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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.
Nghiên cứu khoa học công nghệ 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 INTRODUCTION OF SLIDING CONTROL Consider a system described by the following dynamic equation: (1) x Ax Bu , n n.n n.m m where: 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] where: (2a ) x1 A11 x1 A12 x2 (2b) x2 A21 x1 A22 x2 B2 u nm m x1 R ; x2 R (2) x2 in (2a) can be regarded as the virtual control signal of the system To obtain the system stability, state feedback control is used: x2 C1 x1 (3) where: C1 R m.( 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 C1 x1 Ix2 Cx (4) where: S R m ; C R m n The problem is that, in order to make the system in (1) stable, control signal u needs satisfying condition (4) Condition (4) occurs when: when m (5) S S T When m then S S1 S m Or : S T (CAx CBu ) By defining: when S 0 u (CB ) 1 CAx (CB ) 1 k sgn ( S ) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 - 2017 (6) (7) 71 Điều khiển – Cơ điện tử - Truyền thông sgn( S ) ( sng ( s1 ) sgn( sm ))T , k is optional constant, k , then S T S is determined as: With S T S S T CAx (CB).(CB)1 CAx CB(CB)1 k sgn( S ) k S T sgn( S ) 0, 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 in 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 u so 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 SLIDING MODE FOR SYSTEMS WITH UNCERTAIN PARAMETERS Consider a system with state equation: x A(t ) x B(t )u (8) with x R n , A(t ) A A(t ) R n n , A is the known component of A(t ) , A(t ) is unknown component of A(t ) and varies over time; B (t ) B B (t ) R n m , B is the known component of B , B (t ) is an unknown component of B(t ) Without loss of m.m , with B2 (t ) B2 B2 (t ) R , (other cases using B ( t ) generality, choose B (t ) linear transformation to bring system to this form) [1] With that assumption, system in form (8) can be rewritten as: x1 A11 (t ) x1 A12 (t ) x2 x2 A21 (t ) x1 A22 (t ) x2 B2 (t ).u (9) Similar to section 1, the super sliding surface is selected so that the system in the following equation: x1 A11 (t ) x1 A12 (t ).(C1 ) x1 (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 A11 and A12 The remaining issue is to identify control vector u so the there exist a sliding mode on the selected sliding surface: (11) S Cx C1 x1 I m x2 72 N Vu, T N Binh, H T Thi, “Sliding mode in automatic control systems.” Nghiên cứu khoa học công nghệ For systems with input matrix B as 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 u is determined by the follwing theorem: Theorem 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 if the follwing conditions are satisfied: ˆ (CBˆ ) 1 k sgn( S ) (CBˆ ) 1. sgn( S ) u (CBˆ ) 1 CAx Where sgn( S ) ( sgn( s1 ) sgn( sm ))T (12) Const , ci a j x j k , n (13) of matrix C , aˆ j is colum vector j of matrix A , x j is i Where ci is row vector j thcomponent of state vector ci bj u j max m Where ci is row vector i i , j (14) of matrix C , b j is colum vector j of matrix B , u j max is maximum value of j thcontrol signal Above theorem is proven as: From (12), combining (13) with (14) we have: ˆ CAx ˆ CBˆ (CBˆ ) 1 k sgn( S ) CBˆ (CBˆ ) 1 CAx S Cx CAx ˆ CAx CBu ˆ CBu S Cx CAx ˆ CAx CAx ˆ k sgn ( S ) S T ( sgnS CBu ) (15) S T S S T CAx in Right hand part of equation (15) including two components Consider term CBu second component: m m c b u c b u CBu i j 1 j j m j 1 j j T (16) m Defining: c b u sgn( s ) i j j i i, j 1 From (14), we have ci bj u j m , so i i 1, , m Therefore, the second component can be rewritten as: ) S T sgn( S )(CBu sgn( S ) ) S T sgn( S ).( )T (17) S T ( sgn( S ) CBu m is defined as: Similarly, the first component which includesexpression CAx n n c a x c a x CAx j j m j j j 1 j 1 T n Defining: c a x sgn(s ) k i j j i i j 1 Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 - 2017 73 Điều khiển – Cơ điện tử - Truyền thông From (13), we have: ci a j x j k , so i n i 1, , n Therefore, the first component can be rewritten as: ) S T sgn( S )(CAx sgn( S ) k ) S T sgn( S ).( )T (18) S T ( k sgn( S ) CAx m Because S T k Sgn( S ) S T Sgn( S ) k S T Sgn( S ).Sgn( S ).CAx S T CAx S T Sgn( S ).Sgn( S ) Combining (17) with (18) into (15), obtaining: S T S This means that system will slide stably on the super sliding surfaceS = Theorem has been proven Theorem provides an algorithm to determined the control vector u for systems which 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 below 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: x Ax Bu Qf (t ) (19) Without loss of generality, system in (19) can be written as: x1 A11 x1 A12 x2 Q1 f (t ) , x2 A21 x1 A22 x2 B2 u Q2 f (t ) Where: f (t ) R r is x1 R nm (20) r components disturbance m ; x2 R , Q1 R ( n m ).r , Q2 R mxr , u R m ; matrices A11 , A12 , A21 , A22 , B2 have suitable size The problem for synthesis of control system can be divided into two steps: step one: selects S C1 x1 I m x2 sliding surface so that the system: x1 A11 x1 A12 x2 Qf (t ) (21) is stable under the effect of disturbance and step 2: synthesize controller so that the sliding mode exists on the selected sliding surface Step can be solved by the setting pole method for state feedback system with stable margin so that under the effect of disturbance Qf (t ) , system (21) is always nearly stable Step 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 Control signalis synthersized by equation : (22) u (CB ) 1 CAx (CB ) 1 sgn ( S ) 74 N Vu, T N Binh, H T Thi, “Sliding mode in automatic control systems.” Nghiên cứu khoa học công nghệ ci q j f j (23) n q j is Where: ci is row vector of C matrix, colum vector of Q matrix , f j is the j th component Similar to theorem 1, theorem is proven as following: n Defining c q f sgn( S ) i j j i i j 1 , so i i 1, , n, r n Therefore, we have S T S S T sgn( S ) CQf sgn( S ) From (23) we have ci q j f j S S S sgn( S ) , , r T T T This means that system will slide stably on the super sliding surface S Theorem was proven In summary, when the value of disturbance was estimated by components ci q j f j , the controller is synthersized by theorem 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: x Ax Bu Qf (24) 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: xM AxM Bu , (25) Where AM A A with A is an unkown parameter of system, or the difference between the model and the real system; B Const Let e x xM (26) Qf then, we have: e Ae Ax (27) Qf , we have: Let F Ax F e Ae (28) and system in (24) can be rewritten as: x AM x Bu F (29) The synthesis of the sliding controller for system in (29) can be performed similarly to system in (19) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 - 2017 75 Điều khiển – Cơ điện tử - Truyền thông 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: (30) x (t ) Ax (t ) Bu (t ) B F ( x ) B D (t ) 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: (31) xM (t ) AxM (t ) Bu (t ) And building the system structure diagram which has the total disturbance identifier and the total disturbance compensator according to the following figure 1: Fig The adaptive control system structure diagram with the disturbance identifier and uncertain nonlinearity Defining is difference between model and system, we have: ( x ) BD (t ) A BF (32) ( x) and D (t ) are estimated error of total ( x) F ( x) F (t ) D(t ) D where: F disturbance (disturbance depend on state and disturbance depend on time) of system, ( x) and D (t ) are etimation value of disturbance F 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 76 N Vu, T N Binh, H T Thi, “Sliding mode in automatic control systems.” Nghiên cứu khoa học công nghệ 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) x A x B u B ku (1) (1) (1) y D x (33) And of after backlash is: x (2) A(2) x (2) B (2)u (2) (34) (2) (2) (2) y D x (1) (1) (1) (1) where: x , y , A , B , D (1) are state vectors respectively, output signal and (2) dynamic matrices of before backlash block; x , 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: u (2) ( ) y1 y2 (35) Where, ( ) is less sensitive nonlinear defined as: 0 k1 ( ).sgn( ) ( ) (36) where: k1 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: u2(2) (C (2) B (2) ) 1 C (2) A(2) x (2) sgn S (37) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 - 2017 77 where: is optional positive constant Điều khiển – Cơ điện tử - Truyền thông and u(2) is selected from (37), system (34) will operate in sliding mode, because: (2) S1S2 S2 [C (2) A(2) x C (2) B(2)u (2) ] S2 sgn(S2 ) S2 After determining virtual control signal, input signal for system in (33) is determined as: (2) (2) u2 k u y (2) yt(1) u y (2) u2 k y (2) sgn( y (2) ) u2 After determining yt(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 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 REFERENCES [1] Utkin.V.I, “Sliding Modes and their Application in Variable structure systems”, Moscow, Mir, 1978 [2] Nguyễn Doãn Phước, Phan Xuân Minh, “Điều khiển tối ưu bền vững”, Nhà xuất Khoa học Kỹ thuật, Hà Nội, 2001 [3] Nguyễn Vũ , “Về phương pháp tổng hợp hệ điều khiển bền vững”, Luận án TSKT, Hà Nội, 2005 [4] Cao Tiến Huỳnh, Nguyễn Vũ, “Về phương pháp tổng hợp hệ thống bền vững, rời rạc, có trễ”, Tuyển tập báo cáo khoa học, Hội nghị toàn quốc lần thứ năm Tự động hoá, VICA5, 2002 [5] Cao Tiến Huỳnh, Nguyễn Vũ, Nguyễn Trung Kiên, Ngơ Trí Nam Cường, “Về phương pháp tổng hợp hệ điều khiển thích nghi cho lớp đối tượng phi tuyến tác động nhiễu bên ngồi”, Tạp chí Nghiên cứu Khoa học Công nghệ quân sự, số 17 tháng 02/2012, tr 06 - 15 [6] Nordin M, “Nonlinear backlash compensation for speed controlled elastic systems”, Stockholm, 2000 78 N Vu, T N Binh, H T Thi, “Sliding mode in automatic control systems.” Nghiên cứu khoa học công nghệ [7] Amstrong B and Amin B, “PID control in the pressure of static friction: A Comparison of algebraic and describing function analysis”, Milwankee, 1998 [8] Brandenburg G and SehayerU, “Influence and partial compensation of backlash for position controlled elastic two – mass system”, Grenoble, 1987 [9] Nguyễn Vũ, Tăng Thanh Lâm, Lê Việt Hồng, “Tổng hợp hệ thống điều khiển cho lớp đối tượng phi tuyến dạng khe hở đàn hồi”, Tạp chí Nghiên cứu Khoa học Cơng nghệ quân sự, số tháng 08/2009 [10].Nguyễn Vũ, Lê Việt Hồng, Phạm Thị Phương Anh, “Nâng cao chất lượng hệ thống điều khiển truyền động cho hệ thống truyền động bánh răng”, Tạp chí Nghiên cứu Khoa học Cơng nghệ quân sự, số 22 tháng 12/2012 [11].Chieh Chen, “Backstepping Control Design and its Applications to Vehicle Lateral Control in Automated Highway Systems”, 1996 [12].Cao Tiến Huỳnh, Nguyễn Vũ, “Kết hợp thuật tốn điều khiển đón trước thuật tốn điều khiển có cấu trúc biến đổi cho lớp đối tượng”, Tuyển tập báo cáo khoa học, Hội nghị tồn quốc lần thứ ba Tự động hố, VICA3, 1998 [13] Nguyễn Vũ, Phạm Tiến Dũng, Lê Ngọc Quyết, “Kết hợp điều khiển theo chế độ trượt với điều khiển thích nghi cho hệ có tham số biến đổi dải rộng”, Tạp chí Nghiên cứu Khoa học Công nghệ quân sự, số đặc san TĐH tháng 04/2014 [14].Cao Tiến Huỳnh, “Điều khiển trượt mặt mờ.” Tuyển tập báo cáo khoa học, Hội nghị tồn quốc lần thứ tư Tự động hố, VICA4, 2000 TÓM TẮT CHẾ ĐỘ TRƯỢT TRONG CÁC HỆ THỐNG TỰ ĐỘNG Trong lĩnh vực cơng nghiệp nói chung kỹ thuật quân nói riêng, đặc biệt cải tiến, đại hoá chế tạo loại vũ khí, khí tài, việc tổng hợp hệ thống điều khiển có chất lượng cao yêu cầu thiết Việc đảm bảo chất lượng cao cho hệ thống điều khiển hệ thống hoạt động điều kiện có nhiễu, điều kiện có bất định mơ hình động học đòi hỏi phải xây dựng thuật tốn điều khiển lý thuyết điều khiển đại Song song với công cụ khác lý thuyết điều khiển đại điều khiển tối ưu, điều khiển thích nghi, điều khiển bền vững tuyến tính, điều khiển mờ , điều khiển trượt công cụ tương đối đa năng, dễ thể kỹ thuật có hiệu cao toán thực tiễn Báo cáo trình bày điều khiển trượt ứng dụng cho số lớp đối tượng kỹ thuật quân Từ khóa: Chế độ trượt, Tham số bất định, Khe hở đàn hồi Received date, 02nd May, 2017 Revised manuscript, 10th June, 2017 Published, 20th July, 2017 Author affiliations: The Department of Military Science; The Control, Automation in Production and Improvement of Technology Institute; * Corresponding author: hathihathu@gmail.com Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 - 2017 79 ... following steps presented in section CONCLUSION Sliding mode control systems presented in this paper have demonstrated the versatility of sliding mode control Systems using sliding mode control. .. 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... T Thi, Sliding mode in automatic control systems. ” Nghiên cứu khoa học công nghệ SLIDING MODE FOR MILITARY DRIVING SYSTEMS The process of improvement, modernization Military driving systems