Matlab Tutorial for State Space Analysis and System Identification

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Matlab Tutorial for State Space Analysisand
System Identification

• Vibha Prasad

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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

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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)

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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)

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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-2 . . . CA C
• System observable if observability matrix O is
  invertible

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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

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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 ep/?
• 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

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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

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Examples followed in the tutorial

• SISO System Example
• Tandem Queue
• MIMO System Example
• Apache HTTP Server

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Tandem Queue

• x1(k 1) 0.13x1(k) 0.069u(k)
• x2(k 1) 0.46x1(k) 0.63x2(k)
• y(k) x1(k) x2(k)

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Apache HTTP Server

• x1(k 1) 0.54x1(k) 0.11x2(k) 0.0085u1(k) -
  0.00044u2(k)
• x2(k 1) -0.026x1(k) 0.63x2(k) -
  0.00025u1(k) 0.00028u2(k)
• y1(k) x1(k) y2(k) x2(k)

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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.

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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

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System Identification Procedure

• Experimental design issues
• Which signals to measure?
• How much data is needed?
• Input signal selection
• Sampling period selection

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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

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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.

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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

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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.

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System Identification Toolbox

• System Identification Toolbox provides features
  to build mathematical models of dynamic systems
  based on observed system data.
• MATLAB Example

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Questions?

• Thank You