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In order to deal with unmodeled dynamics in large vehicle systems, which have an ill condition of the state matrix, the use of model order reduction methods is a good approach. This article presents a new construction of the sliding mode controller for singularly perturbed systems. The controller design is based on a linear diagonal transformation of the singularly perturbed model. Furthermore, the use of a single sliding mode controller designed for the slow component of the diagonalized system is investigated. Simulation results indicate the performance improvement of the proposed controllers.

The design requirements for high precision and high maneuverable missile, need it to be slender and long. Using light weight material and thin wall structure to design the motor for these long and thin missiles induced elasticity in the body. The rigid body model does not satisfy the needs of precision controller. Singular Perturbation analysis is used by many researchers to overcome such a problem [

The use of sliding mode control for singularly perturbed systems is explored. A number of articles have been published on this topic. Slotine [

We propose, in this article, to transform the full order singularly perturbed system into block diagonal form [

We also investigate the design of a single sliding mode controller for the full order system. This single sliding mode controller is designed based on the slow dynamics of the transformed system, the fast dynamics are considered as unmodeled high frequency dynamics.

The outline of this paper is as follows. Section 2 describes the singularly perturbed system and the appropriate transformations. Section 3 illustrates the sliding mode controller design for both reduced subsystems obtained using the diagonalization method. Section 4 describes the design of a single sliding mode controller for the full order system. Section 5 shows the simulation results.

The system considered in this paper may be represented in the following form,

x ˙ = A 11 x + A 12 z + B 1 u (1)

ε z ˙ = A 12 x + A 22 z + B 2 u (2)

where x ∈ R n , z ∈ R m , u ∈ R r and εis a small positive parameter. This is a form for standard linear time invariant singularly perturbed systems. Due to the high dimensionality of the full order system, singular perturbation theory is useful in decoupling the system into two reduced order subsystems considering the fact that fast modes are important only during a short initial period. After that period they are negligible and the behavior of the system can be described by its slow modes. The model with fast modes neglected is called the quasi steady state model (or zeroth order model) [

x ˜ ˙ 1 = A 11 x ˜ + A 12 z ˜ + B 1 u ˜ (3)

0 = A 21 x ˜ 1 + A 22 z ˜ + B 2 u ˜ (4)

where x ˜ , z ˜ and u ˜ are tracking vector error.

Thus we can get the following reduced system [

x ˙ s = A 0 x s + B 0 u (5)

where A 0 = A 11 − A 12 A 22 − 1 A 21 and B 0 = B 1 − A 12 A 22 − 1 B 2

The system (5) is called the zeroth order model and is an approximation of the slow varying state of the full order system. Assuming that the slow variables are constant during fast transients i.e. z ˜ = 0 and x ˜ 1 = x s = constant thus the approximated fast subsystem for the system given by Equation (1) and (2) can be defined as,

ε x ˙ f = A 22 x f + B 2 u f (6)

The eigenvalues of A_{0} are good approximations of the slow modes of the full order system for sufficiently small ε [

x ˙ f = z + L ( ε ) x (7)

This transforms the system described by Equations (1) and (2) into upper block diagonal form as shown in Equation (8)

[ x ˙ x ˙ f ] = [ A 11 − A 12 L A 12 R ( L ) A 22 + ε L A 12 ] [ x x f ] + [ B 1 B 2 + ε L B 1 ] u (8)

We required that matrix L satisfy

R ( L ) = A 21 − A 22 L + ε L A 11 − ε L A 12 L = 0 (9)

Then using the second stage of transformation

x s = x 1 − M x f (10)

To obtain,

[ x ˙ s x ˙ f ] = [ A s S ( M ) 0 A f ] [ x s x f ] + [ B s B f ] u (11)

where B s = B 1 − M B 2 − ε M L B 1 , B f = B 2 + ε L B 1 , A s = A 11 − A 12 L and A f = A 22 + ε L A 12 , also we required that n × m matrix M satisfies the linear algebraic equation,

S ( M ) = ε ( A 11 − A 12 L ) M − M ( A 22 + ε L A 12 ) + A 12 = 0 (12)

Thus the system has the required block diagonal or decoupled form,

[ x ˙ s x ˙ f ] = [ A s 0 0 A f ] [ x s x f ] + [ B s B f ] u (13)

where the slow and the fast variables x s and x f can be solved independently of each other. The eigenvalues of A_{s} are exactly the same as the slow poles of the original system. Similarly, the eigenvalues of A_{f} are the same as the fast poles of the full order model. The matrix L is dependent on the perturbation parameter, ε. Letting ε → 0 gives us L = L 0 = A 22 − 1 A 21 which transforms the diagonal model into the same as the quasi steady state model ( A s = A 0 ). Changing the value of the perturbation parameter ε to a considerable value and checking the difference between the eigenvalues of A_{s} and A_{0} gives us an indication if the quasi steady state is a valid approximation or not. One may expected that if A_{s} is very sensitive to the value of ε then using A_{0} instead of A_{s} may cause large errors in the approximation of the slow poles of the original system. Similarly, if A_{21} and A_{2}_{2} are dependent on ε, the approximation of the fast system where we put A f = A 22 may lead to an unstable system or increase the steady state error of the closed loop system.

Based on the pervious analysis the diagonalization method must be used instead of quasi steady state to decouple the full order system especially when the quasi steady state fails to describe the stability properties of the full order model correctly.

Equation (9) and (13) is an asymmetric Riccati equation which need special procedure to be solved reader can found more detail in the literature [

In this section we will design sliding mode controllers based on the reduced subsystems obtained from transforming the systems into block diagonal form. This is different from the methods proposed in the literature [

The slow sliding mode controller is designed using the slow reduced order model in (14). The equivalent control method [

u e q = − [ ( C s B s ) − 1 C s A s ] x s (14)

Using this control law in the slow subsystem obtained from the block diagonalized reduced system (14) we then obtained,

x ˙ e q s = [ I − B s ( C s B s ) − 1 C s ] A s x S (15)

To guarantee local stability for this controller we must ensure that the eigenvalues of the equivalent system given by equation (16) have negative real parts. A Lyapunov function is used to determine the discontinuous control law (u_{N}) that will satisfy the reaching condition [

u s = u e q + u N = − [ ( C s B s ) − 1 C s ] A s x s − ( C s B s ) − 1 η 1 sgn ( S s ) (16)

where u_{N} is a discontinuous control action that drives the state to the sliding surface.

The same argument used to design the slow sliding mode controller can be used to design the fast sliding mode controller. Define the linear fast switching surface S f = C f x f . The equivalent control method is used to determine the control law for the fast subsystem given by Equation (14). The control law is given by,

u f = − [ ( C f B f ) − 1 C f ] A 22 x f − ( C f B f ) − 1 η 2 sgn ( S f ) (17)

which ensures global asymptotic stability for fast subsystem. The control law for the full order model will be the composite of the slow and fast controllers as follows,

u = u s ( x s ) + u f ( x f ) (18)

Since the eigenvalues of the reduced order subsystems is the same as the full order model this means that the linear transformation preserves the stability condition for the closed loop system, and taking the stability analysis proposed by Kokotovic [

In this section, a single sliding mode controller is proposed to control the full order model. The idea is to use the robust properties of the sliding mode controller to counter the effects of the unmodeled high frequency dynamics. We will consider that the unmodeled high frequency dynamics are represented by the fast subsystem. The main assumption is that the matrix A_{2}_{2} is stable while the full order model given by Equation (1) and Equation (2) may be unstable at some specific value of ε. The surface parameter for the single sliding mode controller is chosen based on Slotine’s [

S J = ( d d t + λ ) n − 1 x s (19)

where λ is the bandwidth of the system [

u J = − [ ( ∂ S J ∂ x s B s ) − 1 ∂ S J ∂ x s ] A s x s − ( ∂ S J ∂ x s B s ) − 1 η 1 sgn ( S J ) (20)

An advantage of using the proposed controller is that the control law in this case will depends only on the slow state. As a result there is no need for measurement of the fast state, which is usually difficult to measure.

Sliding mode controller used usually in missiles and spacecraft [

Consider a magnetic tape control system [

[ x ˙ 1 x ˙ 2 ε z ˙ 1 ε z ˙ 2 ] = [ 0 0.4 0 0 0 0 0.345 0 0 − 0.524 − 0.465 0.262 0 0 0 − 1 ] [ x 1 x 2 z 1 z 2 ] + [ 0 0 0 1 ] u (21)

y = [ 1 0 0 0 0 0 1 0 ] [ x 1 x 2 z 1 z 2 ] (22)

This can be put in standard singularly perturbed form as,

x ˙ = A 11 x + A 12 z + B 1 u

ε z ˙ = A 12 x + A 22 z + B 2 u

where

A 11 = [ 0 0.4 0 0 ] , A 12 = [ 0 0 0.345 0 ] ,

A 12 = [ 0 − 0.524 0 0 ] , A 22 = [ − 0.465 0.262 0 − 1 ]

B 1 = [ 0 0 ] , B 2 = [ 0 1 ] and ε = 0.1

The first 2 × 2 transformation matrix L has to satisfy,

R ( L ) = A 21 − A 22 L + ε L A 11 − ε L A 12 L = 0 (23)

This gives us the value ofL as,

L = [ 0 1.2412 0 0 ] (24)

The second 2 × 2 transformation matrix M has to satisfy

S ( M ) = ε ( A 11 − A 12 L ) M − M ( A 22 + ε L A 12 ) + A 12 = 0 (25)

Similarly this gives us the value of M as,

M = [ 0.0862 0.0325 − 0.9094 − 0.2489 ] (26)

The slow subsystem is given by,

x ˙ s = A s x s + B s u s

where A s = A 11 − A 12 L = [ 0 0.4 0 − 0.4282 ] and B s = [ − 0.0325 0.2489 ]

The fast subsystem can be described by,

x ˙ f = A f x f + B f u f (27)

where A f = [ − 0.4222 0.262 0 − 1 ] and B f = [ 0 1 ]

We notice that the eigenvalues of the reduced system are the same as the eigenvalues of the full order model. We design two sliding mode controllers based on the slow and fast subsystems. For the slow subsystem the control law will be,

u s = [ 0 0.4442 ] x s − 4.839 sgn ( S s ) (28)

For the fast subsystem the control law is,

u s = [ − 1.2665 − 0.214 ] x f − 2 sgn ( S f ) (29)

The control law for the full order model will be the composite of the slow and fast controls as follows:

u = u s ( x s ) + u f ( x f ) (30)

Since the fast subsystem is stable then based on the decoupled reduced order system we will design a sliding mode controller for the slow subsystem. The fast subsystem will be considered as unmodeled high frequency dynamics. Choosing the surface parameter as,

C s = [ 2 1 ]

the control law will have the form,

u s = [ 0 2.0217 ] x s − 5.4337 sgn ( S s ) (31)

a single sliding mode controller. It is shown that it is possible to use one sliding mode controller to control the full order model for singularly perturbed system given that the fast subsystem is stable and we choose the right surface parameters that do not cause excitation of the fast subsystem.

Two sliding mode controller designs for singularly perturbed systems have been proposed. The designs are based on a block diagonal transformation of the system into fast and slow subsystems. The first design method proposes using two separate sliding mode controllers, one for the slow subsystem and a second for the fast subsystem. Simulation results indicate improved performance in comparison to previously published design methods since the errors which are produced during the decoupled process have been avoided. In the second proposed design method a single sliding mode controller is designed only for the slow subsystem and the fast subsystem is considered as high frequency unmodeled dynamics. These allow us to avoid measuring fast state, which is usually difficult to measure. Simulation results indicate good performance where the proposed controllers have less control effort compare to dual controller used before.

The authors declare no conflicts of interest regarding the publication of this paper.

Ahmed, A.E.M. and Zohdy, M. (2021) Sliding Mode Control for Singularly Perturbed Systems Using Accurate Reduced Model. International Journal of Modern Nonlinear Theory and Application, 10, 1-12. https://doi.org/10.4236/ijmnta.2021.101001