Chapter 10
State Variable Analysis
Automatic Control Systems, 9th Edition
F. Golnaraghi & B. C. Kuo
Section 10-1, p. 673
10-1 Introduction
Objectives of this chapter:
• Introduce the basic methods of state variables and state
equations.
• Present the closed-form solutions of LTI state equations.
• Establish the relationship between the transfer-function
approach and the state-variable approach.
• Define controllability and observability of linear systems.
10-1
Section 10-2, p. 673
10-2 Block Diagrams, Transfer
Functions, and State Diagrams
Transfer Functions (Multivariable Systems)
• A linear system has
p inputs and q outputs.
• Transfer function between
the jth input & the ith output:
Rk(s) = 0, k = 1,2,…,p, k  j.
• The jth output when all inputs are in action:
10-2
Section 10-2, p. 673
Transfer Func. in Matrix-Vector Form
•
q 1
p 1
• 10-2-2 Block Diagrams and Transfer Functions of
Multivariable Systems
See Section 3-1-5 (pp. 117-119)
q p
10-3
Section 10-2, p. 677
State Diagram
The basic elements of a state diagram are similar to the SFG,
except for the integration operation.
• A state diagram can be constructed directly from the system’s
differential equations.
• A state diagram can be constructed from the system’s
transfer function.
• The state-transition equation may be obtained from the
state diagram by using the SFG gain formula.
• The transfer function of a system can be determined from
the state diagram.
• The state equations and the output equations can be
determined from the state diagram.
10-4
Section 10-2, p. 677
SFG Representation of Integration
10-5
Section 10-2, p. 678
From Differential Equations to State Diagram
Step 1: arrange the nodes
Step 2: connect the branches to portray Eq. (10-25)
Figure 10-5 (a)
Figure 10-5 (b)
10-6
Section 10-2, p. 678
From Differential Equations to State Diagram
Step 3: insert the integrator branches with gains of s1
and add the initial conditions to the outputs of the integrators
Figure 10-5 (c)
The outputs of the integrators are defined
as the state variables, x1, x2, …, xn.
10-7
Section 10-2, p. 679
Example 10-2-2
10-8
Section 10-2, p. 680
From State Diagrams to Transfer Func.
• Using the gain formula and setting all other inputs and
initial states to zero.
• Example 10-2-3:
10-9
Section 10-2, p. 680
From State Diagrams to State/Output Eqs.
•
•
State equation:
Output equation:
1.
2.
Delete the initial states and the integrator branches.
State equations: regard the nodes that represent the
derivatives of the state variables as output nodes.
Output equation: the output y(t) in the output equation is an
output node variable.
Regard the state variables and the inputs as input variables.
Apply the SFG gain formula to the state diagram.
3.
4.
10-10
Section 10-2, p. 680
Example 10-2-4
10-11
Section 10-2, p. 681
Example 10-2-5
10-12
Section 10-3, p. 682
10-3 Vector-Matrix Representation of
State Equations
• State equations:
state vector
input vector
disturbance vector
• Output equations:
• Dynamic equations (LTI systems):
– State equations:
– Output equations:
10-13
Section 10-4, p. 684
10-4 State-Transition Matrix
The state-transition matrix is defined as
a matrix that satisfies
• (t): state-transition matrix 
• x(0): initial state 
 (10-53)
 (10-54)
10-14
Section 10-4, p. 685
Properties of State-Transition Matrix
1.
2.
3.
(t2t1): the transition of the state from t = t1 to t = t2
when the inputs are zero.
4.
10-15
Section 10-5, p. 687
10-5 State-Transition Equation
• State equation:
• Output equation:
10-16
Section 10-5, p. 688
Example 10-5-1
10-17
Section 10-5, p. 689
State-Transition Equation
Determined from State Diagram
(10-89) can be written from the state diagram using the gain
formula with Xi(s), i = 1, 2, …, n, as the output nodes.
• Example 10-5-2:
10-18
Section 10-5, p. 690
Example 10-5-2 (cont.)
10-19
Section 10-5, p. 690
Example 10-5-3
state equation:
10-20
Section 10-5, p. 691
Example 10-5-3 (cont.)
Using the state transition approach:
1.
:
2.
:
10-21
Section 10-6, p. 692
10-6 Relationship between State Equations
and High-Order Differential Equations
• phase-variable canonical form (PVCF) or
controllability canonical form (CCF)
– State equation:
– Output equation:
10-22
Section 10-6, p. 693
Example 10-6-1
state variables
state equations:
output equation:
10-23
Section 10-7, p. 693
10-7 Relationship between State
Equations and Transfer Functions
LTI systems:
• State equations:
• Output equation:
• Transfer function: initial condition x(0) = 0
10-24
Section 10-7, p. 694
Example 10-7-1
10-25
Section 10-7, p. 694
Example 10-7-1 (cont.)
10-26
Section 10-7, p. 695
Example 10-7-1 (cont.)
10-27
Section 10-8, p. 695
10-8 Characteristic Equations,
Eigenvalues, and Eigenvectors
n>m
• Characteristic equation:
• Transfer function:
• State equations:
10-28
Section 10-8, p. 696
Examples 10-8-1 ~ 10-8-3
• From differential equation:
• From transfer function:
• From state equations:
10-29
Section 10-8, p. 697
Eigenvalues & Eigenvectors
• Eigenvalues of A:
the roots of the characteristic equation
• Eigenvector of A associated with the eigenvalue i:
pi: any nonzero vector
• Example 10-8-5: State equation with the coefficient matrix
– Characteristic equation:
– Eigenvalues:
– Eigenvectors:
10-30
Section 10-8, p. 698
Generalized Eigenvectors
Assume that there are q (<n) distinct eigenvalues among
the n eigenvalues of A.
• Eigenvectors corresponded to the q distinct eigenvalues:
i  1,2,...q
• Generalized eigenvectors: the eigenvectors corresponded to
the remaining high-order eigenvalues.
j: the mth order eigenvalue
( m  n  q)
10-31
Section 10-8, p. 698
Example 10-8-6
•
:
•
:
•
:
10-32
Section 10-9, p. 699
10-9 Similarity Transformation
• P: nonsingular matrix
•
10-33
Section 10-9, p. 700
Invariance Properties
• Characteristic Equations, Eigenvalues & Eigenvectors:
• Transfer-Function Matrix:
10-34
Section 10-9, p. 701
Controllability Canonical Form (CCF)
• Characteristic equation:
• CCF transformation matrix:
controllability matrix
• CCF model:
10-35
Section 10-9, p. 702
Example 10-9-1
Coefficient matrices:
i) Controllability matrix:
ii)
iv)
iii)
10-36
Section 10-9, p. 703
Observability Canonical Form (OCF)
• Characteristic equation:
• OCF transformation matrix:
• OCF model:
observability matrix
10-37
Section 10-9, p. 703
Example 10-9-2
Coefficient matrices:
i) observability matrix:
ii) OCF transformation matrix:
iii) OCF model:
10-38
Section 10-9, p. 704
Diagonal Canonical Form (OCF)
• Coefficient matrix A has n distinct eigenvalues, 1, 2, …, n.
• DCF transformation matrix:
pi, i = 1,2,…,n: the eigenvector associated with i.
• DCF model:
10-39
Section 10-9, p. 705
OCF and Example 10-9-3
• If A is of CCF and A has distinct eigenvalues, then
• Example 10-9-3:
10-40
Section 10-10, p. 707
10-10 Decompositions of Transfer Funcs.
Figure 10-13 Block diagram showing the relationships
among various methods of describing linear systems.
10-41
Section 10-10, p. 708
Direct Decomposition to CCF
• The nth-order SISO system:
1. Express the transfer function in negative powers of s.
2. Multiply by a dummy variable X(s).
3.
4. Construct the state diagram using (10-227) and (10-229).
10-42
Section 10-10, p. 709
CCF State Diagram & Dynamic Eqs.
CCF state diagram
10-43
Section 10-10, p. 709
Direct Decomposition to OCF
OCF state diagram
10-44
Section 10-10, p. 710
OCF Dynamic Equations
10-45
Section 10-10, p. 710
Example 10-10-1
• CCF dynamic equation:
• OCF dynamic equation:
10-46
Section 10-10, p. 711
Example 10-10-1 (cont.)
10-47
Section 10-10, p. 712
Cascade Decomposition
• State diagram:
• Dynamic equations:
10-48
Section 10-10, p. 712
Cascade Decomposition: Example
• State diagram:
• Dynamic equations:
10-49
Section 10-10, p. 713
Parallel Decomposition
• State diagram:
• Dynamic equations:
10-50
Section 10-10, p. 714
Example 10-10-2
• State diagram:
• Dynamic equations:
10-51
Section 10-11, p. 714
10-11 Controllability of Control Syst.
• Pole-placement design:
constant gain
Find the feedback matrix K such that the eigenvalues of
(ABK), or of the closed-loop system, are of certain
prescribed values.
• If the system is controllable, then there exists a constant
feedback matrix K that allows the eigenvalues of (ABK) to
be arbitrarily assigned.
10-52
Section 10-11, p. 714
Observer and State Feedback
• Not all the state variables are physically accessible
 Design and construct an observer that will estimate the state
vector from the output vector y(t).
• The condition that such an observer can be designed for the
system is called the observability of the system.
10-53
Section 10-11, p. 716
General Concept of Controllability
• The process is said to be completely controllable if every
state variable of the process can be controlled to reach a
certain objective in finite time by some unconstrained control
u(t), as shown in Fig. 10-23.
• An uncontrollable system  x1(t): controllable
x2(t): uncontrollable
10-54
Section 10-11, p. 716
Definition of State Controllability
LTI system:
x(t): n1 state vector
• The state x(t) is said to be controllable at t = t0 if there exists
a piecewise continuous input u(t) that will drive the state to
any final state x(tf) for a finite time (tf  t0)0.
• If every state x(t0) is controllable, in a finite time interval,
the system is said to be completely state controllable.
Theorem 10-1: For the system described by Eq. (10-261) to be
completely state controllable, the following controllability
matrix has a rank of n.
• controllab le  rank (S)  n
10-55
Section 10-11, p. 718
Examples 10-11-1 ~ 10-11-3
Example 10-11-1:
 uncontrollable
S = [B AB] is singular. Two state equations are dependent.
Example 10-11-2:
 uncontrollable
, which is singular. (see Fig. 10-24)
Example 10-11-3:
 uncontrollable
, which is singular.
10-56
Section 10-12, p. 719
10-12 Observability of Linear Systems
• A system is completely observable if every state variable of
the system affects some of the outputs.
• An unobservable system  x1(t): observable, x1(t) = y(t)
x2(t): unobservable
10-57
Section 10-12, p. 719
Definition of Observability
• The state x(t0) is said to be observable if given any input
u(t), there exists a finite time (tf  t0)0 such that the
knowledge of u(t) for t0ttf, matrices A, B, C, and D; and
the output y(t) for are sufficient to determine x(t0).
• If every state x(t0) is observable for a finite tf, the system is
said to be completely observable.
Theorem 10-4: For the system described by Eq. (10-261) to be
completely observable, the following observability matrix
has a rank of n.
• observable  rank (V)  n
10-58
Section 10-12, p. 720
Example 10-12-1
 unobservable
, which is singular.
10-59
Section 10-13, p. 721
10-13 Relationship among Controllability,
Observability, and Transfer Functions
Theorem 10-7:
If the input-output transfer function of a linear system has
pole-zero cancellation, the system will be uncontrollable or
unobservable, or both, depending on how the state variables
are defined.
If the input-output transfer function does not have pole-zero
cancellation, the system can always be represented by
dynamic equations as a completely controllable and
completely observable.
10-60
Section 10-13, p. 721
Example
10-61
Section 10-13, p. 722
Example 10-13-1
• Decomposition to CCF:
, which is singular.  unobservable
• Decomposition to OCF:
, which is singular.  uncontrollable
10-62
Section 10-14, p. 723
10-14 Invariant Theorem on
Controllability and Observability
Theorem 10-8. Invariant theorem on similarity transformations:
The controllability of [ A, B] and the observability of [ A, C]
are not affected by the transformation.
10-63
Section 10-14, p. 723
Theorem 10-9
Theorem on controllability of closed-loop systems with state
feedback
If the open-loop system
is completely controllable, then the closed-loop system obtained
through state feedback,
so that the state equations becomes
is also completely controllable.
– If [A, B] is uncontrollable, there is no K that will make
the pair [ABK, B] controllable.
– If an open-loop system is uncontrollable, it cannot be
made controllable through state feedback.
10-64
Section 10-14, p. 724
Theorem 10-10
Theorem on observability of closed-loop systems with state
feedback:
If an open-loop system is controllable and observable, then
the state feedback of the form
could destroy observability.
• The observability of open-loop and closed-loop systems due
to state-feedback is unrelated.
10-65
Section 10-14, p. 724
Example 10-14-1
[A, B] is controllable and [A, C] is observable.
• State feedback:
,
• Closed-loop system:
• Observability matrix:
• If k1 and k2 are chosen so that
would be uncontrollable.
, the closed-loop system
unobservable
10-66
Section 10-15, p. 725
10-15 Case Study:
Magnetic-Ball Suspension System
• Dynamic equations:
• State variables:
• Nonlinear state equations:
2
10-67
Section 10-15, p. 726
Magnetic-Ball Suspension System (2/4)
• Linearized equations:
• Characteristic equation:
• Eigenvalues:
• State-transition matrix:
10-68
Section 10-15, p. 727
Magnetic-Ball Suspension System (3/4)
• Transfer function:
• Controllability:
rank (S)  3  completely controllab le
10-69
Section 10-15, p. 727
Magnetic-Ball Suspension System (4/4)
• Observability:
– C*=[1 0 0]:
 completely observable
– C*=[0 1 0]:
 completely observable
– C*=[0 0 1]:
 unobservable
10-70
Section 10-16, p. 728
10-16 State-Feedback Control
Block diagram
state feedback
• State equation:
10-71
Section 10-16, p. 729
Control of a 2nd-Order System
by State Feedback
System with state feedback:
• Tachometer feedback:
• PD control:
10-72
Section 10-17, p. 730
10-17 Pole-Placement Design
through State Feedback
• State equation:
• State-feedback control:
• Closed-loop system with state feedback:
– Characteristic equation:
• CCF model:
10-73
Section 10-17, p. 731
Example 10-17-1
Linearized state model of magnetic-ball system:
Specification:
1. The system must be stable.
2. For any initial disturbance on the position of the ball from
its equilibrium position ( x1(t) = 0.5 m), the ball must return
to the equilibrium position with zero steady-state error.
3. The time response should settle to within 5% of the initial
disturbance in not more than 0.5 sec.
4. The control is to be realized by state feedback:
10-74
Section 10-17, p. 732
Example 10-17-1 (cont.)
• The following characteristic equation roots should satisfy
the design requirement:
• The corresponding characteristic equation:
• The characteristic equation of the closed-loop system with
state feedback:
• The feedback-gain matrix:
10-75
Section 10-17, p. 733
Example 10-17-1 (cont.)
Initial state:
10-76
Section 10-17, p. 734
Example 10-17-2
(a) State diagram of second-order sun-seeker system
(b) State diagram of second-order sun-seeker system with state
feedback
•
•
Zero steady-state error due to a step input: k1 = 2500
 characteristic equation:
The maximum overshoot, rise time, and settling time are all at
minimum when k2 = 75. 
10-77
Section 10-18, p. 735
10-18 State-Feedback with Integral Control
10-78
Section 10-18, p. 736
State Feedback with Integral Control
Design objective:
1.
2. The n+1 eigenvalues of ( A  BK )
are placed at desired locations.
 the pair [ A, B] must be
completely controllable.
10-79
Section 10-18, p. 737
Example 10-18-1
Design objective:
1. The steady-state output must follow a step function with zero error.
2. The rise time and settling time must be less than 0.05 sec.
3. The maximum overshoot of the unit-step response must be less than
5%.
10-80
Section 10-18, p. 737
Example 10-18-1 (cont.)
• The design specification can be satisfied by placing the
roots at
• The desired characteristic equation is
•
10-81
Section 10-18, p. 738
Example 10-18-2
DC-motor control system:
Design objective:
1.
2.
3. The eigenvalues of the closed-loop system with state
feedback and integral control are at s =
10-82
Section 10-18, p. 739
Example 10-18-2 (cont.)
• State feedback with integral control:
10-83
Section 10-18, p. 740
Example 10-18-2 (cont.)
10-84