ELEC8406 Sliding Mode Control of Power Electronics Lecture 3: Introduction to Sliding Mode Control Reference: S.C. Tan, Chapter 1. Sliding Mode Control of Switching Power Converters
History (1) • SM control can be traced back to the 1930s • Earliest forms of SM control for ship-course control 2
History (2) • Development of the SM theory/applications first initiated by Russian engineers in the 1950s • Theoretical framework facilitating the widespread of SM control was reported in Russian literature • SM was subsequently disseminated outside Russia in English written manuscripts by Itkis (1976) and Utkin (1977) • Since then, SM control has aroused a lot of interests of control theoreticians and practicing engineers 3
Characteristics • a kind of nonlinear control developed primarily for variable structure systems • consists of a time-varying state-feedback discontinuous control law • switches at a high frequency from one continuous structure to another • force the dynamics of the system to follow exactly what is desired and pre-determined 4
Advantages • main advantage - guaranteed stability and robustness against parameter uncertainties • high degree of flexibility in its design choices • relatively easy to implement as compared to other nonlinear control methods • highly suitable for applications in nonlinear systems • wide utilization in industrial applications, e.g., electrical drivers, automotive control, etc. 5
General Theory (1) • Consider an equilibrium point O on an imaginary plane in a system with 3-D space. O represents a stable attractor where trajectory touching it will settle upon. 6
General Theory (2) Consider that trajectory of a system is arbitrarily • located far away from the plane. Without any control action, the trajectory will • move according to the natural characteristics of the system. When a control action is given, the trajectory • can be altered in a “preferred way.” The direction in which the trajectory moves is • dependent on the type of control action given. 7
General Theory (3) • A series of different control actions may be given to the system such that regardless of its initial condition, the controlled trajectory will first move toward the plane, and upon reaching the plane, will slide along the plane toward and eventually settle upon O. • A control such as this is known as SM control. 8
General Theory (4) The plane which guides the trajectory is called the sliding plane or sliding surface , or more generally, the sliding manifold . The control actions required for performing the SM control will involve very fast switching between different control functions. The sectors of the space in which the trajectory can be made to perform SM control is called the sliding regimes . 9
Definition For any given system, if a sliding regime exists and the sliding manifold ζ = 0 possesses a stable equilibrium point O , when operated in sliding mode, the feedback tracking trajectory S, regardless of its location, will be driven toward the sliding manifold , and upon hitting the manifold , it will induce the control of the system to switch alternately between two or more discrete control functions U1, U2, . . ., etc., at an infinite frequency , such that the system’s trajectory will be trapped precisely on the sliding manifold such that S = ζ = 0 , and eventually the trajectory will be directed toward the desired equilibrium point O . 10
Operating Mechanism (1) • Entire SM operation can be divided into two phases. • First phase is reaching phase , achieved through the compliance of the so-called hitting condition • Ensures that controlled trajectory is directed toward sliding manifold . 11
Operating Mechanism (2) • When trajectory touches sliding manifold, system enters sliding phase and is in SM operation • Trajectory is trapped on the sliding manifold and is directed toward O and finally settling at O • Possible by satisfying existence condition and stability condition . 12
Properties of Sliding Motion • An ideal control • Practical Limitations and Chattering • Constant Dynamics • Quasi-Sliding Mode Control 13
An Ideal Control • Sliding manifold as a reference path for trajectory flow. • Inherently adopting an infinite control gain which enables it to trap trajectory to slide along the manifold. • No external disturbance or system’s uncertainty can affect the ideal control performance of having a precise tracking, zero-regulation error (infinite DC gain), and very fast dynamic response. • In a certain sense, the SM control is an ideal (optimal) type of control for variable structure systems. 14
Practical Limitations and Chattering (1) • Everything has been based on ideal assumption of infinite switching frequency and perfect components of the control • In practice, there are imperfections of switching devices like time delay, response time constant, presence of dead zone, hysteresis effect, saturation of device switching frequency, etc. • Actual behavior of the sliding motion deviates slightly from that expected for the ideal condition. • In addition, a kind of high frequency oscillation may occur in the control process which is reflected in the actual behavior of the trajectory. • This phenomenon is known as chattering . 15
Practical Limitations and Chattering (2) • Non-ideality of switching does not affect reaching phase and is the same for both ideal and non-ideal conditions • For the sliding phase under the non-ideal condition, trajectory S does not move exactly on the sliding manifold, but instead oscillates within its vicinity at a high frequency while concurrently converging toward O. 16
Practical Limitations and Chattering (3) • Ideal condition: S stops precisely at O upon arrival • Non-ideal condition: S will be trapped in a periodically- oscillating state at a point near O • Ideal condition: no error during steady state • Non-ideal condition : steady-state error. 17
Constant Dynamics • In sliding phase, the movement of the trajectory is confined along the sliding manifold, which means that the motion equation of the trajectory is S = ζ = 0. • The dynamics of a system under SM operation is constant and is independent of the system parameters or disturbance. • Such a property applies only to the sliding phase but not the reaching phase which has a different set of dynamic characteristics for a given operating condition. 18
Quasi Sliding Mode Control • Extreme high-speed switching of SM control may result in excessive losses and wear out, and is a source of noise. • Switching frequency of the control implementation must be confined within a practical range. • Control is now a quasi-sliding mode (QSM) or pseudo- sliding mode (PSM) control, which is an approximation of the ideal SM control. • Consequence is degradation of system’s robustness and deterioration of the regulation properties. 19
Mathematical Formulation 20
Hitting Condition (1) Objective : ensure that control decision will direct the trajectory to approach and reach, within a vicinity δ , the sliding manifold . Initial state of trajectory S i = S ( t = 0) is located at a distance away from the sliding manifold ζ = 0 21
Hitting Condition (2) The necessary and sufficient condition for the system to satisfy the hitting condition is Compliance of inequality signifies that S is continuously being attracted toward the sliding manifold ζ = 0 for t > 0 , and that the choice of u i = u ( t > 0) is supporting this attraction. 22
Hitting Condition (3) Hence, one fundamental aspect of designing the SM control is to first determine, for a desired set of control parameters (sliding coefficients), the suitable discontinuous control action for the system. In other words, design of U + and U − would have to ensure that hitting condition be always satisfied for the system. The inequality is a partial result of the Lyapunov second theorem on stability , of which the Lyapunov function candidate is 23
Existence Condition Ensures that once the trajectory is at locations within the vicinity of the sliding manifold such that 0 < | S | < δ, it is still always directed toward the sliding manifold. Existence condition of the SM operation can be determined by inspecting This can be expressed as 24
Stability Condition (1) • Stability condition ensures trajectory moves toward a stable equilibrium point. • Left figure shows the trajectory stabilizing at O when stability condition is fulfilled. • Right figure shows the same trajectory moving pass O when stability condition is not fulfilled. 25
Stability Condition (2) System with Linear Sliding Manifold System with trajectory made up of state variables and their time derivatives/integrals (so-called in the phase canonical form) has a linear sliding manifold as such Applying Laplace transform, we have By applying Routh-Hurwitz stability criterion , condition for stability can be obtained. For example, for a second-order polynomial, the stability condition would be α 1 > 0 and α 2 > α 3 > 0 . 26
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