Exercise 5.1 Consider the nonlinear system

1. Linearize the system around x₁=0, x₂ = 0 and u = 0.
   
   
2. Is the linear model stable in an open-loop mode?
   
   
3. Verify that the resulting (A, B) of the linear model is stabilizable.
   
   
4. Design a feedback controllers u = k₁x₁ + k₂x₂ such that both poles of the closed-loop system for the linear model are located at s = –2.

Exercise 5.2 Consider the nonlinear system

1. Verify that x* = [1 1]^T, u* = 0 is an equilibrium point of the nonlinear system.
   
   
2. Perform a change of coordinates z = x – x* and rewrite the nonlinear system in the z-coordinates.
   
   
3. Verify that z* = [0 0]^T, u* = 0 is an equilibrium point of the nonlinear system in the z-coordinates.
   
   
4. Linearize the system around the equilibrium point z* = [0 0]^T, u* = 0.
   
   
5. Design a feedback controller u = k₁z₁ + k₂z₂such that the poles of the closed-loop system for the linear model are located at s = –1 ± j.

Exercise 5.3 Use a simulation study to investigate the performance of the linear feedback control law

developed in Exercise 5.2 when applied to the original nonlinear system. Consider several initial conditions close to the equilibrium point x = x* to get a rough idea of how large is the region of attraction around the equilibrium point.

Exercise 5.4 Use a simulation study for the satellite example of Example 5.2. Assume that: . Consider the following cases:

Simulate the differential equation for about 100 s. provide plots of the satellite motion in cartesian coordinates instead of polar coordinates. Interpret your results. Compare the solution of the nonlinear differential equation with that of the linearized model (assume that r₀ = 10; θ₀ = 0; ω = 0.1). Discuss the accuracy of the linearized model as an approximation of the nonlinear system. Plot the trajectories of the satellite motion of both the linear and nonlinear model on the same diagram for comparison purposes.

Exercise 5.5 Consider the nonlinear state equation

with the nominal initial state , and the nominal input u*(t) = 1. Show that the nominal output is y*(t) = 1. Linearize the state equation about the nominal solution.

Exercise 5.6 Consider the following second-order model which represents a field controlled DC motor [248]

where x₁ is the armature current, x₂ is the speed, and u is the field current. It is required to design a speed control system so that y(t) asymptotically tracks a constant reference speed y_d= 100. It is assumed that the domain of operation for the armature current is restricted to x₁ > 0.2.

1. Find the steady-state field current u_ssand steady-state armature current x_1ss(within the domain of operation) such that the output y follows exactly the desired constant speed y_d = 100.
   
   
2. Verify that the control u = u_ssresults in an asymptotically stable equilibrium point.
   
   
3. Using small-signal linearization techniques, design a state feedback control law to achieve the desired speed control.
   
   
4. Using computer simulations, study the performance of the linear controller of part (c) when applied to the nonlinear system. Assume that y_d = 100 and at a certain time it increases (step change) to y_d = 105. Repeat the simulation experiment while gradually increasing the step change to y_d = 110, 115, 120,....

Exercise 5.7 Consider the same field controlled DC motor of Exercise 5.6. Suppose that the speed x₂ is measurable but the armature current x₁ is not measured for feedback control purposes.

1. Repeat part (d) of Exercise 5.6 using an observer to estimate the current; i.e., instead of using x₁ in the feedback control, use where is generated by an observer.
   
   
2. Design a gain scheduling, observer based controller, where the scheduling variable is the measured speed x₂.
   
   
3. Study the performance of the gain scheduling controller using computer simulation. Compare to the performance of the linear controller of part (a) obtained via small-signal linearization and discuss.

Exercise 5.8 Consider the Example 5.4 on page 195, which describes the model of a single-link manipulator with flexible joints.

1. Show that the transformation z = T(x) given by (5.30) is indeed a diffeomorphism, by obtaining the inverse x = T ^–1(z). What is the region in which this diffeomorphism is valid.
   
   
2. Verify the differential equations (5.31).

Exercise 5.9 Consider the system

Convert the system to normal form. Design a feedback linearizing tracking controller so that y(t) tracks the target signal y_d (t) –sin(t).

Exercise 5.10 For the system given in Exercise 5.9, after converting the system to normal form, use standard backstepping to design a tracking controller so that y(t) tracks the target signal y_d (t).

Exercise 5.11 For the system given in Exercise 5.9, use command filtered backstepping to design a tracking controller so that y(t) tracks the target signal y_d (t).

Exercise 5.12 Consider the system

1. Is the system input-output linearizable? Under what conditions? Assuming that these conditions are valid, design a tracking controller.
   
   
2. Assume that , where is known while ε is assumed by the designer to be zero, while in reality it is equal to 0.05. Investigate to what degree this modeling error affects the linearization and the design of the tracking controller.

Exercise 5.13 Design a tracking control algorithm for the system

where the desired output signal is y_d (t) = sin(3t).

Exercise 5.14 Consider Example 5.7 on page 206. Perform a computer simulation study to illustrate the performance of the control system. Similarly, perform a computer simulation for Example 5.8 and compare the differences.

Exercise 5.15 Consider Example 5.9 on page 218. Assume that the actual uncertainty term η is given by

while the bound is given by . Perform a computer simulation study to illustrate the performance or the control system using both the discontinuous algorithm and the continuous approximation obtained using the tanh function, with ε = 0.1.

Exercise 5.16 Consider Example 5.10 on page 220. As in Example 5.15, assume that the actual uncertainty term n is given by η(x) = 1.2 cos(x₁). Let ø and η₀= 1.2 cos(x₁). Perform a computer simulation study to illustrate the performance of the control system obtained using the nonlinear damping method. Repeat the simulation for various values of k. Compare the control performance and control effort with the Lyapunov redesign method of Example 5.15.

Exercise 5.17 Consider Example 5.12 on page 224. Let and θ = 1. Simulate this example for k₁ = k₂ = γ = 2. Plot the tracking error, the control effort and the parameter estimation error. Discuss your results.

Exercise 5.18 For the bounding control of Section 5.4.1 that uses the smoothing approximation, show that e(t) ultimately converges to the set e < δ. Also show that e(t) ≤ δ for .