Feedback Control part 2
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1 Overview Feedback Control part EGR 36 April 19, 017 Concepts from EGR 0 Open- and closed-loop control Everything before chapter 7 are open-loop systems Transient response Design criteria Translate criteria into eignvalues into gain matrix RLC Circuit Example (Matlab) State Feedback Feedback Control Design a feedback controller: Identify which modes need to be controlled Identify which variables, components or subsystems this control translates to Select and calculate the pole placement Change the eigenvalues of the system State vs. output feedback 1
2 Recap Pole Placement Example Introduced two class periods back, so to review... What values of K will force the system to have λ = -1, -10? Compute the state feedback gain matrix that places the poles of the system at -1 and -10. Why are only A and B matrices defined? " x = $ # % " 'x + 1 % $ 'u & # 1 & Shaping the Dynamic Response Design a controller: Translate desired transient response into system eigenvalues Use dominant first- and second-order subsystems as approximations Transient characteristics are: Rise time, t R Peak time, t P Percent overshoot, PO Settling time, t S EGR 0: Form of Response 1. What are the options for the values s 1 and s?. What is the relationship of α and ω o for each type of response? EGR 0: Series RLC Natural Response d i dt + R di L dt + 1 LC i = 0 Assume that i = Ae st, then Ae st (s + R L s + 1 LC ) = 0 (s + R L s + 1 LC ) = 0 s 1, = R L ± " R % $ ' # L & 1 LC = α ± α ω 0
3 EGR 0: Typical Responses Overdamped α > ω 0 The roots, s, are negative, unequal and real Critically damped α = ω 0 The roots, s, are equal and real EGR 36, Chapter 7 (p 38) Pole placement for a nd order system closely resembles the analysis of an RLC circuit. For EGR 36 we have: λ + c m λ + k m = λ + ξ λ + Underdamped α < ω 0 The roots, s, are complex λ 1, = ξ ± ξ 1 10 EGR 36 Step Response Characteristics Simple MSD Example k H(s) = m s + c m s + k m = s + ξ s + ξ = c km = k m λ 1, = ξ ± ξ 1 Similar approach to that of defining α and ω 0 in EGR 0 for RLC circuits 3
4 Simple MSD Example Equate the specific and more general characteristic equations and compare coefficients Define the eigenvalues in terms of the basic system properties of damping ratio and natural frequency λ + c m λ + k m = λ + ξ λ + λ 1, = ξ ± ξ 1 Simple MSD Example From the possible behavior patterns, the underdamped case is the most interesting for control This behavior is the most problematic and so most in need of attention and control action Standard transient performance characteristics for nd order underdamped behavior are: Rise time, t R Peak time, t P Percent overshoot, PO Settling time, t S Dynamic Response Characteristics Parallel RLC Example Observe behavior of parallel RLC circuit Working from Matlab script handout, At tables: Decide upon new performance characteristics Provide values to me to use in the Matlab script. Design the necessary controller. Matlab STEP command, p43. Matlab help. Right-click on figure for property options. 4
5 (1) Determine Desired Performance We have 4 performance characteristics t R, t P, t S, PO We have only system properties: Damping ratio, ξ Undamped natural frequency, Therefore, we can independently determine of the performance characteristics (not all 4) Select the you will determine, and their desired values () Calculate Properties I Using the appropriate performance characteristic expressions, calculate ξ & Note expressions are exact, and are approximations..16ξ t R t s 4 ξ π t P = PO =100e ξπ 1 ξ 1 ξ () Calculate Properties II For example, using your selected PO, the new damping ratio can be determined (see page 4) () Calculate Properties III Next using your selected rise time, peak time or settling time, determine the new undamped natural frequency ξ ' =! ln PO $ # & " 100 % '! π + ln PO $ * ) # &, ( " 100 % + t R.16ξ t s 4 ξ π t P = 1 ξ 5
6 (3) Calculate New λ Using the new ξ and just found, calculate the desired λ λ 1, = ξ ± ξ 1 (4) Calculate the Feedback Gain In order to force your system to have these new, desired λ (that will result in the desired system performance), recall the feedback state equation x = (A BK)x + BFu Therefore, find the elements in the feedback gain matrix, K, that will result in the desired λ Use MATLAB place() command (5) Confirm the New Performance I Calculate ALL the system performance characteristics Including the two that you did not predetermine, but that are nonetheless effected by the new λ Comment on the effect your feedback has had on the system behavior (5) Confirm the New Performance II Simulate the system performance with state feedback to confirm expected behavior Method 1: Define the A, B, C, D matrices, and the STEP() command in Matlab, to plot the reduced order ( nd order) system behavior Method : Create a Simulink model, with the state feedback loop, and plot behavior of full order system (next HW) 6
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