Linear System Theory and Design
Control theory and
control engineering
Group 6: H.Tan, Y.Q.Ran , L.L.Guo (headman)
Chapter 1 Describe State-space of Linear
Control System
According to the number of independent energy sto-rage
elements to determine the number of state variables , but the
choice of state variables is not the only.
According to choice of state variables for setting up the
state equation and output equation.
State equation (first-order differential equation):

x(t)  Ax(t)  Bu(t)
Output equation (algebraic equation):
y(t)  Cx(t)  Du(t)
The state space description of the system:


x (t)  Ax(t)  Bu(t)


 y(t)  Cx(t)
By the system of the state space description can export
the block diagram of a system dynamic equation and transfer function of the system.
The analysis method of linear discrete-time system with
the introduction of linear time-varying systems analysis
method is similar.
Chapter 2 Kinematic Analysis of Linear
System
Ⅰ.Solution
of homogeneous linear time-invariant system

state equations:
x  Ax
Homogeneous equation solution of equation of state:


1 2
1 K
2
K
x(t )  I  A(t  t 0 )  A t  t 0    
A t  t 0    X t 0 
2
!
K
!


For the above formula , the series of right bracketsis is
n×n, the exponential matrix function is denoted by eAt .
e A t  I  At 
namely
1 2 2
1 K K
At 
A t 
2!
K!
x t   e A t t x t 0 
0
Ⅱ.Solution
of inhomogeneous linear time-invariant system state
equations:

x(t)  Ax(t)  Bu(t)
Inhomogeneous equation solution of equation of state:
x(t )  e A(t t ) x(t0 )  t e A(t t ) Bu( )d
0
t
0
0
Ⅲ.The
solution of state transition matrix
1)According to definition , compute φt 
φt   e At  I  At 
2)According
1
1
A2t 2   
Ak t k  
2!
k!
to Laplace transform , compute φt 
φt   eAt  L1[sI  A]1
3)According
to Cayley-Hamilton theorem ,compute φt 
4).According
to linear transformation , compute φt 
Matrix A by the linear change of diagonal matrix Λ ,
compute φt 
0

1

  PAP  



n 
 0
1
Matrix A by the linear change of Jordan-matrix J ,
compute φt 
1
0

1

1
J  PAP  

0

 



1
1 
Matrix A by the linear change of modal matrix M ,
compute φt 
  
M  PAP 1  

   
Chapter 3 Stability of Linear Systems
The nature of Lyapunov function:constructs an energy
function ,and analyses its first-order differential coefficient
sign to get the correlative information of a system's stability.
V(x) is positive-definite and continued,there is a first-order
partial derivation.
Selection of V(x) is not the only, usually choose the quadratic form.
 x  <0, if ||x||→  , as V(x,t)→  , then it is
(1). When V
asymptotically stable in a whole ;or it is asymptotically stable .
 x  ≤0, x≠0, V
 x  
 0, if ||x||→  , as V(x,t)→
(2).When V
 , then it is asymptotically stable in a whole ;or it is asymp-
totically stable in a whole ;or it is asymptotically stable .
 x  ≤0, x≠0, V
 x  ≡0, it is stable .
(3).When V
 x  >0, it is instable .
(4).When V
The geometrical meaning of stability is shown in the following figures:
Fig.3.1 Stable balanced state
Fig.3.2 Asymptotically stable balanced state
Fig.3.3 Asymptotically stable in a whole
balanced state
Fig.3.4 Instable balanced state
The sufficient and necessary condition for linear timevarying system to be congruously asymptotically stable :
For any given positive-definite symmetry matrix Q,there
is an unique symmetry positive-definite matrix P satisfying the following matrix Lyapunov equation:
ATP+PA=-Q
Chapter 4 Controllability and Observability
of Linear Systems
Controllability:If the state vector x is controlled by the input vector u.
Observability :Whether the output y contains all the information about the state vector x .
Ⅰ.Controllability criterions of linear time-invariant systems
1) If all eigenvalues of matrix A are different with each
other,derived the diagonal dynamic equations
λ1

 x  1 Bu
x  

 P

λ n 
Not the whole row of zero
element in matrix B .
2) If the eigenvalues of matrix A are λ1 (m1 times) , λ 2
k
(m2 times), … , λ k (mk times),  m i  n
i 1
 J1

 x  1 Bu
x   

 P

J k 
Not the whole last low elements
of the Jordan block corresponding to matrix B are zero element.
3)The system is completely controllable if and only if
rank B AB ... A n 1B  n
Ⅱ.Observability criterions of linear time-invariant systems
1)The diagonal canonical form
Not the whole rank of zero element in matrix C.
2)The system is completely observable if and only if
rank C
CA
... CA n 1   n
T
Ⅲ.LTI SISO system is output completely controllable if and
only if : Transfer function matrix G(s) =(SI-A)-1B has no polezero cancellation (the zeros and poles are different).
LTI SISO system is output completely observable if and
only if :Transfer function matrix G(s) =C(SI-A)-1 has no
pole-zero cancellation(the zeros and poles are different).
LTI SISO system is output completely controllable and
observable if and only if : Transfer function matrix G(s) =
C(SI-A)-1 B has no pole-zero cancellation(the zeros and
poles are different).
Ⅳ. The controllable canonical form :
~
A  PAP
1
~
C  CP
~
B  PB
1
The observable canonical form :
Aˆ  T 1 AT

T  T1
Cˆ  CT
Bˆ  T 1B
AT1  An1T1

 P1 


P
1A

P
  

n 1 
 P1 A

Chapter 5 State Feedback and State Observer
Key point:with the design of the linear feedback control
law to improve the stability and dynamical qualities of corresponding closed-loop system.
Ⅰ.State Feedback(All the Status Information Available)
By the controlled variable u modify the state matrix A,
to change the pole positions, then improve the stability.
A→A-BK
Ⅱ.State Observer
When some state variables are available, reconstruct
the state vector x by measurable variables y and u, then
get the state feedback.
The existence condition of state observer : The system
is observable .
Fig. 5.1 State feedback system with a state-observer
Ⅲ.The
equation of observer
~
x   A  GC ~
x  Bu  Gy
input
~
x—— the asymptoyion of x
G —— output error feedback matrix of state-observer
Only when t   , state feedback system with a stateobserver is completely equivalent to directly state feedback system .
Assigned a reasonable pole positions by G,to make
 x  ~x   0 easily .