Switched Offline Multiple Model Predictive Control with Polyhedral

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Switched Offline Multiple Model Predictive Control with Polyhedral Invariant Sets Dewen Li, Xiangyuan Tao, Ning Li, and Shaoyuan Li Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01412 • Publication Date (Web): 24 Jul 2017 Downloaded from http://pubs.acs.org on August 3, 2017

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Industrial & Engineering Chemistry Research

Switched Offline Multiple Model Predictive Control with Polyhedral Invariant Sets Dewen Li, Xiangyuan Tao, Ning Li*, Shaoyuan Li Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240 Corresponding author*: [email protected]

Abstract—This paper presents a switched offline multiple model predictive control procedure for nonlinear processes to ease the online computational burden and reduce the number of sub-models. We employ the gap metric to characterize the dynamic difference between linear models and establish a linear model bank to approximate the nonlinear system. Based on the robust MPC algorithm, we develop an offline model predictive controller for each sub-model. The polyhedral invariant set is utilized to expand the work scope of each local controller. In the offline part, a series of discrete states are selected, the corresponding feedback gains are pre-computed and associated polyhedral invariant sets are constructed. In the online implementation, the control input is simply calculated by calling the feedback gain according to the current state. A switching rule is then designed to integrate the sub-models and guarantee the stability of the whole system. Finally, the corresponding simulation example is presented to validate the efficiency of the presented algorithm. Index Terms—Multiple model predictive control; Offline

method; Polyhedral invariant set; Gap metric; Switching stability; I.

INTRODUCTION

Model Predictive Control (MPC), aiming at handling the constrained control problem, is widely adopted in the industrial process control. Nevertheless, nonlinear MPC (NMPC) technique is far away from mature in both theory and application, which limits the existing MPC algorithm to linear or quasi-linear systems. Moreover, the heavy online computation burden causes the poor performance in real-time [1, 2] . Many researches take the technique of linearization to handle the nonlinear process [3, 4, 5]. However, for these nonlinear systems with a wide operating region, one linear MPC cannot be efficient unless the system always works in a small region around the operating point. Multiple model predictive control (MMPC), approximating the nonlinear system by a set of linear model in which the linear MPC is designed, is generally used to deal with the constrained nonlinear process with a large operating region [6, 7 , 8]. Using the divide and conquer strategy, the MMPC technique can effectively decrease the heavy computation burden of NMPC [9] . The decomposition of a nonlinear system and composition of sub-models are two main parts in multiple model method. A complete and non-redundant sub-model bank not only relates to the global stability but also reduces the complexity of the controller design. The selection of an equilibrium point is generally dependent on experience or

previous knowledge of the system [10, 11]. The neural network is also used to identify the linear model bank [12]. However, these selection schemes separate the bank establishment from controller design. Moreover, the selection of previous sub-model has little directive influence on the following one [13] . In conclusion, the above methods cannot guarantee the nonredundancy of the selected model set. The gap metric theory was firstly introduced into control field in [14]. It provides a better measurement of the dynamic difference between linear models than the norm-based metric [15]. Tao et al. proposed a neighborhood estimation algorithm based on gap metric to determine the approximated region of each sub-model [9]. In [16], a gap metric based procedure is presented to partition the operating region by prescribing a distance level. In [17], the gap metric is applied to choose operating points in multimodel method. Other applications based on gap metric are also proposed [18, 19]. The combination of sub-models is the other crucial problem in MMPC procedure. The switching methods and weighting methods are two typical modes for the integration of sub-models. Proper weighting functions are used to unite the local controllers’ information in weighting methods [20, 21, 22]. For switching methods, only one local controller is chosen according to an appropriate performance index [20]. MMPC method transfers the nonlinear MPC to linear MPC and acquires satisfactory effects. However, the frequent switch between sub-models and heavy online computational burden will deteriorate the MMPC performance. Enlarging the feasible region of the local controller and moving the online computational burden offline are the efficient ways to solve the above problems. Enlarging the work scope of local controllers can reduce the switch and optimize the MMPC performance. When designing local model predictive controller, the terminal state constraints are usually used to handle the difficulty in stability analysis [23].The ellipsoidal invariant set is widely used in the robust MPC algorithms to handle the constraints of the states and inputs [24, 25]. Nevertheless, it becomes conservative when the constraints of the process are given as linear inequalities. The polyhedral invariant set is superior in dealing with the linear constraints. In [9], the authors presented a modified IHMPC algorithm with a polyhedral stability region to decrease the number of sub-models. In [26], a fast robust MPC algorithm with polyhedral invariant sets is presented, which expands the feasible region of the controller. The heavy computational burden has partly limited the application of MMPC. The rolling optimization mechanism of MPC method increases the tolerability for the mismatches

ACS Paragon Plus Environment


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and uncertainties of models, however, it also increases the computational burden. Lots of researches have been focusing on high-efficiency MPC algorithms. The tool of linear matrix inequality (LMI) was used to ease the online computational burden [24]. A novel scheme where the predefined feedback laws are interpolated is developed for MPC design, which reduces the computational burden [27]. In [28], a novel interpolation between global Vertex Control and local unconstrained robust control is presented for linear discrete-time systems, which provides a computationally easy solution. The offline design and online synthesis is one of the effective strategies to ease the online computation. In [25], a sequence of ellipsoidal invariant sets is constructed and corresponding feedback gains are obtained in advance, which moves the online calculation burden offline. Based on pre-computed feedback gains, each with its own polyhedral invariant set, two interpolation techniques are developed for the offline model predictive control for the discrete linear time-varying system [29], which efficiently eases the online computation. This paper introduces the offline strategy and polyhedral invariant set to design less-switched MMPC with a low online computational burden. We take the gap-metric-based neighborhood estimation algorithm [9] to construct the linear model set. In the local controller design, a series of nested polyhedral invariant sets are pre-constructed and the corresponding feedback gains are obtained, which enlarges the feasible region of local controllers and reduces the online computation. In the online part, the input is easily calculated by calling the feedback gain in accordance with the system state. Then a switching rule between sub-models is designed to guarantee the stability of the whole system. The rest of this paper is arranged in following manner. Section II states the problem formulation. In section III, we focus on the local offline controller design and give the corresponding proof and simulation. In section IV, we present a switching rule to combine sub-models and give the global stability analysis. Section V gives the conclusion of this paper. II. PROBLEM FORMULATION Consider a nonlinear time-invariant system described as follows: x(k + 1) = f ( x(k ), u (k ))

(1)

where x ∈ X ⊆ ℜ , u ∈ U ⊆ ℜ are the constraints of the states and inputs. The structure of MMPC design for (1) can be described as Figure 1. According to a sequence of chosen equilibrium points, we divide X into n sub-regions X 1 , X 2 ,..., X n such that X = X 1 ∪ X 2 ∪ ∪ X n . In sub-region X i , a linear sub-model is obtained and the local predictive controller #i is designed. In each sampling period, a linear model and corresponding local controller is selected in accordance with a certain switching rule. As mentioned before, the feasible region and online computational burden of the local predictive controller can nx

nu

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greatly influence the performance of MMPC system. In the local MPC design, the ellipsoidal invariant set is usually used to handle the constraints, which is conservative when dealing with asymmetrical constraints. Additionally, it is time-consuming to calculate the input by solving a constrained optimization problem in each sampling time. In this paper, we take the polyhedral invariant set to expand the feasible region of the local predictive controller and use offline strategy to ease the online computational burden. We expect to utilize such strategies to improve the MMPC performance and try to answer following questions: 1. How to enlarge the feasible region of a local predictive controller by using the property of polyhedral invariant set? 2. How to design the offline controller based on offline design and online synthesis strategy to ease the online computation? 3. How to design the switching rules inside and outside of each local controller to guarantee the global stability of the whole system?

Figure 1. Structure of switched MMPC system

III. OFFLINE PREDICTIVE CONTROLLER DESIGN A. Neighborhood estimation using gap metric Ranging from 0 to 1, the gap metric is proved better than ∞-norm in measuring the dynamic distance of two linear systems. A small gap metric indicates the two linear models can be controlled by at least one common feedback controller [15] . Suppose P( s ) be a rational transfer matrix. Let P( s ) takes on a normalized right coprime factorization: ~

~

P( s ) = N ( s ) M −1 ( s ) , with M M + N N = I ~

~

where M ( s ) = M (− s )T , N ( s ) = N (− s )T . The gap metric between two linear systems P1 and P2 is formulated as [15] (2) δ ( P1 , P2 ) = max{δ ( P1 , P2 ), δ ( P2 , P1 )} where


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M  M j 

inf  i  −  δ ( Pi , P=  Q , i, j ∈ {1, 2}, i ≠ j j) Q∈H  Ni   N j  ∞ ∞

The gap defines a metric on the space of (even unstable) linear models [16], and it is superior in comparing the dynamic behaviors of the linearized models at different operating points of the nonlinear system. The gap metric can help to construct the linear model bank in the MMPC method. Algorithm 1 [9]. Neighborhood Estimation 1. Select a desired equilibrium point E p :{x p , u p } , linearize

The model predictive control algorithm is used to handle the tracking problem that drives the considered system from any state within the sub-region to the equilibrium point. Consider the following optimization problem that minimizes the infinite horizon quadratic cost function: min J (k ) u ( k + j | k ), j =0,1,..., ∞

T

 x (k + j k )  Q 0   x (k + j k )  J (k ) = ∑      (k + j k )   0 R  u (k + j k )  j = 0 u where Q > 0 , R > 0 , and subject to ∞

(4)

the nonlinear system (1) at E p and obtain the linear

ur (k + j k ) ≤ ur ,max , j ≥ 0, r = 1, 2,..., nu ，

(5)

model Tp :{ A, B, C , D} .

xr (k + j k ) ≤ xr ,max , j ≥ 0, r = 1, 2,..., nx .

(6)

2. Construct a large region around E p in the operating region of the system, labeled as Φ ={( x, u ) x ∈ Π x , u ∈ Π u } , where Π = = {x | x − x p ≤ δ x} , Π {u | u − u p ≤ δ u} . x u 3. In Φ , equably select N ( N is large) reference points Ei ( xi , ui ), i = 1,2,, N , linearize system (1) at ( xi , ui ) and acquire corresponding linear model set Si :{ Ai , Bi , C , D} ; 4. Initialize X p ={E p } and i = 1 . Prescribe a distance level τ . 5. Calculate the gap metric between Si and Tp , labeled as gap (i ) . If gap (i ) ≤ τ , set X p =X p  E i . 6. If i < N , set i = i + 1 and go to step 5. If i = N , stop and transfer X p to a continuous state region, which can be represented by X p ={x M ( x − x p ) ≤ d } . Remark 1. Algorithm 1 constructs an estimated neighborhood around the selected equilibrium point. That means we can design local feasible feedback controller in the obtained region. Remark 2. The initial region Φ should be large enough such that the linearized model Tp is definitely not able to span the whole range based on the prior knowledge of the system. Once Φ is settled, N is determined according to a chosen sampling interval. Remark 3. The selection of the threshold τ is a trade-off. It is necessary to consider the balance between the approximation and switching effect. The empirical value of τ is given as 0.4 ≤ τ ≤ 0.6 [9,16,17]. B. Constrained MPC based on LMI Consider a nonlinear time-invariant system described as (1). Linearize (1) at the equilibrium point E p :{x p , u p } . Let x= x − x p , u= u − u p , the linear model Tp is given below: x (k += 1) Ax (k ) + Bu (k )

(3)

where nu and nx are the dimensionality of input and state. Following [24], a state feedback control law u (k + j k ) = Kx (k + j k ) is designed to minimize the upper bound of J (k ) . In (3), construct a quadratic cost function of x as follow: (7) V ( x ) = x T Px Assume V ( x ) satisfies the following stability condition: V ( x (k + i + 1| k )) − V ( x (k + i | k )) ≤ (8) 2 2 − [ x (k + i | k ) Q + u (k + i | k ) R ] Suppose the system is asymptotically stable, which indicates V ( x (∞ | k )) = 0 or x (∞ | k ) = 0 . Summing (8) from = i 0 ~ ∞ , we get: (9) J ∞ (k ) ≤ V ( x (k | k )) Define a scalar γ > 0 and matrixes = Qt γ= P −1 , Yt KQ . Let V ( x(k | k )) ≤ γ . Based on Schur complements, the optimization problem (4) is transferred to the following optimization problem via LMIs:

1 x T   ≥0 * Qt 

(10)

Qt Qt AT + Yt T BT Qt Q1/2 Yt T R1/2    0 0  Qt (11) * ≥0 * * 0  γI   * * γI  * The constraints of input and state described by (5) and (6) are guaranteed if the following LMIs are satisfied:  X t Yt  2 (12) 1,..., nu  * Q  ≥ 0, X t , jj ≤ u j , j = t 

Z − Qt ≥ 0, Z rr ≤ xr2,max , r = 1,..., nx

(13)

where X t and Z are symmetric matrixes. Lemma 1[24]. Robust MPC using LMIs For system (3), at each sampling time k , the state feedback control law u (k + j k ) = Kx (k + j k ) is applied to minimize

the

upper −1 t

bound

of

J (k )

,

where K (k ) = Yt (k )Q (k ) . Yt (k ) and Qt (k ) are calculated by ACS Paragon Plus Environment


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solving the following minimization problem: min γ subject to (9),(10),(11) and (12) γ , Qt ,Yt , X t , Z

(14)

The minimization problem (14), if feasible at sampling time k , would be feasible at each time after k . The above MPC algorithm asymptotically stabilizes the closed-loop system. Remark 4. The symmetrical constraints (eqs 5 and 6) are required when using above algorithm. An estimated ellipsoidal stable region is given as Ψ ={x ∈ ℜn ( x − x p )T Qt−1 ( x − x p ) ≤ 1} in each

(16) M 1 x ≤ d and umin ≤ u ≤ umax 2. Construct a symmetrical region described as (5) and (6) in the neighborhood. Estimate the maximum ellipsoidal invariant set with Lemma 2, and the state feedback matrix K p ,0 = Yp ,0 G p−,01 is obtained, where Yp ,0 and G p ,0 are the optimization solution of problem (15). 3. Define matrixes as follow:  M1   d1    M =  K  , d = umax  − K   umin 

x

sampling time. Based on Lemma 1, the ellipsoidal invariant set for system (1) is adopted to delimit the initially feasible region of the control problem. Definition 1[25]. Consider a discrete autonomous system x(k + 1) = f ( x(k )) , if a set Ψ ={x ∈ ℜn xT G −1 x ≤ 1} satisfies:

solution of Qt in following problem:

max log det(Qt ) subject to (11)-(13)

γ , Qt ,Yt X t , Z

(15)

(17)

4. Set j = 1 . 5. Solve the following optimization problem:

x

whenever x(k0 ) ∈ Ψ , x(k ) ∈ Ψ and ∀k > k0 , then Ψ is called an ellipsoidal invariant set. Lemma 2[25]. For the closed-loop system (3), an asymptotically stable ellipsoidal invariant region of infinite-horizon MPC in Lemma 1 is given by Ψ ={x ∈ ℜnx ( x − x p )T G −1 ( x − x p ) ≤ 1} , where G is the

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max W x

s.t. W = M j ( A + BK ) x − d j   Mx ≤ d  

(18)

where M j , d j are the j th row of M and d . 6. If

the optimization solution M   d  , d = . M =   M j ( A + BK ) d j 

W ≥0

,

let

Note that the region constructed by Algorithm 1 is probably asymmetrical. However, a symmetrical sub-region in X must be constructed before using Lemma 1 [9]. Generally, the obtained invariant ellipsoid is much smaller than X , which obviously leads to great conservativeness. Next, we introduce the polyhedral invariant set to expand the feasible region of local controllers.

7. Let j= j + 1 , and go to step 5 until j > Row( M ) ,

C. Offline MPC with Polyhedral Invariant Set

sampling time. If not, add the constraint into M and d . With the iteration, the region described by M and d becomes smaller and smaller. When the iteration finishes, the states in the region will definitely satisfy the property of Mx ≤ d .

The polyhedron is more suitable to approximate the obtained sub-region than ellipsoid. In this section, we propose an offline MPC algorithm based on polyhedral invariant sets. Definition 2[30]. For system x(k + 1) = f ( x(k )) , the set Φ ={x ∈ ℜ

Mx ≤ d } is called an asymptotically stable

nx

polyhedral invariant set, if it satisfies: whenever x(k0 ) ∈ Φ , x(k ) ∈ Φ for all times ∀k > k0 and x(k ) → 0 as k → ∞ . The definition of polyhedral invariant set indicates its property of describing the state region by a set of linear constraints. It is superior in dealing with the asymmetrical constraint. So, we take the polyhedral invariant set to design the local controller. We firstly give an algorithm to construct the polyhedral invariant set based on the feedback gain. [30]

Algorithm 2

. Polyhedral Invariant Set Estimation

1. For the neighborhood obtained by Algorithm 1, write the corresponding constraints of states and inputs as follows:

where Row( M ) is the number of rows in M . Remark 5. The optimization problem (18) checks whether the constraint described by M j and d j is satisfied at the next

Algorithm 3. Offline MPC with Polyhedral Invariant Set Offline 1. For an equilibrium point E p :{x p , u p } in nonlinear system (1), estimate the approximated region of linear system (3) using Algorithm 1. 2. Construct the polyhedral invariant set nx Φ p ,0= {x ∈ ℜ M p ,0 ( x − x p ) ≤ d p ,0 } using Algorithm 2, and the feedback matrix is obtained by K p ,0 = Yp ,0 G p−,01 . 3. In Φ p ,0 , select a series of discrete state points which are gradually close to E p :{x p , u p } , labeled as xlp (l = 1, 2,..., n p ) . Let x lp =xlp − x p (l =1, 2,..., n p ) . Initialize i = 1 . 4. Replace x by x ip in (10), and solve the optimization problem (14) with an additional constraint G p ,i−1 > Qt . Store


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1 2 the feedback matrix K p ,i = Yp ,i G p−,1i in a lookup table, That is to say, (22) V (k + 1) ≤ V (k ) 3 where Yp ,i and G p ,i are the solutions of Yt and Qt in (14). For system (3), the Lyapunov function is strictly 4 5. Using Algorithm 2 (ignore step 1 and 2), estimate the decreasing, with the property of stable polyhedral region, the 5 polyhedral invariant set initial state in Φ 6 p ,i−1 will definitely enter Φ p ,i in finite time. nx based on . K Φ = x ∈ ℜ M x − x ≤ d { ( ) } p , i 7 p ,i p ,i p p ,i This process continues until x(k ) ∈ Φ p , n , and the feedback 8 6. Let i = i + 1 , go to step 4 until i > n . control input = u (k ) K p , n ( x(k ) − x p ) + u p is guaranteed to 9 10 drive the system to the equilibrium point. □ Online 11 Search the maximum index i such that the current state D. Simulation Example 12 satisfies . The input is given x(k ) ∈ Φ p ,i 13 The continuous stirred tank reactor (CSTR) is widely 14 as u= (k ) K p ,i ( x(k ) − x p ) + u p . applied in chemical industry, which can be described as 15 follows [25]: ⋅ Remark 6. 16 q E C = (C Af − C A ) − k0 exp(− )C A A The constraint of G p ,i−1 > Qt guarantees that the obtained 17 V RT (23) 18 ⋅ invariant ellipsoids are constructed one within another. In q E UA −∆H T= (T f − T ) + k0 exp(− )C A + (Tc − T ) 19 T V RT V ρC p ρC p step 4, we initialize = M p ,i  M p ,i −1 K p ,i − K p ,i  and 20 21 The controlled states are the concentration ( C A ) and T , the obtained invariant d p ,i  d p ,i −1 umax −umin  22= temperature ( T ) in the reactor. Tc is the input, which 23 polyhedral sets are also nested, i.e., the index i is unique at represents the temperature of the coolant stream. The 24 each sampling time. corresponding constraints are , 0 ≤ C A ≤ 1mol / L 25 250 K ≤ T ≤ 500 K , 250 K ≤ Tc ≤ 500 K . Other parameters 26 Remark 7. 27 The robust MPC in Lemma 1 gets the optimal solution of are q = 100 L / min , V = 100 L , = k0 4.71×108 min −1 , 28 the feedback gain K in each sampling time, which is , UA 105 J / ( g ⋅ K ) , ρ = 103 g / L , ∆H =−2 ×105 J / mol= 29 time-consuming. This offline method sacrifices optimality = C p 1J / ( g ⋅ K ) , C Af = 1mol / L , E / R = 8000 K , T f = 400 K . 30 somewhat to ease the online computation. 31 In our previous work [9], we choose the equilibrium point T 32 Remark 8. as x p [C= = [0.52,398.8]T . With a sampling time of 18s, A ,T ] p 33 In step 3, the selection of the discrete states can be a the linearization model Tp ,1 is obtained as: 34 trade-off. More discrete states can improve the optimality of (24) x(k += 1) A1 x(k ) + B1u (k ) 35 the control law but it somewhat increases the computational 36 burden of the offline part. Moreover, the selection also relates  −1.095e − 5 0.942 −0.7337e − 3 , B1 =  to the size of the initial feasible region. Generally, more where A1 =  37 .  1.086  0.03129  5.615  discrete states are selected in a bigger feasible region. On the 38 other hand, the selected discrete states must satisfy the 39 1 40 condition of xip+1 = < 1(i 0,1, 2,..., n p − 1) to guarantee the G 0.9 41 switching stability. 0.8 42 43 0.7 Theorem 1. For system (1), if the initial state x(0) ∈ Φ p ,0 , the 44 0.6 45 closed-loop system is asymptotically stable based on the 0.5 46 offline MPC in Algorithm 3. 0.4 47 Proof. Suppose the current state is within Φ p ,i−1 but outside 0.3 48 of Φ p ,i . Consider the Lyapunov function as follows: 49 0.2 (19) V (k ) = x (k )T Px (k ) with P = γ Qt 50 0.1 51 γ and Qt are obtained by solving optimization problem (14), 0 52 180 160 140 120 100 80 60 40 20 0 which indicates the feedback matrix K satisfies : 53 (20) [ A + BK ]T P[ A + BK ] − P + K T RK + Qt ≤ 0 54 Figure 2. Gap metrics between reference models and Tp ,1 T Multiply and at the right and left side of (20) .We x  ( k ) x (k ) 55 Then, we construct a large region around x p , with which 56 get 189 reference points are selected to compute the gap metrics 57 x (k + 1)T Px (k + 1) − x (k )T Px (k ) ≤ (21) 58 δ i , i = 1,2,...,189 between the reference models and Tp ,1 . The − x (k )T [ K T RK + Qt ]x (k ) 59 60 gap

p ,i



threshold value of gap metric is chosen as τ = 0.4 . The relationship between δ i and τ is shown in Figure 2. Based on the obtained efficient neighborhood, we implement the proposed offline MPC algorithm to drive the system from the initial state x0 = [0.375,411.1]T to the

sub-models are required and the switch between sub-models causes worse spiking effects on the performance. Detailed comparison will be discussed in the global control presented in next section. Figure 4(d) gives the switch between polyhedral invariant sets of OPMPC method.

desired state x p = [0.52,398.52]T . Using Lemma 2, the

0.55

feedback gain is given as F1,0 = [ −406.93 −6.58] , and the obtained as , where

states

as

x11 = [0.42, 407.28]T

x = [0.46, 403.95] 3 1

T

x12 = [0.44, 405.61]T

,

x = [0.48, 402.28]

T

4 1

,

0.45

A

0.015 −0.9  G1,0 =  .  −0.9 116.74  Based on Algorithm 2, the estimated polyhedral invariant set Φ1,0 is shown in Figure 3. In Φ1,0 , we choose six discrete

0.5

C (mol/L)

maximum invariant ellipse is T −1 Ψ1,0 = {x | ( x − x p ) G1,0 ( x − x p ) ≤ 1}

, ,

0.35

x = [0.5, 400.62] , x = [0.51,399] . Using Algorithm 3, the polyhedral invariant sets are shown in Figure 3. The corresponding feedback matrixes are F1,1 = [−283.39, −4.95] , T

5 1

6 1

OPMPC OEMPC GM-MMPC OIMPC

0.4

0

50

100

T

F1,2 = [−289.55, −5.17]

,

F1,3 = [−234.52, −4.52]

,

F1,4 = [−171.55, −3.69]

,

F1,5 = [−161.85, −3.55]

,

150 Sampling time

200

250

300

(a) Concentration of reactor-- C A 412 OPMPC OEMPC GM-MMPC OIMPC

410

F1,6 = [−162.82, −3.60] .

408

420

T(K)

406

415

X: 0.375 Y: 411.1

OEMPC

404

OPMPC

410

402 405

400

400

OIMPC

T(K)

395

398 0

GM-MMPC

50

100

390

200

250

300

(b) Temperature of reactor-- T

385 310

380

375 0.2

150 Sampling time

0.3

0.4

0.5

0.6

0.7

0.8

305

0.9

CA (mol/L) 300

Figure 3. Regions of each polyhedral invariant set

x(k ) ∈ Φ1,i , x(k ) ∉ Φ1,i +1  F1,i ( x(k ) − x p ) + u p u (k ) =  ( ( ) ) F x k x u x(k ) ∈ Φ1,6 − + (25)  1,6 p p i = 0,1,...,5 The performance of proposed offline MPC is shown in Figure 3 and Figure 4. Here, the proposed offline algorithm (OPMPC) is compared with the ellipsoidal invariant set based offline MPC (OEMPC), the GM-MMPC algorithm in [9] and the interpolation based offline MPC (OIMPC) using Algorithm 1 in [29]. Compared with other algorithms, OPMPC method provides least number of sub-regions and online computation time. In OEMPC method, two

295 c

Based on Algorithm 3, the input is given as follows:

T (K)

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 6 of 11

290

285 OPMPC OEMPC GM-MMPC OIMPC

280

275

0

50


100

150 Sampling time

200

(c) Temperature of coolant-- Tc

250

300

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satisfies x(k ) ∈ Φ i ,0 . In Φ i ,0 , Search the maximum index j

6

such that the current state satisfies x(k ) ∈ Φ i , j .The input is

5 Number of ployhedral set

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


given as u= (k ) Fi , j ( x(k ) − xi ) + ui . 4

Remark 9. In the third step, we choose the next equilibrium point in the region of (Φ i ,0 − Φ i ,1 ) which is closest to initial point. By

3 2 1 0 0

50

100

150 sampling time

200

250

300

(d) Switch of polyhedral sets Figure 4. Simulation results on CSTR

IV. OFFLINE MMPC FOR NONLINEAR SYSTEM

this way, we can obtain the least sub-regions to cover the operating region. The flowchart of the offline MMPC (offline MMPC with polyhedral invariant sets, or OPMMPC) is explained in Figure 5. A two-layer switching structure is applied in the control mechanism. By judging the current state, the local controller is determined in the outer layer and the feedback law is selected in the inner layer. For any given initial state which satisfies x(0) ∈  ni=1 Φ i ,0 , it can be proved that the system will be driven to the desired state in finite time by using the proposed control method.

A. Offline MMPC with Polyhedral Invariant Set

x= (0)

The utilization of polyhedral invariant sets enlarges the work scope of the local controller to some degree. It still remains a large gap in the application in the nonlinear system which works in a large operating range. In this section, we present an offline MMPC procedure with which a switching rule with stability guaranteed is designed.

x= 0 0,k

x(k ) ∈Φ1?

x(k ) ∈Φ 2 ?

i=1

... ...

No

Yes

x(k ) ∈ Φ n ?

i=2

No

Yes

i=n

x(k ) ∈ Φ i ,ni ?

2. Compute the approximated linear model Tp ,i . Using

Yes

Algorithm 3, design the local offline predictive controller #i for Tp ,i . Meanwhile, the parameter matrixes

No

Inner layer

x(k ) ∈ Φ i ,ni −1?

Yes

u (k ) = Fi ,ni x(k )

number of the selected discrete states in the local controller. The feedback matrixes are computed −1 as Fi , j Y= . The polyhedral invariant sets = G j n ( 0,1,..., ) i, j i, j i are expressed nx as Φ i , =j {x ∈ℜ M i , j ( x − xi ) ≤ di , j }( =j 0,1,..., ni ) .

Outer layer

Yes

Algorithm 4. Offline MMPC with Polyhedral Invariant Sets Offline 1. Let i = 1 . Choose the desired operating point as the first equilibrium point, labeled as E1p :( x1 = xt , u1 ) .

Yi , j , Gi , j ( j = 0,1,..., ni ) are obtained, where ni represents the

No

u (k ) = Fi ,ni −1 x(k )

... ... No x(k ) ∈ Φ i ,0 ?

No

Yes

u (k ) = Fi ,0 x(k )

x ( k + 1) = f ( x ( k ), u ( k )) k= k + 1

3. If the initial state satisfies x(0) ∈Φ i ,0 , set n = i and stop. If x(0) ∉Φ i ,0 point

,

choose i +1 p

E :( xi+1 , ui+1 )

the

next such

equilibrium that

xi+1 ∈ (Φ i ,0 − Φ i ,1 ) and d ( xi+1 , x(0)) = min d ( xR , x(0)) , where (Φ i ,0 − Φ i ,1 ) represents the region within Φ i ,0 but outside of Φ i ,1 , xR means all the equilibrium points in (Φ i ,0 − Φ i ,1 ) and d ( x, y ) denotes the Euclidean distance between x and y . Set i = i + 1 , and go to step 2. Online Search the minimum index i such that the current state

No

x ( k ) = xt ? Yes Stop

Figure 5. Flow chart of OPMMPC algorithm

Theorem 2. For system (1), the closed-loop system is asymptotically stable in the region of  ni=1 Φ i ,0 based on the OPMMPC in Algorithm 4.



Proof. According to Theorem 1, for the state x(k ) ∈ Φ i ,0 , i =1, 2,..., n , the local offline controller #i drives to

0.5

.Suppose the current state x(k ) ∈ (Φ i ,0 − Φ i −1,0 ), i ≠ 1 , the local controller #i will drive xi

the system to the equilibrium point xi ∈Φ i ∩ Φ i−1 , the controller switches to controller # i −1 when x(k ) ∈ Φ i −1,0 . This switching process will continue until x(k ) ∈ Φ1,0 . The

0.45

0.4

A

system

C (mol/L)

the

controller #1 guarantees the convergence to the objective state xt . □

0.35

0.3 OPMMPC OEMMPC GM-MMPC OIMMPC

0.25

B. Simulation Example The previous simulation shows the performance of the presented offline MPC method on the local control problem of CSTR. Next, we implement the proposed OPMMPC method on the global transfer process. The initial state is selected as x0 = [0.189,432.083]T and the objective state

0.2 0

50

100

150

200

250 300 Sampling time

350

400

450

500

(b) Response of state variable-- C A

is x1 = [0.52,398.8] . First, we design the local controller around x1 via Algorithm 3. It is obvious that x0 ∉Φ1,0 . The switching rule T

435

425

choose another three equilibrium points such that the initial state satisfies x0 ∈Φ 4,0 . The four equilibrium states x3 = [0.266,422]T

,

,

420

T(K)

x4 = [0.215,428.3]T

are

OPMMPC OEMMPC GM-MMPC OIMMPC

430

requires the next equilibrium point x1 ∈ (Φ1,0 − Φ1,1 ) . Here, we

x2 = [0.375,411.07]T and x1 = [0.52,398.8]T , respectively. Here, the presented OPMMPC algorithm is compared with the same algorithms in last section with multiple model strategy, which are called OEMMPC, GM-MMPC and OIMMPC, respectively.

415

410

405

400

395

0

100

50

150

200

350


400

450

500

440

(c) Transfer trajectory of state variable-- T 430

420

310

OEMMPC OIMMPC

410

300

400

290

302

OPMMPC

300

GM-MMPC

390

298

280 T (K)

296

C

T(K)

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 11

380

294

270 0.1

0.2

0.3

0.4

0.5 CA (mol/L)

0.6

0.7

0.8

0.9

60

80

100

120

140

260 OPMMPC OEMMPC GM-MMPC OIMMPC

(a) State transfer trajectory 250

240

0

50

100

150

200


350

400

450

(d) Change of control variable-- Tc

Figure 6. Simulation result of OPMMPC on CSTR


500

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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


In this simulation, four local offline controllers based on polyhedral invariant set are designed. Figure 6(a) shows the connected four polyhedrons Φ1 ~ Φ 4 and the state transfer trajectory. To avoid confusion, we did not plot the nested polyhedrons and ellipses in each sub-region. For OEMMPC algorithm, we only give the trajectory and did not give all the six sub-regions. As shown in Figure 6(b~c), the proposed algorithm drives the system from the initial state to the objective point in finite time. Compared with GM-MMPC algorithm, the settling time of Algorithm 4 is longer because we sacrifice the optimality of the feedback gains in Algorithm 4. But the computation time in each sampling period is about 22 times less than that of GM-MMPC method. On the other hand, the proposed method has less local controllers than that of OEMMPC algorithm. The interpolation technique can partly decrease the spiking effect, but a linear programming problem needs to be solved in each sampling period which somewhat increases the online computational burden. The corresponding results can be found in Table 1. The comparison verifies the superiority of the proposed algorithm in lower online computational burden and less sub-models establishment. Table 1. Comparison of simulation results on CSTR Number of local controllers OPMMPC 4 OEMMPC 6 GM-MMPC 4 OIMMPC 4 *tc represents the average computation time of input Algorithm

supported by Shanghai Science and Technology Commission (NO. 15dz1206700). REFERENCES (1) (2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

tc (s) 0.0028 0.0043 0.0613 0.0203 in each sampling

(10)

(11)

period.

V.

(12)

CONCLUSION

This paper presented a switched offline multiple model predictive control procedure for the nonlinear system to ease the online computational burden and reduce the number of sub-models. A set of piecewise linear models are established to approximate the nonlinear system. In the local controller design, a series of state feedback gains are pre-computed offline and the polyhedral invariant sets are constructed to expand the work scope of local controllers. In the online implementation, the feedback gain is selected to calculate the control input according to the current state. The offline technique eases the online computational burden and the polyhedral invariant set enlarges the feasible region of local controllers. The simulation study verifies the efficiency of the proposed procedure in the control of nonlinear systems. The comparison with other algorithms validates the advantages of the proposed method in lower online computation and less sub-model establishment. ACKNOWLEDGMENT

(13)

(14)

(15) (16)

(17)

(18)

(19)

(20)

This work was supported by the National Nature Science Foundation of China (61374109，61590925，61673273), the National Basic Research Program of China (973 Program-2013CB035500), National High Technology R&D Program of China (863 Program-2015AA043102) and partly

(21)

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Switched Offline Multiple Model Predictive Control with Polyhedral

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