R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition
x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form
z U 2 s z Y 4 z 2 s z 2 3 Figure 7.3 Block diagram for Eq. (7.) in modal canonical form
x x x 2 s s U s x 3 x 4 s Y x 3 2 4 Figure 7.4 Block diagram for a fourth-order system in modal canonical form with shading indicating portion in control canonical form
U 2 x 2 s x 2 x s x Y 2 7 Figure 7.5 Observer canonical form
Plant u x Fx Gu x H Y Control law K xˆ Estimator R Matrix of constants State vector estimate Compensation Figure 7.6 Schematic diagram of state-space design elements
u x Fx Gu x H Y u Kx Figure 7.7 Assumed system for control-law design
.0 0.8 0.6 x Amplitude 0.4 0.2 0 0.2 0.4 0.6 0.8 0 u/4 x 2 2 3 4 5 6 7 Time (sec) Figure 7.8 Impulse response of the undamped oscillator with full-state feedback (ω 0 = )
U a(s) b(s) Y (a) U s s s a a 2 a 3 (b) s x c s x 2c s x 3c b 3 b b 2 Y U a a 2 a 3 (c) Figure 7.9 Derivation of control canonical form.
R N u K u Plant x Y R N K u Plant x Y N x (a) (b) Figure 7.0 Block diagram for introducing the reference input with full-state feedback: (a) with state and control gains; (b) with a single composite gain
.0 0.8 x x ss 0.6 Amplitude 0.4 x 2 u ss 0.2 0 u/4 0.2 0 2 3 4 5 6 7 Time (sec) Figure 7. Step response of oscillator to a reference input
R K u x 2 x s s K 2 (a) R N u x 2 x s s K 2 K (b) Figure 7.2 Alternative structures for introducing the reference input. (a) Eq. (7.90); (b) Eq. (7.9)
x 3.2.0 LQR Tape position 0.8 0.6 0.4 Dominant second-order 0.2 0 0 2 4 6 8 0 2 Time (msec) Figure 7.3 Step responses of the tape servomotor designs
T 0.02 0.0 LQR Tape tension 0.02 0.04 0.06 Dominant second-order 0.08 0.0 0.2 0 2 4 6 8 0 2 Time (msec) Figure 7.4 Tension plots for tape servomotor step responses
Im(s) r 0 r 0 r 0 r 0 a a Re(s) Figure 7.5 Symmetric root locus for a first-order system
Im(s) j r 0 Re(s) j Figure 7.6 Symmetric root locus for the satellite.
25 r 0.0 20 z 2 dt 0 5 0 5 r r 00 0 0 5 0 5 20 25 0 u 2 dt Figure 7.7 Design trade-off curve for satellite plant
.5.0 Imaginary axis 0.5 0 0.5.0.5 2.5 2.0.5.0 0.5 0 0.5.0 Real axis Figure 7.8 Nyquist plot for LQR design
Im(s) 3 2 4 3 2 2 3 4 Re(s) 2 3 Figure 7.9 Symmetric root locus for the inverted pendulum
.0 0.9 0.8 0.7 Position, x 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.5.5 2 2.5 3 3.5 4 Time (sec) 4.5 Figure 7.20 Step response for the inverted pendulum
.4.2 r 0 r Tape position, x 3.0 0.8 0.6 0.4 0.2 r 0.0 0 0 2 4 6 8 0 Time (msec) 2 (a) 0.0 Tape tension, T 0.05 0 0.05 0.0 0.5 r 0 r 0.20 r 0.0 0.25 0 2 4 6 8 0 Time (msec) 2 (b) Figure 7.2 (a) Step response of the tape servomotor for LQR designs, (b) Corresponding tension for tape servomotor step responses
u Process (F, G) x H y Model (F, G) xˆ H ŷ Figure 7.22 Open-loop estimator
u(t) Process (F, G) Model (F, G) x(t) ˆx(t) H H ˆ y(t) y(t) L Figure 7.23 Closed-loop estimator
r N u Plant x Fx Gu y Hx x y u ~y Estimator x Fx Gu Ly ~ ˆ ˆ xˆ H ŷ K Figure 7.24 Estimator connected to the plant
3.5 3.0 2.5 2.0 ˆ x 2 Amplitude.5.0 0.5 x ˆ x 0 0.5 x 2.0 0 0.5.0.5 2.0 2.5 3.0 3.5 4.0 Time (sec) Figure 7.25 Initial-condition response of oscillator showing x and ˆx
U b 3 b 2 b s x 3o s x 2o s x o Y a 3 a 2 a Figure 7.26 Block diagram for observer canonical form of a third-order system
y F ba LF aa L u G b LG a ˆ x b Ly x c s ˆ x b Ly x c ˆ x b F bb LF ab Figure 7.27 Reduced-order estimator structure
0 8 6 Amplitude 4 2 ˆ x 2 0 x x 2 2 0 0.5.0.5 2.0 2.5 3.0 3.5 4.0 Time (sec) Figure 7.28 Initial-condition response of the reduced-order estimator
Imaginary axis 5 4 3 2 0 2 3 4 5 q 0 q 0 5 4 3 2 0 2 3 4 5 Real axis Figure 7.29 Symmetric root locus for the inverted pendulum estimator design
w v Plant Sensor u(t) x Fx Gu x(t) H y(t) Control law K u(t) ˆ x(t) Compensator ˆ Estimator ˆ x Fx Gu L(y Hx) ˆ Figure 7.30 Estimator and controller mechanization
Im(s) 6 K 40.4 K 40.4 4 2 8 6 4 2 Re(s) 2 4 6 Figure 7.3 Root locus for the combined control and estimator, with process gain as the parameter
00 40 Phase Magnitude 0 0. 0.0 20 50 80 20 240 270 0. 0 00 v (rad/sec) 53 0.02 0.04 0. 0.2 0.4 0.6 2 4 6 0 20 40 60 00 v (rad/sec) Figure 7.32 Frequency response for G(s) = /s 2 Compensated Uncompensated 20 0 20 db
Y 8.07 0.69 s U 6.4 Figure 7.33 Simplified block diagram of a reduced-order controller that is a lead network
Im(s) 4 2 8 6 4 2 Re(s) Figure 7.34 Root locus of a reduced-order controller and /s 2 process, root locations at K = 8.07 shown by the dots
00 40 Phase Magnitude 0 0. 0.0 20 50 80 0 20 0. 0 00 v (rad/sec) Compensated 55 Uncompensated 20 db 20 0.02 0.04 0. 0.2 0.4 0.6 2 4 6 8 0 20 40 60 00 v (rad/sec) Figure 7.35 Frequency response for G(s) = /s 2 with a reduced-order estimator
U 0 s s s Y 6 0 Figure 7.36 DC Servo in observer canonical form
Im(s) 8 6 4 8 6 4 2 2 Re(s) 4 6 8 Figure 7.37 Root locus for DC Servo pole assignment
Im(s) 6 5 4 3 8 7 6 5 4 2 2 3 Re(s) 3 4 5 Figure 7.38 Root locus for DC Servo reduced-order controller
Im(s) 5 4 3 2 9 8 7 6 5 4 3 2 Controller poles 2 Estimator poles 3 4 5 2 3 4 5 6 7 8 Re(s) Figure 7.39 Symmetric root locus
Im(s) 7 6 4 3 2 0 6 4 2 2 Re(s) 2 Controller poles Estimator poles 3 4 6 7 Figure 7.40 Root locus for pole assignment from the SRL
Continuous controller Plant Step 94.5s 2 992.25s 900.3572 s 3 9.6s 2 50.27s 63.06 0 s 3 0s 2 6s Mux Control Mux Output Discrete controller 5.957z 2 7.2445z 2.0782 z 3.3905z 2 0.7866z 0.472 Plant 0 s 3 0s 2 6s Figure 7.4 Simulink block diagram to compare continuous and discrete controllers
.4.2.0 0.8 Digital controller Continuous controller y 0.6 0.4 0.2 0 0 2 3 4 Time (sec) 5 (a) u 7 6 5 4 3 2 0 Continuous controller 2 0 Digital controller 2 3 4 Time (sec) (b) 5 Figure 7.42 Comparison of step responses and control signals for continuous and discrete controllers: (a) step responses, (b) control signals
Magnitude 0 0 0 5 GM GM GM 2 LQR q 00 q 0 q 0 0 0 0 0 2 0 3 v (rad/sec) Phase (deg) 90 20 50 80 20 240 270 PM PM q LQR q 0 0 0 0 0 0 2 0 3 v (rad/sec) q 00 Figure 7.43 Frequency response plots for LTR design
R D c (s) G(s) Y V Figure 7.44 Closed-loop system for LTR
To workspace4 u Scope2 State-space controller x Ax Bu y Cx Du Integrator s Integrator s Band-limited white noise Scope To workspace3 v Figure 7.45 Simulink block diagram for LTR
Plant r N u y K xˆ Estimator Compensator (a) r e Estimator ˆ x K u Plant y Compensator (b) Figure 7.46 Possible locations for introducing the command input: (a) compensation in the feedback path; (b) compensation in the feedforward path
r N u M Plant y r N u Plant y K xˆ Estimator K xˆ Estimator (a) (b) Plant u y K xˆ Estimator e r (c) Figure 7.47 Alternative ways to introduce the reference input: (a) general case zero assignment; (b) standard case estimator not excited, zeros = α e (s); (c) error-control case classical compensation
R 0.8 e 8.32 s x c u s(s ) Y 3. 3.02 (a) R e (s )(8.32s 0.8) (s 4.08)(s 0.096) u s(s ) Y (b) Figure 7.48 Servomechanism with assigned zeros (a lag network): (a) the two-input compensator; (b) equivalent unity feedback system
0.2 0.08 0.24 0.6 0.08 0.08 0.2 Im(s) 4 3 2 Re(s) Figure 7.49 Root locus of lag-lead compensation
000 00 Magnitude 0 0. 0.0 0.00 0.000 0.0 0. 0.4 2 460 40 00 v (rad/sec) (a) 90 Phase 20 50 80 0.0 0. 0.4 2 4 6 0 40 v (rad/sec) 00 (b) Figure 7.50 Frequency response of lag-lead compensation
.2 y.0 0.8 0.6 0.4 0.2 0 0 2 3 4 5 Time (sec) Figure 7.5 Step response of the system with lag compensation
Continuous controller Plant Step 8.32s 2 9.2s 0.8 s 2 4.0996s 0.08 s 2 s Mux Control Mux Output Discrete controller 8.32z 2 5.8848z 7.574 z 2.6630z 0.6637 Plant s 2 s Figure 7.52 Simulink block diagram to compare continuous and discrete controllers
.4.2 Digital controller.0 y 0.8 0.6 0.4 Continuous controller 0.2 0 0 2 3 4 Time (sec) 5 (a) 0 8 6 u 4 2 0 2 0 Continuous controller Digital controller 2 3 4 Time (sec) (b) 5 Figure 7.53 Comparison of step responses and control signals for continuous and discrete controllers: (a) step responses, (b) control signals
r e x I K u s K 0 Plant x y Figure 7.54 Integral control structure
Plant w r e x I s 25 s 3 y 7 Estimator xˆ Figure 7.55 Integral control example
y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. y y 2 0.0 0 0.5 0.5.5 2 2.5 3 3.5 4 4.5 Time (sec) (b) 0.0 0.0 u 0.0 0.0 u 0.0 0.0 0.0 0.0 u 2 0 0.5 0.5.5 2 2.5 3 3.5 4 4.5 Time (sec) (b) Figure 7.56 Transient response for motor speed system:(a) step responses, (b) control efforts
r e K u s Plant x y K 0 Figure 7.57 Integral control using the internal model approach
r e Compensator K 2 K Plant s s u s(s ) y v 0 2 Internal model K 0 x K 02 x 2 Figure 7.58 Structure of the compensator for the servomechanism to track exactly the sinusoid of frequency ω 0
Phase (deg) Magnitude (db) 250 200 50 00 50 0 0 2 0 0 0 0 v (rad/sec) 00 50 0 50 00 50 200 0 2 0 0 0 0 v (rad/sec) Figure 7.59 Controller frequency response
Phase (deg) Magnitude (db) 50 0 50 00 50 200 0 0 0 0 0 2 v (rad/sec) 300 250 200 50 00 0 0 0 0 0 2 v (rad/sec) Figure 7.60 Sensitivity function frequency response
To workspace r To workspace2 e To workspace y Scope2 Scope r e u x2 x 6.392 Integrator s s s Gain 2.078 s Integrator Integrator2 Gain5 Integrator3 Out Scope Gain Gain4 4.464 Gain2 3.928 Gain3 Figure 7.6 Simulink block diagram for robust servomechanism
.5 r Reference, output 0.5 0 0.5 y.5 0 5 0 5 20 25 Time (sec) Figure 7.62 Tracking properties for robust servomechanism
Disturbance, output w 0.8 0.6 0.4 0.2 y 0 0. 0.4 0.6 0.8 0 5 0 5 20 25 Time (sec) Figure 7.63 Disturbance rejection properties for robust servomechanism
Phase (deg) Magnitude (db) 0 0 0 20 30 40 0 2 0 0 0 0 v (rad/sec) 90 45 45 0 90 35 80 225 270 0 2 0 0 0 0 v (rad/sec) Figure 7.64 Closed-loop frequency response for robust servomechanism
N R 25 s s 3 Y N 7 Figure 7.65 Example of internal model with feedforward
N R e K s u Plant x Y N K 0 Figure 7.66 Internal model as integral control with feedforward
.2.0 0.8 N 8 5 0 Amplitude 0.6 0.4 0.2 0 0 0.5.0.5 2.0 2.5 3.0 Time (sec) Figure 7.67 Step responses with integral control and feedforward
W a w (s) r a r (s) u a r (s) r G(s) e u W G(s) r e u G(s) e ˆr Extended estimator ˆr Extended estimator K xˆ K xˆ (a) (b) (c) Figure 7.68 Block diagram of a system for tracking and disturbance rejection with extended estimator: (a) equivalent disturbance; (b) block diagram for design; (c) block diagram for implementation
W r u e 2 0 xˆ y s 3 y, x, ˆ r ˆ 2 3 4 ˆr ˆr Extended estimator 5 6 2 xˆ 7 0 0.2 0.4 0.6 0.8.0.2.4.6.8 2 Time (sec) (a) (b) Figure 7.69 Motor speed system with extended estimator (a) block diagram; (b) command step response and disturbance step response
R(s) c r (s) d(s) b(s) a(s) Y(s) c y (s) Dynamic controller Figure 7.70 Direct transfer-function formulation
y(t) t d l t (a) Compensator D(s) R D(s) e ls G(s) Y R D(s) e ls G(s) Y G(s) e ls G(s) (b) (c) Figure 7.7 A Smith regulator for systems with time delay
Steam Control valve Flow Product Temperature sensor Steam Figure 7.72 A heat exchanger
u To workspace Scope Scope.25s.45.20 s 2.28s.0925 600s 2 y 70s Step Gain Transfer Fcn Transfer Fcn To workspace 600s 2 70s Transfer Fcn2 Transport delay Transport delay Figure 7.73 Closed-loop Simulink diagram for a heat exchanger
.4 Output temperature, y.2 0.8 0.6 0.4 Closed-loop Open-loop 0.2 0 0 50 00 50 200 250 300 Time (sec) Figure 7.74 Step response for a heat exchanger
7 6 5 Control, u 4 3 2 0 0 50 00 50 200 250 300 Time (sec) Figure 7.75 Control effort for a heat exchanger
Im(s) 0.3 Closed-loop poles 0.2 0.4 0.3 0.2 0. 0. 0.2 0.3 0.4 Re(s) Closed-loop poles 0.2 0.3 Figure 7.76 Root locus for a heat exchanger
d Stable trajectory Unstable trajectory d e e (a) (b) Figure 7.77 Definition of Lyapunov stability
r e u Ts x 2 x s y Figure 7.78 An elementary position feedback system with a nonlinear actuator
r e u x2 x y s s Step Saturation Integrator Integrator Scope Figure 7.79 Simulink diagram for position feedback system
.2 0.8 Output Amplitude 0.6 0.4 0.2 0 0.2 0 5 0 5 x 2 Time (sec) Figure 7.80 Step response for position control system
U G s 4 x 4 G 2 2s x 3 H 2 s x 2 H s x Y Figure 7.8 A block diagram for Problem 7.4
U s 4 s 2 x 2 x x s 3 3 s x Y 5 (a) U s 0 x 4 4 x 2 s 2 x 3 s s 3 x Y (b) Figure 7.82 Block diagrams for Problem 7.5
U s s 2 4 Y Figure 7.83 System for Problem 7.22
a k l u u 2 m u u m Figure 7.84 Coupled pendulums for Problem 7.25
U K Ts s 2 2js Y Figure 7.85 Control system for Problem 7.29
Im(L(jv)) a 2 60 Re(L(jv)) Figure 7.86 Nyquist plot for an optimal regulator
u i L L C v c R R y Figure 7.87 Electric circuit for Problem 7.34
r s K y G G L F s F H x f x p H Figure 7.88 Block diagram for Problem 7.35
y(t) u(t) C R x L x 2 R 2 Figure 7.89 Electric circuit for Problem 7.36
k d g u u 2 M F Gas jet K kd u v 2 u K(u u 2 ) F/ml u 2 v 2 u 2 K(u u 2 ) F/ml F M Figure 7.90 Coupled pendulums for Problem 7.38
u N (s) G (s) D (s) y u u N (s) y u 2 G (s) D (s) N 2 (s) G 2 (s) D2 (s) y u y y y 2 u 2 N 2 (s) G 2 (s) D2 (s) y 2 (a) (b) r u N (s) G (s) D (s) y N 2 (s) G 2 (s) D2 (s) u 2 (c) Figure 7.9 Block diagrams for Problem 7.40. (a) series; (b) parallel; (c) feedback
u x y Reference longitude Desired location on orbit Figure 7.92 Diagram of a station-keeping satellite in orbit
u Figure 7.93 Pendulum diagram for Problem 7.43
Fuselage reference axis Vertical u d Rotor thrust Rotor u Figure 7.94 Helicopter
u k M y Figure 7.95 Simple robotic arm
u h h 2 Figure 7.96 Coupled tanks for Problem 7.5
b Ship motion c d Figure 7.97 View of ship from above
d G c D c c d Figure 7.98 Ship control block diagram
R(s) 0 K (s 4)(s ) Y(s) Figure 7.99 Control system for Problem 7.6