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### CHE 185 – PROCESS CONTROL AND DYNAMICS

State Space Models

Block Diagram Algebra

- THE SEQUENCE OF SIGNAL PROCESSING IN A CONTROL LOOP CAN BE EXPRESSED IN BLOCK DIAGRAMS.
- BLOCK DIAGRAM ADVANTAGES
- EACH STEP IN THE PROCESS IS RETAINED AND CAN BE ADJUSTED INDEPENDENTLY
- CAUSE AND EFFECT RELATIONSHIPS FOR THE OVERALL SYSTEM ARE EASY TO ANALYZE
- THE IMPACT OF EACH COMPONENT IN tHE SYSTEM CAN BE CHECKED WHILE RETAINING A COMPREHENSIVE MODEL

System Algebra and Block Diagram

- Laplace transform – PARALLEL SYSTEM
- time-domain
- h(t) is the overall impulse response

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

System Algebra and Block Diagram

- Laplace transform – Series combination
- time-domain
- h(t) is the overall impulse response

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

System Algebra and Block Diagram

- Laplace transform – Feedback system
- time-domain
- h1(t) is the FORWARD PATH AND h2(t) IS FEEDBACK
- S-DOMAIN
- transfer function

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

System Algebra and Block Diagram

- Laplace transform – Feedback system
- feedback could be either positive or negative
- FOR NEGATIVE
- TRANSFER FUNCTION

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node15.html

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- DEVELOP THE RELATIONSHIPS BETWEEN CA2 AND CA0 FOR TWO STAGE CSTR SYSTEM WITH CONSTANT FLOWS, SAME VOLUME, AND REACTION RATE CONSTANT, k.

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- TRANSFER FUNCTIONS RELATING FLOWS IN AND OUT OF EACH UNIT:
- LINEAR COMPONENT MATERIAL BALANCES:

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- LaPLACE TRANSFORMS FOR MATERIAL bALANCES:

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- FROM THE FIRST OF THESE mass balance EQUATIONS:
- AND FROM THE SECOND EQUATION:

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- DEFINING TRANSFER FUNCTIONS FOR THE CONCENTRATIONS IN/OUT OF EACH REACTOR:

:

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- BLOCK DIAGRAM MATH::
- ALTERNATE DIAGRAM CAN BE DEVELOPED TO REPRESENT THE SAME OVERALL TRANSFER FUNCTION

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- ALTERNATE DIAGRAM CAN BE DEVELOPED TO REPRESENT THE SAME OVERALL TRANSFER FUNCTION

EXAMPLE OF BLOCK DIAGRAM DEVELOPMENT

- FORMAT FOR BLOCK DIAGRAMS
- Use of transfer function:

GENERAL SIGNAL RELATIONSHIPS

- (EX. 5-9), RIGGS

GENERAL SIGNAL RELATIONSHIPS

- (EX. 5-9), RIGGS
- Overall transfer function
- Which compares with the same function derived with laplace transforms in ex. 5-4

Block Diagram Algebra example

- Consider following block diagram:

In-Class Exercise: Block Diagram Algebra

- Develop overall transfer function for

In-Class Exercise: Block Diagram Algebra

- Develop overall transfer function for

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