Robust feedback linearization for nonlinear processes control

ISA Transactions 74 (2018) 155–164

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Robust feedback linearization for nonlinear processes control José de Jesús Rubio Sección de Estudios de Posgrado e Investigación, Esime Azcapotzalco, Instituto Politécnico Nacional, Av. de las Granjas no. 682, Col. Santa Catarina, México D.F., 02250, Mexico

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Article history: Received 18 June 2017 Revised 2 January 2018 Accepted 2 January 2018 Available online 3 February 2018

In this research, a robust feedback linearization technique is studied for nonlinear processes control. The main contributions are described as follows: 1) Theory says that if a linearized controlled process is stable, then nonlinear process states are asymptotically stable, it is not satisfied in applications because some states converge to small values; therefore, a theorem based on Lyapunov theory is proposed to prove that if a linearized controlled process is stable, then nonlinear process states are uniformly stable. 2) Theory says that all the main and crossed states feedbacks should be considered for the nonlinear processes regulation, it makes more difficult to find the controller gains; consequently, only the main states feedbacks are utilized to obtain a satisfactory result in applications. This introduced strategy is applied in a fuel cell and a manipulator. © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Control Robust feedback linearization technique Uniform stability Nonlinear processes

1. Introduction Robust feedback linearization technique is concerned with the local stability of a nonlinear process. It is a formalization of the intuition that a nonlinear process should behave similarly to its linearized approximation for small range motions. Because all physical processes are inherently nonlinear, robust feedback linearization technique serves as the fundamental justification of using linear control strategies in practice, i.e., it shows that stable design by linear control assures stability of the original physical process locally. This research is focused in this interesting issue. There are some investigations about stable controllers. Stability of controllers for delayed processes is introduced in Refs. [1–3]. In Refs. [4–7], stability of some kind of adaptive controls is mentioned. Stability of controllers for linear processes is considered in Refs. [8–11]. The above mentioned investigations show that stable controllers could be directly designed for nonlinear processes; however, in some cases, stable controllers are employed in synthetic models, which is a little far to applications. There is some research about robust control of linearized models. Controllers based on a feedback linearization are designed in Refs. [12–15]. In Refs. [16–19], processes controls based on linearized models are designed. Robust feedback linearization is mentioned in Refs. [20–23]. In Refs. [24–27], controllers of the linearized turbine, pendulum, robotic arm, and rotor are investigated. The aforementioned research shows that in these days a robust linearization tech-

E-mail address: [email protected].

https://doi.org/10.1016/j.isatra.2018.01.017 0019-0578/© 2018 ISA. Published by Elsevier Ltd. All rights reserved.

nique is utilized in applications; therefore, it is an actual and interesting issue. Robust feedback linearization technique is used for the nonlinear processes regulation. Regulation is a kind of control in which all process states should converge to constant references. Robust feedback linearization technique has two main problems which are focused on differences between the theory and applications: 1. Theory says that if a linearized controlled process is stable, then nonlinear process states are asymptotically stable, it means that all process states should converge to zero [28,29]. Nevertheless, it is not exactly satisfied in applications because in some cases, some nonlinear process states only converge to small values. 2. A main state is when a state is utilized by the controller for regulation of the same state, while a crossed state is when a state is utilized by the controller for regulation of a different state. Theory says that feedback of all the main and crossed states should be considered in the controller for the nonlinear process regulation [28,29]. It sometimes makes more difficult to find the controller gains. This investigation proposes a strategy to solve aforementioned problems which is detailed by the following two steps: 1. From Lyapunov theory, uniform stability is stronger than stability because the first is satisfied for any initial time, while the second is satisfied only for a zero initial time. However, uniform stabil-

156

J.J. Rubio / ISA Transactions 74 (2018) 155–164

ity is weaker than asymptotic stability because the first assures the error convergence to a small value, while the second assures the error convergence to zero. This study suggests a theorem to prove that if the linearized controlled process is stable, then nonlinear process states are uniformly stable. It represents better the control applications. 2. In the introduced method, only main states feedbacks are utilized by the controller to obtain a satisfactory result in applications. It makes easier to find the controller gains.

2.3. Stability analysis

From the above research, nonlinear processes with robustness are studied in Refs. [13,14,16,18,23,30,31]. In control theory, robust control is an approach that explicitly deals with uncertainty. Robust methods aim to achieve robust performance or stability in presence of uncertainties. In results, this method is applied to two nonlinear processes with inputs and parameters uncertainties. Finally, the suggested technique is applied to two nonlinear processes: a fuel cell and a manipulator. A fuel cell is applied for the electricity generation from hydrogen fuel. A manipulator is mainly utilized to move objects in the automobile industry. Other sections are focused in the following issues. In Section 2, the proposed controller is introduced for the nonlinear processes regulation. Suggested controller is applied for the fuel cell regulation in Section 3. In Section 4, mentioned controller is applied for the manipulator regulation. Control results for the two processes regulation are shown in Section 5. In Section 6, the conclusion and future research are described.

2.3.1. Closed loop nonlinear process A closed loop nonlinear process of controller applied for the nonlinear processes regulation will be obtained. It will be used for stability analysis. Applying Taylor series to (1) gives the following result:

In this subsection, stability of an introduced controller for the nonlinear processes regulation is analyzed. It is based on four parts, 1) a closed loop nonlinear process is obtained, 2) the controllability is studied, 3) controller gains are determined, and 4) a theorem to analyze the stability of the controller applied to nonlinear processes is introduced.

Ẋ =

) 𝜕 f (X , U ) ( ) 𝜕 f (X , U ) ( X − Xd + U − Ud + r 𝜕X 𝜕U

where Xd are desired states and Ud are desired inputs, Xd and Ud are considered as zero because case, r is a residue. ( it is the regulation ) ( ) Adding and subtracting to (3) gives: Ẋ =

(

+ (

+ ⇒ Ẋ =

(

[

+

Ẋ = f (X , U )

(1)

where X ∈ ℜ are states, U ∈ ℜ are inputs, and f (·) ∈ ℜ are continuous differentiable nonlinear functions. n

m

n

(

)

(

𝜕 f (X , U) || U− 𝜕 U ||X=0,U=0 )

+

In this subsection, nonlinear processes are described. Consider the following nonlinear processes:

)

𝜕 f (X , U) || X− 𝜕 X ||X=0,U=0

𝜕 f (X , U) || X+ 𝜕 X ||X=0,U=0 [

2.1. Nonlinear processes

𝜕f (X,U) | | 𝜕X |X=0,U=0

𝜕 f (X , U ) 𝜕 f (X , U ) X+ U+r 𝜕X 𝜕U

2. Controller of nonlinear processes In this section, nonlinear models are presented, a robust feedback linearization controller is proposed, and a theorem to study the process stability is introduced.

𝜕 f (X , U ) − 𝜕X 𝜕 f (X , U ) − 𝜕U

( (

(

U = −KX

(2)

where K ∈ ℜm×n are controller gains. Fig. 1 shows a proposed controller where U are inputs, X are states, and f (·) are nonlinear functions.

𝜕f (X,U) | | 𝜕 U | X = 0 ,U = 0

U

)

𝜕 f (X , U) || X 𝜕 X ||X=0,U=0 )

𝜕 f (X , U) || U 𝜕 U ||X=0,U=0 )

𝜕 f (X , U) || 𝜕 X ||X=0,U=0 𝜕 f (X , U) || 𝜕 U ||X=0,U=0

(4)

)]

X

)] U+r

Equation (4) can be rewritten as follows: Ẋ = AX + BU + ̃ AX + ̃ BU + r

where:

In this subsection, a controller for the nonlinear processes regulation is studied. The objective of controller is that using inputs, states of nonlinear processes should follow constant references, it is denoted as the states regulation. Consider control functions as follows:

X and

𝜕 f (X , U) || U 𝜕 U ||X=0,U=0

(5)

⇒ Ẋ = AX + BU + 𝛿 2.2. Proposed controller

(3)

(

A=

( B=

𝜕 f (X , U) || 𝜕 X ||X=0,U=0 𝜕 f (X , U) || 𝜕 U ||X=0,U=0

𝜕 f (X , U ) ̃ A= − 𝜕X 𝜕 f (X , U ) ̃ B= − 𝜕U

( (

) )

𝜕 f (X , U) || 𝜕 X ||X=0,U=0 𝜕 f (X , U) || 𝜕 U ||X=0,U=0

)

(6)

)

AX + ̃ BU + r is an unmodelled error which is bounded as foland 𝛿 = ̃ lows ‖𝛿 ‖ ≤ 𝛿 . Substituting the control function (2) in equation (5) gives: Ẋ = AX + B [−KX ] + 𝛿

⇒ Ẋ = AC X + 𝛿 Fig. 1. A proposed controller.

(7)

where AC = A − BK. Equation (7) is the closed loop nonlinear process.

J.J. Rubio / ISA Transactions 74 (2018) 155–164

2.3.2. Controllability First, utilize A and B of the second equation (5) to obtain the controllability matrix ℂ as follows:

[ ℂ= B

A2 B

AB

]

An−1 B

···

(8)

157

following results are proposed: 1.- if the linearized process (7) is stable, i.e., Re𝜆i ≤ 0 for all the eigenvalues of AC , then states of the closed loop nonlinear process (7) are uniformly stable. 2.- If there exists symmetric and positive semi-definite matrices R ∈ ℜn×n , S ∈ ℜn×n , and a small positive scalar 𝜓 ∈ ℜ such that the following equation is solved:

The controllability criteria is based on the controllability matrix ℂ as follows:

ATC R + RAC + 𝜓 R = −S

If rankℂ = n, then all process states are controllable

with AC = A − BK, then states of the closed loop nonlinear process (7) are uniformly stable and satisfy:

If rankℂ ≤ n, then not all process states are controllable

(9)

2.3.3. Controller gains Controller gains are selected by the following process. Consider that AC is defined in (7), 𝜆i are eigenvalues of AC , and Re𝜆i are real parts of eigenvalues 𝜆i . Eigenvalues 𝜆i are selected such as their real parts Re𝜆i are negative or zero, i.e. eigenvalues are selected as follows:

𝜆1 is such that Re𝜆1 ≤ 0 𝜆i is such that Re𝜆i ≤ 0

(10)

⋮

(16)

where Xi are initial conditions of states X, 𝜇 = 𝜂 , 𝜆min (R)𝜑

𝜂=

T

𝜆max (R) , 𝜆min (R)

𝜀=

𝛿 R𝛿 , 𝜑 = 𝜆min (SR−1 ), ‖𝛿‖ ≤ 𝛿 are bounded unmodelled errors, ‖·‖ is the Euclidean norm in ℜn , 0 < 𝜑 ∈ ℜ, 0 < 𝜇 ∈ ℜ, 0 < 𝜀 ∈ ℜ. 3.- States of the closed loop nonlinear process (7) are unstable if Re𝜆i > 0 for one or more of eigenvalues of AC . 1

𝜓

Now, the characteristic polynomial of selected eigenvalues is as follows:

(

)

L = X T RX

(17)

Substituting the closed loop nonlinear process (7) in the derivative of (17) gives:

𝜆n is such that Re𝜆n ≤ 0 )

‖2 ‖X ‖2 ≤ 𝜇e−𝜑t ‖ ‖Xi ‖ + 𝜀

Proof. Consider the following Lyapunov function:

⋮

(

(15)

(

P𝜆 = s − 𝜆1 · · · s − 𝜆i · · · s − 𝜆n

)

= sn + bn−1 sn−1 + · · · + b1 s + b0

(11)

= sn + an−1 (K )sn−1 + · · · + a1 (K )s + a0 (K )

(12)

2X T R𝛿 ≤ 𝜓 X T RX +

1

𝜓

(18)

𝛿 T R𝛿

(19)

[

]

L̇ ≤ X T ATC R + RAC + 𝜓 R X +

P𝜆 = PAC (13)

= sn + an−1 (K )sn−1 + · · · + a1 (K )s + a0 (K )

⇒ L̇ ≤ −X T SX +

1

𝜓

1

𝜓

𝛿 T R𝛿 (20)

T

𝛿 R𝛿

where equation ATC R + RAC + 𝜓 R = −S is satisfied with R ≥ 0 and S ≥

Considering element by element gives the following equations: an−1 (K ) = bn−1

⋮

[ ]T [ ] ⇒ L̇ = AC X + 𝛿 RX + X T R AC X + 𝛿 [ ] ⇒ L̇ = X T ATC R + RAC X + 2X T R𝛿

where 𝜓 > 0 is a small scalar. Substituting (19) into (18) gives:

where K are controller gains defined in (2). Considering the equality between P𝜆 and PAC gives:

⇒ sn + bn−1 sn−1 + · · · + b1 s + b0

T

The last element of past equation satisfies the following inequality:

The characteristic polynomial of Ac is obtained as follows: PAC = det (sI − Ac)

L̇ = Ẋ RX + X T RẊ

(14)

a1 (K ) = b1 a0 (K ) = b0 The solution of equation (14) gives controller gains K.

0 when Re𝜆i ≤ 0 for all eigenvalues of AC , and ‖𝛿‖ ≤ 𝛿 . Considering (17), and multiplying the first element of equation (20) by R−1 R = I with I as the identity matrix, equation (20) can be represented as follows: L̇ ≤ −X T SR−1 RX + 𝜂

(21)

⇒ L̇ ≤ −𝜑L + 𝜂 where 𝜂 =

1

T

𝛿 R𝛿 and 𝜑 = 𝜆min (SR−1 ). From (21), [32,33], it is satis-

Remark 1. Eigenvalues are selected to satisfy the following two characteristics: 1) the real parts of eigenvalues should be in preference negative, but on the other case, some could be zero, 2) eigenvalues should produce controller gains such that the regulation objective is reached, i.e., process states must converge to constant references.

fied that states of the closed loop nonlinear process are uniformly stable. Utilizing (17), the variable L to solve (21) will be found employing the following mathematical approach. Multiplying e𝜑t to both sides of (21), it gives:

2.3.4. Stability The following Theorem is suggested in this investigation to describe the stability analysis of closed loop nonlinear processes.

e𝜑t L̇ + e𝜑t 𝜑L ≤ e𝜑t 𝜂

Theorem 1. Consider the controller (2) for regulation of nonlinear processes (1). A and B are defined in (6), AC is defined in (7), 𝜆i are eigenvalues of AC , and Re𝜆i are real parts of eigenvalues 𝜆i . Therefore, the

𝜓

e𝜑t L̇ ≤ −e𝜑t 𝜑L + e𝜑t 𝜂

Taking into account that e𝜑t L̇ + e𝜑t 𝜑L = d ( 𝜑t ) e L ≤ e𝜑t 𝜂 dt

(22) d dt

( 𝜑t ) e L in (22), it gives: (23)

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J.J. Rubio / ISA Transactions 74 (2018) 155–164

3.- States of the closed loop nonlinear process (7) are unstable if Re𝜆i > 0 for one or more of eigenvalues of AC .

Integrating both sides of (23), it gives: t

d ( 𝜑𝜏 ) e L d𝜏 ≤ 𝜂

∫0 d𝜏

e𝜑𝜏 L||0 ≤ 𝜂 t

e𝜑t L − L

i

≤𝜂

t

∫0

e𝜑𝜏 d𝜏

e𝜑𝜏 ||

t

(24)

𝜑 ||0

[ 𝜑t e

𝜑

−

1

L = X T RX

]

e𝜑 t L ≤ L i + 𝜂

𝜑 𝜂 L ≤ e−𝜑t Li + 𝜑

−

1

T L̇ = Ẋ RX + X T RẊ

]

𝜑

(25)

𝜆min (R) ‖X‖2 ≤ X T RX = L 𝜂 𝜂 = e−𝜑t XiT RXi + 𝜑 𝜑 ‖2 𝜂 ≤ 𝜆max (R)e−𝜑t ‖ ‖Xi‖ + 𝜑 ≤ e−𝜑t Li +

(26)

𝜆max (R) −𝜑t ‖ ‖2 𝜂 e ‖Xi‖ + 𝜆min (R) 𝜆min (R)𝜑 𝜆max (R) , 𝜆min (R)

Proof is established.

𝜀=

(27)

𝜂 , equation (16) is found. Thus, the 𝜆min (R)𝜑

■

Remark 2. From (16), uniform stability is assured when the element

𝜂 can become arbitrarily small. From (16), element 𝜂 =

T

𝛿 R𝛿 can 𝜓 be arbitrarily small by increasing the value of a selected element 𝜓 , 1

or by decreasing the value of a selected matrix R. Remark 3. Please take into account that 𝜓 is not utilized in the controller (2). Please also take into account that 𝜓 should be selected small such as (15) is solved, but it also should be selected such as 𝜂 lets to assure uniform stability. Now, the following Theorem of [28] and [29] is described to explain differences between the previous research and actual study. It is based on considering 𝛿 = 0 in equation (6). Theorem 2. Consider the controller (2) for regulation of nonlinear processes (1) and consider unmodelled errors 𝛿 of equation (6) be equal to zero. A and B are defined in (6), AC is defined in (7), 𝜆i are eigenvalues of AC , and Re𝜆i are real parts of eigenvalues 𝜆j . Therefore, the following results are assured: 1.- if the linearized process (7) is stable, i.e., Re𝜆i ≤ 0 for all eigenvalues of AC , then states of the closed loop nonlinear process (7) are asymptotically stable. 2.- If there exists symmetric and positive semi-definite matrices R ∈ ℜn×n and S ∈ ℜn×n such that the following equation is solved: ATC R + RAC = −S

L̇ = −X T SX

(28)

𝜇e−𝜑t ‖‖Xi‖‖2

(29)

where Xi are initial conditions of states X, 𝜇 =

𝜆min

(SR−1 ), ‖·‖

𝜆max (R) , 𝜆min (R)

𝜑=

is the Euclidean norm in ℜ , 0 < 𝜑 ∈ ℜ, 0 < 𝜇 ∈ ℜ. n

(32)

where the equation + RAC = −S is satisfied with R ≥ 0 and S ≥ 0 when Re𝜆i ≤ 0 for all eigenvalues of AC . Considering (30), and multiplying the first element of equation (32) by R−1 R = I with I as the identity matrix, equation (32) can be represented as follows: ATC R

L̇ = −X T SR−1 RX (33)

where 𝜑 = 𝜆min (SR−1 ). From (33), [28,29], it is satisfied that states of the closed loop nonlinear process are asymptotically stable. Utilizing (30), the variable L to solve (33) will be found using the following mathematical approach. Multiplying e𝜑t to both sides of (33), it gives: e𝜑t L̇ ≤ −e𝜑t 𝜑L

(34)

e𝜑t L̇ + e𝜑t 𝜑L ≤ 0 Taking into account that e𝜑t L̇ + e𝜑t 𝜑L =

d dt

( 𝜑t ) e L in (34), it gives:

d ( 𝜑t ) e L ≤0 dt

(35)

Integrating both sides of (35), it gives: t

d ( 𝜑𝜏 ) e L d𝜏 ≤ 0

∫0 d𝜏

(36)

e𝜑𝜏 L||0 ≤ 0 t

e𝜑 t L − L i ≤ 0 the variable L to solve (33) is found from (36) as follows: e𝜑 t L ≤ L i + 𝜂

[ 𝜑t e

𝜑 𝜂 L ≤ e−𝜑t Li + 𝜑

−

1

]

𝜑

(37)

where Li is the initial condition of L. Now, the solution of X to solve (37) will be found employing the following mathematical approach. Utilizing the function (30) in the last equation of (37), it gives:

𝜆min (R)‖X‖2 ≤ X T RX = L ≤ e−𝜑t Li = e−𝜑t XiT RXi

with AC = A − BK, then states of the closed loop nonlinear process (7) are asymptotically stable and satisfy:

‖X‖2 ≤

(31)

⇒ L̇ ≤ −𝜑L

where Xi is the initial condition of X. Taking the first and last elements of the inequality (26) gives:

By using 𝜇 =

[ ]T [ ] ⇒ L̇ = AC X RX + X T R AC X [ ] ⇒ L̇ = X T ATC R + RAC X (31) can be rewritten as follows:

where Li is the initial condition of L. Now, the solution of X to solve (25) will be found by using the following mathematical approach. Utilizing the function (17) in the last equation of (25), it gives:

‖X‖2 ≤

(30)

Substituting the closed loop nonlinear process (7) with 𝛿 = 0 in the derivative of (30) gives:

𝜑

the variable L to solve (21) is found from (24) as follows:

[ 𝜑t e

Proof. The proof of this Theorem uses a similar procedure to Theorem 1 by utilizing 𝛿 = 0. Consider the following Lyapunov function:

(38)

‖2 ≤ 𝜆max (R)e−𝜑t ‖ ‖Xi‖ where Xi is the initial condition of X. Taking the first and last elements of inequality (38) gives:

‖X‖2 ≤

𝜆max (R) −𝜑t ‖ ‖2 e ‖Xi‖ 𝜆min (R)

(39)

J.J. Rubio / ISA Transactions 74 (2018) 155–164

159

From ideal gas law, it is known that partial pressure of each gas is proportional to the amount of gas in a cell, in which there are three important inputs depending on the gas inlet flow rate, gas consumption, and gas outlet flow rate. Thus, state equations are:

[

( ) PH2 R T dPH2 = m m H2,in − 2Kr Ac i − H2,in − 2Kr Ac i dt VA Pop [

( ) PO2 R T dPO2 = m m O2,in − Kr Ac i − O2,in − Kr Ac i dt VC Pop

] (40)

] (41)

[

( ) PH2 OC dPH2 OC R T = m m H2 OC,in + 2Kr Ac i − O2,in + 2Kr Ac i dt VC Pop where Kr = Fig. 2. A fuel cell.

By using 𝜇 = lished.

𝜆max (R) , equation (29) is found. Thus, the Proof is estab𝜆min (R)

■

Remark 4. Investigations of [28] and [29] consider unmodelled errors 𝛿 of equation (6) equal to zero in the stability analysis as the same as Theorem 2, to obtain the asymptotic stability, while the result of this research considers unmodelled errors 𝛿 small and different to zero in the stability analysis as the same as Theorem 1, to obtain uniform stability. The case of 𝛿 small and different to zero represents better real applications. Remark 5. Please note that in Refs. [28] and [29], authors only consider points 1.- and 3.- of Theorem 2. Therefore, the point 2.- of Theorem 2, and all the points of Theorem 1 are the main contributions of this paper. Remark 6. The difference of the proposed method with the pole placement technique is as follows: the pole placement technique is applied to linear processes where asymptotic stability is assured, while the proposed technique is applied to nonlinear processes where uniform stability is assured. 3. Control of a fuel cell In this section, the controller of a fuel cell is studied. It is based on two parts, the mathematical model and controller design.

3.1. Mathematical model of a fuel cell In this section, a fuel cell mathematical model is described, it will be used in the following subsection for control design. An external load is considered in a fuel cell, and it is included as a voltage subtracted to the generated voltage. Fig. 2 shows a fuel cell. It considers a function which represents cations formed during the reaction in cathode. Freed electrons go through the diffuser and they finally go to cathode. Oxygen is freed in cathode and flows to the diffuser until it arrives to catalyzer, where the hydrogen cations and electrons make reaction to obtain a final product which is water.

Nm , 4Fm

] (42)

Ac is a cell active area, i is a current density, Pop is

an operating pressure, VA is anode volume, VC is cathode volume, Rm is a gas constant, Tm is an operating temperature, Fm is a Faraday constant, Nm is the stack cells number, H2 OC ,in , H2,in , O2,in , are inlet flow rates of the water, hydrogen, and oxygen in cathode, PH2 OC , PH2 , PO2 , are partial pressures of the water, hydrogen, and oxygen inside a cell. 3.1.2. Final fuel cell model The water input H2 OC ,in is not taken into account in state equations because it is not a process input. Defining states as x1 = PH2 , x2 = PO2 , x3 = PH2 OC , inputs as u1 = H2,in , u2 = O2,in , u3 = i. Consequently, the mathematical model of equations (40)–(42) becomes to: ẋ 1 =

Rm Tm VA

+ ẋ 2 =

ẋ 3 =

u u1 + 2Kr Ac 3 Pop Pop

) x1

(

−

u u2 + Kr Ac 3 Pop Pop

)

x2 (43)

Rm Tm R T u2 − m m Kr Ac u3 VC VC

Rm Tm VC

+

−

Rm Tm R T u − m m 2Kr Ac u3 VA 1 VA

Rm Tm VC

+

(

(

−

u2 u − 2Kr Ac 3 Pop Pop

) x3

Rm Tm 2Kr Ac u3 VC

3.2. Controller of a fuel cell In this subsection, the controller and stability of a fuel cell are studied. 3.2.1. Control design The controller objective is that using inputs, states of a fuel cell should follow constant references, it is denoted as states regulation. The process is (43). From equation (43), it can be observed that the process has 3 inputs for the regulation of 3 states; therefore, controllers are as follows: u1 = − k1 X

3.1.1. Mathematical model Partial pressures of hydrogen, oxygen, and water on cathode side are defined as process state variables, and relationship between the inlet and outlet gases is as follows: Anode: H2,in + H2 OA,in = H2,out + H2 OA,out Cathode: N2,in + O2,in + H2 OC ,in = N2,out + O2,out + H2 OC ,out

u 2 = − k2 X

(44)

u 3 = − k3 X where X = [x1

x2

x3 ]T are states, u1 , u2 , and u3 are control inputs,

k1 = [k11 0 0], k2 = [0 controller gains.

k22

0], and k3 = [0

0

k33 ] are

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J.J. Rubio / ISA Transactions 74 (2018) 155–164

3.2.2. Stability analysis Making the linearization of a fuel cell model (43) gives the following result: Ẋ = AX + BU + 𝛿

(45)

where:

⎡0 ⎢ A = ⎢0 ⎢0 ⎣

0 0 0

⎡ Rm Tm ⎢ VA ⎢ B=⎢ 0 ⎢ ⎢ 0 ⎣

0⎤ ⎥ 0⎥ ⎥ 0⎦ 0

Rm Tm 2Kr Ac⎤ VA ⎥ ⎥ R T − m m Kr Ac ⎥ VC ⎥ Rm Tm 2Kr Ac ⎥ ⎦ VC

−

Rm Tm VC 0

(46)

4.1.1. Structural mathematical model The mathematical model of a manipulator is written as follows:

[

]

J13 + (m2 + 4m3 )z212 ż 12 + 2(m2 + 4m3 )z12 z21 z22 + v1

(49)

(m2 + 4m3 )ż 22 − (m2 + 4m3 )z12 z221 = v2

(50)

m3 ż 23 − m3 g = v3

(51)

where z11 = 𝜃1 is an angle of the link 1, z12 = l2 and z13 = l3 are lengths of links 2 and 3, z21 is an angular velocity in the link 1, z22 and z23 are displacements velocities in links 2, and 3, v1 , v2 , and v3 are torques to move links 1, 2, and 3, m1 , m2 , and m3 , are masses of links 1, 2, and 3, J13 = J1 + J2 + J3 , J1 , J2 , and J3 are inertias of links 1, 2, and 3, l1 , l2 , and l3 , are lengths of links 1, 2, and 3, g is the acceleration gravity constant.

A control function (44) can be rewritten as follows: U = −KX

⎡k11 ⎢ where K = ⎢ 0 ⎢ ⎣0

(47) 0 k22 0

0⎤

⎡u1⎤ ⎥ ⎢ ⎥ 0 ⎥, U = ⎢u2⎥. Substituting control functions ⎥ ⎢ ⎥ k33⎦ ⎣u3⎦

in the linearized process produces a closed loop nonlinear process as follows: Ẋ = AX + B [−KX] + 𝛿

⇒ Ẋ = [A − BK] X + 𝛿

4.1.2. Final manipulator model Define state variables as x1 = z11 , x2 = z12 , x3 = z13 , x4 = z21 , x5 = z22 , x6 = z23 , inputs as u1 = v1 , u2 = v2 , u3 = v3 . Consequently, the mathematical model of equations (49)–(51) becomes to: ẋ 1 = x2 2(m2 + 4m3 ) ] x2 x3 x4 ẋ 2 = − [ J13 + (m2 + 4m3 )x22

+ [

(48)

1 ] u1 J13 + (m2 + 4m3 )x22

⇒ Ẋ = AC X + 𝛿

ẋ 3 = x4

where AC = A − BK.

ẋ 4 = x3 x22 +

(52) 1

(m2 + 4m3 )

u2

4. Control of a manipulator

ẋ 5 = x6

In this section, the controller of a manipulator is studied. It is based on two parts, the mathematical model and controller design.

ẋ 6 = g +

4.1. Mathematical model of a manipulator

4.2. Controller of a manipulator

Fig. 3 shows a manipulator. The manipulator of this study is a kind of three links cylindrical robotic arm, where each joint is equipped with a motor to provide the input torque, an encoder is employed to measure the joint position, the first joint is rotational, and the second and third joints are translational.

In this subsection, the controller and stability of a manipulator are studied.

1

(m3 )

u3

4.2.1. Controller design The controller objective is that using inputs, states of a manipulator should follow constant references, it is known as states regulation. The process to control is (52). From equation (52), it can be seen that the process has 3 inputs for regulation of 6 states; therefore, controllers are as follows: u1 = − k1 X u 2 = − k2 X

(53)

u 3 = − k3 X where u3

[0

Fig. 3. A manipulator.

are

X = [ x1 control

x2

x3

inputs,

0 k23 k24 0 are controller gains.

0],

x4

x6 ] T

x5

k1 = [k11 and

are

states,

u1 ,

u2 ,

k12

0

0

0

0], k2 =

k3 = [ 0

0

0

0

k35

k36 ]

J.J. Rubio / ISA Transactions 74 (2018) 155–164

4.2.2. Stability analysis Making a linearization of the manipulator model (52) produces the following result: Ẋ = AX + BU + 𝛿

Table 1 Parameter values for a fuel cell.

(54)

where:

⎡0 ⎢ ⎢0 ⎢0 A=⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0

1

0

0

0

0⎤

0

0

0

0

0⎥

0

0

1

0

0⎥

0

0

0

0

0⎥

0

0

0

0

0

0

0

0

⎡ 0 ⎢ 1 ⎢ ⎢ J13 ⎢ 0 B=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

⎥

(55)

⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎥ m3 ⎦

m2 + 4m3 0 0

U = −KX k12

0

0

0

0

k23

k24

0

0⎤

⎡u1⎤ ⎥ ⎢ ⎥ 0 ⎥, U = ⎢u2⎥. Substituting ⎥ ⎢u ⎥ k36⎦ ⎣ 3⎦

0 0 0 k35 control functions produce a closed loop nonlinear process as follows: Ẋ = AX + B [−KX] + 𝛿

⇒ Ẋ = [A − BK] X + 𝛿

(57)

⇒ Ẋ = AC X + 𝛿

+

x25

1 T f ∫0

Tf

) 12

X d𝜏 2

B

AB

A2 B . Since rank ℂ = 3, 3 process states can be controlled.

0

0⎤

k2

0 ⎥ with k1 = 1,

0

k3⎦

x26

⎥ ⎥

5.1.3. Stability SMC of [12] is employed where initial conditions are xs0 = [5, 5, 5]T , and controller gains are ks1 = [0.175, 0, 0] , ks2 = [0, 0.35, 0] , ks3 = [0, 0, 1.75]. PC of equation (44) is utilized where initial conditions are x0 = [5, 5, 5]T , and controller gains are k1 = [1, 0, 0], k2 = [0, 2, 0], k3 = [0, 0, 10]. Substituting values of Table 1 and controller gains in a matrix AC of (48), the following result is obtained:

⎡−451.88 ⎢ AC = ⎢ 0 ⎢ 0 ⎣

0

112.04⎤

−452.92

28.075⎥

0

⎥

(59)

−56.15⎥⎦

for a manipulator, and Tf is the final time.

0 r22

r13⎤

⎥

r23⎥ with

⎢r ⎥ ⎣ 13 r23 r33⎦ −3 r11 = 1.1066 × 10 , r13 = 2.4410 × 10 , r22 = 1.1041 × 10 , r23 = ⎡s11 0 s13⎤ ⎢ ⎥ −5 −3 6.0901 × 10 , r33 = 9.4306 × 10 , and S = ⎢ 0 s22 s23⎥ with ⎢ ⎥ ⎣s13 s23 s33⎦ s11 = 0.99999, s13 = 2.249 × 10−6 , s22 = 1.0, s23 = −8.2553 × 10−7 , s33 = 1.0, and the scalar as follows 𝜓 = 0.1; therefore, by utilizing −3

−4

points 1 and 2 of Theorem 1 this nonlinear process is assured to be uniformly stable. 5.1.4. Robustness To study the robustness, controller inputs are modified as follows: u1 = −k1 X + 0.2 rand

(58)

X 2 = x21 + x21 + x23 for the fuel cell, and X 2 = x21 + x21 + x23 +

+

◦

⎡r11 ⎢

In this section, the proposed controller called PC will be compared with the sliding mode controller of [12] called SMC for the two nonlinear processes regulation which are a fuel cell and a manipulator. Two selected processes have in common that they have the nonlinear structure described in equation (1). The objective is that process states X should follow a desired behavior, which is given by the tracking to a zero reference, it is understood as a satisfactory result in states regulation. PC is compared with SMC because both have the following two characteristics: 1) PC and SMC are robust to inputs and parameters uncertainties in nonlinear processes, and 2) PC and SMC are uniformly stable. In the following two subsections, root mean square error (RMSE) will be used in comparisons, it is given as follows:

x24

353 K 6.495 cm2 12.96 cm2 35

𝜆3 = −453.0, and matrices as follows R = ⎢ 0

5. Results

where

Value

Tm VA VC Nm

Considering eigenvalues as follows 𝜆1 = −56.15, 𝜆2 = −451.80,

where AC = A − BK.

RMSE =

Parameter

⎡k 1 ⎢ utilized to obtain controller gains K = ⎢ 0 ⎢0 ⎣ k2 = 2, k3 = 10.

(56)

(

◦

5.1.2. Controller gains Eigenvalues are selected as 𝜆1 = −56.15, 𝜆2 = −451.80, 𝜆3 = −453.0, then a procedure similar to the pole placement method is

where J13 = J1 + J2 + J3 . A control function (53) can be rewritten as follows:

⎡k11 ⎢ where K = ⎢ 0 ⎢0 ⎣

8.3144 J/mol K 96485 C/mol 101 × 103 Pa 136.7 cm2

5.1.1. Controllability matrix ℂ is analyzed by obtaining ℂ = [ The controllability ]

0⎤

0 1

Value

Rm Fm Pop AC

Table 1 shows parameter values for a fuel cell.

⎥ 1⎥ ⎥ 0⎦

0

Parameter

5.1. Fuel cell

⎥

0

161

u2 = −k2 X + 0.2 rand u3 = −k3 X + 0.2 rand and the nonlinear model is modified as follows:

162

J.J. Rubio / ISA Transactions 74 (2018) 155–164

Table 2 Results for a fuel cell. Methods

RMSE

SMC PC

2.2727 0.4238

Table 3 Parameter values for a manipulator. Parameter

Value

Parameter

Value

J1 J2 J3 l1 l2

0.04624 kgm2 2 0.02545 kgm 0.03616 kgm2 0.3 m 0.3 m

m1 m2 m3 l3 g

0.46 kg 0.34 kg 0.34 kg 0.3 m 2 9.81 m/s

controller is robust to inputs and parameters uncertainties of Fig. 4 because the states stability is remained. Thus, PC is preferable for the states regulation in a fuel cell. Fig. 4. Inputs for a fuel cell.

5.2. The manipulator ẋ 1 = f1 (X , U ) + 5 rand

Table 3 shows parameter values for a manipulator.

ẋ 2 = f2 (X , U ) + 5 rand ẋ 3 = f3 (X , U ) + 5 rand where rand are random numbers. 5.1.5. Discussion Figs. 4 and 5 show the inputs and states of a fuel cell with controllers for a time from 0 s to 1 s. Table 2 shows RMSE of (58). Note that the control objective is not complex because there are 3 inputs for regulation of 3 states, and the main objective is reached because all states in a fuel cell are regulated, i.e., the constant behavior in static phase must be reached as fast as possible. Dynamic phase only is considered during the first 0.1 s and overshoot is not presented. From Figs. 4 and 5, it can be seen that PC reaches a better regulation than SMC because process signals for the first follow better references than for the second. From Table 2, it can be seen that PC reaches better accuracy than SMC because RMSE for the first is smaller than for the second. States of Fig. 5 show that the proposed

5.2.1. Controllability [ The controllability matrix ℂ ] is analyzed by obtaining ℂ = B AB A2 B A3 B A4 B A5 B . Since rank ℂ = 6, 6 process states can be controlled. 5.2.2. Controller gains Eigenvalues are selected as 𝜆1 = −0.96949, 𝜆2 = −1853.2, 𝜆3 = −1763.9, 𝜆4 = −293.12, 𝜆5 = −1.0175 + 2.8835 × 10−2 i, 𝜆6 = −1.0175 − 2.8835 × 10−2 i, then a procedure similar to the pole placement method is utilized to obtain controller gains K =

⎡k11 ⎢ ⎢0 ⎢0 ⎣

k12

0

0

0

0⎤

0

k23

k24

0

0 ⎥ with k11 = 200, k12 = 200, k23 =

0

0

0

k35

⎥ ⎥

k36⎦

500, k24 = 500, k35 = 600, k36 = 600. 5.2.3. Stability SMC of [12] is employed where initial conditions are xs0 = [5, 5, 5, 5, 5, 5]T , and controller gains are ks1 = [35, 35, 0, 0, 0, 0], ks2 = [0, 0, 87.5, 87.5, 0, 0], ks3 = [0, 0, 0, 0, 105, 105]. PC of equation (53) is utilized where initial conditions are x0 = [5, 5, 5, 5, 5, 5]T , and controller gains are k1 = [200, 200, 0, 0, 0, 0], k2 = [0, 0, 500, 500, 0, 0], k3 = [0, 0, 0, 0, 600, 600]. Substituting values of Table 3 and controller gains in a Matrix AC of (57) the following result is obtained:

⎡ 0 ⎢ ⎢aC21 ⎢ 0 AC = ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0

Fig. 5. States for a fuel cell.

aC12

0

0

0

0 ⎤

aC22

0

0

0

⎥

0

0

aC34

0

0 ⎥ 0 ⎥

0

aC43

aC44

0

0 ⎥

0

0

0

0

0

0

0

aC65

⎥

(60)

⎥ aC56⎥ ⎥ aC66⎦

with aC12 = 1, aC21 = −1854.4, aC22 = −1854.4, aC34 = 1, aC44 = −294.12, aC56 = 1, aC65 = −1764.7, aC43 = −294.12, aC66 = −1764.7. Considering eigenvalues as follows 𝜆1 = −0.96949, 𝜆2 = −1853.2, 𝜆3 = −1763.9, 𝜆4 = −293.12, 𝜆5 = −1.0175 + 2.8835 × 10−2 i, 𝜆6 = −1.0175 − 2.8835 × 10−2 i,

J.J. Rubio / ISA Transactions 74 (2018) 155–164

⎡r11 r12 0 0 0 0⎤ ⎢ ⎥ r 0 0 0 0⎥ r ⎢ 12 22 ⎢0 0 0⎥ 0 r33 r34 ⎥ matrices as follows R=⎢ ⎢0 0 0⎥ 0 r34 r44 ⎢ ⎥ 0 0 0 r55 r56⎥ ⎢0 ⎢ ⎥ 0 0 0 r56 r66⎦ ⎣0 −4 −4 with r11 = 1.0529, r12 = 2.9802 × 10 , r22 = 2.6980 × 10 , r34 = 1.8793 × 10−3 , r44 = 1.7067 × 10−3 , r33 = 1.0545, −4 r56 = 3.1317 × 10 , r66 = 2.8352 × 10−4 , and r55 = 1.0529, ⎡s11 s12 0 0 0 0⎤ ⎢ ⎥ 0 0 0 0⎥ ⎢s12 s22 ⎢0 0 0⎥ 0 s33 s34 ⎥ with s11 = 1.0, s12 = S=⎢ ⎢0 0 0⎥ 0 s34 s44 ⎢ ⎥ 0 0 0 s55 s56⎥ ⎢0 ⎢ ⎥ 0 0 0 s56 s66⎦ ⎣0 −5 3.5606 × 10 , s22 = 1.0, s33 = 1.0, s34 = 2.639 × 10−5 , s44 = 1.0, s55 = 1.0, s56 = 4.7526 × 10−5 , s66 = 1.0, and the scalar as follows 𝜓 = 0.1; therefore, by utilizing points 1 and 2 of Theorem 1 this

163

Fig. 7. States for a manipulator.

nonlinear process is assured to be uniformly stable. Table 4 Results for a manipulator.

5.2.4. Robustness To study the robustness, controller inputs are modified as follows: u1 = −k1 X + 0.2 rand

Methods

RMSE

SMC PC

14.1122 2.0022

u2 = −k2 X + 0.2 rand u3 = −k3 X + 0.2 rand and the nonlinear model is modified as follows: ẋ 1 = x2 ẋ 2 = f2 (X , U ) + 5 rand ẋ 3 = x4 ẋ 4 = f4 (X , U ) + 5 rand ẋ 5 = x6 ẋ 6 = f6 (X , U ) + 5 rand where rand are random numbers.

5.2.5. Discussion Figs. 6 and 7 show inputs and states of a manipulator with controllers for a time from 0 s to 10 s. Table 4 shows RMSE of (58). Note that the control objective is complex because there are 3 inputs for regulation of 6 states; however, the main objective is reached because all states in a manipulator are regulated, i.e., a constant behavior in static phase must be reached as fast as possible. Dynamic phase is considered during the first 4 s and overshoot is not presented, dynamic phase is bigger than in other cases because a manipulator has more states related each other. From Figs. 6 and 7, it can be seen that PC reaches a better regulation than SMC because process signals for the first follow better references than for the second. From Table 4, it can be seen that PC reaches better accuracy than SMC because RMSE for the first is smaller than for the second. States of Fig. 7 show that the proposed controller is robust to inputs and parameters uncertainties of Fig. 6 because the states stability is remained. Thus, PC is preferable for the states regulation in a manipulator. Remark 7. Processes of this study describe realistic examples, the design procedure is described as follows: 1) Figs. 2 and 3, and Tables 1 and 3 show the prototypes and their parameters utilized to validate models of equations (43), (52), and 2) models of equations (43), (52), are utilized to prove the suggested strategy of equations (44), (53). 6. Conclusions

Fig. 6. Inputs for a manipulator.

In this research, a controller was proposed for the nonlinear processes regulation. The proposed controller was based on the combination of the robust feedback linearization technique and Lyapunov method. The suggested controller was compared with a sliding mode controller for the states regulation in a fuel cell and a manipulator producing that the first achieved better accuracy in comparison with the second because the first showed the best tracking of constant references. The fuel cell and manipulator were controllable. The sug-

164

J.J. Rubio / ISA Transactions 74 (2018) 155–164

gested controller applied to two processes was robust to inputs and parameters uncertainties. The studied controller could be applied to other kind of electrical, mechanical, hydraulic, pneumatic, robotic, or mechatronic processes. In the future, other kind of controllers will be designed for the regulation, trajectory tracking, disturbance rejection, measurement noise, and required actuating power. Acknowledgment Author is grateful with the Editor-in-Chief and with reviewers for their valuable comments and insightful suggestions, which can help to improve this research significantly. Author thanks the Instituto Politécnico Nacional, Secretaría de Investigación y Posgrado, Comisión de Operación y Fomento de Actividades Académicas, and Consejo Nacional de Ciencia y Tecnología for their help in this research. References [1] Basin M, Calderon-Alvarez D. Delay-dependent stability for vector nonlinear stochastic systems with multiple delays. Int J Innovat Comput Inf Contr 2011;7(4):1565–76. [2] Yang T, Zhang L, Li Y, Leng Y. Stabilisation of Markov jump linear systems subject to both state and mode detection delays. IET Control Theory & Appl 2014;8(4):260–6. [3] Zhu Y, Zhang Q, Wei Z, Zhang L. Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing 2013;110:44–50. [4] Jafarnejadsani H, Pieper J, Ehlers J. Adaptive control of a variable-speed variable-pitch wind turbine using radial-basis function neural network. IEEE Trans Contr Syst Technol 2013;21(6):2264–72. [5] Pan Y, Zhang J, Yu H. Model reference composite learning control without persistency of excitation. IET Control Theory & Appl 2016;10(16):1963–71. [6] Pan Y, Er MJ, Sun T, Xu B, Yu H. Adaptive fuzzy PD control with stable H∞ tracking guarantee. Neurocomputing 2017;237:71–8. [7] Saifia D, Chadli M, Karimi HR, Labiod S. Fuzzy control for electric power steering system with assist motor current input constraints. J Franklin Inst 2015;352:562–76. [8] Grande A, Hernandez T, Curtidor AV, Paramo LA, Tapia R, Cazares IO, Meda JA. Analysis of fuzzy observability property for a class of ts fuzzy models. IEEE Lat Am Transac 2017;15(4):595–602. [9] Pan Y, Sun T, Yu H. Composite adaptive dynamic surface control using online recorded data. Int J Robust Nonlinear Control 2016;26(18):3921–36. [10] Tong Y, Zhang L, Basin M, Wang C. Weighted H∞ control with d-stability constraint for switched positive linear systems. Int J Robust Nonlinear Control 2014;24:758–74. [11] Yang T, Zhang L, Yin X. Time-varying gain-scheduling 𝜎 -error mean square stabilisation of semi-Markov jump linear systems. IET Control Theory & Appl 2016;10(11):1215–23. [12] Aguilar-Ibañez C. Stabilization of the pvtol aircraft based on a sliding mode and a saturation function. Int J Robust Nonlinear Control 2017;27:843–59.

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