自动控制原理 第十五讲 动态系统的回顾

Transformed system has the same form but the matrices are different ˜ = T AT −1, ˜ = T B, ˜ = CT −1 , ˜ =D A B C D Transfer function and impulse response remain invariant with coordinate transformations.
Two Views on Dynamics
State Models - White Boxes
State Models
Standard representations:
dx = f ( x , u) dt y = ( x , u) d( x − x0 ) = A( x − x0 ) + B (u − u0 ) dt y − y0 = C( x − x0 ) + D (u − u0 )
dx = Ax + Bu dt y = Cx
Algebraic Criteria
The system
dx = Ax + Bu, dt is controllable if the matrix y = Cx
Controllability: Assume that the system is at the origin initially. Can we find a control signal so that the state reaches a given position at a fixed time? Notice we do not require that it stays there! Observability: Can the state x be determined from observations of the output y over some time interval.
The numerator of the transfer function G (s) is the characteristic polynomial of the matrix A.
The numerator of the transfer function G (s) is the characteristic polynomial of the matrix A.
Transfer function
b1 sn−1 + b2 sn−2 + . . . + bn G ( s) = n +D s + a1 s n − 1 + a2 s n − 2 + . . . + a n
Transfer function
b1 sn−1 + b2 sn−2 + . . . + bn G ( s) = n +D s + a1 s n − 1 + a2 s n − 2 + . . . + a n
c
where x and u now denotes deviations from the steady state
Associated concepts
• Observability and controllability
1
K. J. Åström August, 2001
The Concepts
c K. J. Åström August, 2001 2
Coordinate Changes
Coordinate changes are often useful
dx = Ax + Bu dt y = C x + Du z = Tx x = T −1 z dz ˜ + = Az dt ˜ + y = Cz ˜ Bu ˜ Du
c
K. J. Åström August, 2001
3
Kalmans Decomposition
Partitioning of state space • Sco controllable and observable • Sc o ¯ controllable not observable • Sc ¯o not controllable observable • Sc ¯o ¯ not controllable not observable
– Differential equations – Laplace transforms and complex numbers – Linear algebra and matrices
• The standard models • Relations between different representations • Computational aspects • Intuition amplifiers: SysQuake and ICTools
Controllable Canonical Form
−a1 −a2 . . . an−1 −an 1 1 0 0 0 0 dz 0 1 0 0 0 = z + u dt . . . . . . 0 0 1 0 0 y = b1 b2 . . . bn−1 bn z + Du
-Soc u Soc
Matlab, SciLab, Octave and SysQuake
Σ
Soc
y
Soc
One advantage of the matrix formulation is that there is a very good computer support. This makes it easy to solve real problems. Matlab can be viewed as a matrix calculator or a matrix oriented programming environment. It was invented by Cleve Moler. SciLab and Octave are matrix oriented public domain software. SysQuake is a newer product which has been designed for a higher degree of interaction. Matlab has many tool-boxes for special domains. The Control System Toolbox and Simulink are particularly useful for control. You can find out what it contains by typing the command
Standard forms for linear systems G ( s) = b1 s + b2 s + . . . + bns n n − 1 s + b1 s + . . . + an s
n−1 n−2
Linear Time Invariant Systems
dx = Ax + Bu dt y = Cx + Du
Observable Canonical Form
−a1 1 0 −a2 dz . . = . dt −an−1 0 −an 0 y = 1 0 0...
0 1
...
0 0
0 z + Du
0 b1 0 b2 . . z + u . 1 bn−1 0 bn
• A detailed description of the inner workings of the system • The heritage from mechanics • The notion of state and stability • States describe storFra Baidu bibliotekge of mass, energy and momentum
Input-Output Models - Black Boxes
where equilibrium is given by f ( x0 , u0 ) = 0 and y0 = ( x0 , u0 )
dx = Ax + Bu dt y = Cx + Du
• A description of the input output behavior • The heritage of electrical engineering • The notions of transfer function, poles, zeros, minimum phase • The idea of frequency response
−1 ˜ ˜ ˜ At ˜ (t) = Ce B = CT −1 eT AT t T B = Ce At B = (t)
Transfer function
n
G ( s) =
i=1
β iγ i +D s − λi
Notice appearance of eigenvalues of matrix A
and ˜ ( s) = C ˜ (sI − A ˜ )−1 B ˜ = CT −1 (sI −T AT −1)−1 T B = C(sI − A)−1 B = G (s) G
Wc = ( B AB A2 B . . . An−1 B )
has full rank. The system observable if the matrix C CA CA2 Wo = . . .
CAn−1
has full rank.
Input-Output Models
Diagonal Form
β1 λ1 0 β2 λ2 dz = z + u . . . . dt . . 0 λn βn y = γ 1 γ 2 . . . γ n z + Du
Associated concepts
Variables now denote deviations from steady state. Solution
x( t) = e At x(0) +
0 t t
• Impulse response (t) • Frequency response G (iω ) • Bode plots and Nyquist curves • Poles, zeros and gain
e A( t− s) Bu( s) ds e A( t− s) Bu( s) ds + Du( t)
y( t) = Ce At x(0) + C
0
First terms depend on initial condition the second on the input. Transfer function: G (s) = C(sI − A)−1 + D Impulse response: h(t) = CeAt B + Dδ (t)
Lecture 15 - Review of Dynamical Systems
K. J. Åström 1. Introduction 2. Different ways to view dynamics 3. State models 4. Input output models 5. Summary
Theme: Collecting bits and pieces.
Introduction
• Dynamics is a key foundation of control • Linear time invariant systems has been our work horse • A rich field with many concepts and results • Mathematical foundations