Lecture 7: Control of a DC Motor Drive

Transcription

1 / 25 Lecture 7: Control of a DC Motor Drive ELEC-E8405 Electric Drives (5 ECTS) Marko Hinkkanen Aalto University School of Electrical Engineering Autumn 205

2 2 / 25 Learning Outcomes After this lecture and exercises you will be able to: Draw the block diagram of a cascaded control system and explain it Calculate the controller gains based on the model parameters and the desired bandwidth Implement the current and speed controllers in the Simulink software

3 Introduction Modern electric drives automatically identify motor parameters (auto-commissioning, identification run) Model-based controllers are preferred, since they can be automatically tuned based on the known (identified) motor parameters Cascaded control system is commonly used Current or torque controller (fast) Speed controller (slower), not always needed Various control methods exist: our approach is based on two-degrees-of-freedom (2DOF) PI controllers Control systems are typically implemented digitally (but can be often designed in the continuous-time domain) Controllers and tuning principles can be extended to AC drives A cascaded control system may also include a position controller, but we will not cover it in these lectures. 3 / 25

4 4 / 25 Outline Preliminaries Current Control Voltage Saturation and Anti-Windup Cascaded Control System and Speed Control

5 Closed-Loop Control Closed-loop transfer function y(s) r(s) = H(s) = L(s) + L(s) r Controller e K (s) u d Plant G(s) y where L(s) = K (s)g(s) is the loop transfer function Typical control objectives Zero control error in steady state Well-damped and fast transient response r = reference e = control error u = control output d = load disturbance y = output The stability of the closed-loop system H(s) is often evaluated indirectly via the loop transfer function L(s). For example, the gain and phase margins can be read from a Bode plot or a Nyquist plot of L(jω). In these lectures, we mainly analyse the closed-loop transfer function H(s) directly. 5 / 25

6 6 / 25 PI Controller Most common controller type Time domain u = k p e + k i e dt s PI controller k i Transfer function e(s) u(s) = K (s) = k p + k i s r y e k p u

7 7 / 25 Outline Preliminaries Current Control PI Controller IMC Tuning Principle 2DOF PI Controller Voltage Saturation and Anti-Windup Cascaded Control System and Speed Control

8 8 / 25 Current Control System U dc Closed-loop current control enables:. Current limitation 2. Precise and fast torque control,ref Current controller u a,ref PWM q A, q B u a

9 Block Diagram Assumptions: Switching-cycle averaged quantities Ideal voltage production: u a = u a,ref Motor model e a k f ω M T L,ref Current u a controller sl a + R a k f sj T M Switching-cycle averaged symbols will not be marked with overlining in these lectures. 9 / 25

10 0 / 25 Simplified Block Diagram Back-emf e a is a slowly varying load disturbance P controller cannot drive the steady-state error to zero PI controller suffices e a,ref Current u a controller sl a + R a

11 / 25 PI Current Controller Time domain u a = k p e + k i e dt Current controller,ref e u a K (s) Y a (s) e a Transfer function e(s) u a (s) = K (s) = k p + k i s How to tune the gains k p and k i? Y a (s) = sl a + R a

12 2 / 25 Closed-Loop Transfer Function Closed-loop transfer function (s),ref (s) = H(s) = K (s)y a(s) + K (s)y a (s) Desired closed-loop system α c H(s) = s + α c where α c is the bandwidth Time constant of the closed-loop system is τ c = /α c,ref e u a K (s) Y a (s) e a

13 3 / 25 Internal Model Control (IMC) Principle Let us equal the closed-loop transfer function with the desirable one H(s) = Controller K (s) can be solved K (s)y a(s) + K (s)y a (s) = α c K (s)y a (s) = α c s + α c s K (s) = α c sy a (s) = α c s (sl a + R a ) = α c L a + α cr a s Result is a PI controller with the gains k p = α c L a and k i = α c R a Bandwidth α c should be (at least) one decade smaller than the angular sampling frequency 2π/T s

14 4 / 25 Example: Step Response R a = Ω, L a = 0 mh Constant e a = k f ω M = 00 V Open-loop controller Slow (τ a = 0 ms) Inaccurate even in steady state (since R a varies as a function of temperature) Closed-loop PI controller Fast (τ c = 2 ms) Steady-state accuracy depends on the current measurement accuracy (A) u a (V) ,ref Closed-loop control Open-loop control t (ms) u a = k p(,ref ) + k i (,ref )dt u a = R a,ref + e a t (ms)

15 Example: Step Response During Acceleration (A) 60 Some control error during accelaration IMC-tuned PI controller is sensitive to the varying load disturbance e a = k f ω M Causes control error during accelerations Load-disturbance rejection can be improved with an active resistance u a (V) t (ms) e a increases 2 V/ms t (ms) 5 / 25

16 2DOF PI Current Controller: Improved Disturbance Rejection Active resistance r is a controller gain Affects like a physical resistance (without causing losses) (s) u a(s) = Y a(s) = sl a + R a + r Bandwidth of Y a(s) can be selected to equal α c Current controller,ref u a u a K (s) sl a + R a r e a R a + r L a = α c r = α c L a R a This controller structure could also be represented as a state feedback controller with integral action and reference feedforward. 6 / 25

17 7 / 25 2DOF PI Current Controller IMC principle gives K (s) = α c sy a(s) Controller gains = α c L a + α c(r a + r) s,ref e u a K (s) Y a(s) e a r = α c L a R a k p = α c L a k i = α c (R a + r) = α 2 cl a Y a(s) = sl a + R a + r

18 8 / 25 Outline Preliminaries Current Control Voltage Saturation and Anti-Windup Cascaded Control System and Speed Control

19 9 / 25 Voltage Saturation: Control Loop Becomes Nonlinear Maximum converter output voltage is limited: u max u a u max Reference u a,ref may exceed u max for large,ref steps (especially at high rotor speeds due to large back-emf e a ) s k i Saturation e a,ref e k p u a,ref u a sl a + R a r

20 20 / 25 Back-Calculation Anti-Windup Method Current controller /k p This signal is nonzero only if the voltage saturates s k i e a,ref e k p u a,ref u a sl a + R a r

21 Example: Step Responses With and Without Anti-Windup Voltage saturates after the current reference step (due to high e a ) No overshoot if anti-windup is implemented Rise time is longer than the specified one (due to voltage saturation) (A) u a (V) No anti-windup Back-calculation anti-windup method t (ms) u max = 400 V Higher speed higher e a t (ms) 2 / 25

22 22 / 25 Outline Preliminaries Current Control Voltage Saturation and Anti-Windup Cascaded Control System and Speed Control

23 23 / 25 Cascaded Control System Control system U dc ω M,ref Speed controller T M,ref k f,ref Current controller u a,ref PWM q A, q B ω M u a Current controller: Inner (faster) loop Speed controller: Outer (slower) loop Controllers can be designed separately due to different time scales

24 24 / 25 Block Diagram Ideal torque control can be assumed: T M = T M,ref What is the resulting system? Motor model ω M,ref Speed controller T M,ref k f,ref u a,ref Current controller e a sl a + R a k f k f T L sj ω M T M

25 25 / 25 Simplified Block Diagram Assumption on ideal torque control T M = T M,ref Holds well if the current-control bandwidth is (at least) one decade faster than the speed-control bandwidth T L ω M,ref Speed controller T M sj ω M Mechanical and electrical systems are analogous sj + B i.e. J L a, B R a, T M u a, ω M,... sl a + R a Same control structures and tuning principles can be directly used