D(s) G(s) A control system design definition
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1 R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition
2 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
3 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
4 x x x 2 s s U s x 3 x 4 s Y x Figure 7.4 Block diagram for a fourth-order system in modal canonical form with shading indicating portion in control canonical form
5 U 2 x 2 s x 2 x s x Y 2 7 Figure 7.5 Observer canonical form
6 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
7 u x Fx Gu x H Y u Kx Figure 7.7 Assumed system for control-law design
8 x Amplitude u/4 x Time (sec) Figure 7.8 Impulse response of the undamped oscillator with full-state feedback (ω 0 = )
9 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.
10 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
11 x x ss 0.6 Amplitude 0.4 x 2 u ss u/ Time (sec) Figure 7. Step response of oscillator to a reference input
12 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)
13 x LQR Tape position Dominant second-order Time (msec) Figure 7.3 Step responses of the tape servomotor designs
14 T LQR Tape tension Dominant second-order Time (msec) Figure 7.4 Tension plots for tape servomotor step responses
15 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
16 Im(s) j r 0 Re(s) j Figure 7.6 Symmetric root locus for the satellite.
17 25 r z 2 dt r r u 2 dt Figure 7.7 Design trade-off curve for satellite plant
18 .5.0 Imaginary axis Real axis Figure 7.8 Nyquist plot for LQR design
19 Im(s) Re(s) 2 3 Figure 7.9 Symmetric root locus for the inverted pendulum
20 Position, x Time (sec) 4.5 Figure 7.20 Step response for the inverted pendulum
21 .4.2 r 0 r Tape position, x r Time (msec) 2 (a) 0.0 Tape tension, T r 0 r 0.20 r 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
22 u Process (F, G) x H y Model (F, G) xˆ H ŷ Figure 7.22 Open-loop estimator
23 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
24 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
25 ˆ x 2 Amplitude x ˆ x x Time (sec) Figure 7.25 Initial-condition response of oscillator showing x and ˆx
26 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
27 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
28 0 8 6 Amplitude 4 2 ˆ x 2 0 x x Time (sec) Figure 7.28 Initial-condition response of the reduced-order estimator
29 Imaginary axis q 0 q Real axis Figure 7.29 Symmetric root locus for the inverted pendulum estimator design
30 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
31 Im(s) 6 K 40.4 K Re(s) Figure 7.3 Root locus for the combined control and estimator, with process gain as the parameter
32 00 40 Phase Magnitude v (rad/sec) v (rad/sec) Figure 7.32 Frequency response for G(s) = /s 2 Compensated Uncompensated db
33 Y s U 6.4 Figure 7.33 Simplified block diagram of a reduced-order controller that is a lead network
34 Im(s) 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
35 00 40 Phase Magnitude v (rad/sec) Compensated 55 Uncompensated 20 db v (rad/sec) Figure 7.35 Frequency response for G(s) = /s 2 with a reduced-order estimator
36 U 0 s s s Y 6 0 Figure 7.36 DC Servo in observer canonical form
37 Im(s) Re(s) Figure 7.37 Root locus for DC Servo pole assignment
38 Im(s) Re(s) Figure 7.38 Root locus for DC Servo reduced-order controller
39 Im(s) Controller poles 2 Estimator poles Re(s) Figure 7.39 Symmetric root locus
40 Im(s) Re(s) 2 Controller poles Estimator poles Figure 7.40 Root locus for pole assignment from the SRL
41 Continuous controller Plant Step 94.5s s s 3 9.6s s s 3 0s 2 6s Mux Control Mux Output Discrete controller 5.957z z z z z Plant 0 s 3 0s 2 6s Figure 7.4 Simulink block diagram to compare continuous and discrete controllers
42 Digital controller Continuous controller y Time (sec) 5 (a) u Continuous controller 2 0 Digital controller 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
43 Magnitude GM GM GM 2 LQR q 00 q 0 q v (rad/sec) Phase (deg) PM PM q LQR q v (rad/sec) q 00 Figure 7.43 Frequency response plots for LTR design
44 R D c (s) G(s) Y V Figure 7.44 Closed-loop system for LTR
45 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
46 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
47 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
48 R 0.8 e 8.32 s x c u s(s ) Y (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
49 Im(s) Re(s) Figure 7.49 Root locus of lag-lead compensation
50 Magnitude v (rad/sec) (a) 90 Phase v (rad/sec) 00 (b) Figure 7.50 Frequency response of lag-lead compensation
51 .2 y Time (sec) Figure 7.5 Step response of the system with lag compensation
52 Continuous controller Plant Step 8.32s 2 9.2s 0.8 s s 0.08 s 2 s Mux Control Mux Output Discrete controller 8.32z z z z Plant s 2 s Figure 7.52 Simulink block diagram to compare continuous and discrete controllers
53 .4.2 Digital controller.0 y Continuous controller Time (sec) 5 (a) u Continuous controller Digital controller 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
54 r e x I K u s K 0 Plant x y Figure 7.54 Integral control structure
55 Plant w r e x I s 25 s 3 y 7 Estimator xˆ Figure 7.55 Integral control example
56 y y y Time (sec) (b) u u u Time (sec) (b) Figure 7.56 Transient response for motor speed system:(a) step responses, (b) control efforts
57 r e K u s Plant x y K 0 Figure 7.57 Integral control using the internal model approach
58 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
59 Phase (deg) Magnitude (db) v (rad/sec) v (rad/sec) Figure 7.59 Controller frequency response
60 Phase (deg) Magnitude (db) v (rad/sec) v (rad/sec) Figure 7.60 Sensitivity function frequency response
61 To workspace r To workspace2 e To workspace y Scope2 Scope r e u x2 x Integrator s s s Gain s Integrator Integrator2 Gain5 Integrator3 Out Scope Gain Gain Gain Gain3 Figure 7.6 Simulink block diagram for robust servomechanism
62 .5 r Reference, output y Time (sec) Figure 7.62 Tracking properties for robust servomechanism
63 Disturbance, output w y Time (sec) Figure 7.63 Disturbance rejection properties for robust servomechanism
64 Phase (deg) Magnitude (db) v (rad/sec) v (rad/sec) Figure 7.64 Closed-loop frequency response for robust servomechanism
65 N R 25 s s 3 Y N 7 Figure 7.65 Example of internal model with feedforward
66 N R e K s u Plant x Y N K 0 Figure 7.66 Internal model as integral control with feedforward
67 N Amplitude Time (sec) Figure 7.67 Step responses with integral control and feedforward
68 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
69 W r u e 2 0 xˆ y s 3 y, x, ˆ r ˆ ˆr ˆr Extended estimator xˆ Time (sec) (a) (b) Figure 7.69 Motor speed system with extended estimator (a) block diagram; (b) command step response and disturbance step response
70 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
71 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
72 Steam Control valve Flow Product Temperature sensor Steam Figure 7.72 A heat exchanger
73 u To workspace Scope Scope.25s s 2.28s s 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
74 .4 Output temperature, y Closed-loop Open-loop Time (sec) Figure 7.74 Step response for a heat exchanger
75 7 6 5 Control, u Time (sec) Figure 7.75 Control effort for a heat exchanger
76 Im(s) 0.3 Closed-loop poles Re(s) Closed-loop poles Figure 7.76 Root locus for a heat exchanger
77 d Stable trajectory Unstable trajectory d e e (a) (b) Figure 7.77 Definition of Lyapunov stability
78 r e u Ts x 2 x s y Figure 7.78 An elementary position feedback system with a nonlinear actuator
79 r e u x2 x y s s Step Saturation Integrator Integrator Scope Figure 7.79 Simulink diagram for position feedback system
80 Output Amplitude x 2 Time (sec) Figure 7.80 Step response for position control system
81 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
82 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
83 U s s 2 4 Y Figure 7.83 System for Problem 7.22
84 a k l u u 2 m u u m Figure 7.84 Coupled pendulums for Problem 7.25
85 U K Ts s 2 2js Y Figure 7.85 Control system for Problem 7.29
86 Im(L(jv)) a 2 60 Re(L(jv)) Figure 7.86 Nyquist plot for an optimal regulator
87 u i L L C v c R R y Figure 7.87 Electric circuit for Problem 7.34
88 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
89 y(t) u(t) C R x L x 2 R 2 Figure 7.89 Electric circuit for Problem 7.36
90 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
91 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 (a) series; (b) parallel; (c) feedback
92 u x y Reference longitude Desired location on orbit Figure 7.92 Diagram of a station-keeping satellite in orbit
93 u Figure 7.93 Pendulum diagram for Problem 7.43
94 Fuselage reference axis Vertical u d Rotor thrust Rotor u Figure 7.94 Helicopter
95 u k M y Figure 7.95 Simple robotic arm
96 u h h 2 Figure 7.96 Coupled tanks for Problem 7.5
97 b Ship motion c d Figure 7.97 View of ship from above
98 d G c D c c d Figure 7.98 Ship control block diagram
99 R(s) 0 K (s 4)(s ) Y(s) Figure 7.99 Control system for Problem 7.6
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