TRANSFER FUNCTION FORMULATION OF ANALYSIS AND DESIGN PROBLEMS FOR DISCRETE/CONTINUOUS CONTROL THEORY by Bardhyl Prishtina Thesis Advisor: Dr. C.D. Johnson Committee Member: Dr. Peter Gibson Committee Member: Dr. Laurie L. Joiner Date: December 05,

Analog-Type Control Signal ZOH-Type Control Signal Discrete/Continuous Type Control Signal 2

STATE SPACE REPRESENTATION Linear and Time Invariant Continuous-Time Plant Model Equivalent ZOH-Type Discrete Time Plant Model Equivalent D/C-Type Discrete Time Plant Model 3

IDEA OF DISCRETE/CONTINUOUS-TYPE CONTROL  The control signal is allowed to vary continuously and strategically in time between two consecutive sample times.  The D/C control can be viewed as generalization of a classical ZOH-Type discrete-time control.  The control-variation rule is calculated and decided at the beginning at each intersample period.  These control-variations can not be changed until the beginning of next sample period 4

D/C TYPE CONTROL SIGNAL Figure 1.2: A Typical “Smart” D/C Type Discrete-Time Control Signal Variations [8] 5

LINEAR-IN-TIME TYPE D/C CONTROL Figure 1.3: Typical Linear in Time Type D/C Control Variations [8] (1-2) Polynomial-Spline Type Wave Form Model 6

QUADRATIC-IN-TIME TYPE OF D/C CONTROL Figure 1.4: Typical Quadratic in Time Type of D/C Control Variations [8] (1-3) 7

CUBIC-IN-TIME TYPE OF D/C CONTROL Figure 1.5: Typical Cubic-in-Time Type of D/C Control Variations (1-4) 8

EXPONENTIAL-IN-TIME TYPE OF D/C CONTROL, Figure 1.6: Typical Exponential-in-Time Type D/C Control Variations [8] (1-5) 9

EXACT DISCRETIZATION OF CONTINUOUS-TIME- STATE-SPACE EQUATION WITH D/C CONTROL (1-9) (1-10) (1-11) (1-13) (1-12) for 10

BLOCK-DIAGRAM OF D/C CONTROLLED SYSTEM USING REAL-TIME STATE OBSERVER (1-14) 11

REVIEW OF THE TRANSFER-FUNCTION METHOD IN CLASSICAL ZOH-TYPE SYSTEMS ANALYSIS AND CONTROL DESIGN (2-1) 12

State Transition Matrix Control Distribution Matrix ZOH-Type Plant Transfer-Function Matrix (2-7) (2-8) (2-12) 13

DOUBLE INTEGRATOR PLANT-MODEL WITH ZOH- TYPE CONTROL (2-13) (2-14) (2-15) (2-16) By selecting 14

(2-17) (2-18) (2-19) (2-20) (2-21) (2-22) Thus, the characteristic polynomial of A and minimal polynomial of A are the same and hence 15

(2-23) (2-24) (2-25) (2-26) (2-28) 16

(2-12) (2-32) 17

INITIAL CONDITION RESPONSE AND ZOH-TYPE CONTROL SIGNAL FOR DOUBLE INTEGRATOR Initial Conditions: 18

THE HARMONIC OSCILLATOR PLANT-MODEL WITH ZOH-TYPE CONTROL (2-42) If we select: Then the matrix form of the state and output equations are: (2-43) (2-44) 19

(2-45) (2-46) 20 Harmonic Oscillator (continued)

(2-49) (2-50) (2-51) Designsuch that system performs “deadbeat response”. Hence, all eigenvalues ofshould be zero. Also is designed for deadbeat response. All eigenvalues of should be zero. 21 Therefore: Must design value of T such that ;

INITIAL CONDITION RESPONSE AND ZOH-TYPE CONTROL SIGNAL FOR HARMONIC OSCILLATOR Initial Conditions: 22

TRANSFER-FUNCTION ANALYSIS OF A LINEAR PLANT WITH D/C CONTROL (3-1) (3-2) (3-8) (3-15) (3-16) 23 D/C “control state

FIRST ORDER PLANT WITH LIT-TYPE D/C CONTROL (3-17) (3-18) (3-20) (3-19) (3-22) 24 LiT-Type D/C control

The Control gain matrix for a=3, b=5 and T=1 is computed to be: (3-24) (3-26) (3-27) (3-28) (3-29) 25 Assuming: where

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR FIRST ORDER PLANT Initial Condition: 26

D/C DISCRETE-TIME TRANSFER-FUNCTION OF A DOUBLE INTEGRATOR PLANT WITH D/C-TYPE CONTROL (3-33) (3-35) (3-39) 27

(3-41) (3-45) (3-47) (3-48) 28 Since s=n and we have

(3-50) (3-52) (3-53) 29

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR DOUBLE INTEGRATOR Initial Conditions: 30

DOUBLE INTEGRATOR WITH REDUCED ORDER STATE OBSERVER BLOCK (C. D. JOHNSON DESIGN RECIPE) [Ref #13] (3-54) (3-55) 31 Choose T12 and T11 as follows:

(3-67) (3-68) (3-69) 32

INITIAL CONDITION RESPONSE AND D/C CONTROL SIGNAL FOR DOUBLE INTEGRATOR PLANT AND REDUCED ORDER STATE OBSERVER Initial Conditions: 33

HARMONIC OSCILLATOR (D/C) CONTROL (3-70) (3-71) (3-72) 34

(3-73) (3-76) 35Harmonic Oscillator D/C control continues:

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR HARMONIC OSCILLATOR PLANT AND FULL ORDER STATE OBSERVER Initial Conditions: 36

TRANSFER-FUNCTION ANALYSIS OF CLOSED-LOOP DISCRETE/CONTINUOUS CONTROLLED SYSTEM MIMO: SISO: (4-1) (4-2) (4-3) 37

D/C CONTROLLED SYSTEM WITH FULL ORDER OBSERVER (4-4) (4-5) (4-6) 38 Detailed Expressions: MIMO: SISO:

TRANSFER-FUNCTION ANALYSIS OF A D/C CONTROLLED SYSTEM WITH A REDUCED ORDER PLANT-STATE OBSERVER (USING C. D. JOHNSON’S OBSERVER DESIGN RECIPE) [Ref #13] (4-16) (4-21) (4-22) 39

CLOSED LOOP SYSTEM WITH A STATE OBSERVER 40

DESIGN OF D/C DISCRETE TIME “COMPENSATOR” BY TRANSFER-FUNCTION METHOD (5-1)(5-2) (5-3) (5-4) (5-5) (5-6) 41 Candidate Choices for compensator design process.

CLOSED-LOOP SYSTEM WITH DISCRETE-TIME “SERIES COMPENSATOR” 42

D/C COMPENSATOR DESIGN FOR FIRST-ORDER PLANT (3-17) (3-24) (4-3) 43

(5-8) (5-10) (5-9) (5-11) (5-12) (5-13) (5-14) (5-15) 44 Trial Candidate: Design of (k1, k2) for all roots at zero (deadbeat response)

DISCRETE-TIME INITIAL-CONDITION RESPONSE FOR FIRST ORDER PLANT (5-15) Initial Condition: 45

PURE GAIN D/C COMPENSATOR DESIGN FOR THE DOUBLE INTEGRATOR PLANT WITH LIT D/C CONTROL (3-39) (5-8) 46 Trial Candidate Design k1 and k2 to be designed.

(5-16) (5-17) (5-18) (5-19) (5-20) (5-21) 47 Design of k1 and k2 for deadbeat response D/C compensator design

DISCRETE-TIME INITIAL CONDITION RESPONSE FOR DOUBLE INTEGRATOR PLANT AND PURE GAIN D/C COMPENSATOR (5-22) Initial Condition: 48

FIRST ORDER COMPENSATOR DESIGN FOR THE DOUBLE INTEGRATOR PLANT MODEL WITH LIT D/C CONTROL (3-39) (5-23) 49 Trial Candidate Design

(5-35) (5-33) (5-34) (5-28) 50 Design of a, kc1, kc2 for deadbeat response D/C compensator design

DISCRETE-TIME INITIAL CONDITION RESPONSE FOR DOUBLE INTEGRATOR PLANT AND FIRST ORDER D/C COMPENSATOR (5-36) Initial Condition: 51

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK In this Thesis it has been demonstrated that by using conventional discrete-time transfer-function block-diagram methods a D/C Type Discrete-Time Controller for time invariant MIMO systems can be formulated more directly than by using the conventional state- variable technique. The discrete-time block-diagram transfer- function formulation is simple and more direct compared to state- variable methods and revels very accurate results. In addition, we designed transfer-functions type D/C Controllers for some very common problems such as the: First Order, Double Integrator and Harmonic Oscillator plant model. 52 Summary

The conventional transfer-function technique used in this thesis to analyze and design D/C Controllers is a more direct method and should be useful in many applications. By using this technique, we have avoided the introduction of the state-variable, state- equations, state-observers, state “control-law” etc. in the design of D/C Control systems. Recommendations for Further Work Further work in this are should address application of the transfer- function employed here to advanced-forms of D/C controllers, including “bumpless” and “smooth bumpless” D/C Controllers [16]. In addition, the practical application of D/C Control, using available “field-programmable analog-array chips”, would be an important step in demonstrating the practical utility and effectiveness of D/C Control. 53 Conclusions