Development of a Real-Time Model-Prediction-Based Framework for

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Development of a real time model prediction based framework for reset controller design Valiollah Ghaffari, Paknosh Karimaghaee, and Alireza Khayatian Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie5027869 • Publication Date (Web): 28 Aug 2014 Downloaded from http://pubs.acs.org on September 2, 2014

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Industrial & Engineering Chemistry Research

Development of a real time model prediction based framework for reset controller design Valiollah Ghaffari*, Paknosh Karimaghaee, Alireza Khayatian School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran KEYWORDS. Reset controller, Model predictive controller, Reset law, Process control.

ABSTRACT. This paper addresses a real time reset controller design based on the model prediction strategy for process control. Reset control design consists of two main steps: 1) basesystem controller design 2) reset law determination. The base system controller is designed according to the base (no reset) plant dynamics and the reset law is determined by an algebraic condition which is checked throughout the time for controller state resetting. In this paper, based on the model prediction approach, a LMI based formulation is derived for designing the reset controller. When the reset condition is satisfied, the reset law is computed by solving an LMI optimization problem, and then base-system parameters and current controller states are suddenly changed to new values. This result is used in two examples; numerical simulations verify the efficiency of proposed approach.

*

Corresponding author. Tel./fax: +98-711-2303081. Email addresses: [email protected], [email protected] (V. Ghaffari),

[email protected] (P. Karimaghaee), [email protected] (A. Khayatian).

ACS Paragon Plus Environment

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1-Introduction Now days, model prediction based schemes have been increasingly used in industrial control applications; this includes internal model controllers (IMC), model predictive controllers (MPC) and their sub-variants 1, 2. In these schemes, based on a given nominal model, the future behavior of the system is predicted and then a control signal is computed such that a given performance index is optimized

3, 4

. It seems that model prediction based approaches are powerful tool and

they can be applied for real time control system design [5-8]. By using such mechanism, at specific times in real time operation, an optimization problem can be solved and then the controller parameters are tuned 5, 6. Recently, a new field known as reset control system has emerged in control system theory 7. The basic idea behind the reset control systems is that, at certain times, the controller states are reset. Depending on the reset times, the controller state resetting action can either improve or destroy the performance and or stability of systems. It has been shown that there may not exist a continues-time controller to satisfy certain constrains on the performance of a system simultaneously but the reset controller can meet these required performance for that system 8. Thus, control systems with the reset action have attracted the attention of some researchers 8-16. In the reset control systems, an algebraic condition is always checked continuously and when the reset condition is satisfied then the states of the controller are reset. In reset control system theory, the reset condition is indicated by a reset set and is known as reset law. Usually reset law includes an equality which must be satisfied exactly, but in practice it is relaxed with a zero crossing technique 12 or reset bounds 15, 17. In a reset control systems when the reset condition is never satisfied, the system is known as base-system. Thus, the base-system is described by ordinary differential equations (ODE) while the reset control system is represented by impulsive


2

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differential equation (IDE). The impulsive behavior can destroy the well-posedness property of the reset control systems solution. The well-posedness of the solution are investigated in 15, 18. The stability and performance of the base-system cannot be extended easily to the reset control systems. Therefore, fundamental stability analyses of reset control systems have been studied in the control literatures. These include Lyapunov based stability analysis , H β stability condition

14

and stability analysis with discrete-time triggering condition

12

based on the passivity property

13, 19

reset control systems are investigated in

8, 10, 20, 21

9, 10

, stability

, reset-time depended stability

16

. The performance properties of

. Reset controllers have successfully been used

in industrial systems which justify the application of these systems to process control 10, 21-28. Reset control system has an extra degree of freedom in comparison to the base-system which is the reset law. Recently, two optimal approaches have been proposed for reset law design 11, 29. In these methods, the reset law is selected by defining an optimization problem and it is shown that the performance of closed loop system can be improved while the stability of system is guaranteed. In reset control system, reset law design is strongly dependent on the base-system parameters. Therefore, suitable base-system design is a key point to the reset control system design which is the main contribution of this paper. In this paper, for the first time, the reset controller parameters are determined by a model prediction approach whenever an algebraic condition is satisfied; which can also be applied in real time operation. It means that at reset times, both the base-system parameters and the reset law are computed simultaneously to meet a desired performance index. The rest of this paper is organized as following: In section 2, some required mathematical lemmas are stated. In section 3, problem formulation is presented and a systematic LMI based framework for designing reset controller is derived in section 4. Proposed reset controller is


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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

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applied to a typical process control system, in section 5, to meet the required performance. Finally, concluding remark is placed in the last section of paper.

2-Mathematical Preliminary To make the paper self-contained, the following two mathematical lemmas from the literature are required. Note that, throughout the paper, L2 norm is stated with . and also the star symbol * is used in the blocked matrix to indicate that a matrix has symmetric property.

Lemma 1 (Schur complement lemma): The following inequalities are equivalent for any three matrix functions Q( x) , S ( x) and R ( x) :  Q ( x ) S (x )  S T (x ) R (x )  > 0  

if

and

only

⇔

if

R (x ) > 0 Q (x ) − S (x )R −1 (x )S T (x ) > 0  OR  Q (x ) > 0 R (x ) − S T (x )Q −1 (x )S (x ) > 0 

(1)

Lemma 2 (Barbalat's lemma): Let φ (.) : R → R be a uniformly continuous function on [ 0, ∞ ) . t

Suppose that lim ∫ φ (τ )d τ exists and is finite then lim φ (t ) = 0 . t →∞

t → +∞

0

In order to present our approach for rest controller design, in the next section the problem formulation is investigated.

3-Problem Setup Consider a plant is described by the following ordinary differential equation (ODE):

 x& p = A p x p + f p ( x p ) + B p u   y = C p x p


(2)

4

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where x p (t ) ∈ R is the state vector, y (t ) ∈ R p is the output vector and u (t ) ∈ R m is the n

input vector of the plant (2). Assumption 1. The nonlinear term f p (.) in equation (2), although unknown, is assumed to be continuous in its argument, and is zero at zero (i.e. f p (0) = 0 ). Also, for any η , µ ∈ R n , there exists a positive scalar L p such that the following Lipschitz condition on f p (.) is satisfied: (3)

f P (η ) - f P ( µ ) ≤ L p η - µ

Assumption 2. Assume that the plant (2) is stabilizable. That is, there exists a control signal

u (t ) such that the solution of the system x (t ) is converged to its equilibrium point. For the given plant (2), a typical reset controller, which its order is same as the plant (2), can be represented by the following impulsive differential equation (IDE):

x&c = Ac x c + B c e x ∉ M  + (4) x c = ρc ( x ) x ∈ M u = C x + D e c c c  n p where x c (t ) ∈ R is the controller state vector, e (t ) = r − y (t ) ∈ R is the error signal T vector, x =  x p

{

T

x cT  ∈ R 2 n and M is reset set which is defined as the following:

}

(5) M = ξ = ξ Tp ξcT  , ξ p ∈ R n , ξc ∈ R n | C p ξ p = r & ρc (ξ ) ≠ ξc In simple word, the reset controller in (4) and reset set in (5) mean that the controller states are T

reset if the error signal is zero and the corresponding closed loop state does not belong to the null space of the ξc − ρc (ξ ) . Based on relation (4), it is clear that at certain times t = t k , if the reset condition (i.e. x (t k ) ∈ M ) is satisfied then the controller states are instantaneously changed to

ρc ( x ) . The interconnection of a reset controller with the plant can be depicted as shown in figure 1.


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Figure 1. Reset control system interconnection For the plant (2), an optimal approach for designing the reset controller can be addressed by minimization of the following cost function: Jk = ∫

+∞

tk

( xˆ

T p

(τ | t k )Qxˆ p (τ | t k ) + uˆT (τ | t k )Ruˆ (τ | t k ) )dt

(6)

where xˆ p (τ | t k ) and uˆ (τ | t k ) are the predicted states and control signal of the plant (2) such that at reset time t = t k , their values are:

xˆ p (t k | t k ) = x p (t k )

uˆ (t k | t k ) = u (t k )

(7)

The reset controller which was described by equation (4) has two main characteristics: 1) a dynamic part and 2) a reset law. The dynamic part of the reset controller is referred to as basesystem which is described by:  x&c = Ac x c + B c e  u = C c x c + Dc e

(8)

T Based on the augmented state vector x =  x p

T

x cT  , the resultant closed loop system

which consists of the plant (2) and the controller (8) can be compactly written as: x& = Ax + f ( x ) + Br  x c+ = ρc ( x )   y = Cx 

x ∉M x ∈M

(9)


6

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where  A p − B p D cC p A =  − B cC p B D  B = p c  Bc 

B pC c   A p = Ac   0

C = C p

0 

0   0 B p   Ac + 0   I 0  C c

Bc   0  Dc   −C p

I 0 

f (x )  f (x ) =  p p   0 

Remark 1. Based on the assumption 1, it can be easily shown that f ( x) satisfies a similar Lipschitz condition, for any vector α , β ∈ R 2 n , with the same positive constant L p as follows:

f (α ) − f ( β ) ≤ L p α − β

(10)

In the reset controller (4), at reset time t = tk , the controller states are reset to new values

ρc ( x (t k ) ) which is very general. In order to handle the determination of this function more conveniently, in this paper, we assume that the reset law has a structure as given by the following form:

ρc ( x (t k ) ) = Fp (k )x p (t k ) + Fc (k )x c (t k ) + F (k ) Assumption 3. Assume that the reference

(11)

r is a constant signal and also the base-system and

reset law has a same equilibrium point ( x ). That is  A x + f ( x ) + Br = 0   y = r = Cx x = ρ x c ( )  c

If the assumption 3 holds, then by defining a deviation variable ζ = x − x , the closed loop system (9) is converted to a zero input system; Thus in this paper, the regulation problem is studied and the reference signal r (t ) is set to zero.

Remark 2 (Temporal Regularization). The solution of system (9) may contain infinite number of reset in finite time which is called Zeno phenomena. To avoid Zeno phenomena in reset


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control systems, one can limit the resetting rate by defining a positive constant like t ρ . Then, the reset can be programmed to occur only at least after t ρ seconds after a previous reset which is known as temporal regularization constraint. The closed loop system (9) with r = 0 and temporal regulation is written as:

τ& = 1, x& = A x + f ( x ) x ∉ M or τ < t ρ  + + τ = 0, x c = ρc (x ) x ∈ M and τ ≥ t ρ  y = Cx 

(12)

The time interval t ρ is the minimum time between two subsequent reset times.

Assumption 4. For avoiding beating and deadlock phenomena in the solution of system (12), it is assumed that the after reset value of controller states does not belong to the reset set M. That is the controller jumps have the following property: If x (t k ) ∈ M then x (t k+ ) ∉ M .

(13)

Some instants after the reset, the reset condition may still hold again (resetting) or neither flow nor jump is possible (see an example in 9); The assumption 4 guarantees the existence and uniqueness of solution of reset control system (12). The after reset value and the base-system parameters are proposed to be selected based on the minimization of the cost function (6). The cost function (6) can be compactly written as:

) ∞ J (t k ) = ∫ xˆT (τ | t k )Qxˆ (τ | t k )d τ

(14)

tk

where ) Q + C Tp DcT RDcC p Q = T  −C c RDcC p

−C Tp DcT RC c  Q = C cT RC c   0

0  C Tp DcT + 0   −C cT

  R  DcC p 

−C c 

(15)

T

In the cost function (14), xˆ (τ | t k ) =  x Tp (τ | t k ) x cT (τ | t k )  is the prediction of the augmented system state vector at time t = τ when xˆ (t k | t k ) = x (t k ) . Therefore optimizing of the cost


8

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function (6) for the plant (2) is equivalent to finding the optimum value of (14) for the system (9). By this fact, an LMI based procedure for simultaneously determination of the reset law and base-controller parameters Ac , B c , C c and Dc is stated in the next section.

4-Reset Controller Design In this section, the reset law design for the given uncertain system (12) is investigated in real time operation. For this purpose, two theorems are proposed which is the main contribution of this paper. In the Theorem 1, an LMI based approach is addressed for computation of the reset law. Then, by solving such LMI optimization problem, the reset law is founded.

Theorem 1. Consider the uncertain system (12) with assumption 1-4. If at reset time t = t k (i. e. x (t k ) ∈ M ), there exist a symmetric positive definite matrixes Γ ∈ R 2 n ×2 n and two positive scalars γ , δ and triple matrixes Fp ( k ) ∈ R n ×n , Fc ( k ) ∈ R n ×n and F (k ) ∈ R n ×1 such that the following minimization problem is feasible

Min

γ

(16)

subject to

Γ x (t k+ )   >0 * 1  

(17)

 T  A Γ + ΓA  *   * 

(18)

1  2 L p .Γ ΓQˆ 2   −δ I 0  0  

(19)


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x p (t k )   where x (t k+ ) =    Fp (k )x p (t k ) + Fc (k )x c (t k ) + F (k ) 

(20)

Then, at each time instants tk , the reset law x c (t k+ ) = ρc ( x (t k ) ) given as:

ρc ( x (t k ) ) = Fp ( k ) x p (t k ) + Fc (k ) x c (t k ) + F ( k )

(21)

minimizes the performance index (14) and also the solution of closed loop reset system (12) is asymptotically stable.

Proof. Consider a quadratic Lyapunov function as V (x ) = x T Px for the system (12), where the matrix P ∈ R 2 n ×2 n is a positive definite and symmetric matrix. Assume that at each reset time

tk , the time derivative of V ( x ) along the trajectory of (12) has an upper bound like: ) d V ( xˆ (t k + τ | t k ) ) < − xˆ (t k + τ | t k )T Qxˆ (t k + τ | t k ) ∀τ > 0 dτ

(22)

If the inequality (22) is satisfied then

(

)

xˆ (t k + τ | t k )T A T P + PA + Qˆ xˆ (t k + τ | t k ) + 2xˆ (t k + τ | t k )T Pf (xˆ (t k + τ | t k )) < 0

(23)

By using the inequality (10), the second term of (23) has an upper bound like: xˆ (t k + τ | t k )T Pf ( xˆ (t k + τ | t k )) ≤ xˆ (t k + τ | t k ) . P . f (xˆ (t k + τ | t k )) ≤

(24)

≤ xˆ (t k + τ | t k ) . P .L p xˆ (t k + τ | t k ) = L p P xˆ (t k + τ | t k )T xˆ (t k + τ | t k )

and then the inequality (23) implies that A T P + PA + 2 L p P I + Qˆ < 0

(25) 1

Let define Γ = γ P −1 and pre and post multiply the inequality (25) by γ 2 P −1 ΓA T + A Γ + 2 L p Γ Γ −1 Γ + γ −1ΓQˆ Γ < 0

(26)

By using the Schur lemma, the inequality (19) is equivalent to Γ −1 < δ −1 and also the inequality (18) is equivalent to following:


10

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ΓA T + A Γ + 2 L p δ −1Γ 2 + γ −1ΓQˆ Γ < 0

(27)

Then both the inequalities (18) and (19) lead to inequality (26). Therefore two inequalities (18) and (19) simultaneously imply that the inequality (22) is hold. Having satisfied the relation (22), we can find an upper bound for performance index J k . For this, integrate both side of the inequality (22) from τ = 0 + to ∞ : V ( xˆ (t k + ∞ | t k ) ) −V ( xˆ (t k+ | t k ) ) < −J k

(28)

Barbalat’s lemma implies that limV ( xˆ (t k + τ | t k ) ) = 0 , therefore τ →∞

J k 0  1 x p (t k )

 A pY + B p M +Y A pT + M T B pT   *  *   *   *  * 

Y V * Y  * *  * * Y I 

δI 0 Y *

(31)

A p − B p NC p + K T

Y

−M T

2 L p .Y

X A p − LC p + A pT X − C pT LT

I

C pT N T

2 L p .I

* *

−γ Q −1 *

0 −γ R −1

0 0

*

*

*

−δ I

*

*

*

*

0 δ I  >0 V   Y 

2 Lp V .   0   0   0  0   −δ I 

(32)

(33)

I  >0 X 

(34)

where U = ( I − XY )V −1 , then the reset controller (4) is given with the following base-system parameters: Ac = U −1 ( K − X A pY + LC pY − X B p M − X B p NC pY )V

B c = U −1 ( L − X B p N

−1

)

(35) (36)


12

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C c = ( M + NC pY )V

−1

(37)

Dc = N

(38)

and a reset law given as:

x c (t k+ ) = ρc ( x (t k ) ) = Fp ( k ) x p (t k ) + Fc ( k ) x c (t k ) + F ( k )

(39)

minimizes the performance index (6) at each time instants tk and the solution of closed loop reset system (12) is asymptotically stable.

Proof. In the theorem 1, let partition the matrixes Γ and Γ −1 as the following forms: Y Γ= V

V  Y 

X Γ −1 =  U

U X 

(40)

where X ∈ R n ×n , Y ∈ R n ×n , U ∈ R n × n and V ∈ R n × n are symmetric and positive definite matrixes. It is easy to say that the following mathematical relations between these matrixes X ,

Y , U and V are hold: XY +UV = I  XV +UY = 0

(41)

By substituting blocked matrix Γ , in inequalities (17) and (19) , two inequalities (31) and (33) are verified. Then, let pre and post multiply the equation (27) by a nonsingular matrix Θ : ΘT ΓA T Θ + ΘT A ΓΘ + 2Lp δ −1ΘT Γ2Θ + γ −1ΘT ΓQˆ ΓΘ < 0

(42)

It is trivial that two equalities (27) and (42) are equivalent. Now, consider the matrix Θ as the following form: I X  Θ=  0 U  with some mathematical effort, the inverse of Θ is computed as:

 I −X U −1  Θ =  U −1  0 −1

(43)

(44)


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The matrix ΓΘ is computed as: Y ΓΘ =  V

I 0 

(45)

Since the matrix Γ is positive definite, then the congruence transformation of the Γ must be a positive definite matrix. That is: I  Y (46) ΘT ΓΘ =   I X  which the inequality (34) is verified. Next, two terms ΘT A ΓΘ and ΘT ΓQˆ ΓΘ are computed

which used in inequality (42). The congruence transformation of the matrix A Γ is: I ΘT A ΓΘ =  X

0    A p  U    0

0   0 B p   Ac + 0   I 0  C c

Bc   0  D c   −C p

I   Y  0   V

Bc   V  Dc   −C pY

0  −C p 

(47)

I 0 

Then, the equation (47) can be written as:

A p   0 B p   Ac  A pY ΘT A ΓΘ =  +   XA pY XA p  U XB p  C c The equation (48) can be decomposed as:  A pY ΘT A ΓΘ =   0

A p   0 B p   U +  X A p   I 0    0

X B p   Ac I  C c

Bc   V  Dc   −C pY

(48)

0   X A pY + I   0

0  0    I    0    0 −C p 

(49)

If the controller parameters are deliberately selected as −1

0 B c  U X B p   K − X A pY L  V =  −C Y I      Dc   0 I   M N  p  Then the controller parameters are obtained as Ac = U −1 ( K − X A pY + LC pY − X B p M − X B p NC pY )V −1  Ac C  c

B c = U −1 ( L − X B p N

)

C c = ( M + NC pY )V

−1

−1

(50)

(51)

Dc = N

By substitution the equation (50) into the equation (49), the matrix ΘT A ΓΘ is obtained as:  A pY ΘT A ΓΘ =   0

Ap  0 B p   K + XA p   I 0  M

0  A pY + B p M L  I  = K N   0 −C p  


A p − B p NC p  XA p − LC p 

(52)

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The matrix ΘT ΓQˆ ΓΘ is computed as: Y ΘT ΓQˆ ΓΘ =  I

V  Q + C Tp DcT RDcC p  0   −C cT RDcC p

−C Tp DcT RC c  Y  C cT RC c  V

(53)

I 0 

or equivalently V   Q  0    0

Y Θ ΓQˆ ΓΘ =  I T

0  C Tp DcT  +  R  D cC p 0   −C cT  

 Y −C c     V

(54)

I 0 

In the equation (51), the controller parameters are C c = MV

−1

+ NC pY V

−1

and Dc = N . If the

controller parameters are substituted in the equation (53), we have: Y QY ΘT ΓQˆ ΓΘ =   QY

YQ +  −M Q  

T

NC p  R  − M

Y NC p  =   I

−M T C pT N T

 Q    0

0 Y  R   − M

I  NC p 

(55)

Then, the equation (42) is: T

 A pY + B p M A p − B p NC p   A pY + B p M A p − B p NC p   +  + K XA − LC K XA − LC p p p p     T I  Y −M  Q 0   Y V  Y I  −1 Y +γ −1   −M NC  + 2 Lp δ  T T     t ρ ). In the other word, t ρ indicates the closeness of two consecutive reset times. By selecting enough big t ρ , we have enough time for computing the reset controller parameters. The application of Theorem 2 is investigated in the next section to design a control system for a process system.

5-Numerical Simulation In this section, the proposed method is compared with a variety of existing approaches via two examples. In the first example, an off-line controller is computed by using Remark 4, and it is compared with a common MPC approach. In reset control systems, since the selection of basecontroller affects the system performance; hence, design of reset control system only based on the determination of reset law may not be a reasonable way to improve the performance. For


16

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showing this point, in the second example, our approach is used in an isothermal chemical system to show the benefit of simultaneous design of reset law and base controller. Simulations results verify the effectiveness of the proposed approach as compared to other methods.

Example 1: Consider a linear plant with the following transfer function:

G p (s ) =

4 s (s + 2)

(61)

The system (61) can be realized with the following state space model:

 −2 0  −4 Ap =  , B p =   , C p = [ 0 1] , D p = 0   −1 0 0 In this example, for showing the effectiveness of the proposed strategy, the result from Theorem 2 is compared with a model predictive controller (MPC). As we know, in the MPC algorithm, the system prediction is computed at each sampling time instant but in our approach the prediction is computed only at certain times (e.g. either at reset times or sampling times or initial time). In the literature, an LMI based MPC algorithm for the linear discrete-time system is proposed in

30

. In this method, MPC is realized as a static state feedback which its gain is

computed at each sampling time by solving an LMI minimization problem. In order to use the result of 30 for this example, the system (61) is discretized with sampling time T s = 0.2 seconds. The discrete-time equivalent system is:

G p ( z −1 ) =

z −1 ( 0.07032 + 0.06155z −1 ) 1 − 1.67z −1 + 0.6703z −2

(62)

The state space model of the discrete-time system (62) can be written as:  0.6703 0 Ap =   -0.1648 1 

 -0.6594  Bp =   0.07032

C p = [ 0 1] D p = 0

In the numerical simulations, since the proposed method is applied to the continuous-time system (61), the integration increment is set to 0.01 seconds. The MPC is used for the discrete-time


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Page 18 of 28

equivalent system (62) with sampling time T s = 0.2 seconds. The plant states start from

x (0) = [1 1] and the controller states are initialized from zero. Let Q = diag (9,16) and R = 4 T

in the performance index (6). By applying Remark 4, the following continuous-time controller is obtained:

 -0.6845 0.5122  -0.0640  Ac =  , Bc =    , C c = [ -8.7758 0.0001] , -4.7537 -2.8559   0.1362 

Dc = 0

and its transfer function is:

G c (s ) =

0.5614s + 0.9913 s 2 + 3.54s + 4.39

(63)

For the plant (61), two control systems are used: 1) The proposed controller (63) which is an off-line controller 2) The MPC method

30

which is a real time control system. The results of

closed loop system with two control systems are shown in the Figures 2-4. The performance of the proposed method is compared with the MPC and the results are shown in Table 1. In this table, γ b is the upper bound of the cost function (6) and it is computed by solving the minimization problem of the theorem 2. The proposed controller is a dynamic continues-time controller and in Figure 4, it is seen that the control signal start from zero and its amplitude is considerably less than the MPC method.

Table 1: Comparison of performance index in the example 1 performance index

Proposed approach

MPC method 30

γb

∫ (x

15.0390

17.1238

35.1037

+∞

0

T p

(t )Qx p (t ) + u T (t )Ru (t ) )dt

From Figures 2 and 3, it can be seen that the settling time of the plant states are slightly improved for the proposed approach, compared to the MPC method.


18

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Figure 2: The first state of example 1

Figure 3: The second state of example 1


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Page 20 of 28

Figure 4: Applied control signal in example 1

Example 2: In this example, an isothermal continuous stirred tank reactor (CSTR) is considered which the reaction A → B occurs ( A is cyclopentadiene and B is cyclopentenol). In Figure 5, an illustrative schematic of isothermal CSTR is shown 31-33.

Figure 5. An isothermal continuous stirred tank reactor (CSTR)

For each component, the molar rate of formation (per unit volume) is


20

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 rA = − k 1C A − k 3C A2   rB = k 1C A − k 2C B By assuming a constant volume reactor, the balance equations for this reactor is 32:  dC A F 2  dt = V (C Af − C A ) − k 1C A − k 3C A   dC B = − F C + k C − k C B 1 A 2 B  dt V

(64)

(65)

In the balance equation (65), C A is the concentrations of component A , C B is the concentrations of the desired product B , F is volume flow , V is reactor volume and k 1 , k 2 and

k 3 are the rate coefficients. Consider the following parameters given by 31 for a reactor operating at 134.14◦C :

Fs = 0.3138liter .min -1 V = 10.0liter C Afs = 5.1 mol / liter C As = 1.235 mol / liter C Bs = 0.9 mol / liter Note that the subscript s is used to indicate the steady state value and C Afs is the steady state feed concentration. At operating point, the equation (65) is Fs  2 0 = V (C Afs − C A s ) − k 1C As − k 3C A s  0 = − Fs C + k C − k C Bs 1 As 2 Bs  V

(66)

The rate coefficients k 1 and k 2 are equal 34. Then k 1 , k 2 and k 3 are found as: k 1 = 0.0843min -1 , k 2 = 0.0843min -1 , k 3 = 0.0113liter(mol.min) -1

Now, defining deviation variables as the new state:

x1 =

C A − C As C As

x2 =

C B − C Bs C Bs

u=

Fs (C Af − C Afs ) V C As

(67)

It is trivial that two deviation variables x 1 (t ) and x 2 (t ) are dimensionless and also the dimension of u (t ) is unit per minute. Therefore, following state space equation is founded:


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 dx 1 2  dt = u − α1x 1 − α 2 x 1   dx 2 = α x − α x 3 1 4 2  dt

Page 22 of 28

(68)

where

α1 =

Fs + k 1 + 2k 3C As V

α 2 = k 3C As α 3 = k 1

C As C Bs

α4 = k 2 +

Fs V

In the notation of equation (2), we have:

x  0   −α 1  x p =  1  , Ap =  1 B = , p  0  , x 2     α 3 −α 4 

 −α 2 x 12  f p (x p ) =    0 

The reset set M is selected such that reset occurs when x 2 (t ) = 0 (i. e. C p = [ 0 1] ). Let

Q = diag (1,10) and R = 10 for the weight matrixes in the cost function (6). The initial condition of controller states are set to zero (i.e. x c (0) = 0 ) and plant states are initialized from

x 1 (0) = 1 and x 2 (0) = −1 . In reset control systems, the reset law design is strongly depended on the suitable base-controller selection. This point is shown in this example. Hence for comparison purpose, the followings control systems are used for the plant (68): 1- The proposed approach which is a real time reset controller. By using the theorem 2, the base-controller parameters and the reset law are computed at reset times. 2- A base-controller which is obtained by using the remark 4. The remark 4 is a special case of the theorem 2. This controller is denoted with G cb (s ) . 3- Another base-controller which is more aggressive than the previous base-controller. This controller is denoted with G c2 (s ) . 4- Applying the real time reset law to the controller in case 3. This controller is designed only with the ideas in 29.


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It should be noted that in this example, reset law and base-controller are designed simultaneously based on the proposed approach. The base-controller is designed by minimizing the following cost function: J0 = ∫

+∞

0

( xˆ

T p

(τ )Qxˆ p (τ ) + uˆT (τ ) Ruˆ (τ ) )dt

(69)

By applying the remark 4, an off-line controller is found with the following state space representation: 0.0269 -0.2202 Ac =   ,  2.0626 -1.1996

 -0.0881 -5 Bc =   , C c = [ -4.2691 0.0001] , D c = 5.5258 × 10 -0.4118  

The transfer function of this base-controller is

G cb (s ) =

0.3761s + 0.06393 s 2 + 1.173s + 0.422

(70)

Let the 3rd case controller be selected as the following:

G c2 (s ) =

s +1 s2 +s +2

(71)

The concentration deviation of components A and B about their steady state values are shown in Figures 6 and 7. In this example, the controller states are reset when x 2 (t ) = 0 . Thus in the Figure 7, it is seen that the reset controller is reset at time 6.39 minute and the controller 2 with reset law 29 is reset at 4.75 and 17.95 minutes. In real time, by applying the main theorem of this paper, the transfer function of the base-system of reset controller calculated at 6.39 minutes is

G c (s ) =

0.3433s + 0.05805 s 2 + 1.128s + 0.3971

(72)

The control signal u (t ) which is volume flow deviation is shown is Figure 8. In order to compare the performance of proposed approach, the following performance index is computed for each controller:


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J =∫

+∞

0

(x

T p

Page 24 of 28

(t )Qx p (t ) + u T (t ) Ru (t ) )dt

(73)

The performance index of the base-controller, real time reset law 29 and the reset controller are compared in the table 2. Note that the base-controller is computed off-line but reset controller is computed in real time operation. In table 2,

γ b is the upper bound of cost function for designing

the base-controller (73) which is computed off-line. Suitable base-controller design, which is the main idea of this paper, is a major challenge in reset control system. The effectiveness of the proposed method in comparison to the other reset controller, which is only obtained by the idea in 29, are seen in the Figures 6-8 and also in the table 2. In the Figures 6 and 7, it is seen that the response of our real time approach is considerably faster than other controllers. Table 2. Comparison of performance index in the example 2 Reset controller

Base-controller (70)

Controller 2

Controller 2+reset 29

γb

24.9922

25.9378

36.7287

34.9993

42.6897

Figure 6. Concentration deviation of component A in the isothermal CSTR


24

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Figure 7. Concentration deviation of component B in the isothermal CSTR

Figure 8. Volume flow in isothermal CSTR


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Page 26 of 28

6-Conclusion In this paper, a real time approach for the design of reset controller based on the model prediction is investigated. For this purpose, an LMI based sufficient condition is derived. For this, at specific time known as reset time, an optimization problem is solved, and then basesystem and rest law of the reset controller is determined. This real time method is used in two typical control systems and the effectiveness of this approach is shown in simulation results.

Acknowledgement The authors would like to thank Dr. Gabriele Pannocchia with the University of Pisa for his valuable comments that improve the manuscript.

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Development of a Real-Time Model-Prediction-Based Framework for

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