3Asymptotic StabilityA linear system is asymptotically stable if all its trajectories converge to the origin i.e. for any initial state x(k0), x(k) → 0 as k → ∞. Also called internal stability.
4Asymptotic Stability Condition Theorem 8.1 A discrete-time linear system is asymptotically (Schur) stable if and only if all the eigenvalues of its state matrix are inside the unit circle.
5BIBO Stability• Poles of z-transfer function inside the unit circle. • Asymptotic stability implies BIBO stability. • For a minimal system (not in general), BIBO stability implies asymptotic stability (no pole-zero cancellation).
6Example 8.4Test the BIBO stability of the system • All poles are inside the unit circle. • The system is BIBO stable.
7Controllability and Observability The concepts of controllability and observability, introduced first by Kalman, play an important role in both theoretical and practical aspects of modern control. The conditions on controllability and observability essentially govern the existence of a solution to an optimal control problem. This seems to be the basic difference between optimal control theory and classical control theory. In the classical control theory, the design techniques are dominated by trial-and-error methods so that given a set of design specifications the designer at the outset does not know if any solution exists. Optimal control theory, on the other hand, has criteria for determining at the outset if the design solution exists for the system parameters and design objectives.
8Controllability and Observability The condition of controllability of a system is closely related to the existence of solutions of state feedback for assigning the values of the eigenvalues of the system arbitrarily. The concept of observability relates to the condition of observing or estimating the state variables from the output variables, which are generally measurable.
9ControllabilityDefinition A system is said to be controllable if for any initial state x(k0) there exists a control sequence u(k), k = k0 , ..., k0 +1, kf − 1, such that an arbitrary final state x(kf) can be reached in finite kf .
10Controllability Condition The process is said to be completely controllable if every state variable of the process can be controlled to reach a certain objective infinite time by some unconstrained control 'u(f). Intuitively, it is simple to understand that, if any one of the state variables is independent of the control u(t), there would be no way of driving this particular state variable to a desired state in finite time by means of a control effort. Therefore, this particular state is said to be uncontrollable, and, as long as there is at least one uncontrollable state, the system is said to be not completely controllable or, simply, uncontrollable.
11Uncontrollable System As a simple example of an uncontrollable system. Because the control U(t) affects only the state x1(t) the state x2(t) is uncontrollable.
12Controllability Condition Theorem 8.4 A linear time-invariant system is completely controllable iff the products, 𝑊 𝑖 𝑇 Bd, i = 1,2,..., n, are all nonzero where is the ith left eigenvector of the state matrix. Furthermore, modes for which the product is zero are uncontrollable.
13Controllability Rank Condition Theorem 8.5 A LTI system is completely controllable if and only if the n×m.n controllability matrix has rank n.
15Example 8.5Determine the controllability of the following state-equation
16Solution• Controllability matrix has rank 3: controllable. • First 3 columns of matrix linearly independent: sufficient to conclude controllability. • In general, compute more columns until n linearly independent columns are obtained.
17ObservabilityA system is said to be observable if any initial state x(k0) can be estimated from the control sequence u(k), k =k0 , k0 +1, …, kf − 1, and the measurements y(k ), k = k0 , k0+1, …, kf .
18The concept of observability Essentially, a system is completely observable if every state variable of the system affects some of the outputs. In other words, it is often desirable to obtain information on the state variables from the measurements of the outputs and the inputs.
19Unobservable SystemState diagram of a system that is not observable.
20Observability Condition Theorem 8.8 A system is observable iff Cvi is nonzero for i = 1, 2, … , n, where vi is the ith eigenvector of the state matrix. Furthermore, if the product Cvi is zero then the ith mode is unobservable.
21Observability Rank Test Theorem 8.9 A LTI system is completely observable iff the l.n × n observability matrix has rank n.
28Stabilizability & Detectability Definition 8.4 Stabilizability A system is stabilizable if all its uncontrollable modes decay to zero asymptotically. Definition 8.6 Detectability A system is detectable if all its unobservable modes decay to zero asymptotically.
29StabilizabilityA slightly weaker notion than controllability is that of Stabilizability. A system is determined to be stabilizable when all uncontrollable states have stable dynamics. Thus, even though some of the states cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behavior
30Example 8.9Determine the controllability and stabilizability of the systemSolutionThe system is in normal form and its input matrix has one zero row corresponding to its zero eigenvalue. Hence, the system is uncontrollable. However, the uncontrollable mode at the origin is asymptotically stable, and the system is therefore stabilizable.
31DetectabilityA slightly weaker notion is Detectability. A system is detectable if and only if all of its unobservable modes are stable. Thus even though not all system modes are observable, the ones that are not observable do not require stabilization.
32Example 8.12Determine the observability and detectability of the systemSolutionThe system is in normal form, and its output matrix has one zero column corresponding to its eigenvalue -2. Hence, the system is unobservable. The unobservable mode at -2, |-2| > 1, is unstable, and the system is therefore not detectable.
39Example 8.18Obtain the controllable canonical realization of the difference equation using basic principles then show how the realization can be written by inspection from the transfer function or the difference equation
41By Inspectionn = 3, n − 1=2 ⇒ orders of zero matrices •The coefficients of the last row appear in the difference equation with their signs reversed. •The same coefficients appear in the denominator of the transfer function
46Comments• State-space equations can be written by inspection from the difference equation or the transfer function. • Simulation diagram: shows how to implement the system with summers, delays and gains. • For an nth order system we need: n delay elements, two summers and at most 2 n + 1 gains. • Operations can be easily implemented using a microprocessor or DSP chip.
47Example 8.19Write the state-space equations in controllable canonical form for the following transfer functions
48Solution (a)• Same denominator and same state equation as Example • Numerator = (− z + 0 z2) output equation
51Parallel Realization• Positive feedback loop with forward transfer function z−1 (time delay ) & feedback gain pi• Physical realization of the transfer function in terms of constant gains and fixed time delays.
60Observable Form• Obtain from the controllable realization: 1- Transpose the state matrix A. 2- Transpose and interchange the input matrix B and the output matrix C. • Can be written directly from the TF
61Simulation diagram for the observable canonical realization
62Example 8.22Write the state-space equations in observable canonical form for the transfer function of Example 8.19(b).
63SolutionThe transfer function =constant + a transfer function with numerator order < denominator order (Example 7.15(b)).
64DualityThe concepts of controllability (stabilizability) and observability (detectability) are often referred to as duals. This term is justified by the following theorem. Theorem 8.12: The system (A, B) is controllable (stabilizable) if and only if (AT,BT) is observable (detectable). The system (A, C) is observable (detectable) if and only if (AT,CT) is controllable (stabilizable).
65Example 8.23Show that the reducible transfer function has a controllable but unobservable realization and an observable but uncontrollable realization.
67SolutionTransposing the state, output, and input matrices and interchanging the input and output matrices gives the observable realization By duality, the realization is observable but has one uncontrollable mode.