Compensation of Unmatched Disturbances via Sliding-Mode Control

Abstract

We study the disturbance rejection properties of recent and classical sliding-mode approaches with respect to unmatched disturbances. We consider nonlinear systems with arbitrary relative degree subject to various configurations of unmatched disturbances. The disturbances are state-dependent, time-varying, or a combination of both. For our analysis, we choose to transform the system into Byrnes–Isidori form and discuss the impact of the disturbances on the relative degree and stability properties. We investigate the capability of the considered sliding-mode approaches to compensate the disturbances on a given output of interest. In a comprehensive case study, we illustrate the characteristics of each approach on various system configurations. Finally, we isolate several mechanisms that are responsible for the disturbance compensation in each case.

Appendices

Appendix 1: Relative Degree Under Disturbance

A disturbance $\phi :\mathbb R^n \times \mathbb {R} \rightarrow \mathbb R^{n}$ may change the relative degree of the nominal system for a given output y as the following example illustrates. Consider the system

$$\begin{aligned} \dot{x}_1&=x_2+\phi _1(x,t)\\ \dot{x}_2&=x_3\\ \dot{x}_3&=u \\ y&=x_1\,. \end{aligned}$$

For $\phi _1=0$, the output y has relative degree 3, whereas for $\phi _1(x,t)=x_3$ the output y has relative degree 2.

In order to formulate a condition that ensures that the relative degree is not impaired by the disturbance $\phi$, we introduce the following multi-index notation for the Lie derivative of the sum of two vector fields f and $\phi$. Let

$$\begin{aligned} (L_{f}L_{\phi })^{(\alpha ,\beta )}:=L_{f}^{\alpha _1}L^{\beta _1}_{\phi } \dots L_{f}^{\alpha _n}L^{\beta _n}_{\phi } \end{aligned}$$

(9.58)

denote the composition of Lie derivatives according to the combination indicated by $\alpha _i,\beta _i\in \{0,1\}$. Observe that the Lie derivatives $L_{f}$ and $L_{\phi }$ do not commute in general. Therefore, we write for the nth Lie derivative of the vector field $f+\phi$

$$\begin{aligned} L^n_{f+\phi }=\sum _{|{(\alpha ,\beta )}|=n}(L_{f}L_{\phi })^{(\alpha ,\beta )}\,, \end{aligned}$$

(9.59)

where $\alpha ,\beta \in \{0,1\}^n$ and

$$\begin{aligned} |{(\alpha ,\beta )}|:=\sum _{i=1}^{n}\alpha _i+\beta _i\,. \end{aligned}$$

(9.60)

The composition $(L_{f}L_{\phi })^{(\alpha ,\beta )}$ denotes permutations of Lie derivatives of length n.

Consider the system (9.1) with relative degree r, and then the definition (9.3) ensures $L_{g}L_{f}^k h(x)=0$ and $L_{g}L_{f}^{r-1} h(x)\ne 0$ for $k \in \{0,\dots ,r-2\}$. In order to retain the relative degree, the condition

$$\begin{aligned} L_{g}L^k_{f+\phi }h(x)=0 \end{aligned}$$

(9.61a)

$$\begin{aligned} L_{g}L^{r-1}_{f+\phi }h(x)\ne 0 \, \end{aligned}$$

(9.61b)

must hold. Using the proposed notation (9.58)–(9.60), we may write

$$\begin{aligned} L_{g}L^k_{f+\phi }h(x)=L_{g}L_{f}^k h(x) +\sum _{|{(\alpha ,\beta )}|=k, \alpha \ne 1}L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big )\quad \text {for}\quad k \in \mathbb {N}\,. \end{aligned}$$

(9.62)

Thus, with (9.62) we can see that condition (9.61a) is equivalent to

$$\begin{aligned} \sum _{|{(\alpha ,\beta )}|=k, \alpha \ne 1}L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big )=0\quad \text {for}\quad k \in \{1,\dots , r-2\}\,. \end{aligned}$$

(9.63)

Further, we can see that with (9.3b) and (9.62), the condition

$$\begin{aligned} \sum _{|{(\alpha ,\beta )}|=r-1, \alpha \ne 1}L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big ) = 0 \end{aligned}$$

(9.64)

is sufficient but not necessary to ensure (9.61b). These two conditions are fulfilled if

$$\begin{aligned} L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big )&= 0,&\alpha ,\beta \in \{0,1\}^k, |{(\alpha ,\beta )}|=k\end{aligned}$$

(9.65a)

$$\begin{aligned} L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big )&=0,&\alpha ,\beta \in \{0,1\}^{r-1}, |{(\alpha ,\beta )}|=r-1 \end{aligned}$$

(9.65b)

for $k \in \{0, \dots , r-2\}$, $|{(\alpha ,\beta )}|=k$ and $\alpha {\ne } 1$ hold. Finally, this can be expressed by

$$\begin{aligned} L_{g}\big ( (L_{f}L_{\phi })^{(\alpha ,\beta )}h(x)\big ) = 0\, \end{aligned}$$

(9.66)

for $k \in \{0, \dots , r-1\}$, $\alpha ,\beta \in \{0,1\}^k$, $|{(\alpha ,\beta )}|=k$ and $\beta \ne 0$. Note that the nominal term $L_gL_f^k h(x)$ is deliberately excluded in condition (9.66) for all $k \in \{0, \dots , r-1\}$. But mind that for $k \in \{0, \dots , r-2\}$ it is $L_gL_f^k h(x)=0$, while $L_gL_f^{r-1} h(x)\ne 0$ by definition of the relative degree of the nominal system. The proposed condition (9.66) is sufficient (but not necessary) to guarantee that the relative degree is not impaired by the disturbance. However, there are a number of system classes that satisfy this condition, such as systems in nonlinear block controllable form.

Appendix 2: Sliding-Mode Differentiator

As introduced in [26] and used in [4], the sliding-mode differentiator for a function f of order r takes the form

$$\begin{aligned} \dot{\nu }_0&=-\lambda _0 \varLambda ^{\frac{1}{r+1}}\varPsi ^{\frac{r}{r+1}}(\nu _0-f(t))+\nu _1\\ \dot{\nu }_i&=-\lambda _i \varLambda ^{\frac{1}{r+1-i}}\varPsi ^{\frac{r-i}{r-i+1}}(\nu _i-\dot{\nu }_{i-1})+\nu _{i+1}\\ \dot{\nu }_{r}&=-\lambda _{r} \varLambda \varPsi ^{0}(\nu _{r}-\dot{\nu }_{r-1})\,, \end{aligned}$$

where

$$\begin{aligned} \varPsi ^\beta (\zeta )=\begin{bmatrix}|{\zeta _1}|^\beta {{\,\mathrm{sign}\,}}(\zeta _1)&\dots&|{\zeta _n}|^\beta {{\,\mathrm{sign}\,}}(\zeta _n) \end{bmatrix}^{\top } \end{aligned}$$

and $\nu _i$ is the ith derivative of f.

Appendix 3: Quasi-continuous Higher-Order Sliding-Mode Controller

We will consider the quasi-continuous controller from [9] for a switching function with relative degree r as the benchmark HOSM. The control is given by

$$\begin{aligned} u&=-\alpha \varPsi _{r-1,r}(\sigma ,\dot{\sigma },\dots ,\sigma ^{r-1})\,,\\ \varphi _{0,r}&=\sigma , \qquad N_{0,r}=|{\sigma }|, \qquad \varPsi _{0,r}=\frac{\varphi _{0,r}}{N_{0,r}}={{\,\mathrm{sign}\,}}(\sigma )\,,\\ \varphi _{i,r}&=\sigma ^{(i)}+\beta _i N^{\frac{r-i}{r-i+1}}_{i-1,r}\varPsi _{i-1,r}\\ N_{i,r}&=|{\sigma ^{(i)}}|+\beta _i N^{\frac{r-i}{r-i+1}}_{i-1,r}\\ \varPsi _{i,r}(\cdot )&=\frac{\varphi _{i,r}}{N_{i,r}}\,, \qquad i=0,\dots ,r-1\,, \end{aligned}$$

where $\beta _1,\dots , \beta _{r-1}\in \mathbb {R}$. The first-order sliding-mode control law has the form

$$\begin{aligned} u&=\varPsi _{0,1}(\sigma )=-\alpha {{\,\mathrm{sign}\,}}(\sigma )\,, \end{aligned}$$

(9.67)

and the second-order control law takes the form

$$\begin{aligned} u&=-\alpha \varPsi _{1,2}(\sigma ,\dot{\sigma })= -\alpha \frac{\dot{\sigma }+|{\sigma }|^{\frac{1}{2}}{{\,\mathrm{sign}\,}}(\sigma )}{|{\dot{\sigma }}|+|{\sigma }|^{\frac{1}{2}}}\,. \end{aligned}$$

(9.68)

Appendix 4: Controller Design for the Simulation Examples

In this section, we state the control laws implemented for case studies in Sect. 9.6.

1.1 A 4.1 Case: No Internal Dynamics

For $a=0$ and $b=1$, we obtain the system

$$\begin{aligned} \dot{x}_1&=x_2+\phi _1(x,t)\end{aligned}$$

(9.69a)

$$\begin{aligned} \dot{x}_2&=x_3+\phi _2(x,t)\end{aligned}$$

(9.69b)

$$\begin{aligned} \dot{x}_3&=u\end{aligned}$$

(9.69c)

$$\begin{aligned} y&=x_1 \end{aligned}$$

(9.69d)

in controller normal form with y being an output with full relative degree $r=3$.

1.1.1 FOSM and FOSM-D

Define the sliding manifold as in (9.22) by

$$\begin{aligned} \sigma (x)=x_1+2\dot{x}_1+\ddot{x}_1\,. \end{aligned}$$

(9.70)

For FOSM-D $\dot{x}_1$ and $\ddot{x}_1$ are numerically calculated. For FOSM, we use $\dot{x}_1=x_2$ and $\ddot{x}_1=x_3$ and use the sliding manifold proposed in (9.12) with

$$\begin{aligned} \sigma (x)=x_1+2x_2+x_3\,. \end{aligned}$$

(9.71)

Then, choose the control law $w=-\alpha {{\,\mathrm{sign}\,}}(\sigma (x))$ from (9.13)/(9.23) with $\alpha =40$ to obtain the desired control law by using the input transformation (9.14)/(9.24)

$$\begin{aligned} u&=-x_2-2x_3+w\,. \end{aligned}$$

(9.72)

1.1.2 ISMC and ISMC-D

Introduce the integrator v as in (9.31) with $\dot{v}=x_1$ with v(0) so that the initial state lies on the sliding manifold. Then, define the sliding manifold

$$\begin{aligned} \sigma (v)&=v+3\dot{v}+3\ddot{v}+\dddot{v}\,. \end{aligned}$$

For ISMC-D, we use the numerical derivatives of v. For ISMC, we substitute $\dot{v}=x_1$, $\ddot{v}=x_2$, $\dddot{v}=x_3$ and obtain the switching function as in (9.32) with

$$\begin{aligned} \sigma (v,x)&=v+3x_1+3x_2+x_3\,. \end{aligned}$$

Use the control law $w=-\alpha {{\,\mathrm{sign}\,}}(\sigma (v,x))$ with $\alpha =40$ and use the input transformation

$$\begin{aligned} u&=-x_1-3x_2-3x_3+w\, \end{aligned}$$

to obtain the control law as in (9.34).

1.1.3 nHOSM

Define the switching functions and virtual control laws as in (9.43)–(9.45)

$$\begin{aligned} \sigma _1(x)&=x_1\,&\sigma _2(x)&=x_2-w_1\,&\sigma _3(x)&=x_3-w_2\,\\ \dot{u}_{1,1}&=u_{1,2}\,&\dot{u}_{2,1}&=-\alpha _2 \varPsi _{2,3}(\sigma _2,\dot{\sigma }_2)\,&\\ \dot{u}_{1,2}&=-\alpha _1\varPsi _{2,3}(\sigma _1,\dot{\sigma }_1,\ddot{\sigma }_1)\,&\\ w_1&=u_{1,1}\,&w_2&=u_{2,1}\,&w_3&=-\alpha _3 {{\,\mathrm{sign}\,}}(\sigma _3)\, \end{aligned}$$

with $\alpha _1=15$, $\alpha _2=13$, and $\alpha _3=40$. The derivatives $\dot{\sigma }_1$, $\ddot{\sigma }_1$, and $\dot{\sigma }_2$ are numerically calculated. Finally, use the first-order sliding-mode input transformation (9.46a)

$$\begin{aligned} u=-\alpha _2 \varPsi _{2,3}(\sigma _2,\dot{\sigma }_2)+w_3\,. \end{aligned}$$

(9.73)

Note that we use an additional first-order sliding-mode transformation to make this control law more comparable to FOSM, FOSM-D, ISMC, and ISMC-D. This leads to a slightly different notation in comparison to (9.45).

1.1.4 nFOSM

Define the switching functions and virtual control laws as in (9.47)–(9.49)

$$\begin{aligned} \sigma _1(x)&=x_1&\sigma _2(x)&=x_2-w_1&\sigma _3(x)&=x_3-w_2\\ w_1&=-\sigma _1-\phi _1&w_2&=-\sigma _2+\phi _2-x_2-\dot{\phi }_1-\phi _1&w_3&=-\alpha {{\,\mathrm{sign}\,}}(\sigma _3) \end{aligned}$$

with $\alpha = 40$. Use the first-order sliding-mode input transformation

$$\begin{aligned} u&=-2x_3+x_2+w_3\,. \end{aligned}$$

Again, we have introduced the additional first-order sliding-mode transformation which changes the notation in comparison to (9.49) slightly.

1.2 A 4.2 Case: Stable Internal Dynamics

Consider the system (9.51) with $a=-1$

$$\begin{aligned} \dot{x}_1&=x_1+x_3+\phi _1(x_1,t)\end{aligned}$$

(9.74a)

$$\begin{aligned} \dot{x}_2&=-x_2+x_3+\phi _2(x_1,x_2,t)\end{aligned}$$

(9.74b)

$$\begin{aligned} \dot{x}_3&=u\end{aligned}$$

(9.74c)

$$\begin{aligned} y&=x_1\,. \end{aligned}$$

(9.74d)

This system has the internal dynamics

$$\begin{aligned} \dot{\eta }_1&=-\eta _1+\xi _2-\xi _1\,. \end{aligned}$$

(9.75)

It is easy to see that the zero dynamics are asymptotically stable at 0. Thus, it is sufficient to find a control law w for the system (9.54a), (9.54b) to stabilize the system (9.54a)–(9.54c).

1.2.1 FOSM and FOSM-D

For ISMC-D, use the sliding manifold

$$\begin{aligned} \sigma (x)=x_1+\dot{x}_1\, \end{aligned}$$

with numerical derivatives. For ISMC, use the substitution $\dot{x}_1=x_1+x_3\,$ to obtain the sliding manifold

$$\begin{aligned} \sigma (x)=2x_1+x_3\,. \end{aligned}$$

Then, choose $w=-\alpha {{\,\mathrm{sign}\,}}(\sigma )$ with $\alpha = 40$ to obtain the desired control law by using the input transformation

$$\begin{aligned} u&=-2(x_1+x_3)+w\,. \end{aligned}$$

1.2.2 ISMC and ISMC-D

Introduce the integrator v with $\dot{v}=x_1$. For ISMC-D, define the sliding manifold

$$\begin{aligned} \sigma (v)=v+2\dot{v}+\ddot{v}\,. \end{aligned}$$

For ISMC, we use $\dot{v}=x_1$ and $\ddot{v}=x_1+x_3$ to obtain

$$\begin{aligned} \sigma (v,x)=v+2x_1+x_1+x_3 \end{aligned}$$

and choose $w=-\alpha {{\,\mathrm{sign}\,}}(\sigma )$ with $\alpha =40$. Finally, use the input transformation

$$\begin{aligned} u&=x_1+3(x_1+x_3)+w\,. \end{aligned}$$

1.2.3 nHOSM

Define

$$\begin{aligned} \sigma _1&=\xi _1&\sigma _2&=\xi _2-w_1\\ \dot{u}_{1,1}&=-\alpha _1\varPsi _{1,2}(\sigma _1,\dot{\sigma }_1)\\ w_1&=u_{1,1}&w_2&=-\alpha _2 {{\,\mathrm{sign}\,}}(\sigma _2) \end{aligned}$$

with $\alpha _1=2$, $\alpha _2=40$. Use the first-order sliding-mode transformation to obtain

$$\begin{aligned} w&=-\alpha _1\varPsi _{1,2}(\sigma _1,\dot{\sigma }_1)+w_2 \end{aligned}$$

and the final control law

$$\begin{aligned} u&=-x_1-x_3+w\,. \end{aligned}$$

Note that we use an additional first-order sliding-mode transformation to make this control law more comparable to FOSM, FOSM-D, ISMC, and ISMC-D. This leads to a slightly different notation in comparison to (9.45).

1.2.4 nFOSM

Define

$$\begin{aligned} \sigma _1&=\xi _1&\sigma _2&=\xi _2-w_1\\ w_1&=-\phi _1 -\xi _1\,.&w_2&=-\alpha {{\,\mathrm{sign}\,}}(\sigma _2) \end{aligned}$$

with $\alpha =40$. Use the first-order sliding-mode transformation

$$\begin{aligned} w&=-x_1+x_3+w_2\, \end{aligned}$$

and obtain the final control law

$$\begin{aligned} u&=-x_1-x_3+w\, . \end{aligned}$$

1.3 A 4.3 Case: Unstable Internal Dynamics

Consider the system (9.51) with $a=1$

$$\begin{aligned} \dot{x}_1&=-x_1+x_3+\phi _1(x_1,t)\end{aligned}$$

(9.76a)

$$\begin{aligned} \dot{x}_2&=x_2+x_3+\phi _2(x_1,x_2,t)\end{aligned}$$

(9.76b)

$$\begin{aligned} \dot{x}_3&=u\end{aligned}$$

(9.76c)

$$\begin{aligned} y&=x_1\,. \end{aligned}$$

(9.76d)

This system has the internal dynamics

$$\begin{aligned} \dot{\eta }_1&=\eta _1+\xi _2-\xi _1\,. \end{aligned}$$

(9.77)

These internal dynamics are unstable. Thus, it is not sufficient anymore to find a control law w for the system (9.54a), (9.54b) to stabilize the system (9.54a)–(9.54c). This makes FOSM, FOSM-D, nHOSM, and nFOSM not straightforwardly applicable. Thus, we only consider ISMC and discuss its results.

1.3.1 ISMC

Introduce the integrator v with $\dot{v}=x_1$. Consider the new output

$$\begin{aligned} h_{v}(v,x)&=2v+x_1-x_2\, \end{aligned}$$

with full relative degree with respect to the input u. Then define the sliding manifold

$$\begin{aligned} \sigma (v,x)=h_{v}(v,x)+3\frac{dh_{v}(v,x)}{dt}+3\frac{d^2h_{v}(v,x)}{dt^2}+\frac{d^3h_{v}(v,x)}{dt^3}\,. \end{aligned}$$

Substitute $\frac{dh_{v}(v,x)}{dt}=x_1-x_2$ , $\frac{d^2h_{v}(v,x)}{dt^2}=-x_1-x_2$, and $\frac{d^3h_{v}(v,x)}{dt^3}=-x_1-x_2-2x_3$ and define $w=-\alpha {{\,\mathrm{sign}\,}}(\sigma )$ with $\alpha =40$. Finally, use the first-order sliding-mode input transformation

$$\begin{aligned} u=-\frac{8x_2+6x_3-\alpha {{\,\mathrm{sign}\,}}(\sigma )}{2}\,. \end{aligned}$$

(9.78)