High Order Sliding Mode Neurocontrol for Uncertain Nonlinear SISO Systems - Theory and Applications

These principles constitute themain idea in developing one of the most effective approach to identify and con- trol a wide class of uncertain nonlinear systems: Sliding Mode Control SMC [

for Uncertain Nonlinear SISO Systems: Theory and Applications

Isaac Chairez1, Alexander Poznyak1, and Tatyana Poznyak2

1 Department of Automatic Control CINVESTAV-IPN, M´exico, D.F

a new instrument for identification, state estimation and control of many classes

of uncertain systems affected by external perturbations This chapter deals withthe realization of this idea and suggests an adaptive control designing based

on both Differential Neural Network Observation and High Order Sliding Mode

Technique Below this approach is referred to as High Order Sliding Mode Neural Control (HOSMNC).

1.1 Classical and Unconventional Sliding Mode

Many physical systems naturally require the use of discontinuous terms in theirdynamics or in their control actions The keystone of this approach is the the-ory of differential equations with discontinuous right-hand side [5] Based on

it, the switching logic is designed in such a way that a contracting propertydominates the controlled dynamics leading thus to stabilization on a desiredmanifold, which induces desirable trajectories These principles constitute themain idea in developing one of the most effective approach to identify and con-

trol a wide class of uncertain nonlinear systems: Sliding Mode Control (SMC)

[30], [31] The essential feature of this technique is the application of ous feedback laws to reach and maintain the closed-loop dynamics on a certainmanifold in the state space (the switching surface) with some desired properties

discontinu-G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 179–200, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

180 I Chairez, A Poznyak, and T Poznyak

for the system trajectories This control offers many advantages comparing toother identification and control techniques: good transient behavior, the need for

a reduced amount of information in comparison to classical control techniques,unmodeled disturbance rejection capability, insensitivity to plant nonlinearities

or parameter variations, remarkable stability and performance robustness.There are two important drawbacks in real sliding-mode controllers imple-mentations:

- the first one is related to the reaching phase which could induce controlactions with high values, this fact could provoke the real controlled system toreach its maximum operation limits (dissipated energy, physical restrictions, etc.)leading to a bad performance on the close-loop dynamics;

- the second one deals with the actuators which have to cope with the highfrequency bang-bang type of control actions that could produce premature wear

or even breaking

However the last drawback-phenomenon, called chattering, could be reduced

using several techniques such as nonlinear gains, dynamic extensions, dynamicfiltering and high-order (or unconventional) sliding modes [3], [6] This chapterexactly describes the application effect of the last technique

The standard sliding mode (first order or 1-Sliding Mode) may be implemented only if the relative degree of the sliding variable s is 1 with respective to the

measurable variable It is well known that it provokes a high frequency switching(chattering effect) of the designed control Therefore, the traditional notion ofsliding mode control has been extended and the concept of 2-Sliding Mode andhigher order sliding modes (HOSM) has been developed [2], [15] With thesecontrollers, a sliding control of an arbitrary smoothness order can be achieved.Shortly speaking, higher order sliding modes remove the restrictions faced bystandard sliding mode, while keeping its main properties

1.2 Differential Neural Networks

Artificial Neural Networks (ANN) have shown good identification properties in

the presence of some uncertainties or external disturbances There are known two

type of NN: static one, using the, so-called, back-propagation technique [8] and

dynamic neural networks (DNN) [19], [27] The first one deal with the class of

global optimization problems trying to adjust the weights of such NN to minimize

an identification error The second type exploits the feedback properties of theapplied DNN and permits to avoid many problems related to global extremumsearch, transforming the learning process to an adaptive feedback design [21] Ifthe mathematical model of a considered process is incomplete or partially known,the DNN-approach provides an effective instrument to attack a wide spectrum

of problems such as identification, state estimation, trajectories tracking an etc.[17], [21]

Dynamic neuro-observers are studied in [21] In [24] the SMC approach is used

to obtain the algebraic (non differential) weight-learning procedure for on-line

identification of a nonlinear plant (a model design) with completely availablestates DNN observers containing sign-term are considered also in [23].

2.1 Class of Nonlinear Systems

The class of uncertain nonlinear SISO systems considered throughout this

chap-ter is governed by a set of n nonlinear ordinary differential equations (ODE) and

the algebraic state-output mapping given by

˙

x t = f (x t , u t ) + ξ 1,t , y t = Cx t + ξ 2,t (1)

where x t ∈  n is the system state at time t ≥ 0, y t ∈  is the system output,

The vectors ξ 1,t ∈  n and ξ 2,t ∈  represent the state and output deterministic

bounded (unmeasurable) disturbances, i.e.,

where f0(x, u | Θ) is the nominal dynamics while ˜ f t is a vector called the

mod-elling error Here the parameters Θ are suggested to be adjusted to minimize the

approximation of the nominal part In particular and according to the DNN proach [21], the nominal dynamics may be defined within the following nonlinearstructure

182 I Chairez, A Poznyak, and T Poznyak

selected as a stable one and such that the pair (A, C) is observable The functions σ ( ·) := [σ1(·) , , σ l(·)]and ϕ ( ·) := [ϕ1(·) , , ϕ s(·)]are usuallyconstructed with sigmoid function components (following the standard neuralnetworks design algorithms):

A1 A is a stable matrix.

A2 Any of unknown controlled SISO ODE has solution and it is unique, that is(3) and (4) hold

A3 The unmeasured disturbances for the uncertain dynamics ξ 1,tand the output

signal ξ 2,t satisfy (2) and they do not violate the existence of the ODEsolution (1)

A4 Admissible controls satisfy the sector condition (9), and again, do not violatethe existence of the solution to ODE (1)

2.2 DNN Observer with Variable Structure Term

Defined the DNN observer which can be used to reproduce the unknown x t

where ˆx0is fixed and the weight matrices (W j,t , j = 1, 2) are updated by a

nonlinear learning law

˙

to be designed Notice this nonlinear adaptive observer reproduces (as it ally called in the state estimation theory) the nominal plant structure (or itsapproximation) with two additional output based correction terms: one propor-tional to the output error and the second one known as a unitary corrector As a

usu-consequence, when y t = C ˆ x t, the ODE (11) should be attended as a differential

inclusion (see [5]) The pair of correction matrices K1and K2should be selected

as it is described below

2.3 Problem Statement

The main idea is to force the uncertain nonlinear system (1) to track a sired reference signal by means of a sliding mode controller using the estimatedstates provided by a differential neural network observer This problem can beformulated as the solution of the following two subproblems:

averaged estimation error β defined as

would be as less as possible.

would be small enough.

Here x ∗ is the state vector of a suitable desired dynamics given by:

2.4 Adaptive Weights Learning Law with Bounded Dynamics

To adjust the given neuro-observer (11), let us apply the following learning law:

184 I Chairez, A Poznyak, and T Poznyak

The time varying paramters k j,t , j = 1, 2 are such that k j,t ≥ 0 and ˙k j,t ∀t ≥ 0

(see the following subsection) P j , j = 1, 2 are the positive definite solutions for

the following algebraic Riccati equations [21]:

Q2:= 2 ˜f1λmax



+ ˜f2λmax

(17)

2.5 Main Result

Theorem 1 If there exist positive definite matrices [Λ1, Λ2, Q0, Λ K1 , Λ K2 ] and

definite solutions, then the DNN observer (11), supplied by the learning law (16)

state estimation process:

Proof The proofs of this theorem is given in Appendix.

Remark 1 As it can be seen from (18)-(19) the quality of the suggested neural

observer depends on both the power of external perturbations Υ1, Υ2 and theapproximation error ˜f0 as well

Remark 2 If there are no noises in the system dynamics and the output

mea-surements (Υ1 = Υ2 = 0) and if the class of uncertain systems and the control

functions are ”zero-cone” type, i.e., ( ˜ f0 = v0 = 0), then ρ Q=0 and the

asymp-totic error convergence Δ t → 0 (t → ∞) is guaranteed.

Once the upper bound for the estimation process has been derived, it is possible(independently) to consider the stability analysis on the learning laws obtainedduring the observer development Define the Lyapunov function

2z t 2+1

and the constantκ := V2(z t , k 2,t)| t=0

Theorem 2 The weights time profiles (16) are bounded, and moreover, for the

To guarantee the non-increasing property for V 1,t on the trajectories of (11), it

is sufficient to fulfill (ε0> 0, k 1,0 > k 1,min > 0)

that implies ˙V 1,t ≤ 0 The last inequality permits the existence of several learning

laws schemes depending on the k 1,t structure

186 I Chairez, A Poznyak, and T Poznyak

process consists in a suitable approximation (or estimation) of these values It

can be realized off-line (before the beginning of the state estimation) by the best selection of the nominal parameters Θ := [A, W ∗

1, W ∗

1] using some available experimental data (u t k , x t k) | k=1,N and a numerical interpolator algorithm al-lowing to manage these data as a semi-continuous signals Obviously, the datamust be sampled with a fixed supplied frequency to contain enough information

to process a special kind of parametric identification [18] including the

”persis-tent excitation” condition and so on Here we suggests to apply the least-mean

square algorithm (see [18]) as well as the integral sliding mode adjustment toattain this aim

a) Mean Least Square (MLS) application

Eq (5) in its integral form is

The Matrix Least Square estimate Φ iden

b) Integral Sliding Mode application

Define an auxiliary sliding surface as

s P (t, x t ) := χ t+ ¯x t (26)

where χ (t) is some auxiliary variable and ¯ x t is an artificial reconstruction

of the continuous nonlinear dynamics using the classical discrete time niques for sampling, holding and interpolating By direct differentiation of (26)

tech-we get

˙s P (t, δ t) = ˙χ t+

, s P (0, x0) = 0 (27)

The main objective is to enforce the sliding mode to the surface s (t, δ t) = 0

=−x0 and define the relay control v tby

effect in the realization of v if classical sliding controller is applied One of posible

188 I Chairez, A Poznyak, and T Poznyak

techniques to reduce the undesirable chattering performance is to use the output

v (av) t of a low-pass filter

instead of v t , for small enough τ (In practice τ 0.01).

The equation (29), may be rearrange as

So, it can be applied the LSM approach discussed above

3.1 Quasi Separation Principle

For the given uncertain system (1) the tracking performance index is as follows

As it follows from (32), to minimize the upper bound for J t(31) it is sufficient

to minimize the bounds for two terms in (32): the upper bound (18) for the

first term J t,est is already guaranteed by the DNN-method application (see thesubsection 1.2.5) valid for any control satisfying (9); the upper bound for the

second term J t,track in (32) can be minimized by the corresponding design of

the control actions u t in (11) providing that this control is from U adm (9) Thisdesigning can be realized by the high-order sliding mode approach [13] The

relation (32) is referred to as the Quasi Separation Principle.

3.2 High-Order Sliding Mode Neurocontrol

In general, the trajectory tracking problem may be treated as a special constraint

on the tracking error dynamics, i.e., these trajectories must belong to a special

surface S = 0 where S = S (ˆ x t − x ∗

t ) := S (δ t) This surface is closely related

with the generalized performance index in the linear quadratic regulator (LQR)

problem [22]

For nonlinear systems, one of the important structure characteristics is the

relative degree [10] According to [15], the desired sliding surface S can reflect

this property having also the corresponding relative degree r with respect to the tracking error δ t This property is related to the following system of differentialequations:

The high-order sliding mode control as it is suggested in [15] is able to drive

the tracking error δ t to the surfaces (33) in finite time practically without theundesired chattering effect

Remark 4 Such controller design requires the exactly knowledge of relative

de-gree r This information is always available in the case of DNN sliding mode

control since the dynamics, which we are traying to reach for a given DNN, isalways known (we constructed it according to (33))

The controller to be designed is expected to be a discontinuous function of thetracking error if we are going to apply the high-order sliding mode approach that

requires the real-time calculation of the S-successive derivatives S (i) , i = 1 r −1.

The collection of equation (33) represents a manifold The adaptive version ofthe high-order sliding mode control is proposed below as a feedback function of

The task for the desired high-order sliding mode controller is to provide in finite

time the convergence to S = 0 keeping (33) By the definition of the relative degree r of S the control action u t firstly appears only in the r-th derivative of

φ i,r = S (i) + β i N (r −i)/(r−i+1)

i −1,r Ψ i −1,r , N i,r =S (i)+ β i N (r −i)/(r−i+1)

190 I Chairez, A Poznyak, and T Poznyak

a high-order sliding mode controller applicable to all nonlinear systems (1) with

relative degree equal to r For small relative degrees the controllers are as

Since the controllers given above require the direct measurement of  ˙S ,  ¨S ,

etc., this can be realized by the direct application of the high-order sliding mode differentiator (see [14] and [15]) given by (j = 1, , n)

af-fecting the tracking error surface S, there exist positive constants μ i and s i(see

[16]) depending exclusively on the differentiator parameters (λ k , k = 1 n) such

• for systems with relative degree two;

• or for systems with relative degree one introducing an integrator in the loop

(twisting-as-a-filter or super-twisting)

In this chapter, the latter approach is used The control law u t (36) is thecombination of two terms: the first one is defined by its discontinuous timederivative, while the other one (which appears during the reaching phase only)

is a continuous function of the available sliding variable The algorithm isdefined by

3.3 Main Result on Quality of a Sliding Mode Neurocontrol

Once the nonlinear observer and the adaptive (output based) HOSM controller

have been designed, we may formulate the main result on the neuro-

tracking-control for the class (1) of uncertain nonlinear dynamic system subject to state

and output external perturbation

Theorem 3 For the uncertain nonlinear systems (1) under the assumptions

A1-A4, the sliding mode neurocontrol (36), which uses the auxiliary signals ˆx t

generated by DNN observer (11), provides the following quality (on average) of the desired dynamic x ∗

4.1 Second-Order Mechanical System

The most of industrial manipulators are equipped with the proportional andderivative (PD) controller However, their realization has two crucial points: