curs06-07- stability, observability, controllability (1)

Adrian Burlacu, Associate Professor at Dept. of Automatic Control and Applied Informatics, Technical University of Iasi, Romania

System-Theory for Robotics

Systems and Control Master Program 2014 -2015


Part I System Analysis State-space representation of continuous- and discrete-

time systems. Transformations of system models

Linearization of nonlinear mathematical models

Solving continuous- and discrete-time state-space equations. Transfer matrix

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Lyapunov stability analysis of continuous- and discrete-time systems. Stability of a discrete-time system obtained by discretizing a continuous-time system

Controllability and observability of linear dynamical systems. Effects of the discretization

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Stability of linear time-invariant systems

The total response of a linear system is the sum of the natural response and the forced response. Natural response describes the way the system dissipates or acquires energy. The form or nature of this response is dependent only on the system, not the input. On the other hand, the form or nature of the forced response is dependent on the input.

For a control system to be useful, the natural response must (1) eventually approach zero, thus leaving only the forced response, or (2) oscillate. In some systems, however, the natural response grows without bound rather than diminish to zero or oscillate. Eventually, the natural response is so much greater than the forced response that the system is no longer controlled. This condition, called instability, could lead to self-destruction of the physical device if limit stops are not part of the design.


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Using the natural response: A linear, time-invariant system is stable if the natural response approaches zero as time

approaches infinity. A linear, time-invariant system is unstable if the natural response grows without bound as

time approaches infinity. A linear, time-invariant system is marginally stable if the natural response neither decays

nor grows but remains constant or oscillates as time approaches infinity.

Using the total response (BIBO):

A system is stable if every bounded input yields a bounded output. A system is unstable if any bounded input yields an unbounded output


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The poles of a transfer function are - the values of the Laplace transform variable, s, that cause the transfer function to become infinite or - any roots of the denominator of the transfer function that are common to roots of the numerator. The zeros of a transfer function are -the values of the Laplace transform variable, s, that cause the transfer function to become zero, or - any roots of the numerator of the transfer function that are common to roots of the denominator.



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Poles in the left half-plane generate either pure exponential decay or damped sinusoidal natural responses. These natural responses decay to zero as time approaches infinity. Thus, if the closed-loop system poles are in the left half of the plane and hence have a negative real part, the system is stable. That is, stable systems have closed-loop transfer functions with poles only in the left half-plane.

Poles in the right half-plane yield either pure exponentially increasing or exponentially increasing sinusoidal natural responses. These natural responses approach infinity as time approaches infinity. Thus, if the closed-loop system poles are in the right half of the s-plane and hence have a positive real part, the system is unstable.


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Liapunov Stability Analysis


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A choice for the Liapunov function


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Lyapunov stability analysis of continuous- and discrete-time systems. Stability of a discrete-time system obtained by discretizing a continuous-time system

Controllability and observability of linear dynamical systems. Effects of the discretization

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If an input to a system can be found that takes every state variable from a desired initial state to a desired final state, the system is said to be controllable; otherwise, the system is uncontrollable.

The concepts of controllability and observability were introduced by Kalman. They play an important role in the design of control systems in state space. In fact, the conditions of controllability and observability may govern the existence of a complete solution to the control system design problem. The solution to this problem may not exist if the system considered is not controllable. Although most physical systems are controllable and observable, corresponding mathematical models may not possess the property of controllability and observability. Then it is necessary to know the conditions under which a system is controllable and observable


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Controllability

If an input to a system can be found that takes every state variable from a desired initial state to a desired final state, the system is said to be controllable; otherwise, the system is uncontrollable.

Complete State Controllability of Continuous-Time Systems

The system is said to be state controllable at t=t0 if it is possible to construct an unconstrained control signal that will transfer an initial state to any final state in a finite time interval. If every state is controllable, then the system is said to be completely state controllable.


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Controllability


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Controllability

If the system is completely state controllable, then, given any initial state x(0)

This requires that the rank of is n

The result just obtained can be extended to the case where the control vector u is r-dimensional. If the system is described by ,, the matrix

is commonly called the controllability matrix.


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Example

the system is not completely state controllable

the system is completely state controllable


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Controllability

Condition for Complete State Controllability in the s Plane

The condition for complete state controllability can be stated in terms of transfer functions or transfer matrices. It can be proved that a necessary and sufficient condition for complete state controllability is that no cancellation occur in the transfer function or transfer matrix. If cancellation occurs, the system cannot be controlled in the direction of the canceled mode.


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Controllability

Output Controllability

In the practical design of a control system, we may want to control the output rather than the state of the system. Complete state controllability is neither necessary nor sufficient for controlling the output of the system. For this reason, it is desirable to define separately complete output controllability.

The system is said to be completely output controllable if it is possible to construct an unconstrained control vector u(t) that will transfer any given initial output y(t0) to any final output y(t1) in a finite time interval t0 < t < t1 .

is completely output controllable if and only if the mx (n+1)r matrix is of rank m


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Example


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Example


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Observability

The ability to control all of the state variables is a requirement for the design of a controller. State-variable feedback gains cannot be designed if any state variable is uncontrollable. Uncontrollability can be viewed best with diagonalized systems. The signal-flow graph showed clearly that the uncontrollable state variable was not connected to the control signal of the system. A similar concept governs our ability to create a design for an observer. Specifically, we are using the output of a system to deduce the state variables. If any state variable has no effect upon the output, then we cannot evaluate this state variable by observing the output.


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Observability

The system is said to be completely observable if every state x(t0) can be determined from the observation of y(t) over a finite time interval, The system is, therefore, completely observable if every transition of the state eventually affects every element of the output vector. The concept of observability is useful in solving the problem of reconstructing unmeasurable state variables from measurable variables in the minimum possible length of time.

For linear, time-invariant systems, without loss of generality, we can assume that t0=0.


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Observability

Since the matrices A, B, C, and D are known and u(t) is also known, the last two terms on the right-hand side of this last equation are known quantities. Therefore, they may be subtracted from the observed value of y(t). Hence, for investigating a necessary and sufficient condition for complete observability, it suffices to consider the system described by


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Observability

If the system is completely observable the rank of the nmxn matrix is n


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Observability

Example

the system is completely state controllable

the system is completely output controllable 1 01 1

=

C

CA

the system is completely observable

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curs06-07- stability, observability, controllability (1)

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