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 ∈ ℝ 𝑜 , input 𝑣 𝑢 ∈ 𝑉 ⊆ ℝ 𝑛 – Evolution according to differential equation 𝑦 𝑢 = 𝐵 ⋅ 𝑦 𝑢 + 𝐶 ⋅ 𝑣(𝑢) – Bounded set 𝑉 ⊆ ℝ 𝑛 , constant matrices 𝐵 ∈ ℝ 𝑜×𝑜 , 𝐶 ∈ ℝ 𝑜×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 ⊆ ℝ 𝑜 is invariant, if no state in ℝ 𝑜 \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 −3 𝑦 + 𝑣 1 • 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 𝑛 𝐺 𝑡 ≔ 𝐺 0 + 𝑡 𝑗 𝐺 𝑗 ≤ 0 𝑗=1 𝐺(𝑡) ≤ 0 denotes negative semidefiniteness: ∀ 𝑤 ∈ ℝ 𝑜 : 𝑤 𝑈 ⋅ 𝐺 𝑡 ⋅ 𝑤 ≤ 0 • 𝑡 = (𝑡 1 , … , 𝑡 𝑛 ) ∈ ℝ 𝑛 is a variable • 𝑛 ∈ ℝ 𝑜×𝑜 are given matrices 𝐺 0 , … , 𝐺 • {𝑡 ∈ ℝ 𝑛 ∣ 𝐺 𝑡 ≤ 0} is a convex set • Paul Hänsch, RWTH Aachen 6

Finding invariant ellipsoids via LMIs An ellipsoid is given by {𝑦 ∈ ℝ 𝑜 ∣ 𝑦 𝑈 𝑄𝑦 ≤ 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 ∣ 𝑦 𝑈 𝑄𝑦 ≤ 1} is invariant for 𝑇 ” [Boyd et al, 94, §6.1.3], [Hänsch, Kowalewski] Volume of {𝑦 ∣ 𝑦 𝑈 𝑄𝑦 ≤ 1} antiproportional to det 𝑄 • Maximize det 𝑄 subject to 𝐷 𝑄 ≤ 0 • – Gives invariant ellipsoid 𝐹 𝑄 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 𝜇 𝑏 −𝑐 𝑧 𝑢 = ⋅ 𝑧 𝑢 + 𝑅𝐶 ⋅ 𝑣 𝑢 𝑐 𝑏 ⋱ Many independent variables, e.g. 𝑧 1 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 {𝑧 1 } and {𝑧 2 , 𝑧 3 } 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) • 1 1 • Then 𝐽 = [− 𝜇 inf 𝑉, − 𝜇 sup 𝑉] is a tight invariant for 𝑧(𝑢) , since 1 For 𝑧 𝑢 ≥ − 𝜇 sup 𝑉 , it follows 𝑧 𝑢 ≤ 0 1. 1 For 𝑧 𝑢 ≤ − 𝜇 inf 𝑉 , it follows 𝑧 𝑢 ≥ 0 2. It also follows that each interval 𝐾 ⊇ 𝐽 is invariant • Deduce invariant for original system from 𝑧 𝑢 = 𝑅𝑦 𝑢 : • − 1 𝜇 inf 𝑉 ≤ 𝑅 𝑗 𝑦 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 ⋅ 𝑦 𝑢 + 𝑣(𝑢) 𝑐 𝑏 • 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 From now on, 𝑣 ∈ −1,1 𝑛 • • Toy example from before 2 −1 0 −2 𝑦 = 𝑦 + 𝑣 −1 3 1 −4 3 1 −2 3 −3 −4 • Invariants have the same volume Paul Hänsch, RWTH Aachen 13

Example 3 𝑦 = −4 −3 𝑦 + −1 3 −2 𝑣 2 1 1 • 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 −2 𝑦 + 1 𝑦 = −1 1 𝑣 • Contrary to previous example – Input to 𝑦 1 depends on input to 𝑦 2 • Invariants – Directly using LMI: blue ellipsoid – Decomposition: red rectangle • Decomposition would produce the same result for the system −2 𝑦 + 1 0 𝑦 = −1 1 𝑣 0 • Decomposition introduces substantial overapproximation 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 1 0 −1 𝑦 = 𝑦 + 𝑣 1 0 −2 0 1 −3 • Idea: mixed decomposition, consider sets of canonical subsystems that share common inputs Method Volume 1. Directly using LMI 139 2. Full decomposition 133 3. Intersection of 1. and 2. 72 Mixed decomposition { 𝑦 1 , 𝑦 2 }, {𝑦 3 } 4. 60 5. Intersection of 1., 2., and 4. 55 Paul Hänsch, RWTH Aachen 17

Mixed decomposition heuristic 𝑇 1 𝑣 ∈ 𝑉 = 𝑅𝐶 ⋅ −1,1 𝑛 𝑦 = 𝑦 + 𝑣, ⋱ 𝑇 𝑙 𝑙 canonical subsystems and 2 𝑙 sets of canonical subsystems • • Which sets of subsystems give good invariants? – Is it reasonable to consider subsystems 𝑇 𝑗 , 𝑇 𝑘 separately? – Depends on degree of mutual independence of subsystem inputs – Possible measure: Ratio of the volumes of • 𝑄𝑠𝑝𝑘 𝑗𝑘 𝑉 = {[𝑣 𝑗 , 𝑣 𝑘 ] ∣ [… , 𝑣 𝑗 , … , 𝑣 𝑘 , … ] ∈ 𝑉} , and • 𝑄𝑠𝑝𝑘 𝑗 𝑉 × 𝑄𝑠𝑝𝑘 𝑘 𝑉 = 𝑣 𝑗 … 𝑣 𝑗 … ∈ 𝑉 × {𝑣 𝑘 ∣ [… 𝑣 𝑘 … ] ∈ 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