14.3: Exercises

Exercise 14.1 Bounded Perturbation

Recall Example 14.3. In this problem we want to improve the lower bound on \(\gamma(A)\).

(a) To improve the lower bound, we use the information that if \(\Delta\) is real, then poles appear in complex conjugate pair. Define

\[A_{w}=\left(\begin{array}{cc}
A & w I \\
-w I & A
\end{array}\right)\nonumber\]

Show that

\[\gamma(A) \geq \min _{w \in \mathbb{R}} \sigma_{\min }\left[A_{w}\right] \nonumber\]

(b) If you think harder about your proof above, you will be able to further improve the lower bound. In fact, it follows that

\[\gamma(A) \geq \min _{w \in \mathbb{R}} \sigma_{2 n-1}\left[A_{w}\right]\nonumber\]

where \(\sigma_{2 n-1}\) is the next to last singular value. Show this result.

Exercise 14.2

Consider the LTI unforced system given below:

\[\dot{x}=A x=\left(\begin{array}{cccccc}
0 & 1 & 0 & 0 & \ldots & 0 \\
0 & 0 & 1 & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-a_{N-1} & -a_{N-2} & \ldots & \ldots & \ldots & -a_{0}
\end{array}\right) x\nonumber\]

(a) Under what conditions is this system asymptotically stable?

Assume the system above is asymptotically stable. Now, consider the perturbed system

\[\dot{x}=A x+\Delta x,\nonumber\]

where \(\Delta\) is given by

\[\Delta=\left(\begin{array}{cccccc}
0 & 0 & 0 & 0 & \ldots & 0 \\
0 & 0 & 0 & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-\delta_{N-1} & -\delta_{N-2} & \ldots & \ldots & \ldots & -\delta_{0}
\end{array}\right), \quad \delta_{i} \in \mathbb{R}\nonumber\]

(b) Argue that the perturbation with the smallest Frobenius norm that destabilizes the system (makes the system not asymptotically stable) will result in \(A + \Delta\) having an eigenvalue at the imaginary axis.

(c) Derive an exact expression for the smallest Frobenius norm of \(\Delta\) necessary to destabilize the above system (i.e., \(\dot{x} = (A+\Delta)x\) is not asymptotically stable). Give an expression for the perturbation \(\Delta\) that attains the minimum.

(d) Evaluate your answer in part 3 for the case \(N = 2\), and \(a_{0} = a_{1}\).

Exercise 14.3 Periodic Controllers

(a) Show that the periodically varying system in Exercise 7.4 is asymptotically stable if and only if all the eigenvalues of the matrix \(\left[A_{N-1} \ldots A_{0}\right]\) have magnitude less than 1.

(b)

(i) Given the system

\[x(k+1)=\left(\begin{array}{cc}
0 & 1 \\
1 & -1
\end{array}\right) x(k)+\left(\begin{array}{l}
0 \\
1
\end{array}\right) u(k), \quad y(k)=\left(\begin{array}{cc}
1 & 1) x(k)
\end{array}\right.\nonumber\]

write down a linear state-space representation of the closed-loop system obtained by implementing the linear output feedback control \(u(k) = g(k)y(k)\).

(ii) It turns out that there is no constant gain \(g(k) = g\) for which the above system is asymptotically stable. (Optional: Show this.) However, consider the periodically varying system obtained by making the gain take the value \(-1\) for even \(k\) and the value 3 for odd \(k\). Show that any nonzero initial condition in the resulting system will be brought to the origin in at most 4 steps. (The moral of this is that periodically varying output feedback can do more than constant output feedback.)

Exercise 14.4 Delay Systems

The material we covered in class has focused on finite-dimensional systems, i.e., systems that have state-space descriptions with a finite number of state variables. One class of systems that does not belong to the class of finite-dimensional systems is continuous-time systems with delays.

Consider the following forced continuous-time system:

\[y(t)+a_{1} y(t-1)+a_{2} y(t-2)+\ldots+a_{N} y(t-N)=u(t) \quad t \geq N, t \in \mathbb{R}\nonumber\]

This is known as a delay system with commensurate delays (multiple of the same delay unit). We assume that \(u(t) = 0\) for all \(t < N\).

(a) Show that we can compute the solution \(y(t), t \geq N\), if \(y(t)\) is completely known in the interval \([0,N)\). Explain why this system cannot have a finite-dimensional state space description.

(b) To compute the solution \(y(t)\) given the initial values (denote those by the function \(f (t), t \in [0, N)\), which we will call the initial function) and the input \(u\), it is useful to think of every non-negative real number as \(t=\tau+k\) with \(\tau \in[0,1)\) and \(k\) being a non-negative integer. Show that for every fixed \(\tau\), the solution evaluated at \(\tau+k(y(\tau+k))\) can be computed using discrete-time methods and can be expressed in terms of the matrix

\[A=\left(\begin{array}{cccccc}
0 & 1 & 0 & 0 & \ldots & 0 \\
0 & 0 & 1 & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-a_{N} & -a_{N-1} & \ldots & \ldots & \ldots & -a_{1}
\end{array}\right)\nonumber\]

and the initial vector

\[(f(\tau) \quad f(\tau+1) \quad \ldots \quad f(\tau+N-1))^{T}\nonumber\]

Write down the general solution for \(y(t)\).

(c) Compute the solution for \(N = 2, f (t) = 1\) for t \in [0, 2)\), and \(u(t) = e^{-(t-2)}\) for \(t \geq 2\)

(d) This system is asymptotically stable if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all initial functions with \(|f(t)|<\delta, t \in[0, N)\), and \(u = 0\), it follows that \(|y(t)|<\epsilon\), and \(\lim _{t \rightarrow \infty} y(t)=0\). Give a necessary and sufficient condition for the asymptotic stability of this system. Explain your answer.

(e) Give a necessary and sufficient condition for the above system to be BIBO stable (\(\infty\)-stable). Verify your answer.

Exercise 14.5 Local Stabilization

(a) One method for stabilizing a nonlinear system is to linearize it around an equilibrium point and then stabilize the resulting linear system. More formally, consider a nonlinear time-invariant system

\[\dot{x}=f(x, u)\nonumber\]

and its linearization around an equilibrium point \((\tilde{x}, \tilde{u})\)

\[\dot{\delta} x=A \delta x+B \delta u\nonumber\]

As usual,\(\delta x=x-\tilde{x}\) and \(\delta u=u-\tilde{u}\). Suppose that the feedback \(\delta u=K \delta u\) asymptotically stabilizes the linearized system

1. What can you say about the eigenvalues of the matrix A + BK.
2. Show that \(\dot{x} = f (x, Kx)\) is (locally) asymptotically stable around \(\tilde{x}\)

(b) Consider the dynamic system \(S_{1}\) governed by the following differential equation:

\[\ddot{y}+\dot{y}^{4}+\dot{y}^{2} u+y^{3}=0\nonumber\]

where \(u\) is the input.

1. Write down a state space representation for the system \(S_{1}\) and find its unique equilibrium point \(x^{*}\).
2. Now try to apply the above method to the system \(S_{1}\) at the equilibrium point \(x^{*}\) and \(u^{*}=0\). Does the linearized system provide information about the stability of \(S_{1}\). Explain why the method fails.

(c) To find a stabilizing controller for \(S_{1}\), we need to follow approaches that are not based on local linearization. One approach is to pick a positive definite function of the states and then construct the control such that this function becomes a Lyapunov function. This can be a very frustrating exercise. A trick that is commonly used is to find an input as a function of the states so that the resulting system belongs to a class of systems that are known to be stable (e.g. a nonlinear circuit or a mechanical system that are known to be stable). Use this idea to find an input \(u\) as function of the states such that \(S_{1}\) is stable.

Exercise 14.6

For the system

\[\begin{array}{l}
\dot{x}(t)=\sin [x(t)+y(t)] \\
\dot{y}(t)=e^{x(t)}-1
\end{array}\nonumber\]

determine all equilibrium points, and using Lyapunov's indirect method (i.e. linearization), classify each equilibrium point as asymptotically stable or unstable.

Exercise 14.7

For each of the following parts, all of them optional, use Lyapunov's indirect method to determine, if possible, whether the origin is an asymptotically stable or unstable equilibrium point.

(a) \[\begin{array}{l}
\dot{x}_{1}=-x_{1}+x_{2}^{2} \\
\dot{x}_{2}=-x_{2}\left(x_{1}+1\right)
\end{array}\nonumber\]

(b) \[\begin{array}{l}
\dot{x}_{1}=x_{1}^{3}+x_{2} \\
\dot{x}_{2}=x_{1}-x_{2}
\end{array}\nonumber\]

(c) \[\begin{array}{l}
\dot{x}_{1}=-x_{1}+x_{2} \\
\dot{x}_{2}=-x_{2}+x_{1}^{2}
\end{array}\nonumber\]

(d) \[\begin{array}{l}
x_{1}(k+1)=2 x_{1}(k)+x_{2}(k)^{2} \\
x_{2}(k+1)=x_{1}(k)+x_{2}(k)
\end{array}\nonumber\]

(e) \[\begin{array}{l}
x_{1}(k+1)=1-e^{x_{1}(k) x_{2}(k)} \\
x_{2}(k+1)=x_{1}(k)+2 x_{2}(k)
\end{array}\nonumber\]

Exercise 14.8

For each of the nonlinear systems below, construct a linearization for the equilibrium point at the origin, assess the stability of the linearization, and decide (using the results of Lyapunov's indirect method) whether you can infer something about the stability of the equilibrium of the nonlinear system at the origin. Then use Lyapunov's direct method prove that the origin is actually stable in each case; if you can make further arguments to actually deduce asymptotic stability or even global asymptotic stability, do so. [Hints: In part (a), find a suitable Lyapunov (energy) function by interpreting the model as the dynamic equation for a mass attached to a nonlinear (cubic) spring. In parts (b) and (c), try a simple quadratic Lyapunov function of the form \(px^{2} + qy^{2}\), then choose \(p\) and \(q\) appropriately. In part (d), use the indicated Lyapunov function.]

(a) \[\begin{array}{l}
\dot{x}=y \\
\dot{y}=-x^{3}
\end{array}\nonumber\]

(b) \[\begin{array}{l}
\dot{x}=-x^{3}-y^{2} \\
\dot{y}=x y-y^{3}
\end{array}\nonumber\]

(c) \[\begin{aligned}
x_{1}(k+1) &=\frac{x_{2}(k)}{1+x_{2}^{2}(k)} \\
x_{2}(k+1) &=\frac{x_{1}(k)}{1+x_{2}^{2}(k)}
\end{aligned}\nonumber\]

(d) \[\begin{aligned}
\dot{x} &=y(1-x) \\
\dot{y} &=-x(1-y) \\
V(x, y) &=-x-\ln (1-x)-y-\ln (1-y)
\end{aligned}\nonumber\]