Closed Form Tuning Equations for Model Predictive Control of First-Order plus Fractional Dead Time Models Peyman Bagheri and Ali Khaki-Sedigh* Abstract:

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Closed Form Tuning Equations for Model Predictive Control of First-Order plus Fractional Dead Time Models

Peyman Bagheri and Ali Khaki-Sedigh*

Abstract: Many industrial processes can be effectively described with first-order plus fractional dead time models. In the case of plants with a large dead time relative to the time constant, approximations in discretizing the time delay can adversely affect the performance and if the sample time is enforced by system requirements, the fractional nature of the delay should be considered. In this paper, an analytical approach to model predictive control tuning for stable and unstable first-order plus dead time models with fractional delay is presented. The existing tuning methods are based on trial and error or numerical optimization approaches and the available closed form equations are limited to plants with integer delays. In this paper, an analytical approach is adopted and the issues of closed loop stability and achievable performance are addressed. Finally, simulation results are used to show the effectiveness of the proposed tuning strategy.

Keywords: Achievable and feasible performances, analytical tuning, first-order plus dead time model, fractional delay, model predictive control.

1. INTRODUCTION

In the past three decades, Model Predictive Control (MPC) established itself as the most widely used advanced control methodology in the industry. Directly incorporating constraints into the optimization problem is one of the major advantages of MPC in real applications. Many successful implementations of MPC in advanced process control systems are reported [1-4].

In MPC design, model complexity leads to complicated closed loop analysis, which can be a key limitation. However, many of industrial processes can be sufficiently described by First-Order plus Dead Time (FOPDT) models [5]. Hence, in this paper FOPDT models are used to analyze the MPC in detail and closed loop studies lead to effective tuning formulas. In addition, the time delay is considered to be in a general fractional form as a non integer multiple of the sampling time. This can be of practical interest as in many real applications sample time is enforced by system requirements. It can be easily shown that for plants with a large dead time relative to the time constant, delay approximations can deteriorate the closed loop performance.

Typical MPC tuning parameters are prediction and control horizons and the weight matrices used in the cost function. These parameters must be tuned for closed loop stability, performance and robustness characteristics.

Note that, the tuning parameters are related to these characteristics in a complex and nonlinear manner, so the tuning procedure is an intricate problem and active constraints considerably complicates this problem. The issue of MPC tuning is addressed in many research papers [6,7]. In a general classification, different MPC tuning approaches can be grouped as the practical guidelines, the tuning strategies based on the numerical methods and the closed form tuning equations. Typically, practical approaches do not have a theoretical basis.

There are many papers in the second group but they are time consuming. Also, these methods lead to numerical results which are not useful in closed loop studies. Closed form tuning equations with theoretical basis are more appealing as they provide a foundation for analytical closed loop analysis. A MPC tuning strategy is proposed to achieve closed loop robust performance based on sensitivity functions analysis in [8]. A tuning approach developed in [9] based on inverse problem of the controller matching which numerically tunes the weight matrices. An analytical tuning method for Dynamic Matrix Control (DMC) parameters based on the FOPDT model with integer delay is provided in [10]. A modified GPC algorithm and a tuning approach was extended for the Second-Order plus Dead Time (SOPDT) plants in [11]. Closed form tuning equations for DMC parameters are developed in [12] based on the application of Analysis of Variance (ANOVA) and curve fitting for FOPDT processes. In [13] an analytical tuning strategy is developed for single input-single output MPC of FOPDT models when the constraints are inactive. There are two main assumptions in derivations of [13]. These are the open loop stability and the integer delay assumptions.

In this paper, an analytical MPC tuning methodology is proposed for FOPDT models with fractional delay and the open loop stability assumption is relaxed. Also, it is

Manuscript received December 21, 2013; revised April 18, 2014; accepted June 15, 2014. Recommended by Editor Sung Jin Yoo under the direction of Editor Duk-Sun Shim.

Peyman Bagheri and Ali Khaki-Sedigh are with the Center of Excellence in Industrial Control, Department of Electrical Engi- neering, K. N. Toosi University of Technology, Tehran, Iran (e- mails: [email protected], [email protected]).

* Corresponding author.

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shown that for fractional delays certain tuning parameters can lead to closed loop instability. To clarify and resolve this problem, two key concepts are introduced as the achievable gains and the feasible gains.

The paper is organized as follows. In Section 2, the state space MPC formulation for FOPDT models with fractional delay is given and closed loop transfer functions are obtained. Section 3 provides the analytical MPC tuning equations. The efficiency of the proposed tuning algorithms is analyzed through examples in Section 4. Finally, Section 5 concludes the paper.

2. STATE-SPACE MPC FORMULATION FOR FOPDT MODELS WITH FRACTIONAL DELAY

Consider the following FOPDT model and its discretized form with a sampling time of T_s

1

G ( ) ,

1

G ( ) (1 )(1 ) ,

( )

θ s m

m d

s ke τs

b z b

z k a

z z a

−

+

= +

− +

= −

−

(1)

where a e= ⁻^{T τ}^s ,θ T d b= _s( + ),0≤ <b 1 and d is a non negative integer number. Note that the delay is considered to be in the general form as a non integer multiple of the sampling time. The augmented state-space model with an integrator is as follows:

( 1) ( ) Δ ( ),

( ) ( ),

m

n n u n

y n n

+ = +

=

A B

C

x x

x (2)

where Δ 1= −z⁻¹ and

( ) [Δ ( ) Δ ( 1) Δ ( ) (1 ) Δ ( 1) ( )] ,

0 1 0 0 0 0

0 0 1 0 0 0

, (1 ) ,

0 0 1 0 1

0 0 0

0 0 1 1 1

[0 1 1 0 1].

m m m

m T

n y n y n y n d

k a b u n y n d

k a

a b

b

a b

= + +

− − +

⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥

=⎢⎢ ⎥⎥ = − ⎢⎢ − ⎥⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ − ⎥

⎣ ⎦ ⎣ ⎦

= − −

A B

C

x

(3)

Denote the future model output values as

1 1

2 1

ˆ ( | )

ˆ ( 1| )

( ) ( ) Δ ( ),

ˆ ( | )

m m m

m P

y n N n y n N n

n n n

y n N n _×

⎡ + ⎤

⎢ + + ⎥

⎢ ⎥

=⎢ ⎥ = +

⎢ + ⎥

⎢ ⎥

⎣ ⎦

F S

y x u (4)

1

Δ ( ) Δ ( 1)

Δ ( ) ,

Δ ( 1) _M

u n n u n

u n M _×

⎡ ⎤

⎢ + ⎥

⎢ ⎥

=⎢ ⎥

⎢ + − ⎥

⎢ ⎥

⎣ ⎦

u

1 2

( 3)

,

d d

d P P d + +

+ × +

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

CA F CA

CA

1

1 2

0 0 0

0 0 ,

d

d d

d P d P d P M

P M +

+ − + − + −

×

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

CA B

CA B CA B S

CA B CA B CA B

(5) where _N₁_{= +}_d _1,_N₂ _{= +}_{d P} and _y_{ˆ ( | )}_m _⋅_n is the

prediction of model output at instance n, P is the prediction horizon and M is the control horizon.

According to (3) and (5), we have

1 2 3 1

1 1 2 2 3 1

[ ], for 1, 2, ,

[V V V ] , ,

d i P

T

d P d d P

i d

a a

+ ×

+ + + ×

= = =

F F F F F

F F F

0

1 (6)

where V_i =

∑

ⁱ_j₌₁a^j⁻¹, ⁰⁼^[0^0]^T^and¹⁼^[1^{1] .}^T The future plant outputs can be described as

1

1 1

2

ˆ ( | )

ˆ ( 1| )

( ) ( ) ( ) ,

ˆ ( | )

p

p p m P

p

y n N n y n N n

n n d n

y n N n

×

⎡ + ⎤

⎢ + + ⎥

⎢ ⎥

=⎢ ⎥= +

⎢ ⎥

⎢ + ⎥

⎣ ⎦

y y 1 (7)

where y nˆ ( | )_p ⋅ is the prediction of plant output and ( ) _p( ) _m( )

d n =y n −y n (8)

and y n_p( ) is the plant output. The finite optimal control problem is

Δ ( )

[ ( ) ( )] [ ( ) ( )]

min [Δ ( )] [Δ ( )]

p T p

n T

n n n n

n n

⎛ − − ⎞

⎜ ⎟

⎜ + ⎟

⎝ ⎠

Q

u R

w y w y

u u (9)

max

max 1 1 2

s.t. ( | ) , 0, 1 ,..., 1

ˆ ( | ) , , 1, ..., ,

min

min p

u u n i n u i M

y y n i n y i N N N

≤ + ≤ = −

≤ + ≤ = +

where

1 2 3

2 2

1 2

( ) ( ) , diag{1, , , , },

(1 ) diag{ , , , }.

P P P P

M M M

n w n q q q

k a r r r

× ×

×

= =

= −

Q R

…

w 1

(10) In the case of no active constraints, the optimal control effort solution of (9) is

1 1

1 2

3

Δ ( ) ( ) [ ( ( ) ( ))

Δ (ˆ ) (1 ) Δ ( 1)

ˆ ( )].

T T

P

d m d

d m

n w n d n

y n d k a b u n y n d

− ×

+ +

+

= + −

− + − − −

− +

R S QS S Q

F F

F

u 1

(11)

Let

1 1 1

1

[K K ] ( ) [K K ] ( ) 1 [K K ] (1 )( / ).

T T T

x x xM d

T T T

y y yM P

T

u u uM xk a b a

− +

− ×

= = +

= = −

R S QS S QF R S QS S Q

K

K K

(12)

Hence, the optimal control effort solution of (11) leads to Δ ( )un =K_yw n( )−K_xΔ (y n dˆ_p + )–_K_{y p}_{y n d}_{ˆ (} ₊ ₎

(1 )( / )Δ ( 1).

xk a b a u n

−K − − (13)

(3)

) (n u

) ( Δun

1

z−

1 1

1

−z−

Ku1

Kx1

K_y1

1−z⁻1

) (n d y_p + ˆ

) ( Δyˆp n+d

) (n w ) (n d

) (n y_p )

(n y_m

Fig. 1. Block diagram of the closed loop plant.

Now, the current control effort is obtained as

1 1 ˆ

Δ ( ) Ku n = _yw n( ) K Δ (− _x y n d_p + )–K_y₁y n dˆ_p( + ) K (1_x1k a b a u n)( / )Δ ( 1).

− − − (14)

The block diagram of the proposed control configuration is shown in Fig. 1.

Remark 1: In the case of unstable open loop plants, the model (1) cannot be used directly for open loop prediction calculations. The following model decomposi- tion resolves this problem [14]

1 2

G ( ) M ( ) [1 M ( )],_m z = _m z − _m z (15) where _M₁_m and _M₂_m are both stable models. By the

assumption of _{y n}_m_{( )} and _{y n}_p_{( )} matching [14], the block diagrams of Fig. 2 are equivalent and the structure of Fig. 2(c) can be used for unstable models.

Closed loop plant analysis: Let _{d n}_{( ) 0,}₌ that is the plant and model outputs are the same. To obtain the current control effort according to (14), we need

ˆ (_p ).

y n d+ Due to (1), it can be shown that

1 1 1

1

ˆ ( ) ( )

(1 )(1 ) ( ) ( ).

1

p d m

d d

y n d a y n b bz

k a z a z u n

az

− − − −

−

+ =

+ − − + −

−

(16)

Using (8), (14) and (16), it can be shown that the closed loop transfer function is

( ) K [(1 )1 ]

G ( ) ,

( ) Δ ( )

p y

cl cl

y n b z b

z w n z

′ − +

= = (17)

where

2 1

Δ ( )_cl z =z z^d[ + − − +z( 1 a K (1′_x1 +b a( ⁻ −1))

1 1 1 1

K (1 )) (′_y b a K (1′_x b a( ⁻ 1)) K )],′_yb

+ − + − + − +

(18)

1 1 1 1

K′_x =k(1−a)K , K_x ′_y =k(1−a)K ._y (19) Lemma 1: For the open loop model (1), the closed

loop plant (17) is stable if

1

1 1

1

K 0

K (1 ) K ( 0.5) (1 )

K .

(1 ) (1 )

y

y y

x

b a b a

a a

a b a a b a

′ >

′ − − ′ − + +

< ′ <

+ − + −

(20)

Gm y^m

u _u M1m y^m

m

M2

++ u M1m

m

M2 ⁺ +

ym

yp

(a) (b) (c) Fig. 2. The structure of simulator for unstable plants.

Proof: Direct application of the Jury’s test proves

(20).

Note that, Lemma 1 applies to both stable and unstable open loop model (1) with the assumption of a>0.

Remark 2: In the case of _b₌_0, i.e., dead time of system is integer, the results of [13] readily follows

1 1

2 1 1 1

G ( ) K .

( 1 K K ) ( K )

d y cl

x y x

z z

z z a a

′ −

= + − − + ′ + ′ + − ′ (21) Note 1: In the case of higher order plants, the real plant can be described as follows:

1 1

1

( ) N( ) G ( )

( ) D( )

p p

y n z

z u n z

− −

= = − (22)

the model (1) is employed for the predictive control design. Some mathematical manipulations on (1), (8), (16) and (22) gives

1 1

(1 ) D( )

ˆ ( ) [1

1 N( )

( ) (1 )] ( ),

p

d p

b bz z y n d

az z

z z k a y n

− −

− − −

+ = + − +

−

× − −

(23)

equations (14), (19) and (23) lead to

1 1

1

K (1 )N( )

G ( ) ,

Δ ( )

cl y

cl

az z

z z

− −

′ −

= (24)

where

1 1 1 1

1 1

1 1 1

1 1

Δ ( ) D( ) (1 )(1 )(1 ) [1 K ] [K (1 )K ] {N( ) (1 )

D( ) (1 )[(1 ) ]

( )}.

cl

x y x

d

z z k a z az

b z z

a

z az

z k a b bz

z z

− − − −

− −

− − −

= − − −

′ ′ ′

× + + + −

× × −

+ − − +

× −

(25)

The closed loop transfer functions given by (17) and (24) are used in the Section 3 to derive exact tuning formulas for MPC of FOPDT models with fractional delay.

3. TUNING FORMULAS FOR THE MPC OF FOPDT MODELS WITH FRACTIONAL DEAD

TIME

Using closed loop transfer functions (17) or (24), it is possible to achieve the desired closed loop transfer function by properly choosing the desired gains K′_xd₁ and

K .′_yd1 This leads to the closed form equations for tuning parameters and determines the possible attainable per-

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formance for the MPC of FOPDT models. However, it is important to note that not any desired performance is achievable. In addition, some achievable gains do not ensure closed loop stability and are not feasible. In what follows, the key achievability and feasibility concepts are defined and the relevant theorems are introduced.

Definition 1 (Achievable gains): The desired gains K′_xd1 and K′_yd₁ that satisfy (12) and (19) are called the achievable gains.

Definition 2 (Feasible gains): The achievable gains K′_xd1 and K′_yd₁ that ensure the closed loop stability of (17) or (24) are called the feasible gains.

Note that the gains are selected prior to (12) and (19).

If these gains are feasible, there exits proper tuning parameters Q, R, P, M that satisfy these equations and lead to a stable closed loop plant. However, such parameters may not exist in which case the gains would be infeasible.

3.1. The control horizon of one Consider M =1 in (12). We have

1 1 1

1 Z

K ( ) ,

(1 ) X

1 Y

K ( ) 1 ,

(1 ) X

T T

x d

T T

y p

a

k a r

− +

− ×

= + =

− +

= + =

− +

R S QS S QF R S QS S Q

(26)

where

1 2 1 2 1 2

X X= +q_PX , Z Z= +q_PZ , Y Y= +q_PY , (27)

1 1

1 2 1

1 1

1 1 1 2

1 2

1

1 1

2 2 1

X (V ) , Y (V ),

Z V (V ), X (V ) ,

Y V , Z V (V ), 1.

P P

i i

i i i i

i i

P i P

i i i P

i

P P

P P P

q ba q ba

q ba ba

ba ba q

− −

= =

− − −

=

− −

= − = −

= − = − =

∑ ∑

∑

⁽²⁸⁾

Let K′_xd₁ and K′_yd₁ defined in (26) and (19) be the desired gains. To achieve these gains two parameters must be tuned. In MPC tuning problem, the weight matrices are more dominant than other parameters [10].

Hence, the weights on the cost function (9), i.e., r and

2, , , 3 _P

q q … q are chosen as the tuning parameters.

Theorem 1: Let

2 2

diag{1, q q2 3, , , q_P}, R k (1 a r) ,

= = −

Q (29)

where q_i∈R₀₊ for i=2, 3, ...,P−1 are arbitrary selected, q_P,r∈R₊ are the selected tuning parameters and P≥2 , P∈Z₊ and R ,₊ R₀₊ and Z₊ denote the real, non negative real and the positive integer sets of numbers. In the case of no active constraint, the optimization problem (9) for the FOPDT model (1) lead to the closed loop transfer function (17) or (24) with the desired achievable gains K′_xd₁ and K′_yd₁ that satisfy

1 1 1

2 1 1

0 K K , 0 K ,

(1 )

Y K Y ,

Z K Z

xd xd yd

yd xd

a b a a b a

a a

′ +

′ ′

< < <

+ −

< ′ <

′

(30)

and the feasibility conditions of these desired gains in the case of no model mismatch is intersection of (20) and (30). By selecting qP and r as the tuning parameters, the tuning equations for achieving these desired gains are

1 1 1 1

1 2 1 2

1 2 1 1 2 2 1 1 2

1 2 1 1 2 1 2 1 2

K Y K Z K Z K Y ,

[ K (X Y X Y ) (Z Y Z Y ) K (X Z X Z )]/[K Z K Y ].

xd yd

P

yd xd

xd

yd yd xd

q a

a

r a

a a

′ − ′

= ′ − ′

= − ′ − + −

′ ′ ′

+ − −

(31)

Proof: Equation (31) can be derived from (26), (28), (29) and (19) using simple mathematical manipulations.

Note that qP and r should be positive. In addition, using

2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2

X Z X Z ,

X Y X Y (1 )

Z Y Z Y ,

X Y X Y (1 ) b a b a

a a b a

− =

− + −

− =

− + −

it can be shown that the achievable gains area is given by (30).

Remark 3: In the case of high order plants, the stability conditions of the closed loop transfer function given by (24) and the inequalities (30) lead to gains feasibility.

Remark 4: Theorem 1 holds for both stable and unstable open loop model (1) with the assumption of a > 0.

Corollary 1: In Theorem 1, let

2 2

diag{1, 0, 0, , 0, q_P}, R k (1 a r)

= = −

Q (32)

consider P→ ∞ for stable FOPDT models and large enough prediction horizon for unstable open loop models.

These choices lead to the maximum achievable area as

1 1 1

K ,

0 K 0 K ,

(1 )

yd

xd yd

a b a a b a

′ +

′ ′

< < <

+ − (33)

1 1 1 1

1 K 1

for stable model, K

K 1

0 for unstable model.

K

yd xd yd xd

a

a a

a

⎧ − < ′ <

⎪ ′

⎪⎨ ′

⎪ < <

⎪ ′

⎩

(34)

Proof: we can show that

{ }

2 3, , ,max1, Y Z1 1 1

q q qP₋ P =

and

{ }

2 3, , , 1, 2 2

1 0 1

min Y Z

0 1 .

q q qP P

a a

a

−

− < <

= ⎨⎧⎩ ≤

So, due to inequalities of (30) assertion follows.

Corollary 2: In (33) and (34), there exist some values for the tuning parameters that lead to unstable closed loop plant. This can occur for b > 0.5.

Proof: Inequalities (33), (34), and (20) gives

1 1

1 K ( 0.5) (1 ) K 1

K K

(1 ) (1 )

y y

x b a b a y

a a b a a b a

′ − + + ′ +

′ = = = ′

+ − + −

(5)

that leads to K′ =_x₁ 2 ,a K′ =_y₁ 2 and b=0.5.

For a stable FOPDT plant (1), let T_s =0.1 .τ In Fig. 3, the results of maximum achievable, maximum feasibility and stability regions are demonstrated for the gains. Also, results of corollary 2 are illustrated in Fig. 3.

Note that, in the commercial MPC packages weight factor is the only tuning parameter in the cost function.

The industrial advantages of this tuning structure can be appealing to practical MPC users. Corollary 3 gives the tuning equation for this case.

Corollary 3: Let

2 2

, R (1 ) ,

P P_× k a r

= = −

Q I (35)

where r∈R₊ is selected as the tuning parameter and 1,

P≥ P∈Z .₊ In the case of no active constraint, the optimization problem given by (9) for the FOPDT model (1) leads to the closed loop transfer function given by (17) or (24) with the desired achievable gain K_xd′ ₁ that satisfies the following inequality

0 K< ′_xd1<( Z/X)a (36)

and the feasibility conditions are obtained by the intersection of (36) and the close loop stability conditions. By selecting r as the tuning parameter, the tuning equation for achieving the desired gain K_xd′ ₁ is

1 1

( Z XK )/K ._xd _xd

r= a − ′ ′ (37)

Proof: Similar to the proof of Theorem 1.

Note that the assumptions of corollary 3, lead to the key limitation as follows:

1 1

K′_y =K Y/( Z),′_xd a (38)

where X, Y, Z are defined in (27) with the assumption of

i 1

q = for i=1, 2, ..., .P Equation (38) means that it is not possible to choose any desired pair (K ,K )′_xd₁ _xd′ ₁ in the feasibility area. For locating K′_y₁ close to its desired value K ,′_yd₁ it is possible to find a proper prediction horizon such that K′_yd₁≈K (Y/ Z).′_xd₁ a

3.2. The control horizon of two

Consider M =2 in (12) and (19). we have

1 1

11 1 12 11 11

12 22 2 2 2 22 22

K K

X X Z Y

X X K K Z Y ,

x y

r a

′ ′

⎡ ⎤

⎡ + ⎤ ⎡ ⎤

⎢ ⎥=

⎢ + ⎥⎢ ′ ′ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦ (39)

where

11 1 2 1

1

22 1 2 2

2

1 2

12 1

2 11 1

1

22 1 2

2 11 1

1

22 1 2

2

X (V ) , 1,

X (V ) ,

X (V )(V ),

Z V (V ),

Y (V ),

Y (V ).

P i

i i

i

P i

i i

i

P i i

i i i

i

P i

i i i

i

P i

i i i

i

P i

i i

i

P i

i i

i

q ba q

q ba

q ba ba

q ba

−

=

− −

=

− −

= −

−

=

− −

=

−

=

− −

=

= − =

= −

= − −

= −

∑

(40)

Theorem 2: Let diag{1, ,q2 , },q_P

=

Q R=k²(1−a) diag{ , }² r r_{1 2} , (41) where q_i∈R₀₊ for i=2, 3, ...,P−1 and q_P∈R₊ are arbitrary selected, r r_{1 2}, R∈ ₊ are chosen as the tuning parameters and P≥2, P∈Z .₊ In the case of no active constraint, optimization problem given by (9) for the FOPDT model (1) leads to the closed loop transfer function given by (17) or (24) with the desired achievable gains K′_xd₁ and K′_yd₁ that satisfy the following inequalities

11 1 22 11 12 22

1 1 1

Y K X Y X Y ,

Z K (X Z X Z )

0 K K , 0 K ,

(1 )

yd xd

yd

xd yd

a a

a b a a b a

′ −

< <

′ −

′ +

′ ′

< < <

+ −

(42)

and the feasibility conditions of these desired gains in the case of no model mismatch is intersection of (20) and (42). The tuning equations for achieving these desired feasible gains are

1 1 12 11 11 22 22 11 11 22

1 12 11 11 22 22 1 22 1

[ K (X Y X Y ) (Z Y Z Y ) K (X Z X Z )] /[ Z K Y K ],

xd

yd yd xd

r a

a a

= − ′ − + −

′ ′ ′

+ − −

2 [K (X Y_xd1 22 11 X Y )12 22

r = ′ − (43)

1 22 11 12 22 11 1 11 1

K′_yd a(X Z X Z )]/[ Z Ka ′_yd Y K ].′_xd

− − −

Proof: Equation (43) can be derived from (39), (40) and (41). Note that r1 and r2 should be positive. Also, we have

12 11 11 22 12 11 11 22 22 11 11 22 12 11 11 22

X Z X Z ,

X Y X Y (1 )

Z Y Z Y

X Y X Y (1 )

b a b a

a a b a

− =

− + −

− =

− + −

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.50.51.52.53.54.501234

K'_x1 K'^y1

b = 0

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.50.51.52.53.54.501234

K'_x1 K'^y1

b = 0.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.50.51.52.53.54.501234

K'_x1 K'^y1

b = 0.99

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.50.51.52.53.54.501234

K'_x1 K'^y1

b = 0.8

Fig. 3. The maximum feasible/achievable (filled black/

red) and stability (dashed line) regions for gains.

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by some mathematical manipulations achievable gains area given by (42) is obtained.

Corollary 4: In Theorem 2, let diag{1,0, , 0, q_P},

=

Q R=k²(1−a) diag{ , }² r r_{1 2} (44) P→ ∞ for stable FOPDT models and large enough prediction horizon for unstable models. These choices lead to the maximum achievable area (33) and (34).

Proof: Similar to the proof of corollary 1.

Remark 5: The maximum achievable gains area with control horizons of one and two are identical, and therefore the achievable performances are the same.

Deriving similar results for control horizons of larger than 2 requires complicated mathematical manipulations.

However, a simulation based procedure [13] can be used to show that the maximum feasible area in these cases is similar to areas of control horizons of one and two.

4. SIMULATION RESALTS

In this section, simulation examples are presented to illustrate the efficiency of the proposed tuning method.

Example 1: Consider the following model

12 1 min 1

13 1

G ( ) 0.7 G ( )

6 1 0.1075 . 1 0.8465

s s

m s e T m z

s z

z

− = −

−

= ⎯⎯⎯⎯→

+

= −

An MPC is designed to give a desired performance based on the approach of [13]. First, consider the case of no model mismatch. The closed loop response is shown in Fig. 4. Now we consider the dead time of plant is 11.5 and 12.5 min. In Fig. 4, the closed loop responses considering these dead times are shown and the importance of considering fractional delay is illustrated.

Example 2:An Unstable FOPDT Plant Consider an unstable FOPDT plant as

1.2 0.25 sec

5

G ( ) G ( ) G ( )

1

0.0568 0.2272

G ( ) ,

( 1.284)

s s

p m T p

m

s s e z

s z z

z z

− =

= = ⎯⎯⎯⎯⎯→

−

= = +

−

equation (1) gives _k_{= −}_1, a=1.284, b=0.8, d=4.

Let the desired closed loop performance has no overshoot and a settling time of 10 to 20 sec. Using the closed loop transfer function given by (17), the desired gains lie in the region shown in Fig. 5. Then, a proper choice can be K′ =_xd₁ 1.4 and K′ =_yd₁ 0.08. The feasibility conditions of Theorem 1 with q_i =0 for

2, 3, ..., 1

i= P− are considered. It gives P≥7. Let 8,

P= then _N₁₌₅ and _N₂₌_12. The tuning equations according to (31) are q_P =0.159,r=0.924. In Theorem 2, let q_i =1 for i=1, 2, ... , .P According to the conditions given by the inequalities (42), we have

8≤ ≤P 74. Choose _P₌_10.

0 20 40 60 80 100

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (min)

Output

Reference Time Delay = 12 Time Delay = 12.5 Time Delay = 11.5

0 20 40 60 80 100

0 1 2 3 4 5

Time (min)

Control Signal

Time Delay = 12 Time Delay = 12.5 Time Delay = 11.5

Fig. 4. Closed loop responses (example 1).

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

K'_x1 K'^y1

0.5 1 1.5

0 0.05 0.1 0.15

K'_x1 K'^y1

Fig. 5. Stable (dotted line), maximum achievable/

feasible (solid/dashed line) and desired areas of the gains (filled).

Hence, the tuning weights are r₁=15, r₂ =45.29 and due to (9) we have R=diag{1.210, 3.653}. Finally, due to the conditions of corollary 3 we can choose

8.

P= The tuning equation according to (37) is 113.26.

r= Note that due to (38) we have K′ =_y₁ 0.0724 K≈ ′_yd1 and according to Fig. 5, this gain is in the desired area. Suppose the input constraint u n( ) ≤ 3.25. The above tuning parameters lead to the closed loop step tracking responses depicted in Fig. 6. In the case of no active constraints the output has no overshoot and the settling time is about 10.5 sec.

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0 10 20 30 40 50 60 70 -3

-2 -1 0 1 2 3

Time (sec)

Out put

Reference Output Desired Response

0 10 20 30 40 50 60 70

-5 -4 -3 -2 -1 0 1 2 3 4 5

Time (sec)

Con trol Sig

nal Control Signal

Desired u umax umin

Fig. 6. The closed loop responses (example 2).

Example 3:The pH Neutralization Process

The pH neutralization process [15] consists of acid, base and buffer streams that are mixed in a vessel. The control objective is to control the pH value of the outlet stream. pH is measured at a distance from the plant, which introduces a measurement time delay θ. The nominal pH parameters are given in [15]. Let θ = 25 sec.

The control objective is pH step point tracking in the range of pH = 5.5 up to pH = 7.25. It can be shown that in this working range, the plant can be modeled as

0.3 0.9, 90 98

G ( ) , .

1 24 26

θ s

p s k e k τ

τs θ

⎧ − ≤ ≤ ≤ ≤ ⎫

⎪ ⎪

= ⎨⎪⎩ + ≤ ≤ ⎬⎪⎭

10 sec

Ts= is a proper sampling time. Hence the discrete set is as follows:

1

3

0.3 0.9

(1 )

G ( ) (1 ) 0.895 0.903 .

( )

0.4 0.6

p

b z b k

z k a a

z z a

b

−

⎧ ≤ ≤ ⎫

⎪ − + ⎪

=⎨ − ≤ ≤ ⎬

⎪ − ≤ ≤ ⎪

⎩ ⎭

The nominal model is chosen as (1) with k=0.6, 0.9,

a= b=0.5, d=2. The desired performance is defined to have an overshoot of less than 10% and a settling time of 150 to 250 sec. The closed loop transfer function (24) is calculated and the proper gains K′_x₁ and K′_y₁ that satisfy the desired performance are computed. The intersections of the desired areas of the gains that yield to the desired robust performance are illustrated in Fig. 7. Note that the desired performance is in the feasible area and therefore a proper tuning set exists to ensure closed loop stability and robust performance. Using Fig. 7, we choose K_xd′ =₁ 0.6 and

K′ =_yd1 0.14. To obtain these gains we use Theorem 1 with q_i =1 for i=2, 3, ...,P−1. The conditions of this theorem lead to P≥7. Let P=10, employing

tuning equations of Theorem 1 gives q_P =0.835 and 12.379.

r= Let the input constraint 0<u n( ) 30.< A white noise with variance 0.0001 is added as the measurement noise in the pH output. Fig. 8 shows the desired closed loop responses.

5. CONCLUSION

An analytical tuning methodology is developed for single input-single output MPC of FOPDT models with fractional delay when the constraints are inactive. In many applications, sampling time is enforced by system requirements which lead to fractional delays. The closed loop transfer function is obtained and the stability and attainable performances are addressed. Exact closed form tuning formulas to reach the desired performances are derived for both stable and unstable plants. It is shown that for FOPDT models, control horizon of one and two lead to the maximum achievable performance, which simplifies the controller. A significant achievement of this paper is that it can considerably reduce the computational cost. This fact is due to reducing the control horizon to one or two, using a simple FOPDT model and reducing the prediction horizon by using appropriate values for other tuning parameters. Finally, simulation results are used to show the effectiveness of the proposed tuning approach.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

K'_x1 K'^y1 ^0.06^0.4 ^0.5 ^0.6 ^0.7

0.08 0.1 0.12 0.14 0.16 0.18

K'x1 K' y1

Fig. 7. The desired and feasibility areas of the gains.

12505 1500 1750 2000 2250 2500 2750 3000 3250 3500 5.5

6 6.5 7 7.5

Time (sec) pH

Reference pH

12508 1500 1750 2000 2250 2500 2750 3000 3250 3500 10

12 14 16 18

Time (sec) Base

(ml/s ec)

Control Signal

Fig. 8. The closed loop responses (example 3).

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REFERENCES

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Automatica, vol. 25, no. 3, pp. 335-348, 1989.

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3505-3515, 2010.

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Peyman Bagheri was born on December 1984 in Tabriz, Iran. He obtained his B.Sc. degree in Electrical Engineering from the Sahand University of Technolo- gy in 2007, masters in Control Engineer- ing from K. N. Toosi University of Technology in 2009 and he is currently a Ph.D. student at the K. N. Toosi Univer- sity of Technology. The main areas of his interest are model predictive control, controller tuning, multivariable control, process control and pH control.

Ali Khaki-Sedigh is currently a profes- sor of control systems with the Depart- ment of Electrical Engineering, K. N.

Toosi University of Technology, Tehran, Iran. He obtained an honors degree in mathematics in 1983, a master’s degree in control systems in 1985 and a Ph.D. in control systems in 1988, all in the UK.

He is the author and co-author of about 90 journal papers, 170 international conference papers and has published 14 books in the area of control systems. His main research interests are adaptive and robust multivariable control systems, complex systems and chaos control, research ethics and the history of control.