About the classical and structural controllability and observability of a common class of activated sludge plants

In this work, the stability, controllability and observability properties of a class of activated sludge plants are analysed. Specifically, the five biological reactors and the secondary settler in the Benchmark Simulation Model no. 1 (BSM1) are studied. For the task, we represented the plant as a dynamical system consisting of 145 state variables, 13 controls, 14 disturbances and 15 outputs and as a complex networks to study its full-state controllability and observability properties from a structural and a classical point of view. By analysing the topology of the network, we show how this class of systems is controllable but not observable in a structural sense, and thus how it is controllable but not observable in a classical sense for almost all possible realisations. We also show how a linearisation commonly used in the literature is neither full-state controllable nor full-state observable in the classical sense. The control and observation efforts are quantified in terms of energy-and centrality-based based metrics. © 2022TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense


Introduction
Stricter effluent requirements, costs minimisation, sustainable water and energy cycles, recovery of nutrients and other resources, as well as increasing expectations in the public to attain high service standards require wastewater treatment to face unprecedented operational challenges.Because of their wide diffusion, activated sludge processes play a central role in the biological treatment of wastewater and their efficient management has a large technological and societal impact.
Many control strategies for activated sludge plants have been proposed in the industrial and academic literature.More than forty years ago the first specialised conference on Instrumentation, Control and Automation of Water and Wastewater systems was founded with the firm goal of encouraging the application of automation technologies to wastewater treatment plants.Pioneering works, as Olsson et al. [1],Olsson and Andrews [2], inspired numerous researchers and practitioners to approach this specific field.Extensive reviews of the various control solutions can be found in Olsson et al. [3,4].Importantly, many research efforts have been fostered thanks to a number of support tools that provide a simulation protocol for real-world activated sludge processes.The Benchmark Simulation Model no. 1 (BSM1, Gernaey et al. [5]), specifically, singled out as the reference platform for developing and controlling activated sludge processes subjected to typical municipal wastewater influents.
The availability of the BSMs has led to the design of many modelling and control solutions [6][7][8][9][10][11][12][13][14][15][16], which, regretfully, have not yet been digested in a comprehensive review.These works, while remarkable in addressing specific control objectives, highlight the necessity to further understand BSM-like systems from a control theoretical perspective.System properties like stability, controllability and observability of BSMs have been touched only marginally or studied only for simpler subsystems.In Benazzi and Katebi [17], the nonlinear global observability of a single bioreactor from the plant is analysed.Busch et al. [18] discusses optimal measurement configurations that ensure local observability for a BSM1 in which the settler's model is simplified.In Zeng et al. [19], the same simplified model is decomposed into subsystems which are then each tested for observability, individually.Works like Yin and Liu [20],Yin et al. [21],Yin and Liu [22] systematically exploit the notion of observability to either design state estimators, subsystem decompositions, or to test this property for the BSM1 model, though under a rather unrealistically large set of measurements.To the best of our knowledge, no similar studies report on the stability and controllability properties.Under this scenario, this work aims at contributing to the understanding of a large class of activated sludge plants by analysing the stability, controllability and observability properties of this important benchmark.For the analysis, the dynamical system associated with the BSM1 is mapped onto a complex network where its full-state controllability and observability properties are studied.As we have been interested in determining whether the system is controllable and observable under all feasible linearisations [23,24], we couple classical control notions from linear system theory [25] with graph-theoretic tools [26][27][28][29][30][31][32].This allows us to determine such properties from the structure of the graph, regardless of the linearisation.The classical analysis is performed on a linearisation of the BSM1 often encountered in the literature.
Our analyses show that activated sludge plants described using the BSM1 are structurally controllable, but that they are not structurally observable.Because of the generality of this result, we implicitly show also how the BSM1 is full-state controllable but not full-state observable in a classical sense, for almost all possible linearisations of the model.In addition, the conditions to satisfy the more restrictive notions of strong structural controllability and observability are not satisfied.The classical counterpart of this result is verified with Popov-Belevitch-Hautus tests [33].An important result of our analysis is that a common linearisation of the BSM1 is not full-state controllable and not full-state observable.Being controllability and observability notions of the binary type, we complement the analysis by determining also the energy-and centrality-based metrics [34][35][36] which quantify the control and observation efforts that are needed to operate this class of wastewater treatment plants.
The analysis is presented as follows: Section 2 describes the activated sludge plant and its state-space model, Section 3 sets the preliminaries and the notation by overviewing the concepts of stability and the classical and structural notions of controllability and observability.Section 4 discusses our results on the stability, full-state controllability and observability for this class of activated sludge plants.In Section 5 we discuss the results for two alternative configurations of the plant.Model equations, model parameters, and the operating point used for linearisation are reported in Appendix.

The activated sludge plant: Process overview and statespace model
We consider the activated sludge process in a conventional biological wastewater treatment plant.Based on the denitrificationnitrification process, bacteria reduce nitrogen present in the influent wastewater in the form of ammonia into nitrate, which is subsequently reduced into nitrogen gas to be released into the atmosphere.The prototypical process, in Fig. 1, consists of five biological reactors and a settler.
The treatment starts with a first reactor where wastewater from primary sedimentation, return sludge from secondary sedimentation and internal recycle sludge are fed.The outflow from the first reactor is then fed sequentially to the downstream reactors and, eventually, from the fifth reactor to the secondary settler.Mixed liquor from the fifth reactor is recirculated into the first reactor together with the recycle sludge from secondary sedimentation, as mentioned.Excess sludge from the settler can also be directed towards other processes.Oxygen can be added by insufflating air into each reactor.In the aerated reactors, the ammonium nitrogen (NH 4 -N) in the wastewater is oxidised into nitrate nitrogen (NO 3 -N), which is in turn reduced into nitrogen gas (N 2 ) in the anoxic reactors.An additional carbon source can be added to each rector independently.No other chemicals are added to the process.
Each reactor is described by the Activated Sludge Model no. 1 [37].For the settler, a 10-layers non-reactive model by Takács et al. [38] is used.Under this setup, the process corresponds to the Benchmark Simulation Model no. 1 [5], or activated sludge plant (ASP).
The dynamics of each reactor A(r) (r = 1, . . ., 5) are described by 13 state variables, the vector of concentrations and controllable inputs u A(r) EC ), the oxygen transfer coefficient K L a (r) and external carbon source flow-rate EC .The dynamics of each layer S(l) (l = 1, . . ., 10) of the settler are described by 8 state variables, the vector SS , S S(l) I , S S(l) S , S S(l) O , S S(l) NO , S S(l) NH , S S(l) ND , S S(l) ALK ) T . ( The plant is subjected to three additional controllable inputs, the internal and external sludge recycle flow-rates (Q A and Q R , respectively) and the wastage flow-rate Q W , and to 14 disturbances, the influent flow-rate Q IN and its concentrations x A (IN) , all entering the first reactor.Wastewater concentrations in the internal recycle are given by x A (5) , whereas x S (1) are concentrations in the external recycle and wastage.
As for the measurements, we consider a sensor-arrangement consisting of analysers determining the concentrations y = (y A (1) , . . ., y A (5) , X S (10)

S IN S , S A(r) S
, S

S(l) S
Readily biodegradable substrate D, S, S g COD m −3 Biochemical oxygen demand M g COD m −3 COD S (10)  Chemical oxygen demand M g COD m −3

P
).The state-space model for this class of ASPs is given as with state variables x(t) = ((x A (1) , . . ., x A (5) ), (x S (1) , . . ., x S (10) , and disturbances ≥0 , all at time t.The time-invariant dynamics f (•|θ x ) and g(•|θ y ) depend on a set of stoichiometric and kinetic parameters collectively denoted by the vectors θ x and θ y .The state-space model in Eq. ( 5) thus consists of N x = 13 × 5 + 8 × 10 = 145 state variables, N u = 3 + 2 × 5 = 13 controls, N w = 1 + 13 = 14 disturbances and N y = 5 × 2 + 5 = 15 outputs.As N x ≫ N u and N x ≫ N y , the system is both under-actuated and under-observed.We refer to Table 1 for a characterisation of the variables.The complete model equations and parameters (θ x , θ y ) in Eq. ( 5) are reported in Appendix.
The default control strategy proposed in Gernaey et al. [5] for the BSM1 considers two low-level (PI) controllers: • Nitrate and nitrite nitrogen in the second reactor, S A (2)   NO , is controlled by manipulating the internal recycle, Q A ; • Dissolved oxygen concentration in the fifth reactor, S A (5)   O , is controlled by manipulating the oxygen mass transfer coefficient K L a (5) , a proxy to the air flow-rate.The plant's performance is based on flow-weighted and timeaveraged effluent concentrations of total suspended solids (X SS ), biochemical oxygen demand (BOD 5 ), chemical oxygen demand (COD), total nitrogen (N TOT ) and ammonia (S NH ).Typically, the control performance is given in terms of effluent quality by measuring and minimising the effluent concentration of these compounds [5].
Importantly, our state-space configuration includes all control handles suggested in Gernaey et al. [5] that do not require changes to the plant layout depicted in Fig. 1.We consider the possibility of having the default low-level controllers applied on each of the five reactors.As such, our configuration necessarily includes a sensor-arrangement that considers the measurement of S A(r) NO and S A(r) O in all reactors (r = 1, . . ., 5).

Preliminaries: Dynamical properties
We consider the general state-space representation of a deterministic non-autonomous controlled dynamic system The state Eq.(6a), determines the evolution of the state x(t) ∈ ).This allows to set t 0 = 0, though we often and intentionally omit mentioning it.
The structural form of the state-space representation of the system can be written using the usual linear model The structure of matrices A, B, G and C can be defined using inference diagrams in such a way that element A n ′ x ,nx (respectively, B nx,nu , G nx,nw and C ny,nx ) is non-zero, and potentially unknown, whenever component x nx (u nu , w nw and again x nx ) appears in the vector field f n ′ x (•) and algebraic function g ny (•); that is, whenever the (n ′ x , n x )-th element ∂f n ′ x /∂x nx (respectively, ∂f n ′ x /∂u nu , ∂f n ′ x /∂w nw and ∂g ny /∂x nx ) in the Jacobian matrix(es) is non-zero.
When the elements of A, B, G and C are either zeros or unknown, the resulting family of systems is referred to as structured dynamical system [28].Evaluating the Jacobians at a specific point (x ′ , u ′ , w ′ ) leads to an approximated linear time-invariant (LTI) system in which functions A, B, G and C are known.A steady-state point is usually chosen for fixing the linearisation.Certain properties of structured and LTI systems can be studied by mapping the state equation Eq. (7a) onto the digraph where the vertex set of directed edges between state and control vertices.The measurement process is studied by mapping the state and output equations Eq. ( 7) onto digraph By coupling controls to state variables and state variables to measurements, the notions of controllability and observability define the prerequisite for control and state estimation.These conditions can be relaxed in the absence of unstable modes, in favour of the weaker notions of stabilisability and detectability.For linear systems, classical sufficient and necessary controllability and observability tests have been derived [25,33,39,40].When a system is known only structurally, stronger notions of structural controllability and observability [26,27] and associated sufficient and necessary conditions [29][30][31][32] can be used.For LTI systems, the classical notions lead to important tools like the Kalman's canonical decomposition.For the sake of completeness and notational necessity, this section reviews all these concepts.

Stability
We review stability in terms of the conditions under which the system in Eq. (7) subjected to a bounded input produces bounded state-and output-response trajectories [25].These notions are referred to as external and internal stability, the latter being more general.For simplicity and without any loss, we do not distinguish between controls and disturbances and momentarily redefine B ≡ [B|G].
A linear system with impulse response matrix dτ is bounded when u(t) is bounded.This notion of external stability is defined as the existence of a finite gain κ < ∞ such that for all bounded inputs u the following relation holds External stability can be verified at the system level by showing that, under some mild conditions on the smoothness of u and H, the upper bound κ for the induced matrix norm of the input-output map exists and can be used as gain in aforementioned relation; that is, For a LTI system, Eq. (10) specialises and it suffices to show that (i) the impulse response is absolutely integrable and (ii) the transfer function More generally, system (A, B, C ) with state transition matrix For the LTI system in Eq. (7) ) )e Re(λn)t .

Controllability and observability
A system is full-state controllable if it is possible to steer it from any initial state to any final state in finite time, whereas it is full-state observable if it is possible to uniquely determine its initial state from a sequence of measurements over a finite time.These notions are overviewed and classical necessary and sufficient conditions are given for LTI systems.For systems that are controllable and observable, we review a set of energy-based controllability and observability metrics.We also review the more general structural notions of these notions and provide weak and strong validity conditions.

Classical controllability and observability
Let the controllability Gramian of the pair (A, B) be the N x × N x symmetric positive semidefinite matrix A sufficient and necessary controllability condition is Let the observability Gramian of the pair (A, C ) be the N x × N x symmetric positive semidefinite matrix A sufficient and necessary observability condition is Gramian-based criteria in Eqs. ( 15) and ( 17) are straightforward but unpractical.Equivalent criteria can be defined from controllability and observability matrices [39].
N x observability matrix of the system.Sufficient and necessary condition for controllability and observability are The criteria in Eqs.(18a) and Eq.(18b) are more direct and, for low-dimensional systems, their evaluation only requires a small number of matrix multiplications.The computation of C and O can still be troublesome for high-dimensional systems.The limitation is due to numerical over-and under-flows resulting from computing large powers of A and A T .A scalable alternative that overcomes the limitations of both Gramian-based and Kalman's rank criteria is provided by the Popov-Belevitch-Hautus (PBH) rank tests.Necessary and sufficient conditions are given by the two following lemmas: Lemma 2 ([33]).The statement 'the pair (A, C ) is observable' is equivalent to the statements: Based on Lemma 1, the pair (A, B) is controllable if and only if, the columns of B have at least one component in the direction the columns of C have at least one component in the direction are unobservable with the measurements determined by C .Controllability and observability are invariant with respect to nonsingular similarity transformations P ∈ R Nx×Nx .Thus, • the pair (A, B) is controllable if and only if the pair (A ′ , B ′ ) =

Controllability and observability metrics
Full-state controllability and observability are binary properties.Starting from the seminal work by Müller and Weber [34], various scalar metrics have been proposed to quantify the difficulty of control and observation tasks.We overview some energy-related metrics recently proposed by Pasqualetti et al. [35], and Summers et al. [36] for LTI systems.
Define the quadratic control and measurement energies In optimal quadratic regulation, we search for a controller that minimises these energies given positive definite weighting matrices R ∈ R Nx×Nx and Q ∈ R Ny×Ny .When minimised with R = I u and Q = I y , the unweighted energies determine Finite-and infinite-horizon controllability and observability metrics can be derived from Eq. ( 15) and ( 17).The eigenvectors ) correspond to state-space directions that require increasingly larger control energy the smaller λ c nx , whereas the eigenvectors ) correspond to directions of increasingly larger output energy the larger λ o nx .The control and measurement efforts associated with pairs (A, B) and (A, C ) can thus be quantified by single scalars defined from spectra σ ( Infinite-time Gramians, W c (∞) and W o (∞), always exist for Hurwitz systems, Eq. ( 12), and can be efficiently computed by solving Lyapunov equations [41].Thus, we only review a number of infinite-time metrics: Finite-time counterparts are evaluated by integrating Eqs. ( 15) and (17).IV. λ o min (W o (∞)): It is inversely related to the output energy along the least observable eigen-direction.

Definition 1 (Energy-Related Controllability Metrics
The control effort associated with attempting to control the full-state by only controlling one individual state variable x nx at a time is quantified by the average controllability centrality ) . ( This non-negative quantity is computed when a single control acts only on the n x -th state variable, when B = e nx is a unit vector in the standard basis of R Nx .Infinite-horizon Gramians = −e nx e T nx for n x ∈ {1, . . ., N x }.
The measurement effort associated with attempting to reconstruct the full-state by only measuring one state variable x nx at a time is quantified by the average observability centrality ) . ( This non-negative quantity is computed when a single sensor measures directly only the n x -th state variable, when

Stabilisability and detectability
A system is stabilisable if it possible to steer it from any initial state to the zero-state (a steady-state, for linearised systems), whereas it is detectable if its initial state can be asymptotically approximated from a sequence of measurements.Formally, Definition 3 (Stabilisability).The pair (A, B) is said to be stabilisable if, given any initial state x(0), it is possible to design an input u(t) such that x(t) → 0 as t → ∞.

Definition 4 (Detectability). The pair (A, C
) is said to be detectable if, giving any initial state x(0), it is possible to compute a state estimate x(t) from the force-free evolution of y(t), so that (x(t) − x(t)) → 0 as t → ∞.
These properties are often viewed as weaker notions of controllability and observability.Sufficient and necessary conditions for stabilisability and detectability can be derived from Kalman's canonical decomposition.
Lemma 3 (Minimal Realisation, Kalman [40]).Let C and O be, respectively, the controllability and observability matrix of a full- x c ō(t ) x co (t) There exists a nonsingular matrix P c ∈ R Nx×Nx , whose first N xc columns are the linearly independent columns of C, such that the transformation are respectively the controllable and uncontrollable subsystems in Kalman's decomposition (Lemma 3).A sufficient and necessary condition for stabilisability is that Re(λ j ) < 0 for all λ j ∈ σ ] , are respectively the observable and unobservable subsystems in Kalman's decomposition (Lemma 3).A sufficient and necessary condition for detectability is that Re(λ j ) < 0 for all eigenvalues Though straightforward, these criteria are unpractical for highdimensional systems as the design of P c and P o requires the computation of C and O, respectively.Scalable alternatives are the Popov-Belevitch-Hautus tests for stabilisability and detectability: Necessary and sufficient conditions are Lemma 4 ([33]).Let σ (A) = {λ i } Nx i=1 be the spectrum of A and Lemma 5 ([33]).Let σ (A) = {λ i } Nx i=1 be the spectrum of A and rank( Based on Lemma 4, (A, B) is stabilisable if and only if, for each unstable eigenvalue λ i of A (Re(λ i ) ≥ 0 and rank(λ i I − A) < N x ), the columns of B have at least one component in the direction

Structural controllability and observability
Structural analysis aims at assessing a family of systems with the same structure.The dynamics and measurement process of a structured dynamical system (A, B, C ) can be studied by mapping its state and output equations onto the digraph G = (V, E), (24) where the vertex set of directed edges between state and control component vertices, and set of directed edges between state and output component vertices.The structural controllability of the family of systems with dynamics represented by pair (A, B) can be studied through its By duality, the structural observability of the family of systems with measurement process represented by (A, C ) can be studied through its associated di- The pair (A, B) is structurally controllable if the nonzero elements of A and B can be set in such a way that the system is controllable in the classical sense.Pair (A, C ) is structurally observable if the nonzeros of A and C can be set in such a way that the system is observable in the classical sense.Formally, for an arbitrarily small ε > 0, we have the definitions Definition 5 (Structural Controllability, [26]).The pair (A, B) is structurally controllable if and only if there exists a controllable pair ( Ā, B) of the same dimension and structure of (A, B) such that ∥ Ā − A∥ < ε and ∥ B − B∥ < ε.
Definition 6 (Structural Observability, [26]).The pair (A, C ) is structurally observable if and only if there exists an observable pair ( Ā, C) of the same dimension and structure of (A, C ) such that ∥ Ā − A∥ < ε and ∥ C − C ∥ < ε.

be the network associated to the pair (A, B). The pair (A, B) is said to be structurally controllable if and only if the following conditions hold:
• (Accessibility) For every x nx ∈ V A there exists at least one directed path from any u nu ∈ V B to x nx .
• (Dilation-free) For every • (Accessibility) For every x nx ∈ V A there exists at least one directed path from x nx to any y ny ∈ V C .
The first condition in Lemma 6 can be verified by identifying the state vertices that are accessible from each possible origin vertex (a control).Similarly, the first condition in Lemma 7 can be verified by identifying the output vertices that are accessible from each possible origin vertex (a state component).Any graph search algorithm can be used for both tasks [42].The second condition in both lemmas can be verified by forming a maximum matching Lemma 6, or are directly connected to distinct output vertices in The maximum matching problem consists of identifying a (possibly not unique) subset of edges without common vertices that has maximum cardinality.The bipartite graph K = ) is defined by the disjoint and independent vertex sets fails to meet both accessibility and contraction-free conditions: There is no path from x 2 to output y 1 , and the subset S = {x 1 , x 2 } is larger than its neighbourhood set T (S) = {x 1 }.As a result, the system (A, B, C ) is uncontrollable and unobservable also in the conventional sense for all realisation of its nonzero entries.This can be confirmed by verifying that the controllability and observability matrices are of the forms which are always rank-deficient.Note that augmenting the statespace with a control u 2 (respectively, a sensor y 2 ) acting on (measuring) state-variable x 2 leads to a structural controllable (observable) system.

Strong controllability and observability
Based on Lin [26], a structurally controllable (respectively, structural observable) system might, under certain conditions, still admit full-state uncontrollable (respectively, full-state unobservable) realisations [44].This limitation can be overcome with the notions of strong structural controllability and observability [27].
A structural pair (A, B) is strongly controllable in a structural sense if every possible realisation of its nonzero entries leads to a full-state controllable system.By duality, a structural pair (A, C ) is strongly observable in a structural sense if every possible realisation of its nonzero entries leads to a full-state observable system.A strongly structurally controllable (respectively, observable) system is always structurally controllable (observable), the converse is not always true: Definition 7 (Strong Structural Controllability, [27]).The pair (A, B) is strongly structurally controllable if and only if any pair ( Ā, B) of the same dimension and structure of (A, B) is controllable in the classical sense.
As not all nodes x nx ∈ V A can be coloured black, neither graph is colourable and the system (A, B, C ) is thus uncontrollable and unobservable in a strong structural sense, an expected result.Note that augmenting the state-space with a control u 2 (respectively, a sensor y 2 ) acting on (measuring) state-variable x 2 leads to a strongly structural controllable (observable) system.Definition 8 (Strong Structural Observability, [27]).The pair (A, C ) is strongly structurally observable if and only if any pair ( Ā, C) of the same dimension and structure of (A, C ) is observable in the classical sense.
Sufficient and necessary conditions for strong structural controllability of (A, B) can be derived from its associated network The conditions for strong structural observability of (A, C ) result Lemma 8 ([32]).Let G c = (V c , E c ) be the network associated to the pair (A, B) and Gc = (V c , Ẽc ) an alternative graph with edge set Ẽc defined by Eq. (26).The pair (A, B) is said to be strongly structurally controllable if and only if both G c = (V c , E c ) and Gc = (V c , Ẽc ) are colourable.

Lemma 9 ([32]
).Let G o = (V o , E o ) be the network associated to the pair (A, C ) and Go = (V o , Ẽo ) an alternative graph with edge set Ẽo defined by Eq. (27).The pair (A, C ) is said to be strongly structurally observable if and only if both The networks G c and Gc are colourable if all nodes x nx ∈ V A are coloured black according to the following procedure: (1) Initially, colour all vertices v ∈ V white; (2) Colour v j ∈ V black if it is the only white out-neighbour for any fixed v i ∈ V and (v i , v j ) ∈ E; (3) Repeat step 2 until no more colour changes are possible.
By duality, this notion of colourability is similarly defined for the subgraphs G o and Go by inverting the edge direction in step 2. This colouring procedure is illustrated in Fig. 3.

Controllability and observability centralities
The relevance of a node in a graph is quantified by its centrality [45].We overview the centrality of individual state variables as encoded by the subgraph G A = (V A , E A ) of state nodes and their mutual relations.We look at node centralities as basic structural equivalent of control and observation energy-based metrics [46].
The in-degree centrality of state node x nx is defined as where 1 is a N x -th dimensional vector of all ones.k in (n x ) counts incoming edges to node x nx , the number of state variables that directly affect its dynamics.For observability, k in (n x ) quantifies how many state variables are indirectly observed if we were to measure only the n x -th state variable.
The out-degree centrality of state node x nx is defined as k out (n x ) counts the outgoing edges from node x nx , the number of state-variables whose dynamics are directly affected by the n x -th state variable.For controllability, k out (n x ) measures the number of state variables whose evolution is indirectly affected if we were to control only the n x -th state variable.

The activated sludge plant: Structural and classical dynamical properties
In this section, we analyse the full-state controllability and observability properties for the class of activated sludge plants represented by Eq. ( 5) defined in Section 2. The presentation begins with the structural controllability and observability analysis of system (A, B, C ) describing the structure of the ASP.A classical analysis of stability, controllability, and observability, is then performed for a standard linearisation (A SS , B SS , C SS ) of the model.A minimal realisation of this linearisation is also used to discuss the approximated system.

Structural properties
For the activated sludge plant ẋ(t) = f (x(t), u(t), w(t)|θ x ) with measurements y(t) = g(x(t)|θ y ) the structural matrices A ∈ R Nx×Nx , B ∈ R Nx×Nu , and C ∈ R Ny×Nx are obtained from the Jacobians A = ∂f /∂x, B = ∂f /∂u, and C = ∂g/∂x with N x = 145, N u = 13 and N y = 15.The associated digraph G = (V, E) has the vertex and edge sets We discuss the structural controllability and observability of the pairs (A, B) and (A, C ) and associated digraphs, Fig. 4.

Controllability and observability
The structural pair (A, B) associates with the directed subgraph

{S S(l)
I , S S(l) S , S S(l) O , S S(l) NO , S S(l) NH , S S(l) ND , S S(l) ALK } 10 l=7 can be coloured by following the procedure in Definition 9. Being (A, B) not-controllable in a strong structural sense, there exist realisations of A and B for which the system is not controllable in a classical sense.Being structurally controllable, (A, B) is also controllable in a classical sense, for almost all possible realisations of A and B.

The structural pair (A, C ) associates with the subgraph
) is not structurally observable (Lemma 7).As there are no paths from state vertices {S A(r) ALK } 5 r=1 , {S S(l) ALK } 10 l=1 and {S S(l) O } 10 l=7 to any of the output vertices, the accessibility condition is not satisfied.Conversely, the contraction-free condition is satisfied by the same matching M of size |M| = N x as before, obtained by choosing every state vertex's self-loop.The lack of structural observability implies non-observability in a classical sense.
As expected, the topology of G o = (V o , E o ) indicates that pair (A, C ) is also not strongly structurally observable: It is not possible to find a colouring for both graph G o = (V o , E o ) and modified graph Go = (V o , Ẽo ) in which all vertices x nx ∈ V A are coloured (Lemma 9).For instance, only the nodes associated to

{S A(r)
O , S A(r) NO } 5 r=1 and {X S(l) SS , S S(l) NH } 10 l=6 can be coloured in by the procedure in Definition 9.

Controllability and observability centrality
The relevance of each state node in the network can be quantified in terms of their in-degree and out-degree centrality, Eq. ( 28) and (29).Fig. 5 shows how particulate components r=1 are among the state vertices with highest in-degree centralities.This indicates that a large portion of the state-space can be directly observed by a sensor configuration that directly measures those state variables.

S A(1⇝5) O
, are also among the state vertices of highest out-degree centralities.Thus, a large portion of the network can be directly controlled by an input configuration that manipulates those variables.However, note that it is possible to control S A(r) O (through K L a (r) ), while individually controlling (or measuring) any of the concentrations , is practically unfeasible.Moreover, the species in the reactors have higher centralities than those in the settler.
We conclude that, for the activated sludge plant in Eq. ( 5), it is possible to design a control u(•) that transfers the plant to a desired state, in finite time, regardless of the realisation of (A, B).It is not possible, however, to determine an initial state x(t 0 ), and thus neither intermediate states x(t), from measurements y(t 0 ⇝ t f ).As a result, it is not possible to design a full-state observer based on such measurements and model, whatever the realisation of (A, C ). Being of structural nature, the conclusions are valid also in a classical sense.

Classical properties of a common linearisation
We now consider the linearisation (A SS , B SS , C SS ), corresponding to the fixed point SS ≡ (x SS , u SS , w SS , y SS ) considered by Gernaey et al. [5].This linearisation is commonly utilised in the literature and constitutes the default configuration of the BSM1.The matrices  6, is defined by the vertex and edge sets We discuss the stability of (A SS , B SS , C SS ) and the classic controllability and observability properties of (A SS , B SS ) and (A SS , C SS ).
For completeness, we use the approximation to validate the structural results that we reported earlier.

Stability
The spectrum of A SS consists of 69 distinct eigenvalues and as- i=1 , with {λ 1 , . . ., λ 31 } ⊂ R and {(λ 32 , λ * 32 ), . . ., (λ 69 , λ * 69 )} ⊂ C, Fig. 7. Five complex conjugate pairs of eigenvalues have algebraic multiplicity equal to two and two distinct real eigenvalues have algebraic multiplicities equal to two and twenty-eight, respectively.The distribution of eigenvalues in the complex plane shows that most of the modes have relatively slow time constants.As most eigenvalues are close to the real axis, pseudo-oscillatory behaviour is barely noticeable.
Being Re(λ i ) < 0 for all λ i ∈ σ (A SS ), then A SS is Hurwitz and (A SS , B SS , C SS ) is asymptotically stable (Section 3.1).This result can also be visualised through the simulation of individual system modes, shown in Fig. 8.As the unforced evolution of the system, from any x(0), is a linear combination of system modes, the fact that all curves converge to zero further confirms that the system is asymptotically stable.

Regarding the controllability of pair (A SS , B SS ) and associated digraph
is controllable in a structural sense (Lemma 6).Because pair (A SS , B SS ) corresponds to the linear time-invariant approximation of Eq. (5a) about steady-state point SS, it is possible to study its controllability also in a conventional sense.Classical controllability can be verified with a PBH test (Lemma 1), as an accurate computation of the controllability matrix Surprisingly, the PBH test contradicts the structural result and returns that (A SS , B SS ) is not controllable in the classical sense.Specifically, a real eigenvalue with algebraic multiplicity equal to 28 leads to a rank-deficient matrix The twenty-eight associated eigenvectors are shown in Fig. 9, top.Interestingly, the non-zero entries of the eigenvectors correspond to state variables relative to soluble matter in the settler's  last layer, showing that it is not possible to synthesise a control u(•) that enforces a desired profile of soluble matter in the settler.As A SS is Hurwitz the eigenvalue failing the PBH test satisfies Re(λ i ) < 0 indicating that the pair (A SS , B SS ) is stabilisable (Lemma 4).The apparent contradiction between classical and structural controllability results is explained in Section 4.3. .Interestingly, these correspond to concentrations of non-reacting matter.Because A SS is a stable matrix, all eigenvalues failing the PBH test satisfy Re(λ i ) < 0, thus rendering the pair (A SS , C SS ) detectable (Lemma 5).

Observability of the pair (A SS , C SS ) and associated digraph
We conclude that for the approximation (A SS , B SS , C SS ) it is not possible to design a control u(•) that transfers the plant to any state x(t f ) in finite time.Moreover, it is also not possible to determine the initial state x(t 0 ), and thus neither intermediate states x(t), from measurements y(t 0 ⇝ t f ).

Controllability and observability metrics and minimal realisation
To provide a qualitative analysis of controllability and observability, we firstly analyse the effort associated with controlling or observing each state variable individually.Then, we analyse a minimal realisation of linearisation (A SS , B SS , C SS ).
Considering the linearisation (A SS , B SS , C SS ), the average energy that is required to respectively control or reconstruct the full-state by directly controlling or measuring only one individual state variable is quantified by its average controllability and average observability centralities, in Fig. 10.
Our results show that the energy required to reach any point in the entire state-space is among the lowest if we were to actuate on control variables only affecting biomass (X

BA , and X A(r) P
) or particulate inert organic matter (X

A(r) I
) in the reactors.This reflects the fact that such variables are central to the process, but will evolve slowly if not controlled.Conversely, the required energy would be the highest if we were to actuate on controls only affecting dissolved oxygen (S A(r) O ).Again, it is worth mentioning that it is still possible to control S A(r) O (through K L a (r) ), whereas individually controlling any of the concentrations , is practically unfeasible.The analysis also Fig. 8.The normalised modes t k e λ i t /max t t k e λ i t (k = 0, . . ., µ(λ i ) − 1) for the real eigenvalues (top) and complex conjugated pairs of eigenvalues (bottom) from the spectrum σ (A SS ) = {λ i , ν i (λ i )} Nx i=1 .Grouping is based on each mode's time constant τ i = 1/Re[λ i ].
shows that acting directly on most state variables in the reactors is less demanding than acting on any state variables in the settler.
Our results also show that the effort required to reconstruct any point in the entire state-space is the lowest if we were to directly measure suspended solids at the bottom of the settler S(1) SS ).Additionally, the effort required is also among the lowest if we were to directly measure biomass (X

BA , and X A(r) P
) and particulate inert organic matter (X

A(r) I
) in the reactors.Again, this confirms the importance of measuring such variables and how reconstructing the state is more demanding when they are not available.In practice, only dissolved oxygen (S A(r) O ) and nitrate and nitrite nitrogen (S A(r) NO ) in each reactor, along with NH + 4 +NH 3 nitrogen in the effluent (S S (10) NH ), are directly measured.These variables are associated with the highest measurement effort if used individually to reconstruct the entire state of the process.
To complete the analysis of (A SS , B SS , C SS ), we study the compound energy-related metrics for the control and measurement configuration.Being the linearisation both uncontrollable and unobservable, these Gramian-based metrics of (A SS , B SS , C SS ) will obviously conclude that its control or state reconstruction are infinitely demanding.Alternatively, we further analyse (A SS , B SS , C SS ) using a minimal realisation (Lemma 3), as it preserves its input-output behaviour, Re(λ i ) < 0 for all eigenvalues λ i ∈ σ (A co ), and the minimal realisation (A co , B co , C co ) is stable.
We analysed the energy-related metrics defined for the infinite-horizon controllability (W c (∞)) and observability (W o (∞)) Gramians (Lemma 2).As the state matrix is Hurwitz, these Gramians are computed by solving Lyapunov equations 2) reveal and observing system difficult, even for a minimal realisation.Specifically, the fact that λ min (W c (∞)) and λ min (W o (∞)) are virtually zero implies the existence of state-space directions which are unaccessible.
We conclude that, though formally controllable and observable, the realisation (A co , B co , C co ) requires large control and measurement efforts.The cumulative coverage of the minimal statespace is shown in terms of a normalised cumulative sum (Λ W o (∞), Fig. 10.As more than 90% of the coverage is reached with a small number of eigenvalues, the state-space coverage implies that most of the control and output energy are comprised within a small number of directions.Together with Table 2, this shows that the input-output behaviour is mostly described by a small number of state-space directions, some of which being virtually inaccessible.

About the contradiction between structural and classical controllability results
When studying the controllability of (A SS , B SS , C SS ), we have shown that (A SS , B SS ) is controllable in a structural but not in a classical sense, a result that could be anticipated by the fact that (A, B) is not controllable in a strong structural sense.We further explain the apparent contradiction from the analysis of the dilation-free condition on G c SS = (V c SS , E c SS ).Specifically, as the existence of a self-loop for each state vertex is sufficient to satisfy the dilation-free condition, input vertices are needed only to satisfy the accessibility condition.Whenever some of the self-loop weights are equal, the dilation-free condition will underestimate the controls needed for full-state controllability [44].This is the case for (A SS , B SS ), where all reactors' non-reacting components (respectively, all settler's soluble components) from the same unit (layer) always have identical self-dynamics.
Consider non-reacting components S A(r) a (a ∈ {I, ALK }) and (b ∈ {I, P}) in the rth reactor.For all r = 1, . . ., 5, their dynamics in Eq. (5a) are each of the form ṠA(r) r) denoting influent flow-rates and R which is equal for all reactors, regardless of fixed-point SS.
Similarly, the dynamics of the soluble components S for all lower layers, {Q S(l) l=1 , and for the feed layer we have Q S (6) S .The model also assumes constant volume hold-ups V (l) S = 600 m 3 .For the relevant entries in the Jacobian matrix ∂f /∂x, which is equal for all components, whatever the fixed-point.

The activated sludge plant: Properties of certain alternative control configurations
We complete our study with an analysis of two alternative configurations of the activated sludge plant defined in Section 2. We firstly study the ASP when the Takács' model of the settler is replaced by a simpler 3-layer model with the same structure, while retaining our setup for the sensors and actuators.Secondly, we consider the ASP with the full process model under the minimal configuration of sensors and actuators as proposed in the original benchmark.

ASP with a simplified model for the settler
Because of the impossibility to reach any desired concentration profile along the settler with the given controls (Section 4.2.2),we are interested in a potentially controllable representation with a reduced number of identical self-dynamics.
We consider a 3-layer model of the settler for the concentrations of soluble matter: The layers are top, feed, and bottom ones.The variables related to soluble matter in the lth layer are xS(l) = (S S(l) I , S S(l) S , S S(l) O , S S(l) NO , S S(l) NH , S S(l) ND , S S(l) ALK ),  x ) ∈ E As , input-state edges (u nu , x nx ) ∈ E Bs , and state-output edges (x nu , y ny ) ∈ E Cs are dyed to match the corresponding entries in (A s , B s , C s ).Self-loops are omitted.
= {x 1 , . . ., x N x } ∪ {u 1 , . . ., u Nu } ∪ {y 1 , . . ., y Ny }; ) associated to this simplified model.We show that the system is now controllable, but it is still unobservable, in both a structural and in a classical sense.

Controllability:
The structural pair (A s , B s ) associates with subgraph The topology of G s c = (V s c , E s c ) indicates that pair (A s , B s ) is structurally controllable (Lemma 6).The accessibility condition is satisfied since all state vertices are reachable from a control vertex: Specifically, it is possible to see how the state nodes all are reachable through directed paths from either control vertex Q R or Q W .The dilation-free condition is satisfied through a perfect matching M of size |M| = N x formed by choosing every state vertex's self-loop.
The topology of G s c = (V s c , E s c ) also shows that pair (A s , B s ) is not strongly structurally controllable (Example 3.1): It is not possible to find a colouring for both graph G s c = (V s c , E s c ) and modified graph Gs c = (V s c , Ẽs c ) in which all state nodes x nx ∈ V A are coloured (Lemma 8).Being not strongly structurally controllable, there exist certain realisations of A s and B s for which the system is uncontrollable in a classical sense.

Observability: The structural pair (A s , C s ) associates with subgraph
structurally unobservable (Lemma 7).As there are still no paths from state vertices {S A(r) ALK } 5 r=1 , {S S(l) ALK } l= [1,6,10] and S S (10)   O to any of the output vertices, the accessibility condition is not satisfied.The contraction-free condition is still satisfied through a perfect matching M of choosing every state vertex's self-loop.The topology of G s o = (V s o , E s o ) also indicates that the structural pair (A s , C s ) is not strongly structurally observable, an expected result.

Controllability:
The controllability of pair (A SS s , B SS s ) can be verified using the PBH controllability test (Lemma 1), as an accurate computation of the controllability matrix Observability: The observability of (A SS s , C SS s ) can be verified using the PBH observability test (Lemma 2), again because an accurate computation of the observability matrix ] T is unfeasible.The test confirms that (A SS s , C SS s ) is not full-state observable, as ten distinct eigenvalues, including two real values with multiplicities equal to two, a real value with multiplicity seven, and two complex conjugated pairs with multiplicities equal to two, lead to rank-deficient matrices [λ i I − The twenty eigenvectors associated to such eigenvalues are depicted in Fig. 12.The nonzero entries of these eigenvectors relate to the same state variables associated to nonzero entries of the eigenvectors from the PBH test of (A SS , C SS ), Fig. 9.

Results
We conclude that for the activated sludge plant with a simplified model for the settler, it is possible to design a control u(•) that transfers the system to any desired state, in finite time.This is true for almost any possible realisation of (A s , B s ).It is not possible, however, to determine the initial state x(t 0 ), and thus neither intermediate states x(t), starting from a sequence of measurements y(t 0 ⇝ t f ).Thus, it is also not possible to design a full-state-observer based on the existing measurements, no matter what realisation of (A s , C s ) is used.
Being of structural nature, these conclusions are valid also in a classical sense, regardless of the linearisation.For the linearisation that is capable to transfers the system to a desired state x(t f ) in finite time.It is not possible, however, to determine x(t 0 ) and the intermediate states x(t), from y(t 0 ⇝ t f ).

BSM1's actuator and sensor configuration
We further consider the ASP with the minimal set of actuators and sensors proposed for the default low-level control of the BSM1 [5].The state-space model has state variables x(t) = ((x A (1) , . . ., x A (5) ), (x S (1) , . . ., x S (10) NO , X S (10)

5
, COD S (10) , N S( 10) The model consists of the original N x = 145 state variables and N w = 14 disturbances, but it has N ũ = 2 controllable inputs and N ỹ = 7 measurements.

Structural analysis
The structural matrices A d ∈ R Nx×Nx , B d ∈ R Nx×N ũ , and C d ∈ R N ỹ×Nx are obtained from the Jacobians and the digraph G d = (V d , E d ), Fig. 13, has the vertex and edge sets structurally controllable (Lemma 6).The accessibility condition is satisfied with all state vertices reachable from a control vertex, with all state vertexes reachable through directed paths starting from the control vertex Q A .The dilation-free condition is satisfied through a perfect matching M of size |M| = N x formed by choosing every state vertex's self-loop, thus leaving no vertex unmatched.A perfect matching such as M ensures the dilationfree condition and suggests that controls are only needed to ensure accessibility.This implies that the state can be controlled by manipulating only the internal recirculation and then relying on self-dynamics to reach any state in the state-space.Though formally correct, such control strategy is clearly nonviable.

The topology of
) in which all state vertices x nx ∈ V A are coloured (Lemma 8).Being strongly structurally uncontrollable, there exist certain realisations of A d and B d for which the system is not controllable in a classical sense.

Classical analysis
The numerical linearisation The nonzero entries of these eigenvectors relate to the same state variables as the nonzero entries of the eigenvectors previously observed on the PBH test of pair (A SS , C SS ), Fig. 9.This implies that pair (A SS , C SS ), with N y = 15 outputs, and pair

Results
We conclude that for the activated sludge plant given by Eq. ( 34) it is possible to design a control ũ(t) that transfers the plant to a desired state, in finite time, for almost any possible realisation of (A d , B d ).It is not possible, however, to determine the initial state x(0), and thus neither intermediate states x(t), starting from a measurement ỹ(t f ).Thus, it is also not possible to design a full-state-observer based on existing measurements, no

Concluding remarks
The dynamical properties of a model describing a common class of activated sludge plants are analysed, in a structural and a classical sense.We discuss the capabilities and limitations of the control and estimation tasks for activated sludge plants described by the Benchmark Simulated Model no. 1.The analysis is meant to provide a backbone for the design of efficient model-based controllers of activated sludge plants.
Our analyses show that activated sludge plants described by the BSM1 in Eq. ( 5) are full-state controllable but they are not observable, in a structural sense.Our results show that the plant is neither controllable nor it is observable in a strong structural sense.Formally, it is thus possible to determine a sequence of control actions that are capable to transfers the state of the system to any desired point in the state-space, from any initial state, in finite time, and for almost any realisation.However, it is also not possible to determine the initial state of the system, and thus neither its intermediate states, from a finite sequence of measurements.That is, it is also not possible to design a state-observer over the entire state-space.
For a linear approximation of the model which is commonly used in the literature, we studied the stability, controllability and observability, in a conventional sense.Under such realisation, we found that this system is stable, but it is not full-state controllable and it is not full-state observable.However, being stable, the system is both stabilisable and detectable.Energy-related metrics based on this linearisation show that the effort required to control and observe the system is very large.These efforts are among the lowest if biomass and particulate inert matter in the bio-reactors are directly controlled or measured.As actuation and sensing devices capable to directly control or measure these variables are practically unfeasible, these strategies are virtually inviable.Finally, a compound energy-related analysis shows that controlling and observing the BSM1 are high-demanding tasks even for a minimal and yet controllable and observable realisation.
The analysis of this class of activated sludge plants is extended on two alternative configurations of the system.We considered a reduced automation setup and a reduced-order model for the settler.For the first configuration, we found that the system is controllable but not observable in a structural sense, and that it is neither controllable nor observable in a conventional sense.For the second configuration, we found that this system is controllable but it is not observable in a structural sense.In this case, however, the system is controllable in a conventional sense, under the usual linearisation.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

:Definition 2 (
The control effort for (A, B) is quantified by the scalars: I. trace (W c (∞)): It is inversely related to the control effort averaged over all state-space directions; It is related to the control effort averaged over all directions in the state-space; III.log (det(W c (∞))): It is related to the volume of a N x - dimensional hyper-ellipsoid whose points are reachable with one unit or less of control energy; IV. λ c min (W c (∞)): It is inversely related to the control energy along the least controllable eigen-direction.Energy-Related Observability Metrics).Let W o (∞) be the solution of W o (∞)A T + AW o (∞) + C T C = 0.The output effort for (A, C ) is quantified by the scalars: I. trace (W o (∞)): It is inversely related to the output effort averaged over all state-space directions; II.trace ( W † o (∞) ) : It is related to the output effort averaged over all directions in the state-space; III.log (det(W o (∞))): It is related to the volume of a N x - dimensional hyper-ellipsoid whose points are observable with one unit or less of output energy;

Fig. 3 .
Fig. 3. Example 3.1: The colouring procedure applied to the subgraphs G c = (V c , E c ), left, andG o = (V o , E o ), right.As not all nodes x nx ∈ V A can be coloured , and C SS ∈ R Ny×Nx are obtained from the Jacobians evaluated at such equilibrium point A SS = (∂f /∂x)| SS , B SS = (∂f /∂u)| SS , and C SS = (∂g/∂x)| SS .The, now weighted, associated digraph G SS = (V SS , E SS ), shown at Fig.

Fig. 7 .
Fig. 7. Spectrum σ (A SS ): Eigenvalues λ i ∈ σ (A SS ) and associated eigenvectors ν i (λ i ) (left and right panels, respectively).The grid in the complex plane displays lines corresponding to constant damping factors (diagonal lines) and natural frequencies (vertical lines, in rad/days) for the associated modes.

,
not observable in a structural sense, as there is still no directed path from the state vertices {S A(r) ALK } 5 r=1 , {S S(l) ALK } 10 l=1 and {S S(l) O } 10 l=7 to any of the output vertices.Similarly, classical observability of (A SS , C SS ) can be verified only with the PBH test (Lemma 2), because of the limitations in the computation ofO = [C SS T A SS T C SS T • • • (A SS T ) Nx−1 C SS T ] T .The PBH test confirms that the pair (A SS , C SS ) is not observable.Ten distinct eigenvalues, including two real ones with multiplicities equal to two and twenty-eight, respectively, and five complex conjugated pairs with multiplicities equal to two lead to rank-deficient matrices [λ i I − A SS T C SS T ] T .The forty-three associated eigenvectors are shown in Fig. 9, bottom.As before, the non-zero entries of the eigenvectors associated to one of the real eigenvalues refer to state variables relative to effluent soluble matter.From the remaining fifteen eigenvectors, three have non-zero entries only at state variables {X A(r) I the other twelve have non-zero entries only at state variables {S A(r) I , S A(r) ALK } 5 r=1 and {S S(l) I , S S(l) ALK } 10 l=1

Fig. 10 .
Fig. 10.System (A SS , B SS , C SS ): On the left, the average controllability centrality C c (n x ), top, and average observability centrality C o (n x ), bottom, associated to each state variable x nx (n x = 1, . . ., N x ).On the right, the cumulative sum Λ(N) for the eigenvalues of the infinite-horizon controllability Gramian W c (∞), top, and the infinite-horizon observability Gramian W o (∞), bottom.

b
indicating the contribution from reactions.The model assumes equal influent flow-rates, {QA(r)

= 0 .
For the relevant entries in the Jacobian ∂f /∂x, we have

S
(l) c (c ∈ {I, S, O, NO, NH, ND, ALK }) in the lth layer of the settler are each represented in Eq. (5a) by first-order differential equations of the form ṠS(l) c = Q S(l) (S S(l ′ ) c − S S(l) c ), for l = 1, . . ., 10. Q S(l) denotes the influent flow-rate to the lth layer.The model assumes a same influent flow-rate for all upper layers, {Q S(l)

Fig. 11 .
Fig. 11.Network G s = (V s , E s ) (left) associated to (A s , B s , C s ) (right).State vertices x nx ∈ V As are in black, input vertices u nu ∈ V Bs are in blue, and output vertices y ny ∈ V Cs are in red.State-state edges (x nx , x n ′x ) ∈ E As , input-state edges (u nu , x nx ) ∈ E Bs , and state-output edges (x nu , y ny ) ∈ E Cs are dyed to match the corresponding entries in (A s , B s , C s ).Self-loops are omitted.

Fig. 12 .
Fig. 12. Pair (A SS s , C SS s ): Eigenvectors ν i (λ i ) with λ i ∈ σ (A SS s ) failing the PBH observability test (rank([λ i I − A SS T s C SS T s ] T ) < N x).

Fig. 13 .
Fig. 13.Network G d = (V d , E d ) associated to (A d , B d , C d ) (left and right panels, respectively).State vertices x nx ∈ V A are in black, input vertices u nu ∈ V B are in blue, and output vertices y ny ∈ V C are in red.State-state edges (x nx , x n ′ x ) ∈ E A , input-state edges (u nu , x nx ) ∈ E B , and state-output edges (x nx , y ny ) ∈ E C are dyed to match the corresponding entries in (A d , B d , C d ).The state self-loops have been omitted.
) is structurally unobservable (Lemma 7).Similarly to the original configuration, no paths starting from state vertices {S A(r) ALK } 5 r=1 , {S S(l) ALK } 10 l=1 and {S S(l) O } 10 l=7 reach any of the output vertices.The accessibility condition is therefore not satisfied.Conversely, the contraction-free condition is still satisfied through a perfect matching M of size |M| = N x formed by choosing every state vertex's self-loop.As expected, the topology of G d o = (V d o , E d o ) also indicates that (A d , C d ) is also not strongly structurally observable.
Fig.14, bottom.The nonzero entries of these eigenvectors relate to the same state variables as the nonzero entries of the eigenvectors pre- d ), with only N ỹ = 7 outputs, share the same observable subspace.Because A SS d is identical to the stable A SS , all eigenvalues failing the PBH test satisfy Re(λ i ) < 0 so that pair (A SS d , C SS d ) is detectable (Lemma 5).

Fig. 14 .
Fig. 14.System (A d , B SS d , C SS d ): Eigenvectors ν i (λ i associated with λ i ∈ σ (A SS d ) failing the PBH controllability test (rank([λ i I − A SS d B SS d ]) < N x ), top, and with λ i ∈ σ (A SS d )
the union of vertex set V A and vertex set V C = {y 1 , . . ., y Ny } of outputs.Edge set E o = E A ∪E C is the union of set E A and set E C = { (x nx , y ny ) | C ny,nx ̸ = 0 } There exists a nonsingular matrix P o ∈ R Nx×Nx , whose first N xo rows are the linearly independent rows of O, such that transformation [43]tched state vertices linked to distinct control or output vertices form a V −A −perfect matching.The dilation-free condition is guaranteed from the Hall's theorem[43].Example 3.1 illustrates the concepts presented in this section.
Example 3.1.Consider the structured dynamical system (A, B, C ) and associated network G = (V, E) in Fig.2.The pair (A, B) is not structurally controllable, as subgraph G c = (V c , E c ) does not satisfy the dilation-free condition: The subset S = {x 2 , x 3 } is larger than its in-neighbourhood set T (S) = {x 1 }.Similarly, the pair (A, C ) is not structurally observable, as subgraph State vertices x nx ∈ V A are in black, input vertices u nu ∈ V B in blue, and output vertices y ny ∈ V C in red.State-state (x nx , x n ′ x ) ∈ E A , input-state (u nu , x nx ) ∈ E B , and state-output edges (x nx , y ny ) ∈ E C are dyed to match the corresponding entries in (A, B, C ). Self-loops have been omitted.
M ensures the dilation-free condition and suggests that controls are only needed to ensure accessibility.The topology ofG c = (V c , E c ) shows that (A, B) is not strongly structurally controllable: It is not possible to find a colouring for both graph G c = (V c , E c ) and modified graph Gc = (V c , Ẽc ) in which all vertices x nx ∈ V A are coloured (Lemma 8).For G c = (V c , E c ),only the nodes associated to {S A(r) O } 5 r=1 and Fig. 4. Network G = (V, E) (left) for structured system (A, B, C ) (right).

Table 2
System (A co , B co , C co ): Energy-related metrics.trace(W ) trace(W † ) log(det(W )) N u = 13, N w = 14, N y = 15.The structural realisation of matricesA s ∈ R N x×N x , B s ∈ R N x×Nu , and C s ∈ R Ny×N x can be obtained as A s = ∂ f /∂ x, B s = ∂ f /∂u, and C s = ∂g/∂ x.The digraph G s = (V s , E s ) associated to (A s , B s , C s ),Fig.11, has vertex and edge sets RNy≥0 are unchanged: [5]≡ (x SS , u SS , w SS , y SS ) of Gernaey et al.[5]by evaluating the Jacobians:A SS s = ∂ f /∂ x| SS , B SS s = ∂ f /∂u| SS ,and C SS s = ∂g/∂ x| SS .We discuss the properties of the realisations (A s , B s , C s ) and