IB: 2P6: Information

Question 1

What is a block diagram?

Answer 1

Each block has inputs and outputs which are signals (i.e. functions of time). The block itself corresponds to a system, i.e. a differential equation the relates the inputs and outputs.

Question 2

What is in each block of a block diagram?

Answer 2

They represent systems, which can be written as “transfer functions” (often ODE’s)

Question 3

What goes into and out of each block in a block diagram?

Answer 3

Information, not just the flow of “stuff”. The inputs and outputs to each block is a different signal.

Question 4

What is a linear system?

Answer 4

A system is linear if it satisfies the principle of superposition

Question 5

What is a causal system?

Answer 5

Causality is satisfied by all physical systems. It captures the property that “the current output depends only on past inputs and not on future inputs”

Question 6

What is a time-invariant system?

Answer 6

Question 7

What is linearization?

Answer 7

All real systems are actually nonlinear, but many of these behave approximately linearly for small perturbations from equilibrium.

Question 8

When can we use linear systems theory?

Answer 8

If we are going to design a controller to keep a system near equilibrium then we can ensure that perturbations are small (and hence that behaviour is approximately linear). This justifies the use of linear theory for the design. Hence, linear systems theory can be extremely useful, even when the underlying system is actually non-linear.

Question 9

What is the definition of a laplace transform?

Answer 9

Question 10

Is the operation of taking laplace transforms linear or non-linear?

Answer 10

Linear

Question 11

What is a rational function?

Answer 11

A function that can be written as the ratio of two polynomials

Question 12

What is the shift in s theorem?

Answer 12

Question 13

What are the poles and zeros of a rational functions of s?

Answer 13

• The roots of n(s) are called the zeros of G(s)
• The roots of d(s) are called the poles of G(s)

Question 14

What is the final value theorem?

Answer 14

Only valid if signal tends to a constant for t → ∞

Question 15

What is the intial value theorem?

Answer 15

Question 16

What is feedback used for?

Answer 16

To reduce sensitivity

Question 17

What characterises the input-output properties of a Linear, Time invariant, and Causal (LTI) system?

Answer 17

• Impulse response
• Step response
• Transfer function

Each completely characterises the input-output properties of the system

Question 18

Given the input to an LTI system, how can you determine the output?

Answer 18

• In the time domain: as the convolution of the impulse response and the input.
• In the laplace domain: as the multiplication of the transfer function and the laplace transform of the input

Question 19

How is the transfer function linked to the impulse response and step response of the LTI system?

Answer 19

• The transfer function is the laplace transform of the impulse response
• The step response is then the integral of the impulse response

Question 20

What is the unit impulse?

Answer 20

Any function δ(t) which has the property:

Question 21

What is the impulse response of a system?

Answer 21

Question 22

What is the step response of a system?

Answer 22

Question 23

How can you determine the transfer function of a system?

Answer 23

Although the transfer function is strictly defined as the laplace transform of the impulse response, it is usually most easily calculated directly from the system’s differential equations.

Question 24

What is the response of a system dominated by?

Answer 24

The slow transient: the term with the largest time constant

Question 25

How are transfer functions usually reprented?

Answer 25

With a capital letter (rather than a bar above the letter like other laplace transforms). Examples include: * G(s) * F(s) * H(s)

Question 26

How can you determine the overall transfer function of the entire system?

Answer 26

Question 27

What is asymptotic stability?

Answer 27

**g(t) → 0 as t → ∞**

Question 28

How can you determine the type of stability of a system? ## Footnote Note: Systems with proper rational transfer functions

Answer 28

By the pole locations: **Asymptotic stability:** * All the poles lie in the left hand plane (all poles have negative real components) **Marginal stability:** * One or more distinct poles lie on the imaginary axis and are isolated (no repeated poles) * AND, any remaining poles (if any others) lie in the left hand plane (negative real components) **Unstable:** * If any of the poles have a positive real component * If there are repeated poles on the imaginary axis

Question 29

For a system described by an ODE, what are the poles of the transfer function?

Answer 29

The poles of the transfer function are the solutions to the auxilliary equation of the ODE

Question 30

Given the pole locations in the complex plane, how can you determine: * The natural frequency, ωₙ * The damping ratio, ζ * The reciprocal of the time constant of the decay, ωₙζ

Answer 30

* Radial contours of constant damping ratio, ζ * Circles of constant natural frequency, ωₙ * Vertical lines of constant ωₙζ ## Footnote Note: For higher order systems, you can read off the natural frequency and damping ratio for each "mode" of the system (each pair of complex poles). The poles closest to the imaginary axis are often called the dominant poles (their contribution dies away most slowly, and so tends to dominate the response)

Question 31

What are dominant poles?

Answer 31

Higher order systems will have many poles. The dominant poles are the poles closest to the imaginary axis. Their contribution dies away most slowlym and so tends to dominate the response.

Question 32

What is marginal stability?

Answer 32

Question 33

What is an unstable system?

Answer 33

**g(t) → ∞ as t → ∞**

Question 34

What is the transient response?

Answer 34

The initial part of the (time domain) response of a system to a general input (before the "transients" have died out). To a very large extend, the transients are a characteristic of the system itself, rather than the input.

Question 35

How can the impulse response of a system be expressed?

Answer 35

As a sum of terms due to each real pole, or pair of complex poles. The systems response to ANY input will also include these features!

Question 36

How do the poles of a transfer function dictate the transient response of the system?

Answer 36

* The real part of the pole, σ, determines both stability and the time constant, |1/σ| * The imaginary part of the pole, ω, determines the damped natural frequency (actual frequency of oscillation) in rad/sec * The magnitude of the pole determines the natural frequency * The argument of the pole determines the damping ratio

Question 37

What is the frequency response to an asymptotically stable LTI system?

Answer 37

If a pure sinusoid is input to an asymptotically stable LTI system, then the output will also settle down, eventually, to a pure sinusoid. This steady-state output will have the same phase and frequency as the input, but be at a different amplitude and phase. The dependence of this amplitude and phase on the frequency of the input is called the frequency response of the system.

Question 38

What are the gain and phase shift of the frequency response given by?

Answer 38

Question 39

How can you plot the frequency response?

Answer 39

* **The Bode Diagram:** Two separate graphs, one of |G(jω)| vs ω (on log-log axes), and one of ∠G(jω) (lin axis) vs ω (log axis) * **The Nyquist Diagram:** One single parametric plot, of Re(G(jω)) against Im(G(jω)) (on linear axes) as ω varies * **The Nichols Diagram:** One single parametric plot, of |G(jω)| (log axis) against ∠(G(jω)) (lin axis) as ω varies

Question 40

What are the uses of Bode, Nyquist, and Nichols diagrams?

Answer 40

* The Bode diagram is relatively straightforward to sketch to a high degree of accuracy, is compact and gives an indication of the frequency ranges in which different levels of performance are achieved * The Nyquist diagram provides a rigorous way of determining the stability of a feedback system * The Nichols diagram combines some of the advantages of both of these (although it is not quite as good in either specific application) and is widely used in industry. ## Footnote Note: Nichols diagram not in course

Question 41

How do you sketch a Bode diagram?

Answer 41

Sketch the asymptodes!!!! Determine what the amplitude and phase will be for: ω → 0 ω → ∞ ω = 1 / T Draw the asymptodes on the diagram, see where they cross, and use them to sketch the rough shape Sum the different asymptodes from the different terms in the transfer function to obtain the overall plot.

Question 42

How do you sketch the Bode diagram for the term in the transfer function: **a(s) = (sT)ᵏ**?

Answer 42

Question 43

How do you sketch the Bode diagram for the term in the transfer function: **G(s) = (1 + sT)**?

Answer 43

Question 44

How do you sketch the Bode diagram for the term in the transfer function: **G(s) = (1 + 2ζ sT + s²T²)**?

Answer 44

Question 45

Why will Open-loop control not work in practice? Why is closed-loop (feedback) control required?

Answer 45

It requires an exact model of the plant and that there be no disturbances (no uncertainty) as there is no feedback. Therefore feedback is used to combat the effects of uncertainty, this could be unknown parameters, unknown disturbances, or unknown equations.

Question 46

What is the closed-loop transfer function?

Answer 46

The transfer function linking the input to the output. If there are multiple inputs (including disturbances) then there will be multiple closed-loop transfer functions

Question 47

For a single-loop closed-loop transfer function, what is similar between all the closed-loop transfer functions?

Answer 47

They all have the same denominator

Question 48

What is the closed-loop characteristic equation?

Answer 48

The denominator of the closed-loop transfer functions set equal to zero

Question 49

What are the closed-loop poles?

Answer 49

The poles of the closed-loop transfer function. They are the roots of the closed-loop characteristic equation

Question 50

What do the closed-loop poles determine?

Answer 50

* The stability of the clsoed-loop system * Characteristics of the closed-loop system's transient response

Question 51

What is the return ratio? | Also known as loop transfer function

Answer 51

It is the product of all the terms around the loop.

Question 52

What is the closed-loop characteristic equation in terms of the return ratio, L(s)?

Answer 52

1 + L(s) = 0

Question 53

What are the Sensitivity S(s) and the Complementary Sensitivity T(s) closed-loop transfer functions?

Answer 53

* The sensitivity function characterizes the sensitivity of a control system to disturbances appearing at the output of the plant S(s) + T(s) = 1

Question 54

What is an example of when the Sensitivity S(s) and the Complementary Sensitivity T(s) appear?

Answer 54

Question 55

What is the "Steady-state response"?

Answer 55

Question 56

What is the Steady-state error?

Answer 56

It is the final value theorem applied to the difference between the input and the output

Question 57

What is Proportional control?

Answer 57

It is the simplest control policy where the controller is just a multiplication with a constant kₚ.

Question 58

How does the value of kₚ effect the response in proportional control?

Answer 58

A large kₚ leads to a smaller steady-state error, but this is at the expense of a more oscillatory response (or even instability)

Question 59

How can you increase damping?

Answer 59

Use derivative action

Question 60

How can you remove steady-state errors?

Answer 60

Use integral action

Question 61

What is Proportiona + Derivative (PD) control?

Answer 61

In addition to the proportional control, there is a term where the input to the controller is differentiated and multiplied by a constant kd.

Question 62

How does the value of kd effect the response in PD control?

Answer 62

Increasing kd: * Increased damping (less oscillatory) * Greater sensitivity to noise

Question 63

What is Proportional and Integral (PI) control?

Answer 63

Integral action (if stabilizing and the system is asymptotically stable) always results in zero steady-state error, in the presence of constant disturbances and demands. To remove the steady-state error, we must ensure K(0) = ∞

Question 64

What is PID control?

Answer 64

A PID controller can combine the advantages of derivative, integral, and proportional control. Appropriate tuning of the controller parameters is important

Question 65

What is the Nyquist stability criterion?

Answer 65

The Nyquist stability criterion states that, if: * The open-loop system is asymptotically stable (i.e. the return ratio L(s) has all its poles in the LHP) and, * The Nyquist diagram of L(jω) does not enclose the point "-1", then the closed-loop system will be asymptotically stable (i.e. the closed-loop transfer function L(s)/(1+L(s)) will have all its poles in the LHP)

Question 66

What is the real power of the Nyquist stability criterion?

Answer 66

It allows you to determine the stability of the clsoed-loop system from the behaviour of the open-loop stability diagram. This is important from a design viewpoint as it is realtively easy to see how changing K(s) affects L(s) = H(s)G(s)K(s), but difficuly to see how changing K(s) affects L(s)/(1+L(s)) directly. It also allows more detailed information about the behaviour of the closed-loop system to be inferred. For example, gain and phase margins measure how clsoe the Nyquist locus gets to -1, and hence how clsoe the clsoed loop system is to instability

Question 67

How do you plot a Nyquist diagram?

Answer 67

* Evaluate both the modulus and argument of G(jω) * Start sketching points for varying ω. Start with the points as ω → 0 and ω → ∞ and then do a few points inbetween

Question 68

L(jω) encircling or going through the point -1 will mean the closed-loop is not asymptotically stable. Why is L(jω) coming close to -1 without encirlcing it also endesireable?

Answer 68

* It implies that a closed-loop pole will be close to the imaginary axis and that the clsoed-loop system will be oscillatory * If G(s) is the transfer function of an innacurate model, then the "true" Nyquist diagram might encircle -1

Question 69

What are gain and phase margins?

Answer 69

Gain and phase margins are used to measure how close the return ratio L(jω) gets to -1: * The gain margin measures how much the gain of the return ratio can be increased before the celosed-loop system becomes unstable. **Gain margin = 1/α** * The phase margin measures how much phase lag can be added to teh return ratio before the closed-loop system becomes unstable. **Phase margin = θ**

Question 70

How do you determine the gain and phase margins from the Bode plot?

Answer 70

* For the gain margin we find first the frequency where the phase of L(jω) is -180°. We then find |L(jω)| at this frequency. * For the phase margin find first the frequency where |L(jω)| = 1. We then find the phase of L(jω) at this frequency and how much it deviates from -180°.

Question 71

Given a Nyquist diagram of L(s) = kG(s) for k = 1, how do you find the gain and phase margins for k ≠ 1?

Answer 71

Question 72

What is a good feeback property for the sensitivity (1/(1+L(jω))?

Answer 72

Small sensitivity, as this causes: * Rejection of disturbances * Reduction in the effects of uncertainty This occurs when |L(s)| >> 1

Question 73

Feedback reduces the effect of disturbances at low frequencies, up to ω₁. The closed loop will respond to frequencies up to around ω₂. Between these frequencies both disturbances and reference signals are amplified. How can ω₁ and ω₂ be found from the open-loop frequency response?

Answer 73

Question 74

What is the Nyquist diagram of the below return ratio

Answer 74

Question 75

What happens if there are multiple encirclements on a Nyquist diagram?

Answer 75

Question 76

L(s) often has one or more poles at the origin, how is this dealt with when sketching the Nyquist diagram?

Answer 76

Question 77

If L(s) is unstable and has nₚ unstable poles, how is the Nyquist stability theorem modified?

Answer 77

Question 78

When choosing K(s) to shape L(s) = K(s)G(s), what should you aim for?

Answer 78

L(jω) should: * Have high gain at low frequencies * Low gain at high frequencies * Phase should be larger than -180° at ωc (the frequency where the gain is 1). This ensures that the Nyquist stability criterion is satisfied with sufficient gain and phase margin

Question 79

How is this achieved?

Answer 79

Question 80

What is a phase lag compensator?

Answer 80

A phase lag compensator has a transfer function of the form specified below. It is the generalized form of PI action. * A phase lag compensator increases the low frequency gain, and so reduces steady-state errors. * However this is achieved at the expense of introducing a phase lag. However, if the compensator is designed such that the maximum phase lag occurs at a frequency well below ωc, then this phase lag will not affect significantly the phase margin of the system

Question 81

What is a phase lead compensator?

Answer 81

A phase lead compensator has a transfer function of the form specified below. It is the generalized form of PD action. * A phase lead compensator results in an increase in the phase at a certain frequency range. If this occurs at a frequency close to ωc then this will improve the gain margin of the system. * However, it decreases the low frequency gain and icnreases the high frequency gain whichare undesireable properties.