13.4: Exercises

Exercise 13.1

Consider the horizontal motion of a particle of unit mass sliding under the influence of gravity on a frictionless wire. It can be shown that, if the wire is bent so that its height \(h\) is given by \(h(x) = V_{\alpha}(x)\), then a state-space model for the motion is given by

\[\begin{array}{l}
\dot{x}=z \\
\dot{z}=-\frac{d}{d x} V_{\alpha}(x)
\end{array}\nonumber\]

Suppose \(V_{\alpha}(x)=x^{4}-\alpha x^{2}\)

(a) Verify that the above model has \((z, x) = (0 , 0)\) as equilibrium point for any \(\alpha\) in the interval \(-1 \leq \alpha \leq 1\), and it also has \((z, x)=(0, \pm \sqrt{\frac{\alpha}{2}})\) as equilibrium points when \(\alpha\) is in the interval \(0<\alpha \leq 1\).

(b) Verify that the linearized model about any of the equilibrium points is neither asymptotically stable nor unstable for any \(\alpha\) in the interval \(-1 \leq \alpha \leq 1\).

Exercise 13.2

Consider the dynamic system described below:

\[\ddot{y}+a_{1} \dot{y}+a_{2} y+c y^{2}=u+\dot{u}\nonumber\]

where \(y\) is the output and \(u\) is the input.

(a) Obtain a state-space realization of dimension 2 that describes the above system.

(b) If \(a_{1}=3, a_{2}=2, c=2\), show that the system is asymptotically stable at the origin.

(c) Find a region (a disc of non-zero radius) around the origin such that every trajectory, with an initial state starting in this region, converges to zero as \(t\) approaches infinity. This is known as a region of attraction.

Exercise 13.3

Consider the system

\[\dot{x}(t)=-\frac{d P(x)}{d x}\nonumber\]

where \(P (x)\) has continuous first partial derivatives. The function \(P (x)\) is referred to as the potential function of the system, and the system is said to be a gradient system. Let \(\bar{x}\) be an isolated local minimum of \(P (x)\), i.e. \(P(\bar{x})<P(x)\) for \(0<\|x-\bar{x}\|<r\), some \(r\).

(a) Show that \(\bar{x}\) is an equilibrium point of the gradient system.

(b) Use the candidate Lyapunov function

\[V(x)=P(x)-P(\bar{x})\nonumber\]

to try and establish that \(\bar{x}\) is an asymptotically stable equilibrium point.

Exercise 13.4

The objective of this problem is to analyze the convergence of the gradient algorithm for finding a local minimum of a function. Let \(f : \mathbb{R}^{n} \rightarrow \mathbb{R}\) and assume that \(x^{\prime}\) is a local minimum; i.e., \(f (x^{\prime}) < f(x)\) for all \(x\) close enough but not equal to \(x^{\prime}\). Assume that \(f\) is continuously differentiable. Let \(g(t) : \mathbb{R}^{n} \rightarrow \mathbb{R}\) be the gradient of \(f\) :

\[g^{T}=\left(\frac{\partial g}{\partial x_{1}} \quad \ldots \quad \frac{\partial g}{\partial x_{n}}\right)\nonumber\]

It follows from elementary Calculus that \(g\left(x^{*}\right)=0\).

If one has a good estimate of \(x^{*}\), then it is argued that the solution to the dynamic system:

\[\dot{x}=-g(x) \ \tag{13.10}\]

with \(x(0)\) close to \(x^{*}\) will give \(x(t)\) such that

\[\lim _{t \rightarrow \infty} x(t)=x^{*}\nonumber\]

(a) Use Lyapunov stability analysis methods to give a precise statement and a proof of the above argument.

(b) System 13.10 is usually solved numerically by the discrete-time system

\[x(k+1)=x(k)-\alpha\left(x_{k}\right) g\left(x_{k}\right) \ \tag{13.11}\]

where \(\alpha\left(x_{k}\right)\) is some function from \(\mathbb{R}^{n} \rightarrow \mathbb{R}\). In certain situations, \(\alpha\) can be chosen as a constant function, but this choice is not always good. Use Lyapunov stability analysis methods for discrete-time systems to give a possible choice for \(\alpha\left(x_{k}\right)\) so that

\[\lim _{k \rightarrow \infty} x(k+1)=x^{*}\nonumber\]

(c) Analyze directly the gradient algorithm for the function

\[f(x)=\frac{1}{2} x^{T} Q x, Q \text{ Symmetric, Positive Definite}\nonumber\]

Show directly that system 13.10 converges to zero \(\left(=x^{*}\right)\). Also, show that \(\alpha\) in system 13.11 can be chosen as a real constant, and give tight bounds on this choice.

Exercise 13.5

(a) Show that any (possibly complex) square matrix \(M\) can be written uniquely as the sum of a Hermitian matrix \(H\) and a skew-Hermitian matrix \(S\), i.e. \(H^{\prime}=H\) and \(S^{\prime}=S\). (Hint: Work with combinations of \(M\) and \(M^{\prime}\).) Note that if \(M\) is real, then this decomposition expresses the matrix as the sum of a symmetric and skew-symmetric matrix.

(b) With \(M\), \(H\), and \(S\) as above, show that the real part of the quadratic form \(x^{\prime} M x\) equals \(x^{\prime} H x\), and the imaginary part of \(x^{\prime} M x\) equals \(x^{\prime} S x\). (It follows that if \(M\) and \(x\) are real, then \(x^{\prime} M x= x^{\prime} H x \)).

(c) Let \(V(x)=x^{\prime} M x\) for real \(M\) and \(x\). Using the standard definition of \(d V(x) / d x\) as a Jacobian matrix - actually just a row vector in this case - whose \(j\)th entry is \(\partial V(x) / \partial x_{j}\), show that

\[\frac{d V(x)}{d x}=2 x^{\prime} H\nonumber\]

where \(H\) is the symmetric part of \(M\), as defined in part (a).

(d) Show that a Hermitian matrix always has real eigenvalues, and that the eigenvectors associated with distinct eigenvalues are orthogonal to each other.

Exercise 13.6

Consider the (real) continuous-time LTI system \(\dot{x}(t)=A x(t)\).

(a) Suppose the (continuous-time) Lyapunov equation

\[P A+A^{\prime} P=-I \ \tag{3.1}\]

has a symmetric, positive definite solution \(P\). Note that (3.1) can be written as a linear system of equations in the entries of \(P\), so solving it is in principle straightforward; good numerical algorithms exist.

Show that the function \(V (x) = x^{\prime}P x\) serves as a Lyapunov function, and use it to deduce the global asymptotic stability of the equilibrium point of the LTI system above, i.e. to deduce that the eigenvalues of \(A\) are in the open left-half plane. (The result of Exercise 13.5 will be helpful in computing \(\dot{V} (x)\).)

What part (a) shows is that the existence of a symmetric, positive definite solution of (3.1) is sufficient to conclude that the given LTI system is asymptotically stable. The existence of such a solution turns out to also be necessary, as we show in what follows. [Instead of \(-I\) on the right side of (3.1), we could have had \(-Q\) for any positive definite matrix \(Q\). It would still be true that the system is asymptotically stable if and only if the solution \(P\) is symmetric, positive definite. We leave you to modify the arguments here to handle this case.]

(b) Suppose the LTI system above is asymptotically stable. Now define

\[P=\int_{0}^{\infty} R(t) d t, \quad R(t)=e^{A^{\prime} t} e^{A t} \ \tag{3.2}\]

The reason the integral exists is that the system is asymptotically stable - explain this in more detail! Show that \(P\) is symmetric and positive definite, and that it is the unique solution of the Lyapunov equation (3.1). You will find it helpful to note that

\[R(\infty)-R(0)=\int_{0}^{\infty} \frac{d R(t)}{d t} d t\nonumber\]

The results of this problem show that one can decide whether a matrix \(A\) has all its eigenvalues in the open left-half plane without solving for all its eigenvalues. We only need to test for the positive definiteness of the solution of the linear system of equations (3.1). This can be simpler.

Exercise 13.7

This problem uses Lyapunov's direct method to justify a key claim of his indirect method: if the linearized model at an equilibrium point is asymptotically stable, then this equilibrium point of the nonlinear system is asymptotically stable. (We shall actually only consider an equilibrium point at the origin, but the approach can be applied to any equilibrium point, after an appropriate change of variables.)

Consider the time-invariant continuous-time nonlinear system given by

\[\dot{x}(t)=A x(t)+h(x(t)) \ \tag{4.1}\]

where \(A\) has all its eigenvalues in the open left-half plane, and \(h(.)\) represents "higher-order terms", in the sense that \(\|h(x)\| /\|x\| \rightarrow 0\) as \(\|x\| \rightarrow 0\).

(a) Show that the origin is an equilibrium point of the system (4.1), and that the linearized model at the origin is just \(\dot{x}(t) = Ax(t)\).

(b) Let \(P\) be the positive definite solution of the Lyapunov equation in (3.1). Show that \(V(x)=x^{\prime} P x\) qualifies as a candidate Lyapunov function for testing the stability of the equilibrium point at the origin in the system (4.1). Determine an expression for \(\dot{V} (x)\), the rate of change of \(V (x)\) along trajectories of (4.1)

(c) Using the fact that \(x^{\prime} x=\|x\|^{2}\), and that \(\|P h(x)\| \leq\|P\|\|h(x)\|\), how small a value (in terms of \(\|P\|\) of the ratio \(\|h(x)\| /\|x\|\) will allow you to conclude that \(\dot{V}(x(t))<0\) for \(x(t) \neq 0\)? Now argue that you can indeed limit \(\|h(x)\| /\|x\|\) to this small a value by choosing a small enough neighborhood of the equilibrium. In this neighborhood, therefore, \(\dot{V}(x(t))<0\) for \(x(t) \neq 0\). By Lyapunov's direct method, this implies asymptotic stability of the equilibrium point.

Exercise 13.8

For the discrete-time LTI system \(x(k+1)=A x(k)\), let \(V(x)=x^{\prime} P x\), where \(P\) is a symmetric, positive definite matrix. What condition will guarantee that \(V (x)\) is a Lyapunov function for this system? What condition involving \(A\) and \(P\) will guarantee asymptotic stability of the system? (Express your answers in terms of the positive semidefiniteness and definiteness of a matrix.)