State and output equations

The state \(\bm{x}\), input \(\bm{u}\), and output \(\bm{y}\) vectors interact through two equations: $$\begin{align} \frac{d\bm{x}} {d t} &= \bm{f}(\bm{x},\bm{u},t) \label{eq:state_nonlinear} \\ \bm{y} &= \bm{g}(\bm{x},\bm{u},t)\label{eq:output_nonlinear} \end{align}$$ where \(\bm{f}\) and \(\bm{g}\) are vector-valued functions that depend on the system. Together, they comprise what is called a state-space model of a system. Let’s not glide past these equations, which will be our dear friends for the rest of our analytic lives. The first equation is called the state equation. Given state and input vectors at a moment in time, it’s function \(\bm{f}\) describes, how the state is changing (i.e. \(d\bm{x}/d t\)). Clearly, the state equation is a vector differential equation, which is equivalent to a system of first-order differential equations.¹

In accordance with the definition of a state-determined system from , given an initial condition \(\bm{x}(t_0)\) and input \(\bm{u}\), the state \(\bm{x}\) is determined for all \(t\ge t_0\). The state-space model is precisely the “mathematical model” described in the definition of a state-determined system. Determining the state requires the solution—analytic 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 . Since the output \(\bm{y}\) is a vector of variables of interest, the output equation is constructed in two steps: (1) define the output variables and (2) write them in terms of the state variables \(x_i\) and input variables \(u_j\).

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.² 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 model is sufficiently effective.³

A linear, time-invariant (LTI) system has state-space model $$\begin{align} \frac{d\bm{x}} {d t} &= A \bm{x} + B \bm{u} \label{eq:state} \\ \bm{y} &= C \bm{x} + D \bm{u}\label{eq:output} \end{align}$$ where

• \(A\) is an \(n\times n\) matrix that describes how the state \(\bm{x}\) changes the state \(\bm{x}\),

• \(B\) is an \(n\times r\) matrix that describes how the input \(\bm{u}\) changes the state \(\bm{x}\),

• \(C\) is an \(m\times n\) matrix that describes how the state \(\bm{x}\) contributes to the output \(\bm{y}\), and

• \(D\) is an \(m\times r\) matrix that describes how the input \(u\) contributes to the output \(\bm{y}\).

In the next two lectures, we will learn how to derive a state-space model—for linear systems, how to find \(A\), \(B\), \(C\), and \(D\)—for a system from its linear graph. This is the link between the linear graph model and the state-space model.