Stability Analysis of Rule-Based Affine
Systems Using Model Checking
2014 S5
Jonathan Hoffman, Kuldip S. Rattan, and
Matthew Clark
AFRL Aerospace Systems Directorate
Verification and Validation of Complex Systems
14 June 2014
Integrity  Service  Excellence
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Rule-Based Systems
• Natural language is a convenient way of
describing the if-then natural rules for controlling a
dynamic system.
• Fuzzy Logic is a method to represent If-then rules
• Fuzzy systems can be used to approximate the
behavior of a non-linear system and then be
converted into a piecewise linear representation.
• But how do we convert Fuzzy systems into a form
that is analyzable for verification and validation?
• Specifically, how do we analyze the stability of a
Piecewise Affine Fuzzy System
• How scalable is this stability analysis approach to
large piecewise linear systems?
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2
Two State Rule Base Controller
Classic Controller
Fuzzy Controller
X1 NB
X1 NM
X1 NS
X1 Z
X1 PS
X1 PM
X1 PB
X2
NB
NB
NB
NB
NB
NM
NS
Z
RULE BASE
X2 X2 X2 X2 X2
NM NS Z PS PM
NB NB NB NM NS
NB NB NM NS Z
NB NM NS Z PS
NM NS Z PS PM
NS Z PS PM PB
Z PS PM PB PB
PS PM PB PB PB
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X2
PB
Z
PS
PM
PB
PB
PB
PB
3
Two State
Inference System
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Piecewise
Hybrid System Derivation
Sandhu, G. S., Brehm, T., & Rattan, K. S. (1996, May). Analysis and design of a proportional plus derivative fuzzy logic controller. In Proceedings
of Aerospace and Electronics Conference, 1996. (Vol. 1, pp. 397-404). IEEE.
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Equivalent Piecewise
Hybrid Controller Mode
𝑒𝑒
Classic Linear Controller
∆𝑒𝑒
Piecewise Linear Controller
𝑒𝑒
∆𝑒𝑒
Kp𝒆𝒆𝒆𝒆𝒆𝒆
K𝒅𝒅𝒆𝒆𝒆𝒆𝒆𝒆
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
u =Kp𝒆𝒆𝒆𝒆𝒆𝒆 ∗e + K𝒅𝒅𝒆𝒆𝒆𝒆𝒆𝒆 ∗ 𝚫𝚫𝚫𝚫 + 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
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Equivalent Piecewise
Hybrid Controller Mode
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Hybrid Model
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Lyapnuov Function
• For linear systems
, assume a
quadratic function
• System is stable if we can find a P such
that the Lyapnuov equation
𝑃𝑃 ∗ 𝐴𝐴 + 𝐴𝐴𝑇𝑇 ∗ 𝑃𝑃 + 𝑄𝑄 ≤ 0,
𝑄𝑄 > 0
• For a nonlinear system, there is no generic
method. However, a polynomial Lyapnuov
function can be searched using semidefinite programming or S-procedure.
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Lyapnuov Function for
Hybrid Systems
• For a hybrid system
where i indicates the mode the mode selection,
the stability of the switched systems cannot be
inferred from the stability of the individual mode.
• For examples for two modes (m=2) , if both
modes are stable (we can find positive definite
), switching can result in the hybrid system
unstable for some initial conditions.
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Hybrid System
• 𝑥𝑥̇ = 𝐴𝐴𝑖𝑖 𝑥𝑥 + 𝐵𝐵𝐵𝐵
• 𝑦𝑦 = 𝐶𝐶𝐶𝐶 + 𝐷𝐷𝐷𝐷
−1
10
• 𝐴𝐴1 =
−100 −1
−1 100
• 𝐴𝐴2 =
−10 −1
1
• 𝑥𝑥 0 =
0
0
𝐵𝐵 =
1
1
𝐶𝐶 =
0
0
𝐷𝐷 =
0
0
1
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Zero-Input Response
A1 Zero-Input Response
A2 Zero-Input Response
Stable
Marginally Stable
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Hybrid System Switching
State 1
𝐴𝐴1
−1
10
−100 −1
𝑥𝑥1 𝑥𝑥2 < 0
State 2
𝐴𝐴2
𝑥𝑥1 𝑥𝑥2 > 0
−1
−10
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100
1
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Effects of Switching
System 1
State 1
𝐴𝐴1
−1
10
−100 −1
System 2
State 1
𝑥𝑥1 𝑥𝑥2 < 0
𝐴𝐴2
𝑥𝑥1 𝑥𝑥2 > 0
𝑥𝑥1 𝑥𝑥2 < 0
𝐴𝐴2
−1 100
−10
1
State 2
𝑥𝑥1 𝑥𝑥2 > 0
−1 100
−10
1
State 2
𝐴𝐴1
−1
10
−100 −1
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Effects of Switching
System 1
State 1
𝐴𝐴1
−1
10
−100 −1
System 2
State 1
𝑥𝑥1 𝑥𝑥2 < 0
𝐴𝐴2
𝑥𝑥1 𝑥𝑥2 > 0
𝑥𝑥1 𝑥𝑥2 < 0
𝐴𝐴2
−1 100
−10
1
State 2
𝑥𝑥1 𝑥𝑥2 > 0
−1 100
−10
1
Stable
State 2
𝐴𝐴1
−1
10
−100 −1
Unstable
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Lyapnuov Function for
Hybrid Systems (Cont’d)
• If the switching results in a stable systems,
there should exists a common (global) P
matrix. However, finding this P matrix is
difficult.
• On the other hand, if both modes are
unstable, switching can make the overall
system stable.
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Lyapunov Stability
for Hybrid Systems
• Lyapunov Stability Equations:
• 𝑃𝑃 ∗ 𝐴𝐴 + 𝐴𝐴𝑇𝑇 ∗ 𝑃𝑃 + 𝐼𝐼 ≤ 0
• 𝑉𝑉 𝑥𝑥 = 𝑥𝑥 𝑇𝑇 𝑃𝑃𝑃𝑃
• Objective is to determine a positive definite matrix
P to satisfy the Lyapunov equations for a hybrid
system.
• Want to find a single P matrix for whole system.
• Difficulties
• The A matrices change due to switching
• Switching itself can cause instability
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Possible Approaches
• Two possible Approaches
1. Find a P matrix common to each hybrid mode
2. Use simulation data to calculate a ‘Global’ P
• First approach could be very difficult for
hybrid systems with many modes.
• Also, as shown in the previous example, a
system can be stable or unstable depending
on the switching mechanism, regardless of
the individual mode dynamics.
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Finding a Global P
• Hypothesis:
– A global P matrix that solves Lyapunov
equations can be found by running zero-input
simulations of a hybrid system and calculating
a positive definite P matrix to fit a decreasing
energy function, V(x).
– The P matrix can be verified for all initial
conditions (in a specified range) by using a
model checker.
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Lyapunov Stability
for Hybrid Systems
• It has been shown that Lyapunov stability
can be extended to hybrid systems.
• The switching of the hybrid system
separates V(x) into distinct intervals.
• 𝑉𝑉 𝑥𝑥 = 𝑥𝑥 𝑇𝑇 𝑃𝑃𝑃𝑃
• 𝑉𝑉̇ (𝑥𝑥) ≤ 0
Branicky, Michael S. "Multiple Lyapunov functions and other analysis tools for switched and hybrid systems."
Automatic Control, IEEE Transactions on 43.4 (1998): 475-482.
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Lyapunov Stability
for Hybrid Systems
• Two conditions must be met for stability:
– The Lyapunov functions in each interval are
decreasing
– The maximum value of
the Lyapunov function in
the interval of 𝑉𝑉𝑖𝑖 is less
than the maximum value
in the previous interval
of 𝑉𝑉𝑖𝑖
DeCarlo, Raymond A., et al. "Perspectives and results on the stability and stabilizability of hybrid systems." Proceedings of the IEEE 88.7 (2000): 1069-1082.
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Constraints on Energy Equation
• Constraints may be placed on the Energy
Equation.
• 𝑉𝑉 𝑥𝑥 = 𝑥𝑥 𝑇𝑇 𝑃𝑃𝑃𝑃 ≤ 𝑊𝑊(𝑡𝑡)
• 𝑊𝑊(𝑡𝑡) bounds 𝑉𝑉 𝑥𝑥 to be non-increasing
W(t) = constant
W(t) = exponential
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Approach
• Using the previous hybrid
system a simulation is run.
• Knowing the state data over
time the energy equation can
be written as a system of
equations:
• 𝑉𝑉 𝑥𝑥 0
= 𝑥𝑥 𝑇𝑇 0 𝑃𝑃𝑃𝑃 0 ≤ 𝑊𝑊 0
• 𝑉𝑉 𝑥𝑥 𝑛𝑛
= 𝑥𝑥 𝑇𝑇 𝑛𝑛 𝑃𝑃𝑃𝑃 𝑛𝑛 ≤ 𝑊𝑊 𝑛𝑛
• 𝑉𝑉 𝑥𝑥 1
• ⋮
= 𝑥𝑥 𝑇𝑇 1 𝑃𝑃𝑃𝑃 1 ≤ 𝑊𝑊 1
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Solving for P
• The systems of equations can now be solved for
P.
• For the t=0 case where W is a constant:
• 𝑥𝑥 𝑇𝑇 0 𝑃𝑃𝑃𝑃 0 ≤ 𝑊𝑊
𝑃𝑃1 𝑃𝑃2 𝑥𝑥1 0
• 𝑥𝑥1 0 𝑥𝑥2 0
≤ 𝑊𝑊
𝑃𝑃3 𝑃𝑃4 𝑥𝑥2 0
• 𝑥𝑥1,0 𝑃𝑃1 𝑥𝑥1,0 +𝑥𝑥1,0 𝑃𝑃2 𝑥𝑥2,0 +𝑥𝑥2,0 𝑃𝑃3 𝑥𝑥1,0 +𝑥𝑥2,0 𝑃𝑃4 𝑥𝑥2,0 ≤ 𝑊𝑊
• One inequality is produced for each time step of
the simulation.
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Solving for P
• The system of inequalities becomes:
• 𝑥𝑥1,0 𝑃𝑃1 𝑥𝑥1,0 +𝑥𝑥1,0 𝑃𝑃2 𝑥𝑥2,0 +𝑥𝑥2,0 𝑃𝑃3 𝑥𝑥1,0 +𝑥𝑥2,0 𝑃𝑃4 𝑥𝑥2,0 ≤ 𝑊𝑊
• 𝑥𝑥1,1 𝑃𝑃1 𝑥𝑥1,1 +𝑥𝑥1,1 𝑃𝑃2 𝑥𝑥2,1 +𝑥𝑥2,1 𝑃𝑃3 𝑥𝑥1,1 +𝑥𝑥2,1 𝑃𝑃4 𝑥𝑥2,1 ≤ 𝑊𝑊
• ⋮
• 𝑥𝑥1,𝑛𝑛 𝑃𝑃1 𝑥𝑥1,𝑛𝑛 +𝑥𝑥1,𝑛𝑛 𝑃𝑃2 𝑥𝑥2,𝑛𝑛 +𝑥𝑥2,𝑛𝑛 𝑃𝑃3 𝑥𝑥1,𝑛𝑛 +𝑥𝑥2,𝑛𝑛 𝑃𝑃4 𝑥𝑥2,𝑛𝑛 ≤ 𝑊𝑊
• Solving for P is a non trivial problem when there
are many equations.
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Solving for P
• A model checker is a good option to find a P to
satisfy the system of equations.
• Because model checkers are good at finding
counterexamples to logical equations the
statement is asserted:
– There does not exist a P matrix to satisfy the system
of equations.
• If a solution to the system of equations exists, the
model checker will return with one P matrix that
refutes the assertion. This P will satisfy the system
of equations for the simulated state data.
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Stable/Marginally Stable
Switching
State 1
𝐴𝐴1
−1
10
−100 −1
𝑥𝑥1 𝑥𝑥2 < 0
State 2
𝐴𝐴2
𝑥𝑥1 𝑥𝑥2 > 0
−1
−10
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100
1
27
Stable/Marginally Stable
Results
Simulation
Stable/Marginally Stable Modes Hybrid System
• Stable Zero-Input response of the system
• Global P matrix found
Energy
Global P Matrix
𝑃𝑃 =
1.0018
−0.0010
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−0.0010
0.1002
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Two Unstable Modes
with Switching
State 1
𝐴𝐴1
1
−100
10
1
𝑥𝑥1 𝑥𝑥2 < 0
State 2
𝐴𝐴2
𝑥𝑥1 𝑥𝑥2 > 0
1
−10
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100
1
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Two Unstable Mode
Switching Results
Simulation
2 Unstable Modes Hybrid System
• Stable Zero-Input response of the system
• Global P matrix found
Energy
Global P Matrix
𝑃𝑃 =
1.0000
0.0100
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0.0100
0.1000
30
How Scalable?
• So how scalable is this stability analysis
approach to large piecewise linear
systems?
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8 State Proportional Controller
S1
S2
S3
S4
S5
S6
S7
S8
• 8 State Proportional Controller built from fuzzy type ruleset.
• Problem: Need to guarantee that each state is active at some point in the
simulation.
• Solution: Run one Zero-Input Response starting in each state and feed all of
that data to the model checker.
• Potential for optimization later on to minimize the number of simulations
needed and resulting in less data for the model checker to analyze.
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8 State Proportional
Controller Results
Simulations
8 State Hybrid System
• Stable Zero-Input response of the system
beginning from each of the 8 states
• Global P matrix found
Energy
Global P Matrix
𝑃𝑃 =
1.0254
−0.0010
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−0.0010
0.0001
33
36 State Proportional Plus
Derivative Controller
1
7
13
19
25
31
2
8
14
20
26
32
Two State Fuzzy statistics
- 36 Discrete states
- 2 Continous Dynamic Modes
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
X1 NB
X1 NM
X1 NS
X1 Z
X1 PS
X1 PM
X1 PB
X2
NB
NB
NB
NB
NB
NM
NS
Z
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6
12
18
24
30
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RULE BASE
X2 X2 X2 X2 X2
NM NS Z PS PM
NB NB NB NM NS
NB NB NM NS Z
NB NM NS Z PS
NM NS Z PS PM
NS Z PS PM PB
Z PS PM PB PB
PS PM PB PB PB
X2
PB
Z
PS
PM
PB
PB
PB
PB
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36 State Proportional
Plus Derivative Results
Simulations
36 State Hybrid System
• Stable Zero-Input response
of the system beginning from
each of the 36 states
• Global P matrix found
Energy
Global P Matrix
𝑃𝑃 =
1.0018
−0.0010
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−0.0010
0.1002
35
Future Work
•
•
•
•
3+ Continuous Dimensions
Unstable / Marginally Stable systems
Cascading / Multistage Rule Bases
Verification of Learned Behaviors
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Questions?
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