Reference Governor - LIRMM

Reference Governor

1

• Reference governor is an add-on safety supervisor for the

existing/legacy controllers

• Monitors and modifies commands if necessary to ensure

constraints are satisfied

Nominal closed-loop system with

an existing/legacy controller

2

Reference Governor

𝑡 ⋅ 𝑇𝑠

Basic idea: Compute 𝑣(𝑡) so that if constantly applied it would not lead to constraint violations

3

Reference Governor


𝑡 ⋅ 𝑇𝑠

4

Reference Governor


𝑡 ⋅ 𝑇𝑠

5

y(t)

Reference Governor


𝑡 ⋅ 𝑇𝑠

6

v(t+𝑘)=

y(t)

Reference Governor


𝑡 ⋅ 𝑇𝑠

7

v(t+𝑘)=

y(t)

v(t)

Reference Governor


𝑡 ⋅ 𝑇𝑠

8

v(t+𝑘)=

y(t)

v(t)

Reference Governor

EXPERIMENTS Plant: Inverted Pendulum

Control Law: Linear Quadratic Regulator

LQR𝑢

𝑥

𝑣

𝑟

𝑟

𝑣 𝑣

9Slides from 2014 IEEE CDC Workshop by E. Garone, S. Di Cairano, and I.V. Kolmanovsky

EXPERIMENTS Plant: Inverted Pendulum

Control Law: Linear Quadratic Regulator

𝑟

LQR𝑢

𝑥

RG

𝑟𝑣

𝑣

10Slides from 2014 IEEE CDC Workshop by E. Garone, S. Di Cairano, and I.V. Kolmanovsky

subject to

Maximize 𝜅(𝑡)

𝑣(𝑡)𝑥(𝑡)

∈ 𝑃 ⊆ 𝑂∞

𝑣 𝑡 = 𝑣 𝑡 − 1 + 𝜅 𝑡 𝑟 𝑡 − 𝑣 𝑡 − 1 ,

0 ≤ 𝜅(𝑡) ≤ 1

11

Scalar Reference Governor

• 𝑂∞ is the set of safe pairs of initial states, 𝑥 0 , and

constant commands, 𝑣 𝑡 ≡ 𝑣, which do not cause

subsequent constraint violation

𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑣, 𝑦 𝑡 = 𝐶𝑥 𝑡 + 𝐷𝑣 ∈ 𝑌 ⇒

𝑂∞ = ሼ 𝑣, 𝑥(0) : 𝐶𝐴𝑡𝑥(0) + 𝐶 𝐼 − 𝐴𝑡 𝐼 − 𝐴 −1𝐵𝑣 + 𝐷𝑣 ∈ 𝑌,𝑡 = 0,1,⋯ ,∞ }

• Example: For asymptotically stable observable linear system:

12

Safe Set

13

• Finitely determined inner approximation is obtained by

slightly tightening the “steady-state” constraints

෨𝑂∞ = ሼ 𝑣, 𝑥 0 : (𝐶 𝐼 − 𝐴 −1𝐵 + 𝐷)𝑣 ∈ 1 − 휀 𝑌, 𝐶𝐴𝑡𝑥 0 + 𝐶 𝐼 − 𝐴𝑡 𝐼 − 𝐴 −1𝐵 + 𝐷𝑣 ∈ 𝑌,𝑡 = 0,1,⋯ , 𝑡∗} ⊂ 𝑂∞

Implementation based on subsets

14

• If the constraint set is polyhedral, then ෨𝑂∞ is polyhedral

Safe Sets

𝑌 = 𝑦:𝐻𝑦 ≤ ℎ ⇒

෨𝑂∞ = 𝑣, 𝑥 0 :

𝐻𝐶 𝐼 − 𝐴 −1𝐵 + 𝐷 0𝐻𝐷

𝐻𝐶𝐵 + 𝐻𝐷𝐻𝐶𝐻𝐶𝐴

⋮𝐻𝐶 𝐼 − 𝐴𝑘 𝐼 − 𝐴 −1𝐵 + 𝐻𝐷

⋮

⋮𝐻𝐶𝐴𝑘

⋮

𝑣𝑥(0) ≤

1 − 휀 ℎℎℎ⋮ℎ⋮

• Redundant and “almost redundant” inequality constraints are

eliminated while remaining constraints are tightened to obtain

a simply represented 𝑃 ⊆ ෨𝑂∞

Computing 𝑷

Computing 𝜿

17

Example

Model:

𝑥1 𝑡 + 1 = 𝑥1 𝑡 + 0.1𝑥2 𝑡 ,𝑥2(𝑡 + 1) = 𝑥2 𝑡 + 0.1𝑢(𝑡)

Constraints:

|𝑥1| ≤ 1,|𝑥2| ≤ 0.1,

|𝑢| ≤ 0.1

Nominal closed-loop:

𝑢 = −0.917 𝑥1 − 𝑟 − 1.636𝑥2,

Reference command

𝑟 𝑡 = 0.5.

0 20 40 60 80-0.1

0

0.1

0.2

0.3

0.4

0.5

t

x1

x2

u

Response without reference governor

18

Example (cont’d)

𝑢 = −0.917 𝑥1 − 𝑣 𝑡 − 1.636𝑥2 , 𝑣(𝑡) = 𝑅𝐺(𝑣 𝑡 − 1 , 𝑥 𝑡 )

Response with reference governor

19

Example (cont’d)

Cross-sections of 𝑃 = ෨𝑂∞ :

1 DOF spacecraft with flexible appendage

1 DOF spacecraft with flexible appendage

Almost redundant constraint elimination

Vahidi, A., Kolmanovsky, I.V., and Stefanopolou, A., "Constraint handling in a fuel cell system: A fast reference

governor approach," IEEE Transactions on Control Systems Technology, vol. 15, no. 1, pp. 86-98, January, 2007.

More general sets 𝑷, nonlinear systems…

Scalar reference governor

Using 𝑷 without actually computing it

Response prediction based on linear model

Online prediction-based reference governor

Nicotra, M., Garone, E., and Kolmanovsky, I.V., “A fast reference governor for linear systems,” AIAA

Journal of Guidance, Control, and Dynamics, vol. 40, no. 2, pp. 460-464, 2017.

• Linear and nonlinear systems with set-bounded

disturbances and parameter uncertainties can be treated

• Feasibility at initial time implies constraint adherence and

recursive feasibility for all future times

• Finite-time convergence of 𝑣(𝑡) to 𝑟(𝑡) or nearest steady-

state feasible value for constant 𝑟(𝑡)

• Similar convergence results for ``nearly constant’’ and

slowly-varying 𝑟(𝑡)

• Enlarged constrained domain of attraction

28

Remarks on existing theory

Survey paper

Reference governor extensions



Implementing Linear

Design on a

Nonlinear System

33

Adopting linear design to a nonlinear system

12/15/2014

• Consider a disturbance-free nonlinear system

𝛿𝑥 𝑡 + 1 = 𝐴𝛿𝑥 𝑡 + 𝐵𝛿𝑣(𝑡) ∈ 𝑌

𝛿𝑥 𝑡 = 𝑥 − 𝑥𝑜𝑝,

𝛿𝑣 𝑡 = 𝑣 − 𝑣𝑜𝑝,

𝑓 𝑥𝑜𝑝, 𝑣𝑜𝑝 = 0

𝑦𝑙𝑖𝑛 𝑡 = 𝐶 𝛿𝑥 𝑡 + 𝐷 𝛿𝑣 𝑡

• Let a linearization of the nonlinear model at an operating

point (𝑥𝑜𝑝, 𝑣𝑜𝑝, 𝑦𝑜𝑝) be given by

𝑥 𝑡 + 1 = 𝑓 𝑥 𝑡 , 𝑣 𝑡

𝑦𝑛𝑜𝑛𝑙 𝑡 = 𝑔 𝑥 𝑡 , 𝑣 𝑡 ∈ 𝑌

34


• Main idea: Correct the linear model prediction into the future

by a disturbance term by 𝑑(𝑡)

ො𝑦𝑛𝑜𝑛𝑙 𝑡 + 𝑘|𝑡 = 𝑦𝑜𝑝 + 𝑦𝑙𝑖𝑛 𝑡 + 𝑘 𝑡 + 𝑑 𝑡

෨𝑂∞,𝑎𝑢𝑔 = ሼ 𝛿𝑣, 𝛿𝑥 0 , 𝑑 :

𝐶𝐴𝑡𝛿𝑥 0 + 𝐶 𝐼 − 𝐴𝑡 𝐼 − 𝐴 −1𝐵𝛿𝑣 + 𝐷𝛿𝑣 + 𝑑 ∈ 𝑌~ሼ𝑦𝑜𝑝},𝑡 = 0,1,⋯ ,∞ }⋂Γ∞

Vahidi, K, Stefanopoulou, IEEE TCST 15 (1), 86-98 (2007)

• Let 𝑑 𝑡 = 𝑦𝑛𝑜𝑛𝑙 𝑡 − 𝑦𝑙𝑖𝑛 𝑡 − 𝑦𝑜𝑝 be the output deviation

from the output predicted by the linear model at current time

• Define

35


𝑣 𝑡 − 1 + 𝛽 𝑡 𝑟 𝑡 − 𝑣 𝑡 − 𝑣𝑜𝑝𝛿𝑥(𝑡)𝑑(𝑡)

∈ ෨𝑂∞,𝑎𝑢𝑔

𝛽 𝑡 → max 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 0 ≤ 𝛽 𝑡 ≤ 1

• Reference governor logic:

𝑑 𝑡 = 𝑦𝑛𝑜𝑛𝑙𝑖𝑛 𝑡 − 𝑦𝑙𝑖𝑛 𝑡 − 𝑦𝑜𝑝

𝑎𝑛𝑑

36

Discussion

• The proposed technique is motivated by a similar scheme in

MPC

• It is heuristic but has been shown to work well in several

applications

• The study of its theoretical properties remains an open

research problem

• Extensions to command governor and extended command

governor cases are feasible

37

Controller state and reference governor (CSRG)

Controller PlantCSRG

K. McDonough and I.V. Kolmanovsky, “Controller state and reference governors for discrete-time

linear systems with pointwise-in-time state and control constraints,” Proceedings of 2015 American

Control Conference, Chicago, IL, pp. 3607-3612, 2015.

Gas turbine engine

Command

CSRG modified

command

Fan speed

Constraints

Linear model simulations

• Trim point to trim point transition feasibility is

determined based on set of states that can be

recovered by CSRG

• The actual transitions are controlled by CSRG

Envelope-aware flight management system

Di Donato, P.F.A., Balachandran, S., McDonough, K., Atkins, E., and Kolmanovsky, I.V., “Envelope-

aware flight management for loss of control prevention given rudder jam,” AIAA Journal of Guidance,

Control, and Dynamics, vol. 40, pp. 1027-1041, 2017.

Chance constrained reference governor

Kalabic, U., Vermillion, C., and Kolmanovsky, I.V. “Constraint enforcement for a lighter-than-air wind-energy

system: An application of reference governors with chance constraints,” Proceedings of 20th IFAC World

Congress, Toulouse, France, IFAC-PapersOnLine, vol. 50, no. 1, pp. 13258-13263, July 2017.

Formation control

Frey, G., Petersen, C., Leve, F., Garone, E., Kolmanovsky, I.V. and Girard, A., “Parameter governors for

coordinated control of n-spacecraft formations,” AIAA Journal of Guidance, Control, and Dynamics, vol. 40,

no. 11, pp. 3020-3025, November, 2017.

Concluding remarks

Controller PlantReference

governor

• Augment rather than replace nominal controller

• Inactive if no danger of constraint violation

• Easy to implement / fast online computations

• Special properties

• Much room for future research and applications

BACKUP SLIDES

45

The Extended Command

Governor (ECG)

46

Agenda

• Extended Command Governor (ECG)

• Design of ancillary dynamical system

• Response properties

• Interpretation as a form of Model Predictive Controller (MPC)

• Application examples

• Command governor and vector reference governor

47

Extended command governor

(Gilbert and Ong, Automatica, Vol. 47, pp. 334-340, 2011)

Motivation:

• Enlarge constrained region of attraction

• Provide faster response

• Increase robustness to unmodeled dynamics

48


ҧ𝑥 𝑡 + 1 = ҧ𝐴 ҧ𝑥 𝑡

𝜌 𝑡 + 1 = 𝜌 𝑡

𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑣 𝑡

𝑦 𝑡 = 𝐶𝑥 𝑡 + 𝐷𝑣 𝑡 ∈ 𝑌

• Auxiliary “command generating” subsystem:

• System:

𝑣 𝑡 = 𝜌 𝑡 + ҧ𝐶 ҧ𝑥 𝑡

• Requirement: ҧ𝐴 is asymptotically stable (Schur)

Auxiliary command generator

𝜌(0)

𝑣 𝑡 = ҧ𝐶 ҧ𝐴𝑡 𝑥 0 + 𝜌(0)

0 𝑡


𝜌 𝑡 + 1 = 𝜌 𝑡


𝑣 𝑡

• Auxiliary “command generating” subsystem:

50

Augmented system


𝜌 𝑡 + 1 = 𝜌 𝑡

𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑣 𝑡



• Combined system:

• Constraints:

51

Augmented system


• Augmented system with constraints

ҧ𝑥(𝑡 + 1)𝑥(𝑡 + 1)

= 𝐴𝑎ҧ𝑥(𝑡)𝑥(𝑡)

+ 𝐵𝑎 𝜌 𝑡

𝐴𝑎 =ҧ𝐴 0

𝐵 ҧ𝐶 𝐴, 𝐵𝑎 =

0𝐵

𝐶𝑎 = 𝐷 ҧ𝐶 𝐶 𝐷𝑎 = 𝐷

52

Strictly steady-state admissible commands

• Set Γ∞ of steady-state admissible constant references:

Γ∞ = ሼ 𝜌 0 , ҧ𝑥 0 , 𝑥(0) : 𝜌(0) ∈ ℛ}

• Tightened set of steady-state feasible commands:

𝑟 ∈ ℛ ⇒ 𝐻𝑟 ∈ 1 − 𝜖 𝑌

• Static gain 𝐻 = 𝐶 𝐼 − 𝐴 −1𝐵 + 𝐷

0 ∈ 𝑖𝑛𝑡 𝑌, 𝑌 is compact, 0 < 𝜖 < 1

53

The set 𝑶∞

𝑂∞ = ሼ 𝜌 0 , ҧ𝑥 0 , 𝑥(0) : 𝑦(𝑡) ∈ 𝑌, 𝑡 = 0,1,⋯ ,∞}

• Safe set

54

The set ෩𝑶∞

• Safe set is tightened in steady-state

෨𝑂∞ = 𝑂∞ ⋂Γ∞

• Properties (under suitable assumptions):

• 𝑟 ∈ ℛ ⇒ (𝑟, 0, 𝑥𝑠𝑠 𝑟 ) = (𝑟, 0, 𝐻𝑟) ∈ 𝑖𝑛𝑡 𝑂∞

• ෨𝑂∞ is positively invariant for augmented system

• ෨𝑂∞ is finitely-determined

55

• Suppose 𝑌 is a polytope: 𝑌 = ሼ𝑦: Λy ≤ 𝜆}

𝐻𝜌,𝑡 𝜌 + 𝐻 ҧ𝑥,𝑡 ҧ𝑥 0 +𝐻𝑥,𝑡 𝑥 0 ≤ 𝜆

𝐻∞𝜌 ≤ (1 − 𝜖)𝜆

𝐻 ҧ𝑥,𝑡, 𝐻𝑥,𝑡 = Λ𝐶𝑎𝐴𝑎𝑡 ,

𝐻𝜌,𝑡 = Λ(𝐶𝑎 𝐼 − 𝐴𝑎𝑡 𝐼 − 𝐴𝑎

−1𝐵𝑎 + 𝐷𝑎)

𝐻∞ = Λ(𝐶𝑎 𝐼 − 𝐴𝑎−1𝐵𝑎 + 𝐷𝑎)

The set ෩𝑶∞

• Then ෨𝑂∞ is defined by affine inequalities on 𝜌, ҧ𝑥 0 , 𝑥 0 :

• Consider the disturbance-free case (𝑊 = 0)

• Inequalities for all 𝑡 sufficiently large (𝑡 ≥ 𝑡∗) are redundant

and need not be included

(𝑡 = 0,12,⋯ )

(0 < 𝜖 ≪ 1)

56

subject to

𝜌 𝑡 , ҧ𝑥 𝑡 , 𝑥(𝑡) ∈ ෨𝑂∞

𝐽 = 𝜌 𝑡 − 𝑟 𝑡𝑇𝑆(𝜌 𝑡 − 𝑟 𝑡 ) + ҧ𝑥 𝑡 𝑇 ҧ𝑆 ҧ𝑥(𝑡) → 𝑚𝑖𝑛𝜌(𝑡), ҧ𝑥(𝑡)


• Optimization problem:

• Command computation based on ҧ𝑥(𝑡) and 𝜌(𝑡):

𝑣 𝑡 = ҧ𝐶 ҧ𝑥 𝑡 + 𝜌(𝑡)

57


ҧ𝐴𝑇 ҧ𝑆 ҧ𝐴 − ҧ𝑆 < 0, ҧ𝑆𝑇 = ҧ𝑆 > 0

• Assumption 2: The weight ҧ𝑆 in the cost function must satisfy

• Assumption 1: The weight 𝑆 in the cost function satisfies

𝑆𝑇 = 𝑆 > 0

• Observation: If 𝜌 𝑡 − 1 , ҧ𝑥(𝑡 − 1) are feasible at time 𝑡 − 1,

then

𝜌 𝑡 = 𝜌 𝑡 − 1 , 𝑥 𝑡 = ҧ𝐴 ҧ𝑥(𝑡 − 1)

are feasible at time 𝑡 and

𝑥𝑇 𝑡 ҧ𝑆 𝑥 𝑡 ≤ ҧ𝑥𝑇 𝑡 − 1 ҧ𝑆 ҧ𝑥(𝑡 − 1)

58

Observations

• Suppose 𝑟 𝑡 = 𝑟(𝑡 − 1)

• Let 𝐽∗(𝑡) denote the optimal cost. Then:

𝐽∗ 𝑡 = 𝜌 𝑡 − 𝑟 𝑡𝑇𝑆 𝜌 𝑡 − 𝑟 𝑡 + ҧ𝑥 𝑡 𝑇 ҧ𝑆 ҧ𝑥 𝑡

= 𝐽∗ 𝑡 − 1

≤ 𝜌 𝑡 − 𝑟 𝑡𝑇𝑆 𝜌 𝑡 − 𝑟 𝑡 + 𝑥 𝑡 𝑇 ҧ𝑆 𝑥 𝑡

≤ 𝜌 𝑡 − 1 − 𝑟 𝑡 − 1𝑇𝑆 𝜌 𝑡 − 1 − 𝑟 𝑡 − 1

+ ҧ𝑥 𝑡 − 1 𝑇 ҧ𝑆 ҧ𝑥 𝑡 − 1

• The optimal cost is non-increasing, 𝐽∗ 𝑡 ≤ 𝐽∗(𝑡 − 1)

59

Comments [1]

• ECG plans a recovery command sequence as an output of

a stable auxiliary system to avoid constraint violation and

minimize interference with the system operation

• The first element of the recovery sequence, 𝑣 𝑡 , is

applied to the system

• If ෨𝑂∞ is polyhedral, the ECG optimization problem is a

Quadratic Program (QP) with linear inequality constraints

• This QP can be solved online by a QP solver [such as

PQP, GPAD, Qpkwik, CVX,…] or explicitly by multi-

parametric solvers (MPT or hybrid toolbox)

60

Comments [2]

• ECG achieves large constrained domain of attraction

(= 𝑃𝑟𝑜𝑗𝑥 ෨𝑂∞). It is typically larger than that of RG

• ECG achieves faster response, i.e., faster

convergence of 𝑣(𝑡) to 𝑟, in particular, for systems with

actuator rate limits

• Improved robustness to model uncertainty observed in

simulations

61

Theoretical results

• Suppose a feasible solution exists at time 0 and 𝑟 𝑡 = 𝑟𝑠

for all 𝑡 ≥ 𝑡𝑠. Define 𝑟𝑠∗ = argmin

𝑟∈ℛ𝑟 − 𝑟𝑠

2be the

nearest feasible reference

• Then there exists a 𝑡𝑓 ∈ 𝑍+ such that 𝑣 𝑡 = 𝑟𝑠∗ for all 𝑡 ≥ 𝑡𝑓 .

• Given 𝜖 > 0, there exists a 𝑡𝜖 ∈ 𝑍+ such that

𝑥 𝑡 ∈ 𝐹∞ 𝑟𝑠∗ + 𝜖𝐵𝑛 for all 𝑡 ≥ 𝑡𝜖

(Gilbert and Ong, 2011)

62

Sketch of the proof

• 𝐽∗(𝑡) is monotonically non-increasing, hence

𝐽∗ 𝑡 − 1 − 𝐽∗ 𝑡 → 0

• Using the properties of minimum norm projection on a closed

and convex set, it is shown that

ҧ𝐴 ҧ𝑥 𝑡 − 1 − ҧ𝑥 𝑡ҧ𝑆

2+ 𝜌(𝑡 − 1) − 𝜌 𝑡

𝑆

2≤ 𝐽∗ 𝑡 − 1 − 𝐽∗(𝑡)

• Henceҧ𝐴 ҧ𝑥 𝑡 − 1 − ҧ𝑥 𝑡 → 0

𝜌 𝑡 − 1 − 𝜌 𝑡 → 0 as 𝑡 → ∞

63

Sketch of the proof

• Apply Lemma with

𝑧 = ҧ𝐴 ҧ𝑥 𝑡 − 1 , 𝜌 𝑡 − 1 , 𝑧𝑜𝑝 = ҧ𝑥 𝑡 , 𝜌 𝑡 ,

𝑧𝑠 = 0, 𝑟 , 𝑄 = 𝑑𝑖𝑎𝑔( ҧ𝑆, 𝑆)

64

Sketch of the proof

𝑧𝑜𝑝

𝑧𝑠

𝑧𝑎

𝑏𝑐

𝑎2 + 𝑏2 ≤ 𝑐2 since 𝜃 ≥ 𝜋/2

𝑍

𝜃

⇒ 𝑎2 ≤ 𝑐2 − 𝑏2

65

Sketch of the proof

• Thus ҧ𝑥 𝑡 = ҧ𝐴 ҧ𝑥 𝑡 − 1 + 𝜂(𝑡), and 𝜂 𝑡 → 0 as 𝑡 → ∞.

• Since ҧ𝐴 is Schur, ҧ𝑥 𝑡 → 0.

• Thus 𝑣 𝑡 − 𝑣 𝑡 − 1 → 0 and 𝑥 𝑡 → 𝑥𝑠𝑠 𝑡

• The proof is finalized by strict constraint admissibility in

steady-state. Formally, we demonstrate that for large 𝑡and 휀 > 0 sufficiently small,

𝜌 𝑡 − 휀𝜌 𝑡 −𝑟𝑠

𝜌 𝑡 −𝑟𝑠, ҧ𝑥(𝑡) = 0

are feasible.

• This is only possible if 𝜌 𝑡 = 𝑟𝑠 and hence 𝑣 𝑡 = 𝑟𝑠

66

Choices of ഥ𝑨, ഥ𝑪

• Shift register of length 𝑛 ҧ𝑥:

ҧ𝐴 =

0 𝐼0 0

0 ⋯𝐼 ⋯

⋮ ⋮0 0

⋮ ⋮⋯ 𝐼

, ҧ𝐶 = 𝐼 0 ⋯ 0

• Auxiliary system outputs a recovery sequence “stored” in ҧ𝑥(0)

• 𝑣(𝑡) converges to a constant, 𝜌 0 , in 𝑛 ҧ𝑥 steps

67

Example

• Example: 𝑣 ∈ 𝑅1, 𝑛 ҧ𝑥 = 3

• Augmented state

𝑣 𝑛 ҧ𝑥 + 𝑘 = 𝜌 0 , 𝑘 ≥ 0

ҧ𝑥 =ҧ𝑥1ҧ𝑥2ҧ𝑥3

• Generated command sequence:

𝑣 0 = 𝜌 0 + ҧ𝑥1 0 ,𝑣 1 = 𝜌 0 + ҧ𝑥2 0 ,𝑣 2 = 𝜌 0 + ҧ𝑥3 0

68

Choices of ഥ𝑨, ഥ𝑪

• Laguerre Sequence Generator:

ҧ𝐴 =

휀𝐼 𝛽𝐼 −휀𝛽𝐼 휀2𝛽𝐼 ⋯

0 휀𝐼 𝛽𝐼 −휀𝛽𝐼 ⋯00⋮

00⋮

휀𝐼0⋮

𝛽𝐼휀I⋮

⋯⋯⋱

,

ҧ𝐶 = 𝛽 𝐼 −휀I 휀2𝐼 −휀3𝐼 ⋯ −휀 𝑁−1𝐼

where 𝛽 = 1 − 휀2, and 0 ≤ 휀 ≤ 1 is a selectable parameter that

corresponds to the time constant of the fictitious dynamics. Note that

with 휀 = 0, this is the shift register.

(Kalabic et. al., Proc. of 2011 MSC)

69

Command governor

subject to

𝑣 𝑡 , 𝑥(𝑡) ∈ ෨𝑂∞

𝐽 = 𝑣 𝑡 − 𝑟 𝑡𝑇𝑆(𝜌 𝑡 − 𝑟 𝑡 ) → 𝑚𝑖𝑛𝑣(𝑡)

• Command Governor (ECG) is a special case of ECG with 𝑛 ҧ𝑥 =0.


• Lower dimensional optimization problem versus ECG,

smaller constrained domain of attraction

70

Command governor

• Lower dimensional optimization problem versus ECG

• Define constrained domain of attraction as the domain of all

states which can be recovered without constraint violation

• CG has the same domain of recoverable states as RG, which

is smaller than that of ECG

• In cases when 𝑛𝑣 > 1 (multiple command channels that need

to be coordinated), CG is faster than conventional scalar RG

71

Vector Reference Governor (VRG)

𝑣 𝑡 = 𝑣 𝑡 − 1 + Κ 𝑡 (𝑟 𝑡 − 𝑣 𝑡 − 1 )

Κ 𝑡 =

𝜅1(𝑡) 0 00 ⋱ 00 0 𝜅𝑛𝑣(𝑡)

0 ≤ 𝜅𝑖 𝑡 ≤ 1, 𝑖 = 1,⋯ , 𝑛𝑣

• Vector Reference Governor uses a vector gain to

independently adjust each channel

• Optimization problem: 𝑣 𝑡 − 𝑟 𝑡𝑇𝑆(𝑣 𝑡 − 𝑟 𝑡 ) → min

𝑣 t

subject to 𝑣 𝑡 = 𝑣 𝑡 − 1 + Κ 𝑡 (𝑟 𝑡 − 𝑣 𝑡 − 1 )

𝑣 𝑡 , 𝑥 𝑡 ∈ ෨𝑂∞72

MPC format for ECG

• ECG with the shift register can be re-formulated as a

special variant of MPC controller

• Let 𝑣 𝑡 = 𝜌 𝑡 + 𝑢 𝑡

• Introduce notation commonly used in predictive control

𝑣 𝑡 + 𝑘 𝑡 = 𝜌 𝑡 + 𝑢 𝑡 + 𝑘 𝑡 , 𝑘 = 0,⋯ ,𝑁𝑐 − 1

• Consider the disturbance-free case with 𝑊 = 0

73

MPC format for ECG


𝐽 𝑡 = 𝜌 𝑡 − 𝑟 𝑡𝑇𝑆 𝜌 𝑡 − 𝑟 𝑡 +

𝑘=0

𝑁𝑐−1

𝑢 𝑡 + 𝑘 𝑡 𝑇 ҧ𝑆𝑘𝑢 𝑡 + 𝑘 𝑡 → min𝜌 𝑡 ,𝑢(𝑡+⋅|𝑡)

subject to

𝑥 𝑡 + 𝑘 + 1 𝑡 = 𝐴𝑥 𝑡 + 𝑘 𝑡 + 𝐵(𝜌 𝑡 + 𝑢 𝑡 + 𝑘 𝑡 )

𝑥 𝑡 𝑡 = 𝑥 𝑡 ,

𝑦 𝑡 + 𝑘 𝑡 ∈ 𝑌, 𝑘 = 0,1,⋯ ,𝑁𝑐 − 1,

𝜌 𝑡 , 𝑥 𝑡 + 𝑁𝑐 𝑡 ∈ ෨𝑂∞ ( ෨𝑂∞ = ෨𝑂∞𝑅𝐺 )

74

MPC format for ECG

• Condition that must hold: ҧ𝑆𝑘 ≥ ҧ𝑆𝑘−1 ≥ ⋯ ≥ ҧ𝑆0> 0

• ECG theory provides recursive feasibility and finite-time

convergence results for this special class of MPC

controllers, i.e., it guarantees that

𝜌 𝑡 = 𝑟 𝑡 , 𝑢 𝑡 + 𝑘 𝑡 = 0, 𝑘 = 0,⋯ ,𝑁𝑐 − 1

for all 𝑡 ≥ 𝑡𝑠 and for constant, strictly constraint

admissible, 𝑟 𝑡 = 𝑟𝑠.

75

Extended Command Governor (f16aircraft_ecg.m)

% --------- Construct appended system for ECG -----

nh = 5;

[sys_app, sys_full] = ecg_appdyn(nh, ss(A,B,C,D,dT),'laguerre', 0.4);

Abar = sys_app.A;

Cbar = sys_app.C;

nb = size(Abar, 1);

Secg = dlyap(Abar,0.1*eye(nb)); % weight matrix on xbar

…

Recg = diag([1,1]); % weighting matrix on rho

…

for i = 1:t_sim,

...

[v(:,i+1),p(:,i+1),rho(:,i+1)] = gov_ecg(Recg, Secg, Hx, Hp, Hr, h,x_gov(:,i+1),

r(:,i+1), sys_app.c, p(:,i), rho(:,i));

…

end76

Response with Extended Command Governor

Pitch angle

Flight path angle

𝑟𝜃𝑣𝜃𝜃

𝑟𝛾𝑣𝛾𝛾

• Response with tightened rate limits

• ECG is faster than RG

≤ 0 ⇒ 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑

Constraints

77

The Set ෩𝑶∞ (with disturbances, 𝑾 is polyhedral)

𝐻𝜌,𝑡 𝜌 + 𝐻 ҧ𝑥,𝑡 ҧ𝑥 0 +𝐻𝑥,𝑡 𝑥 0 ≤ 𝜆𝑡

𝐻∞𝜌 ≤ 𝜆∞

𝑌𝑡 = 𝑦: Λ𝑦 ≤ 𝜆𝑡 = 𝑌 ∼ 𝐷𝑤𝑊 ∼ 𝐶𝐵𝑤𝑊 ∼ ⋯ ∼ 𝐶𝐴𝑡−1𝐵𝑤𝑊

𝜆𝑡 = 𝜆𝑡−1 −maxሼ𝑊 𝑖

Λ𝐶𝐴𝑡−1𝐵𝑤𝑊𝑖 }, 𝑡 > 1

𝑌∞ = 𝑦: Λ𝑦 ≤ 𝜆∞ , 0 < 𝜆∞ < inf𝑡𝜆𝑡

• ෨𝑂∞ is defined by affine inequalities on 𝜌, ҧ𝑥 0 , 𝑥 0 :

𝜆0 = 𝜆 −maxሼ𝑊 𝑖

Λ𝐷𝑤𝑊𝑖 }

𝑊 = 𝑐𝑜𝑛𝑣ℎሼ𝑊(𝑖), 𝑖 = 1,⋯ , 𝑛𝑤}

(𝑡 = 0,12,⋯ )

78

Reference Governors for

Nonlinear Systems

79

Nonlinear systems with pointwise-in-time

constraints

Control objectives

• Tracking: 𝑧 𝑡 ≈ 𝑟(𝑡)• Satisfy pointwise-in-time state/control constraints: 𝑦 𝑡 ∈ 𝑌• Robustness to disturbances/uncertainties: 𝑤 𝑡 ∈ 𝑊 ∀𝑡 ≥ 0• Optimality

• obstacle avoidance

• actuator limits

• safety limits

• …

𝑦 𝑡 ∈ 𝑌 ∀𝑡 ≥ 0

80

Nonlinear systems with constraints,

disturbances, and commands

• Stationary disturbances 𝑤 𝑡 :

• Nonlinear system with constraints, disturbances and

commands

𝑥 𝑡 + 1 = 𝑓(𝑥 𝑡 , 𝑣 𝑡 , 𝑤 𝑡 )

𝑦 𝑡 ∈ 𝑌 ⇔ (𝑥 𝑡 , 𝑣 𝑡 ) ∈ 𝐶 ∀𝑡 ∈ 𝑍+

Gilbert and Kolmanovsky, Automatica (38) 2063-2073 (2002)

- set-bounded

- set-bounded and rate bounded

- parametric uncertainties

- …

12/14/2014

𝑤 ⋅ ∈ 𝕎 ⇒ 𝑤 ⋅ +𝜎 ∈ 𝕎 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜎 ∈ 𝑍+

81

Functional description of safe set of

states and constant commands

• Safe set* of initial states and constant commands

ത𝑉 𝑥(0), 𝑣(0) ≤ 0 ⇒ 𝑥 𝑡 , 𝑣 0 ∈ 𝐶 ∀𝑡 ∈ 𝑍+

- ത𝑉 is continuous (can be non-smooth)

- Strong returnability:

ത𝑉 𝑥(𝑡), 𝑣(0) ≤ −𝜖 for some 𝑡 ∈ 𝑍+

𝑥 𝑡 = 𝑥 𝑡 𝑥 0 , 𝑣 𝑘 = 𝑣 0 , 𝑘 = 1,⋯ , 𝑡 − 1

• Technical assumptions

ത𝑉 𝑥(0), 𝑣(0) ≤ 0 ⇒

*Safe set is not required to be positively invariant82

Functional description of safe set of

states and constant commands

83

Scalar reference governor

subject to

• Maximize 𝛽(𝑡)

ത𝑉 𝑥(𝑡), 𝑣(𝑡) ≤ 0,

𝑣 𝑡 = 𝑣 𝑡 − 1 + 𝛽 𝑡 𝑟 𝑡 − 𝑣 𝑡 − 1 ,

• 𝛽 𝑡 = 0 if no feasible solution exists

• Accept small increments, 𝑣 𝑡 − 𝑣 𝑡 − 1 , only if ത𝑉 𝑥(𝑡), 𝑣(𝑡 − 1) ≤ −𝜖

0 ≤ 𝛽(𝑡) ≤ 1

• Solution via bisections or grid search, explicit in some cases (e.g., if ത𝑉 is quadratic)

• Solution within a known tolerance is sufficient for subsequent theoretical results to hold.

84

Reference governor: basic approach

ത𝑉 𝑥, 𝑣(𝑡) ≤ 0

ത𝑉 𝑥, 𝑣(𝑡 + 4) ≤ 0

85

Definitions

Π 𝑞 = ሼ𝑥: ത𝑉 𝑥, 𝑞 ≤ 0} is a “safe” set with 𝑣 𝑡 = 𝑞

Π𝜀 𝑞 = ሼ𝑥: ത𝑉 𝑥, 𝑞 ≤ −𝜖}

• Let 𝑆 be a compact and convex set such that for 𝑣 ∈ 𝑆technical assumptions hold

• For 𝑞 ∈ 𝑆 define:

86

Acceptance Logic

• Let 𝛽∗(𝑡) denote the solution to the optimization problem. If

𝛽∗ 𝑡 𝑟(𝑡) − 𝑣 𝑡 − 1∞< 𝛿

whileത𝑉 𝑥 𝑡 , 𝑣 𝑡 − 1 > −휀

⇒ 𝑣 𝑡 = 𝑣 𝑡 − 1 (maintain the last command)

• Practical guidelines:

𝛿 ≤ 10−2max𝑟,𝑣∈𝑆

𝑟 − 𝑣∞

Select 휀 so that Π𝜀 𝑞 is between 0.9 and 0.99 of Π(𝑞)

87

Reference Governor Response Properties

• Constraint satisfaction:

𝑥 0 ∈ Π 𝑣 0 ⇒ 𝑥 𝑡 , 𝑣 𝑡 ∈ 𝐶 ∀𝑡 ∈ 𝑍+

• It is possible to handle any initial state such that

𝑥 0 ∈ ΠIS =ራ

𝑞∈𝑆

Π(𝑞)

• Finite time convergence property for constant reference

inputs

𝑟 𝑡 = 𝑟0 ∈ 𝑆 for all 𝑡 ≥ 𝑡0 ⇒ ∃ ǁ𝑡 ≥ 𝑡0 such that 𝑣 𝑡 = 𝑟0for all 𝑡 ≥ ǁ𝑡

88

Reference Governor Response Properties

• Response properties for non-constant inputs:

𝑟 𝑡 − 𝑟0 ∞≤ 𝛿0 for all 𝑡 ≥ 𝑡0, 𝑟0 ∈ 𝑆, and 0 < 𝛿0 <

1

2𝛿,

Then if 𝛿 is sufficiently small ⇒ 𝑣 𝑡 − 𝑟0 ∞≤ 𝛿0 for all 𝑡

sufficiently large

• Under additional assumptions,

𝑣 𝑡 = 𝑟(𝑡) if 𝑟 𝑡 − 𝑟0 ∞≤ 𝛿0 for all 𝑡 ≥ 𝑡0, r0 ∈ 𝑆

• Can handle additional constraints

𝑣 𝑡 − 𝑣 𝑡 − 1∞≤ 𝛿𝑚𝑎𝑥

89

Constructing ഥ𝑽

• Closed-loop Lyapunov or ISS-Lyapunov functions

- Define ത𝑉 𝑥, 𝑣 = 𝑉 𝑥, 𝑣 − 𝑐 𝑣 , where 𝑉 is a Lyapunov or an ISS-

Lyapunov function of the closed-loop system

- For a given 𝑣, maximize 𝑐 𝑣 subject to sublevel set

Π 𝑣 = ሼ𝑥: ത𝑉 𝑥. 𝑣 ≤ 0} satisfying constraints

- In cases with bounded disturbances, need 𝑐 𝑣 ≥ 𝑐𝑚𝑖𝑛(𝑣) for the

sublevel set Π(𝑣) to be strongly returnable

- Simplifications occur due to positive invariance of sublevel sets

90


• Off-line simulations and machine learning

- ത𝑉 𝑥, 𝑣 = Φ(𝑥, 𝑣) is a classifier separating safe and unsafe initial

conditions and constant commands

- Scenarios (Monte Carlo simulations) are run with respect to 𝑤(⋅)

- Non-smooth classifiers permitted, e.g., ത𝑉 𝑥, 𝑣 = min𝑗ሼΦ𝑗(𝑥, 𝑣)}.

Thus can represent unions of safe regions.

Φ 𝑥(0), 𝑣 ≤ 0 ⇒ (𝑥(0), 𝑣) is safe

- In the disturbance-free case, model simulations are run for

various combinations of 𝑥(0) and 𝑣

91


𝑧 =𝑥𝑣

Φ𝑗 𝑧 = maxሼ𝜂𝑖𝑗𝑇 𝑧 − 𝑐𝑗 , 𝑖 ∈ 𝐼}

ത𝑉 𝑥, 𝑣 = ത𝑉 𝑧 = min𝑗

Φ𝑗(𝑧)

Example: Cover safe initial

conditions and commands

by a union of hyper-

rectangles

92

ഥ𝑽 implicitly defined through on-line prediction

• Constraints:

𝐶 = ሼ 𝑥, 𝑣 : ℎ𝑖 𝑥, 𝑣 ≤ 0, 𝑖 = 1,⋯ , 𝑟}

ത𝑉 𝑥, 𝑣 = 𝑚𝑎𝑥ሼℎ𝑖(𝜙 𝑡, 𝑥, 𝑣, 𝑤 ⋅ , 𝑣), 𝑖 = 1,⋯ , 𝑟, 𝑡 = 0,⋯ , 𝑡0, 𝑤 ⋅ ∈ 𝑊}

where 𝑡0 is sufficiently large so that

ℎ𝑖 𝜙 𝑡, 𝑥, 𝑣, 𝑤 ⋅ , 𝑣 ≤ −𝜖, 𝑖 = 1,⋯ , 𝑟, 𝑤 ⋅ ∈ 𝑊 and 𝑡 ≥ 𝑡0

𝜙 = “solution”

Bemporad, IEEE TAC AC-43(4) 451-461 (1998); Gilbert and K, Automatica, 38, 2063-2073, 2002

• On-line prediction of maximum constraint violation

93

ഥ𝑽 implicitly defined through on-line prediction

Sun and K., IEEE TCST 13 (6), pp. 991-919, 2005

• Simplifications in parametric uncertainty/robust reference

governor case

𝜙 𝑡, 𝑥, 𝑣, 𝑤 ≈ 𝜙 𝑡, 𝑥, 𝑣, 𝑤0 +𝜕𝜙

𝜕𝑤ቚ𝑡,𝑥,𝑣,𝑤0

(𝑤 − 𝑤0)

𝜕𝜙

𝜕𝑤|𝑡,𝑥,𝑣,𝑤0

= solution of sensitivity ODEs

94

Control of EAMSD

• Position and current constraints

Miller, K, Gilbert, Washabaugh, IEEE Control Systems Magazine,2000

𝑑𝑥1𝑑𝑡

= 𝑥2

𝑑𝑥2𝑑𝑡

=𝑘

𝑚𝑥1 −

𝑐

𝑚𝑥2 +

𝛼

𝑚

𝑢

𝑑0 − 𝑥1𝛾 , 𝑢 = 𝑖𝛽

𝑢 =1

𝛼(𝑑0 − 𝑥1 )

𝛾 𝑘𝑣 − 𝑐𝑑𝑥2

𝑑0

𝑥𝑒(𝑣)

ത𝑉 𝑥, 𝑣 =𝑘

2𝑥1 − 𝑣 2 +

𝑚

2𝑥22 − 𝑐 𝑣

Position constraint

𝑥1

𝑥2

Current constraint

ത𝑉 𝑥, 𝑣 ≤ 0

95

Control of EAMSD

Without RG

Experimental Results

With RG

• Position and current constraints

𝑑0

Position response

96

17:00-17:20, Paper FrC15.4 Add to My Program

Constrained Spacecraft Attitude Control on SO(3) Using

Reference Governors and Nonlinear Model Predictive Control

Kalabic, Uros V. Univ. of Michigan

Gupta, Rohit Univ. of Michigan

Di Cairano, Stefano Mitsubishi Electric Res. Lab.

Bloch, Anthony M. Univ. of Michigan

Kolmanovsky, Ilya V. The Univ. of Michigan

16:20-16:40, Paper WeC06.2 Add to My Program

Constraint Enforcement of Piston Motion in a Free-Piston Engine

Zaseck, Kevin Univ. of Michigan

Brusstar, Matthew The United States Environmental Protection

Agency

Kolmanovsky, Ilya V. The Univ. of Michigan

Nonlinear reference governor

applications at 2014 ACC

97

https://css.paperplaza.net/conferences/conferences/2014ACC/program/2014ACC_AuthorIndexWeb.html#82397








Reference Governor for Linear

Systems with Nonlinear

Constraints

98

Linear systems with nonlinear constraints

𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑣 𝑡

𝑦 𝑡 ∈ 𝑌 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ 𝑍+

y 𝑡 = 𝐶𝑥 𝑡 + 𝐷𝑣 𝑡

𝑌 = 𝑦: ℎ𝑖 𝑦 ≤ 0, 𝑖 = 1,⋯ , 𝑟

• Linear system model:

• Treat nonlinear constraints (without polyhedral approximations):

Kalabic et. al., Proc. of 2011 CDC, pp. 2680-2686 99

Comments

• Motivation: Handling constraints for Feedback Linearizable systems

• Theory in Gilbert, K., and Tan (1994,1995) and Gilbert and K. (2002) is

applicable to the case of linear systems and nonlinear constraints

• We discuss computations, heuristics and examples

100

Scalar Reference Governor

𝛽 𝑡 ∈ 0,1 ,

𝑦 𝑡 + 𝑘|𝑡 ∈ 𝑌, 𝑘 = 0,⋯ 𝑡∗

𝛽 𝑡 → 𝑚𝑎𝑥

𝑣 𝑡 = 𝑣 𝑡 − 1 + 𝛽 𝑡 𝑟 𝑡 − 𝑣 𝑡 − 1


1We use the approach of Bemporad (1998) with implicitly defined constraints

2Here 𝑡∗ is a finite determination index or an upper bound on it. We assume it

to be known in all the subsequent developments.

101

Linear model based prediction

𝑦 𝑡 + 𝑘|𝑡 = Γ 𝑘 𝑥 𝑡 + 𝐻 𝑘 𝑣 𝑡 − 1 + 𝛽 𝑡 𝐻 𝑘 (𝑟 𝑡 − 𝑣 𝑡 − 1 )

Γ 𝑘 = 𝐶𝐴𝑘 ,

𝐻 𝑘 = 𝐶 − Γ 𝑘 (𝐼 − 𝐴)−1𝐵 + 𝐷

• Predicted output is an affine function of 𝛽(𝑡)

• Predicted response to a constant command

𝑣 𝑡 + 𝑘|𝑡 ≡ 𝑣 𝑡 = 𝑣 𝑡 − 1 + 𝛽(𝑡)(𝑟 𝑡 − 𝑣 𝑡 )

102

Convex nonlinear constraints

If 𝛽 0 = 0 is feasible, then an admissible interval for the values of 𝛽 𝑡 is of

the form

𝐾 𝑡 = 0, 𝛽𝑚𝑎𝑥 𝑡 , 1 ≥ 𝛽𝑚𝑎𝑥 𝑡 ≥0

and the reference governor sets 𝛽 𝑡 =𝛽𝑚𝑎𝑥 𝑡 . The constraints are satisfied

for all 𝑡 ≥ 0.

• Suppose that

𝑌 = 𝑦: ℎ𝑖 𝑦 ≤ 0, 𝑖 = 1,⋯ , 𝑟

ℎ𝑖 are convex functions

• Proposition

103

Algorithmic implementation

If ℎ𝑖 𝑦 𝑡 + 𝑘 𝑡 > 0 where 𝑦 𝑡 + 𝑘 𝑡 is the predicted response with

𝛽 𝑡 = 𝛼, use bisections to search for a scalar 𝛼+ that approximately

solves

ℎ𝑖 Γ 𝑘 𝑥 𝑡 + 𝐻 𝑘 𝑣 𝑡 − 1 + 𝛼+𝐻 𝑘 𝑟 𝑡 − 𝑣 𝑡 − 1 = 0

• Set 𝛼 = 1.

• For 𝑖 = 1,⋯ , 𝑟, and 𝑘 = 0,⋯ , 𝑡∗, repeat

• Update 𝛼 = 𝛼+

• Apply “constraints active last first” evaluation heuristics (see the

paper)

104

Convex Quadratic Constraints

𝑦𝑇 ෨𝑄𝑦 + ሚ𝑆𝑦 + ሚ𝐶 ≤ 0, ෨𝑄 = ෨𝑄𝑇 ≥ 0

• Suppose that the constraints are of the form

• This is a quadratic function of 𝛽(𝑡). The root finding can be performed by

solving a quadratic equation.

• Then the constraints can be re-written as

𝑥 𝑡 𝑇 𝑣 𝑡 𝑇 + 𝛽(𝑡)(𝑟 𝑡 − 𝑣(𝑡))𝑇 ത𝑄 𝑘𝑥(𝑡)

𝑣 𝑡 + 𝛽(𝑡)(𝑟 𝑡 − 𝑣(𝑡)

+ ҧ𝑆 𝑘𝑥(𝑡)

𝑣 𝑡 + 𝛽(𝑡)(𝑟 𝑡 − 𝑣(𝑡)+ ሚ𝐶 ≤ 0

105

Mixed Logical Dynamic Constraints

𝑔𝑖 𝑦 > 0 → ℎ𝑖 𝑦 ≤ 0, 𝑖 = 1,⋯𝑟,

• We consider a set of constraints of if-then type

where 𝑔𝑖 , ℎ𝑖 are convex functions

• Observations:

• The set of β 𝑡 ∈ [0,1] for which 𝑔𝑖(𝑦 𝑡 + 𝑘 𝑡 ) ≤ 0 is a

(possibly empty) sub-interval of [0,1], 𝐾𝑖(𝑘)

• The set of β 𝑡 ∈ [0,1] for which ℎ𝑖(𝑦 𝑡 + 𝑘 𝑡 ) ≤ 0 is

another (possibly empty) sub-interval of [0,1], 𝐾𝑖(𝑘)

106

Mixed Logical Dynamic Constraints

𝑔𝑖 𝑦(𝑡 + 𝑘|𝑡) > 0 → ℎ𝑖 𝑦(𝑡 + 𝑘|𝑡) ≤ 0

• Then the set of feasible 𝛽 𝑡 ∈ [0,1] for which

is also an interval 0,1 ∩ 𝐾𝑖 𝑘 ∩ 𝐾𝑖(𝑘)

• The recursive feasibility of 𝛽 𝑡 = 0 is preserved by the reference

governor, hence it follows that the feasible values of 𝛽(𝑡) satisfy

β 𝑡 ∈ 0, 𝛽𝑚𝑎𝑥 , 𝛽𝑚𝑎𝑥 ≥ 0

107

Concave Constraints

• Suppose that the constraint functions ℎ𝑖 in Y = 𝑦: ℎ𝑖 𝑦 ≤ 0 , 𝑖 =1,⋯𝑟, are concave

• Approximate the constraints by the dynamically reconfigurable affine

constraints

𝑦(𝑡 + 𝑘|𝑡) ∈ 𝑌𝑐(𝑡)

where

𝑌𝑐 𝑡 = 𝑦: ℎ𝑖 𝑦𝑖,∗ 𝑡 +𝜕ℎ𝑖𝜕𝑦

(𝑦𝑖,∗ 𝑡 )(𝑦 − 𝑦𝑖,∗ 𝑡 ) ≤ 0 ,

𝑖 = 1,⋯ 𝑟,

and 𝑌𝑐(𝑡) ⊆ 𝑌

108

Concave Constraints

If 𝑦𝑖∗ 0 , 𝑖 = 1,⋯ 𝑟 exist such that 𝛽 0 = 0 is feasible, then

𝛽 𝑡 = 0 and 𝑦𝑖∗ 𝑡 =𝑦𝑖∗ 𝑡 − 1 remain feasible for 𝑡 > 0 and

constraints are adhered to for all 𝑡>0.

• Proposition

𝑦 𝑡

ℎ(𝑦)

𝑦𝑖∗ 𝑡

𝑦

109

Spacecraft relative motion example

• Dynamic model is linear

• Hill–Clohessy-Wiltshire equations

• Constraints are nonlinear:

• Approach within LOS half-cone in front of

the docking port (convex, quadratic)

• Thrust/delta-v magnitude squared is

limited (convex, quadratic)

• Soft-docking: Small velocity when close to

docking position (Mixed Logical Dynamic

with quadratic 𝑔 and ℎ)

110

Spacecraft relative motion example

-1000 0 1000-300

-200

-100

0

100

200

300

Ra

dia

l p

ositio

n (

m)

Along-track position (m)-1000 0 1000

-300

-200

-100

0

100

200

300

Cro

ss-t

rack p

ositio

n (

m)

Along-track position (m)

• Reference governor is applied to

guide in-track orbital position set-point

for an unconstrained LQ controller

0 200 400 600 8000

1

2

3

4

5

time (sec)LOS cone

Dockingposition

820 830 840 8500

0.2

0.4

0.6

0.8

1

time (sec)

force (N)

Separation distance

Relative velocity

magnitude

111

Electromagnetically Actuated Mass Spring Damper

• Dynamics are feedback linearizable

• Constraints

- Current limit results in a concave nonlinear constraint

- Overshoot constraint is linear

ሶ𝑥1ሶ𝑥2

=0 1

−𝑘/𝑚 −𝑐/𝑚

𝑥1𝑥2

+0

1/𝑚𝑢

𝑢 = 𝑘𝑣 − 𝑐𝑑𝑥2

0 ≤ 𝑢 ≤𝛼 𝑖𝑚𝑎𝑥

𝛾

𝑑0 − 𝑥1𝛾

𝑥1 ≤ 0.008

force

max. current

112


0 2 4 60

0.002

0.004

0.006

0.008

0.01

0.012

mass position x1

(m)

time (sec)

unconstrained

imax

=0.5342

imax

=0.365

0 1 2 3 4 50

0.2

0.4

0.6

0.8

current (A)

time (sec)

unconstrained

imax

=0.5342

imax

=0.365

113


• Landing control example

• Voltage limits

• MLD constraints on soft-landing

velocity AND magnetic force

exceeding spring force

𝑑𝑧

𝑑𝑡= 𝑞

𝑑𝑞

𝑑𝑡=

1

𝑚(−𝐹𝑚𝑎𝑔 + 𝑘𝑠 𝑧𝑠 − 𝑧 − 𝑏𝑞)

𝑑𝑖

𝑑𝑡=𝑉𝑐 − 𝑟𝑖 +

2𝑘𝑎𝑖(𝑘𝑏 + 𝑧)2

𝑞

2𝑘𝑠𝑘𝑏 + 𝑧

𝐹𝑚𝑎𝑔 =𝑘𝑎

𝑘𝑏 + 𝑧 2𝑖2

114


115

Documents

Reference Governor - LIRMM