Robust and Nonfragile 𝐻_∞ Kalman-Type Filter Design for Parameter-Uncertain Time-Delay Systems: PLMI Approach

Abstract

This paper describes the synthesis of a robust and nonfragile 𝐻_∞ Kalman-type filter design for a class of time-delay systems with polytopic uncertainties, filter-gain variations, and disturbances. We present the sufficient condition for filter existence and the method for designing a robust nonfragile 𝐻_∞ filter by using LMIs (Linear Matrix Inequalities) technique. Because the obtained sufficient condition can be represented as PLMIs (Parameterized Linear Matrix Inequalities), which can generate infinite LMIs, we use a relaxation technique to find finite solutions for a robust nonfragile 𝐻_∞ filter. We show that the proposed filter can minimize estimation error in terms of parameter uncertainties, filter-fragility, and disturbances.

1. Introduction

The Kalman filtering approach is a popular state estimation method for a class of systems presented in linear models and thus has been widely applied in previous research on control engineering and target tracking, among others [1–3]. However, this approach does not provide reliable performance if there exist system uncertainties due to no full knowledge of the system model. So, the robust Kalman filtering for uncertain system has been studied on various issues [4–9]. Another serious issue in filter design is fragility. In practice, a filter designed by robust theory can be fragile because of the uncertainty associated with the implemented filter itself (e.g., limited word lengths and round-off error, etc.) [10]. To solve this problem in [11–13], nonfragile design methods have been investigated using LMIs and Riccati equation. Also, in practical applications, the noise effects must be considered to reduce estimation error and improve filter performance. To deal with this problem, 𝐻_∞ filtering approach was proposed [13–22].

The aforementioned research has motivated our research to construct the filter for practical application with better performance against uncertainties, various disturbances, and time-delay. This paper examines additional methods to design a robust and nonfragile Kalman-type filter for a class of uncertain systems with disturbances and time-delay by taking the PLMIs (parameterized linear matrix inequalities) approach. Here, we represent the uncertain linear system by polytopic uncertainties, filter-gain variation, time-delay, and disturbances. Next, we rewrite the sufficient condition for filter existence as PLMIs and use a relaxation technique to find finite feasible solutions. Finally, a numerical example is provided to verify the performance of the proposed filter design method. The main contributions of this article are summarized as follows.(1)Since uncertainties in the system and filter are designed as structured uncertainties, it is possible to find filter-gain region which guarantees filter performance using PLMI and relaxation technique with less computational efforts.(2)The design method included all of factor increasing estimation error such as system uncertainties, filter-gain variations, disturbances, and time-delay. Therefore, it is more suitable for practical filter applications.

2. Problem Statement

Consider the following linear system with parameter uncertainties and time-varying delay: 𝑥̇(𝑡)=𝐴(𝑡,𝛼)𝑥(𝑡)+𝐴_𝑑(𝑡,𝛼)𝑥(𝑡−𝜏(𝑡))+𝐵_𝑤𝑤(𝑡),𝑦(𝑡)=𝐶(𝑡,𝛼)𝑥(𝑡), (1) where 𝑥(𝑡)∈ℜ^𝑛 is the state, 𝑦(𝑡)∈ℜ^𝑚 is the measured output, 𝜏(𝑡)∈ℜ is the time-varying delay, 𝑤(𝑡) is the disturbance, and 𝐵_𝑤 is constant matrix with appropriate dimension. Here, we define the parameter uncertainties as follows: 𝐴(𝑡,𝛼)=𝐴₀+^𝐿_𝑖=1𝛼_𝑖(𝑡)𝐴_𝑖,𝐴_𝑑(𝑡,𝛼)=𝐴_𝑑0+^𝐿_𝑖=1𝛼_𝑖(𝑡)𝐴_𝑑𝑖,𝐶(𝑡,𝛼)=𝐶₀+^𝐿_𝑖=1𝛼_𝑖(𝑡)𝐶_𝑖,𝛼(𝑡)≥0,^𝐿_𝑖=1𝛼_𝑖(𝑡)=1,𝑖=1,2,…,𝐿. (2)

Here, the value of 𝐿 represents the number of parameter uncertainties for system matrices, and the time-varying delay satisfies 0≤𝜏(𝑡)≤∞,𝜏̇(𝑡)≤𝑑<1. (3)

We can express a full-order filter for system (1) as 𝑥̇_𝑓(𝑡)=𝐴_𝑓𝑥_𝑓(𝑡)+𝐾𝑡,𝛽𝑦(𝑡)−𝐶_𝑓𝑥_𝑓(𝑡,𝑦_𝑓)(𝑡)=𝐶_𝑓𝑥_𝑓(𝑡), (4) where 𝑥_𝑓(𝑡) is the filter state, 𝑦_𝑓(𝑡) is the estimated output, and 𝐴_𝑓 and 𝐶_𝑓 are filter matrices. In addition, we can define the estimation error of filter as 𝑒(𝑡)=𝑥(𝑡)−𝑥_𝑓(𝑡) (5) and calculate next state of estimation error 𝑒(̇𝑡) from (5) as 𝑒̇(𝑡)=𝑥̇(𝑡)−𝑥̇_𝑓(𝑡)=𝐴(𝑡,𝛼)𝑥(𝑡)+𝐴_𝑑(𝑡,𝛼)𝑥(𝑡−𝜏(𝑡))+𝐵_𝑤𝑤(𝑡)−𝐴_𝑓𝑥_𝑓(𝑡)+𝐾𝑡,𝛽𝑦(𝑡)−𝐶_𝑓𝑥_𝑓(𝑡)=𝐴_𝑘𝑡,𝛼,𝛽𝑥(𝑡)+𝐴_𝑑(𝑡,𝛼)𝑥(𝑡−𝜏(𝑡))−𝐴_𝑘𝑓𝑡,𝛽𝑥_𝑓(𝑡)+𝐵_𝑤𝑤(𝑡), (6) where 𝐴_𝑘()𝐶(),𝐴𝑡,𝛼,𝛽=𝐴𝑡,𝛼−𝐾𝑡,𝛽𝑡,𝛼_𝑘𝑓𝑡,𝛽=𝐴_𝑓−𝐾𝑡,𝛽𝐶_𝑓. (7)

Here, we consider a robust and nonfragile filter for the system (4) as 𝐾𝑡,𝛽=𝐾₀+^𝐿_𝑗=1𝐵_𝑗(𝑡)𝐾_𝑗,𝐾_𝑗=𝐾_𝑗−𝐾₀,𝐵_𝑗(𝑡)≥0,^𝐿_𝑗=1𝐵_𝑗(𝑡)=1,𝑗=1,2,…,𝐿, (8) where 𝐾(𝑡,𝛽) is the region of filter-gain variations, 𝐾_𝑗 is the vertex of the polytope, and 𝐾_𝑗 is the measure of nonfragility against filter-gain variation.

3. Robust and Nonfragile 𝐻_∞ Filter Design

In this section, we consider system performance and obtain the filter-gain 𝐾(𝑡,𝛽). We can solve the parameters by using PLMI method because the system (1) and the filter (4) are transformed into matrix inequalities through the Lyapunov method.

Theorem 1. The estimation error (5) is asymptotically minimizable with 𝐻_∞ norm bound 𝛾 with the filter-gain (8) if there are positive definite matrices 𝑃 and 𝑅 such that

𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃+𝑃𝐴_𝑘𝑡,𝛼,𝛽+𝑅𝑃𝐴_𝑑(𝑡,𝛼)−𝑃𝐴_𝑘𝑓𝑡,𝛽−𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃𝑃𝐵_𝑤𝐶^𝑇_𝑓∗−(1−𝑑)𝑅−𝐴^𝑇_𝑑(𝑡,𝛼)𝑃00∗∗𝐴^𝑇_𝑘𝑓𝑡,𝛽+𝑃𝐴_𝑘𝑓𝑡,𝛽−𝑃𝐵_𝑤0∗∗∗−𝛾²𝐼0∗∗∗∗−𝐼<0. (9)

Proof. We define a Lyapunov candidate functional as 𝑉(𝑒,𝑥,𝑡)=𝑒^𝑇(𝑡)𝑃𝑒(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)𝑅𝑥(𝑠)𝑑𝑠. (10)
When assuming the zero input, we have 𝑉̇(𝑒,𝑥,𝑡)∶=𝑒^𝑇̇(𝑡)𝑃𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃𝑒̇(𝑡)+𝑥^𝑇(𝑡)𝑅𝑥(𝑡)−(1−𝑑)×𝑥^𝑇(𝑡−𝜏(𝑡))𝑅𝑥(𝑡−𝜏(𝑡)). (11)
In the next place, assume the zero initial condition and introduce 𝐽=^∞₀𝑦^𝑇(𝑡)𝑦(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡. (12)
Noting 𝐽≤^∞₀𝑦^𝑇(𝑡)𝑦(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)+𝑉̇(𝑒,𝑥,𝑡)𝑑𝑡 (13) and further substituting (11) to (13).
Let 𝜉(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇_𝑓(𝑡)𝑤^𝑇(𝑡)]^𝑇, then 𝐽≤^∞₀𝜉^𝑇(𝑡)𝑍𝜉(𝑡), (14) where 𝑍 is defined as 𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃+𝑃𝐴_𝑘𝑡,𝛼,𝛽+𝑅𝑃𝐴_𝑑(𝑡,𝛼)−𝑃𝐴_𝑘𝑓𝑡,𝛽−𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃−𝑃𝐵_𝑤∗−(1−𝑑)𝑅−𝐴^𝑇_𝑑(𝑡,𝛼)𝑃0∗∗𝐴^𝑇_𝑘𝑓𝑡,𝛽+𝑃𝐴_𝑘𝑓𝑡,𝛽+𝐶^𝑇_𝑓𝐶_𝑓𝑃𝐵_𝑤∗∗∗−𝛾²𝐼. (15)
Therefore, when 𝑍<0, 𝑡≥0, the estimation error is minimized with 𝐻_∞ norm bound 𝛾. The inequality (15) is transformed into (9) using Schur complements [23].

We present the proposed condition for a robust and nonfragile filter by PLMIs, which involve an infinite number of LMIs, and thus transform PLMIs into a finite number of LMIs by using, relaxation technique.

Theorem 2. The estimation error from linear parameter uncertain system (1) and filter (4) with time-delay is asymptotically minimizable with disturbance attenuation 𝛾 and nonfragility whenever there exist matrices 𝑀₀, 𝑀_𝑗(𝑗=1,2,…,𝐿), positive definite matrices 𝑃, 𝑅, and positive constant 𝜌 such that 𝑥^𝑇𝑀₀(𝑧)𝑥+^𝐿_𝑖=1𝛼_𝑖𝑥^𝑇𝑀_𝑖(𝑧)𝑥+^𝐿_𝑗=1𝛽_𝑗𝑥^𝑇𝑁_𝑗(𝑧)𝑥+_{1≤𝑖≤𝑗≤𝐿}max−𝑥^𝑇𝑀_𝑖𝑗(𝑧)𝑥×𝛼_𝑖²+𝛽_𝑗²2−𝛼_𝑖+𝛽_𝑗2,𝑥+0.125_𝑇𝑀_𝑖𝑗(𝑧)𝛼𝑥⋅^𝑖2+𝛽_𝑗²2∀‖𝑥‖𝛼<0,=1,_𝑖,𝛽_𝑗∈vertΓ, (16) where 𝑧 is the decision variable, 𝑀_𝑖(𝑧), 𝑁_𝑗(𝑧), 𝑀_𝑖𝑗(𝑧), are affine symmetric matrix-valued functions of 𝑧. Here, (16) holds for 𝑧, 𝑀₀(𝑧), 𝑀_𝑖(𝑧), 𝑁_𝑗(𝑧), and 𝑀_𝑖𝑗(𝑧) defined below. 𝑀₀(𝑧)=𝜙₁𝑃𝐴_𝑑0𝜙₂𝑃𝐵_𝑤𝐶^𝑇₀∗−(1−𝑑)𝑅−𝐴^𝑇_𝑑0𝑃00∗∗𝜙₃−𝑃𝐵_𝑤0∗∗∗−𝛾²,𝑀𝐼0∗∗∗∗−𝐼_𝑖(𝑧)=𝜙₄𝑃𝐴_𝑑𝑖𝜙₅0𝐶^𝑇_𝑖∗0−𝐴^𝑇_𝑑𝑖,𝑁𝑃00∗∗000∗∗∗00∗∗∗∗0_𝑗(𝑧)=𝜙₆0𝜙₇00∗0000∗∗𝜙₈,𝑀00∗∗∗00∗∗∗∗0_𝑖𝑗(𝑧)=𝜙₉0𝜙₁₀,00∗0000∗∗000∗∗∗00∗∗∗∗0 (17) where 𝑀₀=𝑃𝐾₀,𝑀_𝑗=𝑃𝐾_𝑗,𝜌=𝛾²,𝜙₁=𝐴^𝑇₀𝑃+𝑃𝐴₀−𝐶^𝑇₀𝑀^𝑇₀−𝑀₀𝐶₀𝜙+𝑅,₂=−𝐴^𝑇₀𝑃−𝑃𝐴_𝑓+𝐶^𝑇₀𝑀^𝑇₀+𝑀₀𝐶_𝑓,𝜙₃=𝐴^𝑇_𝑓𝑃+𝑃𝐴_𝑓−𝐶^𝑇_𝑓𝑀^𝑇₀−𝑀₀𝐶_𝑓,𝜙₄=𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖−𝐶^𝑇_𝑖𝑀^𝑇₀−𝑀₀𝐶_𝑖,𝜙₅=−𝐴^𝑇_𝑖𝑃+𝐶^𝑇_𝑖𝑀^𝑇₀,𝜙₆=−𝐶^𝑇₀𝑀^𝑇_𝑗−𝑀_𝑗𝐶₀,𝜙₇=𝐶^𝑇₀𝑀^𝑇_𝑗+𝑀_𝑗𝐶_𝑓,𝜙₈=−𝐶^𝑇_𝑓𝑀^𝑇_𝑗−𝑀_𝑗𝐶_𝑓,𝜙₉=−𝐶^𝑇_𝑖𝑀^𝑇_𝑗−𝑀_𝑗𝐶_𝑖,𝜙₁₀=𝐶^𝑇_𝑖𝑀^𝑇_𝑗. (18)

Proof. The inequality (9) in Theorem 1 is same as follows:𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃+𝑃𝐴_𝑘𝑡,𝛼,𝛽+𝑅𝑃𝐴_𝑑(𝑡,𝛼)−𝑃𝐴_𝑘𝑓𝑡,𝛽−𝐴^𝑇_𝑘𝑡,𝛼,𝛽𝑃𝑃𝐵_𝑤𝐶^𝑇∗−(1−𝑑)𝑅−𝐴^𝑇_𝑑(𝑡,𝛼)𝑃00∗∗𝐴^𝑇_𝑘𝑓𝑡,𝛽+𝑃𝐴_𝑘𝑓𝑡,𝛽−𝑃𝐵_𝑤0∗∗∗−𝛾²𝐼0∗∗∗∗−𝐼<0⇔𝜙₁𝑃𝐴_𝑑0𝜙₂𝑃𝐵_𝑤𝐶^𝑇₀∗−(1−𝑑)𝑅−𝐴^𝑇_𝑑0𝑃00∗∗𝜙₃−𝑃𝐵_𝑤0∗∗∗−𝛾²+𝐼0∗∗∗∗−𝐼^𝐿𝑖=1𝛼_𝑖𝜙₄𝑃𝐴_𝑑𝑖𝜙₅0𝐶^𝑇_𝑖∗0−𝐴^𝑇_𝑑𝑖+𝑃00∗∗000∗∗∗00∗∗∗∗0_𝐿𝑗=1𝛽_𝑗𝜙₆0𝜙₇00∗0000∗∗𝜙₈00+∗∗∗00∗∗∗∗0^𝐿𝑖=1𝛼_𝑖^𝐿_𝑗=1𝛽_𝑗𝜙₉0𝜙₁₀00∗0000∗∗000∗∗∗00∗∗∗∗0<0. (19)(20)The inequality (20) is converted to the (16) using the relaxation technique [24], because the fourth term in (16) is a tight upper bound of the fourth term in (20).

Since the filter implementation accompanied by imprecision inherent in analog-digital and digital-analog conversion, finite word length, and finite resolution measuring instruments and round-off errors in numerical computation, we have to consider a design procedure which has sufficient space for coefficients readjustment. Inequality (16) provides a sufficient condition for the existence of the robust 𝐻_∞ filter under additive filter-gain perturbations.

4. Numerical Example

Consider nominal system matrices for parameter uncertain linear system with 𝐴₀=,𝐶−1013−4₀=11, (21) and assume a system uncertainty range as ±10%. Then, we can obtain the parameter uncertain matrices 𝐴_𝑖 and 𝐶_𝑖 as 𝐴₁=,𝐴−111.13.3−4.4₂=,𝐶−90.92.7−3.6₁=,𝐶1.11.1₂=0.90.9 (22) and the filter system matrices are 𝐴_𝑓=𝐴₀,𝐶_𝑓=𝐶₀. (23)

To solve this problem, we use the LMI toolbox with a parameter relaxation technique and obtain all feasible solutions as follows: ,,𝑀𝜎=0.7368,𝑃=0.5471−0.5194−0.51940.5577𝑅=21.68700.00300.003021.6278₀=,𝑀3.50877.7695₁=,𝑀−7.8882−5.7857₂=.−0.9740−1.0059 (24)

In this way, we obtain robust and nonfragile filter-gain with time-delay and vertices of the perturbation satisfying the nonfragility condition as follows: 𝐾𝛾=0.8584,₀=,𝐾169.6605171.9380₁=,𝐾−209.6399−205.6144₂=.−30.1728−29.9039 (25)

For the simulation, we define the time-varying parameter as follows: 𝐴(𝑡,𝛼)=𝐴₀+𝐴₁sin²𝑎𝑡+𝐴₂cos²𝐴𝑎𝑡,_𝑑(𝑡,𝛼)=𝐴_𝑑0+𝐴_𝑑1sin²𝑏𝑡+𝐴_𝑑2cos²𝐶()𝑏𝑡,𝑡,𝛼=𝐶₀+𝐶₁sin²𝑐𝑡+𝐶₂cos²𝐾𝑐𝑡,𝑡,𝛽=𝐾₀+𝐾₁sin²𝑑𝑡+𝐾₂cos²𝑑𝑡,𝑎∈ℜ,𝑏∈ℜ,𝑐∈ℜ,𝑑∈ℜ,𝑎≠𝑏≠𝑐≠𝑑. (26)

If the initial values of all states are zero and the value of 𝑤(𝑡) is defined as 𝑤₁(𝑡)≤1,if𝑤1≤𝑡≤5,₂(𝑡)≤1,if𝑤(𝑡)3≤𝑡≤7,=0,otherwise. (27)

We simulate that the proposed nominal filter-gain 𝐾₀ for the system (1) with uncertainties and disturbances in Figure 1~Figure 4. It is demonstrated that the proposed nominal filter-gain 𝐾₀ guarantees the asymptotic minimization and disturbance attenuation with 𝛾=0.8584 from the simulation results Figures 2 and 3 because the trajectories of filter (estimated) output 𝑦_𝑓(𝑡) converged to system output 𝑦(𝑡) through computer simulation. Figure 4 shows the minimized estimation error between nominal system and filter.

(a) Trajectories of the 𝑤1(𝑡)

(b) Trajectories of the 𝑤2(𝑡)

Figure 1 

Simulation results of disturbances 𝑤(𝑡).

(a) System state 𝑥1(𝑡) and filter state 𝑥𝑓1(𝑡)

(b) System state 𝑥2(𝑡) and filter state 𝑥𝑓2(𝑡)

Figure 2 

Simulation results of states for the nominal system and filter.

Figure 3 

Simulation results of output for the nominal system and filter.

Figure 4 

Simulation result of estimation error for the nominal system and filter.

Next, it is demonstrated that the practical filter 𝐾₀+𝛽₁(𝑡)𝐾₁+𝛽₂(𝑡)𝐾₂ with filter-gain perturbations has the nonfragility as well as the robustness in Figures 5, 6, and 7. Through the simulation results, the proposed robust and nonfragile 𝐻_∞ filter guarantees the asymptotic minimization of estimation error and disturbance attenuation with 𝛾=0.8584 from the results in Figures 5 and 7. Figure 8 demonstrates that the proposed filter shows good performance even though there exist uncertainties, disturbances, and time-delay in system.

(a) System state 𝑥1(𝑡) and filter state 𝑥𝑓1(𝑡)

(b) System state 𝑥2(𝑡) and filter state 𝑥𝑓2(𝑡)

Figure 5 

Simulation results of states for the practical system and filter.

Figure 6 

Simulation results of output for the practical system and filter.

Figure 7 

Simulation result of estimation error for the practical system and filter.

Figure 8 

Comparisons of estimation error between nominal filter and practical filter.

5. Conclusion

This paper proposes a method for designing a robust and nonfragile 𝐻_∞ Kalman-type filter for linear time-delay systems with parameter uncertainties. We obtain filter regions satisfying the nonfragility condition by taking the relaxation technique based on PLMI approach, and the proposed filter is robust against parameter uncertainties in the system. Finally, even though there exist filter-gain variations in the polytopic region and disturbances, the obtained Kalman-type filter ensures the satisfactory filter performance and disturbance attenuation for the system.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was financially supported by the Ministry of Education (MOE) and National Research Foundation of Korea (NRF) through the Human Resource Training Project for Regional Innovation (No. 2013H1B8A2032081).