DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

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1 EE 5 Homewors Fall 04 Updated: Sunday, October 6, 04 Some homewor assignments refer to the textboos: Slotine and Li, etc. For full credit, show all wor. Some problems require hand calculations. In those cases, do not use MATLAB except to chec your answers. It is OK to tal to people before you start wor. After you start wor, you are expected to WORK ALONE on the homewor WITH NO ASSISTANCE DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

2 EE 5 Homewor Fall 04 State Variable Systems, Computer Simulation, Linearization. Simulate the van der Pol oscillator y" ( y ) y' y 0 using MATLAB for various ICs. Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y(0)=0., y'(0)= 0. a. For = b. For= Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 50 sec. with all initial states equal to 0.5. Plot states versus time, and also mae -D plot of x, x, x using PLOT(x,x,x). ( x x ) bx xx use = 0, r= 8, b= 8/. rx x x x. Obtain the linear model of the system described by y( y ) y y 0 around the equilibrium point. 4. The system of equations axbxx cx dx exx fx describes the growth of two competing species that prey on each other. The constants are positive parameters and it is assumed that the two states are positive. Determine the linear model of the system around the equilibrium point (0,0).

3 Nonlinear Systems and Equilibrium. Consider the Voltera predator-prey system xxx. x x x EE 5 Homewor Fall 04 a. Find the equilibrium points and their nature. b. Simulate the system using MATLAB for various initial conditions. Tae ICs spaced in a uniform mesh in the box x=[-,], x=[-,]. Mae one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].. Consider the system y( y ). y x( yx ) Simulate the system using MATLAB for various initial conditions for the two cases: a. Tae ICs spaced in a uniform mesh in the box x=[-0,0], x=[-0,0]. Mae one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. b. Tae ICs spaced in a uniform mesh in the box x=[-,], x=[-,]. Mae one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].. The system of equations axbxx cx dx exx fx describes the growth of two competing species that prey on each other. The constants are positive parameters. Simulate the system using MATLAB for various initial conditions. Mae one phase plane plot with all the trajectories on it. 4. Duffing s equation is interesting in that it exhibits bifurcation, or dependence of stability properties and number of equilibrium points on a parameter. The undamped Duffing equation is x 0 a. Find the equilibrium points. Show that for 0 there is only one e.p. b. For 0 there are e.p.s Linearize the system and study the nature of these e.p.s c. Simulate the Duffing oscillator for. Mae time plot and phase plane plot.

4 EE 5 Homewor Fall 04 Chaos, Phase Plane. A system that exhibits chaos is the logistic function x ( ) x ( x ) However, chaos only occurs for certain values of sweep through the values using for fixed less than but close to. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of x vs, which was taen for and initial value of 0. Interpret the plot with some discussion in terms of bifurcation theory. Plot also. Show your MATLAB code. It is indeed interesting that the logistic function appears in economic systems and military supply systems.. Rather than try all values of. For Slotine & Li Example. on P. 0- a. Find equilibrium points b. Linearize the system about each equilibrium point. Find poles in each case. c. Simulate the system to find the Region of Attraction., we can. Slotine and Li p. 9 problem.. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the boo. 4. Slotine and Li p. 5 Example.7b. Show that this system has an unstable limit cycle. 4

5 EE 5 Homewor 4 Fall 04 Lyapunov Stability Analysis. Slotine and Li p. 97 problem... Use Lyapunov to show that the system xx x ( x x ) x x x ( x x ) is locally asymptotically stable. Find the Region of Asymptotic Stability. Use Lyapunov to examine the stability of these systems. Simulate time histories from many uniformly spaced ICs to verify your results. a. b. c. xx x xx x sin x x x sin x x x x ( x ) x 4. Use Lyapunov to show that the system x x ( x x ) x x ( x x ) has a stable limit cycle. Simulate the system from many uniformly spaced ICs. 5

6 Lyapunov s Method EE 5 Homewor 5 Fall 04. UUB of system with disturbance. Consider the system on S&L p. 66 with a disturbance d c( x) d 0 Assume that xc( x) ax with a 0 a nown positive constant a. Assume that d is unnown but is bounded by d D with D a nown positive constant. Prove that the system is UUB and find the bound on x(t). b. Assume that d is unnown but is bounded by d D x with D a nown positive constant. Prove that the system is UUB and find the bound on x(t).. UUB Use Lyapunov to show that the system x x x ( x x ) x x x ( x x ) is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin.. Lyapunov Theorem for Control Design. A system is given by x xu x x u a. Use Lyapunov Linearization Method to show that the open-loop system with u(t)= 0 is unstable about the origin. b. Select the nonlinear feedbac control input u x. Find the closed-loop system. Use a Lyapunov extension to show that the nonlinear closed-loop system is UUB. That is, select the quadratic Lyapunov function and find V along the closed-loop system trajectories. Then show that V is negative outside a region (i.e. if x is large enough). If you cannot solve for x such that V ( x) is negative, then plot V ( x) using MATLAB and draw conclusions about stability. c. Discuss the stability. When is the Lyapunov derivative negative? Can you use a LaSalle Extension to show AS? 4. Use Lyapunov Equation on p. 8 to chec the stability of the linear systems 0 a. x b. Ax x 6

7 EE 5 Homewor 6 Fall 04 Redo Exam and turn in as Hw 6. Lyapunov Functions- limit cycle, UUB Use Lyapunov functions to examine the stability of the following systems. Be clear and show all steps. a. Pic a suitable complicated Lyapunov function to study the stability of the limit cycle for the system 4 6 x x( x x ) 4 x x ( x x ) b. Use the standard simple quadratic Lyapunov function to show that this system is UUB. Describe the region outside which the Lyapunov derivative is negative. 4 6 x x( x x ) 4 6 x x x ( x x ). Barbalat s Lemma and LaSalle extension a. Use quadratic Lyapunov Function to show this system is locally AS x x( x ) xx Find a ball within which V 0. This region is contained in the region of attraction. b. Use quadratic Lyapunov Function to show this system is locally SISL x x( x ) x Find a region within which V 0. c. Use Barbalat s Lemma to verify that V 0. Chec uniform continuity of the Lyapunov derivative of the system in part b. d. Use LaSalle s extension to verify that the system in part b is actually AS. Find the equilibrium point.. Equilibrium points and linearization System is x ( x x ) x ( x x ) a. Find all equilibrium points 7

8 b. Find Jacobian c. Find the nature of all e.p.s 4. Lyapunov Equation, AS, SISL a. Use Lyapunov equation to show this system is AS 0 Ax x 8 6 Use Q=I>0. Is the solution P to the Lyapunov equation unique? b. Use Lyapunov equation to show this system is SISL 0 Ax x Use Q 0 0. Now, you have to find ANY positive definite P that solves the Lyapunov equation. Is the solution P unique for this case? 8

9 i/o Feedbac Linearization EE 5 Homewor 7 Fall 04. A system is given by sin x 4 x cos x u with output y() t x (t) a. Design a FB linearization controller to mae the output follow a desired trajectory yd ( t ) That is, find ut () b. Discuss the internal dynamics. Are they a problem?. NMP System A system is given by 4 y zy u z ( y ) z 5 z 5 0 a. Tae output y( t ) and find the FB linearization controller ut ( ) to follow the prescribed trajectory yd ( t ). b. Find the internal dynamics. Set yt ( ) 0 to get the zero dynamics. Are the ZD stable? Does the FB linearization controller wor?. Effect of Output Choice in i/o FB Linearization It is desired to stabilize a system given by x sin x x u x x a. Select the out as y x and use FB lin. design to select the control u(t) to follow the desired trajectory yd ( t ). Chec the internal dynamics. Set y=0 to get the zero dynamics. Is the system minimum phase? b. Select the new output y x. Find the FB lin. controller u(t). Does this wor? What about the internal dynamics? 9

10 Feedbac Linearization, bacstepping EE 5 Homewor 8 Fall 04, Slotine and Li. I/O feedbac linearization. Slotine and Li Problem 6.. System is sin x 4 x cos x u y x Do i/o feedbac linearization to mae output trac desired trajectory yd ( t ).. Bacstepping. The system is sin x u 4 x cos x x Do bacstepping to stabilize this system. Select the desired value x d to yield the first step dynamics of x 0. Compare this to Problem, which uses i/o FB linearization.. Bacstepping. Slotine and Li Problem 6.. Globally stabilize means bacstepping. 4. I/O fb linearization. It is desired to stabilize a system given by x sin x x u x x a. Select the out as y x and use FB lin. design to select the control to mae output go to zero. Is the system minimum phase? b. Select the new output y x. Does this wor? c. Design a bacstepping controller. 0

11 The following homewors are out of date. Do not do homewors until they are assigned. They may change.

12 Nonlinear Systems and Equilibrium EE 5 Homewor Fall 009. Obtain the linear model of the system described by y( y ) y y 0 around the equilibrium point.. The system of equations axbxx cx dx exx fx describes the growth of two competing species that prey on each other. The constants are positive parameters and it is assumed that the two states are positive. Determine the linear model of the system around the equilibrium point. Simulate the system using MATLAB for various initial conditions. Mae phase plane plot.. Determine the equilibrium points and their nature for the predator-prey system xxx. x x x Simulate the system using MATLAB for various initial conditions. Mae phase plane plot. 4. Determine the equilibrium points and their nature for the system y( y ). y x( yx )

13 EE 5 Homewor Fall 009 State Variable Systems, Computer Simulation, Linearization. Simulate the van der Pol oscillator y" ( y ) y' y 0 using MATLAB for various ICs. Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y(0)=0., y'(0)= 0. c. For = d. For= Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 50 sec. with all initial states equal to 0.. Plot states versus time, and also mae -D plot of x, x, x using PLOT(x,x,x). ( x x ) bx xx use = 0, r= 8, b= 8/. rx x x x. A system has transfer function H ( s) s s 4 4s a. Use MATLAB to mae a -D plot of the magnitude of H(s) b. Use MATLAB to mae a -D plot of the phase of H(s) c. Use MATLAB to draw magnitude and phase Bode plots 4. Use separation of variables to verify the formula for x(t) in Slotine & Li ex.. on p Duffing s equation is interesting in that it exhibits bifurcation, or dependence of stability properties and number of equilibrium points on a parameter. The undamped Duffing equation is x 0 d. Find the equilibrium points. Show that for 0 there is only one e.p. e. For 0 there are e.p.s Linearize the system and study the nature of these e.p.s f. Simulate the Duffing oscillator for. Mae time plot and phase plane plot.

14 EE 5 Homewor Fall 009 Chaos, Phase Plane. A system that exhibits chaos is the logistic function x ( ) x ( x ) However, chaos only occurs for certain values of sweep through the values using for fixed less than but close to. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of x vs, which was taen for Interpret the plot with some discussion in terms of bifurcation theory. Plot also. Show your MATLAB code. It is indeed interesting that the logistic function appears in economic systems and military supply systems.. Rather than try all values of. For Slotine & Li Example. on P. 0- a. Find equilibrium points b. Linearize the system about each equilibrium point. Find poles in each case. c. Simulate the system to find the Region of Attraction., we can. Slotine and Li p. 9 problem.. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the boo. 4. Slotine and Li p. 9 problem. Simulate using MATLAB using various initial conditions. Do not do the problem requested in the boo. 4

15 EE 5 Homewor 4 Fall 009 Slotine and Li Lyapunov s Method. Slotine and Li p. 97 problem... Use Lyapunov Equation on p. 8 to prove asymptotic stability of the system 0 x 6 5. Use Lyapunov to show that the system xx x ( x x ) x x x ( x x ) is locally asymptotically stable. Find the Region of Asymptotic Stability 4. UUB Use Lyapunov to show that the system x x x ( x x ) x x x ( x x ) is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin. 5. Lyapunov Theorem for Control Design. A system is given by x xu x x u a. Use Lyapunov Linearization Method to show that the open-loop system with u(t)= 0 is unstable about the origin. b. Select the nonlinear feedbac control input u x. Find the closed-loop system. Use a Lyapunov extension to show that the closed-loop system is UUB. That is, select the quadratic Lyapunov function and find V along the closed-loop system trajectories. Then show that V is negative outside a region (i.e. if x is large enough). c. Discuss the stability. When is the Lyapunov derivative negative? Can you use a LaSalle Extension to show AS? 5

16 EE 5 Homewor 5 Fall 009, Slotine and Li Lyapunov. S&L p. 0, Example 4.. a. For the systems given, prove the stability claimed by verifying the conditions given. b. Integrate the state equations to find the solutions x(t) of the three systems.. S&L p. 05, Example 4.. Integrate the state equation to verify the solution given.. S&L p. 55 problem 4.9, parts a and b. 4. Consider the nonlinear dynamics for an m-lin robot manipulator, M qq Cq, q q Dq gq, m where q, R. M q accounts for the robot inertia. Cq, q q accounts for centrifugal and Coriolis forces. Dq accounts for viscous damping. g q accounts for gravity forces. In addition, we have the following properties: i. M q is a symmetric positive definite matrix of q. ii. M C is a sew symmetric matrix of q, q. iii. g qw q q, where W q is a positive definite function of q. Show that for D 0, the map from to q is passive lossless. And when D is positive definite, the map from to q is passive dissipative. T Hint: Use the total energy V q M qq Pq as a storage function. Select an appropriate Pq, a positive definite function of q. 6

17 Feedbac Linearization, bacstepping EE 5 Homewor 6 Fall 009, Slotine and Li. I/O feedbac linearization. Slotine and Li Problem 6.. Bacstepping. The system is sin x u 4 x cos x x Do bacstepping to stabilize this system. Select the desired value x 0 to yield the first step dynamics of x 0. Compare this to Problem 6., which uses i/o FB linearization.. Bacstepping. Slotine and Li Problem 6.. Globally stabilize means bacstepping. 4. I/O linearization. It is desired to stabilize a system given by x sin x x x x u d. Select the out as y x and use FB lin. design to select the control. Is the system minimum phase? e. Select the new output y x. Does this wor. f. Design a bacstepping controller. 5. Input-State Linearization. Slotine and Li Problem 6.7. a. Write f ( x ), gx, ( ) ad f g. b. Is the system input-state linearizable. c. Chec the given z ( ) x. Does it wor? 7

18 Bifurcations EE 5 Homewor 7 Fall 009, Strogatz boo x. Plot with MATLABthe following vector fields as -D surfaces in the (x,r) plane. Also plot the bifurcation diagrams in the plane. r a. b. c. d. e. f. f( x, r) rx f( x, r) rx f( x, r) rx f( x, r) rx f( x, r) rx f( x, r) rx x 5 8