14 Feedback Control

8/10/2019 14 Feedback Control

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Introductory ControlTheoryI400/B659: Intelligent roboticsKris Hauser


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Control Theory

The use of feedbackto regulate a signal

Controller

Plant

Desiredsignal xd

Signal x Control input u

Error e = x-xd(By convention, xd= 0)

x = f(x,u)


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What might we be interested in?

Controls engineering

Produce a policy u(x,t), given a description of the

plant, that achieves good performance

Verifying theoretical properties Convergence, stability, optimality of a given policy

u(x,t)


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Agenda

PID control

Feedforward + feedback control

Control is a hugetopic, and we wont dive into

much detail


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Model-free vs model-based

Two general philosophies:

Model-free: do not require a dynamics model to be provided

Model-based: do use a dynamics model during computation

Model-free methods:

Simpler Tend to require much more manual tuning to perform well

Model-based methods:

Can achieve good performance (optimal w.r.t. some cost function)

Are more complicated to implement

Require reasonably good models (system-specific knowledge) Calibration: build a model using measurements before behaving

Adaptive control: learn parameters of the model online fromsensors


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PIDcontrol

Proportional-Integral-Derivativecontroller

A workhorse of 1D control systems

Model-free


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Proportional term

u(t) = -Kpx(t)

Negative sign assumes control acts in the same

direction as x

xt

Gain


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Integral term

u(t) = -Kpx(t) - KiI(t)

I(t) =

0 (accumulation of errors)

xt

Residual steady-state errors driven

asymptotically to 0

Integral gain


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Instability

For a 2ndorder system (momentum), P control

xt

Divergence


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Derivative term

u(t) = -Kpx(t)Kdx(t)

x

Derivative gain


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Putting it all together

u(t) = -Kpx(t) - KiI(t) - Kdx(t)

I(t) =

0


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I(t) =

0

Intuition: Kpcontrols the spring stiffness

Kicontrols the learning rate

Kdcontrols the damping


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I(t) =

0

Intuition: Kpcontrols the spring stiffness

Kdcontrols the amount of damping

Kicontrols the learning rate

Control limits If u is bounded in range [umin,umax], need to:

1. Clamp u

2. Bound the magnitude of the I term to prevent overshoot


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Parameter tuning


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Example: Damped Harmonic

Oscillator Second order time invariant linear system, PID controller

x(t) = A x(t) + B x(t) + C + D u(x,x,t)

For what starting conditions, gains is this stable and

convergent?


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Stability and Convergence

System is stableif errors stay bounded

System is convergentif errors -> 0


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Example: Damped Harmonic

Oscillator x = A x + B x + C + D u(x,x)

PID controller u = -KpxKdx Ki I

x = (A-DKp) x + (B-DKd) x + C - D Ki I

Assume Ki=0


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Homogenous solution

Instable if A-DKp > 0

Natural frequency w0= sqrt(DKp-A)

Damping ratio z=(DKd-B)/2w0

If z> 1, overdamped

If z< 1, underdamped(oscillates)


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Example: Trajectory following

Say a trajectory xdes(t) has been designed

E.g., a rockets ascent, a steering path for a car, a planes landing

Apply PID control

u(t) = Kp(xdes(t)-x(t)) - KiI(t) + Kd(xdes(t)-x(t))

I(t) = 0

The designer of xdesneeds to be knowledgeable about the

controllers behavior!

xdes(t)x(t)

x(t)


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Controller Tuning Workflow

Hypothesize a control policy

Analysis:

Assume a model

Assume disturbances to be handled

Test performance either through mathematical analysis, orthrough simulation

Go back and redesign control policy

Mathematical techniques give you more insight to improve

redesign, but require more work


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Example


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Feedforward control

If we knowa model for a system and know how it should

move, why dont we just compute the correct control?

Ex: damped harmonic oscillator

x = A x + B x + C + D u

Calculate a trajectory x(t) leading to x(T)=0 at some point T

Compute its 1stand 2ndderivatives x(t), x(t)

Solve for u(t) = 1/D*(x(t) - A x(t) + B x(t) + C)

Would be perfect!


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Feedforward control

If we knowa model for a system and know how it should

move, why dont we just compute the correct control?

Ex: damped harmonic oscillator

x = A x + B x + C + D u

Calculate a trajectory x(t) leading to x(T)=0 at some point T

Compute its 1stand 2ndderivatives x(t), x(t)

Solve for u(t) = 1/D*(x(t) - A x(t) + B x(t) + C)

Problems

Control limits: trajectory must be planned with knowledge ofcontrol constraints

Disturbances and modeling errors: open loop control leads to

errors not converging to 0


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Handling errors: feedforward +

feedback Idea: combine feedforward control uff with feedback control

ufb e.g., ufbcomputed with PID control

Feedback term doesnt have to work as hard (= 0 ideally)

Plant

uffFeedforward

calculationx(0) +

ufb

Feedback

controller

u

x(t)

xdes


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Application: Feedforward control

Feedback control: let torques be a

function of the current error between

actual and desired configuration

Problem: heavy arms require strong

torques, requiring a stiff system

Stiff systems become unstable relatively

quickly


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Application: Feedforward control

Solution: include feedforward torques

to reduce reliance on feedback

Estimate the torques that would

compensate for gravity and achieve

desired accelerations (inverse

dynamics), send those torques to the

motors


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Handling errors: feedforward +

feedback Idea: combine feedforward control uff with feedback control

ufb e.g., ufbcomputed with PID control

Feedback term doesnt have to work as hard (= 0 ideally)

Plant

uffFeedforward

calculationx(0) +

ufb

Feedback

controller

u

x(t)

xdes

Model based

Model free


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Model predictive control

(MPC) Idea: repeatedly compute feedforward control using model of

the dynamics

In other words, rapid replanning

Plant

Feedforward

calculation u=uff

x(t)

xdes


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MPC to avoid moving obstacles


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Next class: Sphero Lab

You will design a feedback controller for the move command

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14 Feedback Control

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