1. Matlab Tutorial for State Space Analysis and System Identification
CS 851 Feedback Control of Computing Systems
Matlab Tutorial for State Space Analysis and System Identification
Vibha Prasad

2. CS 851 Feedback Control of Computing Systems
Overview
Review of Chapter 7
Review of Chapter 10
Examples Followed in the Tutorial
Specifying LTI systems with space-state models
System response
System Analysis
Controller Design Tools
System identification

3. CS 851 Feedback Control of Computing Systems
Chapter 7 Review
State space models: Scalable approach to model MIMO systems
State vector: x(k)
State Model equations:
x(k +1) = Ax(k) + Bu(k)
y(k) = Cx(k)
State vector: x(k): set of state variables
Steady-state Gain = C(I - A)-1B
Final value theorem: G(1)
Equation:how the system changes
How the system variables/state affects the output

4. Chapter 7 Review (Contd.)
CS 851 Feedback Control of Computing Systems
Chapter 7 Review (Contd.)
System Analysis
Characteristic polynomial, det(zI – A)
Poles = eigenvalues of A
Steady-state Gain = C(I - A)-1B
Equivalence: w(k) = Tx(k)
w(k + 1) = (TAT-1)w(k) + (TB)u(k)
y(k) = (CT-1)w(k)
Many techniques for solving transfer functions apply to state space as well.
Settling time and max overshoot are computed just as they were with transfer function models.
Stesdy state systems have certain characteristics that are not observed in other models

5. Chapter 7 Review (Contd.)
CS 851 Feedback Control of Computing Systems
Chapter 7 Review (Contd.)
Controllability: system can be driven to an arbitrary state by properly choosing a set of inputs.
C = [ An-1B An-2B AB B ]
System controllable if controllability matrix C is invertible
Observability: all the state can be inferred from its outputs.
O = [ CAn-1 CAn CA C ]
System observable if observability matrix O is invertible
Steady state gain can be found using the final value theorem just as before.

6. CS 851 Feedback Control of Computing Systems
Chapter 10 Review
State Feedback Architecture
Static state feedback
Similar to proportional control
Does not include a reference point
Static state feedback with precompensation
Can track reference points
Poor disturbance rejection
Dynamic state feedback
Similar to PI control for SISO systems
Can track reference input and reject disturbance

7. Chapter 10 Review (Contd.)
CS 851 Feedback Control of Computing Systems
Chapter 10 Review (Contd.)
State feedback controller design
Pole placement
Specify max settling time and overshoot (ks* and Mp*)
Obtain desired dominant poles of the closed –loop system
Ks* = - 4/ log(|r|)
Mp* = eπ/|θ|
Construct the desired characteristic polynomial
Other poles should have magnitude less than 0.25r
Construct the modeled characteristic equation
Equate coefficients of the desired and modeled characteristic polynomial and calculate gains

8. Chapter 10 Review (Contd.)
CS 851 Feedback Control of Computing Systems
Chapter 10 Review (Contd.)
State feedback controller design (contd.)
LQR – Linear Quadratic Regulation
Chooses feedback gains to minimize a weighted sum of control error and control effort
J = ½ ∑ [ xT(k)Qx(k) + uT(k)Ru(k)]
Select Q and R
Compute feedback gains K
Predict control system performance or run simulations
Choose new Q and R and repeat the above steps if the performance is not suitable
Consider Control Errors (Δx(k)) and Control Effort (Δu(k)), where x is the state vector and u is control input
Reduce Control Errors: reduce settling time and overshoot
Reduce Control Effort: reduce noise sensitivity
Q: cost of state variation (control error)
R:cost of input variation (control effort)

9. Examples followed in the tutorial
CS 851 Feedback Control of Computing Systems
Examples followed in the tutorial
SISO System Example
Tandem Queue
MIMO System Example
Apache HTTP Server

10. CS 851 Feedback Control of Computing Systems
Tandem Queue
Say something about:
SISO systems
Tandem queue
Control input, K(k)= size of buffer 1
Measured output, R(k) = end-to-end response time
x1(k + 1) = 0.13x1(k) – 0.069u(k)
x2(k + 1) = 0.46x1(k) x2(k)
y(k) = x1(k) + x2(k)

11. CS 851 Feedback Control of Computing Systems
Apache HTTP Server
Measured
CPU
Controller
Apache HTTP
Server
KeepAlive
MaxClients
Reference
MEM
Sya something about:
MIMO systems
Apache HTTP server
x1(k + 1) = 0.54x1(k) – 0.11x2(k) u1(k) u2(k)
x2(k + 1) = x1(k) x2(k) u1(k) u2(k)
y1(k) = x1(k) y2(k) = x2(k)

12. System Identification
CS 851 Feedback Control of Computing Systems
System Identification
The system identification task is one of the most time consuming tasks in advanced control implementation projects.
Problems:
System analyst should have extensive background knowledge about the system, control theory, discrete time systems, optimization, statistics etc.
Large no. of design variables.
Solution:
Understand the various system identification methods and associated decision variables.
Effectively use a priori knowledge regarding the system to be identified and the purpose of the intended controller.
Explain System ID
How to design the experiment? How much data to collect?
How to choose the model structure?
Experience, prior knowledge
How to deal with noise? Data contains noise hence the measurements are unreliable.
Preprocess data, filter
How do we measure the quality of the model?
How will the purpose of the model affect the identification?
How do we handle non-linear and time-varying effects?

13. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Design experiment and collect data.
Examine the data. Preprocess the data.
Detrending, prefiltering, outlier removal
Model structure selection
Depends on the application.
Compute best model in the model structure.
Examine the properties of the model obtained.
Model validation
This in terms of the system identification toolbox.
The flowchart tells us what the text covers and what we started out at the beginning of the semester
Define the model scope:
Stochastics
Model structure
Workloads
Choose the input signal such that the data carries maximum information.
Examine the data. Preprocess the data.
Reduce the influence of noise
Select and define a model structure.
Use prior knowledge
Model Validation: Is the model good enough? Good is subjective and depends on the purpose of the model.
If the model is good enough, then stop;
otherwise go back to step 3 to try another model structure.
You can also try other estimation methods (step 4), or work further on the input-output data (steps 1 and 2). Design an experiment and collect data.

14. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Experimental design issues
Which signals to measure?
How much data is needed?
Input signal selection
Sampling period selection
Choose the input signal such that the data carries maximum information.
Input signal selection:
Range of values of the input signal
Coverage of values within the operating region
Richness in exciting the dynamics of the target system

15. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Examine the data
Plot the data
Preprocess the data
Filter the data
Remove trends in the data
Reduce noise
Remove outliers
Resample the data
Sensitive to noise
Difficult to apply a short impulse big enough such that the response is much larger than the noise

16. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Model Structure Selection
Many standard model structures are available depending on the approach (how to model the influence of the input and the disturbances).
Model structure should suit the actual system.
Finding the best model structure and model order is an iterative procedure.
Model structure should suit the actual system.
clear purpose of the system
Many standard model structures are available depending on the approach (how to model the influence of the input and the disturbances).
using a priori knowledge of the system analyst

17. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Compute the best model in the model structure.
Parameter Estimation
Examine the properties of the model
Poles and zeros
Model Output
Transient response
Compute the best model in the model structure according to input-output data and goodness to fit criterion
General prediction error methods

18. System Identification Procedure
CS 851 Feedback Control of Computing Systems
System Identification Procedure
Model validation techniques
Simulation
Plot the measured output time series versus the predicted output from the model
Crossvalidation
Simulate on a data set different from the one used for parameter estimation.
For the number of different model structures, plot the error and select the minimum.
Is the model good enough? Good is subjective and depends on the purpose of the model.
A model is of no use unless it is validated!
Check the residuals
Pole-zero cancellation
Cross validation

19. System Identification Toolbox
CS 851 Feedback Control of Computing Systems
System Identification Toolbox
System Identification Toolbox provides features to build mathematical models of dynamic systems based on observed system data.
MATLAB Example

20. CS 851 Feedback Control of Computing Systems Thank You
Questions?
Thank You