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Control Systems



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What does a Fourier Transform do to a signal?
Transforms the signal from the time domain to the frequency domain.
When would a full PID compensator be used?
If both transient and steady state responses need to be manipulated
For 2nd order systems with complex poles where are the poles located on a Bode plot?
Corner frequency? Or 1?
When would a PI be a good compensator to use?
Very reduced steady state error with minimal impact on transient response characteristics
At the frequency of the pole in a 1st order system what will the magnitude of the output be?
-3dB
What do the different components of PID control do?
P - proportional controller directly scales the instantaneous error ( difference between the current system output and the desired system output) I - integral controller scales the summation of the past errors to reduce the steady state error D - differential controller scales the rate of change of the error to reduce oscillations, improving the time to steady state.
How would you analyse a Multi input system?
You would try to break the system down into a series of single input systems and analyse them individually combining them afterwards using the principle of superposition.
When will a segment of the real axis be on the root locus?
Left of an odd number of poles
What does pole, or zero, placement mean in the context of designing a control system?
It affects the transient response of the system, the direction of the root locus, whether or not desireable characteristics of a response to a step input are observed, and if the system is stable or not.
What is system linearization and how is it achieved?
system linearisation is finding the linear equation of about a working point on a system higher than fiirst order. it can be done using taylor series expansion. Find an operating point and approximate the system linearly around the operating point
What is the time shift theorem in Laplace Transforms?
e^(-s tau) F(s)
If the transfer function of a system was a 3rd order equation what could the step response look like?
Looks like an underdamped 2nd order
What is the frequency domain design procedure for a phase-lag compensator?
1. Set compensator gain for steady state specs 2. Create Bode plot 3. Find frequency where phase margin is 5-10 deg above target 4. Place 0 at least 1 dec before (3) 5. Place pole so magnitude reduction at (3) is as needed 6. Calculate controller gain to satisfy steady state error
What is the integral theorem in Laplace transforms?
F(s)/s
When would a PD be a good compensator to use?
Gives shorter Tss, reduces steady state error (doesn't get rid of it)
What are the 3 parameters of a sinusoid?
Amplitude, frequency, phase
How would you design a controller for a non-linear system that operated over a range larger than a lineariastion model would be accurate for?
Frequency domain techniques can still be used to determine stability of non-linear systems
How can a 3rd order system (with no delay) be made stable for all system gains?
s^2 coefficient * s coefficient larger than s^3 coeefficient * constant. Could make constant 0
How can a 2nd order system decrease the frenquency range the phase transition occurs over and what affect will this have on the magnitude?
Have repeated roots. Means the magnitude will go down 40dB/dec at that point
Why is the root locus always symetrical about the real axis?
Because the branches of the roots diverge away from each other. This results in branches acting symmetrically about the real axis as they have to move away from each other at the greatest angle (180 degrees).
What shape could the asymtotes of the root locus branches take if the system has 4 poles and no zeroes?
All heading off to different corners
Why did the proportial controller become unstable in practical 1 when the motor is a 1st order system?
The gain value input to the system was large which caused the error to become large and hence oscillation down to a negative amplitude and the back up with an even larger overshoot, as this repeats the system becomes unstable
What are the steady-state step and ramp errors for a type-2 system?
Both: 0
If you needed to reduce the bandwith of a system what sort of compensator would you use and why?
Phase-lag because it decreases the gain at higher frequencies
Why is feedback control used?
Feedback control is used to respond to the disturbances that are present during the operation of the plant and to ensure that the desired output of the system is reached.
What does system approximation mean and when can it be used?
System approximation is when the response of a higher order system is approximated by a lower order system (usually first or second order). The dominant pole/poles of the system determine the approximation. It can be used when the dominant poles are sufficiently faster than the non-dominant poles. A good approximation quality is n>=6
What are the similarities and differences between mechanical and electrical models?
For mechanical motion of objects is explained using mathematics and analysis, for electrical voltage is explained using mathematics and analysis Mechanical focuses on physical properties, as well as interconnections and forces using Equations Of Motion, newtons law Electrical focuses on electronic properties, as well as interconnections and current using KCL, KVL and Ohm’s Law Electrical and Mechanical systems have the same underlying mathematical representation, and can be modelled the same way.
How can required performance be achieved in a system if the goal point is not on the root locus?
Add poles/zeros
What is the difference between SISO and MIMO systems?
Siso systems are single input single output systems that only have one input and only produce one output. Whereas a mimo system is a multi input multi output system which can have two or more inputs to produce two or more outputs
What are the steady-state step and ramp errors for a type-0 system?
Step: 1/(1+constant), ramp: inf
What equations are used to model electrical systems?
Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), Ohm, Component rules
What compensator design methodology is best suited when a system has significant time delay and why?
Phase-lag because it is less susceptible to noise due to the decreased bandwidth, since a delay will increase the effects of noise this cancels it out
Is an open loop stable 2nd order system (with no delay) always stable for all system gains?
Yes. Poles of the open loop system are not dependent on K
How does a step response relate to its transfer function?
A step response will be a scaler of A/s that enters a constant input into the transfer function to determine its response.
Why do we use the Laplace Transform in Control Systems?
Laplace transforms are used as control systems are more focused on controlling aspects of a system relative to the changes required based on previous state of a system and do not require an exact knowledge of when they need to occur. Due to this it is simpler to operate the control system in the frequency domain, however the equations used to derive a system are generally relative to time. This is why it is necessary to perform Laplace transforms to manipulate the derived equations in the frequency domain. Changes derivative equations to linear (easier to work with)
How do you find time to steady state for a 2nd order system?
Using the general form for a 2nd order system, determine the damping ratio from the denominator. And determine the stability of the system based on the damping ratio from this the relative Tss value can be found from the relevant equation.
What shape could the asymtotes of the root locus branches take if the system has 3 poles and no zeroes?
Single pole heading to negative infinity, complex conjugate pair heading away to infinity 60 degrees above and below the real axis
What does it mean for a system to be unstable?
If the poles are in the positive region of the pole-zero map and the damping ratio is less than zero the system will be unstable. This results in a n unbounded output
What is the derivative theorem in Laplace transforms?
sF(s)-f(0)
What is a root locus
A root locus is a plot showing the movement of the pole locations for different gains.
What is the frequency domain design procedure for a phase-lead compensator?
1. Create Bode plot 2. Set gain so that 0dB point is half desired value 3. Create compensated Bode plot 4. Calculate required gain margin 5. Calculate additional phase margin needed 6. Calculate beta 7. Calculate T

What are some pros and cons of Open Loop control?

Pros

Very simple & easy to design. Are generally stable. Easy to make and use

Cons

Less bandwidth in open loop. Inaccurate & unreliable. Output is affected by disturbances

How can a root locus be used to determine the system gains for stability?
System is stable so long as the poles are on the left half plane
What are the three types of partial fractions that will represent the dynamics of any order system?
A/ax+b, A/ax+b + B/(ax+b)^2, (Ax+B)/(ax^2+bx+c)
What type of input would you use to experimentally determine the transfer function of a real 1st or 2nd order system and why?
A step response would be used as in most real systems would require a constant input to be able to control the plant. This is why a step response which inputs a constant value into a transfer function represents the constant input to real systems.
What is the main practical difference between Laplace and Fourier Transforms?
Laplace transforms are used to analyse the transient response of a system whereas Fourier transforms analyse the steady state of a signal. Laplace have a real component, Fourier's are purely complex
How can a Nyquist plot be used to determine the stability margins of a system?
Observe closeness to (-1,0)
If the transfer function of a system was a 1st order equation what would the step response look like?
The step response to a first order system would asymptotically approach its steady state value. Logarithmic growth
Why is the Nyquist plot symetrical about the real axis?
The value at the negative frequency is the complex conjugate, and hence the plot is symmetric about the real axis
Does a lag compensator increase or decrease the overall magnitude of the system output?
Lag compensator decreases magnitude (dB) of the system output
What does the time to steady-state of a system mean?
The time to steady state is the time the response of a system takes to reach and maintain a value that is within 98%-102% of the steady state response of the system.
Why is it not always possible to get the exact calculated performance from a system, even with a PID controller?
Real life issues, like time delays and imperfections
What is a major limitiation in understanding a high order system if you only have the transfer function?
Harder to determine stability
A lag compensator has what net impact on the phase of the system at the phase margin frequency?
No net phase impact, causes the phase margin to increase because it causes it to happen at a lower frequency
What is the root locus angular requirement?
The sum of the angles between the poles and zeros must be equal to 180.
How is a high level of disturbance rejection obtained in a control system?
If C*G*H is high enough disturbances are decreased. Higher C gain = overall control law, G= plant/actual control system, higher H = closed loop, high feedback control.
If a LTI system was excited by a sinusoid what components of the signal could be different, and what must remain the same, at the output?
Same frequency, different amplitude and phase
Will the inclusion of a digital controller make a system more or less stable at high frequency and why?
Less stable because of the time delays inherent in digital systems
If a time delay were inserted into a 2nd order system would the overshoot get larger or smaller?
Larger
What effect will increasing the P component of a PID compensator have on: rise time, overshoot, settling time and steady-state error?
Decreases rise time, increases overshoot, small increase in tss, decreases steady state error and stability
Algebraically how does a transfer system with a time delay compare to one without?
Time delay does not affect the magnitude of the response but introduces a linearly increasing phase shift (delay). This decreases the gain and phase margins, moving system closer to instability.
How does the impulse response of a system relate to its transfer function?
An impulse function sets the output at s=0 to 1. this allows a simple response of the system to be generated. Impulse response is given by transfer function.
What are the two main ways to calculate the stability of a system?
Routh Hurwitz Method and Root locus for time domain Nyquist criterion and Bode plot for frequency domain
What does the location of the poles and zeros on the s-plane tell you about the dynamics of the system?
Determines whether the system is stable or unstable. Poles increase the oscillations and zeros dampen the oscillations. Smaller magnitude of zeros makes the systems response faster and increases overshoot/undershoot. Smaller magnitude of poles makes the system response slower.
What is the damped frequency of a system and how does it relate to the natural frequency?
Damped frequency is the natural frequency multiplied by the square root of 1-zeta
What can a root locus be used for?
The root locus can be used to qualitatively describe the performance of a system. transient properties, systems stability
What is a transfer function of a system?
The transfer function models the systems output for each possible input Found by applying Laplace transform to the differential equation of the system Tells you about the systems performance
Why is it sometimes good to have a high controller gain, irrespective of system dynamics?
to determine when the system becomes unstable, or to have a faster system response. To amplify the system response, which can be useful when passing it into another system.
What is the fundemental component all signals can be decomposed into?
All signals can be decomposed into a series of sinusoids
What is the time domain formula for general 2nd order systems?
The time domain formula for a 2nd order system will be a constant multiplied by an exponential that decays multiplied by a sinusoidal function.
Does a lead compensator increase or decrease the phase at the phase margin frequency and why?
Increases the phase margin, both by increasing the phase

What are the zeros and poles of a system?

The poles and zeros of a system are the points at which the systems response will be calculated as 0.

The poles are found in the denominator of the transfer function and the zeros are found in the numerator of the transfer function.

What are the steady-state step and ramp errors for a type-1 system?
step: 0, ramp: 1/constant
What is the procedure to create a Nyquist plot?
Find the open-loop transfer function, OLTF = G(s)H(s) Find the number of unstable open loop poles, P Set the Nyquist path in the s-plane Draw the Nyquist Diagram in the GH plane Count the number of encirclements of the point (-1,0), N Calculate the number of unstable closed loop poles, Z = N+P If Z=0 the system is stable
What does it mean for a system to be stable?
A system is stable if the poles are in the negative real region of a pole zero map, and the damping ratio is greater than 0. The system converges on the steady state eventually
What are the gain and phase margins of a system and how would you determine them?
Gain margin is the gain value when the frequency crosses the horizontal axis, phase margin is the phase value when the gain crosses the horizontal axis
When designing a controller for a high order system which design method would be more accurate and why?
Routh Hurwitz works for all orders to determine stability without needing to factorise (useful at high orders). Higher order systems can have both temporal performance and steady state error improved by a PID controller
How would you go about creating a digital twin for a mechanical system?
Split the mechanical system into free body diagrams, analyse them, use the similar equations between mech and elec systems to draw it as an RCL circuit
What does the steady-state error of a system mean?
The steady state error of a system is the difference between the magnitude at steady state and the desired response (input) of a system.
What effect will increasing the D component of a PID compensator have on: rise time, overshoot, settling time and steady-state error?
Small decrease in rise time, decreased overshoot and tss, minor change to steady state error, improves stability
To what temporal charateristic is phase margin most closely related?
Overshoot
What are the similarities and differences between an impulse and white noise in the frequency domain?
Both have all frequencies, but impulse has specific phases where white noise has random phases
Without knowing the closed loop system poles what are two ways of determaning system stability and how do they compare?
Open loop poles are the same as the poles of the characteristic equation, Routh array?
As the system gain gets larger what happens to the branches of the root locus?
Head to infinity or to system zeros

What are some pros and cons of Closed Loop control?

Pros

More accurate in presence of non-linearities, Less affected by noise

Cons

Complex systems. The feedback reduces gain of the system. Can become unstable quite easily

What is an operation you can perform to determine system stability without finding system roots?
Routh Array
How do you find the time to steady-state for a 1st order system?
Using the general form for a first order system the time constant tau can be found in the denominator using this value the time to steady state will be found by multiplying it by 4.
Why may you want to create a digital twin for a mechanical system?
The parameters for a digital twin would be simpler to manipulate than an actual mechanical system A digital twin can make the system more efficient. It also enables a better design of the system as the twin creates data about the likely performance outcomes.
What does it mean for a system to be LTI?
A linear time invariant system is a system that produces an output for any arbitrary input while adhering to the principles of superposition and homogeneity, while also ensuring there is no transient change in the system.
How many capacitors would be required to build an op-amp based electronic circuit that functioned as a lead-lag compensator?
4. 2 for lead, 2 for lag
If a 1st order system has a pole at w over what frequency range will the phase transition occur?
1/w to 10w
If you can use MATLAB to find system roots why would you need a Routh Array?
Useful for determining unknown design parameters while maintaining stability and without solving the characteristic equation
How can you tell if a system is stable?
A system can be considered stable if it converges to some final value relative to the systems input. Poles will have a real component <=0
Why is it important to design controllers with possible disturbances in mind?
Because the real world has disturbances, and disturbance rejection is important to account for those and still get the desired response. System needs to be able to adapt to avoid undesired or unstable responses
What is feedback control?
Feedback control is a method of control that feeds back the system output and generates a response for the system to correct the output so that it meets the desired input.
How can you determine the order of a system?
Order of the polynomial in the denominator
What are the three components of PID control?
Three components are as follows: P - the proportional component is a scaler to the error that directly controls the output of the system. I - integral component generates a system response based on the summation of all past errors to minimise the steady state error of the system. D - differential component measures the rate of change of the error to predict the expected changes to the system and dampen the system accordingly.
What affects do lead and lag compensators have on the phase margin frequency?
Lag compensator decreases the phase margin frequency, lead compensator increases it
Why do the block diagram reduction techniques only work for linear systems?
Linear systems are causal, block diagrams rely on causal relationships with the arrow indicating the direction for functions like addition and multiplication which rely on the principlpes of superposition and homogeneity
How does the damping ratio of a 2nd order system affect the Bode plot?
As the damping ratio decreases, both the magnitude peak, and sharpness of the phase change, at critical (natural) frequency gets larger.
How does a time delay affect a system?
Time delay can degrade the performance of a control system or lead to system instability. Time delays can also reduce the information systems use to respond to the environment and requires the system to predict future states based on the past inputs.
What is a block diagram of a system?
Representation of a system using blocks Block Diagrams represent the simultaneous Differential equations that have been converted to the s domain by a Laplace transform, explain the action a system will have on an input. Systems can be reduced to one block.
What affect will a lead compensator have on the time domain response of a system and why?
fast transient response
What is the Nyquist Stability Criterion?
N = Z - P = 0 for stability. Nyquist stability criterion states the number of encirclements about the critical point (1+j0) must be equal to the poles of characteristic equation
When will the time domain formula for a 2nd order system step response have a cosine component?
if there is no s^1 component in the denominator. the denominator should have roots in the form (s+1)(s-1). if there is a zero in the X(s) then the f(t) will have a cosine component
How can you determine the order of a system by looking at the Bode plot?
Number of slope changes (down 20dB/dec for poles, up 20dB/dec for zeros)
What effect will increasing the I component of a PID compensator have on: rise time, overshoot, settling time and steady-state error?
Small decrease in rise time, increase in overshoot and tss, large decrease in steady state error, decreases stability
At the frequency of the pole in a 1st order system what will the phase of the output be?
-45deg
What is system sensitivity and how is it different between open- and closed-loop control?
System sensitivity is the ratio of change in the system TF to the ratio of change in plant TF. Open: 1, Closed: 1/(1+CG)
How many brances will the root locus of a system have?
The number of branches on the root locus will be equal to the number of open loop poles in the system
What is Contour mapping and why would it be used?
Contour map is a contour in the s plane that has been mapped into the F(s) plane using a certain function. It is used to create a Nyquist plot
What is the relationship between Nyquist and Bode plots?
Bode plot shows the same information as the nyquist plot but separated into phase and magnitude, on a frequency scale
Explain how a mass-spring-damper system and a resistor-capacitor-inductor circuit are similar
Mechanical systems generate/dissipate forces, electronic systems generate currents Series Mass->inductor Spring->capacitor Damper->resistor Force->voltage Velocity->current Parallel Mass->capacitor Spring->inductor Damper->resistor Force->current Velocity->voltage
What is the maximum phase change induced by a 3rd order system with 2 zeros?
neg 270 (from the poles, the zeros would act against it)
What equations are used to model mechanical systems?
Newtons laws of motion (F=ma), component laws
What does it mean to find the impulse response of a system?
The impulse response demonstrates the response of a system to an instantaneous pulse. This will assist to determine how the system is to react in the event of single inputs into the system and how the relative damping of the system responds to the pulse.
If a transfer function of a function I s 2nd order what would it look like?
Depends on the damping ratio
What are the modes that make up any order real system?
Any linear system is equivalent to a finite number of real, first and second order systems in parallel. These systems are commonly referred to as the modes
What does it mean to find the step response of a system?
The step response is a constant input. This means that finding the step response to a system is to find the response of a system to a given constant input.
What is the time domain fomula for a general 1st order step response?
The step response to a 1st order equation in the tine domain would be the gain (k) multiplied by 1-e^(-t/tau) all multiplied by u(t) the step response.
What is aliasing and why is it detrimental for system analysis and control?
Aliasing occurs when the sampling frequency is less than double the frequency of the highest frequency component of the signal. It causes information to be lost and can produce a misleading analysis
What is pole cancellation and why is it rarely done in practice?
Placing a zero exactly on a pole to cancel its effects on the system. Rarely done in practice because it has to be perfect otherwise it introduces undesired system dynamics
What does it mean for a system to be critically stable?
At least one pole has a real component of 0, no poles are in the right hand plane. Any increase in gain will affect stability. Oscillating with constant frequency and amplitude
What is the bandwidth of a system?
The bandwidth of the system is the range of frequencies where the magnitude is within 3dB of either the input frequency or some other reference frequency
What are the three main Input signals used for analysis
impulse, Unit step and ramp (1, 1/s 1/s^2)
When will a system have a complex pole?
The system will have complex conjugate poles when the damping ratio is less than 1
How many capacitors would be required to build an op-amp based electronic circuit that functioned as a PID compensator?
2. 1 for PI 1 for PD
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