Homework 6 - Due: 02/29

Problem 1

Show that if all eigenvalues of a matrix \(A\) have real parts strictly less than some \(-\mu<0\), then for every \(Q=Q^T>0\) the equation \[ PA+A^TP+2\mu P=-Q \] has a unique solution \(P=P^T>0\). Show that in this case we have \[|x(t)|\le ce^{-\mu t}|x(0)|\] for some \(c>0\). The number \(\mu\) is called a stability margin.

Problem 2

In class we are focusing on continuous-time systems, but we will occasionally mention discrete-time systems in the exercises. The two cases are usually quite similar but there are some notable differences.

  1. Write the formula for the solution \(x(k)\) at time \(k\) of the discrete-time control system \(x(k+1)=Ax(k)+Bu(k)\) starting from some initial state \(x(0)\) at time 0.
    Hint: Your answer will be the discrete counterpart of the variation-of-constants formula.

  2. Under what conditions on the eigenvalues of \(A\) is the discrete-time system \(x(k+1)=Ax(k)\) with no controls asymptotically stable? stable? Justify your answers. Stability definitions are the same as for continuous-time systems, just replace \(t\) by \(k\).
    Hint: Same as for continuous-time case, consider diagonal and Jordan blocks.

  3. Lyapunov’s second method for the discrete-time system \(x(k+1)=f(x(k))\) involves the difference \[{\Delta} V(x):=V(f(x))-V(x)\] instead of the derivative \(\dot V(x)\); with this substitution, the statement is the same as in the continuous-time case.
    Derive the counterpart of the Lyapunov equation for the LTI discrete-time system \(x(k+1)=Ax(k)\).

Problem 3

Lyapunov’s second method (for asymptotic stability) generalizes to time-varying systems \(\dot x=f(t,x)\) as follows.

Let \(V(t,x)\) be a function such that for some positive definite functions \(W_1(x)\), \(W_2(x)\), and \(W_3(x)\) we have

\[W_1(x)\le V(t,x)\le W_2(x)\] and \[ \dot V(t,x):=\frac{\partial V}{\partial t}(t,x)+\frac{\partial V}{\partial x}(t,x)\cdot f(t,x)\le -W_3(x) \tag{1}\] for all \(t\) and \(x\). Then the system is asymptotically stable (globally if \(W_1\) is radially unbounded).

Now, consider an LTV system \(\dot x=A(t)x\), and let \(V\) be of the form \(V(t,x)=x^TP(t)x\).

Derive the time-varying analogue of the Lyapunov equation, in other words, derive an equation that \(P(t)\) needs to satisfy to guarantee asymptotic stability.

Carefully specify all required properties of the quantities appearing in your equation so that the above stability result, based on Equation 1, is applicable.

Problem 4

Let \(x_1,x_2,x_3\) be positions of thre moving points on the real line - let’s call them “agents”.

Tip

For a slightly more general framework, see formation control in the demo page

For each agent \(i\), let \(N_i\) denote the set of all other agents that agent \(i\) can “see” (meaning it can measure their positions), and consider the consensus equations:

\[ \dot x_i=-\sum\nolimits_{j\in N_i}(x_i-x_j),\qquad i=1,2,3 \]

Assume initially that all agents see each other, so that

\[N_1=\{2,3\},\qquad N_2=\{1,3\}, \qquad N_3=\{1,2\}\]

Define the following \(3\times 3\) matrices:

  • the degree matrix \(D\) is the diagonal matrix with diagonal elements \(D_{ii}\) equal to the number of agents in \(N_i\)
  • the adjacency matrix \(J\) has elements \(J_{ij}=1\) if agents \(i\) and \(j\) see each other, and 0 if they don’t (by assumption, \(J\) is symmetric and its diagonal elements are 0)
  • the Laplacian matrix \(L:=D-J\).
Part 1
  1. Write down the consensus equations as an LTI system \(\dot x = Ax\).
  2. Verify that the matrix \(A\) in part a) equals \(-L\).
  3. Is it true that system equilibria are exactly points in \(\mathbb{R}^3\) with \(x_1=x_2=x_3\) (agents’ positions coincide)?
  4. Are the system equilibria stable? Asymptotically stable?
Part 2

Now assume that agents 2 and 3 don’t see each other. Answer questions a)–d) for this case.

Part 3

Now assume that agent 3 doesn’t see agents 1 and 2 and is not seen by them. Answer a)–d) again.