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โข Reference governor is an add-on safety supervisor for the
existing/legacy controllers
โข Monitors and modifies commands if necessary to ensure
constraints are satisfied
Nominal closed-loop system with
an existing/legacy controller
2
Reference Governor
๐ก โ ๐๐
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
3
Reference Governor
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
๐ก โ ๐๐
4
Reference Governor
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
๐ก โ ๐๐
5
y(t)
Reference Governor
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
๐ก โ ๐๐
6
v(t+๐)=
y(t)
Reference Governor
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
๐ก โ ๐๐
7
v(t+๐)=
y(t)
v(t)
Reference Governor
Basic idea: Compute ๐ฃ(๐ก) so that if constantly applied it would not lead to constraint violations
๐ก โ ๐๐
8
v(t+๐)=
y(t)
v(t)
Reference Governor
EXPERIMENTS Plant: Inverted Pendulum
Control Law: Linear Quadratic Regulator
LQR๐ข
๐ฅ
๐ฃ
๐
๐
๐ฃ ๐ฃ
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
๐
LQR๐ข
๐ฅ
RG
๐๐ฃ
๐ฃ
10Slides from 2014 IEEE CDC Workshop by E. Garone, S. Di Cairano, and I.V. Kolmanovsky
subject to
Maximize ๐ (๐ก)
๐ฃ(๐ก)๐ฅ(๐ก)
โ ๐ โ ๐โ
๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ๐ ๐ก ๐ ๐ก โ ๐ฃ ๐ก โ 1 ,
0 โค ๐ (๐ก) โค 1
11
Scalar Reference Governor
โข ๐โ is the set of safe pairs of initial states, ๐ฅ 0 , and
constant commands, ๐ฃ ๐ก โก ๐ฃ, which do not cause
subsequent constraint violation
๐ฅ ๐ก + 1 = ๐ด๐ฅ ๐ก + ๐ต๐ฃ, ๐ฆ ๐ก = ๐ถ๐ฅ ๐ก + ๐ท๐ฃ โ ๐ โ
๐โ = แผ ๐ฃ, ๐ฅ(0) : ๐ถ๐ด๐ก๐ฅ(0) + ๐ถ ๐ผ โ ๐ด๐ก ๐ผ โ ๐ด โ1๐ต๐ฃ + ๐ท๐ฃ โ ๐,๐ก = 0,1,โฏ ,โ }
โข Example: For asymptotically stable observable linear system:
12
Safe Set
13
โข Finitely determined inner approximation is obtained by
slightly tightening the โsteady-stateโ constraints
เทจ๐โ = แผ ๐ฃ, ๐ฅ 0 : (๐ถ ๐ผ โ ๐ด โ1๐ต + ๐ท)๐ฃ โ 1 โ ํ ๐, ๐ถ๐ด๐ก๐ฅ 0 + ๐ถ ๐ผ โ ๐ด๐ก ๐ผ โ ๐ด โ1๐ต + ๐ท๐ฃ โ ๐,๐ก = 0,1,โฏ , ๐กโ} โ ๐โ
Implementation based on subsets
14
โข If the constraint set is polyhedral, then เทจ๐โ is polyhedral
Safe Sets
๐ = ๐ฆ:๐ป๐ฆ โค โ โ
เทจ๐โ = ๐ฃ, ๐ฅ 0 :
๐ป๐ถ ๐ผ โ ๐ด โ1๐ต + ๐ท 0๐ป๐ท
๐ป๐ถ๐ต + ๐ป๐ท๐ป๐ถ๐ป๐ถ๐ด
โฎ๐ป๐ถ ๐ผ โ ๐ด๐ ๐ผ โ ๐ด โ1๐ต + ๐ป๐ท
โฎ
โฎ๐ป๐ถ๐ด๐
โฎ
๐ฃ๐ฅ(0) โค
1 โ ํ โโโโฎโโฎ
โข Redundant and โalmost redundantโ inequality constraints are
eliminated while remaining constraints are tightened to obtain
a simply represented ๐ โ เทจ๐โ
17
Example
Model:
๐ฅ1 ๐ก + 1 = ๐ฅ1 ๐ก + 0.1๐ฅ2 ๐ก ,๐ฅ2(๐ก + 1) = ๐ฅ2 ๐ก + 0.1๐ข(๐ก)
Constraints:
|๐ฅ1| โค 1,|๐ฅ2| โค 0.1,
|๐ข| โค 0.1
Nominal closed-loop:
๐ข = โ0.917 ๐ฅ1 โ ๐ โ 1.636๐ฅ2,
Reference command
๐ ๐ก = 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 (contโd)
๐ข = โ0.917 ๐ฅ1 โ ๐ฃ ๐ก โ 1.636๐ฅ2 , ๐ฃ(๐ก) = ๐ ๐บ(๐ฃ ๐ก โ 1 , ๐ฅ ๐ก )
Response with reference governor
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.
Online prediction-based reference governor
Nicotra, M., Garone, E., and Kolmanovsky, I.V., โA fast reference governor for linear systems,โ AIAA
Journal of Guidance, Control, and Dynamics, vol. 40, no. 2, pp. 460-464, 2017.
โข Linear and nonlinear systems with set-bounded
disturbances and parameter uncertainties can be treated
โข Feasibility at initial time implies constraint adherence and
recursive feasibility for all future times
โข Finite-time convergence of ๐ฃ(๐ก) to ๐(๐ก) or nearest steady-
state feasible value for constant ๐(๐ก)
โข Similar convergence results for ``nearly constantโโ and
slowly-varying ๐(๐ก)
โข Enlarged constrained domain of attraction
28
Remarks on existing theory
Adopting linear design to a nonlinear system
12/15/2014
โข Consider a disturbance-free nonlinear system
๐ฟ๐ฅ ๐ก + 1 = ๐ด๐ฟ๐ฅ ๐ก + ๐ต๐ฟ๐ฃ(๐ก) โ ๐
๐ฟ๐ฅ ๐ก = ๐ฅ โ ๐ฅ๐๐,
๐ฟ๐ฃ ๐ก = ๐ฃ โ ๐ฃ๐๐,
๐ ๐ฅ๐๐, ๐ฃ๐๐ = 0
๐ฆ๐๐๐ ๐ก = ๐ถ ๐ฟ๐ฅ ๐ก + ๐ท ๐ฟ๐ฃ ๐ก
โข Let a linearization of the nonlinear model at an operating
point (๐ฅ๐๐, ๐ฃ๐๐, ๐ฆ๐๐) be given by
๐ฅ ๐ก + 1 = ๐ ๐ฅ ๐ก , ๐ฃ ๐ก
๐ฆ๐๐๐๐ ๐ก = ๐ ๐ฅ ๐ก , ๐ฃ ๐ก โ ๐
34
Adopting linear design to a nonlinear system
โข Main idea: Correct the linear model prediction into the future
by a disturbance term by ๐(๐ก)
เท๐ฆ๐๐๐๐ ๐ก + ๐|๐ก = ๐ฆ๐๐ + ๐ฆ๐๐๐ ๐ก + ๐ ๐ก + ๐ ๐ก
เทจ๐โ,๐๐ข๐ = แผ ๐ฟ๐ฃ, ๐ฟ๐ฅ 0 , ๐ :
๐ถ๐ด๐ก๐ฟ๐ฅ 0 + ๐ถ ๐ผ โ ๐ด๐ก ๐ผ โ ๐ด โ1๐ต๐ฟ๐ฃ + ๐ท๐ฟ๐ฃ + ๐ โ ๐~แผ๐ฆ๐๐},๐ก = 0,1,โฏ ,โ }โฮโ
Vahidi, K, Stefanopoulou, IEEE TCST 15 (1), 86-98 (2007)
โข Let ๐ ๐ก = ๐ฆ๐๐๐๐ ๐ก โ ๐ฆ๐๐๐ ๐ก โ ๐ฆ๐๐ be the output deviation
from the output predicted by the linear model at current time
โข Define
35
Adopting linear design to a nonlinear system
๐ฃ ๐ก โ 1 + ๐ฝ ๐ก ๐ ๐ก โ ๐ฃ ๐ก โ ๐ฃ๐๐๐ฟ๐ฅ(๐ก)๐(๐ก)
โ เทจ๐โ,๐๐ข๐
๐ฝ ๐ก โ max ๐ ๐ข๐๐๐๐๐ก ๐ก๐ 0 โค ๐ฝ ๐ก โค 1
โข Reference governor logic:
๐ ๐ก = ๐ฆ๐๐๐๐๐๐ ๐ก โ ๐ฆ๐๐๐ ๐ก โ ๐ฆ๐๐
๐๐๐
36
Discussion
โข The proposed technique is motivated by a similar scheme in
MPC
โข It is heuristic but has been shown to work well in several
applications
โข The study of its theoretical properties remains an open
research problem
โข 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, โController state and reference governors for discrete-time
linear systems with pointwise-in-time state and control constraints,โ Proceedings of 2015 American
Control Conference, Chicago, IL, pp. 3607-3612, 2015.
โข Trim point to trim point transition feasibility is
determined based on set of states that can be
recovered by CSRG
โข 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., โEnvelope-
aware flight management for loss of control prevention given rudder jam,โ 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. โConstraint enforcement for a lighter-than-air wind-energy
system: An application of reference governors with chance constraints,โ 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., โParameter governors for
coordinated control of n-spacecraft formations,โ AIAA Journal of Guidance, Control, and Dynamics, vol. 40,
no. 11, pp. 3020-3025, November, 2017.
Concluding remarks
Controller PlantReference
governor
โข Augment rather than replace nominal controller
โข Inactive if no danger of constraint violation
โข Easy to implement / fast online computations
โข Special properties
โข Much room for future research and applications
Agenda
โข Extended Command Governor (ECG)
โข Design of ancillary dynamical system
โข Response properties
โข Interpretation as a form of Model Predictive Controller (MPC)
โข Application examples
โข Command governor and vector reference governor
47
Extended command governor
(Gilbert and Ong, Automatica, Vol. 47, pp. 334-340, 2011)
Motivation:
โข Enlarge constrained region of attraction
โข Provide faster response
โข Increase robustness to unmodeled dynamics
48
Extended command governor
าง๐ฅ ๐ก + 1 = าง๐ด าง๐ฅ ๐ก
๐ ๐ก + 1 = ๐ ๐ก
๐ฅ ๐ก + 1 = ๐ด๐ฅ ๐ก + ๐ต๐ฃ ๐ก
๐ฆ ๐ก = ๐ถ๐ฅ ๐ก + ๐ท๐ฃ ๐ก โ ๐
โข Auxiliary โcommand generatingโ subsystem:
โข System:
๐ฃ ๐ก = ๐ ๐ก + าง๐ถ าง๐ฅ ๐ก
โข Requirement: าง๐ด is asymptotically stable (Schur)
Auxiliary command generator
๐(0)
๐ฃ ๐ก = าง๐ถ าง๐ด๐ก ๐ฅ 0 + ๐(0)
0 ๐ก
าง๐ฅ ๐ก + 1 = าง๐ด าง๐ฅ ๐ก
๐ ๐ก + 1 = ๐ ๐ก
๐ฃ ๐ก = ๐ ๐ก + าง๐ถ าง๐ฅ ๐ก
๐ฃ ๐ก
โข Auxiliary โcommand generatingโ subsystem:
50
Augmented system
าง๐ฅ ๐ก + 1 = าง๐ด าง๐ฅ ๐ก
๐ ๐ก + 1 = ๐ ๐ก
๐ฅ ๐ก + 1 = ๐ด๐ฅ ๐ก + ๐ต๐ฃ ๐ก
๐ฆ ๐ก = ๐ถ๐ฅ ๐ก + ๐ท๐ฃ ๐ก โ ๐
๐ฃ ๐ก = ๐ ๐ก + าง๐ถ าง๐ฅ ๐ก
โข Combined system:
โข Constraints:
51
Augmented system
๐ฆ ๐ก = ๐ถ๐ฅ ๐ก + ๐ท๐ฃ ๐ก โ ๐
โข Augmented system with constraints
าง๐ฅ(๐ก + 1)๐ฅ(๐ก + 1)
= ๐ด๐าง๐ฅ(๐ก)๐ฅ(๐ก)
+ ๐ต๐ ๐ ๐ก
๐ด๐ =าง๐ด 0
๐ต าง๐ถ ๐ด, ๐ต๐ =
0๐ต
๐ถ๐ = ๐ท าง๐ถ ๐ถ ๐ท๐ = ๐ท
52
Strictly steady-state admissible commands
โข Set ฮโ of steady-state admissible constant references:
ฮโ = แผ ๐ 0 , าง๐ฅ 0 , ๐ฅ(0) : ๐(0) โ โ}
โข Tightened set of steady-state feasible commands:
๐ โ โ โ ๐ป๐ โ 1 โ ๐ ๐
โข Static gain ๐ป = ๐ถ ๐ผ โ ๐ด โ1๐ต + ๐ท
0 โ ๐๐๐ก ๐, ๐ is compact, 0 < ๐ < 1
53
The set ๐ถโ
๐โ = แผ ๐ 0 , าง๐ฅ 0 , ๐ฅ(0) : ๐ฆ(๐ก) โ ๐, ๐ก = 0,1,โฏ ,โ}
โข Safe set
54
The set เทฉ๐ถโ
โข Safe set is tightened in steady-state
เทจ๐โ = ๐โ โฮโ
โข Properties (under suitable assumptions):
โข ๐ โ โ โ (๐, 0, ๐ฅ๐ ๐ ๐ ) = (๐, 0, ๐ป๐) โ ๐๐๐ก ๐โ
โข เทจ๐โ is positively invariant for augmented system
โข เทจ๐โ is finitely-determined
55
โข Suppose ๐ is a polytope: ๐ = แผ๐ฆ: ฮy โค ๐}
๐ป๐,๐ก ๐ + ๐ป าง๐ฅ,๐ก าง๐ฅ 0 +๐ป๐ฅ,๐ก ๐ฅ 0 โค ๐
๐ปโ๐ โค (1 โ ๐)๐
๐ป าง๐ฅ,๐ก, ๐ป๐ฅ,๐ก = ฮ๐ถ๐๐ด๐๐ก ,
๐ป๐,๐ก = ฮ(๐ถ๐ ๐ผ โ ๐ด๐๐ก ๐ผ โ ๐ด๐
โ1๐ต๐ + ๐ท๐)
๐ปโ = ฮ(๐ถ๐ ๐ผ โ ๐ด๐โ1๐ต๐ + ๐ท๐)
The set เทฉ๐ถโ
โข Then เทจ๐โ is defined by affine inequalities on ๐, าง๐ฅ 0 , ๐ฅ 0 :
โข Consider the disturbance-free case (๐ = 0)
โข Inequalities for all ๐ก sufficiently large (๐ก โฅ ๐กโ) are redundant
and need not be included
(๐ก = 0,12,โฏ )
(0 < ๐ โช 1)
56
subject to
๐ ๐ก , าง๐ฅ ๐ก , ๐ฅ(๐ก) โ เทจ๐โ
๐ฝ = ๐ ๐ก โ ๐ ๐ก๐๐(๐ ๐ก โ ๐ ๐ก ) + าง๐ฅ ๐ก ๐ าง๐ าง๐ฅ(๐ก) โ ๐๐๐๐(๐ก), าง๐ฅ(๐ก)
Extended command governor
โข Optimization problem:
โข Command computation based on าง๐ฅ(๐ก) and ๐(๐ก):
๐ฃ ๐ก = าง๐ถ าง๐ฅ ๐ก + ๐(๐ก)
57
Extended command governor
าง๐ด๐ าง๐ าง๐ด โ าง๐ < 0, าง๐๐ = าง๐ > 0
โข Assumption 2: The weight าง๐ in the cost function must satisfy
โข Assumption 1: The weight ๐ in the cost function satisfies
๐๐ = ๐ > 0
โข Observation: If ๐ ๐ก โ 1 , าง๐ฅ(๐ก โ 1) are feasible at time ๐ก โ 1,
then
๐ ๐ก = ๐ ๐ก โ 1 , ๐ฅ ๐ก = าง๐ด าง๐ฅ(๐ก โ 1)
are feasible at time ๐ก and
๐ฅ๐ ๐ก าง๐ ๐ฅ ๐ก โค าง๐ฅ๐ ๐ก โ 1 าง๐ าง๐ฅ(๐ก โ 1)
58
Observations
โข Suppose ๐ ๐ก = ๐(๐ก โ 1)
โข Let ๐ฝโ(๐ก) denote the optimal cost. Then:
๐ฝโ ๐ก = ๐ ๐ก โ ๐ ๐ก๐๐ ๐ ๐ก โ ๐ ๐ก + าง๐ฅ ๐ก ๐ าง๐ าง๐ฅ ๐ก
= ๐ฝโ ๐ก โ 1
โค ๐ ๐ก โ ๐ ๐ก๐๐ ๐ ๐ก โ ๐ ๐ก + ๐ฅ ๐ก ๐ าง๐ ๐ฅ ๐ก
โค ๐ ๐ก โ 1 โ ๐ ๐ก โ 1๐๐ ๐ ๐ก โ 1 โ ๐ ๐ก โ 1
+ าง๐ฅ ๐ก โ 1 ๐ าง๐ าง๐ฅ ๐ก โ 1
โข The optimal cost is non-increasing, ๐ฝโ ๐ก โค ๐ฝโ(๐ก โ 1)
59
Comments [1]
โข 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
โข The first element of the recovery sequence, ๐ฃ ๐ก , is
applied to the system
โข If เทจ๐โ is polyhedral, the ECG optimization problem is a
Quadratic Program (QP) with linear inequality constraints
โข This QP can be solved online by a QP solver [such as
PQP, GPAD, Qpkwik, CVX,โฆ] or explicitly by multi-
parametric solvers (MPT or hybrid toolbox)
60
Comments [2]
โข ECG achieves large constrained domain of attraction
(= ๐๐๐๐๐ฅ เทจ๐โ). It is typically larger than that of RG
โข ECG achieves faster response, i.e., faster
convergence of ๐ฃ(๐ก) to ๐, in particular, for systems with
actuator rate limits
โข Improved robustness to model uncertainty observed in
simulations
61
Theoretical results
โข Suppose a feasible solution exists at time 0 and ๐ ๐ก = ๐๐
for all ๐ก โฅ ๐ก๐ . Define ๐๐ โ = argmin
๐โโ๐ โ ๐๐
2be the
nearest feasible reference
โข Then there exists a ๐ก๐ โ ๐+ such that ๐ฃ ๐ก = ๐๐ โ for all ๐ก โฅ ๐ก๐ .
โข Given ๐ > 0, there exists a ๐ก๐ โ ๐+ such that
๐ฅ ๐ก โ ๐นโ ๐๐ โ + ๐๐ต๐ for all ๐ก โฅ ๐ก๐
(Gilbert and Ong, 2011)
62
Sketch of the proof
โข ๐ฝโ(๐ก) is monotonically non-increasing, hence
๐ฝโ ๐ก โ 1 โ ๐ฝโ ๐ก โ 0
โข Using the properties of minimum norm projection on a closed
and convex set, it is shown that
าง๐ด าง๐ฅ ๐ก โ 1 โ าง๐ฅ ๐กาง๐
2+ ๐(๐ก โ 1) โ ๐ ๐ก
๐
2โค ๐ฝโ ๐ก โ 1 โ ๐ฝโ(๐ก)
โข Henceาง๐ด าง๐ฅ ๐ก โ 1 โ าง๐ฅ ๐ก โ 0
๐ ๐ก โ 1 โ ๐ ๐ก โ 0 as ๐ก โ โ
63
Sketch of the proof
โข Apply Lemma with
๐ง = าง๐ด าง๐ฅ ๐ก โ 1 , ๐ ๐ก โ 1 , ๐ง๐๐ = าง๐ฅ ๐ก , ๐ ๐ก ,
๐ง๐ = 0, ๐ , ๐ = ๐๐๐๐( าง๐, ๐)
64
Sketch of the proof
๐ง๐๐
๐ง๐
๐ง๐
๐๐
๐2 + ๐2 โค ๐2 since ๐ โฅ ๐/2
๐
๐
โ ๐2 โค ๐2 โ ๐2
65
Sketch of the proof
โข Thus าง๐ฅ ๐ก = าง๐ด าง๐ฅ ๐ก โ 1 + ๐(๐ก), and ๐ ๐ก โ 0 as ๐ก โ โ.
โข Since าง๐ด is Schur, าง๐ฅ ๐ก โ 0.
โข Thus ๐ฃ ๐ก โ ๐ฃ ๐ก โ 1 โ 0 and ๐ฅ ๐ก โ ๐ฅ๐ ๐ ๐ก
โข The proof is finalized by strict constraint admissibility in
steady-state. Formally, we demonstrate that for large ๐กand ํ > 0 sufficiently small,
๐ ๐ก โ ํ๐ ๐ก โ๐๐
๐ ๐ก โ๐๐ , าง๐ฅ(๐ก) = 0
are feasible.
โข This is only possible if ๐ ๐ก = ๐๐ and hence ๐ฃ ๐ก = ๐๐
66
Choices of เดฅ๐จ, เดฅ๐ช
โข Shift register of length ๐ าง๐ฅ:
าง๐ด =
0 ๐ผ0 0
0 โฏ๐ผ โฏ
โฎ โฎ0 0
โฎ โฎโฏ ๐ผ
, าง๐ถ = ๐ผ 0 โฏ 0
โข Auxiliary system outputs a recovery sequence โstoredโ in าง๐ฅ(0)
โข ๐ฃ(๐ก) converges to a constant, ๐ 0 , in ๐ าง๐ฅ steps
67
Example
โข Example: ๐ฃ โ ๐ 1, ๐ าง๐ฅ = 3
โข Augmented state
๐ฃ ๐ าง๐ฅ + ๐ = ๐ 0 , ๐ โฅ 0
าง๐ฅ =าง๐ฅ1าง๐ฅ2าง๐ฅ3
โข Generated command sequence:
๐ฃ 0 = ๐ 0 + าง๐ฅ1 0 ,๐ฃ 1 = ๐ 0 + าง๐ฅ2 0 ,๐ฃ 2 = ๐ 0 + าง๐ฅ3 0
68
Choices of เดฅ๐จ, เดฅ๐ช
โข Laguerre Sequence Generator:
าง๐ด =
ํ๐ผ ๐ฝ๐ผ โํ๐ฝ๐ผ ํ2๐ฝ๐ผ โฏ
0 ํ๐ผ ๐ฝ๐ผ โํ๐ฝ๐ผ โฏ00โฎ
00โฎ
ํ๐ผ0โฎ
๐ฝ๐ผํIโฎ
โฏโฏโฑ
,
าง๐ถ = ๐ฝ ๐ผ โํI ํ2๐ผ โํ3๐ผ โฏ โํ ๐โ1๐ผ
where ๐ฝ = 1 โ ํ2, and 0 โค ํ โค 1 is a selectable parameter that
corresponds to the time constant of the fictitious dynamics. Note that
with ํ = 0, this is the shift register.
(Kalabic et. al., Proc. of 2011 MSC)
69
Command governor
subject to
๐ฃ ๐ก , ๐ฅ(๐ก) โ เทจ๐โ
๐ฝ = ๐ฃ ๐ก โ ๐ ๐ก๐๐(๐ ๐ก โ ๐ ๐ก ) โ ๐๐๐๐ฃ(๐ก)
โข Command Governor (ECG) is a special case of ECG with ๐ าง๐ฅ =0.
โข Optimization problem:
โข Lower dimensional optimization problem versus ECG,
smaller constrained domain of attraction
70
Command governor
โข Lower dimensional optimization problem versus ECG
โข Define constrained domain of attraction as the domain of all
states which can be recovered without constraint violation
โข CG has the same domain of recoverable states as RG, which
is smaller than that of ECG
โข In cases when ๐๐ฃ > 1 (multiple command channels that need
to be coordinated), CG is faster than conventional scalar RG
71
Vector Reference Governor (VRG)
๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ฮ ๐ก (๐ ๐ก โ ๐ฃ ๐ก โ 1 )
ฮ ๐ก =
๐ 1(๐ก) 0 00 โฑ 00 0 ๐ ๐๐ฃ(๐ก)
0 โค ๐ ๐ ๐ก โค 1, ๐ = 1,โฏ , ๐๐ฃ
โข Vector Reference Governor uses a vector gain to
independently adjust each channel
โข Optimization problem: ๐ฃ ๐ก โ ๐ ๐ก๐๐(๐ฃ ๐ก โ ๐ ๐ก ) โ min
๐ฃ t
subject to ๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ฮ ๐ก (๐ ๐ก โ ๐ฃ ๐ก โ 1 )
๐ฃ ๐ก , ๐ฅ ๐ก โ เทจ๐โ72
MPC format for ECG
โข ECG with the shift register can be re-formulated as a
special variant of MPC controller
โข Let ๐ฃ ๐ก = ๐ ๐ก + ๐ข ๐ก
โข Introduce notation commonly used in predictive control
๐ฃ ๐ก + ๐ ๐ก = ๐ ๐ก + ๐ข ๐ก + ๐ ๐ก , ๐ = 0,โฏ ,๐๐ โ 1
โข Consider the disturbance-free case with ๐ = 0
73
MPC format for ECG
โข Optimization problem:
๐ฝ ๐ก = ๐ ๐ก โ ๐ ๐ก๐๐ ๐ ๐ก โ ๐ ๐ก +
๐=0
๐๐โ1
๐ข ๐ก + ๐ ๐ก ๐ าง๐๐๐ข ๐ก + ๐ ๐ก โ min๐ ๐ก ,๐ข(๐ก+โ |๐ก)
subject to
๐ฅ ๐ก + ๐ + 1 ๐ก = ๐ด๐ฅ ๐ก + ๐ ๐ก + ๐ต(๐ ๐ก + ๐ข ๐ก + ๐ ๐ก )
๐ฅ ๐ก ๐ก = ๐ฅ ๐ก ,
๐ฆ ๐ก + ๐ ๐ก โ ๐, ๐ = 0,1,โฏ ,๐๐ โ 1,
๐ ๐ก , ๐ฅ ๐ก + ๐๐ ๐ก โ เทจ๐โ ( เทจ๐โ = เทจ๐โ๐ ๐บ )
74
MPC format for ECG
โข Condition that must hold: าง๐๐ โฅ าง๐๐โ1 โฅ โฏ โฅ าง๐0> 0
โข ECG theory provides recursive feasibility and finite-time
convergence results for this special class of MPC
controllers, i.e., it guarantees that
๐ ๐ก = ๐ ๐ก , ๐ข ๐ก + ๐ ๐ก = 0, ๐ = 0,โฏ ,๐๐ โ 1
for all ๐ก โฅ ๐ก๐ and for constant, strictly constraint
admissible, ๐ ๐ก = ๐๐ .
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
โฆ
Recg = diag([1,1]); % weighting matrix on rho
โฆ
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));
โฆ
end76
Response with Extended Command Governor
Pitch angle
Flight path angle
๐๐๐ฃ๐๐
๐๐พ๐ฃ๐พ๐พ
โข Response with tightened rate limits
โข ECG is faster than RG
โค 0 โ ๐๐๐๐ ๐ก๐๐๐๐๐ก๐ ๐ ๐๐ก๐๐ ๐๐๐๐
Constraints
77
The Set เทฉ๐ถโ (with disturbances, ๐พ is polyhedral)
๐ป๐,๐ก ๐ + ๐ป าง๐ฅ,๐ก าง๐ฅ 0 +๐ป๐ฅ,๐ก ๐ฅ 0 โค ๐๐ก
๐ปโ๐ โค ๐โ
๐๐ก = ๐ฆ: ฮ๐ฆ โค ๐๐ก = ๐ โผ ๐ท๐ค๐ โผ ๐ถ๐ต๐ค๐ โผ โฏ โผ ๐ถ๐ด๐กโ1๐ต๐ค๐
๐๐ก = ๐๐กโ1 โmaxแผ๐ ๐
ฮ๐ถ๐ด๐กโ1๐ต๐ค๐๐ }, ๐ก > 1
๐โ = ๐ฆ: ฮ๐ฆ โค ๐โ , 0 < ๐โ < inf๐ก๐๐ก
โข เทจ๐โ is defined by affine inequalities on ๐, าง๐ฅ 0 , ๐ฅ 0 :
๐0 = ๐ โmaxแผ๐ ๐
ฮ๐ท๐ค๐๐ }
๐ = ๐๐๐๐ฃโแผ๐(๐), ๐ = 1,โฏ , ๐๐ค}
(๐ก = 0,12,โฏ )
78
Nonlinear systems with pointwise-in-time
constraints
Control objectives
โข Tracking: ๐ง ๐ก โ ๐(๐ก)โข Satisfy pointwise-in-time state/control constraints: ๐ฆ ๐ก โ ๐โข Robustness to disturbances/uncertainties: ๐ค ๐ก โ ๐ โ๐ก โฅ 0โข Optimality
โข obstacle avoidance
โข actuator limits
โข safety limits
โข โฆ
๐ฆ ๐ก โ ๐ โ๐ก โฅ 0
80
Nonlinear systems with constraints,
disturbances, and commands
โข Stationary disturbances ๐ค ๐ก :
โข Nonlinear system with constraints, disturbances and
commands
๐ฅ ๐ก + 1 = ๐(๐ฅ ๐ก , ๐ฃ ๐ก , ๐ค ๐ก )
๐ฆ ๐ก โ ๐ โ (๐ฅ ๐ก , ๐ฃ ๐ก ) โ ๐ถ โ๐ก โ ๐+
Gilbert and Kolmanovsky, Automatica (38) 2063-2073 (2002)
- set-bounded
- set-bounded and rate bounded
- parametric uncertainties
- โฆ
12/14/2014
๐ค โ โ ๐ โ ๐ค โ +๐ โ ๐ ๐๐๐ ๐๐๐ ๐ โ ๐+
81
Functional description of safe set of
states and constant commands
โข Safe set* of initial states and constant commands
เดค๐ ๐ฅ(0), ๐ฃ(0) โค 0 โ ๐ฅ ๐ก , ๐ฃ 0 โ ๐ถ โ๐ก โ ๐+
- เดค๐ is continuous (can be non-smooth)
- Strong returnability:
เดค๐ ๐ฅ(๐ก), ๐ฃ(0) โค โ๐ for some ๐ก โ ๐+
๐ฅ ๐ก = ๐ฅ ๐ก ๐ฅ 0 , ๐ฃ ๐ = ๐ฃ 0 , ๐ = 1,โฏ , ๐ก โ 1
โข Technical assumptions
เดค๐ ๐ฅ(0), ๐ฃ(0) โค 0 โ
*Safe set is not required to be positively invariant82
Scalar reference governor
subject to
โข Maximize ๐ฝ(๐ก)
เดค๐ ๐ฅ(๐ก), ๐ฃ(๐ก) โค 0,
๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ๐ฝ ๐ก ๐ ๐ก โ ๐ฃ ๐ก โ 1 ,
โข ๐ฝ ๐ก = 0 if no feasible solution exists
โข Accept small increments, ๐ฃ ๐ก โ ๐ฃ ๐ก โ 1 , only if เดค๐ ๐ฅ(๐ก), ๐ฃ(๐ก โ 1) โค โ๐
0 โค ๐ฝ(๐ก) โค 1
โข Solution via bisections or grid search, explicit in some cases (e.g., if เดค๐ is quadratic)
โข Solution within a known tolerance is sufficient for subsequent theoretical results to hold.
84
Reference governor: basic approach
เดค๐ ๐ฅ, ๐ฃ(๐ก) โค 0
เดค๐ ๐ฅ, ๐ฃ(๐ก + 4) โค 0
85
Definitions
ฮ ๐ = แผ๐ฅ: เดค๐ ๐ฅ, ๐ โค 0} is a โsafeโ set with ๐ฃ ๐ก = ๐
ฮ ๐ ๐ = แผ๐ฅ: เดค๐ ๐ฅ, ๐ โค โ๐}
โข Let ๐ be a compact and convex set such that for ๐ฃ โ ๐technical assumptions hold
โข For ๐ โ ๐ define:
86
Acceptance Logic
โข Let ๐ฝโ(๐ก) denote the solution to the optimization problem. If
๐ฝโ ๐ก ๐(๐ก) โ ๐ฃ ๐ก โ 1โ< ๐ฟ
whileเดค๐ ๐ฅ ๐ก , ๐ฃ ๐ก โ 1 > โํ
โ ๐ฃ ๐ก = ๐ฃ ๐ก โ 1 (maintain the last command)
โข Practical guidelines:
๐ฟ โค 10โ2max๐,๐ฃโ๐
๐ โ ๐ฃโ
Select ํ so that ฮ ๐ ๐ is between 0.9 and 0.99 of ฮ (๐)
87
Reference Governor Response Properties
โข Constraint satisfaction:
๐ฅ 0 โ ฮ ๐ฃ 0 โ ๐ฅ ๐ก , ๐ฃ ๐ก โ ๐ถ โ๐ก โ ๐+
โข It is possible to handle any initial state such that
๐ฅ 0 โ ฮ IS =แซ
๐โ๐
ฮ (๐)
โข Finite time convergence property for constant reference
inputs
๐ ๐ก = ๐0 โ ๐ for all ๐ก โฅ ๐ก0 โ โ ว๐ก โฅ ๐ก0 such that ๐ฃ ๐ก = ๐0for all ๐ก โฅ ว๐ก
88
Reference Governor Response Properties
โข Response properties for non-constant inputs:
๐ ๐ก โ ๐0 โโค ๐ฟ0 for all ๐ก โฅ ๐ก0, ๐0 โ ๐, and 0 < ๐ฟ0 <
1
2๐ฟ,
Then if ๐ฟ is sufficiently small โ ๐ฃ ๐ก โ ๐0 โโค ๐ฟ0 for all ๐ก
sufficiently large
โข Under additional assumptions,
๐ฃ ๐ก = ๐(๐ก) if ๐ ๐ก โ ๐0 โโค ๐ฟ0 for all ๐ก โฅ ๐ก0, r0 โ ๐
โข Can handle additional constraints
๐ฃ ๐ก โ ๐ฃ ๐ก โ 1โโค ๐ฟ๐๐๐ฅ
89
Constructing เดฅ๐ฝ
โข Closed-loop Lyapunov or ISS-Lyapunov functions
- Define เดค๐ ๐ฅ, ๐ฃ = ๐ ๐ฅ, ๐ฃ โ ๐ ๐ฃ , where ๐ is a Lyapunov or an ISS-
Lyapunov function of the closed-loop system
- For a given ๐ฃ, maximize ๐ ๐ฃ subject to sublevel set
ฮ ๐ฃ = แผ๐ฅ: เดค๐ ๐ฅ. ๐ฃ โค 0} satisfying constraints
- In cases with bounded disturbances, need ๐ ๐ฃ โฅ ๐๐๐๐(๐ฃ) for the
sublevel set ฮ (๐ฃ) to be strongly returnable
- Simplifications occur due to positive invariance of sublevel sets
90
Constructing เดฅ๐ฝ
โข Off-line simulations and machine learning
- เดค๐ ๐ฅ, ๐ฃ = ฮฆ(๐ฅ, ๐ฃ) is a classifier separating safe and unsafe initial
conditions and constant commands
- Scenarios (Monte Carlo simulations) are run with respect to ๐ค(โ )
- Non-smooth classifiers permitted, e.g., เดค๐ ๐ฅ, ๐ฃ = min๐แผฮฆ๐(๐ฅ, ๐ฃ)}.
Thus can represent unions of safe regions.
ฮฆ ๐ฅ(0), ๐ฃ โค 0 โ (๐ฅ(0), ๐ฃ) is safe
- In the disturbance-free case, model simulations are run for
various combinations of ๐ฅ(0) and ๐ฃ
91
Constructing เดฅ๐ฝ
๐ง =๐ฅ๐ฃ
ฮฆ๐ ๐ง = maxแผ๐๐๐๐ ๐ง โ ๐๐ , ๐ โ ๐ผ}
เดค๐ ๐ฅ, ๐ฃ = เดค๐ ๐ง = min๐
ฮฆ๐(๐ง)
Example: Cover safe initial
conditions and commands
by a union of hyper-
rectangles
92
เดฅ๐ฝ implicitly defined through on-line prediction
โข Constraints:
๐ถ = แผ ๐ฅ, ๐ฃ : โ๐ ๐ฅ, ๐ฃ โค 0, ๐ = 1,โฏ , ๐}
เดค๐ ๐ฅ, ๐ฃ = ๐๐๐ฅแผโ๐(๐ ๐ก, ๐ฅ, ๐ฃ, ๐ค โ , ๐ฃ), ๐ = 1,โฏ , ๐, ๐ก = 0,โฏ , ๐ก0, ๐ค โ โ ๐}
where ๐ก0 is sufficiently large so that
โ๐ ๐ ๐ก, ๐ฅ, ๐ฃ, ๐ค โ , ๐ฃ โค โ๐, ๐ = 1,โฏ , ๐, ๐ค โ โ ๐ and ๐ก โฅ ๐ก0
๐ = โsolutionโ
Bemporad, IEEE TAC AC-43(4) 451-461 (1998); Gilbert and K, Automatica, 38, 2063-2073, 2002
โข On-line prediction of maximum constraint violation
93
เดฅ๐ฝ implicitly defined through on-line prediction
Sun and K., IEEE TCST 13 (6), pp. 991-919, 2005
โข Simplifications in parametric uncertainty/robust reference
governor case
๐ ๐ก, ๐ฅ, ๐ฃ, ๐ค โ ๐ ๐ก, ๐ฅ, ๐ฃ, ๐ค0 +๐๐
๐๐คแ๐ก,๐ฅ,๐ฃ,๐ค0
(๐ค โ ๐ค0)
๐๐
๐๐ค|๐ก,๐ฅ,๐ฃ,๐ค0
= solution of sensitivity ODEs
94
Control of EAMSD
โข Position and current constraints
Miller, K, Gilbert, Washabaugh, IEEE Control Systems Magazine,2000
๐๐ฅ1๐๐ก
= ๐ฅ2
๐๐ฅ2๐๐ก
=๐
๐๐ฅ1 โ
๐
๐๐ฅ2 +
๐ผ
๐
๐ข
๐0 โ ๐ฅ1๐พ , ๐ข = ๐๐ฝ
๐ข =1
๐ผ(๐0 โ ๐ฅ1 )
๐พ ๐๐ฃ โ ๐๐๐ฅ2
๐0
๐ฅ๐(๐ฃ)
เดค๐ ๐ฅ, ๐ฃ =๐
2๐ฅ1 โ ๐ฃ 2 +
๐
2๐ฅ22 โ ๐ ๐ฃ
Position constraint
๐ฅ1
๐ฅ2
Current constraint
เดค๐ ๐ฅ, ๐ฃ โค 0
95
Control of EAMSD
Without RG
Experimental Results
With RG
โข Position and current constraints
๐0
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
Linear systems with nonlinear constraints
๐ฅ ๐ก + 1 = ๐ด๐ฅ ๐ก + ๐ต๐ฃ ๐ก
๐ฆ ๐ก โ ๐ ๐๐๐ ๐๐๐ ๐ก โ ๐+
y ๐ก = ๐ถ๐ฅ ๐ก + ๐ท๐ฃ ๐ก
๐ = ๐ฆ: โ๐ ๐ฆ โค 0, ๐ = 1,โฏ , ๐
โข Linear system model:
โข Treat nonlinear constraints (without polyhedral approximations):
Kalabic et. al., Proc. of 2011 CDC, pp. 2680-2686 99
Comments
โข Motivation: Handling constraints for Feedback Linearizable systems
โข Theory in Gilbert, K., and Tan (1994,1995) and Gilbert and K. (2002) is
applicable to the case of linear systems and nonlinear constraints
โข We discuss computations, heuristics and examples
100
Scalar Reference Governor
๐ฝ ๐ก โ 0,1 ,
๐ฆ ๐ก + ๐|๐ก โ ๐, ๐ = 0,โฏ ๐กโ
๐ฝ ๐ก โ ๐๐๐ฅ
๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ๐ฝ ๐ก ๐ ๐ก โ ๐ฃ ๐ก โ 1
โข Optimization problem:
1We use the approach of Bemporad (1998) with implicitly defined constraints
2Here ๐กโ 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
๐ฆ ๐ก + ๐|๐ก = ฮ ๐ ๐ฅ ๐ก + ๐ป ๐ ๐ฃ ๐ก โ 1 + ๐ฝ ๐ก ๐ป ๐ (๐ ๐ก โ ๐ฃ ๐ก โ 1 )
ฮ ๐ = ๐ถ๐ด๐ ,
๐ป ๐ = ๐ถ โ ฮ ๐ (๐ผ โ ๐ด)โ1๐ต + ๐ท
โข Predicted output is an affine function of ๐ฝ(๐ก)
โข Predicted response to a constant command
๐ฃ ๐ก + ๐|๐ก โก ๐ฃ ๐ก = ๐ฃ ๐ก โ 1 + ๐ฝ(๐ก)(๐ ๐ก โ ๐ฃ ๐ก )
102
Convex nonlinear constraints
If ๐ฝ 0 = 0 is feasible, then an admissible interval for the values of ๐ฝ ๐ก is of
the form
๐พ ๐ก = 0, ๐ฝ๐๐๐ฅ ๐ก , 1 โฅ ๐ฝ๐๐๐ฅ ๐ก โฅ0
and the reference governor sets ๐ฝ ๐ก =๐ฝ๐๐๐ฅ ๐ก . The constraints are satisfied
for all ๐ก โฅ 0.
โข Suppose that
๐ = ๐ฆ: โ๐ ๐ฆ โค 0, ๐ = 1,โฏ , ๐
โ๐ are convex functions
โข Proposition
103
Algorithmic implementation
If โ๐ ๐ฆ ๐ก + ๐ ๐ก > 0 where ๐ฆ ๐ก + ๐ ๐ก is the predicted response with
๐ฝ ๐ก = ๐ผ, use bisections to search for a scalar ๐ผ+ that approximately
solves
โ๐ ฮ ๐ ๐ฅ ๐ก + ๐ป ๐ ๐ฃ ๐ก โ 1 + ๐ผ+๐ป ๐ ๐ ๐ก โ ๐ฃ ๐ก โ 1 = 0
โข Set ๐ผ = 1.
โข For ๐ = 1,โฏ , ๐, and ๐ = 0,โฏ , ๐กโ, repeat
โข Update ๐ผ = ๐ผ+
โข Apply โconstraints active last firstโ evaluation heuristics (see the
paper)
104
Convex Quadratic Constraints
๐ฆ๐ เทจ๐๐ฆ + แ๐๐ฆ + แ๐ถ โค 0, เทจ๐ = เทจ๐๐ โฅ 0
โข Suppose that the constraints are of the form
โข This is a quadratic function of ๐ฝ(๐ก). The root finding can be performed by
solving a quadratic equation.
โข Then the constraints can be re-written as
๐ฅ ๐ก ๐ ๐ฃ ๐ก ๐ + ๐ฝ(๐ก)(๐ ๐ก โ ๐ฃ(๐ก))๐ เดค๐ ๐๐ฅ(๐ก)
๐ฃ ๐ก + ๐ฝ(๐ก)(๐ ๐ก โ ๐ฃ(๐ก)
+ าง๐ ๐๐ฅ(๐ก)
๐ฃ ๐ก + ๐ฝ(๐ก)(๐ ๐ก โ ๐ฃ(๐ก)+ แ๐ถ โค 0
105
Mixed Logical Dynamic Constraints
๐๐ ๐ฆ > 0 โ โ๐ ๐ฆ โค 0, ๐ = 1,โฏ๐,
โข We consider a set of constraints of if-then type
where ๐๐ , โ๐ are convex functions
โข Observations:
โข The set of ฮฒ ๐ก โ [0,1] for which ๐๐(๐ฆ ๐ก + ๐ ๐ก ) โค 0 is a
(possibly empty) sub-interval of [0,1], ๐พ๐(๐)
โข The set of ฮฒ ๐ก โ [0,1] for which โ๐(๐ฆ ๐ก + ๐ ๐ก ) โค 0 is
another (possibly empty) sub-interval of [0,1], ๐พ๐(๐)
106
Mixed Logical Dynamic Constraints
๐๐ ๐ฆ(๐ก + ๐|๐ก) > 0 โ โ๐ ๐ฆ(๐ก + ๐|๐ก) โค 0
โข Then the set of feasible ๐ฝ ๐ก โ [0,1] for which
is also an interval 0,1 โฉ ๐พ๐ ๐ โฉ ๐พ๐(๐)
โข The recursive feasibility of ๐ฝ ๐ก = 0 is preserved by the reference
governor, hence it follows that the feasible values of ๐ฝ(๐ก) satisfy
ฮฒ ๐ก โ 0, ๐ฝ๐๐๐ฅ , ๐ฝ๐๐๐ฅ โฅ 0
107
Concave Constraints
โข Suppose that the constraint functions โ๐ in Y = ๐ฆ: โ๐ ๐ฆ โค 0 , ๐ =1,โฏ๐, are concave
โข Approximate the constraints by the dynamically reconfigurable affine
constraints
๐ฆ(๐ก + ๐|๐ก) โ ๐๐(๐ก)
where
๐๐ ๐ก = ๐ฆ: โ๐ ๐ฆ๐,โ ๐ก +๐โ๐๐๐ฆ
(๐ฆ๐,โ ๐ก )(๐ฆ โ ๐ฆ๐,โ ๐ก ) โค 0 ,
๐ = 1,โฏ ๐,
and ๐๐(๐ก) โ ๐
108
Concave Constraints
If ๐ฆ๐โ 0 , ๐ = 1,โฏ ๐ exist such that ๐ฝ 0 = 0 is feasible, then
๐ฝ ๐ก = 0 and ๐ฆ๐โ ๐ก =๐ฆ๐โ ๐ก โ 1 remain feasible for ๐ก > 0 and
constraints are adhered to for all ๐ก>0.
โข Proposition
๐ฆ ๐ก
โ(๐ฆ)
๐ฆ๐โ ๐ก
๐ฆ
109
Spacecraft relative motion example
โข Dynamic model is linear
โข HillโClohessy-Wiltshire equations
โข Constraints are nonlinear:
โข Approach within LOS half-cone in front of
the docking port (convex, quadratic)
โข Thrust/delta-v magnitude squared is
limited (convex, quadratic)
โข Soft-docking: Small velocity when close to
docking position (Mixed Logical Dynamic
with quadratic ๐ and โ)
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)
โข 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
โข Dynamics are feedback linearizable
โข Constraints
- Current limit results in a concave nonlinear constraint
- Overshoot constraint is linear
แถ๐ฅ1แถ๐ฅ2
=0 1
โ๐/๐ โ๐/๐
๐ฅ1๐ฅ2
+0
1/๐๐ข
๐ข = ๐๐ฃ โ ๐๐๐ฅ2
0 โค ๐ข โค๐ผ ๐๐๐๐ฅ
๐พ
๐0 โ ๐ฅ1๐พ
๐ฅ1 โค 0.008
force
max. current
112
Electromagnetically Actuated Mass Spring Damper
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
Electromagnetically Actuated Mass Spring Damper
โข Landing control example
โข Voltage limits
โข MLD constraints on soft-landing
velocity AND magnetic force
exceeding spring force
๐๐ง
๐๐ก= ๐
๐๐
๐๐ก=
1
๐(โ๐น๐๐๐ + ๐๐ ๐ง๐ โ ๐ง โ ๐๐)
๐๐
๐๐ก=๐๐ โ ๐๐ +
2๐๐๐(๐๐ + ๐ง)2
๐
2๐๐ ๐๐ + ๐ง
๐น๐๐๐ =๐๐
๐๐ + ๐ง 2๐2
114