Lecture 5: Basic Dynamical SystemsDynamical SystemsIn One DimensionIn Two DimensionsThe Damped SpringThe Linear Spring ModelQualitative BehaviorsGeneralize to Higher DimensionsQualitative Behavior, AgainNode BehaviorFocus BehaviorSaddle BehaviorSpiral Behavior (stable attractor)Center Behavior (undamped oscillator)The Wall FollowerSlide 16Slide 17Slide 18Tuning the Wall FollowerAn Observer for Distance to WallExperiment with AlternativesZiegler-Nichols TuningZiegler-Nichols ParametersTuning the PID ControllerZiegler-Nichols Closed LoopClosed-Loop Z-N PID TuningLecture 5:Basic Dynamical SystemsCS 344R: RoboticsBenjamin KuipersDynamical Systems•A dynamical system changes continuously (almost always) according to•A controller is defined to change the coupled robot and environment into a desired dynamical system.€ ˙ x =F(x) where x∈ℜn€ ˙ x = F(x,u)y = G(x)u = Hi(y)€ ˙ x = F(x,Hi(G(x)))In One Dimension•Simple linear system•Fixed point•Solution –Stable if k < 0–Unstable if k > 0€ ˙ x =kx€ x =0 ⇒˙ x =0€ x€ ˙ x € x(t)=x0ektIn Two Dimensions•Often, position and velocity:•If actions are forces, causing acceleration:€ x=(x,v)Twhere v=˙ x € ˙ x =˙ x ˙ v ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ =˙ x ˙ ˙ x ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ =vforces⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ €The Damped Spring•The spring is defined by Hooke’s Law:•Include damping friction•Rearrange and redefine constantsxkxmmaF1−===&&xkxkxm&&&21−−=0=++ cxxbx&&&€ ˙ x =˙ x ˙ v ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ =˙ x ˙ ˙ x ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ =v−b˙ x −cx⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟The Linear Spring Model•Solutions are: •Where r1, r2 are roots of the characteristic equation € ˙ ˙ x +b˙ x +cx=0€ λ2+bλ +c =0€ r1,r2=−b± b2−4c2€ x(t)=Aer1t+Ber2t€ c ≠ 0Qualitative Behaviors•Re(r1), Re(r2) < 0 means stable.•Re(r1), Re(r2) > 0 means unstable.•b2-4c < 0 means complex roots, means oscillations.bcunstablestablespiralnodalnodalunstable€ r1,r2=−b± b2−4c2Generalize to Higher Dimensions•The characteristic equation for generalizes to –This means that there is a vector v such that•The solutions are called eigenvalues.•The related vectors v are eigenvectors.€ ˙ x =Ax€ det(A−λI)=0€ Av=λv€ λQualitative Behavior, Again•For a dynamical system to be stable:–The real parts of all eigenvalues must be negative.–All eigenvalues lie in the left half complex plane.•Terminology:–Underdamped = spiral (some complex eigenvalue)–Overdamped = nodal (all eigenvalues real)–Critically damped = the boundary between.Node BehaviorFocus BehaviorSaddle BehaviorSpiral Behavior(stable attractor)Center Behavior(undamped oscillator)The Wall Follower(x,y)€ θThe Wall Follower•Our robot model: u = (v )T y=(y )T   0.•We set the control law u = (v )T = Hi(y)€ ˙ x =˙ x ˙ y ˙ θ ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ =F (x,u) =vcosθvsinθω⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ € e=y−ysetso˙ e =˙ y and˙ ˙ e =˙ ˙ yThe Wall Follower•Assume constant forward velocity v = v0–approximately parallel to the wall:   0.•Desired distance from wall defines error:•We set the control law u = (v )T = Hi(y)–We want e to act like a “damped spring”€ e=y−ysetso˙ e =˙ y and˙ ˙ e =˙ ˙ y € ˙ ˙ e +k1˙ e +k2e=0The Wall Follower•We want •For small values of •Assume v=v0 is constant. Solve for –This makes the wall-follower a PD controller.€ ˙ ˙ e +k1˙ e +k2e=0€ ˙ e =˙ y = vsinθ ≈ vθ˙ ˙ e =˙ ˙ y = vcosθ˙ θ ≈ vω€ u =vω ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=v0−k1θ −k2v0e ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟= Hi(e,θ)Tuning the Wall Follower•The system is •Critically damped is •Slightly underdamped performs better.–Set k2 by experience.–Set k1 a bit less than € ˙ ˙ e +k1˙ e +k2e=0€ k12−4k2=0€ k1= 4k2€ 4k2An Observer for Distance to Wall•Short sonar returns are reliable.–They are likely to be perpendicular reflections.Experiment with Alternatives•The wall follower is a PD control law.•A target seeker should probably be a PI control law, to adapt to motion.•Try different tuning values for parameters.–This is a simple model.–Unmodeled effects might be significant.Ziegler-Nichols Tuning•Open-loop response to a step increase.d TKZiegler-Nichols Parameters•K is the process gain.•T is the process time constant.•d is the deadtime.Tuning the PID Controller•We have described it as:•Another standard form is:•Ziegler-Nichols says:€ u(t) =−kPe(t)−kIedt0t∫−kD˙ e (t)€ u(t) = −P e(t) + TIe dt0t∫+ TD˙ e (t) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥€ P =1.5 ⋅TK ⋅dTI= 2.5 ⋅d TD= 0.4 ⋅dZiegler-Nichols Closed Loop1. Disable D and I action (pure P control).2. Make a step change to the setpoint.3. Repeat, adjusting controller gain until achieving a stable oscillation.•This gain is the “ultimate gain” Ku.•The period is the “ultimate period” Pu.Closed-Loop Z-N PID Tuning•A standard form of PID is:•For a PI controller:•For a PID controller:€ u(t) = −P e(t) + TIe dt0t∫+ TD˙ e (t) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥€ P = 0.45 ⋅KuTI=Pu1.2€ P = 0.6