Systems of ODEs: Introducing State-Space Models

In the next few sections, we will consider sets of linear, time-invariant, first-order ODEs. For an analysis of a system described by a set of ODEs, engineers often use the language of system dynamics. In system dynamics, the set of ODEs is referred to as a state-space model. We briefly introduce the concept of a state-space model here.

For a state-space model, there are three key vector-valued functions of time (often just called “vectors”):

• State \(\bm{x}(t)\), which has components that are a minimal set of variables from which the system’s behavior can be determined (e.g., position, voltage, flowrate)
• Input \(\bm{u}(t)\), which has components that are a set of variables that are from the system’s environment
• Output \(\bm{y}(t)\), which has components that are a set of variables that are of interest

These vector-valued functions interact through two equations: $$\begin{align} \frac{d\bm{x}}{d t} &= \bm{f}(\bm{x},\bm{u},t) \label{eq:ssresp_eq:state_nonlinear2} \\ \bm{y} &= \bm{g}(\bm{x},\bm{u},t)\label{eq:ssresp_eq:output_nonlinear2} \end{align}$$ where \(\bm{f}\) and \(\bm{g}\) are vector-valued functions that describe the system. Together, the equations comprise what is called a state-space model of a system.

In accordance with the definition of a state-determined system, given an initial condition \(\bm{x}(t_0)\) and input \(\bm{u}\), the state \(\bm{x}\) is determined for all \(t\ge t_0\). Determining the state response requires the solution—analytical or numerical—of the vector differential equation .

The second equation is algebraic. It expresses how the output \(\bm{y}\) can be constructed from the state \(\bm{x}\) and input \(\bm{u}\). This means we must first solve the state equation for \(\bm{x}\), then the output \(\bm{y}\) is given by .

Just because we know that, for a state-determined system, there exists a solution to , doesn’t mean we know how to find it. In general, \(\bm{f}:\mathbb{R}^n \times \mathbb{R}^r \times \mathbb{R}\rightarrow\mathbb{R}^n\) and \(\bm{g}:\mathbb{R}^n \times \mathbb{R}^r \times \mathbb{R}\rightarrow\mathbb{R}^m\) can be nonlinear functions.¹ We don’t know how to solve most nonlinear state equations analytically. An additional complication can arise when, in addition to states and inputs, system parameters are themselves time-varying (note the explicit time \(t\) argument of \(\bm{f}\) and \(\bm{g}\)). Fortunately, often a linear, time-invariant (LTI) model is sufficient.

A linear, time-invariant (LTI) state-space model is of the form $$\begin{align} \frac{d\bm{x}}{d t} &= A \bm{x} + B \bm{u} \label{eq:ssresp_eq:state2} \\ \bm{y} &= C \bm{x} + D \bm{u},\label{eq:ssresp_eq:output2} \end{align}$$ where \(A\), \(B\), \(C\), and \(D\) are constant matrices containing system lumped-parameters such as mass or inductance. Consult a system dynamics book for details on the derivation of such models.

In the following sections, we learn to solve for the state response and substitute the result into for the output response.