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Abstract

Model reference adaptive control takes the output of a reference model as the target and ensures that the actual output is consistent with the target. In this scenario, the reference model determines the performance of the control system. Constructing a reference model with specific performance is challenging, and most researchers construct the reference model by experience. As alternative frameworks, we propose two novel reference model construction methods, namely, zero-pole method and frequency response method. The zero-pole method establishes a nonlinear optimization model to ensure that the added zeros and poles are as far away from the imaginary axis as possible. In the case where this cannot be done, the frequency response method matches the system frequency response of the reference model and the given model to the extent possible using a nonlinear optimization model. Experimental results are presented to confirm that both methods can construct a reference model based on a given transfer function. Deviations in response between the reference models constructed using the two methods and the given model are compared, and the reasons for higher accuracy of the frequency response method are discussed.

Keywords

Frequency response method model reference adaptive control reference model construction method zero-pole method

Introduction

Compared with conventional closed-loop negative feedback control, model reference adaptive control (MRAC) offers considerably higher performance. Therefore, MRAC is widely used in complex control systems. An MRAC system consists of the controlled object, controller, reference model, and adaptive mechanism, wherein the controller and controlled object form an inner loop, and the reference model and adaptive mechanism form an outer loop. The goal of the MRAC is to ensure that the output of the controlled system closely tracks the output of the reference model. Therefore, to some extent, the reference model determines the control performance of the control system.

In Whitaker et al.,¹ an adaptive correction mechanism called Massachusetts Institute of Technology (MIT) is proposed, but MIT cannot always guarantee the stability of an MRAC system. In previous works,^2–5 this instability-related limitation is overcome by improving the adaptive control scheme, and an adaptive law is proposed. Since then, the scope of the application of MRAC has broadened greatly. For example, in previous works,^6–8 the classical theory of MRAC is applied to different control systems. In previous works,^9–16 MRAC is further employed to ameliorate the tracking performance of a control system.

Ideal values of control law parameters are used to obtain the reference model in Tárník and Murgaš,⁶ whereas in Li et al.⁸ a lower-order dynamic system with excellent properties is selected as the reference model. In Nair et al.,⁹ a second-order reference model is used, and forward and feedback path gains are adjusted adaptively using the Lyapunov stability theory. In Mola et al.,¹³ all coefficients of the reference model are positive and arbitrary. In Xie and Zhao,¹⁵ the closed-loop reference model is degenerated into the open-loop reference model, and $L_{i}$ gains are introduced to suppress the high-frequency oscillations caused by the open-loop reference model.

MRAC is widely used in complex system control, and its control performance is closely related to the reference model. But it is difficult to build a reference model with satisfactory performance, since the model needs to meet some constraints, such as positive realism, the same relative order as the controlled object, and so on. Hence, we propose the zero-pole method and the frequency response method to make it easier to construct a reference model, and the experimental results show that the reference model constructed using our methods not only satisfies those constraints, but also shows better performance.

The remainder of this article is organized as follows. In section “The reference model construction method,” the reference model construction method is proposed. Guidelines for constructing the reference model are provided in section “Construction guidelines for the reference model.” The algorithms for constructing the reference model using the zero-pole method and frequency response method are proposed in sections “The zero-pole method” and “The frequency response method,” respectively. In section “Experimental results,” the construction algorithms of the two methods are discussed and the feasibility of these algorithms is verified experimentally. In section “Discussion,” the reasons for the higher accuracy of the frequency response method relative to that of the zero-pole method are analyzed. Finally, our conclusions are detailed in section “Conclusion.”

The reference model construction method

Construction guidelines for the reference model

Adaptive law

The adaptive law of the MRAC is designed to minimize the error between the actual output of the controlled object and the expected output of the reference model, based on Lyapunov stability theory. Lyapunov stability theory guarantees the stability of the MRAC by introducing the Lyapunov function, and, in particular, it guarantees the stability of the reference model through the strictly positive real principle. The strictly positive real principle is defined as follows.²

$g (t)$ is a mathematical function with time t being its independent variable, and $G (s)$ is a new function obtained from $g (t)$ by the Laplace transform, where s is a complex variable and $s = σ + j ω$ . If the rational fractional function $G (s)$ satisfies the following conditions:

When s is a real number, $G (s)$ is real;

$G (s)$ has no pole on the right closed plane, and the right half plane is represented as $R_{e} (s) \geq 0$ ;

For any real number $ω$ , $R_{e} [G (j ω)] > 0$ ;

then $G (s)$ is a strictly positive real function.

The control systems of $n - m > 2$ are unstable, and the case of $n - m > 2$ could be approximately simplified as $n - m = 2$ , so the relative order of the controlled object is $n - m = 1$ , or $n - m = 2$ , where $n$ and $m$ are the orders of the denominator and the numerator of the transfer function, respectively.¹⁷ We take $n - m = 1$ as an example to discuss the method of constructing the reference model, and the process for the relative order of $n - m = 2$ is similar. According to the stated definition, we employ the following third-order transfer function as an example to deduce the constraints that satisfy the strictly positive real principle

$G_{3} (s) = \frac{b_{2} s^{2} + b_{1} s + 1}{a_{3} s^{3} + a_{2} s^{2} + a_{1} s + 1}$

where $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}$ are greater than 0. First, when $a_{1}, a_{2}, a_{3}$ are greater than 0, $G_{3} (s)$ has no pole on the right closed plane. According to the definition of the strictly positive real function, judging is as follows

$\begin{matrix} G_{3} (j ω) = \frac{(1 - b_{2} ω^{2}) + b_{1} ω j}{(1 - a_{2} ω^{2}) + (a_{1} ω - a_{3} ω^{3}) j} = \frac{[(1 - b_{2} ω^{2}) + b_{1} ω j] [(1 - a_{2} ω^{2}) - (a_{1} ω - a_{3} ω^{3}) j]}{{(1 - a_{2} ω^{2})}^{2} + {(a_{1} ω - a_{3} ω^{3})}^{2}} \\ = \frac{[1 + (a_{1} b_{1} - a_{2} - b_{2}) ω^{2} + (a_{2} b_{2} - a_{3} b_{1}) ω^{4}] + j [b_{1} ω (1 - a_{2} ω^{2}) - (1 - b_{2} ω^{2}) (a_{1} ω - a_{3} ω^{3})]}{{(1 - a_{2} ω^{2})}^{2} + {(a_{1} ω - a_{3} ω^{3})}^{2}} \end{matrix}$

From this expression, we obtain the following

$R_{e} [G_{3} (j ω)] = [1 + (a_{1} b_{1} - a_{2} - b_{2}) ω^{2} + (a_{2} b_{2} - a_{3} b_{1}) ω^{4}]$

Therefore, when ${\begin{matrix} a_{1} b_{1} - a_{2} - b_{2} \geq 0 \\ a_{2} b_{2} - a_{3} b_{1} \geq 0 \end{matrix}$ , $R_{e} [G_{3} (j ω)] > 0$ , $G_{3} (s)$ is a strictly positive real function.

The constraints of the first- to fourth-order transfer functions for which the relative order is $n - m = 1$ and the positive real principle is satisfied are listed in Table 1.

Table 1.

Strictly positive real constraints for the first- to fourth-order transfer functions.

Order	Transfer function	Strictly positive real constraints
1	$G_{1} (s) = \frac{1}{a_{1} s + 1}$	$a_{1} > 0$
2	$G_{2} (s) = \frac{b_{1} s + 1}{a_{2} s^{2} + a_{1} s + 1}$	${\begin{matrix} a_{1} > 0, a_{2} > 0, b_{1} > 0 \\ a_{1} b_{1} - a_{2} \geq 0 \end{matrix}$
3	$G_{3} (s) = \frac{b_{2} s^{2} + b_{1} s + 1}{a_{3} s^{3} + a_{2} s^{2} + a_{1} s + 1}$	${\begin{matrix} a_{1} > 0, a_{2} > 0, a_{3} > 0 \\ b_{1} > 0, b_{2} > 0 \\ a_{1} b_{1} - a_{2} - b_{2} \geq 0 \\ a_{2} b_{2} - a_{3} b_{1} \geq 0 \end{matrix}$
4	$G_{4} (s) = \frac{b_{3} s^{3} + b_{2} s^{2} + b_{1} s + 1}{a_{4} s^{4} + a_{3} s^{3} + a_{2} s^{2} + a_{1} s + 1}$	${\begin{matrix} a_{1} > 0, a_{2} > 0, a_{3} > 0, a_{4} > 0 \\ b_{1} > 0, b_{2} > 0, b_{3} > 0 \\ a_{3} b_{3} - a_{4} b_{2} \geq 0 \\ a_{4} + a_{2} b_{2} - a_{3} b_{1} - a_{1} b_{3} \geq 0 \\ a_{1} b_{1} - a_{2} - b_{2} \geq 0 \end{matrix}$

Specific performance specification

Different control systems are required to achieve various performance specifications; for instance, some are required for rapidity and others stability. A series of standardized transfer functions have been proposed with excellent control characteristics in some aspects. In this study, we employ two such standardized transfer functions as examples.

One is the optimal integrated time and absolute error (ITAE),^18,19 which is a comprehensive performance specification with the smallest adjustment time and least oscillation. It is expressed as $ITAE = \int_{0}^{T} t | e (t) | dt$ , where $T$ is the adjustment time of the system and $e (t)$ is the difference between the output value of the control system and the target value. The other function is deadbeat response,²⁰ which is the time response with the smallest overshoot. It quickly reaches the allowable range of fluctuation of the steady-state response and continues to remain within the fluctuation range.

When the controlled object is a typical transfer function $G (s) = b_{0} / (s^{n} + b_{n - 1} s^{n - 1} + \dots + b_{1} s + b_{0})$ with $n$ poles and no finite zeros, the step response curves of the two specifications, namely, optimal ITAE and deadbeat, are as shown in Figures 1 and 2, respectively. The typical transfer functions in Figures 1 and 2 are shown in Table 2.²¹

Figure 1.

Step response curve of optimal ITAE.

Figure 2.

Step response curve of deadbeat.

Table 2.

The typical transfer functions in Figures 1 and 2.

Order	Transfer function (Figure 1)	Transfer function (Figure 2)
2	$\frac{1}{s^{2} + 1.4 s + 1}$	$\frac{1}{s^{2} + 1.82 s + 1}$
3	$\frac{1}{s^{3} + 1.75 s^{2} + 2.15 s + 1}$	$\frac{1}{s^{3} + 1.9 s^{2} + 2.2 s + 1}$
4	$\frac{1}{s^{4} + 2.1 s^{3} + 3.4 s^{2} + 2.7 s + 1}$	$\frac{1}{s^{4} + 2.2 s^{3} + 3.5 s^{2} + 2.8 s + 1}$
5	$\frac{1}{s^{5} + 2.8 s^{4} + 5.0 s^{3} + 5.5 s^{2} + 3.4 s + 1}$	$\frac{1}{s^{5} + 2.7 s^{4} + 4.9 s^{3} + 5.4 s^{2} + 3.4 s + 1}$

The zero-pole method

Usually, the transfer function of the controlled object is a high-order model because of the complicated controlled object. Since the first- and second-order transfer functions are relatively mature, we consider adding zeros and poles to the low-order, that is, first- and second-order, transfer function $G_{I} (s)$ to obtain the reference model $G_{R} (s)$ which has the same order as the controlled object.

Assume that $G_{I} (s)$ is as follows

$G_{I} (s) = \frac{s^{m} + B_{m - 1} s^{m - 1} + \dots + B_{1} s + B_{0}}{s^{n} + A_{n - 1} s^{n - 1} + \dots + A_{1} s + A_{0}}$ (1)

where all zeros and poles of the system are in the left half plane and $m \leq n$ .

To make the reference model and controlled object have the same order, it is necessary to add p zeros and q poles to $G_{I} (s)$ , and $G_{R} (s)$ can then be expressed as follows

$G_{R} (s) = \frac{(s^{m} + B_{m - 1} s^{m - 1} + \dots + B_{1} s + B_{0}) (b_{1} s + 1) \dots (b_{p} s + 1)}{(s^{n} + A_{n - 1} s^{n - 1} + \dots + A_{1} s + A_{0}) (a_{1} s + 1) \dots (a_{q} s + 1)}$ (2)

where the parameters $a_{1}, a_{2}, \dots, a_{q} > 0$ and $b_{1}, b_{2}, \dots, b_{p} > 0$ .

To ensure that the response characteristics of $G_{R} (s)$ and $G_{I} (s)$ are consistent to the greatest extent, it is necessary to place the new zeros and poles as far away from the imaginary axis as possible. In addition, $G_{R} (s)$ must adhere to the strictly positive real principle; thus, the following optimization problem mathematical model can be established according to $G_{R} (s)$

$\begin{matrix} min {Z_{1} : = - (\frac{1}{b_{1}} + \frac{1}{b_{2}} + \dots + \frac{1}{b_{p}}) \\ - (\frac{1}{a_{1}} + \frac{1}{a_{2}} + \dots + \frac{1}{a_{q}})} \\ s . t . R_{e} [G_{R} (j ω)] > 0 \end{matrix}$ (3)

The optimization problem can be solved using a multivariable constrained nonlinear programming algorithm.²² The constraints of this method are that the order of the given transfer function must be lower than that of the reference model and the given transfer function must be a positive real function.

The frequency response method

The disadvantage of the zero-pole method is that it constrains the order and positive realism of a given model. To overcome this limitation, we attempt to construct the reference model by matching the frequency response of the given model and the reference model. Inspired by the model simplification,^23,24 we propose the frequency response method.

Assume that the transfer function $G_{I} (s)$ is as follows

$G_{I} (s) = K \frac{d_{p} s^{p} + \dots + d_{1} s + 1}{c_{g} s^{g} + \dots + c_{1} s + 1}$ (4)

where all closed-loop poles of the system are in the left half plane and $p \leq g$ . The pending reference model $G_{R} (s)$ can then be expressed as follows

$G_{R} (s) = K \frac{b_{m} s^{m} + b_{m - 1} s^{m - 1} + \dots + b_{1} s + 1}{a_{n} s^{n} + a_{n - 1} s^{n - 1} + \dots + a_{1} s + 1}$ (5)

To ensure that the reference model and the given model have the same steady-state response, their gain factors must be consistent. The idea is to select the appropriate coefficients $a_{i}$ and $b_{i}$ so that the frequency response of $G_{R} (s)$ is as close to the frequency response of $G_{I} (s)$ as possible.

From the perspective of control, it can be seen from Figure 3 that the relationship between the output $U_{o}$ and the input $U_{i}$ is $U_{o} = U_{i} \cdot G_{I} (s) - U_{i} \cdot G_{R} (s)$ . $G_{I} (s)$ and $G_{R} (s)$ have the close frequency response means that the output $U_{o}$ is close to zero, that is, the value of $G_{R} (s) / G_{I} (s)$ is as close to unity as possible at different frequency points.

Figure 3.

Control block diagram for $G_{I} (s)$ and $G_{R} (s)$ .

We can express $G_{R} (j ω) / G_{I} (j ω) = M (j ω) / N (j ω)$ , and then let $λ (ω) = {| G_{R} (j ω) / G_{I} (j ω) |}^{2} = (M (j ω) M (- j ω)) / (N (j ω) N (- j ω))$ and expand $M (j ω) M (- j ω)$ and $N (j ω) N (- j ω)$ into a Taylor series with $ω = 0$ individually

$λ (ω) = \frac{M (j ω) M (- j ω)}{N (j ω) N (- j ω)} = \frac{M_{0} + M_{2} ω^{2} + M_{4} ω^{4} + \dots}{N_{0} + N_{2} ω^{2} + N_{4} ω^{4} + \dots}$ (6)

In equation (6), because the numerator and denominator of $λ (ω)$ are both even functions, all odd items of $ω$ are eliminated. And $M_{2 q}$ (and similarly $N_{2 q}$ ) in equation (6) is defined as

$M_{2 q} = \frac{1}{(2 q)!} \frac{d^{2 q} [M (j ω) M (- j ω)]}{d ω^{2 q}} |_{ω = 0}$ (7)

According to the Leibniz formula, equation (7) can be further extended to

$eqalign M_{2 q} = \frac{1}{(2 q)!} \sum_{k = 0}^{2 q} (\begin{matrix} 2 q \\ k \end{matrix}) \frac{d^{k} M (j ω)}{d ω^{k}} \frac{d^{2 q - k} M (- j ω)}{d ω^{2 q - k}} |_{ω = 0} = \sum_{k = 0}^{2 q} \frac{1}{k! (2 q - k)!} {(- 1)}^{k} j^{2 q} \frac{d^{k} M (j ω)}{d {(j ω)}^{k}} \frac{d^{2 q - k} M (- j ω)}{d {(- j ω)}^{2 q - k}} |_{ω = 0}, q = 0, 1, 2, \dots$ (8)

Because the derivative is independent of $ω$ , we replace $j ω$ and $- j ω$ with $s$ to obtain the following expression

${\begin{matrix} M_{2 q} = \sum_{k = 0}^{2 q} {(- 1)}^{k + q} \frac{M^{(k)} (s) M^{(2 q - k)} (s)}{k! (2 q - k)!} |_{s = 0}, q = 0, 1, 2, \dots \end{matrix}$ (9)

where $M^{(k)} (s) = (d^{k} / d s^{k}) M (s)$ .

Equation (9) can be simplified as follows

$M_{2 q} = \sum_{k = 0}^{2 q} \frac{{(- 1)}^{k + q} M^{(k)} (0) M^{(2 q - k)} (0)}{k! (2 q - k)!}, q = 0, 1, 2, \dots$ (10)

Similarly

$N_{2 q} = \sum_{k = 0}^{2 q} \frac{{(- 1)}^{k + q} N^{(k)} (0) N^{(2 q - k)} (0)}{k! (2 q - k)!}, q = 0, 1, 2, \dots$ (11)

To ensure $λ (ω) \to 1$ , equation (6) can be written as follows

$M_{0} + M_{2} ω^{2} + M_{4} ω^{4} + \dots = N_{0} + N_{2} ω^{2} + N_{4} ω^{4} + \dots$ (12)

This means that $M_{2 q}$ and $N_{2 q}$ are as close as possible, and equation (12) can be written as

$M_{2 q} - N_{2 q} \to 0, q = 1, 2, \dots$ (13)

The reference model must also satisfy the strictly positive real principle; that is to say, the expressions pertaining to the coefficients $a_{i}$ and $b_{i}$ contain a series of equations and inequalities. To satisfy equation (13), we employ $\sum_{q} {(M_{2 q} - N_{2 q})}^{2}$ as the objective function and establish mathematical models for optimization problems

$\begin{matrix} min {Z_{2} : = \sum_{q} {(M_{2 q} - N_{2 q})}^{2}, q = 1, 2, \dots} \\ s . t . R_{e} [G_{R} (j ω)] > 0 \end{matrix}$ (14)

Experimental results

The zero-pole method

Optimal ITAE

Assume that the reference model is a third-order transfer function that satisfies $n - m = 1$ and the given model is the second-order optimal ITAE standard model $G_{I} (s)$

$G_{I} (s) = \frac{1}{s^{2} + 1.4 s + 1}$ (15)

After adding zeros and poles to the given model, the resulting third-order reference model $G_{R} (s)$ is

$G_{R} (s) = \frac{(b_{1} s + 1) (b_{2} s + 1)}{(a_{1} s + 1) (s^{2} + 1.4 s + 1)}$ (16)

$G_{R} (s)$ must satisfy the positive real principle. Then, the corresponding mathematical model is

$\begin{matrix} min f (x) = - (\frac{1}{b_{1}} + \frac{1}{b_{2}}) - \frac{1}{a_{1}} \\ s . t . {\begin{matrix} a_{1} > 0, b_{1} > 0, b_{2} > 0 \\ b_{1} b_{2} (1.4 a_{1} + 1) - a_{1} (b_{1} + b_{2}) \geq 0 \\ (a_{1} + 1.4) (b_{1} + b_{2}) - (1.4 a_{1} + 1) - b_{1} b_{2} \geq 0 \end{matrix} \end{matrix}$

Using the nonlinear programming function fmincon in MATLAB to solve for the coefficients $a_{i}$ and $b_{i}$ , we obtain the following results

${\begin{matrix} a_{1} = 0.1000 \\ b_{1} = 0.1001 \\ b_{2} = 0.7096 \end{matrix}$

That is, the corresponding reference model is as follows

$\begin{matrix} G_{R} (s) = \frac{(0.1001 s + 1) (0.7096 s + 1)}{(0.1 s + 1) (s^{2} + 1.4 s + 1)} \\ = \frac{0.071 s^{2} + 0.8097 s + 1}{0.1 s^{3} + 1.14 s^{2} + 1.5 s + 1} \end{matrix}$ (17)

Deadbeat response

Similarly, the given model $G_{I} (s)$ is

$G_{I} (s) = \frac{1}{s^{2} + 1.82 s + 1}$ (18)

The resulting third-order reference model $G_{R} (s)$ is

$G_{R} (s) = \frac{(b_{1} s + 1) (b_{2} s + 1)}{(a_{1} s + 1) (s^{2} + 1.82 s + 1)}$ (19)

Finally, the reference model is as follows

$\begin{matrix} G_{R} (s) = \frac{(0.5468 s + 1) (0.1001 s + 1)}{(0.1 s + 1) (s^{2} + 1.82 s + 1)} \\ = \frac{0.0547 s^{2} + 0.6469 s + 1}{0.1 s^{3} + 1.182 s^{2} + 1.92 s + 1} \end{matrix}$ (20)

The frequency response method

Optimal ITAE

Identical to the case of the zero-pole method, the given model $G_{I} (s)$ is

$G_{I} (s) = \frac{{ω_{n}}^{2}}{s^{2} + 1.4 ω_{n} s + {ω_{n}}^{2}} = \frac{1}{s^{2} + 1.4 s + 1}$ (21)

Let the pending reference model $G_{R} (s)$ be

$G_{R} (s) = \frac{b_{2} s^{2} + b_{1} s + 1}{a_{3} s^{3} + a_{2} s^{2} + a_{1} s + 1}$ (22)

According to the positive real principle, the corresponding mathematical model can be expressed as follows

$\begin{array}{l} \min f (x) = {[{(b_{1} + 1.4)}^{2} - 2 (1.4 b_{1} + b_{2} + 1) - ({a_{1}}^{2} - 2 a_{2})]}^{2} \\ + {[2 b_{2} - 2 (b_{1} + 1.4) (b_{1} + 1.4 b_{2}) + {(1.4 b_{1} + b_{2} + 1)}^{2} - ({a_{2}}^{2} - 2 a_{1} a_{3})]}^{2} \\ + {[({(b_{1} + 1.4 b_{2})}^{2} - 2 b_{2} (1.4 b_{1} + b_{2} + 1) - {a_{3}}^{2})]}^{2} + {({b_{2}}^{2})}^{2} \\ s . t . {\begin{cases} a_{1} > 0, a_{2} > 0, a_{3} > 0 \\ b_{1} > 0, b_{2} > 0 \\ a_{1} b_{1} - a_{2} - b_{2} \geq 0 \\ a_{2} b_{2} - a_{3} b_{1} \geq 0 \end{cases} \end{array}$ (23)

Using fmincon to solve for the coefficients $a_{i}$ and $b_{i}$ , the following reference model is obtained

$G_{R} (s) = \frac{0.3152 s^{2} + 1.0494 s + 1}{0.5421 s^{3} + 1.8047 s^{2} + 2.0201 s + 1}$ (24)

Deadbeat response

The given model $G_{I} (s)$ is

$G_{I} (s) = \frac{1}{s^{2} + 1.82 s + 1}$ (25)

Let the pending reference model $G_{R} (s)$ be

$G_{R} (s) = \frac{b_{2} s^{2} + b_{1} s + 1}{a_{3} s^{3} + a_{2} s^{2} + a_{1} s + 1}$ (26)

The reference model is then as follows

$G_{R} (s) = \frac{0.3364 s^{2} + 1.0547 s + 1}{0.7464 s^{3} + 2.3399 s^{2} + 2.5374 s + 1}$ (27)

Discussion

Data comparison between the two methods

The ITAE step response, illustrated in Figure 4(a), shows that the response of the reference model obtained using the frequency response method is closer to that of the given model than using the zero-pole method. It can be seen from Figure 4(b) that the error of the frequency response method is considerably lower than that of the zero-pole method. The deadbeat response shown in Figure 4(c) and (d) shows similar results.

Figure 4.

Comparison of step response between the two methods: (a) optimal ITAE unit step response and (b) the error in this response; (c) deadbeat unit step response and (d) the error in this response.

Table 3 summarizes the ITAE deviations of the two methods. Based on the output of the ITAE model, the ITAE deviation of the frequency response method is 28.4% smaller than that of the zero-pole method. Similarly, based on the output of the deadbeat model, the ITAE deviation of the frequency response method is 33.6% smaller than that of the zero-pole method.

Table 3.

ITAE deviation of the two methods.

Construction	Given model	Reference model	ITAE deviation
Zero-pole	$\frac{1}{s^{2} + 1.4 s + 1}$	$\frac{0.071 s^{2} + 0.8097 s + 1}{0.1 s^{3} + 1.14 s^{2} + 1.5 s + 1}$	1.4048
Frequency response	$\frac{1}{s^{2} + 1.4 s + 1}$	$\frac{0.3152 s^{2} + 1.0494 s + 1}{0.5421 s^{3} + 1.8047 s^{2} + 2.0201 s + 1}$	1.0058
Zero-pole	$\frac{1}{s^{2} + 1.82 s + 1}$	$\frac{0.0547 s^{2} + 0.6469 s + 1}{0.1 s^{3} + 1.182 s^{2} + 1.92 s + 1}$	0.8407
Frequency response	$\frac{1}{s^{2} + 1.82 s + 1}$	$\frac{0.3364 s^{2} + 1.0547 s + 1}{0.7462 s^{3} + 2.3399 s^{2} + 2.5374 s + 1}$	0.5580

ITAE: integrated time and absolute error.

Analysis of experimental results

From section “Data comparison between the two methods,” the error of the reference model constructed using the zero-pole method is larger than that for the frequency response method. The reasons for this difference are analyzed in this section.

The main concept of the zero-pole method is to ensure that the added zeros and poles have the least effect possible on the frequency response. However, the given model is constrained to meet the positive real principle, which limits the positional distribution of the added zeros and poles.

Although the frequency response method demands that the sum of the frequency response deviations between the reference model and the given model be at a minimum, this model offers a wider range of optimal solutions of the reference model.

To compare the optimization space between the two methods, we take the second-order reference model as an example and draw the feasible domain of the two methods.

The given model is selected as the first-order optimal ITAE standard model

$G_{I} (s) = \frac{1}{s + 1}$ (28)

According to the zero-pole method, the reference model is set as follows

$G_{R 1} (s) = \frac{b_{1} s + 1}{(a_{2} s + 1) (s + 1)}$ (29)

To ensure that the reference model satisfies the positive real constraint listed in Table 1 through the added zeros and poles, the coefficients $a_{2}$ and $b_{1}$ are subject to

${\begin{matrix} a_{2}, b_{1} > 0 \\ a_{2} b_{1} + b_{1} - a_{2} > 0 \end{matrix}$

The reference model constructed using the frequency response method is as follows

$G_{R 2} (s) = \frac{b_{1} s + 1}{a_{2} s^{2} + a_{1} s + 1}$ (30)

According to Table 1, the ranges of $a_{1}, a_{2}$ , and $b_{1}$ are

${\begin{matrix} a_{1}, a_{2}, b_{1} > 0 \\ a_{1} b_{1} - a_{2} > 0 \end{matrix}$

Figure 5(a) illustrates the feasible domain of $G_{R 1} (s)$ . There are no upper limits to the values of $a_{2}$ and $b_{1}$ . We set their maximum values to 5 to indicate the feasible domain. Because there are only two coefficients in $G_{R 1} (s)$ , the feasible domain of $G_{R 1} (s)$ is a two-dimensional plane, and the optimization space is small. By contrast, the added zero $- 1 / b_{1}$ and pole $- 1 / a_{2}$ cannot be infinitely far away from the imaginary axis owing to the positive real constraint on the given model, so the error between the reference model and the given model of the zero-pole method is comparatively large.

Figure 5.

Feasible domain of the reference model constructed using the two methods: (a) zero-pole method and (b) frequency response method.

Figure 5(b) shows the feasible domain of $G_{R 2} (s)$ . $G_{R 2} (s)$ contains three coefficients, so its feasible domain is a three-dimensional space. Compared with the zero-pole method, the frequency response method can select the optimal solution over a wider range, and thus the error between the reference model and the given model is smaller when using this method.

Conclusion

Very little work has been conducted on constructing a reference model for MRAC. In this work, we attempted to design general construction methods for the reference model. The specific contributions of this work are as follows.

We deduce the first- to fourth-order positive real constraints of $n - m = 1$ and consider two specific performance specifications as examples: ITAE and deadbeat response. For each specification, we propose the zero-pole method and the frequency response method.

By adding zeros and poles as far away from the imaginary axis as possible, we establish a nonlinear optimization model using the zero-pole method and constrain it using the positive real principle. By setting the frequency response of the reference model to be as close as possible to that of the given model, we also establish a nonlinear optimization model with the frequency response method.

Our experimental results show that both methods are feasible. The zero-pole method is suitable only in cases where the order of the given transfer function is lower than that of the reference model and the given transfer function is positive real, but its algorithm is simple and easy to implement. The frequency response method is more accurate and versatile than the zero-pole method, and the underlying reasons for this have been analyzed.

Footnotes

Handling Editor: Dumitru Baleanu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China (Grant No. 51705214) and the National Quality Technology Infrastructure Construction Project of Zhejiang Quality Supervision System (Grant No. 20180129).

ORCID iD

Shi-jie Su

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