Energy-saving sub-optimal sliding mode control with bounded actuation

The problem under consideration requires a bounded actuation $|u|\leq U$ with only three discrete values, i.e. $u\in\{-U,\,0,\,U\}$, where $U>0$ is a known parameter of the control system. For such systems, we relax the above uncertainty condition (ii) and assume a constant input coupling, meaning $\Gamma_{m}=\Gamma_{M}=g=1$. This implies the maximal control magnitude $U$ and leads to $\alpha=1$ to be substituted instead of (6). Under the above assumptions, the inequalities (8), (9) and (10) result in a set of the parametric conditions

Now, we are in the position to specify an energy-saving sub-optimal SMC with the bounded actuation.

Given the fact of a bounded control authority $U$, we are interested to modify the control (3) with $\alpha=1$ so as to allow for energy-saving control phases with $u=0$. Within each reaching cycle, i.e. between two successive extreme values $\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$ with $i\in\mathbb{N}<\infty$, the new control policy must admit the following. The control is on, i.e. $|u|=U$, on the intervals $\sigma\in[\beta_{1}\sigma_{M_{i}};\sigma_{M_{i}}]$ and $\sigma\in[\sigma_{M_{i+1}};\beta_{2}\sigma_{M_{i}}]$. And the control is off, i.e. $u=0$, on the interval $\sigma\in[\beta_{2}\sigma_{M_{i}};\beta_{1}\sigma_{M_{i}}]$, while $\beta_{2}\neq\beta_{1}$ is required for an energy-saving operation. The case where $\beta_{2}=\beta_{1}=\beta$ should correspond to the sub-optimal SMC (3) with $\alpha=1$, thus satisfying (11) and either (12) or (13). This benchmarking control will be denoted as conventional sub-optimal SMC. For the energy-saving sub-optimal SMC fulfills its task, following must be guaranteed:

• (a)

  convergence in finite time $T_{c}<\infty$, which implies $|\sigma_{M_{i+1}}|<|\sigma_{M_{i}}|$ for all $1\leq i<\infty$;

• (b)

  fuel- and thus energy-saving, which implies

  $$\int\limits_{0}^{T_{c}}\bigl{|}u(\beta_{1},\beta_{2})\bigr{|}dt<\int\limits_{0}^{\hat{T}_{c}}\bigl{|}\hat{u}(\beta_{1})\bigr{|}dt,$$

  where the variables with hat (like $\hat{\circ}$) are used for the conventional sub-optimal SMC.

The proposed modification of the second-order sub-optimal SMC algorithm is

In addition to (11), the thresholds relationship $\beta_{1}>\beta_{2}$ is required, while the convergence and energy-saving conditions on $(\beta_{1},\beta_{2})$ parameters are derived and analyzed in the following. An initializing control action, cf. [7],

is also required for speeding up the reaching of the first extreme value at $t_{M_{1}}$.

Using the general formula

for a double-integrator with $v=\pm U\pm\Phi$, the $(\sigma,\dot{\sigma})$-trajectories can be calculated for an arbitrary initial state $(\sigma_{0},\dot{\sigma}_{0})$. Then, without loss of generality, consider the first extreme value $\sigma_{M_{i}}>0$, while the preceding reaching phase is always ensured by the initialization control (15). After applying first the control value $u=-U$, according to (14), the state trajectory proceeds in the IV-th quadrant, cf. Figure 2. While both limiting cases of the perturbation $f=\pm\Phi$ are effective, we focus here on the worst-case scenario, that is essential for convergence, in which the sign of $f$ is negative. Note that this renders the system to be underdamped. Using $-U-\Phi$ in the denominator of (16) and substituting the threshold value $\beta_{1}\sigma_{M_{i}}$ on the left-hand side of (16) one obtains the first peaking point (labeled by $P^{\prime\prime}$) of the $\dot{\sigma}$-state as

At this point, the control is switching from $-U$ to $0$. Then, evaluating the parabolic trajectory (16) between the $\beta_{1}\sigma_{M_{i}}$ and $\beta_{2}\sigma_{M_{i}}$ points, with $u=0$ and worst-case $f=-\Phi$, one obtains

Substituting (17) into (18) and solving (18) with respect to $\dot{\sigma}$ results in

where the second peaking point $P_{2}$ is the one where the control is switching back from $0$ to $U$. Continuing this line of calculations, one considers also the parabolic trajectory (16) after passing the $\beta_{2}\sigma_{M_{i}}$-threshold as

Here again, $f=-\Phi$ is assumed as worst-case from a convergence point of view. Substituting the left-hand-side of (20) by the next extreme value $\sigma_{M_{i+1}}$, at which $\dot{\sigma}=0$, results in

Note that for convergence, it is strictly required that

for all pairs of the consecutive extreme values, indexed by $i$ and $i+1$. Since $\sigma_{M_{i}}>\sigma_{M_{i+1}}$ is guaranteed for all trajectories in the IV-th and III-rd quadrants, cf. Figure 2, the above convergence inequality can be written, with respect to (21), as

Substituting (19) into (22) yields

which results in the strict parameters condition

Furthermore, one needs to ensure

in order to realize the both control switchings between two successive extreme values. Note that (25) is inherited from the conventional sub-optimal SMC, cf. (7). Here we emphasize that the parametric conditions (24), (25), (26) must be satisfied for the energy-saving sub-optimal SMC to yield the global convergence, for $U>\Phi$ and irrespective the perturbations $|f(t)|<\Phi$.

The imposed $\beta_{1},\beta_{2}$ constraints (24), (25), (26) have an illustrative graphical interpretation as shown in Figure 1.

The $\beta_{1},\beta_{2}$ values admissible for convergence are inside of a grey-shadowed triangle, while on the $\beta_{1}=\beta_{2}$ edge the energy-saving control (14), (15) reduces to the conventional sub-optimal SMC. Obviously, the $\Phi/U$ ratio determines the size of an admissible $\{\beta_{1},\beta_{2}\}$ set, including whether the negative $\beta_{2}$-values are allowed. It is also worth noting that the opposite $\beta_{2}=2\Phi/U-\beta_{1}$ edge maximizes the $\beta_{1}-\beta_{2}$ distance. However, this can largely increase the convergence time (including $T_{c}\rightarrow\infty$), similar as in case of a minimal-fuel control with free response time of an unperturbed double-integrator analyzed in detail in [8].

Consider first the reaching time between two consecutive extreme values $\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$. Recall that a conservative approach is taken so that a worst-case scenario is calculated for all control phases, in terms of a slowest time. The associated trajectories of one reaching phase are schematically drawn in Figure 2. Similar as before, and without loss of generality, consider the first extreme value $\sigma_{M_{i}}>0$ so that the trajectories proceed in the III-rd and IV-th quadrants. We recall that the threshold value $\beta_{2}$ can be either positive or negative, cf. Figure 1. Evaluating first the peaking point $P^{\prime}$ one obtains the corresponding $\dot{\sigma}$-value as

which is reached by the slowest trajectory driven by $-U+\Phi$. For the subsequent phase with control-off, i.e. $u=0$, one needs distinguishing between two boundary trajectories, one driven by $\Phi$ and another driven by $-\Phi$. Also we notice that both control-off phases are considered starting from the same $P^{\prime\prime}$ peaking point. Indeed, one can recognize that starting from $\beta_{1}\sigma_{M_{i}}$, the $\Phi$-driven trajectory reaches the $\sigma$-abscissa later from $P^{\prime\prime}$ than from $P^{\prime}$ peaking point, see Figure 2. For the control-off phase between the $\beta_{1}\sigma_{M_{i}}$ and $\beta_{2}\sigma_{M_{i}}$ thresholds, starting from $P^{\prime\prime}$ and ending in $P_{1}$ peaking point, one obtains

Note that the second term under the square root in (28) must be positive, so here $\beta_{2}$ has to be larger than the point where the $\Phi$-driven trajectory reaches the $\sigma$-abscissa (see the dotted arc from $P_{1}$ in Figure 2).

With the determined $\dot{\sigma}$ values at all relevant peaking points, i.e. $P^{\prime}$, $P^{\prime\prime}$, $P_{1}$ and $P_{2}$, one can analyze the slowest time for each phase of a reaching cycle individually.

The slowest time of the first control-on phase is

In a similar way, one obtains the slowest time of the subsequent control-off phase driven by $\Phi$ as

One can recognize that if $\beta_{2}=\beta_{1}$ (yielding the conventional sub-optimal SMC), then both square root terms in (30) become equal, and $T_{1}^{*}$ becomes zero. One can also recognize that the upper bound of $T_{1}^{*}$ is when the second square root term in (30) becomes zero. This results in the maximal control-off time of this phase equal to

When evaluating the control-off phase driven by $-\Phi$, cf. Figure 2, we write

Substituting (17) and (19) into (32) and solving it with respect to $T^{*}_{2}$ results in

Here, it can also be recognized that $T^{*}_{2}$ has its maximal value when the second square root term in (33) becomes zero. Thus, the maximal control-off time for this phase is

Comparing both maximal time values, one can show that

holds always due to $\beta_{2}<\beta_{1}$. Despite this fact, which determines the slowest control-off phase during one reaching cycle, we recall that the overall convergence time depends also on the number of cycles. Therefore, both (31) and (34) are taken into account in further calculations.

Evaluating both successive control-on phases, starting once from $P_{1}$ and once from $P_{2}$ peaking point, we obtain

and

respectively. Also here one can show, cf. (35), (36), that

holds always due to $\beta_{2}<\beta_{1}$.

For analyzing the total worst-case (i.e. largest) reaching time between two consecutive extreme values $\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$, one needs to distinguish between two pathes: $P^{\prime\prime}\rightarrowtail P_{1}\rightarrowtail\sigma^{*}_{M_{i+1}}$ and $P^{\prime\prime}\rightarrowtail P_{2}\rightarrowtail\sigma^{**}_{M_{i+1}}$, cf. Figure 2. In the first case, the overall maximal reaching time is

In the second case, respectively, the overall maximal reaching time is

Note that in both above equations $\bar{\Omega}$ denote the terms where the control is off, i.e. $u=0$, while $\tilde{\Omega}$ denote the terms where the control is on, i.e. $|u|=U$.

Once the convergence of the energy-saving sub-optimal SMC is guaranteed, being analyzed in Section 3.2, the overall accumulated worst-case reaching time results in

Because the convergence implies

the infinite sum in (39) turns out into a geometric series

which is convergent. This allows concluding that the upper bound of the finite time convergence is

Note that the initial reaching time of the first extreme value is $t_{M_{1}}\leq|\dot{\sigma}(0)|/(U-\Phi)$, cf. (15). Furthermore, we note that depending on the branching of trajectories, as shown above for the control-off phase, also a maximum value $\eta=\max\{\eta_{1},\eta_{2}\}$ must be taken into account in (42).

For determining $\eta_{1}$ and $\eta_{2}$, we evaluate both subsequent extreme values and obtain, based on (16), the equations

and

Comparing (43) and (44) with (40), results in

and

Following the same line of calculations as above, one can evaluate the reaching and convergence time of the conventional sub-optimal SMC, for which $\beta\equiv\beta_{1}$. Considering the $P^{\prime}$ and $P^{\prime\prime}$ peaking points results in

Then, the upper bound of the finite time convergence of the conventional sub-optimal SMC results in

where $0<\hat{\eta}<1$, cf. with (40). For evaluating $\hat{\eta}$, consider the peaking point $P^{\prime\prime}$ and obtain

Note that both (45) and (46) reduce to (49) if assuming $\beta_{2}=\beta_{1}$. Also worth noting is that (49) coincides with the convergence condition (24) if $\beta_{2}=\beta_{1}$ and requires

We will denote $\Omega_{1},\,\Omega_{2},\,\hat{\Omega}$ as amplification factors of the convergence time, and $\eta_{1},\,\eta_{2},\,\hat{\eta}$ as contraction factors of the convergence. Inspecting (42) and (48), one can recognize that the former are proportionally increasing the convergence time while the latter can drastically slow down the overall convergence once $\eta,\hat{\eta}$ become closer to one.

For the energy-saving sub-optimal SMC becomes effective, one needs to ensure that its total energy consumption is less than that of the conventional sub-optimal SMC, cf. condition (b) at the end of Section 2, while assuming the $U$, $\Phi$, and $\beta_{1}$ values are the same. In terms of the both control-on phases, i.e. $u\neq 0$, the amplification factors to be taken into account are

and

cf. (37), (38). Taking out of consideration the initial reaching phase in (42) and (48), we obtain two energy cost functions

and

for the energy-saving and conventional sub-optimal SMC, respectively. In order to find an energy-saving pair of the $(\beta_{1},\beta_{2})$ threshold parameters for with some fixed $U,\Phi$ process values satisfying $U>\Phi$, we formulate the

minimization problem under the hard constraint

In addition, in order to avoid $\beta_{1}\rightarrow 1$, which can lead to $\hat{T}_{c}\rightarrow\infty$, see [8] for details, we specify the upper bound

This implies restriction on the convergence time $\hat{T}_{c}$ and, this way, forces a reasonable $\beta_{1}<1$ in course of solving the minimization problem (55).

The convergence time cost function $\hat{J}$ is exemplary visualized in Figure 3 in dependency of $\beta_{1}$, for two cases assuming $U=1$ and $\Phi=\{0.3,\,0.6\}$. Note that $\hat{J}\rightarrow\infty$ for both boundary values $\beta_{1}\rightarrow\Phi/U$, cf. (50), and $\beta_{1}\rightarrow 1$.

Note that the hard constraint (56) is required to ensure that the total control-on time of the energy-saving sub-optimal SMC is shorter than that of the conventional one. Moreover, one can recognize that (55) ensures the maximal possible energy-saving, due to the control-off phases throughout the convergence, comparing to the conventional sub-optimal SMC, both having the same $\beta_{1}$ value. That means for any fixed $\beta_{1}$ complying with (57), there is an optimal $\beta_{2}$ counterpart. The latter represents the solution of the constrained minimization problem (55). The results of the constrained optimization are exemplary visualized in Figure 4, for the perturbation to control ratio equal to $\Phi/U=0.3$.

It should be noted that the formulated optimization problem does not take explicitly into account those transient phases where the control is off, cf. (37), (38) and (51), (52). However, this does not imply any issues since the finite time convergence of the energy-saving sub-optimal SMC is upper bounded by (42).

The convergence behavior of the energy-saving sub-optimal SMC is exemplary shown for examining the parametric condition (24). The constraining parameters are chosen so that $\Phi/U=0.5$ with $U=1$ and $\mu=0$, while for a worst-case scenario of the bounded perturbation $f=0.5\,\mathrm{sign}(\dot{\sigma})$ is assumed. Note that the assumed $f$ constitutes an always ’co-acting’ perturbation signal of the maximal amplitude $\Phi$, a worst-case scenario that challenges the contraction during each reaching cycle and, at large, the total convergence to $\sigma=\dot{\sigma}=0$ equilibrium. Figure 7 discloses the converging behavior for $\beta_{1}=0.8$, $\beta_{2}=0.25$ versus the diverging behavior for $\beta_{1}=0.8$, $\beta_{2}=0.19$. Note that here, the threshold values distance $\beta_{1}-\beta_{2}>0.6$ is required for (24) is fulfilled.

The energy-saving sub-optimal control behavior is analyzed in comparison to the conventional sub-optimal one, with regard to the recorded convergence time $T_{c}$ and overall energy consumption

In order to determine the numerical convergence instant, the $T_{c}$ value is obtained once the Euclidian norm of the state vector $\bigl{(}\sigma(t),\dot{\sigma}(t)\bigr{)}$ decreases below some assigned low residual constant, i.e. $\bigl{\|}(\sigma,\dot{\sigma})\bigr{\|}(t)<0.004$. The residual constant is determined out from the numerical simulations, with respect to the solver type and step-size and the resulted numerical pattern around $\sigma=0$ after the principal dynamics of $\sigma(t)$ has converged. For the assigned $\Phi/U=0.3$ ratio with $U=1$ and $\mu=0$, three configurations of the perturbing signal are considered: $f=0$, $f=0.3\,\mathrm{sign}(\dot{\sigma})$, $f=-0.3\,\mathrm{sign}(\dot{\sigma})$. Note that the second one is ’co-acting’ to the control signal. This has, however, an adverse impact on contraction of the state trajectories, cf. Figure 2. The third one acts, on the contrary, as a maximal possible system damping and can negatively affect the control-off phases in terms of the reaching time, cf. section 3.3. Two parameter sets, obtained from the constrained optimization, $[\beta_{1}=0.7,\beta_{2}=0.55]$ and $[\beta_{1}=0.83,\beta_{2}=0.32]$ cf. with Figure 4, are used. All simulation results are summarized in Table 1. Recall that for the conventional sub-optimal SMC the $\beta_{2}=\beta_{1}$ is always assigned.

One can recognize a superior performance of the energy-saving sub-optimal SMC in all above given cases, except the last row in Table 1 (see light red shadowed). Here the energy saving is largely achieved, but the convergence time is significantly increased comparing to the conventional sub-optimal SMC. This is related to a relatively slowly ’drifting’ trajectory between the lying apart $\beta_{1}=0.83$ and $\beta_{2}=0.32$, being driven by the $f=-0.3\,\mathrm{sign}(\dot{\sigma})$ perturbation. The situation with $T_{c}$ improves, still guaranteing for energy saving, when both threshold parameters are lying closer to each other like in the shown case $\beta_{1}=0.7$ and $\beta_{2}=0.55$, see Table 1.

For better interpreting the listed results, some of the above configurations are shown below in the plots of $\sigma(t)$ and $u(t)$. All configurations with $f=0$ are depicted in Figures 8 and 9.

One can see that the energy-saving sub-optimal SMC converges faster, correspondingly stops to switch earlier than its conventional counterpart. One can also notice that both controllers here disclose a monotonic convergence without twisting, cf. section 2.

The convergence in case of $f=0.3\,\mathrm{sign}(\dot{\sigma})$ is also exemplary shown in Figure 10. Here, the conventional sub-optimal SMC has a monotonic convergence, in accord with (13). On the contrary, the energy-saving sub-optimal SMC changes into twisting mode here, because it allows for control phases driven only by the perturbation quantity.

For evaluating the residual oscillations (i.e. chattering) owing to a parasitic actuator dynamics, cf. section 4, different parameter sets with $\mu\neq 0$ are compared. Three parameter configurations used are shown in Table 2. The amplitude and frequency of the residual steady-state oscillations are computed based on the harmonic balance analysis provided in section 4, i.e. using (65), (66), and compared with those out from the numerical simulations. Certain, yet acceptable, error of the harmonic balance analysis is not surprising, since DF solely approximates the first main harmonic, cf. with results in [14].

Figure 11 exemplifies the appearance of residual steady-state oscillations when the actuator dynamics is in place. The time response of the sliding variable $\sigma(t)$ is shown above for all three parameter sets, cf. Table 2. The control value $u(t)$, i.e. of the control (14), is exemplary shown below for the parameter set $\mu=0.03$, $\beta_{1}=0.8$, $\beta_{2}=0.2$.

Table 2 and Figure 11 confirm the chattering analysis of the energy-saving sub-optimal SMC, developed in section 4, and disclose how the amplitude and frequency change depending on the control and actuator parameters $\beta_{1}$, $\beta_{2}$, and $\mu$.