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Control Systems Lab SC4070

Lecture 3, February 13, 2012

dr.ir. Alessandro Abate

Delft Center for Systems and Control

Delft University of Technology

The Netherlands

e-mail: [email protected]

tel: 015 27 85606

(slides modified from the original, drafted by Robert Babuska)

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Lecture outline

Overview of control design methods.

Continuous vs. discrete time design. State-feedback control, observers.

Control architectures, nonlinear control.

PID controllers.

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Linear control design methods

P, PD, PI, PID, lead-lag control (classical, in frequency)

state feedback, output feedback (modern, in state-space)

LQR, linear quadratic control (optimal)

model predictive control (optimal, finite-horizon, constrained)

robust control (H, synthesis)

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Nonlinear control techniques

feedback linearization

sliding-mode control

nonlinear model predictive control

passivity-based control

knowledge-based control

adaptive control

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Computer-controlled systems: approach 1

y(t)u(t)Algorithm

Clock

A-D D-A System

Design a continuous-time controller, then make sure that the (dig-

ital) computer implementation approximates the continuous-time

controller as well as possible.

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Computer-controlled systems: approach 2

y(t)u(t)Algorithm

Clock

A-D D-A System

Describe the system from the computers (digital) viewpoint and

design directly a discrete-time controller.

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System from the computers viewpoint

y t( )u t( ) System

Clock

{ ( )}y tk{ ( )}u tk A-DD-A

Describe system (including converters) only over discrete sampling

instants.

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A-D Converter: Zero-Order Hold

Timet t t t t t 0 1 2 3 4 5

u(t) =u(tk), tk t

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Zero-order hold sampling of systems

Continuous-time system:

dx(t)

dt

= Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t)

Discrete-time system:

x(k+ 1) = x(k) +u(k)y(k) = Cx(k) +Du(k)

with (recall solution of non-autonomous ODE):

=eAh , =

h0

eAsdsB, where h is the sampling period

Reflect on eigenvalues placement

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Useful basic MATLAB commands

G = ss(A,B,C,D); % LTI continuous-time state-space model

h = 0.1; % sampling period [s]

H = c2d(G,h); % convert to discrete time (ZOH)

H = c2d(G,h,method); % method = foh, matched, ...

G = d2c(H); % convert to continuous time (ZOH)

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Selection of sampling period (1st order)

Number of samples per rise time: Nr= Tr

h 410

0 5

1

0

1(a)

0 50

1

0 5

1

0

1(b)

0 50

1

0 5

1

0

1(c)

0 50

1

0 5

1

0

1(d)

Time

0 50

1

Time

(a)Nr=1

(b)N

r=2

(c)Nr=4

(d)Nr=8

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Selection of sampling period (2nd order)

Nr= Tr

h 410 corresponds to 0h 0.20.6

0 50

1

(a)

0 50

1

(b)

0 50

1

(c)

Time

0 50

1

(d)

Time

(a)h=0.125(0h=0.23), (b)h=0.250(0h=0.46),

(c)h=0.500(0h=0.92), (d)h=1.000(0h=1.83)

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State feedback in DT: problem formulation

Discretize LTI model choosing a sampling interval

Model : x(k+ 1) = x(k) +u(k)

Linear controller:

u(k) = Lx(k)

Design parameters: closed-loop poles

Evaluation: compare x(k) and u(k) with specifications

(trade-off between control magnitude and speed of response)

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Poles placement: Ackermann formula

Compute L such that (L) has a desired characteristic poly-

nomial P(z). Ackermann formula:

L = (0 . . . 0 1)W1c P()

where P() is the desired characteristic polynomial in

Place poles inside unit ball

In Matlab:

L = acker(Phi,Gamma,Po) (SISO, numeric problems ?)

L = place(Phi,Gamma,Po) (MISO, more robust)

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Alternatively, choice of desired poles in CT

Use the continuous-time 2nd order model, study char. pol.:

s

2

+ 2s +2

,

which leads to z2 +p1z +p2 with

p1 = 2eh cosh

12

p2 = e2h

Thereafter in Matlab use c2d for closed-loop model

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Linear quadratic control: LQR

J=N

k=1

x(k)TQx(k) + u(k)TRu(k),

where (matrices, weights) Q, R are design parameters

A state feedback matrix Lthat gives a minimal Jcan be found by

solving the Riccati equation.

Similar to pole placement, but no need to define poles!

In Matlab: dlqr(Phi,Gamma,Q,R) % state weighting

In Matlab: dlqry(Phi,Gamma,C,D,Q,R) % output weighting

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State estimation: observers

x(k+ 1) = x(k) +u(k)

y(k) = Cx(k)

Assume input and output are available, reconstruct the state:

Direct calculation (requires differentiation)

Luenberger observer (model-based)

Kalman filter (optimal in presence of noise)

Note: the terms observer, estimator, filter are in this con-

text used synonymously

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State reconstruction based on a model

Consider the model: x(k+ 1) = x(k) +u(k)

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Model-based state estimation


Introduce feedback term from measured y(k)

x(k+ 1) = x(k) +u(k) + K[y(k)Cx(k)]

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State reconstruction based on a model


Introduce feedback from measured y(k)

x(k+ 1) = x(k) +u(k) + K[y(k)Cx(k)]

Defineestimation error e=x x

e(k+ 1) = e(k)KCe(k) = [KC]e(k)

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Observer block diagram

K

1z

u k( ) x k+( 1)^

y k( )^

y k( )

Cx k( )^

x k( )^

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Output feedback (observer + state feedback)

x(k+ 1) = x(k) +u(k) + K

y(k)Cx(k)

u(k) = L x(k)

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Poles of the closed-loop system

x(k+ 1) = x(k) +u(k)

e(k+ 1) = (KC)e(k)

u(k) = L(x(k) e(k))

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x(k+ 1) = x(k) +u(k)

e(k+ 1) = (KC)e(k)

u(k) = L(x(k) e(k))

x(k+ 1)e(k+ 1)

=L L

0 KCx(k)

e(k)

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x(k+ 1) = x(k) +u(k)

e(k+ 1) = (KC)e(k)

u(k) = L(x(k) e(k))

x(k+ 1)e(k+ 1)

=L L

0 KCx(k)

e(k)

Separation principle:(Closed-loop) Process poles: Ar(z) = det(zI+L)

Observer poles: Ao(z) =det(zI+ KC)

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Feed-forward (two-degree-of-freedom)

Goal: respond to a reference signal with desired specs.Replace u(k) = L x(k) by: u(k) = L x(k) +Lcuc(k)

ux^

Lc

Observer

Process-L

uc

y

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Feed-forward (two-degree-of-freedom)

Closed-loop system:

x(k+ 1) = (L)x(k) +Le(k) +Lcuc(k)

e(k+ 1) = (KC)e(k)

y(k) = Cx(k)

Transfer function from uc to y (for impulse response):

Hcl(z) = C(zI+L)1Lc=Lc

B(z)

Ar(z)

A li i li

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Application to a nonlinear system

Feedbackcontroller

Reference Nonlinearsystem

y0

y

u0

y0

Feedforwardcontroller

Li i i f li d l

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Linearization of nonlinear models

xnl = f(xnl,unl)

ynl = g(xnl,unl)

Choose x0, y0 or u0 such that 0= f(x0,u0) and y0=g(x0,u0) (equi-

librium). Linearize the above model:

x = fxnl

xnl=x0

(xnlx0) + funl

unl=u0

(unl u0)

ynly0 = g

xnl

xnl=x0

(xnlx0) + g

unl

unl=u0

(unl u0)

Li i ti f li d l

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Linearization of nonlinear models

x = f

xnl

xnl=x0

(xnlx0) + f

unl

unl=u0

(unl u0)

ynly

0 =

g

xnlxfnl=x0 (

xnlx

0) +

g

unl

unl=u0 (u

nlu

0)

x = Ax + Bu

y = Cx + Du

with x=xnlx0, u=unl u0, y=ynly0, etc.

C i f d l th h i l ti

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Comparison of models through simulation

u

y

u0

Linearized

model

Nonlinearsystem

y0

y

u=unl u0 ynl=y +y0

Alt ti l

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Alternatively . . .

u0

Linearizedmodel

Nonlinear

system

y0

u y

unl=u + u0 y=ynly0

Control b local linear controller

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Control by local linear controller

u

u0

Linearcontroller

Nonlinear

system

y0

y

Linearized

model

Model based adaptive control

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Model-based adaptive control

u

Adaptation

Design parameters

Controller

-

Process

Linearmodel

yr

ym

Continuous time PID controller

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Continuous-time PID controller

The textbook version of a PID controller:

u(

t) =

Ke(

t) +

1

Ti

t

e(

s)

ds+

Td

de(t)

dt

Continuous time PID controller

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Continuous-time PID controller

The textbook version of a PID controller:

u(t) =Ke(t) + 1Ti

t

e(s)ds + Td

de(t)

dt

A more realistic PID controller:

U(s) =K

bUc(s)

Y(s) +

1

sTi(Uc(s)

Y(s))

sTd

1 + sTd/NY(s)

Discrete time PID controller

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Discrete-time PID controller

P-term: P(k) =K(buc(k)y(k))

I-term: I(k+ 1) =I(k) +KTie(k)

D-term: D(k) = TdTd+Nh

D(k1) KTdNTd+Nh

(y(k)y(k1))

u(k) =P(k) +I(k) +D(k)

PID tuning

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PID tuning

Pole placement

Root locus

Bode diagram

Tuning rules (Ziegler-Nichols, tuning)

G(s) =et0s Kp

(s + 1) Kc=

Kp(+ t0), Ti= , Td=

t0

2

Cascaded control Example:Inverted Pendulum

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Cascaded control Example:Inverted Pendulum

Reference Position

controller InvertedpendulumAnglecontroller

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