Graphical representations

Several types of graphical representations can be useful. We begin with the venerable diagram or schematic, which should include all important elements of the system, such as those shown in fig. ¿fig:diagrams?.

The mathematical relations in schematics are frequently rather implicit. For instance, the schematics of the two-mass system in fig. ¿fig:diagrams? explicitly specifies certain equations, such that the force applied by the spring \(k\) is equal in magnitude and opposite in direction for each mass element \(m_1\) and \(m_2\). The equations for each element, later called the “elemental equations,” are rarely explicitly given in any of the graphical system representations. In our two-mass example, for instance, each mass element would have Newton’s second law as its elemental equation. This is implied by the fact that it is represented as a mass. However, sometimes additional information is required. In the case of the spring \(k\) connecting the masses, it is reasonable to expect that the force in the spring is monotonically related to its length—this is (as we will see) what it means for an element to behave as a spring. However, from the schematic alone, it would be risky to assume the spring follows Hooke’s law—that is, that its force is proportional to its length. So many diagrams require a narrative supplement.

A more-precise and minimal graphical representation of a dynamic system is the linear graph, the subject of (ch:linear_graph_models?).¹ Some sample linear graphs are shown in fig. ¿fig:linear-graphs?.

Linear graphs are more-precise than schematics in that they explicitly connect elements (graph edges) at graph nodes. Explicit here are the structural relations among elements, which can be transcribed into equations. For instance, the electronic subsystem of the middle linear graph above tells us, à la Kierchhoff’s voltage law, that the voltage across each of the elements sum to zero: \(V_S - v_1 - v_{R} = 0\). Similarly, à la Kierchhoff’s current law, the currents through each of the elements at the node connecting elements \(R\) and \(1\) sum to zero: \(i_R - i_1 = 0\). We generalize these relations for linear graphs in (lec:element_interconnection_laws?) Once again, the elemental equations are implicit.

The final type of graphical system representation we consider extensively in the text is the block diagram, a couple examples of which are shown in fig. ¿fig:block-diagrams?.

Block diagrams are more high-level than schematics and linear graphs, and usually show the interconnection of multiple dynamic systems. In fig. ¿fig:block-diagrams?, the systems represented by blocks \(G\) and \(H\) are concatenated such that the output of \(G\) is the input of \(H\). The contents of the blocks are systems usually interpreted in an explicitly mathematical form that will be introduced in a moment called the transfer function that relates the input and output variables; in our example, \(G\) maps \(Y_1\) to \(Y_2\), \(H\) maps \(Y_2\) to \(Y_3\), and \(T\) maps \(Y_6\) to \(Y_7\). The lines and arrows “carry” variables (\(Y_i\) in our case) among the systems. In addition to system blocks, sometimes summing junctions, as shown on the right of fig. ¿fig:block-diagrams?, sum two variables; in our example, \(Y_6 = Y_4 + Y_5\).

Time-domain (differential) equations

Now we begin to make explicit all the mathematics implied in the graphical representations above. We have already seen how algebraic relationships are implied by the interconnection of elements in schematics, linear graphs, and block diagrams. The “elemental” equations implicit in schematics and linear graphs can also be algebraic, but usually some in every system are differential equations. For example, the elemental equation for a mass element \(m\) is Newton’s second law, which, in one dimension and with applied force \(f\) and coordinate \(x\), is \[\begin{aligned} f = m \frac{\diff^2 x} {\diff t^2}. \end{aligned}\] The time-derivative here makes this a differential equation. When combined with algebraic and other differential equations in the system, the system of equations remains differential.

So a system is described by a system of algebraic and differential equations. There are two common ways to represent this system. The first is as a single differential equation of order \(n\), which is the result of combining all the algebraic and differential equations into a single scalar equation. The result—when the system is ordinary, linear, and time-invariant (all these terms will be described later in this text)—is what we call the input-output differential equation (io ODE) that relates input variable \(u(t)\) and output variable \(y(t)\): \[\begin{aligned} {8} \label{eq:ioode_intro} \frac{d^n y} {d t^n} & {}+{} & a_{n-1} \frac{d^{n-1} y}{d t^{n-1}} & {}+{} & \cdots & {}+{} & a_1 \frac{d y} {d t} & {}+{} && a_0 y =\nonumber \\ b_m \frac{d^m u} {d t^m} & {}+{} & b_{m-1} \frac{d^{m-1} u}{d t^{m-1}} & {}+{} & \cdots & {}+{} & b_1 \frac{d u} {d t} & {}+{} && b_0 u \end{aligned}\] where \(a_i, b_j\) are constants defined in terms of a system’s parameters. The io ODE representation is convenient because, for many common system orders \(n\) and inputs \(u\), the analytic solution for \(y(t)\) is known, or at least a methodical process for deriving the solution is known. (ch:superposition_stability_and_other_lti_system_properties?,ch:qualities_of_transient_response) make extensive use of the io ODE representation and show multiple solution techniques.

The other common representation of the system of equations is the state equation: a system of first-order ODEs in dependent variables \(x_i(t)\) collected into a state vector \(\bm{x}(t)\) and multiple input variables \(u_j(t)\) collected into the input vector \(\bm{u}\) such that $$\begin{align} \frac{d\bm{x}}{d t} &= \bm{f}(\bm{x},\bm{u},t) \label{eq:state_nonlinear_intro} \end{align}$$ where \(\bm{f}\) is a vector-valued function that also depends on the system parameters. This equation can be linear or nonlinear. Note that multiple outputs \(y_k\) can also be found from the $$\begin{align} \bm{y} &= \bm{g}(\bm{x},\bm{u},t)\label{eq:output_nonlinear_intro} \end{align}$$ where \(\bm{g}\) is a vector-valued function that also depends on the system parameters.

Together, eq. ¿eq:state_nonlinear_intro?,eq:output_nonlinear_intro comprise what is called a state-space model of a system. The state-space model can fully represent a system with multiple inputs and multiple outputs, something not possible with an io ODE. For many linear systems, the state equation can be solved analytically, as described in (ch:state_space_response?). Another advantage of the state-space model is that it can also be easily solved numerically, as (ch:state_space_response?) also covers.

The mathematical system representations above include many variables that are presented as explicit functions of time \(t\), which is why we say that they are time-domain representations.

Frequency-domain (algebraic) equations

The remaining system representations are in what is called the frequency domain, which encompasses those that involve functions of not time but either angular frequency \(\omega\) or Laplace transform variable \(s\). In fact, the Laplace and Fourier transforms² and their inverses are the bridges between the time- and frequency-domains.

The frequency domain will be properly introduced in (part:frequency_domain?). For now, let’s define these two remaining system representations in terms of their respective transforms. It is worth noting that the following representations are only defined for linear systems.

The transfer function \(H(s)\) is defined as the ratio of the Laplace transform of the output \(Y(s) \equiv \mathcal{L}(y(t))\) to the Laplace transform of the input \(U(s) \equiv \mathcal{L}(u(t))\); that is, \[\begin{aligned} \label{eq:tf_intro} H(s) &\equiv \frac{Y(s)} {U(s)}. \end{aligned}\] A quick rearrangement yields the output \[\begin{aligned} Y(s) &= H(s) U(s). \end{aligned}\] If we were to inverse Laplace transform \(\mathcal{L}^{-1}\) this, we would get \(y(t)\)! Let’s loop back for a moment to the block diagram representations of fig. ¿fig:block-diagrams?. We said the block \(G\) maps input \(Y_1\) to output \(Y_2\). The specific mapping is now clear: the block \(G\) is usually represented by the transfer function \(G(s)\), so \[\begin{aligned} G(s) = \frac{Y_2(s)} {Y_1(s)} \quad\text{and}\quad Y_2(s) &= G(s) Y_1(s). \end{aligned}\] We see that this means, in the Laplace domain, system blocks are just algebraic products of the input and the transfer function. What’s more, this transfer function thinking is very powerful: the system is understood as operating on an input with a transfer function and yielding an output.

The frequency response function \(H(j\omega)\) is defined similarly, but in terms of the Fourier transform. The Fourier transform \(\mathcal{F}\) is introduced gently in (ch:fourier_series_and_transforms?), but for now just think of it as similar to the Laplace transform (it is). The frequency response function \(H(j\omega)\) is defined as the ratio of the Fourier transform of the output \(Y(\omega) \equiv \mathcal{F}(y(t))\) to the Fourier transform of the input \(U(\omega) \equiv \mathcal{F}(u(t))\); i.e. \[\begin{aligned} \label{eq:frf_intro} H(j\omega) &\equiv \frac{Y(\omega)} {U(\omega)}. \end{aligned}\] A quick rearrangement yields the output \[\begin{aligned} Y(\omega) &= H(j\omega) U(\omega). \end{aligned}\] If we were to inverse Fourier transform \(\mathcal{F}^{-1}\) this, we would get \(y(t)\)! Here we can gain the insight that the frequency response function, like the transfer function, relates the input and output. The difference is that the frequency response function is explicitly dependent on the angular frequency \(\omega\). We will see that, for a sinusoidal input at frequency \(\omega\), \(H(j\omega)\) returns a sinusoid at the same frequency, with only the amplitude and phase altered! This will be one of the key insights that will allow us to understand the performance of a system in terms of its frequency response function.

Summarizing

We’ve considered the three most important graphical and four most important explicitly mathematical dynamic system representations! Most of system dynamics is the study of such representations, their construction, characteristics, and performance. If you’re overwhelmed, keep in mind that this is a high-level view of … the entirety of system dynamics!

Many of these representations can straightforwardly be converted into any other. The representations and their conversion pathways are sketched in fig. ¿fig:relations_among_system_representations_intro?. Use this as a map as we explore the rich landscape of system dynamics!