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

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1 EE 533 Homeworks Spring 07 Updated: Saturday, April 08, 07 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. Some homework assignments refer to the textbooks: Slotine and Li, etc. For full credit, show all work. Some problems require hand calculations. In those cases, do not use MATLAB except to check your answers. It is OK to talk about the homework beforehand. BUT, once you start writing the answers, MAKE SURE YOU WORK ALONE. The purpose of the Homework is to evaluate you individually, not to evaluate a team. Cheating on the homework will be severely punished. The next page must be signed and turned in at the front of ALL homeworks submitted in this course.

2 EE 533 Nonlinear Control Systems Homework Pledge of Honor On all homeworks in this class - YOU MUST WORK ALONE. Any cheating or collusion will be severely punished. It is very easy to compare your software code and determine if you worked together It does not matter if you change the variable names. Please sign this form and include it as the first page of all of your submitted homeworks Typed Name: Pledge of honor: "On my honor I have neither given nor received aid on this homework. e-signature:

3 EE 533 Homework State Variable Systems, Computer Simulation. 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 0.4. Plot states versus time, and also make 3-D plot of x, x, x3 using PLOT3(x,x,x3). ( x x ) 3 bx3 xx use = 0, r= 8, b= 8/3. rx x x x 3. Consider the Voltera predator-prey system xxx. x x x 3 Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a uniform mesh in the box x=[-,], x=[-,]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. 3

4 Nonlinear Systems and Equilibrium. Consider the Voltera predator-prey system xxx. x x x Find the equilibrium points and their nature. EE 533 Homework. 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 3 x 0 a. Find the equilibrium points. Show that for 0 there is only one e.p. b. For 0 there are 3 eps. Linearize the system and study the nature of these 3 e.p.s c. Simulate the Duffing oscillator and make time plot and phase plane plot. Do for a. b. 0. c. 3. Consider the system y( y ). y x( yx ) Simulate the system using MATLAB for various initial conditions for the two cases: a. Take ICs spaced in a uniform mesh in the box x=[-0,0], x=[-0,0]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. b. Take ICs spaced in a uniform mesh in the box x=[-3,3], x=[-3,3]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. 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. Pick a=c=d=f=, b=e=3. Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a uniform mesh in the box x=[-,], x=[-,]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. 4

5 Vector Fields, Flows, First Integrals. Consider the undamped oscillator 0 EE 533 Homework 3 a. Write position-velocity state space form X f( X). b. Plot the trajectories x( t), ( t) vs. time. Use initial conditions of x(0) 0., ( t) 0 f( x, x) c. Plot the vector field f( X) f( x, x) in the phase plane ( x, x ) ( x, ). Plot for points spaced in a uniform mesh in the box x=[-0,0], x=[-0,0]. d. Plot the system trajectories (flows or orbits) in the phase plane. Take ICs spaced in a uniform mesh in the box x=[-0,0], x=[-0,0]. e. Derive the First Integral of Motion F( x, x ) as done in class. Plot the FIM as a 3-D surface over the phase plane on the x=[-0,0], x=[-0,0].. Repeat for the unstable system 0 5

6 EE 533 Homework 4 E.P.s, Lyapunov Stability Analysis. Equilibrium points and linearization System is x( xx ) x ( x x ) a. Find all equilibrium points b. Find Jacobian c. Find the nature of all e.p.s. Use Lyapunov to examine the stability of these systems. Simulate time histories from many uniformly spaced ICs to verify your results. a. b. xx x xx x x ( x ) x 3. Use Lyapunov to show that the system xx x ( x x 3) x x x ( x x 3) is locally asymptotically stable. Find the Region of Asymptotic Stability. Simulate the system from many uniformly spaced ICs. 3. Use Lyapunov to show that the system x x ( x x ) x x ( x x ) is UUB. Simulate the system from many uniformly spaced ICs. 6

7 EE 533 Homework 5 Lyapunov s Method. 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 known positive constant a. Assume that d is unknown but is bounded by d D with D a known positive constant. Prove that the system is UUB and find the bound on x(t). b. Assume that d is unknown but is bounded by d D x with D a known 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 3) x x x ( x x 3) 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. 3. Use Lyapunov Equation to check the stability of the linear systems a. 0 Ax x 6 5 b. 4 Ax x c. 0 Ax x 4 0 7

8 EE 533 Homework 6 Lyapunov Controls Design, Feedback Linearization. On Slotine & Li p. 7 we used the Lyapunov function 4 V( x) ( x x 0) and we obtained Lyapunov derivative V ( x) (4x x)( x x 0) Plot V( x ) and V ( x) using MATLAB. Pick a region for the domains that reveals a nice plot.. A system is given by xsgn( x) xx u Select Lyapunov function candidate V( x) ( x x) Use Lyapunov to design a controller u(x) to make system SISL. 3. For each of these systems, Use Lyapunov to Design feedback control u(x) to make the system i. SISL, and then ii. AS a. b. c. xx x xu xx u xx xx x u 4. Multi-input Control. Use Lyapunov to design controls u, u to make this system i. SISL, and then ii. AS xx u xx u 3 7 8

9 EE 533 Homework 7 I/O Feedback Linearization. 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 make 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 5 z( y) z z 0 a. Take 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 work? 3. 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 output as y x and use FB lin. design to select the control u(t) to follow the desired trajectory yd ( t ). Check 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 work? What about the internal dynamics? 9

10 DO NOT WORK BEYOND THIS PAGE The following hwks have not yet been assigned and may change. 0

11 EE 533 Homework 3 Chaos, Phase Plane. A system that exhibits chaos is the logistic function xk ( k ) xk ( xk ) However, chaos only occurs for certain values of sweep through the k values using k k for fixed less than but close to. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken for and initial value of 0 3. Interpret the plot with some discussion in terms of bifurcation theory. Plot also k. Show your MATLAB code. It is indeed interesting that the logistic function appears in economic systems and military supply systems. k. Rather than try all values of k, we can. Slotine and Li p. 39 problem.. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 3. Slotine and Li p. 35 Example.7b. Show that this system has an unstable limit cycle.

12 EE 533 Homework 4 Lyapunov Stability Analysis 4. Slotine and Li p. 97 problem Use Lyapunov to show that the system xx x ( x x 3) x x x ( x x 3) is locally asymptotically stable. Find the Region of Asymptotic Stability 6. Use Lyapunov to examine the stability of these systems. Simulate time histories from many uniformly spaced ICs to verify your results. c. d. e. xx x xx x sin x x x sin x x x x ( x ) x 7. 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.

13 EE 533 Homework 5 Lyapunov s Method. 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 known positive constant c. Assume that d is unknown but is bounded by d D with D a known positive constant. Prove that the system is UUB and find the bound on x(t). d. Assume that d is unknown but is bounded by d D x with D a known 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 3) x x x ( x x 3) 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. 3. Use Lyapunov Equation to check the stability of the linear systems 0 d. x e. f Ax x 0 Ax x 4 0. 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. 3

14 c. Use Barbalat s Lemma to verify that V 0. Check 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. 4

15 EE 533 Homework 6 Lyapunov Controls Design, I/O Feedback Linearization. A system is given by xsgn( x) xx u Select Lyapunov function candidate V( x) ( x x) Use Lyapunov to design a controller u(t) to make system SISL.. 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 make the output follow a desired trajectory yd ( t ) That is, find ut ( ) b. Discuss the internal dynamics. Are they a problem? 3. 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 c. Select the output as y x and use FB lin. design to select the control u(t) to follow the desired trajectory yd ( t ). Check the internal dynamics. Set y=0 to get the zero dynamics. Is the system minimum phase? d. Select the new output y x. Find the FB lin. controller u(t). Does this work? What about the internal dynamics? 5

16 Feedback Linearization, backstepping EE 533 Homework 7. I/O feedback linearization. Slotine and Li Problem 6.3. System is sin x 4 x cos x u y x Do i/o feedback linearization to make output track desired trajectory yd ( t ).. Backstepping. The system is sin x u 4 x cos x x Do backstepping 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. 3. Backstepping. Slotine and Li Problem 6.. Globally stabilize means backstepping. System is 4 5 y zy u 5 z( y) z z 0 6

17 The following homeworks are out of date. Do not do homeworks until they are assigned. They may change. 7

18 Old hwk Obtain the linear model of the system described by y( y ) y y 0 around the equilibrium point. 9. 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). 8

19 EE 533 Homework Nonlinear Systems and Equilibrium 0. Consider the Voltera predator-prey system xxx. x x x a. Find the equilibrium points and their nature. b. Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a uniform mesh in the box x=[-,], x=[-,]. Make 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: c. Take ICs spaced in a uniform mesh in the box x=[-0,0], x=[-0,0]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. d. Take ICs spaced in a uniform mesh in the box x=[-3,3], x=[-3,3]. Make 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. Make one phase plane plot with all the trajectories on it. 3. 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 3 x 0 d. Find the equilibrium points. Show that for 0 there is only one e.p. e. For 0 there are 3 e.p.s Linearize the system and study the nature of these 3 e.p.s f. Simulate the Duffing oscillator for. Make time plot and phase plane plot. 9

20 EE 533 Homework 3 Chaos, Phase Plane. A system that exhibits chaos is the logistic function xk ( k ) xk ( xk ) However, chaos only occurs for certain values of sweep through the k values using k k for fixed less than but close to. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken for and initial value of 0 3. Interpret the plot with some discussion in terms of bifurcation theory. Plot also k. Show your MATLAB code. It is indeed interesting that the logistic function appears in economic systems and military supply systems. k. 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. k, we can 3. Slotine and Li p. 39 problem.. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 4. Slotine and Li p. 35 Example.7b. Show that this system has an unstable limit cycle. 0

21 EE 533 Homework 4 Lyapunov Stability Analysis 4. Slotine and Li p. 97 problem Use Lyapunov to show that the system xx x ( x x 3) x x x ( x x 3) is locally asymptotically stable. Find the Region of Asymptotic Stability 6. Use Lyapunov to examine the stability of these systems. Simulate time histories from many uniformly spaced ICs to verify your results. f. g. h. xx x xx x sin x x x sin x x x x ( x ) x 7. 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.

22 EE 533 Homework 5 Lyapunov s Method. 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 known positive constant e. Assume that d is unknown but is bounded by d D with D a known positive constant. Prove that the system is UUB and find the bound on x(t). f. Assume that d is unknown but is bounded by d D x with D a known 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 3) x x x ( x x 3) 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. 3. Lyapunov Theorem for Control Design. A system is given by x xu 3 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 feedback 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 check the stability of the linear systems 0 g. x h. Ax x

23 Redo Exam and turn in as Hwk 6. Lyapunov Functions- limit cycle, UUB EE 533 Homework 6 Use Lyapunov functions to examine the stability of the following systems. Be clear and show all steps. a. Pick a suitable complicated Lyapunov function to study the stability of the limit cycle for the system x x( x 3x 3) 4 x x ( x 3x 3) 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 x x( x 3x 3) 4 6 x x x ( x 3x 3). Barbalat s Lemma and LaSalle extension e. 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. f. Use quadratic Lyapunov Function to show this system is locally SISL x x( x ) x Find a region within which V 0. g. Use Barbalat s Lemma to verify that V 0. Check uniform continuity of the Lyapunov derivative of the system in part b. h. Use LaSalle s extension to verify that the system in part b is actually AS. Find the equilibrium point. 3. Equilibrium points and linearization System is x ( x x ) x ( x x ) d. Find all equilibrium points 3

24 e. Find Jacobian f. 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? 4

25 EE 533 Homework 7 i/o Feedback Linearization. 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 make 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. Take 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 work? 3. 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 e. Select the out as y x and use FB lin. design to select the control u(t) to follow the desired trajectory yd ( t ). Check the internal dynamics. Set y=0 to get the zero dynamics. Is the system minimum phase? f. Select the new output y x. Find the FB lin. controller u(t). Does this work? What about the internal dynamics? 5

26 Feedback Linearization, backstepping EE 533 Homework 8 Slotine and Li. I/O feedback linearization. Slotine and Li Problem 6.3. System is sin x 4 x cos x u y x Do i/o feedback linearization to make output track desired trajectory yd ( t ).. Backstepping. The system is sin x u 4 x cos x x Do backstepping 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. 3. Backstepping. Slotine and Li Problem 6.. Globally stabilize means backstepping. 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 make output go to zero. Is the system minimum phase? b. Select the new output y x. Does this work? c. Design a backstepping controller. 6

27 The following homeworks are out of date. Do not do homeworks until they are assigned. They may change. 7

28 Nonlinear Systems and Equilibrium EE 533 Homework 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. Make phase plane plot. 3. 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. Make phase plane plot. 4. Determine the equilibrium points and their nature for the system y( y ). y x( yx ) 8

29 EE 533 Homework 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 0.3. Plot states versus time, and also make 3-D plot of x, x, x3 using PLOT3(x,x,x3). ( x x ) 3 bx3 xx use = 0, r= 8, b= 8/3. rx x x x 3 3. A system has transfer function H ( s) s s 4 4s 3 a. Use MATLAB to make a 3-D plot of the magnitude of H(s) b. Use MATLAB to make a 3-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 3 x 0 g. Find the equilibrium points. Show that for 0 there is only one e.p. h. For 0 there are 3 e.p.s Linearize the system and study the nature of these 3 e.p.s i. Simulate the Duffing oscillator for. Make time plot and phase plane plot. 9

30 EE 533 Homework 3 Fall 009 Chaos, Phase Plane. A system that exhibits chaos is the logistic function xk ( k ) xk ( xk ) However, chaos only occurs for certain values of sweep through the k values using k k for fixed less than but close to. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken 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. k k. 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. k, we can 3. Slotine and Li p. 39 problem.. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 4. Slotine and Li p. 39 problem.3 Simulate using MATLAB using various initial conditions. Do not do the problem requested in the book. 30

31 EE 533 Homework 4 Fall 009 Slotine and Li Lyapunov s Method. Slotine and Li p. 97 problem 3... Use Lyapunov Equation on p. 8 to prove asymptotic stability of the system 0 x Use Lyapunov to show that the system xx x ( x x 3) x x x ( x x 3) is locally asymptotically stable. Find the Region of Asymptotic Stability 4. UUB Use Lyapunov to show that the system x x x ( x x 3) x x x ( x x 3) 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 3 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 feedback 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? 3

32 EE 533 Homework 5 Fall 009, Slotine and Li Lyapunov. S&L p. 03, Example 4.. a. For the 3 systems given, prove the stability claimed by verifying the 3 conditions given. b. Integrate the state equations to find the solutions x(t) of the three systems.. S&L p. 05, Example 4.3. Integrate the state equation to verify the solution given. 3. S&L p. 55 problem 4.9, parts a and b. 4. Consider the nonlinear dynamics for an m-link 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 skew 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. 3

33 Feedback Linearization, backstepping EE 533 Homework 6 Fall 009, Slotine and Li. I/O feedback linearization. Slotine and Li Problem 6.3. Backstepping. The system is sin x u 4 x cos x x Do backstepping to stabilize this system. Select the desired value x 0 to yield the first step dynamics of x 0. Compare this to Problem 6.3, which uses i/o FB linearization. 3. Backstepping. Slotine and Li Problem 6.. Globally stabilize means backstepping. 4. I/O linearization. It is desired to stabilize a system given by sin x 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 work. f. Design a backstepping 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. Check the given z ( ) x. Does it work? 33

34 Bifurcations EE 533 Homework 7 Fall 009, Strogatz book x. Plot with MATLABthe following vector fields as 3-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