Cleveland State University MCE441: Intr. Linear Control Systems

Lecture 2: Control Engineering Cycle Open-Loop and Closed-Loop Positive vs Negative Feedback Mathematical Models Dynamic Modeling of Engineering Systems Prof. Richter 1 / 14

Open Loop and Closed Loop Control Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems



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Open Loop (feedforward): Control input independent of output: Scheduling Useful with no uncertainties, no unpredictable disturbances: Know the future Closed Loop (feedback): Control input depends on output: Fundamental error correction mechanism If judiciously used, can reject disturbances, maintain stability and performance in the face of uncertainties. It can destabilize an open-loop stable system. Details later.

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Feedforward vs Feedback: Example Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Negative vs. Positive Feedback Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Negative Feedback: The fed back output and the reference input are subtracted Negative feedback has an error-compensating effect Still, too much or too little feedback can be destabilizing Positive Feedback: The fed back output and the input are added Positive feedback has a destabilizing effect Still, some systems can tolerate some positive feedback and remain stable. More later...

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Negative Feedback - Usually good Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Reproduced with permission from Dr. Steve Vogel, Duke University, Biology Department.

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Positive Feedback - Usually bad Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Reproduced with permission from Dr. Steven Vogel, Duke University, Biology Department

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Control Design Cycle Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Y Controls Problem

N

Reliable Model?

Modeling

Validation

Model Analysis

Assumptions

Y

Suitable Control T echnique?

N

ControlTheoretical Work

Design and Tuning

Simulation Study

Assumptions

Satisfactory Performance?

N

Deployment

Y

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Classification of Models Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

Distributed Parameters (Infinite-Dimensional)

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Partial Differential Equations (PDE) Example: Fourier's heat conduction law

Discrete-Time Difference Equations Example: Savings account balance with daily capitalization

LINEAR*

Lumped Parameters (Finite-Dimensional)

Continuous-Time Ordinary differential equations*(ODE) Integral equations Integrodifferential equations

Asterisk indicates model types considered for MCE441 Linear, time-invariant, single input, single output systems are denoted as SISO-LTI

NONLINEAR

-State-Space* -Transfer Function* -Time-Varying -Time-Invariant* -Certain -Uncertain* -Deterministic* -Stochastic -Multivariable -Single Input/Output*

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Dynamic Modeling of Electrical Systems Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Linear Resistor

R Va

Vb

Variables: Current i: Amperes (A) Voltage Vab : Volts (V) Parameters: Resistance R: Ohms (Ω) 1 A= 1 V/A

i

C Va

Law:

i = VaR−Vb

Ideal Capacitor

Laws:

Variables: Charge q: Coulomb (C) Parameters: Capacitance C: Farad (F) 1 A=1 F.V/s 1 C=1 F.V

i = C dVdtab

Vb

i

L Va

Vb

q = CVab (integral form)

Ideal Inductor

Law:

Parameters: Inductance L: Henry (H) 1 V=1 H.A/s

di Vab = L dt

i

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electrical systems... Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Other elements: Op-Amps, Diodes (non-linear) presented later, if needed Kirchhoff’s Laws: Voltage Law: “The sum of voltage differences around a loop must equal zero” Current Law: “The sum of currents at a node must be zero” Simple example: Find the input/output differential equation (in terms of Vo and Vi ):

L

Vi

C

R

Vo

C

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Solution Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Another example Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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Find the input/output differential equation (in terms of Vo and Vi ):

Vo

L

R

C

Vi

i

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Solution Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

We can easily write all equations describing the system. Elimination of unwanted variables to produce the desired I/O equation can be difficult and tedious. The Laplace transform will greatly facilitate the process.

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Solution Open Loop and Closed Loop Control Feedforward vs Feedback: Example Negative vs. Positive Feedback Control Design Cycle Classification of Models Dynamic Modeling of Electrical Systems

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