Robust cascade control of electrical drives using discrete-time chattering-free sliding mode controllers with output saturation

Abstract

The paper considers electrical drives control having a hierarchical cascaded structure. This structure has an inner current control loop and outer loops for speed and position control. The design of the control is performed using a discrete-time model of electrical drive. In all the loops, the discrete-time quasi sliding mode control is used for controller design because of its robustness to external and parametric matched disturbances (inherent to electrical drives) and the capability to ensure the desired dynamics. To enhance the robustness to disturbances, a nonlinear disturbance compensator is also implemented. The chattering in sliding mode is eliminated by using a new modified discrete-time super twisting control. The current and the speed controllers are designed for linear discrete-time first-order models, while the position controller is designed for a linear second-order discrete-time model. The axis position is measured by a mechanical sensor (encoder). The speed is estimated from the position measurements using Euler derivative approximation. Alternatively, it can be obtained by an observer. The proposed design is straightforward and results in high-performance, robust control with strong disturbance rejection capability and negligible overshoots. All theoretically obtained claims are demonstrated by experiments on an induction motor drive with a rotor field-oriented control structure.

Appendices

Appendix A: Proof of Proposition 1

One of the DSM stability conditions, proposed in [41], is given with.

$$ \left| {s_{{{\updelta },k + 1}} } \right| < \left| {s_{{{\updelta },k}} } \right|. $$

(A.1)

This means that the system trajectory tends to the sliding surface. If this condition is satisfied in a saturation regime, system trajectories will approach the sliding surface and the control signal must come out of saturation in a finite time.

Let it holds $\left| {u_{\Sigma,k} } \right| = \left| {\left( {k_{r1} + k_{r2} } \right)\frac{{s_{{{\updelta },k }} }}{T} + {\mathbf{c}}_{{\updelta }} {\mathbf{A}}_{{\updelta }} {\mathbf{x}}_{k} } \right| > U_{0}$, $\left| {d_{k} } \right| \le d_{0} < U_{0 } .$ Let $k_{r3} \left( {0 \le k_{r3} < 1} \right)$ be introduced such that $k_{r1} + k_{r2} + k_{r3} = 1$. Then the sliding dynamics, defined by (5), in the saturated regime can be rewritten as

$$ s_{{{\updelta },k + 1}} = \left[ {\left( {k_{r1} + k_{r2} } \right)s_{{{\updelta },k}} + T{\mathbf{c}}_{{\updelta }} {\mathbf{A}}_{{\updelta }} {\mathbf{x}}_{k} } \right]\left( {1 - \frac{{U_{0} }}{{\left| {u_{\Sigma,k} } \right|}}} \right) + k_{r3} s_{\updelta,k} + Td_{k} . $$

(A.2)

Since $\left| {u_{\Sigma,k} } \right| > U_{0}$, then $0 < \left( {1 - \frac{{U_{0} }}{{\left| {u_{\Sigma,k} } \right|}}} \right) < 1$. Also, having in mind that $0 < k_{r3} < 1$ and $\left| {d_{k} } \right| < d_{0} < U_{0}$, the following inequality can be obtained

$$ \begin{aligned} & \left| {s_{{\updelta ,k + 1}} } \right| < \left| {\left( {k_{{r1}} + k_{{r2}} } \right)s_{{\updelta ,k}} + Tu_{{eq,k}} } \right|\left( {1 - \frac{{U_{0} }}{{\left| {u_{{\Sigma ,k}} } \right|}}} \right) + k_{{r3}} \left| {s_{{\updelta ,k}} } \right| + Td_{0} = T\left| {u_{{\Sigma ,k}} } \right|\left( {1 - \frac{{U_{0} }}{{\left| {u_{{\Sigma ,k}} } \right|}}} \right) + k_{{r3}} \left| {s_{{\updelta ,k}} } \right| + Td_{0} \\ & \quad = T\left| {u_{{\Sigma ,k}} } \right| - TU_{0} + k_{{r3}} \left| {s_{{\updelta ,k}} } \right| + Td_{0} \le \left( {k_{{r1}} + k_{{r2}} } \right)\left| {s_{{\updelta ,k}} } \right| + T\left| {u_{{eq,k}} } \right| - TU_{0} + k_{{r3}} \left| {s_{{\updelta ,k}} } \right| + Td_{0} \\ & \quad = \left| {s_{{\updelta ,k}} } \right| + T\left( {\left| {{\mathbf{c}}_{\updelta } {\mathbf{A}}_{\updelta } {\mathbf{x}}_{k} } \right| + d_{0} } \right) - TU_{0} \\ \end{aligned} $$

(A.3)

Hence, it holds

$$ \left| {s_{{{\updelta },k + 1}} } \right| < \left| {s_{{{\updelta },k}} } \right| + T\left( {\left| {u_{eq,k} } \right| + d_{0} } \right) - TU_{0} $$

(A.4)

If

$$ U_{0} > \left| {u_{eq,k} } \right| + d_{0} , $$

(A.5)

(A.1) is fulfilled.

This means that the both $s_{{{\updelta },k}}$ and $x_{k}$ will be decreasing, making $\left| {u_{\Sigma,k} } \right|$ decreasing as well. Therefore, it is inevitable that the control signal becomes smaller than the saturation limit and the system exits saturation. (A.5) must be valid from the initial moment, i.e.

$$ U_{0} > \left| {{\mathbf{c}}_{{\updelta }} {\mathbf{A}}_{{\updelta }} {\mathbf{x}}_{k} } \right| + d_{0} . $$

(A.6)

■

Appendix B: Proof of Proposition 2

After leaving the saturation mode, the control becomes

$$ u_{k} = - {\mathbf{c}}_{{\updelta }} {\mathbf{A}}_{{\updelta }} {\mathbf{x}}_{k} - T^{ - 1} \left( {k_{r1} + k_{r2} } \right)s_{{{\updelta },k}} $$

(B.1)

and

$$ {\updelta }s_{{{\updelta },k}} = - T^{ - 1} \left( {k_{r1} + k_{r2} } \right)s_{{{\updelta },k}} + d_{k} . $$

(B.2)

From (5) and (B.1), it follows

$$ s_{{{\updelta },k + 1}} = \left( {1 - k_{r1} - k_{r2} } \right)s_{{{\updelta },k}} + Td_{k} . $$

(B.3)

After the first sampling period of the unsaturated control, controller gain is reduced by applying $k_{r2}$ = 0 in (B.1) and the compensation control is activated. Now, the control becomes

$$ u_{k} = - {\mathbf{c}}_{{\updelta }} {\mathbf{A}}_{{\updelta }} {\mathbf{x}}_{k} - T^{ - 1} k_{r1} s_{{{\updelta },k}} - u_{c,k} , $$

(B.4a)

$$ u_{c,k} = u_{c,k - 1} + k_{{{{int}}}}^{{\upbeta }} Ts_{{{\updelta },k - 1}} . $$

(B.4b)

Sliding variable dynamics is now defined as

$$ s_{{{\updelta },k + 1}} = \left( {1 - k_{r1} } \right)s_{{{\updelta },k}} + T\left( {d_{k} - u_{c,k} } \right), $$

(B.5a)

$$ u_{c,k + 1} = u_{c,k} + k_{int}^{{\upbeta }} Ts_{\updelta ,k} . $$

(B.5b)

By replacing $z_{k} = d_{k} - u_{c,k}$, previous equations become

$$ s_{{{\updelta },k + 1}} = \left( {1 - k_{r1} } \right)s_{{{\updelta },k}} + Tz_{k} , $$

(B.6a)

$$ z_{k + 1} = - k_{int}^{{\upbeta }} Ts_{{{\updelta },k}} + z_{k} + {\Delta }_{k} . $$

(B.6b)

where ${\Delta }_{k} = d_{k + 1} - d_{k}$.

The characteristic equation of the system (B.6) is

$$ \begin{gathered} \left| {\begin{array}{*{20}c} {z - 1 + k_{r1} } & { - T} \\ {k_{int}^{{\upbeta }} T} & {z - 1} \\ \end{array} } \right| = 0 \Rightarrow F\left( z \right) = a_{2} z^{2} + a_{1} z + a_{0} = 0, \hfill \\ a_{2} = 1 > 0, a_{1} = k_{r1} - 2, a_{0} = 1 - k_{r1} + k_{int}^{{\upbeta }} T^{2} . \hfill \\ \end{gathered} $$

(B.7)

By employing the Jury’s stability test, stability conditions are obtained as

$$ k_{int}^{{\upbeta }} T^{2} > 0, $$

(B.8a)

$$ 4 - 2k_{r1} + k_{int}^{{\upbeta }} T^{2} > 0, $$

(B.8b)

$$ k_{int}^{{\upbeta }} T^{2} < k_{r1} . $$

(B.8c)

Since $0 < k_{r1} < 1$, these conditions will be fulfilled if $0 < k_{int}^{{\upbeta }} T^{2} < k_{r1}$. ■

Remark B1

The modified gain $k_{int}^{{\alpha}}$ from (24) must also satisfy the obtained condition (B.8c).

Remark B2

The condition (B.8c) has been established in [31] in another way for the system whose nonlinearity in (18i) is replaced by the unity gain. This condition is also valid for the system (18) with nonlinearity in (18i) for the stability of the nominal system in a small vicinity of the equilibrium. If the parameter β is set to 0 in (18i) (the nonlinearity becomes signum function), stable oscillations arise even in the nominal system. The exact parameters of the oscillations are very difficult to determine analytically. A practical approach to finding those oscillation parameters is by using the describing function method [37]. For β > 0 oscillations do not occur in the nominal system, but in the system with uncertainty and unmodelled dynamics, oscillations arise. The maximal possible amplitude of the oscillations can be then identified as in the case of β = 0.

Appendix C: Robustness considerations

Let the system be subjected to external disturbance $f_{k}$ and parameter variations, introduced into the initial model as

$$ \updelta{\mathbf{x}}_{k} = \left( {{\mathbf{A}}_{{\updelta}} + {\Delta }{\mathbf{A}}_{\updelta } } \right){\mathbf{x}}_{k} + {\mathbf{b}}_{{\updelta}} \left( {u_{k} + f_{k} } \right), $$

(C.1)

where ${\Delta }{\mathbf{A}}_{\updelta } \in R^{n \times n}$ is the matrix uncertainties that satisfies the matching conditions. Then it holds ${\Delta }{\mathbf{A}}_{\updelta } = {\mathbf{b}}_{{\updelta}} {\mathbf{v}}$, where ${\mathbf{v}} = \left[ {\begin{array}{*{20}c} {v_{1} } & {v_{2} } & \cdots & {v_{n} } \\ \end{array} } \right]$ is an unknown vector with the property $\left| {v_{i} } \right| \le v_{m}$ for $i = 1,2, \cdots ,n$. Using this, the system model becomes (4), where $d_{k} = f_{k} + {\mathbf{vx}}_{k}$ is the overall matched disturbance. It is evident that $\left| {d_{k} } \right| \le \left| {f_{k} } \right| + \left| {{\mathbf{vx}}_{k} } \right| \le f_{0} + v_{m} \left\| {{\mathbf{x}}_{{k }} } \right\|_{1}$, where $\left\| {\mathbf{x}} \right\|_{1}$ denotes 1-norm, i.e. $\left\| {\mathbf{x}} \right\|_{1} = \mathop \sum \nolimits_{i = 1}^{n} \left| {x_{i} } \right|$.

Taking into account the augmented perturbed model (C.1), it is easy to establish that the condition (20) in the Proposition 1 now becomes

$$ ~U_{0} > |{\mathbf{c}}_{\updelta } {\mathbf{A}}_{\updelta } {\mathbf{x}}_{k} | + f_{0} + f_{m} \left\| {{\mathbf{x}}_{k} } \right\|_{1} ,\user2{~}\;\forall k \ge 0. $$

(C.2)

Since the structure of the model (4), used for the stability consideration in the Proposition 2, is identical to the perturbed model (C.1), the convergence conditions (21) remains unchanged under the assumption of bounded disturbance ${\Delta }_{k}$ in (B.6b). However, ${\Delta }_{k}$ in the case of (C.1) becomes ${\Delta }_{k} = d_{k + 1} - d_{k} = f_{k + 1} - f_{k} + {\mathbf{v}}\left( {{\mathbf{x}}_{k + 1} - {\mathbf{x}}_{k} } \right)$. Hence, $\left| {\Delta _{k} } \right| \le \left| {f_{{k + 1}} - f_{k} } \right| + v_{m} \left( {\left\| {{\mathbf{x}}_{{k + 1 }} } \right\|_{1} - \left\| {{\mathbf{x}}_{{k }} } \right\|_{1}} \right)$. For small sampling periods $T$, these differences may be regarded bounded that makes ${\Delta }_{k}$ bounded as well. The presence of external disturbances and parameter perturbations prevents the occurrence of ideal DTSM, so the system motion takes place in some bounded vicinity around the sliding manifold. The width of such a quasi-sliding domain depends on the magnitude of ${\Delta }_{k}$.

This robustness analysis has been conducted under the assumption of the matched parameter perturbations. It is important to point out that the speed contour in the considered cascade controlled ED system is the most disturbance contaminated control loop. Due to the much faster inner current control loop, the speed control loop may be regarded as first-order plant. Fortunately, in the first-order plants the disturbance matching conditions are always fulfilled. This indicates that the requirement for disturbances to be matched is not restrictive.