Another Whack at the Pendulum Pump and Balance the Pendulum

Transcription

1 Pump and Balance the Pendulum Qualitative Hybrid Control of the Pendulum Benjamin Kuipers Subramanian Ramamoorthy [HSCC, ] Another Whack at the Pendulum Pump and Balance the Pendulum Clear intuitions: three distinct modes. Pump Spin Balance Why Use Multiple Models? Local controllers can be simple and intuitive. Global behavior can be analyzed in terms of transitions among local regions. Non-linear controllers can be more effective than linear controllers. Non-linear controllers can be designed by composition of local models. Why Use Qualitative Models? A qualitative differential equation (QDE expresses partial knowledge of a dynamical system. One QDE describes a whole class of ODEs, non-linear as well as linear systems. A QDE can express partial knowledge of a plant or a controller design. QSIM can predict all possible behaviors of all ODEs described by the given QDE. 1

2 Qualitative Design of a Hybrid Controller Design local models with the desired behavior. Identify qualitative constraints to guarantee the right transitions. Provide weak conditions sufficient to guarantee desired behavior. Remaining degrees of freedom are available for optimization by any other criterion. Demonstrate with a global pendulum controller. Local models: Pump, Balance, Spin. QDEs and QSIM Each variable is a reasonable function. Continuously differentiable, etc. Range described by landmark values and intervals. Constraints link variables. ADD, MULT, MINUS, D/DT Monotonic functions: y=f(x for f in M [x] =sign(x Semi-quantitative bounds and envelopes. QSIM predicts all possible behaviors. Temporal logic model-checking can prove theorems about ODEs from QSIM prediction. The Monotonic Damped Spring The spring is defined by Hooke s Law: F = ma = mx & =! k1x Include damping friction m & x =! k x k x& 1! Rearrange and redefine constants && x bx& cx Generalize to QDE with monotonic functions && x f ( x& g( x Lemma 1: The Monotonic Damped Spring Let a system be described by && x f ( x& g( x where f M and [g(x] = [x] Then it is asymptotically stable at (,, with a Lyapunov function: x 1 V ( x, x& = x&! g( x dx Proof in the paper. Lemma : The Spring with Anti-Damping Pendulum Models Suppose a system is described by && x! f ( x& g( x where Near the top. Near the bottom. f M and [g(x] = [x] Then the system has an unstable fixed-point at (,, and no limit cycle. Proof in the paper. & f (&! ksin & f ( ksin!

3 Balance the Pendulum Design the control input u to make the pendulum into a damped spring. & f (&! ksin u(,& Define the Balance controller: = g(! [g( # k sin] =[] Lemma 1 shows that it converges to (,. && f (& g(! ksin The Balance Region If the control action has upper bound u then gravity defines the limiting angle: u = k sin Energy defines imum velocity at top: 1 =! #& g( # k sin # d# Define the Balance region: # & &! 1 Pump the Hanging Pendulum Define the control action u to make the pendulum into a spring with negative damping. Define the Pump controller = h( h f # M gives &&! ( h! f (& k sin Lemma proves it pumps without a limit cycle. Slow the Spinning Pendulum If the pendulum is spinning rapidly, define the Spin control law to augment natural friction: = f ( f M The Pump-Spin Boundary Prevent a limit-cycle behavior that cycles between Pump and Spin regions, overshooting Balance. Define the Pump-Spin boundary to be the separatrix of the undamped pendulum. Pump and Spin create a sliding mode controller. The separatrix leads straight to the heart of Balance. The Separatrix as Boundary A separatrix is a trajectory that begins and ends at the unstable saddle point of the undamped, uncontrolled pendulum: & k sin! Points on the separatrix have the same energy as the balanced pendulum: 1 KE PE = &! k sin d = k Simplify to define the separatrix: s (,& = &! k(1 cos 1 = 3

4 The Sliding Mode Controller The Global Pendulum Controller Differentiate to see how s changes with time: s & = f ( u(!, In the Pump region: s < and s& = #& ( h f (#&! In the Spin region: s > and s& = #& ( f f (#&! Therefore, both regions approach s Pump s&! sliding mode Balance Spin s&! The Global Controller The control law: Constraints: if Balance = g(! [g( # k sin] =[] else if Pump = h( h f # M else Spin (!, ( u = f f M Pendulum Controller Example System : c! & k sin! u(!, Balance : u = ( c k(! # c 11 Spin : u = c! & Pump : u = ( c c 3 1 c.1, k = 1, u = 4 c11.4, c1.3! =.4,.3 = c.5 c 3.5 Pendulum Example, cont. The Controlled Pendulum The switching strategy: If α 1 then Balance else if s < then Pump else Spin! = 1 s =! & k (1 cos!! 4

5 The Controlled Pendulum Related Work Astrom and Furuta, Swinging up a pendulum by energy control Zhao and Spong, 1 Hybrid global control of cart-pole system Chung and Hauser, 1995 Control of swinging pendulum on a cart, stabilization of periodic orbit Conclusions Qualitative modeling allows the designer to specify the controller for a large class of non-linear systems. Identifies weak sufficient conditions required for controller operation. Any instance of QDE will achieve the behavior. So the designer can optimize for any desired criteria. Allows the continuous phase portrait to be abstracted to a transition graph Successful control of cart-pole system (in simulation using qualitative control methodology 5