Integral linear quadratic Gaussian regulator subject to unknown inputs: application in photovoltaic systems

Abstract

This paper describes the development of a new quadratic optimal controller for discrete-time systems with integral action and subject to non-manipulable and possibly non-measurable external inputs or disturbances. The proposed controller is a linear quadratic regulator (LQR) based on an augmented state-space that aims to include the integral of error and disturbance modeling in the solution. For cases where the disturbance is not measured, this controller is applied in conjunction with a specific Kalman filter for estimating non-measurable inputs. The new controller is applied to maximum power point tracking (MPPT) simulations for photovoltaic systems and compared using the perturb and observe method. MPPT is performed by controlling the duty cycle of a DC-DC boost converter connected to the output of the photovoltaic system. The case studies seek to evaluate controller performance regarding variations in temperature and irradiance, measurement noises, and uncertainties in the model. The results show that the new controller is able to increase system efficiency while reducing costs associated with implementing current, voltage, and irradiance filters and sensors.

A proof of theorem 1

The system (1) can be rewritten in augmented form as

$$\begin{aligned} {{{\mathcal {X}}}_{k + 1}} = {{\mathcal {F}}}_k{{{\mathcal {X}}}_k} + {{{\mathcal {G}}}}_k{u_k} \end{aligned}$$

(63)

where

$$\begin{aligned} {{{\mathcal {X}}}_k} := \left[ {\begin{array}{*{20}{c}} {{{x}_k}}\\ {{\varepsilon _k}}\\ {{{ r}_k}}\\ {{{d}_{k}}} \end{array}} \right] ,{{{\mathcal {F}}}_k} := \left[ {\begin{array}{*{20}{c}} {{F_k}}&{}0&{}0&{}{{E_k}}\\ { - {T_s}{H_k}}&{}I&{}{{T_s}}&{}0\\ 0&{}0&{}{{L_k}}&{}0\\ 0&{}0&{}0&{}0 \end{array}} \right] ,{{{\mathcal {G}}}_k} := \left[ {\begin{array}{*{20}{c}} {{G_k}}\\ 0\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$

(64)

Through of (64), the Problem (2) can be rewritten as

$$\begin{aligned}{} & {} \mathop {\min }\limits _{{ u_k}} \left\{ {{\mathcal {X}}}_N^T{{{\mathcal {P}}}_N}{{{\mathcal {X}}}_N} + \sum \limits _{k = 0}^{N - 1} {\left( {{{\mathcal {X}}}_k^T{{{\mathcal {Q}}}_k}{{{\mathcal {X}}}_k} + u_k^T{R_k}{u_k}} \right) } \right\} ,\nonumber \\{} & {} \quad s.a \ {{{\mathcal {X}}}_{k + 1}} = {{\mathcal {F}}}_k{{{\mathcal {X}}}_k} + {{{\mathcal {G}}}}_k{u_k} \end{aligned}$$

(65)

with

$$\begin{aligned} {{{\mathcal {P}}}_k} = \left[ {\begin{array}{*{20}{c}} {P_k^x}&{}{P_k^{x\varepsilon }}&{}{P_k^{xr}}&{}{P_k^{xd}}\\ {P_k^{\varepsilon x}}&{}{P_k^\varepsilon }&{}{P_k^{\varepsilon r}}&{}{P_k^{\varepsilon d}}\\ {P_k^{rx}}&{}{P_k^{r \varepsilon }}&{}{P_k^r}&{}{P_k^{rd}}\\ {P_k^{dx}}&{}{P_k^{d \varepsilon }}&{}{P_k^{dr}}&{}{P_k^d} \end{array}} \right] ,{{{\mathcal {Q}}}_k} = diag(Q_k^x,Q_k^\varepsilon ,0,0) \end{aligned}$$

(66)

where the submatrices of ${{\mathcal {P}}}_k$ are known, symmetric, positive semi-definite weighting matrices with appropriate dimensions.

The problem (65–66) can be solved as a classic LQR problem with the solution given by [14]:

$$\begin{aligned} u_k^*({{\mathcal {X}}}_k)= & {} - {{\left( {{R_k} + {{\mathcal {G}}}_k^T{{{\mathcal {P}}}_{k + 1}}{{{\mathcal {G}}}_k}} \right) ^{ - 1}}{{\mathcal {G}}}_k^T{{{\mathcal {P}}}_{k + 1}}{{\mathcal {F}}}{_k}}{{{\mathcal {X}}}_k} \end{aligned}$$

(67)

$$\begin{aligned} {{{\mathcal {P}}}_k}= & {} {{\mathcal {F}}}_k^T\left( {{{{\mathcal {P}}}_{k + 1}} - {{{\mathcal {P}}}_{k + 1}}{{{\mathcal {G}}}_k}{{\left( {{R_k} + {{\mathcal {G}}}_k^T{{{\mathcal {P}}}_{k + 1}}{{{\mathcal {G}}}_k}} \right) }^{ - 1}}{{\mathcal {G}}}_k^T{{{\mathcal {P}}}_{k + 1}}} \right) \nonumber \\{} & {} {{{\mathcal {F}}}_k} + {{{\mathcal {Q}}}_k} \end{aligned}$$

(68)

1.1 A.1 Control law

Substituting (64) and (66) in (67) results in

$$\begin{aligned} u_k^*= & {} - {\left( {{R_k} + {{\left[ {\begin{array}{*{20}{c}} {{G_k}}\\ 0\\ 0\\ 0 \end{array}} \right] }^T}\left[ {\begin{array}{*{20}{c}} {P_{k + 1}^x}&{}{P_{k + 1}^{x\varepsilon }}&{}{P_{k + 1}^{xr}}&{}{P_{k + 1}^{xd}}\\ {P_{k + 1}^{\varepsilon x}}&{}{P_{k + 1}^\varepsilon }&{}{P_{k + 1}^{\varepsilon r}}&{}{P_{k + 1}^{\varepsilon d}}\\ {P_{k + 1}^{rx}}&{}{P_{k + 1}^{r\varepsilon }}&{}{P_{k + 1}^r}&{}{P_{k + 1}^{rd}}\\ {P_{k + 1}^{dx}}&{}{P_{k + 1}^{d\varepsilon }}&{}{P_{k + 1}^{dr}}&{}{P_{k + 1}^d} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{G_k}}\\ 0\\ 0\\ 0 \end{array}} \right] } \right) ^{ - 1}}\\{} & {} \quad {\left[ {\begin{array}{*{20}{c}} {{G_k}}\\ 0\\ 0\\ 0 \end{array}} \right] ^T}\\{} & {} \left[ {\begin{array}{*{20}{c}} {P_{k + 1}^x}&{}{P_{k + 1}^{x\varepsilon }}&{}{P_{k + 1}^{xr}}&{}{P_{k + 1}^{xd}}\\ {P_{k + 1}^{\varepsilon x}}&{}{P_{k + 1}^\varepsilon }&{}{P_{k + 1}^{\varepsilon r}}&{}{P_{k + 1}^{\varepsilon d}}\\ {P_{k + 1}^{rx}}&{}{P_{k + 1}^{r\varepsilon }}&{}{P_{k + 1}^r}&{}{P_{k + 1}^{rd}}\\ {P_{k + 1}^{dx}}&{}{P_{k + 1}^{d\varepsilon }}&{}{P_{k + 1}^{dr}}&{}{P_{k + 1}^d} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{F_k}}&{}0&{}0&{}{{E_k}}\\ { - {T_S}{H_k}}&{}I&{}{{T_S}}&{}0\\ 0&{}0&{}{{L_k}}&{}0\\ 0&{}0&{}0&{}0 \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{x_k}}\\ {{\varepsilon _k}}\\ {{r_k}}\\ {{d_k}} \end{array}} \right] \end{aligned}$$

which can be simplified to

$$\begin{aligned} u_k^*= & {} - {\left( {{R_k} + G_k^TP_{k + 1}^x{G_k}} \right) ^{ - 1}}G_k^T\\{} & {} \left[ {\begin{array}{*{20}{c}} {\left( {P_{k + 1}^x{F_k} - {T_S}P_{k + 1}^{x\varepsilon }{H_k}} \right) }&{P_{k + 1}^{x\varepsilon }}&{\left( {{T_S}P_{k + 1}^{x\varepsilon } + {L_k}P_{k + 1}^{xr}} \right) }&{P_{k + 1}^x{E_k}} \end{array}} \right] \\{} & {} \left[ {\begin{array}{*{20}{c}} {{x_k}}\\ {{\varepsilon _k}}\\ {{r_k}}\\ {{d_k}} \end{array}} \right] \end{aligned}$$

and reorganized as follow

$$\begin{aligned} u_k^*= & {} - {\left( {{R_k} + G_k^TP_{k + 1}^x{G_k}} \right) ^{ - 1}}G_k^T\left( \left( {P_{k + 1}^x{F_k} - {T_S}P_{k + 1}^{x\varepsilon }{H_k}} \right) {x_k} \right. \\{} & {} \left. + P_{k + 1}^{x\varepsilon }{\varepsilon _k} + \left( {{T_S}P_{k + 1}^{x\varepsilon } + {L_k}P_{k + 1}^{xr}} \right) {r_k} + P_{k + 1}^x{E_k}{d_k} \right) \end{aligned}$$

or

$$\begin{aligned} u_k^* = - \left( {K_k^x{x_k} + K_k^e{e_k} + K_k^r{r_k} + K_k^d{d_k}} \right) \end{aligned}$$

with

$$\begin{aligned} K_k^x= & {} {\Phi _k}\left( {P_{k + 1}^x{F_k} - {T_S}P_{k + 1}^{x\varepsilon }{H_k}} \right) ,\\ K_k^\varepsilon= & {} {\Phi _k}P_{k + 1}^{x\varepsilon },\\ K_k^r= & {} {\Phi _k}\left( {{T_S}P_{k + 1}^{x\varepsilon } + {L_k}P_{k + 1}^{xr}} \right) ,\\ K_k^d= & {} {\Phi _k}P_{k + 1}^x{E_k},\\ {\Phi _k}= & {} {\left( {{R_k} + G_k^TP_{k + 1}^x{G_k}} \right) ^{ - 1}}G_k^T, \end{aligned}$$

as presented in the Theorem 1.

1.2 A.2 Riccati equation

Substituting (64) and (66) in 68) results in

$$\begin{aligned}{} & {} {{{\mathcal {P}}}_k}= {\left[ {\begin{array}{*{20}{c}} {{F_k}}&{}0&{}0&{}{{E_k}}\\ { - {T_S}{H_k}}&{}I&{}{{T_S}}&{}0\\ 0&{}0&{}{{L_k}}&{}0\\ 0&{}0&{}0&{}0 \end{array}} \right] ^T}\left( \left[ {\begin{array}{*{20}{c}} {P_{k + 1}^x}&{}{P_{k + 1}^{x\varepsilon }}&{}{P_{k + 1}^{xr}}&{}{P_{k + 1}^{xd}}\\ {P_{k + 1}^{\varepsilon x}}&{}{P_{k + 1}^\varepsilon }&{}{P_{k + 1}^{\varepsilon r}}&{}{P_{k + 1}^{\varepsilon d}}\\ {P_{k + 1}^{rx}}&{}{P_{k + 1}^{r\varepsilon }}&{}{P_{k + 1}^r}&{}{P_{k + 1}^{rd}}\\ {P_{k + 1}^{dx}}&{}{P_{k + 1}^{d\varepsilon }}&{}{P_{k + 1}^{dr}}&{}{P_{k + 1}^d} \end{array}} \right] \right. \\{} & {} \quad \left. - \left[ \begin{array}{*{20}{c}} {P_{k + 1}^x}&{}{P_{k + 1}^{x\varepsilon }}&{}{P_{k + 1}^{xr}}&{}{P_{k + 1}^{xd}}\\ {P_{k + 1}^{\varepsilon x}}&{}{P_{k + 1}^\varepsilon }&{}{P_{k + 1}^{\varepsilon r}}&{}{P_{k + 1}^{\varepsilon d}}\\ {P_{k + 1}^{rx}}&{}{P_{k + 1}^{r\varepsilon }}&{}{P_{k + 1}^r}&{}{P_{k + 1}^{rd}}\\ {P_{k + 1}^{dx}}&{}{P_{k + 1}^{d\varepsilon }}&{}{P_{k + 1}^{dr}}&{}{P_{k + 1}^d} \end{array}\right] \left[ {\begin{array}{*{20}{c}} {{G_k}}\\ 0\\ 0\\ 0 \end{array}} \right] {{\left( {{R_k} + G_k^TP_{k + 1}^x{G_k}} \right) }^{ - 1}} \right. \\{} & {} \left. {{\left[ \begin{matrix} {{G}_{k}} \\ 0 \\ 0 \\ 0 \\ \end{matrix} \right] }^{T}}\left[ \begin{matrix} P_{k+1}^{x} &{} P_{k+1}^{x\varepsilon } &{} P_{k+1}^{xr} &{} P_{k+1}^{xd} \\ P_{k+1}^{\varepsilon x} &{} P_{k+1}^{\varepsilon } &{} P_{k+1}^{\varepsilon r} &{} P_{k+1}^{\varepsilon d} \\ P_{k+1}^{rx} &{} P_{k+1}^{r\varepsilon } &{} P_{k+1}^{r} &{} P_{k+1}^{rd} \\ P_{k+1}^{dx} &{} P_{k+1}^{d\varepsilon } &{} P_{k+1}^{dr} &{} P_{k+1}^{d} \\ \end{matrix} \right] \right) \\{} & {} \quad \left[ \begin{matrix} {{F}_{k}} &{} 0 &{} 0 &{} {{E}_{k}} \\ -{{T}_{S}}{{H}_{k}} &{} I &{} {{T}_{S}} &{} 0 \\ 0 &{} 0 &{} {{L}_{k}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] +{{\mathcal {Q}}_{k}} \end{aligned}$$

which can be simplified to

$$\begin{aligned} {{\mathcal {P}}_{k}}= & {} \left[ \begin{matrix} F_{k}^{T} &{} -{{T}_{S}}H_{k}^{T} &{} 0 &{} 0 \\ 0 &{} I &{} 0 &{} 0 \\ 0 &{} {{T}_{S}} &{} L_{k}^{T} &{} 0 \\ E_{k}^{T} &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] \left( \left[ \begin{matrix} P_{k+1}^{x} &{} P_{k+1}^{x\varepsilon } &{} P_{k+1}^{xr} &{} P_{k+1}^{xd} \\ P_{k+1}^{\varepsilon x} &{} P_{k+1}^{\varepsilon } &{} P_{k+1}^{\varepsilon r} &{} P_{k+1}^{\varepsilon d} \\ P_{k+1}^{rx} &{} P_{k+1}^{r\varepsilon } &{} P_{k+1}^{r} &{} P_{k+1}^{rd} \\ P_{k+1}^{dx} &{} P_{k+1}^{d\varepsilon } &{} P_{k+1}^{dr} &{} P_{k+1}^{d} \\ \end{matrix} \right] \right. \\{} & {} -\left. \left[ \begin{matrix} P_{k+1}^{x}{{G}_{k}}{{\left( {{R}_{k}}+G_{k}^{T}P_{k+1}^{x}{{G}_{k}} \right) }^{-1}} \\ P_{k+1}^{\varepsilon x}{{G}_{k}}{{\left( {{R}_{k}}+G_{k}^{T}P_{k+1}^{x}{{G}_{k}} \right) }^{-1}} \\ P_{k+1}^{rx}{{G}_{k}}{{\left( {{R}_{k}}+G_{k}^{T}P_{k+1}^{x}{{G}_{k}} \right) }^{-1}} \\ P_{k+1}^{dx}{{G}_{k}}{{\left( {{R}_{k}}+G_{k}^{T}P_{k+1}^{x}{{G}_{k}} \right) }^{-1}} \\ \end{matrix} \right] \right. \\{} & {} \left. \left[ \begin{matrix} G_{k}^{T}P_{k+1}^{x} &{} G_{k}^{T}P_{k+1}^{x\varepsilon } &{} G_{k}^{T}P_{k+1}^{xr} &{} G_{k}^{T}P_{k+1}^{xd} \\ \end{matrix} \right] \right) \\{} & {} \left[ \begin{matrix} {{F}_{k}} &{} 0 &{} 0 &{} {{E}_{k}} \\ -{{T}_{S}}{{H}_{k}} &{} I &{} {{T}_{S}} &{} 0 \\ 0 &{} 0 &{} {{L}_{k}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] +{{\mathcal {Q}}_{k}}. \end{aligned}$$

By defining $\Psi :={{G}_{k}}{{\left( {{R}_{k}}+G_{k}^{T}P{{_{k+1}^{x}}_{k+1}}{{G}_{k}} \right) }^{-1}}G_{k}^{T}={{G}_{k}}{{\Phi }_{k}}$, the above equation become

$$\begin{aligned} {{\mathcal {P}}_{k}}= & {} \left[ \begin{matrix} F_{k}^{T} &{} -{{T}_{S}}H_{k}^{T} &{} 0 &{} 0 \\ 0 &{} I &{} 0 &{} 0 \\ 0 &{} {{T}_{S}} &{} L_{k}^{T} &{} 0 \\ E_{k}^{T} &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] \left( \left[ \begin{matrix} P_{k+1}^{x} &{} P_{k+1}^{x\varepsilon } &{} P_{k+1}^{xr} &{} P_{k+1}^{xd} \\ P_{k+1}^{\varepsilon x} &{} P_{k+1}^{\varepsilon } &{} P_{k+1}^{\varepsilon r} &{} P_{k+1}^{\varepsilon d} \\ P_{k+1}^{rx} &{} P_{k+1}^{r\varepsilon } &{} P_{k+1}^{r} &{} P_{k+1}^{rd} \\ P_{k+1}^{dx} &{} P_{k+1}^{d\varepsilon } &{} P_{k+1}^{dr} &{} P_{k+1}^{d} \\ \end{matrix} \right] \right. \\{} & {} -\left. \left[ \begin{matrix} P_{k+1}^{x}\Psi P_{k+1}^{x} &{} P_{k+1}^{x}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{x}\Psi P_{k+1}^{xr} &{} P_{k+1}^{x}\Psi P_{k+1}^{xd} \\ P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{x} &{} P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{xr} &{} P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{xd} \\ P_{k+1}^{rx}\Psi P_{k+1}^{x} &{} P_{k+1}^{rx}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{rx}\Psi P_{k+1}^{xr} &{} P_{k+1}^{rx}\Psi P_{k+1}^{xd} \\ P_{k+1}^{dx}\Psi P_{k+1}^{x} &{} P_{k+1}^{dx}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{dx}\Psi P_{k+1}^{xr} &{} P_{k+1}^{dx}\Psi P_{k+1}^{xd} \\ \end{matrix} \right] \right) \\{} & {} \left[ \begin{matrix} {{F}_{k}} &{} 0 &{} 0 &{} {{E}_{k}} \\ -{{T}_{S}}{{H}_{k}} &{} I &{} {{T}_{S}} &{} 0 \\ 0 &{} 0 &{} {{L}_{k}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] +{{\mathcal {Q}}_{k}}\end{aligned}$$

which is expanded to

$$\begin{aligned} {{\mathcal {P}}_{k}}= & {} \left[ \begin{matrix} F_{k}^{T} &{} -{{T}_{S}}H_{k}^{T} &{} 0 &{} 0 \\ 0 &{} I &{} 0 &{} 0 \\ 0 &{} {{T}_{S}} &{} L_{k}^{T} &{} 0 \\ E_{k}^{T} &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] \\{} & {} \left( \left[ \begin{matrix} P_{k+1}^{x}-P_{k+1}^{x}\Psi P_{k+1}^{x} &{} P_{k+1}^{x\varepsilon }-P_{k+1}^{x}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{xr}-P_{k+1}^{x}\Psi P_{k+1}^{xr} &{} P_{k+1}^{xd}-P_{k+1}^{x}\Psi P_{k+1}^{xd} \\ P_{k+1}^{\varepsilon x}-P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{x} &{} P_{k+1}^{\varepsilon }-P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{\varepsilon r}-P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{xr} &{} P_{k+1}^{\varepsilon d}-P_{k+1}^{\varepsilon x}\Psi P_{k+1}^{xd} \\ P_{k+1}^{rx}-P_{k+1}^{rx}\Psi P_{k+1}^{x} &{} P_{k+1}^{r\varepsilon }-P_{k+1}^{rx}\Psi P_{k+1}^{x\varepsilon } &{} P_{k+1}^{r}-P_{k+1}^{rx}\Psi P_{k+1}^{xr} &{} P_{k+1}^{rd}-P_{k+1}^{rx}\Psi P_{k+1}^{xd} \\ P_{k+1}^{dx}-P_{k+1}^{dx}\Psi P_{k+1}^{x} &{} P_{k+1}^{d\varepsilon }-P_{k+1}^{dx}\Psi P_{k+1}^{xe} &{} P_{k+1}^{dr}-P_{k+1}^{dx}\Psi P_{k+1}^{xr} &{} P_{k+1}^{d}-P_{k+1}^{dx}\Psi P_{k+1}^{xd} \\ \end{matrix} \right] \right) \\{} & {} \left[ \begin{matrix} {{F}_{k}} &{} 0 &{} 0 &{} {{E}_{k}} \\ -{{T}_{S}}{{H}_{k}} &{} I &{} {{T}_{S}} &{} 0 \\ 0 &{} 0 &{} {{L}_{k}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] +{{\mathcal {Q}}_{k}}. \end{aligned}$$

Performing the two matrices multiplications results in

$$\begin{aligned}{} & {} \left[ \begin{matrix} P_{k}^{x} &{} P_{k}^{xe} &{} P_{k}^{xr} &{} P_{k}^{xd} \\ P_{k}^{ex} &{} P_{k}^{e} &{} P_{k}^{er} &{} P_{k}^{ed} \\ P_{k}^{rx} &{} P_{k}^{re} &{} P_{k}^{r} &{} P_{k}^{rd} \\ P_{k}^{dx} &{} P_{k}^{de} &{} P_{k}^{dr} &{} P_{k}^{d} \\ \end{matrix} \right] =\left[ \begin{matrix} \alpha &{} \phi &{} \varphi &{} \gamma \\ {{\phi }^{T}} &{} \beta &{} \lambda &{} \rho \\ {{\varphi }^{T}} &{} {{\lambda }^{T}} &{} \chi &{} \sigma \\ {{\gamma }^{T}} &{} {{\rho }^{T}} &{} {{\sigma }^{T}} &{} \delta \\ \end{matrix} \right] \\{} & {} +\left[ \begin{matrix} Q_{k}^{x} &{} 0 &{} 0 &{} 0 \\ 0 &{} Q_{k}^{e} &{} 0 &{} 0 \\ 0 &{} 0 &{} Q_{k}^{r} &{} 0 \\ 0 &{} 0 &{} 0 &{} Q_{k}^{d} \\ \end{matrix} \right] \end{aligned}$$

where the auxiliary variables are given by (10) to (19).