Control Systems - Chibum Lee · 2015. 6. 1. · Chibum Lee -Seoultech Advanced Control Systems...
Transcript of Control Systems - Chibum Lee · 2015. 6. 1. · Chibum Lee -Seoultech Advanced Control Systems...
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Control Systems
Compensator
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Advanced Control SystemsChibum Lee -Seoultech
Outline
Combined control law and estimator
Reference tracking input with the estimator and
controller
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Advanced Control SystemsChibum Lee -Seoultech
Regulator with estimated states
Plant:
𝑥 = 𝐴𝑥 + 𝐵𝑢 + 𝐵𝑤𝑤
𝑦 = 𝐶𝑥
Control law:
𝑢 = −𝐾 𝑥
Estimator:
𝑥 = 𝐴 𝑥 + 𝐵𝑢 + 𝐿(𝑦 − 𝐶 𝑥)
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Advanced Control SystemsChibum Lee -Seoultech
Compensator: Combined Control and Estimator
Plant: 𝑥 = 𝐴𝑥 + 𝐵𝑢 + 𝐵𝑤𝑤, 𝑦 = 𝐶𝑥
Estimator: 𝑥 = 𝐴 𝑥 + 𝐵𝑢 + 𝐿(𝑦 − 𝐶 𝑥)
Control law: 𝑢 = −𝐾 𝑥
Combining these equations:
𝑥 = 𝐴𝑥 − 𝐵𝐾 𝑥 + 𝐵𝑤𝑤
𝑥 = 𝐴 𝑥 − 𝐵𝐾 𝑥 + 𝐿𝐶(𝑥 − 𝑥)
• Thus the closed-loop state equations are:
𝑥 𝑥=
𝐴 −𝐵𝐾𝐿𝐶 𝐴 − 𝐵𝐾 − 𝐿𝐶
𝑥 𝑥+
𝐵𝑤0
𝑤
𝑦 = 𝐶 0𝑥 𝑥
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Advanced Control SystemsChibum Lee -Seoultech
Compensator: Combined Control and Estimator
These equations may also be written in terms of the estimation
error 𝑥 = 𝑥 − 𝑥
𝑥 = 𝐴𝑥 − 𝐵𝐾 𝑥 + 𝐵𝑤𝑤
𝑥 = 𝐴 𝑥 − 𝐵𝐾 𝑥 + 𝐿𝐶(𝑥 − 𝑥)
• Subtracting: 𝑥 = 𝐴 − 𝐿𝐶 𝑥 + 𝐵𝑤𝑤
• Substituting: 𝑥 = 𝑥 − 𝑥 in the first equation yields:
𝑥 = 𝐴 − 𝐵𝐾 𝑥 + 𝐵𝐾 𝑥 + 𝐵𝑤𝑤
With this alternative set of state variables the closed-loop system
equations are:
𝑥 𝑥=
𝐴 − 𝐵𝐾 𝐵𝐾0 𝐴 − 𝐿𝐶
𝑥 𝑥+
𝐵𝑤𝐵𝑤
𝑤
𝑦 = 𝐶 0𝑥 𝑥
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Advanced Control SystemsChibum Lee -Seoultech
Compensator: Combined Control and Estimator
The closed-loop characteristic polynomial is thus:
𝛼(𝑠) = det(𝑠𝐼 − 𝐴𝑐𝑙) = det𝑠𝐼 − 𝐴 + 𝐵𝐾 −𝐵𝐾
0 𝑠𝐼 − 𝐴 + 𝐿𝐶
which is block triangular. Hence
𝛼 𝑠 = det 𝑠𝐼 − 𝐴 + 𝐵𝐾 ⋅ det 𝑠𝐼 − 𝐴 + 𝐿𝐶
= 𝛼𝑘 𝑠 ⋅ 𝛼𝐿(𝑠)
• Thus, the poles of the CL system consist of the poles obtained by
full state feedback through 𝐾, together with the estimator poles
determined by the selection of 𝐿
• That is, the control law and the estimator can be designed
independently of each other
The controlled plant poles are not changed by the
design of estimator
Separation Principle
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Advanced Control SystemsChibum Lee -Seoultech
Compensator: Combined Control and Estimator
By including the feedback law 𝑢 = −𝐾 𝑥 , the estimator
equation 𝑥 = 𝐴 𝑥 + 𝐵𝑢 + 𝐿(𝑦 − 𝐶 𝑥) becomes
𝑥 = 𝐴 − 𝐵𝐾 − 𝐿𝐶 𝑥 + 𝐿𝑦
𝑢 = −𝐾 𝑥
Transfer function for compensator:
𝐻 𝑠 =𝑈(𝑠)
𝑌(𝑠)= −𝐾 𝑠𝐼 − 𝐴 + 𝐵𝐾 + 𝐿𝐶 −1𝐿
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Selection of estimator poles
CL char. poly: 𝛼 𝑠 = 𝛼𝑘 𝑠 ⋅ 𝛼𝐿(𝑠)
The estimator poles (roots of 𝛼𝐿 𝑠 = 0) are usually chosen to be
between 2 to 6 times faster than the controller poles (roots of
𝛼𝑘 𝑠 = 0)
Clearly there is no direct effect of choosing faster estimator poles
on control effort. However, consider the effect of sensor noise.
Assume that the sensed output is 𝑦 = 𝐶𝑥 + 𝑣. The estimation error
dynamics then become:
𝑥 = 𝐴 − 𝐿𝐶 𝑥 + 𝐵𝑤𝑤 − 𝐿𝑣
Thus, the higher estimator gains required to produce faster
estimator dynamics will further amplify the sensor noise, thereby
corrupting the state estimates
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Symmetric root locus for SISO system estimator
Estimator dynamics with 'process noise' 𝑤 and 'sensor noise' 𝑣 :
The process noise 𝑤 can represent unknown disturbances and errors
in the plant model parameters
Recall the estimator equations: 𝑥 = 𝐴 𝑥 + 𝐵𝑢 + 𝐿(𝑦 − 𝐶 𝑥)
If 𝑤 is large:
• our plant model is uncertain, and hence a poor predictor
• we would put greater emphasis on the sensor data to correct the model
predictions larger 𝐿, faster estimator poles
If 𝑣 is large:
• our measurements are uncertain, and hence provide poor corrections
• we would put greater emphasis on the model predictions smaller 𝐿,
slower estimator poles
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𝑥 = 𝐴 − 𝐿𝐶 𝑥 + 𝐵𝑤𝑤 − 𝐿𝑣
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Advanced Control SystemsChibum Lee -Seoultech
Symmetric root locus for SISO system estimator
In optimal estimation theory
• the process noise is modeled as white noise with variance 𝜎𝑤2
• the sensor noise is modeled as white noise with variance 𝜎𝑣2
The estimator pole locations which will minimize the variance of the
state estimation error can be found from the solution of the
symmetric root locus (SRL) equation: 1 + 𝜌𝑁𝑒 𝑠 𝑁𝑒(−𝑠)
𝐷 𝑠 𝐷(−𝑠)= 0
where 𝜌 = 𝜎𝑤2/𝜎𝑣2 is a measure of the plant model uncertainty
relative to the measurement uncertainty, and
𝑍(𝑠)
𝑊(𝑠)= 𝐶 𝑠𝐼 − 𝐴 −1𝐵𝑤 =
𝑁𝑒(𝑠)
𝐷(𝑠)
is the plant transfer function from the process noise input to the
measured output
If the process noise 𝑤 and control input 𝑢 are additive (𝐵𝑤 = 𝐵),
the same SRL can be used for controller and estimator pole
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Ex: regulation of water level in two-tank system
The system can be described
Required closed-loop time
constants are 1 = 0.2 s, 2 = 0.05 s.
Assume
For
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Ex: regulation of water level in two-tank system
Controller: For closed-loop time constants 1 = 0.2 s, 2 = 0.05 s; the
desired CL char. poly is
• Demonstrate Ackermann's formula:
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Ex: regulation of water level in two-tank system
Estimator: Try estimator poles = 2 x controller poles;
• i.e., desired CL char. poly is α = (s +10)(s + 40) = s2 + 50s + 400
• Demonstrate Ackermann's formula:
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Ex: regulation of water level in two-tank system
Complete system
• Equivalent feedback compensator:
• Transfer function for compensator:
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Compensator for reduced order estimator
The similar approach can be carried out for reduced order estimator
Control law is
Substituting it into the equation for reduced order estimator
where
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MATLAB design of regulator with state estimator
Select desired closed-loop poles dclp
• e.g. by use of symmetric root locus
Use place or acker to get state feedback gains
• K = place(A, B, dclp)
Select desired estimator poles dep
• e.g. by use of symmetric root locus
Use place or acker to get estimator gains
• L = place(A’, C’, dep)’
Form feedback compensator (regulator)
• H = reg(G, K, L) % or
• H = reg(G,K,L,sensors,known,controls)
Connect regulator to plant
• Gcl = feedback(G, H, +1) % or
• Gcl = feedback(G,H,feedin,feedout,+1)16
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Advanced Control SystemsChibum Lee -Seoultech
Ex. DC Servo-full order Compensator
Use the state-space pole placement method to design a compensator for
the DC servo system with the transfer function
𝐺(𝑠) =10
𝑠(𝑠 + 2)(𝑠 + 8)
Using a state description in OCF, place the control poles at 𝑝𝑐 = [−1.42 −
1.04 ± 2.14𝑗] locations and the full-order estimator poles at 𝑝𝑒 = [−4.25 −
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𝐴 =−10 1 0−16 0 10 0 0
, 𝐵 =0010
𝐶 = 1 0 0 , 𝐷 = 0
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Advanced Control SystemsChibum Lee -Seoultech
Ex. DC Servo-full order Compensator
The estimator gain Lt=place(A’,C’, pe), L=Lt’
𝐿 =0.561.4216
The compensator transfer function
𝐻 𝑠 =𝑈(𝑠)
𝑌(𝑠)= −𝐾 𝑠𝐼 − 𝐴 + 𝐵𝐾 + 𝐿𝐶 −1𝐿
= −190(𝑠 + 0.432)(𝑠 + 2.10)
(𝑠 − 1.88)(𝑠 + 2.94 ± 8.32𝑗)
Compensator has an unstable pole at 𝑠 = 1.88
but CL poles are stable.
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1 + 𝐾10
𝑠(𝑠 + 2)(𝑠 + 8)
(𝑠 + 0.432)(𝑠 + 2.10)
(𝑠 − 1.88)(𝑠 + 2.94 ± 8.32𝑗)= 0
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Advanced Control SystemsChibum Lee -Seoultech
Ex. DC Servo-reduced order compensator
Design a compensator for the DC servo system using the same
control poles at 𝑝𝑐 = [−1.42 − 1.04 ± 2.14𝑗] but the reduced-order
estimator poles at 𝑝𝑒 = −4.24 ± 4.24𝑗 positions (𝜔𝑛 = 6 rad/sec,
𝜁 = 0.707)
Sol)
𝐴𝑎𝑎 𝐴𝑎𝑏𝐴𝑏𝑎 𝐴𝑏𝑏
=−10 1 0−16 0 10 0 0
,𝐵𝑎𝐵𝑏
=0010
The estimator characteristic polynomials det 𝑠𝐼 − 𝐴𝑏𝑏 − 𝐿𝐴𝑎𝑏 = 0
The desired polynomials 𝛼𝑒 𝑠 = 𝑠2 + 2 ⋅ 0.707 ⋅ 6𝑠 + 36
Solving for 𝐿 using place 𝐿 =8.536
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Advanced Control SystemsChibum Lee -Seoultech
Ex. DC Servo-reduced order compensator
The compensator transfer function
𝐻 𝑠 =𝑈(𝑠)
𝑌(𝑠)= 𝐶𝑟 𝑠𝐼 − 𝐴𝑟
−1𝐵𝑟 + 𝐷𝑟
= 20.92(𝑠 − 0.735)(𝑠 + 1.871)
𝑠 + 0.990 ± 6.120𝑗
Compensator has an unstable zero
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1 + 𝐾10
𝑠(𝑠 + 2)(𝑠 + 8)
(𝑠 − 0.735)(𝑠 + 1.871)
𝑠 + 0.990 ± 6.120𝑗= 0