Integrated Multilinear Model Predictive Control of Nonlinear Systems

Subscriber access provided by CARLETON UNIVERSITY

Article

Integrated Multi-linear Model Predictive Control of Nonlinear Systems Based on Gap Metric Jingjing Du, and Tor Arne Johansen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie504170d • Publication Date (Web): 04 May 2015 Downloaded from http://pubs.acs.org on May 9, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Integrated Multi-linear Model Predictive Control of Nonlinear Systems Based on Gap Metric Jingjing Du*, † , Tor Arne Johansen‡ † ‡

School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454000, China Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO 7491 Trondheim,

Norway

ABSTRACT: Two integrated multi-linear model predictive control (MLMPC) algorithms are proposed for nonlinear chemical processes. The gap metric and the gap metric stability margin are employed to select local linear models and design local MPC controllers. Thus the local stability and desired closed-loop performance can be incorporated into the model bank selection process. After that, a gap-metric-based weighting method is used to combine the local MPC controllers into a global MLMPC controller for the nonlinear process. Therefore, the local model selection, the local controller design, and the local controller combination are all completed according to the gap-metric-based criteria. Close connections are established among the three key elements of the multi-linear model predictive control approach. Thereby the design of a MLMPC controller is more systematical, which is found to improve the accuracy and robust performance of a MLMPC controller. Since the gap metric does not consider constraints and the use of linear models in the multi-model approach may not lead to a globally stable control systems, an additional simulation-based criterion is employed to evaluate the overall closed-loop performance. A SISO and a MIMO CSTR processes are studied to demonstrate the effectiveness of the proposed algorithms. Keywords: integrated MLMPC, gap metric, the gap metric stability margin, linear model bank, MIMO

1. INTRODUCTION Model predictive control (MPC) has been the most successful and widely used advanced control strategy applied in process industries due to its significant advantages over traditional control techniques. MPC displays superior performance because it predicts the future dynamic response to compute the current control move, which provides benefit when controlling processes with complex dynamics and hard and soft constraints. Moreover, MPC provides a convenient architecture for handling multivariable control due to the superposition of linear models within the controller.1

*person to whom all correspondence should be addressed: [email protected], [email protected]

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

However, when applied to a nonlinear chemical processes, the linear MPC controller may be limited to a relatively small operating region. Hence, for nonlinear chemical systems with wide range of operating levels or multiple operating conditions, the capabilities of MPC may degrade as the system moves away from its designed operating level. Although nonlinear MPC is a method to deal with nonlinear control problem, nonlinear MPC may be challenging due to the need of a nonlinear model with sufficiently large domain of validity, and the complexity of solving nonlinear (non-convex) optimization problems. To conquer this difficulty and maintain the performance of the MPC controller over a wide operating range, researchers embed the MPC strategy into the multi-model approach framework and have proposed multiple model predictive control (MMPC) strategies, making full use of the advantages of both the MPC strategy and multi-model control approach. The current MMPC methods can roughly be classified into three types in respect to the MMPC controller structure: (1) Multiple local models are interpolated to make a global nonlinear model using weighting functions, and then a (nonlinear) MPC controller is designed based on the global model.2-8 This kind of MMPC method usually involves a nonlinear optimization problem. While the first approach may be the best to deal with strong nonlinearities during transition between operating points, the nonlinear optimization is challenging from an implementation point of view. (2) A bank of linear models are used to approximate the considered nonlinear system, and then a global MPC is designed based on the linear models using the min-max strategy. In the second type of MMPC approach, the objective function is minimized for the worst process model, resulting in a robust MPC.9-14 A possible challenge is that the robust MPC may be too conservative, leading to poor performance. (3) A set of local linear MPC controllers are designed based on local linear models, and then the local MPCs are combined into a global MPC by weighting functions or switching.15-19 We call the third type ‘Multi-linear model predictive control (MLMPC)’. In addition, some researchers have used the MMPC method to address the control problem of hybrid systems.20 In the MLMPC methods, the linear MPC can be directly used to design local controllers, making the control design problem easier and simpler. Therefore, in this paper, we focus on the MLMPC methods. The MLMPC methods generally include three key steps: the model bank determination, the local controller design, and the local controller combination. With existing methods, the connection among the model bank determination, the local controller design, and the local controller combination is rather weak. That is to say, the model bank determination, the local controller design, and the local controller combination are usually handled separately according to different criteria. This makes the MLMPC design process rather problem-dependant and can incur improper linear model selection, complicate the MLMPC controller structure, and thereby increase the computational load and reduce performance. In order to make the MLMPC design process more systematic, an integrated multi-model control framework is proposed by Du & Johansen,21 in which the linear model bank determination and the local controller design are closely connected by the gap metric. Many control techniques can be used in this integrated framework,


Page 2 of 22

Page 3 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


including MPC. However, in ref 21 the controller weighting method does not employ the gap metric criterion, so it is not connected with the first two steps. Besides, the method is only derived for SISO nonlinear systems. In ref 22, a gap-metric-based weighting method is proposed to combine local MPC controllers for MIMO nonlinear systems. In this paper, we integrate and extend the ideas in ref 21 and ref 22 to establish a close connection among the three key steps using the gap metric criteria. Two thoroughly integrated MLMPC algorithms are developed for nonlinear chemical processes to make the MLMPC design more systematic and effective to improve the closed-loop performance. In this work, the presence of constraints on inputs and outputs cannot be accounted for in the gap-metric-based analysis methods, and their impacts are evaluated by an additional simulation-based validation criterion. The proposed algorithms are applied to a SISO CSTR process and a MIMO CSTR process. Closed-loop simulations illustrate that the integrated MLMPC methods are superior to state-of-the-art MLMPC methods. The paper is organized as follows. In Section 2, relevant background is shortly reviewed. Section 3 describes the basics of the multi-linear model predictive control. In Section 4, the proposed integrated MLMPC design algorithms are detailed. Section 5 presents some simulation results to illustrate the effectiveness of the proposed methods, and Section 6 concludes the paper.

2 GAP METRIC AND STABILITY MARGIN In this section, relevant theoretical background will be briefly reviewed. 2.2 Gap metric The gap metric between two finite-dimensional linear systems P1 and P2 with the same number of inputs and outputs can be computed by: 23 M  M 

M  M 

δ ( P1 , P2 ) = max{ inf  1  −  2  Q , inf  2  −  1  Q } Q∈H  N1   N 2  ∞ Q∈H  N 2   N1  ∞ ∞

(1)

∞

where P1 = N1M 1−1 and P2 = N 2 M 2 −1 are the normalized right coprime factorizations of P1 and P2 respectively, with M i* M i + N i* Ni = I and M i* ( s ) = M i T (− s ) , Ni* ( s ) = Ni T ( − s ) , i = 1, 2. The gap metric, which is bounded between 0 and 1, can be thought of as an extension of the ∞-norm. El-Sakkary24 showed that for the purpose of robust stabilization the gap metric was more suitable to measure the distance between two linear systems than a metric based on norms. If the gap metric between two systems is close to 0, it means that there exists at least one common feedback controller stabilizing both systems and the distance between the closed-loop systems is small in the ∞-norm sense.24 On the contrary, if the gap is close to 1, it will be difficult to find a controller stabilizing both systems. Recently, the gap metric has been applied to the multi-model control of nonlinear systems by quite a few researchers, either for model bank selection21, 25-31 or local



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 22

controller combination.22,32 By linearizing a nonlinear system about a set of operating points, the nonlinear system is approximated by multiple linear subsystems, and then the gap metric is applicable.

2.2 The gap metric stability margin The gap metric stability margin (also called the normalized coprime stability margin) of a linear system P and its stabilizing controller K is defined as:33 bP , K

I   =   ( I + PK ) −1  I  K 

 P 

−1

∞

I   =   ( I + KP ) −1  I  P

 K 

−1

(2) ∞

where I is the identity matrix. The maximum gap metric stability margin of P is defined as:33

 bopt ( P) :=  inf K stabilizing

[

I  −1   K  ( I + PK )  I   

~ ~ = 1− N M

]

2

 P 

   ∞

−1

(3)

δ ( P, P0 )

(4)

3 MULTI-LINEAR MODEL PREDICTIVE CONTROL 3.1 Local MPC controller Design Consider a nonlinear process described by Eq.(5)  x& = f ( x, u )   y = g ( x, u )

(5)

where x ∈ R n is the state vector, u ∈ R r is the control input vector, and y ∈ R m is the output vector.

f (·) and g (·) are nonlinear differentiable functions.


Page 5 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Let θ be the vector of the scheduling variables of plant (5). The scheduling variables contain the variables that characterize the operating behavior and uniquely defines the equilibrium points of the plant, which should vary slowly and capture plant nonlinearities according to the principles of gain scheduled control design. 34 Generally, the vector θ will contain a subset of the states, inputs, outputs and disturbances, and may also contain other model variables such as references and parameters. Let Ф be the variation range of θ, i.e., θ ∈ Φ . Ф is called the scheduling space of system (5). Suppose the nonlinear system is approximated by Nm local linear systems with the subregions Φi , i = 1, ..., N m . Φ i satisfies Φi ⊆ Φ , Φ1 ∪ ... ∪ Φ N m = Φ and 3 Φ i ∩ Φ j = ∅ for i ≠ j. The ith local linear system Pi is built by linearizing and

discretizing Eq. (5) around the ith operating point (xoi, uoi, yoi) = (xo(θi) uo(θi) yo(θi)),, which is an equilibrium point of system Eq. (5) defined by a selected θi in Фi. The ith local linear system Pi is described as Eq. (6): ∆x(t + 1) = Ai ∆x(t ) + Bi ∆u (t ) ,  ∆y(t ) =C i ∆x(t ) + Di ∆u (t )

i = 1, …, Nm

(6)

where t ∈ Ζ , ∆x(t ) = x(t ) − xoi , ∆y(t ) = y(t ) − yoi , ∆u(t ) = u(t ) − uoi , and Ai, Bi, Ci, Di are the discretized system matrices. Note that the scheduling variables θ are selected such that there exists only one equilibrium point for every value of θ. Therefore, only one linearization model is set up for every θ. One of our goals in this work is to select a proper local linear model bank from a number of linearized models like Eq. (6) to approximate the nonlinear system (5), which will be solve in Section 4. Based on the local linear system Pi, a linear MPC controller can be designed by considering the following cost function: Ny

Nu

J i = ∑ r (t + j ) − y (t + j | t ) Q + ∑ ∆ui (t + j − 1)) Q j =1

2

y

j =1

2

(7)

u

with constraints:

 u min ≤ u i ( t ) ≤ u max   ∆ u i , min ≤ ∆ u i ( t ) ≤ ∆ u i , max y  min ≤ y ( t ) ≤ y max

(8)

where Qy, Qu are the weighting matrices, Ny is the prediction horizon, Nu is the control horizon, and r is the reference signal. Eqs.(7)-(8) form a Quadratic Programming problem. The selection of model bank, and the parameters Qy, Qu, Ny, and Nu will be tuned to satisfy a given gap-metric-stability-margin condition so that the MPC will have enough robust performance in its subregion, which will be detailed in Section 4. Note that the computation of the gap metric stability margin bP,K according to Eq. (2) only depends on the parameters Qy, Qu, Ny, and Nu, while the constraints have no



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

influence on the computation. This is because the gap metric and the gap metric stability margin are based on linear and unconstrained systems. Thus the closed-loop stability may not be guaranteed when there are constraints. In order to account for the effects of constraints, an additional simulation-based criterion is used if their effects are expected to be significant.

3.2 Multi-linear MPC controller combination After the design of local MPC controllers, the next step is to combine these local linear MPC controllers into a global one to apply to the nonlinear system (5). For controller combination, basically, there are two approaches: the switching method (hard switching) and the weighting method (soft switching). In the switching method, local controllers are scheduled according to a certain switching condition. At every sample period, only one local linear controller is used in the feedback loop. In the weighting method, weighting functions (functions of scheduling variables) are employed to combine local controllers into a global one. The output of global multi-linear MPC controller is a weighted sum of the local MPC controllers’ output(s). Generally speaking, the weighting method is more promising for controller combination, as the weighted sum of local controllers makes the controller output smooth, but it is also more challenging since unintended interactions between controllers may occur. Moreover, it is not trivial to choose a proper weighting function, because usually there are tuning parameters to be determined by a prior knowledge and experience, and different choices may lead to different weights. In ref 22, a gap-metric-based weighting method is proposed to combine local controllers. Similar to Gaussian and trapezoidal functions, the gap-metric-based weights depend only on the scheduling variables and can be calculated off-line and stored in a look-up table. Thereby the computational load can be significantly reduced. On the other hand, the gap-metric-based weighting method has only one tuning parameter. It is a conspicuous advantage over Gaussian and trapezoidal weighting methods, which means that the complex tuning work can be reduced, especially for nonlinear systems with multiple scheduling variables. The gap metric is used to combine local controllers, making the controller combination connected with the model bank selection and local controller design. Although another gap-metric-based weighting method is proposed in ref 32, the weights based on closed-loop gap in ref 32 have to be calculated on-line, which involves more computation. Moreover, the closed-loop gap based method in ref 32 is only applicable to the cases where the setpoint is equal to one of the operating points (see the definition of closed-loop gap metric in ref 32), which is not applicable to the cases in our paper. So in this work, the gap-metric-based soft weighting method from ref 22 is chosen to realize local linear MPC controller combination. At time instant t, the value of θ is denoted as θt; the equilibrium point corresponding to θt is denoted as ( x ot , u ot , y ot ) ; and the discretized model of the


Page 6 of 22

Page 7 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


linearization of the nonlinear system (5) around ( x ot , u ot , y ot ) is denoted as Pt. Then the weighting function of the ith local MPC controller at time t is formulated as follows: φi (θt ) =

(1 − γ i (θt )) ke , i = 1,…, Nm.

Nm

∑ (1 − γ

j

(9)

(θt )) ke

j =1

where γ i (θ t ) = δ ( Pi , Pt ) is the gap metric between the discretized model of the linearization of nonlinear system (5) and the ith local linear system (6) at time instant t; ke ≥ 1 is a tuning parameter, which can be used to adjust the weights if necessary. The normalized weights φi satisfies

Nm

∑ φ (θ ) = 1 . i

t

i =1

Let ui(t) be the output of the ith local MPC controller at time instant t, then the output of the global multi-linear MPC controller is Nm

u (t ) = ∑ φi (θ t )ui (t )

(10)

i =1

4. INTEGRATED MLMPC DESIGN In our previous work (ref 29), a systematic multi-linear model decomposition method is proposed for MIMO nonlinear systems. However, the controller design is not connected with the linear model selection. In ref 21, two general integrated multi-model control algorithms are proposed, in which the linear model bank selection and the local controller design are closely connected by the gap metric. However, the algorithms are limited to SISO nonlinear systems. Besides, the local controller combination is not connected with the model bank selection or the local controller design. In ref 22, the gap metric is used to formulate weighting functions for multi-model control of nonlinear systems, as is reviewed in Section 3.2. In this paper, we integrate the ideas in ref 29 and ref 21-22, extend the integrated multi-model control methods in ref 21 to multivariable systems, and propose two thoroughly integrated MLMPC methods. Without loss of generality, we suppose the nonlinear system (5) has two scheduling variables, α and β, i.e. θ = (α , β )T . If there exist more scheduling variables, the algorithms can be extended to fit such situations. The gap metric based gridding algorithm,29 which is a kind of dichotomy gridding algorithm, is employed here in the first step of the decomposition. The basic idea is summarized as follows. Firstly, divide the range of each scheduling variable into several intervals that must be small such that changes in dynamics due to nonlinearities are well captured by the gridding. For each interval, two equilibrium points are allocated at the ends. Secondly, linearize the nonlinear system around each equilibrium point to get a linearized model. Finally, compute the gaps between two neighbor linearized models. If the gap between two neighbor models is bigger than a prescribed threshold value, insert a new point between them. Repeat the above



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 22

process until the gap between any two neighbor points is less than the threshold value. Details of the gridding algorithm are referred to ref 29. Suppose we get α = [α1, α2, …, αm]，β = [β1, β2, …, βn] after gridding by the gap metric based method. Every combination (αi, βj) corresponds to an equilibrium point. We linearize the nonlinear system (5) about (αi, βj) to obtain a linear system P(αi, βj), which is denoted as P(i, j) for simplicity, i = 1, 2, …, m, j = 1, 2, …, n. So n × m linearized models are built in total. In order to choose a proper nominal local model for a series of linearized models in a given range, we compute the gaps between the linearized models in this range. For example, in the range {(α, β)| αa ≤ αi ≤ αb, βc ≤ βj ≤ βd}, we compute the gaps between P(i, j) (a ≤ i ≤ b, c ≤ j ≤ d) and the other linearized models P(k, l) (a ≤ k ≤ b, c ≤ l ≤ d, k ≠ i, l ≠ j. Define ∆ ij := max δ ( P (i, j ), P ( k , l )) . Then ∆ ij is the biggest a≤ k ≤b ,k ≠i c≤l ≤ d ,l ≠ j

gap corresponding to P(i, j). For the (b - a + 1) × (d - c + 1) linearized models, we get a biggest-gap-matrix, Λ := [∆ ij ](b−a+1)×( d −c+1) . Then we choose the nominal local model P* of this range {(α, β)| αa ≤ αi ≤ αb, βc ≤ βj ≤ βd} according to Eq. (11):

P* = {P(v, h) : ∆ vh = min(Λ)}

(11)

The biggest gap corresponding to P* is denoted as δ max :

δ max := min( Λ )

(12)

Then the integrated MLMPC design procedures are summarized as follows.

Algorithm 1: Integrated MLMPC based on stability margin (Extension of Algorithm 1 in ref 21) Step 1. Set i0 = 1, j0 = 1. Step 2. Set i = i0, j = j0. Step 3. Set i = i + 1. Choose the nominal local model P* among the (i - i0 + 1) × (j - j0 + 1) linearized models P(k, l) (i0 ≤ k ≤ i and j0 ≤ l ≤ j ) according to Eq. (11) and get the biggest gap δ max according to Eq. (12) Step 4. For the local linear model P*, design an unconstrained linear MPC controller K according to Eqs. (7) - (8). If K satisfies both bP , K > δ max and the desired *

constrained control performance requirements evaluated using simulation cases with the constrained MPC, go directly to Step 5. If a controller with both bP ,K > δ max and *

desired closed-loop performance is not found, set i = i − 1 and go to Step 5. Step 5. Set j = j + 1. Choose the nominal model P* among the (i - i0 + 1) × (j j0 + 1) linearized models P(k, l), i0 ≤ k ≤ i and j0 ≤ l ≤ j according to Eq. (11) and get the biggest gap δ max according to Eq. (12) Step 6. For the local linear model P*, design an unconstrained linear MPC


Page 9 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


controller K according to Eqs. (7) - (8). If K satisfies both bP* , K > δ max and the desired constrained control performance requirements, go directly to Step 3. If a controller with both bP* ,K > δ max and desired closed-loop performance is not found, set j = j − 1 and go to Step 3. Step 7. Repeat Step 3 – Step 6 until in neither Step 4 nor Step 6 can an acceptable MPC controller be found. Step 8. Then the (i - i0 + 1) × (j -j0 + 1) linearized models P(k, l) (i0 ≤ k ≤ i, j0 ≤l ≤ j) are classified into one group. Namely, the operating range {(α, β)| αi0 ≤ α ≤ αi, βj0 ≤ β ≤ βj } is a new subregion with its local linear model P* and local MPC controller K. Step 9. If i ≤ m or j ≤ n, choose a new starting point, such as i0=i, j0=1, which is a vertex or an edge point of the former subspace, and go back to Step 2 to determine a new subregion. If i > m and j > n, then the integrated multi-linear model decomposition and local linear MPC design of the nonlinear system is completed. Step 10. Suppose the nonlinear system is decomposed into Nm subsystems, i.e. Nm local linear models/controllers are designed after Step 9. The Nm local linear controllers are combined into one global MPC controller by the gap metric based weighting method, as discussed in Section 3. The schematic diagram of Algorithm 1 (A1) is shown in Figure 1. The local model and MPC controller pair is handled one by one in succession. Then the local MPC controllers are combined into a global MPC controller using the gap-metric-based weighting method. The gap metric is applied through the whole design process, connecting the local model selection, local MPC controller design and the final local MPC controller combination.



Nonlinear System δ(Pi ,Pj), bP,K Local model P1 bP1,K1>δmax 1 st

δ(Pi ,Pj), bP,K

Local model Pi

bPi,Ki>δmax i

Local MPC controller Ki

ith

...

Local model Pn bPn,Kn>δmax n

δ(Pi ,Pj), bP,K

Gap metric based weighting method

1

Local MPC controller K1

...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 22

Global MLMPC Controller K

Local MPC controller Kn

nth

Figure 1. Design scheme of Algorithm 1

Algorithm 2: Integrated MLMPC using the maximum stability margin (Extension of Algorithm 2 in ref 21) S1. Set i0 = 1, j0 = 1. S2. Choose a design parameter ε. S3. Set i = i0, j = j0. S4. Set i = i + 1. Choose the nominal model P* among the (i - i0 + 1) × (j - j0 + 1) linearized models P(k, l) (i0 ≤ k ≤ i, j0 ≤ l ≤ j ) according to Eq. (11), and compute the biggest gap δ max according to Eq. (12) and the maximum stability margin bopt(P*) according to Eq (3). Then the threshold value is set by Eq. (13): γ t = min(bopt ( P* ), ε )

(13)

S5. If δ max ≤ γ t , go directly to S6. If δ max > γ t , set i = i − 1 and go to S6. S6. Set j = j + 1. Choose the nominal model P* among the (i - i0 + 1) × (j - j0 + 1) linearized models P(k, l) (i0 ≤ k ≤ i, j0 ≤ l ≤ j ) according to Eq. (11), and compute the biggest gap δ max according to Eq. (12) and the maximum stability margin bopt(P*) according to Eq. (3). Then the threshold value is set by Eq. (13). S7. If δ max ≤ γ t , go directly to S4. If δ max > γ t , set j = j - 1 and go to S4. S8. Repeat S4 – S7 until δ max > γ t in both S5 and S7.


Page 11 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


S9. For the local linear model P*, design an unconstrained linear MPC controller K according to Eqs. (7)-(8). If a controller with both bP* ,K > δ max and desired closed-loop constrained control performance is not found, go back to S2 and adjust ε. If the controller K satisfies both bP* ,K > δ max and the desired constrained control performance, then the (i - i0 + 1) × (j - j0 + 1) linearized models P(k, l) (i0 ≤ k ≤ i, j0 ≤ l ≤ j) are classified into the subregion {(α, β)| αi0 ≤ α ≤ αi, βj0 ≤ β ≤ βj } with its local linear model P* and local MPC controller K. S10. If i≤m or j≤n, choose a new starting point and go back to S3 to determine a new subregion. If i>m and j>n, then the integrated multi-linear model decomposition and local linear MPC design of the nonlinear system is completed. S11. Local linear controllers are combined into one global controller by the gap metric based weighting method. In Algorithm 2 (A2), the minimum of ε and the maximum stability margin is used as threshold values. The gap metric is used throughout the design process as well. The schematic diagram of Algorithm 2 is omitted, as it is similar to Figure 1 except ε and bopt(P*) are added into the scheme. Remark 1: In both A1 and A2, it is clear that the normalized coprime stability margin of the local model with its stabilizing controller is bigger than the biggest gap between the local model and the other linearized models in its subregion. Therefore, the controller can stabilize all of the linearized models in the subregion according to Proposition 1. So the local robust stability is guaranteed. Note that the stability here denotes stability of the system around its equilibrium points and does not consider constraints, which are included through an additional evaluation of the closed-loop control performance using simulation. The global stability of the nonlinear system is not guaranteed, which is a limitation common to many multi-model control approaches applied to nonlinear systems.12 Remark 2: Obviously, in the decomposition methods, the subregions are all rectangles for a nonlinear system with two scheduling variables. Different choice of a new starting point in Step 9 and S10 may lead to a different decomposition result. Remark 3: ε can be different for different subregions. ε is typically chosen between 0.3 and 0.8. If ε is properly tuned, Algorithm 1 and 2 may result in the same decomposition. Remark 4: Algorithm 1 is more complicated than Algorithm 2, because a linear controller needs to be designed and tested every time a linearized model is added into a subregion. However, Algorithm 1 is simpler than many empirical methods for three reasons. (a) We grid the system using the dichotomy method, so the number of linearized models is low. (b) For a succession of linearized models it is usually very easy to design and test the controllers, because the parameters of the controllers are similar. Once a satisfactory controller is designed for a neighbor linearized model, only small changes are typically needed to get controllers for others. (c) All the design and test work is done offline; no computational load is added in the online MLMPC implementation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 22

5 CASE STUDIES In this section, both a SISO and a MIMO nonlinear chemical systems are studied to demonstrate the effectiveness of the proposed algorithms. Here we focus on the performance of the integrated MLMPC controllers, in which the three key elements of the multi-model approach are unified by the gap metric criteria. In the following simulation study, we were able to tune ε to make A2 have the same decomposition results as A1, so we present simulation results for only one algorithm. In the following simulations, we use the MPC Toolbox in Matlab to solve the Quadratic Programming problem formed by Eqs.(7)-(8).

Figure 2. Steady-state input-output map for the exothermal CSTR.

5.1 Case 1: A SISO CSTR Consider a benchmark exothermal continuous stirred tank reactor (CSTR) process26 described by Eq. (14).

 x&1 = − x1 + Da ⋅ (1 − x1 ) ⋅ exp[ x2 /(1 + x2 / γ )]   x&2 = − x2 + B ⋅ Da ⋅ (1 − x1 ) ⋅ exp[ x2 /(1 + x2 / γ )] + η ⋅ (u − x2 ) y = x 2 

(14)

where x1 is the reagent conversion, x2 is the reactor temperature (output), and u is the coolant temperature (input). All variables are dimensionless. The nominal values for the constants are Da = 0.072, γ = 20, B = 8 and η = 0.3, respectively. The operating range is { y | y ∈ [0, 6]} .

This exothermal CSTR system exhibits strong output multiplicity, as can be seen from Figure 2. The output y is chosen as the scheduling variable since it characterizes the nonlinear behavior and uniquely defines the operating condition of the exothermal CSTR process. Applying the proposed integrated MLMPC methods (A1, and A2 with ε = 0.75) to the exothermal CSTR system, we get 3 subsystems with 3 linear MPC controllers as summarized in Table 1. The parameters Qy, Qu, Ny, and Nu for every


Page 13 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


linear MPC are tuned to satisfy the local stability and control performance. The three local linear MPCs have different parameters in this example. The threshold value in the gap-metric-based gridding algorithm is 0.15, and 42 gridding points are obtained. Table 1 gives the operating points of the local linear models, the subrange of each subsystem, the biggest gap corresponding to each local linear model, and the gap metric stability margin of each local MPC. Simulations confirm that constraints do not degrade stability and performance significantly. Table 1. Decomposition result of the exothermal CSTR according to A1, or A2 with ε= 0.75 Subregion

1st

2nd

3rd

Operating point of local linear model (x1, x2, u)

(0.2159,1.4375, 0.473)

(0.6077,3.625, -0.4937)

(0.7393,4.5,-0.2149)

Local linear model

P1* =

δmax

δ max ( P1* ) = 0.7405

δ max ( P2* ) = 0.4322

δ max ( P3* ) = 0.5249

Subrange

0 ≤ y≤ 2.1

2.1 < y ≤ 4

4

Integrated Multilinear Model Predictive Control of Nonlinear Systems

Recommend Documents