Scalable Control of Positive Systems Anders Rantzer LCCC Linnaeus Center Lund University Sweden The Dujiangyan Irrigation System (5 B.C.) Water Still a Control Challenge! A scarce resource with different qualities for different needs: Drinking Washing Toilets Irrigation Industrial cooling... Many producers, many consumers in a complex network. The Power Grid Needs Control Traffic Networks Need Control [Source: www.motorauthority.com] [Source: geospatial.blogs.com] Control challenges: Throughput. Safety. Environmental footprint. Control challenges: More producers. Variable capacity. Limited storage. Flexible components. Communication Networks Need Control Towards a Scalable Control Theory [Source: www.hindawi.com] Process Controller Challenges: Service. Accessibility. Resource efficieny. [ P C P P P P C C P Riccati equations use O (n3 ) flops, O (n ) memory Model Predictive Control requires even more Today: Exploiting monotone/positive systems
Linear Positive Systems Transportation networks Vehicle formations Nonlinear Monotone Systems Voltage stability HIV/cancer treatment Frequency domain: Positively Dominated Systems Open problems and Conclusions Positive systems A linear system is called positive if the state and output remain nonnegative as long as the initial state and the inputs are nonnegative: dx = Ax+ Bu y= Cx Equivalently, A, B and C have nonnegative coefficients except possibly for the diagonal of A. Examples: Probabilistic models. Economic systems. Chemical reactions. Ecological systems. Positive Systems and Nonnegative Matrices Classics: Mathematics: Perron (97) and Frobenius (9) Economics: Leontief (936) Example : A Transportation Network Books: Nonnegative matrices: Berman and Plemmons (979) Large Scale Systems: Siljak (978) Positive Linear Systems: Farina and Rinaldi () Recent work on control of positive systems Examples: Biology inspired theory: Angeli and Sontag (3) Synthesis by linear programming: Rami and Tadeo (7) Switched systems: Liu (9), Fornasini and Valcher () Distributed control: Tanaka and Langbort () Robust control: Briat (3) Transportation Network in Practice ẋ l 3 l x w ẋ l l 3 l 3 x l 3 l 3 l 3 l 43 l 34 w w 3 ẋ 4 l 43 4 l 34 x 4 w 4 How do we selectl ij to minimize some gain from w to x? Example : A Vehicle Formation Irrigation systems Power systems Traffic flow dynamics Communication/computation networks Production planning and logistics Example : A Vehicle Formation Stability of Positive systems Suppose the matrix A has nonnegative off-diagonal elements. Then the following conditions are equivalent: ẋ = x +l 3 (x 3 x )+w ẋ =l (x x )+l 3 (x 3 x )+w ẋ 3 =l 3 (x x 3 )+l 34 (x 4 x 3 )+w 3 ẋ 4 = 4x 4 +l 43 (x 3 x 4 )+w 4 (i) (ii) The system dx = Ax is exponentially stable. There is a diagonal matrix P such that A T P+ PA (iii) There exists a vector ξ> such that Aξ<. (The vector inequalities are elementwise.) (iv) There exits a vector z> such that A T z<. How do we selectl ij to minimize some gain from w to x?
Lyapunov Functions of Positive systems Solving the three alternative inequalities gives three different Lyapunov functions: A T P+ PA Aξ< A T z< V(x)= x T Px V(x)=max (x k/ξ k ) V(x)= z T x k A Distributed Search for Stabilizing Gains a l a a 4 Suppose a +l a l a 3 a 3 +l a 33 a 3 forl,l [, ]. a 4 a 43 a 44 A Scalable Stability Test x x x 3 x 4 Stability of ẋ= Ax follows from existence of ξ k > such that a a a 4 ξ a a a 3 ξ a 3 a 33 a 3 ξ 3 < a 4 } a 43 {{ a 44 } ξ 4 A The first node verifies the inequality of the first row. The second node verifies the inequality of the second row.... Verification is scalable! Optimal Control of Transportation Networks For stabilizing values ofl,l, find µ k ξ k such that a a a 4 ξ a a a 3 ξ a 3 a 33 a 3 ξ 3 µ < µ a 4 a 43 a 44 ξ 4 and setl = µ /ξ andl = µ /ξ. Every row gives a local test. Distributed synthesis by linear programming (gradient search). ẋ l 3 l x ẋ l l 3 l 3 x l 3 l 3 l 3 l 43 l 34 Bw ẋ 4 l 43 4 l 34 x 4 How do we selectl ij [, ] to minimize some gain from w to Cx? Performance of Positive systems Example : Transportation Networks Suppose that G(s)= C(sI A) B+Dwhere A R n n is Metzler, while B R+ n, C R n + and D R +. Define G = sup ω G(iω). Then the following are equivalent: ẋ l 3 l x w ẋ l l 3 l 3 x l 3 l 3 l 3 l 43 l 34 w w ẋ 4 l 43 4 l 34 x 4 w y= Cx (i) The matrix A is Hurwitz and G < γ. (ii) A B The matrix is Hurwitz. C D γ C= [ ] C= [ ] C= [ ] Example : Vehicle Formations ẋ l 3 l 3 x ẋ l l l 3 l 3 x l 3 l 3 l 34 l 34 Bw ẋ 4 l 43 4 l 43 x 4 Linear Positive Systems Transportation networks Vehicle formations Nonlinear Monotone Systems Voltage stability HIV/cancer treatment B= B= B= Frequency domain: Positively Dominated Systems Open problems and Conclusions 3
Nonlinear Monotone Systems The system ẋ(t)= f(x(t), u(t)), x()=a is a monotone system if its linearization is a positive system. Max-separable Lyapunov Functions Let ẋ= f(x) be a globally asymptotically stable monotone system with invariant compact set X R+. n Then there exist strictly increasing functions V k : R + R + for k=,..., n with d V(x(t))= V(x(t)) in X where V(x) is equal to V(x)=max{V (x ),..., V n (x n )}. x() x() y() y() x t= t= x(3) x() y(3) y() t=3 t= x Voltage Stability Convex-Monotone Systems The system generator currents load currents i G (t) i L (t) Y GG Y = G L Y LG Y LL }{{} network impedances u G (t) u L (t) generator voltages load voltages ẋ(t)= f(x(t), u(t)), x()=a is a monotone system if its linearization is a positive system. It is a convex monotone system if every row of f is also convex. di L k (t)= p k u L k (t) il k(t) k=,..., n Voltage stabilization is an important large-scale control problem. Monotonicity be exploited for control synthesis! Theorem. For a convex monotone system ẋ= f(x, u), each component of the trajectory φ t (a, u) is a convex function of(a, u). Combination Therapy is a Control Problem Example Evolutionary dynamics: ( ẋ= A i u i D i ) x Optimized drug doses:.9.8.7.6 u u u3 Total virus population: 3 time-varying constant.5 Each state x k is the concentration of a mutant. (There can be hundreds!) Each input u i is a drug dosage. A describes the mutation dynamics without drugs, while D,..., D m are diagonal matrices modeling drug effects. Determine u,..., u m with u + +u m such that x decays as fast as possible!.4.3.. 4 6 8 4 6 8 time [days] Experiments on mice: - - 4 6 8 4 6 8 time [days] [Hernandez-Vargas, Colaneri and Blanchini, JRNC ] [Jonsson, Rantzer,Murray, ACC 4] Platoon of Vehicles with Inertia Linear Positive Systems Transportation networks Vehicle formations Nonlinear Monotone Systems Voltage stability HIV/cancer treatment Frequency domain: Positively Dominated Systems Open problems and Conclusions (s +.s)x = C (s)x + w (s +.s)x = C (s)[λ (x x )+λ 3 (x 3 x )]+w (s + s)x 3 = C 3 (s)[λ 3 (x x 3 )+λ 34 (x 4 x 3 )]+w 3 (s + s)x 4 = C 4 (s)x 4 + w 4 Negative feedback destroys positivity in second order models. Is there a scalable approach to controller design? 4
General Approach C(s) P(s) T(s) Λ I Λ. First design the local controllers to make T = PC[ I PC] positively dominated: ptk (iω )p Tk () Linear Positive Systems Nonlinear Monotone Systems for all k, ω Transportation networks Vehicle formations Voltage stability HIV/cancer treatment Frequency domain: Positively Dominated Systems Open problems and Conclusions. Then use scalable methods to optimize the weights Λ. Caveat: Not advisable for resonant systems. Open Problems For Scalable Control Use Positive Systems!. Sometimes the same H -performance can be attained using distributed positivity based controllers as with Riccati based centralized controllers. Find out when! Verification and synthesis scale well Distributed controllers by linear programming No need for global information Use local controllers to get positive dominance. Overflow channels in waterways (and capacity limits in traffic networks) give rise to piecewise linear monotone systems. Find scalable methods to optimize their closed loop performance. 3. For convex-monotone systems, develop methods for design of controllers that give a monotone closed loop system with a separable Lyapunov function. Li Bing Our Control Ancestor Thanks! Enrico Lovisari 5 Vanessa Jonsson Daria Madjidian Christian Grussler