Additive state decomposition

Additive state decomposition occurs when a system is decomposed into two or more subsystems with the same dimension as that of the original system. A commonly used decomposition in the control field is to decompose a system into two or more lower-order subsystems, called lower-order subsystem decomposition here. In contrast, additive state decomposition is to decompose a system into two or more subsystems with the same dimension as that of the original system. x ˙ = f ( t , x , u ) , x ( 0 ) = x 0 {displaystyle {dot {x}}=f(t,x,u),x(0)=x_{0}}     (1) x ˙ p = f p ( t , x p , u p ) , {displaystyle {dot {x}}_{p}=f_{p}(t,x_{p},u_{p}),} x p ( 0 ) = x p , 0 {displaystyle x_{p}(0)=x_{p,0}}     (2) x s = x − x p , {displaystyle x_{s}=x-x_{p},} u s = u − u p . {displaystyle u_{s}=u-u_{p}.}     (3) x ˙ s = f ( t , x p + x s , u p + u s ) {displaystyle {dot {x}}_{s}=f(t,x_{p}+x_{s},u_{p}+u_{s})} − f p ( t , x p , u p ) , x s ( 0 ) = x 0 − x p , 0 , {displaystyle -f_{p}(t,x_{p},u_{p}),x_{s}(0)=x_{0}-x_{p,0},}     (4) x ˙ ( t ) = ( A + Δ A ( t ) ) x ( t ) + A d x ( t − T ) + B r ( t ) {displaystyle {dot {x}}(t)=left(A+Delta A(t) ight)x(t)+A_{d}x(t-T)+Br(t)}     (5) x ˙ p ( t ) = A x p ( t ) + A d x p ( t − T ) + B r ( t ) {displaystyle {dot {x}}_{p}(t)=Ax_{p}(t)+A_{d}x_{p}(t-T)+Br(t)}     (6) x ˙ s ( t ) = ( A + Δ A ( t ) ) x s ( t ) + A d x s ( t − T ) + Δ A ( t ) x p ( t ) {displaystyle {dot {x}}_{s}(t)=left(A+Delta A(t) ight)x_{s}(t)+A_{d}x_{s}(t-T)+Delta A(t)x_{p}(t)}     (7) x ˙ = A x + b u + ϕ ( y ) + d , x ( 0 ) = x 0 {displaystyle {dot {x}}=Ax+bu+phi (y)+d,x(0)=x_{0}} y = c T x {displaystyle y=c^{T}x}     (8) x ˙ p = A x p + b u p + ϕ ( r ) + d , x p ( 0 ) = x 0 {displaystyle {dot {x}}_{p}=Ax_{p}+bu_{p}+phi (r)+d,x_{p}(0)=x_{0}} y p = c T x p {displaystyle y_{p}=c^{T}x_{p}}     (9) x ˙ s = A x s + b u s + ϕ ( c T x p + c T x s ) − ϕ ( r ) , x s ( 0 ) = 0 {displaystyle {dot {x}}_{s}=Ax_{s}+bu_{s}+phi left(c^{T}x_{p}+c^{T}x_{s} ight)-phi (r),x_{s}(0)=0} y s = c T x s {displaystyle y_{s}=c^{T}x_{s}}     (10) Additive state decomposition occurs when a system is decomposed into two or more subsystems with the same dimension as that of the original system. A commonly used decomposition in the control field is to decompose a system into two or more lower-order subsystems, called lower-order subsystem decomposition here. In contrast, additive state decomposition is to decompose a system into two or more subsystems with the same dimension as that of the original system. Taking a system P for example, it is decomposed into two subsystems: Pp and Ps, where dim(Pp) = np and dim(Ps) = ns, respectively. The lower-order subsystem decomposition satisfies