State Space Control D R. T A R E K A. T U T U N J I

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1 State Space Control D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I V E R S I T Y J O R D A N P R E S E N T E D A T H O C H S C H U L E B O C H U M G E R M A N Y M A Y 19-21,

2 State Space Description Transfer functions concentrates on the inputoutput relationship only. But, it hides the details of the inner workings. To get a better insight into the system s behavior, variables states are introduced.

3 State Space Description State variables describe the complete dynamic behavior of a system State variables change as a function of time and form a trajectory in dimensional space (referred to statespace)

4 Block Diagram Example

5 Properties of States Memory. The state summarizes the past. Dynamics. The effect of the input is directly connected to the derivative (the change) in the state vector. Not unique. The state representation is not unique.

6 Ordinary Differential Equations The state of a system is a collection of variables that summarize the past of a system for the purpose of predicting the future A system can be represented by the differential equation x state variable, u input, y output f and h are functions

7 Linear Systems where A, B, C and D are constant matrices. Such a system is said to be linear and time-invariant, or LTI for short. Matrix A is called the dynamics (or system) matrix Matrix B is called the control (or input-gain) matrix Matrix C is called the sensor (or output-gain) matrix Matrix D is called the direct term.

8 State Space Matrices The system matrix captures the internal structure of the system and determines many fundamental properties. The input-gain and output-gain matrices can be modified by adding, modifying or deleting some actuators (to control) or sensors (to measure) from the process.

9 State-Vector Differential Equation

10 Example: Spring-Mass with Damping

11 Example: Circuit

12 Two Mass Example

13 To: Out(2) Amplitude To: Out(1) System Response using MATLAB >> k1=1; k2=1; c=0.2; m1=5; m2=2; >> A=[ ;-(k1+k2)/m1 -c/m1 k2/m1 0; ;k2/m2 0 -k2/m2 0]; >> B=[0 1/m1 0 0]'; >> C=[ ; ]; >> D=0; >> sys=ss(a,b,c,d); >> step(sys); Step Response Time (seconds)

14 Alternative Problem Derive the state-space equations using three states x 1 = y 1, x 2 = y 1, x 3 = y 2 and two outputs y 1 and y 2

15 State-Space and Transfer Functions Direct Canonical Form

16 State Space and Transfer Functions

17 Example

18 Controllability A System is controllable if a control vector u(t) exists that will transfer the system from any initial state x(t0) to some final state x(t) Controllability Matrix MATLAB Command M=ctrb(A,B) If Full Rank Controllable

19 Observability A system is observable if the system states x(t) can be exactly determined from the measured output y(t) Observability Matrix MATLAB Command N=obsv(A,C) If Full Rank Observable

20 Controllability Flow x x y x x x 1 x u( t) 0 uncontrollable x (0 2 ) s x (0 1 ) s u x 2 1 s x 2 1 x 1 1 s x 1 1 y controllable

21 Observability Flow x x y x x x 1 x u( t) 1 x (0 2 ) s x (0 1 ) s u 1 1 x 2 s x 2 1 x 1 s x 1 1 y unobservable observable

22 MIMO Example >> M=ctrb(A,B) M = >> r=rank(m) r = 3

23 Practical Example: Orbiting Satellite Reference: Mauricio de Oliveira

24 State-Feedback Control AND

25 Regulator Design via Pole Placement MATLAB Command K=place(A,B,P)

26 Example

27 Motor State-Space Model

28 Motor Control Example

29 Motor Control Example

30 Motor Control Example If desired poles at -2, -3, -4 In general MATLAB k=place(a,b,[-2; -3 ;-4]) k = [ ]

31 Full-State Observer

32 Example

33 Closed-Loop Control with Observer AND

34 Reduced State-Observer A full-order state observer estimates all state variables In practice, some states are already measured. Then, we use a reduced-state observer. Consider the case with three states: x1, x2, and x3 Assume x1 is measured. Then, need to estimate x2 and x3 only

35 Example Desired Char. Eq. for the Controller Desired Char. Eq. Observer

36 Controller-Observer

37 Controller-Observer

38 References Advanced Control Engineering (Chapter 8: State Space Methods for Control System Design) by Roland Burns 2001 Modern Control Engineering (Chapters 9 and 10 Control System Analysis and Design in State Space) by Ogata 5th edition 2010 Modern Control Engineering (Chapter 10: State Space Design Methods) by Paraskevopoulos 2002 Feedback Systems: An Introduction to Scientists and Engineers (Chapter 8: System Models) by Astrom and Muray 2009