### 2. Time-Domain State-Space Models

##### 2.1) Continuous-Time State-Space Models

The equations of motion for a finite-dimensional linear-dynamic system are a set of $$n$$ first-order differential equations \eqref{Eq: continuous time-invariant linear system x_dot} along with an initial condition $$\boldsymbol{x}(t_0)$$. The $$n$$-dimensional state $$\boldsymbol{x}(t)$$ is most often related to the output through the measurement equation \eqref{Eq: continuous time-invariant linear system y}.

\begin{align} \label{Eq: continuous time-invariant linear system x_dot} \dot{\boldsymbol{x}}(t) &= A_c\boldsymbol{x}(t)+B_c\boldsymbol{u}(t)\\ \label{Eq: continuous time-invariant linear system y} \boldsymbol{y}(t) &= C\boldsymbol{x}(t)+D\boldsymbol{u}(t). \end{align}

This system of equations constitutes a continuous-time state-space model of a dynamical system. Given the initial condition $$\boldsymbol{x}(t_0)$$ at some $$t=t_0$$, solving for $$\boldsymbol{x}(t)$$ for $$t>t_0$$ yields

\begin{align} \boldsymbol{x}(t) = e^{A_c(t-t_0)}\boldsymbol{x}(t_0)+\int_{t_0}^te^{A_c(t-\tau)}B_c\boldsymbol{u}(\tau)\mathrm{d}\tau. \end{align}

Without loss of generality, we will consider that $$t_0 = 0$$.

##### 2.2) Discrete-Time State-Space Models

Dynamic systems are typically modeled by continuous-time or discrete-time equations. A close approximation of a continuous-time model can be obtained by a discrete one provided that the sampling rate is sufficiently high. A linear discrete system is most commonly described by an $$n^{\text{th}}$$ order difference equation, the weighting sequence, or a discrete state-space model. Let $$\Delta t$$ be a constant time interval and $$f=1/\Delta t$$ the sampling rate. Continuous versions of the $$A$$ and $$B$$ matrices are

\begin{align} \label{Eq: A Discrete_time} A &= e^{A_c\Delta t},\\ \label{Eq: B Discrete_time} B &= \int_0^{\Delta t}e^{A_c\tau}\mathrm{d}\tau B_c,\\ \boldsymbol{x}(k+1) &= \boldsymbol{x}((k+1)\Delta t),\\ \boldsymbol{u}(k) &= \boldsymbol{u}(k\Delta t). \end{align}

The discrete-time matrices $$A$$ and $$B$$ in \eqref{Eq: A Discrete_time} and \eqref{Eq: B Discrete_time} may be computed by the following series expansions:

\begin{align} \label{Eq: A series} A &= e^{A_c\Delta t} = \sum_{i=0}^{\infty}\dfrac{1}{i!}\left[A_c\Delta t\right]^i,\\ \label{Eq: B series} B &= \int_0^{\Delta t}e^{A_c\tau}\mathrm{d}\tau B_c = \left[\sum_{i=0}^{\infty}\dfrac{1}{i!}A_c^i\left(\Delta t\right)^{i+1}\right]B_c. \end{align}

A sufficient condition for these series expansions to converge is that the continuous-time state matrix $$A_c$$ is asymptotically stable in the sense that the real parts of all its eigenvalues are negative. If none of the eigenvalues of $$A_c$$ are zero, the discrete-time matrix $$B$$ may also be computed by

\begin{align} B = \left[A-I\right]A_c^{-1}B_c. \end{align}

Therefore, a discrete-time invariant linear system can be represented by

\begin{align} \label{Eq: discrete time-invariant linear system x(k+1)} \boldsymbol{x}(k+1) &= A\boldsymbol{x}(k)+B\boldsymbol{u}(k)\\ \boldsymbol{y}(k) &= C\boldsymbol{x}(k)+D\boldsymbol{u}(k) \end{align}

together with an initial state vector $$\boldsymbol{x}(0)$$, where $$\boldsymbol{x}$$, $$\boldsymbol{u}$$ and $$\boldsymbol{y}$$ are the state, control input and output vectors respectively. The constant matrices $$A$$, $$B$$, $$C$$ and $$D$$ with appropriate dimensions represent the internal operation of the linear system, and are used to determine the system's response to any input.

##### 2.3) Weighting Sequence Description and Markov Parameters

Solving for the state $$\boldsymbol{x}(k)$$ and the output $$\boldsymbol{y}(k)$$ with arbitrary initial condition $$\boldsymbol{x}(0)$$ in terms of the previous inputs $$\boldsymbol{u}(i)$$, $$i=0, 1, \ldots, k$$, yields

\begin{align} \boldsymbol{x}(k) &= A^k\boldsymbol{x}(0) + \sum_{i=1}^kA^{i-1}B\boldsymbol{u}(k-i),\\ \boldsymbol{y}(k) &= CA^k\boldsymbol{x}(0) + \sum_{i=1}^kCA^{i-1}B\boldsymbol{u}(k-i) + D\boldsymbol{u}(k). \end{align}

It appears naturally that the constant matrices sequence

\begin{align} \begin{split} h_0 &= D,\\ h_1 &= CB,\\ h_2 &= CAB, \\ & \hspace{0.5em} \vdots \\ h_k &= CA^{k-1}B,\\ & \hspace{0.5em} \vdots \end{split} \end{align}

plays an important role in identifying a mathematical model for linear dynamical systems. In fact, with zero initial condition $$\boldsymbol{x}(0)=\boldsymbol{0}$$, considering the response to a pulse sequence such that for $$j=1, 2, \ldots, r$$,

\begin{align} u_j(i) = \left\lbrace\begin{array}{ll} 1 & \text{for } i = 0\\ 0 & \text{for } i = 1, 2, \ldots \end{array}\right. \end{align}

the $$r$$ corresponding outputs $$\left\lbrace\boldsymbol{y}^{(j)}(i)\right\rbrace_{i=1, 2, \ldots}$$ can be assembled at each time step to recover the sequence $$\left\lbrace h_i\right\rbrace_{i=1, 2, \ldots}$$ as follows:

\begin{align} h_i = \begin{bmatrix} \boldsymbol{y}^{(1)}(i) & \boldsymbol{y}^{(2)}(i) & \cdots & \boldsymbol{y}^{(r)}(i) \end{bmatrix}. \end{align}

The constant matrices in the sequence $$\left\lbrace h_i\right\rbrace_{i=1, 2, \ldots}$$ are known as system Markov parameters or, in short, Markov parameters. It is obvious that the matrices $$A, B, C, D$$ are embedded in the Markov parameter sequence; undeniably, the determination of Markov parameters should be tantamount to system identification. The general form of the Markov parameters is thus

\begin{align} \label{Eq: Markov parameters} h_{i} = \left\lbrace\begin{array}{ll} D & i = 0,\\ CA^{i-1}B & i \geq 1,\\ 0 & i < 0. \end{array}\right. \end{align}