SLIDING MODE CONTROL

(4.20) VSS were discovered to have properties independent of the dynamics of the original systems in

the structure when switched at high frequency between the structures following a dynamic switching condition called the switching surface. When this occurs, the VSS is said to be in a sliding mode. The basic idea behind sliding mode control is to deliberately introduce sliding modes into a system by making it variable structure using a discontinuous control action[39]. The discontinuity in the control action is created by switching the control law based on the condition of a pre-specified switching surface . The control law is designed such that the system is driven towards the chosen surface and into a sliding mode on the surface in finite time.

In sliding mode, the system inherits the dynamics of the switching surface and becomes invariant to any external disturbance occurring in the same direction as the control input. The control design effort in sliding mode control consists therefore in the design of the switching surface so the system when on the surface has the desired dynamics(i.e. the surface dynamics) and the

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design of a discontinuous control that will drive the system to the surface and keep it there upon intersection [7]. The system does not actually slide on the surface when in sliding mode but switches at high frequency around the vicinity of the surface. This high frequency switching of the system on the surface leads to the problem of chattering. Chattering is a disadvantage in the application of sliding mode control as it can lead to damages in the actuator of physical systems if left unchecked.

4.2.1 SLIDING SURFACE DESIGN

The sliding surface is designed to have a reduced order from the system and a desired dynamics.

The switching surface is linear time invariant and exponentially stable. The switching surface is defined based on the error between the system states and the reference value of the states if a tracking control is desired. where state variable and reference of

state and where is the order of the original system. Then the switching surface can be defined according to [6] as:

(4.21)

where is a tuning parameter that set certain desired properties in the dynamics of the surface like the time constant of the surface.

4.2.2 SLIDING CONTROL DESIGN

The design of the sliding control action is based on the stability theory due to Lyapunov. The sliding mode control action is designed such that the distance from the surface goes to zero in finite time. A Lyapunov function based on the distance from the surface is defined thus:

(4.22) The control action is designed such that the derivative of the Lyapunov function is negative

definite. This according to the stability theory of Lyapunov is necessary if the distance from the surface is to approach zero and therefore drive the system states to the surface, .From (4.22), . For to be negative definite, and must be of opposite signs. This is a fundamental condition for the system to reach the sliding surface and therefore the existence of a sliding mode with any designed discontinuous control action [39]. Various laws exist that meet this reaching condition and they are called the reaching laws. Common reaching laws include the following: [40]

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 Constant rate reaching law which has the form with and drives the switching variable towards the switching surface at a constant rate, .

 Exponential rate reaching law with the form or with which drives the switching variable to the surface exponentially.

 Power rate reaching law having the form or

with and drives the switching variable very fast when far from the surface but slower when close to the surface thereby reducing chattering.

With a chosen reaching law , designing the sliding mode control action involves evaluating and equating it to the reaching law. The control action can be obtained by solving the resulting equation for This design process follows the method of equivalent control[39].

4.2.3 DEALING WITH CHATTERING [41]

As stated earlier, chattering is a common phenomena that plagues sliding mode control. To reduce chattering , continuous functions such as saturation and relay functions that approximate the discontinuous sign function are used, that is, .

4.2.4 SLIDING MODE CONTROL DESIGN FOR THE APPROXIMATELY LINEARIZED INVERTED PENDULUM(FL/SMC)

Consider now the approximately linearized inverted pendulum system (4.19) of section 4.1.4.

Let , and the surface is defined thus:

(4.24)

Evaluating the error derivatives in (4.24) for a constant reference yields the surface equation as

(4.25) To design the control action, an exponential reaching law is chosen.

Thus . (4.26) Evaluating from (4.25) we obtain (4.27)

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Equating (4.26) and (4.27) and solving for the control action , we obtain the sliding mode control action as in (4.28)

(4.28)

The control action applied to the plant is therefore according to the transformations done in section 4.1.3 computed thus:

and

. (4.29) To reduce chattering in the control law of (4.29) above, the following modifications are made:-

(1) A power reaching law, is used instead.

(2) is replaced by .

The control action is thus computed:

(4.30) 4.2.5 TUNING AND SIMULATION OF FEEDBACK LINEARISATION WITH SLIDING MODE CONTROLLER

The controller designed above is implemented in MATLAB/Simulink as shown in Figs A3.1-A3.2(Appendix A). In tuning the controller, the Integral of Square Error(ISE) and Mean Absolute Control Action(MACA) are used as indices to judge the relative performance of each parameter chosen. Knowing that determines the rate of convergence to zero of the error on the surface, a value of is first selected to get a time constant of about on the surface. is first set at to have a short surface reaching time and chosen to reduce chattering. Further changes are then made to the parameters using the performances indices as guide as shown in Table 4.1

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Table 4.1: Tuning of sliding mode controller

CONTROL PARAMETERS PERFORMANCE INDICES COMMENT

ISE is low but MACA is high and results in chattering.

ISE and MACA both low. Good transient performance and control action. Low chattering.

Poorer Control and transient

The tuning parameters and are therefore selected . Figure 4.1 shows an output from the approximately linearized system tracking a cart reference of 0.3m and balancing the pendulum at zero degrees(upper equilibrium) from an initial position of (0.2rads) using the parameters selected above.

Figure 4.1 Approximate feedback linearization with sliding mode control(Power Law)

From Fig. 4.1 above, it is observed that the controller satisfies the design goals. The cart has a rise time of about 2s and a settling time of 3s.The pendulum balances in about 3.3secs. No steady state error or overshoot is observed. The control action is below and cart displacement is less than , both satisfying the physical constraints on the system. It is also observed that the maximum angular displacement that can be given to the system and still obtain satisfactory

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performance meeting all constraints is about . However, ignoring all constraints the controller maintains satisfactory performance up to about from the balance point. Figure 4.2 demonstrates the tracking ability of the controller in the presence of white noise and disturbance. The noise power is about 0.02units and the a disturbance of 0.2 on both outputs occurs at 10s, 20s and 35s. It can be observed from figure 4.5 that the controller has good tracking and good recovery from disturbance even in the presence of noise. The control action and cart distance remain within the constraints even with noise and disturbance being present.

Figure 4.2: Tracking, Noise and Disturbance Rejection of Approximate feedback linearization with SMC(Power Law)