Optimal control approach for nonlinear chemical processes with uncertainty and application to a continuous stirred-tank reactor problem

3.1 Properties of problem (2.1a)-(2.1e)

Note that the system state variable $x\left(t\right)$ depends on the control input $u\left(t\right)$ and the random variable $\theta$ . Then, a more brief mathematical expression for the stochastic constraints described by (2.1d) can be achieved as follows:

3.2 Approximation of stochastic constraints

Under the premise of ensuring the feasibility, the main task of approximating the stochastic constraints described by (2.1d) is to construct q continuous functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right):\left(0,+\infty \right)\times R^n\to \left[0,+\infty \right),i=1,\cdots ,q$ , satisfying the following three properties:(1) $E\left[A\left(g_i\left(u\left(t\right),\theta \right)\right)\right]⩽{\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , for any $\alpha >0$ and $u\left(t\right)\in U$ ;(2) $\underset{\alpha >0}{\inf }{\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right)=E\left[A\left(g_i\left(u\left(t\right),\theta \right)\right)\right],i=1,\cdots ,q$ , for any $u\left(t\right)\in U$ ;(3) ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , are non-decreasing functions with respect to the parameter $\alpha$ .

By using the properties (2)-(3) and the continuity of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , respect to the parameter $\alpha$ , it follows that

Suppose that the following assumption is true:

Assumption 3.1. Let ${∊}_i\in [0.5,1),i=1,\cdots ,q$ . The stochastic constraints described by (2.1d) is called to be regular if for any $u\left(t\right)\in U$ with $\mathit{Prob}\left\{g_i\left(x\left(t\right),u\left(t\right),\theta \right)⩽0\right\}={∊}_i,i=1,\cdots ,q$ , there exists a sequence ${\left\{u_k\right(t\left)\right\}}_{k\in N}$ such that $\underset{k\to +\infty }{\lim }{u}_k\left(t\right)=u\left(t\right)$ and $\mathit{Prob}\left\{g_i\left(x\left(t\right),u_k\left(t\right),\theta \right)⩽0\right\}>{∊}_i,i=1,\cdots ,q$ .

Then, the relationship between $\mathrm{\Delta },\mathrm{\Omega }$ , and $\mathrm{\Gamma }$ can be provided by the following theorem:

Theorem 3.1. The compactness of the set U, the continuity and monotonicity of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , with respect to the parameter $\alpha$ , and the properties (1)-(3) of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , indicate that for any sequence ${\left\{{\alpha }_k\right\}}_{k\in N}$ , the following equality is true:

Proof. Note that $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ . By using the property (3) of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , we can obtain that the sequence ${\left\{\mathrm{\Gamma }\left({\alpha }_k\right)\right\}}_{k\in N}$ satisfies $\mathrm{\Gamma }\left({\alpha }_k\right)\subset \mathrm{\Gamma }\left({\alpha }_{k+1}\right),k=1,2,\cdots$ , due to the compactness of the set U and the continuity of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ . Further, it follows that

Suppose that $u\left(t\right)\in U$ . Then, there exist the following two cases:

Case 1: $E\left[A\left(g_i\left(u\left(t\right),\theta \right)\right)\right]<1-{∊}_i,i=1,\cdots ,q$ .

Note that $E\left[A\left(g_i\left(u\left(t\right),\theta \right)\right)\right]=\underset{\alpha >0}{\inf }{\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , and the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , are continuous and non-decreasing with respect to the parameter $\alpha$ . Then, there exists a $\overline{k}$ such that the following inequality is true:

Case 2: $E\left[A\left(g_i\left(u\left(t\right),\theta \right)\right)\right]=1-{∊}_i,i=1,\cdots ,q$ .

By using Assumption 3.1, there exists a sequence ${\left\{u_k\left(t\right)\right\}}_{k\in N}$ in the set U such that $\underset{k\to +\infty }{\lim }{u}_k\left(t\right)=u\left(t\right)$ and $\mathit{Prob}\left\{g_i\left(x\left(t\right),u_k\left(t\right),\theta \right)⩽0\right\}>{∊}_i,i=1,\cdots ,q$ . This indicates that $E\left[A\left(g_i\left(u_k\left(t\right),\theta \right)\right)\right]<1-{∊}_i,i=1,\cdots ,q$ . Further, by using the property (2) of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , it follows that for any $u_k\left(t\right)$ , there exists a sufficiently small parameter ${\alpha }_k$ such that $E\left[A\left(g_i\left(u_k\left(t\right),\theta \right)\right)\right]<{\mathrm{\Upsilon }}_i\left({\alpha }_k,u_k\left(t\right)\right)<1-{∊}_i,i=1,\cdots ,q$ . Thus, $u_k\left(t\right)\in \mathrm{\Gamma }\left({\alpha }_k\right)$ for all k, which indicates that $u\left(t\right)\in \underset{k\to +\infty }{\lim \sup }\mathrm{\Gamma }\left({\alpha }_k\right)$ .

The properties (2)-(3) of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , imply that the parameter ${\alpha }_k$ can be selected such that ${\left\{{\alpha }_k\right\}}_{k\in N}$ is a monotonically decreasing sequence and satisfies $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ . Thus, $\underset{k\to +\infty }{\lim }\mathrm{\Gamma }\left({\alpha }_k\right)=\mathrm{\Delta }=\mathrm{\Omega }$ . This completes the proof of Theorem 3.1.

It should be pointed out that the equality $\underset{{\alpha }_k\to 0^{+}}{\lim \sup }\mathrm{\Gamma }\left({\alpha }_k\right)=\underset{{\alpha }_k\to 0^{+}}{\lim }\mathrm{\Gamma }\left({\alpha }_k\right)=\mathrm{\Omega }=\mathrm{\Delta }$ is also true because of the compactness of the set $\mathrm{\Gamma }\left({\alpha }_k\right)$ and the monotony property $\mathrm{\Gamma }\left({\alpha }_k\right)\subset \mathrm{\Gamma }\left({\alpha }_{k+1}\right)$ for all $0<{\alpha }_{k+1}<{\alpha }_k$ . The so-called concentration of measure inequalities from probability theory can be used as a background to construct the smooth approximation functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , which satisfy properties (1)-(3) (Pinter, 1989). For example, the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , satisfying properties (1)-(3) can be defined as ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right)=E\left[\mathrm{B}\left(\alpha ,g_i\left(u\left(t\right),\theta \right)\right)\right],i=1,\cdots ,q$ , where $B:\left(0,+\infty \right)\times R\to \left[0,+\infty \right)$ is a given function. Further, it provides a feasible technical route for constructing a tractable smooth approximation of problem (2.1a)-(2.1e). There exists many functions $B\left(\alpha ,z\right)$ such that the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ obtained by ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right)=E\left[\mathrm{B}\left(\alpha ,g_i\left(u\left(t\right),\theta \right)\right)\right],i=1,\cdots ,q$ satisfy properties (1)-(3). For example,

• $B_1\left(\alpha ,z\right)=e^{\frac{z}{\alpha }},\alpha >0$ ;

• $B_2\left(\alpha ,z\right)=\frac{1}{\alpha }{\int }_{-\infty }^0\mathrm{\Psi }\left(\frac{s+z}{\alpha }\right)\mathit{ds},\alpha >0$ , where $\mathrm{\Psi }\left(z\right)$ is a non-negative integrable and symmetric function (i.e. $\mathrm{\Psi }\left(z\right)=\mathrm{\Psi }\left(-z\right)$ and ${\int }_{-\infty }^{+\infty }\mathrm{\Psi }\left(z\right)\mathit{dz}=1$ ).

Unfortunately, the functions $B_1\left(\alpha ,z\right)$ and $B_2\left(\alpha ,z\right)$ can not be used directly to the computation of general non-convex stochastic constraints (Pinter, 1989). To tackle this issue, this paper use the following function

Theorem 3.2. If $0<v_2⩽v_1$ and $\alpha >0$ , then the function $B\left(\alpha ,z\right)$ possesses the following two properties:(1) $B\left(\alpha ,z\right)>0$ , for any value of z;(2) $B\left(\alpha ,z\right)⩾1$ , for $z⩾0$ .

Proof. (1) Note that $v_1>0,v_2>0$ , and $\alpha >0$ . Then, from $e^{-\frac{1}{\alpha }z}>0$ with $z\in R$ , it follows that $B\left(\alpha ,z\right)>0$ , for any value of z.(2) Clearly, one can obtain that

Theorem 3.3. If $v_1$ and $v_2$ satisfy the condition $0<v_2⩽\frac{v_1}{1+v_1}$ , then the function $B\left(\alpha ,z\right)$ possesses the following three properties:(1) $B\left(\alpha ,z\right)$ is strictly monotonically increasing with respect to $z\in R$ ;(2) $B\left(\alpha ,z\right)$ is non-decreasing with respect to $\alpha \in (0,1)$ ;(3) $B\left(\alpha ,z\right)$ satisfies the following equality:

Proof. (1) Note that the function $e^{-\frac{1}{\alpha }z}$ is strictly decreasing with respect to $z\in R$ . Then, we can obtain that $B\left(\alpha ,z\right)$ is strictly monotonically increasing with respect to $z\in R$ .(2) By using the definition $B\left(\alpha ,z\right)$ described by (3.12), the partial derivative of $B\left(\alpha ,z\right)$ with respect to $\alpha$ can be given by $$\frac{\partial \mathrm{B}\left(\alpha ,z\right)}{\partial \alpha }=\frac{v_1}{1+\alpha v_2{e}^{-\frac{1}{\alpha }z}}+\frac{1+\alpha v_1}{{\left(1+\alpha v_2{e}^{-\frac{1}{\alpha }z}\right)}^2}\left[-v_2{e}^{-\frac{1}{\alpha }z}-v_2\left(\frac{z}{\alpha }\right)e^{-\frac{1}{\alpha }z}\right]$$

Assumption 3.2. For any fixed $u\left(t\right)\in U,\mathit{Prob}\left(\mathrm{\Xi }\left(u\left(t\right)\right)\right)=0$ , where $\mathit{Prob}\left\{·\right\}$ presents the probability and $\mathrm{\Xi }\left(u\left(t\right)\right)=\left\{\left.\theta \in \mathrm{\Lambda }\right|g_i\left(u\left(t\right),\theta \right)=0,i=1,\cdots ,q\right\}$ .

Now, by using the properties (2)-(3) of the functions ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , the definition of $B\left(\alpha ,z\right)$ , the definition of $A\left(g_i\left(u\left(t\right),\theta \right)\right),i=1,\dots ,q$ , and Theorem 3.3, the following corollary can be obtained directly.

Corollary 3.1. If $v_1$ and $v_2$ satisfy the condition $0<v_2⩽\frac{v_1}{1+v_1}$ , Assumption 3.2 is true, and the variable z is replaced by using $g_i\left(u\left(t\right),\theta \right),i=1,\dots ,q$ , in Theorem 3.3, then

Close

Fig. 3.1

$A\left(z\right)$ and $B\left(\alpha ,z\right)$ with $v_1=1,v_2=0.3$ , and different values of $\alpha$ .

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4.1 Convergence analysis of feasible regions

By using the property (2) of ${\mathrm{\Upsilon }}_i\left({\alpha }_k,u\left(t\right)\right),i=1,\cdots ,q$ , if follows that $\mathrm{\Gamma }\left(\alpha \right)\subset \mathrm{\Delta }$ , for any $\alpha \in \left(0,1\right)$ . Particularly, the sequence ${\left\{\mathrm{\Gamma }\left({\alpha }_k\right)\right\}}_{k=1}^{+\infty }$ satisfies $\underset{k\to +\infty }{\lim \sup }\mathrm{\Gamma }\left({\alpha }_k\right)\subset \mathrm{\Delta }$ , for any sequence ${\left\{{\alpha }_k\right\}}_{k=1}^{+\infty }$ with $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ . Further, $\underset{k\to +\infty }{\lim }\mathrm{\Gamma }\left({\alpha }_k\right)=\mathrm{\Delta }$ is guaranteed under Assumption 3.1 (see Theorem 3.1). Next, the following Theorem will show that the feasible regions $\mathrm{\Gamma }\left({\alpha }_k\right)$ of problem (4.1a)-(4.1d) with $k\in \{1,2,\cdots \}$ are uniformly convergent to the feasible region $\mathrm{\Delta }$ of problem (2.1a)-(2.1e).

Theorem 4.1. Suppose that ${\left\{{\alpha }_k\right\}}_{k=1}^{+\infty }\subset \left(0,1\right)$ is a sequence satisfying $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ . Then, Assumption 3.1 indicates that

Proof. Note that ${\left\{\mathrm{\Gamma }\left({\alpha }_k\right)\right\}}_{k=1}^{+\infty }$ is a sequence consisting of compact sets, $\mathrm{\Gamma }\left({\alpha }_k\right)\subset U$ , the set U is compact, $\underset{k\to +\infty }{\lim }\mathrm{\Gamma }\left({\alpha }_k\right)=\mathrm{\Delta }$ (see Theorem 3.1), and the set $\mathrm{\Delta }$ is also compact. Then, by using the definition of the Hausdorff distance, we can obtain that $${\displaystyle \underset{k\to +\infty }{\lim }}D\left(\mathrm{\Gamma }\left({\alpha }_k\right),\mathrm{\Delta }\right)=0\text{.}$$ This completes the proof of Theorem 4.1.

Remark 4.1. Note that the results of Theorem 3.1 and 4.1 is true for any sequence ${\left\{{\alpha }_k\right\}}_{k=1}^{+\infty }$ with $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ and ${\alpha }_{k+1}<{\alpha }_k,i=1,2,\cdots$ . Then, these results can be summarized as follows:

In generally, it is challenging to obtain the exact analytic representation for the stochastic constraints described by (2.1d). In addition, it is also very difficulty to solve directly problem (2.1a)-(2.1e) numerically on $\mathrm{\Delta }$ . Fortunately, the above analysis and discussion show that $\mathrm{\Gamma }\left(\alpha \right)$ can be as close as possible to $\mathrm{\Delta }$ and equal to $\mathrm{\Delta }$ in the limit. This implies that we can obtain the numerical solution of problem (2.1a)-(2.1e) on $\mathrm{\Gamma }\left({\alpha }_k\right)$ with $\underset{k\to +\infty }{\lim }{\alpha }_k=0$ .

4.2 Convergence analysis of approximate solutions

From the definition of problem (4.1a)-(4.1d) and the properties (1)-(3) of ${\mathrm{\Upsilon }}_i\left(\alpha ,u\left(t\right)\right),i=1,\cdots ,q$ , it follows that any solution $u_k\left(t\right)$ of problem (4.1a)-(4.1d) is a feasible solution of problem (3.6a)-(3.6e). Further, for any ${\alpha }_k$ , the objective function’s optimal value of problem (4.1a)-(4.1d) is an upper bound for the objective function’s optimal value of problem (2.1a)-(2.1e). Next, the following Theorem will show that any limit point of the local optimal solution sequence ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ of problem (4.1a)-(4.1d) is also a local optimal solution of problem (2.1a)-(2.1e).

Theorem 4.2. Suppose that ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ is a sequence and $u_k\left(t\right)$ is a local optimal solution of problem (4.1a)-(4.1d), for any $k\in \{1,2,\cdots \}$ . Then, there exists a subsequence ${\left\{u_{k_j}\left(t\right)\right\}}_{k_j=1}^{+\infty }$ of ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ such that $\underset{k_j\to +\infty }{\lim }{u}_{k_j}\left(t\right)=u^{*}\left(t\right)$ and there exists an open ball $C\left(u^{\ast }\right(t\left)\right)$ around $u^{\ast }\left(t\right)$ such that $u^{*}\left(t\right)\in \mathrm{\Delta }\cap C\left(u^{*}\left(t\right)\right)$ and $u^{\ast }\left(t\right)$ is a local optimal solution of problem (2.1a)-(2.1e). Inversely, if $\tilde{u}\left(t\right)$ is a strict local optimal solution of problem (2.1a)-(2.1e), then there exists a local optimal solution sequence ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ of problem (4.1a)-(4.1d) such that $\underset{k\to +\infty }{\lim }{u}_k\left(t\right)=\tilde{u}\left(t\right)$ .

Proof. Note that the set U is compact and ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }\subset U$ . Thus, there exists a subsequence ${\left\{u_{k_j}\left(t\right)\right\}}_{k_j=1}^{+\infty }$ of ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ such that $\underset{k_j\to +\infty }{\lim }{u}_{k_j}\left(t\right)=u^{*}\left(t\right)$ . Since $u_{k_j}\left(t\right)\in \mathrm{\Gamma }\left({\alpha }_{k_j}\right)$ and $\underset{k\to +\infty }{\lim }\mathrm{\Gamma }\left({\alpha }_k\right)=\mathrm{\Delta }$ (see Theorem 3.1), one can obtain that $u^{\ast }\left(t\right)\in \mathrm{\Delta }$ . Further, for a sufficiently large $j\in \{1,2,\cdots \}$ , there exists a ball $C\left(u^{\ast }\right(t\left)\right)$ such that $u_{k_j}\left(t\right)\in \mathrm{\Delta }\cap C\left(u^{*}\left(t\right)\right)$ is a local optimal solution of problem (4.1a)-(4.1d).

Suppose that $u^{\ast }\left(t\right)\in \mathrm{\Delta }\cap C\left(u^{*}\left(t\right)\right)$ is not a local optimal solution of problem (2.1a)-(2.1e). This indicates that there exists a $\overline{u}\left(t\right)\in \mathrm{\Delta }\cap C\left(u^{*}\left(t\right)\right)$ such that

Suppose that $\tilde{u}\left(t\right)$ is a local optimal solution of problem (2.1a)-(2.1e). Let $J\left(u\left(t\right)\right)>J\left(\tilde{u}\left(t\right)\right)$ , for any $u\left(t\right)\in \overline{C}\left(\tilde{u}\left(t\right)\right)$ , where $\overline{C}$ denotes the closure of C and the inequality constraints $G_i\left(u\right(t\left)\right)⩾{∊}_i,i=1,\cdots ,q$ , hold true, for a bounded ball $C\left(\tilde{u}\left(t\right)\right)$ . Further, let $J\left(u\left(t\right)\right)⩾J\left(u_k\left(t\right)\right),k=1,2,\cdots$ , for any $u\left(t\right)\in \overline{C}\left(\tilde{u}\left(t\right)\right)$ with $1-{\mathrm{\Upsilon }}_i\left({\alpha }_k,u\left(t\right)\right)⩾{∊}_i,i=1,\cdots ,q$ . By using the compactness of $\overline{C}$ , it follows that there exists a sequence ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ such that $\underset{k\to +\infty }{\lim }{u}_k\left(t\right)=u^{*}\left(t\right)$ . Note that $\underset{k\to +\infty }{\lim }D\left(\mathrm{\Gamma }\left({\alpha }_k\right)\cap \overline{C}\left(\tilde{u}\left(t\right)\right),\mathrm{\Delta }\cap \overline{C}\left(\tilde{u}\left(t\right)\right)\right)=0$ . Thus, by using the continuity of the objective function $J(·)$ , it follows that $J\left(u\left(t\right)\right)⩾J\left(u^{*}\left(t\right)\right)$ for any $u\left(t\right)\in \overline{C}\left(\tilde{u}\left(t\right)\right)$ with $G_i\left(u\left(t\right)\right)⩾{∊}_i,i=1,\cdots ,q$ . Thus, $J\left(u^{*}\left(t\right)\right)=J\left(\tilde{u}\left(t\right)\right)$ . If $u^{*}\left(t\right)\ne \tilde{u}\left(t\right)$ , then this is in contradiction with $\tilde{u}\left(t\right)$ is a strict local optimal solution of problem (2.1a)-(2.1e). Thus, the sequence ${\left\{u_k\left(t\right)\right\}}_{k=1}^{+\infty }$ converges to $\tilde{u}\left(t\right)$ , where $u_k\left(t\right)$ is the local optimal solution of problem (4.1a)-(4.1d). This completes the proof of Theorem 4.2.