2. The state at time t and the inputs togeth their derivatives at time t determine uniquely the value of any system variable at the time t The state may be regarded as a vector, the components of which are state variables. Network variables that are candidates for the state variables are the branch currents and voltages. Our problem is to choose state variables in order to formulate the state equations. Like the nodal, cutset, or loop system of equations, the stat equations are formulated from the primary system of equations For our purposes, we shall focus our attention on how to obtain state equations for linear systems. 7.2 State Equations in Normal Form For a linear network containing k energy storage elements and h independent sources, our objective is to write a system of k first-order differential equations from the primary system of equations, as follows (t) x()+∑b( In matrix notation, Eq. (7. 1)becomes Lx(o」 a2() or, more compact (t)=ax(t)+ Bu(t) (73) The real functions x,(o), x(t),.x (r) of the time t are called the state variables, and the k-vector x(r) formed by the state variables is known as the state vector. The h-vector u(t)formed by the h known forcing functions or excitations u(r) is referred to as the input vector. The coefficient matrices A and B, depending only upon the network parameters, are of orders k x k and k x h, respectively. Equation(7.3)is usually called the state equation in normal form. The state variables x, may or may not be the desired output variables. We therefore must express the desired output variables in terms of the state variables and excitations. In general, if there are q output variables y(r) G=1, 2,.,g) and h input excitations, the output vector y(t) formed by the g output variables y r) can be expressed in terms of the state vector x(t)and the input vector u(t)by the matrix equation r()=Cx()+ du(t) e 2000 by CRC Press LLC© 2000 by CRC Press LLC 2. The state at time t and the inputs together with some of their derivatives at time t determine uniquely the value of any system variable at the time t. The state may be regarded as a vector, the components of which are state variables. Network variables that are candidates for the state variables are the branch currents and voltages. Our problem is to choose state variables in order to formulate the state equations. Like the nodal, cutset, or loop system of equations, the state equations are formulated from the primary system of equations. For our purposes, we shall focus our attention on how to obtain state equations for linear systems. 7.2 State Equations in Normal Form For a linear network containing k energy storage elements and h independent sources, our objective is to write a system of k first-order differential equations from the primary system of equations, as follows: (7.1) In matrix notation, Eq. (7.1) becomes (7.2) or, more compactly, (7.3) The real functions x1(t), x2(t), ..., xk(t) of the time t are called the state variables, and the k-vector x(t) formed by the state variables is known as the state vector. The h-vector u(t) formed by the h known forcing functions or excitations uj (t) is referred to as the input vector. The coefficient matrices A and B, depending only upon the network parameters, are of orders k ¥ k and k ¥ h, respectively. Equation (7.3) is usually called the state equation in normal form. The state variables xj may or may not be the desired output variables. We therefore must express the desired output variables in terms of the state variables and excitations. In general, if there are q output variables yj (t) (j = 1, 2, . .., q) and h input excitations, the output vector y(t) formed by the q output variables yj (t) can be expressed in terms of the state vector x(t) and the input vector u(t) by the matrix equation (7.4) x t ˙ ( ) a x (t) b u (t), i , , ... , k) i ij j k j ij j h = + j = = = Â Â 1 1 ( 1 2 ˙ ( ) ˙ ( ) . . . ˙ ( ) . . . . . . . . . . . . . . . . . . . . . x t x t x t a a a a a a a a a k k k k k kk 1 2 11 12 1 21 22 2 1 2 È Î Í Í Í Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ = È Î Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ È Î Í Í Í Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ + ( ) ( ) . . . ( ) . . . . . . . . . . . . . . . . x t x t x t b b b b b b k h h 1 2 11 12 1 21 22 2 . . . . . ( ) ( ) . . . ( ) b b b u t u t u t k k kh h 1 2 1 2 È Î Í Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ È Î Í Í Í Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ x˙( )t = + Ax( )t Bu(t) y( )t t = + Cx( ) Du(t)