Abstract—This paper considers a hybrid approach to control of linear dynamic and impulse controllable continuous-time disturbed linear descriptor systems. The first step in control is a design of continuous state feedback that makes system impulse-free. The obtained system is represented as a state space system of relative order zero. Based on this model, a full order discrete-time sliding mode control providing given pole placement is designed. The reaching control is completely decentralized and chattering-free. Simulations show a very good suppression of slow disturbances. All design steps and simulations require only standard MATLAB Toolbox.

I. INTRODUCTION

HE main characteristic of continuous-time (CT) linear descriptor models is that the derivative of coordinate x

cannot be calculated from control inputs and their present values. These models either mirror reality of a system, or are an intentional result of modeling. Linear electric circuits, for example, have descriptor models, as well as mechanical systems with mass matrix practically singular. Sometimes the design of the mechanical subsystem is a descriptor, as e.g. robot window washer. Singular model arises when outputs of state space sub-systems are inputs to other state space sub-systems, making a large complex system. The descriptor systems are more complicated than regular state-space systems and they require separate methods for control synthesis. Detailed presentation of differences between regular state- space systems and descriptor (or singular) systems are given in [1].

The properties and design control methods (mostly linear) of descriptor systems are well covered in publications [2]-[5]. Recently published book [6] extensively covers properties of descriptor systems and methods of their analysis and control design. Design methods are concentrated to desired dynamics, impulse elimination and regularization, while the disturbance effects are rarely considered.

This work was supported in part by the Ministry of Education, Science and Technological Development of the Republic of Serbia under Project Grant III44004.

B. Draženović is with the Faculty of Electrical Engineering, University of Sarajevo, Sarajevo, Bosnia and Herzegovina (e-mail: brana_p@hotmail.com).

Č. Milosavljević is with the Faculty of Electrical Engineering, University of Istočno Sarajevo, Sarajevo, Bosnia and Herzegovina (e-mail: milosavljevic@elfak.ni.ac.rs).

B. Veselić is with the Faculty of Electronic Engineering, University of Niš, Niš, Serbia (e-mail: boban.veselic@elfak.ni.ac.rs).

The sliding mode control can eliminate matched disturbances [7], and alleviate the effects of unmatched ones. The application of sliding mode control to CT descriptor systems is not widely treated. The most important contribution so far is [8] showing how to obtain chattering free control and disturbance elimination by applying second order integral sliding mode control. The design is performed on generic descriptor model.

Discrete-time (DT) descriptor systems controlled by sliding mode approach are rarely present in publications. Two contributions [9, 10] should be mentioned here. In [9] the so called sliding sector control method is proposed to control of DT single input descriptor system. This method produces in the reaching mode a control signal as series of pulses, and the free motion in the sliding sector. No design example was given therein. Paper [10] is dedicated to control of DT multivariable descriptor system with state delay. There was propounded that a serious chattering problem arose due to the sign function in DT reaching law control [11]. Then a new reaching law, based on least squares support vector machine (LS-SVM), was proposed in order to solve the chattering problem. Two examples are presented giving comparison between the two methods [10] and [11].

In this paper a different approach to avoid chattering and suppress disturbances is taken. A hybrid system consists of a linear continuous state feedback applied in the inner control loop, which has as a task to eliminate impulses. Once the system is made impulse-free, DT integral sliding mode (DISM), based on some chattering-free control algorithm such as [12-14], should be applied to achieve desired dynamics and suppress disturbances. The design steps are very simple and may be performed using standard MATLAB Control toolbox. No specialized software is needed to verify results by simulation. The design is easily implemented using contemporary technology.

The remainder of the paper is organized as follows. Section 2 describes mathematical model of CT linear descriptor systems and offers a state transformation method resulting in a standard state space form. Derivation of the DT model is given in Section 3. A suitable control design is presented in Section 4, which effectiveness is demonstrated in Section 5 on a numerical example. The paper ends with some concluding remarks and references.

Discrete-Time Chattering-Free Integral Sliding Mode Control of Continuous-Time Descriptor Systems

Branislava Draženović, Senior Member, IEEE, Čedomir Milosavljević, Boban Veselić, Member, IEEE

T

13th IEEE Workshop on Variable Structure Systems,VSS’14, June 29 -July 2, 2014, Nantes, France.

978-1-4799-5566-4/14/$31.00@2014 IEEE

II. DESCRIPTOR SYSTEM MODEL IN DYNAMIC FORMS The general form of a regular LTI descriptor system is

given by the following equation BuAxxE , (1)

where kx and mu are the state and control vectors, respectively. The matrices E and A are square and the polynomial )det( AsE is not identically equal to zero. This guaranties regularity that is the existence and uniqueness of solutions to descriptor linear systems. For a descriptor system rank(E)=n<k. Here k, the size of matrices E and A, is known as descriptor system order, and n is known as dynamic system order. Due to the rank deficiency of E, x cannot be calculated from (1) by inversion of the matrix E. The other requirements are that the system is dynamically and impulse controllable. This means that the dynamical subsystem, obtained after a suitable decomposition of a descriptor system, is fully controllable, as well as there exists a possibility for impulse elimination. These issues will be discussed later.

The question that must be answered in sliding mode application to descriptor systems is what types of sliding mode control are acceptable. The original model (1) is less suitable for this analysis than so named dynamic form, which will be used in this paper. The dynamic form is given by the following two sets of equations

uBwAzAz zzwzz , (2) uBwAzA wwwwz0 , (3)

where nz represents n dynamic coordinates and nkw represents (k-n) algebraic coordinates. The

dynamic system form may be obtained by a suitable nonsingular transformation from (1). A very convenient way [4,9] is the application of the singular value decomposition (SVD), supported by MATLAB. Namely, let

xVw

z T , (4)

where V is an unitary matrix defined by SVD of E, i.e.

TUSVE , 00

0S , nn . (5)

Here is a nonsingular diagonal matrix whose elements are non-zero singular values of E. Then

ITBTU

B

BAVTU

AA

AA T

w

zT

wwwz

zwzz

0

0,,

1

. (6)

A very important feature of descriptor systems is possible occurrence of impulses. Impulses may arise due to the changes in control, or due to the initial conditions. Impulses obviously are not acceptable in any practical application, and any control must first attend to this problem. The indication that the system is impulsive is the singularity of the matrix

wwA in (3). However, if the system is impulse controllable, a linear state feedback wKu ww may remove impulses. The system is impulse controllable if the following condition is

satisfied nkBA wwwrank . (7)

Here rank(B)=m. When a suitable wK is found, the correctional control is added as a term to original control. The obtained system has the same form as one given by (2) and (3), except that two original matrices wwA and zwA are replaced by )( wwww KBA and )( wzzw KBA respectively.

If matrix wwA is invertible the vector w may be expressed by z and u

uBAzAAw wwwwzww11 . (8)

If this value of w is replaced in the dynamic part (2), one obtains

uBAABzAAAAz wwwzwzwzwwzwzz )()( 11 . (9) The equations (9) and (8) have the form of an ordinary

state space system of relative order zero, if the coordinates w are considered as outputs. Rewrite these equations as

uBzAz cc , (10) uDzCw cc . (11)

This form may be represented as SYS structure, which allows application of Control toolbox from MATLAB.

Obviously, a sliding mode control may be applied in the space of coordinates z. This control in principle may be of reduced order or full order, using CT or DT control. First consider application of a switching control in CT SM. When such control is applied, coordinates z will react to step-like changes of control with change in slope only, producing typical zig-zag trajectories. However, the equation (11) indicates that a step in control will result in a step in coordinates w. Since original coordinates x are a linear function of z and w, any chattering in control will show as a chattering in descriptor system coordinates. Therefore, a switching sliding mode control does not seem to be feasible approach. Alternative approaches as having no infinite derivatives (as for example boundary layer control) or second order sliding mode control may result in a an acceptable sliding mode-like dynamics.

The next issue is to explore the DT sliding mode control. If we assume that control changes only in sampling instants, the model defined by (10) and (11) may be seen as a state space system and it may be converted into a DT system using any available software, as e.g. c2d command in MATLAB. The DT sliding mode control may be established in one step only and then maintained using equivalent control which is linear in coordinates z. If the initial value of z is far away from the sliding subspace, an excessive value of u may be needed. Such value of u may cause high changes in coordinates. But, there are a variety of reaching control methods described in [12,13, 15] that may require more than one step to reach sliding subspace, so the step changes needed to maintain switching mode will not be high and in the equilibrium point control may be zero if there are no disturbances. Moreover, in full order (integral) sliding mode

[16,17], the sliding mode starts immediately due to the adjustment of integrator initial values, and may be maintained using the equivalent control.

However due to the intrinsic properties of digital sliding mode, even a switching control can not completely suppress disturbances. First, any change in disturbance value in intervals between samples cannot be suppressed since control keeps a fixed value. Second, even in sample points, a complete suppression requires knowledge of disturbance. But, if the disturbances are slow, as it is often the case in industrial and electro-mechanical systems, it is possible to achieve a reasonably good disturbance elimination, using basically the estimated value of disturbance and fast sampling rate [18,19]. Based on these considerations, in this paper we design a DT integral (full order) sliding mode control of an impulse controllable descriptor system, and show by simulations that such a control is chattering-free and capable to deal with slow matched disturbances.

III. DISCRETE-TIME MODEL OF CONTINUOUS LTI DESCRIPTOR SYSTEM

The initial step in the design of a digital control for a CT system is to obtain its DT model. Such state space models show coordinate values in sampling instants, while it is understood that the value of the system coordinates will not exhibit very quick changes during the sampling interval. That may be achieved if the sampling frequency is high enough. This is not true for all descriptor systems. If the system is impulsive, its response to abrupt changes and even to some initial values may contain impulse functions [6]. There is no sampling frequency which is high enough to guarantee a usable DT model. Therefore, the digital control of CT descriptor system is limited to impulse-free systems only.

For DT control design purposes, let a regular LTI descriptor system, subjected to a bounded matched disturbance vector d, be considered. In such case the initial model (1) can be rewritten as

)( duBAxxE . (12) If the system is impulse-free, or made impulse-free, it can always be transformed into the form (10)-(11), which in the presence of a disturbance becomes

)( duBzAz cc , (13) )( duDzHw cc . (14)

Since (13) and (14) represents a standard CT state space model, DT counterpart can be easily found by imposing zero-order hold on the control. The obtained DT model is

)()()()1( kfkuBkzAkz dd , (15) )]()([)()( kdkuDkzHkw cc , (16)

where T

cA

T

cA

dTA

d TkdBekfBeBeA ccc

00

d))1(()(,d, .

(17)

IV. CONTROL DESIGN The first step in design is to check is the descriptor system

impulse controllable, and if so to find a state feedback that guarantees the impulse-free behavior. In available references the problem was treated as trivial, and it was solved heuristically for a low order system. We present here a simple algorithm that either declares that the system is not impulse controllable, or finds a suitable feedback that makes the system impulse-free.

Suppose that the rank deficiency of wwA is r. Since the matrix wB has m columns, r<m. The aim is to find in wwA r columns that are linearly dependent of other columns, denoted iwwA , and replace such column with the sum of

iwwA , , i=1...k-n, and one of columns of wB . The algorithm

starts with 1,wwA and proceeds to nkwwA , . In each step a new

column of wwA is added to the left side of matrix. If this extension increases the rank, it is left in place. If not, this is a linearly dependent column, and we subtract from it the next available new column from wB . If we find a linearly dependent column, and there are no available columns in

wB , the system is not impulse controllable. If such procedure ending with last column of wwA shows a full rang of built up matrix, the system is impulse controllable. The corresponding feedback matrix wK has ones in place (i,j) if j-th column of wB is added to i-th column of wwA . All the other elements are zero.

The resulting system matrix after closing the feedback has the form )( wwww KBA and it is invertible now. Adding some kind of linear feedback may be beneficial even if the system is impulse free, but the matrix wwA is ill conditioned or its determinant is very small. If the matrix wwA is ill conditioned, its inverse will not be dependable, and it will impede further design process. This problem may be resolved taking several approaches. An interesting one is given in [20] where condition and algorithm for singular value assignment are given. This problem is an interesting topic for a more detailed investigation. The next step is the design of DISM control.

According to the equations (15) and (16) it is obvious that sliding mode control strategy comprises only state coordinates z. If integral (full order) SM is to be organized then DT integral sliding manifold [17] should be defined as

),1()1()(

),()0()()(

kzEkk

kzDkzDkg

i

ii (18)

where mg , are the switching functions and integrator outputs, respectively and mBD di )(rank . Constant matrices iD and iE have full rank. To ensure that the motion starts the sliding subspace initial conditions of must ensure that g(0)=0. The equivalent control equ that

provides g(k+1)=0 can be found from (18) and (15) as

).0()(

)]()()()[()()(1

1

zDBD

kfDkkzEADBDku

idi

iididieq (19)

Obviously, such equivalent control is not feasible since it requires exact knowledge of disturbance f(k), which is generally unavailable in practice. Instead, a disturbance estimate )(

~kf can be employed in the feasible real control.

Then the switching function differs from zero )(

~)()1( kfDkfDkg ii , (20)

and depends on the disturbance estimate quality. Hence, a feasible control can be obtained from (19), using the identity

)0()()()( zDkzDkgk ii , in the form

)}(~

)()(])({[)()( 1 kfDkgkzEIADBDku iididi . (21)

Substitution of (21) into (15) gives integral sliding mode dynamics

),(~

)()(

)()()()()1(1

1

kfDBDBkf

kgBDBkzKBAkz

idid

diddd (22)

where ])([)( 1ididi EIADBDK . If the pair ),( dd BA

is controllable then a matrix K can be found that provides desired eigenvalue spectrum of the matrix )( KBA dd , which defines closed-loop poles. This implies that the matrix

iD can be arbitrary selected providing a full rank of di BD , while iE should ensure the desired dynamics and can be calculated as

)( KBIADE ddii (23) A suitable choice of iD made in [21] is to satisfy the

condition mdi IBD , which implies iD to be left pseudo-

inverse of dB , i.e. di BD . Such choice of iD results in a fully decupled control in which every sliding variable is controlled by its own control input.

V. NUMERICAL EXAMPLE AND SIMULATION RESULTS Consider a regular impulsive, but impulse controllable

descriptor system (12) defined by following matrices [6]

.010000

010001

,

100001

010000

001000

001010

000001

000100

,

000001

000000

010000

000100

000010

000001

T

B

AE

(24)

The system is made impulse-free by applying the linear CT feedback xK x , where

)(,100000

0010000 2

T IKVKK wwx .(25)

By applying SVD on (24) supplied with this linear feedback, the system is transformed into the form (13)-(14) defined by the matrices

.5.00

5.01,

5.05.005.0

5.05.005.0

,

5.01

5.01

00

5.00

,

5.05.005.0

5.05.015.0

0001

5.05.005.0

cc

cc

DH

BA

(26)

Initial conditions of the transformed system are given with T]21.034[)0(z and T]00[)0(w . The DT model

(15)-(16), needed for digital control design, in case of T=0.01s is defined by

.005038.0005038.0105.2005012.0

01005.001005.000

,

005.1005012.0105.2005038.0

005038.0005.1010025.0004987.0

105.2105.21010025.0

005012.0005012.0105.2005.1

T

5

5

55

5

d

d

B

A

(27) DT closed-loop poles of the subsystem (15) are set by the desired spectrum },,,{ T4T3T2T eeeep , which can be ensured by the conventional DT state feedback controller with the following gain matrix

41216413160552762314

26676208100761483446

....

....K . (28)

Matrices iD and iE that define DISM controller (21),

calculated according to di BD and (23), are obtained as

.412.15413.1505.27613.13

2667.62078.9574.138269.6

,00166.000166.099665.05.199

75.49751.4949956.0996.99

i

i

E

D

(29)

A. Simulations To validate the designed DISM controller for the given

descriptor system (24), simulations have been performed using standard MATLAB Simulink. Unable to build the simulation model in the original form (12), the transformed model (13)-(14) has been simulated. The original coordinates x is observed using inverse transformation

w

zVx . (30)

The system is subjected to a disturbance vector

)6(5

)12()5sin(5)(

th

thttd . (31)

DISM controller (21) without disturbance compensation ( 0)(

~kf ) has been tested, Fig. 1. To evaluate its

performance a comparison has been made with the performance of the conventional digital linear state feedback controller (28), designed to provide the same dynamics, Fig. 2. Obviously, the designed DISM controller provides much better response than the conventional state feedback. The corresponding control signals are given in Fig. 3 and Fig. 4, respectively. From these figures it is evident that the obtained SM control is chattering-free and similar in nature to the conventional DT feedback control.

Fig. 1. System response with the designed DT ISM controller.

Fig. 2. System response with the DT state feedback controller (28).

Fig. 3. Control signals in case of DT ISM controller.

Fig. 4. Control signals in case of DT state feedback controller.

Fig. 5. Switching functions dynamics

Separate states comparisons between the proposed and conventional controllers are depicted in Fig. 6. It can be seen from the simulation results that DISM control is an effective method for control of linear descriptor systems since control is without chattering. Barring that, DISM control is very effective in compensation of slow varying disturbances. Constant type disturbances are fully rejected while the impacts of other types of slow disturbances are significantly compensated.

Switching functions dynamics are depicted in Fig. 5. It is obvious that matched disturbances are directly reflected in switching functions channels g1 and g2. Also, it is evident that the control components 1u and 2u have separate influence on sliding variables g1 and g2 respectively, i.e. the above mentioned decupling is completely realized.

Further accuracy improvement can be accomplished by employing disturbance estimate )(

~kf in control (21). A very

useful disturbance estimation method are presented in [19] using only switching function measurement instead applying one step delayed disturbance estimator based on DT state-space model [17,18].

VI. CONCLUSION This paper investigates the problems of sliding mode control of uncertain LTI descriptor systems. It is shown that impulse controllable CT descriptor systems can be successfully controlled by using a hybrid two stage control. The first stage is continuous-time feedback control that eliminates impulsiveness. The second stage is discrete-time integral sliding mode (DISM) control. The DISM control is chattering-free method enabling fully rejection of constant type matched disturbances and significant compensation of other types of slowly varying matched disturbances. The design method is fully supported by existing software. The proposed control system has been validated through a numerical example of a multivariable impulsive descriptor system. Also, a comparison with a conventional linear state feedback control is given. The simulations show that the recommended control algorithm provides smoothness of the state variables and high steady-state accuracy under action of slowly varying matched disturbances. Further improvements of the proposed design approach may be obtained by implementation of disturbance compensation, based on the switching functions measurement.

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Fig. 6. State responses for DT ISM and conventional state feedback controllers.