Prescribed-time Control for Linear Systems in Canonical Form Via Nonlinear Feedback

We restrict our analysis to the following system in canonical form with uncertainties/disturbances

where $A=J_{n}+\mathcal{L}_{n}(\mathbf{a})$ is the system matrix, $B=[0,\cdots,0,1]^{\top}$, $C=[b_{0},b_{1},\cdots,b_{n-1}]$ are coefficient vectors, $D=[0,\cdots,0,d(\mathbf{x},t)]^{\top}$ with $d(\mathbf{x},t):\mathbf{R}^{n}\times[0,\infty)\rightarrow\mathbf{R}$ modeling the unknown nonvanishing uncertainties/disturbances of the system, ${\mathbf{x}}(t)=[x_{1},\cdots,x_{n}]^{\top}\in\mathbf{R}^{n}$ is the vector of system states, and $u(t)\in\mathbf{R}$ is the control input. $(A,B)$ is controllable and $(A,C)$ is observable. The control objective is to design a feedback control $u(t)$ to stabilize (1) within prescribed-time $t_{p}$, i.e., $\mathbf{x}(t)\in\ell_{\infty}[0,t_{p})$ and $\lim_{t\rightarrow t_{p}}\{x_{i}(t)\}_{i=1}^{n}=0.$ We are particularly interested in making use of the feedback information $x$ through a nonlinear way to construct the control scheme.

Instead of using $x$ directly, we process the feedback information $x$ through the following hyperbolic-tangent-like function $h(x):(-\infty,+\infty)$ $\rightarrow$ $\left(-\frac{1}{b},\frac{1}{a}\right)$ as:

where $a$ and $b$ are design parameters, which becomes the standard hyperbolic tangent function for $a=b=1$. Such nonlinear feedback exhibits two salient properties.

Property 1: The function $h(x)$ is $\mathcal{C^{\infty}}$ on $\mathbf{R}$ and $h(x)=0$ if and only if $x=0$. Under $0<b\leq a$, the inequality $0\leq|x|h(|x|)\leq xh(x)$ holds.

Proof: Define a continuous function $F(x)=xh(x)-|x|h(|x|)$. For $\forall x\geq 0$, we have

For $\forall x<0$, it follows that

Thus, under $0<b\leq a$, the inequality $F(x)\geq 0$ holds for $\forall x\in(-\infty,+\infty)$, implying that $0\leq|x|h(|x|)\leq xh(x)$ for $\forall x\in(-\infty,+\infty)$. $\hfill\blacksquare$

Property 2: Function $h(x)$ is strictly monotonically increasing with respect to (w.r.t.) $x$, its upper and lower bounds are $1/a$ and $-1/b$, respectively. By selecting different design parameters $a$ and $b$, various functions can be obtained from $h(x)$, In particular, if choosing $a$ and $b$ small enough, then $h(x)=x$.

Proof: The upper and lower bounds of $h(x)$ are

When $a$ and $b$ are sufficiently small, for $~{}x\in(-\infty,+\infty)$, by using L’H$\hat{\text{o}}$pital’s Rule, we have

implying that the expanded/compressed signal $h(x)$ reverses back to the regular signal $x$. $\hfill\blacksquare$

The interesting feature behind this nonlinear feedback is that it includes regular (direct) feedback of $x$ as a special case, and it allows the linear regular feedback control and nonlinear feedback control to be unified through such function, providing a variety of ways to making use of $x$ for control development. By using properties of the hyperbolic-tangent-like function, we establish the following lemma, which is crucial to our later technical development.

Proof: Consider the analytical expression of (3):

Let $V_{*}=e^{aV}-e^{-bV}$, where ${V_{*}}_{0}=e^{aV(0)}-e^{-bV(0)}$. Then we have $V_{*}\geq 0$, and

holds for $t\in[0,t_{p})$. Hence we derive $V_{*}\leq\beta({V_{*}}_{0},\mu(t))$ according to Lemma 1 in [18], namely $V_{*}\in\ell_{\infty}[0,t_{p})$ and $\lim_{t\rightarrow t_{p}}V_{*}=0$. The same result can be established for $(e^{aV}-e^{-bV})$. It follows from the fact $V$ tends to $+\infty$ “slower” than $(e^{aV}-e^{-bV})$ as $V\rightarrow+\infty$ that $V(t)\in\ell_{\infty}[0,t_{p})$ and $\lim_{t\rightarrow t_{p}}V(t)=0$. In addition, the inequality (4) can be transformed into the following form:

where $C_{1}=\left(e^{aV_{0}}-e^{-bV_{0}}\right)/{t_{p}^{k}}$ is the integral constant. In fact, one can easily verify the following calculations

Furthermore, from (6.3) we have $(a+b)\dot{V}\leq 0$ and

Indeed, notice that $\dot{V}$ is a continuous function and $\dot{V}=-k\mu h(V)$ for $V=0$, implying that $\dot{V}\in\ell_{\infty}[0,t_{p})$ and $\dot{V}(t)\rightarrow 0$ as $t\rightarrow t_{p}$. This completes the proof. $\hfill\blacksquare$

As discussed earlier, when $a$ and $b$ are sufficiently small, we have $\lim_{a,b\rightarrow 0}h(V)=V$. Lemma 1 is therefore equivalent to the Corollary 1 in [18].

By using the time-varying scaling function and the hyperbolic-tangent-like function as introduced in Section 2, we construct the vectors as

where $\mu=1/(t_{p}-t)$ and $h(\cdot)$ is defined in (2). In addition, we introduce the following two auxiliary vectors

where $\mathbf{z}=[z_{1},\cdots,z_{n}]^{\top}\in\mathbf{R}^{n}$, $\Phi=[\phi_{1},\cdots,\phi_{n}]^{\top}\in\mathbf{R}^{n}$. Note that $J_{n}^{\top}$ is lower triangular, thus both $\mathbf{z}$ and $\Phi$ can be easily calculated recursively (see (28) for specific example of computing $z_{i}$ and $\phi_{i}$). It is interesting to see that the convergence property of the closed-loop system only depends on the parameters $a$, $b$, $k$ and $r$ in vector $\Gamma(\mathbf{z})$.

Proof: By the definition of $J_{n}$, $B$ and $\mathcal{L}_{n}(\mathbf{a})$, it is readily verified that $J_{n}J_{n}^{\top}+BB^{\top}=I_{n}$ and $BB^{\top}\mathcal{L}_{n}(\mathbf{a})=\mathcal{L}_{n}(\mathbf{a})$, therefore, it holds that

1) proof of GUAS result: we prove that the closed-loop system under the control law (10) with $r=0$ (in this case, $\Gamma(\mathbf{z})=kH(\mathbf{z})$) is GUAS. To this end, we define the error vector between $H(\mathbf{z})$ and $\mathbf{z}(t)$ as

where $\mathbf{E}=[e_{1},\cdots,e_{n}]^{\top}\in\mathbf{R}^{n}$ is a smooth function satisfying $\lim_{\alpha,\beta\rightarrow 0}\mathbf{E}=\mathbf{0}$ and $\mathbf{E}$ is bounded as long as $\mathbf{z}(t)$ is bounded. Consider a positive definite function $V_{1}=\mathbf{z}^{\top}\mathbf{z}/2$, the time derivative of $V_{1}$ along (11) is

where the fact that ${\mathbf{z}}^{\top}\left(J_{n}-J_{n}^{\top}\right){\mathbf{z}}=0,~{}\forall{\mathbf{z}}\in\mathbf{R}^{n}$ is used since $J_{n}-J_{n}^{\top}$ is a skew symmetric matrix. It follows from (13) that $\dot{V}_{1}=0$ if and only if $\mathbf{z}=\mathbf{0}$, thus the transformed system (11) is asymptotically stable on $[0,+\infty)$, establishing the same to system (1) according to the converging-input converging-output property of the corresponding auxiliary vectors.

From (13), it can be further shown that

where $\Delta\coloneqq\sup\{\mathbf{\|E\|}\}$. By integrating both sides of the inequality (14), we obtain $V_{1}(t)\leq V_{1}(0)e^{-kt/2}+\Delta^{2}(1-e^{-kt/2})$. If we further choose the design parameters $a$ and $b$ in $H(\mathbf{z})$ small enough, one can obtain $\lim_{a,b\rightarrow 0}\Delta=0$, yielding $V_{1}(t)\leq V_{1}(0)e^{-kt/2}$, thus we have $\|\mathbf{z}(t)\|\leq\|\mathbf{z}(0)\|{e^{-kt/2}}$ and the transformed system (11) is exponentially stable. In addition, it follows from (9) that $x_{1}(t)=z_{1}(t)$, therefore exponential output regulation to zero of (1) can be achieved. The word “exponential” actually means that “near exponential”, because the parameters $a$ and $b$ can only be selected as sufficiently small, not zero.

2) Proof of PT-GUAS result: we now show that the closed-loop system under the control law (10) with $r=1$ (in this case, $\Gamma(\mathbf{z})=k\mu H(\mathbf{z})$) is PT-GUAS. For $t\in[0,t_{p}),$ there exists a continuous positive function $W(\mathbf{z},t)=\sqrt{\mathbf{z}^{\top}{\mathbf{z}}/n}$ such that $W(\mathbf{z},t)=0$ as $\mathbf{z}=\mathbf{0}$ and

In addition, its time derivative can be shown as

Using Property 1 and (15), we have

Since we select $k>n$, then $k/n>1$. By virtue of Lemma 1, one can prove that $W(t)\in\ell_{\infty}[0,t_{p})$, $\dot{W}(t)\in\ell_{\infty}[0,t_{p})$ and

Hence the closed-loop signals $\{z_{i}\}_{i=1}^{n}\in\ell_{\infty}[0,t_{p})$ and $\{\dot{z}_{i}\}_{i=1}^{n}\in\ell_{\infty}[0,t_{p})$, and meanwhile converge to zero as $t$ tends to $t_{p}$. Using the property of the hyperbolic-tangent-like function, we can proceed to prove that $\Gamma(\mathbf{z})\in\ell_{\infty}[0,t_{p})$ and the convergence of which to zero at the prescribed time.

By means of the auxiliary vectors as introduced in (9), one can find that $x_{1}=z_{1}$ and the following closed-loop $z_{1}$-dynamics holds

where $z_{2}$ is a bounded function, which also can be treated as a vanishing disturbance. When $t\rightarrow t_{p}$, the equivalent form of (19) is

It follows from (4)-(6) and (20) that

where $C_{2}=\left(e^{ax_{1}(0)}-e^{-bx_{1}(0)}\right)/t_{p}^{k}$. Then taking time derivative on both sides of (21), we have

Observe that (22) means that $x_{2}\rightarrow 0$ as $t\rightarrow t_{p}$. Continue, using the analysis similar to that used in (21)-(22), by taking the $i$-th $(i=2,\cdots,n)$ derivative of both sides of (21), we can generalize that $\{x_{i}\}_{i=2}^{n}$ and $\{\dot{x}_{i}\}_{i=1}^{n}$ converge to zero as $t\rightarrow t_{p}$ (this is the reason for $k>n$). Therefore, $\lim_{t\rightarrow t_{p}}\|\mathbf{x}(t)\|=0$ and $\lim_{t\rightarrow t_{p}}\|\dot{\mathbf{x}}(t)\|=0$ hold.

In addition, it follows from (9) that $J_{n}^{\top}\dot{\Phi}=\dot{\mathbf{z}}-\dot{\mathbf{x}}$, therefore $\Phi=\dot{\mathbf{z}}-\dot{\mathbf{x}}+J_{n}^{\top}\mathbf{z}+\Gamma(\mathbf{z})$, then

Consequently, from (10) we have

Note that each term in the third line of (24) is bounded on $[0,t_{p})$ and converges to zero as $t\rightarrow t_{p}$. Therefore, $u(t)\in\ell_{\infty}[0,t_{p})$ and $\lim_{t\rightarrow t_{p}}|u(t)|=0.$ This completes the proof. $\hfill\blacksquare$