State Estimators for Uncertain Linear Systems with Different Disturbance/Noise Using Quadratic Boundedness

Abstract

This paper designs state estimators for uncertain linear systems with polytopic description, different state disturbance, and measurement noise. Necessary and sufficient stability conditions are derived followed with the upper bounding sequences on the estimation error. All the conditions can be expressed in the form of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the approach.

1. Introduction

In many control systems, the state variables are usually not accessible for direct connection. In this case, it is necessary to design a state estimator, so that its output will generate an estimate of those states. Generally speaking, there are two kinds of estimators for dynamic systems: observers and filters. The former is under the supposition of the perfect knowledge of system and measurement equations, and the latter can be applied to the system with disturbance. Many literatures focus on the design of state estimators for linear system, for example, a sliding mode and a disturbance detector for a discrete Kalman filter [1], the quantized measurement method [2], the least squares estimation for linear singular systems [3], stochastic disturbances and deterministic unknown inputs on linear time-invariant systems [4], and the bounded disturbances on a dynamic system [5].

The concept of quadratic boundedness (QB) is first defined for an uncertain nonlinear dynamical system [6], and then its necessary and sufficient conditions for a class of nominally linear systems [7] and a class of linear systems which contain norm-bounded uncertainties [8] are obtained. For discrete system, QB is applied mainly two regions: the receding horizon control (RHC) and the design of estimator. In RHC research, Ding utilizes QB to characterize the stability properties of the controlled system [9–13]. Alessandri et al. find the upper bounds on the norm of the estimation error by means of invariant sets, and these upper bounds can be expressed in terms of linear matrix inequalities [14]. Paper in [5] designs a filter by searching a suitable tradeoff between the transient and asymptotic behaviors of the estimation error. The designed filter is for the linear discrete systems with the identical state disturbance and measurement noise. For the discrete linear systems, the disturbance and noise are different in general. Nevertheless, little work has been done on the design of the state estimators for uncertain linear systems with different disturbance/noise. So how to design state estimators for uncertain linear systems with different state disturbance and measurement noise is important work.

The existing research work on state estimation usually constructs a filter for uncertain systems with bounded disturbance/noise, with no consideration of input or state constraint. Since the disturbance and noise are not assumed exactly identical, the stability station of the estimator is different from that in the paper [5]. For the above reasons, the situation becomes more complicated and the extension of the method is not straightforward.

This paper designs state estimators for uncertain linear systems with polytopic description. The problem is constructed in the form of linear matrix inequalities (LMIs). The organization of the paper is as follows. The earlier results are presented in the Section 2. The new robust estimator for uncertain linear systems with different disturbance/noise is designed in Section 3. A numerical simulation example is followed in Section 4. And some conclusions are given in the end.

Notations. For any vector 𝑥 and a positive-defined matrix 𝑄,ℰ_𝑄 is the ellipsoid which is defined as {𝑥∣𝑥^′𝑄𝑥≤1};𝑄^′ is the transpose of matrix 𝑄. ||𝑥|| is the Euclid norm of vector 𝑥. The symbol * induces a symmetric structure in LMIs.

2. Earlier Results

In this section, some results presented by Alessandri et al. [5, 14] are briefly introduced.

For a given discrete-time dynamic system,𝑥_𝑡+1=𝐴_𝑡𝑥_𝑡+𝐺_𝑡𝑤_𝑡,𝑡=0,1,…, (2.1) where 𝑥_𝑡∈𝑅^𝑛 is the state vector and 𝑤_𝑡∈ℰ_𝑄⊂𝑅^𝑝 is the noise vector. The definition of strictly quadratically bounded with a common Lyapunov matrix of a system and positively invariant set are defined by Alessandri et al. [5, 14], and the following theorem is proved.

Theorem 2.1 (see [14]). The following facts are equivalent:(i)System (2.1) is strictly quadratically bounded with a common Lyapunov matrix 𝑃>0 for all allowable 𝑤_𝑡∈ℰ_𝑄 and (𝐴_𝑡,𝐺_𝑡)∈𝜑,𝑡=0,1,…, where 𝜑 is a known bounded set.(ii)The ellipsoid ℰ_𝑃 is a positively invariant set for system (2.1) for all allowable 𝑤_𝑡∈ℰ_𝑄 and (𝐴_𝑡,𝐺_𝑡)∈𝜑,𝑡=0,1,….(iii) There exists 𝛼_𝑡∈(0,1) such that for any (𝐴_𝑡,𝐺_𝑡)∈𝜑,𝐴^′_𝑡𝑃𝐴_𝑡−𝑃+𝛼_𝑡𝑃𝐴^′_𝑡𝑃𝐺_𝑡∗𝐺^′_𝑡𝑃𝐺_𝑡−𝛼_𝑡𝑄≤0. (2.2)
For a discrete-time linear system with the same state disturbance and noise, 𝑥_𝑡+1=𝐴_𝑡𝑥_𝑡+𝐵_𝑡𝑤_𝑡,𝑦_𝑡=𝐶_𝑡𝑥_𝑡+𝐷_𝑡𝑤_𝑡,𝑧_𝑡=𝐿_𝑡𝑥_𝑡, (2.3) for 𝑡=0,1,…, where 𝑥_𝑡∈𝑅^𝑛,𝑦_𝑡∈𝑅^𝑚,𝑧_𝑡∈𝑅^𝑟 are the state vector, the measured output and the signal will be estimated, respectively, and 𝑤_𝑡∈ℰ_𝑄⊂𝑅^𝑝 is the disturbance/noise vector. 𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡,𝐿_𝑡 are the system matrixes with the proper dimensions. We consider the disturbance/noise to be unknown, and the system matrixes are supposed to be unknown and time varying but belonging to a polytopic set 𝒫, that is, (𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡,𝐿_𝑡)∈𝒫,𝑡=0,1,…, where 𝒫≜(𝐴,𝐵,𝐶,𝐷,𝐿)=^𝑁_𝑖=1𝜆_𝑖𝐴^(𝑖),𝐵^(𝑖),𝐶^(𝑖),𝐷^(𝑖),𝐿^(𝑖);^𝑁_𝑖=1𝜆_𝑖=1,𝜆_𝑖≥0,𝑖=1,2,…𝑁. (2.4) Here (𝐴^(𝑖),𝐵^(𝑖),𝐶^(𝑖),𝐷^(𝑖),𝐿^(𝑖))𝑖=1,2,…𝑁 are vertexes of the polytope 𝒫.

Definition 2.2 (see [5]). A sequence of vectors 𝜉_𝑡 is said to be exponentially bounded with constants 𝛽∈(0,1),𝑘₁≥0, and 𝑘₂>−𝑘₁ if 𝜉_𝑡≤𝑘₁+𝑘₂1−𝛽^𝑡,𝑡=0,1,…. (2.5)
It is easy to see that 𝛽 determines the convergence speed and 𝑘₁^1/2 represents an upper bound of the sequence 𝜉_𝑡.

Theorem 2.3 (see [5]). Consider two scalars 𝛼∈(0,1) and 𝛾>0. The following facts are equivalent.(i)There exist 𝐴,𝐵,𝐿, and 𝑃>0 such that the following conditions are satisfied for any (𝐴,𝐵,𝐶,𝐷,𝐿)∈𝒫: 𝐶𝑃⁻¹𝐶^′−𝛾²𝐴𝐼<0,_′𝑃𝐴−𝑃+𝛼𝑃𝐴^′𝑃𝐺∗𝐺^′𝑃𝐺−𝛼𝑄<0. (2.6)(ii)There exist 𝑉,𝑊,𝑋>0,𝑌>0, and 𝑍 such that 𝛾²()𝑋()𝐼𝐿−𝑊𝐿∗𝑋𝑋∗∗𝑌>0,1−𝛼1−𝛼𝑋0𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′+𝑉^′∗(1−𝛼)𝑌0𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′∗∗𝛼𝑄𝐵^′𝑋𝐷^′𝑍+𝐵^′𝑌∗∗∗𝑋𝑋∗∗∗∗𝑌>0, (2.7)
for (𝐴,𝐵,𝐶,𝐷,𝐿)=(𝐴^(𝑖),𝐵^(𝑖),𝐶^(𝑖),𝐷^(𝑖),𝐿^(𝑖)),𝑖=1,2,…𝑁.
Then, the cost 𝐽(𝑟,𝛼)≜𝜇𝛾−(1−𝜇)𝛼 can be minimized over 𝑉,𝑊,𝑋>0,𝑌>0,𝑍,𝛼∈(0,1) and 𝛾>0 under the constrains (2.7).

3. A Robust Estimator for Uncertain Linear Systems with Different Noises

Let us consider the discrete-time linear system with different disturbance/noise:𝑥_𝑡+1=𝐴_𝑡𝑥_𝑡+𝐵_𝑡𝑤_𝑡,𝑦_𝑡=𝐶_𝑡𝑥_𝑡+𝐷_𝑡𝑣_𝑡,𝑧_𝑡=𝐿_𝑡𝑥_𝑡, (3.1) where the matrixes and vectors are the same as (2.3) except 𝑤_𝑡∈ℰ_𝑄1,𝑣_𝑡∈ℰ_𝑄2, which are the vectors of the state disturbance and measurement noise, respectively.

To estimate the signal 𝑧_𝑡, the linear filter is introduced which has the following form:𝑥_𝑡+1=𝐴𝑥_𝑡+𝐵𝑦_𝑡,𝑧_𝑡=𝐿𝑥_𝑡, (3.2) for 𝑡=0,1,…, where 𝑥_𝑡∈𝑅^𝑛 is the filter state vector and 𝑧_𝑡∈𝑅^𝑟 is the estimation of the signal 𝑧_𝑡.

Define the estimation error 𝑒_𝑡, the augmented state vector, and the augmented disturbance/noise as𝑒_𝑡≜𝑧_𝑡−𝑧_𝑡,𝑥_𝑡≜𝑥_𝑡𝑥_𝑡,𝑤_𝑡=𝑤_𝑡𝑣_𝑡 (3.3) and the dynamic system associated with the estimation error𝑥_𝑡+1=𝐴0𝐵𝐶_𝑡𝐴𝐴𝑥_𝑡+𝐵_𝑡00𝐵𝐷_𝑡𝐵𝑤_𝑡,𝑒_𝑡=𝐿_𝑡−𝐿𝐶𝑥_𝑡, (3.4)(3.5) for 𝑡=0,1,….

The objective is to find an estimate 𝑧_𝑡 of the signal 𝑧_𝑡 such that the estimation error 𝑒_𝑡=𝑧_𝑡−𝑧_𝑡 is exponentially bounded for any 𝑥₀∈𝑅^𝑛,𝑤_𝑡∈ℰ_𝑄1,𝑣_𝑡∈ℰ_𝑄2, and (𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡,𝐿_𝑡)∈𝒫,𝑡=0,1,…. Then the following problem has to be solved.

Problem 1. Find matrices 𝐴,𝐵,𝐿 such that, for any 𝑥₀∈𝑅^𝑛,𝑤_𝑡∈ℰ_𝑄1,𝑣_𝑡∈ℰ_𝑄2 and (𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡,𝐿_𝑡)∈𝒫,𝑡=0,1,…; the estimation error 𝑒_𝑡 is exponentially bounded with constants 𝛽∈(0,1),𝑘₁≥0, and 𝑘₂>−𝑘₁.
In order to solve Problem 1, we now exploit the results on quadratic boundedness. More specifically, the following proposition holds.

Proposition 3.1. Suppose there exist matrices 𝐴,𝐵,𝐿, a symmetric matrix 𝑃>0, and two scalars 𝛾>0 and 𝛼∈(0,1) such that, for any (𝐴,𝐵,𝐶,𝐷,𝐿)∈𝒫, 𝐶𝑃⁻¹𝐶^′−𝛾²𝐴𝐼<0,_′𝑃𝐴−𝑃+𝛼𝑃𝐴^′𝑃𝐵∗𝐵^′𝑃𝐵−𝛼𝑅<0, (3.6)(3.7) where 𝑅=diag{𝑄₁,𝑄₂}. Then, for any 𝑥₀∈𝑅^𝑛,𝑤_𝑡∈ℰ_𝑄1,𝑣_𝑡∈ℰ_𝑄2, and (𝐴_𝑡,𝐵_𝑡,𝐶_𝑡,𝐷_𝑡,𝐿_𝑡)∈𝒫,𝑡=0,1,…, the estimation error is exponentially bounded with constants 𝛽=𝛼,𝑘₁=𝛾²,𝑘₂=𝛾²𝑥^′₀𝑃𝑥₀−1. (3.8) Hence the matrices 𝐴,𝐵, and 𝐿 are a solution of Problem 1.

Remark 3.2. We can see that condition (3.7) ensures system (3.1) is strictly bounded with a common Lyapunov matrix 𝑃, and from Corollary 2 of paper [5], it is clearly, 𝛾 is a bound. But its feasibility cannot be easily verified. The following theorem transposes them into the equivalent LMI conditions.

Theorem 3.3. Consider two scalars 𝛼∈(0,1) and 𝛾>0. The following facts are equivalent.(i)There exist 𝐴,𝐵,𝐿, and 𝑃>0 such that conditions (3.6) and (3.7) are satisfied for any (𝐴,𝐵,𝐶,𝐷,𝐿)∈𝒫.(ii)There exist 𝑉,𝑊,𝑋>0,𝑌>0, and 𝑍 such that𝛾²()𝑋()𝐼𝐿−𝑊𝐿∗𝑋𝑋∗∗𝑌>0,1−𝛼1−𝛼𝑋00𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′+𝑉^′∗(1−𝛼)𝑌00𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′∗∗𝛼𝑄₁0𝐵^′𝑋𝐵^′𝑌∗∗0𝛼𝑄₂0𝐷^′𝑍∗∗∗∗𝑋𝑋∗∗∗∗∗𝑌>0. (3.9)(3.10)

Proof. The proof of (3.6)⇔(3.9) is the same as Theorem 2 of reference [5]; here it is omitted for brevity.
(3.7)⇒(3.10) suppose the condition (3.7) is satisfied. The matrix 𝑃 and matrix 𝑃⁻¹ are partitioned as 𝑃𝑃=¹¹𝑃₁₂𝑃₂₁𝑃₂₂,𝑃⁻¹=𝑆₁₁𝑆𝑆₂₁𝑆, (3.11) with 𝑃₁₁∈𝑅^𝑛×𝑛 and 𝑆₁₁∈𝑅^𝑛×𝑛. Clearly 𝑃𝑃⁻¹=𝑃⁻¹𝑃=𝐼, so we have 𝑆₁₂𝑃^′₁₂=𝐼−𝑆₁₁𝑃₁₁,𝑆₁₁𝑃₁₂+𝑆₁₂𝑃₂₂=0. (3.12) Moreover, (3.7) is strict inequality, and we can assume, without loss of generality, that 𝐼−𝑆₁₁𝑃₁₁ is invertible [15]. Hence 𝑆₁₂ and 𝑃₁₂ are invertible. Using the Schur complement, we can rewrite (3.7) as (1−𝛼)𝑃0𝐴^′𝑃0𝛼𝑅𝐵^′𝑃∗∗𝑃>0. (3.13) Define 𝐻𝑇≜^′000𝐼000𝐻^′, (3.14) where 𝐻^′=_𝑆^𝐼𝐼′_12𝑆⁻¹₁₁₀, so 𝑇^′(1−𝛼)𝑃0𝐴^′𝑃0𝛼𝑅𝐵^′𝑃()𝐴∗∗𝑃𝑇=𝐻000𝐼000𝐻1−𝛼𝑃0^′𝑃0𝛼𝑅𝐵^′𝑃𝐻∗∗𝑃^′000𝐼000𝐻^′=(1−𝛼)𝐻𝑃0𝐻𝐴^′𝑃0𝛼𝑅𝐵^′𝑃𝐻𝐻𝑃𝐴𝐻𝑃𝐵𝐻𝑃^′000𝐼000𝐻^′=(1−𝛼)𝐻𝑃𝐻^′0𝐻𝐴^′𝑃𝐻^′0𝛼𝑅𝐵^′𝑃𝐻^′𝐻𝑃𝐴𝐻^′𝐻𝑃𝐵𝐻𝑃𝐻^′,𝐻𝑃𝐻^′=𝐼𝑆⁻¹₁₁𝑆₁₂𝑃𝐼0¹¹𝑃₁₂𝑃₂₁𝑃₂₂𝑆𝐼𝐼_′12𝑆⁻¹₁₁0=𝑃₁₁+𝑆⁻¹₁₁𝑆₁₂𝑃₂₁+𝑃₁₂+𝑆⁻¹₁₁𝑆₁₂𝑃₂₂𝑆^′₁₂𝑆⁻¹₁₁𝑃₁₁+𝑆⁻¹₁₁𝑆₁₂𝑃₂₁𝑃₁₁+𝑃₁₂𝑆^′₁₂𝑆⁻¹₁₁𝑃₁₁. (3.15) Using condition (3.12), we can get 𝐻𝑃𝐻^′=𝑆⁻¹₁₁𝑆⁻¹₁₁𝑆⁻¹₁₁𝑃₁₁=𝑋𝑋𝑋𝑌, (3.16) where 𝑋≜𝑆⁻¹₁₁,𝑌≜𝑃₁₁: 𝐻𝐴^′𝑃𝐻^′=𝐼𝑆⁻¹₁₁𝑆₁₂𝐴𝐼0^′𝐶^′𝐵^′0𝐴^′𝑃₁₁𝑃₁₂𝑃₂₁𝑃₂₂𝑆𝐼𝐼_′12𝑆⁻¹₁₁0=𝐴^′𝐶^′𝐵^′+𝑆⁻¹₁₁𝑆₁₂𝐴^′𝐴^′𝐶^′𝐵^′𝑃₁₁+𝑃₁₂𝑆^′₁₂𝑆⁻¹₁₁𝑃₁₁𝑃₂₁+𝑃₂₂𝑆^′₁₂𝑆⁻¹₁₁𝑃₂₁=𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′+𝑉^′𝐴^′𝑋𝐴^′𝑌+𝐶^′𝑍^′. (3.17) Here we define 𝑉≜𝑃₁₂𝐴𝑆^′₁₂𝑆⁻¹₁₁,𝑍≜𝑃₁₂𝐵: 𝐵^′𝑃𝐻^′=𝐵^′00𝐷^′𝐵^′𝑃₁₁𝑃₁₂𝑃₂₁𝑃₂₂𝑆𝐼𝐼_′12𝑆⁻₁₁0=𝐵^′𝑃₁₁𝐵^′𝑃₁₂𝐷^′𝐵^′𝑃₂₁𝐷′𝐵′𝑃₂₂𝑆𝐼𝐼_′12𝑆⁻₁₁0=𝐵^′𝑃₁₁+𝐵^′𝑃₁₂𝑆^′₁₂𝑆⁻¹₁₁𝐵^′𝑃₁₁𝐷^′𝐵^′𝑃₂₁+𝐷^′𝐵^′𝑃₂₂𝑆^′₁₂𝑆⁻¹₁₁𝐷^′𝐵^′𝑃₂₁=𝐵^′𝑋𝐵^′𝑌0𝐷^′𝑍^′. (3.18) So we can get condition (3.10).
(3.10)⇒(3.7) suppose there exist 𝑉,𝑋>0,𝑌>0 and 𝑍 satisfied condition (3.4), as condition (3.4) holds at every vertex (𝐴^(𝑖),𝐵^(𝑖),𝐶^(𝑖),𝐷^(𝑖),𝐿^(𝑖)) of the polytope 𝒫, and it also holds for every system matrice (𝐴,𝐵,𝐶,𝐷,𝐿)∈𝒫.
We can obtain ^𝑋𝑋_𝑋𝑌>0 from condition (3.10), and based on the Schur complement, the result of 𝐼−𝑋⁻¹𝑌<0 can be deduced. Then there exist two square invertible matrixes 𝑀 and 𝑁 such that 𝑀^′𝑁^′=𝐼−𝑋⁻¹𝑌. Choosing 𝑃₁₁=𝑌,𝑆₁₁=𝑋⁻¹,𝑆^′₁₂=𝑀 and 𝑃₁₂=𝑁, condition (3.7) can be obtained by premultiplying and postmultiplying condition (3.10) by (𝑇^′)⁻¹and 𝑇⁻¹. If we apply the change of variable 𝐴=𝑁⁻¹𝑉𝑋⁻¹𝑀⁻¹,𝐵=𝑁⁻¹𝑍,𝐿=𝑊𝑋⁻¹𝑀⁻¹𝑁,𝑃=𝑌𝑁_′−𝑁^′𝑋⁻¹𝑀⁻¹. (3.19) so, we can get the linear filter (3.2).

Remark 3.4. In paper [5], the estimators for uncertain systems propose that the state disturbance and measurement noise are identical with the time. Ordinarily, they are different in practice, so the result in our paper is the general case.

4. A Numerical Example

Let us consider the system [12] in the form of (3.1) with𝐴_𝑡=0.3850.330.21+𝑎_𝑡0.59,𝐵_𝑡=,𝐶0.30.3_𝑡=0.20.2+𝑎_𝑡,𝐷_𝑡=0.3,𝐿_𝑡=[10], (4.1) where 𝑎_𝑡 is an uncertain parameter satisfying |𝑎_𝑡|≤0.11. Suppose the state disturbance satisfies the condition |𝑤_𝑡|≤0.25 and the measurement noise satisfies |𝑣_𝑡|≤0.1 (i.e., 𝑄₁=16,𝑄₂=100).

As this kind of uncertainty is of the polytopic type described in Section 3, the proposed method is used to obtain a linear filter. In the context, we will refer to this filter as the “filter with different disturbance/noise” (FDDN). Choose two sets of initial states: 𝑥₀={[86]^𝑇,[−117]^𝑇},𝑥₀={[107]^𝑇,[−9−6]^𝑇}. The resulting state trajectories are shown in Figure 1 by the marked solid line, followed with the estimator state trajectories shown by the marked dotted line. Figure 1 indicates that the designed estimator can track the systems’ states effectively.

Figure 1 

The state and estimator trajectories.

The performance of the filter can be further studied by using an average measure of the estimation error, such as the expected quadratic estimation error. Comparison was then made between the FDDN and the “filter with identity disturbance/noise” (FIDN), to evaluate the performance achieved when the different disturbance/noise is taken into account in the synthesis of the filter. At each time instant, the uncertain parameters were chosen to be within, with equal probability, one of their limit values. We assumed 𝑥₀,𝑤_𝑡 and 𝑣_𝑡,𝑡=0,1,…, to be independent random vectors, and the initial states are in the ball of radius 10 (i.e., ||𝑥₀||≤10). Figure 2 shows the plots of the “root mean square error” (RMSE), computed over 10³ randomly chosen simulations, for the considered filters. The performance of the FDDN turns out to be better from the point of view of the asymptotic behavior when there is large difference between the disturbance and noise.

Figure 2 

Plots of the RSME for the considered filters.

5. Conclusions

The main contribution of this work is the method of constructing an estimator for the uncertain system with the different state disturbance and measurement noise. The stability of the estimator is analyzed using quadratic boundness. Moreover, the estimator can be got by LMI procedures.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grant 60974051, Natural Science Foundation of Beijing under Grant 4122071, Fundamental Research Funds for the Central Universities under Grant 12MS143, and the Construction Project from Beijing Municipal Commission of Education. The authors would like to thank the reviewers for their pertinent comments.