Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Remarks Necessary condition for parameter estimation Essential for parameters with practical significance Prerequisite to experiment design Identifiability does not guarantee Good fit to experimental data Good fit only with unique vector of parameters Unidentifiable implies infinite number of parameter vectors will give same fit (even for perfect data) Many techniques for linear systems Laplace transform or transfer function Taylor series of output Similarity transformation (exhaustive modelling) Taylor series and similarity transformation approaches are applicable for nonlinear systems Differential algebra Rational systems with differentiable inputs/outputs Heavily dependent on symbolic computation MJ Chappell University of Warwick July 2016 Structural identifiability 12/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Laplace Transform Approach MJ Chappell University of Warwick July 2016 Structural identifiability 13/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach General linear system x ( t , p ) = A ( p ) x ( t , p ) + B ( p ) u ( t ) , x ( 0 , p ) = x 0 ( p ) , ˙ y ( t , p ) = C ( p ) x ( t , p ) , where A ( p ) is an n × n matrix of rate constants B ( p ) is an n × m input matrix C ( p ) is an l × n output matrix Assume that x 0 = 0 (not essential) & take Laplace transforms: s Q ( s ) = A ( p ) Q ( s ) + B ( p ) U ( s ) Y ( s ) = C ( p ) Q ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) U ( s ) MJ Chappell University of Warwick July 2016 Structural identifiability 14/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Laplace Transform Approach This gives relationship between LTs of input & output: Y ( s ) = G ( s ) U ( s ) , where the matrix G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) is the transfer (function) matrix Measurements for G ( s ) assumed known Coefficients of powers of s in numerators & denominators uniquely determined by input-output relationship MJ Chappell University of Warwick July 2016 Structural identifiability 15/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: Transfer function: G ( s ) = MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) = MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) = b 1 c 1 s + a 01 MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) = b 1 c 1 s + a 01 So b 1 c 1 and a 01 globally identifiable MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) = b 1 c 1 s + a 01 So b 1 c 1 and a 01 globally identifiable But b 1 and c 1 unidentifiable MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment b 1 u ( t ) a 01 y = c 1 q 1 1 Input: impulse: b 1 u ( t ) = b 1 n 0 δ ( t ) ; b 1 unknown, n 0 known Output: y = c 1 q 1 , where c 1 unknown. System equations: ˙ q 1 = − a 01 q 1 + b 1 u ( t ) , q 1 ( 0 ) = 0 , y = c 1 q 1 Transfer function: G ( s ) = C ( p ) ( s I n − A ( p )) − 1 B ( p ) = b 1 c 1 s + a 01 So b 1 c 1 and a 01 globally identifiable But b 1 and c 1 unidentifiable So model is unidentifiable unless b 1 or c 1 known (then SGI ) MJ Chappell University of Warwick July 2016 Structural identifiability 16/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 2 Compartments I = bu ( t ) a 12 y = c 1 x 1 1 2 a 01 Model is: � ˙ � � − a 01 � � x 1 � � 0 � x 1 a 12 = + u ( t ) ˙ − a 12 x 2 0 x 2 b � � x 1 � � y = c 0 x 2 Transfer function: � − 1 � 0 � � s + a 01 � − a 12 bca 12 � G ( s ) = c 0 = 0 s + a 12 b ( s + a 01 )( s + a 12 ) MJ Chappell University of Warwick July 2016 Structural identifiability 17/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Locally identifiable example Transfer function: bca 12 G ( s ) = ( s + a 01 )( s + a 12 ) and so the following are unique: bca 12 , a 01 + a 12 and a 01 a 12 Yields two possible solutions for a 01 and a 21 If b (or c ) known then two possible solutions for c (or b ) hence locally identifiable If neither b nor c known then unidentifiable If both b and c known then globally identifiable MJ Chappell University of Warwick July 2016 Structural identifiability 18/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Taylor series approach MJ Chappell University of Warwick July 2016 Structural identifiability 19/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Generally applied when there is a single input (eg, 0 or impulse) Outputs y i ( t , p ) expanded as Taylor series about t = 0: y i ( 0 , p ) t 2 ( 0 , p ) t k y i ( t , p ) = y i ( 0 , p )+ ˙ y i ( 0 , p ) t + ¨ 2 ! + · · · + y ( k ) k ! + . . . i where ( 0 , p ) = d k y i y ( k ) ( k = 1 , 2 , . . . ) . i d t k t = 0 Taylor series coefficients y ( k ) ( 0 , p ) unique for particular output i Approach reduces to determining solutions for p that give: y i ( 0 , p ) , y ( k ) ( 0 , p ) ( 1 ≤ i ≤ l , k ≥ 1 ) . i Notice that we have a possibly infinite list of coefficients: y 1 ( 0 , p ) , . . . , y l ( 0 , p ) , ˙ y 1 ( 0 , p ) , . . . , ˙ y l ( 0 , p ) , ¨ y 1 ( 0 , p ) , . . . , ¨ y l ( 0 , p ) , . . . For linear systems: at most 2 n − 1 independent equations needed MJ Chappell University of Warwick July 2016 Structural identifiability 20/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: First coefficient: y ( 0 , p ) = y ( 0 , p ) = Second coefficient: ˙ MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: First coefficient: y ( 0 , p ) = y ( 0 , p ) = Second coefficient: ˙ MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = y ( 0 , p ) = Second coefficient: ˙ MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = Second coefficient: ˙ MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = − a 01 b 1 c 1 n 0 Second coefficient: ˙ MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = − a 01 b 1 c 1 n 0 Second coefficient: ˙ So b 1 c 1 & b 1 c 1 a 01 unique MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = − a 01 b 1 c 1 n 0 Second coefficient: ˙ So b 1 c 1 & b 1 c 1 a 01 unique (ie b 1 c 1 & a 01 globally identifiable) MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = − a 01 b 1 c 1 n 0 Second coefficient: ˙ So b 1 c 1 & b 1 c 1 a 01 unique (ie b 1 c 1 & a 01 globally identifiable) But b 1 and c 1 unidentifiable MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 1 Compartment y = c 1 q 1 a 01 1 Input: impulse in I.C.s: q 1 ( 0 ) = b 1 n 0 ; b 1 unknown, n 0 known. Output: y = c 1 q 1 , where c 1 unknown. System equations: q 1 = − a 01 q 1 , ˙ q 1 ( 0 ) = b 1 n 0 , y = c 1 q 1 First coefficient: y ( 0 , p ) = b 1 c 1 n 0 y ( 0 , p ) = − a 01 b 1 c 1 n 0 Second coefficient: ˙ So b 1 c 1 & b 1 c 1 a 01 unique (ie b 1 c 1 & a 01 globally identifiable) But b 1 and c 1 unidentifiable So model unidentifiable unless b 1 &/or c 1 known (then SGI ) MJ Chappell University of Warwick July 2016 Structural identifiability 21/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 2 Compartments a 21 y = c 1 q 1 1 2 a 12 a 01 Input: bolus intravenous injection of drug (unknown amount) Output: concentration of drug in the plasma System equations: q 1 ( t , p ) = ˙ q 2 ( t , p ) = ˙ y ( t , p ) = MJ Chappell University of Warwick July 2016 Structural identifiability 22/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 2 Compartments a 21 y = c 1 q 1 1 2 a 12 a 01 Input: bolus intravenous injection of drug (unknown amount) Output: concentration of drug in the plasma System equations: q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = ˙ y ( t , p ) = MJ Chappell University of Warwick July 2016 Structural identifiability 22/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 2 Compartments a 21 y = c 1 q 1 1 2 a 12 a 01 Input: bolus intravenous injection of drug (unknown amount) Output: concentration of drug in the plasma System equations: q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = MJ Chappell University of Warwick July 2016 Structural identifiability 22/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: 2 Compartments a 21 y = c 1 q 1 1 2 a 12 a 01 Input: bolus intravenous injection of drug (unknown amount) Output: concentration of drug in the plasma System equations: q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) MJ Chappell University of Warwick July 2016 Structural identifiability 22/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: Second coefficient: Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 Second coefficient: Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − c 1 ( a 01 + a 21 ) b 1 Second coefficient: ˙ Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − c 1 ( a 01 + a 21 ) b 1 Second coefficient: ˙ Third coefficient: y ( 2 ) 1 ( t , p ) = c 1 ( − ( a 01 + a 21 ) ˙ q 1 ( t , p ) + a 12 ˙ q 2 ( t , p )) Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − c 1 ( a 01 + a 21 ) b 1 Second coefficient: ˙ Third coefficient: y ( 2 ) 1 ( t , p ) = c 1 ( − ( a 01 + a 21 ) ˙ q 1 ( t , p ) + a 12 ˙ q 2 ( t , p )) ( a 01 + a 21 ) 2 b 1 + a 12 a 21 b 1 � � y ( 2 ) 1 ( 0 , p ) = c 1 = ⇒ Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − c 1 ( a 01 + a 21 ) b 1 Second coefficient: ˙ Third coefficient: y ( 2 ) 1 ( t , p ) = c 1 ( − ( a 01 + a 21 ) ˙ q 1 ( t , p ) + a 12 ˙ q 2 ( t , p )) ( a 01 + a 21 ) 2 b 1 + a 12 a 21 b 1 � � y ( 2 ) 1 ( 0 , p ) = c 1 = ⇒ Fourth coefficient: y ( 3 ) 1 ( t , p ) = ( a 01 + a 21 ) 2 ˙ � � q 1 − a 2 q 1 − a 12 ( a 01 + a 21 ) ˙ q 2 + a 12 a 21 ˙ 12 ˙ c 1 q 2 MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach q 1 ( t , p ) = − ( a 01 + a 21 ) q 1 ( t , p ) + a 12 q 2 ( t , p ) , q 1 ( 0 , p ) = b 1 ˙ q 2 ( t , p ) = a 21 q 1 ( t , p ) − a 12 q 2 ( t , p ) , q 2 ( 0 , p ) = 0 ˙ y ( t , p ) = c 1 q 1 ( t , p ) First coefficient: y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − c 1 ( a 01 + a 21 ) b 1 Second coefficient: ˙ Third coefficient: y ( 2 ) 1 ( t , p ) = c 1 ( − ( a 01 + a 21 ) ˙ q 1 ( t , p ) + a 12 ˙ q 2 ( t , p )) ( a 01 + a 21 ) 2 b 1 + a 12 a 21 b 1 � � y ( 2 ) 1 ( 0 , p ) = c 1 = ⇒ Fourth coefficient: y ( 3 ) 1 ( t , p ) = ( a 01 + a 21 ) 2 ˙ � � q 1 − a 2 q 1 − a 12 ( a 01 + a 21 ) ˙ q 2 + a 12 a 21 ˙ 12 ˙ c 1 q 2 � � − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � ⇒ y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 MJ Chappell University of Warwick July 2016 Structural identifiability 23/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: Second coefficient: Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: a 01 + a 21 unique Third coefficient: Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: a 01 + a 21 unique Third coefficient: a 12 a 21 unique Fourth coefficient: MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: a 01 + a 21 unique Third coefficient: a 12 a 21 unique Fourth coefficient: a 12 unique MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: a 01 + a 21 unique Third coefficient: a 12 a 21 unique Fourth coefficient: a 12 unique So a 21 and then a 01 unique MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach y 1 ( 0 , p ) = c 1 b 1 y 1 ( 0 , p ) = − b 1 c 1 ( a 01 + a 21 ) ˙ � ( a 01 + a 21 ) 2 + a 12 a 21 � y ( 2 ) 1 ( 0 , p ) = b 1 c 1 − ( a 01 + a 21 ) 2 − 2 a 12 a 21 � � � � y ( 3 ) 1 ( 0 , p ) = b 1 c 1 − a 2 ( a 01 + a 21 ) 12 a 21 First coefficient: b 1 c 1 unique Second coefficient: a 01 + a 21 unique Third coefficient: a 12 a 21 unique Fourth coefficient: a 12 unique So a 21 and then a 01 unique Same result as before MJ Chappell University of Warwick July 2016 Structural identifiability 24/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Similarity transformation/exhaustive modelling approach MJ Chappell University of Warwick July 2016 Structural identifiability 25/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Generates set of all possible linear models: ( A ( p ) , B ( p ) , C ( p )) with same I/O behaviour as given one: ( A ( p ) , B ( p ) , C ( p )) Consider the model given by q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ (1) y ( t , p ) = C ( p ) q ( t , p ) , and suppose that following are satisfied: Controllability rank condition: B ( p ) A ( p ) B ( p ) A ( p ) n − 1 B ( p ) � � rank . . . = n C ( p ) C ( p ) A ( p ) Observability rank condition: rank = n . . . C ( p ) A ( p ) n − 1 If both are satisfied model is minimal. MJ Chappell University of Warwick July 2016 Structural identifiability 26/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = ˙ z ( 0 ) = Tq 0 ( p ) , = y ( t , p ) = C ( p ) q ( t , p ) = has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = B ( p ) = C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ z ( 0 ) = Tq 0 ( p ) , = y ( t , p ) = C ( p ) q ( t , p ) = has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = B ( p ) = C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ = TA ( p ) T − 1 z ( t ) + TB ( p ) u ( t ) , z ( 0 ) = Tq 0 ( p ) , y ( t , p ) = C ( p ) q ( t , p ) = has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = B ( p ) = C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ = TA ( p ) T − 1 z ( t ) + TB ( p ) u ( t ) , z ( 0 ) = Tq 0 ( p ) , y ( t , p ) = C ( p ) q ( t , p ) = C ( p ) T − 1 z ( t ) . has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = B ( p ) = C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ = TA ( p ) T − 1 z ( t ) + TB ( p ) u ( t ) , z ( 0 ) = Tq 0 ( p ) , y ( t , p ) = C ( p ) q ( t , p ) = C ( p ) T − 1 z ( t ) . has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = TA ( p ) T − 1 , B ( p ) = C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ = TA ( p ) T − 1 z ( t ) + TB ( p ) u ( t ) , z ( 0 ) = Tq 0 ( p ) , y ( t , p ) = C ( p ) q ( t , p ) = C ( p ) T − 1 z ( t ) . has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = TA ( p ) T − 1 , B ( p ) = TB ( p ) , C ( p ) = for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Then there exists invertible n × n matrix T such that, if z = Tq : z ( t ) = T ˙ q ( t , p ) = TA ( p ) q ( t , p ) + TB ( p ) u ( t ) ˙ = TA ( p ) T − 1 z ( t ) + TB ( p ) u ( t ) , z ( 0 ) = Tq 0 ( p ) , y ( t , p ) = C ( p ) q ( t , p ) = C ( p ) T − 1 z ( t ) . has identical input-output behaviour. Therefore, if p ∈ Ω gives rise to a model: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = q 0 ( p ) , ˙ y ( t , p ) = C ( p ) q ( t , p ) , with identical input-output behaviour as the initial one (1), then A ( p ) = TA ( p ) T − 1 , B ( p ) = TB ( p ) , C ( p ) = C ( p ) T − 1 , for some invertible n × n matrix T . MJ Chappell University of Warwick July 2016 Structural identifiability 27/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Sometimes easier to deal with: A ( p ) T = TA ( p ) , (2) B ( p ) = TB ( p ) , (3) C ( p ) T = C ( p ) . (4) If only solution is T = I n then p = p and the system is SGI If T can take any of a finite set (with more than 1 element) of possibilities, then the system is SLI Otherwise, ( T can take any of a infinite set of possibilities) then the system is unidentifiable MJ Chappell University of Warwick July 2016 Structural identifiability 28/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: Two-compartment model. I = b 1 u ( t ) a 21 y = c 1 q 1 1 2 a 12 a 01 System equations: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = 0 ˙ y ( t , p ) = C ( p ) q ( t , p ) where A ( p ) = , B ( p ) = , C ( p ) = MJ Chappell University of Warwick July 2016 Structural identifiability 29/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: Two-compartment model. I = b 1 u ( t ) a 21 y = c 1 q 1 1 2 a 12 a 01 System equations: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = 0 ˙ y ( t , p ) = C ( p ) q ( t , p ) where � − ( a 01 + a 21 ) � a 12 A ( p ) = , B ( p ) = , C ( p ) = a 21 − a 12 MJ Chappell University of Warwick July 2016 Structural identifiability 29/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: Two-compartment model. I = b 1 u ( t ) a 21 y = c 1 q 1 1 2 a 12 a 01 System equations: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = 0 ˙ y ( t , p ) = C ( p ) q ( t , p ) where � − ( a 01 + a 21 ) � � b 1 � a 12 A ( p ) = , B ( p ) = , C ( p ) = a 21 − a 12 0 MJ Chappell University of Warwick July 2016 Structural identifiability 29/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach Example: Two-compartment model. I = b 1 u ( t ) a 21 y = c 1 q 1 1 2 a 12 a 01 System equations: q ( t , p ) = A ( p ) q ( t , p ) + B ( p ) u ( t ) , q ( 0 , p ) = 0 ˙ y ( t , p ) = C ( p ) q ( t , p ) where � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 a 21 − a 12 0 MJ Chappell University of Warwick July 2016 Structural identifiability 29/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � � B ( p ) A ( p ) B ( p ) = Observability: C ( p ) � � � � = C ( p ) A ( p ) Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � B ( p ) A ( p ) B ( p ) = 0 Observability: C ( p ) � � � � = C ( p ) A ( p ) Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = 0 b 1 a 21 Observability: C ( p ) � � � � = C ( p ) A ( p ) Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � = C ( p ) A ( p ) Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = C ( p ) A ( p ) Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 So model is minimal Equation (3): B ( p ) = TB ( p ) and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 So model is minimal Equation (3): � � � t 11 � � b 1 � b 1 t 12 B ( p ) = = TB ( p ) = t 21 t 22 0 0 and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 So model is minimal Equation (3): � � � t 11 � � b 1 � � t 11 b 1 � b 1 t 12 B ( p ) = = TB ( p ) = = t 21 t 22 0 t 21 b 1 0 and so MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 So model is minimal Equation (3): � � � t 11 � � b 1 � � t 11 b 1 � b 1 t 12 B ( p ) = = TB ( p ) = = t 21 t 22 0 t 21 b 1 0 and so t 21 = 0 and MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � b 1 � � � a 12 A ( p ) = , B ( p ) = , C ( p ) = c 1 0 − a 12 a 21 0 Controllability: � � � b 1 � − b 1 ( a 01 + a 21 ) B ( p ) A ( p ) B ( p ) = rank 2 0 b 1 a 21 Observability: C ( p ) � � � � c 1 0 = rank 2 C ( p ) A ( p ) − c 1 ( a 01 + a 21 ) c 1 a 12 So model is minimal Equation (3): � � � t 11 � � b 1 � � t 11 b 1 � b 1 t 12 B ( p ) = = TB ( p ) = = t 21 t 22 0 t 21 b 1 0 and so t 21 = 0 and t 11 = b 1 / b 1 MJ Chappell University of Warwick July 2016 Structural identifiability 30/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � � � b 1 / b 1 t 12 C ( p ) T = = C ( p ) c 1 0 = c 1 0 0 t 22 and so Equation (2): A ( p ) T = = TA ( p ) = = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so Equation (2): A ( p ) T = = TA ( p ) = = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and Equation (2): A ( p ) T = = TA ( p ) = = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and b 1 c 1 = b 1 c 1 Equation (2): A ( p ) T = = TA ( p ) = = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and b 1 c 1 = b 1 c 1 Equation (2): � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 0 A ( p ) T = a 21 − a 12 0 t 22 = TA ( p ) = = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and b 1 c 1 = b 1 c 1 Equation (2): � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 0 A ( p ) T = a 21 − a 12 0 t 22 � � � − ( a 01 + a 21 ) � b 1 / b 1 0 a 12 = TA ( p ) = a 21 − a 12 0 t 22 = MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and b 1 c 1 = b 1 c 1 Equation (2): � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 0 A ( p ) T = a 21 − a 12 0 t 22 � � � − ( a 01 + a 21 ) � b 1 / b 1 0 a 12 = TA ( p ) = a 21 − a 12 0 t 22 � − b 1 � b 1 ( a 01 + a 21 ) t 22 a 12 = = b 1 − a 12 t 22 b 1 a 21 MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 t 12 A ( p ) = , C ( p ) = , T = � � c 1 0 a 21 − a 12 0 t 22 Equation (4): � � � � C ( p ) T = = C ( p ) b 1 c 1 / b 1 c 1 t 12 = c 1 0 and so t 12 = 0 and b 1 c 1 = b 1 c 1 Equation (2): � − ( a 01 + a 21 ) � � � a 12 b 1 / b 1 0 A ( p ) T = a 21 − a 12 0 t 22 � � � − ( a 01 + a 21 ) � b 1 / b 1 0 a 12 = TA ( p ) = a 21 − a 12 0 t 22 � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = = b 1 − a 12 t 22 a 21 t 22 − a 12 t 22 b 1 a 21 MJ Chappell University of Warwick July 2016 Structural identifiability 31/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: so (1,2) component: (2,1) component: (1,1) component: So: MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: (2,1) component: (1,1) component: So: MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: (1,1) component: So: MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: a 21 = a 21 (1,1) component: So: MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: a 21 = a 21 (1,1) component: a 01 = a 01 So: MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: a 21 = a 21 (1,1) component: a 01 = a 01 So: a 01 , a 12 and a 21 all globally identifiable MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: a 21 = a 21 (1,1) component: a 01 = a 01 So: a 01 , a 12 and a 21 all globally identifiable combination b 1 c 1 globally identifiable MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Laplace transform approach Structural identifiability Taylor series approach Techniques for nonlinear models Similarity transformation/exhaustive modelling approach � − b 1 � � � − b 1 b 1 b 1 ( a 01 + a 21 ) t 22 a 12 b 1 ( a 01 + a 21 ) b 1 a 12 = b 1 − a 12 t 22 − a 12 t 22 a 21 t 22 b 1 a 21 (2,2) component: a 12 = a 12 so (1,2) component: t 22 = b 1 / b 1 (2,1) component: a 21 = a 21 (1,1) component: a 01 = a 01 So: a 01 , a 12 and a 21 all globally identifiable combination b 1 c 1 globally identifiable individual b 1 and c 1 unidentifiable MJ Chappell University of Warwick July 2016 Structural identifiability 32/49
Motivation Taylor series approach Structural identifiability Observable normal form Techniques for nonlinear models Techniques for nonlinear models MJ Chappell University of Warwick July 2016 Structural identifiability 33/49
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