Matlab Tutorial for State Space Analysis and System Identification

1. Matlab Tutorial for State Space Analysisand System Identification Vibha Prasad

2. 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. 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)

4. 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)

5. Chapter 7 Review (Contd.) • Controllability: system can be driven to an arbitrary state by properly choosing a set of inputs. • C = [ An-1BAn-2B . . . AB B ] • System controllable if controllability matrix C is invertible • Observability: all the state can be inferred from its outputs. • O = [ CAn-1CAn-2 . . . CA C ] • System observable if observability matrix O is invertible

6. 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.) • 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.) • 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

9. Examples followed in the tutorial • SISO System Example • Tandem Queue • MIMO System Example • Apache HTTP Server

10. 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)

11. Reference CPU Measured CPU KeepAlive Controller Apache HTTP Server MaxClients Reference MEM Measured MEM 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)

12. 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.

13. 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 System Identification Procedure

14. System Identification Procedure • Experimental design issues • Which signals to measure? • How much data is needed? • Input signal selection • Sampling period selection

15. 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

16. 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.

17. 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

18. 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.

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

20. Questions? Thank You