Sensitivity matrices as keys to local structural system properties of large-scale nonlinear systems

Abstract

Sensitivities are shown to play a key role in a highly efficient algorithm, presented in this paper, to establish three fundamental structural system properties: local structural identifiability, local observability, and local strong accessibility. Sensitivities have the advantageous property to be governed by linear dynamics, also if the system itself is nonlinear. By integrating their linear dynamics over a short time period, and by sampling the result, a sensitivity matrix is obtained. If this sensitivity matrix satisfies a rank condition, then the local structural system property under investigation holds. This rank condition will be referred to in this paper as the sensitivity rank condition (SERC). Applying a singular value decomposition (SVD) to the sensitivity matrix not only determines its rank but also pinpoints exactly the system components causing a possible failure to satisfy the local structural system property. The algorithm is highly efficient because integration of linear systems over short time-periods and computation of an SVD are computationally cheap. Therefore, it allows for the handling of large-scale systems in the order of seconds, as opposed to conventional algorithms that mostly rely on Lie series expansions and a corresponding Lie algebraic rank condition (LARC). The SERC and LARC algorithms are mathematically and computationally compared through a series of examples.

Appendices

Appendix 1: Dynamic model of the combined heat and power system of a hotel building [33, 34] used in Example 5

$$ f\left( {x,u,\theta } \right) = \left[ {\begin{array}{*{20}c} {k_{1} \left( {x_{2} - x_{1} } \right) + \left( {\frac{{b_{1} }}{{V_{1} }} - \frac{{x_{1} }}{{V_{1} }}} \right)u_{1} } \\ {k_{2} \left( {x_{1} - x_{2} } \right) + \left( {\frac{{x_{4} }}{{V_{2} }} - \frac{{x_{2} }}{{V_{2} }}} \right)u_{2} } \\ {\left( {\frac{{x_{2} }}{{V_{3} }} - \frac{{x_{3} }}{{V_{3} }}} \right)u_{2} + \left( {\frac{{x_{4} }}{{V_{3} }} - \frac{{x_{3} }}{{V_{3} }}} \right)u_{3} } \\ {\left( {\frac{{x_{6} }}{{V_{4} }} - \frac{{x_{4} }}{{V_{4} }}} \right)u_{4} } \\ {\left( {\frac{{x_{3} }}{{V_{5} }} - \frac{{x_{5} }}{{V_{5} }}} \right)u_{4} } \\ {\left( {\frac{{x_{7} }}{{V_{6} }} - \frac{{x_{6} }}{{V_{6} }}} \right)u_{7} + \left( {\frac{{x_{9} }}{{V_{6} }} - \frac{{x_{6} }}{{V_{6} }}} \right)u_{9} + \left( {\frac{{x_{12} }}{{V_{6} }} - \frac{{x_{6} }}{{V_{6} }}} \right)u_{5} } \\ {k_{7} \left( {x_{8} - x_{7} } \right) + \left( {\frac{{x_{5} }}{{V_{7} }} - \frac{{x_{7} }}{{V_{7} }}} \right)u_{7} } \\ {k_{8} \left( {x_{7} - x_{8} } \right) + \left( {\frac{{b_{2} }}{{V_{8} }} - \frac{{x_{8} }}{{V_{8} }}} \right)u_{8} } \\ {k_{9} \left( {x_{10} - x_{9} } \right) + \left( {\frac{{x_{5} }}{{V_{9} }} - \frac{{x_{9} }}{{V_{9} }}} \right)u_{9} } \\ {k_{10} \left( {x_{9} - x_{10} } \right) + \left( {\frac{{b_{3} }}{{V_{10} }} - \frac{{x_{10} }}{{V_{10} }}} \right)u_{10} } \\ {\left( {\frac{{x_{5} }}{{V_{11} }} - \frac{{x_{11} }}{{V_{11} }}} \right)u_{5} + \left( {\frac{{x_{12} }}{{V_{11} }} - \frac{{x_{11} }}{{V_{11} }}} \right)u_{12} } \\ {\left( {\frac{{x_{11} }}{{V_{12} }} - \frac{{x_{12} }}{{V_{12} }}} \right)u_{11} } \\ {k_{13} \left( {x_{14} - x_{13} } \right) + \left( {\frac{{x_{12} }}{{V_{13} }} - \frac{{x_{13} }}{{V_{13} }}} \right)u_{11} } \\ {k_{14} \left( {x_{13} - x_{14} } \right) + \left( {\frac{{b_{4} }}{{V_{14} }} - \frac{{x_{14} }}{{V_{14} }}} \right)u_{6} } \\ \end{array} } \right]\,,\, $$

(A1)

$$ \theta = \left[ {V_{1} ,\,V_{2} ,..,V_{14} ,\,\,k_{1} ,k_{2} ,k_{7} ,k_{8} ,k_{9} ,k_{10} ,k_{13} ,k_{14} ,b_{1} ,b_{2} ,b_{3} ,b_{4} } \right]^{T} $$

(A2)

Appendix 2: Proof of generic equivalence between SERC and LARC for local strong accessibility and local observability

2.1 Proof for local strong accessibility

Concerning local strong accessibility, the examples in this paper show that the rank given by SERC equals the one given by LARC, provided the selected trajectory (2.3) of system (2.1) is nonsingular, as shown in Fig. 12. For local strong accessible systems (2.1), it is proved in [37] that trajectories (2.3) of system (2.1) are generically nonsingular. Then, LARC gives full rank for $x_{L}$ in Fig. 12 while SERC generically does, namely if the selected trajectory (2.3) of system (2.1) containing $x_{L}$ is nonsingular. In this appendix, we will prove that if the rank obtained from LARC is not full, SERC still generically produces the same rank for $x_{L}$ in Fig. 12. To that end, we will first summarize the proof given in [37], when the rank obtained from LARC is full.

Theorem 2 in [37] states that for local strong accessible systems, trajectories are generically nonsingular. Its proof relies mostly on proposition 2.1 in [37]. This proposition concerns an observability-related property developed in [39]. Recall that LARC for local strong accessibility is fully determined by the dynamics (2.1). The proof of theorem 2 in [37] requires extension of these dynamics with

$$ \frac{{{\text{d}}Q\left( t \right)}}{{{\text{d}}t}} = A\left( t \right)Q\left( t \right) + Q\left( t \right)A\left( t \right) + B\left( t \right)B^{T} \left( t \right),\,\,\,t \ge t_{0} ,\,\,Q\left( {t_{0} } \right) = 0 $$

(A3)

$$ A\left( t \right) = \frac{{{\text{d}}f}}{{{\text{d}}x}}\left( {x\left( t \right),u\left( t \right)} \right),\,\,B\left( t \right) = \frac{{{\text{d}}f}}{{{\text{d}}u}}\left( {x\left( t \right),u\left( t \right)} \right),\,\,Q\left( t \right) \in R^{n \times n} $$

(A4)

In addition, the following output equation is required:

$$ y_{e} \left( t \right) = h_{e} \left( {x\left( t \right),Q\left( t \right)} \right) = \det \left( {Q\left( t \right)} \right). $$

(A5)

Note that, $Q\left( t \right)$, $t > t_{0}$ in (A3) is the controllability matrix of the time-varying linear system (2.18) that describes the propagation of sensitivities. If $Q\left( t \right)$, $t > t_{0}$ is not full rank, the linear dynamics (2.18) is not controllable and the trajectory is singular. In that case, the determinant of $Q\left( t \right)$, $t > t_{0}$, being the output $y_{e} \left( t \right)$ in (A5), equals zero. System (2.1), extended with (A3), (A5), when written in the standard state-space format becomes

$$ \begin{aligned} \frac{{{\text{d}}x_{e} \left( t \right)}}{{{\text{d}}t}} = &\, f_{e} \left( {x_{e} \left( t \right),u\left( t \right),\theta } \right) = \left[ {\begin{array}{*{20}c} {f\left( {x\left( t \right),u\left( t \right),\theta } \right)} \\ {st\left( {A\left( t \right)Q\left( t \right) + Q\left( t \right)A\left( t \right) + B\left( t \right)B^{T} \left( t \right)} \right)} \\ \end{array} } \right], \\ x_{e} \left( t \right) = & \left[ {\begin{array}{*{20}c} {x\left( t \right)} \\ {st\left( {Q\left( t \right)} \right)} \\ \end{array} } \right], \\ \end{aligned} $$

(A6)

$$ y_{e} \left( t \right) = h_{e} \left( {x_{e} \left( t \right)} \right) = \det \left( {Q\left( t \right)} \right), $$

(A7)

where $st$ denotes the stack operator that stacks all columns of the matrix argument, starting with the first one. The proof of theorem 2 in [37] comes down to show, by means of proposition 2.1, that trajectories having initial state $x_{e} = \left[ {\begin{array}{*{20}c} x \\ 0 \\ \end{array} } \right]$, generically, do not produce a zero output $y_{e} \left( t \right)$ in (A5), and therefore are nonsingular. To obtain this result, local strong accessibility of system (2.1) is required to guarantee that every state $x_{e} = \left[ {\begin{array}{*{20}c} x \\ 0 \\ \end{array} } \right]$ of the extended system belongs to the set $G$, mentioned in proposition 2.1 of [37].

Here, we would like to establish generic equivalence for any rank given by LARC, for any $x_{L}$ in Fig. 12. This can be achieved since $f$ in (2.1) is assumed analytic. Then, the Hermann Nagano theorem [5] holds, according to which the state space separates into one or more manifolds having equal or different dimensions. Then by Chow’s theorem [5], a realization of each manifold exists, that is local strong accessible, having a state dimension equal to the rank given by LARC. Given their local strong accessibility, we are allowed again to apply the proof given in [37], to these realizations. This proves the generic equivalence between SERC and LARC for any rank, and for any state $x_{L}$ in Fig. 12.

2.2 Proof for local observability

For local observability, to show that the rank from LARC generically equals the one from SERC, we can again apply proposition 2.1 from [37]. Now, LARC is obtained from Lie differentiation instead of Lie bracketing [5], [7], [8] and nonsingular trajectories are those leading to an observable linear system (2.14). Therefore, we must extend the dynamics (2.1) with dynamics describing the observability matrix propagation of the time-varying linear system (2.14),

$$ \frac{{{\text{d}}Q\left( t \right)}}{{{\text{d}}t}} = A^{T} \left( t \right)Q\left( t \right) + Q\left( t \right)A^{T} \left( t \right) + C^{T} \left( t \right)C\left( t \right),\,\,\,t \ge t_{0} ,\,\,Q\left( {t_{0} } \right) = 0, $$

(A8)

$$ A\left( t \right) = \frac{{{\text{d}}f}}{{{\text{d}}x}}\left( {x\left( t \right),u\left( t \right)} \right),\,\,C\left( t \right) = \frac{{{\text{d}}h}}{{{\text{d}}x}}\left( {x\left( t \right),u\left( t \right)} \right),\,\,Q\left( t \right) \in R^{n \times n} , $$

(A9)

while taking as the output,

$$ y_{e} \left( t \right) = h\left( {x\left( t \right),Q\left( t \right)} \right) = \det \left( {Q\left( t \right)} \right). $$

(A10)

If $Q\left( t \right)$, $t > t_{0}$ is not full rank, the linear dynamics (2.14) are not observable and the trajectory is singular. In that case, the determinant of $Q\left( t \right)$, $t > t_{0}$, being the output $y_{e} \left( t \right)$ in (A10), equals zero. The state-space representation of the extended system now becomes

$$ \begin{aligned} \frac{{{\text{d}}x_{e} \left( t \right)}}{{{\text{d}}t}} = &\, f_{e} \left( {x_{e} \left( t \right),u\left( t \right),\theta } \right) = \left[ {\begin{array}{*{20}c} {f\left( {x\left( t \right),u\left( t \right),\theta } \right)} \\ {st\left( {A^{T} \left( t \right)Q\left( t \right) + Q\left( t \right)A^{T} \left( t \right) + C^{T} \left( t \right)C\left( t \right)} \right)} \\ \end{array} } \right], \\ x_{e} \left( t \right) = &\, \left[ {\begin{array}{*{20}c} {x\left( t \right)} \\ {st\left( {Q\left( t \right)} \right)} \\ \end{array} } \right], \\ \end{aligned} $$

(A11)

$$ y_{e} \left( t \right) = h_{e} \left( {x_{e} \left( t \right)} \right) = \det \left( {Q\left( t \right)} \right). $$

(A12)

Again the proof follows from proposition 2.1 of [37] if all states $x_{e} = \left[ {\begin{array}{*{20}c} x \\ 0 \\ \end{array} } \right]$ of the extended system belong to the set $G$, mentioned in proposition 2.1. Now, this requires the original system (2.1), (2.2) to be local observable. That this is so follows from application of the same proposition 2.1, now taking (2.1) as the system and (2.2) as the output. So if LARC gives full rank for any $x_{L}$ in Fig. 12, SERC generically does, namely if the selected trajectory (2.3) of system (2.1) is nonsingular.

To establish generic equivalence between LARC and SERC for any rank, we rely on our assumption that $f$ in (2.1) and $h$ in (2.2) are analytic. Then, the Hermann Nagano and Chow theorems [5] provide us with local accessible realizations of each manifold in which the state space separates. Moreover, these realizations each preserve the observation space, the dimension of which equals the rank obtained from LARC. Next, for each of those realizations, we may obtain a minimal realization that also preserves the observation space [7]. This minimal realization is locally observable and has a state dimension equal to the one of the observation space given by the rank obtained from LARC. Because these minimal realizations are locally observable, we may apply proposition 2.1 of [37] again, which proves the generic equivalence between LARC and SERC for any rank, and any state $x_{L}$ in Fig. 12.