**1. Introduction**

The elevator statement about sliding mode control (SMC) is that it is one of the robust control design techniques which is mathematically well-structured and assures performance in the presence of certain class of disturbance and uncertainties. Due to this it is used for controlling practical uncertain systems. It is originated from the concept of variable structure control (VSC). The name VSC itself describes that there is more than one structure defining a system which describes the complete behavior of the variable structure systems. In VSC, the control input is logically so chosen that the final closed-loop system behavior becomes stable regardless of the natures of the substructures (stable or unstable). This gives rise to a new system behavior not a part of any of the substructures. This phenomenon of getting a new system behavior is called sliding mode in the domain of variable structure control [1–4].

The design procedure of SMC consists of two steps. The first step is to design a sliding surface appropriately which decides the behavior of the system during sliding. Then a control action is designed so that all the state trajectories are steered to the sliding surface in finite time and then forced to stay on the surface. Once the sliding is established, i.e., the trajectories are on the sliding surface, the system becomes invariant to modelling inaccuracies and exogenous disturbances. The term "invariant" is stronger than robustness as it satisfies certain conditions additionally. The whole design procedure can be observed in three modes or phases, i.e., reaching mode, sliding mode, and steady-state mode. Reaching mode is the phase where the state trajectories are driven to the sliding surface. It is also known as hitting mode or

certain clock cycles. A lot of works have been done in the field of chattering elimination and reduction. Schemes like continuous approximation around the sliding surface (quasi-sliding mode) [1–3], higher-order sliding mode [5–8],

whose theory is rich and well-structured.

*DOI: http://dx.doi.org/10.5772/intechopen.91245*

*Discrete Time Sliding Mode Control*

**2. Discrete-time sliding mode control**

**3. Control problem formulation**

input single-output system.

*3.1.1 Design procedure*

reaching law [9].

**103**

**3.1 Controller design by Gao's reaching law**

step is to choose a sliding variable as

Consider an uncertain discrete-time system:

discrete-time sliding modes are a few way outs for the process of chattering. Here in this chapter, the concept of discrete-time sliding mode (DTSM) design is discussed. Readers can explore more in the field of continuous time higher-order sliding mode

Control system designs are streaming from continuous to discrete design with the invention of digital circuitry. High-performance computing devices, portable microprocessors, and plug and play features make the sophisticated design easy to implement. Discrete-time sliding mode control is the obvious transformation from the continuous time sliding mode control for the real-time application. Like continuous time sliding mode control, DTSMC is also easy to design and also well-suited for implementation.

where the states *x k*ð Þ<sup>∈</sup> *<sup>n</sup>*, control input *<sup>u</sup>*<sup>∈</sup> *<sup>m</sup>*, *<sup>f</sup>* <sup>∈</sup> *<sup>m</sup>*, and the output *<sup>y</sup>*<sup>∈</sup> . *f k*ð Þ is the disturbance coming from an exogenous system and is upper bounded by *fm*. *A*, *B* are system matrix and input matrix, respectively, and are having appropriate dimensions. Here the problem is either to stabilize the system, i.e., lim *<sup>k</sup>*!∞*x k*ð Þ¼ 0, or to track a time-varying trajectory, i.e., lim *<sup>k</sup>*!∞*x k*ð Þ¼ *xd*ð Þ*k* , where *xd*ð Þ*k* is the desired trajectory. But tracking can be treated as error stabilization mathematically, i.e., by making lim *<sup>k</sup>*!∞*e k*ð Þ¼ 0 where *e k*ð Þ¼ *x k*ð Þ� *xd*ð Þ*k* . The system (1) will be transferred to error space *e k*ð Þ¼ þ 1 *Ae k*ð Þþ *Bu k*ð Þþ *f k*ð Þþ *Axd*ð Þ� *k xd*ð Þ *k* þ 1 . So here in this chapter, only stabilization is addressed for single-

Here the aim is to design a control law *u k*ð Þ such that lim *<sup>k</sup>*!<sup>∞</sup>*x k*ð Þ¼ 0. The first

*s k*ð Þ¼ *c*

of *α* and *β* is given in the next section. Using Eqs. (1)–(3), one can write

where *c* is a sliding variable design parameter. Next step is to choose Gao's

where *α* ∈ð Þ 0, 1 and *β* >0 and *d k*ð Þ are assumed to be the same as the uncertain quantity *<sup>c</sup>TBf k*ð Þ and are bounded by *dm* <sup>¼</sup> <sup>∣</sup>*cTBf <sup>m</sup>*∣. A detailed selection procedure

*x k*ð Þ¼ þ 1 *Ax k*ð Þþ *Bu k* ½ � ð Þþ *f k*ð Þ (1)

*Tx k*ð Þ (2)

*s k*ð Þ¼ þ 1 *αs k*ð Þ� *βsign s k* ð ð Þþ *d k*ð Þ (3)

#### **Figure 1.**

*Trajectories of ideal variable structure systems.*

**Figure 2.** *Trajectories of practical variable structure systems.*

non-sliding mode. In sliding mode, the trajectories are restrained and kept moving along the surface towards the equilibrium point or reference point. Finally, in steady-state mode, the system reaches its final state, which would be zero-error state, constant offset state, or limit-cycle state. Different modes of VSC are shown in **Figures 1** and **2**.

SMC is always being judged by its steady-state mode, more specifically for chattering. Chattering is a high-frequency oscillation around the equilibrium point which arises due to the discontinuous nature of the control action. Due to this, the well-designed control action stands unsuitable for many practical applications. This behavior creates a problem of wear and tears in the mechanical parts, vibrations in the machines or flapping of wing vanes in aerospace, hitting effect, etc. Hence, it is unwanted in the light of implementation. The discontinuous nature demanded by the control action cannot be delivered by any real physical actuator due to its finite bandwidth. The numerical computation done by a computer is also limited by

*Discrete Time Sliding Mode Control DOI: http://dx.doi.org/10.5772/intechopen.91245*

certain clock cycles. A lot of works have been done in the field of chattering elimination and reduction. Schemes like continuous approximation around the sliding surface (quasi-sliding mode) [1–3], higher-order sliding mode [5–8], discrete-time sliding modes are a few way outs for the process of chattering. Here in this chapter, the concept of discrete-time sliding mode (DTSM) design is discussed. Readers can explore more in the field of continuous time higher-order sliding mode whose theory is rich and well-structured.
