Modelling and control of nonlinear resonating processes: part I—system identification using orthogonal basis function

Abstract

Resonating systems show oscillatory characteristics. System identification of resonating systems and design of their model based control strategy always draw special attention. This work presents a Wiener type system identification technique for nonlinear resonating systems. Orthogonal basis function (henceforth termed as OBF) is employed to capture the linear dynamic part of the Wiener structure while the static nonlinear mapping is described by two means, viz., wavelet decomposition and least squares support vector machine. Use of OBF leads to a parsimonious nature in the resulting nonlinear model. Two types of OBFs have been used in this work viz. Laguerre filter and Kautz filter. The Kautz filter has capability of modelling systems with complex conjugate poles. A case study has been performed with continuous stirred tank reactor (henceforth referred as CSTR) which is a reasonably nonlinear resonating systems. Degree of nonlinearity as well as resonance increases with series-connected CSTR. Simulations are carried out using $\hbox {MATLAB}^\circledR$ software in order to evaluate the performances of various Wiener structures, and identify the OBF–Wiener model best suited for designing a model based controller.

Appendix: Detailed derivation of Kautz representation

Using Theorem 1 the state space Kautz model given in Eqs. (8) and (9) can be derived in the following state space form (q is shift operator).

$$\begin{aligned} \varphi _{1}(k)&=\Psi _{1}( q) u(k) \nonumber \\&=C_{1}^{( 1) }\left( 1-a_{1}^{( 1) }q\right) \Gamma ^{( 1) }( q) u(k) \nonumber \\&=C_{1}^{( 1) }\left( 1-a_{1}^{( 1) }q\right) \dfrac{1}{(q-\beta _{1}) \left( q-\beta _{1}^{*}\right) }u(k) \nonumber \\&=C_{1}^{( 1) }\left( q^{-1}-a_{1}^{( 1) }\right) \dfrac{q}{q^{2}+h_{1}^{( 1) }q+h_{2}^{(1) }}u(k) \end{aligned}$$

(70)

where

$$\begin{aligned} h_{1}^{( 1) }&=-\left( \beta _{1}+\beta _{1}^{*}\right) \end{aligned}$$

(71)

$$\begin{aligned} h_{2}^{( 1) }&=\beta _{1}\beta _{1}^{*} \end{aligned}$$

(72)

Define the following states

$$\begin{aligned} x_{1}(k)&=\dfrac{q}{q^{2}+h_{1}^{( 1) } q+h_{2}^{( 1) }}u(k) \end{aligned}$$

(73)

$$\begin{aligned} x_{2}(k)&=x_{1}( k-1) \end{aligned}$$

(74)

Using Eqs. (73) and (74) in Eq. (70) one obtains

$$\begin{aligned} \varphi _{1}(k) =C_{1}^{( 1) }\left[ x_{2}( k) -a_{1}^{( 1) }x_{1}(k) \right] \end{aligned}$$

(75)

Similarly

$$\begin{aligned} \varphi _{2}(k)&=\Psi _{2}\left( q\right) u(k) \nonumber \\&=C_{2}^{\left( 1\right) }\left( 1-a_{2}^{\left( 1\right) }q\right) \Gamma ^{\left( 1\right) }\left( q\right) u(k) \nonumber \\&=\cdots =C_{2}^{( 1) }\left[ x_{2}(k) -a_{2}^{\left( 1\right) }x_{1}(k) \right] \end{aligned}$$

(76)

Further

$$\begin{aligned} \varphi _{3}(k)&=\Psi _{3}\left( q\right) u(k)\nonumber \\&=C_{1}^{\left( 2\right) }\left( 1-a_{1}^{\left( 2\right) }q\right) \Gamma ^{\left( 2\right) }\left( q\right) u(k) \nonumber \\&=C_{1}^{\left( 2\right) }\left( 1-a_{1}^{\left( 2\right) }q\right) \dfrac{\left( 1-\beta _{1}q\right) \left( 1-\beta _{1}^{*}q\right) u(k) }{\left( q-\beta _{2}\right) \left( q-\beta _{2}^{*}\right) \left( q-\beta _{1}\right) \left( q-\beta _{1}^{*}\right) }\nonumber \\&=C_{1}^{\left( 2\right) }\left( q^{-1}-a_{1}^{\left( 2\right) }\right) \nonumber \\&\quad \times \left( \dfrac{h_{2}^{\left( 1\right) }q^{2}+h_{1}^{\left( 1\right) }q+1}{q^{2}+h_{1}^{\left( 2\right) }q+h_{2}^{\left( 2\right) }}\right) \left( \dfrac{q}{q^{2}+h_{1}^{\left( 1\right) }q+h_{2}^{\left( 1\right) }}\right) u(k) \nonumber \\&=C_{1}^{\left( 2\right) }\left( q^{-1}-a_{1}^{\left( 2\right) }\right) \dfrac{h_{2}^{\left( 1\right) }q^{2}+h_{1}^{\left( 1\right) }q+1}{q^{2}+h_{1}^{\left( 2\right) }q+h_{2}^{\left( 2\right) }} x_{1}(k) \end{aligned}$$

(77)

where

$$\begin{aligned} h_{1}^{( 2) }&=-\left( \beta _{2}+\beta _{2}^{*}\right) \end{aligned}$$

(78)

$$\begin{aligned} h_{2}^{( 2) }&=\beta _{2}\beta _{2}^{*} \end{aligned}$$

(79)

Define the following states

$$\begin{aligned} x_{3}(k)&=\dfrac{h_{2}^{(1) }q^{2} +h_{1}^{( 1) }q+1}{q^{2}+h_{1}^{(2) } q+h_{2}^{(2) }}\times x_{1}(k) \end{aligned}$$

(80)

$$\begin{aligned} x_{4}(k)&=x_{3}( k-1) \end{aligned}$$

(81)

Using Eqs. (80) and (81) in Eq. (77), one obtains the regressors, similar to Eqs. (75) and (76) as

$$\begin{aligned} \varphi _{3}(k)&=C_{1}^{(2) }\left[ x_{4}(k) -a_{1}^{(2) }x_{3}(k) \right] \end{aligned}$$

(82)

$$\begin{aligned} \varphi _{4}(k)&=C_{2}^{(2) }\left[ x_{4}(k) -a_{2}^{(2) }x_{3}(k) \right] \end{aligned}$$

(83)

The derivation of regressors can further be continued and a generalized expression can be given as in Eqs. (11)–(16).

Further, upon applying inverse transformation on the Eq. (73) and thereupon using Eq. (74), following discrete time domain state space expression can be obtained

$$\begin{aligned} x_{1}( k+1)&=-h_{1}^{(1)}x_{1}(k) -h_{2} ^{(1)}x_{1}\left( k-1\right) +u(k)\nonumber \\&=-h_{1}^{(1)}x_{1}(k) -h_{2}^{(1)}x_{2}(k) +u(k) \end{aligned}$$

(84)

$$\begin{aligned} x_{2}\left( k+1\right)&=x_{1}(k) \end{aligned}$$

(85)

Similarly, upon applying inverse transformation the above Eq. (80) and thereupon using Eqs. (81) and (84)

$$\begin{aligned} x_{3}\left( k+1\right)&=-h_{1}^{\left( 2\right) }x_{3}\left( k\right) -h_{2}^{\left( 2\right) }x_{3}\left( k-1\right) +h_{2} ^{(1)}x_{1}\left( k+1\right) \nonumber \\&\quad +h_{1}^{(1)}x_{1}(k) +x_{1}\left( k-1\right) \nonumber \\&=\cdots =h_{1}^{(1)}\left( 1-h_{2}^{(1)}\right) x_{1}(k) +\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} x_{2}(k) \nonumber \\&-h_{1}^{\left( 2\right) }x_{3}(k) -h_{2}^{(2)} x_{4}(k)+h_{2}^{(1)}u(k) \end{aligned}$$

(86)

$$\begin{aligned} x_{4}\left( k+1\right)&=x_{3}(k) \end{aligned}$$

(87)

Similar to Eqs. (80) and (81),

$$\begin{aligned} x_{5}(k)&=\dfrac{h_{2}^{(2)}q^{2}+h_{1}^{(2)}q+1}{q^{2}+h_{1}^{\left( 3\right) }q+h_{2}^{\left( 3\right) }}\times x_{3}(k) \end{aligned}$$

(88)

$$\begin{aligned} x_{6}(k)&=x_{5}\left( k-1\right) \end{aligned}$$

(89)

and applying inverse transformation on Eq. (88) and thereupon using Eqs. (89) and (86)

$$\begin{aligned} x_{5}\left( k+1\right)&=h_{1}^{\left( 1\right) }h_{2}^{(2)}\left( 1-h_{2}^{\left( 1\right) }\right) x_{1}(k) \nonumber \\&\quad + h_{2}^{(2)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} x_{2}(k)+h_{1}^{(2)}\left( 1-h_{2}^{(2)}\right) x_{3}(k) \nonumber \\&\quad +\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} x_{4}(k)-h_{1} ^{\left( 3\right) }x_{5}(k) -h_{2}^{\left( 3\right) } x_{6}(k) \nonumber \\&\quad +h_{2}^{(2)}h_{2}^{(1)}u(k) \end{aligned}$$

(90)

$$\begin{aligned} x_{6}\left( k+1\right)&=x_{5}(k) \end{aligned}$$

(91)

Likewise,

$$\begin{aligned} x_{7}\left( k+1\right)&=\left[ \begin{array}[c]{c} h_{1}^{\left( 1\right) }h_{2}^{(2)}h_{2}^{(3)}\left( 1-h_{2}^{\left( 1\right) }\right) x_{1}(k) \\ \quad +h_{2}^{(2)}h_{2}^{(3)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} x_{2}(k)\\ \quad +h_{1}^{(2)}h_{2}^{(3)}\left( 1-h_{2}^{(2)}\right) x_{3}(k) \\ \quad +h_{2}^{(3)}\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} x_{4}(k)\\ \quad +h_{1}^{\left( 3\right) }\left( 1-h_{2}^{(3)}\right) x_{5}(k) \\ \quad +\left\{ 1-\left( h_{2}^{\left( 3\right) }\right) ^{2}\right\} x_{6}(k) \\ \quad -h_{1}^{\left( 4\right) }x_{7}(k) -h_{2}^{\left( 4\right) }x_{8}(k) \\ \quad +h_{2}^{(3)}h_{2}^{(2)}h_{2}^{(1)}u(k) \end{array} \right] \end{aligned}$$

(92)

$$\begin{aligned} x_{8}\left( k+1\right)&=x_{7}(k) \end{aligned}$$

(93)

and

$$\begin{aligned} x_{9}\left( k+1\right)&=\left[ \begin{array}[c]{c} h_{1}^{\left( 1\right) }h_{2}^{(2)}h_{2}^{(3)}h_{2}^{(4)}\left( 1-h_{2}^{\left( 1\right) }\right) x_{1}(k) \\ \quad + h_{2}^{(2)}h_{2}^{(3)}h_{2}^{(4)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} x_{2}(k)\\ \quad +h_{1}^{(2)}h_{2}^{(3)}h_{2}^{(4)}\left( 1-h_{2}^{(2)}\right) x_{3}\left( k\right) \\ \quad + h_{2}^{(3)}h_{2}^{(4)}\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} x_{4}(k)\\ \quad + h_{1}^{\left( 3\right) }h_{2}^{(4)}\left( 1-h_{2}^{(3)}\right) x_{5}(k)\\ \quad + h_{2}^{(4)}\left\{ 1-\left( h_{2}^{\left( 3\right) }\right) ^{2}\right\} x_{6}(k)\\ h_{1}^{(4)}\left\{ 1-h_{2}^{(4)}\right\} x_{7}(k)\\ \quad + \left\{ 1-\left( h_{2}^{\left( 4\right) }\right) ^{2}\right\} x_{8}(k) \\ -h_{1}^{\left( 5\right) }x_{9}(k) -h_{2}^{\left( 5\right) }x_{10}(k) \\ \quad + h_{2}^{(4)}h_{2}^{(3)}h_{2}^{(2)}h_{2}^{(1)}u(k) \end{array} \right] \end{aligned}$$

(94)

$$\begin{aligned} x_{10}\left( k+1\right)&=x_{9}(k) \end{aligned}$$

(95)

The above expressions in Eqs. (84)–(90), (91)–(95) can further be represented in a compact matrix form as,

$$\begin{aligned} \begin{bmatrix} x_{1}(k+1)\\ x_{2}(k+1)\\ x_{3}(k+1)\\ x_{4}(k+1)\\ x_{5}(k+1)\\ x_{6}(k+1)\\ x_{7}(k+1)\\ x_{8}(k+1)\\ x_{9}(k+1)\\ x_{10}(k+1) \end{bmatrix} =\mathbf {\digamma }\times \begin{bmatrix} x_{1}(k)\\ x_{2}(k)\\ x_{3}(k)\\ x_{4}(k)\\ x_{5}(k)\\ x_{6}(k)\\ x_{7}(k)\\ x_{8}(k)\\ x_{9}(k)\\ x_{10}(k) \end{bmatrix} + \begin{bmatrix} 1\\ 0\\ h_{2}^{(1)}\\ 0\\ h_{2}^{(1)}h_{2}^{(2)}\\ 0\\ h_{2}^{(1)}h_{2}^{(2)}h_{2}^{(3)}\\ 0\\ h_{2}^{(1)}h_{2}^{(2)}h_{2}^{(3)}h_{2}^{(4)}\\ 0 \end{bmatrix} \times u(k) \end{aligned}$$

and

$$\begin{aligned} \mathbf {\digamma }=\left[ \begin{array}[c]{cccc} \digamma _{1}&\digamma _{2}&\digamma _{3}&\digamma _{4} \end{array} \right] \end{aligned}$$

where

$$\begin{aligned} \digamma _{1}= \begin{bmatrix} -h_{1}^{(1)}&\quad -h_{2}^{(1)}\\ 1&\quad 0\\ h_{1}^{\left( 1\right) }\left( 1-h_{2}^{\left( 1\right) }\right)&\quad 1-\left( h_{2}^{(1)}\right) ^{2}\\ 0&\quad 0\\ h_{1}^{\left( 1\right) }h_{2}^{(2)}\left( 1-h_{2}^{\left( 1\right) }\right)&\quad h_{2}^{(2)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} \\ 0&\quad 0\\ h_{1}^{\left( 1\right) }\prod \limits _{i=2}^{3}h_{2}^{(i)}\left( 1-h_{2}^{\left( 1\right) }\right)&\quad \prod \limits _{i=2}^{3}h_{2} ^{(i)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} \\ 0&\quad 0\\ h_{1}^{\left( 1\right) }\prod \limits _{i=2}^{4}h_{2}^{(i)}\left( 1-h_{2}^{\left( 1\right) }\right)&\quad \prod \limits _{i=2}^{4}h_{2} ^{(i)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} \\ 0&\quad 0 \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} \digamma _{2}= \begin{bmatrix} 0&\quad 0\\ 0&\quad 0\\ -h_{1}^{(2)}&\quad -h_{2}^{(2)}\\ 1&\quad 0\\ h_{1}^{(2)}\left( 1-h_{2}^{(2)}\right)&\quad 1-\left( h_{2}^{(2)}\right) ^{2}\\ 0&\quad 0\\ h_{1}^{(2)}h_{2}^{(3)}\left( 1-h_{2}^{(2)}\right)&\quad h_{2}^{(3)}\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} \\ 0&\quad 0\\ h_{1}^{(2)}\prod \limits _{i=3}^{4}h_{2}^{(i)}\left( 1-h_{2}^{(2)}\right)&\quad \prod \limits _{i=3}^{4}h_{2}^{(i)}\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} \\ 0&\quad 0 \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} \digamma _{3}= \begin{bmatrix} 0&\quad 0\\ 0&\quad 0\\ 0&\quad 0\\ 0&\quad 0\\ -h_{1}^{\left( 3\right) }&\quad -h_{1}^{\left( 3\right) }\\ 1&\quad 0\\ h_{1}^{(3)}\left( 1-h_{2}^{(3)}\right)&\quad 1-\left( h_{2}^{\left( 3\right) }\right) ^{2}\\ 0&\quad 0\\ h_{1}^{\left( 3\right) }h_{2}^{(4)}\left( 1-h_{2}^{(3)}\right)&\quad h_{2}^{(4)}\left\{ 1-\left( h_{2}^{\left( 3\right) }\right) ^{2}\right\} \\ 0&\quad 0 \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} \digamma _{4}= \begin{bmatrix} 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ -h_{1}^{\left( 4\right) }&\quad -h_{2}^{\left( 4\right) }&\quad 0&\quad 0\\ 1&\quad 0&\quad 0&\quad 0\\ h_{1}^{(4)}\left\{ 1-h_{2}^{(4)}\right\}&\quad 1-\left( h_{2}^{\left( 4\right) }\right) ^{2}&\quad -h_{1}^{\left( 5\right) }&\quad -h_{2}^{\left( 5\right) }\\ 0&\quad 0&\quad 1&\quad 0 \end{bmatrix} \end{aligned}$$

The complete state space model can be derived as given in Eq. (20) whose $(2n-1)$th row will be

$$\begin{aligned} x_{2n-1}\left( k+1\right)&=\left[ \begin{array}[c]{c} h_{1}^{\left( 1\right) }\prod \limits _{i=2}^{n-1}h_{2}^{(i)}\left( 1-h_{2}^{\left( 1\right) }\right) \\ \prod \limits _{i=2}^{n-1}h_{2}^{(i)}\left\{ 1-\left( h_{2}^{(1)}\right) ^{2}\right\} \\ h_{1}^{(2)}\prod \limits _{i=3}^{n-1}h_{2}^{(i)}\left( 1-h_{2}^{(2)}\right) \\ \prod \limits _{i=3}^{n-1}h_{2}^{(i)}\left\{ 1-\left( h_{2}^{(2)}\right) ^{2}\right\} \\ h_{1}^{\left( 3\right) }\prod \limits _{i=4}^{n-1}h_{2}^{(i)}\left( 1-h_{2}^{\left( 3\right) }\right) \\ \prod \limits _{i=4}^{n-1}h_{2}^{(i)}\left\{ 1-\left( h_{2}^{(3)}\right) ^{2}\right\} \\ \vdots \\ h_{1}^{\left( n-1\right) }\left( 1-h_{2}^{\left( n-1\right) }\right) \\ 1-\left( h_{2}^{\left( n-1\right) }\right) ^{2}\\ -h_{1}^{\left( n\right) }\\ -h_{2}^{\left( n\right) } \end{array} \right] ^{T}\left[ \begin{array}[c]{c} x_{1}(k) \\ x_{2}(k) \\ x_{3}(k) \\ x_{4}(k) \\ x_{5}(k) \\ x_{6}(k) \\ \vdots \\ x_{2\left( n-1\right) -1}(k) \\ x_{2\left( n-1\right) }(k) \\ x_{2n-1}(k) \\ x_{2n}(k) \end{array} \right] \nonumber \\&\quad +\left\{ \prod \limits _{i=1}^{n-1}h_{2}^{(i)}\right\} u(k) \end{aligned}$$

(96)

and

$$\begin{aligned} x_{2n}\left( k+1\right) =x_{2n-1}(k) \end{aligned}$$

(97)

which would eventually yield the Eqs. (24)–(26).