Closed-Loop Control Strategy for Simulated Smoke Concentration in Aircraft Cargo Compartment Mock-Up

Abstract

In airworthiness verification flight test on fire smoke detection systems in aircraft cargo compartments, simulated smoke from a smoke generator that can replicate the characteristics of actual fire smoke should be used. In current studies, most of the boundary conditions of smoke generators are adjusted by open-loop control to fulfill the equivalence of the actual fire smoke. However, this method consumes a large amount of resources and has difficulty achieving optimal results from the complex turbulent flow field of smoke in an aircraft cargo compartment. To solve this problem, a computational fluid dynamics (CFD) model was first established to simulate smoke in the full-scale cargo compartment mock-up. Then, a state-space model with the exit parameters of the smoke generator as the input and the smoke concentration in the full-scale cargo compartment mock-up as the state variable was constructed using the data generated by the CFD model. Subsequently, a closed-loop simulated smoke concentration control strategy is proposed. In this strategy, the sensor in the full-scale cargo compartment mock-up collects the simulated smoke concentration signal and feeds it back to the controller. The controller processes the control target and feedback signal in real time. It realizes the automatic closed-loop control of the simulated smoke concentration by automatically adjusting the boundary condition parameters of the smoke generator. In the control law design, the control input constraint and the error between the linear state-space model and the real flow are considered, making the linear state-space model a nonlinear model. Finally, we prove the stability of the control system. The simulation shows that the designed control law can realize the closed-loop control of smoke concentration. The findings of this study can expand the control of simulated smoke flow field in airworthiness verification flight test from open loop to closed loop, and offer a new theoretical approach for smoke simulation technology.

Appendix: Control Law Design and Mathematical Derivation of the Stability Proof for the Closed-Loop Control of Simulated Smoke Concentration

1.1 Control Law Design

After the model identification is completed, we first transform the state-space model of simulated smoke obtained from the identification in Sect. 2.3 into a controllable standard type, which can be written as follows:

$$\begin{array}{*{20}l} \dot{x}_{1} = x_{2} \hfill \\ \dot{x}_{2} = x_{3} \hfill \\ \dot{x}_{3} = \varphi (x) + u \hfill \\ y = x_{1} , \hfill \\ \end{array}$$

(11)

$${\text{where}}{\kern 1pt} \,\varphi (x) = - b_{0} x_{1} - b_{1} x_{2} .$$

(12)

$b_{0}$, $b_{1}$ are all positive constants. The state variable $x_{1}$ represents the smoke concentration in the full-scale cargo compartment and is also the system output. The variable $u$ represents the system input, which is the outlet flow rate of the smoke generator.

Here, we consider the input of the control system to be limited and the flow rate of the smoke generator changes in a limited scope. Therefore, we used a hard saturation function $H(u)$ for limited control input (Fig. 14). As shown in Equation (13), $u_{m1}$ and $u_{m2}$ are the two saturated inflection points of the control input signal $u(t)$, which are positive constant. The parameters $\overline{\delta } > 0$ and $- \underline{\delta } < 0$ are the upper and lower bounds of the control input restricted hard saturation function, respectively. The slope $k(t)$ is a bounded time-varying function, where we define the greater of $\overline{\delta }$ and $\underline{\delta }$ as $\Psi = \max \left\{ {\overline{\delta } ,\underline{\delta } } \right\}$. As the outlet flow of the smoke generator cannot be negative, we set $\underline{\delta }$, $\overline{\delta }$ and $k(t)$ as 0 g/s, 10 g/s, and 1 to simplify the calculation.

$$H(u) = \left\{ {\begin{array}{*{20}l} {\overline{\delta } ,u > u_{m1} } \hfill \\ {k(t)u, - u_{m2} \le u \le u_{m1} } \hfill \\ { - \underline{\delta } ,u < - u_{m2} } \hfill \\ \end{array} } \right..$$

(13)

Figure 14

Asymmetric saturation function H (u) with dead zones and its approximation function S (u)

Because $H(u)$ is an asymmetric and non-smooth hard saturation function, which increases the difficulty of system control law design, we choose a soft saturation function $S(u)$ to approximate the asymmetric non-smooth hard saturation function $H(u)$. This soft saturation function $S(u)$ is expressed as follows:

$$S(u) = \frac{{\overline{\delta } {\text{e}}^{{(r + \beta {\text{u}})}} - \underline{\delta } {\text{e}}^{{ - (r + \beta {\text{u}})}} }}{{{\text{e}}^{{(r + \beta {\text{u}})}} + {\text{e}}^{{ - (r + \beta {\text{u}})}} }},$$

(14)

where parameter ${\text{r}}$ is defined as ${\text{r}} = {0}{\text{.5ln}}\left( {\frac{{\underline{\delta } }}{{\overline{\delta } }}} \right)$, and parameter $\beta$ represents the design parameters. Therefore, the hard saturation function can be expressed as follows:

$$H(u) = S(u) + D(u),$$

(15)

where $D(u)$ in represents the difference between the hard saturation function $H(u)$ and soft saturation function $S(u)$. As shown in Figure 13, when different values of $\beta$ are taken, the approximate degrees of $S(u)$ and $H(u)$ are different. Figure 7 also shows that function $S(u)$ is a bounded function.

Considering an error between the prediction results of the approximate linear state space of the simulated smoke turbulent flow field and the prediction results of the CFD model of the turbulent flow field, we use variable $d(t)$ to represent the error between the third-order linear state-space model and the real turbulent flow field. In summary, the linear state-space model of the system becomes the following nonlinear model:

$$\begin{array}{*{20}l} \dot{x}_{1} = x_{2} \hfill \\ \dot{x}_{2} = x_{3} \hfill \\ \dot{x}_{3} = \varphi (x) + H(u) + d(t). \hfill \\ \end{array}$$

(16)

$${\text{The system output is}}\,y = x_{1} .$$

(17)

To realize the equivalence to the real fire smoke flow field based on the closed-loop control method is to realize the target tracking control of the system output $y$. We define the expected tracking signal of the system as $y_{d}$, and assume its N + 1 derivative $y_{d}^{(n)}$, where $i = 1,2,...,n + 1$, are known and bounded functions of time. Our ultimate goal is to design a control law that can solve the control problem of nonlinear systems under the condition of asymmetric constrained control inputs, and to prove the designed control law theoretically by using Lyapunov stability theory.

We define the residual $e = x_{1} - y_{d}$ between the simulated smoke concentration at the monitoring position and the desired target. By taking the derivative of the left and right sides of this equation for several times, we can obtain

$$\begin{array}{*{20}l} e = x_{1} - y_{d} \hfill \\ e^{(1)} = \dot{e} = \dot{x}_{1} - \dot{y}_{d} = x_{2} - y_{d}^{(1)} \hfill \\ e^{(2)} = x_{3} - y_{d}^{(2)} \hfill \\ e^{(3)} = \varphi (x) + H(u) + d(t) - y_{d}^{(3)} . \hfill \\ \end{array}$$

(18)

To facilitate the control law design, we define a variable $s$ to represent the tracking error signal of the nonlinear system (Equation 16). Its mathematical expression is shown as follows:

$$s = \lambda_{1} e + \lambda_{2} e^{(1)} + e^{(2)} ,$$

(19)

where $\lambda_{1}$, $\lambda_{2}$ are constants. We can observe that when variable $s \to 0$, the tracking error signal $e$, $e^{(1)}$, $e^{(2)}$ approaches 0, indicating that the target tracking control of simulated smoke concentration has been realized. Taking the derivative of variables $s$, we can obtain

$$\begin{aligned} \dot{s} = & \lambda_{1} e^{(1)} + \lambda_{2} e^{(2)} + e^{(3)} \\ = & \varphi (x) + H(u) - y_{d}^{(3)} + \lambda_{1} e^{(1)} + \lambda_{2} e^{(2)} + d(t). \\ \end{aligned}$$

(20)

Let $\eta = - y_{d}^{(3)} + \lambda_{1} e^{(1)} + \lambda_{2} e^{(2)}$, and the above equation can be written as

$$\begin{aligned} \dot{s} = & \varphi (x) + H(u) + \eta + d(t) \\ = & \varphi (x) + H(u) + \eta + d(t) + c_{1} s - c_{1} s + S(u) - (H(u) - D(u)) \\ = & - c_{1} s + (c_{1} s + \varphi (x) + S(u) + \eta ) + H(u) + (d(t) + D(u)). \\ \end{aligned}$$

(21)

We define variables $z$ and $\overline{d}$ as follows:

$$\left\{ \begin{array}{*{20}l} z = c_{1} s + \varphi (x) + S(u) + \eta \hfill \\ \overline{d} = d(t) + D(u) \hfill \\ \end{array} \right.,$$

(22)

where variable $d(t)$ satisfies the inequality $d(t) \le d_{\max } < \infty$ and $d_{\max }$ is the maximum difference between the calculation results of the third-order state space approximation model and the CFD model. Substituting Equation (22) into Equation (21), we can express the variable as follows:

$$\dot{s} = - c_{1} s + z + H(u) + \overline{d}.$$

(23)

Taking the derivative of variables $z$, we can obtain

$$\dot{z} = c_{1} \dot{s} + \dot{\varphi }(x) + \dot{S}(u) + \dot{\eta }.$$

(24)

The soft saturation function $S(u)$ is a time-smooth function, so its derivative can be obtained as follows:

$$\dot{S}(u) = \frac{\partial S}{{\partial u}}\dot{u} = g\dot{u}.$$

(25)

The variable $g$ in Equation (25) can be solved according to the description of $S(u)$ in Equation (14):

$$\begin{aligned} g = & \frac{{(\overline{\delta } \beta {\text{e}}^{{(r + \beta {\text{u}})}} + \underline{\delta } \beta {\text{e}}^{{ - (r + \beta {\text{u}})}} )({\text{e}}^{{(r + \beta {\text{u}})}} + {\text{e}}^{{ - (r + \beta {\text{u}})}} )}}{{({\text{e}}^{{(r + \beta {\text{u}})}} + {\text{e}}^{{ - (r + \beta {\text{u}})}} )^{2} }} \\ & - \frac{{(\overline{\delta } {\text{e}}^{{(r + \beta {\text{u}})}} - \underline{\delta } {\text{e}}^{{ - (r + \beta {\text{u}})}} )(\beta {\text{e}}^{{(r + \beta {\text{u}})}} - \beta {\text{e}}^{{ - (r + \beta {\text{u}})}} )}}{{({\text{e}}^{{(r + \beta {\text{u}})}} + {\text{e}}^{{ - (r + \beta {\text{u}})}} )^{2} }} \\ = & \frac{{2\beta (\overline{\delta } + \underline{\delta } )}}{{({\text{e}}^{{(r + \beta {\text{u}})}} + {\text{e}}^{{ - (r + \beta {\text{u}})}} )^{2} }}. \\ \end{aligned}$$

(26)

Meanwhile, we let the control signal $u$ generated by an equivalent control input signal $v$ get through a first-order filter. Their relationship is shown as follows:

$$\dot{u} = - \alpha u + v,$$

(27)

where $\alpha$ is a positive constant.

Continuing to process the $\dot{\varphi }(x)$ in Equation (24), we can obtain

$$\begin{aligned} \dot{\varphi }(x) = & \frac{\partial \varphi }{{\partial x_{3} }}\dot{x}_{3} + \sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} \dot{x}_{i} \\ = & \frac{\partial \varphi }{{\partial x_{3} }}(\varphi (x) + H(u) + d(t)) + \sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} . \\ \end{aligned}$$

(28)

Taking the derivative of $\eta$ in Equation (24), we can obtain

$$\begin{aligned} \dot{\eta } = & - y_{d}^{(4)} + \lambda_{1} e^{(2)} + \lambda_{2} e^{(3)} \\ = & - y_{d}^{(4)} + \lambda_{1} e^{(2)} + \lambda_{2} (\varphi (x) + H(u) + d(t) - y_{d}^{(3)} ). \\ \end{aligned}$$

(29)

$${\text{We define}}\,\eta_{1} = - y_{d}^{(4)} - \lambda_{2} y_{d}^{(3)} + \lambda_{1} e^{(2)} .$$

(30)

Then, Equation (29) can be changed into

$$\dot{\eta } = \lambda_{2} (\varphi (x) + H(u) + d(t)) + \eta_{1} .$$

(31)

Substituting Equations (25)–(31) into Equation (24), the derivative of variable $z$ can be further expressed as

$$\begin{aligned} \dot{z} = & c_{1} (\varphi (x) + H(u) + \eta + d(t)) + \sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} + g( - \alpha u + v) \\ & + \frac{\partial \varphi }{{\partial x_{3} }}(\varphi (x) + H(u) + d(t)) + \lambda_{2} (\varphi (x) + H(u) + d(t)) + \eta_{1} \\ = & c_{1} \varphi (x) + c_{1} H(u) + c_{1} \eta + c_{1} d(t) + \sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} - g\alpha u + gv + \frac{\partial \varphi }{{\partial x_{3} }}\varphi (x) \\ & + \frac{\partial \varphi }{{\partial x_{3} }}H(u) + \frac{\partial \varphi }{{\partial x_{3} }}d(t) + \lambda_{2} \varphi (x) + \lambda_{2} H(u) + \lambda_{2} d(t) + \eta_{1} . \\ \end{aligned}$$

(32)

We define function $A(\cdot)$ as

$$\begin{aligned} A(\cdot) = & c_{1} \varphi (x) + c_{1} H(u) + c_{1} \eta + c_{1} d(t) + \sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} + \frac{\partial \varphi }{{\partial x_{3} }}\varphi (x) \\ & + \frac{\partial \varphi }{{\partial x_{3} }}H(u) + \frac{\partial \varphi }{{\partial x_{3} }}d(t) - g\alpha u + \lambda_{2} \varphi (x) + \lambda_{2} H(u) + \lambda_{2} d(t) + \eta_{1} . \\ \end{aligned}$$

(33)

Substituting Equation (33) into Equation (32), we can obtain the expression of $\dot{z}$ as follows:

$$\dot{z} = gv + A(\cdot).$$

(34)

We scale the function $A(\cdot)$ to obtain the following expression:

$$\begin{aligned} A(\cdot) \le & c_{1} \left\| {\varphi (x)} \right\| + c_{1} \Psi + c_{1} \left| \eta \right| + c_{1} d_{\max } + \left| {\frac{\partial \varphi }{{\partial x_{3} }}} \right|\left\| {\varphi (x)} \right\| + \left| {\frac{\partial \varphi }{{\partial x_{3} }}} \right|\Psi \\ & + \left| {\sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} } \right| + \left| {\frac{\partial \varphi }{{\partial x_{3} }}} \right|d_{\max } + \alpha \left| {gu} \right| + \lambda_{2} \left\| {\varphi (x)} \right\| \\ & + \lambda_{2} \Psi + \lambda_{2} d_{\max } + \left| {\eta_{1} } \right| \le aF(\cdot), \\ \end{aligned}$$

(35)

where $a$ is a positive known constant whose expression is given by Equation (36):

$$a = \max \left\{ {\begin{array}{*{20}c} {c_{1} ,} & {c_{1} \Psi ,} & {c_{1} ,} & {c_{1} d_{\max } ,} & {\Psi ,} & {} \\ {1,} & {d_{\max } ,} & {\alpha ,} & {\lambda_{2} ,} & {\lambda_{2} \Psi ,} & {\lambda_{2} d_{\max } } \\ \end{array} } \right\}.$$

(36)

The definition of the scalar function $F(\cdot)$ in Equation (35) is shown as

$$F(\cdot) = 4 + 2\left\| {\varphi (x)} \right\| + \left| \eta \right| + \left| {\eta_{1} } \right| + \left| {gu} \right| + \left| {\frac{\partial \varphi }{{\partial x_{3} }}} \right|\left\| {\varphi (x)} \right\| + 2\left| {\frac{\partial \varphi }{{\partial x_{3} }}} \right| + \left| {\sum\limits_{{{\text{i}} = 1}}^{2} {\frac{\partial \varphi }{{\partial x_{i} }}} x_{i + 1} } \right|.$$

(37)

Thus, the inequality can be further written as follows

$$A(\cdot) \le aF(\cdot).$$

(38)

Substituting Equation (38) into Equation (34), we can further obtain the following expression

$$\dot{z} \le gv + aF(\cdot).$$

(39)

For the above nonlinear system with limited control input, the equivalent control input signal $v$ we designed can be expressed as follows:

$$v = \frac{1}{g}[ - \frac{{azF^{2} }}{\left| z \right|F + \varepsilon } - s - c_{2} z],$$

(40)

where $\varepsilon$ is an arbitrary small constant and $c_{2}$ is the control law parameter to be designed.

1.2 Stability Proof

We define a Lyapunov function expression (41) as follows:

$$V = \frac{1}{2}s^{2} + \frac{1}{2}z^{2} .$$

(41)

Taking the derivative of the Lyapunov function $V(t)$ and substituting Equations (23) and (34), we can obtain

$$\begin{aligned} \dot{V} = & s\dot{s} + z\dot{z} = s( - c_{1} s + z + H(u) + \overline{d}) + z(gv + A) \\ \le & s( - c_{1} s + z + H(u) + \overline{d}) + zgv + \left| z \right|aF. \\ \end{aligned}$$

(42)

According to Equation (40),

$$\begin{aligned} zgv + \left| z \right|aF = & - \frac{{az^{2} F^{2} }}{\left| z \right|F + \varepsilon } - zs - c_{2} z^{2} + \left| z \right|aF \\ = & - c_{2} z^{2} - zs - \frac{{az^{2} F^{2} }}{\left| z \right|F + \varepsilon } + \frac{\left| z \right|aF(\left| z \right|F + \varepsilon )}{{\left| z \right|F + \varepsilon }} \\ \le & - c_{2} z^{2} - zs - \frac{{az^{2} F^{2} }}{\left| z \right|F + \varepsilon } + \frac{{az^{2} F^{2} }}{\left| z \right|F + \varepsilon } + \frac{\left| z \right|Fa\varepsilon }{{\left| z \right|F + \varepsilon }} \\ \le & - c_{2} z^{2} - zs + a\varepsilon . \\ \end{aligned}$$

(43)

Substituting Equation (43) in Equation (42), we can obtain

$$\dot{V} \le - c_{1} s^{2} + sz + sH(u) + s\overline{d} - c_{2} z^{2} - zs + a\varepsilon .$$

(44)

According to Young’s inequality, some terms in the expression of Lyapunov derivative function $\dot{V}$ are scaled, and the following can be obtained:

$$\begin{gathered} sH(u) \le \frac{1}{2}s^{2} + \frac{1}{2}H^{2} (u) \le \frac{1}{2}s^{2} + \frac{1}{2}\Psi^{2} \hfill \\ s\overline{d} \le \frac{1}{2}s^{2} + \frac{1}{2}\overline{d}^{2} \le \frac{1}{2}s^{2} + \frac{1}{2}d^{2}_{\max } . \hfill \\ \end{gathered}$$

(45)

Substituting Equation (45) into Equation (44), we can obtain

$$\dot{V} \le - c_{1} s^{2} - c_{2} z^{2} + \frac{1}{2}s^{2} + \frac{1}{2}\Psi^{2} + \frac{1}{2}s^{2} + \frac{1}{2}d^{2}_{\max } + a\varepsilon .$$

(46)

We define constant variables $C = \frac{1}{2}\Psi^{2} + \frac{1}{2}d^{2}_{\max } + a\varepsilon$ and substitute it into Equation (46) to obtain

$$\dot{V} \le - (c_{1} - 1)s^{2} - c_{2} z^{2} + C.$$

(47)

According to Young’s inequality, Equation (47) is further processed, and the following equation can be obtained:

$$\begin{aligned} \dot{V} \le & - (c_{1} - 1)s^{2} - c_{2} z^{2} + C \\ = & - 2(c_{1} - 1)\frac{1}{2}s^{2} - 2c_{2} \frac{1}{2}z^{2} + C \\ \le & - \frac{1}{2}\lambda_{0} s^{2} - \frac{1}{2}\lambda_{0} z^{2} + C \\ = & - \lambda_{0} \left( {\frac{1}{2}s^{2} + \frac{1}{2}z^{2} } \right) + C \\ = & - \lambda_{0} V + C, \\ \end{aligned}$$

(48)

where $\lambda_{0} = \min \left\{ {2(c_{1} - 1),2c_{2} } \right\}$. The design of parameters $c_{1}$ in the control law needs to satisfy $c_{1} - 1 > 0$ to ensure the stability of the closed-loop control system.

Continuing to deal with Equation (48), we multiply both sides of the inequality by ${\text{e}}^{{\lambda_{0} t}}$ and obtain

$$\dot{V}(t){\text{e}}^{{\lambda_{0} t}} \le - \lambda_{0} V(t){\text{e}}^{{\lambda_{0} t}} + C{\text{e}}^{{\lambda_{0} t}} ,$$

(49)

that is,

$$\dot{V}(t){\text{e}}^{{\lambda_{0} t}} + \lambda_{0} V(t){\text{e}}^{{\lambda_{0} t}} \le C{\text{e}}^{{\lambda_{0} t}} .$$

(50)

When Equation (50) is written in differential form, we can obtain

$$\frac{d}{dt}(V(t){\text{e}}^{{\lambda_{0} t}} ) \le C{\text{e}}^{{\lambda_{0} t}} .$$

(51)

We continue to integrate the left and right sides of Equation (51) and obtain

$$V(t){\text{e}}^{{\lambda_{0} t}} \le V(0) + C\int_{0}^{t} {{\text{e}}^{{\lambda_{0} \tau }} } d\tau .$$

(52)

Thus, we can further obtain

$$V(t) \le V(0){\text{e}}^{{ - \lambda_{0} t}} + \frac{C}{{\lambda_{0} }}(1 - {\text{e}}^{{ - \lambda_{0} t}} ) = \frac{C}{{\lambda_{0} }} + \left( {V(0) - \frac{C}{{\lambda_{0} }}} \right){\text{e}}^{{ - \lambda_{0} t}} .$$

(53)

According to the preceding equation, Lyapunov function $V(t)$ is bounded. When time $t \to \infty$, the formula for calculating the limit of Lyapunov function $V(t)$ is as follows

$$\mathop {\lim }\limits_{t \to \infty } V(t) \le \frac{C}{{\lambda_{0} }}.$$

(54)

According to the Lyapunov function expression (41), the following can be obtained:

$$\frac{1}{2}s^{2} \le V(t).$$

(55)

Therefore,

$$\mathop {\frac{1}{2}\lim }\limits_{t \to \infty } s^{2} (t) \le \frac{C}{{\lambda_{0} }}.$$

When time $t \to \infty$, the following inequality holds:

$$\mathop {\lim }\limits_{t \to \infty } s(t) \le \sqrt {\frac{2C}{{\lambda_{0} }}} .$$

(56)

The preceding analysis shows that the variable $s$ characterizing the tracking error of the simulated smoke concentration tends to a compact set with the mathematical description $\Omega : = \left\{ {s:\left| s \right| \le \sqrt {\frac{2C}{{\lambda_{0} }}} } \right\}$. Therefore, the tracking error variables of simulated smoke concentration $e$ are also globally ultimately bounded. By increasing the value of the parameters $\lambda_{0}$, set $\Omega$ can be adjusted to a sufficiently small range. This result means that we can achieve the global uniformly bounded closed-loop control of the tracking error of the simulation smoke concentration by adjusting the control law parameters $c_{1} ,c_{2}$.