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**ECE 8443 – Pattern RecognitionECE 3163 – Signals and Systems
**

• **Objectives:
Useful Matrix Properties
**

**Time-Domain Solutions
**

**Relationship to Convolution
**

**Laplace Transform Solution
**

• **Resources:
SHS: State Equation Solutions
**

**MIT: State Equation Solutions
**

**LECTURE 39: SOLUTIONS OF THE STATE EQUATIONS
**

**Audio:URL:**

**ECE 3163: Lecture 39, Slide 1
**

• **Recall our state equations:
**

• **To solve these equations, we will need a few mathematical tools. First:
**

**where I is an ***N*x*N ***identity matrix. A***k ***is simply A**x**A**x**…A.
**

• **For any real numbers ***t ***and ****:
**

• **Further, setting ** = -*t***:
**

• **Next:
**

• **We can use these results to show that the solution to
**

**is:
**

**Preliminaries
**

)()()(

)()()(

*ttxt
*

*ttt
*

**DvCy
**

**BvAxx
**

!3!2

3322 *tt
te t
*

**AA
AI
**

**A
**

**AAA ***eee tt * )(

**I
AAA ** *ttt eee *)(

*t
*

*t
*

*e
tt
*

*t
*

*t
t
*

*tt
t
*

*dt
*

*d
e
*

*dt
*

*d
*

**A
**

**A
**

**A
AA
**

**AIA
**

**A
AA
**

**AA
AI
**

!3!2

!2!3!2

3322

23 2

3322

)()( *tt ***Axx **

0),0()( *tet t***xx A**

**ECE 3163: Lecture 39, Slide 2
**

• **If:
**

• **is referred to as the state-transition matrix.
**

• **We can apply these results to the state equations:
**

• **Note that:
**

• **Integrating both sides:
**

**Solutions to the Forced Equation
**

*te***A
**

)()0()0()0()( *tee
dt
*

*d
e
*

*dt
*

*d
t
*

*dt
*

*d ttt
***AxxAxxx
**

**AAA **

0),0()( *tet t***xx A
**

)()()(

)()()(

)()()(

*tette
*

*ttt
*

*ttt
*

*tt
***BvAxx
**

**BvAxx
**

**BvAxx
**

**AA **

)()(

)()()()()()()(

*tete
dt
*

*d
*

*ttetetete
dt
*

*d
tete
*

*dt
*

*d
*

*tt
*

*tttttt
*

**Bvx
**

**AxxxAxxxx
**

**AA
**

**AAAAAA
**

0,)()0()(

)()0()(

0

0

*tdeet
*

*dete
*

*t
*

*tt
*

*t
*

*t
*

**Bvxx
**

**Bvxx
**

**AA
**

**AA Generalization of our
**

**convolution integral**

**ECE 3163: Lecture 39, Slide 3
**

• **Recall:
**

• **Using the definition of the unit impulse:
**

• **Recall our convolution integral for a single-input single-output system:
**

• **Equating terms:
**

**Solution to the Output Equation
**

0),()()0(

0),()()0()()()(

0

0

*ttdee
*

*ttdeettxt
*

*t
*

*tt
*

*t
*

*tt
*

**DvBvCxC
**

**DvBvxCDvCy
**

**AA
**

**AA
**

0,)()()0()( 0

*tdteet
*

*t
*

*t
*

*t
*

*t
*

*t
*

*zs
*

*zi *

**y
**

**A
**

**y
**

**A vDBvCxCy **

*teth
*

*dvthdvtve
*

*t
*

*t t
*

*t
*

**DBC
**

**DBC
**

**A
**

**A
**

)(

)()()()( 0 0

0,)()()(*)( 0

*tdvthtvthty
t
*

*zs *

**The impulse response can be computed
**

**directly from the coefficient matrices.**

**ECE 3163: Lecture 39, Slide 4
**

**Solution via the Laplace Transform
**

• **Recall our state equations:
**

• **Using the Laplace transform on the first equation:
**

• **Comparing this to:
**

**reveals that:
**

• **Continuing with the output equation:
**

• **For zero initial conditions:
**

)()()(

)()()(

*ttxt
*

*ttt
*

**DvCy
**

**BvAxx
**

)()0()(

)()0()(

)()()0()(

11
*ssss
*

*sss
*

*ssss
*

**BVAIxAIX
**

**BVxXAI
**

**BVAXxX
**

0,)()0()( 0

*tdeet
*

*t
*

*tt * **Bvxx AA
**

11 **AIA ***se t ***L
**

)()0( )()()(

)()()(

11
*ssxs
*

*sss
*

*ttxt
*

**VDBAICAI
**

**DVCXY
**

**DvCy
**

**DBAICHXHY ** 1)(where)()()( *sssss
*

**The transfer function
**

**can be computed
**

**directly from the
**

**system parameters.**

**ECE 3163: Lecture 39, Slide 5
**

• **The discrete-time equivalents of the state equations are:
**

• **The process is completely analogous to CT systems, with one exception:
**

**these equations are easy to implement and solve using difference equations **

• **For example, note that if ***v*[n] = 0 **for ***n *≥ 0**,
**

**and A***n ***is the state-transition matrix for the discrete-time case.
**

• **The output equation can be derived following the procedure for the CT case:
**

• **The impulse response is given by:
**

**Discrete-Time Systems
**

*nnxn
*

*nnn
*

**DvCy
**

**BvAxx
**

1

1,0 1

0

1

*nin
n
*

*i
*

*inn
***BvAxAx
**

0**xAx ***nn *

1,][0

][

1

0

1

][

*nnin
*

*n
*

*n
*

*i
*

*in
*

*n
*

*n
*

*zs
*

*zi *

**y
**

**y
**

**DvBvCAxCAy
**

1,

0,
1 *n
*

*n
nh
*

*n
***BCA
**

**D**

**ECE 3163: Lecture 39, Slide 6
**

• **Just as we solved the CT equations with the Laplace transform, we can solve
**

**the DT case with the z-transform:
**

• **Note that historically the CT state equations were solved numerically using a
**

**discrete-time approximation for the derivatives (see Section 11.6), and this
**

**provided a bridge between the CT and DT solutions.
**

• **Example:
**

**Solutions Using the z-Transform
**

)(

)(

11

11

)(]0[)(

)(]0[)(

)()(]0[)(

1

*z
*

*z
*

*zs
*

*zzzzz
*

*nnxn
*

*zzzzz
*

*zzzzz
*

*nnn
*

**Y
**

**H
**

**VDBAICxAICY
**

**DvCy
**

**BVAIxAIX
**

**BVAXxX
**

**BvAxx
**

][][][][

][][

][][]1[

][][]1[

][][][]1[

2312

21

323

212

2121

*nvnxnxny
*

*nxny
*

*nvnxnx
*

*nvnxnx
*

*nvnvnxnx
*

**ECE 3163: Lecture 39, Slide 7
**

**Summary
**

• **Introduced some useful properties of matrices that allowed us to solve the
**

**state equations.
**

• **Demonstrated a solution to the forced equation.
**

• **Discussed how this is a generalization of the single-input single-output
**

**analysis approach previously studied (e.g., convolution).
**

• **Applied the Laplace transform to solve the state equations using algebra.
**

**Derived an expression for the transfer function.
**

• **Extended the state equations to the discrete-time case.
**

• **Demonstrated a DT signal flow graph implementation of the DT solution.
**

• **Next: We will work some examples involving circuit analysis using the state
**

**variable approach.**