Gain and Phase Lag

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[2019-10-25 Fri 12:58]

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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

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

$p(D) = D^3 + 1$, $p(D)x = 2 \cos(t)$