117. 9-7 Design with Lead-Lag Controller
Section 9-
7, p. 574
9-7 Design with Lead-Lag Controller
Transfer function of a simple lead-lag (or lag-lead) controller:
The phase-lead portion is used mainly to achieve a shorter rise time and higher bandwidth, and the phase-lag portion is brought in to provide major damping of the system.
Either phase-lead or phase-lag control can be designed first.
lead
lag

118. Example 9-7-1: Sun-Seeker System
Section 9-
7, p. 575
Example 9-7-1: Sun-Seeker System
Example 9-5-3: two-stage phase-lead controller design
Example 9-6-1: two-stage phase-lag controller design
Phase-lead control: From Example  a1 = 70 and T1 =
Phase-lag control:

119. Section 9-
7, p. 575
Example (cont.)

120. 9-8 Pole-Zero-Cancellation Design: Notch Filter
Section 9-
8, p. 576
9-8 Pole-Zero-Cancellation Design: Notch Filter
The complex-conjugate poles, that are very close to the imaginary axis of the s-plane, usually cause the closed-loop system to be slightly damped or unstable.
 Use a controller to cancel the undesired poles
Inexact cancellation:

121. Inexact Pole-Zero Cancellation
Section 9-
8, p. 577
Inexact Pole-Zero Cancellation
K1 is proportional to 11, which is a very smaller number. Similarly, K2 is also very small.
Although the poles cannot be canceled precisely, the resulting transient-response terms will have insignificant amplitude, so unless the controller earmarked for cancellation are too far off target, the effect can be neglected for all practical purpose.

122. Section 9-
8, p. 577

123. Section 9-
8, p. 578

124. Second-Order Active Filter
Section 9-
8, p. 579
Second-Order Active Filter

125. Frequency-Domain Interpretation
Section 9-
8, p. 580
Frequency-Domain Interpretation
“notch” at the resonant frequency n.
Notch controller do not affect the high- and low-frequency properties of the system

126. Section 9-
8, p. 581
Example 9-8-1

127. Example 9-8-1 (cont.) Loop transfer function:
Section 9-
8, p. 582
Example (cont.)
Loop transfer function:
Resonant frequency  1095 rad/sec
The closed-loop system is unstable.

128. Example 9-8-1: Pole-Zero Cancellation
Section 9-
8, p. 582
Example 9-8-1: Pole-Zero Cancellation
Pole-Zero-Cancellation Design with Notch Controller
Performance specifications:
The steady-state speed of the load due to a unit-step input should have an error of not more than 1%
Maximum overshoot of output speed  5%
Rise time  0.5 sec
Settling time  0.5 sec
Notch controller:
 to cancel the undesired poles 47.66  j1094
The compensated system:

129. Example 9-8-1 (cont.) G(s): type-0 system: Step-error constant:
Section 9-
8, p. 583
Example (cont.)
G(s): type-0 system:
Step-error constant:
Steady-state error:
ess  1%  KP  99
Let n = 1200 rad/sec and p = 15,000
Maximum overshoot = 3.7%
Rise time tr = sec
Settling time ts = sec

130. Example 9-8-1: Two Stage Design
Section 9-
8, p. 583
Example 9-8-1: Two Stage Design
Choose n = 1000 rad/sec and p = 10 the forward-path transfer function of the system with the notch controller:  maximum overshoot = 71.6%
Introduce a phase-lag controller or a PI controller to Eq. (9-167) to meet the design specification given.

131. Example 9-8-1: Phase-Lag Controller
Section 9-
8, p. 584
Example 9-8-1: Phase-Lag Controller
Second-Stage Phase-Lag Controller Design
Phase-lag controller:

132. Example 9-8-1: PI Controller
Section 9-
8, p. 584
Example 9-8-1: PI Controller
Second-Stage PI Controller Design
PI controller:
Phase-lag controller (9-169) 
KP = and KI/KP = 20  KI = 0.1 maximum overshoot = 1% rise time tr = sec settling time ts = sec

133. Example 9-8-1: Pole-Zero Cancellation
Section 9-
8, p. 585
Example 9-8-1: Pole-Zero Cancellation
Sensitivity due to Imperfect Pole-Zero Cancellation
Transfer function: maximum overshoot = 0.4% rise time tr = 0.17 sec settling time ts = sec
Notch Controller

134. Example 9-8-1: Unit-Step Responses
Section 9-
8, p. 585
Example 9-8-1: Unit-Step Responses

135. Example 9-8-1: Freq.-Domain Design
Section 9-
8, p. 586
Example 9-8-1: Freq.-Domain Design
attenuation =  dB

136. Example 9-8-1: Notch Controller
Section 9-
8, p. 586
Example 9-8-1: Notch Controller
PM = 13.7°
Mr = 3.92

137. Section 9-
8, p. 587
Example (cont.)

138. Example 9-8-1: Notch-PI Controller
Section 9-
8, p. 586
Example 9-8-1: Notch-PI Controller
Desired PM = 80°
New gain-crossover frequency:
(9.32) 
(9.25) 

139. Section 9-
8, p. 588
Example (cont.)

140. 9-9 Forward and Feedforward Controllers
Section 9-
9, p. 588
9-9 Forward and Feedforward Controllers
Forward compensation:

141. Section 9-
9, p. 589
Example 9-9-1
Second-order sun-seeker with phase-lag control (Ex ):
Time-response attributes:
maximum overshoot = 2.5%, tr = sec, ts = sec
Improve the rise time and the settling time while not appreciably increasing the overshoot  add a PD controller Gcf(s) to the system (forward) add a zero to the closed-loop transfer function while not affecting the characteristic equation  maximum overshoot = 4.3%, tr = , ts =

142. Feedforward controller
Section 9-
9, p. 590
Example (cont.)
Forward controller
Feedforward controller

143. 9-10 Design of Robust Control Systems
Section 9-
10, p. 590
9-10 Design of Robust Control Systems
Control-system application: 1. the system must satisfy the damping and accuracy specifications. 2. the control must yield performance that is robust (insensitive) to external disturbance and parameter variations
d(t) = 0 
r(t) = 0 

144. Section 9-
10, p. 591
Sensitivity
Disturbance suppression and robustness with respect to variations of K can be designed with the same control scheme.

145. Section 9-
10, p. 591
Example
Second-order sun-seeker with phase-lag control (Ex )
Phase-lag controller  low-pass filter the sensitivity of the closed-loop transfer function M(s) with respect to K is poor
a = 0.1 T = 100

146. Section 9-
10, p. 592

147. Section 9-
10, p. 593

148. Section 9-
10, p. 594
Example (cont.)
Design strategy: place two zeros of the robust controller near the desired close-loop poles
According to the phase-lag-compensated system, s =  j9.624
Transfer function of the controller:
Transfer function of the system with the robust controller:

149. Section 9-
10, p. 595

150. Section 9-
10, p. 596
Example (cont.)

151. Section 9-
10, p. 596

152. Section 9-
10, p. 597
Example (cont.)

153. Section 9-
10, p. 597

154. Section 9-
10, p. 597
Example
Third-order sun-seeker with phase-lag control (Ex )
Phase-lag controller: a = 0.1 and T = 20 (Table 9-19)  roots of characteristic equation: s =   j164.93
Place the two zeros of the robust controller at 180  j 
Forward controller:

155. Section 9-
10, p. 598
Example (cont.)

156. Section 9-
10, p. 599
Example (cont.)

157. Section 9-
10, p. 599
Example
Design a robust system that is insensitive to the variation of the load inertia.
Performance specifications: 0.01  J  0.02
Ramp error constant Kv  200
Maximum overshoot  5%
Rise time tr  0.05 sec
Settling time ts  0.05 sec
s = 50  j86.6
s = 50  j50

158. Section 9-
10, p. 600
Example (cont.)
Place the two zeros of the robust controller at 55  j45 
K = 1000 and J = 0.01: K = 1000 and J = 0.02:
Forward controller:

159. Section 9-
10, p. 600
Example (cont.)

160. 9-11 Minor-Loop Feedback Control
Section 9-
11, p. 601
9-11 Minor-Loop Feedback Control
Rate-Feedback or Tachometer-Feedback Control
Transfer function:
Characteristic equation:
The effect of the tachometer feedback is the increasing of the damping of the system.

161. Steady-State Analysis
Section 9-
11, p. 602
Steady-State Analysis
Forward-path transfer function:  type 1 system
For a unit-ramp function input: tachometer feedback  ess = (2+Ktn)/n PD control  ess = 2/n
For a type 1 system, tachometer feedback decrease the ramp-error constant Kv but does not affect the step-error constant KP.

162. Section 9-
11, p. 603
Example
Second-order sun-seeker system:

163. Example 9-11-1 Characteristic equation:
Section 9-
11, p. 604
Example
Characteristic equation:
Kt = 0.02: maximum overshoot = 0 tr = sec ts = sec tmax = 0.4 sec

164. 9-12 A Hydraulic Control System
Section 9-
12, p. 605
9-12 A Hydraulic Control System
double-acting single rod linear actuator
two-stage electro-hydraulic valve

165. Modeling Linear Actuator
Section 9-
12, p. 606
Modeling Linear Actuator
The applied force: f: the force efficiency of the actuator
Volumetric efficiency:
for an ideal case 

166. Four-Way Electro-Hydraulic Valve
Section 9-
12, p. 606
Four-Way Electro-Hydraulic Valve
Two-stage control valve:
The first stage is an electrically actuated hydraulic valve, which controls the displacement of the spool of the second stage of the valve.
The second stage is a four-way spool valve, which controls the fluid flow and pressure into and out of ports A and B of the actuators.
At the nominal operating conditions:

167. Orifice Equation The classic orifice equation for the fluid flow:
Section 9-
12, p. 607
Orifice Equation
The classic orifice equation for the fluid flow:
The discharge coefficient:

168. Liberalized Flow Equations for Four-Way Valve
Section 9-
12, p. 608
Liberalized Flow Equations for Four-Way Valve
Orifice equation:
Taylor series: Take x0 = 0 and  Kq: flow gain Kc: pressure-flow coefficient

169. Rectangular valve-port geometry
Section 9-
12, p. 609
Rectangular valve-port geometry
For the rectangular geometry:
For the open-center valve:

170. Equations for Four-Way Valve
Section 9-
12, p. 610
Equations for Four-Way Valve
Volumetric flow rates into and out of the actuator: for critical centered valves 

171. Input Voltage & Main Spool Displacement
Section 9-
12, p. 611
Input Voltage & Main Spool Displacement
Flow equation of pilot spool: (the control valve is critically centered)
Assuming an incompressible fluid:
Displacement of pilot spool:

172. Transfer Function of Two-Stage Valve
Section 9-
12, p. 611
Transfer Function of Two-Stage Valve

173. Modeling the Hydraulic System
Section 9-
12, p. 612
Modeling the Hydraulic System

174. Mathematical Equations
Section 9-
12, p. 612
Mathematical Equations
Force balance equation for an ideal linear actuator:
Expressing pressure level PA and pressure level PB:
Expressing the pressure difference two sides of linear actuator:
General equation:

175. Applications: Translational Motion
Section 9-
12, p. 614
Applications: Translational Motion
The voltage fed back from the actuator displacement z:
The input voltage to the two-stage valve Verror:
Desired input zdesired and desired voltage Vdesired:
General equation:
Transfer function of two-stage valve:

176. Transfer Func. of Translational System
Section 9-
12, p. 615
Transfer Func. of Translational System

177. Applications: Rotational System
Section 9-
12, p. 616
Applications: Rotational System

178. Transfer Function of Rotational System
Section 9-
12, p. 615
Transfer Function of Rotational System
The translational displacement of the rod in terms of the angular displacement:
Main valve displacement:
System transfer function:

179. Applications: Variable Load
Section 9-
12, p. 617
Applications: Variable Load

180. Section 9-
13, p. 617
9-13 Control Design
P Control:

181. P Control: transfer function
Section 9-
13, p. 618
P Control: transfer function
Simplified hydraulic system transfer function: (neglecting the pole at 142350)
Apply a P controller: closed-loop transfer function:

182. P Control: root loci & responses
Section 9-
13, p. 619
P Control: root loci & responses

183. Section 9-
13, p. 621
PD Control
PD control
Closed transfer function:

184. PD Control: design Steady-state error for a unit-ramp input:
Section 9-
13, p. 622
PD Control: design
Steady-state error for a unit-ramp input:
ess   KP  5
Damping ratio for KP = 5:
 = 1  KD =
Setting time: ts   KD 
Stability requirement: KP  0 and KD  

185. PD Control: root loci & root contour
Section 9-
13, p. 622
PD Control: root loci & root contour
Characteristic equation (KD = 0):

186. Section 9-
13, p. 623
PD Control: responses

187. Section 9-
13, p. 626
PI Control
PI control

188. Section 9-
13, p. 626
PI Control: design
Steady-state error for a ramp input: ess  0.2  KI  0.015
Characteristic equation: stable 
KI/KP = 5 

189. PI Control: root loci & responses
Section 9-
13, p. 627
PI Control: root loci & responses

190. PI Control: attributes
Section 9-
13, p. 627
PI Control: attributes PI

191. Section 9-
13, p. 628
PID Control
Transfer function:

192. PID Control: design PD controller: Table 9-28  KD1 = 0.0066, KP1 = 5
Section 9-
13, p. 629
PID Control: design
PD controller: Table 9-28  KD1 = , KP1 = 5
PI controller: KI2/KP2 = 5 

193. PID Control: attributes
Section 9-
13, p. 629
PID Control: attributes

194. PID Control: responses
Section 9-
13, p. 630
PID Control: responses