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Method in nonlinear control theory

In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set valued control signal) that forces the system to "slide" along a cross section of the system's normal behavior. The state feedback control law is not a continuous function of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of the boundaries is called the sliding (hyper)surface. In the context of modern control theory, any variable structure system, like a system under SMC, may be viewed as a special case of a hybrid dynamical system as the system both flows through a continuous state space but also moves through different discrete control modes.

Figure 1 shows an example trajectory of a system under sliding mode control. The sliding surface is described by , and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the theoretical description of sliding modes, the system stays confined to the sliding surface and need only be viewed as sliding along the surface. However, real implementations of sliding mode control approximate this theoretical behavior with a high frequency and generally non deterministic switching control signal that causes the system to "chatter" in a tight neighborhood of the sliding surface. Chattering can be reduced through the use of deadbands or boundary layers around the sliding surface, or other compensatory methods. Although the system is nonlinear in general, the idealized (i. e., non chattering) behavior of the system in Figure 1 when confined to the surface is an LTI system with an exponentially stable origin. One of the compensatory methods is the adaptive sliding mode control method proposed in

which uses estimated uncertainty to construct continuous control law. In this method chattering is eliminated while preserving accuracy (for more details see references [2] and [3]). The three distinguished features of the proposed adaptive sliding mode controller are as follows: (i) The structured (or parametric) uncertainties and unstructured uncertainties (un modeled dynamics, unknown external disturbances) are synthesized into a single type uncertainty term called lumped uncertainty. Therefore, a linearly parameterized dynamic model of the system is not required, and the simple structure and computationally efficient properties of this approach make it suitable for the real time control applications. (ii) The adaptive sliding mode control scheme design relies on the online estimated uncertainty vector rather than relying on the worst case scenario (i. e., bounds of uncertainties). Therefore, a priory knowledge of the bounds of uncertainties is not required, and at each time instant, the control input compensates for the uncertainty that exists. (iii) The developed continuous control law using fundamentals of the sliding mode control theory eliminates the chattering phenomena without trade off between performance and robustness, which is prevalent in boundary layer approach. Intuitively, sliding mode control uses practically infinite gain to force the trajectories of a dynamic system to slide along the restricted sliding mode subspace. Trajectories from this reduced order sliding mode have desirable properties (e. g., the system naturally slides along it until it comes to rest at a desired equilibrium). The main strength of sliding mode control is its robustness. Because the control can be as simple as a switching between two states (e. g., "on"/"off" or "forward"/"reverse"), it need not be precise and will not be sensitive to parameter variations that enter into the control channel. Additionally, because the control law is not a continuous function, the sliding mode can be reached in finite time (i. e., better than asymptotic behavior). Under certain common conditions, optimality requires the use of bang–bang control; hence, sliding mode control describes the optimal controller for a broad set of dynamic systems. One application of sliding mode controller is the control of electric drives operated by switching power converters. Because of the discontinuous operating mode of those converters, a discontinuous sliding mode controller is a natural implementation choice over continuous controllers that may need to be applied by means of pulse width modulation or a similar technique of applying a continuous signal to an output that can only take discrete states. Sliding mode control has many applications in robotics. In particular, this control algorithm has been used for tracking control of unmanned surface vessels in simulated rough seas with high degree of success. Sliding mode control must be applied with more care than other forms of nonlinear control that have more moderate control action. In particular, because actuators have delays and other imperfections, the hard sliding mode control action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics. Roughly speaking, the resulting closed loop system moving along is approximated by the smooth dynamics however, this smooth behavior may not be truly realizable. Similarly, high speed pulse width modulation or delta sigma modulation produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different nonlinear control design method that produces a continuous controller. In some cases, sliding mode control designs can be approximated by other continuous control designs. where and are constants.

Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods. A reusable computing paradigm such as a genetic algorithm can, however, be utilized to transform a 'unsolvable problem' of optimal design into a practically solvable 'non deterministic polynomial problem'. This results in computer automated designs for sliding model control.