Radial basis function neural network-based adaptive control of uncertain nonlinear systems

Radial Basis Function Neural Network-based

Adaptive Control of Uncertain Nonlinear Systems

Hamou AIT ABBAS

and Boubakeur ZEGNINI

des Mat´eriaux Semi-conducteurs et Di´electriques, Universit´e Amar Telidji - Laghouat,

BP G37 Route de Ghardaia (03000 Laghouat)Alg´erie Email: hamouconf@gmail.com

bakzegni@yahoo.fr

Mohammed BELKHEIRI

(2)_{Laboratoire de T´el´ecommunications Signaux et Syst`emes,}

Universit´e Amar Telidji - Laghouat,

BP G37 Route de Ghardaia (03000 Laghouat)Alg´erie Email: m.belkheiri@lagh-univ.dz

Abdelhamid RABHI

(4)_{Laboratoire de Mod´elisation Information et Syst´emes,}

Universit´e de Picardie Jules Verne, 33 rue Saint Leu 80000 Amiens, France

Email: abdelhamid.rabhi@u-picardie.fr

Abstract—We aim to design in the present paper an adaptive

output feedback control scheme to address the tracking problem of an uncertain system having full relative degree in the presence of neglected dynamics and modelling errors. Then, the obtained controller is augmented by an online radial basis function neural network (RBF NN) that is used to adaptively compensate for the nonlinearity existing in the uncertain systems. A linear observer is introduced to generate an error signal for the adaptive laws. Ultimate boundedness is proven through Lyapunov’s direct method. The forcefulness of the theoretical results is demon-strated through computer simulations of a nonlinear second-order system.

I. INTRODUCTION

The nonlinear behavior exists in a large range of physical systems and devices, such as electromagnetism, mechanical actuators, electronic relay circuits, chaotic systems and other areas. Unfortunately, such nonlinearity often limits system performance.

In recent decades, adaptive control design has been sig-nificantly advanced for nonlinear systems, e.g., Brunovsky systems [4], [11], [14], feedback-linearized systems [2], [3], [22], and strict-feedback systems [23], [25], [26]. A key objective for these research efforts in control theory was the development of systematic design strategies for controlling uncertain nonlinear systems in order to have its outputs track given reference signals. Unfortunately, for many control applications, exact models are not available, too expensive to derive or their parameters are not accurately known which prevent the error signals from tending to zero [7].

Various adaptive state feedback and output feedback control algorithms are expected to exhibit more excellent performance. They have been developed for a large class of nonlinear systems under unstructured uncertainties (e.g., see [6], [3], [25]). Thus, an output feedback control using a high-gain observer for nonlinear systems was presented in [1] and [13]. The author in [21] gives a solution to the output feedback stabilization problem for systems in which nonlinearities de-pend only upon the available measurement. The authors in [16] present backstepping-based approaches to adaptive output

feedback control of uncertain systems that are linear with respect to unknown parameters. Therefore, development of an the intelligent tracking control approach to handle the effect of uncertainty for nonlinear systems is highly desirable.

Neural network systems [3], [5], [25] and fuzzy logic systems [20], [24], [26] have been successfully applied in recent years for several engineering applications to approx-imate nonlinearities of dynamic systems. Specifically, NN techniques have shown an excellent promise as competitive methods for nonlinear control due to their universal features including approximation ability, efficient nonlinear mapping between inputs and outputs without an exact knowledge of the mathematical model [2]. In [12], the author presents a new adaptive controller which cancels the effect of uncer-tainties to achieve precise tracking performance of nonlinear systems using NNs, whereas a radial basis function neural network (RBF NN) augmented backstepping controller for the nonlinear induction motor is applied in [5] to gain from the approximation ability of NNs and guarantee the stability of the closed loop system by an Augmented Lyapunov Function. Thus, to overcome the effect of parametric uncertainty and unmodelled dynamics for highly uncertain nonlinear systems, an other method that augments feedback linearization control using single hidden layer NNs can be found in [6], and good performances on the tracking accuracy were achieved.

In this paper, we consider a single-input-single-output (SISO) nonlinear systems -tunnel diode circuit- (TDC), and we contribute to design an intelligent adaptive output feedback tracking controller in order to address the tracking problem of the uncertain nonlinear system (TDC) in the presence of neglected dynamics, modelling errors and arbitrary complexity uncertainties. In this note, one implication is to employ feed-back linearization, coupled with a RBF NN to handle the effect of uncertainty. A signal, comprised of a linear combination of the tracking error and the compensator states, is used as a teaching signal for the NN. Current and past input/output data represent the input vector to the NN. Numerical simulations of the nonlinear system, tunnel diode circuit model having full

relative degree, are used to demonstrate the effectiveness of the proposed approach.

The rest of this paper is organized as follows: The control problem is formulated in Section II. Section III proposes the output feedback controller design applied to control the uncertain nonlinear system. Section IV develops the adaptive controller, in which NN augmentation is well detailed. In Sec-tion V, rigorous stability analysis is presented to guarantee the boundedness of the tracking error elements. The effectiveness of the proposed control system is demonstrated throughout simulation computer in Section VI.

II. PROBLEMFORMULATION

Consider the following observable nonlinear SISO system: ˙𝑥 = 𝑓(𝑥, 𝑢),

𝑦 = ℎ(𝑥). (1)

Where 𝑥 ∈ ℜ𝔫is the state, 𝑢 ∈ ℜ, and 𝑦 ∈ ℜ are the input and output of the plant, respectively.

𝐴𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛1. The functions 𝑓 : ℜ𝑛+1 _{−→ ℜ}𝑛 _{and ℎ :}

ℜ𝑛 _{−→ ℜ are sufficiently smooth partially known, and the}

output is assumed to have full relative degree𝑟 for all (𝑥, 𝑢) ∈ Ω × ℜ where Ω ⊂ ℜ𝑛_.

There exists a mapping that transforms the system in (1) into the so-called normal form [[15]:

˙𝜉𝑖 = 𝜉𝑖+1, 𝑖 = 1, ..., 𝑟 − 1

˙𝜉𝑟= ℎ(𝜉, 𝑢)

𝜉1= 𝑦.

(2)

where ℎ(𝜉, 𝑢) = 𝐿(𝑟)_𝑓 ℎ are the Lie derivatives, and 𝜉 =

[𝜉1 ... 𝜉𝑟]𝑇. The objective is to synthesize a feedback

control law that utilizes the available measurement 𝑦 so that

𝑦(𝑡) tracks a smooth bounded reference trajectory 𝑦∗_{(𝑡) with}

bounded error.

III. ERRORDYNAMICS ANDCONTROLLERDESIGN

A. Feedback Linearization

Feedback linearization is approximated by defining the following control input signal:

𝑢 = ˆℎ−1_{(𝑦, 𝑣) (3)}

where𝑣 is referred to as a pseudocontrol. The function ˆℎ(𝑦, 𝑢) represents the best available approximation of ℎ(𝑦, 𝑢). Then, the system dynamics can be expressed as

𝑦(𝑟)_{= 𝑣 + 𝛿 (4)}

where

𝛿(𝜉, 𝑣) = ℎ(𝜉1, ˆℎ−1(𝜉1, 𝑣)) − ˆℎ(𝜉1, ˆℎ−1(𝜉1, 𝑣)) (5)

is the inversion error. The pseudo-control is chosen to have the form

𝑣 = 𝑦∗(𝑟)_{+ 𝐿}𝑐

𝑑− 𝑎𝑠𝑐 (6)

where 𝑦∗(𝑟) is the 𝑟𝑡ℎ derivative of the input signal 𝑦∗, generated by a stable command filter, 𝐿𝑐_𝑑 is the output of a dynamic compensator, 𝑎𝑠_𝑐 is the adaptive control signal designed to overcome𝛿.

With (6), the dynamics in (4) reduce to

𝑦(𝑟)_{= 𝑦}∗(𝑟)_{+ 𝐿}𝑐

𝑑− 𝑎𝑠𝑐+ 𝛿 (7)

From (5), notice that𝛿 depends on 𝑎𝑠_𝑐 through 𝑣, whereas

𝑎𝑠

𝑐 has to be designed to approximately cancel 𝛿. B. Linear Compensator and Error Dynamics

Define the output tracking error as (𝑒 = 𝑦∗− 𝑦). Then the dynamics in (7) can be rewritten as:

𝑒(𝑟)_{= −𝐿}𝑐

𝑑+ 𝑎𝑠𝑐− 𝛿. (8)

Note that the adaptive term𝑎𝑠_𝑐 is not required when (𝛿 = 0). Consequently, the error dynamics in (8) reduce to

𝑒(𝑟)_{= −𝐿}𝑐

𝑑. (9)

The following linear compensator is introduced to stabilize the dynamics in (9):

{

˙𝜒 = 𝐴𝑚𝜒 + 𝑏𝑚𝑒,

𝐿𝑐

𝑑= 𝑐𝑚𝜒 + 𝑑𝑚𝑒. 𝜒 ∈ ℜ𝑟−1 (10)

Note that𝜒 needs to be at least of dimension (𝑟−1) [8]. This follows from the fact that (9) corresponds to error dynamics that have 𝑟 poles at the origin. One could elect to design a compensator of dimension≥ r as well. In the future, we will assume that the minimum dimension is chosen.

Returning to (8), notice that the vector 𝑒_𝑟 = [𝑒 ˙𝑒 ... 𝑒(𝑟−1)_]𝑇 _{mutually with the compensator state 𝜒} will obey the following dynamics, referred to as tracking error dynamics:

{

˙𝐸 = 𝐴𝑏𝐸 + 𝑏𝑏[𝑎𝑠𝑐− 𝛿]

𝑧 = 𝐶𝑏𝐸

(11) where 𝑧 is the vector of available measurements.

Reminder that: 𝐴𝑏= [ 𝐴 − 𝑑𝑚𝑏𝑐 −𝑏𝑐𝑚 𝑏𝑚𝑐 𝐴𝑚 ] , 𝑏𝑏 = [ 𝑏 0 ] , 𝑐𝑏= [ 𝑐 0 0 𝐼 ] . (12) and a new vector

𝐸𝑑 = [ 𝑒𝑇𝑟 𝜒𝑇 ]_𝑇 . (13) where 𝐴 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 ⋅ ⋅ ⋅ 0 0 0 1 ⋅ ⋅ ⋅ 0 .. . . .. ... ... ... 0 0 . .. ⋅⋅⋅ 1 0 0 0 ⋅ ⋅ ⋅ 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , 𝑏 = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 .. . 1 ⎤ ⎥ ⎥ ⎥ ⎦, 𝑐 = ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 .. . 0 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑇 .

Noting that𝐴_𝑚, 𝑏_𝑚, 𝑐_𝑚and𝑑_𝑚in (10) should be designed such that𝐴_𝑏 is Hurwitz.

C. Observer Design for the Error Dynamics

For the full-state feedback application [9], [2], [18], Lyapunov-like stability analysis of the error dynamics in (11) results in update laws for the adaptive control parameters in terms of the error vector 𝐸. In [15], and [10], adaptive state observers are used to provide the necessary estimates in the adaptation terms. In this note, we propose a simple linear observer for the tracking error dynamics in (11) and show through Lyapunov’s direct method that the adaptive part of the control signal(𝑎𝑠_𝑐) can compensate for the inversion error

𝛿, if the output of this observer is used as an error signal for

the adaptive laws.

A minimal-order observer of dimension (𝑟 − 1) may be designed for the dynamics in (11). In what follows, we consider the case of a full-order observer of dimension(2𝑟−1) [15].

To this end, consider the subsequent linear observer for the tracking error dynamics in (11):

{ ˙ˆ_{𝐸 = 𝐴𝑏𝐸 + 𝐾(𝑧 − ˆ𝑧),}ˆ

ˆ𝑧 = 𝐶𝑏𝐸.ˆ

(14) where 𝐾 is a gain matrix, and should be chosen such that (𝐴𝑏− 𝐾𝐶𝑏) is asymptotically stable, and 𝑧 is defined in (11).

The following remark will be useful in the sequel.

Remark 1: . Equation (14) provides estimates only for the

states that are feedback linearized with the transformation and not for the states that are associated with the internal dynamics.

Let ˜

𝐴 = 𝐴𝑏− 𝐾𝐶𝑏, 𝐸 = ˆ˜ 𝐸 − 𝐸, ˜𝑧 = ˆ𝑧 − 𝑧. (15)

Then, the observer error dynamics can be written

{ ˙˜_{𝐸 = ˜𝐴 ˜𝐸 − 𝑏𝑏[𝑎}𝑠

𝑐− 𝛿] ˜𝑧 = 𝑐𝑏𝐸˜

(16) IV. ADAPTIVE CONTROLSIGNAL

Following [15], given a compact set𝒟 ⊂ 𝑅𝑛+1and𝜖∗> 0, the model inversion error𝛿(𝜉, 𝑣) can be approximated over 𝒟 by a radial basis function neural network (RBF NN)

𝛿(𝜉, 𝑣) = 𝑀𝑇_{𝜙(𝜚) + 𝜖(𝑑, 𝜚), ∣𝜖∣ < 𝜖}∗_{. (17)}

using the input vector

𝜚(𝑡) = [𝑣𝑇 𝑑(𝑡) 𝑦𝑇𝑑(𝑡)]𝑇 ∈ 𝒟, ∥𝜚∥ ≤ 𝜚∗, 𝜚∗> 0. (18) Note that 𝑣𝑇 𝑑(𝑡) = [𝑣(𝑡) 𝑣(𝑡 − 𝑑) ... 𝑣(𝑡 − (𝑛1− 𝑟 − 1)𝑑)]𝑇, 𝑦𝑇 𝑑(𝑡) = [𝑦(𝑡) 𝑦(𝑡 − 𝑑) ... 𝑦(𝑡 − (𝑛1− 1)𝑑)]𝑇.

with 𝑛1 ≥ 𝑛, 𝑑 > 0 denoting time-delay and 𝜚∗ being a

uniform bound for all(𝜉, 𝑣) ∈ 𝒟.

The adaptive signal is designed as follow

𝑎𝑠

𝑐= ˆ𝑀𝑇𝜙(ˆ𝜚). (19)

where ˆ𝑀 is the estimate of 𝑀 that is updated according to the following adaptation law:

˙ˆ

𝑀 = −𝛽𝑀[2𝜙(ˆ𝜚) ˆ𝐸𝑇𝑃 𝑏𝑏+ 𝜆𝑀(ˆ𝑀 − 𝑀0)]. (20)

in which 𝑀₀ is the initial value of 𝑀, 𝑃 is the solution of the Lyapunov equation

𝐴𝑇

𝑏𝑃 + 𝑃 𝐴𝑏= −𝑄. (21)

for some𝑄 > 0, 𝑘 > 0, 𝛽𝑀 is the adaptation gain matrices, and ˆ𝜚 is an implementable input vector to the NN on the compact setΩ_𝜚ˆ, defined as ˆ𝜚 = [𝑣𝑇_𝑑(𝑡) ˆ𝑦_𝑑𝑇(𝑡)]𝑇 ∈ Ω_𝜚ˆ, ˆ𝑦_𝑖 =

ˆ

𝐸𝑖+ 𝑦∗(𝑖−1), 𝑖 = 1, ..., 𝑟 − 1. Notice that in (19), there is an algebraic loop, sinceˆ𝜚, by definition, depends upon 𝑎𝑠_𝑐through

𝑣, see (18). However, with bounded squashing functions, this

algebraic loop has at least one fixed-point solution as long as

𝜙(.) is made up of bounded basis functions.

Using (17) and (19), we can write the mismatch between the adaptive signal and the real NN as:

𝑎𝑠

𝑐− 𝛿 = ˆ𝑀𝑇𝜙(ˆ𝜚) − 𝑀𝑇𝜙(𝜚) − 𝜖

= ˜𝑀𝑇_{𝜙 + 𝑀ˆ} 𝑇_{𝜙 − 𝜖˜} (22)

where ˜𝑀 = ˆ𝑀 − 𝑀, ˆ𝜙 = 𝜙(ˆ𝜚), ˜𝜙 = 𝜙(ˆ𝜚) − 𝜙(𝜚).

Using (22, the error dynamics in (11) and the observer error dynamics in (16) can be formulated as

˙𝐸 = 𝐴𝑏𝐸 + 𝑏𝑏[˜𝑀𝑇𝜙 + 𝑀ˆ 𝑇𝜙 − 𝜖]˜ (23) ˙˜𝐸 = ˜𝐴˜𝐸 + 𝑏𝑏[˜𝑀𝑇𝜙 + 𝑀ˆ 𝑇𝜙 − 𝜖]˜ (24) Notice that for radial basis function and many other activa-tion funcactiva-tions that satisfy ∣𝜙_𝑖∣ ≤ 1, 𝑖 = 1, ..., 𝑁, there exists an upper bound over the set𝒟

∥𝜙(𝜚)∥ ≤ 𝜛, 𝜛 = max

𝜚∈𝒟∥𝜙(𝜚)∥ (25)

where 𝜛 remains of the order one, even if 𝑁 is large. With this, we have the following upper bound:

∣𝑀𝑇_{𝜙∣ ≤ 2∥𝑀∥𝜛.˜ (26)}

V. STABILITY ANALYSIS

In the current section, we confirm through Lyapunov’s direct method that if the initial errors of the variables 𝐸𝑇, ˜𝐸𝑇, ˜𝑀 and belong to a prescribed compact set, then they are ulti-mately bounded.

Lemma 1: . Following [10], if𝐴 is any asymptotically stable

matrix, then given any positive definite symmetric matrix𝑄 > 0, there exists a unique positive definite symmetric matrix 𝑃 > 0 such that

𝐴𝑇

𝑏𝑃 + 𝑃 𝐴𝑏= −𝑄.

Remark 2: . Consider the compact set𝐷 and initial

condi-tions inΩ. The feedback control law given by

where 𝑣 = 𝑦∗(𝑟)_{+ 𝐿}𝑐 𝑑− 𝑎𝑠𝑐 𝐿𝑐 𝑑= 𝑐𝑚𝜒 + 𝑑𝑚𝑒 𝑎𝑠 𝑐= ˆ𝑀𝑇𝜙(ˆ𝜚)

with the following adaptive laws ˙ˆ

𝑀 = −𝛽𝑀[2𝜙(𝜚) ˆ𝐸𝑇𝑃 𝑏𝑏+ 𝜆𝑀(ˆ𝑀 − 𝑀0)].

for 𝛽_𝑀 > 0, 𝜆_𝑀 > 0, 𝑃2 is the solution to the following

Lyapunov equation

𝐴𝑇

𝑏𝑃2+ 𝑃2𝐴𝑏= −𝑄2.

for some 𝑄2> 0, guarantees that all the error signals in the closed loop system are ultimately bounded (refer to [19] for more details).

VI. APPLICATION

To make clear the performance of the proposed adaptive controller in the presence of uncertainties, we consider the tunnel diode circuit example

⎧  ⎨  ⎩ ˙𝑥1= _𝐶1𝑥2−_𝐶1ℎ(𝑥1) ˙𝑥2= −𝑅_𝐿𝑥2−_𝐿1𝑥1+_𝐿𝑢 (27)

in which 𝑥1 is the voltage across the capacitor 𝐶 and 𝑥2 is the current through the inductor 𝐿. The element values of the circuit are 𝑅 = 1.5𝑘Ω, 𝐿 = 1𝑛𝐻, and 𝐶 = 2𝑝𝐹 , and the initial conditions were set as 𝑥1(0) = 0.1, 𝑥₂(0) = 0.006. Notice that the output 𝑦 = 𝑥₁ has a full relative degree (𝑛 =

𝑟 = 2). The function ℎ : ℜ −→ ℜ represents the characteristic

curve of the tunnel diode,

ℎ(𝑥1) = 𝑥1+ 3𝑥21+ 2𝑥31− 𝑥41− 4𝑥51 (28)

The command signals 𝑦∗ and 𝑦∗(2) are generated through a second -order command filter with natural frequency of 1𝑟𝑎𝑑/𝑠 and damping of 0.7. The following dynamic

compen-sator: {

˙𝜒 = −5.2𝜒 + 2𝑒

𝐿𝑐

𝑑= −20.4𝜒 + 9.16𝑒

(29) places the poles of the closed-loop error dynamics in (9) of the nonlinear systems at−2.8, −1.2±𝑗. The observer dynamics in (16) were designed so that its poles are four times faster than those of the error dynamics. A radial basis function NN with five neurons was used in the adaptive control. The functional form for each RBF neuron was defined by

𝜙𝑖(𝜅) = 𝑒−(𝜅−𝜅𝑐𝑖)𝑇(𝜅−𝜅𝑐𝑖)/𝜎2, 𝜎 = 1, 𝑖 = 1, 2, 3, 4, 5. (30) The centers 𝜅_𝑐𝑖, 𝑖 = 1, 2, 3, 4, 5, were arbitrarily selected over a grid of possible values for the vector𝜅. The adaptation gains were set to 𝛽_𝑀 = 2.1, with sigma modification gain

𝜆𝑀 = 0.009.

The contribution of this paper is to design an adaptive control component using a radial basis function neural network

(RBF NN) that compensates adaptively for the nonlinearities that exist in the tunnel diode circuit model, what bring to force the system response to track a given reference trajectory with bounded errors. First, setting the output 𝑦 = 𝑥1. Then, we employ feedback linearization, coupled with an on-line NN to handle the inversion errors, according to the equation (7). The dynamic compensator, described in (10) and (29), is designed to stabilize the linearized system. A signal, constituted of a linear combination of the measured tracking error and the compensator states, such as presented in (23), is used to adapt the NN weights [2]. 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time [sec] Tracking whitout NN y* y

Fig. 1. Tracking without RBF NN.

0 10 20 30 40 50 60 70 80 90 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time [sec] Tracking with NN y* y

Fig. 2. Tracking with the aid of RBF NN.

Figure(1) compares the system measurement𝑦 without NN augmentation (dashed line) with the reference model output

𝑦∗ _{(solid line), clearly demonstrating the almost unstable} os-cillatory behavior caused by the nonlinear elements(𝛿) in the tunnel diode model. While, with the aid of NN augmentation, Figure(2) shows that the effect of these nonlinearities is suc-cessfully eliminated. This is due essentially for the excellent identification of the model inversion error (𝛿) (dashed line)

by the adaptive term (𝑎𝑠_𝑐) (solid line), which is illustrated in Figure(3). 0 10 20 30 40 50 60 70 80 90 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 Time [sec] δ and a c s a_cs δ

Fig. 3. Identification of uncertainties(𝛿) by NN (𝑎𝑠_𝑐).

0 10 20 30 40 50 60 70 80 90 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 Time [sec]

Control effort without and with RBF NN

(y*−y) with RBF NN (y*−y) without RBF NN

Fig. 4. Control effort without and with RBF NN.

Figure(4) compares the control efforts(𝑦∗− 𝑦) without and with adaptation, in which the NN based adaptive controller (𝑎𝑠

𝑐) exhibits a steady state tracking error. This error can be minimized by designing an excellent linear compensator, according to the equation (8). Moreover, we must choose a good structure of the network of neuron in order to avoid the phenomenon of on training which deteriorates the control performance. The NN controller weights history are shown in figure(5).

As expected, the RBF NN improves the tracking perfor-mance due to its ability to ”model” nonlinearities. Conse-quently, simulation results show that the NNs augmented adaptive output feedback controller compensates successfully for the unstructured uncertainties.

VII. CONCLUSION

In this paper, a new control approach is proposed for adaptive output feedback control of an uncertain nonlinear

0 10 20 30 40 50 60 70 80 90 −0.5 0 0.5 1 Time [sec] NN Weight history

Fig. 5. NN weights history.

system using RBF NNs. However, the control performance of the given nonlinear system still influenced by the term of nonlinearity, and to compensate for these uncertainties an adaptive output feedback control is proposed. Then, the obtained controller is then augmented by a RBF NN used as an approximator for the unstructured uncertainties that exist in the nonlinear system. The derivatives of the tracking error are estimated by the simple linear observer. These estimates are used in the adaptation laws for the NN parameters. Ultimate boundedness of the tracking error and observation error are proven using Lyapunov’s direct method. The designed control scheme is applicable for observable and stabilizable systems of unknown but bounded dimension when the relative degree is known. Through computer simulation, we were able to demonstrate the practical potential of the proposed approach, and excellent tracking performance was succeeded.

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