1. Chapter 7 Stability and Steady-State Error Analysis
§ Stability of Linear Feedback Systems
§ Routh-Hurwitz Stability Test
§ System Types and Steady-State Error
§ Time-Domain Performance Indices

2. § 7.1 Stability of Linear Feedback Systems (1)
Basic Concepts:
(1) Equilibrium States,
(2) Stable System
System response can restore to initial equilibrium state under small
disturbance.
(3) Meaning of Stable System
Energy sense – Stable system with minimum potential energy.
Signal sense – Output amplitude decays or grows with different meaning.
Lyapunov sense – Extension of signal and energy sense for state evolution
in state space.

3. § 7.1 Stability of Linear Feedback Systems (2)
Plant Dynamics:
Regular pendulum (Linear)
Inverted pendulum (Linear)
Natural response
Natural response

4. § 7.1 Stability of Linear Feedback Systems (3)
Closed-loop System:
Natural behavior of a control system, r(t)=d(t)=0
Equilibrium state
Initial relaxation system, I.C.=0
No general algebraic solution for 5th-order and above polynomial equation
(Abel , Hamilton)

5. § 7.1 Stability of Linear Feedback Systems (4)
Stability Problems:
Stabilization of unstable system
Destabilization Effect on stable system
G(s) : Unstable plant
Closed-loop : Stable
G(s) : stable plant
Closed-loop : Unstable

6. § 7.1 Stability of Linear Feedback Systems (5)
Stability Definition
(1) Asymptotic stability
Stable system if the transient response decays to zero
(2) BIBO stability
Stable system if the response is bounded for bounded input signal
The impulse response of a system is absolutely integrable.

7. § 7.1 Stability of Linear Feedback Systems (6)
(3) S-domain stability
System Transfer Function : T(s)
Stable system if the poles of T(s) all lies in the left-half s-plane.
The definitions of (1), (2), and (3) are equivalent for LTI system.

8. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (1)
Characteristic Polynomial of Closed-loop System
Hurwitz polynomial
All roots of D(s) have negative real parts stable system
Hurwitz’s necessary conditions: All coefficients (ai) are to be positive.
Define
Note: Any zero root has been removed in D(s).

9. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (2)
Routh Tabulation (array)
Routh-Hurwitz Stability Criterion
(1) The polynomial D(s) is a stable polynomial if are all positive, i.e.
are all positive.
(2) The number of sign changes in is equal to the
number of roots in the RH s-plane.
(3) If the first element in a row is zero, it is replaced by a small
and the sign changes when are counted after completing the
array.
(4) If all elements in a row are zero, the system has poles in the RH plane
or on the imaginary axis.

10. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (3)
For entire row is zero
Identify the auxiliary polynomial The row immediately above the zero row.
The original polynomial is with factor of auxiliary polynomial.
The roots of auxiliary polynomial are symmetric w.r.t. the origin:

11. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (4)
Ex: For a closed-loop system with transfer function T(s)
Ex: Find stability condition for a closed-loop system with
characteristic polynomial as
Sol:

12. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (5)
Ex: For a colsed-loop system with characteristic polynomial
Determine if the system is stable
Sol:
Ex: For , determine if the system is stable
Sol:

13. § 7.2 Routh-Hurwitz Stability Test (1865-1905) (6)
Absolute and Relative Stability
Absolute Stability
Relative Stability
Characteristic equation
R-H Test on D(s)
Characteristic equation
R-H Test on D ’(p)

14. § 7.3 System Types and Steady-State Error (1)
Steady-state error for unity feedback systems
For nonunity feedback systems

15. § 7.3 System Types and Steady-State Error (2)
Fundamental Regulation and Tracking Error
Regulation s.s. error
Tracking s.s. error

16. § 7.3 System Types and Steady-State Error (3)
Open-loop System Types

17. § 7.3 System Types and Steady-State Error (4)
Position Control of Mechanical Systems
(1) Command signal
Region 1 and 3: Constant acceleration and deceleration
Region 2: Constant speed
Region 4: Constant position
(2) Error constants

18. § 7.3 System Types and Steady-State Error (5)
(3) Systems control with non-zero steady-state position error
Constant position for Type 0 system
Constant velocity for Type 1 system
Constant acceleration for Type 2 system

19. § 7.3 System Types and Steady-State Error (6)
Steady-state position errors for different types of system and input signal
Output positioning in feedback control is driven by the dynamic positional error.
System nonlinearities such as friction, dead zone, quantization will introduce steady-state error in closed-loop position control.

20. § 7.3 System Types and Steady-State Error (7)
Ex: Find the value of K such that there is 10% error in the steady state
Sol: System G(s) is Type 1
s.s. error in ramp input
For velocity error constant

21. § 7.4 Time-Domain Performance Indices (1)
Performance of Control System
Stability
Transient Response
Steady-state Error
Performance Indices (PI)
A scalar function for quantitative measure of the performance specifications of a control system.
error
command
state
output
Use P.I. To trade off transient response and steady-state error with sufficient stability margin.

22. § 7.4 Time-Domain Performance Indices (2)
Systems Control
(1) Classical control
Plant: Input-Output Model
Controller: PID Control
P-Proportional control:
I-Integral control:
D-Differential control:
(2) Modern control
Plant: State-space Model
P. I.: Usually Quadratic functional
Controller: States feedback control

23. § 7.4 Time-Domain Performance Indices (3)
Optimal Control
Given: Plant model
Control configuration (Usually feedback)
Controller structure (Usually linear)
Design constraints
Objective: Minimize P. I. (P. I. to be selected)
Find: Optimal parameters in controller
Ex: Design optimal proportional control system
Find optimal K to minimize P. I.