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Robust Output Feedback Control of a Class of Nonlinear Systems Using a Disturbance Observer Zi-Jiang Yang, Seiichiro Hara, Shunshoku Kanae, Member, IEEE, and Kiyoshi Wada, Member, IEEE

Abstract—This paper considers the output-tracking control problem of feedback linearizable nonlinear systems in the presence of external disturbances and modeling errors. A robust output feedback nonlinear controller is designed to achieve excellent output-tracking performance. By exploiting the cascade features of backstepping design, a simple disturbance observer (DOB) is proposed to suppress the effects of the uncertainties, and a high-gain observer (HGOB) is applied to estimate the unmeasureable states of the system. Although the DOB-based controllers are usually designed according to linear control theory, in this study, strict analysis of the nonlinear control system is given. Experimental results on a magnetic levitation system are provided to support the theoretical results and to verify the advantages of the proposed controller. Index Terms—Backstepping design, disturbance observer (DOB), feedback linearizable nonlinear system, high-gain observer (HGOB), input-to-state stability (ISS), nonlinear damping term.

I. INTRODUCTION

T

HE development of feedback linearization techniques provides a powerful tool for nonlinear system control. Under some geometric conditions, the state variable model of a class of single-input single-output (SISO) nonlinear systems can be rendered to be the normal form as described in (2) [1]–[3]. To handle the uncertainties of the nonlinear functions, adaptive control schemes based on function approximation techniques have been being studied extensively. While having been extensively studied in both theories and applications, the adaptive control approach using neural or fuzzy networks may require heavy computational burden and may exhibit unsatisfactory transient performance. The system model (2) can be obtained by performing coordinate transformation on the SISO nonlinear system (1) Manuscript received January 04, 2009; revised June 25, 2009; accepted April 06, 2010. Manuscript received in final form May 03, 2010. Date of publication May 24, 2010; date of current version February 23, 2011. Recommended by Associate Editor C.-Y. Su. Z.-J. Yang is with the Department of Intelligent Systems Engineering, College of Engineering, Ibaraki University, Ibaraki 316-8511, Japan. S. Hara and K. Wada are with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka 812-8581, Japan. S. Kanae is with the Department of Electrical, Electronics, and Computer Engineering, Fukui University of Technology, Fukui 910-8505, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2049998

with relative degree , under some geometric conditions [3]. , and , and are some apHere, propriate nonlinear functions of the state variables and external disturbances which may include some uncertainties. There also have been some works on more general systems with a lower and a stable zero dynamics [4]–[6]. Since relative degree the application example in this work is of full relative degree and since the basic idea can be straightforwardly extended to these systems, in this work, we confine our study to the system model (2). The model (2) may also arise in models of mechanical or electromechanical systems where the position is measured while their derivatives (velocity, acceleration) are not measured. In many practical situations, even when the state variables of the original system (1) are exactly measurable, the transformed state variables may not be exactly calculated due to modeling uncertainties. Therefore, there has been strong practical necessities to construct a high performance output feedback controller using only the output measurement. The HGOB approach has been considered as a powerful tool to tackle the problems of output feedback control [4]–[15]. The design procedure of a HGOB-based output feedback controller is summarized as follows. 1) transform the system model into the normal form if possible; 2) design a state feedback controller as if the state variables are known; 3) design a HGOB to estimate the states so that the controller is implementable using only the output measurement; 4) saturate the controller outside the domain of interest so that the controller is globally bounded; 5) prove that the output feedback controller asymptotically recovers the performance achieved by the state feedback controller based on analysis of the singularly perturbed system. For the basic theory, the readers are invited to see [9]–[11]. Typical control techniques combined with the HGOB are adaptive control [4], [5], [12], sliding mode control [13], [14], and robust control by nonlinear damping terms [15]. However, the sliding mode control approach may lead to large control efforts and chattering control response, whereas the adaptive control approach may require much more computational burden and may exhibit unsatisfactory transient performance. Therefore, attention should be paid to overcome these drawbacks. As an alternative popular approach for compensating external disturbances and model mismatch, a DOB is often included into a controller [16]–[20]. The DOB-based controllers have been widely accepted in the industrial side, due to their simplicity and transparency of design, and excellent disturbance compen-

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YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

sation ability. In [20], a state feedback nonlinear robust controller with DOB for a voltage-controlled magnetic levitation system has been proposed where the stability of the nonlinear control system is analyzed rigorously. However, the DOB may be vulnerable to the measurement noise when the velocity is calculated by pseudodifferentiation. Moreover, due to the modeling errors, the model is not exactly transformed to the normal form (2) so that the resultant controller is complicated and the control results are not so good as those by the adaptive controllers [21]. In this paper, instead of the popular adaptive control approach, a robust output feedback controller using a DOB is proposed for the output-tracking control problem of the linearizable nonlinear system (2). By exploiting the cascade features of backstepping design, our idea is to introduce the th subsystem rather than to estimate or DOB to the approximate the uncertainties in the th subsystem in the model (2), so that the controller design related to the DOB is much simpler. The essence of the idea is that, by virtue of backstepping design, the control error signal of the th subsystem is treated as a disturbance term affecting the th subsystem. Although in most works of the DOB-based controllers the control design is based on the linear control theory, in this study, through cascade analysis of backstepping design, it is shown that the stability of the overall nonlinear control system is estrack the tablished and the DOB helps to make the output reference trajectory with small tracking error. Then by using a HGOB that estimates the state variables, the designed controller is implementable as an output feedback controller. It is shown that the output feedback controller asymptotically recovers the performance achieved by the state feedback controller based on analysis of the singularly perturbed system. Experimental results are given to support the theoretical results.

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and are bounded away from zero Assumption 1: , where is the domain with the same known sign, for of interest. Assumption 2: There exist finite positive, but not necessarily , , , and known continknown constants and such that the following uous functions : inequalities hold for (4) (5) and are appropriate known functions Here, (bounding functions) which will be used for construction of nonlinear damping terms [22]. The inequalities in (5) mean that and the error function do the nominal function itself. not grow in a higher order than III. ROBUST NONLINEAR STATE FEEDBACK CONTROLLER Due to its flexibility for systematic desirable modifications of the controller such as compensation for modeling errors or external disturbances, the backstepping control design techniques have received great attention in the last decade [22]. We here design the proposed controller in a backstepping manner. To this end, we first define the output-tracking error signal as (6) Then we will show the concrete design procedure. A. Steps Define the error signal

II. PROBLEM STATEMENT

(7)

Consider the following SISO feedback linearizable nonlinear system in normal form:

where is the virtual input to stabilize Then from (2) we have subsystem

.

(8) .. .

The virtual input

is designed as (9)

(2) is the vector of state variables; is the where and are modelable nonlinear functions control input; denotes unmodeled with known nominal functions; and nonlinearities and disturbances. Denoting the nominal nonlinearities based on the prior and , we have knowledge as (3) and denote the modelling errors. where We impose the following standing assumptions.

where

.

B. Step Define the error signal (10) where is the virtual input to stabilize . Then we have the dynamical equation of subsystem

as (11)

It is well known that can be made sufficiently small at the next step of design by using the adaptive control technique

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which directly estimate the nonlinearities. However, the adaptive control approach using neural or fuzzy networks may require heavy computational burden and may exhibit unsatisfactory transient performance. Our policy is to design a moderate is bounded and robust controller at step of design so that not too large. Then we introduce a disturbance observer to the to compensate the effects of on the subsubsystem system . That is, is considered as a disturbance term to . the subsystem , we have Viewing as a disturbance to the subsystem

where

, and

(17) To stabilize the subsystem controller:

, we design the following

(18)

(12) cannot be used exHowever, usually, the next error signal plicitly at the current step of backstepping design [22], since we at the next step design if includes have to calculate explicitly. Therefore, instead of itself, we can use its so that is computable. To this end, low-passed estimate to obtain the estiwe pass (12) through a low-pass filter as mate of (13) This idea is very similar to the so called DOB studied extensively in the literature [16]–[19]. In this paper, we adopt a simple low-pass filter as the following: (14) is a time-constant. Since where used at the next step of design, we should have is sufficient. found empirically We design the virtual controller as

where

(19) ; is a feedback proportional where is a nonlinear controller with model compensation; ; is a nonlinear damping damping term to suppress ; is a nonlinear damping term term to suppress to suppress . The nonlinear damping terms are designed such that their time-varying gains grows at the same order as their corresponding uncertain terms. Further analysis will be given in Section IV.

will be . We have

IV. STABILITY ANALYSIS OF THE STATE FEEDBACK CONTROL SYSTEM A. Steps The subsystems expressed as

controlled by the virtual input

can be

(15)

(20)

; is a small positive number, ex., ; is a nonlinear damping term that enhance the is relatively large. Usually, we can exdamping effect when within the pass-band of . The estimation error pect beyond the pass-band of can be suppressed sufficiently by and . relatively large can be ob1) Step : The dynamics of the subsystem tained as

is made uniformly ultimately bounded at the Assuming next step, we can derive

where

(21) and (22) Finally, we have

(16)

(23)

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

The derivation of (23) is analogous to the proof of Lemma 6.20 in [22]. Furthermore, we have

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and hence

(24) Integrating both sides of (24) and taking mean value of time, we have (31)

(25)

Equation (23) characterizes the input-to-state stability (ISS) property [22] of , and (25) characterizes the mean-square performance. B. Step Applying the virtual input have

to the subsystem

C. Step Applying the control input have

to the subsystem

(16), we

, we

(26) is made uniformly ultimately bounded at the Assuming next step, we can derive

(32) Similar to the previous analysis, we can derive (33)

(27) where

where (28)

When in the numerator of grows, the nonlinear damping term in the denominator grows accordingly such that can be suppressed sufficiently, and the effects of hence we have (29) Notice that owing to the DOB, is likely small than in most cases especially within the pass-band of the DOB. It is helps to improve the error signal . clear that , we furthermore have Provided the ISS property of

(34) It can be verified that each uncertain term in the numerator of is suppressed by the corresponding nonlinear damping term, i.e., when a term in the numerator grows, the corresponding term in the denominator grows at the same order such is uniformly bounded. Then similar to the previous that step, we have the following ISS property: (35) Provided the ISS property of

(30)

, we furthermore have

(36)

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and hence

(37)

D. Overall Error System

, Define a compact set where is relatively small (not necessarily very small as will be explained in Remark 1) by an appropriate set of reference trajectory, initial states and design parameters, according to the in, and if the control gains are designed equality (39). If sufficiently large, then we can find an appropriate set of initial , so that we can make , conditions where is defined according to (39) as the following:

The aforementioned ISS properties of the three subsystems can be summarized as follows:

.. .

(38) These results imply that the overall error system can be expressed as the cascade of ISS systems. We therefore can conclude based on Lemma C.4 in [22] that the overall error system is also ISS. . Define the error signal vector as Then by repeated use of [22, Lemma C.4], we have the following results. The derivation process is quite lengthy and is therefore omitted. Interested readers may contact the authors for detailed proof. (39)

(41) As long as the error vector is relatively small, the system output may track a smooth reference trajectory with acceptable accuracy and the state variables do not escape from the domain of interest . Thus, we impose the following assumption. Assumption 4: If is made such that , then . Remark 1: The inequality (39) characterizes the ISS property of the overall error system, which ensures the boundedness of is sufficiently the error signals. According to (28), small within the pass-band of the DOB, owing to the DOB. of the final subsystem may not On the other hand, itself may not be so small. be so small. Therefore, It is only required that . However, our final purpose sufficiently small. is to make the output-tracking error and hence According to (38), we can conclude that can be made sufficiently small within the pass-band of the DOB. We can also explain the effects of the DOB by considering the mean-squares of the error signals. The mean-squares of the error signals of the subsystems derived previously are summarized as follows:

where

.. . (40a) (40b) (40c) (40d) and is the impulse response of . However, due to the natural constraint of the system under study in the case of a real application, we should verify if the system states are constrained in an appropriate domain of interest , where the results obtained here are valid. To ensure the controller feasible, we should verify if there is a compact set such that we can make by the designed controller. To this end, we impose the following assumption. function of Assumption 3: The reference trajectory is a time (continuously differentiable with respect to time up to th order) and , where and is an appropriate compact subset of .

(42)

By applying the Parseval’s theorem, we have

(43) is the frequency response of the low-pass filter where of the DOB, and is the Fourier transformation of over the time interval . Notice that for

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

the frequencies below the cutoff frequency of . Therefore, is sufficiently broad, the mean-square if the pass-band of , and hence those of can be made suffiof ciently small. The results of the state feedback controller are summarized in the following theorem. Theorem 1: Let Assumptions 1 4 hold. And let the initial conditions . If the state feedback robust nonlinear controller is applied to the nonlinear system (2) under study, the following results hold. such that 1) There exists a compact set , where is the domain of interest. 2) The overall error system is ISS such that

261

where .. . (47)

.. .. .

..

..

.

..

.

.

..

.

.

..

.

.. . (48)

. Then (2) Define the estimation error of HGOB as and (44) lead to the dynamics of the estimation error of HGOB with . 3) The mean-squares of the error signals satisfy (42). However, our problem setting is that only the output signal, i.e., is available. We have to estimate the system states by a HGOB so that an output feedback controller is implementable.

(49) where ..

.

.

..

.

.

..

.

V. CONSTRUCTION OF THE OUTPUT FEEDBACK CONTROLLER A. Construction of the HGOB To implement the state feedback controller by output feedback, we use the following HGOB [4]–[15]:

.. . .. .

.. .. .

..

.. .

.. .

(50)

(51) Scale the observer estimation error

.. . (44) where and

is a small positive constant specified by the designer are selected such that the roots of

(52) . Then (46) and (49) lead to the and define closed-loop system in the standard singularly perturbed form (53a) (53b)

(45) are in the open left-half -plane. In this paper, is obtained. chosen such that a polynomial

are

B. Saturation of the Controller of the HGOB may inSince the output troduce incorrect peaking signals especially in the first transient phase, we have to saturate the signals outside of the domain of interest according to the physical limitation of the system under study [4]–[15]. We will denote the saturated system estimates , and the saturated control input as . as

.. . .. .

.. .. .

..

..

.

.

..

.

.

..

.

.. . (54)

The -scaling (52) causes an impulsive-like behavior in [4], [5], [8]–[15], but since enters the slow system (53a) through , the slow system signal vector does the saturated input not exhibit a similar impulse-like-behavior.

C. Singularly Perturbed Form Equations (20), (26), and (16) lead to the closed-loop system of the subsystems (46)

VI. STABILITY ANALYSIS OF THE OUTPUT FEEDBACK CONTROL SYSTEM The first part of our analysis is to establish that there is a short transient period during which the fast signal decays to

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level, while, due to the difference in speed, the slow signal still remains within the domain of interest. Similar to the discussions in Section IV, we can find an ap. Since propriate set of initial conditions is a low-passed version of , is . Then we have for some bounded as long as as long as . Let be the solution of , where , and define . Taking the time derivative of along the trajectory of (53a), we have

(55) where

, , and hence (56)

Since

, we have

(57) If the initial conditions are chosen such that for some according to the definition of given in the , independent previous section, then there exists a time of , such that over the time interval . We next investigate the behavior of the fast system (53b) over . According to (6), (7), (10) and (12), the time interval can be expressed by and since included in over the time interval , we have for some . be the solution of , where Let , and define . Taking the time derivative of along the trajectory of (53b), we have

tends to zero as , there is a small such that for all . Hence we deduce that there such that for all , we exists a time have , where is the first time that escapes from can be made to be the set , and as will be claimed later, . governed We now go back to studying the behavior of by the slow system (53a) over the time interval . Notice . Since that over this time interval, and hence are of the signals remain in the domain of interest, saturation of the controller does not occur. Since the observer’s output is only injected to the th subsystem of the slow system (53b), it is sufficient to investigate the behavior of the th subsystem owing to the cascade nature of the backstepping design. To proceed the analysis, we impose here one more assumption. Assumption 5: The nominal nonlinear functions and , and the bounding functions and employed in the controller are locally Lipschitz on the domain of interest . Considering the locally Lipschitz property of the dependent , we have functions in the control input

Since

(61) for some . Substituting system (16), we can rewrite (32) as

into the sub-

(62) and (33) is rewritten as (63) where, owing to (61),

in (34) is rewritten as (64)

for some

. Hence (35) is rewritten as

(58)

(65)

, . is bounded From (52), we deduce that by for some . It follows that as long as over the time interval , we have

For any , (65) characterizes the input-to-state practical stability property (ISpS which is an extension of the concept of ISS, see [24] for definition). Notice that the discussions and thus the starting time of the ISpS are for property characterized by (65) is . The behavior of for is characterized in (57). Finally, (39) is rewritten as

for where

(59) Thus the time

when

can be calculated as (60)

(66) for some

.

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

Similar to defined in (41), according to (57) and (65) we can find an appropriate set of initial conditions , so that we can make provided a sufficiently small . Here, is defined as the following:

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3) The mean-squares of the error signals satisfy

.. .

(67)

where, according to (57), we have

(68) for Therefore, we can make the escape time , we can rewrite (36) as Provided

.

for

.

VII. GUIDELINES FOR CONTROLLER PARAMETER DESIGN

(69)

. And hence (37) can be rewritten as

for some

(70)

Finally, the performance of the output feedback controller is summarized as the following theorem. Theorem 2: Let Assumptions 1 5 hold. And let the initial . If the output feedback robust conditions nonlinear controller is applied to the nonlinear system (2) under such that for all , the study, there exists following results hold. such that 1) There exists a compact set , where is the domain of interest. 2) The overall error system is ISpS such that

The guidelines for design of the controller parameters are drawn here. Since the nonlinear damping terms are nonlinear functions of some internal signals, it is recommendable to to avoid noisy or large choose modest control efforts. In contrast, the parameters of the nominal which also contribute to achieve fast controller transient phase and small error signals can be chosen relatively large, without causing large amplitude of the control input. that is closer A small time-constant of DOB leads to a noisy, to . However, too small a may make especially in the presence of measurement noise. Empirically, since the other parts such as the nonlinear damping terms also contribute to suppress the effects of uncertainties, the control performance is not so sensitive with respective to the value of . Usually, is chosen to be very small. Too small a value of may require a very small sampling interval, and also may cause noisy output of the HGOB. Empirically, for a fast electromechanical system such as the magnetic levitation system which . This is will be studied in this paper, is of the order of comparable to the time-constant of the low-pass filter of pseudodifferentiation for the same system in our previous works [20], [21], [23]. Usually, the performance of a high-gain observer-based controller (not limited to our present controller) is relatively sensitive to the value of . However, in practice, we have found that the selection of the time-constant of the lowpass filter of pseudodifferentiation is even much more sensitive. VIII. APPLICATION TO A MAGNETIC LEVITATION SYSTEM

with

, and .

tends to zero as

To demonstrate the advantages of the proposed robust nonlinear output feedback controller with DOB over our previously proposed state feedback adaptive or robust controllers [20], [21], [23], extensive experiments have been studied on a magnetic levitation system which is strongly nonlinear, unstable

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To convert the original nonlinear system model into the normal form, we adopt the following nonlinear coordinate transformation (74) Then through straightforward calculations, the system model (71) is transformed into

(75) where (76a) (76b) Fig. 1. Diagram of the magnetic levitation system.

(76c)

and subjected to external disturbance and errors of physical parameters. In this section, we will show how the experimental results reflect the theoretical analysis. A. Model of the System

(76d)

Consider a magnetic levitation system shown in Fig. 1, in which an electromagnet exerts attractive force to levitate a steel ball. The system dynamics can be described in the following equations [20], [21], [23]:

(76e)

(71)

Notice that owing to the relation has been replaced as the following: (74),

defined in

(77)

(72) where is the state variable vector. And, : air gap (vertical position) of the steel ball; : coil current; : gravity acceleration; : mass of the steel ball; : electrical , and : positive resistance; : voltage control input; constants determined by the characteristics of the coil, magnetic : a bounded external disturbance with core and steel ball; unknown bound. is given In this study, the bounded external disturbance as (73) for experimental studies. It is artificially added to the command value of the voltage control signal .

The nominal functions and are obtained by reand by their nomplacing the physical parameters in , , , , and . inal values , Remark 2: The gravity acceleration is known in most cases. However, an external constant (or slowly varying) mechanical disturbance can be treated as the bias of equivalently. In this paper, we allow the nominal value of is biased from . Also, notice that according to (77), it is required that and . B. Experimental Setup The physical parameters of the setup shown in Fig. 1 are given in Table I. The levitated steel ball is controlled by a digital control system that consists of a PC loaded with Windows 2000 OS, 12 bits analog-to-digital (A/D) and digital-to-analog (D/A) converters, and a controllable voltage source. The control algorithm is coded in Borland C++ language and discretized with ms . The physically allowable a sampling interval of operating region of the steel ball shown in Fig. 1 is limited to

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should be saturated such that V V; denotes a saturated signal. where 3) Reference Trajectory Initialization: Theorems 1 or 2 imply that the initial conditions of the error signals can influence the transient performance significantly, and it is thus desirable to make them small. and Suppose the steel ball is initially at rest with , i.e., the steel ball is held on the steel plate shown in Fig. 1 before the feedback controller’s start. Then if we choose the initial conditions of the reference trajectory and , we have such that , according to (6), (7), and (10). •

TABLE I PARAMETERS OF THE MAGNETIC LEVITATION SYSTEM

m m . The vertical position of the steel ball is measured by a laser distance sensor. And the output of the V V. controllable voltage source is limited to To apply the proposed controller to the magnetic levitation system, we first define the domain of interest as

D. Controller Design (78) where mm is the maximally allowable position of the steel ball. The domain of interest is due to the natural constraint of the setup. That is, the steel ball can only move within a narrow vertical domain, and the electromagnet can only exert attractive pulling-up force so that the downward acceleration cannot be larger than the gravity acceleration itself. C. Application of the Proposed Controller 1) Verification of the Assumptions: To make the theoretical results feasible, we should first verify if the imposed assumptions are satisfied. It is trivial to see that Assumption 1 is satisfied. To satisfy Assumption 2, we choose the bounding functions employed in (19) as

The following nominal system parameters with considerable errors were used for experimental studies, to verify the robust performance of the proposed controller: kg m

m/s

H

Hm (80)

Along the guidelines of controller design drawn in Section VIII-C, the proposed controller was designed as follows. (1) Robust nonlinear feedback controller without DOB

(81) Notice that when the DOB is not used, the nonlinear damping term corresponding to the DOB is not neces. sary and hence we can set (2) Robust nonlinear feedback controller with DOB

(79) and then we can verify that Assumption 2 is satisfied. Let the position command be a rectangular wave that changes between 4 and 12 mm, and pass it through a low-pass s . Then we obtain a smooth reference filter trajectory that satisfies Assumption 3. As long as the error vector is controlled to be relatively small, the steel ball may track a smooth reference trajectory with acceptable accuracy and does not escape from the domain of interest . Therefore, Assumption 4 is satisfied in generic cases. Inspections of the nonlinear functions in (76), (77) and (79) imply that Assumption 5 is satisfied. 2) Saturation of the Signals: Since the output of the HGOB may introduce incorrect peaking signals especially in the first transient phase, we have to saturate the signals outside of the domain of interest according to the physical limitation of the setup [4]–[15]. According to the discussions given so far, we saturate the signals as follows: • •

should be saturated such that should be saturated such that

is well defined; ;

(82) Notice that for this case the experiments were performed for three different values of . E. Comments on the Experimental Results The experimental results are shown in Figs. 2–5, where from the top to the bottom are the measured position , position error , velocity error , acceleration error , estimated position , estimated velocity , estimated acceleration , and the control input . Notice that and are calculated based on the estimated states , not on the true . It can be found in Fig. 2 that in the absence of the DOB, the boundedness is still ensured by the nonlinear damping terms, in spite of the considerably large errors of the physical parameters. However, the position-tracking performance is not acceptable, and we can even see the oscillations due to the sinusoidal disturbance.

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Fig. 2. Experimental results of the robust nonlinear controller without DOB ( = 0:0020).

Fig. 4. Experimental results of the robust nonlinear controller with DOB ( =

0:0020).

Fig. 5. Experimental results of the robust nonlinear controller with DOB ( =

Fig. 3. Experimental results of the robust nonlinear controller with DOB ( = 0:0015).

The results shown in Figs. 3–5 indicate the high performance of the proposed controller. It can be verified that introducing a simple DOB at step 2 of design brings significant improvement.

0:0025).

We can find the HGOB works for some different values of . Also, as discussed in Remark 1, we need not pay great efforts to suppress , and thus it is allowable that the amplitude of is not so small. However, owing to the DOB employed at step 2,

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

we can see that is suppressed significantly and consequently is suppressed to be very small. Due to the limitation of space, we will not reproduce here the results by the state feedback controllers such as the robust control without DOB [23] or with DOB [20], and the adaptive control [21] for the same setup. However, we will give some comparative comments. It should be noticed here that only in the present work, a sinusoidal disturbance is imposed on the voltage input signal. In our previous works [20], [21], [23], however, we did not consider the external disturbance in the problem setting. Comparing the results of the present work with those of our previous works, we can directly see that the present controller yields better results (even in the presence of disturbance) than our previous works (in the absence of disturbance). In [20], [21], and [23], the model transformation (74) was calculated based on the nominal model provided the measurements of state variables . This resulted in an extra, unmatched untertain term in the second subsystem of (75). That is, the system could not be completely transformed into the normal form. And hence by the robust controller without DOB in [23], the uncertainties were not suppressed sufficiently. By the robust controller with DOB in [20], the control performance was improved. However, since the velocity was calculated by pseudodifferentiation, we found that the signals were very sensitive to the noise when the pass-band of the DOB was broad. Therefore, the performance improvement was limited. The adaptive controllers in [21] delivered better position-tracking performance than the robust controllers [20], [23], owing to model learning by neural networks. However, the adaptive controllers were much more complicated than the robust controllers. Moreover, the adaptive controllers could not handle effectively an unparameterized external disturbance. In contrast, we have found the present controller is simple in form, and yields better position-tracking performance compared to our previous works, while only requiring the position measurement.

IX. CONCLUSION In this paper, we have proposed a robust output feedback controller for feedback linearizable nonlinear systems in the presence of uncertainties. Instead of the popular adaptive control approach, a robust output feedback controller using a DOB was proposed for the nonlinear system under study. By exploiting the cascade features of backstepping design, a simple DOB was employed to suppress the effects of the uncertainties, and an HGOB was applied to estimate the unmeasureable states of the system. Strict analysis was performed based on the singularly perturbed system, and the ISpS property of the error system was established. The theoretical results were verified through extensive experiments on a magnetic levitation system, and it was confirmed that the present method works better than the previous methods studied by the authors on the same experimental setup. Although the proposed controller works quite well for the magnetic levitation system with continuous and smooth uncertainties, we should notice here that this does not imply that the proposed robust controller outperforms the adaptive control

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approach in all cases. For the systems with fast-changing or discontinuous uncertain nonlinearities that affect the system significantly beyond the pass-band of the DOB, the adaptive control approach may still be necessary. REFERENCES [1] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2002. [2] L. X. Wang, A Course in Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1997. [3] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer, 1995. [4] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 177–188, Feb. 1996. [5] S. Seshagiri and H. K. Khalil, “Output feedback control of nonlinear systems using RBF neural networks,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 69–79, Jan. 2000. [6] S. Tong, T. Wang, and J. T. Wang, “Fuzzy adaptive output tracking control of nonlinear systems,” Fuzzy Sets Syst., vol. 111, pp. 169–182, 2000. [7] M. I. El-Hawwary, A. L. Elshafei, H. M. Emara, and H. A. A. Fattah, “Output feedback control of a class of nonlinear systems using direct adaptive fuzzy controller,” IEE Proc. Control Theory Appl., vol. 151, pp. 615–625, 2004. [8] J. Alvarez-Ramidrez, “Adaptive control of feedback linearizable systems: A modelling error compensation approach,” Int. J. Robust Nonlinear Control, vol. 9, pp. 361–377, 1999. [9] F. Esfandiari and H. K. Khalil, “Output feedback stabilization of fully linearizable systems,” Int. J. Control, vol. 56, pp. 1007–1037, 1992. [10] H. K. Khalil, “High-gain observers in nonlinear feedback control,” in New Directions in Nonlinear Observer Design, H. Nijmeijer and T. I. Fossen, Eds. New York: Springer-Verlag, 1999. [11] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2002. [12] K. W. Lee and H. K. Khalil, “Adaptive output feedback control of robot manipulators using high-gain observers,” Int. J. Control, vol. 67, pp. 869–886, 1997. [13] S. Oh and H. K. Khalil, “Nonlinear output feedback tracking using high-gain observer and variable structure control,” Automatica, vol. 33, pp. 1845–1856, 1997. [14] E. G. Strangas, H. K. Khalil, B. Aloliwi, L. Laubinger, and J. Miller, “Robust tracking controllers for induction motors without rotor position sensor: Analysis and experimental results,” IEEE Trans. Energy Conv., vol. 14, no. 4, pp. 1448–1458, Dec. 1999. [15] R. B. Arreola, “Output feedback nonlinear control for a linear motor in suspension mode,” Automatica, vol. 40, no. 12, pp. 2153–2160, 2004. [16] X. Chen, C. Y. Su, and T. Fukuda, “A nonlinear disturbance observer for multivariable systems and its application to magnetic bearing systems,” IEEE Trans. Control Syst. Technol., vol. 12, no. 4, pp. 569–577, Jul. 2004. [17] Y. Fujimoto and A. Kawamura, “Robust servo-system based on twodegree-of-freedom control with sliding mode,” IEEE Trans. Ind. Electron., vol. 42, no. 3, pp. 272–280, May/Jun. 1995. [18] S. Komada, N. Machii, and T. Fukuda, “Control of redundant manipulators considering order of disturbance observer,” IEEE Trans. Ind. Electron., vol. 47, no. 2, pp. 413–420, Mar./Apr. 2000. [19] Z. J. Yang, H. Tsubakihara, S. Kanae, K. Wada, and C. Y. Su, “A novel robust nonlinear motion controller with disturbance observer,” IEEE Trans. Control Syst. Technol., vol. 16, no. 1, pp. 137–147, Jan. 2008. [20] Z. J. Yang, H. Tsubakihara, S. Kanae, K. Wada, and C. Y. Su, “Robust nonlinear control of a voltage-controlled magnetic levitation system with disturbance observer,” presented at the IEEE Multi-Conf. Syst. Control, Singapore, 2007. [21] Z. J. Yang, K. Miyazaki, S. Kanae, and K. Wada, “Adaptive robust dynamic surface control for a magnetic levitation system,” presented at the 42nd IEEE Conf. Decision Control, Hawaii, 2003. [22] M. L. Krstic, Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. Hoboken, NJ: Wiley, 1995. [23] Z. J. Yang, K. Miyazaki, S. Kanae, and K. Wada, “Robust position control of a magnetic levitation system via dynamic surface control technique,” IEEE Trans. Ind. Electron., vol. 51, no. 4, pp. 26–34, Jul./ Aug. 2004. [24] Z. P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,” Automatica, vol. 34, pp. 825–840, 1998.

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Zi-Jiang Yang was born in 1964. He received the Dr.Eng. degree from Kyushu University, Fukuoka, Japan, in 1992. From 1996 to 2000, he was an Associate Professor with Kyushu Institute of Technology, Japan. From 2000 to 2009, he was an Associate Professor with Kyushu University. Since 2009, he has been a Professor with the Department of Intelligent Systems Engineering, Faculty of Engineering, Ibaraki University, Ibaraki, Japan. His research interests include system identification and nonlinear system control.

Seiichiro Hara was born in 1984. He received the B.S. degree from the Faculty of Engineering, Kyushu University, Fukuoka, Japan, in 2007, where he is currently pursuing the M.S. degree from the Department of Electrical and Electronic System Engineering, Graduate School of Information Science and Electrical Engineering. He is engaged in research on nonlinear system control.

Shunshoku Kanae (M’02) was born in 1961. He received the Dr.Eng. degree from Kyushu University, Fukuoka, Japan, in 1995. Since 2009, he has been an Associate Professor with Fukui Institute of Technology, Fukui, Japan. His research interests include system identification, mechatronics system control, and soft computing, etc.

Kiyoshi Wada (M’95) was born in 1947. He received the B.E., M.E., and D.E. degrees in electrical engineering from Kyushu University, Fukuoka, Japan, in 1970, 1972, and 1978, respectively. He is currently a Professor with the Department of Electrical and Electronic Systems Engineering, Kyushu University. His research interests include the areas of system identification, digital signal processing, and adaptive control, etc.

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Robust Output Feedback Control of a Class of Nonlinear Systems Using a Disturbance Observer Zi-Jiang Yang, Seiichiro Hara, Shunshoku Kanae, Member, IEEE, and Kiyoshi Wada, Member, IEEE

Abstract—This paper considers the output-tracking control problem of feedback linearizable nonlinear systems in the presence of external disturbances and modeling errors. A robust output feedback nonlinear controller is designed to achieve excellent output-tracking performance. By exploiting the cascade features of backstepping design, a simple disturbance observer (DOB) is proposed to suppress the effects of the uncertainties, and a high-gain observer (HGOB) is applied to estimate the unmeasureable states of the system. Although the DOB-based controllers are usually designed according to linear control theory, in this study, strict analysis of the nonlinear control system is given. Experimental results on a magnetic levitation system are provided to support the theoretical results and to verify the advantages of the proposed controller. Index Terms—Backstepping design, disturbance observer (DOB), feedback linearizable nonlinear system, high-gain observer (HGOB), input-to-state stability (ISS), nonlinear damping term.

I. INTRODUCTION

T

HE development of feedback linearization techniques provides a powerful tool for nonlinear system control. Under some geometric conditions, the state variable model of a class of single-input single-output (SISO) nonlinear systems can be rendered to be the normal form as described in (2) [1]–[3]. To handle the uncertainties of the nonlinear functions, adaptive control schemes based on function approximation techniques have been being studied extensively. While having been extensively studied in both theories and applications, the adaptive control approach using neural or fuzzy networks may require heavy computational burden and may exhibit unsatisfactory transient performance. The system model (2) can be obtained by performing coordinate transformation on the SISO nonlinear system (1) Manuscript received January 04, 2009; revised June 25, 2009; accepted April 06, 2010. Manuscript received in final form May 03, 2010. Date of publication May 24, 2010; date of current version February 23, 2011. Recommended by Associate Editor C.-Y. Su. Z.-J. Yang is with the Department of Intelligent Systems Engineering, College of Engineering, Ibaraki University, Ibaraki 316-8511, Japan. S. Hara and K. Wada are with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka 812-8581, Japan. S. Kanae is with the Department of Electrical, Electronics, and Computer Engineering, Fukui University of Technology, Fukui 910-8505, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2049998

with relative degree , under some geometric conditions [3]. , and , and are some apHere, propriate nonlinear functions of the state variables and external disturbances which may include some uncertainties. There also have been some works on more general systems with a lower and a stable zero dynamics [4]–[6]. Since relative degree the application example in this work is of full relative degree and since the basic idea can be straightforwardly extended to these systems, in this work, we confine our study to the system model (2). The model (2) may also arise in models of mechanical or electromechanical systems where the position is measured while their derivatives (velocity, acceleration) are not measured. In many practical situations, even when the state variables of the original system (1) are exactly measurable, the transformed state variables may not be exactly calculated due to modeling uncertainties. Therefore, there has been strong practical necessities to construct a high performance output feedback controller using only the output measurement. The HGOB approach has been considered as a powerful tool to tackle the problems of output feedback control [4]–[15]. The design procedure of a HGOB-based output feedback controller is summarized as follows. 1) transform the system model into the normal form if possible; 2) design a state feedback controller as if the state variables are known; 3) design a HGOB to estimate the states so that the controller is implementable using only the output measurement; 4) saturate the controller outside the domain of interest so that the controller is globally bounded; 5) prove that the output feedback controller asymptotically recovers the performance achieved by the state feedback controller based on analysis of the singularly perturbed system. For the basic theory, the readers are invited to see [9]–[11]. Typical control techniques combined with the HGOB are adaptive control [4], [5], [12], sliding mode control [13], [14], and robust control by nonlinear damping terms [15]. However, the sliding mode control approach may lead to large control efforts and chattering control response, whereas the adaptive control approach may require much more computational burden and may exhibit unsatisfactory transient performance. Therefore, attention should be paid to overcome these drawbacks. As an alternative popular approach for compensating external disturbances and model mismatch, a DOB is often included into a controller [16]–[20]. The DOB-based controllers have been widely accepted in the industrial side, due to their simplicity and transparency of design, and excellent disturbance compen-

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YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

sation ability. In [20], a state feedback nonlinear robust controller with DOB for a voltage-controlled magnetic levitation system has been proposed where the stability of the nonlinear control system is analyzed rigorously. However, the DOB may be vulnerable to the measurement noise when the velocity is calculated by pseudodifferentiation. Moreover, due to the modeling errors, the model is not exactly transformed to the normal form (2) so that the resultant controller is complicated and the control results are not so good as those by the adaptive controllers [21]. In this paper, instead of the popular adaptive control approach, a robust output feedback controller using a DOB is proposed for the output-tracking control problem of the linearizable nonlinear system (2). By exploiting the cascade features of backstepping design, our idea is to introduce the th subsystem rather than to estimate or DOB to the approximate the uncertainties in the th subsystem in the model (2), so that the controller design related to the DOB is much simpler. The essence of the idea is that, by virtue of backstepping design, the control error signal of the th subsystem is treated as a disturbance term affecting the th subsystem. Although in most works of the DOB-based controllers the control design is based on the linear control theory, in this study, through cascade analysis of backstepping design, it is shown that the stability of the overall nonlinear control system is estrack the tablished and the DOB helps to make the output reference trajectory with small tracking error. Then by using a HGOB that estimates the state variables, the designed controller is implementable as an output feedback controller. It is shown that the output feedback controller asymptotically recovers the performance achieved by the state feedback controller based on analysis of the singularly perturbed system. Experimental results are given to support the theoretical results.

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and are bounded away from zero Assumption 1: , where is the domain with the same known sign, for of interest. Assumption 2: There exist finite positive, but not necessarily , , , and known continknown constants and such that the following uous functions : inequalities hold for (4) (5) and are appropriate known functions Here, (bounding functions) which will be used for construction of nonlinear damping terms [22]. The inequalities in (5) mean that and the error function do the nominal function itself. not grow in a higher order than III. ROBUST NONLINEAR STATE FEEDBACK CONTROLLER Due to its flexibility for systematic desirable modifications of the controller such as compensation for modeling errors or external disturbances, the backstepping control design techniques have received great attention in the last decade [22]. We here design the proposed controller in a backstepping manner. To this end, we first define the output-tracking error signal as (6) Then we will show the concrete design procedure. A. Steps Define the error signal

II. PROBLEM STATEMENT

(7)

Consider the following SISO feedback linearizable nonlinear system in normal form:

where is the virtual input to stabilize Then from (2) we have subsystem

.

(8) .. .

The virtual input

is designed as (9)

(2) is the vector of state variables; is the where and are modelable nonlinear functions control input; denotes unmodeled with known nominal functions; and nonlinearities and disturbances. Denoting the nominal nonlinearities based on the prior and , we have knowledge as (3) and denote the modelling errors. where We impose the following standing assumptions.

where

.

B. Step Define the error signal (10) where is the virtual input to stabilize . Then we have the dynamical equation of subsystem

as (11)

It is well known that can be made sufficiently small at the next step of design by using the adaptive control technique

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which directly estimate the nonlinearities. However, the adaptive control approach using neural or fuzzy networks may require heavy computational burden and may exhibit unsatisfactory transient performance. Our policy is to design a moderate is bounded and robust controller at step of design so that not too large. Then we introduce a disturbance observer to the to compensate the effects of on the subsubsystem system . That is, is considered as a disturbance term to . the subsystem , we have Viewing as a disturbance to the subsystem

where

, and

(17) To stabilize the subsystem controller:

, we design the following

(18)

(12) cannot be used exHowever, usually, the next error signal plicitly at the current step of backstepping design [22], since we at the next step design if includes have to calculate explicitly. Therefore, instead of itself, we can use its so that is computable. To this end, low-passed estimate to obtain the estiwe pass (12) through a low-pass filter as mate of (13) This idea is very similar to the so called DOB studied extensively in the literature [16]–[19]. In this paper, we adopt a simple low-pass filter as the following: (14) is a time-constant. Since where used at the next step of design, we should have is sufficient. found empirically We design the virtual controller as

where

(19) ; is a feedback proportional where is a nonlinear controller with model compensation; ; is a nonlinear damping damping term to suppress ; is a nonlinear damping term term to suppress to suppress . The nonlinear damping terms are designed such that their time-varying gains grows at the same order as their corresponding uncertain terms. Further analysis will be given in Section IV.

will be . We have

IV. STABILITY ANALYSIS OF THE STATE FEEDBACK CONTROL SYSTEM A. Steps The subsystems expressed as

controlled by the virtual input

can be

(15)

(20)

; is a small positive number, ex., ; is a nonlinear damping term that enhance the is relatively large. Usually, we can exdamping effect when within the pass-band of . The estimation error pect beyond the pass-band of can be suppressed sufficiently by and . relatively large can be ob1) Step : The dynamics of the subsystem tained as

is made uniformly ultimately bounded at the Assuming next step, we can derive

where

(21) and (22) Finally, we have

(16)

(23)

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

The derivation of (23) is analogous to the proof of Lemma 6.20 in [22]. Furthermore, we have

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and hence

(24) Integrating both sides of (24) and taking mean value of time, we have (31)

(25)

Equation (23) characterizes the input-to-state stability (ISS) property [22] of , and (25) characterizes the mean-square performance. B. Step Applying the virtual input have

to the subsystem

C. Step Applying the control input have

to the subsystem

(16), we

, we

(26) is made uniformly ultimately bounded at the Assuming next step, we can derive

(32) Similar to the previous analysis, we can derive (33)

(27) where

where (28)

When in the numerator of grows, the nonlinear damping term in the denominator grows accordingly such that can be suppressed sufficiently, and the effects of hence we have (29) Notice that owing to the DOB, is likely small than in most cases especially within the pass-band of the DOB. It is helps to improve the error signal . clear that , we furthermore have Provided the ISS property of

(34) It can be verified that each uncertain term in the numerator of is suppressed by the corresponding nonlinear damping term, i.e., when a term in the numerator grows, the corresponding term in the denominator grows at the same order such is uniformly bounded. Then similar to the previous that step, we have the following ISS property: (35) Provided the ISS property of

(30)

, we furthermore have

(36)

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and hence

(37)

D. Overall Error System

, Define a compact set where is relatively small (not necessarily very small as will be explained in Remark 1) by an appropriate set of reference trajectory, initial states and design parameters, according to the in, and if the control gains are designed equality (39). If sufficiently large, then we can find an appropriate set of initial , so that we can make , conditions where is defined according to (39) as the following:

The aforementioned ISS properties of the three subsystems can be summarized as follows:

.. .

(38) These results imply that the overall error system can be expressed as the cascade of ISS systems. We therefore can conclude based on Lemma C.4 in [22] that the overall error system is also ISS. . Define the error signal vector as Then by repeated use of [22, Lemma C.4], we have the following results. The derivation process is quite lengthy and is therefore omitted. Interested readers may contact the authors for detailed proof. (39)

(41) As long as the error vector is relatively small, the system output may track a smooth reference trajectory with acceptable accuracy and the state variables do not escape from the domain of interest . Thus, we impose the following assumption. Assumption 4: If is made such that , then . Remark 1: The inequality (39) characterizes the ISS property of the overall error system, which ensures the boundedness of is sufficiently the error signals. According to (28), small within the pass-band of the DOB, owing to the DOB. of the final subsystem may not On the other hand, itself may not be so small. be so small. Therefore, It is only required that . However, our final purpose sufficiently small. is to make the output-tracking error and hence According to (38), we can conclude that can be made sufficiently small within the pass-band of the DOB. We can also explain the effects of the DOB by considering the mean-squares of the error signals. The mean-squares of the error signals of the subsystems derived previously are summarized as follows:

where

.. . (40a) (40b) (40c) (40d) and is the impulse response of . However, due to the natural constraint of the system under study in the case of a real application, we should verify if the system states are constrained in an appropriate domain of interest , where the results obtained here are valid. To ensure the controller feasible, we should verify if there is a compact set such that we can make by the designed controller. To this end, we impose the following assumption. function of Assumption 3: The reference trajectory is a time (continuously differentiable with respect to time up to th order) and , where and is an appropriate compact subset of .

(42)

By applying the Parseval’s theorem, we have

(43) is the frequency response of the low-pass filter where of the DOB, and is the Fourier transformation of over the time interval . Notice that for

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

the frequencies below the cutoff frequency of . Therefore, is sufficiently broad, the mean-square if the pass-band of , and hence those of can be made suffiof ciently small. The results of the state feedback controller are summarized in the following theorem. Theorem 1: Let Assumptions 1 4 hold. And let the initial conditions . If the state feedback robust nonlinear controller is applied to the nonlinear system (2) under study, the following results hold. such that 1) There exists a compact set , where is the domain of interest. 2) The overall error system is ISS such that

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where .. . (47)

.. .. .

..

..

.

..

.

.

..

.

.

..

.

.. . (48)

. Then (2) Define the estimation error of HGOB as and (44) lead to the dynamics of the estimation error of HGOB with . 3) The mean-squares of the error signals satisfy (42). However, our problem setting is that only the output signal, i.e., is available. We have to estimate the system states by a HGOB so that an output feedback controller is implementable.

(49) where ..

.

.

..

.

.

..

.

V. CONSTRUCTION OF THE OUTPUT FEEDBACK CONTROLLER A. Construction of the HGOB To implement the state feedback controller by output feedback, we use the following HGOB [4]–[15]:

.. . .. .

.. .. .

..

.. .

.. .

(50)

(51) Scale the observer estimation error

.. . (44) where and

is a small positive constant specified by the designer are selected such that the roots of

(52) . Then (46) and (49) lead to the and define closed-loop system in the standard singularly perturbed form (53a) (53b)

(45) are in the open left-half -plane. In this paper, is obtained. chosen such that a polynomial

are

B. Saturation of the Controller of the HGOB may inSince the output troduce incorrect peaking signals especially in the first transient phase, we have to saturate the signals outside of the domain of interest according to the physical limitation of the system under study [4]–[15]. We will denote the saturated system estimates , and the saturated control input as . as

.. . .. .

.. .. .

..

..

.

.

..

.

.

..

.

.. . (54)

The -scaling (52) causes an impulsive-like behavior in [4], [5], [8]–[15], but since enters the slow system (53a) through , the slow system signal vector does the saturated input not exhibit a similar impulse-like-behavior.

C. Singularly Perturbed Form Equations (20), (26), and (16) lead to the closed-loop system of the subsystems (46)

VI. STABILITY ANALYSIS OF THE OUTPUT FEEDBACK CONTROL SYSTEM The first part of our analysis is to establish that there is a short transient period during which the fast signal decays to

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level, while, due to the difference in speed, the slow signal still remains within the domain of interest. Similar to the discussions in Section IV, we can find an ap. Since propriate set of initial conditions is a low-passed version of , is . Then we have for some bounded as long as as long as . Let be the solution of , where , and define . Taking the time derivative of along the trajectory of (53a), we have

(55) where

, , and hence (56)

Since

, we have

(57) If the initial conditions are chosen such that for some according to the definition of given in the , independent previous section, then there exists a time of , such that over the time interval . We next investigate the behavior of the fast system (53b) over . According to (6), (7), (10) and (12), the time interval can be expressed by and since included in over the time interval , we have for some . be the solution of , where Let , and define . Taking the time derivative of along the trajectory of (53b), we have

tends to zero as , there is a small such that for all . Hence we deduce that there such that for all , we exists a time have , where is the first time that escapes from can be made to be the set , and as will be claimed later, . governed We now go back to studying the behavior of by the slow system (53a) over the time interval . Notice . Since that over this time interval, and hence are of the signals remain in the domain of interest, saturation of the controller does not occur. Since the observer’s output is only injected to the th subsystem of the slow system (53b), it is sufficient to investigate the behavior of the th subsystem owing to the cascade nature of the backstepping design. To proceed the analysis, we impose here one more assumption. Assumption 5: The nominal nonlinear functions and , and the bounding functions and employed in the controller are locally Lipschitz on the domain of interest . Considering the locally Lipschitz property of the dependent , we have functions in the control input

Since

(61) for some . Substituting system (16), we can rewrite (32) as

into the sub-

(62) and (33) is rewritten as (63) where, owing to (61),

in (34) is rewritten as (64)

for some

. Hence (35) is rewritten as

(58)

(65)

, . is bounded From (52), we deduce that by for some . It follows that as long as over the time interval , we have

For any , (65) characterizes the input-to-state practical stability property (ISpS which is an extension of the concept of ISS, see [24] for definition). Notice that the discussions and thus the starting time of the ISpS are for property characterized by (65) is . The behavior of for is characterized in (57). Finally, (39) is rewritten as

for where

(59) Thus the time

when

can be calculated as (60)

(66) for some

.

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

Similar to defined in (41), according to (57) and (65) we can find an appropriate set of initial conditions , so that we can make provided a sufficiently small . Here, is defined as the following:

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3) The mean-squares of the error signals satisfy

.. .

(67)

where, according to (57), we have

(68) for Therefore, we can make the escape time , we can rewrite (36) as Provided

.

for

.

VII. GUIDELINES FOR CONTROLLER PARAMETER DESIGN

(69)

. And hence (37) can be rewritten as

for some

(70)

Finally, the performance of the output feedback controller is summarized as the following theorem. Theorem 2: Let Assumptions 1 5 hold. And let the initial . If the output feedback robust conditions nonlinear controller is applied to the nonlinear system (2) under such that for all , the study, there exists following results hold. such that 1) There exists a compact set , where is the domain of interest. 2) The overall error system is ISpS such that

The guidelines for design of the controller parameters are drawn here. Since the nonlinear damping terms are nonlinear functions of some internal signals, it is recommendable to to avoid noisy or large choose modest control efforts. In contrast, the parameters of the nominal which also contribute to achieve fast controller transient phase and small error signals can be chosen relatively large, without causing large amplitude of the control input. that is closer A small time-constant of DOB leads to a noisy, to . However, too small a may make especially in the presence of measurement noise. Empirically, since the other parts such as the nonlinear damping terms also contribute to suppress the effects of uncertainties, the control performance is not so sensitive with respective to the value of . Usually, is chosen to be very small. Too small a value of may require a very small sampling interval, and also may cause noisy output of the HGOB. Empirically, for a fast electromechanical system such as the magnetic levitation system which . This is will be studied in this paper, is of the order of comparable to the time-constant of the low-pass filter of pseudodifferentiation for the same system in our previous works [20], [21], [23]. Usually, the performance of a high-gain observer-based controller (not limited to our present controller) is relatively sensitive to the value of . However, in practice, we have found that the selection of the time-constant of the lowpass filter of pseudodifferentiation is even much more sensitive. VIII. APPLICATION TO A MAGNETIC LEVITATION SYSTEM

with

, and .

tends to zero as

To demonstrate the advantages of the proposed robust nonlinear output feedback controller with DOB over our previously proposed state feedback adaptive or robust controllers [20], [21], [23], extensive experiments have been studied on a magnetic levitation system which is strongly nonlinear, unstable

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To convert the original nonlinear system model into the normal form, we adopt the following nonlinear coordinate transformation (74) Then through straightforward calculations, the system model (71) is transformed into

(75) where (76a) (76b) Fig. 1. Diagram of the magnetic levitation system.

(76c)

and subjected to external disturbance and errors of physical parameters. In this section, we will show how the experimental results reflect the theoretical analysis. A. Model of the System

(76d)

Consider a magnetic levitation system shown in Fig. 1, in which an electromagnet exerts attractive force to levitate a steel ball. The system dynamics can be described in the following equations [20], [21], [23]:

(76e)

(71)

Notice that owing to the relation has been replaced as the following: (74),

defined in

(77)

(72) where is the state variable vector. And, : air gap (vertical position) of the steel ball; : coil current; : gravity acceleration; : mass of the steel ball; : electrical , and : positive resistance; : voltage control input; constants determined by the characteristics of the coil, magnetic : a bounded external disturbance with core and steel ball; unknown bound. is given In this study, the bounded external disturbance as (73) for experimental studies. It is artificially added to the command value of the voltage control signal .

The nominal functions and are obtained by reand by their nomplacing the physical parameters in , , , , and . inal values , Remark 2: The gravity acceleration is known in most cases. However, an external constant (or slowly varying) mechanical disturbance can be treated as the bias of equivalently. In this paper, we allow the nominal value of is biased from . Also, notice that according to (77), it is required that and . B. Experimental Setup The physical parameters of the setup shown in Fig. 1 are given in Table I. The levitated steel ball is controlled by a digital control system that consists of a PC loaded with Windows 2000 OS, 12 bits analog-to-digital (A/D) and digital-to-analog (D/A) converters, and a controllable voltage source. The control algorithm is coded in Borland C++ language and discretized with ms . The physically allowable a sampling interval of operating region of the steel ball shown in Fig. 1 is limited to

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

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should be saturated such that V V; denotes a saturated signal. where 3) Reference Trajectory Initialization: Theorems 1 or 2 imply that the initial conditions of the error signals can influence the transient performance significantly, and it is thus desirable to make them small. and Suppose the steel ball is initially at rest with , i.e., the steel ball is held on the steel plate shown in Fig. 1 before the feedback controller’s start. Then if we choose the initial conditions of the reference trajectory and , we have such that , according to (6), (7), and (10). •

TABLE I PARAMETERS OF THE MAGNETIC LEVITATION SYSTEM

m m . The vertical position of the steel ball is measured by a laser distance sensor. And the output of the V V. controllable voltage source is limited to To apply the proposed controller to the magnetic levitation system, we first define the domain of interest as

D. Controller Design (78) where mm is the maximally allowable position of the steel ball. The domain of interest is due to the natural constraint of the setup. That is, the steel ball can only move within a narrow vertical domain, and the electromagnet can only exert attractive pulling-up force so that the downward acceleration cannot be larger than the gravity acceleration itself. C. Application of the Proposed Controller 1) Verification of the Assumptions: To make the theoretical results feasible, we should first verify if the imposed assumptions are satisfied. It is trivial to see that Assumption 1 is satisfied. To satisfy Assumption 2, we choose the bounding functions employed in (19) as

The following nominal system parameters with considerable errors were used for experimental studies, to verify the robust performance of the proposed controller: kg m

m/s

H

Hm (80)

Along the guidelines of controller design drawn in Section VIII-C, the proposed controller was designed as follows. (1) Robust nonlinear feedback controller without DOB

(81) Notice that when the DOB is not used, the nonlinear damping term corresponding to the DOB is not neces. sary and hence we can set (2) Robust nonlinear feedback controller with DOB

(79) and then we can verify that Assumption 2 is satisfied. Let the position command be a rectangular wave that changes between 4 and 12 mm, and pass it through a low-pass s . Then we obtain a smooth reference filter trajectory that satisfies Assumption 3. As long as the error vector is controlled to be relatively small, the steel ball may track a smooth reference trajectory with acceptable accuracy and does not escape from the domain of interest . Therefore, Assumption 4 is satisfied in generic cases. Inspections of the nonlinear functions in (76), (77) and (79) imply that Assumption 5 is satisfied. 2) Saturation of the Signals: Since the output of the HGOB may introduce incorrect peaking signals especially in the first transient phase, we have to saturate the signals outside of the domain of interest according to the physical limitation of the setup [4]–[15]. According to the discussions given so far, we saturate the signals as follows: • •

should be saturated such that should be saturated such that

is well defined; ;

(82) Notice that for this case the experiments were performed for three different values of . E. Comments on the Experimental Results The experimental results are shown in Figs. 2–5, where from the top to the bottom are the measured position , position error , velocity error , acceleration error , estimated position , estimated velocity , estimated acceleration , and the control input . Notice that and are calculated based on the estimated states , not on the true . It can be found in Fig. 2 that in the absence of the DOB, the boundedness is still ensured by the nonlinear damping terms, in spite of the considerably large errors of the physical parameters. However, the position-tracking performance is not acceptable, and we can even see the oscillations due to the sinusoidal disturbance.

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Fig. 2. Experimental results of the robust nonlinear controller without DOB ( = 0:0020).

Fig. 4. Experimental results of the robust nonlinear controller with DOB ( =

0:0020).

Fig. 5. Experimental results of the robust nonlinear controller with DOB ( =

Fig. 3. Experimental results of the robust nonlinear controller with DOB ( = 0:0015).

The results shown in Figs. 3–5 indicate the high performance of the proposed controller. It can be verified that introducing a simple DOB at step 2 of design brings significant improvement.

0:0025).

We can find the HGOB works for some different values of . Also, as discussed in Remark 1, we need not pay great efforts to suppress , and thus it is allowable that the amplitude of is not so small. However, owing to the DOB employed at step 2,

YANG et al.: ROBUST OUTPUT FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS USING A DISTURBANCE OBSERVER

we can see that is suppressed significantly and consequently is suppressed to be very small. Due to the limitation of space, we will not reproduce here the results by the state feedback controllers such as the robust control without DOB [23] or with DOB [20], and the adaptive control [21] for the same setup. However, we will give some comparative comments. It should be noticed here that only in the present work, a sinusoidal disturbance is imposed on the voltage input signal. In our previous works [20], [21], [23], however, we did not consider the external disturbance in the problem setting. Comparing the results of the present work with those of our previous works, we can directly see that the present controller yields better results (even in the presence of disturbance) than our previous works (in the absence of disturbance). In [20], [21], and [23], the model transformation (74) was calculated based on the nominal model provided the measurements of state variables . This resulted in an extra, unmatched untertain term in the second subsystem of (75). That is, the system could not be completely transformed into the normal form. And hence by the robust controller without DOB in [23], the uncertainties were not suppressed sufficiently. By the robust controller with DOB in [20], the control performance was improved. However, since the velocity was calculated by pseudodifferentiation, we found that the signals were very sensitive to the noise when the pass-band of the DOB was broad. Therefore, the performance improvement was limited. The adaptive controllers in [21] delivered better position-tracking performance than the robust controllers [20], [23], owing to model learning by neural networks. However, the adaptive controllers were much more complicated than the robust controllers. Moreover, the adaptive controllers could not handle effectively an unparameterized external disturbance. In contrast, we have found the present controller is simple in form, and yields better position-tracking performance compared to our previous works, while only requiring the position measurement.

IX. CONCLUSION In this paper, we have proposed a robust output feedback controller for feedback linearizable nonlinear systems in the presence of uncertainties. Instead of the popular adaptive control approach, a robust output feedback controller using a DOB was proposed for the nonlinear system under study. By exploiting the cascade features of backstepping design, a simple DOB was employed to suppress the effects of the uncertainties, and an HGOB was applied to estimate the unmeasureable states of the system. Strict analysis was performed based on the singularly perturbed system, and the ISpS property of the error system was established. The theoretical results were verified through extensive experiments on a magnetic levitation system, and it was confirmed that the present method works better than the previous methods studied by the authors on the same experimental setup. Although the proposed controller works quite well for the magnetic levitation system with continuous and smooth uncertainties, we should notice here that this does not imply that the proposed robust controller outperforms the adaptive control

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approach in all cases. For the systems with fast-changing or discontinuous uncertain nonlinearities that affect the system significantly beyond the pass-band of the DOB, the adaptive control approach may still be necessary. REFERENCES [1] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, MA: Kluwer, 2002. [2] L. X. Wang, A Course in Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1997. [3] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer, 1995. [4] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 177–188, Feb. 1996. [5] S. Seshagiri and H. K. Khalil, “Output feedback control of nonlinear systems using RBF neural networks,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 69–79, Jan. 2000. [6] S. Tong, T. Wang, and J. T. Wang, “Fuzzy adaptive output tracking control of nonlinear systems,” Fuzzy Sets Syst., vol. 111, pp. 169–182, 2000. [7] M. I. El-Hawwary, A. L. Elshafei, H. M. Emara, and H. A. A. Fattah, “Output feedback control of a class of nonlinear systems using direct adaptive fuzzy controller,” IEE Proc. Control Theory Appl., vol. 151, pp. 615–625, 2004. [8] J. Alvarez-Ramidrez, “Adaptive control of feedback linearizable systems: A modelling error compensation approach,” Int. J. Robust Nonlinear Control, vol. 9, pp. 361–377, 1999. [9] F. Esfandiari and H. K. Khalil, “Output feedback stabilization of fully linearizable systems,” Int. J. Control, vol. 56, pp. 1007–1037, 1992. [10] H. K. Khalil, “High-gain observers in nonlinear feedback control,” in New Directions in Nonlinear Observer Design, H. Nijmeijer and T. I. Fossen, Eds. New York: Springer-Verlag, 1999. [11] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2002. [12] K. W. Lee and H. K. Khalil, “Adaptive output feedback control of robot manipulators using high-gain observers,” Int. J. Control, vol. 67, pp. 869–886, 1997. [13] S. Oh and H. K. Khalil, “Nonlinear output feedback tracking using high-gain observer and variable structure control,” Automatica, vol. 33, pp. 1845–1856, 1997. [14] E. G. Strangas, H. K. Khalil, B. Aloliwi, L. Laubinger, and J. Miller, “Robust tracking controllers for induction motors without rotor position sensor: Analysis and experimental results,” IEEE Trans. Energy Conv., vol. 14, no. 4, pp. 1448–1458, Dec. 1999. [15] R. B. Arreola, “Output feedback nonlinear control for a linear motor in suspension mode,” Automatica, vol. 40, no. 12, pp. 2153–2160, 2004. [16] X. Chen, C. Y. Su, and T. Fukuda, “A nonlinear disturbance observer for multivariable systems and its application to magnetic bearing systems,” IEEE Trans. Control Syst. Technol., vol. 12, no. 4, pp. 569–577, Jul. 2004. [17] Y. Fujimoto and A. Kawamura, “Robust servo-system based on twodegree-of-freedom control with sliding mode,” IEEE Trans. Ind. Electron., vol. 42, no. 3, pp. 272–280, May/Jun. 1995. [18] S. Komada, N. Machii, and T. Fukuda, “Control of redundant manipulators considering order of disturbance observer,” IEEE Trans. Ind. Electron., vol. 47, no. 2, pp. 413–420, Mar./Apr. 2000. [19] Z. J. Yang, H. Tsubakihara, S. Kanae, K. Wada, and C. Y. Su, “A novel robust nonlinear motion controller with disturbance observer,” IEEE Trans. Control Syst. Technol., vol. 16, no. 1, pp. 137–147, Jan. 2008. [20] Z. J. Yang, H. Tsubakihara, S. Kanae, K. Wada, and C. Y. Su, “Robust nonlinear control of a voltage-controlled magnetic levitation system with disturbance observer,” presented at the IEEE Multi-Conf. Syst. Control, Singapore, 2007. [21] Z. J. Yang, K. Miyazaki, S. Kanae, and K. Wada, “Adaptive robust dynamic surface control for a magnetic levitation system,” presented at the 42nd IEEE Conf. Decision Control, Hawaii, 2003. [22] M. L. Krstic, Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. Hoboken, NJ: Wiley, 1995. [23] Z. J. Yang, K. Miyazaki, S. Kanae, and K. Wada, “Robust position control of a magnetic levitation system via dynamic surface control technique,” IEEE Trans. Ind. Electron., vol. 51, no. 4, pp. 26–34, Jul./ Aug. 2004. [24] Z. P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,” Automatica, vol. 34, pp. 825–840, 1998.

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Zi-Jiang Yang was born in 1964. He received the Dr.Eng. degree from Kyushu University, Fukuoka, Japan, in 1992. From 1996 to 2000, he was an Associate Professor with Kyushu Institute of Technology, Japan. From 2000 to 2009, he was an Associate Professor with Kyushu University. Since 2009, he has been a Professor with the Department of Intelligent Systems Engineering, Faculty of Engineering, Ibaraki University, Ibaraki, Japan. His research interests include system identification and nonlinear system control.

Seiichiro Hara was born in 1984. He received the B.S. degree from the Faculty of Engineering, Kyushu University, Fukuoka, Japan, in 2007, where he is currently pursuing the M.S. degree from the Department of Electrical and Electronic System Engineering, Graduate School of Information Science and Electrical Engineering. He is engaged in research on nonlinear system control.

Shunshoku Kanae (M’02) was born in 1961. He received the Dr.Eng. degree from Kyushu University, Fukuoka, Japan, in 1995. Since 2009, he has been an Associate Professor with Fukui Institute of Technology, Fukui, Japan. His research interests include system identification, mechatronics system control, and soft computing, etc.

Kiyoshi Wada (M’95) was born in 1947. He received the B.E., M.E., and D.E. degrees in electrical engineering from Kyushu University, Fukuoka, Japan, in 1970, 1972, and 1978, respectively. He is currently a Professor with the Department of Electrical and Electronic Systems Engineering, Kyushu University. His research interests include the areas of system identification, digital signal processing, and adaptive control, etc.