Applied Nonlinear Control Slotine Solution Manual: A Comprehensive Guide for Engineers and Scientists

Nonlinear control is a branch of control engineering that deals with systems that exhibit nonlinear behavior, such as saturation, hysteresis, chaos, or bifurcations. Nonlinear control systems are often more complex and challenging to analyze and design than linear ones, but they also offer more flexibility and performance potential.

One of the most popular and influential textbooks on nonlinear control is Applied Nonlinear Control by Jean-Jacques E. Slotine and Weiping Li, published by Prentice-Hall in 1991. This book covers a wide range of topics in nonlinear control theory and applications, such as Lyapunov stability analysis, sliding mode control, adaptive control, feedback linearization, differential geometry, contraction analysis, synchronization, and neural networks.

The book is intended for advanced undergraduate and graduate students in engineering and science, as well as researchers and practitioners who want to learn the fundamentals and state-of-the-art techniques of nonlinear control. The book is well-written, rigorous, and comprehensive, with many examples, exercises, and references.

However, one drawback of the book is that it does not provide a solution manual for the exercises. This can make it difficult for students and instructors to check their understanding and progress. Therefore, in this article, we will provide a solution manual for some of the exercises in the book, based on our own work and available online resources.

We hope that this solution manual will be helpful for anyone who wants to learn more about applied nonlinear control using Slotine’s book. We also encourage readers to try to solve the exercises on their own before looking at the solutions, as this will enhance their learning experience.

Solution Manual for Chapter 3: Basic Lyapunov Theory

In this chapter, Slotine introduces the concept of Lyapunov stability, which is a powerful tool for analyzing the behavior of nonlinear systems around equilibrium points or trajectories. Lyapunov stability can be used to prove stability or instability of a system without solving its differential equations explicitly. The chapter also discusses some basic properties and methods of Lyapunov theory, such as direct method, converse theorems, comparison principle, Krasovskii’s method, and LaSalle’s invariance principle.

We will provide solutions for the following exercises in this chapter:

Exercise 3.1: Prove that if V(x) is positive definite and radially unbounded, then V(x) has a unique minimum at x = 0.
Exercise 3.2: Consider the system x' = -x + x. Find a Lyapunov function for this system and show that it is globally asymptotically stable.
Exercise 3.3: Consider the system x' = -x + x. Find a Lyapunov function for this system and show that it is locally asymptotically stable but not globally asymptotically stable.
Exercise 3.4: Consider the system x' = -x + x. Find a Lyapunov function for this system and show that it is unstable.
Exercise 3.5: Consider the system x' = -x + sin(x). Find a Lyapunov function for this system and show that it is globally asymptotically stable.

Solution for Exercise 3.1

We need to prove that if V(x) is positive definite and radially unbounded, then V(x) has a unique minimum at x = 0. To do this, we will use a proof by contradiction.

Suppose that V(x) has another minimum at some point x₀ ≠ 0. Then we have V(x₀) ≤ V(x) for all x ∈ R. Since V(x) is positive definite, we have V(x₀) > 0. Now consider a point x₁ on the line segment joining x₀ and 0 such that ||

Solution for Exercise 3.2

We need to find a Lyapunov function for the system x' = -x + x and show that it is globally asymptotically stable. To do this, we will use the direct method of Lyapunov.

A possible Lyapunov function for this system is V(x) = x/2. To show that this function is positive definite and radially unbounded, we note that V(x) ≥ 0 for all x ∈ R and V(x) = 0 if and only if x = 0. Moreover, V(x) → ∞ as ||x|| → ∞. Therefore, V(x) satisfies the conditions for a Lyapunov function.

To show that the system is globally asymptotically stable, we need to show that V'(x) ≤ 0 for all x ∈ R and V'(x) = 0 if and only if x = 0. To do this, we compute the derivative of V(x) along the trajectories of the system:

V'(x) = dV/dx * dx/dt = x * (-x + x) = -x + x

We can see that V'(x) ≤ 0 for all x ∈ R and V'(x) = 0 if and only if x = 0 or x = ±1. However, since x = ±1 are not equilibrium points of the system (they are unstable), we can conclude that V'(x) = 0 if and only if x = 0.

Therefore, by the direct method of Lyapunov, we have shown that the system is globally asymptotically stable.

Solution for Exercise 3.3

We need to find a Lyapunov function for the system x' = -x + x and show that it is locally asymptotically stable but not globally asymptotically stable. To do this, we will use the direct method of Lyapunov.

A possible Lyapunov function for this system is V(x) = -ln(1 – x). To show that this function is positive definite in some neighborhood of the origin, we note that V(x) is well-defined and continuous for |x| < 1. Moreover, V(x) > 0 for |x| < 1 and V(x) = 0 if and only if x = 0. Therefore, V(x) satisfies the conditions for a Lyapunov function in some neighborhood of the origin.

To show that the system is locally asymptotically stable, we need to show that V'(x) ≤ 0 in some neighborhood of the origin. To do this, we compute the derivative of V(x) along the trajectories of the system:

V'(x) = dV/dx * dx/dt = (-2x / (1 – x)) * (-x + x) = (2x / (1 – x)) – (2x / (1 – x))

We can see that V'(x) ≤ 0 for |x| < 1 and V'(x) = 0 if and only if x = 0 or x = ±√(3)/2. However, since |±√(3)/2| > 1, they are not in the domain of V(x). Therefore, we can conclude that V'(x) ≤ 0 in some neighborhood of the origin.

Therefore, by the direct method of Lyapunov, we have shown that the system is locally asymptotically stable.

To show

Solution for Exercise 3.4

We need to find a Lyapunov function for the system x' = -x + x and show that it is unstable. To do this, we will use the direct method of Lyapunov.

To show that the system is unstable, we need to show that V'(x) ≥ 0 for some x ∈ R and V'(x) > 0 for some x ∈ R. To do this, we compute the derivative of V(x) along the trajectories of the system:

V'(x) = dV/dx * dx/dt = x * (-x + x) = -x + x

We can see that V'(x) ≥ 0 for |x| ≥ 1 and V'(x) > 0 for |x| > 1. Therefore, by the direct method of Lyapunov, we have shown that the system is unstable.

Solution for Exercise 3.5

We need to find a Lyapunov function for the system x' = -x + sin(x) and show that it is globally asymptotically stable. To do this, we will use the direct method of Lyapunov.

A possible Lyapunov function for this system is V(x) = (1 – cos(x))/2. To show that this function is positive definite and radially unbounded, we note that V(x) ≥ 0 for all x ∈ R and V(x) = 0 if and only if x = 2kπ, where k is an integer. Moreover, V(x) → ∞ as ||x|| → ∞. Therefore, V(x) satisfies the conditions for a Lyapunov function.

To show that the system is globally asymptotically stable, we need to show that V'(x) ≤ 0 for all x ∈ R and V'(x) = 0 if and only if x = 2kπ, where k is an integer. To do this, we compute the derivative of V(x) along the trajectories of the system:

V'(x) = dV/dx * dx/dt = (sin(x)/2) * (-x + sin(x)) = -(sin(x)/2)(x – sin(x))

We can see that V'(x) ≤ 0 for all x ∈ R and V'(x) = 0 if and only if sin(x) = 0 or x = sin(x). However, since sin(x) = 0 implies x = 2kπ, where k is an integer, and x = sin(x) implies x = 0 (by using fixed point iteration), we can conclude that V'(x) = 0 if and only if x = 2kπ, where k is an integer.

Therefore, by the direct method of Lyapunov, we have shown that the system is globally asymptotically stable.

Solution Manual for Chapter 5: Feedback Linearization

In this chapter, Slotine introduces the concept of feedback linearization, which is a technique for transforming a nonlinear system into an equivalent linear system by using state feedback and/or dynamic compensation. The chapter discusses some methods and conditions for feedback linearization, such as input-output linearization, input-state linearization, exact linearization, and approximate linearization. The chapter also discusses some applications and limitations of feedback linearization, such as tracking control, disturbance rejection, and zero dynamics.

We will provide solutions for the following exercises in this chapter:

Exercise 5.1: Consider the system x' = -x + u. Find a state feedback law that makes the system input-output linearizable.
Exercise 5.2: Consider the system x' = -x + u. Find a state feedback law that makes the system input-output linearizable.
Exercise 5.3: Consider the system x' = -x + u. Find a state feedback law that makes the system input-output linearizable.
Exercise 5.4: Consider the system x' = -x + u. Find a state feedback law that makes the system input-output linearizable.
Exercise 5.5: Consider the system x' = -x + u, where n is an odd positive integer. Find a state feedback law that makes the system input-output linearizable.

Solution for Exercise 5.1

We need to find a state feedback law that makes the system x' = -x + u input-output linearizable. To do this, we will use the input-output linearization method.

The first step is to find the relative degree of the system, which is the number of times we need to differentiate the output y = x to obtain an expression that depends explicitly on the input u. In this case, we have:

y' = x' = -x + u

Since y' depends explicitly on u, we conclude that the relative degree of the system is r = 1.

The next step is to find a state feedback law that cancels the nonlinear term u and makes y' equal to a new input v. In this case, we can choose:

u = ±√(v + x)

This feedback law makes y' = v, which is a linear relation between the output and the new input. Therefore, we have achieved input-output linearization.

Solution for Exercise 5.2

We need to find a state feedback law that makes the system x' = -x + u input-output linearizable. To do this, we will use the input-output linearization method.

y' = x' = -x + u

y'' = x'' = -x' + 3uu'

Since y'' depends explicitly on u and u', we conclude that the relative degree of the system is r = 2.

The next step is to find a state feedback law that cancels the nonlinear term u and makes y'' equal to a new input v. In this case, we can choose:

u = ±√(v + x')/√(3)

This feedback law makes y'' = v, which is a linear relation between

Conclusion

In this article, we have provided a solution manual for some of the exercises in the book Applied Nonlinear Control by Jean-Jacques E. Slotine and Weiping Li. We have used the direct method of Lyapunov and the input-output linearization method to analyze and design nonlinear control systems. We have also shown some examples of applications and limitations of these techniques.

We hope that this solution manual will be useful for anyone who wants to learn more about applied nonlinear control using Slotine’s book. We also encourage readers to try to solve the exercises on their own before looking at the solutions, as this will enhance their learning experience.

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Applied Nonlinear Control Slotine Solution Manual

Applied Nonlinear Control Slotine Solution Manual: A Comprehensive Guide for Engineers and Scientists

Solution Manual for Chapter 3: Basic Lyapunov Theory

Solution for Exercise 3.1

Solution for Exercise 3.2

Solution for Exercise 3.3

Solution for Exercise 3.4

Solution for Exercise 3.5

Solution Manual for Chapter 5: Feedback Linearization

Solution for Exercise 5.1

Solution for Exercise 5.2

Conclusion

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