Chapter 1 First and second order scalar differential equations

1.1 First order scalar differential equations

As a simple example we will often consider the following first order differential equation:

\begin{equation} \label {eq:firstorder} T\dot {x}(t)+x(t)=Ku(t), \end{equation}

where \(T,K>0\). The parameter \(T\) is called the time constant and the parameter \(K\) is called the steady state gain. The function \(u:[0,\infty )\phantom {]}\to \mR \) is called the input and the function \(x:[0,\infty )\phantom {]}\to \mR \) is called the state. Usually an initial value \(x^0\in \mR \) is given and the initial condition

\[ x(0)=x^0, \]

is added to (1.1) to make the solution \(x\) unique for a given \(u\).

  • Remark 1.1. The parameters \(T\) and \(K\) in fact scale out of the problem (so that we can without loss of generality set them equal to \(1\)). This is seen as follows. If we re-scale time by defining \(\theta :=t/T\) then we obtain (using \(\frac {d}{dt}=\frac {d\theta }{dt}\frac {d}{d\theta }\))

    \[ T\frac {1}{T}\frac {dx}{d\theta }+x=Ku, \]

    i.e.

    \[ \frac {dx}{d\theta }+x=Ku, \]

    which we can re-write as

    \[ \frac {1}{K}\frac {dx}{d\theta }+\frac {x}{K}=u, \]

    If we then re-scale \(x\) by defining \(\tilde {x}:=x/K\) then we obtain

    \[ \frac {d\tilde {x}}{d\theta }+\tilde {x}=u, \]

    which is the same as (1.1), but now with \(T=K=1\).

    If we interpret \(t\) as time, then \(T\) also has the dimension of time and \(\theta \) is dimensionless. Effectively, \(T\) determines the scale on the horizontal axis. Similarly, if \(u\) is dimensionless, \(K\) has the same physical dimension as \(x\) (with \(\tilde {x}\) being dimensionless) and determines the scale on the vertical axis.

1.2 Second order scalar differential equations

We will also often consider the following second order differential equation:

\begin{equation} \label {eq:secondorder} T^2\ddot {q}(t)+2\zeta T\dot {q}(t)+q(t)=Ku(t), \end{equation}

where \(T,K>0\) and \(\zeta \geq 0\). The parameter \(T\) is called the time constant and \(K\) the steady state gain. The parameter \(\zeta \) is the damping ratio. The equation can alternatively by written as

\[ \ddot {q}(t)+2\zeta \omega _0\dot {q}(t)+\omega _0^2q(t)=K\omega _0^2u(t), \]

where \(\omega _0=\frac {1}{T}\) is called the natural frequency. The function \(u:[0,\infty )\phantom {]}\to \mR \) is called the input and the function \(q:[0,\infty )\phantom {]}\to \mR \) is the unknown. To obtain a unique solution \(q\) for a given input \(u\), we need initial values for both \(q\) and \(\dot {q}\). Therefore the state \(x:[0,\infty )\phantom {]}\to \mR ^2\) of the second order equation (1.2) is

\[ x:=\bbm {q\\\dot {q}}, \]

since we obtain a unique solution (for a given \(u\)) once an initial value \(x^0\in \mR ^2\) is given and the initial condition

\[ x(0)=x^0, \]

is added to (1.2).

  • Remark 1.2. If we re-scale \(\theta :=t/T\) and \(\tilde {q}:=q/K\), then we obtain

    \[ \frac {d^2\tilde {q}}{d\theta ^2}+2\zeta \frac {d\tilde {q}}{d\theta }+\tilde {q}=u. \]

    In contrast to the first order case, the parameter \(\zeta \) still appears in the re-scaled differential equation and the nature of the solutions crucially depend on \(\zeta \).

  • Remark 1.3. The characteristic polynomial for the re-scaled equation

    \begin{equation} \label {eq:secondorder:rescaled} \ddot {q}(t)+2\zeta \dot {q}(t)+q(t)=u(t), \end{equation}

    is \(s^2+2\zeta s+1=0\). There are three cases:

    • • If \(\zeta \in [0,1)\phantom {]}\), then \(s^2+2\zeta s+1\) has two complex conjugate (non-real) roots. This is called the underdamped case. The solutions when \(u=0\) will be of the form \(q(t)=C_1\exp (\alpha t)\cos (\omega t)+C_2\exp (\alpha t)\sin (\omega t)\) where \(s=\alpha \pm i\omega \) are the roots of the characteristic polynomial.

    • • If \(\zeta =1\), then \(s^2+2\zeta s+1\) has a double (real) root. This is called the critically damped case. The solutions when \(u=0\) will be of the form \(q(t)=(C_1+C_2t)\e ^{\alpha t}\) where \(\alpha \) is the unique root of the characteristic polynomial.

    • • If \(\zeta >1\), then \(s^2+2\zeta s+1\) has two distinct real roots. This is called the overdamped case. The solutions when \(u=0\) will be of the form \(q(t)=C_1\exp (\alpha _1 t)+C_2\exp (\alpha _2 t)\) where \(\alpha _1\) and \(\alpha _2\) are the roots of the characteristic polynomial.

    In each case, the constants \(C_1\) and \(C_2\) can be uniquely determined from the initial condition \(x(0)=x^0\).

    The above also holds for the unscaled case (only the polynomial is different): the case of two complex conjugate (non-real) roots is the underdamped case, the case of a double (real) root is the critically damped case and the case of two distinct real roots is the underdamped case. The general form of the solution remains the same (by scaling).

Depending on the physical situation, we might not be interested in the whole state \(x\), but we might only be interested in \(q\) or in \(\dot {q}\) (or some linear combination of them). In such cases we specify an output \(y:[0,\infty )\to \mR \). The two cases we will mostly consider for second order scalar differential equations are

\[ y=q\quad \text {and}\quad y=\dot {q}. \]

1.3 Real-world examples*

  • Example 1.4. Consider a particle with mass \(M>0\) and velocity \(x\). Assume that a damping force \(-dx\) where \(d>0\) acts on the particle and that an additional force \(u\) acts on the particle. By Newton’s law we then have

    \[ M\dot {x}=-dx+u, \]

    which we can re-write as

    \[ \frac {M}{d}\dot {x}+x=\frac {1}{d}u. \]

    This is therefore a first order scalar differential equation with time constant \(T:=\frac {M}{d}\) and steady state gain \(K:=\frac {1}{d}\).

  • Example 1.5. Consider a particle with mass \(M>0\) and position \(q\) which is attached through a spring to a fixed point. Assume that a damping force \(-d\dot {q}\) where \(d>0\) acts on the mass, that the force due to the spring is \(-kq\) where \(k>0\) and that an additional force \(u\) acts on the mass. By Newton’s law we then have

    \[ M\ddot {q}=-d\dot {q}-kq+u, \]

    which we can re-write as

    \[ \frac {M}{k}\ddot {q}+\frac {d}{k}\dot {q}+q=\frac {1}{k}u. \]

    This is therefore a second order scalar differential equation with time constant \(T:=\sqrt {\frac {M}{k}}\) (and therefore natural frequency \(\omega _0:=\sqrt {\frac {k}{M}}\)) and steady state gain \(K:=\frac {1}{k}\). The damping ratio is \(\zeta =\frac {d}{2\sqrt {kM}}\).