Dynamic high-gain observer approach with sliding mode control for an arc-shaped shape memory alloy compliant actuator

Abstract

In this research, we present a dynamic high gain observer (HGO) combined with a sliding mode controller (SMC) for a novel shape memory alloy (SMA) actuator. The dynamic model of the SMA actuator contains nonlinearities with hysteresis, which makes the estimation of velocity and acceleration of the actuator challenging. We first design a dynamic HGO to estimate velocity and acceleration of the SMA actuator by considering nonlinearities with hysteresis in the dynamics. We then develop a SMC for the actuator using the estimated velocity and acceleration. The proposed SMC in conjunction with the dynamic HGO is tested experimentally for the actuator. For comparison, trajectory tracking experiments are also conducted by using PID without an observer, SMC without an observer, and SMC with a non-dynamic HGO. Our results confirm that the dynamic HGO combined with SMC can enhance trajectory tracking accuracy while reducing the chattering phenomenon.

Appendices

Appendix A

Convergence of the dynamic high gain observer-based sliding mode controller

We choose the following Lyapunov function v and differentiate it to design the SMC,

$$V=\frac{1}{2}{s}^{T}s$$

(13)

$$\dot{V}=s\dot{s}$$

(14)

Where s is the sliding surface (9). Using (2) and (10)

$$\dot{s}=\dddot{\widehat{e}}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}$$

(15)

$$= {\dddot{x}}_{1}-{\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}$$

(16)

$$=f+gu-{\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}$$

(17)

If we choose the control law (10),

$$u =-\frac{1}{{\widehat{g}}_{x}}\left(\widehat{f}- {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}+ {K}_{{s}_{1}}{\text{sign}}\left(s\right)+{K}_{{s}_{2}}s\right)$$

(18)

then, we can write eq. (15) by using boundary conditions (11) and (12) as,

$$\dot{s} =f+gu-{\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}=\left(f-\left(1+\Delta \right)\left(\widehat{f}-{\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}+ {K}_{{s}_{1}}{\text{sign}}\left(s\right)+{K}_{{s}_{2}}s\right)\right)- {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}$$

(19)

Applying the upper bound of ∆ with D, we have,

$$\dot{s}\le \left(f-\left(1+D\right)\left(\widehat{f}-{\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}+ {K}_{{s}_{1}}{\text{sign}}\left(s\right)+{K}_{{s}_{2}}s\right)\right)- {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}}$$

(20)

Simplifying above equation, we get,

$$\dot{s}\le \widetilde{f}-D\widehat{f}-D\left( {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}} \right)-\left(1+D\right)\left({K}_{{s}_{1}}{\text{sign}}\left(s\right)+{K}_{{s}_{2}}s\right)$$

(21)

where $f- \widehat{f}=\widetilde{f} \left(x\right)$ represents model uncertainties and is bounded by $\left|\widetilde{f} \right|\le F$. Considering eq. (15), it can be shown that

$$\dot{V}\le s\left(F-D\widehat{f}-D\left( {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}} \right)-\left(1+D\right)\left({K}_{{s}_{1}}{\text{sign}}\left(s\right)+{K}_{{s}_{2}}s\right)\right)\le s\left(F-D\widehat{f}-D\left( {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}})\right) -\left(1+D\right)\left({K}_{{s}_{1}}s+{K}_{{s}_{2}}{s}^{2}\right)\right)$$

(22)

Thus, if we choose ${K}_{{s}_{1}}$ and ${K}_{{s}_{2}}$ large to satisfy following condition,

$$s\left(F-D\widehat{f}-D\left( {\dddot{x}}_{d}+{c}_{{s}_{1}}\ddot{\widehat{e}}+{c}_{{s}_{2}}\dot{\widehat{e}} )\right)<\left(1+D\right)\left({K}_{{s}_{1}}s+{K}_{{s}_{2}}{s}^{2}\right)\right)$$

(23)

Then, we have,

$$\dot{V}<0.$$

(24)

which shows that SMA system with proposed controller would always dissipate energy and the system will become stable by the Lyapunov stability theorem. Once, $\dot{V}<0$ is ensured, $s$ converges to zero. Thus, on the sliding surface where $s=0$, we have,

$$s=0=\ddot{\widehat{e}}+{c}_{{s}_{1}}\dot{\widehat{e}}+{c}_{{s}_{2}}\widehat{e}$$

(25)

which infers that

$$\ddot{\widehat{e}}=-{c}_{{s}_{1}}\dot{\widehat{e}}-{c}_{{s}_{2}}\widehat{e}$$

(26)

We can express (24) as follows,

$$\left[\begin{array}{c}\dot{\widehat{e}}\\ \ddot{\widehat{e}}\end{array}\right]=\left[\begin{array}{c}{\dot{\widehat{e}}}_{1}\\ {\dot{\widehat{e}}}_{2}\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& -{c}_{{s}_{2}}& -{c}_{{s}_{1}}\end{array}\right]\left[\begin{array}{c}{\widehat{e}}_{1}\\ {\widehat{e}}_{2}\end{array}\right]$$

(27)

Now we can choose ${c}_{{s}_{1}}$ and ${{c}_{s}}_{2}$ to place poles to achieve our desired performance. Gains, ${K}_{{s}_{1}}$ and ${K}_{{s}_{2}}$, are used to attract the system states to the sliding surface (ensured by Lyapunov convergence). Parameters, ${c}_{{s}_{1}}$ and ${c}_{{s}_{2}}$, ensure convergence of the state errors to zero along the sliding surface.