Blending Two Feedback Laws

Abstract

In this chapter, the problem under consideration is the synthesis of a stabilizing feedback law for a class of nonlinear systems. These systems have a structural obstacle for the design of a single continuous controller employing the backstepping technique. However, when some boundedness conditions on the vector field are satisfied, a feedback law that practically stabilizes a compact set can be designed. By ensuring that this set is contained in the basin of attraction of a controller that locally stabilizes the equilibrium of the origin, a hybrid feedback law that globally stabilizes the origin can be designed by blending these feedback laws according to appropriate domains of the state space.

Appendix of Chap. 2

2.1.1 The Backstepping Procedure

The backstepping is a well-known method to design a feedback law rendering a class of cascaded systems asymptotically stable. This procedure is recalled here. For more details, the interested reader may address [13, 15,16,17] and references therein.

Consider the system

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{x}_1&{}=&{}f_1(x_1,x_2)\\ \dot{x}_2&{}=&{}f_2(x_1,x_2)u \end{array} \right. \end{aligned}$$

(2.37)

where the functions $f_1:\mathbb {R}^n\rightarrow \mathbb {R}^{n-1}$ and $f_2:\mathbb {R}^n\rightarrow \mathbb {R}_{\ne 0}$ are of class $\mathscr {C}^1$.

Assume that, there exists a feedback law $\phi _1\in \mathscr {C}^1(\mathbb {R},\mathbb {R})$ with $\phi _1(0)=0$ for the $x_1$-subsystem rendering the equilibrium of the origin globally asymptotically stable for

$$\begin{aligned} \dot{x}_1=f_1(x_1,\phi _1(x_1)). \end{aligned}$$

From the converse Lyapunov theorem,⁷ there exist a proper function $V_1\in (\mathscr {C}^1\cap \mathbf P )(\mathbb {R}^{n-1},\mathbb {R}_{\ge 0})$ and a function $\alpha _1\in \mathscr {K}_\infty$ such that the Lie derivative of $V_1$ along the vector field $f_1$ yields the inequality $L_{f_1}V_1(x_1,\phi _1(x_1))\le -\alpha _1(|x_1|)$, for every $x_1\in \mathbb {R}^{n-1}$.

Fix the pair $(x_1,x_2)\in \mathbb {R}^{n-1}\times \mathbb {R}$ and consider the function

$$\begin{aligned} \begin{array}{rrcl} \eta _{x_1,x_2}:&{}[0,1]&{}\rightarrow &{}\mathbb {R}\\ &{}s&{}\mapsto &{}sx_2+(1-s)\phi _1(x_1). \end{array} \end{aligned}$$

Since for every pair $(x_1,x_2)\in \mathbb {R}^{n-1}\times \mathbb {R}$, the function $f_2$ is nonzero, by letting , system (2.37) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{x}_1&{}=&{}f_1(x_1,\phi _1(x_1))+(x_2-\phi _1(x_1))\displaystyle \int \limits _0^1 \dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds\\ \dot{x}_2&{}=&{}v, \end{array} \right. \end{aligned}$$

where $v\in \mathbb {R}$.

Consider the change of variables $e:=x_2-\phi _1(x_1)$. The time derivative of e yields the differential equation $\dot{e}=\dot{x}_2-L_{f_1}\phi _1(x_1,x_2)$. System (2.37) rewritten in the new variable e is given by

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{x}_1&{}=&{}f_1(x_1,\phi _1(x_1))+e\displaystyle \int \limits _0^1 \dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds\\ \dot{e}&{}=&{}w, \end{array}\right. \end{aligned}$$

(2.38)

where $w=v-L_{f_1}\phi _1(x_1,x_2)$. System (2.38) is denoted by $\dot{x}=f(x)$.

Consider the candidate Lyapunov function for system (2.38) given by

$$\begin{aligned} V(x_1,e)=V_1(x_1)+\dfrac{e^2}{2}. \end{aligned}$$

Its Lie derivative along the vector field f yields the inequality

$$\begin{aligned} \begin{array}{rcl} L_fV(x_1,e)\le & {} \alpha _1(|x_1|)+e\left[ \dfrac{\partial V_1}{\partial x_1}(x_1)\cdot \displaystyle \int \limits _0^1\dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds+w\right] . \end{array} \end{aligned}$$

Consider the feedback law defined by the equation

$$\begin{aligned} \phi (x_1,e)=-\dfrac{\partial V_1}{\partial x_1}(x_1)\cdot \int \limits _0^1\dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds-Ke, \end{aligned}$$

where $K>0$ is a constant value. Letting $w=\phi (x_1,e)$, the Lie derivative $L_fV$ yields the inequality

$$\begin{aligned} L_fV(x_1,e)\le -\alpha _1(|x_1|)-Ke^2 \end{aligned}$$

which holds for every $(x_1,e)\in \mathbb {R}^{n-1}\times \mathbb {R}$. Thus, the equilibrium of the origin is globally asymptotically stable for system (2.38) in closed loop with $\phi$. Hence, it is also asymptotically stable for (2.37).

Since $\phi (x_1,e)=w=v-L_{f_1}\phi _1(x_1,x_2)$, it follows that

$$\begin{aligned} v = -\dfrac{\partial V_1}{\partial x_1}(x_1)\cdot \int \limits _0^1\dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds-(x_2-\phi _1(x_1))+L_{f_1}\phi _1(x_1,x_2). \end{aligned}$$

Therefore, the feedback law defined, for every $(x_1,x_2)\in \mathbb {R}^{n-1}\times \mathbb {R}$, by⁸

$$\begin{aligned} \varphi _b(x_1,x_2)=&\dfrac{1}{f_2(x_1,x_2)}\bigg [-\dfrac{\partial V_1}{\partial x_1}(x_1)\cdot \int \limits _0^1\dfrac{\partial f_1}{\partial \eta _{x_1,x_2}}(x_1,\eta _{x_1,x_2}(s))\,ds-(x_2-\phi _1(x_1))\\&\ +L_{f_1}\phi _1(x_1,x_2)\bigg ]. \end{aligned}$$

rends the equilibrium of the origin globally asymptotically stable for (2.37) in closed loop.

2.1.2 The Schur’s Complement

The Schur’s complement is employed in this chapter to design a linear feedback law rendering the origin locally asymptotically stable. For further reading on the Schur’s complement, the interested reader is invited to address [14, 28]. Here, some of the basic concepts are recalled.

Consider the matrices $A\in \mathbb {R}^{p\times p}$, $B\in \mathbb {R}^{p\times q}$, $C\in \mathbb {R}^{q\times p}$, $D\in \mathbb {R}^{q\times q}$ and the block matrix $M\in \mathbb {R}^{(p+q)\times (p+q)}$ given by

$$\begin{aligned} M=\begin{bmatrix} A&B\\ C&D \end{bmatrix}, \end{aligned}$$

and assume that $\det (A)\ne 0$. Consider a vector $z=(x,y)\in \mathbb {R}^p\times \mathbb {R}^q$. The linear system $Mz^T=0$, i.e., the system

$$\begin{aligned} \left\{ \begin{array}{rclrl} Ax&{}+&{}By&{}=&{}0,\\ Cx&{}+&{}Dy&{}=&{}0. \end{array}\right. \end{aligned}$$

Multiplying the first equation by $-CA^{-1}$, on the left, and adding it to the second one, the x-component of the vector is eliminated, and the linear system is given by

$$\begin{aligned} (D-CA^{-1}B)y=0. \end{aligned}$$

The matrix $S=D-CA^{-1}B$ is called Schur complement of A in M (cf. [14]).

The next result is recalled from [28] and adapted to the context of this book.

Theorem 2.25

Let $M\in \mathbb {R}^{(p+q)\times (p+q)}$ be a symmetric matrix given by

$$\begin{aligned} M=\begin{bmatrix} A&B\\ B^T&D \end{bmatrix}, \end{aligned}$$

where $A\in \mathbb {R}^{p\times p}$ and $\det (A)\ne 0$. Then, $M\succ 0$ if and only if $A\succ 0$ and $(D-B^T A^{-1}B)\succ 0$.

From [14, Proposition 1], under the hypothesis of Theorem 2.25, $M\succ 0$ if and only if $D\succ 0$ and $A-B D^{-1}B^T\succ 0$.

2.1.3 A Remark on the Lyapunov Sufficient Conditions for Practical Stability

Recall the concept of global practical asymptotic stabilizability stated in Definition 2.16.

Under Assumptions 2.1 and 2.2, for every value $a>0$, there exists (cf. Proposition 2.4, above.) a feedback law $\varphi _g:\mathbb {R}^n\rightarrow \mathbb {R}$ such the set $\mathbf B _{\le a}(\mathbf A )$ contains a compact invariant set that is globally asymptotically stable for the closed-loop system ($\Sigma _h(\varphi _g)$), where $\mathbf A$ is the set given by

$$\begin{aligned} \mathbf A =\{(x_1,x_2)\in \mathbb {R}^{n-1}\times \mathbb {R}:V_1(x_1)\le M,x_2=\psi _1(x_1)\}. \quad (2.8) \end{aligned}$$

Because of the value $K_V$, the feedback law $\varphi _g$ is parametrized by a (cf. the proof of Proposition 2.4). Consequently, the closed-loop system ($\Sigma _h(\varphi _g)$) and the candidate Lyapunov function

$$\begin{aligned} \begin{array}{rcl} V:\mathbb {R}^{n-1}\times \mathbb {R}&{}\rightarrow &{}\mathbb {R}\\ (x_1,x_2)&{}\mapsto &{}V_1(x_1)+\dfrac{K_V}{2}(x_2-\psi _1(x_1))^2 \end{array}\quad (2.24) \end{aligned}$$

are also parametrized by a.

From now on, the dependence of aforementioned functions on the parameter a is highlighted by adding it as subscript, for instance, ($\Sigma _h(\varphi _{g,a})$).

The following theorem is presented in [3, Theorem 7.5] and it has been reformulated to the context of this chapter. It provides a sufficient condition for the stability of a compact invariant set when the candidate Lyapunov function and the feedback law depend on the parameter.

Theorem 2.26

(Lyapunov sufficient conditions for global practical asymptotic stability) Let $\mathbf A \subset \mathbb {R}^n$ be a compact set. Suppose that, given any $a>0$, there exist a continuous differentiable Lyapunov function $V_a:\mathbb {R}^n\rightarrow \mathbb {R}_{\ge 0}$, and functions $\underline{\alpha }_a$, $\overline{\alpha }_a$ and $\alpha _a$ of class $\mathscr {K}_\infty$ such that, for every $x\in \mathbf B _{\ge a}(\mathbf A )$, the following inequalities

$$\begin{aligned} \underline{\alpha }_a(|x|_\mathbf A )\le V_a(x)\le \overline{\alpha }_a(|x|_\mathbf A ), \end{aligned}$$

(2.39a)

$$\begin{aligned} L_{f_h}V_a(x,\varphi _{g,a})\le -\alpha _a(|x|_\mathbf A ), \end{aligned}$$

(2.39b)

$$\begin{aligned} \lim _{a\rightarrow 0} \underline{\alpha }_a^{-1}\circ \overline{\alpha }_a(a)=0 \end{aligned}$$

(2.39c)

hold. Then, the set $\mathbf A$ is globally practically asymptotically stable for the closed-loop system $\Sigma _h(\varphi _g)$.