A Pragmatic Approach to Robust Gain Scheduling
A Pragmatic Approach to Robust Gain Scheduling
Greg Stewart, PhD Fellow – Research and Development Honeywell Automotive Software
Outline • Traditional gain scheduling design • How things can go wrong in practice (or “awkward moments as a control engineer”)
• Model uncertainty & closed-loop stability • A constructive design procedure • Summary
2
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Standard Gain Scheduling Procedure Begin with a nonlinear plant y = G(u,w): Create LPV Model
G(θ,s)
Design Linear Control
K(θ,s)
Gain Schedule
A(θ ) B(θ ) K = , θ = g ( y, w) C (θ ) D(θ ) w
Analyze Performance
G(u,w)
u
y
K(y,w) 3
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
w
The Problem Create LPV Model
Design Linear Control
Gain Schedule
Analyze Performance
• Steps 1 and 2 (linear modeling and control) are reliably followed in industrial practice. • However, ad hoc techniques are often applied for scheduling the control (Step 3): - Interpolation of controller coefficients e.g. PID tuning parameters - Interpolation of controller outputs - Etc.
e
K1(s)
λ
K2(s)
1-λ
K +
u
• Thus nonlocal stability and performance (Step 4) are often not guaranteed in practice
Practitioners need nonlinear control design techniques! 4
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Practitioners need nonlinear control design techniques Consider set of finite-gain L2 stable plants y=G(u,w) yτ
L2
≤ γ gu uτ
L2
+ γ gw wτ
L2
+ β g for all τ ∈ [0, ∞)
all < ∞ Control design requirements: 1. Robust stability: closed-loop L2 stable for w - all exogenous signals w ∈ L2e and - realistic plant-model mismatch u 2. Performance: closed-loop performance better than K=0. 3. Within designers’ capability: can K(y,w) be designed using linear control techniques? 5
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
G(u,w) y
K(y,w)
w
Model Uncertainty & Closed-Loop Stability
6
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Model Uncertainty & Closed-loop Stability If ∆ and Q are both finite-gain L2 stable: ∆(u , w)τ
L2
≤ γ ∆u uτ
Q(e, w)τ
L2
≤ γ qe eτ
L2 L2
+ γ ∆w wτ
+ γ qw wτ
L2 L2
+ β∆
+ βq
Then closed-loop with G and K is finite-gain L2 stable if: γ ∆uγ qe < 1
• Which may be interpreted as a tradeoff between: - model uncertainty γ∆u and - controller aggressiveness γqe 7
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Gain Scheduling with Stability Guarantees: A Constructive Design Procedure
8
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 1. Create LPV model • Consider a simple plant: x (t ) = a(θ ) x(t ) + b(θ )u (t ) y (t ) = x(t )
Create LPV Model
Design Linear Control
3 2
Gain Schedule
1
b(θ)
0 -1
Analyze Performance
-2
-80
-60
-40
-20
0
θ
20
40
60
80
100
-80
-60
-40
-20
0
20
40
60
80
100
(note gain sign change)
-5 -5.5
a(θ)
-6
-6.5 -7
θ
In practice, designer may not know the underlying nonlinear structure. 9
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 1. Create LPV model
Create LPV Model
• Partition space and assign local linear models: 3
Design Linear Control
2 1
b(θ)
0 -1
Gain Schedule
-2
Analyze Performance
-80
-60
-40
-20
θ
0
20
40
60
80
100
80
100
-5 -5.5
a(θ)
j=1
-6
j=2
j=3
-6.5 -7
-80
−2 ˆ G1 ( s ) = s+5 10
-60
-40
-20
0
θ
20
40
Gˆ 2 ( s ) = 0 (best approx for a gain sign change)
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
60
3 ˆ G3 ( s ) = s+7
Design procedure: 2. Design Linear Control • Design linear controllers for each model: Create LPV Model
Design Linear Control
−2 ˆ G1 ( s ) = s+5
Gˆ 2 ( s ) = 0
3 ˆ G3 ( s ) = s+7
K 2 (s) = 0
I3 K 3 ( s ) = P3 + s
Gain Schedule
Analyze Performance
I1 K1 ( s ) = P1 + s
(PID and other familiar structures are popular in industrial practice.)
Question: how should we switch between controllers? 11
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 3. Gain Schedule
Create LPV Model
• Form the linear Youla-Kucera parameters for each of the modes j: −1 ˆ Q ( s) = K ( s) 1 + G (s) K (s) j
Design Linear Control
j
(
j
)
• And construct the gain scheduled controller:
Gain Schedule
Analyze Performance
Uses only simple components: • banks of linear Qj(s) and Ĝj(s) • signal switches σ(t)
12
j
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 4. Analyze Performance Robust stability Create LPV Model
Design Linear Control
Gain Schedule
• Given finite-gain L2 stable plant y=G(u,w) • For any selection of stable linear plant models Ĝj(s) and switching strategy σ(t) • It is always possible to design corresponding linear controllers Kj(s) • Such that K(y,w) is finite-gain L2 stabilizing for G(u,w) Q is finite-gain L2 stable with
Analyze Performance
Outline of Proof
γ qe ≤
∆ is finitegain L2 stable with γ∆u < ∞ 13
m
∑ j =1
Q j (s)
2 H∞
Thus it is always possible to make γ∆u γqe < 1 ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 4. Analyze Performance Closed-loop performance Create LPV Model
• Given robustly stable closed-loop (γ∆uγqe < 1) • True closed-loop performance in a given mode:
Design Linear Control
r
Kj(s)
G(u,w)
yp
Gain Schedule
Analyze Performance
is related to nominal performance r
By the relation:
ynom − y p
L2
(
Kj(s)
)
−1 ˆ ≤ 1 + G j (s) K j (s)
γ ∆uγ qe r 1 − γ ∆uγ qe 14
⋅ H∞
L2
ynom
Ĝj(s)
So yp→ynom for ∆→0
γ ∆w β ∆ wL + + 1 − γ ∆u γ qe 1 − γ ∆u γ qe
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
2
Design procedure: 4. Analyze Performance Closed-loop performance Create LPV Model
Design Linear Control
Gain Schedule
Analyze Performance
• Achievable performance depends in part on model uncertainty • For example, integral control* in each Kj(s) is possible for all modes if uncertainty is bounded by:
γ ∆u
Greg Stewart, PhD Fellow – Research and Development Honeywell Automotive Software
Outline • Traditional gain scheduling design • How things can go wrong in practice (or “awkward moments as a control engineer”)
• Model uncertainty & closed-loop stability • A constructive design procedure • Summary
2
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Standard Gain Scheduling Procedure Begin with a nonlinear plant y = G(u,w): Create LPV Model
G(θ,s)
Design Linear Control
K(θ,s)
Gain Schedule
A(θ ) B(θ ) K = , θ = g ( y, w) C (θ ) D(θ ) w
Analyze Performance
G(u,w)
u
y
K(y,w) 3
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
w
The Problem Create LPV Model
Design Linear Control
Gain Schedule
Analyze Performance
• Steps 1 and 2 (linear modeling and control) are reliably followed in industrial practice. • However, ad hoc techniques are often applied for scheduling the control (Step 3): - Interpolation of controller coefficients e.g. PID tuning parameters - Interpolation of controller outputs - Etc.
e
K1(s)
λ
K2(s)
1-λ
K +
u
• Thus nonlocal stability and performance (Step 4) are often not guaranteed in practice
Practitioners need nonlinear control design techniques! 4
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Practitioners need nonlinear control design techniques Consider set of finite-gain L2 stable plants y=G(u,w) yτ
L2
≤ γ gu uτ
L2
+ γ gw wτ
L2
+ β g for all τ ∈ [0, ∞)
all < ∞ Control design requirements: 1. Robust stability: closed-loop L2 stable for w - all exogenous signals w ∈ L2e and - realistic plant-model mismatch u 2. Performance: closed-loop performance better than K=0. 3. Within designers’ capability: can K(y,w) be designed using linear control techniques? 5
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
G(u,w) y
K(y,w)
w
Model Uncertainty & Closed-Loop Stability
6
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Model Uncertainty & Closed-loop Stability If ∆ and Q are both finite-gain L2 stable: ∆(u , w)τ
L2
≤ γ ∆u uτ
Q(e, w)τ
L2
≤ γ qe eτ
L2 L2
+ γ ∆w wτ
+ γ qw wτ
L2 L2
+ β∆
+ βq
Then closed-loop with G and K is finite-gain L2 stable if: γ ∆uγ qe < 1
• Which may be interpreted as a tradeoff between: - model uncertainty γ∆u and - controller aggressiveness γqe 7
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Gain Scheduling with Stability Guarantees: A Constructive Design Procedure
8
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 1. Create LPV model • Consider a simple plant: x (t ) = a(θ ) x(t ) + b(θ )u (t ) y (t ) = x(t )
Create LPV Model
Design Linear Control
3 2
Gain Schedule
1
b(θ)
0 -1
Analyze Performance
-2
-80
-60
-40
-20
0
θ
20
40
60
80
100
-80
-60
-40
-20
0
20
40
60
80
100
(note gain sign change)
-5 -5.5
a(θ)
-6
-6.5 -7
θ
In practice, designer may not know the underlying nonlinear structure. 9
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 1. Create LPV model
Create LPV Model
• Partition space and assign local linear models: 3
Design Linear Control
2 1
b(θ)
0 -1
Gain Schedule
-2
Analyze Performance
-80
-60
-40
-20
θ
0
20
40
60
80
100
80
100
-5 -5.5
a(θ)
j=1
-6
j=2
j=3
-6.5 -7
-80
−2 ˆ G1 ( s ) = s+5 10
-60
-40
-20
0
θ
20
40
Gˆ 2 ( s ) = 0 (best approx for a gain sign change)
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
60
3 ˆ G3 ( s ) = s+7
Design procedure: 2. Design Linear Control • Design linear controllers for each model: Create LPV Model
Design Linear Control
−2 ˆ G1 ( s ) = s+5
Gˆ 2 ( s ) = 0
3 ˆ G3 ( s ) = s+7
K 2 (s) = 0
I3 K 3 ( s ) = P3 + s
Gain Schedule
Analyze Performance
I1 K1 ( s ) = P1 + s
(PID and other familiar structures are popular in industrial practice.)
Question: how should we switch between controllers? 11
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 3. Gain Schedule
Create LPV Model
• Form the linear Youla-Kucera parameters for each of the modes j: −1 ˆ Q ( s) = K ( s) 1 + G (s) K (s) j
Design Linear Control
j
(
j
)
• And construct the gain scheduled controller:
Gain Schedule
Analyze Performance
Uses only simple components: • banks of linear Qj(s) and Ĝj(s) • signal switches σ(t)
12
j
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 4. Analyze Performance Robust stability Create LPV Model
Design Linear Control
Gain Schedule
• Given finite-gain L2 stable plant y=G(u,w) • For any selection of stable linear plant models Ĝj(s) and switching strategy σ(t) • It is always possible to design corresponding linear controllers Kj(s) • Such that K(y,w) is finite-gain L2 stabilizing for G(u,w) Q is finite-gain L2 stable with
Analyze Performance
Outline of Proof
γ qe ≤
∆ is finitegain L2 stable with γ∆u < ∞ 13
m
∑ j =1
Q j (s)
2 H∞
Thus it is always possible to make γ∆u γqe < 1 ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
Design procedure: 4. Analyze Performance Closed-loop performance Create LPV Model
• Given robustly stable closed-loop (γ∆uγqe < 1) • True closed-loop performance in a given mode:
Design Linear Control
r
Kj(s)
G(u,w)
yp
Gain Schedule
Analyze Performance
is related to nominal performance r
By the relation:
ynom − y p
L2
(
Kj(s)
)
−1 ˆ ≤ 1 + G j (s) K j (s)
γ ∆uγ qe r 1 − γ ∆uγ qe 14
⋅ H∞
L2
ynom
Ĝj(s)
So yp→ynom for ∆→0
γ ∆w β ∆ wL + + 1 − γ ∆u γ qe 1 − γ ∆u γ qe
ROCOND'12 - Aalborg, Denmark, June 20-22, 2012
2
Design procedure: 4. Analyze Performance Closed-loop performance Create LPV Model
Design Linear Control
Gain Schedule
Analyze Performance
• Achievable performance depends in part on model uncertainty • For example, integral control* in each Kj(s) is possible for all modes if uncertainty is bounded by:
γ ∆u
