1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems

1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems Switching Signals General Form Switching Path Time-driven Switching Law Event-driven Switching Law Pure-state/output-feedback Switching Law

1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems

Different Relation Between to Switching Signals 1Time Transition 2Radial Transition 3Transition Invariant 4Sub Path 5Concatenation

Time Transition

The Switching Signal is said to be time-invariant over [t 0,t 1 ) if it is time invariant at each state in R n [t0,t1) The Switching Signal is said to be completely time-invariant over any time interval

Radial Transition

Example Any switching path is radial transition Any time driven switching law is radial transition Event driven switching signal may not be radial transition Is not radial transition in general

Transition Invariant Transition Invariant = Time Transition + Radial Transition

Sub Path Consider θ 1 : [ t 1, s 1 )  M θ 2 : [ t 2, s 2 )  M θ 2 is said to be a sub-path of θ 1 on [ t 2, s 2 ) denoted by θ 2 = θ 1[t2,s2) if [ t 1, s 1 ) [ t 2, s 2 )

Concatenation Consider θ 1 : [ t 1, s 1 )  M θ 2 : [ t 2, s 2 )  M The concatenation of θ 1 with θ 2 on [ t 2, s 2 ) denoted by θ2θ2 θ1θ1 Is a new switching path defined on [ t 1, s 1 +s 2 -t 2 ) with

Concatenation Concatenation of two switching signals via a given region suppose σ 1 on [ t 1, s 1 ) and σ 2 on [ t 2, s 2 ) are two switching signals Then we define the concatenation of σ 1 with σ 2 at ( x 0,t 1,[ t 2, s 2 )) via Ω If It generates a unique switching path θ [t1,s1+s2-t2) with s 1 =min { t ≥ t 1 : x(t) є Ω }

Concatenation

1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems

Well-defined ness A switching signal is Well-defined on [t1,t2) if 1- It is defined in [t1,t2) 2- For all t in [t1,t2) right and left limθ(s) at t exist (for t1 just right limit) 3- Finite jump instants in any finite time sub-interval of [t1,t2) (no Zeno phenomena)

Well-defined ness

Well-posed ness A switched system is said to be well-posed at x 0 over [t 0,t 1 ) w.r.t. switching signal σ, if for any given piecewise continuous and locally integrable input u, 1- The switching signal σ is well-defined at x 0 over [t 0,t 1 ) w.r.t. switching system 2- The switched system admits a unique solution Via the switching signal x0 over [t 0,t 1 )

Well-posed ness A switched system is said to be well-posed over [t 0,t 1 ) w.r.t. switching signal σ, if for any x 0 it is well-posed A switched system is said to be (completely) well-posed w.r.t. switching signal σ if for any x 0 and any time interval

1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems

Any jump instant tє(t 0,t 1 ) is said to be a switching time. For continuous-time switching path a switching time t must satisfy Note that a switching time must be a discontinuous time.

For a well-defined path θ, let s 1,s 2,…,s l be the ordered switching time in [t 0,t 1 ) with or simply(when the interval time [t 0,t 1 ) is clear) is said to be the switching time sequence over [t 0,t 1 ) of θ and is defined by

For a well-defined path θ, let s 1,s 2,…,s l be the ordered switching time in [t 0,t 1 ) with or simply(when the interval time [t 0,t 1 ) is clear) is said to be the switching index sequence of θ over [t 0,t 1 ) and is defined by

For a well-defined path θ, let s 1,s 2,…,s l be the ordered switching time in [t 0,t 1 ) with or simply(when the interval time [t 0,t 1 ) is clear) Is said to be the switching sequence of θ over [t 0,t 1 ) and is defined by

In switching sequence of θ over [t 0,t 1 )

For a well-defined path θ, let s 1,s 2,…,s l be the ordered switching time in [t 0,t 1 ) with or simply(when the interval time [t 0,t 1 ) is clear) Is said to be the switching duration sequence of θ over [t 0,t 1 ) and is defined by Let

If the switching path θ generated by a switching signal at x 0 over [t 0,t 1 ) then the switching sequence is in the form

Example: Consider the following switched linear system The switch signal is event driven so it is initial state dependent switched sequence is infinite

Dwell time for any two consecutive switching timeIf is said to be dwell time Any switching signal with positive Dwell time Well-defined Any switching signal with positive Dwell time Well-defined Is completely well-defined over [0,∞) but it does not permit a dwell time

1 Classification of Switching Signals 2 Operation on Switching Signals 3 Well-defined ness and Well-posed ness 4 Switching Sequences 5 Solution of Switched Linear Systems

Initialized at x(t 0 )=x 0 Suppose the switching signal is well-defined and its switching sequence is

Suppose the i 0 th subsystem is active during [t 0,t 1 )

During period [t 1,t 2 ), the i 1 th subsystem is active

General response of system

Let we define The state transition matrix is given by General response of system

Conclusion 1- For a switched linear system, if the switching signal is well-defined and the input is globally integrable, then the system always permits a unique solution for the forward time space. 2- The solution is usually not continuously differentiable at the switching instants, even if the input is smooth. 3- The state transition matrix is a multiple multiplication of matrix function of the form e At. Accordingly, properties of functions in this form play an important role in the analysis of switched linear systems.

For linear discrete systems The state transition matrix is: And so

Conclusion 1- For a switched linear system, the system permits a unique solution for the forward time space. Hence, any discrete-time switched system is well- posed. 2- The state transition matrix is a multiple multiplication or matrices. Accordingly, properties of matrix multiplication play an important role in analyzing the switched system.

Is the set of inputs which are piece wise continuous over [t 0, ∞ ) Is a well-defined switching path The set of state attainable from x(t0)=x0 via the switching path θ is If the switching path is transition invariant by proposition 1.6

And by proposition 1.7 By applying all switching path we have By choosing x 0 from a set Ω

It can be seen that sets are independent of t 0 and are radially linear. As a result, if Ω is a neighborhood of the origin, then

Some fundamental features of linear switched systems are: i)If the property holds at some t0, then it also holds at any other time ii)If such a property can be achievable via a well defined switching signal, then it can also be achievable via a well defined switching signal that is transition invariant

Let Some other features of linear switched systems are: