- Captured On
- Source
- Gain and Phase Lag | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare
1 When can we say that a system is stable?
1.1 Front
When can we say that a system is /stable/?
1.2 Back
If the systems’s long-term behaviour does not depend significantly on the initial conditions.
2 Give an example of stable system in mechanics?
2.1 Front
Give an example of stable system in mechanics?
2.2 Back
Any system of masses connected by springs (damped or undamped)
3 Give an example of stable system in network theory?
3.1 Front
Give an example of stable system in network theory?
3.2 Back
Any RLC-network gives a stable system
4 What part of the solution of a linear equation with constants coefficients…
4.1 Front
What part of the solution of a linear equation with constants coefficients…
must look to know if the system is stable or not?
4.2 Back
You need to look the complementary function of the general solution. It’s the solution of the associated homogeneous equation (null input)
5 On which depends the complementary function of a general solution?
5.1 Front
On which depends the complementary function of a general solution?
5.2 Back
The complementary solution has form \(y_h(t) = c_1 y_1 + c_2 y_2\), where \(c_1\) and \(c_2\) depends on the initials conditions
6 How are called the 2 parts of the solution if this ODE is stable?
6.1 Front
How are called the 2 parts of the solution if this ODE is stable?
\(a_0 y’’ + a_1 y’ + a_2 = p(t)\)
6.2 Back
\(y = y_p + c_1 y_1 + c_2 y_2\)
- \(y_p\) is the steady-state solution
- \(c_1 y_1 + c_2 y_2\) is the transient
7 Which are the conditions to determine is this ODE is stable?
7.1 Front
Which are the conditions to determine is this ODE is stable?
\(a_0 y’’ + a_1 y’ + a_2 = p(t)\)
7.2 Back
If and only if for every choice of \(c_1, c_2\), \(c_1 y_1 + c_2 y_2 \to 0\) as \(t \to \infty\)
The characteristic equation is \(a_0 r^2 + a_1 r + a_2 = 0\). The roots of this equation decide the stability of the system
| roots | solutions to ODE | condition for stability |
|---|---|---|
| \(r_{1} \neq n_{2}\) | \(c_{1}e^{r_{1}t} + c_{2}e^{r_{2}t}\) | \(r_{1} \lt 0\), \(r_{2} \lt 0\) |
| \(r_{1} = n_{2}\) | \(e^{r_{1}t}(c_{1} + c_{2}t)\) | \(r_{1} \lt 0\) |
| \(a \pm ib\) | \(e^{at}(c_{1}\cos bt + c_{2} \sin bt)\) | \(a \lt 0\) |
8 Has the input influence on the stability of this ODE?
8.1 Front
Has the input influence on the stability of this ODE?
\(a_0 y’’ + a_1 y’ + a_2 = p(t)\)
8.2 Back
No, only depends on the behaviour of the solution of the associated homogeneous equation
9 What is the stability criterion for second-order ODE’s - root form?
9.1 Front
What is the stability criterion for second-order ODE’s - root form?
\(a_0 y’’ + a_1 y’ + a_2 = p(t)\)
9.2 Back
\(\text{Is stable} \Leftrightarrow \text{all roots of } a_0r^2 + a_1 r + a_2 = 0 \text{ have negative real part}\)
10 What is the stability criterion for second-order ODE’s - coeeficient form?
10.1 Front
What is the stability criterion for second-order ODE’s - coeeficient form?
Assume \(a_0 \gt 0\)
10.2 Back
\(\text{Is stable} \Leftrightarrow a_0, a_1, a_2 \gt 0\)
11 For an k-fold repeated root from characteristic polynomial?
11.1 Front
For an k-fold repeated root from characteristic polynomial
Which solutions has?
11.2 Back
Multiply the root by \(1, t, t^2, \cdots, t^{k-1}\)
12 Can we apply the stability criterion for higher-order ODE’s?
12.1 Front
Can we apply the stability criterion for higher-order ODE’s?
Both, root form and coefficient form?. For example for, 9th order ODE
Assume \(a_0 \gt 0\)
12.2 Back
In root-form, you can apply it, all the real roots are negative, and all the complex roots have negative real part.
But in coefficient form, is not so simple. You can say that
\(\text{ODE(9) is stable } \implies a_0, a_1, \cdots, c_9 \gt 0\)
From coefficient, you need to check Routh-Hurwitz conditions for stability
13 What is the operator for differentiation applied to function of \(t\)?
13.1 Front
What is the operator for differentiation applied to function of $t$?
13.2 Back
\(D = \dv{t}\)
14 What does mean \(D^{3}\) as operator?
14.1 Front
What does mean $D^{3}$ as operator?
14.2 Back
\(D^3 = \dv[3]{t}\)
15 How can we write this ODE with operators?
15.1 Front
How can we write this ODE with operators?
\(a_nx^{(n)} + a_{n-1}x^{(n-1)} + \dots + a_1 x’ + a_0 x = q(t)\)
15.2 Back
\(p(D) = a_n D^n + a_{n-1} D^{n-1} + \dots + a_1 D + a_0\)
\(p(D)x = q\), and its associated homogeneous ODE \(p(D)x = 0\)
16 What is the definition of phase lag in a linear ODE?
16.1 Front
What is the definition of phase lag in a linear ODE?
Where input \(p(D)x = B \cos(\omega t)\)
16.2 Back
It’s the angle by which the output sinusoid is shifted relative to the input sinusoid. Only depends on the \(\omega\) of the input signal.
It comes from the particular solution \(x_p(t) = \frac{B}{\abs{p(i\omega)}} e^{i(\omega t - \phi)}\)
\(\phi = \operatorname{Arg}(p(i \omega))\)
17 What is the definition of gain in a linear ODE?
17.1 Front
What is the definition of gain in a linear ODE?
Where input \(p(D)x = B \cos(\omega t)\)
17.2 Back
It’s the ratio of the amplitude of the output sinusoid to the amplitude of the input sinusoid. Only depends on the \(\omega\) of the input signal.
It comes from the particular solution \(x_p(t) = \frac{B}{\abs{p(i\omega)} } e^{i(\omega t - \phi)}\)
\({\displaystyle g = \frac{1}{\abs{p(i \omega)}}}\)
18 What is the complex gain in a linear ODE?
18.1 Front
What is the complex gain in a linear ODE?
Where \(p(D)x = B \cos(\omega t)\)
18.2 Back
Convert to complex numbers: \(p(D)z_p = Be^{i \omega t}\)
\(g = \frac{1}{p(i \omega)}\)
19 If damping constant \(b\) starts at \(1\) and is increased, …
19.1 Front
If damping constant $b$ starts at $1$ and is increased, …
what happens to the amplitude and phase lag of the solution?
Consider the equation \(\ddot{x} + b \dot{x} + 2x = \cos(t)\)
19.2 Back
\(p(D)x = (D^2 + bD + 2)x = \cos(t) = e^{it}\)
where \(\omega = i\), \(p(i) = i^2 + bi + 2 = 1 + bi\)
\(\phi = \operatorname{Arg}(1 + bi)\), as \(b\) increasing then \(\phi\) increasing too.
\(g = \frac{1}{\abs{1 + bi}} = \frac{1}{\sqrt{1 + b^2}}\), as \(b\) increasing the gain is decreasing, so the output amplitude is decreasing too.
20 What is the \(p(D)\) from \(x’’’ + x = 2 \cos(t)\)
20.1 Front
What is the $p(D)$ from $x’’’ + x = 2 \cos(t)$
20.2 Back
\(p(D) = D^3 + 1\), \(p(D)x = 2 \cos(t)\)