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
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 9-5-3 a1 = 70 and T1 = 0.00004 Phase-lag control:
Section 9- 7, p. 575 Example 9-7-1 (cont.)
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:
Inexact Pole-Zero Cancellation Section 9- 8, p. 577 Inexact Pole-Zero Cancellation K1 is proportional to 11, 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.
Section 9- 8, p. 577
Section 9- 8, p. 578
Second-Order Active Filter Section 9- 8, p. 579 Second-Order Active Filter
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
Section 9- 8, p. 581 Example 9-8-1
Example 9-8-1 (cont.) Loop transfer function: Section 9- 8, p. 582 Example 9-8-1 (cont.) Loop transfer function: Resonant frequency 1095 rad/sec The closed-loop system is unstable.
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:
Example 9-8-1 (cont.) G(s): type-0 system: Step-error constant: Section 9- 8, p. 583 Example 9-8-1 (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 = 0.1879 sec Settling time ts = 0.256 sec
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.
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:
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 = 0.005 and KI/KP = 20 KI = 0.1 maximum overshoot = 1% rise time tr = 0.1380 sec settling time ts = 0.1818 sec
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 = 0.2323 sec Notch Controller
Example 9-8-1: Unit-Step Responses Section 9- 8, p. 585 Example 9-8-1: Unit-Step Responses
Example 9-8-1: Freq.-Domain Design Section 9- 8, p. 586 Example 9-8-1: Freq.-Domain Design attenuation = 44.86 dB
Example 9-8-1: Notch Controller Section 9- 8, p. 586 Example 9-8-1: Notch Controller PM = 13.7° Mr = 3.92
Section 9- 8, p. 587 Example 9-8-1 (cont.)
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)
Section 9- 8, p. 588 Example 9-8-1 (cont.)
9-9 Forward and Feedforward Controllers Section 9- 9, p. 588 9-9 Forward and Feedforward Controllers Forward compensation:
Section 9- 9, p. 589 Example 9-9-1 Second-order sun-seeker with phase-lag control (Ex. 9-6-1): Time-response attributes: maximum overshoot = 2.5%, tr = 0.1637 sec, ts = 0.2020 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 = 0.1069, ts = 0.1313
Feedforward controller Section 9- 9, p. 590 Example 9-9-1 (cont.) Forward controller Feedforward controller
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
Section 9- 10, p. 591 Sensitivity Disturbance suppression and robustness with respect to variations of K can be designed with the same control scheme.
Section 9- 10, p. 591 Example 9-10-1 Second-order sun-seeker with phase-lag control (Ex. 9-6-1) 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
Section 9- 10, p. 592
Section 9- 10, p. 593
Section 9- 10, p. 594 Example 9-10-1 (cont.) Design strategy: place two zeros of the robust controller near the desired close-loop poles According to the phase-lag-compensated system, s = 12.455 j9.624 Transfer function of the controller: Transfer function of the system with the robust controller:
Section 9- 10, p. 595
Section 9- 10, p. 596 Example 9-10-1 (cont.)
Section 9- 10, p. 596
Section 9- 10, p. 597 Example 9-10-1 (cont.)
Section 9- 10, p. 597
Section 9- 10, p. 597 Example 9-10-2 Third-order sun-seeker with phase-lag control (Ex. 9-6-2) Phase-lag controller: a = 0.1 and T = 20 (Table 9-19) roots of characteristic equation: s = 187.73 j164.93 Place the two zeros of the robust controller at 180 j166.13 Forward controller:
Section 9- 10, p. 598 Example 9-10-2 (cont.)
Section 9- 10, p. 599 Example 9-10-2 (cont.)
Section 9- 10, p. 599 Example 9-10-3 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
Section 9- 10, p. 600 Example 9-10-3 (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:
Section 9- 10, p. 600 Example 9-10-3 (cont.)
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.
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+Ktn)/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.
Section 9- 11, p. 603 Example 9-11-1 Second-order sun-seeker system:
Example 9-11-1 Characteristic equation: Section 9- 11, p. 604 Example 9-11-1 Characteristic equation: Kt = 0.02: maximum overshoot = 0 tr = 0.04485 sec ts = 0.06061 sec tmax = 0.4 sec
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
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
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:
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:
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
Rectangular valve-port geometry Section 9- 12, p. 609 Rectangular valve-port geometry For the rectangular geometry: For the open-center valve:
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
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:
Transfer Function of Two-Stage Valve Section 9- 12, p. 611 Transfer Function of Two-Stage Valve
Modeling the Hydraulic System Section 9- 12, p. 612 Modeling the Hydraulic System
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:
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:
Transfer Func. of Translational System Section 9- 12, p. 615 Transfer Func. of Translational System
Applications: Rotational System Section 9- 12, p. 616 Applications: Rotational System
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:
Applications: Variable Load Section 9- 12, p. 617 Applications: Variable Load
Section 9- 13, p. 617 9-13 Control Design P Control:
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:
P Control: root loci & responses Section 9- 13, p. 619 P Control: root loci & responses
Section 9- 13, p. 621 PD Control PD control Closed transfer function:
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 0.00061 KP 5 Damping ratio for KP = 5: = 1 KD = 0.0066 Setting time: ts 0.005 KD 0.0044 Stability requirement: KP 0 and KD 0.00307
PD Control: root loci & root contour Section 9- 13, p. 622 PD Control: root loci & root contour Characteristic equation (KD = 0):
Section 9- 13, p. 623 PD Control: responses
Section 9- 13, p. 626 PI Control PI control
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
PI Control: root loci & responses Section 9- 13, p. 627 PI Control: root loci & responses
PI Control: attributes Section 9- 13, p. 627 PI Control: attributes PI
Section 9- 13, p. 628 PID Control Transfer function:
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 = 0.0066, KP1 = 5 PI controller: KI2/KP2 = 5
PID Control: attributes Section 9- 13, p. 629 PID Control: attributes
PID Control: responses Section 9- 13, p. 630 PID Control: responses