Various performance criteria in a general framework

## 1. Introduction

Research on control of non-linear systems over the years has produced many results: control based on linearization, global feedback linearization, non-linear

In this chapter, we address the non-linear state feedback control design of both continuous-time and discrete-time non-linear fuzzy control systems using the Linear Matrix Inequality (LMI) approach. We characterize the solution of the non-linear control problem with the LMI, which provides a sufficient condition for satisfying various performance criteria. A preliminary investigation into the LMI approach to non-linear fuzzy control systems can be found in [7, 8, 13]. The purpose behind this novel approach is to convert a non-linear system control problem into a convex optimization problem which is solved by a LMI at each time. The recent development in convex optimization provides efficient algorithms for solving LMIs. If a solution can be expressed in a LMI form, then there exist optimization algorithms providing efficient global numerical solutions [3]. Therefore if the LMI is feasible, then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria [4, 6, 15, 16, 17, 18, 19, 20]. We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic sub-optimality with inherent stability property in combination with dissipative type of disturbance attenuation.

In the following sections, we first introduce the Takagi-Sugeno fuzzy modelling for non-linear systems in both continuous time and discrete time. We then propose the general performance criteria in section 3. Then, the LMI control solutions are derived to characterize the optimal and robust fuzzy control of continuous time and discrete time non-linear systems, respectively. The inverted pendulum system control is used as an illustrative example to demonstrate the effectiveness and robustness of our proposed approaches.

The following notation is used in this work:

## 2. Takagi-Sugeno system model

The importance of the Takagi-Sugeno fuzzy system model is that it provides an effective way to decompose a complicated non-linear system into local dynamical relations and express those local dynamics of each fuzzy implication rule by a linear system model. The overall fuzzy non-linear system model is achieved by fuzzy “blending” of the linear system models, so that the overall non-linear control performance is achieved. Both of the continuous-time and the discrete-time system models are summarized below.

### 2.1. Continuous-time Takagi-Sugeno system model

The**Model Rule**

**If**

is

is

**Then** the input-affine continuous-time fuzzy system equation is:

where

It is assumed that the premises are not the function of the input vector

where

for all time

Since, the following properties hold

We have

for all time

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

Substituting this into the system and performance output equation, we have

Using the notation

then the system equation becomes

### 2.2. Discrete-time Takagi-Sugeno system model

At time step

**Model Rule**

**If**

**Then** the input-affine discrete-time fuzzy system equation is:

where

It is assumed that the premises are not the function of the input vector

where

for all

Since, the following properties hold

We have

for all

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

Substituting this into the system and performance output equation, we have

Using the notation

then the system equation becomes

## 3. General performance criteria

In this section, we propose the general performance criteria for non-linear control design, which yields a mixed Non-Linear Quadratic Regular (NLQR) in combination with

### 3.1. Continuous-time general performance criteria

Consider the quadratic Lyapunov function

for the following difference inequality

with

Note that upon integration over time from 0 to

By properly specifying the value of weighing matrices

which is a mixed

Other possible performance criteria which can be used in this framework with various design parameters

### 3.2. Discrete-time general performance criteria

Consider the quadratic Lyapunov function

for the following difference inequality

with

Note that upon summation over

By properly specifying the value of weighing matrices

which is a mixed

Other possible performance criteria which can be used in this framework with various design parameters

Performance Criteria | |||

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## 4. Fuzzy LMI control of continuous time non-linear systems with general performance criteria

The main results of this chapter are summarized in section 4 and section 5. The following theorem provides the fuzzy LMI control to the continuous time non-linear systems with general performance criteria.

**Theorem 1** Given the system model and performance output (2) and control input (8), if there exist matrices

where

using the notation

then inequality (24) is satisfied.

**Proof**

By applying system model and performance output (2)(11), and state feedback input (8), the performance index inequality (24) becomes

Inequality (34) is equivalent to

where

Inequality (35) can be rewritten as

where

By applying Schur complement to inequality (37), we have

Similarly, inequality (39) can also be written as

where

By applying Schur complement again to (40), we have

Equivalently, we have

where

Therefore, we have the following LMI

By multiplying both sides of the LMI above by the block diagonal matrix

we obtain

where

By applying Schur complement again, the final LMI is derived

where

Hence, if the LMI (49) holds, inequality (24) is satisfied. This concludes the proof of the theorem.

Remark 1: For the chosen performance criterion, the LMI (49) need to be solved at each time to find matrices

## 5. Fuzzy LMI control of discrete time non-linear systems with general performance criteria

This section summarizes the main results for fuzzy LMI control of discrete time non-linear systems with general performance criteria:

**Theorem 2:** Given the closed loop system and performance output (13), and control input (19), if there exist matrices

where

and

where

**Proof**

The performance index inequality (28) can be explicitly written as

Equivalently,

which can be written, after collecting terms, as

where

Equivalently, we have

By applying Schur complement, we obtain

By applying Schur complement again, we obtain

Equivalently, the following inequality holds

where

By applying Schur complement one more time, we have

By factoring out the

where

By pre-multiplying and post-multiplying the matrix with the block diagonal matrix

where

where

## 6. Application to the inverted pendulum system

The inverted pendulum on a cart problem is a benchmark control problem used widely to test control algorithms. A pendulum beam attached at one end can rotate freely in the vertical 2-dimensional plane. The angle of the beam with respect to the vertical direction is denoted at angle

where

Due to the system non-linearity, we approximate the system using the following two-rule fuzzy model:

**continuous-time fuzzy model**

*Rule 1:* If

Then

*Rule 2:* If

Then

where

**discrete-time fuzzy model**

*Rule 1:* If

Then

*Rule 2:* If

Then

where

The following values are used in our simulation:

and the initial condition of

The feedback control gain can be found from (31)(51) by solving the LMI at each time. The following design parameters are chosen to satisfy:

Mixed

Mixed

The mixed criteria control performance results are shown in the Figs.2-4. From these figures, we find that the novel fuzzy LMI control has satisfactory performance. The mixed

## 7. Summary

This chapter presents a novel fuzzy control approach for both of continuous time and discrete time non-linear systems based on the LMI solutions. The Takagi-Sugeno fuzzy model is applied to decompose the non-linear system. Multiple performance criteria are used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be obtained by solving LMI at each time. The inverted pendulum is used as an example to demonstrate its effectiveness. The simulation studies show that the proposed method provides a satisfactory alternative to the existing non-linear control approaches.