§1—2 State-Variable Description The concept of state The input-output description of a system is applicable only when the system is initially relaxed. If a system is not relaxed at time t0, then the equation does not hold. For example, whose solution is Obviously, if the initial condition yc(0) is unknown, we cannot determine the output uniquely even the input signal u is known.
Example. Consider the following system described by an nth order differential equation: If we choose as a set of states at time t0, then together with the input u, the output y(t) can be determined uniquely.
Definition: State of a system at t0 is the amount of information at t0 which together with determines uniquely the behavior of the system for all t≥t0. The state at time t0 is the laconic but sufficient representation of the passed activity of the system, which with , can determine the behavior of the system sufficiently for all t≥t0. The set of the initial conditions of y (t0), y(t0), …… , y (n1)(t0) in the above example is such an information. The information updated with t≥t0 is said to be state variables and can be written as
Example: Consider the system which is relaxed at t0 It is not sufficient if we only choose y(t0) as the information. However, if we choose y(t0), , as the information, is redundant since it can be expressed as the linear combination of other terms. Hence, we can choose the sate at time t0 as follows The corresponding state variables are
Example 1-5 A unit time delay system is such system that the output y(t) is equal to the input u(t-1) for all t. For such a system, in order to determine uniquely by input , one should know the information over [t0-1, t0]. y t0 t01 u Hence, the information can be regard as the state of the system at t0. Unlike the Example1-4, in this example the state at t0 is composed by infinite numbers.
Example 1-4 (Un-uniqueness of the state variables）Consider system + + u y C Figure1-5 where, R=3, L=1H, C=0.5F. The transfer function from u to y is Hence the impulse response of the network is
If the network is relaxed at t0, the output is given by If the network is not relaxed at t0, the output is
(1) The effect on y(t) at due to the input . If x1(t0) and x2 (t0) are known, the response can be determined uniquely.
The information at t0 can be expressed as x1(t0) and x2(t0) The information at t0 can be expressed as x1(t0) and x2(t0). Therefore, the state variables
(2) y(t): Voltage across the capacitor y(t): Current passing through the inductor Hence, the derivative at time t0 is
Note that i.e. Therefore, we can choose and as the state at t0.
Dynamical equations Linear dynamical equations The set of equations that describes the unique relation among the input, output, and state is called a dynamical equation. state equation (1－33a) (1－33b ) output equation
If f, g are linear functions of x and u, then Equation(1-33) is a linear dynamical equation: state equation （1-34a） output equation （1-34b） y x u D(t) ∫ C(t) B(t) A(t) where A(t), B(t), C(t) and D(t) are n×n, n×p, q×n and q×p matrices, respectively.
Example: Consider the system Then,
The dynamical equation consists of state equation and output equation. Remark: In section 1, the linearity is defined under the condition that the system is relaxed, i.e. and satisfies With state space, the mapping becomes Then, the definition of linearity becomes
x Linear time invariant dynamical equations If the matrices A, B, C and D do not change with time, then the linear dynamical equation is called linear time invariant dynamical equation. The form of n-dimensional linear time invariant dynamical equation is as follows (1-35) where A, B, C and D are n×n, n×p, q×n and q×p real constant matrices. y x u D ∫ C B A
For time invariant dynamical system, it is easy to verify that Therefore, without loss of general, we can choose the initial time as zero, and the time interval becomes .
The transfer function of time invariant system Taking the Laplace transform on the time invariant dynamical equation, we have (1-40a) (1-40b) From the above equation, we have If x0=0，then
where is called the transfer function matrix of the dynamical equation. Write as Note that ——matrix exponent
Some Important Equations 1. Cayley-Hamilton Theorem Theorem (Cayley-Hamilton): Let the characteristic polynomial of A be then
Proof: For any n×n matrix A, can be written as where R0, R1, R2…Rn-1 are n×n constant matrices. Hence,
Equating the same power of s on both sides of the above equation yields Finally, equating s0 of both sides of the equation yields That is, Q.E.D
2. adj(sIA) and eAt (1) adj(sIA) where
(2) eAt From (1-45), we have Taking the Laplace transform and letting eAt can be expressed as
3. Properties of
4. Algorithms for computing 1) 2) 3) Jordan canonical form
where J is a Jordan canonical form matrix.
Note that
From the theorem we can get the above result. In particular, if Then
Example: Compute eAt of