Problem I: Solution of State Equation (Chapters 4-7)
Search for the function of x=x
( )
t,u , i.e. the time-dependent characteristics of the state parameters, depending on the control vector u.We have to differentiate between three cases:
1. Non-linear State Equations
In this case, the solution of the state equations is either impossible or only approximately determinable.
2. Linear time-variant System
There is mostly only a formal approach to the solution. The difficulties are very similar to those occurring in the case 1.
3. Linear time-invariant State Equations
Only in this case we have an explicit solution. But difficulties also occur with systems of higher order.
Because the explicit solution of the state equations is only seldom possible, research on the system has to be done in another way and, thus, we have to concentrate on the following particular questions:
Problem II: Stability (Chapter 10)
A technical system is neither allowed to run away nor to explode. The state vector must therefore be finite. In this context, one deals with the question for which parameter values there can be “stability” in a broad sense of the definition :
const
→
x for t→ ∞
alternatively, when a system becomes unstable , i.e.
→ ∞
x for t→ ∞.
The question of stability must also be answered without solving the equation. We might face difficulties by doing this with non-linear and time variant systems whereas it is easy to make a stability statement for linear systems.
Problem III: Controllability (not be covered in the following)
The user must be interested to design a simple control of the system. In this context, the question emerges whether a system is controllable or not, i.e. whether the control value u can be chosen in such a way that the system can be transferred from an arbitrary state
( )
t0x into a target state x
( )
t1 . This problem is solvable for linear systems, whereas we still have great difficulties with non-linear systems.Definition 2.6:
A system of n-th order is completely controllable, if for any initial condition x
( )
0 =x0 and anystate x1, there can be assigned an input function u( )
t defined at a finite point in time1 0
t > and within the time interval
[ ]
0,t1 , such that the solution trajectory with t=t1,which started in x0, satisfies the value x1.Kalman’s criteria (Kern (2002)) can be provided in order to control this characteristic.
Problem IV: Optimisation (not be covered in the following)
The problem of controlling a system from an initial state to a given final state can usually be solved through different control programs. Here, the control u should be chosen in such a way that we reach a process which is as cheap and as fast as possible. The outcomes are optimality criteria on which the optimal control should be based upon.
Problem V: Control (s. lecture: Automatic Control)
There exist two ways in which to determine the control method:
• u as a function of time: u
( )
t• u as a function of state: u x
( )
The first case refers to control in narrow sense whereas in the second case we speak of feedback, and u x
( )
will be produced by a controller. The determination of a controller which is as simple as possible and optimal in certain sense is the main problem of automatic control.Problem VI: Simulation (Chapter 4 and 7)
A simulation (from the Latin term simulatio = feint) is an imitation of a real system. The act of simulating is not based on the analysis of the real system but alternatively a model of the system. In literature, the term simulation (in narrow sense) refers to the solution (mostly numerical) and interpretation of system equations.
Problem VII: Identification (Chapter 10)
The theoretical approach of system analysis (deductive modelling; deductive = inference from the general to the special) is mostly insufficient because the system parameters are either difficult to determine or completely unknown. In these cases it is necessary either to experimentally identify the whole structure or the parameters of the mathematical model of the analysed system.
3 Differential-Algebraic-Equation-Systems and Multiport Method
3.1 Differential-Algebraic-Equation-Systems (DAE-Systems)
Hitherto, system equations were always explicitly given, which means that the rate of change x of the state vector could be calculated by a clear calculation rule f consisting of the state x and the state quantity u.
In many applications, system equations according to (2.3) are only existent in implicit form:
(
, ( ), ( ),)
i t t t =
F x x u 0 (3.1)
In this case, x cannot be calculated by a simple analysis of the system function but (3.1) must be solved for x
DAE- systems represent an important special case: Here, the state vector x is composed of two partial vectors xa and xd, ref. Section 3.2.2:
where xd refers to the vectors of the variables, whose derivatives are also part of the system equation.
The state vector xa summarizes all state quantities, whose derivatives do not appear.
A special case which often appears is the following system:
d d d a
These special forms of differential-algebraic equations are also referred to as Hessenberg form.
If f (x x u)a d, a, =0 is solvable for xa, then xa can be inserted and transferred into an ODE-system (ODE = Ordinary Differential Equation)
Frequently this is exactly not the case: It could be, for example, that fa does not depend on xa. In this case xa can of course not be eliminated by fa. Here, xa can only be eliminated by differentiating (3.3) once or several times in respect to time.
Example 3.1: Non-linear Simple Pendulum as DAE-System
A mass point m, which is only movable on the xy-plane, is suspended without friction and rotates at point 0 with a massless beam with fixed length l.
The distance between the mass point m to point 0 will be x in the horizontal and y in the vertical direction.
Fig. 3.1: Non-linear simple pendulum.
The principal of linear momentum in x− and y− direction is as follows:
With the auxiliary quantity
x Sy
S
mx my
λ = = − (3.5)
one obtains the following equation of motion:
x
One obtains a differential equation system of first order through substitution of :
One then adds the kinematic constraint
2 2 2
0
x +y − =l (3.8)
and subsequently the DAE results in:
2 2 2 On separation of the state quantities one yields
[ ]
The system equation (3.9) must of course be transferred into an ordinary differential equation system (ODE). This can be achieved by differentiating and converting the equation (3.9) in respect to time in an appropriate way. By differentiating, e.g. the last equation in (3.9) in respect to time and by substituting it in the first two equations, we obtain:
0= x vx+ y vy. (3.11)
Further differentiation of this equation yields:
2 2 2 2
By differentiating a third time with respect to time, one further obtains
2
and therefore the ODE
2
By differentiating the DAE (3.9) three times with respect to time, one has transferred a DAE to an ODE (3.14).
The number of differentiations which are needed to transfer a DAE into its proper ODE is also called the Index of the DAE.
Definition 3.1:
The Index of the DAE is the number of differentiations that are needed to transfer a DAE into an ODE.
Annotation 3.1:
1. The Index describes so-to-say the “distance” between a DAE and its illustration as ODE.
The index of the mathematical pendulum therefore is 3.
2. An ODE has therefore the index 0.
3. DAE’s with an index higher than 1 are also referred to as DAE’s with a higher index.
4. The index of a DAE can change along the solution (local index).
5. The obstacles which occur during the numerical solution increase according to the level of the index.
The numerical approach, especially of DAE with higher index is often very difficult and complex. Thus, one should attempt to reduce the index of the system before solving it (see Chapter 8). The method shown is also referred to as descriptor form. However, we could also use equation (3.12) to calculate λ
2 2 2
1 (vx vx g y) λ =l + −
and, therefore, to eliminate λ from the remaining equations.
In this case we obtain the ODE:
( )
Furthermore, in this illustration the state quantities x and y are not dependent from one another and can (apart from their algebraic sign) autonomously be calculated
2 2
y= ± l −x
This state space representation can therefore also be called non-minimal.
Annotation 3.2:
This example seems to be more complex when illustrated in the descriptor form or even in the non-minimal system form than when illustrated in the method of minimal coordinates (2.1) and (2.2). Nevertheless, these special state space representations are frequently picked out because
• the representation of complex systems in minimal coordinates is sometimes very laborious
• the equations in minimal coordinates are sometimes very complex and error-prone.
A great disadvantage of non-minimal illustrations is that the auxiliary condition (in the mathematical pendulum for e.g. x2+ y2 − =l2 0) is only considered in the differentiated form. The initial values of the (non-minimal) state equations have necessarily been chosen is such a way that they satisfy the kinematic auxiliary condition. Furthermore, during the numerical solution of non-minimal differential equations the auxiliary conditions normally drift away. This makes a stability procedure necessary, which corrects the state quantities in the course of time so that the algebraic auxiliary conditions are again exactly satisfied.
The ability to efficiently and accurately solve this kind of equation is a requirement for the application of the methods described in the following section.