*2.2. Linear Difference Systems*

Recall that linear difference equations can be used to study discrete-time linear (sampled) systems, which have the following quite general form:

$$R\_0 w + R\_1(\sigma w) + \dots + R\_N(\sigma^N w) = 0,\tag{1}$$

where the discrete time function *<sup>w</sup>* : <sup>Z</sup><sup>+</sup> <sup>→</sup> <sup>R</sup>*<sup>q</sup>* maps time instants into physical amounts (or measurements); the maximum degree of the shift operator *σ* is represented by *N*; and *Ri* <sup>∈</sup> <sup>R</sup>*p*×*<sup>q</sup>* (*<sup>i</sup>* <sup>=</sup> 0, 1, ..., *<sup>N</sup>*). The linear difference system (1) can be compactly expressed as:

$$R(\sigma)w = 0 \; ; \; \tag{2}$$

where *R*(*σ*) is a *p* × *q* polynomial matrix in *σ*, and represents the laws of the physical system with respect to *w*. The components of *w* can be classified as either inputs or outputs. Input functions, denoted by *u*, are independent (v.g., control variables), and output functions, denoted by *y*, are results due to the inputs (v.g., state variables). These variables can be accommodated as an *input/output partition*, that is, *w* := col(*u*, *y*).

#### *2.3. Quadratic Difference Forms (QdFs)*

Functionals, such as Lyapunov functions, have been traditionally used to study stability and other important properties of linear difference systems. In the present case, we use the notion of quadratic difference forms (QdFs), which are functionals of the discrete-time function *w* and its time-shifts, that is,

$$Q\_{\Psi}(w) = \begin{bmatrix} w^{\top} & \sigma w^{\top} & \cdots & \sigma^{N} w^{\top} \end{bmatrix} \Psi \begin{bmatrix} w \\ \sigma w \\ \vdots \\ \sigma^{N} w \end{bmatrix} \,\tag{3}$$

where <sup>Ψ</sup> <sup>∈</sup> <sup>R</sup>*Nq*×*Nq* is referred to as the coefficient matrix of *<sup>Q</sup>*Ψ. The rate of change of functional *Q*Ψ, denoted as ∇*Q*<sup>Ψ</sup> (an analogous to a continuous-time derivative), is given by

$$
\nabla Q\_{\Psi}(w)(t) := \sigma Q\_{\Psi}(w)(t) - Q\_{\Psi}(w)(t) \;. \tag{4}
$$

Stability for autonomous systems represented by (2) can thus be studied by means of QdFs. A system is autonomous if the polynomial matrix *R*(*σ*) in (2) is square and nonsingular (see [25]). In the present case, we will see that this characteristic is easily achieved since the resultant closed-loop system under study is autonomous. An autonomous linear difference system is asymptotically stable if

$$\lim\_{t \to \infty} w(t) = 0 \,, \quad \forall w \text{ satisfying (2)}.$$

A system described by (2) is asymptotically stable, according to the Lyapunov approach, if a QdF *Q*<sup>Ψ</sup> exists and is such that, ∀*w* satisfying (2), the following holds:


This QdF *Q*Ψ that satisfies the above inequalities is referred to as the *Lyapunov function*.
