Invariants for LTI systems with uncertain input Paul Hnsch - - PowerPoint PPT Presentation

Invariants for LTI systems with uncertain input

Paul Hänsch Embedded Software Lab RWTH Aachen University, Germany

Roadmap

• Basic definitions and assumptions
• Finding invariant ellipsoids via LMIs
• Finding invariants via canonical decomposition
• Examples
• Mixed decomposition
• Conclusion
• Related Work
• Literature

Paul Hänsch, RWTH Aachen 2

LTI system with uncertain input

• Linear time-invariant, or simply LTI system 𝐵, 𝐶, 𝑉

– System state 𝑦 𝑢 ∈ ℝ𝑜, input 𝑣 𝑢 ∈ 𝑉 ⊆ ℝ𝑛 – Evolution according to differential equation 𝑦 𝑢 = 𝐵 ⋅ 𝑦 𝑢 + 𝐶 ⋅ 𝑣(𝑢) – Bounded set 𝑉 ⊆ ℝ𝑛, constant matrices 𝐵 ∈ ℝ𝑜×𝑜, 𝐶 ∈ ℝ𝑜×𝑛

• State 𝑨 is reachable from 𝑧 if

∃ input function 𝑣: ℝ → 𝑉 and duration 𝜀 ≥ 0 such that solution 𝑦 𝑢 to above diff. eq. satisfies 𝑦 0 = 𝑧 and 𝑦 𝜀 = 𝑨

• A set 𝑌 ⊆ ℝ𝑜 is invariant, if no state in ℝ𝑜\X is reachable from a state in 𝑌
• Let 𝑌 be invariant and 𝑍 ⊆ 𝑌 and, then 𝑌 is an overapproximation of the

states reachable from 𝑍

• We assume the LTI system to be stable, i.e. for each initial state 𝑦(0), the

solution 𝑦(𝑢) converges to 0 for 𝑣 ≡ 0

Paul Hänsch, RWTH Aachen 3

Toy example

𝑦 = −2 1 1 −3 𝑦 + 𝑣

• Phase portrait and sample trajectory

for 𝑣 ≡ 0

• Trajectory for a specific input

function with 𝑣 𝑢 ∈ −1,1 2

• Which states are reachable from 0?
• Invariants: ellipsoid, rectangle
• Intersection of invariants is again an

invariant!

Paul Hänsch, RWTH Aachen 4

Roadmap

• Basic definitions and assumptions
• Finding invariant ellipsoids via LMIs
• Finding invariants via canonical decomposition
• Examples
• Mixed decomposition
• Conclusion
• Related Work
• Literature

Paul Hänsch, RWTH Aachen 5

LMIs

• A linear matrix inequality (LMI) has the form

𝐺 𝑡 ≔ 𝐺0 + 𝑡𝑗𝐺𝑗

𝑛 𝑗=1

≤ 0

• 𝐺(𝑡) ≤ 0 denotes negative semidefiniteness: ∀ 𝑤 ∈ ℝ𝑜: 𝑤𝑈 ⋅ 𝐺 𝑡 ⋅ 𝑤 ≤ 0
• 𝑡 = (𝑡1, … , 𝑡𝑛) ∈ ℝ𝑛 is a variable
• 𝐺0, … , 𝐺

𝑛 ∈ ℝ𝑜×𝑜 are given matrices

• {𝑡 ∈ ℝ𝑛 ∣ 𝐺 𝑡 ≤ 0} is a convex set

Paul Hänsch, RWTH Aachen 6

Finding invariant ellipsoids via LMIs

• An ellipsoid is given by {𝑦 ∈ ℝ𝑜 ∣ 𝑦𝑈𝑄𝑦 ≤ 1}, 𝑄 > 0
• Each stable LTI system has invariant ellipsoids follows from [Hirsch, Smale] Thm 1, p.145
• For each LTI system 𝑇 one can construct an LMI 𝐷 𝑄 ≤ 0 which implies

“{𝑦 ∣ 𝑦𝑈𝑄𝑦 ≤ 1} is invariant for 𝑇”

[Boyd et al, 94, §6.1.3], [Hänsch, Kowalewski]

• Volume of {𝑦 ∣ 𝑦𝑈𝑄𝑦 ≤ 1} antiproportional to det 𝑄
• Maximize det 𝑄 subject to 𝐷 𝑄 ≤ 0

– Gives invariant ellipsoid 𝐹𝑄 with minimum volume among those which satisfy 𝐷 𝑄 ≤ 0 – Solvers available, e.g. CVX (SeDuMi) for Matlab

Paul Hänsch, RWTH Aachen 7

Roadmap

• Basic definitions and assumptions
• Finding invariant ellipsoids via LMIs
• Finding invariants via canonical decomposition
• Examples
• Mixed decomposition
• Conclusion
• Related Work
• Literature

Paul Hänsch, RWTH Aachen 8

LTI system in real canonical form

𝑦 𝑢 = 𝐵 ⋅ 𝑦 𝑢 + 𝐶 ⋅ 𝑣(𝑢)

• Change of bases 𝑧 𝑢 = 𝑅 ⋅ 𝑦(𝑢) yields new system representation

𝑧 𝑢 = 𝑅𝐵𝑅−1 ⋅ 𝑧 𝑢 + 𝑅𝐶 ⋅ 𝑣(𝑢)

• Real canonical representation looks like

𝑧 𝑢 = 𝜇 𝑏 −𝑐 𝑐 𝑏 ⋱ ⋅ 𝑧 𝑢 + 𝑅𝐶 ⋅ 𝑣 𝑢

• Many independent variables, e.g. 𝑧1 depends on no other state variable
• 𝑅 can be constructed [Hirsch, Smale] §6, [Perko] §1.8
• Works for almost all LTI systems follows from [Hirsch, Smale] Thm 2, p.157
• Consider canonical subsystems separately, e.g. those induced by {𝑧1} and

{𝑧2, 𝑧3}

Paul Hänsch, RWTH Aachen 9

One-dimensional subsystems

• Let 𝑧 𝑢 = 𝜇 ⋅ 𝑧 𝑢 + 𝑣 𝑢 , 𝑣 𝑢 ∈ 𝑉
• Assume 𝜇 < 0 (equivalent to stability)
• Then 𝐽 = [−

1 𝜇 inf 𝑉, − 1 𝜇 sup 𝑉] is a tight invariant for 𝑧(𝑢), since

1. For 𝑧 𝑢 ≥ −

1 𝜇 sup 𝑉, it follows 𝑧 𝑢 ≤ 0

2. For 𝑧 𝑢 ≤ −

1 𝜇 inf 𝑉, it follows 𝑧 𝑢 ≥ 0

• It also follows that each interval 𝐾 ⊇ 𝐽 is invariant
• Deduce invariant for original system from 𝑧 𝑢 = 𝑅𝑦 𝑢 :

− 1 𝜇 inf 𝑉 ≤ 𝑅𝑗𝑦 𝑢 ≤ − 1 𝜇 sup 𝑉

• Geometrical interpretation: 𝑦 𝑢 bounded by two parallel hyperplanes

Paul Hänsch, RWTH Aachen 10

Two-dimensional subsystems

𝑦 𝑢 = 𝑏 −𝑐 𝑐 𝑏 ⋅ 𝑦 𝑢 + 𝑣(𝑢)

• Find invariant ellipsoid via LMIs
• Alternative method [Hänsch, Kowalewski] based on

– Boundary trajectory – Real algebra (Redlog) – Smaller invariants for exotic 𝑉 – No numerical issues – Slower than LMI

Paul Hänsch, RWTH Aachen 11

Roadmap

• Basic definitions and assumptions

 Finding invariant ellipsoids via LMIs  Finding invariants via canonical decomposition

• Examples
• Mixed decomposition
• Conclusion
• Related Work
• Literature

Paul Hänsch, RWTH Aachen 12

Examples 1 and 2

𝑦 = −2 −4 3 −3 −4 𝑦 + 2 −1 −1 3 1 1 −2 3 𝑣

• Invariants have the same volume

Paul Hänsch, RWTH Aachen 13

• From now on, 𝑣 ∈ −1,1 𝑛
• Toy example from before

Example 3

𝑦 = −4 −3 2 1 𝑦 + −1 3 1 −2 𝑣

• Invariants

– Directly using LMI: blue ellipsoid – With decomposition: red parallelotope

• Transform system to canonical form to see

why decomposition pays off 𝑦 = −1 −2 𝑦 + 1 1 𝑣

• Subsystem inputs are mutually

independent!

Paul Hänsch, RWTH Aachen 14

Example 4

𝑦 = −1 −2 𝑦 + 1 1 𝑣

• Contrary to previous example

– Input to 𝑦1 depends on input to 𝑦2

• Invariants

– Directly using LMI: blue ellipsoid – Decomposition: red rectangle

• Decomposition would produce the

same result for the system 𝑦 = −1 −2 𝑦 + 1 1 𝑣

• Decomposition introduces substantial
• verapproximation of input restraint set

Paul Hänsch, RWTH Aachen 15

Roadmap

• Basic definitions and assumptions
• Finding invariant ellipsoids via LMIs
• Finding invariants via canonical decomposition
• Examples
• Mixed decomposition
• Conclusion
• Related Work
• Literature

Paul Hänsch, RWTH Aachen 16

Mixed decomposition

𝑦 = −1 −2 −3 𝑦 + 1 1 1 𝑣

• Idea: mixed decomposition, consider sets of canonical subsystems that

share common inputs

Paul Hänsch, RWTH Aachen 17

Method Volume 1. Directly using LMI 139 2. Full decomposition 133 3. Intersection of 1. and 2. 72 4. Mixed decomposition {𝑦1, 𝑦2}, {𝑦3} 60 5. Intersection of 1., 2., and 4. 55

Mixed decomposition heuristic

𝑦 = 𝑇1 ⋱ 𝑇𝑙 𝑦 + 𝑣, 𝑣 ∈ 𝑉 = 𝑅𝐶 ⋅ −1,1 𝑛

• 𝑙 canonical subsystems and 2𝑙 sets of canonical subsystems
• Which sets of subsystems give good invariants?

– Is it reasonable to consider subsystems 𝑇𝑗, 𝑇

𝑘 separately?

– Depends on degree of mutual independence of subsystem inputs – Possible measure: Ratio of the volumes of

• 𝑄𝑠𝑝𝑘𝑗𝑘 𝑉 = {[𝑣𝑗, 𝑣𝑘] ∣ [… , 𝑣𝑗, … , 𝑣𝑘, … ] ∈ 𝑉}, and
• 𝑄𝑠𝑝𝑘𝑗 𝑉 × 𝑄𝑠𝑝𝑘𝑘 𝑉 =

𝑣𝑗 … 𝑣𝑗 … ∈ 𝑉 × {𝑣𝑘 ∣ [… 𝑣𝑘 … ] ∈ 𝑉} – Projection and volume computation can be expensive but both can be approximated efficiently with sufficient accuracy

Paul Hänsch, RWTH Aachen 18

Conclusion and outlook

• Facing a stable LTI system and looking for good invariants?
• Take a look at the system‘s canonical form
• Is this in some way applicable to nonlinear systems?

Paul Hänsch, RWTH Aachen 19

Related work

• Iterative overapproximation of reachable states [Girard, Le Guernic 2010]

– Bounded time horizon only – Reachable set represented by large number of shapes – Invariants are a useful complement

• Overapproximation based on decomposition [Gayek 86]
• Decidability results based on eigenstructure

– Reachability in systems with special system matrices and inputs

[Lafferriere et al 2001] [Anai, Weispfenning 2001]

– Point-to-point reachability, limit sets in systems without inputs [Hainry 2008]

• Overapproximation of reachable states in systems without inputs [Tiwari 2003]

Paul Hänsch, RWTH Aachen 20

Literature

• Hänsch, Kowalewski: Invariants for LTI systems with uncertain inputs. RP 2012
• Boyd, Vandenberghe: Convex optimization. 2009
• Boyd, Ghaoui, Feron, Balakrishnan: LMI in system and control theory. 1994
• Boyd, Vandenberghe: Semidefinite programming. 1996
• Gayek: Approximating reachable sets for a class of linear control systems. Int. J. Control 1986
• Le Guernic, Girard: Reachability analysis of linear systems using support functions. Nonlinear

Analysis: Hybrid Systems 2010

• Lafferriere, Pappas, Yovine: Symbolic reachability computation for families of linear vector
• fields. J. Symb. Comp. 2001
• Anai, Weispfenning: Reach set computations using real quantifier elimination. HSCC 2001
• Hainry: Computing omega-limit sets in linear dynamical systems. UC 2008
• Hainry: Reachability in linear dynamical systems. CiE 2008
• Tiwari: Approximate reachability for linear systems. HSCC 2003
• Hirsch, Smale: Differential equations, dynamical systems, and linear algebra. 1974
• Perko: Differential equations and dynamical systems. 1991

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