6. Discrete Time Systems

First we introduce some form of discrete systems:

Nonlinear system: \(x(k+1) = f(x(k), u(k)), x \in \mathbb R^n, u \in \mathbb R^n\)
Autonomous system: \(x(k+1) = f(x(k))\)
Linear system: \(x(k+1) = Ax(k) + Bu(k)\)

6.1 Equilibrium of Linear Systems

Let \(u(k) = \bar u\), \(k \geq 0\), \(\bar x = f(\bar x, \bar u)\), where \((\bar x, \bar u)\) is an equilibrium pair, for linear systems, we have:

\[ \bar x = A \bar x + B \bar u \]

if \(A\) has no eigenvalue \(\lambda = 1\), \(\Rightarrow\) \(\bar x = -(\mathbf I - A)^{-1}B\bar u\).

6.2 Stability of Linear Systems

Consider the stability of \((\bar x, \bar u)\) of the equilibrium pair:

\(\forall \varepsilon\), \(\exists \delta > 0\), such that \(\forall x_0 \in B_\delta(\bar x)\), it holds that \(x_{x_0}(k) \in B_\varepsilon (\bar x)\), \(\forall k \geq 0\)

\((\bar x, \bar u)\) is an equilibrium pair if:

\((\bar x, \bar u)\) is stable
\(\exists \delta > 0\), such that \(\lim_{k\to \infty}||x_{x_0}(k)-\bar x = 0||\), \(\forall x_0 \in B_\delta(\bar x)\)

Quote

Stability depends on the specific equilibrium pair \((\bar x, \bar u)\)
A.S. is a local property

For linear systems:

Stability is a property of the system
A.S. \(\Leftrightarrow\) G.A.S.

A N&S condition for a linear system to be A.S. is that all eigenvalues of \(A\) satisfy \(|\lambda| < 1\)

6.3 Stability of Nonlinear Systems

Consider a nonlinear system \(x(k+1) = f(x(k), u(k)), f \in C^1\), \((\bar x, \bar u)\) is an equilibrium pair, we can do linearization for this system:

\[ \delta x(k+1) = A\delta x(k) + B\delta u(k) \]

Where, \(A = \frac{\partial f}{\partial x}|_{x=\bar x, u=\bar u}\), \(B = \frac{\partial f}{\partial u}|_{x=\bar x, u=\bar u}\). \(\lambda\) is the eigenvalue of matrix \(A\), the stability can be judged in following conditions:

\(|\lambda_i| < 1, \forall \lambda_i \in \lambda\) \(\Rightarrow\) \((\bar x, \bar u)\) is A.S.
\(\exists |\lambda_i| > 1, \lambda_i \in \lambda\) \(\Rightarrow\) \((\bar x, \bar u)\) is unstable
\(|\lambda_i| \leq 1, \forall \lambda_i \in \lambda\) and \(\exists |\lambda_j| = 1, \lambda_j \in \lambda\) \(\Rightarrow\) could not tell the stability

6.4 Lyapunov Methods

Considering the perturbance of the discrete system:

\[ x(k+1) = \varphi(x(k)) \]

\(\bar x\) is an equilibrium, \(\bar x = \varphi (\bar x)\), \(\varphi\) is Lipchitz, \(\Delta V(x) = V(\varphi(x)) - V(x)\)

if \(\exists V, V \in C^1, V(\bar x) \succ 0\), and \(\Delta V(\bar x) \preceq 0\) \(\Rightarrow\) \(\bar x\) is stable
if \(\Delta V(\bar x) \prec 0\) \(\Rightarrow\) \(\bar x\) is A.S.
if \(\exists V, V \in C^1, V \in \mathbb R^n \to \mathbb R\), and \(V(\bar x) \succ 0, \Delta V(\bar x) \prec 0, \forall \bar x\) \(\Rightarrow\) \(\bar x\) is G.A.S

6.5 Krasowski - La Salle Theorem:

Suppose \(\exists V, V(\bar x) \succ 0, \Delta V(\bar x) \preceq 0, \bar x \in D\), if the set: \(S = \{x\in D: \Delta V(x) = 0\}\) does not contain any perturbed trajectory of the system, \(\Rightarrow\) \(\bar x\) is A.S.

Note

if \(D=\mathbb R^n\) and \(V \in \mathbb R^n \to \mathbb R\) \(\Rightarrow\) \(\bar x\) is G.A.S. for the above condition

For a discrete linear system, the N&S condition of the A.S. of the system is that \(\forall Q = Q^T \succ 0\), \(\exists P = P^T \succ 0\) that satisfying

\[ A^TPA - P = -Q \]

Info

Proof (Sufficiency):

\(\bar u = 0 \to \bar x = 0\), \(Q\succ0 \to P \succ 0\)

\[ \begin{aligned} V(x) &= x^TPx \\ \delta V(x) &= \underbrace{(Ax)^TP(Ax)}_{V(\dot \varphi(x))} - \underbrace{x^TPx}_{V(x)} = -x^TQx \end{aligned} \]

\(\Delta V(\bar x) \prec 0\)