Week 4-5 1.Real time state machines 2.LTI systems– the [A,B,C,D] representation 3.Differential equation-approximation by discrete- time system 4.Response–

Week 4-5

1. Real time state machines2. LTI systems– the [A,B,C,D] representation3. Differential equation-approximation by

discrete- time system

4. Response– total response = zero-input response + zero-state response

5. Convolution-zero-state response = convolution of

impulse response and input signal6. Impulse response

Recall function-and-set description of state machines

States, Inputs, Outputs, initialState, update function

s(0) = initialStates(n+1) = nextState(s(n), x(n))y(n) = output(s(n), x(n)) }

(s(n+1), y(n)) = update(s(n),x(n))

Real time state machines

Above (Ch 4) n represents step

We now consider machines in which n representsreal time, eg. seconds, micro-seconds, etc.

The only difference this makes is that we cannot have the absent input

We consider machines in which inputs, outputs andstates are represented as tuples of real numbers

Delay3

D D Dx(n) x(n-1) x(n-2) x(n-3)

s1(n) s2(n) s3(n)

y(n)

nextState s1(n+1) = x(n)

s2(n+1) = s1(n)

s3(n+1) = s2(n)

y(n) = s3(n)output

4pt Moving average

D D Dx(n) x(n-1) x(n-2) x(n-3)

s1(n) s2(n) s3(n)

y(n)1/4 +1/4

1/41/4

nextState s1(n+1) = x(n)

s2(n+1) = s1(n)

s3(n+1) = s2(n)

y(n) = s1(n) + s2(n) + s3(n) + x(n) output 14

14

14

14

x1(n)

xM(n)

s1(n)

sN(n)

y1(n)

yK(n)

.

...

.

.

x(n)RM s(n)RN y(n)RK

MIMO system

SISO system if M = K =1

LTI systems

Infinite state systems with linear update function

System = (RN, RM, RK, update, initialState)

States = RN, Inputs = RM, Outputs = RK, initialState = s(0)

update: RN RM RN RK is a linear function

so there are matrices A (N N), B(N M), C (K N), D(K M)such that

s(n+1) = A s(n) + B x(n) y(n) = C s(n) + Dx(n)

v(t)

i(t)

q(t)

R

C

Differential equations

Response

to input signalx = (x(0), x(1), …) [Ints0 R]

State response is

s = (s(0), s(1), …) [Ints0 R]

s(0) = initialStates(n+1) = a s(n) + b x(n), n 0

s(1) = as(0) + bx(0)s(2) = as(1) + bx(1) = a2s(0) + abx(0) + bx(1)…s(n) = ans(0) + an-1bx(0) + an-2bx(1) + … + bx(n-1)

Output response

y = (y(0), y(1), …) [Ints0 R]

is obtained from state response and y(n) = cs(n) + dx(n)

zero-inputresponse

zero-stateresponse

(total)response

= +

Zero-state response is

Define

Then the zero-state response is the convolution sum

h: Integers0 R is the (zero-state) impulse response

The Kronecker delta or impulse at time k is the inputsignal

nk

graph of impulse at k

Notes/Responses/Echo

v(t)

i1(t)

q1(t)

R1

C1

R2

C2q2(t)

i2(t)

Define a state machine by

Substitute from differential equation above to get

Note: A is 22, b is 21, cT is 12, d is 11

Recall Zero-state response is

Define

Then the zero-state response is the convolution sum

h: Integers0 R is the (zero-state) impulse response

The Kronecker delta or impulse at time k is the inputsignal

nk

graph of impulse at k

The impulse signal

1

If k = 0, write instead of 0, and call it the impulse

Recall the general formula for the zero-state responseto any input signal x:

So the response to the impulse is obtained by setting x = , which gives

That is why h is called the (zero-state) impulse response.

Suppose the input signal is k , impulse at k.Substitution in the general formula gives its(zero-state) response as

which is the impulse response delayed by k

0-state response

0 n0

k k+nk

Time-invariance

50 1

graph ofm h(m)

m

0-1-5m

10-4m

graph ofm h(4- m)

40m

-1

graph ofm h(-m) flip

graph ofm h(1- m)

flip & drag

Convolution mechanics by flip and drag

0 1 2 3

x(m)

m 0 1 2

h(m)

m

1

2

1/32/3

1

0-1-2m

h(0-m)

y(0) = 1/3 x 1 = 1/3

10-1m

h(1-m)y(1) = 2/3 x 1 + 1/3 x 2 = 4/3

210m

h(2-m)y(2) = 1 x 1 + 2/3 x 2 + 1/3 x 1= 8/3

Convolution, time-invariance and linearity