– the 1st COE postdoctoral technical presentation –A Reference Governor Control Approachto Closed-loop Systems
with State and/or Control Constraints

1– the 1st COE postdoctoral technical presentation –Table of Contents

Motivation

Constrained system• Closed-loop systems with state and/or control constraints.

1. Motivation• A real control system necessarily has these constraints.

2. Constrained system

actuator bounds

physical limitsz(t)3. Reference governor

The 1st COE Postdoctoral and Doctoral Researchers Technical Presentation
April 22nd , 2004
Nara Institute of Science and Technology

Σp– degrade a system performance,– lead to instability of the designed control system.

Systems and Control Laboratory

2– the 1st COE postdoctoral technical presentation –3– the 1st COE postdoctoral technical presentation –Motivation
uc(t)
u(t)

w
reference & output

reference

0-1

y(t)

-2

z1

=

0
1

10xpu



, Σc :ẋcuc

• Constraint condition about control input−2.25 ≤ u ≤ 2.25


=

030

30−3−30





 w 

4xc
z1

123time [s]45Σp0z 1 w/o constraintz 1 w constraint
01

23time [s]4560-2uuc

1

2

3time [s]45Control output w/o constraint

6Reference
Governorw(t)• Manage a time-varying reference w into r to fuﬁll the constraint.
Gilbert and Kolmanovsky, 2002. Oh-hara and hirata, 2002

• Consider a tracking performance under a constant reference w̄.
(Suitable for mechanical systems)

20– Sugie and Yamamoto, 2001. Kogiso and Hirata, 2003.

-20r(t)– Kapasouris and others, 1988. Hirata and Fujita, 1999.

bounds on
42Σcw

6uu(t)y(t)-16control & boundẋp

6controlΣp :

Closed Loop System (Cosntrained System)
z(t)

1-20

• Plant Σp and controller Σc in state-space description

Reference governor(RG) approach

21w(t)

ΣcMotivation

Numerical example of constraint violation2

4– the 1st COE postdoctoral technical presentation –Motivation

Numerical example of constraint violation

w(t)• If such constraints are violated,

Kiminao KOGISO

ΣpΣcy(t)4. Simulation & experimental result
5. Summary & future planz1(t)u(t)

0123time [s]45Control output w/ constraint6• A few reports on experimental validation of the eﬀectiveness.

5– the 1st COE postdoctoral technical presentation –6– the 1st COE postdoctoral technical presentation –Goals in this study

Reference Governor• Linear discrete-time system: Σx = [ xp xc ] ∈ n , x(0) = x0

• A feasibility of a reference management (optimization) problem• A reference governor in a piecewise state aﬃne function over thespeciﬁed state region( i.e., r = F x + g ∀ x ∈ CR ).

r̂ Ty (t)u (t)Σp||z||2,P7

Reference GovernorO (w)∗r̂ Txs.t. z0 (τ ) ∈ Z, τ = 0, · · · , T − 1x(T ) ∈ O∞ (w̄)||z||2,P = z  P z,

=Q1 x0 + Q2 r̂T ，
Maximal Output Admissible set: O∞ (w̄)
MOA set for Σ has the following properties:i) x(T ) ∈ O∞ (w̄) ⇔ z0 (τ ) ∈ Z, ∀ τ ≥ T ,ii) a convex polyhedral set.
x(T)

∗∗

s.t. x(T ∗ ) ∈ O∞ (w̄)

Vz (x) = min z  Hz

Reference managingr(0), r(1), ::: r(T-1)
w x0?

z∗r̂

=FT∗

x+gT∗

9where r̂ k = F k x(t) + g k .
F 4x + g 4s.t. Gz ≤ N + Sx

∀ x ∈ CRT∗

2. → 3. Bemporad’s explicit LQR, Automatica 2002

O (w)
CR

4

F 1x + g1
CR 1F 2x + g2∗

3. Using KKT condition of MpQP, calculate F T and g T that gives∗∗
the optimal solutoin r̂T to RMP for a measured state x ∈ CRT .
T∗?

0Reference management rule: r̂ k (1) if x(t) ∈ CRk , k = 1, · · · , T ∗r(t) = w̄
if x(t) ∈ O∞ (w̄)

2. Multi-parametric quadratic programming problem(also feasible):∗∗a variable z = r̂ T + H −1 f (x) ∈ p1 T , a state parameter x ∈ n , x

RG in a piecewise state aﬃne fucntion∗z0 (τ ) ∈ Z τ = 0, · · · , T ∗ − 1wr(0), r(1), ::: r(T-1)

Reference Governor

ˆ T ||2,P + ||r̂ T − w̄
ˆ T ||2,Q
min||ẑ1T − w̄∗r̂ T

Reference managing– the 1st COE postdoctoral technical presentation –1. RMP(is feasible for given T ∗ ):

ˆ T ||2,P + ||r̂ T − w̄
ˆ T ||2,Qmin ||ẑ1T − w̄wii) a convex polyhedral set.
w

8Reference Governor

RG in a piecewise state aﬃne fucntion

Reference management problem(RMP)=i) x(0) ∈ O∞ (w̄) ⇔ z0 (τ ) ∈ Z, ∀ τ ∈ Z ,A full state x of Σ is measurable, and a reference is constant w̄ ∈ intW .

– the 1st COE postdoctoral technical presentation –− 1) ] ，ẑ1T
P = P  0+• Assumptions:x(T)P = P  0MOA set for Σ has the following properties:

z0 (t) ∈ Z := {z0 ∈ p0 | Mz z0 ≤ mz } ∀t ∈ Z+

– the 1st COE postdoctoral technical presentation –= z  P z,Maximal Output Admissible set: O∞ (w̄)

• Pointwise-in-time constraint condition:• A validation of the eﬀectiveness using a real control system.

[ r(0) · · · r(Tr̂T = [ r(0) · · · r(T − 1) ] ，ẑ1T = Q1 x0 + Q2 r̂T ，z0 (t) = C0 x(t) + D0 w(t)We easily expect that the proposing method
can be more robust for noises and unmodelled dynamics than the previous work...

z0 (τ ) ∈ Z, τ = 0, · · · , T − 1w (t)Σcxs.t. z0 (τ ) ∈ Z, τ = 0, 1, · · · , ∞Σ++z1(t)z1 (t) = C1 x(t)O (w)

ˆ T ||2,P + ||r̂ T − w̄
ˆ T ||2,Q
min ||ẑ1T − w̄State and Control
Constraintsz0(t)x(t + 1) = Ax(t) + Bw(t)Improvements in this studyr̂T

Reference management problem(RMP)

Formulation of constrained system (tracking ctrl.)• Constructing a reference governor for improvement of control
performance by achieving constraint fulﬁllment• Experimental validation of the proposed method

7– the 1st COE postdoctoral technical presentation –Constrained system

Motivation

F x3+ g3

CR 3x0

CR 2 x1

x2

O-1(w)

O-2(w)
O-3(w)

 11– the 1st COE postdoctoral technical presentation – 10– the 1st COE postdoctoral technical presentation –Model of DC position servomechanism
EncoderL

State and Control
Constraints++Σpz1(t)
y (t)

u (t)

Vr (t) ReferenceGovernor

ΣcLoadL• Identiﬁcation: ζ = 0.7, wn = 7.0(by data of 90 [deg] step response)

JLw (t) = w

MotorMShaftJM

GeDesign and implementation of PI-controllerarx (t)
M



Gear



The constrained system equipped with RG

For a measured state x(t), RG manages the constant reference w̄ intor and achieves the fulﬁllment of the constraint, z0 (t) ∈ Z, ∀ t ∈ Z+ ,of Σ. Furthermore, the output z1 has a good tracking performance.


 13

– the 1st COE postdoctoral technical presentation –0.5 ≤ t < 2.0 [s],w̄2 ,2.0 ≤ t [s],2w̄2 = −0.7853 [rad]

1.5

21

14

bounds oncontroller output

3volt [V]angle [rad]0.5referenceplant output

−100

−3
0.5

1

1.5

2time [s]

2.5

3

3.5

−4
00−1

reference
managedreferenceplant output
0.5

1

1.5

2time [s]

2.5

3

3.5

0.5

1

1.5

2time [s]

2.5

3

3.5

(a) Reference w and controlled output (b) Controller output u and the conθL
straint |u| ≤ 2.4 [V]

10 [ms]Real PlantHold
Circuit

1.521

10.50−10.5

11.5

2time [s]2.5

310 [ms]

10 [ms]Controller

Reference
Governor

3.50bounds oncontroller output

0−1

reference
managedreferenceplant output−0.5−2−3032
bounds on
controller output(a) Managed reference r and controlled (b) Controller output u andoutput θL
the constraint |u| ≤ 2.4 [V]

−2−1
00

1−10

0.5

−0.5

21

Experimental result with RGw̄1 = 1.5708 [rad]21.5

Simulation & experimental result

angle [rad]w̄1 , 15– the 1st COE postdoctoral technical presentation –3

volt [V]

>
>:

 14

Simulation result with RG

angle [rad]Reference w(t) =0 ≤ t < 0.5 [s],• Implemented to a computer (RT-Linux v3.1 Interl 3 733MHz 256MB)

Simulation & experimental result

Experimental result
without RG
80,• Discretized by sampling time 10 [ms] with a zero-holder• Control object:
Tracking control of θL to a reference w̄ fulﬁlling the constraint– the 1st COE postdoctoral technical presentation –1Tc s• Stabilization and tracking: Kc = 3，Tc = 3• Control u = V , output z1 = θL , and constraint |u| ≤ 2.4 [V]Simulation & experimental result>><Kc 1 +(b) A plant picture(a) A model illustration

−0.5

wn2
θL (s)=V (s)
s(s + 2ζwn )RΣvolt [V]z0(t)Simulation & experimental result

DC position servomechanism

RG in a piecewise state aﬃne fucntion

 12– the 1st COE postdoctoral technical presentation –Simulation & experimental result

Reference Governor

0.5

11.5

2time [s]−22.533.5−300.511.5

2time [s]2.53(a) Managed reference r and controlled (b) Controller output u andoutput θL
the constraint |u| ≤ 2.4 [V]
10 [ms]Real Plant

HoldCircuit

10 [ms]

10 [ms]Controller

ReferenceGovernor

3.5

 16

– the 1st COE postdoctoral technical presentation – 17– the 1st COE postdoctoral technical presentation –Summary & future plan

Experimental results by the most fundamental RGa

Summary120• We have clariﬁed the state region, which gives a feasiblity to thereference management problem.

controller output
bounds on voltage90300−30−60−90

reference
managed reference
heli direction−12005−0.38

100510time [s]Voltage [volt]

helicopter direction [deg]300−30−90

reference
managed reference
heli direction−120050 19

Summary & future plan

Future plan
• Control approach to constrained systems which includes networks.

reference

Internetobservation

Pointwise-in-time
constraint

Controller 1

output+Plant

SwitchController ncontrol– Dynamical control approach to networks– Simulation and experimental validation ofeﬀectiveness and practicability• Control applications for educational goals over networks

234567 t Step

01234

– Sampling time of Σ: Ts−0.38

10100510time [s]“A performance improving oﬀ-line reference management for systems with stateand control constraints”, IEEE CDC 2001

– the 1st COE postdoctoral technical presentation –w
r=30.3860

time [s]awr=1voltagebounds on voltage

90−60

Reference

0120

RC helicopter:
Control of yaw angle
in hovering.

• Multi-sampled implementation of RG
Referencetime [s]• To validate the eﬀectiveness, we have performed a simulation andexperiment. This experimental result has neither conservatism nor
violation for the constriaint fulﬁllment at all, and demonstrates thevalidity of this proposed method under a modeling error.

Another study

0.3860voltage [volt]

helicopter direction [deg]• We have constructed a reference governor in a state aﬃne function
which is derived from the reference management (optimization)
problem.

 18– the 1st COE postdoctoral technical presentation –Summary & future plan

Summary & future plan

– Reference management period: νTs– Constraint fulﬁllment for each Ts sampled data
– Reduction of implemented data of RG

567t Step