Sliding mode control

In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by application of 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. x ˙ ( t ) = f ( x , t ) + B ( x , t ) u ( t ) {displaystyle {dot {mathbf {x} }}(t)=f(mathbf {x} ,t)+B(mathbf {x} ,t),mathbf {u} (t)}     ( 1) { x ∈ R n : σ k ( x ) = 0 } {displaystyle left{mathbf {x} in mathbb {R} ^{n}:sigma _{k}(mathbf {x} )=0 ight}}     ( 2) V ( σ ( x ) ) = 1 2 σ ⊺ ( x ) σ ( x ) = 1 2 ‖ σ ( x ) ‖ 2 2 {displaystyle V(sigma (mathbf {x} ))={frac {1}{2}}sigma ^{intercal }(mathbf {x} )sigma (mathbf {x} )={frac {1}{2}}|sigma (mathbf {x} )|_{2}^{2}}     ( 3) σ ( x ) ≜ s 1 x 1 + s 2 x 2 + ⋯ + s n − 1 x n − 1 + s n x n {displaystyle sigma (mathbf {x} ) riangleq s_{1}x_{1}+s_{2}x_{2}+cdots +s_{n-1}x_{n-1}+s_{n}x_{n}}     ( 4) σ ˙ ( x ) = ∂ σ ( x ) ∂ x x ˙ ⏞ σ ˙ ( x ) = ∂ σ ( x ) ∂ x ( f ( x , t ) + B ( x , t ) u ) ⏞ x ˙ = [ s 1 , s 2 , … , s n ] ⏞ ∂ σ ( x ) ∂ x ( f ( x , t ) + B ( x , t ) u ) ⏞ x ˙ ⏟ ( i.e., an  n × 1  vector ) {displaystyle {dot {sigma }}(mathbf {x} )=overbrace {{frac {partial {sigma (mathbf {x} )}}{partial {mathbf {x} }}}{dot {mathbf {x} }}} ^{{dot {sigma }}(mathbf {x} )}={frac {partial {sigma (mathbf {x} )}}{partial {mathbf {x} }}}overbrace {left(f(mathbf {x} ,t)+B(mathbf {x} ,t)u ight)} ^{dot {mathbf {x} }}=overbrace {} ^{frac {partial {sigma (mathbf {x} )}}{partial {mathbf {x} }}}underbrace {overbrace {left(f(mathbf {x} ,t)+B(mathbf {x} ,t)u ight)} ^{dot {mathbf {x} }}} _{{ ext{( i.e., an }}n imes 1{ ext{ vector )}}}}     ( 5) In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by application of 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 s = 0 {displaystyle s=0} , 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. In fact, although the system is nonlinear in general, the idealized (i.e., non-chattering) behavior of the system in Figure 1 when confined to the s = 0 {displaystyle s=0} surface is an LTI system with an exponentially stable origin. 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.:'Introduction' 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.:554–556 Continuous control design methods are not as susceptible to these problems and can be made to mimic sliding-mode controllers.:556–563 Consider a nonlinear dynamical system described by