Linear Dynamical Systems - University of Minnesota

1 Chapter 1. Linear Dynamical Systems system classifications and descriptions A system is a collection of elements that interacts with its environment via a set of input variables u and output variables y. Systems can be classified in di erent ways. Continuous time versus Discrete time A continuous time system evolves with time indices t , whereas a discrete time system evolves with time indices t Z = {.. , 3, 2, 1, 0, 1, 2, ..}. Usually, the symbol k instead of t is used to denote discrete time indices. Examples of continuous time Systems are physical Systems such as pendulums, servomechanisms, etc. An examples of a discrete time system is mutual funds whose valuation is done once a day. An important class of Systems are sampled data Systems . The use of modern digital computers in data processing and computations means that data from a continuous time system is sampled.

2 At regular time intervals. Thus, a continuous time system appears as a discrete time system to the controller (computer) due to the sampling of the system outputs. If inter-sample behaviors are essential, a sampled data system should be analyzed as a continuous time system . Static versus dynamic The system is a static system if its output depends only on its present input. (there exists). a function f (u, t) such that for all t T , y(t) = f (u(t), t). ( ). An example of a static system is a simple light switch in which the switch position determines if the light is on or not. A static time invariant system is one with y(t) = f (u(t)) for all t. To determine the output of a static system at any time t, the input value only at t is needed. Again, a light switch is a static time invariant system . A static time-varying system is one with time-varying parameters such as external disturbance signals.

3 An example is a flow control valve ( ), whose output flow rate Q is given as . 2Ps (t). Q = uA ( ).. 3. 4 c Perry P. A uA. Q. Figure : Flow control valve: a static time invariant system Fs x P. Q. A uA. Figure : Flow control valve: a dynamic time invariant system where u [0, 1] is the input, A is orifice area, Ps is flow pressure and is fluid density. Here Ps is a time-varying parameter that a ects the static output Q. In contrast, a (causal) dynamic system requires past input to determine the system output. to determine y(t) one needs to know u( ), ( , t]. An example of a dynamic time invariant system is the flow control valve shown in Fig. The fluid pressure Ps is constant. However, the flow rate history is a function of the force F (t). acting on the valve. It is necessary to know the time history of the forcing function F (t) in order to determine the flow rate at any time.)

4 The position x(t) of the valve is governed by the di erential equation = F (t) bx kx x ( ). where k is the spring constant and b is the damping factor. For a circular pipe of radius R, the flow rate is then given by: . x2 2Ps (t). Q = 2A ( ). R . University of Minnesota ME 8281: Advanced Control Systems Design, 2001-2012 5. Orbiting pendulum Earth Figure : Example of a dynamic time varying system A dynamic time varying system is shown in Fig. Here a pendulum of length l and mass m orbits around the earth in an elliptical path. The gravitational acceleration g on the pendulum is a function of distance from the center of the earth, which in turn is a function of time, r(t). GMearth g=. r2 (t). where G is the universal gravitational constant. Hence, the frequency of oscillations executed by the pendulum is also a dynamic function of time.

5 G(t). (t) =. l As another example, consider the bank account as the system . Let the streams of deposits and withdrawals be the inputs to the bank account and the balance be the output. It is a dynamic system because knowing the deposits and withdrawals today is not enough to know the bank balance today. One needs to know all the past deposits and withdrawals. Alternatively, one can know the so called state at one time .. Time varying versus time invariant system The examples above illustrate that a system can be both static and time varying, or dynamic and time invariant, or static and time invariant, or dynamic and time varying. In some sense, if the laws of physics are considered to be fixed and time invariant, if su cient details of the system is modeled, all Systems are time invariant since any time variation (including the input function) are due to dynamics of a larger system .

6 State Determined Dynamical Systems The state of a dynamic system at time t0 , x(t0 ), is the extra piece of information needed, so that given the input trajectory u( ), t0 , one is able to determine the behavior of the system for all times t t0 . The behaviors are usually captured by defining appropriate outputs y(t). Note that information about the input before t0 is not necessary. State is not unique. Two di erent pieces of information can both be valid states of the system . What constitutes a state depends on what behaviors are of interest. Some authors require a state 6 c Perry to be a minimal piece of information. In these notes, we do not require this to be so. Example: Consider a car with input u(t) being its acceleration. Let y(t) be the position of the car. 1. If the behavior of interest is just the speed of the car, then x(t) = y(t).

7 Can be used as the state. It is qualified to be a state because given u( ), [t0 , t], the speed at t is obtained by: t v(t) = y(t). = x(t0 ) + u( )d . t0.. y(t). 2. If the behavior of interest is the position of the car, then xa (t) = 2 can be used as y(t).. the state.. y(t) + 2y(t).. 3. An alternate state might be xb (t) = . Obviously, since we can determine the y(t).. old state vector xa (t) from this alternate one xb (t), and vice versa, both are valid state vectors. This illustrates the point that state vectors are not unique. Remark 1. If y(t) is defined to be the behavior of interest, then by taking t = t0 , the definition of a state determined system implies that one can determine y(t) from the state x(t) and input u(t), at time t. there is a static output function h( , , ) so that the output y(t) is given by: y(t) = h(x(t), u(t), t).

8 H : (x, u, t) y(t) is called the output readout map. 2. The usual representation of continuous time Dynamical system is given by the form: x = f (x, u, t). y = h(x, u, t). and for discrete time system , x(k + 1) = f (x(k), u(k), k). y(k) = h(x(k), u(k), k). 3. Notice that a state determined dynamic system defines, for every pair of initial and final times, t0 and t1 , a mapping (or transformation) of the initial state x(t0 ) = x0 and input trajectory u( ), [t0 , t1 ] to the state at a time t1 , x(t1 ). In these notes, we shall use the notation: s(t1 , t0 , x0 , u( )) to denote this state transition mapping 1 . x(t1 ) = s(t1 , t0 , x0 , u( )). if the initial state at time t0 is x0 , and the input trajectory is given by u( ). 1. In class (Spring 2008), we might have used the notation s(x0 , u( ), t0 , t1 ). As long as you are consistent, either way is okay.

9 Please be careful. University of Minnesota ME 8281: Advanced Control Systems Design, 2001-2012 7. off on light bulb Figure : Toggle switch : a discrete system text3. x0 text1 x1 text2 x2. Figure : Semi-group property. Replace text1 by s(t1 , t0 , x0 , u), u( [t0 , t1 ]); text2 by s(t2 , t1 , x1 , u), u( [t1 , t2 ]), and text3 by s(t2 , t0 , x0 , u), u( [t0 , t2 ]). 4. Sometimes, one encounters the term discrete system . Precisely speaking, this means that the state variables can take on discrete values ( x {on, off, 0, 1, 2} as opposed to continuous values ( x ). If the state consists of discrete variables and continuous variables, the system is called a hybrid system . A toggle switch (Fig. ) is an example of a discrete (on, off) system . A state transition map must satisfy two important properties: State transition property For any t0 t1 , if two input signals u1 ( ) and u2 ( ) are such that u1 (t) = u2 (t) t [t0 , t1 ], then s(t1 , t0 , x0 , u1 ( )) = s(t1 , t0 , x0 , u2 ( )).)

10 If x(t0 ) = x0 , then the final state x(t1 ) depends only on past inputs (from t1 ) that occur after t0 , when the initial state is specified. Systems like this are called causal because the state does not depend on future inputs. Semi-group property(Fig. ) For all t2 t1 t0 T , for all x0 , and for all u( ), s(t2 , t1 , x(t1 ), u) = s(t2 , t1 , s(t1 , t0 , x0 , u), u) = s(t2 , t0 , x0 , u). Thus, when calculating the state at time t2 , we can first calculate the state at some interme- diate time t1 , and then utilize this result to calculate the state at time t2 in terms of x(t1 ). and u(t) for t [t1 , t2 ]. Example: Consider a system represented as: x = f (x, u, t). y = h(x, u, t). 8 c Perry with initial condition x0 and control u( ), [t0 , t1 ]. Then the state transition map is given as: t1. x(t1 ) = s(t1 , t0 , x0 , u( )) = x0 + f (x( ), u( ), ( ))d ( ).