Compartmental Systems - definition. Compartmental models: system analysis and control issues. Compartmental Systems - general model

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1 Compartmental Systems - definition Compartmental models: system analysis and control issues Compartmental system Finite number of compartments which exchange material with each other in such a way that the quantity of material in each compartment can be described by a first order ODE. Paula Rocha universidade de aveiro theoria poiesis praxis Modelling, Automation and Control of Physiological Variables 27, Porto Compartmental models: analysis and control Compartmental Systems - definition Compartmental system Finite number of compartments which exchange material with each other in such a way that the quantity of material in each compartment can be described by a first order ODE. The compartments may occupy: Compartmental Systems - general model General model n compartments ( to n) + environment (compartment ) x i amount of material in compartment i, i =,..., n x := [... x n ] T f ij flow rate from compartment i to compartment j different physical spaces (flow of material between different locations) the same physical space (transformations of one substance into another) ẋ i = j i j= f ji (x) total inflow rate j i j= f ij (x) total outflow rate i =,... n Compartmental models: analysis and control Compartmental models: analysis and control Compartmental Systems - general model Total system mass: M(x) = x n = [... ]x = x Linear time invariant (single-input) model Total mass variation: Ṁ(x) = x i = (f i f i ) f ij (x) = k ij x i, i j =,..., n f i = b i u, u input nonnegative coefficients f i = q i x i b i b j Closed system f i =, f i =, i =,..., n mass conservative to/from other compartments i k i j k j i j to/from other compartments q i q j Compartmental models: analysis and control Compartmental models: analysis and control

2 Equations Equations ẋ i = k ji x j k ij x i q i x i + b i u j i j= j i j= i =,... n ẋ i = k ji x j k ij x i q i x i + b i u j i j= j i j= i =,... n In matrix form... ẋ = Ax + Bu with A ij = k ji, j i A ii = q i n j i j= k ij B = [b... b n ] T Compartmental models: analysis and control Compartmental models: analysis and control Example - system with two compartments ẋ ẋ = k 2 q k 2 + b k 2 k 2 q 2 b 2 } {{ } A x B Example - system with two compartments ẋ ẋ = k 2 q k 2 + b k 2 k 2 q 2 b 2 } {{ } A x B Total mass variation Ṁ(x) = ẋ = A x + B u Ṁ(x) = q q 2 (b + b 2 )u Compartmental models: analysis and control Compartmental models: analysis and control Example - system with two compartments ẋ ẋ = k 2 q k 2 + b k 2 k 2 q 2 b 2 } {{ } A x B Total mass variation Ṁ(x) = ẋ = A x + B u Ṁ(x) = q q 2 (b + b 2 )u Example - Compartmental model for the effect of atracurium Closed system q = q 2 =, b = b 2 = Ṁ(x) = (mass conservation) Drug used to induce neuromuscular blockade Compartmental models: analysis and control Compartmental models: analysis and control

3 Features of the system matrix A. A is a Metzler matrix (nonzero matrix with nonnegative off-diagonal elements) 2. A has non-positive diagonal elements (A ii ) 3. A is (column-wise) diagonally dominant A ii l i,l= A li } {{ } q i Features of the system matrix A. A is a Metzler matrix (nonzero matrix with nonnegative off-diagonal elements) 2. A has non-positive diagonal elements (A ii ) 3. A is (column-wise) diagonally dominant A ii l i,l= A li } {{ } q i Any square matrix with properties, 2 and 3 is called a compartmental matrix Compartmental models: analysis and control Compartmental models: analysis and control Positivity of the solutions x() u(t) x(t) Positivity of the solutions x() u(t) x(t) Because... when x(t) reaches the boundary of the positive orthant, say at time t = t if x i (t ) = then ẋ i (t ) = n j i j= k jix j (t ) + b i u(t ) thus x i does not decrease and remains nonnegative. Compartmental models: analysis and control Compartmental models: analysis and control Stability Recall Stability Recall System ẋ = Ax System ẋ = Ax σ(a) spectrum of A = set of all the eigenvalues of A System stable R(λ), λ σ(a) + condition on the eigenvalues λ such that R(λ) = System asymptotically stable R(λ) <, λ σ(a) Compartmental models: analysis and control Compartmental models: analysis and control

4 Location of the eigenvalues of A? Location of the eigenvalues of A? A diagonally dominant and A ii (Ger sgorin Theorem) R(σ(A)) A Metzler matrix there is a unique eigenvalue whose real part is maximal; this eigenvalue is real. Compartmental models: analysis and control Compartmental models: analysis and control Location of the eigenvalues of A? A diagonally dominant and A ii (Ger sgorin Theorem) R(σ(A)) Yet another consequence Asymptotical stability σ(a) A is invertible A Metzler matrix there is a unique eigenvalue whose real part is maximal; this eigenvalue is real. Consequence A has at most one eigenvalue λ with R(λ ) =. In case it exists, λ =. All the other eigenvalues λ are such that R(λ) <. Compartmental models: analysis and control Compartmental models: analysis and control Yet another consequence Asymptotical stability σ(a) A is invertible In physical terms... Total system mass variation For u = : Ṁ(x) = ẋ = A x = q i x i Asymptotical stability fully outflow connected system (every compartment has an escape path to another compartment with positive outward flow) Compartmental models: analysis and control Compartmental models: analysis and control

5 Total system mass variation Goal to bring the system mass M(x) to a set point M For u = : Ṁ(x) = ẋ = A x = q i x i Consequence: a compartmental system with no mass inflow is stable Obvious from the physical point of view! Compartmental models: analysis and control Compartmental models: analysis and control Goal to bring the system mass M(x) to a set point M Without input positivity constraints u(x) = ( b i ) ( q i x i + µ(m M(x)), µ > Goal to bring the system mass M(x) to a set point M Without input positivity constraints u(x) = ( b i ) ( q i x i + µ(m M(x)), µ > Closed-loop compartmental equations ẋ = Ax + B(( b i ) ( q i x i + µ(m M(x))) Compartmental models: analysis and control Compartmental models: analysis and control Goal to bring the system mass M(x) to a set point M Without input positivity constraints u(x) = ( b i ) ( q i x i + µ(m M(x)), µ > With input positivity constraints u(x) = max(, ũ(x)) nonlinear control law! ũ(x) = ( b i ) ( q i x i + µ(m M(x)), µ > Closed-loop compartmental equations ẋ = Ax + B(( b i ) ( q i x i + µ(m M(x))) Closed-loop mass equation Ṁ(x) = µ(m(x) M ) M(x) goes to M Compartmental models: analysis and control Compartmental models: analysis and control

6 With input positivity constraints u(x) = max(, ũ(x)) nonlinear control law! ũ(x) = ( b i ) ( q i x i + µ(m M(x)), µ > Lyapunov analysis V (x) = (M(x) M ) 2 /2 V (x) V (x) = M(x) = M Along the system trajectories V (x) = (M(x) M µ(m(x) M ) 2 u = ũ )Ṁ(x) = (M(x) M )( q i x i ) u = V (x) Therefore... (if A is invertible) the system trajectories tend to the isomass Ω M = {x R n + : M(x) = M } (consequence of LaSalle s invariance principle) and the control goal is achieved. Remark Ω M system. is invariant under the dynamics of the closed-loop Compartmental models: analysis and control Compartmental models: analysis and control Example - Atracurium model k2 =.928, k3 =.7, k2 =.556, q =.47, q2 =, q3 =.836 M =.69, µ =.2 2 system mass.8 desired mass bounds of I(M*).6 A closer look at the atracurium model ẋ k 2 k 3 q k 2 = k 2 k 2 q 2 ẋ 3 k 3 q 3 x Mass (µg/kg) Time (min) Compartmental models: analysis and control Compartmental models: analysis and control A closer look at the atracurium model ẋ k 2 k 3 q k 2 = k 2 k 2 q 2 ẋ 3 k 3 q 3 x 3 + A closer look at the atracurium model ẋ k 2 k 3 q k 2 = k 2 k 2 q 2 ẋ 3 k 3 q 3 x 3 + Restricting the closed-loop sytem to the isomass Ω M u(x) = ũ(x) = q + q 2 + q 3 x 3 x Ω M x 3 = M Restricting the closed-loop sytem to the isomass Ω M u(x) = ũ(x) = q + q 2 + q 3 x 3 x Ω M x 3 = M Replacing in the closed-loop equations yields... Compartmental models: analysis and control Compartmental models: analysis and control

7 ẋ = k 2 k 3 k 2 + q 2 q 3 k 2 k 2 q 2 This is equivalent to: M ([ ẋ = k 2 k 3 q 3 k 2 + q 2 q 3 k 2 k 2 q 2 } {{ } Ã for appropriate [x x 2 ]T (easy, but long formulas) ] [ x x 2 ]) It turns out that R(σ(Ã)) < This implies that (t) x x 2 Thus, for trajectories starting in the isomass Ω M x 3 (t) x x 2 x 3, with x 3 such that x + x 2 + x 3 = M Compartmental models: analysis and control Compartmental models: analysis and control Relevance for the control of neuromuscular blockade r(t) neuromuscular blockade level r(t) = Hill C5,γ(x 3 (t)) = Cγ 5 C γ 5 +(x3(t))γ Relevance for the control of neuromuscular blockade r(t) neuromuscular blockade level r(t) = Hill C5,γ(x 3 (t)) = Cγ 5 C γ 5 +(x3(t))γ Thus, given a desired neuromuscular blockade level r Compute the corresponding x 3 via the Hill equation Compute the corresponding x and x 2 via the easy, but long formulas Compute the corresponding mass M = x + x 2 + x 3 Apply the nonlinear control law with this M Compartmental models: analysis and control Compartmental models: analysis and control It follows from the previous analysis that: It follows from the previous analysis that: x x ; x 3 x 3 ; r r x x ; x 3 x 3 ; r r Compartmental models: analysis and control Compartmental models: analysis and control

8 Control Issues - Parameter uncertainties Major problem parameter uncertainties Question How does the control system behave if the nominal model differs from the real one? Control Issues - Parameter uncertainties Major problem parameter uncertainties Question How does the control system behave if the nominal model differs from the real one? Answer system mass desired mass bounds of I(M*).4 Mass (µg/kg) Time (min) Compartmental models: analysis and control Compartmental models: analysis and control Bibliography Bibliography Basics on compartmental models Application to the control of the neuromuscular blockade Godfrey, K. (983). Compartmental models and their application. Academic Press Control of Compartmental Systems Haddad, W., Hayakawa, T. and Bayley (26). Adaptive control for non-negative and compartmental systems with applications to general anesthesia. International Journal of Adaptive Control and Signal Processing 7 (3), Bastin, G. and A. Provost (22). Feedback stabilization with positive control of dissipative compartmental systems. In: Proceedings of the 5th International Symposium on Mathematical Theory of Networks and Systems. Notre Dame, Indiana, USA. Magalhães, H., Mendoça, T. and Rocha, P.(25). Identification and Control of Positive and Compartmental Systems applied to Neuromuscular Blockade. In: Preprints of the 6th IFAC World Congress. Prague, Czech Republic. Control of Compartmental Systems with parameter uncertainties Sousa, C., Mendonça, T. and Rocha, P. Control of uncertain compartmental systems. Proceedings of the 27 Mediterranean Control Conferece, MED 27, Athens, Greece, June 27. (Available in the Attachments) Compartmental models: analysis and control Compartmental models: analysis and control