Stability and Control of Nonlinear Systems

Stability and Control of Nonlinear outlinesThe future(a) schema was provided to direct intimately passivity, asymptotic stability, and commentary to recount stability properties at curbs.The precondition corpse of dissimilarial equation for analysis is given belowAlso, The convey post type of the carcass is as follows. Let,Similarly therefrom found on the arranging equation is given bySimilarly, based on the state billet representation The state of system , where n=4, where m=2 and where p=2. Hence p=m. A dissipative system with respect to supply rate is express to be passive, ifThe Lyapunov function for the system is given asHence with respect to description of state variables, it bed be rewritten asHence,Also, based on the definition of S,For the given system , hence the lossless system, is passive from u to y.The PD feedback controller of the system with and is represented asHence the state space representation of the system, is given byThe modified Lyapu nov function with potential energy is given byIt shtup be detect that V(x) is divergentiable (V R4 R and a C1 function).Based on the equation of V(x) it can be observed that,The term and all other terms are quadratic in nature. Hence where Hence V(x) is irresponsible definite.Let V(x) is bounded by V(x)M, where M R, then it implies that and and and and Hence V(x) is radially measureless.The derivative of V(x) can be obtained as followsUpon substitution and solving the equations,At It can be observed thatBased on the supra equation, it can be observed thatIt can be observed that , has only quadratic terms with a prohibit sign prefixed hence where Hence is negative definite.The proportion organize of the system at is given byHence origin is the only equilibrium point of the system.Based on the in a higher place observations, it can be concluded that the system is world-widely asymptotically stable at the origin.The given systems were simulated for differen t values of and , modified one at a snip with other disturbance set to zero and the sign coach set at origin.The following observations can be found from subplots of and Hence the disturbance in two the coordinates of the system are additive in nature. It can be observed that, whenever initial state of the system set to origin and disturbance is induced in one of the coordinate ( or ), the other coordinate of the system is not disturbed. frame of reference 1 enjoin of System with disturbance at origin with rd=0Figure 2 State of System with disturbance at origin with thd=0The Figure 3 indicates the state of the system, when is modified from -5 to 5 with , with the initial condition as x = 7,3,5,1. The settling time of the system varies with the magnitude of disturbance and the initial condition. Also, it can be observed from the plot of that the system settles to a point which is offset from the origin (equilibrium) by the value of disturbance. Also, the settling time of the system is more(prenominal) for d=-c, when compared to d=c. Also, disturbance in one of the coordinate (), has its effect in another coordinate.Figure 3 State of System with rd=0 at x = 7,3,5,1The observations of disturbance induced in when , is applicable for the disturbance induced in with Also, it can be observed from Figure 4 that the settling time of the system is higher when a disturbance is induced in r-coordinate, when compared to -coordinate.Figure 4 State of System with thd = 0 at x = 7,3,5,1The effect of having both and was observed by simulating the system response for and . Also, it can be observed that settling time of the system is similar to disturbance induced only in the r-coordinate.Figure 5 State of System with thd = -5, rd=5 at x = 7,3,5,1In all the above plots, it can be observed from the subplot of that the settling point of state as t, and , indicating that the state of the system tracks the input in the respective coordinate.It can also be observed from the previous plots for d=0, system exhibits the property of global asymptotic stability to the origin (equilibrium point). Also, , the state implies the Bounded Input Bounded State property of the system.The input to state stability of the unlikable loop system with respect to and for the system was validated by adding a destabilizing feedback with and .The function k(x) of the disturbance is selected, such that the power transferred to the system is maximized, which can be performed when .From the above equation, it can be observed that the power transferred to the system can be maximized by choosing same sign of with c0. The nature of system response for different range of c is listed in the Table 1 below.Table 1 System Response for Variation in c at initial condition of 7,3,5,1Value of cObservationc 1.99The energy of the system decreases initially, indicated by the plot of Lyapunov function shown in Figure 6 and the same result can be observed on the plot of r and , w here the magnitude decreases initially and oscillates with the bounded magnitude, for the bounded input indicated in plot of theta-d.c1.99For c=5, the energy of the system increases, indicated by the plot of Lyapunov function shown in Figure 7 and the plot of r and indicates that the magnitude continues to increases resulting in unbounded state for the bounded input indicated in plot of theta-d. Also, it can be observed that the rate of increase in energy of the system, decreases with time.Figure 6 State of System at c=1.75The system is not Input to state stable (ISS) for c1.99 and Figure 7 indicates a system which is not ISS for c=5. The value of transition from bounded state to unbounded state was observed at c=1.93 for an initial point of 1,2,1,2. Based on the above observation, the transition value of c is dep balanceent of initial condition (energy) of the system.Figure 7 State of System at c=5The PD control used in the r-coordinate is modified asThe simulations were carried o ut, to identify the properties of ISS convenient by the system, with respect to and as inputs. All the simulations were carried out with respect to the initial condition x0 = (7,3,5,1) intend 1The system is evaluated with zero disturbance and , the result is indicated in Figure 8.Figure 8 System with Zero fretfulnessFor the no disturbance conditions, it can be observed that the system is asymptotically stable about the origin (equilibrium), indicating the Global asymptotic stability of the system about the origin. Also from the plot of Lyapunov function, it can be observed that the energy of the system settles down to zero.Condition 2The destabilizing feedback input used in question 5 for the system was fed to the system and it its response is indicated in figureFigure 9 State of System at c=5 with modified PD ControlThe following observations can be made with respect to figureFor an input , the state , indicating bounded input bounded state property of the system. It can be ob served that, though the energy of the system increases initially, but upper bounded everywhere a period. The energy and the state of the system gets bounded over period of simulation. Hence for the bounded input, state of the system is bounded. Also, the system exhibits property of asymptotic gain, since the state of the system is upper bounded by disturbance with gain of the system.Also, it was observed that though the system is ISS for the c=5, as the value of c increases energy of the system increases (example for c=10, v(x) is upper bounded to 10,000). Hence modifying the PD control, makes the system ISS for a larger range of disturbances, when compared to earlier control.Condition 3The system was fed with the inputFigure 10 State of system rd=0 and theta d=5*exp(t)It can be observed from the plot that d(t) 0 as , also aysmptotically. Hence the system indicates the property of converging input, converging state.The response of the system was evaluated with different possible i nputs for , such as, the state of the system x1, x3 was chosen based on observations made in earlier simulations (q5) where predominantly these states grew out of boundSimilarly, the above input conditions were simulated with =0 and defined as one of the input, few combinations of the above input disturbances and few possible system interconnections such as positive feedback interconnection, negative feedback interconnection, series interconnection.System response for various types of disturbance never-ending DisturbanceThe disturbance of the system is set to constant values, as indicated in Figure 5Figure 11 State of system at theta d=-5 rd=5It can be observed from the plot of Figure 11 and Figure 5 that the settling time of system in r-coordinate has reduced almost by half, when compared to previous control.Positive Feedback InterconnectionThe disturbance input condition is mentioned below and the system response is shown in Figure 12Figure 12 System Response for Positive Feedbac k InterconnectionThe state of the system indicates the converging nature, also it can be observed that after the transient period system follows the input. serial InterconnectionThe system is connected in series, with the following disturbance input configuration for each of the subsystem and the plot for the same is shown in Figure 13.Figure 13 System Response for Series InterconnectionIt can be observed that the behavior of the system is similar with respect to condition 2, but the energy of the system settles down at a higher level when compared to the similar condition with System with different disturbances acting simultaneouslyThe type of disturbance added to the system is given below and the response of the system is shown in figureFigure 14 System response of simultaneous time varying disturbanceIt can be observed that the system exhibit the property of bounded input bounded state, even if the disturbance is of time varying.In all the above simulation conditions, it was obse rved that the system exhibits bounded state nature for a wider range of inputs with higher magnitude, when compared to the PD control implemented earlier. This phenomenon can be attributed to the cubic terms with the negative sign, as it can reduce the rate at which energy of the system increases, before it goes out of bound.APPENDIXCode Used for Generation of PlotsContentsQ4 Constant Value of Theta-d and r-dQ5 for ISSQ6 for ISS with new u2Q4 Constant Value of Theta-d and r-dclcclear allclose allglobal x1dglobal x3dts=500 %Duration for solvingip=7,3,5,1options=odeset(AbsTol,1e-7,RelTol,1e-5)thd=-5rd=5for i=1size(thd,2) for j=1size(rd,2)%-293031 x1d=thd(i) %x1d is Theta-d x3d=rd(j) %x3d is r-d t,x=ode23(deeqn,0 ts,ip,options) figure(1) subplot(2,2,1) hold on plot(t,x(,1)) title(Plot of Theta) xlabel(Time) ylabel(Theta) grid on grid minor subplot(2,2,2) hold on plot(t,x(,2)) title(Plot of Theta-dot) xlabel(Time) ylabel(Theta-dot) grid on grid minor subplot(2,2,3) hold on plot(t,x(, 3)) title(Plot of r) xlabel(Time) ylabel(r) grid on grid minor subplot(2,2,4) hold on plot(t,x(,4)) title(Plot of r-dot) xlabel(Time) ylabel(r-dot) grid on grid minor endendQ5 for ISSclcclose allglobal x1dglobal x3dts=10000 %Duration for solvingip=7,3,5,1options=odeset(AbsTol,1e-7,RelTol,1e-5)x1=ip global c cval=1.92 %1.993 is transition point for i=1size(cval,2) c=cval(i) %4.0125 x1d=0 %x1d is Theta-d x3d=0 %x3d is r-d t,x=ode23(deeqnvx,0 ts,ip,options) figure(2) subplot(2,3,1) hold on plot(t,x(,1)) title(Plot of Theta) xlabel(Time) ylabel(Theta) grid on grid minor subplot(2,3,2) hold on plot(t,x(,2)) title(Plot of Theta-dot) xlabel(Time) ylabel(Theta-dot) grid on grid minor subplot(2,3,4) hold on plot(t,x(,3)) title(Plot of r) xlabel(Time) ylabel(r) grid on grid minor subplot(2,3,5) hold on plot(t,x(,4)) title(Plot of r-dot) xlabel(Time) ylabel(r-dot) grid on grid minor subplot(2,3,3) hold on thdin=c.*sign(x(,2)) plot(t,thdin) title(Plot of theta-d) xlabel(Time) ylabel(theta-dot) grid on grid minor subplot(2,3,6) hold on vxfn=(1/2).*(((x(,3).2)+1).*(x(,2).2)+(x(,4).2)+(x(,1).2)+(x(,3).2)) plot(t,vxfn) title(Plot Lyapunov mesh) xlabel(Time) ylabel(v(x)) grid on grid minor endQ6 for ISS with new u2clcclose allglobal x1dglobal x3dts=100 %Duration for solvingip=7,3,5,1options=odeset(AbsTol,1e-7,RelTol,1e-5)x1=ip global c cval=5 %1.993 is transition point for i=1size(cval,2) c=cval(i) %4.0125 x1d=0 %x1d is Theta-d x3d=0 %x3d is r-d t,x=ode23(deeqnr,0 ts,ip,options) figure(3) subplot(2,3,1) hold on plot(t,x(,1)) title(Plot of Theta) xlabel(Time) ylabel(Theta) grid on grid minor subplot(2,3,2) hold on plot(t,x(,2)) title(Plot of Theta-dot) xlabel(Time) ylabel(Theta-dot) grid on grid minor subplot(2,3,4) hold on plot(t,x(,3)) title(Plot of r) involvement for constant disturbancefunction dx = deeqn(t,x)% turn tail for system model% Argument function for ODE Solverglobal x1dglobal x3ddx=x(2) (-2*x(3)*x(4)*x(2)-x(2)-x(1)+x1d)/((x(3).2)+1)x(4)x(3)*(x(2).2)-x(4)-x(3)+x 3dendSystem with Destabilizing Feedbackfunction dx = deeqnvx(t,x)% Function for system model% Argument function for ODE Solverglobal x1dglobal x3dglobal cx1d=c.*sign((+1).*x(2))dx=x(2) (-2*x(3)*x(4)*x(2)-x(2)-x(1)+x1d)/((x(3).2)+1) x(4) x(3)*(x(2).2)-x(4)-x(3)+x3dendFunction with new u2 and old u1function dx = deeqnr(t,x)% Function for system model% Argument function for ODE Solverglobal x1dglobal x3dglobal cx1d=x(4)%c.*sign((+1).*x(2))x3d=x(2)dx=x(2) (-2*x(3)*x(4)*x(2)-x(2)-x(1)+x1d)/((x(3).2)+1)x(4)x(3)*(x(2).2)-x(4)-x(3)+x3d-(x(3).3)+(x3d.3)endPublished with MATLAB R2016b