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Statespace approach Contents: State space representation Pole placement by state feedback LQR (Linear Quadratic Regulator) Observer design Kalman Filter LQG Separation Principle Spillover Frequency Shaped LQG HACLAC strategy
(Ch.7, p.138)
Transfer function approach:
State variable form:
State space Equation:
Plant noise Measurement noise Feedthrough
The choice of state variable is not unique
Example: s.d.o.f. oscillator: 1.
A
B
Acceleration Output:
C 2.
D (feedthrough)
A is dimensionally homogene
Inverted Pendulum
Equation of motion: Linearization: Change of variable:
with
(natural frequency Of the pendulum)
State variable form:
Output equation:
System transfer function
s.d.o.f. oscillator:
Inverted Pendulum:
For SISO systems, one can write:
Zeros poles
such that, for some initial condition, the free response is Free response:
. Poles:
are the eigenvalues of A, solution of
2.Zeros:
An input produces no output:
applied from appropriate initial conditions
The state vector has the form: If:
That is if:
Then: Y=0 if
(1)
(2)
(1) And (2)
dtm [ ] = 0
Pole placement by state feedback
State feedback
If the system is controllable, the closedloop poles can be placed arbitrarily in the complex plane. The gain G can be chosen such that
Closedloop characteristic equation
Selected arbitrarily
Example: s.d.o.f. oscillator (1)
Relocating the poles Deeper in le lefthalf plane
Example: s.d.o.f. oscillator (2)
Statespace equation:
State feedback: Closedloop characteristic equation:
Desired behaviour:
Linear Quadratic Regulator (SISO)
u such that the performance index J is minimized
Controlled variable:
Control force: u Weighing coefficient
Solution: The closedloop poles are the stable roots of:
where
Characteristic equation:
Identical to that of:
Weighing On the control

Symmetric root locus
·Symmetric with respect to the imaginary axis As well as the real axis ·Only the roots in the left half plane have to be considered
Example: Inverted pendulum (1)
Controlled Variable:
Selected poles
Example: Inverted pendulum (2) 1. Select the poles on the left side of the Symmetric root locus 2. Compute the gains so as to match the desired poles:
Observer design
Full stat observer (Luenberger observer):
(perfect modeling !!)
Duplicates the system
Innovation Error equation:
Error:
If the system is observable, the poles of the Error equation can be assigneg arbitrarily by Appropriate choice of ki In practice, the poles of the observer should Be taken 2 to 6 times faster than the regulator poles
In practice, there are modeling errors and measurement noise; These should be taken into account in selecting the observer gains One way to assign the observer poles: KALMAN filter (minimum variance observer) Scalar White noise processes The optimal poles location minimizing the variance of the Measurement error are the stable roots of the symmetric root locus:
Where
is the T.F. between w and y and Plant noise intensity (w) a Measurement noise intensity (v)
Example: Inverted pendulum (1)
1. Assume that the noise enters the system at the input (E = B) proportional to The same root locus can be used for the regulator and the observer design
Example: Inverted pendulum (2)
2. Assume
Observer poles
Note (SISO design) LQR =
Controlled variable z Input u
Kalman filter =
Output measurement y Plant noise w
Assuming that z = y (H = C) and that the noise enters the plant at the input (E = B
The design of the regulator and the observer can be completed with the same Symmetric root locus corresponding to the openloop transfer function G(s)
Separation Principle
Closedloop equations:
Reconstructed state
Compensator
2n state variables With
Block triangular
the eigenvalues are decoupled
Transfer function of the compensator
The poles of the compensator H(s) are solutions of the characteristic equation:
·They have not been specified anywhere in the design ·They may be unstable ·H(s) is always of the same order as the system
The twomass problem (1) u
X3
X1 = y
Statespace equation:
u
LQG design with symmetric root locus based on
Twomass problem (2): Symmetric rootlocus
Openloop poles
Twomass problem (3) Design procedure:
·Select the regulator poles on the locus ·Compute the corresponding gains G ·Select the observer poles (2 to 6 times faster) ·Compute the corresponding gains K ·Compute the compensator H(s)
One finds:
Notch filter !
Twomass problem (4): Root locus of the LQG controller
Compensator
Notch filter
Optimum design for g= 1
Twomass problem (5): robustness analysis Effect of doubling the natural frequency
Unstable loop ! This frequency has been doubled The notch filter becomes useless
Effect of lowering the natural frequency by 20%
Twomass problem (6): Robustness analysis
Pole/zero Flipping ! The notch is unchanged
(Ch.9, p.206)
Spillover (1)
Phase stabilized
Crossover
Gain stabilized
...
Bandwidth
The residual modes Near crossover may Be destabilized by Spillover
Spillover (2): mechanism
Flexible structure dynamics Controlled modes Actuators Residual modes Sensors
Spillover (3): Equations
Structure dynamics: Controlled modes: Residual modes: Output: Full state observer: Observation Spillover Control Spillover
Full state feedback:
Spillover (4): Eigen value problem
Control spillover
Observation spillover
If either B r=0 or Cr=0, the eigen values remain decoupled If both Br and Cr exist, there is Spillover
Spillover (5): Closedloop poles
The residual modes have a small stability margin (damping !) and can be destabilized by Spillover
(Ch.9, p.211)
Integral control with state feedback
Constant disturbance
Nonzero steady state error on y Introduce the augmented state p such that : State feedback: Closed loop equation:
If G and Gp are chosen so as to stabilize the system, without knowledge of the disturbance w
(Ch.9, p.212)
Frequency Shaped LQR (1)
LQR:
Parseval's theorem:
Frequency independent
Frequencyshaped LQR:
To achieve P + I action At low frequency To increase the rolloff At high frequency
Frequency shaped LQR (2): weight specification P+I Increased rolloff
Frequency shaped LQR (3): Augmented system
Frequency independent cost functional
State space realization of the augmented system
Frequency shaped LQR (4): Augmented system
The state feedback of the augmented system is designed with the frequency independent Cost functional:
Frequency shaped LQR (5): Architecture of the controller
Only the states of the Structure must be reconstructed
Augmented states
(Ch.13, p.295)
HAC / LAC strategy (1)
The control system consists of tho imbedded loops: 1) The inner loop (LAC: Low Authority Control) consists of a decentralized active damping with collocated actuator/sensor pairs (no model necessary). The outer loop (HAC: High Authority Control) consists of a modelbased noncollocated controller (based on a model of the actively damped structure).
2)
HAC / LAC strategy (2)
Advantages: The active damping extends outside the bandwidth of the HAC (reduces the settling time of the modes beyond the bandwidth) The active damping makes it easier to gainstabilize the modes outside the bandwidth of the HAC loop (improved gain margin) The larger damping of the modes within the controller bandwidth makes them more robust to parametric uncertainty (improved phase margin)
HAC / LAC strategy (3): Example Wideband position control of a truss
Setup
Openloop FRF of the HAC For various gains g of the LAC
HAC / LAC strategy (4): Example Wideband position control of a truss
Bode plot of the controller H
Openloop FRF of the design model: GH
HAC / LAC strategy (5): Example Wideband position control of a truss
Openloop FRF of the full system: G*H Nyquist plot
High frequency dynamics
Step response t (sec)
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Module8LQG
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