University of Virginia Proportional Control Spring 2015 Jack Stankovic University of Virginia

Outline Definition and Value of Proportional Control (Review) Desirable Properties of a Controller Analyzing Proportional Control –Root Locus P-Controller Design / Pole Placement Design

University of Virginia Proportional Control Control Law –It quantifies how to set control input to the target system Proportional Control Law u(k) = K p e(k) K p is a constant that is chosen when designing the proportional controller

University of Virginia KpKp G(z) R(z) E(z)Y(z) U(z) D(z) Proportional Control Feedback Control Using P Control (set the control proportional to the error) Controller Target System

University of Virginia Note: Simulation is in time domain System you want to control System starts at zero Goal Steady State Gain G(1) = y(ss)/u(ss) and if goal is y(ss) = 10 => u(ss) = 12 OPEN LOOP

University of Virginia Why is u(ss) = 12 G(1) = y(ss)/u(ss) G(1) =.47/(1-.43) =.82 y(ss)/u(ss) = 10/u(ss) =.82 (goal is 10) 12 = u(ss)

University of Virginia Disturbance Output not at desired Value y(ss) = 26.3

University of Virginia Why is y(ss) = 26.3 G(1) = y(ss)/[u(ss) + d(ss)].82 = y(ss)/( ) * recall from before.47/(1-.43) = * 32 = y(ss) 26.3 = y(ss)

University of Virginia Let Kp = 2 Just guess Trial and Error

University of Virginia Output

University of Virginia Proportional Controller KpKp G(z) R(z) E(z)T(z) U(z) D(z) Feedback using Proportional Control in the presence of measurement noise + + N(z) Y(z)

University of Virginia Simulation of a P Controller with disturbance and noise G(z)=0.47/(z-0.43), and K p =2 Steady State Error

University of Virginia Advantages of Proportional Control Does not require precise analytical model of the system being controlled Simple implementation Proper for applications with simple requirements (Overshoot, settling time, stability)

University of Virginia Deficiency of Proportional Control May cause steady-state error to be nonzero –Disturbance input is non zero –Noise input May cause oscillations Generally, one or more of the requirements for response time, overshoot, and oscillation may be impossible to fulfill at any proportional gain setting

University of Virginia Review - Desirable Properties of Controllers SASO S: Stability A: Accurate, steady-state error S: Settling Time O: Maximum Overshoot

University of Virginia Desirable Properties of Controllers SASO Properties in the step response of a closed-loop system. Settling time Overshoot Controlled variable Time Reference  % Steady StateTransient State Steady state error

University of Virginia Analyze Properties of Controllers Stability –For any bounded input over any amount of time, the output will also be bounded. “BIBO” –In P-Control, stability is assessed by determining if the poles of the closed-loop transfer function have a magnitude less than 1 (pole must be in the unit circle)

University of Virginia Analyze Properties of Controllers Accuracy –Quantified by the magnitude of the steady state control error. –In P-Control, accuracy is assessed by computing the steady-state gain. »There is a zero steady-state error if and only if this gain is 1.

University of Virginia Analyze Properties of Controllers Accuracy

University of Virginia Analyze Properties of Controllers Accuracy Note also that steady-state error can be reduced by using a larger K p. (see handout for derivation)

University of Virginia Analyze Properties of Controllers Settling Time –Settling time is a function of the closed-loop poles and is estimated using the following equation by employing the dominant pole approximation.

University of Virginia Analyze Properties of Controllers Overshoot –Overshoot is a property of the response to a step change in the reference input. –Overshoot is also suggestive of oscillatory behavior since a large overshoot is typically followed by a large undershoot. –Analyzed by an approximation

University of Virginia Analyze Properties of Controllers Maximum Overshoot - M p M p is used to denote the maximum overshoot to a unit step input. M p = |y(max) – y(ss)| / |y(ss)|

University of Virginia Maximum Overshoot For the first order STABLE system –By definition M p =0 if y(ss) > y(max) –M p =|p 1 | otherwise

University of Virginia Analyze Properties of Controllers Second order STABLE system (Approx.) if real dominant pole p 1 >=0 if real dominant pole p 1 <0 if dominant poles Note: Poles are in unit circle if stable

University of Virginia Analyze Properties of Controllers K(z) G(z) R(z) E(z)T(z) U(z) D(z) Closed-loop system + + N(z) Y(z) H(z) W(z)

University of Virginia Analyze Properties of Controllers Closed-loop Transfer Functions

University of Virginia Method to analyze p-controller From the example, three transfer functions have the same poles, which are the solutions to the characteristic equation To visualize the relation between Kp and closed-loop poles, we introduce the root locus of the system

University of Virginia Root Locus The root locus is a graphical procedure for determining the poles of a closed-loop system given the poles and zeros of a forward-loop system. Graphically, the locus is the set of paths in the complex plane traced by the closed-loop poles as the root locus gain is varied from zero to infinity.

University of Virginia Root Locus In p-control, the root locus is the locations of all possible roots of the characteristic equation as Kp varies from zero to infinity

University of Virginia IBM Lotus Domino Server with measurement delay K(z) G(z) R(z) E(z)+ - U(z) P control with Measurement Delay Y(z) H(z) = Z -2 W(z) Example

University of Virginia Example IBM Lotus Domino Server with measurement delay Pole at.43 Two poles at zero

University of Virginia H(z) = 1/z*z Example of Root Locus for 3 Pole System Given Below Two poles At zero Pole At K(z)G(z)H(z)

University of Virginia Analyze Properties of Controllers In P-Control, the closed-loop system is stable at the values of Kp that make all the poles lie inside the unit circle. Any system that has at least one zero at infinity will always become unstable for a large enough Kp.

University of Virginia Root Locus/Matlab Specify transfer function >>G = tf(np,dp) where np = numerator polynomial dp = denominator polynomial Plot root locus figure of transfer function G >>rlocus(G)

University of Virginia Design of Proportional Controllers So far, we analyzed properties of a P controller based on a specific value of Kp (like Kp=2) or found all values of Kp that make system stable. Instead, design the controller, choosing K p, so that the resulting system has desirable properties. (SASO) Pole Placement Design: –Select controller parameters to get desired closed-loop poles.

University of Virginia Pole Placement Design Step 1. Determine the required SASO properties of the closed-loop system. Step 2. Construct the closed-loop transfer function as a function of the proportional gain K P. Step 3. Solve for the closed-loop poles in terms of K P. Step 4. Derive K P from the equation that modeled poles are equal to the desired poles. (Take intersection of solutions) Step 5. Check Robustness

University of Virginia Example for Pole Placement Design For Proportional Control of the IBM Locus Domino Server Step 1. Decide Design Goals Stability e ss <0.1 k s <10 M P <0.1 Note: May not be satisfied! (new control needed or relax requirements) Recall: obtained by system ID

University of Virginia Pole Placement Step 2. Closed loop transfer function. This is first order so one pole.

University of Virginia Pole Placement Step 3. Consider 4 goals separately. Must be in unit circle

University of Virginia Pole Placement Step 4. Solve for the conditions on Kp – one at a time Stability: < K P < 3.0

University of Virginia Step 4 continued Steady State error: e ss 10.9 Recall – this is derived from the final value theorem and only valid if system is stable AND K P must be less than 3.0 to be stable BUT greater than 10.9 for this error Hence, this e(ss) cannot be satisfied

University of Virginia Step 4 continued Setting time requirement k s < 10 Depends on magnitude of largest closed loop pole With absolute value we get two inequalities < K P < 2.3

University of Virginia Step 4 continued Overshoot requirement M P < 0.1: K P < 1.1

University of Virginia Step 4 continued Overall Solution: –Intersection of equations from 1 to 4 –Bound on steady state error is not possible –But if < K P < 1.1 (meets other requirements)

University of Virginia Step 5: Robustness of P-Control System A controller is robust if its behavior does not change much if there are errors in estimating parameters of the target system. For stability,

University of Virginia Robustness of P-Control to errors in estimating the open-loop pole and gain of a first-order target system with a=0.43, b=0.47. Step 5: Robustness

University of Virginia Summary Basics of P Control How to analyze properties of a P controller (for a given Kp) How to use pole placement design to select the right parameter for the P controller to meet requirements Introduction to Root Locus

University of Virginia Extra Slides These following slides were not presented in class

University of Virginia Design of Proportional Controllers Another example IBM Lotus Domino Server with a Moving-Average Filter The stochastic nature of resource consumption in computing systems typically results in substantial variability, especially for performance metrics such as response times, queue lengths, and utilizations. So a MOVING-AVERAGE FILTER is used to mitigate the effect of the stochastics.

University of Virginia Moving-Average Filter The input of the moving-average filter is the “raw” signal and its output is a smoothed signal. Y(k) is the unfiltered signal and w(k) is the filtered signal. w(k+1) = cw(k) + (1-c)y(k) The filter has a transfer function

University of Virginia Moving-Average Filter w(k+1) = cw(k) + (1-c)y(k) C is a constant, 0 <= c < 1 If c = 0, the output has the same time- domain characteristics as the input signal, although it is delayed by one time step. Value of c greater than 0 specifies the degree of smoothing.

University of Virginia Moving Average Filter Effect of a moving average filter on a sine wave for different values of the filter parameter c.

University of Virginia Design of Proportional Controllers KpKp R(z) E(z)Y(z)+ - U(z) Moving Average Filter used in IBM Lotus Domino Server Controller Lotus Server W(z)

University of Virginia Design of Proportional Controllers Two parameters Kp and c need to be set We plot the relations between Kp/c and desirable properties of controllers.

University of Virginia Design of Proportional Controllers Effect of a moving average filter constant c and the proportional gain Kp

University of Virginia Design of Proportional Controllers From the plots c does not affect steady-state error, A large kp decreases e ss When c is small, k s increases when kp increases; when c is large, k s decreases as kp increases. When c is small, Mp increases when kp increases; when c is large, Mp decreases as kp increases. Finally, c = 0.95 and Kp = 2 is good!

University of Virginia Limit Cycle How can CPU utilization be unbounded? –It must vary somewhere between zero and one (so not unbounded) –Implies – can’t be unstable??? –For these types of physical systems where the “physics” of the system do not permit BIBO stability issues »Instability is manifested as a limit cycle CPU utilization bounces between zero and one