dr ian r. manchester - university of...

Dr Ian R. Manchester

Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Amme 3500 : Review

Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics No Friday Tutorial

10 Apr BREAK 6 17 Apr Root Locus 7 24 Apr Root Locus 2 No Wed/Thu Tut 8 1 May Bode Plots Assign 2 due 9 8 May Bode Plots 2 10 15 May State Space Design Assign 3 Due 11 22 May State Space Design 2 12 29 May Advanced Control/Review 1 13 5 June Review 2 Assign 4 Due


•  4 Assignments (40%) – Assignment 1 : Due Week 4 (5%) – Assignment 2 : Due Week 6 (10%) – Assignment 3 : Due Week 10 (10%) – Assignment 4 : Due Week 13 (15%)

•  Final Exam (60%)*

* Note you are expected to pass the exam to pass this course


•  To introduce the methods used for the analysis and design of feedback controllers for linear time invariant (LTI) systems


•  System 1 affects system 2, which affects system 1, which affects system 2….

System 1 (e.g. Controller)

System 2 (e.g. Process)


–  Vehicle Control and Design (land, sea, air, space) – understanding and controlling how the system responds to external disturbances

–  Biomedical (cardiac system, dialysis machine) – design and control of systems that interact with the human body.

–  Manufacturing Processes – controlled conditions for high-performance materials, pharmaceuticals, microsystems.

–  Biological – feedback systems that regulate pressures, concentrations, balance, etc

•  Design the dynamics – Sluggish systems become quick to respond – Unstable systems become stable and predictable

•  “Robustness” – Reject disturbances acting on the system – Same response with large variations in the system


•  Instability: any feedback loop allows for the possibility of instability. The question of stability has long been central in control theory

•  Measurement noise gets feed back into the actual system response.



•  Many control systems can be characterised by these components

Sensor

Actuator Process Control

Reference r(t)

Output y(t)

- +

Error e(t)

Control Signal

u(t)

Plant

Disturbance

Sensor Noise

Feedback


•  A block diagram is made up of signals, systems, summing junctions and pickoff points


•  A system model is one or more equations that describe the relationship between the system variables – often the input(s) and output(s) of the system

•  For physical systems, these equations are derived from study of the physical properties of the system such as mechanics, fluids, electrical, thermodynamics, etc.


•  Force-velocity, force-displacement, and impedance translational relationships for springs, viscous dampers, and mass


•  Torque-angular velocity, torque-angular displacement, and impedance rotational relationships for springs, viscous dampers, and inertia


Voltage-current, voltage-charge, and impedance relationships for capacitors,

resistors, and inductors


•  Starting with an impulse response, h(t), and an input, u(t), find y(t)

u(t), h(t)

U(s), H(s)

convolution y(t)

L L-1

Multiplication, algebraic manipulation

Y(s)


•  We normally use tables of Laplace transforms rather than solving the Laplace equations directly

•  This greatly simplifies the transformation process


•  The Laplace Transform is a linear transformation between functions in the t domain and s domain


•  Second order systems are quite common and are generally written in the following standard form

•  Many systems of interest are of higher order


•  We can relate the natural frequency and damping ratio to the s-plane

θ


•  Damping ratio determines the characteristics of the system response


•  Rise time, settling time and peak time yield information about the speed of response of the transient response

•  This can help a designer determine if the speed and nature of the response is appropriate


•  For a second order system with no finite zeros, the transient response parameters are approximated by – Rise time :

– Overshoot :

– Settling Time (2%) :


•  Find allowable region in the s-plane for the poles of a transfer function to meet the requirements

tr ≤ 0.6 sec

Mp ≤ 10%

ts ≤ 3 sec

sin-1ζ

σ

ωn

Im(s)

Re(s)


•  This table shows the relationship between input, system type, error constants and steady-state error


•  We can easily find the root locations for a second order system

•  What about for a general, possibly higher order, control system?

•  Poles exist when the characteristic equation (denominator) is zero


•  The location of the roots, and hence the nature of the system performance, are a function of the system gain K

•  In order to solve for this system performance, we must factor the denominator for specific values of K

•  We define the root locus as the path of the closed-loop poles as the system parameter varies from 0 to ∞


•  The CL roots move from the OL towards the zeros

•  Additional poles move towards infinity along well defined asymptotes


•  We saw that the steady state response for an LTI system excited by a sinusoid with unit amplitude and frequency ω0 will also exhibit a sinusoidal output of frequency ω0 with magnitude M(ω0) and a phase φ(ω0) where

•  Notice that both the magnitude and the phase of the response on dependent on the frequency of the input ω0


•  We often plot the magnitude and phase of the system response as a function of the input frequency

•  The magnitude is normally plotted in dB=20logM(ω) vs. log(ω)

•  The phase is plotted in degrees vs. log(ω) •  The resulting graph is called the Bode plot


•  Normalized and scaled Bode plots for a. G(s) = s; b. G(s) = 1/s; c. G(s) = (s + a); d. G(s) = 1/(s + a)


•  The complete three-term controller is described by

Kp+Ki/s+Kds -

+

R(s) E(s) C(s) G(s)


•  The ideal derivative compensator effectively adds a pure differentiator to the forward path of the control system

•  This is effectively equivalent to an additional zero

•  As you should by now be aware, the location of the open loop poles and zeros affects the root locus and hence the transient response of the closed loop system


•  Consider a simple second order system whose root locus looks like this (roots -1, -2)

•  Adding a zero to this system drastically changes the shape of the root locus

•  The position of the zero will also change the shape and hence the nature of the transient response

Zero at -3

Zero at -5


•  The Bode plot for a PD controller looks like this

•  The stabilizing effect is seen by the increase in phase at frequencies above the break frequency

•  However, the magnitude grows with increasing frequency and will tend to amplify high frequency noise


•  For compensation using passive components, a pole and zero will result

•  If the pole position is selected such that it is to the left of the zero, the resulting compensator will behave like an ideal derivative compensator

•  The name Lead Compensation reflects the fact that this compensator imparts a phase lead


•  First find a point in the s-plane that we’d like to have on the root locus

•  Place the lead pole at 20 and solve for the position of the zero

x

θ1=120 θ2=112.4 θ3=20.2 τηερεφορε θc=72.6


•  The Bode plot for a Lead compensator looks like this

•  The frequency of the phase increase can be designed to meet a particular phase margin requirement

•  The high frequency magnitude is now limited


•  If we rewrite the transfer function for the integral compensator we find

•  This is simply a pole at the origin and a zero at some other position to be selected based on our design requirements – normally close to the origin to minimize the angular contribution of the compensator


•  The Bode plot for a PI controller looks like this

•  The break frequency is usually located at a frequency substantially lower than the crossover frequency to minimize the effect on the phase margin


•  As with the lead compensation, using passive components results in a pole and zero

•  If the pole position is selected such that it is to the right of the zero near the origin, the resulting compensator will behave like an ideal integral compensator although it will not increase the system type

•  The name Lag Compensation reflects the fact that this compensator imparts a phase lag


•  The Bode plot for a Lag compensator looks like this

•  This compensator effectively raises the magnitude for low frequencies

•  The effect of the phase lag can be minimized by careful selection of the centre frequency


•  System equations are described in matrix form. •  Most fundamental are the state variables: A set of parameters

that completely describes the current state of the system. –  For example position and velocity of a moving body

•  Using measurements of the outputs, the system state can be computed - observers

•  Using the system state, control strategies can be devised to achieve desired performance. –  Pole placement –  Optimal controllers

•  Attractive for multi-input multi-output (MIMO) systems


•  Using the state space approach, we represent a system by a set of n first-order differential equations:

•  The output of the system is expressed as:

x - state vector y - output vector u - input vector

A - state matrix B - input matrix C - output matrix D - feedthrough or feedforward matrix (often zero)

•  We can draw a block diagram describing the general State Space Model



•  We can represent a general state space system as a Block Diagram.

•  If we feedback the state variables, we end up with n controllable parameters.

•  State feedback with the control input

u=-Kx +r.

* N.S. Nise (2004) “Control Systems Engineering” Wiley & Sons


•  We can then control the pole locations by finding appropriate values for K

•  This allows us to select the position of all the closed loop system roots during our design.

•  There are a number of methods for selecting and designing controllers in state space, including pole placement and optimal control methods via the Linear Quadratic Regulator algorithm.

•  Setting u=-Kx+r yields

•  Rearranging the state equation and taking LT yields

•  Select values of K so that the eigenvalues (root locations) of (A-BK) are at a particular location


•  Controllability and Observability are fundamental concepts.

•  For output feedback controllers, the “separation principle” tells us that we can design an observer and a controller separately and the combined controller is stabilizing.



•  You should be familiar with the concepts reviewed in this lecture – Modelling of dynamic systems – Specification of second order systems – Root Locus – Bode Plots – Design and properties of PID (and variants), Lead

and Lag controllers – State Space Modelling and Design


•  You will not be required to find roots of polynomials higher than second order.

•  You will be provided with a selected set of equations you may require for solving the problems

•  In order to prepare I would suggest that you –  Review the assignment questions, making sure you

understand the material covered this semester –  Look over previous years’ exams


•  In order to understand system performance, we must be able to model these systems

•  The study of control provides us with a process for analysing, understanding and deisgning for the behaviour of a system

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