Asymptotic Stability of the Golden-Section Control Law for Multi-Input and Multi-Output Linear Systems

Abstract

This paper is concerned with the problem of the asymptotic stability of the characteristic model-based golden-section control law for multi-input and multi-output linear systems. First, by choosing a set of polynomial matrices of the objective function of the generalized least-square control, we prove that the control law of the generalized least square can become the characteristic model-based golden-section control law. Then, based on both the stability result of the generalized least-square control system and the stability theory of matrix polynomial, the asymptotic stability of the closed loop system for the characteristic model under the control of the golden-section control law is proved for minimum phase system.

1. Introduction

Since the late 1950s, the first adaptive control system was presented by the Massachusetts Institute of Technology, the adaptive control has been attracted extensively. Fruitful results have been achieved in theory and application. However, the adaptive control in practice has not been widely used. The reason is that there are some problems of the existing adaptive control theory in practical engineering applications, such as the following: the transient response is very poor, the number of parameters need to be estimated is too many, the convergence of parameter estimation is difficult to be guaranteed, and parameters that need to be adjusted artificially are too many [1].

To solve the above problems, Wu et al. [1, 2] presented an integrated and practical all-coefficient adaptive control theory and method based on characteristic models, which has been gradually improved in the application course of more than 20 years. This theory and method provide a new approach for the modeling and control of the complex systems. It should be mentioned that the theory and method have already been applied successfully to more than 400 systems belonging to nine kinds of engineering plants in the field of astronautics and industry. In particular, the engineering key points of the method are creatively applied to the reentry adaptive control of a manned spaceship, which the accuracy of the parachute-opening point of the spaceship reaches the advanced level of the world.

The method given by Wu is simple in design, easy to adjust and with strong robustness, and in some ways solves the above-mentioned problems. This method includes the following three aspects: (1) all-coefficient adaptive control method; (2) golden-section adaptive control law; (3) characteristic model [1].

It is worth mentioning that the golden-section control law is a new control law, which can solve the problem that adaptive control cannot guarantee the stability of a closed-loop system during the transient process when the parameters have not converged to their “true value.” The so-called golden-section control law means that the golden section ratio (0.382/0.618) is used to controller designs, see [1, 2] or Section 3 of this paper in detail.

For a second-order single-input and single-output (SISO) invariant linear system, Xie et al. [3, 4] proved that the golden-section control law has strong robust stability. The sufficient conditions for the stability of the closed-loop system based on golden-section control law for SISO and 2-input-2-output invariant linear systems have been reported; see, Qi et al. [5], Sun and Wu [6, 7], and Meng et al. [8]. Recently, Sun [9] gave sufficient conditions for the stability of the golden-section control system for 3-input-3-output invariant linear systems, but these conditions are difficult to verify. Among these references, we notice that the closed-loop control properties based on the golden-section control law are given by using the stability results of the generalized least-square control system and Jury stability criteria, aiming at the characteristic model of a second-order continuous SISO invariant linear system.

In summary, for the MIMO system, the stability of golden-section control systems is still an open question.

In this paper, for the MIMO linear system, we investigate the stability of the characteristic model-based golden-section control law by using the stability results of the multivariable generalized least-square control system.

2. Preliminaries

2.1. Introduction to Generalized Least-Square Controller

Consider the following system described by the linear vector difference equation:𝐴𝑧⁻¹𝑌(𝑘)=𝑧^−𝑑𝐵𝑧⁻¹𝑈(𝑘)+𝐶𝑧⁻¹𝜉(𝑘),(2.1) where 𝑈(𝑘) and 𝑌(𝑘) are the 𝑛×1 input and output vectors, respectively, and 𝜉(𝑘) is the 𝑛×1 zero-mean white-noise vector with covariance matrix 𝐸(𝜉(𝑘)𝜉^𝑇(𝑘))=𝑟_𝜉, 𝑑 denotes the system time delay, and 𝐴,𝐵, and 𝐶 are polynomial matrices in backward shift operator 𝑧⁻¹ given by𝐵𝐴𝑧𝑧⁻¹₋₁=𝐼+𝐴₁𝑧⁻¹+⋯+𝐴_𝑛𝑎𝑧^−𝑛𝑎,𝐶𝑧=𝐵₀+𝐵₁𝑧⁻¹+⋯+𝐵_𝑛𝑏𝑧^−𝑛𝑏,𝐵₀nonsingular,₋₁=𝐼+𝐶₁𝑧⁻¹+⋯+𝐶_𝑛𝑐𝑧^−𝑛𝑐,(2.2)

here 𝐴_𝑖, 𝐵_𝑖, and 𝐶_𝑖 are matrix coefficients. The cost function to be considered is described by𝐽=𝐸‖‖𝑃𝑧⁻¹𝑌(𝑘+𝑑)−𝑅𝑧⁻¹𝑌_𝑟(𝑘)‖‖²+‖‖𝑄^′𝑧⁻¹𝑈(𝑘)‖‖²,(2.3) where 𝑌_𝑟(𝑘) is an 𝑛-vector defining the known reference signal. 𝑃,𝑅, and 𝑄^ are 𝑛×𝑛 weighting polynomial matrices. The notation ‖𝑋‖²=𝑋^T𝑋 has been used.

The optimal control law is [10]𝐻𝑧⁻¹𝑈(𝑘)=−𝐸𝑧⁻¹𝑌_𝑟(𝑘)−𝐺𝑧⁻¹𝑌(𝑘),(2.4) where𝐻𝑧⁻¹𝐸=𝐹𝑧𝑧⁻¹𝐵𝑧⁻¹+𝐶𝑧⁻¹𝑄𝑧⁻¹,(2.5)₋₁𝑄𝑧=−𝐶𝑧⁻¹𝑅𝑧⁻¹,(2.6)₋₁=𝐶𝑧𝑃₀𝐵₀^T⁻¹𝑄^′(0)^T𝑄^′𝑧⁻¹,𝐵₀=𝐵(0),𝑃₀=𝑃(0),(2.7)₋₁𝐶𝑃𝑧𝑧⁻¹=𝐹𝑧⁻¹𝐴𝑧⁻¹+𝑧^−𝑑𝐺𝑧⁻¹,(2.8)₋₁𝐹𝑧⁻¹=𝐹𝑧⁻¹𝐶𝑧⁻¹,(2.9) here the order of the polynomial matrix 𝐹 is equal to 𝑑−1.

2.2. Two Important Lemmas

Let an (𝑚×𝑚) nonsingular real matrix polynomial of 𝑛th-order 𝑃(𝑧) be given by𝑃(𝑧)=𝑎_𝑛𝑧^𝑛+𝑎_𝑛−1𝑧^𝑛−1+𝑎_𝑛−2𝑧^𝑛−2+⋯+𝑎₁𝑧+𝑎₀,(2.10) where 𝑎_𝑛,𝑎_𝑛−1,𝑎_𝑛−2,…,𝑎₁, and 𝑎₀ are (𝑚×𝑚) real matrices, 𝑚≥1.

We can construct an (𝑚𝑛×𝑚𝑛) symmetric matrix 𝐶=[𝑐_𝑖𝑗] by the Christoffel-Darboux formula as follows. 𝑐_𝑖𝑗’s are defined as𝑐_𝑖𝑗=^𝑖_𝑘=1𝑎𝑐^T_{𝑛+𝑘−𝑖}𝑎_{𝑛+𝑘−𝑗}−𝑎^T_𝑗−𝑘𝑎_𝑖−𝑘,𝑖≤𝑗,_𝑗𝑖=𝑐^T_𝑖𝑗,𝑖>𝑗(2.11) for 𝑖,𝑗=1,2,…,𝑛.

For example, when 𝑛=2 [11],C=⎡⎢⎢⎣𝑎^T2𝑎₂−𝑎^T₀𝑎₀𝑎^T₂𝑎₁−𝑎^T₁𝑎₀𝑎^T₁𝑎₂−𝑎^T₀𝑎₁𝑎^T₂𝑎₂−𝑎^T₀𝑎₀⎤⎥⎥⎦.(2.12)

Lemma 2.1 (see [11, Theorem  1]). If 𝐶=[𝑐_𝑖𝑗] defined in (2.11) is positive definite, then all the roots of the determinant of the matrix polynomial (2.10) lie inside the unit circle.

It is well known that the following lemma holds.

Lemma 2.2. Let 𝐴 be an (𝑛×𝑛) real symmetric matrix and 𝐼 be an (𝑛×𝑛) identity matrix. Then there exists a sufficiently small positive number 𝜀 such that 𝐼+𝜀𝐴 becomes a positive definite matrix.

3. Problem Formulation

Suppose that a second-order multi-input and multi-output dynamic process can be expressed as𝑌⁽²⁾(𝑡)+𝐴₁𝑌⁽¹⁾(𝑡)+𝐴₀𝑌(𝑡)=𝐵₁𝑈⁽¹⁾(𝑡)+𝐵₀𝑈(𝑡),(3.1) where 𝑈(𝑘) and 𝑌(𝑘) are the 𝑛×1 input and output vectors, respectively, and 𝐴₀, 𝐴₁, 𝐵₀, and 𝐵₁ are polynomial matrices. By using forward difference method, the corresponding difference equation can be given as𝑌(𝑘+1)=𝐴₁𝑌(𝑘)+𝐴₂𝑌(𝑘−1)+𝐵₀𝑈(𝑘)+𝐵₁𝑈(𝑘−1)+𝑒(𝑘+1),(3.2) where𝐴₁=2𝐼−𝐴₁Δ𝑡,𝐴₂=−𝐼+𝐴₁Δ𝑡−𝐴₀(Δ𝑡)²,𝐵₀=−𝐵₁Δ𝑡,𝐵₁=−𝐵₁Δ𝑡+𝐵₀Δ𝑡²,(3.3) and 𝑒(𝑘) is the modeling error vector.

For a linear multi-input and multi-output constant high-order plant, if sampling period Δ𝑡 is sufficiently small, when the control requirement is position keeping or tracking, we can prove that its characteristic model can be also expressed with (3.2).

It is easily to be seen that (3.2) can be written as (2.1), that is,𝐴𝑧⁻¹𝑌(𝑘)=𝑧⁻¹𝐵𝑧⁻¹𝑈(𝑘)+𝐶𝑧⁻¹𝑒(𝑘),(3.4) where𝐵𝐴𝑧𝑧⁻¹₋₁=𝐼−𝐴₁𝑧⁻¹−𝐴₂𝑧⁻²,𝐶𝑧₋₁=𝐵₀+𝐵₁𝑧⁻¹,=𝐼.(3.5)

The characteristic model-based golden-section control law is designed as follows:𝑈(𝑘)=(1+𝜆)⁻¹𝐵⁰⁻¹𝑌^𝑟(𝑘)−𝐵₁𝑈(𝑘−1)−𝑙₁𝐴₁𝑌(𝑘)−𝑙₂𝐴₂𝑌(𝑘−1),(3.6) where 𝜆≥0 is a constant; 𝑙₁=0.382 and 𝑙₂=0.618 are golden-section coefficients.

4. Main Result and Proof

In this section, by selecting a set of polynomial matrices in the objective function (2.3), for the characteristic model (3.2), we find the generalized least-square control law under the objective function. By comparison, we can see that this control law is just the same law as the golden-section control law based on characteristic model. Finally, based on the stability result of generalized least-square control law, we will give the stability result of the closed-loop of the golden-section control law.

For the objective function (2.3), we choose that𝑃𝑧⁻¹=𝐼−𝑙₂𝐴₁𝑧⁻¹−𝑙₁𝐴₂𝑧⁻²,𝑅𝑧⁻¹=𝐼,𝑄^′𝑧⁻¹=√𝜆𝐵₀,𝐹𝑧⁻¹=𝐼.(4.1)

Note that 𝑑=1, and hence 𝐹(𝑧⁻¹)=𝐼. By 𝐶(𝑧⁻¹)=𝐼, 𝐹(𝑧⁻¹)=𝐼, and (2.9), we get 𝐶(𝑧⁻¹)=𝐼. Using 𝐶(𝑧⁻¹)=𝐼, 𝑅(𝑧⁻¹)=𝐼, and (2.6), 𝐸(𝑧⁻¹)=−𝐼.

From 𝑃₀=𝐼, 𝐵(0)=𝐵₀, 𝑄^(𝑧⁻¹)=√𝜆𝐵₀, and (2.7), it follows that𝑄𝑧⁻¹=𝑃₀𝐵₀^T⁻¹𝑄^′(0)^T𝑄^′𝑧⁻¹=𝐵^T₀⁻¹√𝜆𝐵₀^T√𝜆𝐵₀=𝐵^T₀⁻¹𝐵^T₀𝜆𝐵₀=𝜆𝐵₀.(4.2)

By using Diophantine equation (2.8), we obtain that𝐼−𝑙₂𝐴₁𝑧⁻¹−𝑙₁𝐴₂𝑧⁻²=𝐼−𝐴₁𝑧⁻¹−𝐴₂𝑧⁻²+𝑧⁻¹𝐺𝑧⁻¹.(4.3)

Therefore,𝐺𝑧⁻¹=𝑙₁𝐴₁+𝑙₂𝐴₂𝑧⁻¹.(4.4)

Using (2.5), we have𝐻𝑧⁻¹==𝐹𝑧⁻¹𝐵𝑧⁻¹+𝐶𝑧⁻¹𝑄𝑧⁻¹𝐵₀+𝐵₁𝑧⁻¹+𝐼𝜆𝐵₀=𝐵₀+𝐵₁𝑧⁻¹+𝜆𝐵₀=(1+𝜆)𝐵₀+𝐵₁𝑧⁻¹.(4.5)

According to (2.4), it follows that(1+𝜆)𝐵₀+𝐵₁𝑧⁻¹𝑈(𝑘)=𝐼𝑌^𝑟(𝑘)−𝑙₁𝐴₁+𝑙₂𝐴₂𝑧⁻¹𝑌(𝑘).(4.6)

That is,𝑈(𝑘)=(1+𝜆)⁻¹𝐵⁰⁻¹𝑌^𝑟(𝑘)−𝐵₁𝑈(𝑘−1)−𝑙₁𝐴₁𝑌(𝑘)−𝑙₂𝐴₂𝑌(𝑘−1).(4.7)

The above control law obtained by generalized least-square control law is exactly the characteristic model-based golden-section control law designed in (3.6). Hence, the stability of the closed-loop system based on the characteristic model-based golden-section control law is determined by the distribution of zero points on 𝑧-plane of the following equation:det𝐻𝑧⁻¹det𝐴𝑧⁻¹+𝑧^−𝑑𝐵𝑧⁻¹𝐻⁻¹𝑧⁻¹𝐺𝑧⁻¹=0.(4.8)

We note that𝐻𝑧⁻¹=𝐹𝑧⁻¹𝐵𝑧⁻¹+𝐶𝑧⁻¹𝑄𝑧⁻¹=𝐼𝐵𝑧⁻¹+𝐼𝜆𝐵₀=𝐵𝑧⁻¹+𝜆𝐵₀.(4.9)

Now, taking 𝜆=0, we have 𝐻(𝑧⁻¹)=𝐵(𝑧⁻¹), and, furthermore,𝐴𝑧=⁻¹𝐴+𝑧𝑧^−𝑑₋₁𝐵𝑧⁻¹𝐻⁻¹𝑧⁻¹𝐺𝑧⁻¹=𝐴𝑧₋₁+𝐵𝑧⁻¹𝐵𝑧⁻¹⁻¹𝐶𝑧⁻¹𝑃𝑧⁻¹−𝐹𝑧⁻¹𝐴𝑧⁻¹=𝐴𝑧₋₁+𝐼𝑃𝑧⁻¹−𝐼𝐴𝑧⁻¹=𝑃𝑧₋₁+𝑃𝑧⁻¹−𝐴𝑧⁻¹.(4.10)

Then, when 𝐵(𝑧⁻¹) is stable, the stability of the closed-loop system formed by the characteristic model-based golden-section control law is determined by the stability of 𝑃(𝑧⁻¹).

Theorem 4.1. Assume that (3.2) is a minimum phase system and 𝜆=0. Then, the closed-loop system involving the characteristic model-based golden-section control law (3.6) is asymptotic stable.

Remark 4.2. Since the corresponding relationship between the zero positions of continuous controlled objects and that of the discrete-time systems is complex, here the minimum phase system means that the zero points of the characteristic model (3.2) lie inside the unit circle.

Proof. First, we notice that 𝑃(𝑧⁻¹)=𝐼−𝑙₂𝐴₁𝑧⁻¹−𝑙₁𝐴₂𝑧⁻², and take 𝑎₂=𝐼,𝑎₁=−𝑙₂𝐴₁,𝑎₀=−𝑙₁𝐴₂.(4.11) By (2.12), we have 𝐶=⎡⎢⎢⎣𝐼−𝑙²¹𝐴^T₂𝐴₂−𝑙₂𝐴₁−𝑙₁𝑙₂𝐴^T₁𝐴₂−𝑙₂𝐴^T₁−𝑙₁𝑙₂𝐴^T₂𝐴₁𝐼−𝑙²¹𝐴^T₂𝐴₂⎤⎥⎥⎦.(4.12)
Now, we show that 𝐶is positive definite when Δ𝑡 is sufficiently small.
It is easy to see that 𝐶=⎡⎢⎢⎣𝑁𝐵𝑁_T𝛼𝐼⎤⎥⎥⎦+⎡⎢⎢⎣𝛼𝐼−𝑙𝑂²¹𝐴^T₂𝐴₂𝑂𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂⎤⎥⎥⎦,(4.13) where 𝐵=𝛼𝐼, 𝑁=−𝑙₂𝐴₁−𝑙₁𝑙₂𝐴^T₁𝐴₂, 0.764=2𝑙₁<𝛼<1−𝑙²¹=0.8541, and 𝛼=1−𝛼.
To show 𝐶 be positive definite, we will prove that ⎡⎢⎢⎣𝑁𝐵𝑁_T𝛼𝐼⎤⎥⎥⎦,(4.14) and 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂ are all positive definite when Δ𝑡 is sufficiently small as follows.
First, we notice that ⎡⎢⎢⎣−𝑁𝐼_T𝑂𝐵⁻¹𝐼⎤⎥⎥⎦⎡⎢⎢⎣𝑁𝐵^T𝛼𝑁𝐼⎤⎥⎥⎦⎡⎢⎢⎣𝐼−𝐵⁻¹=⎡⎢⎢⎣𝐵−𝐵𝐵₋₁𝑂𝐼^T𝑁⎤⎥⎥⎦𝑂𝛼𝐼−𝑁^T_T𝑁+𝑁𝐵⁻¹𝑁⎤⎥⎥⎦=⎡⎢⎢⎣𝑂𝐵𝛼𝐼−𝑁𝑂^T𝐵⁻¹𝑁⎤⎥⎥⎦=⎡⎢⎢⎣𝛼𝑂𝐼𝛼𝐼−𝑁𝑂^T𝐵⁻¹𝑁⎤⎥⎥⎦.(4.15) We claim that 𝛼𝐼−𝑁^T𝐵⁻¹𝑁 is positive definite when Δ𝑡 is sufficiently small. In fact, since 𝑁^T𝐵⁻¹𝑁==𝑙−₂𝑙₂𝐴^T₁−𝑙₁𝑙₂𝐴^T₂𝐴₁𝐵⁻¹−𝑙²𝐴₁−𝑙₁𝑙₂𝐴^T₁𝐴₂=1𝐴^T₁+𝑙₁𝑙₂𝐴^T₂𝐴₁𝐵⁻¹𝑙²𝐴₁+𝑙₁𝑙₂𝐴^T₁𝐴₂𝛼𝑙²𝑙₂𝐴^T₁+𝑙₁𝑙₂𝐴^T₂𝐴₁𝑙²𝐴₁+𝑙₁𝑙₂𝐴^T₁𝐴₂,𝐴^T₁+𝑙₁𝑙₂𝐴^T₂𝐴₁==𝑙2₂𝑙₂2𝐼−𝐴^T₁Δ𝑡+𝑙₁𝑙₂−𝐼+𝐴^T₁Δ𝑡−𝐴^T₀(Δ𝑡)²2𝐼−𝐴₁Δ𝑡=2𝑙₂𝐼−𝑙₂𝐴^T₁Δ𝑡−2𝑙₁𝑙₂𝐼+𝑙₁𝑙₂𝐴₁Δ𝑡+𝑙₁𝑙₂𝐴^T₁Δ𝑡−𝐴^T₀(Δ𝑡)²2𝐼−𝐴₁Δ𝑡=2𝑙₂₂1−𝑙₁𝐼−𝑙₂𝐴^T₁Δ𝑡+𝑙₁𝑙₂𝐴₁Δ𝑡+𝑙₁𝑙₂𝐴^T₁Δ𝑡−𝐴^T₀(Δ𝑡)²2𝐼−𝐴₁Δ𝑡=2𝑙₂₂𝐼−𝑙₂𝐴^T₁Δ𝑡+𝑙₁𝑙₂𝐴₁Δ𝑡+𝑙₁𝑙₂𝐴^T₁Δ𝑡−𝐴^T₀(Δ𝑡)²2𝐼−𝐴₁Δ𝑡=2𝑙₂₂𝐼−𝑙₂𝐴^T₁Δ𝑡+𝑙₁𝑙₂𝐴₁Δ𝑡+𝑙₁𝑙₂𝐴^T₁Δ𝑡−𝑙₁𝑙₂𝐴^T₀(Δ𝑡)²2𝐼−𝐴₁Δ𝑡𝐼−𝑙₂𝐴^T₁Δ𝑡+𝑙₁𝑙₂𝐴₁Δ𝑡+2𝑙₁𝑙₂𝐴^T₁Δ𝑡−2𝑙₁𝑙₂𝐴^T₀(Δ𝑡)²−𝑙₁𝑙₂𝐴^T₁𝐴₁(Δ𝑡)²+𝑙₁𝑙₂𝐴^T₀𝐴₁(Δ𝑡)³=2𝑙²²𝐼+−𝑙₂+2𝑙₁𝑙₂𝐴^T₁Δ𝑡+𝑙₁𝑙₂𝐴₁Δ𝑡−2𝑙₁𝑙₂𝐴^T₀+𝑙₁𝑙₂𝐴^T₁𝐴₁(Δ𝑡)²+𝑙₁𝑙₂𝐴^T₀𝐴₁(Δ𝑡)³,(4.16) we get 𝑁^T𝐵⁻¹𝑁=1𝛼4𝑙⁴²𝛼𝐼−𝑁_T𝐼+2𝑙²²−𝑙₂+3𝑙₁𝑙₂𝐴₁+𝐴^T₁Δ𝑡+⋯+𝑙₁𝑙₂²𝐴^T₀𝐴₁𝐴^T₁𝐴₀(Δ𝑡)⁶,𝐵⁻¹𝑁=𝛼𝐼−1𝛼4𝑙⁴²=𝛼−4𝑙₄₂𝐼+2𝑙²²−𝑙₂+3𝑙₁𝑙₂𝐴₁+𝐴^T₁Δ𝑡+⋯+𝑙₁𝑙₂²𝐴^T₀𝐴₁𝐴^T₁𝐴₀(Δ𝑡)⁶𝛼𝐼−𝛼12𝑙²²−𝑙₂+3𝑙₁𝑙₂𝐴₁+𝐴^T₁Δ𝑡+⋯+𝑙₁𝑙₂²𝐴^T₀𝐴₁𝐴^T₁𝐴₀(Δ𝑡)⁶.(4.17) Using 𝛼>2𝑙₁=2𝑙²² yields 𝛼−4𝑙⁴²/𝛼=(𝛼²−4𝑙⁴²)/𝛼>0. Thus, we have =𝛼𝐼−𝛼𝑁^T₂𝐵⁻¹𝑁−4𝑙⁴²𝛼𝐼+𝛼²Δ−𝑡4𝑙⁴²−2𝑙²²−𝑙₂+3𝑙₁𝑙₂𝐴₁+𝐴^T₁−⋯−𝑙₁𝑙₂²𝐴^T₀𝐴₁𝐴^T₁𝐴₀(Δ𝑡)⁵.(4.18) By Lemma 2.2, it is follows that 𝛼𝐼−𝑁^T𝐵⁻¹𝑁 is positive definite when Δ𝑡 is sufficiently small. Therefore, there exists nonsingular matrix 𝐷 such that 𝐷^T(𝛼𝐼−𝑁^T𝐵⁻¹𝑁)𝐷=𝐼. Furthermore, we have ⎡⎢⎢⎢⎣1√𝑂𝛼𝐼𝐷𝑂_T⎤⎥⎥⎥⎦⎡⎢⎢⎣𝛼𝑂𝐼𝛼𝐼−𝑁𝑂_T𝐵⁻¹𝑁⎤⎥⎥⎦⎡⎢⎢⎢⎣1=⎡⎢⎢⎢⎣𝛼√𝑂𝛼𝐼𝐷𝑂⎤⎥⎥⎥⎦√𝑂𝛼𝐼𝐷_𝑇𝑂𝛼𝐼−𝑁^T𝐵⁻¹𝑁⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣1√𝑂𝛼𝐼𝐷𝑂⎤⎥⎥⎥⎦=⎡⎢⎢⎣𝑂𝐼𝑂𝐼⎤⎥⎥⎦.(4.19) That is, by the congruent transformation twice, ⎡⎢⎢⎣𝑁𝐵𝑁_T𝛼𝐼⎤⎥⎥⎦(4.20) can be transformed into the identity matrix. From this, it is positive definite when Δ𝑡 is sufficiently small.
We also claim that 𝛼𝐼−𝑙₁₂𝐴^T₂𝐴₂ is positive definite when Δ𝑡 is sufficiently small. In fact, it can be seen that 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂==𝛼𝐼−𝑙²¹−𝐼+𝐴^T₁Δ𝑡−𝐴^T₀(Δ𝑡)²−𝐼+𝐴₁Δ𝑡−𝐴₀(Δ𝑡)²𝛼−𝑙²¹𝐼−𝑙²¹−𝐴₁Δ𝑡+𝐴₀(Δ𝑡)²−𝐴^T₁Δ𝑡+𝐴^T₁𝐴₁(Δ𝑡)²−𝐴^T₁𝐴₀(Δ𝑡)³+𝐴^T₀(Δ𝑡)²−𝐴^T₀𝐴₁(Δ𝑡)³+𝐴^T₀𝐴₀(Δ𝑡)⁴.(4.21) Using 𝛼<1−𝑙²¹ and 𝛼=1−𝛼, we have 𝛼−𝑙²¹>0. Thus, 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂=𝛼−𝑙²¹𝐼−𝑙²¹Δ𝑡𝛼−𝑙²¹−𝐴₁−𝐴^T₁+𝐴₀Δ𝑡+𝐴^T₁𝐴₁Δ𝑡−𝐴^T₁𝐴₀(Δ𝑡)²+𝐴^T₀Δ𝑡−𝐴^T₀𝐴₁(Δ𝑡)²+𝐴^T₀𝐴₀(Δ𝑡)³.(4.22) By Lemma 2.2, 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂ is also positive definite when Δ𝑡 is sufficiently small.
Thus, it follows from (4.13) that 𝐶 is positive definite when Δ𝑡 is sufficiently small. By Lemma 2.1, 𝑃(𝑧⁻¹) is a stable matrix polynomial. Hence, the asymptotic stability of the closed-loop system follows immediately. This completes the proof.

Remark 4.3. It is well known that an eigenvalue of a matrix is a continuous function with respect to elements of the matrix. From this, (4.18), and (4.22), we can also see that 𝛼𝐼−𝑁^T𝐵⁻¹𝑁 and 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂ are positive definite when Δ𝑡 is sufficiently small.

Remark 4.4. By using a constructive proof of Lemma 2.2, we can determine minimum sampling period Δ𝑡 so that 𝛼𝐼−𝑁^T𝐵⁻¹𝑁 and 𝛼𝐼−𝑙²¹𝐴^T₂𝐴₂ are all positive definite.

Conjecture 4.5. If (3.2) is a nonminimum phase system, we may investigate the stability of the closed-loop system involving the characteristic model-based golden-section control law (3.6) by using the approach of the root locus in linear multivariable systems.

Acknowledgment

This work was supported by the National Nature Science Foundation of China under Grant 60874055, the Key Subject Construction Project, and the Funds for Creative Research Groups of Hebei Normal University of Science and Technology (no. CXTD2010-05).