On unknown input observers designs for discrete-time LPV systems with bounded rates of parameter variation
Introduction
Linear parameter-varying (LPV) systems received significant attention from the control community in recent decades. One interesting feature of such class is the ability to represent nonlinear systems using a finite set of linear models on a convex hull [28]. In the LPV framework, the system dynamics depends on a known time-varying parameter vector and may be represented in a polytopic form, which is a useful approach to obtain design conditions using Lyapunov functions in terms of Linear Matrix Inequalities (LMIs). Therefore, many works regarding this system class have been proposed over the last years, in particular regarding observer designs [7], [13], [20], [30]. It is important to highlight that, besides the traditional state estimation problem, the design of observers for LPV systems may also consider the presence of unknown inputs (UIs). While these exogenous input signals cannot be appropriately measured, the UIs estimation and adequate handling plays an important role in many different applications [10], [24], [27]. One significant example is in fault detection and isolation (FDI) schemes, where the fault may be modeled as an unknown input to be properly monitored and isolated [12]. As the presence of a fault, or an unknown input, in a system may affect its normal functioning, FDI and fault-tolerant control designs that present tolerance to such effects are quite important [16], [17].
An interesting strategy to deal with UIs is the so called unknown input observer (UIO), which is able to achieve better results in real applications than conventional observers [3]. There are two main approaches used to design an UIO: estimation approaches and decoupling approaches. The first requires assumptions over the unknown input dynamics [15] and the second casts the estimated states as a structure that does not depend on the UI [8], [14]. While UIO works can be found since early 80s and 90s [8], [33], many works regarding this topic have been published in recent years for different systems classes, such as: descriptor systems [24], nonlinear systems [4], [22], Takagi-Sugeno systems [17] (whose analysis and design conditions are frequently related to LPV systems [21], [27], [28]) and linear parameter-varying systems for both continuous-[14], [21], [29] and discrete-time frameworks [10], [27], [31], [34]. For the continuous-time framework, we may cite interesting works that present decoupling unknown input observers such as [14], [21], [29]. These works presented existence and design conditions for proportional unknown input observers (P-UIOs) structures. Their formulation, however, considered that the system output matrix does not depend on a time-varying parameter and the observer syntheses were obtained using quadratic Lyapunov functions. Moreover, the information regarding the parameter rate of variation is not used in the design condition.
Concerning the discrete-time framework, we have different works that addressed unknown inputs for LPV systems using the decoupling approach. In [10], [17], [34], the UIOs synthesis conditions were obtained using conservative quadratic Lyapunov functions. On the other hand, [27], [31, Ch.5] and [11] made use of parameter-dependent Lyapunov (PDL) functions to generate the unknown input observers synthesis conditions. Most of these papers, however, do not cope with the possibility of both state and output matrices dependency on the time-varying parameters [10], [27], [31], [34]. Moreover, the possibility of bounded rates of parameter variation is not encompassed. It is well known that taking into account a priori information regarding the rate of variation bounds in the design conditions may lead to less conservative results than adopting only arbitrarily fast parameter variations [1]. Such conservative results occur due to the lack of restrictions on how fast the parameters may vary. Another issue regarding these works is that, while they make use of the decoupling approach to formulate the UIOs to estimate the discrete-time LPV system states and handle the UI effect over the them, they do not address the possibility of estimating the unknown inputs, which is a significant feature of the UIOs [8], [14].
It is worth mentioning that general designs for discrete-time LPV systems, including controller and filter designs, e.g. [5], [6], [13], [25], [26] [31, Ch.5], also do not cope with the possibility of both state and output matrices of the state-space representation subject to time-varying parameters. Such difficulty may occur due to the dependency of the LPV system state-space representation matrices on the time-varying parameters, which can lead to nonconvex problems [11]. The rate of the parameter variation is also not addressed in such works. And while some controller design works deal with some of these problems [9], [23], we highlight that it may be challenging to use dual state-feedback conditions to address observer designs. For example, stability conditions that make use of poly-quadratic approaches lack the duality property between controller and observer designs [26], which may prevent established techniques to obtain linearized design conditions to be appropriately applied.
In this context, this paper proposes a novel unknown input observer design for discrete-time LPV systems with bounded rates of parameter variation. The design encompasses the synthesis of two different UIO structures: the more traditional proportional UIO [3], [14], [31] and the proportional-integral unknown input observer (PI-UIO). Differently from the cited approaches [27], [31], [34], we define existence conditions for the UIOs structures and present a procedure to determine the observers parameterization, which are obtained from design conditions given in terms of LMIs. Moreover, the observers designs regard systems with relative degree one, satisfying classical rank conditions for the outputs and unknown inputs [11], [18], [21]. Parameter-dependent Lyapunov functions are used to address new synthesis conditions such that states and outputs matrices of the state space description may be affected by time-varying parameters. Furthermore, in order to encompass the possibility of bounded rates of parameter variation, we make use of a particular structure to transform the PDL function, and, consequently, the decision variables and the systems matrices. This structure is defined through a specific modeling of the time-varying parameter and its respective variation, which can be seen in similar works [9], [23]. Therefore, a prescribed bound on the parameter rate of variation can be used to obtain improved design conditions and better states and unknown input estimation performance. The effectiveness of the proposed stability and induced norm performance conditions are evaluated through numerical examples. As far as the authors know, this is the first time that such sufficient unknown input observer design conditions are proposed for discrete-time LPV systems with time-varying parameters in all state-space matrices and subject to bounded rates of variation. We highlight that, while most works regarding this last topic aim for stability margins and synthesis performance criteria [1], [9], here we are interested in the practical performance of the observer, investigating the direct improvement of the rate of variation bound over the signals estimations. We may summarize the main novelties and contributions of this paper as follow:
- (i)
The problem of unknown input observer design for discrete-time LPV systems with state-space matrices subject to time-varying parameters and bounded rates of variation is investigated;
- (ii)
A comparison between two different UIO structures and an investigation over the time-varying parameter rate of variation improvements in the observers performance are done;
- (iii)
New sufficient stability and induced norm performance conditions for the UIOs based on a poly-quadratic formulation and in terms of LMIs are presented, which is an established tool to solve control problems.
The remaining of this paper is structured as follows. In Section 2, we state the problem addressed by this work and some preliminary concepts, detailing the discrete-time LPV systems description and its time-varying parameter modeling. In Section 3 we present the main results of this paper. We first detail the P-UIO and the PI-UIO structure and after the respective existence conditions. Then, we provide sufficient stability and induced norm performance conditions to obtain both unknown input observers, including the presence of bounded rates of parameter variation. In Section 4, we address numerical examples to demonstrate the proposed conditions and formulations. Finally, in Section 5 we conclude the paper, summarizing the main ideas and most important results presented.
Throughout the paper, the following notation is used: is the n × n dimensional Euclidean space; AT and denotes, respectively, the transpose and the inverse of matrix A; denotes the pseudo-inverse of matrix A; represents a diagonal matrix with the specified elements; the symbol • represents the transpose elements in the respective symmetric positions; for the sake of brevity, sometimes we may address a parameter-dependent matrix A(ρ(k)) as A(ρ); A≻0 means that A is a positive-definite matrix.
Section snippets
Preliminaries and problem statement
The present paper is focused on the following discrete -time LPV system class:where and are, respectively, the states vector, the inputs, the measurable and exogenous outputs, the disturbance and the unknown inputs of the system. Matrices A( · ), B( · ), C( · ), E( · ) and F( · ) are fixed functions of the time-varying parameter ρ(k). The
UIO for discrete-time LPV systems
We now proceed to the main contributions of this paper, where we first address the unknown input observers existence conditions, then we proceed to the synthesis designs, in terms of LMIs, for both proportional and proportional-integral UIOs. In order to simplify the next discussions, let us assume from now on that in system (1). As presented in [10], [14], one may note that this condition is not restrictive, as the same approach may be directly applied for systems with u(k) ≠ 0.
Numerical results
In order to illustrate the effectiveness of the proposed unknown input observers designs, we now present four numerical examples. These examples address both the stability and the induced norm performance synthesis conditions. The proportional and the proportional-integral UIOs performances will be compared under for both states and unknown input estimation in a discrete-time LPV system. Moreover, we also present an analysis regarding the effect of considering a bound in the parameter rate
Conclusion
This paper concerned the design of a novel unknown input observer strategy for discrete-time LPV systems with bounded rates of parameter variation. The proposed synthesis conditions used a less conservative formulation to obtain proportional and a proportional-integral unknown input observers. Such conditions rely on the use of parameter-dependent Lyapunov functions to generate sufficient poly-quadratic stability and induced norm performance conditions for both UIOs structures. The observer
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.
Acknowledgments
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.
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