ee342 exp7 state variable feedback

8/12/2019 EE342 Exp7 State Variable Feedback

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EE 342 Lab Manual Exp. 7 - 1 Version 4.4

Experiment 7

State Variable Feedback

Introduction

This experiment demonstrates the design of control systems using the methods of modern control

theory. If all state variables of a system are accessible or estimable, these may be gain weighted

and summed to providefull state feedbackcompensation of the system. Theoretically, this

feedback permits the arbitrary placement of the closed loop system poles, programmable via the

feedback gain constants.

In this experiment we will investigate the compensation of the MCSL (without the inverted

pendulum that we attached in Experiment 6) via full state feedback. As in the previous

experiment, the objective is to achieve the best possible step response from the closed loopsystem. The feedback of all states of the plant through gain factors will permit the location of the

closed loop system poles at any desired values (within the linear limitations of the system).

Both states of the second order MCSL system (position and velocity) are accessible for feedback

via appropriate transducers. The position state variable is sensed in the form of the position

feedback voltage vp . The velocity state variable is sensed in the form of the tachometer

feedback voltage vT . These voltages are gain weighted and summed with the input voltage v i

via the resistances R2 and R3 connected to the summing junction of the op amp, and the op amp

feedback resistor Rf . The state feedback gains K2 and K3 are programmed by selection of R2

and R3 relative to Rf . This is illustrated in Figure 1.

The state variable feedback design problem is formulated beginning with the matrix state

equation of the plant. The plant (including the motor, power amplifier and speed reducer) has a

transfer function of the form:

o

o

m

V

K

s s

=

+

1 (1)

where

=KK K

N

a m

m

The corresponding differential equation is:

om

o oK v+ = 1

(2)


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EE 342 State Variable Feedback Experiment 7


Figure 1. Full State Feedback: System Block Diagram and Corresponding Schematic

Define the state variables,

x x xx

xo o1 2

1

2

= = =

(3)

the input variable

u vo= (the output of the op amp)

and the output

y o= The state equations are:

x x

x x K um

1 2

2 2

1

=

= +

(4)

Plant(Motor and Power Amp)

x1 x2

K1

K3

K2

Position Velocity

+

+

+

BlockDiagram

viy = x1

+

_

vp

vi

Rf

R3

R2

R1

KT

KP

Ka

K

s

m

m

m

+1

1

s

1

N

Plant

vT x2

Velocityx1

Posit

SystemSchematic


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In matrix form,

x Ax b= + u where

=

m

10

10

A

b=

0

K

The full state feedback control law is:

u K v K v K v

K v K K K K

i p T

i p o T

= + +

= + +1 2 3

1 2 3 (5)

Note that the tachometer measurement is taken at the motor shaft rather than after the pulley

speed reducer. Therefore,

m o m oN N Nx= = = =, 2 Thus,

u K v K K x K K Nxi p T= + +1 2 1 3 2

or in matrix form,

[ ]u K K K K N K vp T i= +2 3 1x (6)

Applying this input to the plant, we get the closed loop system with the modified matrices:

A=

0 1

12 3K K K K K K Np T

m

b=

0

1K K

Since the system matrix is in Controller Canonical Form, the characteristic equation may be

written by inspection of the bottom row ofA :

( )( )s K K K N s K K K s p s pTm

p

2

3 2 1 2

1

=

(7)


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The values of roots (poles) p1 and p2 may therefore be programmed by appropriate selection of

K2 and K3 . The objective of this experiment is to use this compensation method to locate the

closed loop system poles p1 and p2 such that a desirable step response is obtained.

Recall that for underdamped systems the characteristic equation may be written in terms of the

damping factor and the natural frequency n :

( )( )s s s p s pn n2 2

1 22+ + = (8)

We select K2 and K3 to obtain the desired values of and n , which will characterize the stepresponse of the system.

Procedure

1. Set up the MCSL apparatus the same as in Experiment 2, Configuration 3, including

tachometer feedback via R R K T3 10= + . Let R R K f= =1 30 . (This sets K1 1= .)

2. Using your model parameters from Experiment 1, select appropriate values for K2 and

K3 (and therefore R2 and R3 ) such that = 0 6. and n is at least as large as thatobtained using cascade compensation in Experiment 5. Report the resistor values and

show all calculations. Limit your resistor selection to reasonable values within the ranges

provided by the decade boxes. Since K2 effects the steady state gain of the closed-loop

system, try to keep K2 in the range of 0.5 to 2.0. What are the calculated closed loop

pole locations?

3. Implement your design using the MCSL apparatus and the selected values of the feedback

resistors. Apply a voltage step input of appropriate height and observe the time response

using the digital storage oscilloscope. Note that, depending on the steady state gain you

ended up with, a 6 volt step may be too large, causing the output shaft angle to exceed the

limits of the position sensor pot. Import the data into Sim342 and print the stepresponse plot. Compare your observations with the expected rise time, settling time, peak

time, and peak overshoot for your design values of and n . Report calculated vs.experimental values of each. Identify each transient response metric on the printed

oscilloscope data plot. Note that since Kp is negative, vp changes in the opposite sense

of v i . The steady state output voltage vp is equal to K

Kv i

1

2

. Derive this relationship.

If zero steady state error is desired (at least theoretically), how might you change K1 as a

function of K2 ? Would this affect the system response in any way other than scaling the

closed loop gain? If so, how?


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4. Adjust R2 and R3 iteratively until the desired step response characteristics are obtained.

To obtain reasonable values of these resistances, first determine the effect of varying each

individually on the steady state vp , and rand Mp . Report your final values of R2 and

R3 and the corresponding values of K2 and K3 . Using Sim342, import and print the

actual step response showing the values of the rise time, peak time, settling time and

peak overshoot. Also report the final experimentally determined closed loop pole

locations.

Compare your experimental with your calculated values of K2 3and K . How does the

performance of the state feedback compensation compare with the lead compensation of

Experiment 5?

Prelab Assignment

1. Read the experiment. Review state variable descriptions of linear systems in your EE 302

textbook (Dorf & Bishop,Modern Control Systems, 7th

ed., Chapter 3). Although the

material in this experiment is intended to be self-contained, it would also be helpful to

review/preview Chapter 11 in the text on The Design of State Variable Systems.

2. Using the design technique described in this experiment, select R2and R3such that both

poles of your closed-loop system are at s = -2 . Use the model parameter values for your

system. (Find K2and K3first, then use these to find R2and R3.)

ee342 exp7 state variable feedback

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