Illustrations Chapter 5 – The Performance of Feedback Control Systems The ability to adjust the transient and steady-state response of a feedback control.

Illustrations

Chapter 5 – The Performance of Feedback Control Systems

The ability to adjust the transient and steady-state response of a feedback control system is a beneficial outcome of the design of control systems.

One of the first steps in the design process is to specify the measures of performance.

In this chapter we introduce the common time-domain specifications such as percent overshoot, settling time, time to peak, time to rise, and steady-state tracking error.

We will use selected input signals such as the step and ramp to test the response of the control system.

The correlation between the system performance and the location of the system transfer function poles and zeros in the s-plane is discussed.

We will develop valuable relationships between the performance specifications and the natural frequency and damping ratio for second-order systems.

Relying on the notion of dominant poles, we can extrapolate the ideas associated with second-order systems to those of higher order.

Illustrations

Introduction

Steady-State: exists a long time following any input signal initiation

Transient Response: disappears with time

Design Specifications: normally include several time-response indices for a specified input command as well as a desired steady-state accuracy.

Illustrations

Test Input Signals

A unit impulse function is also useful for test signal purposes. It’s characteristics are shown to the right.

Illustrations

Performance of a Second-Order System

Y s( )K

s2

p s KR s( )

Y s( )n

2

s2

2 n s n2

1 2

with a unity step input

cos1

Y s( )n

2

s2

2 n s n2

s

y t( ) 11

e

n t sin n t

Illustrations

Performance of a Second-Order System

Illustrations

Performance of a Second-Order System

Illustrations

Performance of a Second-Order System

Rise Time, TrPeak Time, ToPercentage Overshoot, P.O.Settling Time, TsNormalized Rise Time Tr1

Illustrations

Performance of a Second-Order System

Standard performance measures are usually defined in terms of the step response of a system. The transient response of a system may be described using two factors, the swiftness and the closeness of the response to the desired response.

The swiftness of the response is measured by the rise time (Tr) and the peak time (Tp).

Underdamped systems: 0-100% rise time is usedOverdamped systems: 10-90% rise time is used

The closeness is measured by the overshoot and settling time. Using these measurements the percent overshoot (P.O.) can be calculated.

Illustrations

Performance of a Second-Order System

POMpv fv

fv100

Ts4

n

Tp

n 1 2

Mpv 1 e

1 2

PO 100e

1 2

Illustrations

Performance of a Second-Order System

Naturally these two performance measures are in opposition and a compromise must be made.

Illustrations

Performance of a Second-Order System

Illustrations

Performance of a Second-Order System

Illustrations

Performance of a Second-Order System

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

T s( )1

s2 2 s 1 s 1

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

Example 5.1 - Parameter Selection

Select the gain K and the parameter p so that the percent overshoot is less than 5% and the settling time (within 2% of the final value) should be less than 4 seconds.

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

Example 5.1 - Parameter Selection

Ts4

n4sec

n 1

When the closed-loop roots are chosen as:

r1 1 j 1r2 1 j 1

We have Ts 4sec and an overshoot of 4.3%.

Therefore, 1

2and n

1

2

T s( )G s( )

1 G s( )K

s2 p s K

n2

s2 2 n s n2

K n2

2 P 2 n 2

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

Example 5.2 – Dominant Poles of T(s)

Y s( )

R s( )T s( )

n2

as a( )

s2 2 n s n2 1 s

For n 3, 0.16 , and a 2.5:

T s( )62.5 s 2.5( )

s2 6 s 25 s 6.25( )

Illustrations

Effects of a Third Pole and Zero on the Second-Order System

Example 5.2 – Dominant Poles of T(s)T s( )62.5 s 2.5( )

s2 6 s 25 s 6.25( )

As a first approximation, we neglect the real pole and obtain:

T s( )10 s 2.5( )

s2 6s 25 We now have 0.6 and n 5 for dominant poles with one

accompanying zero for which a

n0.833

Using the previously mentioned charts (Figure 5.13a), we find that the percent overshoot is 55%. We expect the setting time to within 2% of the final value to be:

T s( )4

n

4

0.6 51.33sec

Using computer simulations the actual percent overshoot is equal to 38% and the settling time is 1.6 seconds.

Thus, the effect of the damping of the third pole of T(s) is to dampen the overshoot and increase the settling time (hence the real pole cannot be neglected.

Illustrations

The s-Plane Root Location and The Transient Response

Illustrations

Steady-State Error of Feedback Control Systems

For

Step Input - Position Error Constant

Ramp Input - Velocity Error Constant

Acceleration Input - Acceleration Error Constant

Illustrations

The Steady-State Error of Nonunity Feedback Systems

For a system in which the feedback is not unity (Fig 5.21), the units of the output are usually different from the output of the sensor. In Fig. 5.22, K1 and K2 convert from rad/s to volts.

Illustrations

The Steady-State Error of Nonunity Feedback Systems

T s( )K1 G s( )

1 K1 G s( )

E s( ) R s( ) Y s( ) 1 T s( )( ) R s( ) ess0s

s E s( )lim

1

1 K1 G 0( )

E s( )1

1 K1 G s( )R s( )

Illustrations

Performance Indices

A performance index is a quantitative measure of the performance of a system and is chosen so that emphasis is given to the important system specifications.

A system is considered an optimum control system when the system parameters are adjusted so that the index reaches an extremum value, commonly a minimum value.

Illustrations

Performance Indices

There are several performance indices:

(1) Integral of the square of the error, ISE

(2) Integral of the absolute magnitude of the error, IAE

(3) Integral of time multiplied by absolute error, ITAE

(4) Integral of time multiplied by the squared error, ITSE

ISE0

T

te2 t( )

d

IAE0

T

te t( )

d

ITAE0

T

tt e t( )

d

ITSE0

T

tt e2 t( )

d

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

System Performance Using MATLAB and Simulink

Illustrations

Exercises and Problems

Chapter 5 – E5.5, E5.16, DP5.4 – Select 3 more problems of your choice. Submit One Set of Multiple Choices, and Matching Concepts