Properties of Open-Loop Controllers Sven Laur University of Tarty 1 Basics of Open-Loop Controller Design Two most common tasks in controller design is regulation and signal tracking. Regulating controllers try to keep system stably in a fixed state. Balancing an inverted pendulum or keeping the speed a car at constant level are most obvious examples of regulation problems. All kinds of servomechanisms, such as autopilots and robotic arms, are the examples of signal tracking problems. An open-loop controller is a controller that does not use any feedback from a systems. For obvious reasons, open-loop controllers can be used only for signaltracking tasks. If we can keep a system in a desired state without measuring the system state, there is no reason to build a controller we can just hardwire the corresponding inputs into the system. As there is no feedback in open-loop controllers, we can always assume that the system output y is a set of signals or internal states that we want to control. For instance, we might want to control the position (measurable output) and speed (internal state) of a car. Any open-loop control task can be formalised as follows. Design a controller that transfers a reference signal r to an input signal u such that the output signal u tracks the reference signal as closely as possible: u[k] u[k]. For time-invariant linear systems, the design of open-loop controllers is relatively straightforward. Assume that we want to design a linear controller that tracks a reference signal with a delay l. The resulting compound system is depicted in Figure 1. Let Ĉ[z] and Ĝ[z] be the corresponding transfer functions. Then the zero-state response of the system can be expressed as ŷ zs = Ĝ[z]û = Ĝ[z]Ĉ[z]ˆr. Since we want signal tracking with a delay l, the following equation Ĝ[z]Ĉ[z] = 1 z l I (1) must hold. Consequently, we must solve the resulting linear equation and then find out a corresponding realisation of the transfer matrix Ĉ[z]. In many cases, the reference signal might contain high frequency components, which can cause high peaks in control signals and put the system under unnecessary stress. Then it makes sense to smooth the reference signal by a low-pass filter. For instance, we can use moving average ĝ[z] = 0.5+0.5z 1. In such cases, the right hand side of the equation (1) must be replaced with an appropriate transfer function, but the general approach remains valid.
r[k] Controller Ĉ[z] u[k] System Ĝ[z] y[k] Fig.1. Basic structure of an open-loop controller. 2 Bounded-Input Bounded-Output Stability Although the design of open-loop controllers is relatively simple, there are some drawbacks that limit the applicability of such controllers. First, it is often impossible to find out the exact parameters in the transfer function. Second, there are no perfect realisations of the controller matrices. Thus, we might loose control over the system even if the equality (1) is formally satisfied. A linear system is BIBO stable (bounded-input bounded-output stable) if for all bounded inputs u the resulting zero-state response y zs is also bounded: m > 0 M > 0 : ( k 0 : u[k] m) ( k 0 : y zs [k] M). In brief, if a system is initially in a zero state, then no bounded input signal can cause unbounded response. Hence, errors made in the identification phase and disturbances occurring in operation can seriously reduce the ability to track reference signals, but we still have some control over the system. Now, let us consider only SISO systems and state some important results about BIBO stability and signal tracking. The first theorem provides a theoretical tool to determine BIBO stability 1 and the second describes how a BIBO stable system reacts to most common signals. T1. A SISO system is BIBO stable iff the impulse response sequence g[ ] is absolutely summable: g[0] + g[1] + g[2] + + g[k] + <. T2. Assume that a system with impulse response g[ ] is BIBO stable and consider the asymptotic behaviour in the process k. (a) Then the output y zs [k] exited by u a approaches a ĝ[1]. (b) Then the output y zs [k] exited by a sinus signal u[k] = sin(ω 0 k) approaches to a sinus signal with the same frequency: y zs [k] ĝ[e iω0 ] sin(ω 0 k + )g[e iω0 ]). Note that the theorem T2 gives some limits on the effect of system identification errors to signal tracking. Indeed, assume that some errors are made so 1 Intuitively, T1 states that a system is BIBO stable if it does not have an external energy source and energy is continuously dissipating form the system.
1 ĝ[z] = z 0.50 ĉ[z] = z 0.55 z ĝ [z] = z 0.55 z 2 0.5z 0.0 0.4 0.8 0 20 40 60 80 Fig. 2. The effect of system identification errors on BIBO stable system. A reference signal is denoted by a red line and the actual output is denoted by a blue line. The true output is indeed ĝ [1] = 0.9 times smaller than the reference signal. that the equation 1 is not satisfied, but we can still compute or estimate the resulting transfer function ĝ [z]. Then the value ĝ [1] determines how much the output signal y zs [k] is different from a reference signal r[k], when the reference signal has remained constant long enough, see Figure 2. The second half of the theorem T2 describes how much the input signal u[k] must be varied to change the zero-state response y zs [k]. If the module of ĝ[e iω ] is small for high frequencies ω, then the open-loop controller must vary u[k] a lot to follow abrupt changes in the reference signal r[k], see Figure 3. Many systems cannot handle large input values and thus the reference signal should be filtered with a low-pass filter when the system damps high frequencies. Note that the BIBO stability depends only on the transfer function and not on how the realisation of a transfer function. Hence, it possible to characterise BIBO stability in terms of the poles of the transfer function, see Figure 4. T3.C. A continuous time-invariant linear system is BIBO stable iff every pole ĝ(s) lies in the left-half plane: ĝ(s) = R(s) < 0. T3.D. A discrete time-invariant linear system is BIBO stable iff every pole ĝ[z] lies inside the unit circle: ĝ[z] = z < 1. 3 Marginal and Asymptotic Stability As a system does not have to be initially in the zero state, the zero-input response y zi can seriously affect the overall response y. In particular, if for a certain input state x 0 the zero input response y zi grows unboundedly, then y zs [k] y zi [k] after long enough initial period and the system becomes uncontrollable. Formally, the stability of zero-input response is defined in terms of a state space description. The zero-input response of the state-space equation { x[k + 1] = Ax[k] + Bu[k] (2) y[k] = Cx[k] + Du[k]
0.0 0.5 1.0 1.5 2.0 ĝ[z] = z + 1 z 2 2 1 0 1 2 0 5 10 15 20 25 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 ĝ[z] = z 1 z 2 2 1 0 1 2 0 5 10 15 20 25 2 4 6 8 10 12 14 Fig. 3. Spectral properties of the simplest low and high pass filters. The response amplitude as a function of a period is depicted on the left and the corresponding responses to cosine waves with unit amplitude are depicted on the right. I(s) I(z) BIBO stable R(s) BIBO stable R(z) Fig.4. Connection between pole location and BIBO stability. A stable region for continuous system on the left and stable region for discrete systems on the right.
is marginally stable if every initial state x 0 R n excites a bounded response x[k] M x0. The zero-input response x[k] is asymptotically stable if every initial state x 0 excites a bounded response x[k] that approaches 0 as k. It is easy to see that every nonzero initial state x 0 causes a bounded zeroinput response y zi when a system is marginally stable. If a system is asymptotically stable then for every initial state x 0 the zero-input response y zi [k] approaches 0 as k. Oddly enough, the opposite implications do not hold, since large changes in the system state do not always manifest in the output. For instance, if the output is independent from some state variable x i, then the zero-input response y zi [k] might converge to 0 even if x i. Hence, it is advantageous consider stability of the output, as well. We say that a zero input response y zi is marginally stable if every initial state x 0 excites a bounded response y zi [k] M. The zero input response is asymptotically stable if every initial state x 0 excites a bounded response y zi [k] that approaches 0 as k. Now it is straightforward to see a close connection between BIBO stability and stability of zero input response. L1. If every state x 1 is reachable from zero state x 0 = 0 in finite number of steps, then BIBO stability implies marginal stability of the non-zero output. Proof. For the sake of contradiction assume that a system is not marginally stable. In other words, there exists an initial state x 1 that creates an unbounded zero input response y zi. As every system state is reachable from x 0 = 0, there exists a finite input signal u 0 that takes system into the state x 1 in l steps and then stays zero: u 0 [l + k] = 0. The latter yields a desired contradiction, since this bounded signal u 0 creates an unbounded output y[l + k] = y zi [l + k]. Secondly, note that marginal stability of the internal state x[k] is equivalent to the marginal stability of the output y zi [k] provided that all input states have non-zero impact on the output. L2. If every state x 1 causes non-zero output, then marginal stability of the output y zi [k] implies also the marginal stability of the internal state x[k]. Proof. Let f be the correspondence bewtween internal state x[k] and the corresponding zero input response y zi [k]. Then by linearity y zi [k] = f(x[k]) = x[k] f(e[k]) where e[k] is the unit vector in the direction of x[k]. By our assumption c := min e f(e) > 0 or otherwise there exists 2 a non-zero unit vector that is mapped to 0. Hence, y zi [k] c x[k] x[k] 1 c y zi[k] 2 Here, we formally use some basic properties from functional analysis such as compactness of finite dimensional spaces and continouty of linear transformations.
and thus the claim must hold. Whenever the assumptions of lemmata L1 and L2 are satisfied, we can establish the stability of internal state by observing the poles of the corresponding transfer function. However, if we know the state space description then we can evalute the stability directly from the eigenvalues of the matrix A. T4.C. The equation ẋ(t) = Ax(t) is asymptotically stable iff all eigenvalues λ 1,...,λ n of A satisfy R(λ i ) < 0. The equation ẋ(t) = Ax(t) is marginally stable iff all eigenvalues satisfy R(λ i ) 0 and all eigenvalues with R(λ i ) = 0 are simple roots of the minimal polynomial of A. T4.D. The equation x[k+1] = Ax[k] is asymptotically stable iff all eigenvalues λ 1,...,λ n of A satisfy λ i < 1. The equation x[k + 1] = Ax[k] is marginally stable iff all eigenvalues satisfy λ i 1 and all eigenvalues with λ i = 1 are simple roots of the minimal polynomial of A. Proof (Sketch). Let us consider first a simplified case, when there exists a basis e 1,..., e n consisting of eigevectors of A. If all eigenvalues satisfy λ i < 1 then x[k] = A k (ξ 1 e 1 + + ξ n e n ) = λ k 1 ξ 1e 1 + + λ k 1 ξ 1e n 0. For the general proof, note that not all matrices have n linearly independent eigenvectors. Nevertheless, it is possible to complement eigenvectors with generalised eigenvectors to form a complete basis in C n. In brief, for each eigenvector e 1 there exists a chain of generalised eigenvectors e 2,...,e k such that e i + λe i+1 = Ae i. Again, let us now consider a simplified case, where the basis consists of a single chain of generalised eigenvectors e 1,...,e n. Then we can express x[k] = A k (ξ 1 e 1 + + ξ n e n ) = A k 1( (λξ 1 + ξ 2 )e 1 + + λξ n e n ) = A k 2( (λ 2 ξ 1 + 2λξ 2 + ξ 3 )e 1 + + λξ n e n ) ( n 1 ( ) ( 0 ( ) k k = )λ k i ξ 1+i e 1 + + )λ k i ξ n+i e n. i i i=0 Note that all coefficients before the powers of λ are bounded by k n. Since exponent function decreases faster than any polynomial all coordinates converge to zero when λ < 1. For the case λ > 1, note that the initial state x 0 = e 1 yields unbounded response. For the case λ = 1, note that the initial state x 0 = e 1 +e 2 yields also an unbounded response x[k] = λ k (e 1 + e 2 ) + kλ k 1 e 1 k e1 e 1 + e 2. The precise argument is a bit more complicated if we assume that initial state x 0 must be a real valued vector, since our construction does not guarantee i=0
x 0 R n. The generalisation to multiple eigenvalues is straightforward. Again, handling complex eigenvalues requires more detailed analysis. The analysis of continuous systems is analogous, since any solution to ẋ(t) = Ax(t) can be decomposed to eigenfunctions and generalised eigenfunctions. The different convergence criterion comes from the nature of eigenfunctions. Finally, note that there is a close correspondence between the eigenvalues of the matrix A and the poles of the corresponding transfer function. L3. Each pole of the transfer function is an eigenvalue of the matrix A. 4 Asymptotic Stability and Signal Tracking Most important properties of open-loop controllers are determined by the poles of the transfer function. In particular, recall that theorem T2 assures that a constant input signal excites a constant output signal when a system is BIBO stable and the initial state x 0 is zero. However, the initial state is rarely zero and thus the zero-input response y zi plays an important role in the beginning and then decays rapidly when the system is asymptotically stable. After that the basic open-loop controller track signal without remarkable errors. Secondly, the zero-input response y zi come into play when the system gets external disturbances that shift it from the original state x[l] to close-by state x[l] + x[l]. Then the corresponding error term y l can be expressed as a zero-input response, since we can express the behaviour as a sum { u[k] = u0 [k] x[0] = x 0 + + { u[k] = 0 x[l] = x[l] + where u 0 [k] is the controllers output. Consequently, marginal stability is necessary to prevent catastrophic consequences. Nevertheless, the precision in signal tracking can gradually deteriorate, when the system is only marginally stable, since error terms can accumulate. Asymptotically stable systems behave significantly better, since error terms y l [k] converge to zero. See Figure 5 Another unexpected failure point is the controller itself. The direct solution to the equation 1 can lead to a controller design that is not BIBO stable. For instance, a minimal realisation of the transfer function ĝ[z] = z 2 z 2 0.25 is both BIBO and marginally stable. However, the corresponding controller ĉ[z] = z2 0.25 z(z 2) is not BIBO nor marginally stable. Hence, a finite input sequence can cause unbounded control signal u[k] that will destroy the system in practice. Similarly,
small disturbances in the controller logic can lead to large errors u l [k] in the control signal that again cause catastrophic changes in the output, see Figure 6. To summarise, open-loop controllers are usable only if both the system and the corresponding controller are both BIBO and marginally stable. 2 1 0 1 2 y zs 0 10 20 30 40 50 60 2 1 0 1 2 y zi 0 10 20 30 40 50 60 2 1 0 1 2 y 0 10 20 30 40 50 60 2 1 0 1 2 y = y zs + y zi + y 0 10 20 30 40 50 60 Fig.5. Due to the linearity, the output of a system can be decomposed into the zerostate response zero-input response, and reactions to external disturbances.
0 40000 80000 ĉ[z] = z2 0.25 z(z 2) 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 ĝ[z] = z 2 z 2 0.25 0 10 20 30 40 0e+00 4e+04 8e+04 ĉ[z] = z2 0.25 z(z 2) 0 10 20 30 40 0.0 0.5 1.0 1.5 ĝ[z] = z 2 z 2 0.25 0 10 20 30 40 Fig. 6. An unstable controller can produce an unbounded control signal. Also, the effect of disturbances in the controller are not cancelled out, although their effect can be marginal compared to unbounded growth of the controller signal. The effect of disturbances is illustrated on two lower graphs.