Robust adaptive control using a universal approximator ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 1, FEBRUARY 2000

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Robust Adaptive Control Using a Universal Approximator for SISO Nonlinear Systems Hyeongcheol Lee and Masayoshi Tomizuka

Abstract—This paper is concerned with the robust adaptive control of a class of nonlinear systems in the presence of parametric uncertainties and dominant uncertain nonlinearities. The proposed controller utilizes the robust adaptive control (RAC) to guarantee uniform boundedness and convergence of tracking errors. In addition, an adaptive fuzzy logic system is used as a universal approximator to reduce the model uncertainties coming from uncertain nonlinearities and to improve tracking performance. The approach does not require the matching condition imposed on control systems by using the backstepping design procedure, and provides boundedness of tracking errors under poor parameter adaptation. The method can be applied to a class of single-input single-output (SISO) nonlinear systems, transformable to a parametric-strict-feedback form. Index Terms—Backstepping design, robust adaptive control, smooth projection, stable adaptive fuzzy control, universal approximator.

I. INTRODUCTION

T

HE focus of adaptive control research in the late 1980's to early 1990's was on performance properties and on extending the results of the linear systems to a certain class of nonlinear systems with unknown parameters. The adaptive control of nonlinear systems has undergone rapid development in this period [1]–[4] leading to global stability and tracking results for reasonably large classes of nonlinear systems. By using the backstepping design procedure, Kanellakopoulos, et al. [1], [2], [5] have presented a systematic design of globally stable and asymptotically tracking adaptive controllers for a class of nonlinear systems transformable to a parametric-strict-feedback system. The robustness problem due to the integral type adaptation law and uncertain nonlinearities has been overcome by using a nonlinear damping [2], [6] or by combining backstepping adaptive control with conventional sliding-mode control [7], [8]. However, in the presence of uncertain nonlinearities, the controller guarantees that the output tracking error exponentially converges not to a point (asymptotic tracking), but to a ball-type error residual set (asymptotic bounding). As the size of the uncertain nonlinearities increases, the size of the residual set may also increase. Therefore, these control methods may not be allowable if the uncertain nonlinearities dominate the whole

Manuscript received March 27, 1998; revised March 17, 1999. The authors are with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720 USA. Publisher Item Identifier S 1063-6706(00)01620-9.

system dynamics to the extent that the achievable residual set is beyond the allowable range. The application of fuzzy set theory to control problems has been the focus of numerous studies [9]–[12]. The motivation is often that the fuzzy set theory provides an alternative way to the traditional modeling and design of control systems when system knowledge and dynamic models in the traditional sense are uncertain and time varying. Although achieving many practical successes, fuzzy control has not been viewed as rigorous due to the lack of formal synthesis techniques, which guarantee the basic requirements for control systems such as global stability. Recently, some research has been focused on the use of the Lyapunov synthesis approach to construct stable adaptive fuzzy controllers [13]–[18]. Several adaptive neural control schemes have been also developed based on the Lyapunov synthesis approach and applied to simple classes of nonlinear [18]. Especially, based on the universal approximation theorem [16], an adaptive fuzzy logic control (AFLC) method can provide stabilizing controllers (in Lyapunov sense) even for nonlinear systems with dominant uncertain nonlinearities by using sufficiently complex approximation functions [15], [16]. However, AFLC is applicable only to some structurally restricted systems, i.e., matched systems in which the control input and the uncertainties appear in the same equation. Another drawback of AFLC is that the reference input can hardly satisfy the required persistent excitation (PE) condition because of an excessive number of unknown parameters. In this paper, robust adaptive control using a universal approximator (RACUA) method is proposed in order to design a high-performance robust controller in the presence of parametric uncertainties and dominant uncertain nonlinearities. The approach effectively combines the design techniques of robust adaptive control (RAC) by backstepping and adaptive fuzzy logic control (AFLC), and provides several attractive features. This is achieved by including a robustness term in the adaptive backstepping algorithm by approximating unknown nonlinearities and by selecting parameter adaptation properly. The robustness term counteracts the effect of model uncertainties and parameter adaptation errors. To make the analysis general, only several required conditions on the robustness term are imposed instead of specifying them. The adaptive fuzzy logic system is used as a universal approximator to reduce the size of unknown nonlinearities. Thus, the size of the residual set of tracking errors can be reduced without using a high gain in the feedback loop. A simple smooth projection technique is used in the adaptation law to bound parameter estimation errors.

1063–6706/00$10.00 © 2000 IEEE

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RACUA is constructed for a class of SISO nonlinear systems, transformable to a parametric-strict-feedback form. II. ADAPTIVE FUZZY LOGIC SYSTEM There are many different interpretations for the fuzzy IF–THEN rules which result in different mappings of the fuzzy inference engine [10], [12]. There are also many different types of fuzzifiers and defuzzifiers. Many combinations of these fuzzy inference engines, fuzzifiers, and defuzzifiers may constitute useful fuzzy logic systems. Computational efficiency and easiness of adaptation are important for real-time adaptive control problems. Therefore, we consider a fuzzy logic system (FLS) consisting of the product-inference rule, singleton fuzzifier, center average defuzzifier, and Gaussian membership function. If the fuzzy rule base consists of a collection of fuzzy IF–THEN rules IF

is

and

and

is

THEN

is

(1)

and are the input and output of the fuzzy logic system, and are labels of fuzzy sets in the universes respectively, and , respectively, and is the of discourse number of rules in the rule base. Then, the resulting fuzzy logic system can be represented as [16]

where

uniformly approximating any real continuous nonlinear function over to any degree of accuracy if is compact. Let be the set of all the FBF expansions (3) with FBF's given by (4), be the sup-metric; and is a metric space. then Theorem 1 (Universal Approximation Theorem): For any given real continuous function on a compact set and arbitrary , there exist a FLS in the form of (3) with . (5) such that Proof:: A proof of this theorem is given in [16]. III. CONTROL OF FIRST-ORDER UNCERTAIN SYSTEM The system equation considered in this section is the firstorder nonlinear system described by (6) is the control input, is a known is a known shape function, is an unknown parameter vector, and is an unknown nonlinear function. Although the exact may not be known, their ranges can values of and often be predicted in advance. Thus, we can make the following are reasonable and practical assumptions that and bounded by some known parameters or known functions (see the following). Assumption 1: where function,

(2) (7) , and 's are fuzzy memwhere achieves its bership functions, and is the point at which 's and view 's as adjustable maximum value. If we fix parameters, then (2) can be written as

(3)

and are known vectors and is a known bounded function for all and . The for two vectors is performed in terms of the operation means corresponding elements of the vectors (e.g., ). In this paper, all functions involved in that . the design are assumed to be finite for all and From Theorem 1, there exists a fuzzy logic system in the form of (3) with (5) such that

where

is a parameter vector. The pawhere 's rameter vector is the collection of the points at which , achieve their maximum values, 's are the fuzzy basis functions (FBF's) defined by and

(8) where

(4) Our first choice for the membership function is the following Gaussian function: (5)

and is the for arbitrary approximation error satisfying . From (6) and (8), the system equation can be rewritten as (9) and

where .

, and are fixed parameters. where One of the most important advantages of FLS is that the FLS has the capability to approximate nonlinear mappings. The following theorem shows that the FLS (3) with (5) is capable of

be the desired output, which is assumed to be Let bounded with bounded derivatives up to a sufficient order. The control objective can be stated as that of designing a bounded such that under Assumption 1, control law for the input

LEE AND TOMIZUKA: ROBUST ADAPTIVE CONTROL USING UNIVERSAL APPROXIMATOR

the system is either globally uniformly bounded stable or . asymptotically stable and the output tracks An adaptive algorithm will be used to estimate the uncertain parameters in vector . From Assumption 1, the unknown parameters are expected to have bounded value. However, it was shown that the estimated parameters by a conventional adaptive scheme designed for a disturbance-free plant model may not be bounded in the presence of small unknown nonlinearities [7]. An effective method for eliminating parameter drift and keeping the parameter estimates within some priori defined bounds is to modify the adaptive scheme by using projection. Projection constrains the parameter estimates to lie inside some known convex bounded set in the parameter space that contains the unknown . There are several different types of modifications using different projection functions [19], [20], [7], [21], [22]. Since the derivatives of the control components are recursively required at each step to design a controller by the backstepping procedure for th-order nonlinear systems, a smooth projection [7], [22] is considered in this study. by Define a smooth projection of

where

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is any bounded time-varying positive scalar, i.e., and is any known bounded function

satisfying (16) where estimation error. Noting

is the parameter

(17) an example of

is (18)

and . Applying (14) to (9), we obtain the error equation

where (10) is a real-valued sufEach projection operator ficiently smooth nondecreasing function with bounded derivatives up to order and defined by

(19) Specify the adaptive law as

is known compact set and where , , and Define

(11)

(20)

is a design value.

Now, we state the main theorem. Theorem 2: With the control law (14), the adaptation law (20), and a robustness term satisfying Condition 1, the following results hold for the system described by (9) if Assumption 1 is satisfied: , then the bounded control input (14) guarantees If global uniform boundedness of the tracking error and its convergence to a compact residual set. Furthermore, the exponential converging rate and the size of the compact residual set can be freely adjusted by the controller parameters in a known form. (Global Uniform Boundedness) , that is, from the universal approximation theorem, If is large enough to inthe searching space for clude and if the desired trajectory satisfies the persistent excitation (PE) condition so that the estimated parameter converges to its true value, i.e., when , then in addition to the result in A, the system output tracks the when . desired output asymptotically, i.e., (Global Asymptotic Tracking.) Proof: . The time Let a positive definite function as is derivative of

(12) Then, and

is positive definite with regard to

for each

(13) . This function will be utiwhere lized as part of a composite Lyapunov function for closed-loop system. The RACUA control law for the nonlinear system in (9) is suggested to be (14) and . The control law is augmented with where to guarantee that remains bounded a robustness term and . The robustness term must satisfy for all bounded the following condition. Condition 1: 1) 2)

(15)

(21)

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2) Saturation function (continuous modification)

Therefore

for (22) . We see that is where . Since is a negative whenever is also bounded for all , we bounded disturbance and is negative outside the compact residual set conclude that (23) , we conclude that decreases Recalling that whenever is outside the set and, hence, is bounded. and are bounded and all the terms involved are Since bounded functions with regard to , we can conclude that the control input (14) is bounded. Moreover, from (22), converges to the compact set . to the compact set The exponentially converging rate and the bound of the tracking error (24) can be freely adjusted. From (13) (25) , noting (19), (20), and (25), the time derivative When is of the positive definite function

(26) and . From (14), This implies that and, thus, is uniformly continuous. By Barbalat's lemma and asymptotic tracking is achieved. Furthermore, [23], from (14), since all terms except are uniformly continuous, is uniformly continuous. Applying Barbalat's lemma again, . From (14), . Thus, the PE condition will . guarantee that Four well known examples of the robustness term satisfying Condition 1 are as follows [24]. 1) Sign function

for for

(27)

for

(28)

3) Hyperbolic tangent function (smooth modification) (29) 4) Nonlinear damping (smooth modification)

(30) When the sign function is used as a robustness term, the control law (14) becomes sliding-mode control [23]. The control law (14) with the sign function guarantees that the system (9) is exponentially stable and its output tracks the desired trajectory asymptotically, even in the presence of both parametric uncertainties and unknown nonlinear functions. However, the control law (14) is usually discontinuous across the sliding surface. This leads to control chattering in practice. The saturation function is often used to overcome the control chattering. This continuous modification of control essentially assigns a low-pass filter structure to the local dynamics of the variable , thus eliminating chattering [23]. This achieves a tradeoff between tracking precision and robustness to unmodeled dynamics. The hyperbolic tangent function is used to smoothen the saturation function at the bound. This smooth modification is especially necessary in the backstepping design procedure for general relative degree system, which recursively requires the derivatives of the control components at each step [7]. Some high-order nonlinear damping, such as , is highly recommended in the control of human related systems such as the control of passenger cars in intelligent vehicle highway systems (IVHS), to maximize ride comfort and safety at the same time. This nonlinear damping term exhibits lower gains at small tracking errors to give ride comfort and higher gains at large tracking errors to give safety than the other robustness terms. Therefore, the control law (14) with the nonlinear damping has the gain scheduling effect. We have to be and to avoid actuator careful in choosing the coefficients saturation when we use the nonlinear damping. Fig. 1 shows some examples of the modification functions and the corresponding possible maximum error bounds . As shown in the figure, the controller with sign function as the robustness term guarantees asymptotic tracking. However, the controllers with other robustness terms guarantee only uniform boundedness. The controller with the nonlinear damping gives the fastest converging rate to the final error bound.


Fig. 1. Robustness term and possible maximum tracking errors 1 1 1; sign function (s(z ) = (z + " )).

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0000; saturation function 010101; tanh function __; nonlinear damping

IV. RACUA OF SISO NONLINEAR SYSTEMS

. Then, (31)

The RACUA technique can be generalized to a class of SISO nonlinear systems with arbitrary known relative degrees. This generalization is achieved by combining the RACUA design technique with the well-known adaptive backstepping algorithms [2]. Consider the SISO nonlinear system transformable to the following parametric-strict-feedback form

can be rewritten as an augmented system:

(34) , , and . be the desired output trajectory, which is assumed Let to be bounded with bounded derivatives up to th order. To improve transient performance, the reference output trajectory is designed from by trajectory initialization [2]. is also assumed to be known and bounded with bounded derivatives up to th order. The control objective is to design a bounded control law for the input such that under Assumption 2, the system is stable and the output asymptotically tracks . We will start by adaptively stabilizing the first equation of to be its control. At each subsequent step, (34) considering we will augment the designed subsystem by one equation. At the th step, an th order subsystem is stabilized with respect to by the design of a stabilizing function a Lyapunov function and a tuning function . The update law for the parameter and the adaptive feedback control are designed estimate at the final step. The design modifies the recursive backstepping procedure in [1], [2], [5], and [7], which proceeds in the following steps. Step 1: The first equation of (34) can be rewritten as

where

(31) where are known functions, are the known shape funcis tions which are assumed to be sufficiently smooth, , a known bounded nonzero function, is the vector of unknown constant parameters and and unknown nonlinear functions, respectively. We make the following assumptions on and . Assumption 2:

(32) , , and are known bounded . functions for all and as the control input of the dynamics, the If we treat dynamics depends only on the states of its previous dynamics, . In other words, only the feedback signals dei.e., termine the dynamics. Such a form is called parametric-strictfeedback form and is studied in [2], [5]. From Theorem 1, there exists a fuzzy logic system in the form of (3) with (5) such that

where

where

for arbitrary , and

(35) as a virtual control we can design a In (35), by viewing such that tracks its desired trajectory stabilizing function . From (14), is given by

(33)

(36)

,

and is the first robustness term. The where will be given in Condition 2. required conditions for

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(a)

(b) Fig. 2.

Fuzzy membership functions.

If were the real-control input, then the adaptation law would be given by (20) and the design would be finished. Since it is not the case, we postpone the choice of the adaptation law and use the first tuning function

Substituting (36) and (38) into (35), the first-error subsystem becomes (39) where

. Choose (40)

(37) to denote the essential part of the adaptation law (20). Define the and its desired difference between the actual value of to be the second-error variable value

where

(38)

where

is a weighting. From (39), its time derivative is (41) .


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Fig. 3. Tracking performance.

Fig. 4. Control input.

Step : In the following, we treat and , i.e.,

Choose the stabilizing function

as

as the function of

(42) where

and

(43)

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Fig. 5. Estimated parameters, # and # .

Step : , the last equation of (34) has the By letting . Therefore, the same form as the intermediate step general form (43) to (47) applies to Step . Since is the real , we can choose it as control and

where

(48)

Then, the th error subsystem may be assumed to be

(44)

where law as

is given by (43) with

. Specify the adaptation (49)

Then the th error subsystem (45) becomes (45) where

(50) and the derivative of the augmented p.d. function by

is given

The augmented p.d. function is (51) (46) The robustness term isfy the following condition. Condition 2: 1)

and its derivative is given by

(47)

2) where

where

.

is required to sat-

(52) is any bounded time-varying positive scalar, i.e., and is any


Fig. 6.

Unknown nonlinearity

1.

known bounded function with continuous partial derivatives up th order and satisfies to

(53) can be expressed by

The term

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(54) where

Theorem 3: With the control law (48), the adaptation law (49), and a robustness term satisfying Condition 2, the following results hold for the system (34) if Assumption 2 is satisfied. If , then the bounded control input (48) guarantees global uniform boundedness of the tracking error and its convergence to a compact residual set. Furthermore, the exponential converging rate and the size of the compact residual set can be freely adjusted by the controller parameters in a known form. (Global uniform boundedness.) , that is from universal approximation theorem, If , is large the searching space for and, if the desired trajectory satisfies enough to include the persistent excitation (PE) condition so that the estimated when parameter converges to its true value, i.e., , then, in addition to the result in A, the system output when tracks the desired output asymptotically, i.e., . (Global asymptotic tracking.) Proof:: From (44)

(55) using (53) since is a known compact set We can calculate is known. One example of and the bounding function of is

(57) (51) becomes

(56) where , and

, .

(58)

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Fig. 7.


Unknown nonlinearity

1. freely adjusted by the controller parameters and . , A of the theorem is Since thus proved. , noting (13), (49), and (51), the time derivative When is of the positive definite function

From (52) and (53)

(59) and

Let

, then (60) (63)

is bounded above by Since is bounded. So, theorem is proved.

,

, (63) implies that . It is also easy to check that by Barbalat's lemma and B of the

(61) V. EXAMPLE where outside the compact residual set

. Since

is negative

Consider the following relative degree two nonlinear system:

(62) , we conclude that recalling that decreases whenever is outside the set and, hence, is bounded. Since and , are bounded and all the terms involved are bounded functions with regard to , we can conclude that the control input (48) is bounded. converges to the compact set . Moreover, from (61), to the compact set The exponentially converging rate and the bound of the final tracking error , can be

(64) where

satisfy

(65)


From the size of unknown nonlinear functions, we know that the system dynamics is dominated by the unknown nonlinear functions. This system can be rewritten by

(66) where

(67) The initial condition is assume to be . The desired output is reference input is calculated by

and and the designed (68)

Actual plant parameters are

(69) We define five fuzzy sets with labels negative big (NB), negative small (NS), zero (ZO), positive small (PS), and positive big (PB). Fuzzy membership functions for these labels are defined as For

if if

then then

(70)

then then

(71)

For

if if

The approximated unknown nonlinearities can be calculated by

where and 's are estimation parameters, and and 's are given by (4), (70), and (71). Fig. 2 shows these fuzzy membership functions defined over the state space. From (48) and (49), the control law and parameter update law can be determined. A nonlinear damping is utilized as the robustness term. Two controllers are run for comparison.

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RACUA: The controller described in the above. RAC: The same control law as in RACUA, but without using the AFLS to approximate the unknown nonlinearities. This controller is designed by setting the parameter adaptation gain corresponding to the AFLS to zero. Fig. 3 shows the tracking performance of both the controllers. The control inputs are shown in Fig. 4. The tracking performance of RACUA is much better than that of RAC. This difference results from the ability to approximate the unknown nonlinearities. As shown in the Figs. 6 and 7, RACUA approximates the unknown nonlinearities very well. Fig. 5 shows parameter estimation for and . For the RAC case, the estimated parameters hit the limit values repeatedly and cannot approach their actual value. It is well known that the integral type adaptation law (49) may suffer from parameter drifting in the presence of even a small disturbance. VI. CONCLUSION In this paper, a systematic design of robust adaptive controllers using a universal approximator (RACUA) for a class of SISO nonlinear systems was presented for systems with parametric uncertainties and dominant uncertain nonlinearities. The approach effectively combined the design techniques of robust adaptive control (RAC) by backstepping and adaptive fuzzy logic control (AFLC) and improved performance by preserving the advantages of both RAC and AFLC. This was achieved by including a robustness term in the adaptive backstepping algorithm, by approximating unknown nonlinearities, and by selecting parameter adaptation properly. The robustness term in the controller structure counteracted the effect of model uncertainties and parameter adaptation errors enhancing the transient performance and final tracking accuracy. Adaptive fuzzy logic system was used as a universal approximator to reduce the size of unknown nonlinearities. A simple smooth projection technique was used to modify the adaptation law. This ensured the boundedness of parameter estimation errors and improved the tracking performance. The control schemes were derived for a class of SISO nonlinear systems which is transformable to a parametric-strict-feedback form. Simulation results validated the analysis. REFERENCES [1] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizqable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, Nov. 1991. [2] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, New York: Wiley, 1995. [3] R. Marino and P. Tomei, “Global adaptive output-feedback control of nonlinear systems, part i: Linear parameterization—Part II: Nonlinear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 17–49, Jan. 1993. [4] J. B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 729–740, June 1992. [5] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, “Adaptive nonlinear control without overparametrization,” Syst. Contr. Lett., vol. 19, pp. 177–185, 1992. [6] I. Kanellakopoulos, “Passive adaptive control of nonlinear systems,” Int. J. Adaptive Contr. Signal Processing, vol. 7, pp. 339–352, 1993. [7] B. Yao, “Adaptive robust control of nonlinear systems with application to control of mechanical systems,” Ph.D. dissertation, Univ. California, Berkeley, CA, 1996.

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[20] S. Sastry, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. [21] D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. Reading, MA: Addison-Wesley, 1984. [22] A. R. Teel, “Adaptive tracking with robust stability,” in Proc. 32nd Conf. Decision Contr., San Antonio, TX, Dec. 1993, pp. 570–575. [23] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [24] H. Lee, “Robust adaptive control using a universal approximator with application to vehicle motion control for IVHS,” Ph.D. dissertation, Univ. California, Berkeley, CA, 1997.

Hyeongcheol Lee was born in Seoul, Korea, on November 22, 1965. He received the B.S. and M.S. degrees from the Seoul National University, Korea, in 1988 and 1990, respectively, and the Ph.D. degree from the University of Calfornia, Berkeley, in 1997. Since 1998, he has been with Ford Motor Company. His research interests include adaptive and nonlinear control, applications to automotive vehicle controls, and vehicle dynamics.

Masayoshi Tomizuka, photograph and biography not available at time of publication.