使用计算整定PI参数的方法 1 PI Controller Design A more convincing way, not necessarily the best way. This example comes from Wu Si’s paper on optimal reset controller. For a typical first order system, take the PMSM (or IM) current loop as an example. The system is written as PI controller is written as G (s) = 1 R + Ls C (s) = Kp + Ki 1 s The open loop overall system is as ∗ 1 s 1 R + Ls C (s) ∗ G (s) = Kp + Ki The closed loop function is written as Simplify it, iq i∗ q = id i∗ d = H (s) = C (s) ∗ G (s) 1 + C (s) ∗ G (s) H (s) = kps + ki Ls2 + (R + kp) s + ki (1) (2) (3) (4) (5) The design of controller kp and ki various depend on the orders of the desired system. Orders and Types • System Orders 1 

The order of the system is defined by the number of independent energy storage elements in the system, and intuitively by the highest order of the linear differential equation that describes the system. In another word, the highest exponent in the denominator. • System Types The system type is defined as the number of pure integrators in the forward path of a unity-feedback system. That is, the system type is equal to the value of n when the system is represented as in the following figure. It does not matter if the integrator are part of the controller or the plant. First order system This section is based on a milestone report. There is a paper that can be taken as reference. Telford D, Dunnigan M W, Williams B W. Online identification of induction machine electrical parameters for vector control loop tuning. In- dustrial Electronics, IEEE Transactions on, 2003, 50(2): 253-261. Q-axis transfer function is obtained as isq vsq = 1 σLs( Rs σLs + s) Machine PI controller is As vsq ierr sq = kps + ki s Next, cancel the machine pole ki kp = Rs σLs The closed loop transfer function becomes isq i∗ sq = kp σLs( kp σLs + s) Define system bandwidth to be ωbw, the PI gain becomes (6) (7) (8) 2 

图 1: damping ratio kp = σLsωbw ki = Rsωbw (9) Second order system The ideal characteristic equation of the closed loop control system should be D (s) = s2 + 2ξωns + ω2 n For a given ξ and ωn, the kp and ki value can be found as kp = 2ξωnL − R ki = ω2 nL (10) (11) (12) Next step is to decide the value of ξ and ωn. ξ is the damping ratio, the value is undamped (ξ = 0), underdamped (ξ < 1) through critically damped (ξ = 1) to overdamped (ξ > 1). System response is like Figure 1. ωn here is the undamped natural frequency. The value of ωn will decide the system bandwidth. Generally speaking the larger the system bandwidth is, the faster the system responses. Also, as the Gain for signal with higher frequency is very small, it can be considered that those signal are filtered. 3 

The find the exact value for system bandwidth, one can build the sys- tem transfer function in matlab. With, s = tf (s) D (s) = 1 s2 + 2ξωns + ω2 n (13) (14) Next, draw the bode diagram and find the cross of −3dB. Or use band- width(D). In this paper, ξ = 1 and ωn = 450. In Matlab it is calculated that the bandwidth is 289.0309rad/s. Approximately 46 Hz. In another document from infineon. Find in the following document. ”Setting the P-I Controller Parameters, KP and KI Application Note” The ξ value is set to 0.707, it is claimed to have a step response with fast settling time and reasonable overshoot. ωn is calculated with ωn = fP W M 5ξ (15) Here fP W M is the sample frequency of the controller. ωn is considered as five times greater than the frequency. Actually 5 times is quite aggressive, it is fair to take ten or higher times of the base frequency. ωn = fP W M 10ξ (16) Bandwidth Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz. For the low pass filter, the bandwidth is equal to the cut-off frequency. How to find bandwidth/ cut-off frequency? On bode plot, the bandwidth is the frequency when the magnitude drops to −3dB. Normally for low pass filter, the maximal gain of the system would be 0 dB, However, if there is no −3dB, i.e. the bode diagram begins at lower magnitude, that is due to the system gain. In that case, the low frequency gain will be considered as base, the cut-off frequency is calculated 4 

as . cut − of f f requency = base gain − 3dB Why 3 dB? The decibel (symbol: dB) is a unit of measurement used to express the ratio of one value of a physical property to another on a logarithmic scale, called the level. LG = 20 log10( Vout Vin ) dB (17) When LG = −3 dB, Vout Vin = 0.7079, that is the value where its spectral density is half of its maximal value. 0.7079 = √ 2 2 Rise Time vs. Bandwidth and Applications There is another sample method to find bandwidth with rising time of a step response. Rise time (Tr) is usually specified as the transition time for a signal to go from the 10% to the 90% of its maximal value. Meanwhile, the system bandwidth can be calculated with B = 0.35/Tr (M Hz) 5