Sliding mode in automatic control systems

In industrial sector as well as military technology - especially in the improvement, modernization and manufacture of new weapons and technical equipment, the synthesis of high quality control systems is always an urgent demand. The assurance of quality for control systems operating under disturbance and uncertainty conditions of dynamic model requires the development of new control algorithms based on the modern control theory.

SLIDING MODE IN AUTOMATIC CONTROL SYSTEMS

Nguyen Vu1*, Tran Ngoc Binh2, Ha Thi Thi2

Abstract: In industrial sector as well as military technology - especially in the

improvement, modernization and manufacture of new weapons and technical equipment, the synthesis of high quality control systems is always an urgent demand The assurance of quality for control systems operating under disturbance and uncertainty conditions of dynamic model requires the development of new control algorithms based on the modern control theory Along with other tools of modern control theory such as optimal control, adaptive control, linear robust control, fuzzy control , sliding mode control is a relatively universal tool, easy in technical realization and have high effectiveness in practice This report presents sliding mode control features and its application for some classes in military technology

Keywords:Sliding mode, Sliding control, Uncertain parameters, Elastic backlash

1 INTRODUCTION OF SLIDING CONTROL

Consider a system described by the following dynamic equation:

xR AR BR uR

The idea of sliding mode is derived from transforming equation (1) into a set of equations [1]

(2 )

x A x A x a

x A x A x B u b









x R  x R

2

x in (2a) can be regarded as the virtual control signal of the system To obtain the system stability, state feedback control is used:

where:C1 Rm n m.(  )

However, because x2 is virtual control signal, equation (3) can not be always exist Then, the equation (3) can be written as:

;

SR CR

The problem is that, in order to make the system in (1) stable, control signaluneeds satisfying condition (4)

Condition (4) occurs when:

When m  1 then S S1 S mT

With ( ) ( ( ) 1 ( ))T

m sgn S  sng s sgn s ,

kis optional constant, k  0, then T

S S is determined as:

S S S CAx CB CB  CAx CB CB  k sgn S k S sgn S S



So, with control signal defined in (7), condition (4) will be met and system (1) is stable

In other words, when control signal is synthesized in the way that sliding condition (4) is satified, system (1) will be stable This is the nature of sliding mode control, including two-steps: first step is to select the super sliding surface S Cx0in the way that when state of the system is drawn into this super sliding surface, they will be going to the origin

of coordinate And step two is to synthesize the control signal uso that the state of the system is always drawn to the selected super sliding surface Control signal can be selected according to (7) or can be synthesized by other algorithms but all aim to ensure the existence of inequality (5)

Because in (7) u is selected with an optional constant k, so depending on the selection

of constant k, the stable margin of the system can be low or high That is the important feature of the sliding mode control, which makes systems using sliding mode control become stable against uncertain of system model as well as against any disturbance on the system This is proven when considering the specific object classes below

2 SLIDING MODE FOR SYSTEMS WITH UNCERTAIN PARAMETERS

Consider a system with state equation:

with xR n, ( )A t  A A t( )R n n. ,A is the known component of A t( ), A t( ) is

known component of B, B t( ) is an unknown component of B t( ) Without loss of generality, choose

2

0 ( )

( )

B t

B t

linear transformation to bring system to this form) [1]

With that assumption, system in form (8) can be rewritten as:

x A t x A t x

x A t x A t x B t u









Similar to section 1, the super sliding surface is selected so that the system in the following equation:

x 1 A t x11( ) 1 A t12( ).(  C x1) 1 (10)

is stable

This can be done by the setting pole method [2] so that the stability margin of system

is suficient to ensure the stable operation of system in full variable range of A11and A12 The remaining issue is to identify control vector uso the there exist a sliding mode on the selected sliding surface:

For systems with input matrixBas constant matrix, this issue has been solved quite completely in[3] For systems that have both input matrix and state matrix are variable and unstable, in order to obtain the existance of sliding mode control, control vector uis determined by the follwing theorem:

Theorem 1 Given system in form (8) where A t( )AˆA B t, ( )BˆB System in

(8) has the existance of sliding mode on the super sliding surface S Cx0 if the follwing conditions are satisfied:

1 u (CBˆ) 1CAxˆ (CBˆ) 1k sgn S. ( ) (CBˆ)  1 sgn S( )

m

Const

   

2 c i a j x j k

n



Where ciis row vector i of matrix C, aˆjis colum vector jof matrix A, x jis

jthcomponent of state vector

m



Where ciis row vector i of matrix C, b jis colum vector j of matrix B, u jmax is maximum value of jthcontrol signal

Above theorem is proven as:

From (12), combining (13) with (14) we have:

SCxCAx CAx CBu   CBu

S S S CAxCAxCAxk sgn S S  sgnS CBu

Right hand part of equation (15) including two components Consider term CBu in

second component:

T

i j j m j j

CBu c b u c b u

Defining:

1

m

j

c b u sgn s  



 

From (14), we have c b u i j j

m





    Therefore, the second component can be rewritten as:

1

m

S  sgn S CBu S sgn S CBu sgn S  S sgn S    (17)

Similarly, the first component which includesexpression CAx is defined as:

1

T

j j m j j

CAx c a x c a x

Defining:

1

n

j

c a x sgn s k 



 

 

From (13), we have: i j j k

c a x

n



Therefore, the first component can be rewritten as:

1

m

S  k sgn S  CAx   S sgn S CAx sgn S   k  S sgn S    (18)

Because

 ( )  ( ).

T

S k Sgn S S Sgn S k

S CAx S Sgn S Sgn S CAx

S Sgn S Sgn S





Combining (17) with (18) into (15), obtaining:

T

S S 

This means that system will slide stably on the super sliding surfaceS = 0 Theorem has

been proven

have variable parameters in a narrow range as well as systems that contain unstable components that satisfy conditions (13) and (14) Application of silding mode to synthersize systems effected by disturbance will be presented in section 3 below

3 SLIDING MODE FOR SYSTEMS EFFECTED BY DISTURBANCE

In order to synthesize slide controllers for systems operating under the influence of disturbance, there are two problems that arise: synthesize systems of intercepted or known disturbance and estimate disturbance values to compensate for disturbance effects

3.1 Synthesis of sliding mode control system for system under disturbances

Given the system in the following form:

( )

Without loss of generality, system in (19) can be written as:

( ) , ( )

x A x A x Q f t

x A x A x B u Q f t









Where:f t( )R r is rcomponents disturbance

( ).

1 n m; 2 m, 1 n m r, 2 mxr, m

x R  x R Q R  Q R u R

suitable size The problem for synthesis of control system can be divided into two steps: step one: selects S  C x1 1 I xm 2  0 sliding surface so that the system:

is stable under the effect of disturbance and step 2: synthesize controller so that the sliding mode exists on the selected sliding surface

Step 1 can be solved by the setting pole method for state feedback system with stable margin so that under the effect of disturbanceQf t( ), system (21) is always nearly stable Step 2 can be solved based on the following theorem:

Theorem 2:

System in form x AxBuQf t( ) will slide on a super sliding surface if the following conditions are satisfied

1 Control signalis synthersized by equation :

u CB CAx CB   sgn S

2 c i q j f j

n





(23) Where:ciis row vector of Cmatrix, qjis colum vector of Qmatrix , f j is the jth component

Similar to theorem 1, theorem 2 is proven as following:

Defining

1

.sgn( )

n

j





From (23) we have c q f i .j j

n



 , so i  0   i 1, , , n r

 1 

r

S S   S S   

was proven

In summary, when the value of disturbance was estimated by components

c q f , the controller is synthersized by theorem 2 will assure sliding mode for system and make the system nearly stable It is important to estimate the range of disturbance and if possible, determine exactly the value of disturbance to compensate effects of disturbance to systems

3.2 Algorithms define effects of disturbance for system

The method of determining the disturbance impact on the system is to use the standard model of the object Based on the difference in output between the model and the real system, the estimator will give the total value of the disturbance and the system model error [4, 5]

The identifier of total disturbance and modeling error refer to state of system was built

in [3], in there, the object is described by the following dynamic equation:

To determine the impact of Qf to system, we need to know exact model of system However, in many cases, system model does not accurately reflect the dynamic of system:

Where AM  A   A with A is an unkown parameter of system, or the difference

Let F  Ax   Qf , we have:

and system in (24) can be rewritten as:

M

The synthesis of the sliding controller for system in (29) can be performed similarly to system in (19)

In many cases, it would be easier to synthesize the system if the total disturbance and model error were returned to the system input Then, the control signal u is synthesized from two components, one eliminating all effects of disturbance and one model error, the other one is responsible for controlling the system with the standard model of the desired trajectory [5]

Given systems in form as:

x t  Ax t Bu t B F x B D t (30) Where A B, matricies are constant matricies , F x( ) is uncertain component of system referred to as an input and also known as system state dependant disturbance D t( )is effect of disturbance to system and referred to as an input

Using standard model of system as:

And building the system structure diagram which has the total disturbance identifier and the total disturbance compensator according to the following figure 1:

Fig 1 The adaptive control system structure diagram with the disturbance identifier

and uncertain nonlinearity

Defining is difference between model and system, we have:

( ) ( )

where: F x ( )  F x ( )  F x  ( ) and D t  ( )  D t ( )   D t ( ) are estimated error of total

disturbance (disturbance depend on state and disturbance depend on time) of system,

( )

F x and D t ( )are etimation value of disturbance

To identify the total disturbance, an adaptive identifier based on the three layer RFB neural network was proposed in [5]

Thus, with the direct transformation or use of an adaptive identifier, total effect of disturbance and modeling error has been determined By the direct compensation method

or compensation in sliding controller, these disturbance are completely eliminated, the sythesis of system is performed similarly to systems which have clear and invariant model

4 SLIDING MODE FOR MILITARY DRIVING SYSTEMS

The process of improvement, modernization Military driving systems leads to electrical driving system includes a nonlinear elastic backlash To overcome this situation, there are many methods proposed such as using PID controller, adaptive control systems That method have problems such as there exists oscillation around an equilibrium point and long transition time Using sliding mode for synthesizing elastic backlash systems is an effective solution that is able to overcome the above mentioned problems [9, 10] Using sliding mode for synthesizing elastic backlash systems will be present below These systems are divided into two blocks: before and after backlash

The block before backlash receives control moment as the input signal In case of the position between of two block lies in the backlash, this whole moment is the input of system, when the backlash is closed, a part of momentum is transmitted to the second block, which is after backlash This is the momentum transmitted to the second block The dynamic equation of before backlash block is:

(1) (1) (1) (1) (1) ( 2)

(1) (1) (1)











(33)

And of after backlash is:

( 2) ( 2) (2) (2) ( 2)

(2) ( 2) (2)











where: x(1), y(1), A(1), B(1), D(1) are state vectors respectively, output signal and dynamic matrices of before backlash block; x(2),y(2),A(2),B(2),D(2) are state vectors, output signal and dynamic matrices of after backlash block, respectively; u(2)is virtual control signal of second block, defined as:

(2)

( )

u

y y

 







(35) Where, ( ) is less sensitive nonlinear defined as:

1

0 ( )

khi

 

 



 

where: k 1 is hardness factor of the elastic part, is the wide of the backlash

In order to synthesize system (33) and (34) in [9] proposes using sliding mode based on standard model of two blocks: before and after backlash An effective method to synthesize systems with virtual control signal is to use the rolling method [11]

The algorithm for synthesizing sliding controller based on rolling method for elastic systems in (33) and in (34) will be present below

Firstly, synthesizing system in (34), determine u(2) so that sliding mode on sliding surface S2 c(2)x(2) is existed Using equivalent control [1], u(2)is determined as:

(2) (2) (2) (2) 1 (2) (2)

u C B  C A x  S

where:2is optional positive constant

and (2)

u is selected from (37), system (34) will operate in sliding mode, because:

(2) (2) (2) (2) (2) (2)

S S    S C A x  C B u   S  S   S 

After determining virtual control signal, input signal for system in (33) is determined as:

(2) (2)

2

2

2

1

0 1

0

t

k

k

















After determining y t(1), the problem is to synthesize the prediction control system for system (33) with input measured disturbance ku(2) For synthesizing this system, using the sliding mode and following steps presented in section 3

5 CONCLUSION

Sliding mode control systems presented in this paper have demonstrated the versatility

of sliding mode control Systems using sliding mode control guarantees technical specifications as desired Results of using sliding mode control systems are presented and illustrated in many reports [9, 10]

Sliding mode is not only efficient in systems with uncertain parameters, in systems with the time varying disturbance and state disturbance, in nonlinear elastic backlash systems but it is also efficient in another systems, specifically when combining the sliding mode with other control algorithms such as prediction control [12], adaptive control [13], and fuzzy control [14]

Incorporating sliding mode control with other control algorithms will be a very interesting research direction and will bring high efficiency to modern control systems

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TÓM TẮT

CHẾ ĐỘ TRƯỢT TRONG CÁC HỆ THỐNG TỰ ĐỘNG

Trong các lĩnh vực công nghiệp nói chung và trong kỹ thuật quân sự nói riêng, đặc biệt là trong cải tiến, hiện đại hoá và chế tạo mới các loại vũ khí, khí tài, việc tổng hợp các hệ thống điều khiển có chất lượng cao luôn là một yêu cầu bức thiết Việc đảm bảo chất lượng cao cho các hệ thống điều khiển khi hệ thống hoạt động trong điều kiện có nhiễu, trong điều kiện có bất định trong mô hình động học đòi hỏi phải xây dựng các thuật toán điều khiển mới trên nền lý thuyết điều khiển hiện đại Song song với các công cụ khác của lý thuyết điều khiển hiện đại như điều khiển tối ưu, điều khiển thích nghi, điều khiển bền vững tuyến tính, điều khiển mờ , điều khiển trượt là một công cụ tương đối đa năng, dễ thể hiện kỹ thuật và có hiệu quả cao trong các bài toán thực tiễn Báo cáo sẽ trình bày về điều khiển trượt và ứng dụng của nó cho một số lớp đối tượng trong kỹ thuật quân sự

Từ khóa: Chế độ trượt, Tham số bất định, Khe hở đàn hồi

Received date, 02 nd May, 2017 Revised manuscript, 10 th June, 2017 Published, 20 th July, 2017

Author affiliations:

1

The Department of Military Science;

2

The Control, Automation in Production and Improvement of Technology Institute; *

Corresponding author: hathihathu@gmail.com