Unidad Orgánica: Dirección General de Defensa Pública y Acceso a la Justicia Dirección de Conciliación Extrajudicial y Mecanismos Alternativos de Solución de Conflictos
I SEM III TRIM IV TRIM ANUAL I SEM III TRIM IV TRIM ANUALIV TRIM
In control theory methods and approaches are developed to influence a system such that it behaves in a desirable manner. The simplest way to influence the system’s behaviour is to apply a signal at the input of the system such that the output behaves in a certain prespecified way. This can be done either without measured information about the output, resulting in open-loop or feedforward control, or with measurement of the system output leading to closed-loop or feedback control.
Usually it is not trivial to formulate precisely what we want with the control system. Or the description of the requirements is such that it is difficult to use it as a basis for the control design. Usually in control theory three desirable aspects of the control system are pursued:
1. stability, 2. performance, 3. robustness.
These aspects are further elaborated in the sequel of this section.
Stability
The first requirement of any system is stability, i.e. the signals must remain bounded at all times. Stability can be defined in many ways. We use the following definitions for stability and asymptotic stability [7]. Consider the discrete time state space equation (possibly nonlinear and time-varying)
x(t + 1) = f (x(t), t) (2.3)
and let x0(t), x(t) be solutions of (2.3) when the initial conditions are x0(t0), x(t0) respec- tively.
Definition 2.2.12 Stability. The solution x0(t) is stable if for a given ε > 0, there
exists a δ(ε, t0) > 0 such that all solutions with x(t0)− x0(t0) < δ are such that x(t) −
x0(t) < ε for all t ≥ t0.
Definition 2.2.13 Asymptotic stability. The solution x0(t) is asymptotically stable
if it is stable and if x(t)−x0(t) → 0 when t → ∞ provided that x(t0)−x0(t0) is small
enough.
Note that stability is defined for a particular solution and not for the system. Various methods exist to test stability points of a system. An important method that plays a role in this thesis is Lyapunov’s method to assess stability .
Lyapunov’s second method is originally developed to determine stability of nonlinear dynamic systems. It is developed for differential equations but similar results can be obtained for difference equations. This version is given here [7].
Definition 2.2.14 A functional V (x) is a Lyapunov function for the system
x(t + 1) = f (x(t)), f (0) = 0
if
1. V (x) is continuous in x and V (0) = 0. 2. V (x) > 0 for x= 0.
3. ∆V (x) := V (f (x))− V (x) < 0 for x = 0.
The last condition implies that the dynamics of the system are such that the value of the function V is decreasing for each new time instant until it reaches the origin. It thus seems reasonable that the existence of such a function ensures asymptotic stability of the system around the origin. This is made more precise in the next theorem.
Theorem 2.2.15 Lyapunov stability theorem. The solution x(t) = 0 is asymptotically
stable if there exists a Lyapunov function to the system x(t + 1) = f (x(t)). Further if
0 < ϕ( x ) < V (x)
where ϕ( x ) → ∞ as x → ∞, then the solution is asymptotically stable for all initial conditions.
Hence, if we are able to find a function with this properties that is sufficient for stability of the system. Usually it is a difficult task for general nonlinear systems to find such a function but for model predictive control it can be constructed in an elegant way as will be discussed in the sequel of this thesis.
Also for linear systems of the form x(t + 1) = Ax(t) a Lyapunov function can easily be found. The positive definite quadratic function is defined by
V (x) = xTXx
with a specific matrix X that can be calculated. The derivative of this function along solutions of the linear autonomous system is given by
∆V (x) = xT[ATXA− X]x
The matrix
ATXA− X = −Q
should be negative definite to guarantee that the origin is an asymptotic stable equilib- rium point of the linear system. For a given positive definite matrix Q, a matrix X can be calculated such that V (x) is a Lyapunov function.
Performance
The second requirement is that the controlled system performs better than the uncon- trolled system. The definition of the desired behaviour of the control system is not always obvious. It is a mix of, usually subjective and sometimes conflicting, goals as safety, speed, accuracy, reliability, economic profit etc. Performance has two branches:
• The quality with which the effect of disturbances can be suppressed. This is denoted
with the regulator behaviour of the control system.
• The quality with which the control system is able to follow a predeterminded signal r. This is denoted with servo behaviour of the control system.
Performance can be measured in several ways. A way to measure performance for linear control systems that has become popular in the control community since the start of the development of robust control theory [172][35] is to use a system norm or a weighted ver- sion thereof to asses the closed loop transfer from one signal to another. For this purpose the generalized plant is a powerful concept in the design of feedback controllers. Two important performance measures in this class are the H2 and H∞ performance measures. Let the transfer from the unmeasured disturbance w to the performance output z in fig- ure 2.1 be given by Gwz. Then a performance measure that is relevant for disturbance attenuation can be defined by
Gwz 2
which is theH2 norm. A deterministic interpretation of this is that it signifies the 2-norm of the pulse response of the system. A probabilistic interpretation is that is equal to the root mean square (rms) value of the output of the system G in response to white
noise excitation of w. This performance measure can be minimized to obtain a feedback controller that is H2-optimal. This problem can be solved with e.g. state space techniques [35].
If the focus is on robustness properties rather than disturbance attenuation the following performance measure can be defined. Let the uncertainty in the model be represented by an uncertain dynamical transfer function ∆ that has an ∞-norm bounded by γ and connects the output z with in input w. Now if we can find a controller that connects the measured output y and the controlled input u such that the infinity norm transfer from w to z is smaller than 1/γ then the controlled system is robust for this class of uncertainties according to the small gain theorem [175]. The performance measure applied here can be defined by
Gzw ∞
This norm has many interpretations. First, it is the worst-case steady-state gain for sinusoidal inputs at any frequency. Furthermore, it is the induced 2-norm in the time- domain [28]
G(z) ∞ = maxw=0 w z 2
2 = maxw2=1 z 2
The H∞-norm is also equal to the induced power (rms) norm and also has an interpreta- tion as an induced norm in terms of the expected values of stochastic signals. Especially in robust control this performance measure is widely used [175].
Also a combination of several performance specifications can be made such that a so-called multi-objective performance measure is used [138]. This is a way to deal with the several conflicting specifications that a control system sometimes has to satisfy.
All these interpretations and the availability of good numerical tools to synthesize feedback controllers that are optimal in terms of these norms have made them useful in engineering applications.
Robustness
A third requirement on the control system is robustness, i.e. the capability of the control system to deal with changes of the system that can occur e.g. due to wear, ageing or a mismatch between the system and the model on the basis of which the controller is designed.