τ cjj,1 s + 1)
(11)
Here, in (11), the overall IC controller is defined as in (12) I ijo ( s ) = I ij ( s ) (1 + k cii ,1k ii ,1 )
⎛ λii ,1 − 1 ⎞ ⎛ θ ii − 2τ ii o ⎟⎜ k cii ,1 = ⎜ ⎜ λ ⎟⎜ ⎝ ii ,1 ⎠ ⎝ 2θ ii K pii
β ij ⎛ λ jj,1 − 1 ⎞ ⎛ θ jj − 2τ jj ⎜ ⎟⎜ λii,1 ⎜⎝ λ jj,1 ⎟⎠ ⎜⎝ 2θ jj K pjj
h ij = (13)
Alternatively, the IC controller can be expressed in the form of
(20)
⎞⎛ α ii s + 1 ⎞ ⎟⎜ ⎟ τ s +1⎟ ⎠ ⎠⎝ cii
(21)
Likewise, substituting (17) and (20) into (16) results in
Note that, one may choose a suitable form for the IC controller, e.g., one of the forms is as follows: I ij ( s ) = β ij (τ cjj ,1 s + 1) (α jj s + 1)
⎞ ⎟ ⎟ ⎠
By substituting (17) and (20) into (15), one can obtain (21)
hij ( s ) = (12)
(19)
β ij ⎛ λ jj ,1 − 1 ⎞ ⎛ θ jj − 2τ jj ⎜ ⎟⎜ λii ,1 ⎜⎝ λ jj,1 ⎟⎠ ⎜⎝ 2θ jj K pjj
⎞ ⎟ ⎟ ⎠
(22)
Let us consider that a PI controller is chosen to control the outermost mode, i.e., the outermost sub-controller is given by
K ii ,0 ( s ) = k cii,0 [1 + 1 (τ Iii,0 s )]
(23) K C ij
where the outermost sub-controller gain is
k cii,0 = (λii,0 − 1) kii,0 : λii,0 > 1
(24)
Note that, the MSC parameter λii ,0 is defined in a similar manner as that of λii ,1 in (18). Let us define another MSC parameter γ ii as proposed in [18], i.e.:
γ ii = τ I ii,0 τ ii : γ ii > 0
(25)
The overall diagonal controller can now be written as follows
Gcii ( s ) = K ii,0 ( s )hii ( s) S (k cii,1 )
(26)
where S (k cii ,1 ) denotes the sign of the inner sub-controller, included to obtain the correct overall controller gain. It has been shown in [19], [20] that the multi-scale controller (26) can be converted into an equivalent PID controller in (27):
⎞ ⎞⎛ ⎛ 1 1 ⎟ Gc ii ( s) = K C ii ⎜⎜1 + + τ D ii s ⎟⎟ ⎜⎜ ⎟ ⎠ ⎝ τ f ii s + 1 ⎠ ⎝ τ I ii s
(27)
In (27), the PID controller parameters are given in terms of the model parameters and the MSC tuning parameters:
K Cii
⎡ ⎛ ( 2τ ii − θ ii ) 2 ⎞ ⎟ ⎢ ⎜ ⎛ (λii ,0 − 1)(λii ,1 − 1) ⎞ ⎢ ⎜⎝ K pii K pii ⎟⎠ ⎟⎢ = ⎜⎜ ⎟ ⎛ (2τ + θ )τ θ 4 γ ii λii ,1 ii ii ii ii ⎝ ⎠⎢ ⎜ ⎢ ⎝ 2γ iτ ii +θ ii ⎣
⎤ ⎥ ⎥ ⎞⎥ ⎟⎥ ⎠ ⎥⎦
(28)
(29)
τ D i = γ ii θ ii τ ii (2γ ii τ ii + θ ii )
(30)
(31)
Next, consider the off-diagonal controller which is given by
Gcij ( s ) = K jj,0 ( s )hij ( s )
τ Iij = γ jj τ jj + 0.5θ ii
(34)
τ D ij = γ jj τ jj θ ii (2γ jj τ jj + θ ii )
(35)
τ f ij = τ cii ,1 = θ ii (2λii ,1 )
(36)
If the IC controller is chosen to be in the form of (14), then one obtains the following off-diagonal controller:
Gcij ( s ) = K jj,0 ( s )h ij
(32)
Gcij ( s ) = K C ij (1 + 1 (τ I ij s ))
(38)
The PI controller parameters in (38) are given as follows:
⎛ β ij (λ jj ,0 − 1)(λ jj,1 − 1) K Cij = ⎜⎜ ⎝ 2λii,1λ jj ,1 ( 2τ jj + θ jj )θ jj
⎞⎛ ( 2τ jj − θ jj ) 2 ⎞ ⎟⎟ ⎟⎟⎜⎜ ⎠⎝ K pjj K pjj ⎠
(39)
(40)
Finally, the ICD-MSC structure in Fig. 1 can be converted into an equivalent multivariable control structure shown in Fig. 3. The multivariable controller matrix for the 2x2 system is ⎡ G c11 ( s ) G c12 ( s ) ⎤ G C (s) = ⎢ ⎥ ⎣⎢G c 21 ( s ) G c 22 ( s )⎦⎥
(41)
where the vector of controlled outputs can be written as
Y(s) = P(s)GC (s)E(s) In Fig. 3, the multivariable controller matrix is given as
Again, (32) can be rearranged into (27). In the case where the IC controller takes the form of (13), the off-diagonal PID controller parameters are as in (33) – (36):
(37)
Since the outermost sub-controller is in the form of a PI controller and the off-diagonal inner-layer transfer function takes a static form, the overall off-diagonal controller is simply equivalent to a PI controller overall, i.e.:
τ I ij = γ jj τ jj
τ I ii = γ ii τ I ii + 0.5θ ii
τ f ii = τ cii ,1 = θ ii (2 λii ,1 )
⎡ ⎛ ( 2τ jj − θ jj ) 2 ⎞ ⎤ ⎟ ⎥ ⎢ ⎜ ⎛ β ij (λ jj ,0 − 1)( λ jj ,1 − 1) ⎞ ⎢ ⎜⎝ K pjj K pjj ⎟⎠ ⎥ = ⎜⎜ (33) ⎟⎟ 4γ jj λ jj ,1 λii ,1 ⎝ ⎠ ⎢ ⎛⎜ τ jj θ jj ( 2τ jj + θ jj ) ⎞⎟ ⎥ ⎢⎜ ⎟⎥ ⎣ ⎝ 2γ jj τ jj + θ ii ⎠ ⎦
(42)
0 ⎤ ⎡ h11 ( s ) h12 ( s ) ⎤ ⎡ K11,0 ( s ) Gc (s) = ⎢ ⎥ ⎥⎢ K 22,0 ( s )⎦⎥ ⎣⎢h21 ( s ) h22 ( s )⎦⎥ ⎣⎢ 0
(43)
Step 2.4: Obtain the off-diagonal controller parameters using (33) – (36) if I ij takes the form of (13), otherwise, if the form in (14) is selected, use (39) – (40) instead (i.e., a PI controller). The entire control system design is completed.
III. DESIGN PROCEDURE A. Tuning of Inter-Communicative Controller Fig. 2, the interaction effect of C j on Yi assuming Ci = 0 Yi ( s ) = [h jj ( s ) g ij ( s ) + hij ( s ) g ii ( s )]C j ( s )
(44)
Let the overall interaction transfer function be in (45) H ij ( s ) = h jj ( s ) g ij ( s ) + hij ( s ) g ii ( s )
(45)
A simple choice is to set (45) at the steady-state as follows H ij (0) = h jj (0) g ij (0) + hij ( s ) g ii (0)
Step 2.3: The overall control system performance can be refined (to meet the target performance) by adjusting the intercommunicative static gain, i.e. fine tuning using (48).
(46)
Remark 1: These initial values often give stable response, otherwise, choose other initial values. Remark 2: The proposed design procedure is quite general (it can be further refined in future study). A more rigorous tuning procedure using the constructed tuning relations can be performed via optimization techniques, e.g., genetic algorithm. Remark 3: We propose robustness 14 dB ≤ GM * ≤ 16 dB for gain margin range and 60 0 ≤ PM * ≤ 80 0 for phase margin range. It is crucial to ensure sufficient robustness margins to accommodate the effect of process uncertainties.
IV. ILLUSTRATIVE EXAMPLE Consider a TITO process given by
Hence, one obtains the IC controller gain in (47):
β ij = −λii ,1 (K pij K pii )
⎡ ⎤ 1.2e −3s e −s ⎢ ⎥ 2 10 s 1 + 20s + 2 s + 1 ⎥ G( s) = ⎢ ⎢ (1 − s )e − s − 1.1e − 4 s ⎥ ⎢ ⎥ 9s + 1 ⎣ (11s + 1)(5s + 1) ⎦
(47)
The actual IC controller gain can be adjusted using a tuning parameter f ij in the following manner
β ij = f ij β ij :
f ij > 0
(48)
Here, f ij may be used as a tuning parameter, or use f ij = 1 .
Note that, feasible decouplers cannot be obtained via the conventional decoupling techniques, i.e., improper decoupler transfer functions. Nevertheless, one can still use non-ideal (i.e., static decouplers) decouplers for such a process. The severity of the process interaction can be quantified using the well-known Bristol’s Relative Gain Array (RGA) [20].
B. Tuning Algorithm for PI/PID Controller A general two-step design procedure using the constructed tuning relations (Section 2) is proposed. First, assume that the desired GM and PM are GM * and PM * respectively. Step 1: Design the individual PID controller Gcii for the ith control loop using (28) – (31).
Step 1.1: Initial setting: λii ,0 = 1.2, λii ,1 = 10, γ ii = 0.8 . Step 1.2: If GM * and PM * are achieved, then the design is completed, otherwise, go to Step 1.3. Step 1.3: Gradually increase λii ,0 until one approximately achieves the specified gain/ phase margin criteria.
⎡0.569 0.431⎤ Λ=⎢ ⎥ ⎣⎢ 0.431 0.569⎦⎥
Step 2.2: Calculate the IC controller static gain β ij using (47).
(50)
The RGA in (50) indicates that strong interactions exist between the two loops, i.e. diagonal element closed to 0.5 indicates very severe loop interactions. Three different control strategies are designed: (a) completely decentralized PID controllers, (b) decentralized plus static decouplers, and (c) proposed multivariable control scheme. Note that, for all control strategies, we use similar diagonal PID plus filter controllers. TABLE I.
Step 2: Design the inter-communicative controller.
Step 2.1: Choose the form for the controller I ij , either in the form of (13) or of (14).
(49)
Controller Gc11 Gc22 Gc12 Gc21
KC 0.777 -0.635 0.414 0.595
PI/PID CONTROLLER PARAMETERS 9.5 9.2 7.2 8.0
τI
τD 1.263 1.565 1.6
0.15 0.20 0.20
τf
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Y1
2
-1
Y2
0
50
100
150 Time
200
250
300
0.5 0 -0.5 0
50
100
150 Time
200
250
300
Proposed: IAE = 32 Decentralized: IAE = 58 Decentralized-Decoupler: IAE = 45
Fig. 4. Closed-loop responses under the nominal condition.
Table 1 shows the PI/PID controller parameters. The PID controllers are obtained using the relations given in (28) to (31) where λ11,0 = 1.5 , λ11,1 = λ22,1 = 10 , γ 11 = γ 22 = 0.8 , and λ22,0 = 1.6 . These settings lead to gain and phase margins within the ranges given in Remark 3. Fig. 4 shows the closedloop responses for the 3 different control strategies when subjected to sequential setpoint changes of 1 unit each in Y1 and Y2; the proposed method leads to improved response. V. CONCLUSIONS A new method for the synthesis of multivariable PI/PID controllers for TITO processes has been presented. The method essentially converts the inter-communicative multi-loop multiscale controllers in [15] to equivalent multivariable PI/PID controllers. The proposed method showed improvement over the purely decentralized PID controllers without and with static decouplers. Some possible future research directions include: (a) tuning of the MSC parameters using some optimization algorithms, and (b) synthesis of the multivariable PID controllers using other structures of the ICD-MSC scheme. ACKNOWLEDGEMENT This work had been conducted under the Intelligent System, Design & Control (ISDCON) Research Area at Curtin University Sarawak, and partly supported by FRGS funding (JPT.S Jld.13(28)) of MOHE and CSRI grant. REFERENCES
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