230 The general steps to design the nonlinear controller are shown below: 1.Calculate the vector fields 14,P 1(x:-x) g,adg,..,ad g (10) P3= T。 y u=EFD 2.Check the controllability and involutivity Hence,Eqn.(1)can be rewritten in terms of the vector The controllability matrix is expressed as: fields fx)and g(x). =f(x)+g(x)u (3) g,ad 8,ad,'g (11) In order to proceed with the feedback linearization,it is 3.Find T (x) necessary to review some basic notations and concepts from differential geometry [2]. a=0, aT ≠0 (12) Definition 1 (Lie derivative):The Lie derivative of a 8x2 a0, ax real-valued function h with respect to the vector field f From (12)it is easy to see that the simplest solution to is defined as the above equation is T (x)=x,then the nonlinear (x) state transformation can be expressed as L,h=Vh.f (4) dx z=T(x)=T LT LT (13) f(x) 4.Derive the control law Definition 2 (Lie bracket):The Lie bracker of two v-L'T vector fields f and g is denoted by adg and defined u=a(x)+B(x)v= (14) L,, as Then,under the nonlinear state transformation (13)and ad 8=f.8=Vgf-Vf g (⑤ the control law (14),the original nonlinear system (3) Repeated Lie bracketfand g can then be defined can be transformed into a simple linear system: recursively as i1=z2 adg=g i2=; (15) (6 ad,g=[,ad,g]i=l,2… i3=v Definition 3 (Relative degree):A single-input single- 5.Controller design Using the state equation (15),we can easily design output nonlinear system is said to have relative degree r controller for either stabilization or tracking purpose. in a region U if,for all xeU,LL,'h(x)=0 With the equation: (0≤i<r-l)and V=-k131-k222-k323 (16) LL,h(x)≠0. one can place the poles of the closed-loop system to (7) desired locations by choosing the proper feedback gains Definition 4 (Complete integrability):A linearly such that the original nonlinear system is stabilized. independent set of vector fieldsf,f2,f is III.Feedback Linearization of MIMO Systems said to be completely integrable if,and only if,there exist n-m scalar functions (h,h2,,h-m In practice,the power system consists of many generators.The technique of feedback linearization for satisfying the system of partial differential equations SISO systems can not be used directly for MIMO Vh·f;=0 (8) systems.Fortunately,the concepts developed for SISO where1≤i≤n-m,I≤j≤m,and the gradients systems can be extended to MIMO systems.However, Vh;are linearly independent. the corresponding definitions and theorems for feedback Definition 5 (Involutivity):A linearly independent set linearization of MIMO systems will be more complex than those for SISO systems of vector fields [f,f2,,f is said to be involutive if,and only if,there exist such that In our study of the multi-machine system,we will restrict our analysis to the systems with the same U.ef,小-a=(.因 寸i,j(9) numbers of inputs and outputs,which can be described in the following state space form:230 Hence, Eqn.(l) can be rewritten in terms of the vector fieldsflx) and g(x). In order to proceed with the feedback linearization, it is necessary to review some basic notations and concepts from differential geometry [2]. Definition 1 (Lie derivative): The Lie derivative of a real-valued function h with respect to the vector field f is defined as x = f(x)+ g(x)u (3) Definition 2 (Lie bracket): The Lie bracket of two vector fields f and g is denoted by adfg and defined as ad,g=[f4]=Vgf-W (5) Repeated Lie bracket f and g can then be defined recursively as Definition 3 (Relative degree): A single-input single￾output nonlinear system is said to have relative degree r in a region u if, for all XEU, L,L,'h(x) = O (0 5 i < I - 1) and L,L,'-'h(x) # 0. (7) Definition 4 (Complete integrability): A linearly independent set of vector fields [ f, , f, , - -., f, } is said to be completely integrable if, and only if, there exist n-m scalar functions {hl, h,, -1.) ha-,} satisfying the system of partial differential equations where 15 i I n - m, 1 I j 5 m, and the gradients Vh; are linearly independent. Definition 5 (Involutivity): A linearly independent set of vector fields (f,, f2, ---, f, } is said to be involutive if, and only if, there exist allk such that Vh, . f, =O (8) 'd i, j (9) The general steps to design the nonlinear controller are shown below: 1. Calculate the vector fields g, ad,g, ..., adf"-'g (10) 2. Check the controllability and involutivity The controllability matrix is expressed as: 3. Find TI (x) From (12) it is easy to see that the simplest solution to the above equation is TI ( x ) = x1 , then the nonlinear state transformation can be expressed as z = T(x) = [TI L,T, Lf2TIr (13) 4. Derive the control law v-Lf3T, u = a(')+ P(')V = (14) Then, under the nonlinear state transformation (13) and the control law (14), the original nonlinear system (3) can be transformed into a simple linear system: Vf2 i, = 2, i, = z3 (15) 2, =v 5. Controller design Using the state equation (15), we can easily design controller for either stabilization or tracking purpose. With the equation: one can place the poles of the closed-loop system to desired locations by choosing the proper feedback gains such that the original nonlinear system is stabilized. III. Feedback Linearization of MIMO Systems In practice, the power system consists of many generators. The technique of feedback linearization for SISO systems can not be used directly for MIMO systems. Fortunately, the concepts developed for SISO systems can be extended to MIMO systems. However, the corresponding definitions and theorems for feedback linearization of MIMO systems will be more complex than those for SISO systems. v =-k,z, -k,z2 -k3z3 (16) In our study of the multi-machine system, we will restrict our analysis to the systems with the same numbers of inputs and outputs, which can be described in the following state space form: