Design of Nonlinear Selectively Invariant Control Systems Based on Quasilinear Models

Abstract

An original analytical method is developed to design the selectively invariant control systems for nonlinear plants with differentiable nonlinearities. To solve the design problem, the method of designing nonlinear control systems is applied on the base of a quasilinear model of nonlinear plants and the internal models principle of external impacts is used, taking into account the requirements for the relative order of the control device and the fast response of the designed system. The system of linear algebraic equations is solved to determine the parameters of the nonlinear control device. The suggested method can be applied to design the control systems for nonlinear plants of various purposes, operating under conditions of regular external impacts of the known form.

APPENDIX

Mathematical models of impacts are homogeneous differential equations of a certain order, may be in combination with the algebraic ones [7–13, 27]. For example, the model of impact  f(t) = f₀1(t) will be the equations ${{\dot {x}}_{f}}$(t) = 0, x_f(0) = f₀,  f(t) = x_f(t), where x_f(0) is an initial condition. The model of the harmonic impact f(t) = f_m sin(ω_f t + ϕ_f) with frequency ω_f, arbitrary amplitude  f_m and phase ϕ_f will be the equations ${{\dot {x}}_{{f1}}}$ = $- \omega _{f}^{2}{{x}_{{f2}}}$, ${{\dot {x}}_{{f2}}}$ = x_f1,  f = r₁x_f 1 + r₂x_f 2 with the initial conditions x_f 10 and x_f 20. Here, r₁, r₂ are some constants.

To parry the effect of the external impact on the system error, it is sufficient to have only its spectral model in system, which unambiguously describes its shape, using only its spectrum. In the general case, the spectral model of the impact g(t) can be represented either by the equation in state variables ${{\dot {x}}_{g}}$ = Gx_g, where G and x_g are numerical matrix and state vector, or by K_p-image, i.e., by the polynomial G(p) = det(pE – G), where p = d/dt. We emphasize that the polynomial G(p) at p = D is a Kulebakin K(D)-image of this impact [7], i.e., the representations of the spectral model by the K_p-image or by the equations in the Cauchy form are equivalent [27].

An important property of the impact’s K_p-image is that the product of the K_p-image on this impact is equal to zero [7] for all t ≥ 0. For example, if the impact φ₁(t) = φ₀ exp(λ_φt), then its K_p-image Φ₁(p) = p – λ_φ, and product Φ₁(p)φ(t) = (p – λ_φ)φ₀ exp(λ_φt) = φ₀[(d exp(λ_φt)/dt) – λ_φ exp(λ_φt)] ≡ 0 for the bounded φ₀, since d exp(λ_φt)/dt = λ_φ exp(λ_φt).

The equation ${{\dot {x}}_{{\tilde {f}}}}$ = $\tilde {F}{{x}_{{\tilde {f}}}}$, where the matrix $\tilde {F}$ = diag{0 ${{\lambda }_{{\tilde {f}}}}$} is a spectral model of the impact $\tilde {f}$(t) = ${{\tilde {f}}_{0}}1$(t) + ${{\tilde {f}}_{e}}$ exp(${{\lambda }_{{\tilde {f}}}}t$), 0 ≤ t < ∞, where ${{\tilde {f}}_{0}}$ and ${{\tilde {f}}_{e}}$ are bounded constants. The K_p-image of this impact is a polynomial $\tilde {F}$(p) = p² – ${{\lambda }_{{\tilde {f}}}}p$. It is easy to verify that (p² – ${{\lambda }_{{\tilde {f}}}}p)\tilde {f}(t)$ ≡ 0. It follows from the above examples that the K_p-image of the sum of impacts is equal to the product of the K_p-images of each of them. Note also that the K_p-image of the impact f(t) can be easily found from the table of the Laplace images [25, p. 29]: it is equal to the denominator of its image  f(s) at s = p. The coefficients of the K_p-images or the coefficients of equations in the Cauchy form of impacts are the spectrum-setting parameters of their models.

Derivation of the “input–output” equation of the closed system. Differential Eq. (7) in the operator form can be written as [pE – H(x)]w = h(x)g + h_f(x)f. From here w = [pE – H(x)]^–1 × {h(x)g + h_f  (x)f }. Taking into account the equality [pE – H(x)]^–1 = adj[pE – H(x)]/det[pE – H(x)] and substituting this expression into second Eq. (7), we obtain Eq. (9), where

$$H(p,x) = \det [pE - H(x)],$$

(A.1)

$${{H}_{g}}(p,x) = \left[ {{{c}^{{\text{T}}}}(x)\;{{{{\mathbf{\bar {0}}}}}^{{\text{T}}}}} \right]\operatorname{adj} [pE - H(x)]h(x),$$

(A.2)

$${{H}_{f}}(p,x) = \left[ {{{c}^{{\text{T}}}}(x)\;{{{{\mathbf{\bar {0}}}}}^{{\text{T}}}}} \right]\operatorname{adj} [pE - H(x)]{{h}_{f}}(x).$$

(A.3)

Here the matrix pE – H(x) is defined by the expression

$$pE - H(x) = \left[ {\begin{array}{*{20}{c}} {pE - A(x)}&{ - b(x){{k}^{{\text{T}}}}(x)} \\ {\Pi (x)}&{pE - R(x)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\tilde {A}}&{\tilde {B}} \\ {\tilde {C}}&{\tilde {D}} \end{array}} \right].$$

(A.4)

Let us show that the operators of Eq. (9) are directly related by expressions (10)–(13) with operators (14)–(16) of the equations “input–output” of quasilinear models (1) and (3). Expressions (14)–(16) are derived from indicated Eqs. (1) and (3) in exactly the same way as the above derivation of Eq. (9). In the general case, NCD output Eq. (3) can have the form u = k^T(x)z + λ_r(x)y + $\sum\nolimits_{i = 1}^q {{{{\tilde {\lambda }}}_{{ir}}}(x){{{\tilde {x}}}_{i}}}$. In this case, the calculations below will become much more complicated, but their meaning will not change [22, pp. 349–353]. Therefore, for greater clarity, it is further assumed that λ_r(x) ≡ 0 and ${{\tilde {\lambda }}_{{ir}}}$(x) ≡ 0, i = $\overline {1,q}$.

Derivation of operator H(p, x) (10). Accordance to the formula (A.8), given in [28, p. 223], the expression: H(p, x) = det[pE – H(x)] = $\det \tilde {A}\det (\tilde {D}$ – $\tilde {C}{{\tilde {A}}^{{ - 1}}}\tilde {B})$ follows from (A.4). From here, taking into account the notation (A.4), we derive the equality:

$$H(p,x) = \det [pE - A(x)]\det \left\{ {pE - R(x) + \Pi (x){{{[pE - A(x)]}}^{{ - 1}}}b(x){{k}^{{\text{T}}}}(x)} \right\}.$$

Since [pE – A(x)]^–1 = adj[pE – A(x)]/det[pE – A(x)], then, taking into account (14), (16) and the notation Π(x), we have

$$H(p,x) = A(p,x)\det \left[ {pE - R(x) + {{\psi }_{l}}(p,x)l(x){{k}^{{\text{T}}}}(x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{l}_{i}}(x){{k}^{{\text{T}}}}(x)} } \right].$$

(A.5)

Here, it is indicated that

$${{\psi }_{l}}(p,x) = B(p,x){\text{/}}A(p,x),\quad {{\psi }_{i}}(p,x) = {{V}_{i}}(p,x){\text{/}}A(p,x).$$

(A.6)

Applying identity (A.25) from [28, p. 233] to the second factor in (A.5) and taking into account (15), we get:

$$H(p,x) = A(p,x)\left[ \begin{aligned} R(p,x) + {{\psi }_{l}}(p,x){{k}^{{\text{T}}}}(x)\operatorname{adj} [pE - R(x)]l(x) \\ + \;\sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{k}^{{\text{T}}}}(x)\operatorname{adj} [pE - R(x)]{{l}_{i}}(x)} \\ \end{aligned} \right].$$

From here, with taking into account the notation (A.6), (15), operator (10) follows.

Derivation of operator H_g(p, x) (11). For this purpose, we use formula (A.12) from [28, p. 223], which for the block matrix (A.4) allows us to write the equality:

$$\operatorname{adj} [pE - H(x)] = \left[ {\begin{array}{*{20}{c}} {\det M\operatorname{adj} \tilde {A} + {{\alpha }^{{ - 1}}}(\operatorname{adj} A)\tilde {B}(\operatorname{adj} M)\tilde {C}(\operatorname{adj} \tilde {A})}&{ - (\operatorname{adj} \tilde {A})\tilde {B}(\operatorname{adj} M)} \\ { - (\operatorname{adj} M)\tilde {C}(\operatorname{adj} \tilde {A})}&{\alpha \operatorname{adj} M} \end{array}} \right],$$

(A.7)

where α = det $\tilde {A}$ ≠ 0, M = $\tilde {D}$ – $\tilde {C}{{\tilde {A}}^{{ - 1}}}\tilde {B}$. Substituting expressions (A.7) and vector h(x) from (8) into (A.2), and taking into account notation (14), we obtain the following equality:

$${{H}_{g}}(p,x) = {{c}^{{\text{T}}}}(x)\operatorname{adj} [pE - A(x)]b(x){{k}^{{\text{T}}}}(x)\operatorname{adj} Mq(x) = B(p,x){{k}^{{\text{T}}}}(x)\operatorname{adj} Mq(x).$$

(A.8)

Since the matrix M = $\tilde {D}$ – $\tilde {C}{{\tilde {A}}^{{ - 1}}}\tilde {B}$, then, taking into account the notation (A.4), we derive

$$M = pE - R(x) + {{A}^{{ - 1}}}(p,x)\left\{ {l(x){{c}^{{\text{T}}}}(x) + \sum\limits_{i = 1}^q {{{l}_{i}}(x){{e}_{i}}\operatorname{adj} [pE - R(x)]} } \right\}b(x){{k}^{{\text{T}}}}(x).$$

Opening the curly brackets here and taking into account the notation (A.6), we obtain

$$M = pE - R(x) + {{\psi }_{l}}(p,x)l(x){{k}^{{\text{T}}}}(x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{l}_{i}}(x){{k}^{{\text{T}}}}(x).} $$

(A.9)

Consequently, the product k^T(x)adj M q(x) in Eq. (A.8) has the form

$${{k}^{{\text{T}}}}(x)\operatorname{adj} Mq(x) = {{k}^{{\text{T}}}}(x)\operatorname{adj} \left[ {pE - R(x) + {{\psi }_{l}}(p,x)l(x){{k}^{{\text{T}}}}(x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{l}_{i}}(x){{k}^{{\text{T}}}}(x)} } \right]q(x).$$

Hence, by formula (A.27) from [28, p. 233] and third notation (15), we have

$${{k}^{{\text{T}}}}(x)\operatorname{adj} Mq(x) = {{k}^{{\text{T}}}}(x)\operatorname{adj} [pE - R(x)]q(x) = Q(p,x).$$

(A.10)

Substituting this equality into expression (A.8), we obtain operator (11).

Derivation of operator H_f  (p, x) (12). From expressions (A.3) and (A.7), we deduce

$${{H}_{f}}(p,x) = {{c}^{{\text{T}}}}(x)\left\{ {(\det M)\operatorname{adj} \tilde {A} + {{\alpha }^{{ - 1}}}(\operatorname{adj} \tilde {A})\tilde {B}(\operatorname{adj} M)\tilde {C}\operatorname{adj} \tilde {A}} \right\}{{b}_{f}}(x).$$

(A.11)

Opening the brackets here and substituting the value $\tilde {B}$ from (A.4), we obtain

$${{H}_{f}}(p,x) = {{c}^{{\text{T}}}}(x)\operatorname{adj} \tilde {A}{{b}_{f}}(x)\det M - {{\alpha }^{{ - 1}}}{{c}^{{\text{T}}}}(x)\operatorname{adj} \tilde {A}b(x)\Lambda ,$$

(A.12)

where it is indicated that

$$\Lambda = {{k}^{{\text{T}}}}(x)(\operatorname{adj} M)\tilde {C}(\operatorname{adj} \tilde {A}){{b}_{f}}(x).$$

(A.13)

Taking into account equalities $\tilde {A}$ = pE – A(x) and (14), we find

$${{c}^{{\text{T}}}}(x)\operatorname{adj} \tilde {A}{{b}_{f}}(x) = {{B}_{f}}(p,x),\quad {{c}^{{\text{T}}}}(x)\operatorname{adj} \tilde {A}b(x) = B(p,x).$$

(A.14)

Applying formula (A.25) from [28, p. 233] to (A.9) and taking into account (15), (16), and (A.6), we have

$$\begin{gathered} \det M = \det \left\{ {pE - R(x) + {{\psi }_{l}}(p,x)l(x){{k}^{{\text{T}}}}(x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{l}_{i}}(x){{k}^{{\text{T}}}}(x)} } \right\} \hfill \\ \quad \quad \quad \quad \quad \quad = \det [pE - R(x)] + {{\psi }_{l}}(p,x){{k}^{{\text{T}}}}(x)\operatorname{adj} [pE - R(x)]\,l(x) \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\;\; + \;\sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x)\left[ {{{k}^{{\text{T}}}}(x)\operatorname{adj} [pE - R(x)]\,{{l}_{i}}(x)} \right]} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; = R(p,x) + {{\psi }_{l}}(p,x)L(p,x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{L}_{i}}(p,x).} \hfill \\ \end{gathered} $$

(A.15)

Substituting $\tilde {C}$, $\tilde {A}$ from (A.4) into (A.13), and opening the brackets, taking into account (A.9), we obtain:

$$\Lambda = \left[ {{{k}^{{\text{T}}}}(x)\operatorname{adj} Ml(x)} \right]{{B}_{f}}(p,x) + \sum\limits_{i = 1}^q {\left[ {{{k}^{{\text{T}}}}(x)\operatorname{adj} M{{l}_{i}}(x)} \right]{{e}_{i}}\operatorname{adj} [pE - A(x)]{{b}_{f}}(x).} $$

(A.16)

In accordance with the third expression of (16): e_i adj [pE – A(x)]b_f(x) = W_i(p, x); by analogy with (A.10) and taking into account (15) we find k^T(x) adj M l(x) = L(p, x), k^T(x) adj M l_i(x) = L_i(p, x). Then from (A.16) the equality follows:

$$\Lambda = L(p,x){{B}_{f}}(p,x) + \sum\limits_{i = 1}^q {{{L}_{i}}(p,x){{W}_{i}}(p,x).} $$

(A.17)

Substituting expressions (A.14), (A.15), and (A.17) into (A.12), we have

$$\begin{gathered} {{H}_{f}}(p,x) = {{B}_{f}}(p,x)R(p,x) + {{\psi }_{l}}(p,x)L(p,x){{B}_{f}}(p,x) + \sum\limits_{i = 1}^q {{{\psi }_{i}}(p,x){{L}_{i}}(p,x){{B}_{f}}(p,x)} \\ - \;{{\psi }_{l}}(p,x)L(p,x){{B}_{f}}(p,x) - {{\psi }_{l}}(p,x)\sum\limits_{i = 1}^q {{{L}_{i}}(p,x){{W}_{i}}(p,x).} \\ \end{gathered} $$

Taking into account (A.6) here, grouping the sums and taking the factor A^–1(p, x) out of the bracket, we obtain

$${{H}_{f}}(p,x) = {{B}_{f}}(p,x)R(p,x) + \sum\limits_{i = 1}^q {{{L}_{i}}(p,x)\left\{ {{{V}_{i}}(p,x){{B}_{f}}(p,x) - B(p,x){{W}_{i}}(p,x)} \right\}{{A}^{{ - 1}}}(p,x).} $$

Finally, taking into account notation (13) here, we obtain operator (12).