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 0 1 5
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.
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)
Block Diagram Example
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.
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
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.
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.
State-Vector Differential Equation
Example: Spring-Mass with Damping
Example: Circuit
Two Mass Example
To: Out(2) Amplitude To: Out(1) System Response using MATLAB >> k1=1; k2=1; c=0.2; m1=5; m2=2; >> A=[0 1 0 0;-(k1+k2)/m1 -c/m1 k2/m1 0;0 0 0 1;k2/m2 0 -k2/m2 0]; >> B=[0 1/m1 0 0]'; >> C=[1 0 0 0;0 0 1 0]; >> D=0; >> sys=ss(a,b,c,d); >> step(sys); 2 1.5 1 0.5 Step Response 0 3 2 1 0-1 0 50 100 150 200 250 300 350 400 450 500 Time (seconds)
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
State-Space and Transfer Functions Direct Canonical Form
State Space and Transfer Functions
Example
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
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
Controllability Flow x x y 1 2 2 0 x x 1 1 0 2 1 x 1 x 1 2 1 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 1 1 2 controllable
Observability Flow x x y 1 2 2 0 x x 1 1 0 2 0 x 1 x 1 2 3 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 3 1 2 unobservable observable
MIMO Example >> M=ctrb(A,B) M = 0 0 1 3 3 9 1 3 3 9 7 21 0 1 1 6 11 42 >> r=rank(m) r = 3
Practical Example: Orbiting Satellite Reference: Mauricio de Oliveira
State-Feedback Control AND
Regulator Design via Pole Placement MATLAB Command K=place(A,B,P)
Example
Motor State-Space Model
Motor Control Example
Motor Control Example
Motor Control Example If desired poles at -2, -3, -4 In general MATLAB k=place(a,b,[-2; -3 ;-4]) k = [6.0 4.5 2.0]
Full-State Observer
Example
Closed-Loop Control with Observer AND
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
Example Desired Char. Eq. for the Controller Desired Char. Eq. Observer
Controller-Observer
Controller-Observer
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