Indagar sobre otros factores de su participación en el PEB que los profe

The synthesis method used in this section is the PWA state feedback method, primarily based on [28]. For this purpose, a common quadratic Lyapunov function V = xT_{Px is}

required such that V > 0 and ˙V < −αV , where P is a 3 × 3 symmetric matrix and α > 0. These conditions guarantee the exponential stability of the PWA system. The reader is 1_{FOCP is an acronym standing for Fourth Order Concave Polynomial Approximation introduced in Sub-}

section 4.4.1.A.

PWA Approximation of f1(x) with 5 Regions

APPROXIMATION METHOD e_max CONTINUITY

Intersection-based FOCP 8.53 Continuous

Voronoi-based 9.88 Discontinuous

Uniform Grid 10.65 Continuous

PWA Approximation of f2(x) with 2 Regions

Intersection-based CONS 3.46 Continuous

Voronoi-based 5.58 Discontinuous

Uniform Grid 4.89 Continuous

Table 5.2: A brief summary of the PWA approximation of f1(x) and f2(x) using different

methods.

encouraged to refer to [28] for more detail. Following [28], the next BMI has to be solved for determining the state feedback gains

  G₁ G₂ GT₂ G₃  < 0 (5.13) where G1= (AiQ+BiYi)T+(AiQ+BiYi)+αQ+QTE_iTΛiEiQ, G2= (bi+Bimi)+QTE_iTΛiei

and G3= eTi Λiei. In the above inequality Q= P−1and Yi= KiQ, where Kiis the state feed-

back gain for the ith region, miis the state feedback affine term and Λiis an m × m matrix

with non-negative elements. For more detail the reader is referred to [28].

Solving the inequality (5.13) by using PENBMI [49] with YALMIP [82] interface, the following Lyapunov matrix is obtained

P=       701.1853 −265.5056 1.2700 −265.5056 103.7864 −0.5107 1.2700 −0.5107 0.0039       (5.14)

The resulting state feedback gains are given by K1= [ 736.89 −115.61 42.91 ], m1= 0 K_{2= [ 786.65 −125.01 47.05 ], m2}= −2.8603 K3= [ 720.50 −112.82 42.03 ], m3= 3.1026 K_{4= [ 1027.7 −162.13 60.64 ], m4}= −2.0473 K_{5= [ 724.66 −113.45 42.21 ], m5}= −1.6368

By selecting β = 1/√2 and α= 7 taken from [16], the simulation results for two different sets of initial conditions, namely x0= [0.4, − 1.4, 0.5]T and x0= [1.04, 0.55, 0.5]T, are

shown in Figures 5.7a and 5.7b . In simulations, by applying the state feedback gains to the intersection-based PWA MG model, it follows that the oscillatory nonlinear model (5.1) is damped out within a short time period. As can be seen form Figures 5.7a and 5.7b, the settling times corresponding to x0= [0.4, − 1.4, 0.5]T and x0= [1.04, 0.55, 0.5]T are

2.4 and 0.8 seconds, respectively.

5.2.3

Conclusions

In this section, the intersection-based PWA approximation method, as a new and powerful method in the PWA approximation of nonlinear systems was developed. Using the algo- rithms provided in Section 4.4.1, the PWA models can be constructed for a wide range of nonlinear functions. The method was successfully applied to the Moore-Greitzer axial compressor surge and stall dynamics, as a safety-critical system. This model had never been stabilized by using the PWA method in the literature before. However, due to the computational complexities involved solving the BMIs are difficult in certain cases. For instance, using PENBMI we were not be able to obtain the state feedback gains for the PWA model obtained by the Voronoi-based and the uniform grid PWA approximation methods.

0 0.5 1 1.5 2 2.5 3 −8 −6 −4 −2 0 2 t x 1 , x 2 , x 3

(a) The initial condition is x0= [0.4, − 1.4, 0.5]T.

0 0.5 1 1.5 2 2.5 3 −8 −6 −4 −2 0 2 t x 1 , x 2 , x 3

(b) The initial condition is x0= [1.04, 0.55, 0.5]T.

Figure 5.7: Simulation results of the closed-loop MG model using the state feedback PWA controller. The dashed, dash-dot and the solid lines represent x1, x2and x3, respectively.

Chapter 6

Conclusions and Future Research

6.1

Conclusions

This chapter summarizes the main contributions of this research. In Chapter 2, the axial compressor that is used in gas turbines were introduced. Then rotating stall and surge were discussed. They are two important instabilities in a compressor that may cause se- vere mechanical damage and engine flame-out. Following the work done by Moore and Greitzer in [3], a second order surge and a third order surge and rotating stall dynamics were provided.

In Chapter 3, pseudo Euler-Lagrange systems that were introduced in [25] were ex- tended to a system of the same order, but with additional nonlinear terms. The proposed method enables one to fit a more general class of second order systems for performing and solving Lyapunov-based control synthesis problems, such as the second order axial compression surge phenomenon in jet engines. Accordingly, the stability analysis of the extended pseudo Euler-Lagrange systems and the technique used to formulate a second order system as an extended pseudo Euler-Lagrange system were addressed. The pro- posed technique was applied to the stabilization problem of the no-stall Moore-Greitzer axial compressor model. In order to compare the stabilization properties of the proposed

method, a feedback linearization and a backstepping controllers were also designed as benchmark methods. The main advantages of the pseudo Euler-Lagrange method for the suppression of the Moore-Greitzer model oscillation were found to be the following:

• The pseudo Euler-Lagrange technique provides the designed controller with a broad margin for choosing the control law coefficients such that the stability of the closed- loop system is guaranteed for a range of disturbances.

• During the design process, nonlinearities that may enhance the stability quality of the system are not canceled out as it is done in the feedback linearization technique.

In Chapter 4, the Intersection-based Piecewise Affine (IPWA) approximation method was proposed. After a brief review on PWA systems, the approximation theory for func- tions of one variable was first addressed. The theory of the proposed methodology was then extended to functions of n-variables. Using the intersection-based Piecewise Affine (IPWA) models, the following properties can be achieved:

• Continuity of the vector fields,

• Optimality of the linearization of the nonlinear function relative to the maximum approximation error,

• Increased reduction of the approximation error for a fixed number of regions can be achieved for the motivating Moore-Greitzer system in Section 5.2 (as compared to the Voronoi-based and the uniform grid PWA models),

• Consistency of the derivative of the nonlinear function with the derivative of its approximation at the linearization points.

The obtained and derived IPWA approximation technique results were used to approximate a third order axial compressor stall and surge system. The PWA control synthesis tools [28] are then used to design controllers for the obtained IPWA model.

In Chapter 5, the control synthesis problem of the Moore-Greitzer axial compres- sor model describing the two major compressor instabilities − namely the rotating stall and surge − was addressed by using both the pseudo Euler-Lagrange and the piecewise affine methodologies. The associated closed-loop system simulations corresponding to each nonlinear technique were provided. The obtained IPWA model of the third order stall and surge system, and a PWA synthesis technique [28] is then used to control the closed-loop system.