Functional observability and the design of minimum order linear functional observers

The design of a minimum-order linear functional observer for linear time-invariant systems has been an open problem for over four decades. This technical note provides a solution to this problem. The technical note also introduces the concept of Functional Observability/Detectability and shows that the well-known concept of Observability/Detectability is a special case of Functional Observability/Detectability.

Functional observability

A simple theorem for Functional Observability is presented considering the observable and unobservable states of a system based on Kalman decomposition. The proposed theorem is also consistent with two other theorems on Functional Observability which was based on eigen decomposition [6]. The paper also reports a new definition for Functional Observability which is consistent with previously reported definitions and theorems [4], [5], [6].

Numerical implementation of a functional observability algorithm : a singular value decomposition approach

The paper outlines a numerical algorithm to implement the concept of Functional Observability introduced in [6] based on a Singular Value Decomposition approach. The key feature of this algorithm is in outputting a minimum number of additional linear functions of the state vector when the system is Functional Observable, these additional functions are required to design the smallest possible order functional observer as stated in [6].

Existence conditions for functional observability from an eigenspace perspective

Two theorems on conditions for nonexistence and for existence, of built functional observers, from an eigenspace perspective are presented and proved. One more theorem on Functional Observability in terms of constructed products of matrices A,C and L0 is also presented. This theorem provides an easy way to check Functional Observability before proceeding with the design of functional observers. The existence and the nonexistence theorems are used to unify previously reported theorems on Functional Observability by showing their equivalence. The connection between the concept of Functional Observability and the well known concept of State Observability is also presented.

On unknown-input functional observability of linear systems

Finding the least possible order of a stable Unknown-Input Functional Observer (UIFO) has always been a challenge in observer design theory. A practical recursive algorithm is proposed in this technical note to design a minimal multi-functional observer for multi-input multi-output (MIMO) linear time-invariant (LTI) systems with unknown-inputs. The concept of unknown-input functional observability is introduced,and it is used as a certificate of the convergence of our algorithm. The proposed procedure looks for a number of additional auxiliary functions to be augmented to the original functions desired for reconstruction. The resulting UIFO is proper, and minimal (of minimum possible order). Moreover, the algorithm does not need the system to be unknown-input observable. A numerical example shows the procedure as well as the effectiveness of the proposed algorithm.

Design of reduced-order scalar functional observers

In the existing literature, the existence conditions and design procedures for scalar functional observers are available for the cases where the observers’ order p is either p=1 or p=(v-1), where v is the observability index of the matrix pair (C,A). Therefore, if an observer with an order p=1 does not exist, the other available option is to use a higher order observer with p=(v-1). This paper shows that there exists another option that can be used to design scalar linear functional observers of the order lower than the well-known upper bound (v-1). The paper provides the existence conditions and a design procedure for scalar functional observers of order 0≤ p ≤2, and demonstrates the presented results with a numerical example.

Existence and design of reduced-order scalar functional observers

In the existing literature, the existence conditions and design procedures for scalar functional observers are available for the cases where the observers’ order p is either p=1 or p=(v-1), where v is the observability index of the matrix pair (C,A). Therefore, if an observer with an order p=1 does not exist, the other option is to use a high-order observer with p=(v-1). This paper provides the existence conditions and a design procedure for scalar functional observers of order 0≤p≤2, and demonstrates the presented results with a numerical example. where K, M, E, H and G are constant matrices to be designed. The problem of observing a scalar functional or multi functionals (z(t)∈Rk , k>1) of the state vector has been the subject of numerous papers, and different algorithms have been proposed (see, [1]-[13] and references therein). There are also papers that deal with the order reduction of multi-dimensional functional observers [9,10,12,13]. For scalar functional observers, a well-known Luenberger’s classic result [1] provides an upper bound on the order with p=(v-1). It is interesting to note here that, except for a recent result of Darouach [12,13], little results have been reported on the order reduction for scalar functional observers.

Auditor independence: regulation, oversight and inspection

Much of what auditors do is unobservable. Indeed, what goes on in an audit has been described as ‘secret audit business’. Audits in this context are of financial reports and those financial reports are the representations of the management of those companies, not the auditors. The audits of financial reports are of value in that they provide a competent and independent (of auditee management) attestation of the validity of those management representations. This attestation lowers the ‘information risk’ for the users of these financial reports. There has been a marked increase in activity to regulate matters relating to independence. The proposals outlined in CLERP 9 are one example of this. The requirements in the United States under the Sarbanes-Oxley Act are a further example.

Audit firms operate in a highly regulated yet highly competitive market. Evidence exists to suggest that audit firms are active competitors in respect of audit pricing and competency, including specialist industry expertise. Until recently, there has been little or no observable evidence that audit firms compete in respect of independence. The issues as they relate to audit independence are complex. One issue is that threats to independence are frequently subtle and difficult to observe and measure. Hence, controlling the decisions that relate to them cannot rely solely on regulation which itself inevitably relies on crude definitions and imprecise measures. Additionally, further regulation may not achieve the desired end without other processes being but in place in tandem.

This paper argues that:

1. auditors of certain classes of companies (in particular, those that are publicly traded) should be provided with incentives or requirements to have observable processes on independence
2. the means of observability should be in the form of an inspection and review process focussing on issues critical to the audit, such as independence
3. expert persons not having a current or past financial interest in the firm or in the commercial outcomes of the review should be used in the inspection and review process
4. the review process should have wide-ranging powers of inspection to examine the policies, processes, structures and ‘culture’ of audit firms
5. the report of the inspection and review should be made public, unedited and in full, and in a timely fashion. The primary objectives of this proposal are to (1) make more transparent to the market for information the characteristics of the audit firms and their process to ensure audit independence, and (2) provide a rigorous oversight of independence decision-making by persons who have no commercial interest in the outcome of the decision.

Comparing towed and baited underwater video techniques for assessing temperate marine fishes

Accurate estimates of fish species occurrence are important to any species’ assessments and distribution model. With increasing emphasis on nondestructive sampling, underwater video techniques are commonly used without a thorough understanding of their advantages and disadvantages. This study compared data collected from baited remote underwater stereo-video systems (stereo BRUVS) and towed-video systems to determine; (1) the differences between these video techniques in terms of fish assemblages, functional groups (i.e. pelagic carnivore, epibenthic carnivore/omnivore or herbivore) and observability (i.e. conspicuous or cryptic), and (2) what impact do these two techniques have on the interpretation of spatially-explicit, predictive models. We found stereo BRUVS and towedvideo techniques recorded very different assemblages, functional groups and observability categories across structurally complex benthic biological habitats (i.e. macroalgae dominated habitats). However, as the habitat complexity became less (e.g. seagrass and areas with no visible macro-biota) both techniques appeared to provide similar fish assemblage information. We also found considerable differences in the predicted extents of habitat suitability between the two video techniques.

A fault detection technique for dynamical systems

This paper shows how a functional observer can be utilized to detect faults in LTI MIMO systems. The fault detection technique is designed so that the functional observer based fault indicator asymptotically converges to a fault indicator that can be derived based on the nominal system. The asymptotic value of the proposed fault indicator is not dependent on the functional observer parameters; moreover, by choosing appropriate functional observer parameters the convergence rate of the fault indicator can be altered. Observability of the system is not a requirement for the design of the fault detection scheme.

A system decomposition approach to the design of functional observers

This paper reports a system decomposition that allows the construction of a minimum-order functional observer using a state observer design approach. The system decomposition translates the functional observer design problem to that of a state observer for a smaller decomposed subsystem. Functional observability indices are introduced, and a closed-form expression for the minimum order required for a functional observer is derived in terms of those functional observability indices. © 2014 Taylor & Francis.

A new algorithm to design minimal multi-functional observers for linear systems

Designing minimum possible order (minimal) observers for Multi-Input Multi-Output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for Linear Time-Invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

Design of state observers for interconnected time-delay systems via a coordinate transformation approach

This paper considers the design of state observers for interconnected time-delay systems using a coordinate transformation method. Through such a transformation, the system that has interconnection and state delays is metamorphosed into a new system that injects time-delay information into its input and output terms, before reintroducing them back into the latter system, effectively coupling the delay terms into the IO injection terms and eliminating the delay values from the state variables. Next, full-order and reduced-order observers are designed based on the transformed system. Finally, the observed states of the transformed system that correspond to the original system is used to deduce the estimates of the original system. A numerical example is provided of an interconnected time-delay system.