54 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2013 1066-033X/13/$31.00©2013IEEE

Understanding Neighborhood of Linearization in Undergraduate Control Education

The first undergraduate control course is usually on automatic control theory. Correctly understanding the concepts in that course has far-reaching implications

for students. Linearization is a standard topic covered in this introductory course. A nonlinear measurement-based approach is presented here to aid in the teaching of linear-ization. The pedagogical objective is to help students un-derstand the concept of the neighborhood of linearization. The pedagogical approach is illustrated by measuring the nonlinearity of the cart-pole system, an example commonly used in control education. Drawing on student surveys in two consecutive academic years, we recommend combin-ing demonstration software visualization with relevant mathematic introduction.

INTRODUCTION Automatic control theory (ACT), also commonly called linear system analysis, is a basic introductory course on the analysis and design of automatic control systems [1], [2]. This course is usually the first time that undergraduates are exposed to control systems. The course covers many concepts [21], and correctly understanding these concepts has far-reaching implications for the students’ subsequent control-related courses.

Linearization is a standard part of control education [1]– [3] because it is a powerful tool to analyze system behavior, such as system stability, disturbance rejection, and refer-ence tracking. Generally speaking, the mathematical foun-dation of linearization is the Taylor series expansion [4]. After expanding a nonlinear function in a Taylor series at an operating point, the function is linearized by neglecting terms of an order greater than one. To ensure the applica-bility of the linearized model, it is strictly assumed that the nonlinear system is operated within a small neighborhood of the operating point [3]. Reference [5] notes that students very often forget the operating point of linearization and suggests an effective method to improve the teaching of linearization, which is to use a coordinate system in which the operating point is not at the origin.

In the School of Control and Computer Engineering at the North China Electric Power University (NCEPU), Bei-jing, P.R. China, about 180 students per year take the ACT course. The teaching method in [5] has been used in two consecutive academic years, 2010–2011 and 2011–2012. The course has a project after all of the lectures have been completed for the semester. The project is on the stabiliza-tion control of a cart-pole system, which is one of the most

enduringly popular and important laboratory models for teaching control systems engineering [6]. The purpose of the project is a synthesis of the main concepts and meth-ods of the course, including system modeling, lineariza-tion, controller design, and simulation. The project objec-tive is to balance the pole in its inverted (upward) position with small angular deviations from the upright position by applying a driving force to the cart, with or without con-sidering the cart position. By employing root locus com-pensation, frequency response compensation, or closed-loop pole-zero assignment, more than 95% of the students can complete the controller design and realize the project objective in matlab after modeling the system and linear-izing it around its upward position. The simulation files are downloadedable [7].

Some of the more inquisitive students extend the project objective and gradually increase the initial angular devia-tion of the pole to verify the effects of nonlinearity. A sur-prising fact puzzles them: the linearized model-based con-troller can actually balance the pole even when an initial deviation is large. Sometimes, the controller can stabilize the pole with an initial deviation as large as /3!r rad. In Taylor series-based linearization, often called “local lin-earization,” The cart-pole system is linearized at the top inverted position, so it is not surprising that the linearized model-based controller can stabilize the system in a small neighborhood of that position. The fact that the controller is able to work well in a large range around the upright posi-tion seems contrary to the technical assumptions of linear-ization. This observation motivates the question of how we should teach the neighborhood of linearization when edu-cating students.

Although neighborhood is a simple concept in math-ematics, it is not correctly understood by many students, especially undergraduates taking a control course for the first time. The students often assume that the neighbor-hood in which a linearized model is useful is very small. One reason for this assumption is that Taylor series-based linearization is presented loosely in most textbooks, for instance in [1]–[3]. After presenting the linearization approach, books often emphasize the fact that the lin-earized model at an operating point is able to accurately describe the dynamics of the nonlinear system only when the states of the nonlinear dynamical system are very close to the operating point. The initial purpose of this emphasis is to ensure that the linearization method is very accurate, but this emphasis may mislead students into thinking that a controller designed based on a linearized model can only work in a small neighborhood of the operating point.

DIANwEI QIAN, JIANQIANG YI, and ShIwEN TONG

Digital Object Identifier 10.1109/MCS.2013.2258767Date of publication: 12 July 2013

AUGUST 2013 « IEEE CONTROL SYSTEMS MAGAZINE 55

To help students correctly understand the significance of the neighborhood of linearization, we introduce the stu-dents to the gap-metric-based nonlinearity measure. Due to the nonlinearity measure being too advanced of a concept to be taught in its entirety to undergraduates, a case-driven pedagogical approach is used. Surveys in two consecutive academic years suggest a recommended teaching practice. To the best of the authors’ knowledge, this is the first time that the gap metric has been successfully used for teaching undergraduates about nonlinearity. To help undergradu-ates understand control concepts and lay a solid founda-tion for subsequent courses, a case-driven pedagogical approach is used that combines demonstration software and a relevant mathematical introduction.

In the next section, the gap-metric-based nonlinearity measure is introduced. The subsequent section applies the nonlinearity measure to a cart-pole system and discusses the implications. Then the impact of the proposed method is assessed via student survey results. The final section has some conclusions.

GAP-METRIC-BASED NONLINEARITY MEASURE As mentioned, students find their controller designed based on a linearized model is able to control the pole in a large range around the inverted position. The contradiction between their simulations and the technical assumptions of linearization encourages both teachers and students to find a rational explanation.

One answer could be that the difference between the real nonlinear dynamical system and the linearized sys-tem may be treated as a disturbance, and that the controller has the ability to reject the disturbance. Such a qualitative answer could be proposed as a partial explanation, but a quantitative answer is more convincing and more inter-esting. Here we introduce the teaching of the gap-metric-based nonlinearity measure to give students a quantitative answer.

A nonlinearity measure is a powerful tool to assess the degree of inherent nonlinearity of a system, instead of roughly judging the system as being linear or nonlin-ear [8], [9]. The gap-metric-based nonlinearity measure is one of the most extensively used measures for nonlinear-ity [10]. In short, the gap metric is an extension of the com-mon measure of the 3-norm of the difference between two systems. The gap-metric-based nonlinearity measure is to measure the gap between a linearization of a nonlinear system at its operating point and a fixed linear system. In [11], the method was reported to assess the nonlinearity of a boiler-turbine unit in industry. The gap metric has also been employed to measure the nonlinearity of a chemical process [12].

Definition 1The gap-metric-based nonlinearity measure go is defined as [10]

( , ),L NP Lg p0o d= (1)

where L NPp0 is the linearized system of a nonlinear system NP at its operation point ,p L0 is a linear system, and ( , )$ $d is the gap between two linear systems P1 and P2 defined by [13]

( , ) || ( ) ( ) ( ) || ,P P I P P P P I P P1 2 2 2 21

2 1 1 1 21

d = + - + 3- - (2)

where , , ,P i 1 2i = and || ||$ 3 denote complex conjugate and 3-norm, respectively.

Remark 1As defined in (2), this measure go is bounded between zero and one [13]. If go between two systems is close to zero, then the two systems have similar dynamics in the expected operating space. On the other hand, if go is close to one, then the two systems behave quite differently.

Remark 2One of the main features encountered in the gap metric is that it is not only applicable to stable systems, but also to integrating and unstable systems [11]. For example, consider

,

.,

Ps

Ps

1

0 11

1

2

=

=+

where s is a complex variable in the Laplace domain. The 3norm of the difference between P1 and P2 is infinite, while the distance in the sense of gap metric is ( , ) . .P P 0 09951 2d =

Remark 3The reason why the gap metric applies to integrating and unstable systems is that it measures the distance in the closed-loop sense instead of the open-loop sense [12]. Even though the open-loop systems may have quite different dynamics, their distance in terms of the gap metric can be close. For example, consider

,

.

Ps

Ps

2 1100

2 1100

1

2

=+

=-

The two open-loop systems have very different dynamics, since P1 is stable and P2 is unstable. However, the gap met-ric between P1 and P2 is ( , ) . ,P P 0 02051 2d = which shows that they are very close in terms of their closed-loop behav-ior. In fact, the closed-loop transfer function for P1 and P2 with unity feedback are close:

.,

..

PP

s

PP

s

1 50 550

1 55049

1

1

2

1

+=+

+=+

56 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2013

The theoretical underpinnings of the nonlinearity measure in Definition 1 are profound and far beyond an undergraduate’s education. When the pedagogical approach was first presented in 2010, it was controversial among our colleagues at NCEPU but the idea was still put into practice. During the fall semester of 2010, the gap metric was introduced in detail. The feedback from students indicated that they did not appreciate its com-plicated mathematical derivation. The student comments suggested that the basic concepts of linear systems pro-vided sufficient background for the students to compre-hend the main idea and application of the nonlinearity measure, provided that the measure was introduced to them with a proper pedagogical approach.

In the fall 2011 semester, a change was made in the instruction. The detailed derivation of the gap metric was abandoned. The general idea of the gap metric was introduced by contrasting the gap metric with the idea of a norm from linear algebra. Since students knew that a norm is defined for measuring the distance between two points, the gap metric could be simply treated as a tool to measure the distance between two linear systems. After the concept of the gap metric was introduced, how to calculate the gap metric was described. Actually, it was hardly possible to calculate the measure without its mathematical definition. Fortunately, matlab software offered a gapmetric command to compute the gap metric between two systems in the form of transfer functions [18]. The command, acting as a calculator, made it possible for students to directly produce a desired result rather than having to focus on the numerical algorithms used to pro-duce the result. The teachers’ role was to show how to cor-rectly use this calculator.

NONLINEARITY MEASURE fOR ThE CART-POLE SYSTEM

DynamicsSince the 1960s, the cart-pole system has been an excellent test bed for research and education because it is simple enough for complete dynamic analyses and experiments, while having strong nonlinearities and dynamic couplings [14], [15]. Its stabi-lization also became the de facto benchmark for verifying the feasibility of novel methods in control research and illustrating the significance of concepts of control education [16].

Shown in Figure 1, the cart-pole system is made up of the two main components suggested by its name, a cart and a pole. The symbols in Figure 1 are the cart mass M, the pole mass m, the half length of the pole l, the cart position x with respect to the origin, the driving force f that is positive in the direction of the positive x-axis, and the pole angle i with respect to its upright position, where i is positive in the clockwise direction.

The uncontrolled pole has two equilibria. The stable equilibrium is in its downward position with ( , )i i =o( rad, 0 rad/s),r and the unstable equilibrium is in its inverted position with ( , ) 0 rad, 0 rad/si i =o ^ h [17]. The objective of its stabilization control is to keep the pole angle

0i = rad by applying a control input to the cart. The objec-tive can be further classified in terms of whether the cart position x is considered. In the course project, either variant on the control problem is acceptable.

Under the standard assumptions of the cart with a point mass, the rigid pole with uniform mass, no friction, etc., the dynamics of the cart-pole system [19] can be written as

( ) ,sin cosJ ml mgl mlx 0p2 i i i+ - + =p p (3)

( ) ,cos sinml ml M m x f2 2i i i i- + + + =p o p (4)

where /J ml 3p2= is the moment of inertia around the cen-

troid and . m/sg 9 81 2= is the gravitational acceleration. The equations of motion of the pole and cart are (3) and (4), respectively. If the driving force f is chosen as the control input u [17], then

,,

x uu

F B

F B

1 1

2 2i

= +

= +pp' (5)

where the expressions for the nonlinear functions are

( ),

( )

( ),

( ),

cos

sin cos sin

cos

sin cos sin

cos

M m m

mg ml

M m m

M m g ml

M m m

34

34

34

34

34

F

F

B

12

2

22

2

12

i

i i i i

i

i i i i

i

=+ -

-

=+ -

+ -

=+ -

o

o

x

i

m

l

fM

Origin

Figure 1 Schematic of the cart-pole system.

AUGUST 2013 « IEEE CONTROL SYSTEMS MAGAZINE 57

and

( ).

cos

cos

M m l ml34

B22i

i=+ -

On the other hand, if the cart acceleration xp is defined as the control input u, then the dynamics of the cart-pole system from (3) in another form [17] can be described by

,

x u

uF Bi

=

= +

p

p) (6)

where

sinlg

43

F i=

and

.cosl4

3B i=

Remark 4The form of the expressions (5) and (6) show that the cart-pole system is inherently nonlinear. Although the two equations are in different forms, both are equivalent in the sense that they both describe the system dynam-ics. However, the models do not behave the same when a control input u is applied. While the pole dynamics in both (5) and (6) are nonlinear, the cart dynamics in (5) are nonlinear and are linear in (6). Thus, the pole component may be regarded as integrating the nonlinearity of the cart-pole system and is a window for investigation of the system nonlinearity.

Measuring Nonlinearity of Cart-Pole SystemSince the pole position is an integral of the system nonlin-earity, the nonlinear pole model in (6) is linearized to mea-sure the extent of nonlinearity of the cart-pole system. The operating interval of the pole is ( , ] .rad radr r- To linear-ize the pole model, we divide the interval into 360 equal parts every /180r rad in the interval to obtain 360 operat-ing points. For any operating point 0i the linearized pole model at that point is derived from (6) by

.

xx

xx

u

F F

F FB

x x x x

x x x xx x

0 00

0 00

0 00

0 00

0 00

0 0

0 00

22

22

22

22

i

ii

ii

= + +

+ +

i i i i i i

i i i i i ii i i

= =

= =

= =

= =

= =

= =

= =

= =

= =

= =

po

o

oo

oo

oo

oo

oo

oo

(7)

The transfer function of the linearized model at the operat-ing point 0i is

u

o

( )( )

,ls cos

cosss

g432

0

0

i

i=

- (8)

where s is a complex variable in Laplace domain and o and u are Laplace transforms of i and u, respectively. For all the operating points in ,( rad, radr r- @ there are 360 lin-earized pole models in the form of a transfer function from (8). Since the desired position of the pole is inverted at the top position, the linearized pole model at 00i = in transfer function form is adopted as a benchmark for all the linear-ized models.

For . ml 0 25= in (5), Figure 2 shows the gap metric between each linearized model and the benchmark. As seen in Figure 2, the gap metric go changes dramatically near / rad,20 !i r= which is the horizontal position of the pole, which indicates the real system dynamics below the horizontal line are completely different from the linear-ized model at rad.00i = moreover, Figure 2 shows that

go is very small for all [ / ,3 rad0 li r- / ],rad3r with .vmax 0 05g = at / .3 rad0 !i r= This analysis indicates

the real system model in [ / ,3 radr- / ]3 radr is almost the same as the linearized model at 00i = rad. This analysis explains why the designed controller for the linearized model at 00i = rad is able to solve the control problem for all [ / , / ],rad rad3 30

0 li r r- where 00i is an initial angu-

lar deviation of the pole—the plants within this range are quantifiably close to the nominal plant in terms of their closed-loop behavior.

As far as the two equilibria of the pole are concerned, it is straightforward from Figure 2 to see that v 1g = at

0i r= rad, which means that the linearized models at the two equilibria are completely different. Thus, linear system design methods based on the linearized model at 00i = cannot be employed to stabilize all initial angular devia-tions for all ( rad, rad]0

0 li r r- . The matlab commands to create Figure 2 are available for download [20].

Remark 5When designing a controller based on a linearized model to stabilize the cart-pole system, one of the standard assump-tions is that the operating region of the system is subject to 1%i rad, as mentioned in [1]. The constraint indicates that the operating region for the linearized model-based controller should be around the inverted upward posi-tion of the pole. The initial purpose of this constraint is to

Gap

Met

ric (v g

) 10.80.60.40.2

0

0

i (rad)

-r r3r4

-3r4

r2

-r2

r4

-r4

Figure 2 Gap metric between the linearized models and the benchmark.

58 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2013

ensure the open-loop accuracy of the linearized model, but emphasizing this constraint makes it easy to mislead stu-dents into thinking the linearized model-based controller can only work in a small neighborhood of the operating point because the controller is designed based on the lin-earized model at .00i = With the introduction of the gap metric, visualization as in Figure 2 can be used to reduce the chances that students gain this misconception.

STUDENT ASSESSMENTS In the School of Control and Computer Engineering at NCEPU, about 180 students per year take ACT. The stan-dard control theory textbooks [1]-[3] are recommended as reference books. Before the lesson on the gap metric, most students have the same misunderstanding about the neighborhood of linearization, that is, they assume that the neighborhood of linearization is just a small region about an operating point and are puzzled by the fact that a linear-ized model-based controller is able to stabilize the cart-pole system in a large range around the operating point.

When we adopted the approach in [5] for the first time in the fall 2010 semester, the gap-metric-based nonlinear-ity measure was also introduced into the ACT course for undergraduates. To verify the effectiveness of the adopted method, students taking the class were divided into two groups. For the first group, the neighborhood concept of linearization was taught according to the gap metric-based nonlinearity measure. For the second group, the neighbor-hood concept of linearization was taught according to the standard control theory textbooks.

A survey was taken after all the course material had been covered. The survey questions are shown in Table 1. The five questions covered five areas of instruction: Q1 for the complexity of the course content, Q2 for understanding and teaching effectiveness, Q3 for higher level conceptual thinking, Q4 for student interest, and Q5 for student confi-dence in applying the methods to other problems.

The first group was made up of 86 students in the fall 2010 semester, and all of them took part in the survey. Stu-dents ranked the statements in Table 1 with strongly dis-agree (SD) = 1, disagree (D) = 2, neutral (n) = 3, agree (A) = 4,

and strongly agree (SA) = 5. Table 2 shows the students’ responses to the survey questions in detail. The mean val-ues indicate that the students’ responses were not very posi-tive. Without question, the undergraduate students did not have the mathematical background to fully understand and appreciate the gap-metric-based nonlinearity measure. So we had to reflect on the teaching process. At the time, we had introduced too many theoretical derivations about the gap metric and nonlinearity measures. Our initial purpose was to give students a lot of background knowledge, but the additional instruction actually made it more difficult for the students to comprehend the material and most likely con-fused them more.

Within the first group, more than 80% of the students had trouble understanding the content on the gap metric and its use as a nonlinearity measure. Comparisons with a second group of students indicated that the approach of teaching the gap metric did not seem to be very effective or attractive to the students. To our surprise, one student in the first group, who changed his major from applied math-ematics to control, became very interested in our method, which suggested that students needed a solid mathematics background to appreciate the complicated mathematical derivations of the gap metric. This survey was contrary to our expectation. We learned that too many mathematical derivations actually confused the students and made stu-dents feel the content was too complex to be understood.

The next fall semester, we adopted the combination of the approach of [5] and our approach. Students were again divided into two groups, and there were still 86 students in the first group. With the lesson from the previous fall semester, we voluntarily avoided all the complicated deri-vations about gap metric and nonlinearity measures. Only the basic ideas were introduced to help students from being confused by complicated mathematical derivations. With-out the detailed mathematical foundation, calculating this gap-metric-based nonlinearity measure was not easy. We resorted to computer software. matlab, which is widely used for control education [18], includes a gapmetric com-mand to compute the value of the gap metric between two systems in transfer function form. With the help of this tool, it was possible to directly obtain the desired result rather than focusing on how to get the result.

Table 1 Survey for teaching evaluation.

Q1 I feel the extended technical contents are complex.

Q2 I can understand the contents by teacher’s explanation and my review.

Q3 The contents can help me think at a higher level and understand difficult concepts.

Q4 I enjoy the contents more than the textbooks.

Q5 The contents increase my confidence in my solutions to other contents.

Table 2 Students’ responses of the first group in 2010; STD = standard deviation.

Sa a N D SD Mean STD

Q1 64 15 7 0 0 4.66 0.625

Q2 1 7 37 37 4 2.58 0.758

Q3 2 14 67 3 0 3.17 0.513

Q4 1 13 65 6 1 3.08 0.557

Q5 1 8 27 46 4 2.49 0.774

AUGUST 2013 « IEEE CONTROL SYSTEMS MAGAZINE 59

In fall 2011, the gap metric was briefly introduced by contrasting it with the concept of a norm in linear algebra. Drawing on this background knowledge, we chiefly dem-onstrated how to use the gapmetric command for plotting Figure 2 and explained the meaning of the curve in the fig-ure. The same survey was completed by the students after all course material was covered in 2011. Table 3 shows the students’ responses to the 2011 survey. The mean values indicate that the students had a highly positive attitude toward this new teaching approach. The conclusion that can be drawn is that visual depiction of the gap metric shown in Figure 2 with a very brief theoretical introduction made the neighborhood concept easy to correctly understand.

Within the first group, more than 80% of the students could understand the presented pedagogical approach. Based on the mean response to Q5, the approach extended their knowledge and helped them correctly comprehend related material in the ACT course. On the other hand, students in the second group still had only a hazy notion of neighborhood and did not completely understand the subject.

Figure 3 compares the mean values for all the five ques-tions in the 2010 and 2011 surveys. Unlike in 2010, the stu-dents in 2011 did not feel that the material (gap metric and nonlinear measures) was too complex. most students in 2011 positively evaluated the method and considered the material good for their future. In the light of our presented case-driven pedagogical approach, students more eas-ily learned that whether or not a linearized model-based controller can realize its control objective for the cart-pole system is decided by the extent of closed-loop similarity between the real system and the linearized model, rather than by the size of a neighborhood. The combination of the approach in [5] and ours was effective for students to com-prehend the technical concepts on linearization.

Figure 4 compares the standard deviations for all five questions in the 2010 and 2011 surveys. The standard devia-tions are very similar for Q1, Q2, and Q4 for the two sur-veys. The standard deviations for Q3 and Q5 are obviously different. Q3 is an assessment by the students of whether the course helped the students in higher level thinking and understanding of difficult concepts. In 2010, most students

did not understand the gap metric and nonlinearity mea-sure (see Q2 in Figure 3), so most of the students felt no ben-efits were obtained in their intellectual development (see Q3 in Figure 3). In 2011, most students could understand the method (see Q2 in Figure 3), but students at different intellectual levels could have different degrees of benefits, leading to a larger standard deviation for Q3 in Figure 4. Q5 is concerned with increasing student confidence in applying the methods to other problems. The mean score for Q5 in 2011 was so high, as seen in Q5 in Figure 3, that the standard deviation in the scores was much lower than in 2010, as seen in Q5 in Figure 4.

CONCLUSION The concept of linearization is a standard part of under-graduate control education but is often not correctly understood by students. Students assume that the neigh-borhood in which a linearization is useful for controller design is small. To help students correctly understand and quantify the neighborhood for which a linearized model is useful for controller design, the gap-metric-based non-linearity measure was introduced to undergraduates’ control education. Since a thorough description of the underlying mathematics of the gap metric is beyond the ability of most undergraduates, a case-driven pedagogical approach was developed. Its effectiveness was evaluated based on two consecutive years of experience and sur-veys in the School of Control and Computer Engineering at NCEPU.

Q1 Q2 Q3 Q4 Q50

1

2

3

4

5

Questions

Mea

n V

alue

Mean in 2011 Mean in 2010

Figure 3 Mean values for all questions in 2010 and 2011.

Q1 Q2 Q3 Q4 Q50

0.2

0.4

0.6

0.8

1

Questions

Sta

ndar

d D

evia

tion STD in 2010 STD in 2011

Figure 4 Standard deviations for all questions in 2010 and 2011.

Table 3 Students’ responses of the first group in 2011; STD = standard deviation.

Sa a N D SD Mean STD

Q1 4 12 64 5 1 3.15 0.642

Q2 16 57 8 5 0 3.97 0.719

Q3 12 55 12 6 1 3.81 0.744

Q4 14 61 9 2 0 4.01 0.603

Q5 11 66 9 0 0 4.02 0.485

60 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2013

The effectiveness of the approach was apparent and acceptable for undergraduates in an introductory control course. With the help of a visual software demonstration and relevant mathematical introduction, the approach was carried out by measuring the inherent nonlinearity of a cart-pole system. The application of the approach shows students are not required to know all the complex back-ground knowledge and are able to correctly comprehend the significance and quantification of the neighborhood of linearization. The presented case-driven pedagogical approach is effective for inspiring students’ interests in automation science and technology. The main contribu-tions are 1) an important concept in nonlinear control theory is introduced into undergraduate control education and 2) an effective case-driven pedagogical approach was presented.

ACKNOwLEDGMENTThis work was supported by the NSFC Projects under grants 60904008 and 61273149, the Natural Science Founda-tion of Hebei Province under grant F2012502023, the Inno-vation method Fund of China under grant 2012Im010200, and the Fundamental Research Funds for the Central Uni-versities under grant 12TD02.

The authors would like to express their gratitude to the anonymous reviewers for their constructive comments.

AUThOR INfORMATIONDianwei Qian received the B.E. degree from the Hohai University, Nanjing, China, in 2003. He received the m.E. degree from Northeastern University, Shenyang, China, and the Ph.D. degree from the Institute of Automation, Chinese Academy of Sciences, Beijing, China, in 2005 and 2008, respectively. Currently, he is an associate professor with the School of Control and Computer Engineering, North China Electric Power University, Beijing, China. His research interests are in the theory and application of intel-ligent and nonlinear control.

Jianqiang Yi received the B.E. degree from the Beijing Institute of Technology, in 1985 and the m.E. and Ph.D. degrees from the Kyushu Institute of Technology, Kitaky-ushu, Japan, in 1989 and 1992, respectively. He worked as a Research Fellow with the Computer Software Develop-ment Company, Tokyo, Japan, from 1992 to 1994. From 1994 to 2001, he was a chief researcher and chief engineer with mYCOm, Inc., Kyoto, Japan. Currently, he is a full profes-sor with the Institute of Automation, Chinese Academy of Sciences, Beijing. His research interests include the theory and application of intelligent control, intelligent robotics, underactuated system control, sliding-mode control, and flight control.

Shiwen Tong received the B.E. degree in chemical engi-neering from the University of Petroleum (East China), Shandong, China, in 1999, the m.E. degree in control theory and control engineering from the University of

Petroleum (Beijing), Beijng, in 2003, and the Ph.D. degree from the Institute of Automation, Chinese Academy of Sciences, Beijing, in 2008. He was an operator with Liaohe Oil Feild Petrochemical Refinery from 1999 to 2002, an engineer with Bejing Anwenyou Science and Technology Company, Ltd. from 2003 to 2005, and an instrumentation senior engineer with China Tianchen Engineering Cor-poration from 2008 to 2012. He is currently on staff with the College of Automation at Beijing Union University. His research interests include intelligent control, networked control, proton exchange membrane fuel cells, and their industrial applications.

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