lecture 3 - university of hong kongelec8406/slides/lecture3.pdflecture 3: introduction to sliding...

Lecture 3:

Introduction to Sliding Mode Control

ELEC8406 Sliding Mode Control of Power Electronics

Reference: S.C. Tan, Chapter 1. Sliding Mode Control of Switching Power Converters

History (1)

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• SM control can be traced back to the 1930s

• Earliest forms of SM control for ship-course control

History (2)

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• 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

Characteristics

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• 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

Advantages

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• 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.

General Theory (1)

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• 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.

General Theory (2)

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• 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.

General Theory (3)

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•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.

General Theory (4)

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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.

Definition

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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.

Operating Mechanism (1)

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•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.

Operating Mechanism (2)

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• 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.

Properties of Sliding Motion

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• An ideal control

• Practical Limitations and Chattering

• Constant Dynamics

• Quasi-Sliding Mode Control

An Ideal Control

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• 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.

Practical Limitations and Chattering (1)

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• 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.


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• 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.


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• 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.

Constant Dynamics

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• 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.

Quasi Sliding Mode Control

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• 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.

Mathematical Formulation

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Hitting Condition (1)

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Objective: ensure that control decision will direct the trajectory to approach and reach, within a vicinity δ, the sliding manifold . Initial state of trajectory Si = S(t = 0) is located at a distance away from the sliding manifold ζ = 0


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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 ui = u(t > 0) is supporting this attraction.


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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

Existence Condition

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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

Stability Condition (1)

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• 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.


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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 α1 > 0 and α2 > α3 > 0.


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System with Non-Linear Sliding Manifold (pp 27-32) For SM-controlled system with nonlinear sliding manifold, a different approach based on the equivalent control method is adopted to derive the stability condition. This approach involves first deriving the ideal sliding dynamics of the system, and then performing a stability analysis on its equilibrium point.


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Ideal Sliding Dynamics Discontinuous control action u(t) (in page 20) is replaced by a continuous control action ueq(t) converts a switching SM system into an average continuous SM system The control action ueq(t), which is the equivalent control derived from the so-called equivalent control method (to be discussed in the following section), is a solution of dS/dt = 0.


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ueq(t) is a function f(.) of the dynamics of the state variables and the sliding coefficients, and from page 20, it can be given as Substituting this equation into previous equation gives which represents the ideal sliding dynamics of the system during SM operation and is independent of the control signal.


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Equilibrium Point Assume there exists a stable equilibrium point on the sliding manifold on which the ideal sliding dynamics eventually settled. Then, the state equations during ideal sliding dynamics (page 29) can be solved to give the steady-state operating point (x1(ss), x2(ss), ..., xm(ss)) during SM operation by putting x˙(t) = 0.


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Linearization of Ideal Sliding Dynamics The linearization of the ideal sliding dynamics around the equilibrium point (x1(ss), x2(ss), ..., xm(ss)) gives where represents the linearized small-signal ideal sliding dynamics around the steady-state operating point, and are the small-signal ac equivalent components of and , respectively.


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Arranging the equation in matrix form where A is the Jacobian square matrix of . The characteristic equation of the linearized sliding dynamics is where det(.) is the determinant function, I is the identity matrix, and λ is the eigenvalue of the system. Application of Routh-Hurwitz stability criterion to the characteristic equation gives a set of conditions that will ensure stability of the ideal sliding dynamics and confirm the presence of a stable equilibrium point on the sliding manifold.

Equivalent Control (1)

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• Ideal SM control operates the system at an infinite switching frequency

• Trajectory moves precisely on the sliding manifold

• Practical limitations of devices induce a low-amplitude high-frequency oscillation (chattering)

• Treated at two components in the trajectory, namely a “fast-moving” (high-frequency) component and a “slow-moving” (low-frequency) component.


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• High-frequency component oscillates between the +ve and −ve directions • Low-frequency component moves along the sliding plane


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• Possible to relate low-frequency component of the trajectory to respectively a low-frequency continuous switching action ulow(t)

• High-frequency component to a high-frequency discontinuous switching action

• Overall control signal is


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• uhigh(t) produces high-frequency trajectory component ulow(t) produces low-frequency trajectory component.

• By ignoring high-frequency component, the motion of trajectory is then solely determined by low-frequency component.

• ulow(t) produces a trajectory that is nearly equivalent to an ideal SM-controlled trajectory.

• This is the so-called equivalent control of the system, i.e., ueq(t), and is actually the low-frequency continuous switching action ulow(t).


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Types of Implementation (1)

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Relay and Signum Functions Conventional method of implementing SM control is realized using a switch relay, which is realized through analog or digital computation of instantaneous trajectory S(t).


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In applications involving only a positive-or-negative decision, signum function can be used for the relay, i.e., where the signum function sgn(.) is defined as For applications involving only digital logic, function is replaced by


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• Implementation is straightforward and simple

• However, switched at a very high frequency giving an unwanted chattering effect in the system

• Unsuitable for some applications which see this as an undesired high-frequency noise

• Possible to restrict the range of the operating frequency by using a hysteresis function


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Hysteresis Function Implementation of SM control through hysteresis function is easily accomplished using where Δ is an arbitrarily small value. Introduction of a hysteresis band with the boundary conditions S = Δ and S = −Δ provides a form of control to the switching frequency of the system.


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As a result, the trajectory S of the system will operate precisely in the vicinity of ± Δ of the sliding manifold with a controlled oscillation as illustrated in the figure. The chattering effect now becomes controllable and is a function of Δ.


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Equivalent Control Function • The equivalent control function ueq(t) of a system is the ideal averaged control function required during the SM phase and is bounded by • Based on the invariance conditions , equivalent control is derived by simply solving • Implementation of the equivalent control function

results in a system that operates with ideal SM control without high-frequency chattering


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• Derivation of equivalent control using only part of the invariance conditions ˙ would be sufficient for implementing the SM control. • Reason is when equivalent function evolves on S, there is interdependency among the state variables of the ideal sliding system (with ) and therefore, one of the equations is redundant. • When a system enters into the SM phase, i.e., S = 0, the implementation of a control law that ensures will automatically imply S = 0, and vice versa.


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lecture 3 - university of hong kongelec8406/slides/lecture3.pdflecture 3: introduction to sliding...

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