Building controlbuilding/docs/Bldg_control_lecture01small.pdfIntegral: PI MPC 2019-3-14 1.3 Building...

Building controlLecture 1: Feedback

Roy Smith

2019-3-14 1.1

Building control concepts

Objectives

I Comfort

I Efficiency

I Cost

Controlled (and measured) quantities

I Room temperatures

I CO2 levels

I Illumination (and glare)

I Humidity

Actuation

I Ventilation air temperature and flow

I Heating and cooling (HVAC)

I Blinds

I Lighting

2019-3-14 1.2

Building control

Feedback control structure

Buildingdynamics

thermal dynamicsair flow dynamics

Measurementsystem

y(t)

room temps: T (t)CO2 conc.

Actuationsystem

HVAC systemblinds

u(t)

disturbances: d(t)ambient temperatureradiation

ControllerADC+

noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue

Proportional: PIntegral: PI

MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


ControllerADC+

noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


ControllerADC+

noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


Controller

ADC+noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


ControllerADC+

noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


ControllerADC+

noise

analogueto digital

references: r(t)

and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control


Buildingdynamics


Measurementsystem

y(t)


Actuationsystem

HVAC systemblinds

u(t)


ControllerADC+

noise

analogueto digital

references: r(t)and predictions

DAC

digital toanalogue


MPC

2019-3-14 1.3

Building control

Control hierarchies

Actuationsystem

Buildingthermaldynamics

Roomtemps.

disturbances: d(t)

BoilerBoiler

controller

TABS water temp: TTABS

u(t) = TTABS

reference

2019-3-14 1.4

Simple feedback control

Unity gain negative feedback

G K +y(t) u(t) r(t)

−

Components: Feedback equations:

“Plant”: G y(t) = Gu(t)

Controller: K u(t) = K(r(t)− y(t))

2019-3-14 1.5

Plant models

A single zone building example

heating/coolingcommand: u(t)

heat loss/gain to ambientat temperature: Tamb(t)

zone temperaturemeasurement: Tz(t)

Plant model

Guy

2019-3-14 1.6

Plant models

Differential equation description

m air mass in zonec specific heat capacity for zone volumek0 thermal transmittance × zone area (to ambient)

mcd

dtTz(t) = k0 (Tamb(t)− Tz(t)) + u(t)

State-space representation

d

dtx(t) =

−k0mc

x(t) +

[k0mc

1

mc

] [Tamb(t)u(t)

]

Tz(t) = x(t)

State: x(t) = Tz(t)

A =−k0mc

, B =

[k0mc

1

mc

], C = 1, D = 0.

2019-3-14 1.7

Open-loop ambient response

0

7.07.6

16.417.0

23

Time(hours)

Temperature ◦C

0 6 12 18 24 30 36 42

Tz(t)Tamb(t)

Model parameters

Zone volume Vz 60 m3 Specific heat capacity c 1.01 J/g/K

Air density ρ 1.225 kg/m3 Total thermal transmittance k0 10 W/K

Air mass m 1000ρVz g

2019-3-14 1.8

Open-loop heat input response

Temperature ◦C

0

7

17

23

35

Time(hours)

0 6 12 18 24 30 36 42

Power [Watts]

0

100

200

Time(hours)

0 6 12 18 24 30 36 42

Tz(t)

Tamb(t)

u(t)

2019-3-14 1.9

Closed-loop proportional (P) control

Feedback loop

G K +y(t) u(t) r(t)

−

Proportional control law

u(t) = KP (r(t)− y(t)) r(t) is the zone temperature reference.

Closed-loop system equations

d

dtx(t) = −

(k0mc

+KP

mc

)x(t) +

k0mc

Tamb(t) +KP

mcr(t)

Tz(t) = x(t)

2019-3-14 1.10

Closed-loop proportional (P) control

Open-loop (uncontrolled) equilibrium

Tz = Tamb +u

k0.

Closed-loop (controlled) equilibrium

Tz =k0

k0 +KPTamb +

KP

k0 +KPr, KP > 0.

Equilibrium error

r − Tz =−k0

k0 +KPTamb +

k0k0 +KP

r, KP > 0.

2019-3-14 1.11

Closed-loop proportional control

Temperature ◦C

0

7

1720.25

23

Time(hours)

0 6 12 18 24 30 36 42

Power [Watts]

0

100

200

Time(hours)

0 6 12 18 24 30 36 42

Tz(t)

Tamb(t)

r(t)

u(t)

2019-3-14 1.12

Closed-loop proportional plus integral (PI) control

Feedback loop

G K +y(t) u(t) r(t)

−

Proportional plus integral control law

Define e(t) = r(t)− y(t)

The PI controller is,

u(t) = KP e(t) +KI

∫ t

0

e(t) dt

︸︷︷︸.

K e(t)

2019-3-14 1.13

Closed-loop proportional plus integral (PI) control

Dynamic controller

Define a controller state, xK(t) =

∫ t

0

e(t) dt.

Then,

d

dtxK(t) = e(t).

Controller state-space representation

d

dtxK(t) = 0xK(t) + 1 e(t)

u(t) = KI xK(t) + KP e(t).

Ke(t)u(t)

2019-3-14 1.14

Closed-loop state-space interconnection

AK BK

CK DK

KA B1 B2

C D1 D2

G

+y(t)

Tamb(t)

u(t) r(t)

−

State-space representation

d

dt

[x(t)xK(t)

]=

[A−B2DKC B2CK

−BKC AK

] [x(t)xK(t)

]+

[B1 B2DK

0 BK

] [Tamb(t)r(t)

]

y(t) =[C 0

] [ x(t)xK(t)

]+[0 0

] [Tamb(t)r(t)

]

Assumes D1 = D2 = 0 for simplicity.

2019-3-14 1.15

PI control properties

Stability

Check the eigenvalues of the closed-loop “A” matrix:

[A−B2DKC B2CK

−BKC AK

]=

−

k0 +KP

mc

KI

mc

−1 0

Steady-state error

dxK(t)

dt= 0 =⇒ 0 = −1x + 0xK + 1 r.

And so limt−→∞

e(t) = r(t)− y(t) = 0.

2019-3-14 1.16

Closed-loop integral control

Temperature ◦C

0

7

17

23

Time(hours)

0 6 12 18 24 30 36 42

Power [Watts]

0

100

200

Time(hours)

0 6 12 18 24 30 36 42

Tz(t)

Tamb(t)

r(t)

u(t)

PI control gains: KP = 10, KI = 0.003

2019-3-14 1.17

Closed-loop integral control

Temperature ◦C

0

7

17

23

Time(hours)

0 6 12 18 24 30 36 42

Power [Watts]

0

100

200

300

400

Time(hours)

0 6 12 18 24 30 36 42

Tz(t)

Tamb(t)

r(t)

u(t)

PI control gains: KP = 10, KI = 0.01

2019-3-14 1.18

Closed-loop integral control (comparions)

Temperature ◦C

0

7

17

23

Time(hours)

0 6 12 18 24 30 36 42

Power [Watts]

0

100

200

300

400

500

Time(hours)

0 6 12 18 24 30 36 42

Tz(t) (K1)

Tz(t) (K2)Tz(t) (K3)

Tamb(t)

r(t)

u(t) (K1)

u(t) (K2)

u(t) (K3)

PI control gains: KP = 10, KI = 0.0025 (K1)KI = 0.01 (K2)KI = 0.025 (K3)

2019-3-14 1.19

Feedback limitations

Poor performance for some signals

There are always exogeneous input signal (here r(t) or Tamb(t)) for which theclosed-loop system will perform worse than having no control at all.

These signals are usually faster than the typical response of the closed-loopsystem.

2019-3-14 1.20

Closed-loop integral control: disturbance response

21

22

23

24

25

26

Time(hours)

Temperature ◦C

12 15 18 21 24

Tamb

Tz (PI control)

Tz (open-loop)

2019-3-14 1.21

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