SLIDING PI CONTROLLER DESIGNS WITH INTERGRAL SLIDING SURFACE FOR a CLASSS OF NONLINEAR SYSTEMS

ĐẠI HỌC QUỐC GIA THÀNH PHỐ HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC BÁCH KHOA KHOA CƠ KHÍ BỘ MƠN CƠ ĐIỆN TỬ  o0o  BÁO CÁO CUỐI KHĨA MƠN HỌC ĐIỀU KHIỂN NÂNG CAO GVHD: PGS.TS Từ Diệp Công Thành HVTH: Phạm Thanh Quyền MSHV: 1870622 THÀNH PHỐ HỒ CHÍ MINH THÁNG 01/2019  P a g e | 14 SLIDING PI CONTROLLER DESIGNS WITH INTERGRAL SLIDING SURFACE FOR A CLASSS OF NONLINEAR SYSTEMS Abstract: The Lyapunov stable theory and L2_gain control performance are employed to interpret a sliding controller design with the proportional-integral (PI) control form and the integral sliding surface First, a theoretical analysis of the considered sliding PI controller is provided Our analysis reveals that the proportional control term of the sliding PI controller, i.e., sliding proportional controller, can lead the energy of system state error to be bounded and thus, the system fundamental robustness can be established The integral control term of the sliding PI controller, i.e., sliding integral controller, can guarantee the system stability further Secondly, a direct adaptive design is further proposed to estimate the optimal proportional gain of the sliding proportional controller, in which the sliding integral controller is employed to handle the total uncertainty with unknown upper bound in the adaptive control design Therefore, the selection problem of the proportional gain is resolved and the valuable robust control property of the sliding integral controller is illustrated again Finally, three simulations are provided to demonstrate these results I SLIDING PI CONTROLLER DESIGN WITH L2_GIAN CONTROL PERFORMANCE The following nonlinear system is considered in the paper  x( n )  f ( X )  g ( X )u  d  f  gu  d (1)  y x   Where: f _is a bounded unknown continuous function g _is a bounded unknown continuous gain function X  [ x, x, , x( n1) ]T  [x1 , x2 , , xn ]T _the state vector t s(e)   x( n1)   ( ym( n )  ( n1) e( n1)  1e  0e)dt _The integral sliding surface ym  [ ym , ym , , ym( n1) ]T _The reference trajectory P a g e | 14 e  ym  y _The state tracking error e  [e( n1) , , e, e]T _The state tracking error vector Theorem 1: The following sliding PI controller can let the system (1) having the L2-gain control performance t us  u p  uI  CP sgn( g )s  CI sgn( g )  s( )d Where: CP _is a nonzero proportional gain CI _is an integral gain and CI  II DIRECT ADAPTIVE DESIGN OF SLIDING PROPORTIONAL CONTROLLER Theorem 2: The following adaptive law with a nonzero bounded adaptive rate  P can be used to estimate the optimal proportional gain CP  sgn( g ) P s t The sliding integral controller, as uI  CI sgn( g )  s( )d with a proper integral gain CI can lead the adaptive control system to be asymptotical stable in a finite time III SIMULATION The inverted pendulum car system is the plant as follows  x  x2   x2  f  gu  d Where: mt Lx22 sin x1 cos x1 mc  mt m cos x1 L(  t ) mc  mt g r sin x1  f  cos x1 mc  mt g m cos x1 L(  t ) mc  mt P a g e | 14 d  3cos(5t )  2sin[t (t  1)] _The disturbance Where: x1 (rad ) _is the angle of the pole x2 (rad / s) _is the angular velocity of the pole g r _is the acceleration due to gravity 9.8m / s mc _is the mass of the cart (1.0kg ) mt _is the mass of the pole (0.lkg) L=0.5m_is the length of the pole The common used parameters: 1  150 , 2  15 , x1 (0)  30(deg) , x2 (0)  , ym  sin(t ) Source codes: clear all clc close all %reference control t=0:0.01:10; Ts=0.01; ym=sin(t); ymd(1)=0; ym2d(1)=0; ym2d(2)=0; for i=2:length(t) ymd(i)=(ym(i)-ym(i-1))/Ts; end for i=3:length(t) ym2d(i)=(ymd(i)-ymd(i-1))/Ts; end %the system parameters gr=9.8; mc=1; mt=0.1; L=0.5; lamda1=15; lamda2=150; %Controller parameters P a g e | 14 % Cp=30; Ci=10; ap=5; %the initial conditions x1=30*pi/180; x2=0; u=0; eqk=0; sn=0; ssum=0; Y=[]; F=[]; U=[]; s=0; G=[]; ts=0; Cp=0; CCp=[]; for i=1:length(t) ts=ts+0.01; d=3*cos(5*ts)+2*sin(ts*(ts+1)); % d=0; G=[G d]; A=gr*sin(x1)-((mt*L*(x2^2)*sin(x1+s)*cos(x1))/(mc+mt)); B=L*(4/3-(mt*(cos(x1))^2)/(mc+mt)); C=cos(x1)/(mc+mt); D=L*(4/3-(mt*((cos(x1))^2)/(mc+mt))); % -f=A/B; g=C/D; % -x1dot=x2; x2dot=f+g*u+d; %update y=x1dot*Ts+x1; %y new Y=[Y y]; e=ym(i)- y; ed=(e-eqk)/Ts; Es=lamda2*e+lamda1*ed; sn=sn+(ym2d(i)+Es)*Ts; % the sliding surface s=-x1dot+sn; ssum=ssum+s*Ts; P a g e | 14 %control law dcp=sign(g)*ap*s^2; Cp=Cp+dcp*Ts; up=Cp*sign(g)*s; ui=Ci*sign(g)*ssum; uss=up+ui; us=30*sign(g)*s+ui; u=us; %update eqk=e; x1=y; x2=x2dot*Ts+x2; F=[F f]; U=[U u]; CCp=[CCp Cp]; end figure(1) plot(t, ym,' b') hold on plot(t, Y, 'r') title('The tracking control','FontSize',16); xlabel('Time(s)'); ylabel('y and ym (rad)'); legend('ym','y'); figure(2) plot(t, U, 'r') title('The corresponding control','FontSize',16); xlabel('Time(s)'); ylabel('U(Nt-m)'); figure(3) plot(t, G, 'r') title('The used disturbance','FontSize',16); xlabel('Time(s)'); ylabel('Disturbance (Nt-m)'); figure(4) plot(t, CCp, 'r') title('The update history of simulation 2','FontSize',16); xlabel('Time(s)'); ylabel('Cp (Nt-m)'); P a g e | 14 A Simulation The robust tacking control capability of the sliding PI controller: t us  u p  uI  CP sgn( g )s  CI sgn( g )  s( )d The disturbance: d  3cos(5t )  2sin[t (t  1)] The used proportional gain is CP  30 , and the used integral gain is CI  10 The simulation time is 10 seconds Simulation results: Figure 1: The used disturbance P a g e | 14 Figure 2: The tracking control performance of simulation Figure The corresponding control action of Fig P a g e | 14 B Simulation The robust tracking control capability of the adaptive design: CP  sgn( g ) P s t us  u p  uI  CP sgn( g )s  CI sgn( g )  s( )d The initial proportional gain is CP (0)  , the integral gain C1  10 is still used, and the adaptive rate is  P  The simulation time is 10 seconds Figure The tracking control performance of simulation P a g e | 14 Figure The corresponding control action of Fig Figure The update history of simulation P a g e 10 | 14 C Simulation As mentioned above, in order to remove the system gain function (g) in the adaptive law design We replace the adaptive rate by a arbitrary bounded constant  P The simulation will show that the effect of the different adaptive rate (  P ) In order to show the effect clearly, we let d = Other simulation conditions are same as those in simulation Figure The tracking control results of simulation P a g e 11 | 14 Figure The corresponding control actions of Fig.7 P a g e 12 | 14 IV DISCUSSIONS - Mô cho ta thấy điều khiển Sliding_PI làm cho vector error giảm nhanh chóng, đảm bảo cho hệ thống ổn định tiệm cận thời gian hữu hạn - Tuy nhiên kết mô khác nhiều so với kết từ báo, lý simulation 1, kết từ tín hiệu điều khiển u(Nt-m) (hình 3) cho thấy khoảng thời gian đầu (0-0,5s) tín hiệu bị vọt q 100Nt-m tín hiệu u từ báo khoảng thời gian khoảng < 20 Nt-m Chính điều làm cho tín hiệu y bám khơng tốt khoảng thời gian đầu (0-0,5s) hình Bên cạnh đó, để ước tính hệ số CP tối ưu điều khiển Sliding_P, người ta thiết kế thích ghi trực tiếp điều khiển Sliding_P thông qua hệ số  P , hình độ tốc độ thích ứng khác hệ số  P khác - - Ý tưởng thiết kế điều khiển tích hợp Sliding _PI đưa điều khiển mạnh để thay cho điều khiển phải dựa giới hạn thực P a g e 13 | 14 So sánh simulation simulation V REFERENCES [1] Yao-Chu Hsueh, Shun-Feng Su, Sliding PI controller designs with integral sliding surface for a class of nonlinear systems P a g e 14 | 14 .. .SLIDING PI CONTROLLER DESIGNS WITH INTERGRAL SLIDING SURFACE FOR A CLASSS OF NONLINEAR SYSTEMS Abstract: The Lyapunov stable theory and L2_gain control performance are employed to interpret a. .. thực P a g e 13 | 14 So sánh simulation simulation V REFERENCES [1] Yao-Chu Hsueh, Shun-Feng Su, Sliding PI controller designs with integral sliding surface for a class of nonlinear systems P a g... _is a nonzero proportional gain CI _is an integral gain and CI  II DIRECT ADAPTIVE DESIGN OF SLIDING PROPORTIONAL CONTROLLER Theorem 2: The following adaptive law with a nonzero bounded adaptive