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State-space 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 HAC-LAC 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 closed-loop poles can be placed arbitrarily in the complex plane. The gain G can be chosen such that

Closed-loop characteristic equation

Selected arbitrarily

Example: s.d.o.f. oscillator (1)

Relocating the poles Deeper in le left-half plane

Example: s.d.o.f. oscillator (2)

State-space equation:

State feedback: Closed-loop 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 closed-loop 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 open-loop transfer function G(s)

Separation Principle

Closed-loop 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 two-mass problem (1) u

X3

X1 = y

State-space equation:

u

LQG design with symmetric root locus based on

Two-mass problem (2): Symmetric root-locus

Open-loop poles

Two-mass 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 !

Two-mass problem (4): Root locus of the LQG controller

Compensator

Notch filter

Optimum design for g= 1

Two-mass 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%

Two-mass 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): Closed-loop 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

Non-zero 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

Frequency-shaped LQR:

To achieve P + I action At low frequency To increase the roll-off At high frequency

Frequency shaped LQR (2): weight specification P+I Increased roll-off

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 model-based non-collocated 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 gain-stabilize 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 Wide-band position control of a truss

Set-up

Open-loop FRF of the HAC For various gains g of the LAC

HAC / LAC strategy (4): Example Wide-band position control of a truss

Bode plot of the controller H

Open-loop FRF of the design model: GH

HAC / LAC strategy (5): Example Wide-band position control of a truss

Open-loop FRF of the full system: G*H Nyquist plot

High frequency dynamics

Step response t (sec)

Information

Module8-LQG

43 pages

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