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Feedback Control of Linear SISO Systems

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Feedback Control of Linear SISO Systems Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-LoopOpen-Loop ProcessProcess The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals 2 Closed-LoopClosed-Loop SystemSystem In study and design of control systems, we are concerned with the dynamic behavior of a controlled or Closed-loop Systems Feedback Control System 3 FeedbackFeedback ControlControl Control is meant to provide regulation of process outputs about a reference, , despite inherent disturbances Controller System Feedback Control System The deviation of the plant output, ,from its intended reference is used to make appropriate adjustments in the plant input, 4 FeedbackFeedback ControlControl Process is a combination of sensors and actuators Controller is a computer (or operator) that performs the required manipulations Computer Actuator + + + - Process Sensor e.g. Classical one degree-of-freedom feedback control loop 5 Closed-LoopClosed-Loop TransferTransfer FunctionFunction Block Diagram of Closed-Loop Process Computer Actuator + + + - Process Sensor - Open-Loop Process Transfer Function - Controller Transfer Function - Sensor Transfer Function - Actuator Transfer Function 6 Closed-LoopClosed-Loop TransferTransfer FunctionFunction For analysis, we assume that the impact of actuator and sensor dynamics are negligible Closed-loop process reduces to the block diagram: Feedback Control System 7 Closed-loopClosed-loop TransferTransfer FunctionsFunctions The closed-loop process has Two inputs The reference signal The disturbance signal Two outputs The manipulated (control) variable signal The output (controlled) variable signal We want to see how the inputs affect the outputs Transfer functions relating , and , 8 Closed-loopClosed-loop TransferTransfer functionfunction There are four basic transfer functions They arise from three so-called sensitivity functions Highlights the dilemma of control system design Only one degree of freedom to shape the three sensitivity functions 9 Closed-loopClosed-loop TransferTransfer FunctionsFunctions Sensitivity functions: The sensitivity function: The complementary sensitivity function: The control sensitivity function: 10 Closed-loopClosed-loop TransferTransfer FunctionsFunctions Overall transfer function for the output: SERVO REGULATORY RESPONSE RESPONSE Servo response is the response of the output to setpoint change Regulatory response is the response of the output to disturbance changes 11 Closed-loopClosed-loop TransferTransfer FunctionsFunctions Servo mechanism requires that: Regulatory response requires that: Since The two objectives are complementary 12 Closed-loopClosed-loop TransferTransfer FunctionsFunctions Note that or requires that the controller is large This leads to large control sensitivity 13 PIDPID ControllerController Most widespread choice for the controller is the PID controller The acronym PID stands for: P - Proportional I - Integral D - Derivative PID Controllers: greater than 90% of all control implementations dates back to the 1930s very well studied and understood optimal structure for first and second order processes (given some assumptions) always first choice when designing a control system 14 PIDPID ControlControl PID Control Equation Proportional Derivative Action Action Integral Controller Action Bias PID Controller Parameters Kc Proportional gain Integral Time Constant Derivative Time Constant Controller Bias 15 PIDPID ControlControl PID Controller Transfer Function or: Note: numerator of PID transfer function cancels second order dynamics denominator provides integration to remove possibility of steady-state errors 16 PIDPID ControlControl Controller Transfer Function: or, Note: Many variations of this controller exist Easily implemented in MATLAB/SIMULINK each mode (or action) of controller is better studied individually 17 ProportionalProportional FeedbackFeedback Form: Transfer function: or, Closed-loop form: 18 ProportionalProportional FeedbackFeedback Example: Given first order process: for P-only feedback closed-loop dynamics: Closed-Loop Time Constant 19 ProportionalProportional FeedbackFeedback Final response: Note: for “zero offset response” we require Tracking Error Disturbance rejection Possible to eliminate offset with P-only feedback (requires infinite controller gain) Need different control action to eliminate offset (integral) 20 Proportional Feedback Servo dynamics of a first order process under proportional feedback increasing controller gain eliminates off-set 21 Proportional Feedback High-order process e.g. second order underdamped process increasing controller gain reduces offset, speeds response and increases oscillation 22 ProportionalProportional FeedbackFeedback Important points: proportional feedback does not change the order of the system started with a first order process closed-loop process also first order order of characteristic polynomial is invariant under proportional feedback speed of response of closed-loop process is directly affected by controller gain increasing controller gain reduces the closed-loop time constant in general, proportional feedback reduces (does not eliminate) offset speeds up response for oscillatory processes, makes closed-loop process more oscillatory 23 IntegralIntegral ControlControl Integrator is included to eliminate offset provides reset action usually added to a proportional controller to produce a PI controller PID controller with derivative action turned off PI is the most widely used controller in industry optimal structure for first order processes PI controller form Transfer function model 24 PIPI FeedbackFeedback Closed-loop response more complex expression degree of denominator is increased by one Assuming the closed-loop system is stable, we get 25 PIPI FeedbackFeedback Example PI control of a first order process Closed-loop transfer function Note: offset is removed closed-loop is second order 26 PIPI FeedbackFeedback Example (contd) effect of integral time constant and controller gain on closed-loop dynamics (time constant) natural period of oscillation damping coefficient integral time constant and controller gain can induce oscillation and change the period of oscillation 27 PI Feedback Effect of integral time constant on servo dynamics Small integral time constant induces oscillatory (underdamped) 28 closed-loop response PI Feedback Effect of controller gain on servo dynamics affects speed of response increasing gain eliminates offset quicker 29 PI Feedback Effect of integral action of regulatory response reducing integral time constant removes effect of disturbances makes behavior more oscillatory 30 PIPI FeedbackFeedback Important points: integral action increases order of the system in closed-loop PI controller has two tuning parameters that can independently affect speed of response final response (offset) integral action eliminates offset integral action should be small compared to proportional action tuned to slowly eliminate offset can increase or cause oscillation can be de-stabilizing 31 DerivativeDerivative ActionAction Derivative of error signal Used to compensate for trends in output measure of speed of error signal change provides predictive or anticipatory action P and I modes only response to past and current errors Derivative mode has the form if error is increasing, decrease control action if error is decreasing, decrease control action Usually implemented in PID form 32 PIDPID FeedbackFeedback Transfer Function Closed-loop Transfer Function Slightly more complicated than PI form 33 PIDPID FeedbackFeedback Example: PID Control of a first order process Closed-loop transfer function 34 PID Feedback Effect of derivative action on servo dynamics Increasing derivative action leads to a more sluggish servo response 35 PID Feedback Effect of derivative action on regulatory response increasing derivative action reduces impact of disturbances on controlled variable slows down servo response and affects oscillation of process 36 PDPD FeedbackFeedback PD Controller Proportional Derivative Control is common in mechanical systems Arise in application for systems with an integrating behaviour Example : System in series with an integrator 37 PDPD FeedbackFeedback Transfer Function Closed-loop Transfer Function Slightly more complicated than PI form 38 PDPD FeedbackFeedback DC Motor example: In terms of angular velocity (velocity control) In terms of the angle (position control) 39 PDPD FeedbackFeedback Closed-loop transfer function Simplifying Notice that Same effect as a PID controller. 40 DerivativeDerivative ActionAction Important Points: Characteristic polynomial is similar to PI derivative action does not increase the order of the system adding derivative action affects the period of oscillation of the process good for disturbance rejection poor for tracking the PID controller has three tuning parameters and can independently affect, speed of response final response (offset) servo and regulatory response derivative action should be small compared to integral action has a stabilizing influence difficult to use for noisy signals usually modified in practical implementation 41 Closed-loop Stability Every control problem involves a consideration of closed-loop stability General concepts: Bounded Input Bounded Output (BIBO) Stability: An (unconstrained) linear system is said to be stable if the output response is bounded for all bounded inputs.
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