Embedded Control: Applications and Theory

Transcription

1 Embedded Control: Applications and Theory IEEE Rock River Valley Section Ramavarapu RS Sreenivas UIUC 30 September 2010 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

2 Outline Two typical applications of Embedded Control (Demos). Embedded control in the Engineering Curriculum at UIUC. Some lessons learned from teaching embedded control at UIUC. Need to grapple with logical/symbolic/algebraic issues. In addition to the usual control-theoretic issues. Why is Embedded Control difficult? An illustration of Supervisory Control for Livelock Freedom. Broad-brush description of ongoing research. My Team at UIUC. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

3 Embedded Control Application #1: Furuta Pendulum Furuta Pendulum 2 Link, Inverted Pendulum. Link #1 is connected to a motor (it is actuated). Link #2 is connected to Link 1 (angular joint) and is un-actuated. Optical-encoders measure the joint-angles. t Link 1 t Link 2 Mass DC Motor Torque (t) Figure: Furuta Pendulum Schematic & The Model built at UIUC. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

4 Embedded Control Application #1: Furuta Pendulum System Details The control-signal = torque ( current-input to DC motor). TI TMS320C6713 DSP controller. Objective Balance link #2 about the origin, starting from rest. Method : Two controllers (1) a swing-up controller, and (2) a (conventional) state-variable-feedback balance-controller, i.e. a Proportional + Derivative (PD) controller. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

5 Embedded Control Application #1: Furuta Pendulum Balance Control The equations of motion p + Ml 2 d 2 dt 2 θ 2(t) «d 2 dt θ 1(t) sin θ 1 (t) cos θ 1 (t)! + Mrl gl d 2! dt 2 θ 1(t) cos θ 2 (t) M + m «sin θ 2 (t) = 0 2 Mrl d2 dt 2 θ 2(t) cos θ 2 (t) Mrl d 2 dt 2 θ 2(t)! 2 sin θ 2 (t) + 2( p + ml 2 ) d dt θ 1(t) d dt θ 2(t) sin θ 2 (t) cos θ 2 (t) + ( + mr 2 + Mr 2 + ( p + ml 2 ) sin 2 θ 2 (t))θ 1 (t) = τ(t) where τ(t): torque exerted by the motor on link 1, l: length of link 2, M: mass attached to the end of link 2, m: mass of link 2, : moment of inertia of link 2 about its end, p: moment of inertia of link 1 about its end, and r: length of link 2. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

6 Embedded Control Application #1: Furuta Pendulum These equations are linearized about the origin for the Furuta pendulum in front of you. State-Equation d dt 0 θ 1 (t) θ 2 (t) d dt θ 1(t) d dt θ 2(t) 1 C A = C B {z } A 0 + θ 1 (t) θ 2 (t) d dt θ 1(t) d dt θ 2(t) 1 {z } B 1 C A C A τ(t) Eigenvalues of A (i.e. poles) are at [0 0 unstable {}}{ ] T Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

7 Embedded Control Application #1: Furuta Pendulum There are optical-encoders on this pendulum that provide the values of θ 1 (t) and θ 2 (t) directly. The output equation is Output-Equation ( ) y(t) = }{{} C θ 1 (t) θ 2 (t) d dt θ 1(t) d dt θ 2(t) This system is controllable (i.e. rank([b AB A 2 B A 3 B]) = 4), and observable (i.e. rank([c T A T C T (A T ) 2 C T (A T ) 3 C T ]) = 4). We can relocate poles anywhere we want (by state-variable feedback) and estimate angular-velocities from optical encoder data. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

8 Embedded Control Application #1: Furuta Pendulum unstable {}}{ The original poles/eigenvalues of A were at [ ] T. Let us suppose we wish to relocate them to [ ] T ( stable, 2% settling time of 4 9 = 0.44secs). We would need a state-variable feedback gain 1 of τ(t) = ( ) }{{} K θ 1 (t) θ 2 (t) d dt θ 1(t) d dt θ 2(t) Since system is observable, we can build linear-estimators for the unmeasured state-variables (the angular-velocities) from optical encoder data. Make sure observer poles are faster, and by separation-principle this design will balance Link #2. 1 MATLAB: K=place(A, B, P), where P is the array of desired pole-locations. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

9 Embedded Control Application #1: Furuta Pendulum Swing-Up Controller We use a square-waveform (of very low frequency) for the current into the motor. This will jerk the motor back-and-forth. Pumping energy into link #2. Like a child, whose legs cannot touch the ground, gets the swing going in a visit to the park. At some point link #2 gets to be close to the vertical position. Shut the swing-up controller off and turn on the state-variable feedback controller designed earlier. And... it works! Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

10 Embedded Control Application #2: The SegBoy 2 Wheeled, self-balancing robot inspired by the Segway that uses an Analog Devices imems Accelerometer (ADXL 103) for acceleration measurement, an InvenSense Integrated Dual-Axis Gyro (IDG 300) for X- and Y-axis angular acceleration, Left- and Right- IR sensors (for wall-following & formation-movement). Sonar-ranging sensor (for wall-following & formation-movement). TI TMS DSP controller. Figure: The Segway & The SegBoy built at UIUC. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

11 Embedded Control Application #2: The SegBoy 1 The SegBoy is programmed to balance itself, and then use a wall-follower algorithm to find its way around a maze. 2 The balance-controller is on all the time. The sensors issue interrupts when obstacles are sensed. The wall-following algorithm reads like a typical program in a CS-course. 3 Issues: Multi-tasking issues (avoided by over-design) Figure: The Segway & The SegBoy built at UIUC. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

12 Embedded Control & Control Curricula at UIUC College of Engineering Control Systems Laboratory (COECSL) 1 Four departments in the College of Engineering at the UIUC that have control theory in their undergraduate curricula pooled their resources to build a central laboratory facility. Started in Several satellite laboratories Hydraulic Control Laboratory, Mechatronics Laboratory, etc. 3 These labs see about 800 students each year. Embedded control is taught mainly in GE423: Mechatronics and in the COECSL Mechatronics Laboratory Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

13 Lessons Learned in the Pedagogy of Embedded Control at UIUC 1 Control Theory is quite mature and most students have a firm grasp of its design principles (We Learn Em good.) 2 A significant amount of time is used/lost in designing/debugging real-time code used in control applications. 1 All of this when we are not even pushing/using the processors to their fullest capacity! 3 Control-Engineers have to deal with things that have a Symbolic/Logical/Algebraic value. 1 In addition to the material in Ogata/Kuo/Chen/etc. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

14 Why is Embedded Control Difficult? Multitasking Scheduling Issues 1 Assume one processor many concurrent threads being executed (in different stages of computation). 2 The scheduler decides which thread gets executed at any time. 1 Threads/jobs can be preempted-and-resumed-later. 3 Rate-Monotonic Schedule (RMS) 1 m periodic tasks, with period T 1 < T 2 <... < T m. 2 Eg. periodic sampling have to complete the job before the next sample arrives. 3 Task with the higher rate (i.e. lower T i ) should get priority over others. 4 There are conditions you can check to see if RMS works. 4 In Deadline-Driven Scheduling, the task with the earliest deadline, has the highest priority. This yields a dynamic priority scheme, and there are conditions you can check to see if this form of scheduling works. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

15 Why is Embedded Control Difficult? Consider a task set = { 1, 2,..., 9 }, sorted by decreasing priorities (3) 2 (2) 3 (2) 4 (2) 9 (9) P 1 P 2 P 3 1 (3) 2 (2) 3 (2) (2) 4 9 (9) (a) Precedence Graph (b) Optimal Schedule Figure: Precedence graph of tasks { 1,..., 9 }, computation time is within the parenthesis and their optimal schedule. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

16 Why is Embedded Control Difficult? What would happen if we added an extra processor? 1 (3) 2 (2) 3 (2) 4 (2) 9 (9) (a) Precedence P 1 P 2 P 3 P 4 (3) 1 (2) 2 (2) 3 (2) 4 P 1 P 2 P 3 1 (3) 2 (2) 3 (2) (2) (9) 5 6 (b) 3 Processor Schedule 15 (c) 4 Processor Schedule (9) Figure: Things get worse when more processors are added! Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

17 Why is Embedded Control Difficult? OK, let s switch back to 3 processors, but what happens if we increase their speed? P 1 P 2 P 3 1 (3) 2 (2) 3 (2) 4 (2) 9 (9) (a) Precedence 1 (2) 2 (1) 3 (2) P 1 P 2 P 3 1 (3) 2 (2) 3 (2) (2) (9) 5 6 (b) Slow 3 Processor Schedule 5 (3) 8 (3) 4 (1) (3) 9 (8) 6 (3) 7 13 (c) Faster 3 Processor Schedule 7 8 Figure: Things get worse when processor speed is increased! Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

18 Why is Embedded Control Difficult? Deadlocks and Livelocks. 1 Deadlock is the situation in which multiple concurrent threads of execution in a system are blocked permanently because of resource requirements that can never be satisfied. 1 There are four necessary and sufficient conditions for this to occur. 2 Difficult to implement in a minimally restrictive fashion. 2 Livelock is more insidious some process/thread is put into suspended animation, while others seems to progress as if nothing happened. 1 Not well-understood. 2 Not easy to control. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

19 An Instance of Livelock Figure: An all-too-familiar instance of Livelock. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

20 An Instance of Livelock How is this handled today? Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

21 Livelock Avoidance: The Control Perspective p 1 t 1 t 2 p 2 t 3 p 3 t 4 t 5 t 6 p 4 p 5 t 7 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

22 Livelock Avoidance: The Control Perspective p 1 t 1 t 2 p 2 t 3 p 3 t 4 t 5 t 6 p 4 p 5 t 7 Event Sequence: t 1 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

23 Livelock Avoidance: The Control Perspective p 1 t 1 t 2 p 2 t 3 p 3 t 4 t 5 t 6 p 4 p 5 t 7 Event Sequence: t 1 t 3 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

24 Livelock Avoidance: The Control Perspective p 1 t 1 t 2 p 2 t 3 p 3 t 4 t 5 t 6 p 4 p 5 t 7 Event Sequence: t 1 t 3 t 6 (Deadlock) Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

25 Livelock Avoidance: The Control Perspective Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

26 Livelock Avoidance: The Control Perspective The event t 1 will not be permitted as ( ) T / Ω. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

27 Livelock Avoidance: The Control Perspective Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

28 Livelock Avoidance: The Control Perspective Event Sequence: t 2 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

29 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

30 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

31 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

32 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 t 2 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

33 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 t 2 t 4 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

34 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 t 2 t 4 t 1 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

35 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 t 2 t 4 t 1 t 7 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

36 Livelock Avoidance: The Control Perspective Event Sequence: t 2 t 3 t 6 t 5 t 2 t 4 t 1 t 7. All events appear at least once. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

37 Livelock Avoidance: Broad-Brush Description of our Research 1 Normative guidelines of how real-time code (for control applications) should be structured. 2 A minimally restrictive scheduler that guarantees there will no livelocked applications for code that adhere to our guidelines. 3 We can characterize the learning process that the supervisor can adopt to prevent livelocks based on the Microsoft-style reports. 4 Hierarchical models. 5 Valuable for safety-critical software for control and signal-processing applications. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38

38 My Team at UIUC 1 D. Block (COECSL Manager) my implementation guru. Taught us everything implementation-related. 2 R. Gummadi (PhD, ECE) & N. Somnath (MS, Aero) theoretical underpinnings of the work presented here. 3 M. Michelloti (MS, SEE) inverting a haptic-interface to obtain a 3DOF platform. 4 Y-M Ahn (MS, Mech) visual servoing and formation-flight for the Parrot AR Drone ( 5 C. Moline (MS, SEE) real-time marinade-quality monitor for a protein-producing sponsor. Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September / 38